Progress in Optics, Volume 50

Dedication This volume is dedicated to the memory of two members of the Editorial Advisory Board of Progress in Optics...

Author: Emil Wolf

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Dedication

This volume is dedicated to the memory of two members of the Editorial Advisory Board of Progress in Optics Professor Lorenzo Narducci and Professor Herbert Walther two distinguished scientists and friends of many of our readers. Their recent deaths represent a profound loss to the whole scientific community.

Preface It is a great pleasure to present to our readers the fiftieth volume of Progress in Optics, a series which commenced publication in 1961, close to the time when the laser was invented. Since then optics has been evolving very rapidly. The nearly 300 articles that appeared in these volumes faithfully reflect most of the exciting developments in optics and in related subjects that have taken place since that time. To celebrate this milestone all the articles contained in the present volume are devoted to historical developments of optics. Emil Wolf Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, NY 14627, USA August 2007

vii

E. Wolf, Progress in Optics 50 © 2007 Elsevier B.V. All rights reserved

Chapter 1

From millisecond to attosecond laser pulses by

N. Bloembergen College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(07)50001-6 1

Contents

Page § 1. From millisecond to nanosecond pulses . . . . . . . . . . . . . . . . .

3

§ 2. From nanosecond to femtosecond pulses . . . . . . . . . . . . . . . .

4

§ 3. The attosecond regime . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2

§ 1. From millisecond to nanosecond pulses Maiman [1960] achieved the first operating laser, utilizing a ruby crystal pumped by a xenon flash lamp. The light flash lasted about one millisecond and the laser output fluctuated during this time interval. The population inversion in the fluorescent levels of the chromium ion changed by the competition between the pumping mechanism and the stimulated emission processes into various laser modes. This led to relaxation-type oscillations and fluctuations in the laser output. Hellwarth [1961] achieved Q-switched pulses of less than one microsecond duration within the first year of ruby laser operations. The quality factor Q of the laser cavity was kept low by the use of a polarization-switching cell with nitrobenzene, subjected to an electric field causing birefringence by the Kerr effect. A large population inversion in the ruby was established, as the light with orthogonal polarization was switched out of the laser cavity with a Nicol prism. When the electric field on the Kerr cell was suddenly switched off, a powerful laser pulse was created by the high gain from the large inverted population. Reproducible pulses with a duration between ten and one hundred nanoseconds were obtained. In this early period Q-switching was also obtained by mounting a mirror or a cubic totally reflecting glass prism on a turbine-driven dentist drill. The laser oscillation could only build up when the rotating mirror was nearly parallel to the fixed mirror of the laser cavity. Clearly this mechanical method had no mode control and became rapidly obsolete, although it survived longer for the Q-switching of high-power CO2 lasers operating near 10 µm wavelength. This cursory overview is intended for non-specialists who wish to become acquainted with the remarkable developments in time-resolved optical techniques. More comprehensive reviews may be found in the references. The next section outlines the evolution from the nanosecond to the femtosecond regime. It is based on familiar nonlinear optical processes, including saturable absorption, intensity-dependent index of refraction, self-focusing and self-phase modulation. The transition to the attosecond regime is described in the third, final section. It requires “extreme” nonlinear processes, including tunneling and the creation of quasi-free electrons, which enable a transition from the visible to the (soft) X-ray spectral region. 3

4

From millisecond to attosecond laser pulses

[1, § 2

§ 2. From nanosecond to femtosecond pulses This section is based on an earlier brief overview (Bloembergen [1999]) and on a comprehensive review by Brabec and Krausz [2000]. Mode-locking of a large number of longitudinal modes all activated by the gain profile of the laser medium can lead to one short pulse traveling back and forth in the laser cavity of length l. The frequency spacing between the modes is c/2l and the output is a train of pulses, separated by the time interval 2l/c. Active mode-locking was first achieved for a helium–neon laser by Hargrove, Fork and Pollack [1964]. They acoustically modulated the index of refraction at the frequency c/2l. Since the gain–bandwidth profile of the gas-laser transition is rather small, the pulse duration remained much longer than ten nanoseconds. Passive mode-locking of a ruby laser was first demonstrated by Mocker and Collins [1965]. They used a saturable absorber in the form of a liquid film or jet with a bleachable dye near one of the mirrors in the laser cavity. The generation of a short pulse output may be qualitatively understood as follows. Initially, spontaneous emission processes are amplified and lead to a stochastic light intensity. A peak in this output is subsequently amplified the most, as it causes more initial bleaching of the saturable absorber. Thus, intervals of higher intensity are attenuated less on passage through the bleachable dye film. After many round trip passages in the laser cavity the energy transfer of the stimulated emission processes is concentrated in a shorter time interval, which is eventually limited by the inverse of the gain–bandwidth product of the laser. For the ruby laser this product was not large enough to break the nanosecond barrier. Pulses shorter than a nanosecond, marking the entrance into the picosecond domain, were first obtained by De Maria, Stetser and Heinan [1966]. They used a neodymium glass laser. This material as well as neodymium–yttrium–aluminum garnet (Nd-YAG) have sufficient gain–bandwidth product to permit the generation of laser pulses of 10 ps duration. These lasers, operating near 1.06 µm wavelength, were widely used for experiments in time-resolved spectroscopy. Other frequencies could be obtained by harmonic generation, stimulated Raman scattering and self-phase modulation. Shapiro [1977] has edited an early review dedicated to picosecond pulse generation. The first entry into the sub-picosecond or femtosecond regime was accomplished by Shank and Ippen [1974]. They used a broad gain dye-laser medium in combination with a saturable dye-absorber film. An analysis by New [1972] had shown that it is possible to obtain pulses shorter than the characteristic relaxation times of the dye media, since the pulse duration is determined by the time that the saturable gain exceeds the saturable absorption. The use of a ring

1, § 2]

From nanosecond to femtosecond pulses

5

dye-laser system with counter-propagating laser pulses, which cross each other in the passage through a saturable dye film, only a few microns thick, led to even shorter pulses. A limit is then set by the dispersion in the group velocity which causes different Fourier components of the short pulse to propagate at different speeds. This produces chirping and stretching of a short pulse. The chirping produced by self-phase modulation is often more important. As the intensity rises during the leading edge, the index of refraction rises, and the light frequency is shifted toward the red. A blue shift occurs at the trailing edge of the pulse. Chirping can be compensated by introducing negative group-velocity dispersion. This may be accomplished by prism or grating configurations which increase the optical path length for red-shifted light relative to that for blue-shifted light. Fork, Britto Cruz, Becker and Shank [1987] reported pulses of 6 fs duration by a prism configuration in the laser cavity. The linearly polarized light beams were incident near Brewster’s angle to avoid reflection losses at the prism surfaces. These femtosecond dye laser systems required frequent adjustments, and alignment of the components was critical. Ippen [1994] has reviewed the theory and experiments of passive mode locking of laser pulses. A true revolution in femtosecond generation occurred when Spence, Kaen and Sibbett [1991] discovered that a titanium-aluminum oxide (Ti-sapphire) laser crystal would yield very short pulses without the use of a saturable absorber. Self-focusing of the laser beam in the Ti-sapphire crystal occurs because of its intensity-dependent index of refraction. In combination with a suitably placed aperture, the self-focusing can cause a larger fraction of the pulse energy to pass through the aperture at higher intensity. Thus, less absorption and more gain occurs for more intense parts of the stochastic amplified spontaneous emission. This so-called Kerr-lens self-focusing is a purely dispersive effect. It is independent of material relaxation times and can be as fast as the inverse of the frequency detuning from the absorption edge of the material. Compensation of group-velocity dispersion in the laser crystal and the chirping by self-phase modulation is again essential. Besides prism and grating configurations, negative group-velocity dispersion may also be obtained from chirped mirrors. These are layered dielectric films in which red-shifted components penetrate deeper than blue-shifted components, as the thickness of alternating films causing Bragg reflection increases with increasing depth. The compensation of self-phase modulation by negative group-velocity dispersion leads to the formation of soliton-like optical pulses. Nonlinear effects are essential not only in the generation of picosecond and femtosecond pulses, but also in their measurement and evaluation. The short pulse is split into two pulses and a variable time delay is introduced between the first (or pump) pulse and the second (or probe) pulse. The two pulses are recombined

6


[1, § 2

in a thin nonlinear crystal and the second harmonic generated in the crystal by the combination of the two pulses is observed as a function of the time delay. Armstrong [1967] first used this technique for picosecond pulses. Note that one femtosecond corresponds to a differential of 0.3 µm in the path lengths of the two pulses. The intensity of the second harmonic as a function of path length differential yields the autocorrelation of the intensity of the laser pulse. For a complete characterization of the pulse the temporal behavior of the phase of the light field must also be determined. This information may be obtained by analyzing the temporal behavior of individual Fourier components in the generated second harmonic signal. It is spectrally analyzed as a function of time delay. Trebino and Kane [1993] introduced this technique, which they called Frequency Resolved Optical Gating or FROG. Other variations were subsequently introduced but they are all based on a comparison of spectrally resolved components of the pulse and its delayed or advanced replica. Conversely, pulses may be generated with prescribed amplitude and phase variation. A femtosecond pulse is spectrally resolved by a grating and different Fourier components may be reflected by different segments of piezoelectric crystals. Voltages applied to the segments will produce different path lengths or phases to individual Fourier components on reflection. These components may be recombined by the same grating to produce a modified pulse with a different temporal phase variation. The different Fourier components may, of course, also be attenuated by adjustable factors. These changes may also be induced by passage through a liquid crystal array with a configuration of electrodes. The latter technique has been used by Shverdin, Walker, Yavuz, Yin and Harris [2005] to generate a halfcycle optical wave form of about 1.5 fs duration. A 5 fs pulse of a Ti-sapphire laser is used to generate several orders of anti-Stokes and Stokes components of vibrational and rotational Raman transitions in hydrogen gas (H2 or D2 ). The components cover a frequency range of more than two octaves. By adjusting the amplitudes and phases of these components before recombining them it was possible to obtain constructive interference over one half cycle, and nearly complete destructive interference outside that time interval. Femtosecond pump-probe techniques have led to many applications in timeresolved spectroscopy in chemistry, biology and solid state physics, but their discussion falls outside the scope of this review. Zewail [2000] received the 1999 Nobel Prize for chemistry for this work on femtochemistry, the time-resolved spectroscopy of a large variety of chemical reactions. Lobastov, Shrinivas and Zewail [2005] have recently extended femtosecond time resolution to the field of electron diffraction. They call it 4D ultrafast electron microscopy. A weak femtosecond “probe” pulse is used to liberate on the average about one photoelectron

1, § 2]

From nanosecond to femtosecond pulses

7

from the cathode of a standard electron microscope with a de Broglie wavelength of 0.335 nm at 120 keV energy. Distortion of the electron orbits due to spacecharge effects by a Coulomb repulsion between electron pairs is thus avoided. A pulse train of ten million pulses per second builds up the diffraction of a target that has been excited by a femtosecond “pump” pulse at an adjustable earlier time. Thus, the variation in the diffraction pattern with sub-nanometer spatial resolution and 10 fs temporal resolution can be observed. The output of a Kerr-lens mode-locked tabletop Ti-sapphire laser with a crystal volume of a few cc typically produces a train of 10 fs pulses with about one nanojoule per pulse with a repetition rate of 10 MHz. Such a pulse train can be amplified by a factor of 103 or 104 by the chirped pulse amplification (CPA) technique described by Mourou, Barty and Perry [1998]. The pulse is first deliberately lengthened by a factor of 103 to 104 by an antiparallel grating configuration before being amplified. This is necessary to avoid damage by light-induced dielectric breakdown in the amplifying medium. The amplified pulse is then re-compressed by a matching configuration of grating pairs. The CPA technique can generate a train of 10 fs pulses with a pulse energy of one microjoule, or even higher. If such a pulse is focused by a microscope objective to an area of 10−8 cm2 , the resulting power flux density would be 1016 W/cm2 , with a corresponding light-field amplitude of 109 V/cm. This is approximately equal to the Coulomb field at the Bohr orbit in the ground state of the hydrogen atom, or the field responsible for the binding of a valence electron in a molecule. It is physically obvious that any material subjected to such a field is immediately transformed to a fully ionized plasma. If the light intensity is lowered by one or two orders of magnitude, in the range of 1014 to 1015 W/cm2 , there is still a probability for ionization by tunneling rather than by multi-photon ionization. The first experiment on two-photon induced photoelectric emission from an alkali-metal cathode was carried out by Teich, Schoer and Wolga [1964] with a ruby laser. The effect of thermionic emission due to heating during the long laser pulse had to be carefully eliminated. Bechtel, Smith and Bloembergen [1975] demonstrated four-photon emission from tungsten by a picosecond Nd-glass laser. The ionization of Kr atoms by an eleven-photon process at 1.06 µm was reported by Mainfray [1978]. The ionization rate and the formation of Kr+ ions increased as the eleventh power of the laser intensity. This probably represents the limit of high-order nonlinear optical perturbation theory. Keldysh [1965] had already discussed the transition from multi-photon to tunneling ionization. He introduced a dimensionless parameter, the square root of the ratio of the ionization potential Ip to the quiver energy or ponderomotive energy Up = e2 E 2 /2mω2 of a free electron oscillating in a laser field of amplitude E and circular frequency ω. For Up > Ip tunneling predominates. This occurs when a

8


[1, § 3

femtosecond train of pulses is focused on a stream of noble gas atoms emanating from a nozzle at intensities in the range of 1014 to 1015 W/cm2 . The presence of quasi-free electrons created by tunneling ionization leads to new phenomena, including high-harmonic generation in the soft X-ray regime. The creation of these quasi-free electrons has made it possible to explore the attosecond regime and the measurement of time intervals shorter than one optical cycle.

§ 3. The attosecond regime Corkum [1993] introduced the recollision model which describes the phenomena occurring in a fraction of an optical cycle. A valence electron tunnels out of its atomic or molecular orbital preferentially when the linearly polarized femtosecond optical field is near its maximum in either direction. The liberated electron is accelerated in the next quarter cycle. When the direction of the laser field reverses at the first zero crossing, the electron is decelerated and reverses direction during the next half cycle and returns to the ion it had left behind. It reaches this ion approximately at the second zero crossing after its liberation. Detailed classical calculations of the equation of motion of the free electron in the optical cycle for a range of moments of its creation by tunneling show that the maximum kinetic energy is 3.2Up , where Up is the ponderomotive energy introduced previously. The electron has a probability to fall back into the valence orbital it had left behind with the emission of a photon with a maximum energy Ip + 3.2Up . If it does not recombine, it may recollide again after half of an optical cycle. If the femtosecond pulse is relatively long and contains many optical cycles, the recollisions spaced by one half optical cycle lead to a time series of photon pulses. In the frequency domain this corresponds to a series of odd harmonics, spaced by 2ω. Such high harmonics, often in the range of ten to a few hundred times the optical frequency, have been observed by many investigators. Kapteyn, Murnane and Christov [2005] have reviewed this generation of extreme UV and soft X-rays in targets which usually consist of a jet of noble gas atoms, into which a train of femtosecond optical pulses are focused. The harmonics are mostly emitted in a narrow cone parallel to the direction of the optical pulse. Phase matching between the harmonics and the infrared pulse is possible because the generated plasma lowers the optical index of refraction. The energy distribution over the harmonic spectrum depends sensitively on the intensity and phase variation in the incident laser pulse. The generation of particular harmonics may be optimized by controlling the phase variation as described previously (Kapteyn, Murnane and Christov [2005]).

1, § 3]

The attosecond regime

9

The classical picture also predicts that the recollision mechanism is not operative for elliptical or circular polarization of the laser pulses. In this case the electron does not return to the ion along a linear path. Experimentally Dietrich, Burnett, Ivanov and Corkum [1994] have demonstrated that a ten percent ellipticity largely suppresses high-harmonic generation. Corkum, Burnett and Ivanov [1994] have proposed a sub-femtosecond switch by using two orthogonally polarized laser pulse trains with two wavelengths differing by ten percent. Linear polarization of the resultant light field would only occur during a fraction of an optical cycle and so would the recollisions. The classical model is substantiated by a fully quantum-mechanical calculation (Lewenstein, Balcou, Ivanov, L’Huiller and Corkum [1994]). The laser field admixes a quasi-free electron wave-packet, oscillating coherently with the laser field. The radiation emitted by the motion of the electron, described by the solution of the time-dependent Schrödinger equation, is determined by the second time derivative of the large induced oscillating dipole moment. While the spectrum of odd harmonics is indicative of a series of subfemtosecond soft X-ray pulses, it does not represent a measurement of their duration. Unfortunately, the technique of second-harmonic generation used for femtosecond pulse durations cannot be used here because the intensity of the softray pulses is much too weak. Itatani, Quéré, Yudin, Ivanov, Krausz and Corkum [2002] have described an alternative method, the attosecond streak camera. Attention is focused on the high-energy tail of emitted harmonics by filtering out harmonics with lower energy by filters of metal foils. The high-energy end, generated in a fraction of a complete optical cycle, is used to liberate photoelectrons in another gas jet exposed simultaneously to the fundamental optical pulse. Photoelectrons are collected in a direction perpendicular to the linear laser polarization, which causes a deflection and a change in energy of the emitted photo-electrons (Hentshel, Klenberger, Spielmann, Reider, Milosevich, Brabec, Corkum, Heinzmann, Drescher and Krausz [2001]). Reproducible quantitative results require the use of a very short reproducible femtosecond pulse. Such a pulse contains only a few optical cycles in the wavelength range of 0.7 to 1 µm from a Ti-sapphire laser. The relative phase between the carrier field and the envelope becomes important (Brabec and Krausz [2000]). When this phase is zero, the laser-field maximum coincides with the maximum in the envelope. At the two adjacent field extrema, a half optical cycle away, the envelope is already substantially reduced. If the intensity is carefully chosen, significant tunneling and emission of the highest-energy harmonics may be limited to a fraction of the optical cycle around the maximum. In this case, the high-frequency tail drops smoothly to zero. When the carrier-envelope

10


[1, § 3

phase is π/2, the laser field passes through zero at the envelope maximum. In this case tunneling may be limited to two short time intervals, one quarter of an optical cycle away on either side of the envelope maximum. The high-energy harmonics emitted from these two time intervals may interfere either constructively or destructively. The high-energy tail of X-rays shows oscillations in this case. The stabilization of the carrier-envelope phase is a prerequisite for attosecond time-resolved measurements (Bucksbaum [2003]). The method to achieve this (Baltuska, Udem, Uberacker, Hentschel, Goultelmakis, Gohle, Hellwarth, Yakovlev, Scrinzi, Hänsch and Krausz [2003]) is the same one as used by Hänsch [2006] for frequency stabilization in an optical time standard. A supercontinuum laser source (Alfano [2006]) can produce spectral broadening by self-phase modulation over more than one octave. The Fourier spectrum consists of millions of equally spaced frequency components, separated by c/2l. An audible beat note can be obtained between a component at ν + N c/2l and the second harmonic 2ν generated by the laser pulse in a nonlinear crystal. The carrier envelope usually shifts because of the difference in group and phase velocity in the laser cavity. This phase difference can however be fixed at a multiple of 2π by simultaneously stabilizing the fundamental and second-harmonic frequency in the same interferometer. Single attosecond soft X-ray pulses of 250 attosecond duration have been generated, and recently a coherent source of 1.3 keV X-rays has been reported by Seres, Seres, Verhoeft, Tempea, Strelill, Wobranskii, Yakovlev, Scrinzi, Spielmann and Krausz [2005]. Attosecond technology may be used for investigations of collision physics with Auger processes in atoms with inner-shell vacancies and for molecular collision processes. Niikura, Légaré, Hasbani, Ivanov, Villeneuve and Corkum [2003] have studied the recollision of an electron with a molecular ion D+ 2 . The hydrogen molecules were first aligned with the laser field before tunnel ionization. The modular ion was left in a vibrational wave packet, which could be studied at different times by choosing different laser wavelengths between 0.6 and 1.8 µm. Future extensions of attosecond spectroscopy to muonic atoms and nuclear excitations have been proposed. A recent authoritative review of attosecond science and technology (Levesque and Corkum [2005]) emphasizes the wave-like nature and coherence of the electron driven by a femtosecond pulse. The understanding of the attosecond processes of soft X-ray emission and its inverse of photo-ionization can be formulated in terms of electron interferometry.

1]

References

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Acknowledgements This author thanks Professor Emil Wolf for the invitation to contribute to the 50th volume of the series Progress in Optics, which he has edited since its inception. This invitation encouraged me to become better acquainted with the developments in attosecond physics. I acknowledge my indebtedness to Professors P.B. Corkum, S.E. Harris, F. Krausz and A.H. Zewail who supplied me with reprints of their important recent contributions.

References Alfano, R.R., 2006, The Supercontinuum Laser Source, 2nd edition, Springer, Berlin. Ch. 17, p. 512. Armstrong, J.A., 1967, App. Phys. Lett. 10, 16. Baltuska, A., Udem, Th., Uberacker, M., Hentschel, M., Goultelmakis, E., Gohle, Ch., Hellwarth, R., Yakovlev, V.S., Scrinzi, A., Hänsch, T.W., Krausz, F., 2003, Nature (London) 421, 611. Bechtel, J.H., Smith, W.L., Bloembergen, N., 1975, Opt. Comm. 13, 56. Bloembergen, N., 1999, Rev. Mod. Phys. 71, S283. Brabec, T., Krausz, F., 2000, Rev. Mod. Phys. 72, 545. Bucksbaum, P.H., 2003, Nature (London) 421, 593. Corkum, P.B., 1993, Phys. Rev. Lett. 71, 1994. Corkum, P.B., Burnett, N.H., Ivanov, M.Y., 1994, Opt. Lett. 19, 1870. De Maria, A.J., Stetser, D.A., Heinan, H., 1966, App. Phys. Lett. 8, 174. Dietrich, P., Burnett, N.H., Ivanov, M., Corkum, P.B., 1994, Phys. Rev. A 50, R3585. Fork, R.L., Britto Cruz, C.H., Becker, P.C., Shank, C.V., 1987, Opt. Lett. 12, 483. Hänsch, T.W., 2006, Rev. Mod. Phys. 78, 1297. Hargrove, L.E., Fork, R.L., Pollack, M.A., 1964, App. Phys. Lett. 5, 4. Hellwarth, R.W., 1961, in: Singer, J.R. (Ed.), Advances in Quantum Electronics, Columbia University, New York, p. 334. Hentshel, M., Klenberger, R., Spielmann, Ch., Reider, G.A., Milosevich, N., Brabec, T., Corkum, P., Heinzmann, U., Drescher, M., Krausz, F., 2001, Nature (London) 414, 509. Ippen, E.P., 1994, App. Phys. B. Laser Opt. 58, 159. Itatani, J., Quéré, F., Yudin, G.I., Ivanov, M.Yu., Krausz, F., Corkum, P.B., 2002, Phys. Rev. Lett. 88, 173903. Kapteyn, H.C., Murnane, M.M., Christov, I.P., 2005, Physics Today 58 (3), 39. Keldysh, L.V., 1965, Soviet Physics TETP 20, 1307. Levesque, J., Corkum, P.B., 2005, Can. Journal Phys. 99, 1. Lewenstein, M., Balcou, Ph., Ivanov, M.Yu., L’Huiller, A., Corkum, P.B., 1994, Phys. Rev. A 48, 2117. Lobastov, V.A., Shrinivas, R., Zewail, A.H., 2005, Proc. Nat. Acad. Sci. USA 102, 7069. Maiman, T.H., 1960, Nature (London) 187, 493. Mainfray, G., 1978, in: Eberly, J.H., Lambropoulos, P. (Eds.), Multiphoton Processes, Wiley, New York, p. 253. Mocker, H.W., Collins, R.J., 1965, App. Phys. Lett. 7, 270. Mourou, G.A., Barty, C.P.J., Perry, M.D., 1998, Physics Today 51 (1), 22. New, G.H.C., 1972, Opt. Comm. 6, 188. Niikura, H., Légaré, F., Hasbani, R., Ivanov, M.Yu., Villeneuve, D.M., Corkum, P.B., 2003, Nature (London) 421, 826.

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[1

Seres, J., Seres, E., Verhoeft, A.J., Tempea, G., Strelill, C., Wobranskii, P., Yakovlev, V., Scrinzi, A., Spielmann, C., Krausz, F., 2005, Nature (London) 453, 596. Shank, C.V., Ippen, E.P., 1974, App. Phys. Lett. 24, 375. Shapiro, S.L., 1977, in: Shapiro, S.L. (Ed.), Ultrashort Light Pulses, Springer, Berlin, p. 1. Spence, D.E., Kaen, P.N., Sibbett, W., 1991, Opt. Lett. 16, 42. Shverdin, M.Y., Walker, D.H., Yavuz, D.D., Yin, G.Y., Harris, S.E., 2005, Phys. Rev. Lett. 94, 033904. Teich, M.C., Schoer, J.M., Wolga, G.J., 1964, Phys. Rev. Lett. 13, 611. Trebino, R., Kane, D.J., 1993, J. Optical Soc. Amer. A 10, 110. Zewail, A.H., 2000, Femtochemistry, in: Les Prix Nobel 1999, Almquist, Wiksell Int., Stockholm, pp. 110–203.


Chapter 2

Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics by

M.V. Berry, M.R. Jeffrey H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(07)50002-8 13

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

§ 2. Preliminaries: electromagnetism and the wave surface . . . . . . . . .

18

§ 3. The diabolical singularity: Hamilton’s ray cone . . . . . . . . . . . .

20

§ 4. The bright ring of internal conical refraction . . . . . . . . . . . . . .

23

§ 5. Poggendorff’s dark ring, Raman’s bright spot . . . . . . . . . . . . .

26

§ 6. Belsky and Khapalyuk’s exact paraxial theory of conical diffraction .

31

§ 7. Consequences of conical diffraction theory . . . . . . . . . . . . . . .

34

§ 8. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

§ 9. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

43


45

Appendix A: Paraxiality . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Appendix B: Conical refraction and analyticity . . . . . . . . . . . . . . .

47

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

14

§ 1. Introduction The bicentenary of the birth in 1805 of William Rowan Hamilton (fig. 1) comes at a time of renewed interest in the phenomenon of conical refraction (Born and Wolf [1999], Landau, Lifshitz and Pitaevskii [1984]), in which a narrow beam of light, incident along the optic axis of a biaxial crystal, spreads into a hollow cone within the crystal, and emerges as a hollow cylinder. The prediction of the effect in 1832 (Hamilton [1837]), and its observation by Lloyd (fig. 1) soon afterwards (Lloyd [1837]), caused a sensation. For a non-technical account, see Lunney and Weaire [2006]. Our purpose here is to describe how the current understanding of conical refraction has been reached after nearly two centuries of theoretical and experimental study. Although our treatment will be roughly historical, we do not adhere to the practice, common in historical research, of describing each episode using only sources and arguments from the period being studied. For the early history of conical refraction, this has already been done, by Graves [1882] and O’Hara [1982]. Rather, we will weave each aspect of the theory into the historical development in ways more concordant with the current style of theoretical physics, hoping thus to bring out connections with other phenomena in mathematics and physics. There are many reasons why conical refraction is worth revisiting: (i) It was an early (perhaps the first) example of a qualitatively new phenomenon predicted by mathematical reasoning. By the early 1800s, it was widely appreciated that mathematics is essential to understanding the natural world. However, the phenomena to which mathematics had been applied were already familiar (e.g., tides, eclipses, and planetary orbits). Prediction of qualitatively new effects by mathematics may be commonplace today, but in the 1830s it was startling. (ii) With its intimate interplay of position and direction, conical refraction was the first non-trivial application of phase space, and of what we now call dynamics governed by a Hamiltonian. (iii) Its observation provided powerful evidence confirming that light is a transverse wave. 15

16

Conical diffraction and Hamilton’s diabolical point

[2, § 1

Fig. 1. Dramatis personae.

2, § 1]

Introduction

17

(iv) Recently, it has become popular to study light through its singularities (Berry [2001], Nye [1999], Soskin and Vasnetsov [2001]). In retrospect, we see conical refraction as one of the first phenomena in singular polarization optics; another is the pattern of polarization in the blue sky (Berry, Dennis and Lee [2004]). (v) It was the first physical example of a conical intersection (diabolical point) (Berry [1983], Uhlenbeck [1976], Berry and Wilkinson [1984]) involving a degeneracy. Nowadays, conical intersections are popular in theoretical chemistry, as spectral features indicating the breakdown of the Born–Oppenheimer separation between fast electronic and slow nuclear freedoms (Cederbaum, Friedman, Ryaboy and Moiseyev [2003], Herzberg and Longuet-Higgins [1963], Mead and Truhlar [1979]). By analogy, conical refraction can be reinterpreted as an exactly solvable model for quantum physics in the presence of a degeneracy. (vi) The effect displays a subtle interplay of ray and wave physics. Although its original prediction was geometrical (Sections 3 and 4), there are several levels of geometrical optics (Sections 5 and 7), of which all except the first require concepts from wave physics, and waves are essential to a detailed understanding (Section 6). That is why we use the term conical diffraction, and why the effect has taken so long to understand. (vii) Analysis of the theory (Berry [2004b]) led to identification of an unexpected universal phenomenon in mathematical asymptotics: when exponential contributions to a function compete, the smaller exponential can dominate (Berry [2004a]). (viii) There are extensions (Section 8) of the case studied by Hamilton, and their theoretical understanding still presents challenges. Effects of chirality (optical activity) have only recently been fully understood (Belsky and Stepanov [2002], Berry and Jeffrey [2006a]), and further extensions incorporate absorption (Berry and Jeffrey [2006b], Jeffrey [2007]) and nonlinearity (Indik and Newell [2006]). (ix) Conical diffraction is a continuing stimulus for experiments. Although the fine details of Hamilton’s original phenomenon have now been observed (Berry, Jeffrey and Lunney [2006]), predictions of new structures that appear in the presence of chirality, absorption and nonlinearity remain untested. (x) The story of conical diffraction, unfolding over 175 years, provides an edifying contrast to the current emphasis on short-term science. Although all results of the theory have been published before, some of our ways of presenting them are original. In particular, after the exact treatment in

18


[2, § 2

Sections 2–4 we make systematic use of the simplifying approximation of paraxiality. This is justified by the small angles involved in conical diffraction (Appendix A), and leads to what we hope are the simplest quantitative explanations of the various phenomena.

§ 2. Preliminaries: electromagnetism and the wave surface Hamilton’s prediction was based on a singular property of the wave surface describing propagation in an anisotropic medium. Originally, this was formulated in terms of Fresnel’s elastic-solid theory. Today it is natural to use Maxwell’s electromagnetic theory. For the physical fields in a homogeneous medium, we write plane waves with wavevector k and frequency ω as Re {Dk , Ek , Bk , Hk } exp i(k · r − ωt) , (2.1) in which the vectors Dk , etc., are usually complex. From Maxwell’s curl equations, ωDk = −k × Hk ,

ωBk = k × Ek .

(2.2)

A complete specification of the fields requires constitutive equations. For a transparent nonmagnetic nonchiral biaxial dielectric, these can be written as Ek = ε−1 Dk ,

Bk = μ0 Hk .

(2.3)

ε−1 is the inverse dielectric tensor, conveniently expressed in principal axes as ⎛ ⎞ 0 0 1/n21 1 ⎝ 0 ε−1 = (2.4) 1/n22 0 ⎠, ε0 0 0 1/n23 where ni are the principal refractive indices, all different, with the conventional ordering (2.5)

n 1 < n2 < n3 .

For biaxiality (all ni different), the microscopic structure of the material must have sufficiently low symmetry; in the case of crystals, this is a restriction to the orthorhombic, monoclinic or triclinic classes (Born and Wolf [1999]). Conical refraction depends on the differences between the indices, so we define α≡

1 1 − 2, n21 n2

β≡

1 1 − 2. n22 n3

(2.6)

2, § 2]

Preliminaries: electromagnetism and the wave surface

19

It will be convenient to express the wavevector k in terms of the refractive index nk , through the definitions ω k ≡ kek ≡ k0 nk ek , k0 ≡ , (2.7) c incorporating c= √

1 . ε0 μ0

(2.8)

It is simpler to work with the electric vector D rather than E, because D is always transverse to the propagation direction ek . Then eqs. (2.2) and (2.3) lead to the following eigenequation determining the possible plane waves:

1 Dk = −ek × ek × ε0 ε−1 · Dk . 2 nk

(2.9)

This involves a 3 × 3 matrix whose determinant vanishes, and the two eigenvalues λ± of the 2 × 2 inverse dielectric tensor transverse to k give the refractive indices 1 . nk± = (2.10) λ± (ek ) Of several different graphical representations (Born and Wolf [1999], Landau, Lifshitz and Pitaevskii [1984]) of the propagation governed by nk± , we choose the two-sheeted polar plot in direction space; this is commonly called the wave surface, though there is no universally established terminology. The wave surface has the same shape as the constant ω surface in k space, that is [cf. eq. (2.7)], the contour surface of the dispersion relation ω(k) ≡

ck ; nk

(2.11)

ω(k) is the Hamiltonian generating rays in the crystal, with k as canonical momentum. An immediate application of the wave surface, known to Hamilton and central to his discovery, is that for each of the two waves with wavevector k, the ray direction, that is, the direction of energy transport, is perpendicular to the corresponding sheet of the surface. This can be seen from the first Hamilton equation, according to which the group (ray) velocity is vg = ∇k ω(k)

(2.12)

(for this case of a homogeneous medium, the second Hamilton equation simply asserts that k is constant along a ray). Alternatively (Landau, Lifshitz and

20


[2, § 3

Pitaevskii [1984]), the ray direction can be regarded as that of the Poynting vector, S = Re E∗ × H,

(2.13)

because for any displacement dk with ω constant – that is, any displacement in the wave surface – 1 S · dk = ω Re E∗ · dD − dE∗ · D + H · dB∗ − dH · B∗ = 0, (2.14) 2 where the first equality is a consequence of Maxwell’s equations and the second follows from the linearity and Hermiticity of the constitutive relations (2.3). (This argument is slightly more general than that of Landau, Lifshitz and Pitaevskii [1984], because it includes transparent media that are chiral as well as biaxial.)

§ 3. The diabolical singularity: Hamilton’s ray cone Figure 2 is a cutaway representation of the wave surface, showing four points of degeneracy where nk+ = nk− , located on two optic axes in the k1 , k3 plane in k space. Each intersection of the surfaces takes the form of a double cone, that is, a diabolo, and since these are the organizing centres of degenerate behaviour we call them diabolical points, a term adopted in quantum (Ferretti, Lami and

Fig. 2. Wave surface for n1 = 1.1, n2 = 1.4, n3 = 1.8, showing the four diabolical points on the two optic axes.

2, § 3]

The diabolical singularity: Hamilton’s ray cone

21

Fig. 3. The ray cone is a slant cone normal to the wave cone.

Villani [1999]) and nuclear physics (Chu, Rasmussen, Stoyer, Canto, Donangelo and Ring [1995]) as well as optics. In fact, such degeneracies are to be expected, because of the theorem of Von Neumann and Wigner [1929] that degeneracies of real symmetric matrices [such as that in (2.9)] have codimension two, and indeed we have two parameters, representing the direction of k. Hamilton’s insight was that at a diabolical point the normals (rays) to the surfaces are not defined, so there are infinitely many normals (rays), not two as for all other k. These normals to the wave cone define another cone. This is the ray cone (fig. 3), about whose structure – noncircular and skewed – we can learn by extending an argument of Born and Wolf [1999]. We choose k along an optic axis, and track the Poynting vector as D rotates in the plane transverse to k. From (2.13) and Maxwell’s equations, S can be expressed as c Re E∗ · Dek − E∗ · ek D . S= (3.1) nk Using coordinates kx , ky , kz , with ez along k and ex in the k1 , k3 plane (i.e., ey = e2 ), ⎛ ⎞ ⎛ ⎞ Ex Dx 1 −1 ε D. D = ⎝ Dy ⎠ , (3.2) E = ⎝ Ey ⎠ = ε0 0 Ez

22


[2, § 3

Here ε is the rotated dielectric matrix, determined by four conditions: rotation about the y axis does not change components involving the 2 axis, the choice of k along an optic axis (degeneracy) implies that the xx and yy elements are the same, and the trace and determinant of the matrices ε and ε are the same. Thus √ ⎞ ⎛ ⎞ ⎛ αβ 1 0 0 0 0

−1 1 ε0 ε (3.3) = 2 ⎝0 1 0⎠ + ⎝ 0 0 0 ⎠. √ n2 αβ 0 α − β 0 0 1 It follows that, in eq. (3.1), 1

|Dx |2 + |Dy |2 , 2 ε0 n2 1 Ez = E∗ · ek = αβDx . ε0

E∗ · D =

(3.4)

Denoting the direction of D in the kx , ky plane by φ, that is Dx = D cos φ,

Dy = D sin φ,

(3.5)

we obtain a φ-parametrized representation of the surface swept out by S: S=

1 − + e cos 2φ + e sin 2φ) . tan 2A(e e z x x y 2 ε0 n32 cD 2

(3.6)

This is the ray surface, in the form of a skewed noncircular cone (fig. 3), with half-angle A given by tan 2A = n22 αβ, (3.7) From eq. (3.6) it is clear that in a circuit of the ray cone (2φ = 2π), the polarization direction φ rotates by half a turn, illustrating the familiar ‘fermionic’ property of degeneracies (Berry [1984], Silverman [1980]). The direction of the optic axis, that is, the polar angle θc in the xz plane (fig. 2) can be determined from the rotation required to transform ε into ε . Thus ε = RεR−1 ,

(3.8)

where ⎛

sin θc ⎝ R= 0 cos θc

0 1 0

⎞ − cos θc ⎠, 0 sin θc

(3.9)

2, § 4]

The bright ring of internal conical refraction

whence identification with eq. (3.3) fixes θc as α . tan θc = β

23

(3.10)

§ 4. The bright ring of internal conical refraction In a leap of insight that we now recognise as squarely in the spirit of singular optics, Hamilton [1837] realised that the ray cone would appear inside a crystal slab, on which is incident a narrow beam directed along the optic axis. The hollow cone would refract into a hollow cylinder outside the slab (fig. 4). This is a singular situation because a beam incident in any other direction would emerge, doubly refracted, into just two beams, not a cylinder of infinitely many rays. This is internal conical refraction, so-called because the cone is inside the crystal. Hamilton also envisaged external conical refraction, in which a different optical arrangement results in a cone outside the crystal. This is associated with a circle of contact between the wave surface and a tangent plane, “somewhat as a plum can be laid down on a table so as to touch and rest on the table in a whole circle of contact” (Graves [1882]). Since the theory is similar for the two effects,

Fig. 4. Schematic of Hamilton’s prediction of internal conical refraction.

24


[2, § 4

our emphasis henceforth will be on internal, rather than external, conical refraction. Hamilton’s research attracted wide attention. According to Graves [1882], Airy called conical refraction “perhaps the most remarkable prediction that has ever been made”; and Herschel, in a prescient anticipation of singular optics, wrote “of theory actually remanding back experiment to read her lesson anew; informing her of facts so strange, as to appear to her impossible, and showing her all the singularities she would observe in critical cases she never dreamed of trying”. In particular, the predictions fascinated Lloyd [1837], who called them “in the highest degree novel and remarkable”, and regarded them as “singular and unexpected consequences of the undulatory theory, not only unsupported by any facts hitherto observed, but even opposed to all the analogies derived from experience. If confirmed by experiment, they would furnish new and almost convincing proofs of the truth of that theory.” Using a crystal of aragonite, Lloyd succeeded in observing conical refraction. The experiment was difficult because the cone is narrow: the semi-angle A is small. If the slab has thickness l, the emerging cylinder, with radius R0 = Al,

(4.1)

is thin unless l is large. To see the cylinder clearly, R0 must be larger than the width w of the incident beam. Lloyd used a beam narrowed by passage through small pinholes with radii w 200 µm. As we will see, this interplay between R0 and w is important. However, large l brings the additional difficulty of finding a

Table 1 Data for experiments on conical diffraction Experiment

n1 , n2 , n3

A (◦ )

l (mm)

w (µm)

Lloyd [1837] (aragonite) Potter [1841] (aragonite) Raman, Rajagopalan and Nedungadi [1941] (naphthalene) Schell and Bloembergen [1978a] (aragonite) Mikhailychenko [2005] (sulfur) Fève, Boulanger and Marnier [1994] (sphere of KTP = KTiOPO4 ) Berry, Jeffrey and Lunney [2006] (MDT = KGd(WO4 )2

1.5326, 1.6863, 1.6908 1.5326, 1.6863, 1.6908

0.96 0.96

12 12.7

200 12.7

1.0 16.7

1.525, 1.722, 1.945 1.530, 1.680, 1.695

6.9 1.0

2 9.5

0.5 21.8

500 7.8

data not provided

3.5

1.7636, 1.7733, 1.8636

0.92

2.02, 2.06, 2.11

1.0

30 2.56 25

17

ρ0

56

53.0

1210

7.1

60

For the experiments of Lloyd and Raman, w is the pinhole radius; for the other experiments, w is the 1/e intensity half-width of the laser beam.

2, § 4]

The bright ring of internal conical refraction

25

Fig. 5. The transition (a–d) from double to conical refraction as the incident beam direction approaches the optic axis. In (d) two rings are visible, separated by the Poggendorff dark ring studied in Section 5. This figure is taken from Berry, Jeffrey and Lunney [2006] (see also Table 1).

clean enough length of crystal; Lloyd describes how he explored several regions of his crystal before being able to detect the effect. Nowadays it is simpler to use a laser beam with waist width w rather than a pinhole with radius w. Nevertheless, observation of the effect remains challenging. Table 1 summarises the conditions of the several experiments known to us; we will describe some of them later. Figure 5 illustrates how double refraction transforms into conical refraction as the direction of the incident light beam approaches an optic axis. Of this transformation, Lloyd [1837] wrote: “This phenomenon was exceedingly striking. It looked like a small ring of gold viewed upon a dark ground; and the sudden and

26


[2, § 5

Fig. 6. Simulation of rings for incident beam linearly polarized horizontally, with polarization directions superimposed, showing the ‘fermionic’ half-turn of the polarization.

almost magical change of the appearance, from two luminous points to a perfect luminous ring, contributed not a little to enhance the interest.” Lloyd also observed a feature that Hamilton had predicted: the half-turn of polarization round the ring (fig. 6).

§ 5. Poggendorff’s dark ring, Raman’s bright spot Hamilton was aware that his geometrical theory did not give a complete description of conical refraction: “I suspect the exact laws of it depend on things yet unknown” (Graves [1882]). And indeed, as fig. 5(d) shows, closer observation reveals internal structure in Hamilton’s ring, that he did not predict and Lloyd did not detect: there is not one bright ring but two, separated by a dark ring. This was first reported in a brief but important paper by Poggendorff [1839] (fig. 1), who noted “. . .a bright ring that encompasses a coal-black sliver” (“einem hellen Ringe vereinigen, der ein kohlschwarzes Scheibchen einschliefst”). The existence of two bright rings, rather than one, was independently discovered soon afterwards by Potter [1841] (Table 1). After more than 65 years, the origin of Poggendorff’s dark ring was identified by Voigt [1905a] (fig. 1) (see also Born and Wolf [1999]), who pointed out that Hamilton’s prediction involved the idealization of a perfectly collimated infinitely narrow beam. This is incompatible with the wave nature of light: a beam

2, § 5]

Poggendorff’s dark ring, Raman’s bright spot

27

of spatial width w must contain transverse wavevector components (i.e., kx , ky ) extending over at least a range 1/w (more, if the light is incoherent); this is just the optical analogue of the uncertainty principle. Therefore the incident beam will explore not just the diabolical point itself but a neighbourhood of the optic axis in k space. In fact Hamilton knew about the off-axis waves: “it was in fact from considering them and passing to the limit that I first deduced my expectation of conical refraction”. Lloyd too was aware of “the angle of divergence produced by diffraction in the minutest apertures”. Voigt’s observation was that the strength of the light generated bythe off-axis waves inside and outside the cylinder is proportional to the radius kx2 + ky2 of the circumference of contributing rings on the wave cone. At the diabolical point this vanishes, so the intensity is zero on the geometrical cylinder itself. Thus, the Poggendorff dark ring is a manifestation of the area element in plane polar coordinates in k space. The elementary quantitative theory of the Poggendorff ring is based on geometrical optics, incorporating the finite k width of the beam – which of course is a consequence of wave physics. Rigorous geometrical-optics treatments were given by Ludwig [1961] and Uhlmann [1982]; here, our aim is to obtain the simplest explicit formulae. To prepare for later analysis of experiments, we formulate the theory so as to describe the light beyond the crystal. Appropriately enough, we will use Hamilton’s principle. We begin with the optical path lengths from a point ri on the entrance face z = 0 of the crystal to a point r beyond the crystal at a distance z from the entrance face, for the two waves with transverse wavevector components kx , ky : path length = kx (x − xi ) + ky (y − yi ) + l k02 n2k± − kx2 − ky2 + (z − l) k02 − kx2 − ky2 . (5.1) The exit face of the crystal is z = l, but all our formulae are also valid for z < l, corresponding to observations, made with lenses outside the crystal, of the virtual field inside the crystal, as was appreciated long ago by Potter [1841], and later by Raman [1941] and Raman, Rajagopalan and Nedungadi [1941]. For the geometrical theory and all subsequent analysis, we will use the following dimensionless variables (fig. 7): 1 {x + R0 , y}, w κ = {κx , κy } = κ{cos φκ , sin φκ } ≡ w{kx , ky }, R0 ρ0 ≡ . w ρ = {ξ, η} = ρ{cos φ, sin φ} ≡

(5.2)

28


[2, § 5

Fig. 7. Dimensionless coordinates for conical diffraction theory.

Here, transverse position ρ and transverse wavevectors κ are measured in terms of the beam width w, with ρ measured from the axis of the cylinder. Especially important is the parameter ρ0 , giving the radius of the cylinder in units of w; this single quantity characterises the field of rays – and also of waves, as we shall see – replacing the five quantities n1 , n2 , n3 , l, w. Well-developed rings correspond to ρ0 1. Near the diabolical point, κ is small, so we can write the sheets of the wave surface as

A(−κx ± κ) . nk± = n2 1 + (5.3) k0 n 2 w Now comes an important simplification, to be used in all subsequent analysis: because all angles are small (Table 1), we use the paraxial approximation (Appendix A) to expand the square roots in eq. (5.1). This leads to path length = k0 (n2 l + z − l) + Φ± (κ, ρ, ρ i ),

(5.4)

where 1 Φ± (κ, ρ, ρ i ) ≡ κ · (ρ − ρ i ) ± κρ0 − κ 2 ζ 2

(5.5)

2, § 5]

Poggendorff’s dark ring, Raman’s bright spot

29

and l + n2 (z − l) . (5.6) n 2 k0 w 2 Here ζ is a dimensionless propagation parameter, measuring distance from the ‘focal image plane’ z = l(1 − 1/n2 ) (fig. 7) where the sharpest image of the incident beam (pinhole or laser waist) would be formed if the crystal were isotropic (i.e., if A were zero). The importance of the focal image plane ζ = 0 was first noted in observations by Potter [1841]. By Hamilton’s principle, rays from {ρ i , 0} to {ρ, z} correspond to waves with wavenumber κ for which the optical distance is stationary. Thus ζ ≡

∇κ Φ = ρ − ρ i ± ρ0 eκ − κζ = 0,

(5.7)

that is ρ − ρ i = (κζ ∓ ρ0 )eκ .

(5.8)

Consider for the moment rays from ρ i = 0. Squaring eq. (5.8) and using ρ > 0, κ > 0 leads to (ρ + ρ0 ) eρ , +: two solutions: (a): κ = ζ (ρ0 − ρ) eρ (ρ ρ0 ), (b): κ = (5.9) ζ (ρ − ρ0 ) −: one solution: (c): κ = eρ (ρ ρ0 ). ζ For each ρ, there are two solutions: (a); and either (b) (if ρ ρ0 ) or (c) (if ρ ρ0 ). As fig. 8 illustrates, these correspond to minus the slopes of the surface 1 2 κ ζ ∓ κρ0 . 2

Fig. 8. The ‘Hamiltonian’ surface (5.10) whose normals generate the paraxial rays.

(5.10)

30


[2, § 5

A ray bundle dκ will reach dρ at ζ with intensity proportional to |J |−1 where J is the Jacobian

dρ ∂(ξ, η) ρ0 J = det (5.11) = det =ζ ζ∓ . dκ ∂(κx , κy ) κ The ray equation (5.8) gives |J |−1 =

κ |ρ ± ρ0 | , = ζρ ζ 2ρ

(5.12)

which for the minus sign vanishes linearly on the cylinder ρ = ρ0 , where, from (5.9), the contributing wavevector is the diabolical point κ = 0. Thus the Poggendoff dark ring is an ‘anticaustic’. This geometrical theory also explains an important observation made by Raman, Rajagopalan and Nedungadi [1941] (fig. 1), who emphasized that the ring pattern changes dramatically beyond the crystal; in our notation, the pattern depends strongly on the distance ζ from the focal image plane. Raman saw that as ζ increases, a bright spot develops at ρ = 0. This is associated with the factor 1/ρ in the inverse Jacobian (5.12), corresponding to a singularity on the cylinder axis ρ = 0 – a line caustic, resulting from the ring of normals where the surface (5.10) turns over at κ = ρ0 /ζ (fig. 8). At the turnover, the surface is locally toroidal, so the Raman spot is analogous to the optical glory (Nussenzveig [1992], van de Hulst [1981]) − another consequence of an axial caustic resulting from circular symmetry. Raman, Rajagopalan and Nedungadi [1941] pointed out that the turnover is related to the circle of contact in external conical refraction, so that the two effects discovered by Hamilton cannot be completely separated. The central spot had been observed earlier by Potter [1841], who however failed to understand its origin. Since the factor 1/ρ in (5.12) applies to all z, why does the Raman spot appear only as ζ increases? The reason is that in the full geometrical-optics intensity Igeom , the inverse Jacobian must be modulated by the angular distribution of the incident beam. Let the incident transverse beam amplitude (assumed circular) have Fourier transform a(κ). Important cases are a pinhole with radius w, and a Gaussian beam with intensity 1/e half-width w, for which [in the scaled variables (5.2)] circular pinhole: ap (κ) =

J1 (κ) , κ

1 2 Gaussian beam: aG (κ) = exp − κ . 2

(5.13)

2, § 6]

Belsky and Khapalyuk’s exact paraxial theory of conical diffraction

Incorporating the Jacobian and the ray equations gives

2 1 Igeom (ρ, ζ ) = 2 |ρ − ρ0 |a |ρ − ρ0 |/ζ 2ζ ρ

2 + (ρ + ρ0 )a (ρ + ρ0 )/ζ .

31

(5.14)

The first term represents the bright rings, separated by the Poggendorff dark ring; in this approximation, the rings are symmetric. The second term is weak except near ρ = 0, where it combines with the first term to give the Raman central spot, with strength proportional to 2 1 a(ρ0 /ζ ) , (5.15) 2 ρζ which (for a Gaussian beam, for example) is exponentially small unless ζ approaches ρ0 , and decays slowly with ζ thereafter. Although Igeom captures some essential features of conical refraction, it fails to describe others. Like all applications of geometrical optics, it neglects interference and polarization, and fails where there are geometrical singularities. Here the singularities are of two kinds: a zero at the anticaustic cylinder ρ = ρ0 , and focal divergences on the axial caustic ρ = 0 and in the focal image plane ζ = 0. Interference and polarization can be incorporated by adding the geometrical amplitudes (square roots of Jacobians) rather than intensities, with phases given by the values of Φ± (from eq. (5.5)) at the contributing κ values. We will return to improvements of geometrical-optics theory in Section 7, and compare a more sophisticated version with exact wave theory (see fig. 12 later). The singularity at ζ = 0 is associated with our choice of ρ i = 0. Thus, although Igeom incorporates the effect of w on the angular spectrum of the incident beam, it neglects the more elementary lateral smoothing of the cylinder of conical refraction. One possible remedy is to average Igeom over points ρ i , that is, across the incident beam. In the focal image plane, where the unsmoothed I is singular, such averaging would obscure the Poggendorff dark ring. However, averaging over ρ i is correct only for incoherent illumination, which does not correspond to Lloyd’s and later experiments, where the light is spatially coherent. The correct treatment of the focal image plane, and of other features of the observed rings, requires a full wave theory.

§ 6. Belsky and Khapalyuk’s exact paraxial theory of conical diffraction Nearly 40 years after Raman, the need for a full wave treatment, based on an angular superposition of plane waves, was finally appreciated. Following (and

32


[2, § 6

correcting) an early attempt by Lalor [1972], Schell and Bloembergen [1978a] supplied such a theory, but this was restricted to the exit face of the crystal (that is, it did not incorporate the ζ dependence of the ring pattern); moreover, it was unnecessarily complicated because it did not exploit the simplifying feature of paraxiality. The breakthrough was provided by Belsky and Khapalyuk [1978], who did make use of paraxiality and gave definitive general formulae, obtained after “quite lengthy calculations”, which they omitted. We now give an elementary derivation of the same formulae. The field D = {Dx , Dy } outside the crystal is a superposition of plane waves κ, each of which is the result of a unitary 2 × 2 matrix operator U(κ) acting on the initial vector wave amplitude a(κ). Thus 1 dκ exp{iκ · ρ}U(κ)a(κ). D= (6.1) 2π U(κ) is determined from two requirements: its eigenphases must be the ρ-in-

dependent part of the phases Φ± (5.5), involving the refractive indices n± (κ), and because the diabolical point at κ = 0 is a degeneracy, its eigenvectors (the eigenpolarizations) must change sign as φκ changes by 2π. These are satisfied by U(κ) = exp{−iF(κ)},

(6.2)

where

1 2 sin φκ 1 0 cos φκ F(κ) = κ ζ + ρ0 κ sin φκ − cos φκ 0 1 2 1 = κ 2 ζ 1 + ρ0 κ · S, 2 in which the compact form involves two of the Pauli spin matrices κ · S = σ 3 κx + σ 1 κy . Evaluating the matrix exponential (6.2) gives the explicit form 1 sin ρ0 κ U(κ) = exp − iκ 2 ζ cos ρ0 κ 1 − i κ ·S . 2 κ

(6.3)

(6.4)

(6.5)

We learned from A. Newell of a connection between the evolution associated with U(κ) and analytic functions. Although we do not make use of this connection, it is interesting, and we describe it in Appendix B. It is not hard to show that U(κ) possesses the required eigenstructure, namely U(κ)d± (κ) = λ± (κ)d± (κ),

(6.6)

2, § 6]

Belsky and Khapalyuk’s exact paraxial theory of conical diffraction

33

where

1 λ± (κ) = exp i − κ 2 ζ ± ρ0 κ , 2

1 cos 2 φκ sin 12 φκ d+ (κ) = , d− (κ) = . sin 12 φκ − cos 12 φκ

(6.7)

Without significant loss of generality, we can regard the incident beam as uniformly polarized and circularly symmetric, that is

d0x = a(κ)d0 , a(κ) = a(κ) (6.8) d0y where a(κ) is the Fourier amplitude introduced in the previous section [cf. eq. (5.13)], related to the transverse incident beam profile D0 (ρ) by ∞ D0 = d0 D0 (ρ) = d0

dκ κJ0 (κρ)a(κ).

(6.9)

0

Combining eqs. (6.1), (6.5) and (6.9), we obtain, after elementary integrations, and recalling ρ = ρ{cos φ, sin φ},

B1 sin φ B0 + B1 cos φ d0 , D= (6.10) B0 − B1 cos φ B1 sin φ where ∞ B0 (ρ, ζ ; ρ0 ) =

1 2 dκ κa(κ) exp − iζ κ J0 (κρ) cos(κρ0 ), 2

0

∞ B1 (ρ, ζ ; ρ0 ) =

1 dκ κa(κ) exp − iζ κ 2 J1 (κρ) sin(κρ0 ). 2

(6.11)

0

These are the fundamental integrals of the Belsky–Khapalyuk theory. For unpolarized or circularly polarized incident light, eq. (6.10) gives the intensity as I = D∗ · D = |B0 |2 + |B1 |2 .

(6.12)

A useful alternative form for D, in terms of the eigenvectors d± evaluated at the direction φ of ρ, and involving A+ ≡ B0 + B1 ,

A− ≡ B0 − B1 ,

(6.13)

34

is


1 1 D = A+ d0x cos φ + d0y sin φ d+ (ρ) 2 2

1 1 + A− d0x sin φ − d0y cos φ d− (ρ). 2 2

[2, § 7

(6.14)

D is a single-valued function of ρ, although d± (ρ) change sign around the origin. The corresponding intensity, 1 1 2 2 I = |A+ | d0x cos φ + d0y sin φ 2 2 (6.15) 1 1 2 2 + |A− | d0x sin φ − d0y cos φ , 2 2 contains no oscillations resulting from interference between A+ and A− , because d+ and d− are orthogonally polarized. But, as we will see in the next section, the exact rings do possess oscillations, from A+ and A− individually. Associated with the polarization structure of the beam (6.14) (see also fig. 6) is an interesting property of the angular momentum (Berry, Jeffrey and Mansuripur [2005]): for well-developed rings (large ρ0 ), the initial angular momentum, which is pure spin, is transformed by the crystal into pure orbital, and reduced in magnitude, the difference being imparted to the crystal.

§ 7. Consequences of conical diffraction theory We begin our explanation of the rich implications of the paraxial wave theory by obtaining an explicit expression for the improved geometrical-optics theory anticipated in Section 5. The derivation proceeds by replacing the Bessel functions in eq. (6.11) by their asymptotic forms, and then evaluating the integrals by their stationary-phase approximations. The stationary values of κ specify the rays (5.9), and the result, for the quantities A± , is: √

|ρ0 − ρ| |ρ0 − ρ| (ρ0 − ρ)2 1 if ρ < ρ0 A+geom = a exp i × , √ −i if ρ > ρ0 ζ ρ ζ 2ζ

√ ρ0 + ρ ρ0 + ρ (ρ0 + ρ)2 a exp i . A−geom = −i √ (7.1) ζ ρ ζ 2ζ For an incident beam linearly polarized in direction γ , that is

cos γ , d0 = sin γ

(7.2)

2, § 7]

Consequences of conical diffraction theory

the resulting electric field (6.14) is

1 1 D = A+geom cos φ − γ d+ (φ) + A−geom sin φ − γ d− (φ), 2 2

35

(7.3)

and the intensity is Igeom1 = D∗ · D

1 1 = |A+geom |2 cos2 φ − γ + |A−geom |2 sin2 φ − γ . (7.4) 2 2 The elementary geometrical-optics intensity Igeom (5.14) is recovered for unpolarized light by averaging over γ , or by superposing intensities for any two orthogonal incident polarizations. Without such averaging, the first term in Igeom1 gives the lune-shaped ring structure observed by Lloyd (fig. 6) with linearly polarized light, and both terms combine to give the unpolarized geometrical central spot. Next, we examine the detailed structure of the Poggendorff rings, starting with the focal image plane ζ = 0 where the rings are most sharply focused. Figure 9

Fig. 9. Emergence of double ring structure in the focal image plane ζ = 0 as ρ0 increases, computed from eq. (6.10) for an unpolarized Gaussian incident beam. (a) ρ0 = 0.5; (b) ρ0 = 0.8; (c) ρ0 = 2; (d) ρ0 = 4.

36


[2, § 7

shows how the double ring emerges as ρ0 increases. Significant events, corresponding to sign changes in the curvature of I at ρ = 0, are: the birth of the first ring when ρ0 = 0.627; the birth of a central maximum when ρ0 = 1.513; and the birth of the second bright ring when ρ0 = 2.669. As ρ0 increases further, the rings become localized near ρ = ρ0 , but their shape is independent of ρ0 and depends only on the form of the incident beam. To obtain a formula for this invariant shape, we cannot use geometrical optics for the largeρ0 asymptotics. The reason is that although eq. (7.1) is a good approximation to A− , it fails for the function A+ that determines the rings, because the ray contribution comes from the neighbourhood of the end-point κ = 0 of the integrals. So, although the Bessel functions can still be replaced by their asymptotic forms, the stationary-phase approximation is invalid. Near the rings, with ρ ≡ ρ − ρ0 ,

(7.5)

Bessel asymptotics gives A+ as 1 A+rings = √ f (ρ, ζ ), ρ

(7.6)

where f (ρ, ζ ) =

2 π

∞ dκ

√

1 1 κa(κ) cos κρ − π exp − iκ 2 ζ . 4 2

(7.7)

0

In the focal image plane ζ = 0, f can be evaluated analytically for a pinhole incident beam, in terms of elliptic integrals E and K. With the conventions in Mathematica (Wolfram [1996]), f0p (ρ) ≡ f (ρ, 0) ⎧ √

⎪ 2 ρ 2 2 2 ⎪ ⎪ 1 − ρE + K √ ⎪ ⎪ π 1 − ρ 1 − ρ ⎪ 1 − ρ ⎪ ⎪ ⎪ ⎪ (ρ < −1), ⎨

1 1 = 2 ⎪ −K (1 − ρ) + 2E (1 − ρ) ⎪ ⎪ π 2 2 ⎪ ⎪ ⎪ ⎪ (|ρ| < 1), ⎪ ⎪ ⎪ ⎩ 0 (ρ > 1).

(7.8)

Figure 10(a) shows the corresponding intensity. To our knowledge, this unusual focused image has not been seen in any experiment.

2, § 7]


37

Fig. 10. Intensity functions f02 (ρ) of rings in the focal image plane, for (a) a circular pinhole (7.8), and (b) a Gaussian beam (7.9).

For a Gaussian beam, the focal image can be expressed in terms of Bessel functions: f0g (ρ) ≡ f (ρ, 0)

|ρ|3/2 exp − 14 ρ 2 1 2 1 2 K3 ρ + sgn ρK 1 ρ = √ 4 4 4 4 2 2π

√ 1 2 1 2 + π 2(−ρ) I 3 (7.9) ρ − I 1 ρ . 4 4 4 4

Figure 10(b) shows the corresponding intensity, and fig. 11 shows how the approximation f0g gets better as ρ0 increases. The focal image functions f0p and f0g have been discussed in detail by Belsky and Stepanov [1999], Berry [2004b] and Warnick and Arnold [1997]; of several equivalent representations, eqs. (7.8) and (7.9) are the most convenient. Away from the focal image plane, that is as ζ increases from zero, secondary rings develop, in the form of oscillations within the inner bright ring [the solid curves in fig. 12(b–e)]; these were discovered in numerical computations by Warnick and Arnold [1997]. Using asymptotics based on eq. (7.7), the secondary

38


[2, § 7

Fig. 11. Ring intensity in the focal image plane for a Gaussian incident beam, calculated exactly (solid curves), and in the local approximation (7.6) and (7.9) (dashed curves), for (a) ρ0 = 5; (b) ρ0 = 10; √ (c) ρ0 = 20. [The divergences at ρ = 0 for the approximate rings come from the factor 1/ ρ in eq. (7.6).]

rings can be interpreted as interference between a geometrical ray and a wave scattered from the diabolical point in k space (Berry [2004b]); the associated mathematics led to a surprising general observation in the asymptotics of competing exponentials (Berry [2004a]). All the features so far discussed are displayed in the simulated image and cutaway in fig. 13, which can be regarded as a summary of the main results of conical diffraction theory. The parameters (ρ0 = 20, ζ = 8) are chosen to display the two bright rings, the Poggendorff dark ring, the nascent Raman spot (whose intensity will increase for larger ζ ), and the secondary rings.

2, § 7]


39

Fig. 12. Intensity for a Gaussian incident beam for ρ0 = 20 and (a) ζ = 1; (b) ζ = 4; (c) ζ = 6; (d) ζ = 8; (e) ζ = 10; (f) ζ = 20. Solid curves: exact theory; dashed curves: refined geometrical-optics theory (7.13).

As ζ increases further, the inner rings approach the Raman spot [solid curves in fig. 12(e,f)], and are then described by Bessel functions (Berry [2004b]): 2 ρ0 π 1 ρ0 ρρ0 exp i − π a J , 0 2ζ 4 ζ ζ 2ζ 3 2 ρ0 π 3 ρ0 ρρ0 B1 (ρ, ζ ; ρ0 ) ≈ ρ0 exp i − π a J , 1 2ζ 4 ζ ζ 2ζ 3

B0 (ρ, ζ ; ρ0 ) ≈ ρ0

(ρ ρ0 , ζ 1).

(7.10)

Figure 14 illustrates how well the approximation J02 +J12 approximates the true intensity. The weak oscillations (shoulders at zeros of J1 (ρρ0 /ζ )) are the result of interference involving the next large-ρ0 correction term to the geometrical-optics approximation (7.1).

40


[2, § 7

Fig. 13. (a) Density plot and (b) cutaway 3D plot, of conical diffraction intensity for ρ0 = 20, ζ = 8, showing bright rings B, Poggenforff dark ring P, Raman spot R and secondary rings S.

In the important special case of a Gaussian incident beam, the conical diffraction integrals (6.11) for general ζ can be obtained by complex continuation from the focal image plane ζ = 0. This useful simplification is a variant of the complexsource trick of Deschamps [1971]. It is based on the observation 1 2 1 2 1 2 exp − κ exp − iκ ζ = exp − κ (1 + iζ ) , 2 2 2

(7.11)

2, § 8]

Experiments

41

Fig. 14. Weak oscillations decorating the Raman spot for a Gaussian incident beam; thick curve: exact intensity; thin curve: Bessel approximation (7.10), for ρ0 = 20, ζ = 20.

and leads to

1 ρ ρ0 B0,1 (ρ, ζ ; ρ0 ) = B0,1 √ , 0, √ . 1 + iζ 1 + iζ 1 + iζ

(7.12)

In geometrical optics, the same trick leads to a further refinement, incorporating a(κ) into the stationary-phase approximation (so that the rays are complex): A+geom1 √ |ρ0 − ρ| (ρ0 − ρ)2 = √ exp iζ (ζ − i) ρ 2(ζ 2 + 1) (ρ0 − ρ)2 1 if ρ < ρ0 , × × exp − −i if ρ > ρ0 2(ζ 2 + 1) √ ρ0 + ρ (ρ0 + ρ)2 (ρ0 + ρ)2 A−geom1 = −i exp − . √ exp iζ (ζ − i) ρ 2(ζ 2 + 1) 2(ζ 2 + 1)

(7.13)

Figure 12 shows how accurately this reproduces the oscillation-averaged rings and the Raman spot when ζ is not small. Near the focal plane, however, the geometrical approximation is only rough, even though the refinement eliminates the singularity at ζ = 0.

§ 8. Experiments Experimental studies of conical refraction are few, probably because of the difficulty of finding, or growing, crystals of sufficient quality and thickness. Table 1

42


[2, § 8

lists the investigations known to us. We have already mentioned the pioneering observations of Lloyd, Poggendorff, Potter and Raman. Lloyd [1837] stated that he used several pinholes, but gave only the size of the largest (which he used as a way of measuring A), so we do not know the values of w, and hence ρ0 , corresponding to his rings. Poggendorff [1839] gave no details of his experiment, except that it was performed with aragonite. Potter [1841] gave a detailed description of his experiments with aragonite, but misinterpreted his observation of the double ring as evidence against the diabolical connection of the sheets of the wave surface, leading to polemical criticisms of Hamilton and Lloyd. Raman, Rajagopalan and Nedungadi [1941] chose naphthalene, whose crystals have a large cone angle A and which they could grow in sufficient thickness. They emphasize that they did not see the Poggendorff dark ring in their most sharply focused images; but in the focal image plane the two bright rings are very narrow (comparable with w ≈ 0.5 µm), so they might not have resolved them. Schell and Bloembergen [1978a] compared measured ring profiles with numerically integrated wave theory and with the stationary-phase (geometrical optics) approximation, in a plane corresponding to the exit face of their aragonite crystal. From the data in Table 1, it follows that this corresponds to ζ ≈ 1.3, a regime in which there is no central spot and no secondary rings, and, as they report, geometrical optics is a reasonable approximation. Perkal’skis and Mikhailychenko [1979] and Mikhailychenko [2005] report large-scale demonstrations of internal and external conical refraction with rhombic sulfur. In an ingenious investigation, Fève, Boulanger and Marnier [1994] used KTP in the form of a ball rather than a slab. The theory of Section 6 applies, provided the parameters are interpreted as follows:

d ρ0 → ρball = ρ0 1 − 2(n2 − 1) , l l + d(2 − n2 ) ζ → ζball = (8.1) , n2 kw 2 where l is the diameter of the ball and d is the distance between the exit face of the ball and the observation plane. In this case, a cone emerges from the ball, giving rings whose radius wρball depends on d; the radius vanishes at a point, close to the ball, where the generators of the cone (that is, the rays) cross. They obtain good agreement between the measured ring profile and the geometrical-optics intensity, which is a good approximation in their regime of enormous effective ρ0 and relatively modest ζ (Table 1). The many wave-optical and geometrical-optical phenomena predicted by the detailed theory of conical diffraction have recently been observed by Berry, Jef-

2, § 9]

Concluding remarks

43

Fig. 15. Theoretical and experimental images for a monoclinic double tungstate crystal illuminated by a Gaussian beam (Berry, Jeffrey and Lunney [2006]), for ρ0 = 60. (a,b) ζ = 3; (c,d) ζ = 12; (e,f) ζ = 30; (a,c,e) theory; (b,d,f) experiment.

frey and Lunney [2006] in a crystal of the monoclinic double tungstate material KGd(WO4 )2 , obtained from Vision Crystal Technology (VCT [2006]). The agreement, illustrated in figs. 15 and 16, is quantitative as well as qualitative. Their measurements confirm predictions for the ζ -dependence of the radii and separation of the main rings, and of the sizes of the interference fringes: the secondary rings decorating the inner bright ring, and the rings decorating the Raman spot.

§ 9. Concluding remarks Although conical diffraction exemplifies a fundamental feature of crystal optics, namely the diabolical point, it can also be regarded as a curiosity, because the effect seems to occur nowhere in the natural universe, and no practical application seems to have been found. To forestall confusion, we should immediately qualify these assertions. It is Hamilton’s idealized geometry (collimated beam, parallel-sided crystal slab, etc.) that does not occur in nature. In generic situations, for example an anisotropic medium that is also inhomogeneous, it is likely for a ray to encounter a

44


[2, § 9

Fig. 16. As fig. 15, with (a) ζ = 3; (b) ζ = 12; (c) ζ = 30. Solid curves: theory; dashed curves: angular averages of experimental images. (After Berry, Jeffrey and Lunney [2006].)

point where its direction is locally diabolical, corresponding to a local refractiveindex degeneracy (Naida [1979]); or, in quantum mechanics, a coupled system with fast and slow components (e.g., a molecule) can encounter a degeneracy of the adiabatically evolving fast sub-system. In understanding such generic situations, the analysis of the idealized case will surely play a major part. On the practical side, it is possible that the bright cylinder of conical refraction can be applied to trap and manipulate small particles; this is being explored, but the outcome is not yet clear. And a related phenomenon associated with the diabolical point, namely the conoscopic interference figures seen under polarized

2, Appendix A]

Paraxiality

45

illumination of very thin crystal plates, is a well-established identification technique in mineralogy (Liebisch [1896]). As we have tried to explain, our understanding of the effect predicted by Hamilton is essentially complete. We end by describing several generalizations that are less well understood. Although the theory of Section 6 applies to any paraxial incident beam, detailed explorations have been restricted to pinhole and Gaussian beams. A start has been made by King, Hogervorst, Kazak, Khilo and Ryzhevich [2001] and Stepanov [2002] in the exploration of other types of beam, for example Laguerre–Gauss and Bessel beams. A further extension is to materials that are chiral (optically active) as well as biaxially birefringent. This alters the mathematical framework, because chirality destroys the diabolical point by separating the two sheets of the wave surface, reflecting the change of the dielectric matrix from real symmetric to complex Hermitian. There have been several studies incorporating chirality (Belsky and Stepanov [2002], Schell and Bloembergen [1978b], Voigt [1905b]), leading to the recent identification of the central new feature: the bright cylinder of conical refraction is replaced by a ‘spun cusp’ caustic (Berry and Jeffrey [2006a]). So far this has not been seen in any experiment. The introduction of anisotropic absorption (dichroism) brings a more radical change: each diabolical point splits into two branch-points, reflecting the fact that the dielectric matrix is now non-Hermitian (Berry [2004c], Berry and Dennis [2003]). The dramatic effects of absorption on the pattern of emerging light have been recently described by Berry and Jeffrey [2006b]. The combined effects of dichroism and chirality have been described by Jeffrey [2007]. The final generalization incorporates nonlinearity. Early results were reported by Schell and Bloembergen [1977] and Shih and Bloembergen [1969], and the subject has been revisited by Indik and Newell [2006].

Acknowledgements Our research is supported by the Royal Society. We thank Professor Alan Newell for permission to reproduce the argument in Appendix B.

Appendix A: Paraxiality The paraxial approximation requires small angles, equivalent to replacing cos θ by 1 − θ 2 /2, that is, to assuming θ 4 /24 1 for all wave deflection angles θ . In

46


[2, Appendix A

conical refraction, deflections are determined by the half-angle of the ray cone, which from eq. (3.7) is

1 A = arctan n22 αβ . (A.1) 2 This is indeed small in practice, because of the near-equality of the three refractive indices. For the Lloyd [1837] experiment on aragonite, the Berry, Jeffrey and Lunney [2006] experiment on MDT, and the Raman, Rajagopalan and Nedungadi [1941] experiment on naphthalene (whose cone angle is the largest yet reported), the data in Table 1 give 1 4 = 3.3 × 10−9 , A 24 Lloyd 1 4 = 9.1 × 10−9 , A (A.2) 24 Berry 1 4 = 8.5 × 10−6 . A 24 Raman As is often emphasized, internal and external conical refraction are associated with different aspects of the geometry of the wave surface. But in the paraxial regime the difference between the cone angles (A and Aext respectively) disappears. To explore this, we first note that (Born and Wolf [1999])

1 Aext = arctan n1 n3 αβ . (A.3) 2 In terms of the refractive-index differences n 2 − n1 n 3 − n2 μ1 ≡ (A.4) , μ3 ≡ , n2 n2 we have, to lowest order, √ A ≈ Aext ≈ μ1 μ3 , (A.5) which is proportional to the refractive-index differences. The difference between the angles is 1 2 √ 2 A − Aext ≈ μ1 μ3 μ1 − μ3 + 3μ1 − 2μ1 μ3 + 3μ3 , (A.6) 4 which is proportional to the square of the index differences, except when the two differences are equal, when it is proportional to the cube. For the aragonite, MDT and naphthalene experiments, ALloyd − Aext,Lloyd = 8.9 × 10−2 ALloyd , ABerry − Aext,Berry = −4.4 × 10−3 ABerry , ARaman − Aext,Raman = −2.7 × 10

−4

ARaman .

(A.7)

2, Appendix B]

Conical refraction and analyticity

47

Appendix B: Conical refraction and analyticity We seek the paraxial differential equation for the evolution of the wave inside the crystal. Within the framework of Section 6, the wave can be described by setting z = l and regarding ζ = l/k0 w 2 as a variable, enabling the evolution operator (6.2) and (6.3) to be written as

1 2 (B.1) U(κ) = exp −iζ κ 1+Γκ ·S , 2 where Γ ≡ Ak0 w.

(B.2)

Writing κ as the differential operator κ = −i∇ = −i{∂ξ , ∂η },

(B.3)

and differentiating eq. (B.1) with respect to ζ leads to

1 2 1 0 ∂η Dξ ∂ξ Dξ i∂ζ = − ∇ − iΓ . Dη ∂η −∂ξ Dη 0 1 2

(B.4)

The differential operator connects the components of D in the same way as the Cauchy–Riemann conditions for analytic functions. The relation is clearer when D is expressed in a basis of circular polarizations and ξ and η are replaced by complex variables, as follows 1 1 D− ≡ √ (Dξ + iDη ), D+ ≡ √ (Dξ − iDη ), 2 2 w+ ≡ ξ + iη, w− ≡ ξ − iη. Then eq. (B.4) becomes

D+ 1 0 0 i∂ζ = −2 ∂w+ ∂w− + iΓ D− ∂w− 0 1

∂w+ 0

(B.5)

D+ D−

. (B.6)

Thus D+ and D− propagate unchanged if D+ is a function of w+ alone and D− is a function of w− alone, that is if the field components are analytic or anti-analytic functions. For such fields, there is no conical refraction, and (for example) a pattern of zeros in the incident field propagates not conically but as a set of straight optical vortex lines parallel to the ζ direction. However, these analytic functions do not represent realistic optical beams, which must decay in all directions φ as ρ → ∞.

48


[2

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Indik, R.A., Newell, A.C., 2006, Conical refraction and nonlinearity, Optics Express 14, 10614– 10620. Jeffrey, M.R., 2007, The spun cusp complexified: complex ray focusing in chiral conical diffraction, J. Opt. A. 9, 634–641. King, T.A., Hogervorst, W., Kazak, N.S., Khilo, N.A., Ryzhevich, A.A., 2001, Formation of higherorder Bessel light beams in biaxial crystals, Opt. Commun. 187, 407–414. Lalor, E., 1972, An analytical approach to the theory of internal conical refraction, J. Math. Phys. 13, 449–454. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P., 1984, Electrodynamics of Continuous Media, Pergamon, Oxford. Liebisch, T., 1896, Grundriss der Physikalischen Krystallographie, Von Veit & Comp, Leipzig. Lloyd, H., 1837, On the phenomena presented by light in its passage along the axes of biaxial crystals, Trans. R. Ir. Acad. 17, 145–158. Ludwig, D., 1961, Conical refraction in crystal optics and hydromagnetics, Comm. Pure. App. Math. 14, 113–124. Lunney, J.G., Weaire, D., 2006, The ins and outs of conical refraction, Europhysics News 37, 26–29. Mead, C.A., Truhlar, D.G., 1979, On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei, J. Chem. Phys. 70, 2284–2296. Mikhailychenko, Y.P., 2005, Large scale demonstrations on conical refraction, http://www.demophys. tsu.ru/Original/Hamilton/Hamilton.html. Naida, O.N., 1979, “Tangential” conical refraction in a three-dimensional inhomogeneous weakly anisotropic medium, Sov. Phys. JETP 50, 239–245. Nussenzveig, H.M., 1992, Diffraction Effects in Semiclassical Scattering, University Press, Cambridge. Nye, J.F., 1999, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, Institute of Physics Publishing, Bristol. O’Hara, J.G., 1982, The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd (1832–1833), Proc. R. Ir. Acad. 82A, 231–257. Perkal’skis, B.S., Mikhailychenko, Y.P., 1979, Demonstration of conical refraction, Izv. Vyss. Uch. Zav. Fiz. 8, 103–105. Poggendorff, J.C., 1839, Ueber die konische Refraction, Pogg. Ann. 48, 461–462. Potter, R., 1841, An examination of the phaenomena of conical refraction in biaxial crystals, Phil. Mag. 8, 343–353. Raman, C.V., 1941, Conical refraction in naphthalene crystals, Nature 147, 268. Raman, C.V., Rajagopalan, V.S., Nedungadi, T.M.K., 1941, Conical refraction in naphthalene crystals, Proc. Ind. Acad. Sci. A 14, 221–227. Schell, A.J., Bloembergen, N., 1977, Second harmonic conical refraction, Opt. Commun. 21, 150–153. Schell, A.J., Bloembergen, N., 1978a, Laser studies of internal conical diffraction. I. Quantitative comparison of experimental and theoretical conical intensity dirstribution in aragonite, J. Opt. Soc. Amer. 68, 1093–1098. Schell, A.J., Bloembergen, N., 1978b, Laser studies of internal conical diffraction. II. Intensity patterns in an optically active crystal, α-iodic acid, J. Opt. Soc. Amer. 68, 1098–1106. Shih, H., Bloembergen, N., 1969, Conical refraction in second harmonic generation, Phys. Rev. 184, 895–904. Silverman, M., 1980, The curious problem of spinor rotation, Eur. J. Phys. 1, 116–123. Soskin, M.S., Vasnetsov, M.V., 2001, Singular optics, in: Progress in Optics, vol. 42, North-Holland, Amsterdam, pp. 219–276. Stepanov, M.A., 2002, Transformation of Bessel beams under internal conical refraction, Opt. Commun. 212, 11–16.

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Uhlenbeck, K., 1976, Generic properties of eigenfunctions, Am. J. Math. 98, 1059–1078. Uhlmann, A., 1982, Light intensity distribution in conical refraction, Commn. Pure. App. Math. 35, 69–80. van de Hulst, H.C., 1981, Light Scattering by Small Particles, Dover, New York. VCT, 2006, Home page of Vision Crystal Technology, http://www.vct-ag.com/. Voigt, W., 1905a, Bemerkung zur Theorie der konischen Refraktion, Phys. Z. 6, 672–673. Voigt, W., 1905b, Theoretisches unt Experimentelles zur Aufklärung des optisches Verhaltens aktiver Kristalle, Ann. Phys. 18, 645–694. Von Neumann, J., Wigner, E., 1929, On the behavior of eigenvalues in adiabatic processes, Phys. Z. 30, 467–470. Warnick, K.F., Arnold, D.V., 1997, Secondary dark rings of internal conical refraction, Phys. Rev. E 55, 6092–6096. Wolfram, S., 1996, The Mathematica Book, University Press, Cambridge.


Chapter 3

Historical papers on the particle concept of light by

Ole Keller Institute of Physics and Nanotechnology, Skjernvej 4, DK-9220 Aalborg Øst, Denmark e-mail: [email protected]

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(07)50003-X 51

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

§ 2. Einstein’s light quanta . . . . . . . . . . . . . . . . . . . . . . . . . .

56

§ 3. Guiding fields for light quanta . . . . . . . . . . . . . . . . . . . . .

65

§ 4. Light quanta and matter waves . . . . . . . . . . . . . . . . . . . . .

70

§ 5. Photon wave mechanics . . . . . . . . . . . . . . . . . . . . . . . . .

79

§ 6. Eikonal equation for the photon . . . . . . . . . . . . . . . . . . . .

91

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

52

§ 1. Introduction In a letter to the Editor of Nature, Gilbert N. Lewis in 1926 advanced the hypothesis that the light quantum of Albert Einstein was a new kind of atom, uncreatable and indestructible (Lewis [1926]). He proposed the name photon for his new atom, which not only acted as the carrier of radiant energy but after absorption in an atom was assumed to persist as an essential constituent of this atom until it later was sent out on a new journey towards another atom. The name photon almost immediately became popular, but the hypothesis of absolute photon conservation was untenable. Elementary textbooks often leave the impression that the photon, once created in an emission process, is a small spherical or pointlike glob of energy moving through space with the vacuum speed of light. This impression perhaps dates back to Einstein’s proposal of the light (energy) quantum concept in 1905 (Einstein [1905]). Einstein’s light quantum idea did not receive wide acceptance from leading physicists for almost twenty years. Only around 1921–22 the opposition to the light quantum hypothesis begin to wane. When Arthur H. Compton [1923] and Peter Debye [1923] in 1923, independently of each other, with great success had used the light quantum model to analyze the X-ray scattering from electrons the pendulum swung in favor of the particle concept of light, and Arnold Sommerfeld went so far as to say that it “sounded the death knell” of the wave theory. Niels Bohr rejected the existence of light quanta for many years, and as late as in 1924, i.e., after Compton’s and Debye’s remarkable explanation of the change of wavelength of radiation scattered by free electrons, he (together with John C. Slater and Hendrik A. Kramers) made a last desperate attempt to avoid the particle concept of light (Bohr, Kramers and Slater [1924]). To account for interference, diffraction, dispersion, etc., it was necessary right from the beginning to work with a wave–particle duality principle. Although a wave-mechanical formalism involving a statistical description of the possibilities for observing (detecting) the light quantum at a definite position in space–time could be established, the point-particle picture of light definitely had to give way to the quantum field theory. According to this theory a photon is an “elementary” quantum excitation of the electromagnetic field. 53

54

Historical papers on the particle concept of light

[3, § 1

In the present article I shall briefly describe and discuss important stages in the historical development of the particle concept of light, starting with Max Planck’s radiation law (Planck [1900c]) and Einstein’s light-quantum hypothesis (Einstein [1905]) and ending with Julius R. Oppenheimer’s wave mechanical description of light quanta from 1931 (Oppenheimer [1931]). Most of the material presented here is based on a reading of theoretical papers on light quanta, and aspects related to these, from the period ∼1900–1931. To keep the length of this article within the frames for this anniversary volume of Progress in Optics a number of important historical papers are only described en passant, or not mentioned at all. My personal interest in the history of the particle concept of light is partly related to my theoretical studies of the possibilities for spatial localization of (quantized) light fields, but since I am not a historian of physics the following description should be regarded as the writings of a dedicated amateur. The first third of the article (on Einstein’s light quanta and Planck’s radiation law) is well known to many physicists; the second third (on guiding fields for light quanta, the Bohr–Kramers– Slater theory, and de Broglie’s world vector relation between the energymomentum and frequency-wave vector four-vector and the relativistic paradox which led him to the general statement of the wave–particle duality) perhaps is best known among historians of physics; and the final third (on photon wave mechanics) is unknown to most scientists nowadays, but deserves renewed attention. In Section 2, Einstein’s light quantum idea is presented. Einstein concluded that monochromatic radiation of low density behaved thermodynamically as though it consisted of independent point particles of energy E = hν, and he came to this conclusion by a study of the entropy of blackbody radiation in the high-frequency (Wien) limit. Before the presentation of the light quantum hypothesis I briefly discuss the considerations on the entropy of independent harmonic oscillators which led Planck to his famous radiation law. In the Wien limit the mean number of photons is small and the particle aspects of radiation therefore particularly pronounced. Four years after the introduction of the light quantum hypothesis Einstein used a certain relation between the variance of the fluctuations in the field energy and the second derivative of the entropy to determine the energy fluctuations in the blackbody radiation (Einstein [1909]). The resulting fluctuation formula perhaps gave physicists the first glimpse of the wave–particle duality of light. Further analyses of Einstein’s 1909 paper of the fluctuations in the radiation pressure indicate that the light quantum carries a momentum of magnitude p = hν/c0 . This expression for the light quantum momentum later appeared in Einstein’s fundamental 1917 paper, entitled “Zur Quantentheorie der Strahlung”, in connection with a study of the momentum transfer between gas molecules and the radiation field (Einstein [1917a]). In the same paper Einstein also presented

3, § 1]

Introduction

55

his famous derivation of Planck’s radiation law via the introduction of the three elementary processes of interaction between gas molecules and radiation (spontaneous and stimulated emission, absorption). In one of the pioneering papers on quantum mechanics Max Born, Werner Heisenberg and Pascual Jordan [1926] derived Einstein’s fluctuation formula for blackbody radiation by quantization of the eigenvibrations of a cavity field. That the field behaves like a sum of independent harmonic oscillators had been anticipated twenty years earlier by Paul Ehrenfest [1906]. The fact that the energy of an eigenvibration of frequency ν apart from an additive constant, the so-called zero-point energy, should be an integer of hν gives a strong association with the light quantum problem. In spite of this the authors point out that the most important aspect of this problem “namely the phenomenon of coupling of distant atoms, (for this problem) does not enter at all into the formulation of our questions regarding the vibrations of a cavity”. The work of Born, Heisenberg and Jordan, known as the Drei-Männer-Arbeit, marks the birth of quantum electrodynamics (QED), and in a broader context of quantum field theory. In its fully developed form QED is capable of treating rigorously “the phenomenon of coupling of distant atoms” as it was called. Born, Heisenberg and Jordan do make a link between the statistics of cavity vibrations and light quanta. An early attempt to derive the light quanta statistics is due to Debye [1910], and the general theory was developed by Satyendra N. Bose [1924] and Einstein [1924a, 1924b]. The Bose–Einstein statistics of quantized light is not discussed in this article. If light consists of pointlike particles carrying energy and momentum how do such particles move in space, and how does one establish the connection to the classical wave theory which had proven to be so successful in many respects? In the period of 1923–1924 important attempts were made to answer these questions by setting up different kinds of guiding fields for light quanta. Thus Slater, as described in Section 3, suggested that even atoms in stationary states communicate with each other via a so-called virtual field (Slater [1924]). The virtual field, which can be calculated formally from the Maxwell theory, was assumed to guide the discrete light quanta of Einstein. In Copenhagen the virtual field became so popular that Bohr and Kramers, who did not believe in the existence of the light quantum, after a discussion with Slater (!), and together with him, suggested as already mentioned the framework for a quantum theory of radiation (BKS) in which the virtual field played a crucial role but the light quantum was eliminated. As explained briefly in Section 3, the BKS suggestion implied an abandoning of the strict conservation of energy and momentum. In Section 4, I discuss how Louis de Broglie in 1923–1924 seeking harmony between special relativity and the quantum energy hypothesis, E = hν, came to the conclusion that the light corpuscle

56


[3, § 2

was accompanied by a so-called phase wave whose group velocity was identical to the velocity of the moving particle (De Broglie, L. [1923a, 1923b, 1923c, 1924]). In his Doctoral Thesis de Broglie established the now famous proportionality between the world vectors of the energy-momentum and frequency-wavevector (De Broglie, L. [1925]). The proportionality factor is Planck’s constant (divided by 2π). De Broglie’s insight helped Erwin Schrödinger to establish his famous (nonrelativistic) wave equation for massive particles (electrons) in 1926 (Schrödinger [1926a, 1926b, 1926c, 1926d]). In his search for a suitable wave equation Schrödinger made use of the fact that the Hamilton–Jacobi equation for the characteristic function in classical point-particle mechanics has a mathematical similarity to the eikonal equation of geometrical optics (Schrödinger [1926b]). But the wave mechanics for a punctiform electron did not easily lead to photon wave mechanics. To uphold the idea of a pointlike light quantum one had to establish a corresponding wave equation in which the appearing wave function in one way or another could describe the statistical probabilities for finding (detecting) a pointlike photon in the various space–time points. In Section 5, I present and discuss the photon wave mechanical theories of Lev D. Landau and Rudolf E. Peierls [1930], and Oppenheimer [1931]. Although much effort is devoted in both these papers to establishing the connection to the previously established quantum field theory of light, I mainly treat the wave mechanics of the free photon. The Landau– Peierls theory results in a less satisfactory spatially nonlocal relation between the photon wave function and the electromagnetic field, whereas Oppenheimer’s approach leads to a spatially local relation between the aforementioned quantities. Oppenheimer’s theory was satisfactory from a relativistic point of view and incorporated the photon helicity concept in a rigorous manner. Although the Schrödinger equation can be rewritten in the form of a simple quantum Hamilton– Jacobi equation, a Hamilton–Jacobi approach to photon wave mechanics was not discussed by Landau–Peierls, nor Oppenheimer. In Section 6, I finish my article with a brief discussion of this approach for the photon. § 2. Einstein’s light quanta In 1905 Einstein published in the Annalen der Physik an epoch-making paper entitled “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” (Einstein [1905]). In this paper, first translated to English sixty years later (Arons and Peppard [1965]), Einstein presents the line of thought and the facts which led him to the idea that (in translation from the introduction) the energy of a light ray spreading out from a point source is not continuously distributed over an ever increasing volume, but consists of a finite number

3, § 2]

Einstein’s light quanta

57

of energy quanta, which are localized at points in space, which move without dividing, and which can be absorbed or emitted as a whole. In the paper Einstein uses alternatively the words energy quantum (Energiequant) and light quantum (Lichtquant). Einstein emphasizes that the wave theory of light (Maxwell theory) has worked well in the representation of purely optical phenomena such as diffraction, reflection, refraction, dispersion, etc., but nevertheless it seems to him that it is still conceivable that the Maxwell theory may lead to contradictions with experience when it is applied to the phenomena of generation and conversion of light. From Einstein’s account it appears that he was worried not least by the impossibility of fitting the blackbody radiation spectrum into the classical Maxwell theory. As evidence in favor of his corpuscular theory he appealed to fluorescence, photoelectricity, and photo-ionization data. In most modern textbooks the 1905 paper is usually referred to as “Einstein’s paper on the photoelectric effect”. In fact, Einstein’s discussion of the photoelectric effect covers less than three pages of the seventeen-page long paper, and the effect does not play a central role in the paper. After a brief discussion of the blackbody radiation law, and Planck’s derivation of the law on the basis of Boltzmann’s relation between entropy and probability, we shall follow the reasoning which led Einstein to the light quantum idea.

2.1. Planck’s radiation law From a statistical-mechanical point of view the entropy S of a system is related to the distribution function fν by the equation S = −k ν fν ln fν , where k is Boltzmann’s constant and the summation is over all (quantum) states of the ensemble (Morse [1964]). Normalization requires that ν fν = 1 for a system with a finite number W of microstates. The entropy is in thermal equilibrium given by (Boltzmann’s principle) S = k ln W

(2.1)

where W = fν−1 . The relation in eq. (2.1) between S and W [which can be obtained by maximizing the general S subject to the constraint W ν=1 fν = 1 (microcanonical ensemble)] was the starting point for Planck. According to Kirchhoff’s law the radiation field is independent of the nature of the body it is in equilibrium with. Planck chose for the black body a model of independent harmonic oscillators (N of frequency ν, N of frequency ν , . . . , with all N large numbers), and proceeded as follows. Let N be the assumed fixed energy of the N oscillators of frequency ν and let be the average energy of the individual oscillators. Thus, N = N . The total entropy of the N independent oscillators is given by SN = N S, where S is the entropy of a single oscillator. To

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[3, § 2

find W , Planck considered N as a quantity which can only be divided into a finite number P of equal parts , i.e., N = P ; and as Planck notes “this is the most essential point of the whole calculation”. The number of ways W , in which one can divide the P indistinguishable equal parts of the energy over the N distinguishable oscillators is from the theory of permutations given by W = (N +P −1)!/[(N −1)!P !] ≈ (N +P )!/[N !P !] since N ,P 1. Each of the ways of distribution Planck calls a “complexion” using an expression by Boltzmann. With the help of Stirling’s formula x! ≈ x x /ex , we get for the entropy of the N oscillators an expression SN = k[(N + P ) ln(N + P ) − N ln N − P ln P ]. The entropy of a single oscillator therefore is given by ln 1 + − ln , S =k 1+ (2.2) and its derivative with respect to by k dS = ln 1 + . d

(2.3)

If eq. (2.3) is combined with the thermodynamic relation dS/d = T −1 one obtains the following expression for the mean energy of the oscillator: −1 = exp (2.4) . −1 kT It is clear from the foregoing discussion that Planck essentially made the assumption that the energy of the given oscillator can take on only one of the energy values 0, , 2, . . . P . Only if kT , Planck gets the classical value = kT for the average energy of the harmonic oscillator at a temperature T . In equilibrium the average rate at which the oscillator absorbs radiation from the electromagnetic field must balance the average rate at which it gives energy back to the field. A classical calculation of this balance leads to the relation u(ν, T ) =

8πν 2 c03

(2.5)

between the field energy density u(ν, T ) [per unit frequency range] and the energy of an oscillator with frequency ν. The quantity c0 is the speed of light in vacuum. Equation (2.5) was first derived by Planck [1900a] and is called Planck’s equation. In order to get agreement with Wien’s law (see Section 2.2) must be proportional to ν, i.e., = hν.

(2.6) 10−27

erg sec) is the famous constant of The proportionality factor h( 6.62 × Planck. In his epoch-making paper (Planck [1900b]), presented to the German

3, § 2]


59

physical society in Berlin on December 14, 1900, Planck calls h a constant of nature. By combining eqs. (2.4)–(2.6) one obtains the blackbody radiation law of Planck, viz., u(ν, T ) =

8πhν 3 c03

1 . hν −1 exp kT

(2.7)

Already on October 19, 1900, Planck had announced his radiation law at a meeting of the Prussian Academy of Science (Planck [1900b]). In the October 19 paper Planck came to his formula by an ad hoc argument based on an explicit expression (guess) for d2 S/d2 . Planck [1900b] himself writes about this expression (in translation): It is by far the simplest of all expressions which lead to S as a logarithmic function of – which is suggested from probability considerations – and which moreover reduces to Wien’s expression for small values of . Ludwig Boltzmann himself did not introduce “Boltzmann’s constant k” nor did he ever write down the relation between S and W in the form given in eq. (2.1); see Klein [1973]. The Boltzmann [1877] paper however contains a section entitled (in translation) “The relation of the entropy to the quantity which I have called partition probability.” The partition probability is essentially ln W and Boltzmann notes that ln W is identical with the entropy up to a constant factor and an additive constant, i.e., S = k ln W + constant. The aforementioned relation appears for the first time in a paper by Planck [1901] written a few weeks after the presentation in Berlin of his epoch-making article (Planck [1900b]). In a paper published a few years before his death (Planck [1943]), and entitled “Zur Geschichte der Auffindung des physikalischen Wirkungsquantums”, Planck also comments on eq. (2.1) and its importance for his studies of the blackbody radiation. Planck’s way of counting the partitions of indistinguishable energy quanta prefigures the Bose [1924] – Einstein [1924b] light quanta counting, and Planck knew the hypothetical character of his way of counting. Thus, he wrote (Planck [1901]): “. . . irgend eine bestimmte Complexion ist ebenso wahrscheinlich, wie irgend eine andere bestimmte Complexion. Ob diese Hypothese in der Natur wirklich zutrifft, kan in letzter Linie nur durch die Erfahrung geprüft werden”.

2.2. The radiation laws of Rayleigh and Wien It may readily be shown that Planck’s radiation law reduces to u(ν, T ) =

8πν 2 c03

kT

(2.8)

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[3, § 2

when hν kT , i.e., in the limit of low frequencies and high temperatures. The result in eq. (2.8) is known as the Rayleigh–Jeans law. A few months before Planck discovered the correct blackbody radiation law Lord Rayleigh (John W. Strutt) showed on classical grounds that u(ν, T ) = c1 ν 2 T (Rayleigh [1900]). In a letter to Nature (Rayleigh [1905]), submitted on May 6 and published on May 18, 1905, Rayleigh computes the constant c1 . His result for c1 was off by a factor of 8 as shown a month later by James H. Jeans [1905]. The Rayleigh–Jeans law, which is the classical limit of the blackbody radiation law, follows from eq. (2.5) if the classical value = kT is taken for the average energy of the harmonic oscillator. In fact this route to the derivation of eq. (2.8) appears in Einstein’s light-quanta paper (Einstein [1905], which was submitted on March 17, 1905, and hence before the submission of Rayleigh’s calculation of c1 . When hν kT , i.e., in the limit of high frequencies and low temperatures, Planck’s radiation law reduces to Wien’s radiation law, viz., hν 8πhν 3 exp − . u(ν, T ) = (2.9) kT c03 Originally, Willy Wien proposed an explicit form u(ν, T ) = (8π/c0 )αν 3 exp(−βν/T ) with two constants α and β which had to be determined empirically (Wien [1896]). It follows from eq. (2.9) that these constants are α = h/c2 and β = h/k. The radiation laws of Rayleigh and Wien, which agreed with the available experimental data only in the appropriate limits (Paschen [1897], Lummer and Pringsheim [1900], Rubens and Kurlbaum [1900]), both satisfy Wien’s displacement law u(ν, T ) = (8π/c0 )ν 3 F (ν/T ), according to which u(ν, T ) is proportional to the product of the third power of the frequency and some function F (ν/T ) of the ratio of the frequency to the absolute temperature. Wien’s displacement law can be derived from general thermodynamic considerations and was established by Wien [1893]. In his 1905 paper on the light quantum idea, Einstein starts with the point of view taken in the classical wave theory. He cites Planck [1900a] for having derived the equilibrium relation in eq. (2.5) between the energy density u(ν, T ) and the average energy per degree of freedom, , of an electron oscillator. With = kT (in Einstein’s and Planck’s papers k appears in the ratio of the gas constant R and Avogadro’s number N, i.e., k = R/N) Einstein reaches the Rayleigh–Jeans law [eq. (2.8)], but he does not cite Rayleigh for the result u(ν, T ) = c1 ν 2 T . He notes that Planck’s radiation law, quoted in the original form with two constants α and β, i.e., u(ν, T ) = αν 3 /[exp(βν/T ) − 1], leads in the limit where T /ν is

3, § 2]


61

large to the classical formula. In passing one might notice that β here is the same as Wien’s β, and that multiplication of α by 8π/c0 gets one Wien’s α, cf. the text just below eq. (2.9). Einstein finally mentions the difficulty that the wider the range of oscillator frequencies the greater becomes the radiation energy and in the limit one gets ∞ u(ν, T ) dν = 0

8π c03

∞ ν 2 dν = ∞

kT

(2.10)

0

in the classical theory. This divergent result for the energy was historically referred to as the “ultraviolet catastrophe” of classical physics.

2.3. Entropy of radiation To support the point of view that light may consist of localized energy quanta Einstein uses Boltzmann’s principle to study the entropy of the radiation field, in particular the changes which might occur in this entropy due to random fluctuations in the volume occupied by the radiation field. From a comparison to the case where a fixed number of point-like molecules moves around randomly in a fixed volume Einstein reached via Wien’s radiation law the conclusion that monochromatic radiation (frequency ν) of low density behaves thermodynamically as though it consists of independent energy quanta of magnitude Rβν/N, i.e., in modern notation hν. Einstein considered radiation occupying a volume V and assumed that (i) the observable properties of the radiation are completely determined when the radiation density u = u(ν, T ) is given for all frequencies and (ii) radiations of different frequencies are independent of each other. The total entropy of the radiation field hence is given by ∞ S=V

σ (u, ν) dν,

(2.11)

0

where σ (u, ν) is the entropy density per unit frequency and volume. In the equilibrium state u(ν, T ) is such a function of ν that the entropy is a maximum. At this point Einstein further assumed that the total energy in the radiation field is fixed, not only its statistical mean value. With this constraint it follows that ∂σ/∂u is independent of ν in equilibrium (Einstein [1905]), and thermodynamics considerations then imply that 1 ∂σ (u, ν) = . ∂u T

(2.12)

62


From Planck’s radiation law one immediately gets k 8πhν 3 1 , = ln 1 + 3 T hν c0 u(ν, T )

[3, § 2

(2.13)

and elimination of T among eqs. (2.12) and (2.13) enables one to determine the entropy density by integration, setting σ = 0 for u = 0. The result may conveniently be written in the form σ (u, ν) = k

u(ν, T )

ln(1 + N ) + N ln 1 + N −1 , N hν

(2.14)

where −1 hν . −1 N = exp kT

(2.15)

In his 1905 paper Einstein only calculated σ (u, ν) in the Wien (W ) limit (hν kT ) where N 1. In this limit one obtains u(ν, T ) (1 − ln N ), N 1, (2.16) hν neglecting terms of first (and higher) order in N in the parenthesis. In the Rayleigh–Jeans (RJ) limit (hν kT ), where N 1, the entropy is given by σW (u, ν) = k

σRJ (u, ν) = k

u(ν, T ) (1 + ln N ), N hν

N 1,

(2.17)

omitting terms of order N −1 (and higher powers of N −1 ) in the parenthesis.

2.4. Hypothesis of light quanta With the expression for σW (u, ν) [eq. (2.16)] established Einstein went on to calculate the relation between the entropy SW (ν) and energy E(ν) of monochromatic radiation of frequency ν contained in a volume V . Using delta-function notation we thus write SW (ν)δ(ν − ν ) = V σW (u(ν , T ), ν ) and E(ν)δ(ν − ν ) = V u(ν , T ), and from eqs. (2.11) and (2.16) we then obtain Einstein’s result for the entropy, viz., E(ν)c03 1 E(ν) 1 − ln . SW (ν) = k (2.18) hν 8πhν 3 V Einstein now calculates the entropy difference between two states of the system, having the same frequency ν and the same energy E(ν), but different volumes,

3, § 2]


63

say V and V0 . From eq. (2.18) one readily finds

E(ν) V ln , SW (ν, E, V ) − SW (ν, E, V0 ) = k hν V0

(2.19)

and thus SW (ν, E, V ) − SW (ν, E, V0 ) = k ln

V V0

E(ν)/(hν) .

(2.20)

To interpret the result in eq. (2.20) physically, Einstein considers an ideal gas with a fixed number n of point-like molecules confined in a volume V0 and randomly moving. The probability that these particles are located in a smaller part V of V0 is W = (V /V0 )n , and from Boltzmann’s principle [eq. (2.1)], the difference in entropy will be given by n V . S(V ) − S(V0 ) = k ln (2.21) V0 A comparison of eq. (2.20) with Boltzmann’s principle led Einstein to the conclusion (in translation): if monochromatic radiation of frequency ν and energy E is enclosed by reflecting walls in a volume V0 , the probability that the total radiation energy will be found in a volume V (part of the volume V0 ) at any randomly chosen instant is W = (V /V0 )E/(hν) . From a comparison between eqs. (2.20) and (2.21), Einstein further concluded that: monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of independent energy quanta of magnitude hν. To reach the above-mentioned conclusions the correctness of eq. (2.19) is crucial. There is nothing in the derivation of eq. (2.19) which associates this equation with random fluctuations in the volume occupied by the radiation. Einstein’s postulate that eq. (2.19) describes the entropy changes associated with statistical volume fluctuations is difficult to agree with unless one a priori ascribes particlelike properties to monochromatic radiation. Thus, when the frequency ν and the energy E(ν) unfounded are assumed to be constant during random volume fluctuations the radiation densities u(ν, T ) and u0 (ν, T ) at V and V0 , respectively, are related by the geometric factor V /V0 , i.e., u(ν, T ) = (V0 /V )u0 (ν, T ). In the limit V → 0 we thus have u(ν, T ) → ∞, a result I only can associate with a pointparticle picture of monochromatic radiation. Although it is sometimes assumed (Knight and Allen [1983]) that the volume of the enclosure itself is changing in the step from eq. (2.18) to eq. (2.19) there is no evidence for this in Einstein’s paper.

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[3, § 2

2.5. A glimpse of the wave–particle duality Let us now consider the field energy E = u(ν, T )V ν and entropy S = σ (u, ν)V ν at some fixed instant of time in the frequency range (ν, ν + ν) and in a small volume V inside the cavity. Fluctuations in the field energy E around its statistical mean value E are accompanied by fluctuations in the entropy. A Taylor series expansion of S(E) around S(E) gives 2 1 ∂ 2 S E − E + · · · , S(E) = S(E) + (2.22) 2 ∂E 2 E=E since ∂S/∂E|E=E = 0. From Boltzmann’s relation S(E) = k ln W (E) it now appears that the probability to second order in E − E is given by 2 1 ∂ 2 S E − E W (E) = C exp (2.23) 2k ∂E 2 E=E where C is a (normalization) constant. Note that as S(E) is a maximum ∂ 2 S/∂E 2 |E=E is negative. The simplest measure of the energy fluctuations is the variance (E − E)2 given by ∞ 2 (E − E)2 W (E) dE

2 ∂ SE −1 E − E = −∞ ∞ (2.24) = −k , ∂E2 −∞ W (E) dE having extended the lower limit of the integrals from 0 to −∞. In a paper with the title “Zum gegenwärtigen Stand des Strahlungsproblems” Einstein in 1909 used the relation between the variance and the second derivative of the entropy to determine the energy fluctuations in the blackbody radiation (Einstein [1909]). In equilibrium, where u(ν, T ) = E/(V ν) is given by Planck’s radiation law [eq. (2.7)], the quantity N of eq. (2.15) may be written in the form E N E = Khν

(2.25)

with K = 8πν 2 νV /c03 . For the entropy one then gets via eq. (2.14) the result E E Khν S E = kK ln 1 + (2.26) + ln 1 + . Khν Khν E By combining eqs. (2.24) and (2.26) one obtains Einstein’s fluctuation formula for blackbody radiation, viz.,

2 E2 . E − E = hνE + (2.27) K The classical electromagnetic wave theory would give for the variance just the second term on the right-hand side of this equation, i.e., (E − E)2 wave =

3, § 3]

Guiding fields for light quanta

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E2 /K. Einstein showed in his 1909 paper that the classical result – apart from a possible dimensionless multiplication factor – can be obtained from dimensional analysis, and he mentions that the exact result follows from somewhat complex mathematical considerations. In 1916, Hendrik A. Lorentz showed this in an appendix to his book on the theory of statistics and thermodynamics (Lorentz [1916]). Starting from the Rayleigh–Jeans expression for the equilibrium entropy, S(E)RJ = kK{1 + ln[E/(Khν)]}, the wave-theory result may readily be derived. About the first term on the right-hand side of eq. (2.27) Einstein [1905] writes: . . . das erste Glied, wenn es allein vorhanden wäre, eine solche Schwankung der Strahlungsenergie liefern, wie wenn die Strahlung aus voneinander unabhängig beweglichen, punktförmigen Quanten von der Energie hν bestünde. Einstein came to the light quantum hypothesis from analyses based on Wien’s law, and from the expression for the equilibrium entropy in the Wien limit, viz. S(E)W = [kE/(hν)]{1 − ln[E/(Khν)]}, and eq. (2.24), one obtains (E − E)2 particle = hνE. Einstein’s fluctuation formula for the blackbody radiation energy gives a glimpse of the wave–particle duality that later followed in a consistent manner from quantum mechanics (Born, Heisenberg and Jordan [1926]).

§ 3. Guiding fields for light quanta Einstein’s light quantum hypothesis did not receive wide acceptance from leading physicists for almost twenty years. Thus, when Max Planck, Herman W. Nernst, Heinrich Rubens and Emil Warburg in 1913 submitted their recommendation for the appointment of Einstein to the Prussian Academy of Sciences it was stated (Kirsten and Körber [1975]) “that Einstein may occasionally have missed the mark in his speculations, as, for example, with his hypothesis of light quanta, ought not to be held too much against him, for it is impossible to introduce new ideas,even in the exact sciences, without taking a risk”. Before 1921 the clearest evidence for light quanta came from Robert A. Millikan’s experimental results for the photoelectric effect (Millikan [1914, 1915, 1916a, 1916b]). But acceptance of Einstein’s law for the photoelectric effect did not lead to an acceptance of the light quantum hypothesis. By late 1921 the opposition to the light quantum idea began to wane not least because of the results of Maurice de Broglie’s X-ray photoelectric measurements (De Broglie, M. [1921a, 1921b, 1921c]). When he reported his results to the Third Solvay Congress he said that the radiation “must be corpuscular, or if it is undulatory, its energy must be concentrated in points on the surface of the wave”; see Wheaton [1983], p. 270. In May 1923, Compton pub-

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lished a paper entitled “A quantum theory of the scattering of X-rays by light elements” (Compton [1923]). He assumed that an incident X-ray beam of frequency ν consists of localized X-ray quanta (particles) of energy hν and momentum hν/c (in the direction of the beam), and that a given X-ray particle is scattered from a single electron of the target. By analyzing each interaction process according to the kinematics of an elastic two-particle relativistic collision, Compton was led to his soon famous formula h λ = (3.1) (1 − cos θ ) m0 c0 for the increase in wavelength, λ, of X-ray quanta scattered at an angle θ from the momentum direction of the incident X-ray quanta. By a comparison to experimental data on X-ray scattering from graphite excellent agreement was found. Shortly before Compton’s paper was published an essentially identical explanation of the wavelength shift had been given by Debye [1923]. To gain deeper insight of the quantum laws connection with wave optics, Debye made use of the “needle radiation” (Nadelstrahlung) picture of the light quantum. In this picture the quantum possessed a longitudinal extension, designed to allow for interference effects. In the wake of Compton’s discovery no physicist could ignore the light quantum any longer. Bohr had rejected the existence of light quanta for many years, and only after an unsuccessful attempt (see Section 3.2) to dispose of the quanta in 1924 (Bohr, Kramers and Slater [1924]) he finally accepted the photon. Although the Compton effect, as Sommerfeld expressed it sounded the death-knell of the wave theory (see Wheaton [1983], p. 286), the seeming incompatibility of the wave and particle representations still existed.

3.1. Slater’s virtual radiation field To obtain a harmony between the physical pictures of the wave theory of light and the theory of light quanta Slater came up with the idea that even in a stationary state an atom may be supposed to communicate with other atoms by means of a so-called virtual field of radiation. Slater only published a short note on his ideas in Nature on March 1st, 1924 (Slater [1924]). More information on Slater’s physical ideas are reported in his autobiographical notes from 1968; see Dresden [1987]. The virtual field, which was assumed to originate from harmonic oscillators having the Bohr frequencies of the possible quantum transitions in the atom, is not a classical electromagnetic field because it transports neither energy nor momentum. Formally, the virtual field can be calculated classically. The function of the virtual field is “to provide for statistical conservation of energy and momentum

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by determining the probabilities for quantum transitions” as Slater writes (Slater [1924]). The virtual field originating from a given atom is supposed to induce a probability that this atom lose energy spontaneously. It is interesting to reflect on the fact that in the modern self-field approach an atom will decay “spontaneously” due to the interaction of the atom with its own self-field (Barut and Huele [1985], Passante and Power [1987], Barut and Dowling [1987], Barut and Blaire [1992]). The self-field approach leads to the same conclusion as quantum electrodynamics (QED) which ascribes the spontaneous emission to the atom’s interaction with the zero-point fluctuations in the photon vacuum (Weiskopf and Wigner [1930, 1931]). Virtual radiation from other atoms induces additional probabilities that the given atom gain or lose energy, cf. Einstein’s suggestion (Einstein [1916a, 1916b, 1917a]). The discontinuous transition marks the change from an old to a new virtual field. Slater says in his Nature article that the idea of the activity of the stationary states came to him in the course of an attempt to set up a field to guide the discrete light quanta, which might move, for example, along the direction of Poynting’s vector. By the probabilistic interpretation Slater kept the idea of coexistence of a discontinuous atomic transition in time and the emission of an electromagnetic wavetrain of finite duration time. If the cycle-averaged Poynting vector of the field produced by an oscillator of frequency ν is S ν , the Slater probability, pν , that the atom will emit a light quantum of energy hν per second is given by 1 S ν · dA, pν = (3.2) hν where dA is the area element on an arbitrary closed surface surrounding the atom. Equation (3.2) leads to the guess that the probability to localize a light quantum at a given point is proportional to the local energy density (0 /2)(E 2 + c02 B 2 ) of the virtual field in this point.

3.2. The Bohr–Kramers–Slater theory: virtual fields without light quanta Around 1925, also Einstein considered the possibility of a guiding field (Führungsfeld) for light quanta, or as he phrased it to Bohr, a ghost field (Gespenster-feld), (Einstein [1949]), but he never published his ideas on this subject. In 1926–27, de Broglie had the idea that the photon was a singularity in the wave field. In this so-called particle-and-pilot-wave theory, recorded in his report to the Fifth Solvay Congress, de Broglie assumes that the wave is real, occupying a certain region in space, and that the light corpuscle is, as he states it later (De Broglie, L. [1971], p. 186), “a material point having a certain position in the wave”. The

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energy density of the wave at any point is proportional to the probability that the corpuscle is present there. Despite the fact that serious objections could be raised against the pilot-wave theory and de Broglie understands that the concept of the wave as a kind of pilot at best is a temporary picture, the approach is attractive to him because there was “[no] need to abandon classical ideas too sweepingly” (De Broglie, L. [1971], p. 186). Bohr rejected the light quantum hypothesis for many years, and in 1924 he was prepared to give up the classical claim of conservation of energy and momentum to avoid Einstein’s light quantum. As soon as Bohr and Kramers became aware of Slater’s ideas they recognized that the virtual-field concept provided a possible means to establish a quantum-mechanical radiation theory without photons. When Slater came to Copenhagen by the end of December 1923, it was evident that he accepted the real existence of the photon. Exactly one month after Slater’s arrival in Copenhagen, the significant Bohr–Kramers–Slater (BKS) theory on a new quantum theory of radiation was submitted to the Philosophical Magazine (Bohr, Kramers and Slater [1924]), and also a German version appeared. The virtual field plays a central role in the BKS paper, but the photons have disappeared. The authors recognize that the photoelectric effect had shown the great heuristic value of Einstein’s light quantum hypothesis, but they emphasize that the theory of light quanta cannot be considered as a satisfactory solution of the problem alone from the fact that the radiation “frequency” ν appearing in E = hν is defined by experiments on wave (interference) phenomena. In the BKS theory an atom in a stationary state i is accompanied by a virtual field which “originates” in the various harmonic oscillators of frequencies νij which the Bohr theory associates with the possible quantum transitions from the stationary state i to the state j . The virtual field from other atoms defines the probability of real transitions in a given atom by laws which are analogous to those which in Einstein’s theory (Einstein [1917a]) hold for the induced transitions. The spontaneous emission the authors ascribe to the interaction of an atom with its own virtual field. The unobservable virtual field carries no energy nor momentum and in this way Einstein’s light quanta were avoided. If an atom A performs a real transition the virtual field will change everywhere in space and therefore also at the location of an atom B. The probability that atom B makes a transition from its initial state to another state is now changed. The real coupling between atoms A and B therefore is determined by probabilistic laws (the mathematics of the virtual transition process is that of classical electromagnetics). The Compton effect which was widely believed to demonstrate the existence of point-like photons was published before the BKS paper was written, and it was therefore important for Bohr, Kramers, and Slater to show that it was possible to

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account for Compton’s result without the help of the photon concept. Compton had earlier suggested that the observed wavelength shift was basically a Doppler effect, and he envisaged the phenomenon as the result of a two-step process. In the first step the incident X-rays would cause the electron to move (with speed v), and in the second step the moving electron would reradiate X-rays. In the laboratory system these X-rays would have undergone a Doppler shift. A simple classical calculation gives a wavelength shift λ =

λβ (1 − cos θ ), 1−β

(3.3)

β = v/c0 . To obtain agreement with the experimental data it is necessary to make the identification λβ/(1 − β) = h/(m0 c0 ). This gives the electron the effective velocity veff = c0 λC /(1 + λC ), where λC = h/(m0 c0 ) is the Compton wavelength. The effective electron velocity as Compton called it is not related in a direct way to the electron velocity. BKS incorporated this Doppler mechanism in their explanation of the wavelength shift. In their theory the incident virtual radiation gives rise to a reaction on the virtual oscillators and makes it move with a velocity different from that of the illuminated electrons themselves. The most conspicious feature of the BKS description of the Compton effect is that the virtual oscillator giving rise to the emitted radiation moves with a velocity different from the electron’s velocity. About this unusual feature BKS write: “That in this case the virtual oscillator moves with a velocity different from that of the illuminated electrons themselves is certainly a feature strikingly unfamiliar to the classical conceptions. In view of the fundamental departures from the classical space–time description, involved in the very idea of virtual oscillators, it seems at the present state of science hardly justifiable to reject a formal interpretation as that under consideration as inadequate”. Just as in Compton’s theory the illuminated electron possesses a certain probability of taking up momentum in various directions in the BKS theory. Because of the independence of the electron recoil and the emission of the scattered wave, BKS predict that there would not be a oneto-one correspondence between the scattering and recoil events. Furthermore, one would in general expect a time delay between these events, and only a statistical relation between the X-ray scattering and electron recoil angles. Within a year after the presentation of the BKS theory the coincidence experiments of Walther Bothe and Hans Geiger [1924, 1925a, 1925b], and the angle-relation experiments of Arthur Compton and Francis Simon [1925] appeared. The experimental result were incompatible with the BKS suggestion, and showed unequivocally that the predictions of the photon theory were correct.

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3.3. Photons – a new kind of atoms? In a letter to the Editor of Nature, Lewis in 1926 made the hypothesis that the light quantum was a new kind of atom (Lewis [1926]), and he wrote: “It would seem inappropriate to speak of one of these hypothetical entities as a particle of light, a corpuscle of light, a light quantum, or a light quant, if we are to assume that it spends only a minute fraction of its existence as a carrier of radiant energy, while the rest of the time it remains as an important structural element within the atom. It would also cause confusion to call it merely a quantum, for later it will be necessary to distinguish between the number of these entities present in an atom and the so-called quantum number. I therefore take the liberty of proposing for this hypothetical new atom, which is not light but plays an essential part in every process of radiation, the name photon”. Nothing has survived of Lewis’ “atom of light” hypothesis except the word “photon”! For the photon Lewis postulates the following properties: (i) In any isolated system the total number of photons is constant, (ii) all radiant energy is carried by photons, (iii) all photons are intrinsically identical, (iv) the energy of the photon, divided by the Planck constant, gives the frequency of the photon, (v) all photons are alike in one property which has the dimensions of angular momentum, and is invariant to relativity transformations, and (vi) the condition that the frequency of a photon emitted from a system be equal to some physical frequency existing within that system, is not in general fulfilled, but comes nearer to fulfilment the lower the frequency is. The conservation of photons was in obvious conflict with existing notions of the radiation process even at Lewis’ time. Thus, for an atom decaying from a given initial state to a final state either directly or via two intermediate levels, the loss is one and three photons, respectively, assuming as Lewis did that exactly one photon is lost in each elementary process. To uphold the conservation of photons one must then assume that at least either the initial state or the final state must be multiple. Even if the inner (spin) quantum number and the “external” quantum numbers are given the atomic states must still be regarded as not completely specified, a hypothesis which turned out to be wrong.

§ 4. Light quanta and matter waves In a series of papers (De Broglie, L. [1923a, 1923b, 1923c, 1924]) submitted in the autumn of 1923 de Broglie tried to bring harmony between the special theory of relativity and the light quantum hypothesis of Einstein. In his attempt to reconcile the wave theory and the light quantum model de Broglie was led to a synthesis of

3, § 4]

Light quanta and matter waves

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particle and wave which applied to atoms of light (quanta de lumière) as well as to atoms of matter. The efforts of de Broglie culminated with the hypothesis of proportionality between the energy-momentum four vector and the wave four-vector, the proportionality factor being Planck’s constant. The formal analogy between the principle of least action in particle dynamics and Fermat’s principle in geometrical optics had played an important role in de Broglie’s work, but it was through Schrödinger’s extended studies of the formal analogy between Hamiltonian mechanics of particles and wave optics in 1925 that wave mechanics of massive particles was born in 1926. Although the light quantum hypothesis had played a central role for de Broglie and Schrödinger, the founders of wave mechanics ended up with a wave equation for material particles. As we shall realize in Section 5, the struggle towards the establishment of a wave equation for the de Broglie waves of the light quantum was left to Landau and Peierls and to Oppenheimer.

4.1. De Broglie’s phase wave According to relativistic dynamics the internal energy of a particle with eigenmass (rest mass) m0 is E = m0 c02 . The velocity of light (c0 ) de Broglie preferred to call the “limiting velocity of energy”. The basic idea of 1923 quantum theory, namely that to a fragment of energy one must assign a certain frequency ν0 led de Broglie to the relation hν0 = m0 c02 .

(4.1)

In a sense de Broglie considers ν0 to be the frequency of the particle’s “own clock”. If the particle is in uniform motion with a speed v = βc 0 relative to an observer (O), this observer will measure a frequency ν1 = m0 c02 1 − β 2 / h of the particle’s clock; this is the well-known phenomenon of the relativisticslowingdown of clocks. For O the energy of the moving particle is m0 c02 / 1 − β 2 , and according to the quantum theory the associated frequency must be ν = m0 c02 /(h 1 − β 2 ). The two frequencies ν1 and ν are different and this was the quantum-relativistic difficulty which occupied de Broglie for a long time, according to himself (De Broglie, L. [1925, 1971]). Before explaining how de Broglie solved this problem and thereby shed light on the paradox of wave and particle let me briefly return to de Broglie’s “atom of light” concept. In his Philosophical Magazine paper from 1924 (De Broglie [1924]), entitled “A Tentative Theory of Light Quanta” de Broglie assumed the real existence of light quanta. He further assumed that all light quanta are identical and that only their velocities are different, yet always very nearly equal to “Einstein’s limiting

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velocity c0 ”. Since β is very close to unity one gets β ≈ 1 − [m0 c02 /(hν)]2 /2. The inability to discriminate the light quanta velocities from c0 led de Broglie to estimate that the rest mass of the atom of light should be less than 10−50 g. Already in his first paper on the subject (De Broglie, L. [1923a]) de Broglie wrote down eq. (4.1) for a particle of finite mass called le mobile and afterwards applied his analysis to atoms of light with finite mass. De Broglie was inspired by a suggestive work published a few years earlier by Marcel Brillouin [1919]. Brillouin assumed that his particle, called a point mobile, performs a quasi-periodic circular motion and has an associated spherical wave. De Broglie solved the quantum-relativistic difficulty mentioned above by his theorem of phase agreement, which can be stated as follows [cf. Annales de Physique 3, 22 (1925)]: “The periodicphenomenon fixed to the moving particle, which has frequency ν1 = m0 c02 1 − β 2 / h with respect to the observer at rest appears to this observer always in phase with a wave of frequency ν = m0 c02 /(h 1 − β 2 ) propagating in the direction of motion of the moving particle with the velocity V = c0 /β”. It was very simple for de Broglie to prove this theorem. Thus, if it is assumed that there is phase agreement between the periodic phenomenon associated with the moving particle and the wave at time t = 0, phase agreement at time t = t implies that x ν1 t = ν t − (4.2) , V where x = βc0 t is the distance travelled by the moving particle in the time interval (0|t). Insertion of the relevant expressions for ν1 and ν in eq. (4.2) immediately leads to the conclusion that the velocity of the wave must be V = c0 /β. The fact that V > c0 implies that the particle-associated wave cannot be an energy transferring wave; it is a phase wave. It appears from the relations V = c0 /β and ν = m0 c02 /(h 1 − β 2 ) that 2 1/2 ν ν (4.3) − λ−2 = C V c0 where λC = h/(m0 c0 ) is the Compton wavelength of the particle (possibly atom of light). Since V is frequency dependent the group velocity U , given by d ν −1 U = (4.4) , dν V deviates from the phase velocity, and from eq. (4.3) it is easy to show that U = v, the important result found by de Broglie. Outside absorption regions the velocity of energy transport in optics is just the group velocity, and this here coincides with the velocity of the moving particle.

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4.2. De Broglie’s world vector relation, Jμ = h¯ Oμ The considerations in Section 4.1 apply for a particle moving uniformly in a straight line. In his Doctoral Thesis [reprinted in Annales de Physique 3, 22 (1925)] de Broglie now posed the question: “How does a phase wave assigned to a moving particle propagate when this particle moves non-uniformly in a field of force?” To address this question de Broglie studies the relativistic motion of an electrically charged particle in an electromagnetic field. Although such a study does not directly indicate how the phase wave can be assigned to the atom of light it leads, as we shall realize, to a relativistic generalization of the quantum hypothesis E = hν which also turns out to hold for the light corpuscle. Guided from the very beginning of his investigations by the idea of a close relationship between the principle of least action and the Fermat principle de Broglie was led to the assumption that for a given value of the total particle energy, and therefore of the frequency of its phase wave, the mechanically possible paths of the particle coincide with the possible rays of the phase wave. Fermat’s principle for the trajectories of light rays is based on the assumption of geometrical optics, viz. that the optical wavelength is small compared to the dimension of any change in the refractive index. The mathematical equivalence of the Hamilton–Jacobi (classical mechanics) and eikonal (geometrical optics) equations was first realized by William R. Hamilton in 1834. To describe in classical terms interference and diffraction phenomena one has to go beyond geometrical optics to optical wave theory, where the optical wavelength plays a central role. The recognition that classical mechanics was only a geometrical optics approximation to a wave theory came when Clinton J. Davisson and Lester Germer’s electron diffraction experiments on crystals revealed the (finite) wavelength of the associated wave (Davisson and Germer [1927]). The wave theory of mechanics established by Schrödinger [1926a, 1926b, 1926c, 1926d] showed that the Hamilton–Jacobi equation was the short-wavelength limit (h → 0 and therefore λ → 0) of the (time-dependent) Schrödinger equation. A valuable introduction to the Hamilton–Jacobi theory can be found in Herbert Goldstein’s book on classical mechanics (Goldstein [1980]). Let us now briefly follow the reasoning which led de Broglie to his covariant particle-wave duality. If we denote the space coordinates by x 1 , x 2 , x 3 and the time coordinate c0 t by x 0 the invariant “length element” is ds ≡ c0 dτ =

dx 0

2

2 2 2 1/2 − dx 1 − dx 2 − dx 3 ,

(4.5)

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where dτ is the proper time element. The world velocity vector divided by c0 , {uμ }, μ = 0, 1, 2, 3, which μ’s contravariant component is given by 1 dx μ (4.6) c0 dτ is now introduced. At each point on the given world line {uμ } is in the direction of the tangent to this line. For a particle moving on the world line with velocity v = (vx , vy , vz )[v = βc0 ], the covariant and contravariant normalized world velocity vectors are {uμ } = (1, −v/c0 )(1 − β 2 )−1/2 and {uμ } = (1, v/c0 )(1 − β 2 )−1/2 , respectively. The scalar product of these vectors is uμ uμ = 1 (with the “summation over repeated indices” convention in force). With the purpose of describing the dynamics of an electron (charge e) in an electromagnetic field de Broglie introduces the potential world vector {Aμ } ≡ (ϕ/c0 , A), [{Aμ } = (ϕ/c0 , −A)], where A and ϕ are the vector and scalar potentials. Then a third world vector {Jμ } given by the relation uμ =

Jμ = m0 c0 uμ + eAμ ,

μ = 0, 1, 2, 3,

(4.7)

is defined. In space–time the principle of least action can be expressed in terms of {Jμ } as follows: Q Jμ dx μ = 0.

δ

(4.8)

P

The left-hand side of eq. (4.8) describes the variation (δ) in the line integral of {Jμ } along the world line between the space–time points P and Q. Equation (4.8) which states that the line integral has a stationary value is of central importance in de Broglie’s extension of the quantum hypothesis E = hν. Although the name of the theorem in eq. (4.8) should be the principle of stationary action, the historical name, the principle of least action, is almost always used in the literature. Since dx μ = uμ ds = uμ c0 (1 − β 2 )1/2 dt, eq. (4.8) can be written in the form t2 δ

1/2 (m0 c0 uμ + eAμ )uμ c0 1 − β 2 dt = 0

(4.9)

t1

in which the times t1 and t2 correspond to the points P and Q. Since uμ uμ = 1, it is apparent that the principle of least action as given in eq. (4.8) can be stated in the familiar form t2 L dt = 0,

δ t1

(4.10)

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where L = −m0 c02 (1 − β 2 )1/2 − eφ + ev · A is the fully relativistic Lagrangian. With addition of the contribution from the electromagnetic field to L, eq. (4.10) can serve as the basis for classical electron electrodynamics. The contravariant world vector {J μ } may also be written as follows: μ E J = (4.11) ,p , c0 where E = m0 c02 (1 − β 2 )−1/2 + eφ, and p = m0 (1 − β 2 )−1/2 v + eA are the relativistic electron energy in the scalar potential φ, and the canonical momentum in the vector potential A. The {J μ }-vector therefore is the well-known energymomentum four-vector. De Broglie next turns to the question of the propagation of the phase wave. From the outset he limits himself to a consideration of the phase ϕ of a sine wave, sin ϕ. If, as above, P and Q are two space–time points, a ray passing through Q the two points must be associated with a stationary value of the line integral P dϕ, that is, Q dϕ = 0,

δ

(4.12)

P

since otherwise phase agreement at P would be lost at Q. For a monochromatic ˆ and locally plane light wave the local wave vector is given by q = (2πν/V )q, where V is the local phase velocity, ν is the local frequency and qˆ = q/q. If dl denotes the infinitesimal line element along the ray in three-dimensional space the infinitesimal phase shift can be written in covariant form, i.e., dϕ = 2π[ν dt − (ν/V )qˆ · dl] = Oμ dx μ , where ω {Oμ } = (4.13) , −q c0 is the contravariant wave four-vector. In de Broglie’s original notation (De Broglie, L. [1925]) the phase shift is written dϕ = 2πOμ dx μ so that his definition of Oμ is the one given in eq. (4.13) divided by 2π. The principle of stationary phase [eq. (4.12)] de Broglie therefore writes as follows: Q Oμ dx μ = 0.

δ

(4.14)

P

If the local frequency is constant, eq. (4.14) is reduced to Fermat’s law B n dl = 0,

δ A

(4.15)

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where A and B are the space points corresponding to P and Q, and n = c0 /V is the local refractive index. De Broglie has now reached the most important point in his search for a generalization of the quantum relation E = h¯ ω. It is apparent from eqs. (4.11) and (4.13) that this quantum relation can be written in the form J0 = h¯ O0 . De Broglie then takes the decisive step towards a generalization of the old quantum hypothesis and assumes that 1 (4.16) {Jμ }. h¯ To the movement of every point particle de Broglie thus has associated the propagation of a certain phase wave in such a manner that the local frequency and wave vector can be calculated from the point particle’s energy and canonical momentum via eq. (4.16). De Broglie admits the somewhat hypothetical character of his new quantum relation, but emphasizes that the new quantum relation is much more satisfactory than the old one, since it is represented by the equality of two world vectors. For the special case of a free particle eq. (4.16) gives the famous de Broglie relations E = h¯ ω and mv = h¯ q, with E = mc02 and m = m0 (1−β 2 )−1/2 . The arguments of de Broglie also make the Bohr–Sommerfeld stability condition in the coordinate invariant form given by Einstein [1917b, 1917c] (quantum condition for the closed electron orbits) plausible because the wave–particle duality relation in eq. (4.16) links this condition to the resonance condition for a wave circulating a closed path, i.e., {Oμ } =

1 h

3 i=1

pi dqi =

ν dl = n, V

(4.17)

where n is a positive integer, and the qi ’s and pi ’s are the generalized coordinates and conjugate momenta of the particle. The particle action integral in eq. (4.17) is often called the Maupertuis action after the original (but vague) statement of the principle by Pierre de Maupertuis in 1744. A brief but recommendable discussion the relations between the Lagrange action, L dt, and the Maupertuis action, of i pi dqi , has been given by Goldstein [1980].

4.3. Wave equation for light corpuscles? In a series of four communications all published in the Annalen der Physik in 1926 Schrödinger succeeded in completing the wave–particle picture mathematically (Schrödinger [1926a, 1926b, 1926c, 1926d]). Schrödinger’s route from point

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mechanics to wave mechanics, which finally led to his famous wave equation for a massive particle (electron), took as starting point the Hamilton–Jacobi equation for Hamilton’s principal function S in the special case where the Hamiltonian H does not depend explicitly on time, viz., ∂S ∂S + H qi , = 0, (4.18) ∂t ∂qi where qi is a representative for the generalized particle position coordinates. With the substitution S = −Et + W (qi ), where the integration constant E here is the energy of the particle, one obtains the equation ∂W (qi ) =E H qi , (4.19) ∂qi for the characteristic function W (qi ). In Cartesian coordinates eq. (4.19) takes the form (∇W ) · (∇W ) = 2m0 (E − V ), where V is the potential energy of the particle, and p = ∇W its momentum. The Hamilton–Jacobi equation in Cartesian coordinates has a similarity to the eikonal equation of geometrical optics, viz., (∇L) · (∇L) = n2 , where n is the (slowly varying) refractive index and the function L(r) is the eikonal (Born and Wolf [1999]). The eikonal is associated with the general representations E(r; ω) = E 0 (r) exp(iq0 L(r)) and B(r; ω) = B 0 (r) exp(iq0 L(r)) of the complex monochromatic electric and magnetic field vectors. When these representations are inserted into the lossless macroscopic Maxwell equations the eikonal equation appears in the short-wavelength limit, i.e., for q0 = ω/c0 → ∞ (Born and Wolf [1999]). Schrödinger searched a wave equation for which the Hamilton–Jacobi equation represents the short-wavelength limit. By considering a sine wave sin ϕ, with ϕ = h¯ −1 (W −Et +const.) he makes the wave–particle association ω = E/h¯ and q = ∇W/h¯ . This suggests that the wave amplitude associated with the mechanical particle motion should have the form ψ(r, t) = ψ0 eiS(r,t)/h¯ .

(4.20)

The phase speed of the wave ω/q = E/|∇W | now agrees with the speed of the constant-S surfaces (Schrödinger [1926b], Goldstein [1980]). The considerations above do not determine the wave equation unambiguously, but “a striving for simplicity” as Schrödinger said [1926b] led him to the guess ∂ h¯ 2 2 ∇ + V ψ(r, t) = ih¯ ψ(r, t). − (4.21) 2m0 ∂t If one substitutes eq. (4.20) into the Schrödinger equation one obtains ∂S ih¯ 2 1 (∇S) · (∇S) + V + ∇ S. = (4.22) 2m0 ∂t 2m0

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[3, § 4

Equation (4.22) may be called the quantum-mechanical Hamilton–Jacobi equation. In the limit h¯ → 0 it reduces to the classical equation (4.18) since ∇S = ∇W = p. Although de Broglie from the outset had assumed that the light quanta have finite rest mass his final wave–particle duality relation [eq. (4.16)] holds even for m0 = 0, remembering that E = pc0 for the photon. Since the Schrödinger equation in (4.21) requires that the particle has a finite rest mass, and only works in the nonrelativistic regime, a wave equation for the photon had not yet been established. In his book Matière et Lumière, published in 1937 and translated into English in 1939 under the title Matter and Light, The New Physics (De Broglie, L. [1971]), de Broglie in several places discusses the search for a photon wave equation. Although the perhaps most important steps were taken by Landau and Peierls [1930], and Oppenheimer [1931], the photon wave mechanics proposed by these authors (and discussed in some detail in Section 5) is not mentioned in de Broglie’s book. In revealing the undulatory aspect of the electron the quantum theories of matter and light were brought together but a wave equation for the photon was still missing. The Schrödinger equation is a development of Newtonian, not of Einsteinian mechanics, and photon wave mechanics must necessarily by relativistic. Furthermore, the Schrödinger equation does not contain an element corresponding to polarization. De Broglie also insists that the new theory must include the possibility of photon annihilation in order to be able to describe the photoelectric effect. Another obstacle stems from the fact that elementary corpuscles of matter, electrons for example, follow Fermi–Dirac statistic when they form a group, whereas photons obey Bose–Einstein statistics. In his reflections on a new photon theory de Broglie turns towards Paul M. Dirac’s theory of the electron. He finds the Dirac equation attractive because it is relativistic and because the spin of the electron has a certain affinity with polarization. In passing it should be mentioned that already in 1927 Jordan had suggested to use a twocomponent equation involving the Pauli spin vector to account for the polarization of the light corpuscle; see Section 5.2. For any corpuscle obeying Dirac’s equation it had turned out that there must be a corresponding anti-corpuscle, and that pair annihilation can occur. De Broglie came up with the idea that the photon might consist of two Dirac corpuscles each having negligible mass and charge. The photon would then obey Bose–Einstein statistics and photon annihilation should be possible in the presence of matter. In case, the annihilation process would constitute the photoelectric effect. De Broglie goes even further in his speculations on the photon concept: he suggests that the massless neutrino is a kind of semiphoton. In a state of isolation, i.e., when not accompanied by an anti-neutrino, de Broglie guessed that the neutrino would have no electromagnetic field since

3, § 5]

Photon wave mechanics

79

no interaction with matter had been observed at that time. United with an antineutrino a photon would be formed and it would have an electromagnetic field of the Maxwellian type. In wave mechanics the point corpuscle is still the basic unit and the role of the associated configuration space wave function is to present, through its absolute square, the respective probabilities that the corpuscle shall be present at different points in space. To reconcile the corpuscular structure of light with the phenomena of interference and diffraction de Broglie held the view that the energy density in the classical field theory gives the probability (density) that a photon will be detected at the space point in question. This view is closely related to that of Oppenheimer [1931], see Section 5.2, and to the modern energy wave function description of the free photon. Apparently, de Broglie did not realize that the transverse character of the electromagnetic field puts a severe limitation on the possibilities for strong spatial localization of the light corpuscle. Roughly speaking a photon emitted from an atom in an electric dipole transition cannot be better localized in space than the extension of the near-field zone of the transition dipole. For the electron corpuscle the spatial localization can be much better. For a massive particle it is the Compton wavelength which determines the lower limit for the linear extension of the localization volume, qualitatively speaking.

§ 5. Photon wave mechanics The birth of quantum field theory, developed in the first instance in connection with radiation in the years 1925–30 through the works of Born and Jordan [1925], Born, Heisenberg and Jordan [1926], Dirac [1926, 1927], Jordan and Pauli [1928], and Heisenberg and Pauli [1929, 1930], did not mean that the particle concept of light was given up. The founders of wave mechanics, Louis de Broglie and Erwin Schrödinger, had suggested the possibility that the dynamics of material particles could be described in terms of waves, and that the results of Heisenberg’s matrix mechanics (Heisenberg [1925]) could be rederived from Schrödinger’s nonrelativistic wave equation. For the old light quantum one had the Einstein relations E = hν and p = (hν/c0 )qˆ between the corpuscular energy (E)/momentum (p) ˆ of the associated monochroand the frequency (ν)/wave vector (q = (2πν/c0 )q) matic plane wave, and it seemed therefore natural to seek to establish a quantummechanical wave equation for the light quantum dynamics. A photon wave function concept in configuration space (coordinate representation) was introduced originally by Landau and Peierls [1930], and the associated Schrödinger-like wave equation for the light quantum was established starting from the free-space Maxwell equations. A substantial part of the Landau–Peierls

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[3, § 5

paper is devoted to the establishment of a many-photon wave mechanical description and the interaction between the light quanta and a Dirac electron. In the last part of the paper the authors show that their wave mechanical formalism is equivalent to the quantum field theory of Heisenberg and Pauli [1929, 1930]. The Landau–Peierls theory leads to a spatially nonlocal relation between the photon wave function and the electromagnetic field, and in consequence the electromagnetic field can act on charged particles located in points in space where the probability of finding the photon vanishes. The vanishing of the Landau–Peierls photon wave function at definite points had “no direct physical significance” to Pauli as he formulated it in his 1933 article on “Prinzipien der Quantentheorie” (Pauli [1933]). A more, but not completely satisfactory theory of photon wave mechanics in configuration space was established by Oppenheimer [1931]. The formalism of Oppenheimer gives a spatially local relation between the photon wave function and the electromagnetic field, and Oppenheimer’s wave equation for the photon is Lorentz invariant. The light quantum theory of Oppenheimer may be characterized as a generalization of Waller’s [1930] relativistic extension of Dirac’s light quantum theory from 1927. An important part of Oppenheimer’s paper is devoted to the study of the connection between the quantum field theory of Heisenberg and Pauli [1929, 1930] and the Dirac–Waller–Oppenheimer light quantum theory. In the following I shall describe and discuss certain aspects of the early history of photon wave mechanics, paying particular attention to the light quantum theories of Landau and Peierls [1930] and of Oppenheimer [1931].

5.1. The light quantum theory of Landau and Peierls In free space the Maxwell equations show that the dispersion relation for monochromatic (angular frequency ω) plane (wave vector q) waves has two branches given by ω = ±c0 q,

(5.1)

where q = |q|. The dispersion depends only on the magnitude of q because of the isotropy of free space. With the assumption that the light-quantum energy is always positive, i.e., hω ¯ > 0, positive-frequency solutions to the Maxwell equations and superposition of these are of primary interest in photon wave mechanics. If one decomposes the electric field in empty space into two spatial Fourier integrals, i.e.,

3, § 5]


1 E(r, t) = 2

∞ E

(+)

−∞

iq·r

(q, t)e

d3 q 1 + 3 2 (2π)

81

∞ E (−) (q, t)eiq·r −∞

d3 q , (2π)3 (5.2)

where E (±) (q, t) = E(q, ±c0 q)e∓ic0 qt ,

(5.3)

it appears that the first integral on the right-hand side of eq. (5.2), viz. E

(+)

1 (r, t) = 2

∞ E(q, c0 q)ei(q·r−c0 qt) −∞

d3 q (2π)3

(5.4)

represents the most general positive-frequency solution to the free-space Maxwell equations. Landau and Peierls consider E (+) (r, t) to be the wave function of the light quantum. The wave function E (+) (r, t) is a complex quantity whereas the general solution, E(r, t), to the Maxwell equations in a classical context must always be real. A Schrödinger-like wave equation for E (+) (r, t) can be obtained taking the time-derivative of eq. (5.4). Landau and Peierls now in√ troduce a nonlocal operator , defined √ by its action on a function F (r) = ∞ 3 q/(2π)3 . Thus, transfers F (r) into F (q) exp(iq · r) d −∞ ∞ √ d3 q F (r) ≡ i qF (q)eiq·r . (2π)3

(5.5)

−∞

√ √ is reasonable because repeated use of √gives an opThe symbolic notation √ √ erator identical to the Laplace operator. With the help of the -operator one obtains the following unnormalized wave equation for the wave function of the light quantum: √ ∂ (+) E (r, t) = −c0 E (+) (r, t). ∂t

(5.6)

Equation (5.6) is Landau–Peierls’ unnormalized wave equation for the light particle. To be in agreement with the Maxwell equation ∇ · E(r, t) = 0, the free-space wave function must satisfy the transversality condition ∇ · E (+) (r, t) = 0.

(5.7)

Besides the transversality condition ∇ · B (+) (r, t) = 0, the positive-frequency part of the magnetic field, B (+) (r, t), satisfies an equation form-identical to (5.6),

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[3, § 5

viz., √ ∂ (+) (5.8) B (r, t) = −c0 B (+) (r, t), ∂t an equation which does not appear in the Landau–Peierls paper, however. It is interesting to reflect on the fact that the unnormalized wave equation in (5.6), possibly in a linear combination with eq. (5.8), is closely related to all other and newer photon wave functions, among which perhaps those based on normal variables (∼E − c0 (q/q) × B), complex field vectors (∼E + ic0 B), and the (transverse) vector potential (∼A) are the most prominent (Keller [2005]). It is the manner in which the normalization problem is addressed that makes the Landau–Peierls approach to photon wave mechanics different from modern approaches related rather closely to the Oppenheimer theory. Recent review articles on the theories of photon wave mechanics are by Bialynicki-Birula [1996] and the present author (Keller [2005]). It should perhaps not come as a surprise to the reader that different wave functions may be used to describe free-photon dynamics. Observational phenomena in electrodynamics are always related to the field– matter interaction, and essential differences between the various choices therefore only show up when photon and matter wave mechanics are coupled. In statistical optics the complex analytical signal plays a particular role (Mandel and Wolf [1995]), and this signal satisfies a spatially nonlocal equation which is a firstorder differential equation in time. This equation is usually named Sudarshan’s equation referring to Sudarshan’s derivation from 1969 (Sudarshan [1969]). It is noteworthy that the Landau–Peierls equation for the photon wave function (the analytical signal for the electric field) established much earlier (in 1930) is identical to Sudarshan’s equation. The wave mechanical form of eq. (5.6) may be emphasized by rewriting eq. (5.6) in the nonlocal form ∂ i h¯ E (+) (r, t) = ∂t

∞

H |r − r | E (+) (r , t) d3 r ,

(5.9)

−∞

where H (R) =

∗ ih¯ c0 d2 (+) δ (R) − δ (+) (R) 2πR dR 2

(5.10)

is a singular Hamilton density operator, δ (+) (R) being the positive-wave number part of the Dirac delta function, δ(R). To express that the wave mechanical description is about precisely one light quantum Landau and Peierls introduce a certain normalization condition on their

3, § 5]


83

wave function in essentially the following manner. With the help of the Parseval– Plancherel identity, eq. (5.2) [with E(−q, −c0 q) = E ∗ (q, c0 q) inserted], and the relation c0 B(q, c0 q) = qˆ ×E(q, c0 q) it is possible to rewrite the classical ∞ expression for the energy in the electromagnetic field, i.e., W = (0 /2) −∞ [E(r, t) · E(r, t) + c02 B(r, t) · B(r, t)] d3 r, in the form 0 W = 2

∞

E(q, c0 q) · E ∗ (q, c0 q)

−∞

d3 q , (2π)3

(5.11)

that is, as a superposition of the energy in the individual positive-frequency planewave modes of which the field is composed. Since the energy of a light quantum of frequency ω = c0 q (>0) is h¯ c0 q, Landau and Peierls use a normalization condition ∞ 0 1 d3 q 1= (5.12) E(q, c0 q) · E ∗ (q, c0 q) 2 h¯ c0 q (2π)3 −∞

for their wave function. Landau and Peierls now introduce a new operator, −1/2 , defined via ∞ d3 q −1/2 −1 (5.13) F (r) ≡ i q −1 F (q)eiq·r . (2π)3 −∞

The folding theorem shows that

−1/2

1 F (r) = 2π 2 i

∞ −∞

F (r ) d3 r , |r − r |2

(5.14)

−1/2

and hence is an integral (nonlocal) operator. By means of this operator one may show that the normalization condition in eq. (5.12) can be given the forms i0 1= 2h¯ c0

∞

∗ E (+) (r, t) · −1/2 E (+) (r, t) d3 r

−∞

0 = 4π 2 h¯ c0

∞

∗ |r − r |−2 E (+) (r, t) · E (+) (r , t) d3 r d3 r.

(5.15)

−∞

Landau and Peierls note that the quantity E (+) · −1/2 E (+) cannot be considered as a probability density because it is not positive definite and then they add (in translation): “we have not yet succeeded in finding the correct expression for the probability density”.

84


[3, § 5

It is possible, however, to replace the wave function E (+) (r, t) by another one Φ LP (r, t), usually called the Landau–Peierls (LP) photon wave function, which fulfils the standard normalization condition ∞ (5.16) Φ ∗LP (r, t) · Φ LP (r, t) d3 r = 1 −∞

and which relates to E (+) (r, t) in such a manner that eqs. (5.6) and (5.7) are still satisfied, as they must be. The Landau–Peierls photon wave function Φ LP (r, t) and E (+) (r, t) are related nonlocally in space as follows: Φ LP (r, t) =

0 π √ 2 h¯ c0

∞

−5/2

2π|r − r |

E (+) (r , t) d3 r ,

(5.17)

−∞

and the translational invariance of this relation leads to a local connection 0 π Φ LP (q, t) = (5.18) q −1/2 E (+) (q, t) √ 2 h¯ c0 in wave-vector space. From the Fourier transform of eq. (5.6), ih¯ ∂E (+) (q, t)/∂t = (+) (q, t), one therefore obtains ih∂Φ (q, t)/∂t = c hqΦ (q, t), and c0 hqE ¯ LP ¯ 0¯ LP hence a Schrödinger-like photon wave equation ∂ ih¯ Φ LP (r, t) = ∂t

∞

H |r − r | Φ LP (r , t) d3 r

(5.19)

−∞

form-identical to the one for E (+) (r, t) [eq. (5.9)]. The LP-wave function is a transverse vector field, i.e., q · Φ LP (q, t) = 0. After having established a wave equation for a single light quantum, Landau and Peierls consider a non-interacting system consisting of one Dirac electron and N light quanta. The wave equation now is an equation in a (3N + 3)-dimensional space and the simplest solutions F N are product solutions of the free Dirac equation and equations of the type in (5.6) [with (5.7)] for the individual light quanta (labelled by the number ν). The wave function F N therefore is a quantity with 4 × 3N components which satisfies the equation N 1 ∂ im0 c0 (5.20) ν F N = 0, +α·∇+ β+ c0 ∂t h¯ ν=1 where m0 is the electron rest mass, and α1 , α2 , α3 [α = (α1 , α2 , α3 )], and β are √ the Dirac matrices. The differential operators ∇ and ν act on the space coordinates of the electron and the νth light quantum, respectively. Since the light

3, § 5]


85

quanta are bosons the wave function of the given physical state is the linear combination of F N ’s which is completely symmetric with respect to permutation of the light particles. Since the light quanta do not interact with the electron, so far, their number is conserved. In analogy to eq. (5.15) the Landau–Peierls normalization condition thus becomes JN ≡

i0 2h¯ c0

N ∞

FN

−∞

N † ν=1

−1/2 F N d3 r ν

N

d3 rν = 1.

(5.21)

ν=1

When the electron–photon interaction is taken into account the number of light quanta is no longer conserved, and JN now gives the probability that precisely N quanta are present at a given time. To describe the quantum mechanics of the light quanta and a single electron in the presence of the electrodynamic interaction Landau and Peierls set up a new system of coupled equations for the various F N ’s, and show that the condition ∞ N=0 JN = 1, which must always be fulfilled in their approach, puts strong restrictions on the equations. The general formalism becomes quite complicated, and below I shall only indicate how Landau and Peierls modify the free Dirac equation in the presence of a single light quantum. In the relativistic equation for an electron (charge e) in an electromagnetic field described by the vector and scalar potentials A and ϕ, viz., 1 ∂ im0 c0 ie ie (5.22) ϕ+ +α· ∇− A + β F 0 = 0, c0 ∂t h¯ h¯ c0 h¯ Landau and Peierls aim at a replacement of the terms proportional to AF 0 and ϕF 0 by terms containing F 1 . In the Coulomb gauge (∇ · A = 0) where ∇ · E = −ϕ and ∇ × B = −A, one obtains with the operator notation −1 F = −1/2 (−1/2 F ) [see eq. (5.13)] the following expressions for the potentials: ϕ (+) = −−1 (∇ · E (+) ) and A(+) = −−1 (∇ × B (+) ). From the transverse (subscript T) positive-frequency Maxwell equation ∇ × B (+) = (+) −2 (+) = −c−1 1/2 E (+) if the transμ0 J (+) T + c0 ∂E T /∂t one obtains ∇ × B T 0 (+) verse current density J T is neglected and eq. (5.6) [which holds for the transverse part of the electric field, cf. eq. (5.7)] is used. One may show that −1 (+) )], and altogether the considerations above results E (+) T = −∇ × [∇ × ( E in the relation

A(+) = −c0−1 ∇ × ∇ × −3/2 E (+) . (5.23) The vector and scalar (ϕ (+) = −−1 (∇ · E (+) )) potentials have now been expressed in terms of E (+) , and the final step for Landau and Peierls is a replacement of the terms α · AF 0 and ϕF 0 in the Dirac equation (5.19) by

86


[3, § 5

1 1 α · {−c0−1 rot1 [rot1 (1 F 1 )]/2} and −div1 (−1 1 F )/2, respectively. The 2 factors originate in the fact that the classical potential energy eϕ is eϕ/2 per electron. In the Landau–Peierls quantum-mechanical theory the Dirac equation in (5.22) hence is replaced by 1 ∂ ie div1 rot1 rot1 1 im0 c0 0 −α· +α·∇+ F = 0, β F + 3/2 c0 ∂t 2h¯ c0 1 h¯ 1 (5.24) 3/2

remembering that the two terms involving F 1 must be evaluated at the position of the electron, i.e., F 1 (r 1 ; r) ⇒ F 1 (r; r).

5.2. Oppenheimer’s note on light quanta A general quantum theory of the electromagnetic field was constructed by Heisenberg and Pauli in 1929–30 by a method in which the values of the electromagnetic potentials in all space points are considered as variables (Heisenberg and Pauli [1929, 1930]). Independently, Enrico Fermi proposed another less general method of field quantization starting from a Fourier analysis of the potentials (Fermi [1932]). Quantum electrodynamics, as the Heisenberg–Pauli formalism is called, gives a unitary treatment of electrostatic and radiative fields, and in the total energy of the field–particle system in addition to the kinetic energy of the charges there are terms giving (i) the electrostatic interaction energy of the charges with each other and with themselves, (ii) the energy of the quantized light field (including the infinite zero-point energy), and (iii) the interaction energy of the charges and the radiation field. The terms (ii) and (iii) are quite similar to those which appear in Waller’s relativistic extension (Waller [1929, 1930]) of Dirac’s wave mechanical radiation theory (Dirac [1927]). The behavior of the Dirac scheme under space rotations had not been investigated before Oppenheimer took up this problem in 1931 and established a new corpuscular light theory in which the angular momentum of the light quanta took a central position (Oppenheimer [1931]). Although an important purpose of Oppenheimer’s work was to study the connection between Dirac’s wave mechanical theory and the quantum electrodynamics field theory of Heisenberg and Pauli I shall here concentrate on a presentation and a discussion of Oppenheimer’s note on the quantum mechanics of free light quanta. In his search for a wave equation for the de Broglie waves of the light quantum, Oppenheimer first notes that the “natural” second-order relativistic equation ( − ∂t2 )ψ = 0 [with ∂t ≡ c0−1 ∂/∂t] which might serve for a free light quantum of vanishing mass is unsatisfactory because it cannot describe the polarization of

3, § 5]


87

the quantum and does not lead to a suitable density-flux vector with vanishing divergence. Also, the corresponding angular momentum operator L = −ih¯ r × ∇ cannot be the correct one because for a strictly plane wave function it gives the eigenvalue zero for the component of L parallel to the vector of propagation, in disagreement with the available experimental data. Just as for the electron Oppenheimer therefore seeks a linear equation in space–time. In 1927 Jordan had suggested to use a two-component equation (σ · ∇ + ∂t )ψ = 0, where σ is the Pauli spin vector, to account for the polarization of the light quanta (Jordan [1927]). It is impossible to associate such a spinor of the first rank (ψ) with an electrodynamic field strength (or potential), and Jordan’s suggestion leads to a half-integer angular momentum of the field. Oppenheimer therefore tries a three component theory, i.e., ψ = (ψ1 , ψ2 , ψ3 ), and replaces σ by an angular momentum vector operator τ = (τ1 , τ2 , τ3 ), whose components are 3×3 Hermitian matrices. Oppenheimer’s wave equation for the light quantum hence takes the form (τ · ∇ + ∂t )ψ = 0,

(5.25)

where the τi ’s (i = 1, 2, 3) satisfy the angular momentum commutator rules τ × τ = iτ .

(5.26)

The eigenvalues for the τ -components hence are 0, ±1. From eq. (5.25) and its Hermitian conjugate (superscript †) one may derive the light quantum conservation law ∂ † ψ ψ + ∇ · ψ † c0 τ ψ = 0 (5.27) ∂t between the probability density ψ † ψ and probability current density ψ † c0 τ ψ. In the Oppenheimer representation of the light quantum wave mechanics c0 τ in some respect plays the role of a photon velocity operator whose noncommuting components have the eigenvalues 0, ±c0 . Perhaps not unexpected, it turns out that the eigenvalue 0 is related to the irrotational (longitudinal) solutions of Maxwell’s equations, and these solutions do not describe the free light quantum dynamics. The longitudinal field dynamics relates to the scalar and longitudinal photons and is of importance not only in high-energy electrodynamics but also in near-field quantum optics (Keller [2005]). In the Dirac representation of electron wave mechanics, the noncommuting components of the velocity operator have the eigenvalues ±c0 . The physics behind this from a nonrelativistic point of view surprising result was clarified by Foldy and Wouthuysen, who, by performing a certain unitary transformation on the Dirac equation, made the link to nonrelativistic wave mechanics (Foldy and Wouthuysen [1950]). Among other things they showed that the “position” operator r in the Pauli equation is a sort of mean

88


[3, § 5

position operator within a volume which linear extension is of the order of the Compton wavelength, λC = h/(m0 c0 ). By considering the transformation of the vector field ψ = (ψ1 , ψ2 , ψ3 ) under infinitesimal rotations in space Oppenheimer obtains the expression J = −ih¯ r × ∇ + h¯ τ

(5.28)

for the angular momentum operator of the field. Oppenheimer notes that eq. (5.25) is invariant under space rotations and the components of J thus are constants of the motion. In the basis in which the operators τ 2 and τz are diagonal (spherical representation) the τi ’s may be represented by the matrices ⎛ ⎞ ⎛ ⎞ 0 1 0 0 −i 0 1 ⎝ 1 τx = √ 1 0 1 ⎠ ; τy = √ ⎝ i 0 −i ⎠ ; 2 2 0 1 0 0 i 0 ⎛ ⎞ 1 0 0 τz = ⎝ 0 0 0 ⎠ . (5.29) 0 0 −1 Oppenheimer uses this standard representation for the spin of the vector field (yet with an interchange of the τx and τy matrices). The spherical (unitary) basis 2 vectors ηm (m = 1, 0, √ −1), eigenvectors of τ and τz , are up to a phase factor η±1 = (1, ±i, 0)/ 2 and η0 = (0, 0, 1). In the Cartesian representation the “spin”-components of the vector field are given by h¯ (τi )j k =

h¯ ij k , i

(5.30)

where ij k is the Levi-Civita tensor. For plane monochromatic waves, ψ = u exp[i(q · r − ωt)], the component of the angular momentum operator along the wave-vector (q) direction commutes with the momentum operator and is just qˆ · J = h¯ qˆ · τ . The eigenvalues of the related helicity operator, h ≡ qˆ · τ , are 0, ±1. It is apparent from eq. (5.25) that c0 phu = h¯ ωu,

(5.31)

where p is the magnitude of the momentum. This means that the eigenvalue 0 for h cannot occur except when ω = 0. It is required that “true” light quanta have ω > 0, and in consequence only one solution to eq. (5.31) survives, namely the one with the eigenvalue +1 for h. If we wish to exclude negative energies eq. (5.31) gives us only one kind of polarization. To obtain all solutions needed Oppenheimer suggests that eq. (5.25) is supplemented by an equation where τ is

3, § 5]


89

replaced by an τ operator satisfying the vectorial commutator relation τ × τ = −iτ , i.e., (τ · ∇ + ∂t )ψ = 0.

(5.32)

To each q and for ω > 0, we now have two solutions [one for eq. (5.25) and one for eq. (5.32)], with respectively the eigenvalues 1 and −1 for the helicity operator. Oppenheimer mentions the possibility of using a six-component light quantum wave function Φ O ≡ (ψ, ψ ) satisfying the block-matrix equation 0 τ · ∇ + ∂t Φ O = 0, (5.33) 0 −τ · ∇ + ∂t and to exclude negative energies he suggests to use the auxiliary condition √ ∂t Φ O = − Φ O , (5.34) √ where is the nonlocal Landau–Peierls operator [see eq. (5.5)]. The Oppenheimer six-component wave function for the light quantum, Φ O , is closely related to the photon energy wave function used by several scientists in recent years (Bialynicki-Birula [1996], Keller [2005]). Although eq. (5.25) is covariant under space rotations it is not Lorentz covariant. This suggests that eq. (5.25) is a degenerate form of a set of four simultaneous equations for a four-component wave function. Oppenheimer therefore replaces the τi -matrices by certain 4 × 4 ρi -matrices of which the τi ’s are submatrices. The ρi -matrices must have eigenvalues ±1, each twice, and he extends the τi -matrices in such a manner that the four-divergence of the new wave function vanishes. Oppenheimer presents his ρi -matrices in a form which relates to the spherical representation of the τi -matrices [eqs. (5.29)]. An analogous but less cumbersome set of ρi -matrices can be obtained by extending the Cartesian representation of the τi ’s [eq. (5.30)]. Below, we shall use the extended Cartesian representation, originally introduced by Ohmura [1956] and Moses [1959]. In the Dirac notation, the Cartesian (|x, |y, |z) and spherical (| + 1, | − 1, |0) bases may be related via (Cohen-Tannoudji, Dupont-Roc and Grynberg [1989])

| ± 1 = ∓2−1/2 |x ± i|y ; |0 = |z, (5.35) and from the matrix elements of τ , m|τ |n, in the spherical representation (|m, |n) the Cartesian (|i, |j , . . .) matrix elements of the Hermitian operator O ≡ τ · ∇ + ∂t are obtained from the formula i|mm|τ |nn|j · ∇ + δij ∂t , i|O|j = (5.36) m,n

90


[3, § 5

where δij is the Kronecker delta. For definiteness we thus take ρi ’s in the form ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 0 1 0 ⎜1 0 0 0 ⎟ ⎜0 0 0 i ⎟ ⎟ ⎜ ⎟ ρ1 = ⎜ ⎝ 0 0 0 −i ⎠ ; ρ2 = ⎝ 1 0 0 0 ⎠ ; 0 0 i 0 0 −i 0 0 ⎛ ⎞ 0 0 0 1 ⎜ 0 0 −i 0 ⎟ ⎟ ρ3 = ⎜ (5.37) ⎝0 i 0 0⎠. 1 0 0 0 Since ρi2 = 1 (i = 1, 2, 3), the eigenvalues of the ρi ’s are ±1, each twice. The Oppenheimer wave equation for the light quanta thus is (ρ · ∇ + ∂t )Ψ = 0.

(5.38)

The ρi ’s also satisfy the anticommutator relations [ρi , ρj ]+ = 2δij , and since therefore (ρ · ∇ − ∂t )(ρ · ∇ + ∂t ) = ∇ 2 − ∂t2 each component of ψ satisfies a wave equation analogous to the one obeyed by the various Cartesian components of the electromagnetic field vectors in vacuum. The system in eq. (5.38) is Lorentz invariant, and from the first component of eq. (5.38) it appears that the four-divergence of Ψ vanishes, i.e., ∂i Ψ i = 0,

(5.39)

with the usual summation convention in force. The ρi ’s do not satisfy the angular momentum commutator rule, and to incorporate the basic light quantum wave equation in eq. (5.25) in the framework of eq. (5.38) Oppenheimer sets ψ0 = 0. In consequence of eq. (5.39) the three-divergence of the four-column wave function Ψ = (0, ψ1 , ψ2 , ψ3 ) = (0, ψ) vanishes, that is ∇ · ψ = 0. This condition in turn eliminates the longitudinal (electrostatic) solution (ω = 0 ⇒ h = 0) of eq. (5.31). In the absence of electrically charged particles longitudinally polarized field quanta do not appear. After having discussed the four-component formalism Oppenheimer shows that his approach is related to the free-space Maxwell equations written in spinor form. The mathematical apparatus of spinor calculus was given by van der Waerden [1929], and he also showed how to write the Dirac equation in spinor form. Shortly afterwards, Laporte and Uhlenbeck (1931) demonstrated that also the Maxwell equations could be written in spinor form, and a few months after the publication of the Laporte–Uhlenbeck paper, Oppenheimer linked his light quantum theory [eq. (5.22) plus the transversality condition ∇ · ψ = 0] to the spinorial form of Maxwell’s equations. The connection between Oppenheimer’s threecomponent wave function ψ = (ψ1 , ψ2 , ψ3 ), with the auxiliary transversality

3, § 6]

Eikonal equation for the photon

91

condition ∇ · ψ = 0, and the transverse electric (subscript T), E T , and magnetic, B, vacuum field vectors is given by 0 (E T + ic0 B). ψ ≡ ψ+ = (5.40) 2 Since all light quanta have positive energy (ω > 0) one must select the solution to eq. (5.25) which has positive helicity (h = 1), as we have seen. Let us denote the (+) corresponding light quantum wave function by ψ + . To include also the negative helicity state (h = −1) we now just use Oppenheimer’s suggestion of a sixcomponent wave function Φ O , satisfying eq. (5.33). Thus, we are led to (+) (+) ψ+ 0 E T + ic0 B (+) = ΦO = (5.41) (+) (+) , 2 E (+) ψ− T − ic0 B where a superscript (+) has been added to the various quantities to emphasize that only the positive-frequency parts of the field vectors are used in the light quantum theory, cf. the Landau–Peierls auxiliary condition [eq. (5.34)]. The wave function Φ O in eq. (5.41) may be recognized by the reader as the photon energy wave function (Bialynicki-Birula [1996], Keller [2005]), often used in modern studies. After proper normalization of Φ O and subsequent insertion into eq. (5.33) one obtains the so-called Schrödinger equation for the photon energy wave function (Bialynicki-Birula [1996], Keller [2005]). Although a number of subsequent works have contributed to the development and understanding of the photon energy wave function formalism, and have shown its usefulness in various contexts, the most essential part of the formalism was born with Oppenheimer’s 1931 article entitled “Note on Light Quanta and the Electromagnetic Field”.

§ 6. Eikonal equation for the photon It was mentioned in Section 4.3 that the Hamilton–Jacobi equation for the characteristic function of a classical point particle has a similarity to the eikonal equation in geometrical optics, and that this similarity helped Schrödinger to establish his famous wave-mechanical equation. By means of a certain transformation [eq. (4.20)] it was realized that the Schrödinger equation could be rewritten as a so-called quantum-mechanical Hamilton–Jacobi equation, and in the limit h¯ → 0 this equation describes Newtonian dynamics of a classical point particle. Since the photon wave mechanics of Oppenheimer can be obtained from the free-space Maxwell equations it is not surprising that the eikonal equation in vacuum appears from Oppenheimer’s theory in the short-wavelength limit. Although Oppenheimer did not make this connection I would like to finish this article with a very brief

92


[3, § 6

introduction to the quantum-physical Hamilton–Jacobi formalism of the free photon. In the presence of light–matter interactions the Hamilton–Jacobi equations of the electron and photon are coupled in the near-field zone of matter. For the following calculation it is useful to rewrite Oppenheimer’s wave equations for photons of positive [eq. (5.25)] and negative [eq. (5.32)] helicity in the forms ∂ (+) (+) ψ (r, t) = ±c0 ∇ × ψ ± (r, t), (6.1) ∂t ± remembering that the superscript (+) indicates that the photon wave functions consist of positive-frequency components only. By insertion of the Fourier integral expressions i

(+) ψ ± (r, t)

1 = 2π

∞

F ± (r; ω)eiq0 S± (r;ω) e−iωt dω,

(6.2)

0

where S± (r; ω) is a real scalar function, in eqs. (6.1) one obtains ∇ × F ± (r; ω) = q0 ±F ± (r; ω) + iF ± (r; ω) × ∇S± (r; ω) .

(6.3)

(+)

From the auxiliary condition ∇ · ψ ± (r, t) = 0 one gets ∇ · F ± (r; ω) = −iq0 F ± (r; ω) · ∇S± (r; ω).

(6.4)

It is easy to show that eq. (6.4) is contained in eq. (6.3). Equations (6.3) and (6.4) constitute for each helicity basic equations among the three components of the complex “amplitude” vector F ± (r; ω) and the gradient of the complex “phase” function S± (r; ω). In the short-wavelength limit (q0 → ∞) the factor to q0 on the right-hand side of eq. (6.3) must be zero. In matrix notation this demand can be written in the form ↔ U ∓τ · ∇S± (r; ω) · F ± (r; ω) = 0, (6.5) ↔

where the τi ’s are given by eq. (5.30), and U is the unit matrix. For a given helicity the set of linear scalar equations in (6.5) has a nontrivial solution only if the associated determinant vanishes. This condition gives after a few elementary calculations ∇S± (r; ω) · ∇S± (r; ω) = 1.

(6.6)

The equations in (6.6) are the eikonal equations associated with short-wavelength photons of positive and negative helicities. It can be shown that the mean photon momenta of the two helicity states, P ± , for purely monochromatic states are given

3]

References

93

by P± =

c0−1

∞

∗ F ± (r; ω) · F ± (r; ω) ∇S± (r; ω) d3 r

(6.7)

−∞

in the eikonal approximation. The directions of the local photon momenta are given by the unit vectors e± (r; ω) = ∇S± (r; ω).

References Arons, A.B., Peppard, M.B., 1965, Am. J. Phys. 33, 367. Barut, A.O., Blaire, B., 1992, Phys. Rev. A 45, 2810. Barut, A.O., Dowling, J.P., 1987, Phys. Rev. A 36, 649. Barut, A.O., Van Huele, J.F., 1985, Phys. Rev. A 32, 3187. Bialynicki-Birula, I., 1996, in: Wolf, E. (Ed.), Progress in Optics, vol. XXXVI, North-Holland, Amsterdam, p. 245. Bohr, N., Kramers, H.A., Slater, J.C., 1924, Phil. Mag. 47, 785 (German version: Zeitschr. Phys. 24, 69). Boltzmann, L., 1877, Wiener Ber. 76, 373. Born, M., Jordan, P., 1925, Zeitschr. Phys. 34, 858. Born, M., Heisenberg, W., Jordan, P., 1926, Zeitschr. Phys. 35, 557. Born, M., Wolf, E., 1999, Principles of Optics, 7th expanded edition, Cambridge Univ., Cambridge. Bose, S.N., 1924, Zeitschr. Phys. 26, 178. Bothe, W., Geiger, H., 1924, Zeitschr. Phys. 26, 44. Bothe, W., Geiger, H., 1925a, Naturwissenschaften 13, 440. Bothe, W., Geiger, H., 1925b, Zeitschr. Phys. 32, 639. Brillouin, M., 1919, Compt. Rend. 168, 1318. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G., 1989, Photons and Atoms, Wiley, New York. Compton, A.H., 1923, Phys. Rev. 21, 483. Compton, A.H., Simon, F., 1925, Phys. Rev. 26, 189. Davisson, C.J., Germer, L., 1927, Nature 119, 558. De Broglie, L., 1923a, Compt. Rend. 177, 507. De Broglie, L., 1923b, Compt. Rend. 177, 548. De Broglie, L., 1923c, Compt. Rend. 177, 630. De Broglie, L., 1924, Phil. Mag. 47, 446. De Broglie, L., 1925, Annales de Physique 3, 22. This is the text of de Broglie’s 1924 Doctoral Thesis. De Broglie, L., 1971, Matter and Light, The New Physics, Dower, New York. De Broglie, M., 1921a, Compt. Rend. 172, 275. De Broglie, M., 1921b, Compt. Rend. 172, 806. De Broglie, M., 1921c, Compt. Rend. 172, 1157. Debye, P., 1910, Ann. Phys. 33, 1427. Debye, P., 1923, Phys. Zeitschr. 24, 161. Dirac, P.M., 1926, Proc. Roy. Soc. A 112, 661. Dirac, P.M., 1927, Proc. Roy. Soc. A 114, 243. Dresden, M., 1987, H.A. Kramers. Between Tradition and Revolution, Springer, Berlin. Chapter 13, Section III. Ehrenfest, P., 1906, Phys. Zeitschr. 7, 528.

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Einstein, A., 1905, Ann. Phys. 17, 132. Einstein, A., 1909, Phys. Zeitschr. 10, 185. Einstein, A., 1916a, Verh. Deutsch. Phys. Ges. Berlin 18, 318. Einstein, A., 1916b, Mitt. Phys. Ges. Zürich 16, 47. Einstein, A., 1917a, Phys. Zeitschr. 18, 121. Einstein, A., 1917b, Verh. Deutsch. Phys. Ges. Berlin 19, 82. Einstein A., 1917c, Sitzungsberichte, Preussische Akademie der Wissenschaften, p. 606. Einstein, A., 1924a, Sitzungsberichte, Preussische Akademie der Wissenschaften. Phys.-math. Kl., p. 261. Einstein, A., 1924b, Sitzungsberichte, Preussische Akademie der Wissenschaften. Phys.-math. Kl., p. 3. Einstein, A., 1949, Autobiographisches, in: Schilpp, P. (Ed.), Albert Einstein: Philosopher–Scientist, Tudor, New York. Fermi, E., 1932, Rev. Mod. Phys. 4, 87. Foldy, L.L., Wouthuysen, S.A., 1950, Phys. Rev. 78, 29. Goldstein, H., 1980, Classical Mechanics, Addison-Wesley, London. Heisenberg, W., 1925, Zeitschr. Phys. 33, 879. Heisenberg, W., Pauli, W., 1929, Zeitschr. Phys. 56, 1. Heisenberg, W., Pauli, W., 1930, Zeitschr. Phys. 59, 168. Jeans, J.H., 1905, Phil. Mag. 10, 91. Jordan, P., 1927, Zeitschr. Phys. 44, 292. Jordan, P., Pauli, W., 1928, Zeitschr. Phys. 47, 151. Keller, O., 2005, Phys. Rep. 411, 1. Kirsten, G., Körber, H., 1975, Physiker über Physiker, Akademie Verlag, Berlin, p. 201. Klein, M., 1973, in: Cohen, E.G.D., Thirring, W. (Eds.), The Boltzmann Equation, Springer, Berlin, p. 53. Knight, P.L., Allen, L., 1983, Concept of Quantum Optics, Pergamon, Oxford, p. 4. Landau, L., Peierls, R., 1930, Zeitschr. Phys. 62, 188. Laport, O., Uhlenbeck, G.E., 1931, Phys. Rev. 37, 1380. Lewis, G.N., 1926, Nature 118, 874. Lorentz, H.A., 1916, Les théories statistiques en thermodynamique, Tübner, Leipzig. Lummer, O., Pringsheim, E., 1900, Verh. Deutsch. Phys. Ges. Berlin 2, 163. Mandel, L., Wolf, E., 1995, Optical Coherence and Quantum Optics, Cambridge Univ., Cambridge. Millikan, R.A., 1914, Phys. Rev. 4, 73. Millikan, R.A., 1915, Phys. Rev. 6, 55. Millikan, R.A., 1916a, Phys. Rev. 7, 18. Millikan, R.A., 1916b, Phys. Rev. 7, 355. Morse, P.M., 1964, Thermal Physics, Benjamin, Amsterdam. Moses, H.E., 1959, Phys. Rev. 113, 1670. Ohmura, T., 1956, Progr. Theoret. Phys. (Kyoto) 16, 684. Oppenheimer, J.R., 1931, Phys. Rev. 38, 725. Paschen, W., 1897, Ann. Phys. 60, 662. Passante, R., Power, E.A., 1987, Phys. Rev. A 35, 188. Pauli, W., 1933, Prinzipien der Quantentheorie, in: Handbuch der Physik, vol. 24, Springer, Berlin, p. 189. Planck, M., 1900a, Ann. Phys. 1, 69. Planck, M., 1900b, Verh. Deutsch. Phys. Ges. Berlin 2, 202. Planck, M., 1900c, Verh. Deutsch. Phys. Ges. Berlin 2, 237. Planck, M., 1901, Ann. Phys. 4, 553. Planck, M., 1943, Naturwiss. 31, 153.

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Rayleigh, J.W.S., 1900, Phil. Mag. 49, 539. Rayleigh, J.W.S., 1905, Nature 72, 54. Rubens, H., Kurlbaum, F., 1900, Sitzungsberichte, Preussische Akademie der Wissenschaften, p. 929. Schrödinger, E., 1926a, Ann. Phys. 79, 361. Schrödinger, E., 1926b, Ann. Phys. 79, 489. Schrödinger, E., 1926c, Ann. Phys. 79, 734. Schrödinger, E., 1926d, Ann. Phys. 81, 109. Slater, J.C., 1924, Nature 113, 307. Sudarshan, E.C.G., 1969, J. Math. and Phys. Sci. (Madras) 3, 121. van der Waerden, B., 1929, Göttingen Nachrichten, p. 100. Waller, I., 1929, Zeitschr. Phys. 58, 75. Waller, I., 1930, Zeitschr. Phys. 61, 837. Weisskopf, V., Wigner, E.P., 1930, Zeitschr. Phys. 63, 54. Weisskopf, V., Wigner, E.P., 1931, Zeitschr. Phys. 65, 18. Wheaton, B.R., 1983, The Tiger and the Shark, Cambridge Univ., Cambridge. Wien, W., 1893, Sitzungsberichte, Preussische Akademie der Wissenschaften, Ber. Berlin Akad. Wiss., p. 55. Wien, W., 1896, Ann. Phys. 58, 662.


Chapter 4

Field quantization in optics by

Peter W. Milonni 104 Sierra Vista Dr., Los Alamos, NM 87544, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(07)50004-1 97

Contents


99

§ 2. Background basics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

§ 3. Coherence theory: Classical and quantum . . . . . . . . . . . . . . .

104

§ 4. Semiclassical radiation theory . . . . . . . . . . . . . . . . . . . . . .

111

§ 5. Non-classical light . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

§ 6. Quantum noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

§ 7. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

98

§ 1. Introduction After the development of quantum electrodynamics (QED) it seemed that little would be left to say about the fundamental nature of light. But beginning in the 1950s new kinds of experiments on the statistical properties of light, together with the advent of lasers, raised questions that QED as then practiced was ill-suited to address. What was needed and achieved was a better characterization of the quantum and coherence properties of light, along with a better appreciation of the effectiveness as well as the limitations of semiclassical radiation theory in which matter is treated quantum-mechanically while light is described classically. It is obviously impossible here to cover all the important developments over the past half-century of quantum optics, or to present a history with all the analysis and attention to detail that a proper history entails. The intent is to survey some key ideas and results, with emphasis on physics rather than formulations and without much concern about citations and credits. The literature has grown enormously in the past two decades in particular; wherever possible and appropriate, citations are to books or reviews where references to some of the original research papers may be found. Progress since 1960 has not required any fundamental change in the quantum theory of radiation. In the following section some of the most basic features of the theory are briefly discussed in the context of quantum optics, and Section 3 reviews the development of the quantum theory of optical coherence based on field correlation functions that have a strong formal resemblance to those introduced in classical coherence theory (Born and Wolf [1994]). Section 4 focuses on the range of validity of semiclassical radiation theory, a subject much debated in the 1970s, and Section 5 reviews some developments in those aspects of light that are distinctly quantum in nature, including those invoked in comparing hidden-variable theories to quantum mechanics. Quantum optics has become increasingly important in communications, especially in the analysis of quantum noise. In Section 6 we discuss some developments in this area related to the quantum noise properties of lasers. Section 7 offers a brief view of the present status of field quantization in optics. 99

100

Field quantization in optics

[4, § 2

§ 2. Background basics In his 1905 paper that is best known for the prediction of the linear relation between radiation frequency and stopping potential in the photoelectric effect, Einstein presented the new idea that in the emission and absorption of light “the energy of a light ray . . . consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed in complete units”. He justified his “heuristic point of view” by comparing the entropy change with volume of thermal radiation of frequency ω with that of an ideal gas of N particles undergoing the same (isothermal) change in volume. The two expressions for the entropy had the same form if the radiation energy in the Wien limit h¯ ω kT was assumed to be N h¯ ω. Einstein concluded that radiation satisfying Wien’s law “behaves thermodynamically as though it consisted of a number of independent energy quanta of magnitude [h¯ ω].” Using the formula ΔE 2 = kT 2 ∂E/∂T for the variance in energy of a system in thermal equilibrium at temperature T , and E = ρ(ω)V dω for the average energy of thermal radiation in a volume V and a frequency interval [ω, ω + dω], we obtain, from the Planck formula for ρ(ω), the fluctuation formula presented by Einstein in 1909 (Wolf [1979]): π 2 c3 2 2 (2.1) ΔE = hωρ(ω) + 2 ρ (ω) V dω. ¯ ω If instead we use the (classical) Rayleigh–Jeans formula for ρ(ω), we obtain ΔE 2 =

ω2 π 2 c3 2 2 (kT ) V dω = ρ (ω)V dω. π 2 c3 ω2

(2.2)

In other words, the second term in brackets in eq. (2.1) can be associated with the classical wave theory of thermal radiation. Now let us regard thermal radiation as a collection of particles of energy h¯ ω. Since there are (ω2 /π 2 c3 )V dω modes of the field (or states of a particle) in the volume V and the frequency interval [ω, ω + dω], we assume E = n(ω)h¯ ω ×

ω2 V dω ≡ ρ(ω)V dω. π 2 c3

(2.3)

If we also assume that the particle numbers belonging to different modes fluctuate independently according to a Poisson distribution, so that Δn2 = n2 − n2 = n,

(2.4)

4, § 2]

Background basics

101

we deduce that ω2 ω2 V dω = n(ω)(h¯ ω)2 2 3 V dω 2 3 π c π c = h¯ ωρ(ω)V dω,

ΔE 2 = Δn2 (h¯ ω)2

(2.5)

which is the first term in eq. (2.1). This result also follows from the argument leading to eq. (2.1), but using the Wien formula for ρ(ω). These results are important as early indicators of wave–particle duality. It is also noteworthy that, while the photoelectric effect is neatly explained by Einstein’s heuristic, “photon” description of radiation, it can in fact be explained quite adequately without quantizing the field (Section 4). Writing eq. (2.1) in terms of the fluctuations in n(ω), we have the variance associated with Bose–Einstein statistics: Δn2 = n + n2 , or, equivalently, n2 − n = 2n2 .

(2.6)

As will be discussed in Section 3, the factor 2 is associated with the photon bunching of thermal radiation. This effect in intensity correlations was observed in the experiments of Hanbury Brown and Twiss that marked the beginning of modern quantum optics (Brown and Twiss [1958], Mandel and Wolf [1995]). The formal quantum theory of the electromagnetic field was developed by Dirac and others in the 1920s. For a monochromatic field in vacuum the quantized electric field, for instance, has the form 1/2 a(t)F(r − a † (t)F∗ (r) . E(r, t) = i(2π hω) (2.7) ¯ E(r, t), a(t), and a † (t) are operators, whereas F(r) is a classical transverse (∇ · E = 0) and normalized ( d3 r |F|2 = 1) mode function satisfying the Helmholtz equation (∇ 2 F + (ω2 /c2 )F = 0) and whatever boundary conditions are appropriate. a and a † are respectively the lowering and raising operators for a harmonic oscillator of frequency ω: [a(t), a † (t)] = 1, a˙ = −iωa. The Hamiltonian for the (single-mode) field is

1 † 2 1 1 3 2 † † d r E + H = h¯ ω aa + a a = hω HF = , (2.8) ¯ a a+ 8π 2 2 and the energy eigenvalues are (n + 12 )h¯ ω, n = 0, 1, 2, . . . . The quantum theory of the field accounts naturally for the observed “wave–particle duality” of light: the wave properties of the field (2.7) are described by F(r), while the particle aspects are evident in the fact that the eigenvalues of the photon number operator a † a are the positive integers and the allowed energies differ by integral multiples of hω. ¯ This duality is seen more explicitly in an analysis of the Young

102


[4, § 2

two-slit experiment in the case of a single incident photon. The probability of registering a photon count at points on the observation screen is given by the (appropriately normalized) classical interference pattern, and is zero, for instance, at points where there is complete destructive wave interference. The entire classical interference pattern is reproduced when the experiment is repeated a very large number of times, or when it is done once with a very large number of incident photons. Such observations led to Dirac’s famous statement in his Principles of Quantum Mechanics that “Each photon . . . interferes only with itself. Interference between two different photons never occurs”. As discussed below, this statement from the early years of quantum theory has caused considerable confusion, and should not be taken too seriously outside its historical context. A thermal field mode is described by the density operator ρ=

∞

p(m)|mm|

with p(m) = 1 − e−h¯ ω/kT e−mh¯ ω/kT ,

m=0

and the fluctuation formula (2.6) deduced by Einstein follows easily when we identify the quantum expectation value Δn2 for the quantum thermal state with the classically defined thermal average Δn2 . Note also that the normally ordered (a’s appearing to the right of a † ’s) expectation value 2

† † † † a a aa = a aa − 1 a = a † a − a † a

= n2 − n = 2n2 , (2.9) in agreement with (2.6). For reasons reviewed in the following section, normal ordering has played a central role in the development of the quantum theory of photon counting and optical coherence. The variance in the photon number can be written as

2 † † † 2 † † 2 Δn = a aa a − a a = a a a + 1 a − a † a

(2.10) = Δn2 particles + Δn2 waves , where Δn2 particles = a † a = n is the variance associated with the Poisson statistics of particles and Δn2 waves ≡ a † a † aa − a † a2 ; the latter is closely related to the Q-parameter introduced by Mandel (Mandel and Wolf [1995]); see Section 5. For a thermal field, Δn2 waves = n2 , or n2 = 2n2 . With these identifications we can say that the photon bunching observed by Hanbury Brown and Twiss arises from wave fluctuations. On the other hand, for a field state that is an eigenstate of the lowering operator a – a coherent state – we have Δn2 waves = 0 and Δn2 = Δn2 particles , i.e., the photon number variance is attributable to particle fluctuations and there is no photon bunching (Section 3).

4, § 2]

Background basics

103

Although single-mode fields and their quantum properties have been investigated experimentally (Section 4), it is of course generally necessary to use the multimode generalization of eq. (2.7): 1/2 E(r, t) = i (2.11) aβ (t)Fβ (r) − aβ† (t)F∗β (r) , (2π hω ¯ β) β

where ωβ is the frequency of the mode labelled by β. The mode functions Fβ (r) have the properties stated above for F(r), and are also assumed to be orthogonal and to form in their entirety a complete set. Any two photon annihilation and creation operators aβ , aβ† commute when they refer to different modes (Loudon [2000], Mandel and Wolf [1995], Scully and Zubairy [1997]). It has become customary in quantum optics to assume the atom–field interaction −d · E rather than the “minimal coupling” form involving the vector potential for electric-dipole transitions, the most important transitions by far for the interaction of light and matter (Ackerhalt and Milonni [1984]). Introducing the operators σij = |ij | and the dipole matrix elements dij = i|d|j , we can write the Hamiltonian for the interaction of the field with an atom at r as

1 H = Ei σii + h¯ ωβ aβ† aβ + 2 β i † ∗ 1/2 (2.12) −i (2π hω ¯ β ) dij σij · aβ Fβ (r) − aβ Fβ (r) , ij

β

where Ei is the energy of the (unperturbed) atomic state i. Radiative atomic transition rates and level shifts are found to depend on |Fβ (r)|2 . In the case of an atom in free space the mode functions Fβ (r) may be taken to vary as eikβ ·r , and of course the transition rates and level shifts are then independent of r. But for an atom inside a cavity, for instance, the functions |Fβ (r)|2 are not independent of r, and quantities like the spontaneous emission rate or the Lamb shift are found to depend on the position of the atom. Beginning as early as the 1960s, experiments have demonstrated that the spontaneous emission rate can be enhanced or suppressed, depending on the environment in which an atom is placed. It has been shown, for example, that spontaneous emission can be completely inhibited if an atom is placed in a cavity such that there is no allowed mode at the atom’s transition frequency (Haroche and Kleppner [1989]). Similarly it has been shown that there is a position dependence of the radiative level shift of an atom near a conducting wall, the so-called Casimir–Polder effect. The study of these and related effects in “cavity quantum electrodynamics” (cavity QED) has become an active area of research (Barton [1994], Berman [1994]).

104


[4, § 3

The (generally infinite) zero-point energy β 21 h¯ ωβ is usually deleted from the Hamiltonian (2.12) on the grounds that such an additive constant in the Hamiltonian has no physical consequences (e.g., it commutes with every operator and therefore does not affect Heisenberg equations of motion). However, changes in this zero-point field energy due to the presence of material bodies in otherwise free space can have measurable effects. The most famous of these effects is the Casimir force between two parallel, perfectly conducting plates: the (infinite) zero-point field energy when the plates are separated by a distance d is different from the (infinite) zero-point field energy when the plates are infinitely far apart. The calculated difference implies an attractive force per unit area that varies as 1/d 4 , and during the past decade the measurement of this force has been reported by several groups (Lamoreaux [2005]). We will not discuss such vacuum QED effects further, except to note that they can be derived in different ways that do not explicitly invoke zero-point field energy (Milonni [1994]).

§ 3. Coherence theory: Classical and quantum The atom–field interaction may be written in the form −d · E = −d · E(+) (r, t) + E(−) (r, t)

(3.1)

for a single atom located at r, where 1/2 (2π hω E(+) (r, t) = i ¯ β ) Fβ (r)aβ (t),

(3.2)

β

E(−) (r, t) = −i

† 1/2 ∗ (2π hω ¯ β ) Fβ (r)aβ (t).

(3.3)

β

As noted earlier, normally ordered field products, in which E(+) operators act on state vectors before E(−) operators, are of fundamental importance in quantum optics. An example in which a normally ordered field product appears is the absorption process in which an atom makes a transition from the initial, lowerenergy state |1 (E1 = 0) to a final, higher-energy state |2 (E2 = h¯ ω0 ), while the field goes from an initial state |I to a final state |F . The transition probability amplitude for this process (|i ≡ |1|I → |f ≡ |2|F ) is i afi (t) = e−iEf t/h¯ d21,μ h¯

t

dt F |Eμ (r, t )|I eiω0 t ,

(3.4)

0

where summation over μ = 1, 2, 3 is implicit. For sufficiently long times t only the “positive-frequency” (e−iωt ) part of Eμ (r, t ) contributes significantly

4, § 3]

Coherence theory: Classical and quantum

105

to afi (t); in other words, the energy-conserving transition probability is afi (t)2 = 1 d21,μ d12,ν h¯ 2 t t

× dt dt

I Eν(−) (r, t )F F Eμ(+) (r, t

)I eiω0 (t −t ) , 0

(3.5)

0

where we have used the fact that F |E(+) (r, t)|I ∗ = I |E(−) (r, t)|F . Summing over all possible final states of the field (assuming that no observations are made to discriminate among these states) and using the completeness relation F |F F | = 1, we obtain the absorption transition probability p(t) =

1

t dt

d21,μ d12,ν h2 ¯

0

t

dt

Eν(−) (r, t )Eμ(+) (r, t

) ,

(3.6)

0

the expectation value being over the initial field state |I . In a practical photodetector there is effectively a continuum of final electron states |f, and the photodetection rate dp/dt is obtained by summing over the possible final electron states, weighting them according to their probabilities for being recorded by the photodetector. In the case of a point, fast “ideal broadband detector”, the photon counting rate is directly proportional to the first-order field correlation function1 (Glauber [1963a, 1965], Mandel and Wolf [1965], Mandel and Wolf [1995])

(−) (+) G(1) (3.7) μν (r1 , t1 ; r2 , t2 ) = Eν (r1 , t1 )Eμ (r2 , t2 ) evaluated at r1 = r2 = r and t1 = t2 = t. But regardless of whether the photodetector is “ideal” or not, the photodetection rate involves a normally ordered field correlation function. The general first-order field correlation function (3.7) describes the interference effects observed, for instance, in Michelson and Young interferometers. In particular, temporal coherence and spatial coherence are described and quantified by G(1) μν (r1 , t1 ; r2 , t2 ) with r1 = r2 and t1 = t2 , respectively. In similar fashion we can consider photodetectors at different locations and calculate the probability that each absorbs a photon in a certain time interval. 1 There is a trivial and long-standing difference in the terminology used in defining these field correlation functions. Mandel and Wolf [1995] and others refer to eqs. (3.7) and (3.8) as second- and fourth-order correlation functions, respectively. The usage here follows that of, for instance, Glauber [1963a, 1963b] and Loudon [2000].

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This leads us to define higher-order field correlation functions such as (2)

Gμνσ δ (r1 , t1 ; r2 , t2 ; r3 , t3 ; r4 , t4 )

(+) = Eμ(−) (r1 , t1 )Eν(−) (r2 , t2 )Eσ(+) (r3 , t3 )Eδ (r4 , t4 ) .

(3.8)

The field operators again appear in normal order, and for the same reason: the photodetectors are assumed to register photoelectron counts via absorption. Normal ordering emerges as a simple consequence of this assumption, which of course is appropriate in the case of all practical photodetectors. If we had photodetectors that functioned by stimulated emission instead of absorption, the appropriate field correlation functions would be antinormally ordered (Mandel [1966]). If we calculate the field from a point source that is turned on at r = t = 0, we find that E(r, t) vanishes for t < r/c. The fields E(±) (r, t) individually, however, do not have this “causal” property, and for this reason the theory of optical coherence based on normally ordered correlation functions has sometimes been criticized as being in violation of relativistic causality. In fact the theory does not violate causality because it proceeds from an interaction such as (3.1) that involves the full, properly retarded electric field E(r, t); normally ordered field correlation functions arise as an approximation in which energy-non-conserving atom–field processes are ignored. If we make this approximation after including the causal step functions associated with the full fields, we are led easily to a manifestly causal theory in which normally ordered correlation functions are multiplied by these step functions (Milonni, James and Fearn [1995]). This modification of the standard theory outlined in the preceding paragraphs is not very important as a practical matter. Field correlation functions such as (3.7) and (3.8) have a close formal resemblance to the correlation functions introduced in the classical theory of partial coherence. The positive-frequency part E(+) (r, t) of the electric field operator in the quantum theory replaces the analytic signal of the classical theory, the quantummechanical expectation value replaces a classical average, and the corresponding correlation functions share formally the same propagation equations as well as various other properties (Born and Wolf [1994], Mandel and Wolf [1995]). One defines first, second and higher degrees of coherence in quantum theory in formally the same way as in classical theory (Wolf [1963]), and these degrees of coherence have the same physical significance in interference experiments as in classical theory. The difference, of course, is that in the quantum theory the E(±) that appear in these definitions are operators that do not in general commute. Phenomena described by second-order correlation functions G(2) have played a particularly important role in elucidating the differences between classical and

4, § 3]


107

quantum properties of light. Prior to the experiments of Hanbury Brown and Twiss the temporal and spatial “coherence” of an optical field generally referred to firstorder interference effects characterized by G(1) , and coherence was usually synonymous with monochromaticity. Experiments, however, reveal that two effectively monochromatic fields can have different second-order coherence properties, even though they might both be “coherent” in the (first-order) sense of exhibiting maximal fringe visibilities in a Michelson interferometer, for instance. Thus the fields from a single-mode laser and a spectrally filtered thermal source have different second-order coherence properties, even though they might have the same narrow bandwidth and fringe visibility. And such sources do not exhaust the variety of second-order interference properties that are predicted (and observed) in quantum optics. G(2) (r1 , t1 ; r2 , t2 ; r2 , t2 ; r1 , t1 ) determines the average of the product of the photon counting rates at two ideal photodetectors at r1 and r2 at times t1 and t2 , respectively. In the simplest case of a single-mode field, we see that the quantity G(2) (r, t; r, t; r, t; r, t), for instance, is proportional to a † a † aa, which for a thermal field is given by eq. (2.9). The factor 2 on the right-hand side of eq. (2.9) indicates a statistical tendency for photons to arrive in pairs – the photon bunching effect observed in intensity correlations by Hanbury Brown and Twiss (fig. 1). In other words, the average of the product of the photon counting rates (a † a † aa) is greater than the product of the average counting rates (a † a2 ). The difference a † a † aa−a † a2 is the photon variance associated earlier with wave fluctuations (eq. (2.10)). Much has been written about the experiments of Hanbury Brown and Twiss (Mandel and Wolf [1970]), and papers published around the time at which they were reported indicate considerable confusion and controversy over the interpretation of the experiments. This was rooted in large part in Dirac’s statement quoted earlier, because the experiments seemed to some to imply that photons were in-

Fig. 1. “Photon bunching” near τ = 0 in the average over time t of the intensity product I (t)I (t + τ ).

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[4, § 3

terfering with each other, given the tendency for them to be counted in pairs. Of course this surmise is incorrect: we should think not in terms of photons interfering but rather – as in all quantum-mechanical interference phenomena – probability amplitudes for indistinguishable processes (Feynman [1961]). Moreover, the discussion earlier concerning the Einstein fluctuation formula shows that the bunching effect observed by Hanbury Brown and Twiss can be understood from the classical wave contribution to Einstein’s fluctuation formula. In fact a simple calculation based on the superposition of classical waves with randomly distributed phases shows that the square of the intensity I averaged over the random phases is equal to twice the square of the average intensity (Purcell [1956]). More generally I n = n!I n , a consequence of the Gaussian statistics of a thermal field (Mandel and Wolf [1995]). It should also be emphasized that bunching is not simply a consequence of the fact that photons are bosons, for it is possible for (non-thermal) light to exhibit antibunching or neither bunching nor antibunching (Section 5). Dirac’s remark also caused confusion when observations of interference between light originating from two independent sources were reported (Forrester, Gudmundsen and Johnson [1955], Magyar and Mandel [1963]). One experiment (Pfleegor and Mandel [1967]) demonstrated that when the fields from two different single-mode lasers are superposed, “interference takes place even under conditions in which the light intensities are so low that, with high probability, one photon is absorbed before the next one is emitted by one or the other source”. Such interference, if interpreted (incorrectly) as an interference between photons, contradicts Dirac’s statement. From the perspective of wave interference and the fact that in quantum theory a wave represents a probability amplitude for photon detection, the results of these elegant experiments are hardly surprising: “Some of the confusion about coherence seemed a little strange even in the early days, and will seem remarkable at this time, but was real” (Townes [1995]). Interference effects involving one- or two-photon fields are typically highly non-classical (Mandel [1999]). A relatively simple example is provided by a nonabsorbing beam splitter with two one-photon inputs. Let a1 , a2 , a3 , and a4 denote the photon annihilation operators for the plane-wave modes indicated in fig. 2. Using the commutation relations for these operators and their Hermitian conjugates, and the relations between the reflection and transmission coefficients for the (symmetric) beam splitter, we obtain straightforwardly (Mandel and Wolf [1995])

† † a3 a4 a4 a3 = (T − R)2 , (3.9) where T and R are respectively the power transmission and reflection coefficients of the beam splitter. a3† a4† a4 a3 is proportional to the joint probability of detecting

4, § 3]


109

Fig. 2. A beam splitter with inputs along directions 1 and 2 and outputs along directions 3 and 4.

one photon in each mode 3 and 4; we have assumed that modes 1 and 2 are each initially in a state of exactly one photon. Equation (3.9) shows that, for a 50/50 beam splitter, the probability of detecting a photon in mode 3 and a photon in mode 4 is zero: either two photons are detected in mode 3 or two photons are detected in mode 4. Because of the phase differences of the transmitted and reflected fields, the probability amplitudes for one photon to be found in either of the modes 3 and 4 interfere destructively. If in the same example there are no photons in mode 2 while the initial state of mode 1 is arbitrary, we obtain

† †

a3 a4 a4 a3 = RT a1† a1† a1 a1 = n21 − n1 . (3.10) This vanishes if there is exactly one photon initially in mode 1, a result that cannot be explained classically: a beam splitter cannot “split” a photon – or, more precisely, there is zero probability of counting photoelectrons in both paths 3 and 4. States of definite photon number (“Fock states”) generally display such highly non-classical properties. At the other extreme are the coherent states which are generally regarded as the most “classical-like” states of the field. A coherent state |α of a single-mode field is an eigenstate of the photon annihilation operator: a|α = α|α, where α is a complex number, and |α can be expressed in terms of Fock states |n as 1

|α = e− 2 |α|

2

∞ αn √ |n. n! n=0

(3.11)

In other words, |n|α|2 = (|α|2n /n!)e−|α| : in a coherent state an n-photon state has an occupation probability given by a Poisson distribution with the mean value |α|2 . For a coherent state, a † a = |α|2 , a † a † aa = |α|4 , and the photon number variance associated with wave fluctuations, a † a † aa − a † a2 , vanishes. For a coherent state, then, the photon number variance Δn2 = Δn2 particles , and the fact that there are no “wave” contributions suggests that a coherent state comes 2

110


[4, § 3

closest to the notion of a deterministic, classical stable wave for which there are no statistical fluctuations and therefore no distinction between the average of the square of the intensity and the square of the average intensity. That is, there are no photon bunching or antibunching effects in a coherent state of the field. The fact that a coherent state is an eigenstate of a photon annihilation operator implies that E(+) (r, t)|ψ = E(r, t)|ψ

(3.12)

for a general (multimode) coherent state |ψ of the field, where E(r, t) is a complex vector field. For a coherent state of the field, therefore, all the correlation functions G(n) have a factored form, e.g., (2)

Gμνσ δ (r1 , t1 ; r2 , t2 ; r2 , t2 ; r1 , t1 ) = Eμ∗ (r1 , t1 )Eν∗ (r2 , t2 )Eσ (r2 , t2 )Eδ (r1 , t1 ),

(3.13)

so that a coherent state is “fully coherent” and has the same interference properties as a deterministic, “classical stable wave”. This classical-like property of a coherent state holds regardless of how small might be the average photon number. A Fock state, by contrast, has non-classical interference properties even if the average photon number is very large. Based on eq. (3.12) and the factorization of the G(n) , it should not be surprising that coherent states play a major role in the quantum theory of optical coherence (Glauber [1963a, 1963b], Mandel and Wolf [1965, 1995]). The fact that the expectation value of a normally ordered field operator product f (a, a † ) reduces to f (α, α ∗ ) for a coherent state is the basis for the optical equivalence theorem that formally relates classical and quantum averages (Sudarshan [1963]). There is another reason for the importance of coherent states in quantum optics: the field from a single-mode laser operating far enough above threshold can be described for many purposes as a coherent state (Sargent III, Scully and Lamb Jr [1974], Loudon [2000], Mandel and Wolf [1995]). The laser field differs from a coherent state |α because its phase drifts randomly, albeit slowly, meaning among other things that it must have a finite frequency bandwidth (Section 6). But the photon counting statistics and other properties make the light from a single-mode laser practically indistinguishable from that described by a coherent state. Like interference effects, photon counting – or, more precisely, photoelectron counting – can reveal differences between different quantum states of the field. Expressions for the probability distribution of photon counts in a fixed time interval have been derived in semiclassical theory (Mandel [1958]) as well as quantized-field theory (Kelley and Kleiner [1964]). In the latter, for the simplest

4, § 4]

Semiclassical radiation theory

111

case of a single-mode field, the probability of m counts in a time interval T is (ξ a † a)m −ξ a † a p(m) = : (3.14) : , e m! where ξ (∝ T ) is the quantum efficiency of the detector and the colons are used to indicate that the operators between them are to be normally ordered; normal ordering appears because, once again, the basic detection process is assumed to be the absorption of light (and the creation of photoelectrons). For a coherent state, p(m) is given by a Poisson distribution. Experiments in the 1960s demonstrated the Poisson distribution for the photon counting statistics of a laser (Loudon [2000], Mandel and Wolf [1995]): p(m) = mm e−m /m!, where m is the mean photoelectron number in the counting time interval. For a filtered thermal field, by comparison, it was confirmed that p(m) = mm /(m + 1)m+1 . For some purposes (e.g., quantum cryptography) light sources described by Fock states are desirable, and considerable progress has recently been made in producing such sources by cavity-QED and other methods (Walther [2004]).

§ 4. Semiclassical radiation theory Semiclassical radiation theory, in which light is treated classically and matter quantum-mechanically, has arguably been more widely applicable than might have been anticipated from a reading of the QED literature before around 1960. We have alluded several times to the fact that the photon bunching observed by Hanbury Brown and Twiss can be explained without field quantization. One more way to see this is to note again that the photon bunching is associated with the wave fluctuation term in eq. (2.10), whereas it is the particle term that arises from the non-commutativity of the field operators a and a † . These experiments, which were so important for the development of quantum optics and which seemed to some on the basis of Dirac’s remark about “photon interference” to violate quantum theory, demonstrated an effect that is essentially classical in nature. The photoelectric effect, which of course is the operating principle behind the detectors employed in photon counting experiments, has traditionally been regarded as evidence for the quantum nature of light; this circumstance obviously stems from Einstein’s 1905 paper (Section 2). However, the photoelectric effect was eventually recognized as one of the myriad effects that could be explained without field quantization (Scully and Sargent III [1972]). Recall the most significant observations in the photoelectric effect as discussed in textbooks: (1) the kinetic energy of the photoelectrons is a linear function of the light frequency:

112


[4, § 4

E = hω ¯ − w, where w is the “work function”; (2) the photocurrent or rate of ejection of electrons is proportional to the light intensity; and (3) electrons are ejected immediately upon exposure of the surface to light. These observations are nicely explained in terms of photons of energy h¯ ω, but in fact they can all be explained by considering an absorptive transition between two states 1 and 2 (E2 − E1 = hω ¯ 21 ) of an electron assumed to be irradiated by a classical monochromatic field E0 cos ωt. The perturbative calculation of the transition probability p12 (t) for this process is a trivial exercise in elementary quantum theory: t 2 t 1

p12 (t) ∼ (4.1) = 2 |d12 · E0 |2 dt ei(ω21 −ω)t + dt ei(ω21 +ω)t . 4h¯ 0

0

For long enough times the term with the + sign is negligible compared to the term with the − sign: p12 (t) ∼ =

2 1 2 sin 2 (ω21 − ω)t |d · E | . 12 0 4h¯ 2 [ 12 (ω21 − ω)]2

1

(4.2)

For a continuous distribution of possible final states 2, as occurs in photoionization (or photoconductivity), we integrate over all possible final states to obtain the transition rate from state 1. Using the assumption that d12 and the density of final states ρE are relatively flat functions of ω21 compared to the sharply peaked sinc function for long enough times, and ∞ dω21 0

sin2 21 (ω21 − ω)t [ 12 (ω21 − ω)]2

∼ =

∞ dω21 −∞

sin2 12 (ω21 − ω)t [ 12 (ω21 − ω)]2

= 2πt,

(4.3)

we obtain the transition rate d π (4.4) p12 (t) = |d12 · E0 |2 ρE (E1 + hω), ¯ dt 2h¯ which of course is just Fermi’s Golden Rule (No. 2) for this simple example. The essential features of the photoelectric effect are all contained in this result. Thus the fact that the kinetic energy of the photoelectrons is a linear function of the light frequency follows from the evaluation of the final density of states at E = h¯ ω + E1 in eq. (4.4): our semiclassical derivation of eq. (4.4) leads automatically to an energy conservation condition such that the change in the electron energy is h¯ ω. Observation (2) follows from the fact that the rate (4.4) is proportional to E02 and therefore to the light intensity: as the light intensity increases the rate of ejection of electrons increases proportionately. And finally observation (3) is just a consequence of eq. (4.4) being a rate. We did require the time t in our R12 =

4, § 4]


113

derivation of eq. (4.4) to be “long enough”, but only long enough to enforce energy conservation; the same requirement is imposed in the quantized-field theory of the photoelectric effect. This explanation of the photoelectric effect suggests that semiclassical radiation theory can account for a huge variety of optical effects. As a practical tool it is indeed very successful in describing the principal operating features of lasers, including things like threshold behavior, hole burning and the Lamb dip, Q-switching, and mode-locking (Sargent III, Scully and Lamb Jr [1974]), as well as in explaining in great detail such “coherent” effects as self-induced transparency. Inevitably, perhaps, questions about the range of validity of semiclassical radiation theory were hotly debated (Mandel [1976], Milonni [1976]). The debate centered largely around spontaneous emission. The simple approach to absorption leading to eq. (4.4) is easily extended to the case of stimulated emission, but in the case of spontaneous emission there is no externally applied field. Even without any calculation it is evident that spontaneous emission poses a challenge for classical electromagnetic theory: An atom in an excited state has a non-vanishing mean-square electric dipole moment, while the average dipole moment is zero. The expectation value of the electric field of the atom therefore vanishes, and so spontaneous emission cannot be accounted for without allowing for the quantum fluctuations of the dipole moment and the field. Evidently semiclassical radiation theory fails to explain spontaneous emission. In fact Dirac’s derivation of the Einstein A coefficient for spontaneous emission was considered at the time a major success of the new quantum electrodynamics; it “confirmed the universal character of quantum mechanics” by showing it could handle particle (photon) creation (Weinberg [1977]). It is interesting in this connection to return again to Einstein’s work on the spectrum of thermal radiation. Einstein considered a two-level atom of mass m undergoing absorption and emission in a thermal field and showed that it experiences an average drag force (Milonni [1994])

ω dρ hω ¯ v = −Rv, F = − 2 (p1 − p2 )B ρ(ω) − (4.5) 3 dω c where p1 and p2 are the lower- and upper-level occupation probabilities, respectively, v is the atom’s velocity, and B is the Einstein coefficient for absorption and stimulated emission. (For simplicity it is assumed that the two levels are nondegenerate and that the atom is constrained to move in one direction.) The change in the atom’s linear momentum in a time interval δt is then given by mv(t + δt) = mv(t) + Δ − Rv(t)δt,

(4.6)

114


[4, § 4

where Δ is the atom’s recoil momentum due to absorption and emission in the time interval δt. Squaring both sides of this equation, taking expectation values, and assuming δt is small enough that terms of second order in it can be ignored, we obtain Δ2 /δt = 2RkT

(4.7)

when it is assumed that Δ is equally likely to be positive or negative, so that v(t)Δ = 0, and that 12 mv 2 (t) = 12 kT , as required by the equipartition theorem for atoms in thermal equilibrium at temperature T . Assuming that each emission and absorption event imparts a linear momentum h¯ ω/c to an atom, and that successive events are statistically independent, we have

2 1 hω ¯ Ap2 + (p1 + p2 )Bρ(ω) Δ2 /δt = 3 c

2 2 hω ¯ Bp1 ρ(ω). = (4.8) 3 c Here A is the Einstein coefficient for spontaneous emission, and in the second equality we have used the absorption-emission equilibrium condition Ap2 + Bρ(ω)p2 = Bρ(ω)p1 . When eq. (4.8) and the relations A/B = (h¯ ω3 /π 2 c3 ) and p2 /p1 = e−h¯ ω/kT are used in eq. (4.7), the solution of the resulting differential equation for ρ(ω) with ρ(0) = 0 is the Planck spectrum. The point of this digression is brought out by using the various relations noted to write the differential equation (4.7) in the form π 2 c3 A ω dρ 2 = ρ(ω) + ρ (ω) . ρ(ω) − (4.9) 3 dω 3ω2 kT B The first term in brackets on the right-hand side would obviously vanish if there were no recoil in spontaneous emission (A → 0). The solution of eq. (4.9) would then be the Rayleigh spectrum of classical physics. In other words, the Planck spectrum requires that an atom recoils not only when it is stimulated to absorb or emit radiation, but also when it radiates spontaneously. Semiclassical radiation theory fails to predict the Planck spectrum because it does not properly describe spontaneous emission. In particular, a classical treatment of the field does not allow a point radiator to recoil because the field is a continuous wave with no single preferred direction of propagation. In this sense, according to Einstein, “Outgoing radiation in the form of spherical waves does not exist”. Recoil in spontaneous emission was observed by Frisch in 1933, and observations involving deflections of atomic beams with lasers confirm that atoms do in fact recoil in spontaneous emission as well as in absorption and stimulated emission. More

4, § 4]


115

Fig. 3. An atom undergoes a single spontaneous emission event in the presence of detectors D1 and D2 . There is a finite probability for a photon count to occur at D1 and a finite probability for a photon count at D2 , but zero probability for counts at both D1 and D2 .

recently these recoil processes have been the basis for laser cooling of atoms, and in particular the spontaneous recoils act to limit the degree of cooling. In the scenario indicated in fig. 3, semiclassical radiation theory predicts that part of the radiation from a spontaneously radiating atom can be measured at D1 and part of it can be measured at D2 , resulting in a non-vanishing probability of photodetection at both D1 and D2 . In quantized-field theory, however, the probability of photodetection at both D1 and D2 is zero. Experiment confirms the prediction of quantum theory (Clauser [1974]). There is one more point worth noting about eq. (4.9). Using again the relation A/B = (h¯ ω3 /π 2 c3 ), we can write A ω2 π 2 c3 2 2 (4.10) ρ(ω) + ρ (ω) = 2 3 h¯ ωρ(ω) + 2 ρ (ω) , B π c ω and the factor in brackets on the right-hand side is identical to that in eq. (2.1). We infer therefore that the particle term in the Einstein fluctuation formula is attributable to spontaneous emission. And since the particle term is related to the non-commutativity of field operators, this provides further evidence for the conclusion that spontaneous emission cannot be correctly described semiclassically. This is not to say that all aspects of spontaneous emission elude semiclassical explanations. In the version of semiclassical radiation called “neoclassical” by Jaynes, the rate coefficient derived for spontaneous emission is the correct Ein2 ω3 /3hc3 ), but the atomic population difference z(t) = stein A coefficient (4d12 21 ¯ p2 (t) − p1 (t) evolves according to the nonlinear equation z˙ = −(A/2)[1 − z2 ], in contrast to the equation z˙ = −A(1 + z) obtained in quantized-field theory.2 2 In neoclassical theory spontaneous emission is caused by the radiation reaction field of the atom. In the quantized-field theory there is another field acting on the atom – the vacuum field having a zero-point energy 12 h¯ ωβ per mode (Section 2). The relative roles of the radiation reaction and vacuum fields in spontaneous emission, Casimir forces and other phenomena have been discussed by several authors (Milonni [1994]).

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[4, § 4

(Both equations are obtained using a Markovian or “Weisskopf–Wigner” approximation.) The neoclassical equation for z(t) implies that an initially excited atom (z(0) = 1) remains in the excited state; however, this initial condition is trivially shown to be unstable against small perturbations, and so neoclassical theory could not be immediately dismissed. Experiments were carried out to check this and related predictions of neoclassical theory, but the results were deemed by some to be ambiguous. The strongest evidence against the theory, in the end, was that its semiclassical nature put it in conflict with effects known to be inexplicable in any classical electromagnetic theory; we have already mentioned some of these and will consider others in the following section. The exactly solvable Jaynes–Cummings model was originally studied in order to compare the semiclassical and fully quantum theories of the atom–field interaction (Jaynes and Cummings [1963]). In this model a single two-level atom interacts with a single mode of the quantized electromagnetic field in the electricdipole approximation and in the “rotating-wave approximation” in which energynonconserving interaction terms are ignored. The Hamiltonian (2.12) can be simplified in this model and written as H =

† 1 † h¯ ω0 σz + h¯ ωa † a + i hC ¯ a σ −σ a , 2

(4.11)

where the σ operators are the Pauli two-state (spin- 12 ) operators in the standard notation, C is the (real) atom–field coupling constant, and ω0 and ω are the atom transition frequency and the field frequency, respectively. Different techniques, such as the dressed-state formalism originally used by Jaynes and Cummings, have been used to calculate various quantities of interest in this model (Knight and Milonni [1980]). In the Heisenberg picture, for instance, it follows from eq. (4.11) that, for ω = ω0 , σ¨ z (t) + 2C 2 σz (t) + 4C 2 σz (t)N = 0,

(4.12)

where 1 (4.13) σz (t) + a † (t)a(t) 2 is a constant of motion in the rotating-wave approximation. If at t = 0 the atom is in the√upper state and the field is in a Fock state of n photons, then σz (t) = cos(2C n + 1t). Using σz (t) = p2 (t) − p1 (t), we have √ √ p1 (t) = sin2 C n + 1t , p2 (t) = cos2 C n + 1t , N≡

and

√ a † (t)a(t) = n + sin2 C n + 1t .

4, § 5]

Non-classical light

117

∼ cos2 Ωt, p1 (t) = ∼ sin2 Ωt, and a † (t)a(t) = ∼ n, with For n 1, p2 (t) = √ Ω ≡ C n the Rabi frequency. In this limit the field may be regarded as having a prescribed, fixed intensity, just as in semiclassical radiation theory. In the limit n → 0, however, semiclassical theory fails because it does not properly describe spontaneous emission; in that limit p2 (t) = cos2 Ct, p1 (t) = a † (t)a(t) = sin2 Ct. The coupling constant C is sometimes called the “vacuum Rabi frequency”, as it is the frequency with which the initial excitation oscillates back and forth between the atom and the field when the field is initially in its vacuum state. If the field is initially in a thermal state, there is a photon number probability distribution Pn = nn /(n + 1)−(n+1) , and in this case p2 (t) =

∞

√ Pn cos2 C n + 1 t

(4.14)

n=0

and

a (t)a(t) = n + †

∞

√ Pn sin2 C n + 1 t

n=0

if again the atom is initially in the upper state. The discrete summations over n in these expressions result from the quantization of the field, i.e., from the “granularity” associated with quantization. A result of this discreteness is a “collapse and revival” behavior (Eberly, Narozhny and Sanchez-Mondragon [1980]) related to the quantum recurrence theorem for quasiperiodic systems (see, for instance, Milonni and Singh [1991]). Experimental evidence for such behavior has been reported (Rempe, Walther and Klein [1987]).

§ 5. Non-classical light Non-classical properties of light can be traced to the quantum mechanics of light sources. Consider, for instance, the experiment indicated in fig. 3. For a “twolevel atom” the second-order, normally ordered field correlation function giving the two-photon coincidence rate at the two detectors is found straightforwardly to † † be proportional to σ12 σ12 σ12 σ12 , in which all four operators are evaluated at the same time. Because of the orthogonality of the two atomic eigenstates |1 and |2, † † σ12 σ12 = |12|12| = 0 and therefore σ12 σ12 σ12 σ12 = 0. The two-photon coincidence rate vanishes, i.e., the photon emitted by the atom cannot be “split”. A similar situation arises in resonance fluorescence, where a strong, resonant laser field drives the atom up and down between the two states of the transition

118


[4, § 5

(Knight and Milonni [1980]). In this case the second-order field correlation function G(2) (τ ) for a single point detector is proportional to

† † σ12 (t)σ12 (t + τ )σ12 (t + τ )σ12 (t) under steady-state driving conditions. This intensity correlation function vanishes at τ = 0, for the same reason as in the preceding example. Thus, as opposed to the photon bunching in the radiation from a thermal source, the field from a twolevel atom in resonance fluorescence exhibits antibunching for small delay times τ between the counting of successive photons (Mandel and Wolf [1995], Loudon [2000]). It is clear physically why this must be so: the probability of photon emission in a given time interval is proportional to the upper-state probability and, having emitted a photon and dropped to its lower state, the two-state atom cannot emit another photon until it has been pumped back into its upper state. The observation of photon antibunching in resonance fluorescence was first reported in 1977 (Kimble, Dagenais and Mandel [1977]). The relation (2.10), which can be written as

2

Δn = n + Δn2 waves , (5.1) leads to a useful measure of non-classicality. For a Fock state, for example, Δn2 = 0 and therefore Δn2 waves = −n < 0, a condition that cannot be satisfied by any field having a non-negative phase-space probability distribution: Δn2 waves < 0 is characteristic of a quantum state of the field that has no classical analog. The Mandel Q parameter is defined as (Mandel [1979], Mandel and Wolf [1995]) Q=

Δn2 waves Δn2 − n = n n

(5.2)

and when it is negative the state of the field is considered to be non-classical, the photon statistics being sub-Poissonian. The largest negative value of Q, −1, occurs for a Fock state. In steady-state resonance fluorescence from a two-level atom, the largest negative value of Q for the antibunched field is −3/4 (Mandel [1979]). Quantum-optical experiments have loomed large in work on the conceptual foundations of quantum theory. The most significant of these experiments during the past three decades have tested the predictions of local hidden-variable theories against the predictions of quantum theory. One of the first and most conclusive tests has involved the photon polarization correlations in the spontaneously emitted light from atomic cascade transitions.

4, § 5]

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119

Fig. 4. (a) J = 0 → 1 → 0 atomic cascade with transition frequencies ωA and ωB ; (b) arrangement for measuring polarization correlations of ωA and ωB photons.

Figure 4(a) shows a three-level atomic cascade with transition frequencies ωA and ωB . We imagine an atom undergoing such a cascade and an experiment [fig. 4(b)] to measure polarization correlations of the two photons emitted when the single-atom cascade is repeated many times. Apparatus for doing this includes filters that transmit light of one or the other frequency and linear polarizers along the ±z directions. For simplicity we restrict ourselves to the consideration of single field modes of frequency ωA and ωB . The atomic states |1 and |4 have magnetic quantum number M = 0, whereas |2 and |3 are states with M = 1 and M = −1, respectively. (We can exclude the intermediate J = 1 state with M = 0 because it is not involved in the generation of transversely polarized light at detectors along the ±z directions.) The linear polarizers are oriented to transmit light polarized at angles φA and φB with respect to some fixed axis normal to zˆ , as indicated in fig. 4(b). The quantity describing the polarization correlations may be taken in our simplified model to be the second-order correlation func† (t)aB† (t)aB (t)aA (t), the expectation value referring to the initial state in tion aA which the atom is in state |4 and the field is in the vacuum state, and the photon annihilation operators aA and aB corresponding to modes of frequency ωA and ωB and polarization direction φA and φB , respectively. From the Hamiltonian (2.12) we easily obtain the Heisenberg equations of motion for the operators aA (t) and aB (t). Ignoring “counter-rotating” terms that do

120


[4, § 5

not oscillate near the frequency ωA in the equation for aA (t), and terms that do not oscillate near the frequency ωB in the equation for aB (t), we write3

2πωA 1/2 ∗ a˙ A = −iωA aA + FA · [d24 σ24 + d34 σ34 ], h¯

2πωB 1/2 ∗ a˙ B = −iωB aB + (5.3) FB · [d12 σ12 + d13 σ13 ]. h¯ For our purposes the only significant feature of the mode functions FA and FB is their polarization: FA ∝ xˆ cos φA + yˆ sin φA , FB ∝ xˆ cos φB + yˆ sin φB ,

(5.4)

where xˆ and yˆ are orthogonal unit vectors in a plane transverse to the z axis. We now write the formal solutions to equations (5.3) and, from these solutions, the expectation value of the second-order field correlation function † (t)aB† (t)aB (t)aA (t). Since the initial field state is the vacuum state of no aA photons, the homogeneous solutions of eqs. (5.3) make no contribution to this † (0) = 0]. The time evoluexpectation value [e.g., aA (0)|vacuum = vacuum|aA tion of the σij operators may be well approximated by the evolution given by the “Weisskopf–Wigner” approximation, or more simply we can just replace σ24 (t), for instance, by σ24 (0)e−iωA t in a short-time approximation; the time dependence of the field correlation function does not play any significant role for our purposes here, and consequently there is no need to indicate it explicitly. We obtain

† † a A a B aB aA ∝ |d24 · FA |2 |d12 · FB |2 σ42 σ21 σ12 σ24 + (d24 · FA )∗ (d34 · FA )(d12 · FB )∗ (d13 · FB )σ42 σ21 σ13 σ34 + (d13 · FB )∗ (d34 · FA )∗ (d24 · FA )(d12 · FB )σ43 σ31 σ12 σ24 + |d34 · FA |2 |d13 · FB |2 σ43 σ31 σ13 σ34 .

(5.5)

Recalling the definition σij = |ij |, and the normalization i|i = 1, we see that all four expectation values in this expression are equal to σ44 (0); this is the initial occupation probability of state |4, and is unity. Therefore 2

† † aA aB aB aA ∝ (d24 · FA )(d12 · FB ) + (d34 · FA )(d13 · FB ) . (5.6) 3 From the Heisenberg equations of motion for the σ operators it follows that σ (t) and σ (t) ij 24 34 vary in time approximately as e−iωA t , while σ12 (t) and σ13 (t) vary approximately as e−iωB t .

4, § 5]

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121

√ √ Using eq. (5.4) and the fact that d12 ∝ (1/ 2)(xˆ − iy), ˆ d13 ∝ (1/ 2)(xˆ + iy), ˆ and similarly for d24 and d34 , we have finally

† † aA aB aB aA ∝ cos2 (φA − φB ) (5.7) for the second-order field correlation function that is proportional to the joint rate for counting A photons and B photons in the experimental setup indicated in fig. 4. The operator character of the atomic operators σij , and therefore of the field operators aA and aB , is crucial to the derivation of eq. (5.7). In semiclassical radiation theory, where of course no operator relations appear, the σ ’s are effectively replaced by ordinary numbers. Then it is found that Rmin /Rmax , where Rmin and Rmax are the minimum and maximum joint counting rates, is never smaller than 1/3, whereas in the quantized-field theory it is zero (when φA − φB = π/2). The quantized-field result for Rmin /Rmax is non-zero when one takes into account polarizer efficiencies and the finite angles subtended by the photodetectors, but experiments clearly rule out the prediction of classical field theory. In other words, the result (5.7) is a consequence of the quantum nature of light, and in particular the quantum nature of photon polarization. The relation (5.6) can be interpreted as follows. (d24 · FA ) is (proportional to) the probability amplitude for the atomic transition |4 → |2 with the emission of a photon of frequency ωA , and (d12 · FB ) is the amplitude for the transition |2 → |1 with the emission of a photon of frequency ωB .4 Their product is the amplitude for the process |4 → |2 → |1 with the emission of a photon of frequency ωA and a photon of frequency ωB . Similarly (d34 ·FA ) is the amplitude for the process |4 → |3 with the emission of a photon of frequency ωA , (d13 ·FB ) is the amplitude for the process |3 → |1 with the emission of a photon of frequency ωB , and their product is the amplitude for the process |4 → |3 → |1 with the emission of a photon of frequency ωA and a photon of frequency ωB . Then the sum (d24 · FA )(d12 · FB ) + (d34 · FA )(d13 · FB ) is the total probability amplitude for the process |4 → |1 with the emission of ωA and ωB photons when we cannot distinguish whether the atom took the path |4 → |2 → |1 or |4 → |3 → |1. We could therefore have written the result (5.7) “immediately” 4 What we always mean by shorthand references to “the emission of a photon of frequency ω” is something like “the radiation accompanying the atomic transition can result in the counting of a single photoelectron”. The “buckshot” picture of photons (Scully and Sargent III [1972]) propagating independently of any photodetection process can lead to the sort of muddle caused by Dirac’s remark about photon interference. But it can get very tiresome to always use precise language when it comes to photons!

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[4, § 5

using the two rules: (i) the probability amplitude for two successive processes is the product of the amplitudes for the individual processes, and (ii) the amplitude for a process that can occur in different indistinguishable ways is the sum of the amplitudes for the individual ways (Feynman [1961]). In our example the paths |4 → |2 → |1 and |4 → |3 → |1 are indistinguishable because a measurement of linear polarization does not distinguish between left- and right-circular polarization. Our derivation of (5.7) was based on the non-commutative, operator character of the quantized field, but “far more fundamental” for the difference between classical and quantum theory in general was “the discovery that in nature the laws of combining probabilities were not those of the classical probability theory of Laplace” (Feynman [1951]). The polarization state of the two-photon pair produced in the J = 0 → 1 → 0 cascade may be written in a linear polarization basis as 1 |ψ = √ |xA |xB + |yA |yB , 2

(5.8)

where x and y refer to orthogonal directions in a plane normal to the z axis. This is an entangled state that implies correlations of the type considered by Einstein, Podolsky and Rosen (EPR) in their argument that quantum theory does not provide a “complete” description of physical “reality” (Clauser and Shimony [1978]). It is also the type of state considered by Bell and others in analyses of hidden-variable theories (Bell [1964]). Hidden-variable theories assume that physical quantities have objective values regardless of whether they are measured or whether quantum theory allows certain of these quantities (like position and momentum) to have simultaneously precise values; these objective values are determined by unknown “hidden variables”. The most palatable of such theories are “local”, which, very loosely speaking, means that they are consistent with the impossibility of two separated systems having instantaneous effects on one another. Bell’s theorem says that no local hiddenvariable theory can be in complete agreement with the statistical predictions of quantum theory. In particular, certain “Bell inequalities” are satisfied by local hidden-variable theories but not by quantum theory. Polarization correlation experiments involving atomic cascades have confirmed the predictions of quantum theory to accuracies of tens of standard deviations (Aspect [1984], Clauser and Shimony [1978], Fry and Walther [2000], Steinberg, Kwiat and Chiao [2006]). Suppose that in the experiment indicated in fig. 4(b) the polarizers have two output channels: + for light polarized parallel to the orientation direction of the polarizer and − for polarization perpendicular to this direction. Let P++ (a, b) be the probability that photons are counted in the + channels of both polarizers

4, § 5]

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123

in fig. 4, P+− (a, b) be the probability that a photon is counted in the + channel of the polarizer on the right and a photon is counted in the − channel of the polarizer on the left, etc. a and b specify the orientations of the polarizers on the right and on the left, respectively, in the figure. According to quantum theory, P++ (a, b) = P−− (a, b) = 12 cos2 θ and P+− (a, b) = P−+ (a, b) = 12 sin2 θ, where θ is the angle between a and b.5 In terms of the correlation coefficient E(a, b) = P++ (a, b) + P−− (a, b) − P+− (a, b) − P−+ (a, b),

(5.9)

the following Bell inequality satisfied by a local hidden-variable theory can be derived (Clauser, Horne, Shimony and Holt [1969]): |S| ≡ E(a, b) − E(a, b ) + E(a , b) + E(a , b ) 2. (5.10) A test of this inequality involves measurements of four correlation coefficients, with two orientations for each of the two polarizers. The correlation coefficient is defined from the experimental data by (Aspect [1984]) E(a, b) =

N++ (a, b) + N−− (a, b) − N+− (a, b) − N−+ (a, b) , N++ (a, b) + N−− (a, b) + N+− (a, b) + N−+ (a, b)

(5.11)

where the N’s are measured coincidence rates. These have been measured for polarizer orientations giving the greatest difference between the Bell inequality (5.10) and the predictions of quantum theory, e.g., (a, b) = (b, a ) = (a , b ) = 22.5◦ and (a, b ) = 67.5◦ ; in this case quantum theory predicts √ S = cos 45◦ − cos 135◦ + cos 45◦ = 2 2 = 2.828. When polarizer inefficiencies and the finite angles subtended by detectors are taken into account, this becomes SQM = 2.70 ± 0.05, compared to the experimental result 2.697 ± 0.015 (Aspect [1984]). Experiment thus confirms the prediction of quantum theory and violates the Bell inequality satisfied by a local hidden-variable theory. The two-photon downconversion sketched in fig. 5 provides another way of producing correlated photons. Laser radiation of frequency ω3 is incident on a birefringent crystal in which the induced polarization has a nonlinear component Pi = χij k Ej Ek , were χij k is the nonlinear susceptibility tensor and E is the electric field of the laser (Boyd [2003]). This results in the “spontaneous” pairwise

5 For example, P 2 ++ (φA , φB ) = |φA |φB |ψ| , where |ψ is given by eq. (5.8) and |φA = cos φA |xA + sin φA |yA and |φB = cos φB |xB + sin φB |yB . Thus P++ (φA , φB ) = 12 cos2 (φA − φB ).

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[4, § 5

Fig. 5. Two-photon downconversion in which light of frequency ω3 incident on a nonlinear crystal results in the generation of photon pairs, one photon with frequency ω1 and the other with frequency ω2 = ω3 − ω1 .

generation of photons of frequency ω1 and ω2 with ω1 + ω2 = ω3 . The conical angle between the ω1 and ω2 photons is determined by the phase-matching condition k3 = k1 + k2 , where ki = n(ωi )ωi /c, n(ωi ) being the crystal’s (linear) refractive index at frequency ωi . The angle between the ω1 and ω2 photons is therefore determined by the frequencies and refractive indices, or the angle can be fixed by pinholes to determine the frequencies of the generated photons. In type-I phase matching, for example, the incident laser beam is linearly polarized as an extraordinary wave incident at 90◦ to the optic axis, and the ω1 and ω2 photons correspond to ordinary waves with the same linear polarization. Using a second crystal rotated by 90◦ with respect to the first, a correlated polarization state of the form (5.8) is generated. This technique has been used to demonstrate a 200σ violation of a Bell inequality in a measurement time of less than a second (Steinberg, Kwiat and Chiao [2006]). Mandel has reviewed other quantum features of interference experiments with two-photon downconverters (Mandel [1999]). It is noteworthy that the time dependence of the second-order correlation function in such experiments leads to techniques for measuring time delays with resolution times of a few femtoseconds. One of the interesting features of entangled states like (5.8) is the “action at a distance” that their correlated nature seems to imply. Suppose that Alice and Bob each observe one member of the photon pair. If Alice measures x (y) polarization, then the state of Bob’s photon is immediately reduced to x (y), suggesting instantaneous action at a distance. However, this correlation cannot be used for instantaneous communication of information, or signaling, simply because Alice cannot choose whether to “send” an x or a y to Bob; she has a 50/50 chance of getting an x or a y herself. She does, of course, have a choice as to polarization basis. For √ example, she can use the circular polarization states |R = (1/ 2)(|x − i|y)

4, § 5]

Non-classical light

125

√ and |L = (1/ 2)(|x + i|y) instead of |x and |y. In a circular polarization basis the state (5.8) takes the form 1 |ψ = √ |RA |RB + |RA |RB . 2

(5.12)

So if Alice chooses to work in the x, y basis, her measurement reduces Bob’s photon state to x or y; if she chooses to work in the R, L basis, her measurement reduces Bob’s photon state to R or L. But the important point is that her choice of basis cannot serve to send information to Bob: given a single photon, Bob cannot distinguish between linear or circular polarization. The impossibility of superluminal (instantaneous) communication of information implies the impossibility of any device that can measure the polarization parameters of a single photon. Were such a device possible, entangled polarization states could be used for instantaneous communication. The density matrix ρB describing Bob’s photon can be obtained by tracing over Alice’s states. We obtain 1 ρB = √ |xB xB | + |yB yB | 2

(5.13)

in the x, y basis and 1 ρB = √ |RB RB | + |LB LB | 2

(5.14)

in the R, L basis. Since these are merely different ways of writing the same reduced density matrix, it is clear that Alice’s choice of whether to measure linear or circular polarization cannot affect Bob’s measurements and therefore cannot be used to transmit information. Suppose, however, that Bob has many particles in the same state. Then he can determine that state with a high degree of accuracy by, for example, letting N (1) photons pass through polarization-dependent beam splitters and thereby determining whether Alice measured linear or circular polarization for her EPRcorrelated photon. This implies that, if the single photon at Bob’s end can be sent through an amplifier to produce a large number of photons in the same (arbitrary) state, Alice would be able to superluminally communicate to Bob whether she measured linear or circular polarization. Such a scheme fails because photons cannot be perfectly cloned: any amplifier will produce photons with polarization different from the incident photon by spontaneous emission; we cannot have stimulated emission without spontaneous emission, and therefore we cannot amplify an arbitrary polarization. Much more generally, quantum theory does not permit

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[4, § 6

the cloning of a single quantum (Wootters and Zurek [1982], Milonni [2005], Scarani, Iblisdir, Gisin and Acín [2005]). Similarly, the impossibility of superluminal communication implies that it must be impossible to determine the polarization state of one member of a polarizationentangled photon pair without making measurements on the other. More generally, it is impossible to locally determine the eigenstate of a single particle, or to have an apparatus that faithfully preserves the polarization of a photon entering it with arbitrary linear polarization while reversing the helicity of a photon entering with circular polarization (Milonni [2005]). The quantum properties of light evidently act to enforce the requirement of special relativity that superluminal communication is impossible. Entangled photon states have recently been of great interest in connection with quantum computation and information (Nielsen and Chuang [2000]), quantum cryptography (Gisin, Ribordy, Tittle and Zbinden [2002]), and related studies.

§ 6. Quantum noise Quantum fluctuations in optics come from the quantum properties of particles, usually electrons, and of the field. In a laser, for example, spontaneous emission noise results in a finite linewidth of the laser radiation, no matter how small might be the “technical noise” due to mirror vibrations and other effects. Spontaneous emission, and therefore spontaneous emission noise, may be attributed in varying degrees to quantum fluctuations of the radiators or of the field (Milonni [1994]). Spontaneous emission noise is especially important in applications involving amplification of radiation, since it can be amplified along with an input signal. Phase noise limits the sensitivity of coherent optical communications and sensor systems based on optical interference. In addition to the phase noise associated with the laser linewidth, laser intensity noise must be kept small in analogmodulation transmission systems, and “mode partition noise” causing singlemode lasing to randomly switch to different modes can increase bit-error rates in high-bandwidth digital communications. In bit-error analyses it is also necessary to take into account the shot noise associated with the absorption process in photodetection. The subject of quantum noise is obviously a broad one. In order to avoid some of the more specialized aspects of the subject as it pertains to optical communications and photonics, we will concentrate our attention here on some of the basic physics involved in the quantum noise of lasers, especially the quantum lower limit to the laser linewidth. This “Schawlow–Townes” laser linewidth due

4, § 6]

Quantum noise

to spontaneous emission noise is

p2 hω ¯ γ 2, ΔωST = p2 − p1 Pout c

127

(6.1)

where ω is the (single-mode) laser oscillation frequency, p2 and p1 are respectively the occupation probabilities of the upper and lower levels of the lasing transition, Pout is the output laser power, and γc is the cavity power damping rate; for a resonator of length d and mirror power reflectivities R1 and R2 , γc = −(c/2d) log(R1 R2 ). ΔωST is typically very much smaller than the “natural” linewidth of the lasing transition, the linewidth traditionally connected with the uncertainty principle. Townes has remarked that “an electrical engineer accustomed to the almost monochromatic oscillation produced by an electron tube with positive feedback would perhaps not have given the problem [of the laser linewidth being smaller than the natural linewidth] a second thought”. However, both Niels Bohr and John von Neumann “immediately questioned how such a narrow frequency could be allowed by the uncertainty principle” (Townes [1995]). In fact ΔωST can easily be derived using the (energy–time) uncertainty principle. We have noted that laser radiation can be described approximately by a coherent state, for which Δn2 /n2 = 1/n 1 for large n. The fluctuation in the energy E = nh¯ ω in the laser field may then be approximated by ΔE = nh¯ Δω, with the photon number n assumed to be constant. Then the relation ΔEΔt > h¯ for the uncertainty in the energy measured in a time interval Δt implies Δω > 1/nΔt for the uncertainty in the laser frequency. Now we argue that 1/Δt must be at least as large as the rate Rsp of spontaneous emission into the lasing mode, so that Δω >

Rsp , n

(6.2)

and Rsp = N2 Rst ,

(6.3)

where N2 (N1 ) is the number of atoms in the upper (lower) laser level and Rst is the stimulated emission (and absorption) rate into that mode when the photon number is 1. In steady-state laser oscillation the rate of growth of the photon number, Rst (N2 −N1 ), is equal to the rate γc at which photons leave the resonator. Then Rsp =

N2 γc p2 γ c = . N 2 − N1 p2 − p1

(6.4)

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[4, § 6

Furthermore the photon number n in the resonator is obviously related to the laser output power: Pout = γc nh¯ ω. Hence Δω >

p2 γc γc h¯ ω = ΔωST . p2 − p1 Pout

(6.5)

This laser linewidth due to spontaneous emission noise is usually extremely small, and it can be seen to increase when the laser resonator loss increases, i.e., when γc increases. However, it has been found for lossy resonators that there is “excess spontaneous emission noise”, over and above that given by eq. (6.1): Δω = KΔωST ,

(6.6)

where K (> 1), called the Petermann factor (Petermann [1979]), approaches unity in the “good cavity” limit. The theory and interpretation of K has in recent years attracted considerable attention especially in the context of semiconductor lasers. (See Henry and Kazarinov [1996] and references therein.) Consider a laser with mirrors having power reflection and transmission coefficients R1 , R2 , T1 , T2 (fig. 6). Let the operator for the intracavity photon annihilation part of the field propagating to the right along the z axis be denoted by AR (z, t)e−iωt , where AR is slowly varying. The field at the right-hand mirror satisfies the relation

2d AR d< , t + = GR1 G AL (d< , t) + T1 G AR,vac (−d, t) c + T2 GR1 G AL,vac (d> , t) + Asp (d< , t). (6.7) d< and d> respectively denote points just inside and outside the mirror at z = d, and G is the power gain factor for a single pass through the gain medium. Subscripts R and L label right- and left-propagating fields, while “vac” indicates the source-free, vacuum (zero-point) field and “sp” the field due to spontaneous emission. The first term on the right-hand side of eq. (6.7) arises from √ the propagation of the left-propagating field at d< through the gain medium ( √G), reflection off √ the left mirror ( R1 ), and a second pass through the medium ( G). The remain-

Fig. 6. A laser with mirror reflection and transmission coefficients Rj and Tj , respectively, and a gain medium of length d filling the space between the mirrors.

4, § 6]

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129

ing terms have similar origins; eq. (6.7) is just a kinematical identity. We can use

2d ∼ 2d ˙ AR (d< , t), AR d< , t + (6.8) = AR (d< , t) + c c which can be justified a posteriori based on the fact that gain nearly balances the loss in steady-state oscillation, to replace eq. (6.7) by c c A˙ R (d< , t) ∼ GT1 AR,vac (−d, t) G R1 R2 − 1 AR (d< , t) + = 2d 2d c + G R1 T2 AL,vac (d> , t) + (6.9) Asp (d< , t), 2d √ where we have also used AL (d< , t) = R2 AR (d< , t). Note that we are allowing for the vacuum fields to enter the resonator from the outside and to be transmitted, reflected, and amplified. The vacuum fields make no contribution to the normally ordered expectation value A†R (t)AR (t) for the initial state in which there are no photons in the field. In this case, from eq. (6.9),

† AR (t)AR (t) =

c 2d

2 t dt 0

t

dt

A†sp (t )Asp (t

) eγ (t +t −2t) ,

(6.10)

0

− 1] ∼ where γ = (c/2d)[(R1 R2 = (d/c)(cg − γc ) in the limit of small ∼ output coupling (R1 , R2 = 1) and g = (1/d) log G, γc = −(c/2d) log(R1 R2 ). Using (Goldberg, Milonni and Sundaram [1991a])

†

1 2d 2 γc p2

Asp (t )Asp (t ) = (6.11) δ(t − t

), 2 c p 2 − p1 )1/2 egd

we obtain the steady-state value

γc γc 1 p2 p2 A†R AR = = . 4γ p2 − p1 2 p2 − p1 γc − cg

Similarly

† AR (t)AR (t + τ ) = A†R AR e−γ τ

(6.12)

(6.13)

in steady-state oscillation, which implies the linewidth (full width at half maximum) Δω = 2γ = γc − cg =

γc 1 p2 . † 2 p2 − p1 A AR

(6.14)

R

Using A†R AR = A†L AL = 12 n, where n is the steady-state intracavity photon number, and Pout = γc nh¯ ω, we obtain the Schawlow–Townes linewidth (6.1).

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[4, § 6

The result (6.11) upon which this derivation of the fundamental laser linewidth is based does not take into account the amplification of the spontaneously emitted radiation in the gain medium. This is not too difficult to do, but it is instructive instead to reformulate the problem using a non-normally-ordered field correlation function, in which case the vacuum fields entering the resonator from the outside make an explicit contribution. We allow these fields to be amplified in the gain medium, as noted earlier. We now show that this leads to an extremely simple derivation of the Petermann K factor for the resonator indicated in fig. 6 (Goldberg, Milonni and Sundaram [1991b]). This derivation is based on the fact that spontaneous emission can be regarded, from one viewpoint (Milonni [1994]), as emission induced by the source-free (vacuum) field plus emission associated with the source field, these contributing in equal measures. The effect of the amplifying medium on spontaneous emission can therefore be obtained by just considering how the vacuum field is amplified. Part of the vacuum field entering the resonator from the left in fig. 6, for instance, exits at z = d after amplification, and part of it can be reflected at z = d and, after another pass through the gain medium, emerge from the resonator as a left-going field at z = 0. The amplification of the vacuum intensity due to these two paths is given by the factor T1 GT2 + T1 GR2 GT1 . Similarly, part of the vacuum field entering the resonator from the right exits at z = 0 after amplification, and part of it is reflected at z = 0 and emerges after amplification as a right-going field at z = d; the amplification factor for these two paths is T2 GT1 + T2 GR1 GT2 . Paths involving additional passes through the gain medium do not contribute to any further modification of the vacuum noise intensity because the amplification is exactly balanced by the loss in steady-state laser oscillation. Then, using the condition G2 R1 R2 = 1 for steady-state oscillation, the sum of the factors for the √ √ four contributing paths may be written as (T1 / R1 + T2 / R2 )2 , and the laser linewidth is seen from the approach outlined above to be

2

c T2 2 T1 h¯ ω Δω = (6.15) = KΔωST , √ +√ 2d Pout R1 R2 where

K=

T1 T2 √ +√ R1 R2

2

(log R1 R2 )2 .

(6.16)

This is identical to the K factor calculated for this case using the so-called adjoint modes (Siegman [1989a, 1989b], Hamel and Woerdman [1989]) for a lossy resonator; experimental evidence for this expression for the linewidth enhancement factor has been reported (Hamel and Woerdman [1990]). It should be noted that

4, § 6]

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131

K can be very large for unstable (Doumont and Siegman [1989]) and misaligned (Firth and Yao [2005]) resonators. The Petermann factor was originally interpreted as an enhancement of the spontaneous emission rate. However, the spontaneous emission rate is not enhanced by K, which is instead attributable to single-pass amplification, as the preceding discussion implies (Goldberg, Milonni and Sundaram [1991a], Henry and Kazarinov [1996]). The Petermann factor played an important role in explaining why gain-guided semiconductor lasers oscillate on many longitudinal modes whereas index-guided lasers oscillate strongly on just one or two modes. The relation γc p2 n=K (6.17) p2 − p1 γc − cg for the photon number in the resonator implies that, for a given laser power, a larger value of K requires a larger value of γc − cg, which determines the mode selectivity. As K increases the relative differences in γc − cg for different modes decreases and therefore so does the mode selectivity. Since gain-guided lasers have much larger values of K than index-guided lasers, therefore, they oscillate on many more modes (Petermann [1979], Streifer, Scifres and Burnham [1982], Henry and Kazarinov [1996]). There is a considerable literature on quantum noise in amplifiers and nonlinear optical processes (Henry and Kazarinov [1996], Gardiner and Zoller [2000]). A relatively simple example is the quantum description of single-mode nonlinear phase conjugation, described by b = Ca † + f,

(6.18)

where a and b are the photon annihilation operators for the incident and phaseconjugated fields, respectively, and the constant C depends on the particular wavemixing process responsible for the phase conjugation. The “noise operator” f also depends on the particular process under consideration, but from the commutation relations [a, a † ] = 1 and [b, b† ] = 1 we must have [f, f † ] = |C|2 +1 in any case, as long as [f, a] = [f † , a] = 0. Using eq. (6.18) and the commutation relations, it is found that

2 2 Δnb = nb − nb 2 = b† bb† b − b† b > nb , (6.19) i.e., the phase-conjugated field will always have “super-Poissonian” photon statistics, even if the input field is sub-Poissonian (Δn2a < na ) (Gaeta and Boyd [1988]). There is also a large literature on measurement techniques aimed at mitigating effects of quantum noise. Consider again a coherent state of a single-mode field

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and define the Hermitian operators a1 = a + a † and a2 = i(a − a † ) corresponding to different (electric and magnetic) components of the field. These operators satisfy the uncertainty relation Δa12 Δa22 1 for any state of the field, and for a coherent state we have Δa12 = Δa22 = 1, which corresponds to the smallest simultaneous uncertainty in the field components a1 and a2 ; in other words, in a coherent state both a1 and a2 have their smallest possible quantum uncertainties. However, there are states in which, for instance, Δa12 = μ < 1 and Δa22 = 1/μ > 1. Such squeezed states, in which one of the components of a has a dispersion smaller than 1, have been of interest because they could in principle lead to the development of extremely low-noise optical communication systems or to the very precise measurements required, for instance, for the detection of gravitational waves (Loudon and Knight [1987]). However, propagation and especially amplification of light act to increase quantum noise effects, and consequently some conceivable applications might actually benefit little from squeezed light sources (Loudon [1997]). § 7. Remarks Despite its long history, going back to Thomas Young at the beginning of the 19th century, optical interference still challenges our understanding, and the last word on the subject probably has not yet been written. With the development of experimental techniques for fast and sensitive measurements of light, it has become possible to carry out many of the Gedanken experiments whose interpretation was widely debated in the 1920s and 1930s in the course of the development of quantum mechanics. – Leonard Mandel [1999]

To forestall the objections of specialists whose favorite examples of the importance of field quantization have not been mentioned, we emphasize that this survey was hardly meant to be exhaustive. The choice of topics was made with certain prejudices as to the concepts that were important historically and that will likely have lasting significance. And there was no intention of describing mathematical or computational techniques that have been found useful in problem solving, that part of the field being too broad to broach here (Barnett and Radmore [2002]). Regarding the current status of field quantization in optics, it seems fair to say that questions as to whether or when field quantization is necessary have largely been settled. Many predicted non-classical features of light have been demonstrated in experiments, and these experiments have played a central role in fundamental tests of quantum theory and local hidden-variable theories. Nonclassical properties such as those associated with entangled polarization states, for instance, are now being applied in quantum cryptography and in quantum information studies. Cavity-QED experiments investigating or applying non-classical

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light have become practically routine. Semiclassical radiation theory remains of course a very useful tool, but its limitations are much more widely appreciated than was the case in the decade or so following the advent of the laser. Advances in semiconductor lasers and optical communications in particular have required consideration of distinctly quantum aspects of noise in both the generation and detection of light: quantum-uncertainty limits on the precision of measurements have become practical concerns. Careful analyses of quantum noise and related “vacuum” effects have generally required consideration of the quantum fluctuations of both light and matter. Progress in photon detection and related technologies has been crucial not only for basic quantum optics, but also for a wide variety of photon counting applications ranging from astronomy to the mapping of the human genome. The more subtle, quantum properties of light involving second- and higher-order field correlation functions will have some practical value, but it remains to be seen whether they will be broadly utilized.

Acknowledgement My interest in many of these topics began when I attended Emil Wolf’s Physics 531 course at the University of Rochester in the early 1970s. It seems appropriate in this volume to acknowledge his lapidary lectures that, alongside his enormous scientific contributions, have inspired so many generations of students.

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Pfleegor, R.L., Mandel, L., 1967, Phys. Rev. 159, 1084. Purcell, E.M., 1956, Nature 178, 1449. Rempe, G., Walther, H., Klein, N., 1987, Phys. Rev. 58, 353. Sargent III, M., Scully, M.O., Lamb Jr, W.E., 1974, Laser Physics, Addison-Wesley, Reading, MA. Scarani, V., Iblisdir, S., Gisin, N., Acín, A., 2005, Rev. Mod. Phys. 77, 1225. Scully, M.O., Sargent III, M., 1972, Physics Today (March), 38. Scully, M.O., Zubairy, M.S., 1997, Quantum Optics, Cambridge University Press, Cambridge. Siegman, A.E., 1989a, Phys. Rev. A 39, 1253. Siegman, A.E., 1989b, Phys. Rev. A 39, 1264. Steinberg, A.M., Kwiat, P.G., Chiao, R.Y., 2006, in: Drake, G.W.F. (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer, Leipzig. Streifer, W., Scifres, D.R., Burnham, R.D., 1982, Appl. Phys. Lett. 40, 305. Sudarshan, E.C.G., 1963, Phys. Rev. Lett. 10, 277. Townes, C.H., 1995, Making Waves, American Institute of Physics, New York. Walther, H., 2004, J. Mod. Opt. 51, 1859. Weinberg, S., 1977, Proc. Am. Acad. Arts Sci. 106, 22. Wolf, E., 1963, in: Fox, J. (Ed.), Proceedings of the Symposium on Optical Masers, John Wiley & Sons, New York. Wolf, E., 1979, Optics News 5, 24. Wootters, W.K., Zurek, W.H., 1982, Nature 299, 802.


Chapter 5

The history of near-field optics by

Lukas Novotny The Institute of Optics, University of Rochester, Rochester, NY 14627, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(07)50005-3 137

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139

§ 2. The diffraction limit

. . . . . . . . . . . . . . . . . . . . . . . . . .

142

§ 3. Synge and Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . .

145

§ 4. First developments . . . . . . . . . . . . . . . . . . . . . . . . . . .

150

§ 5. Surface plasmons and surface enhanced Raman scattering . . . . . .

152

§ 6. Studies and applications of energy transfer . . . . . . . . . . . . . .

153

§ 7. First developments of near-field optical microscopy . . . . . . . . .

155

§ 8. Theoretical near-field optics . . . . . . . . . . . . . . . . . . . . . .

161

§ 9. Near-field scattering and field enhancement

. . . . . . . . . . . . .

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§ 10. Near-field optics and antenna theory . . . . . . . . . . . . . . . . . .

170

§ 11. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction Near-field optics is the study of non-propagating inhomogeneous fields and their interaction with matter. Optical near-fields are localized to the source region of optical radiation or to the surfaces of materials interacting with free radiation (secondary sources). In many situations, optical near-fields are explored for their ability to localize optical energy to length scales smaller than the diffraction limit of roughly λ/2, with λ being the wavelength of light. This localization is being explored for ultrasensitive detection (Fischer [1985, 1986], Levene, Korlach, Turner, Foquet, Craighead and Webb [2003]) and for high-resolution optical microscopy and spectroscopy (e.g., Novotny and Stranick [2006]). Optical near-fields can have physical properties that are drastically different from their free-propagating counterparts. Examples are spatial and temporal coherence (Carminati and Greffet [1999], Apostol and Dogariu [2003], Roychowdhury and Wolf [2003]), the polarization state (Setälä, Kaivola and Friberg [2002], Ellis, Dogariu, Ponomarenko and Wolf [2005]), thermal energy density (Shchegrov, Joulain, Carminati and Greffet [2000]), and the very nature of the light–matter interaction (Carniglia and Mandel [1971], Agarwal [1975], Keller [2000b], Zurita-Sanchez, Greffet and Novotny [2004], Henkel [2005]). An angular spectrum representation of electromagnetic fields yields a decomposition into homogeneous plane waves and inhomogeneous evanescent waves. This decomposition depends on the particular choice of a reference axis (optical axis). The near-field region is generally characterized by the region in space where the evanescent waves cannot be neglected. The most elementary near-field is the one associated with a single evanescent wave as generated, for example, by total internal reflection at the surface of a dielectric material. Mathematically, the evanescent wave is a solution of the Helmholtz equation in free space. However, because of its exponential distance dependence the evanescent wave would yield an infinite energy at a distance far away from its mathematical origin. Therefore, on physical grounds, the evanescent wave cannot exist in free space and is restricted to material boundaries, making it impossible to decouple the evanescent wave from its source. Consequently, an evanescent wave cannot exist in the absence of other waves in space. This property makes a theoretical understand139

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The history of near-field optics

[5, § 1

ing of light–matter interactions in the optical near-field challenging. For example, evanescent waves cannot be quantized because they do not form an orthonormal set of solutions. Only when the evanescent waves are complemented by other solutions (e.g., exciting and reflected plane waves) can a quantization be accomplished (Carniglia and Mandel [1971]). Unlike free propagating radiation, evanescent waves are not purely transverse (∇ · E is not zero everywhere) and hence longitudinal fields enter the light–matter interaction in the near-field. For example, already in 1891 Paul Drude and Walther Nernst investigated experimentally the excitation of fluorescence by a standing evanescent wave (Drude and Nernst [1891]). The study confirmed that maxima of the field coincide with the maxima of the fluorescence yield, i.e., with the strength of the light–matter interaction. While today this study might seem trivial, other fundamental aspects of optical near-fields still pose a great theoretical challenge. For example, because an evanescent field is coupled to its source, the light–matter interaction in the near-field influences the very nature of the source. When two atoms A and B interact over a short distance their physical properties become coupled and causality makes it impossible to define either of them as the source of the interaction (Power and Thirunamachandran [1997], Keller [2000b]). Near-field optics deals with optical interactions on a subwavelength scale. In this sense, nonradiative interactions are of key interest. However, nonradiative interactions are found in so many different fields of study that it is rather difficult to incorporate them consistently into the field of near-field optics. For example, optical near-fields are key ingredients in Van der Waals attraction and in Förster energy transfer. The importance of near-fields was also realized in Arnold Sommerfeld’s analysis of dipole radiation over lossy ground (Sommerfeld [1909]) and in Jonathan A.W. Zenneck’s and Demetrius Hondros’ study of guided electromagnetic waves on metal surfaces (Zenneck [1907], Hondros [1909]). It would be a long haul to describe these developments in detail. We will touch on them only marginally in order to bring near-field optics into the proper historical context. A short account of the history of near-field optics can be found in the 1993 proceedings of the first conference on near-field optics held in Arc-et-Senans (Pohl [1993]). Another historical review written by Dieter W. Pohl summarizes the developments of the decade 1984–1994 (Pohl [2004]). Research in the field of near-field optics was vitally important for the advance of the more general field of nano-optics (Novotny and Hecht [2006]) and the now independent fields of single-molecule spectroscopy (Xie and Trautman [1998]) and nanoplasmonics (Xia and Halas [2005]). Over the past ten years, developments in near-field optics and near-field optical microscopy have been summarized in several review articles and books (Girard and Dereux [1996],

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Paesler and Moyer [1996], Fillard [1996], Fischer [1998], Dunn [1999], Ohtsu and Hori [1999], Courjon [2003], Kawata, Ohtsu and Irie [2002], Richards and Zayats [2004], Wiederrecht [2004], Hong, Swan and Erramilli [2004], Prasad [2004], Keller [2005], Novotny and Stranick [2006], Novotny and Hecht [2006], Bouhelier [2006]). As with all review articles and historical perspectives, it is not possible to account for all the individual contributions in the field. The purpose of this article is to provide a rough chronological outline of developments that have led to the field of near-field optics as it is known today. More recent developments are touched upon only peripherally. This article is organized as follows: Following this short introduction I shortly review the classical diffraction limit and its consequences, and discuss the basic ideas behind near-field optical microscopy. Section 3 then outlines the very first ideas behind near-field microscopy, which date back to the prophetic letters Edward Hutchinson Synge wrote to Albert Einstein in 1928. Over the years, Synge’s ideas were reinvented several times, and his papers resurfaced only in the mid of the 1980s. After reviewing these very early proposals, Section 4 concentrates on the first experimental developments, starting with acoustical experiments performed in 1956 by Albert V. Baez, followed by the microwave experiments of Ash and Nicholls in 1972. A section on surface plasmons and surface enhanced Raman scattering (SERS) reviews research in the 1970s and the early 1980s related to the development of a theoretical understanding of the SERS effect. The theoretical models developed in the SERS community brought important inspiration to the field of near-field optics as evidenced by the 1985 paper of John Wessel. With a similar purpose, Section 6 touches upon studies of energy transfer processes and reviews Hans Kuhn’s ideas of making use of optical near-fields for contact printing. First experimental developments of near-field optical microscopy are the subject of Section 7. Starting with the 1982 IBM patent by Dieter W. Pohl and the first measurements by his group in the same year we review independent parallel developments such as that by Ulrich Ch. Fischer and Hans Kuhn at the Max-Planck-Institute in Göttingen and that by the group of Aaron Lewis at Cornell University. The following years are characterized by technical improvements and first applications of near-field optical microscopy. New modalities, such as the photon scanning tunneling microscope (PSTM), were developed and it was realized that the interpretation of the contrast in the recorded images is not an easy task. Section 8 briefly reviews early theoretical studies, outlines the challenges associated with light–matter interactions in the near-field, and discusses the problem of inverse scattering. Scattering-based methods as well as tip-enhancement methods are then reviewed in Section 9. For metal tips and scattering particles, the similarity between near-field optics and antenna theory is particularly evident,

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[5, § 2

and this analogy is discussed in Section 10. In the same section, I also shortly reflect on Benjamin Franklin’s invention of the lightning rod and its ability to localize static fields. The article concludes with some final remarks in Section 11.

§ 2. The diffraction limit Near-field optics has its origin in the effort of overcoming the diffraction limit of optical imaging. At the end of the nineteenth century, Abbe and Rayleigh derived a criterion for the minimum distance Δx between two point sources at which they can still be unambiguously distinguished as two separate sources (Abbe [1873], Lord Rayleigh [1896]). Abbe’s criterion is illustrated in fig. 1(a) and reads as1 Δx = 0.61λ/NA.

(2.1)

Here, NA = n sin Θmax is the numerical aperture, a property of the optical system. n is the index of refraction of the surrounding medium and Θmax is the maximum collection angle of the optical system. The best possible NA is NA = n which, for optical glasses, is NA ≈ 1.5 and hence Δx ≈ λ/3. The resolution can be increased further by the use of two opposing objectives as demonstrated by Hell and co-workers (Schrader and Hell [1996]). The diffraction limit is often associated with Heisenberg’s uncertainty principle (Vigoureux and Courjon [1992]). From fig. 1(a) we recognize the maximum transverse wavenumber as Max[k ] = ±k sin Θmax which defines the bandwidth of spatial frequencies as Δk = 2Max[k ] = 4πNA/λ. Inserting into eq. (2.1) we obtain ΔxΔk = 0.614π. In agreement with the uncertainty principle this product is larger than 1/2. At first sight, the uncertainty product associated with the diffraction limit could be increased by a factor of ∼4π. To accomplish this, the Airy function defining eq. (2.1) would need to be replaced by the minimum uncertainty function, i.e., a Gaussian function. However, a Gaussian function has a Gaussian spectrum and hence involves evanescent waves which do not propagate through the optical system. It is the hard cut-off of spatial frequencies which gives rise to the Airy function and to a weaker uncertainty product. It is important to note that Abbe’s and Rayleigh’s criteria make no use of any information that is available on the properties of the two emitters. Furthermore, these criteria assume that the light–matter interaction is linear. However, by taking into account knowledge

1 This result is based on a scalar paraxial approximation (sin Θ max ≈ Θmax ) but it provides satisfactory accuracy even for high-NA systems (Novotny and Hecht [2006]).

5, § 2] The diffraction limit 143

Fig. 1. Schematic comparison of (a) diffraction-limited optical microscopy and (b,c) near-field optical microscopy. In (a) the point-spread function is defined by the wavelength λ and the numerical aperture NA = n sin Θmax , whereas in (b,c) it depends on the size and proximity of a local probe (e.g., aperture or particle). Near-field optical imaging is a sampling technique, i.e., the sample is probed point-by-point by raster-scanning the local probe over the sample surface and recording for every image pixel a corresponding optical signature. The local probe functions as an optical antenna, converting localized energy into radiation, and vice versa.

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about the excitation properties or the spectral properties of the sample it is possible to stretch the resolution limit beyond the diffraction limit (Toraldo di Francia [1955]). For example, the distance between a red emitting molecule and a green emitting molecule can be measured with nanometer accuracy by using spectral filters to only pass the radiation from one molecule at a given time. In this case, resolution is reduced to the problem of position accuracy (Novotny and Hecht [2006]). Recent work has demonstrated that resolution can be extended beyond Abbe’s limit by driving the light–matter interaction into saturation (Klar, Jakobs, Dyba, Egner and Hell [2000], Willig, Rizzoli, Westphal, Jahn and Hell [2006]) or by recording higher harmonics of the sample’s spatial frequencies (Gustafsson [2005]). In near-field optics, the bandwidth of spatial frequencies is no longer limited by Δk = 4πNA/λ. Instead, the dependence on the wavelength λ is replaced by a dependence on a characteristic length d (e.g., aperture diameter or tip diameter) of a local probe. As illustrated in fig. 1(b,c), in near-field optical microscopy the local probe ensures a confined photon flux between the probe and the sample surface. The probe is raster-scanned over the sample surface and for predefined positions (x, y) of the probe a remote detector acquires the optical response. In this way an optical contrast image can be recorded. A reproduction of an early near-field optical scan trace is shown in fig. 2. It was recorded on 22 October 1982.

Fig. 2. Early near-field optical scan trace recorded with an aperture-type probe. The data are from the laboratory book of Winfried Denk then working with Dieter Pohl at the IBM Research Laboratory in Switzerland.

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§ 3. Synge and Einstein It is well known that the original idea of using a tiny aperture in an opaque screen for performing near-field optical imaging was published in 1928 by Edward Hutchinson Synge (Synge [1928]), an Irishman living in Dublin. Synge’s contributions and his later life have been summarized in a beautiful article by McMullan (McMullan [1990]). Here, we shall revisit Synge’s key ideas and outline their relationship to the developments in the field of near-field optics. In his 1928 publication, Synge writes “The idea of the method is exceedingly simple, and it has been suggested to me by a distinguished physicist that it would be of advantage to give it publicity, even though I was unable to develop it in more than an abstract way.” Today, we know that the distinguished physicist to whom Synge was referring is Albert Einstein. In a letter dated 22 April 1928 Synge describes to Einstein a microscopic method in which not the field penetrating through a tiny aperture is used as a light source but the scattered field from a tiny particle. Synge states that “By means of the method the present theoretical limitation of the resolving power in microscopy seems to be completely removed and everything comes to depend upon technical perfection.” In the same letter, Synge provides a sketch of the apparatus. Figure 3(a) shows an adaptation of this sketch. Synge writes “If a small colloidal particle, e.g. of gold, be deposited upon a quartz slide placed above a Zeiss cardioid condenser of NA 1.05, then, all rays of light from the condenser which reach the surface of the slide will be totally reflected by the surface, except those which strike the surface at the base of the particle. These will be scattered in all directions and if the objective of a microscope is suitably arranged above the slide, a proportion of the rays so scattered will come to a focus in the eye of an observer, or upon a photographic plate, or a photo-electrical cell suitably placed.” Synge describes here what is called today dark-field microscopy, a technique invented at the turn of the twentieth century by the Austrian chemist Richard Adolf Zsigmondy. In his letter, Synge then proposes to place a very thin stained biological section onto a quartz cover glass and to raster scan it in close distance over the irradiated particle. He argues that the amount of light received from the particle and collected by the objective will depend upon the relative opacity of the different parts of the section. In the remainder of the letter, Synge addresses different technical difficulties, describes the scanning process, and also proposes to embed the particle into the end face of the quartz slide. He also realizes a potential difficulty: “I am not sure how near the biological section could be brought to the surface of the quartz plate without impairing the totality of the reflection.” He thinks this is the only potential theoretical limitation and that everything else depends only

146 The history of near-field optics [5, § 3

Fig. 3. (A) Synge’s original proposal of near-field optical microscopy based on using the scattered light from a particle as a light source. The figure is adapted from Synge’s letter dated 22 April 1928 sent to Einstein. (B) Wessel’s proposal from 1985 showing the following elements: (a) particle, (b) sample surface, (c) optically transparent probe-tip holder, and (d) piezoelectric translators (Wessel [1985]). (C) 1988 experiment by Fischer and Pohl. The near-field probe consists of a gold-coated polystyrene particle (Fischer and Pohl [1989]). (D) Figure 6 from Ueyanagi’s 2001 patent (Ueyanagi [2001]): (1) optical head, (2b) parallel laser beam, (2c) convergent light, (5) objective lens, (6) transparent condensing medium, (6a) incident surface, (6b) light condensed surface, (8) micro metal member, (9) light spot, (10) near-field light, (12) disk, (120) disk-like plastic plate, (121) recording medium.

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on perfecting the technique. Synge also writes that there is no institution in his country that could carry out the necessary experiments and he suggests that the experiments could be undertaken by the Institute of Physics in Berlin. On 3 May 1928 Einstein replies with a short letter written in German and sent from Berlin. He states that he believes that Synge’s basic idea is correct but that his particular implementation seems to be of no use (“prinzipiell unbrauchbar”). He argues that if the distance between the sample surface and the surface supporting the particle becomes small then there will be considerable light leakage (frustrated total internal reflection) and the total image field will become bright. Instead, Einstein suggests using the light that penetrates through a tiny hole in an opaque layer as a light source. He also states that he couldn’t read many parts of Synge’s letter. This could be related to Synge’s challenging handwriting (which improved in Synge’s second letter). Only five days later, on 9 May 1928 (imagine the efficiency of the postal service at that time), Synge replies to Einstein’s letter and states “It was my original idea to have a very small hole in an opaque plate, as you suggest, and it was in this form that I had mentioned it to several people.” In the same letter Synge suggests what later became the most standard way of fabricating aperture probes used in near-field optical microscopy: “A better way could be, if one could construct a little cone or pyramid of quartz glass having its point P brought to a sharpness of order 10−6 cm. One could then coat the sides and point with some suitable metal (e.g. in a vacuum tube) and then remove the metal from the point, until P was just exposed. I do not think such a thing would be beyond the capacities of a clever experimentalist.” In a following paragraph Synge states that he is “sure that some idea of the kind will be made use of ultimately, but it obviously requires to drop into the brain of an experimental genius.” A couple of decades later this prophecy was indeed verified. Einstein’s reply from 14 May 1928 does not address the new thoughts of Synge. Instead, Einstein refers to the original idea of using the scattered light from a tiny particle as a light source and he reiterates the problem of total internal reflection. Einstein states that he believes that it is practically impossible to generate a tiny light source through total internal reflection and scattering. He adds that he neither believes in the promise of the other proposals for generating a tiny light source and that it makes him hesitant to recommend it to an experimentalist. Einstein suggests that Synge publish the idea in a scientific journal and highlight the technical difficulties associated with a practical implementation. This encouragement forms the decisive impetus that leads Synge to publish his idea of aperture-based near-field optical microscopy (Synge [1928]). On 29 August 1928 Synge writes his last letter to Einstein. In this letter he proposes an arrangement for an X-ray in-

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terferometer and he mentions that he contacted Sir W. Bragg but that he couldn’t get a satisfactory response. There appears to be no record of a response from Einstein. Synge’s original proposals to overcome the diffraction limit of light microscopy were reinvented over the years and form the basis for both ‘aperture’ and ‘apertureless’ scanning near-field optical microscopy. Synge was also the first to propose the principle of scanning which forms today a key ingredient in a wide range of technologies ranging from television to scanning electron microscopy (SEM). Before Synge’s time the idea of manipulating the position of a source relative to a target in an imaging apparatus and thereby improving its imaging capabilities was not known (McMullan [1990]). In a follow-up paper in 1932, Synge suggested the use of piezo-electric quartz crystals for rapidly and accurately scanning the specimen (Synge [1932]). He estimated that a translation of 5 µm can be accomplished with a voltage of 250 V. This estimate matches perfectly the sensitivity of present-day piezo-electric transducers used in scanning probe microscopy. The idea of using piezo-electric transducers to control the gap between the probe (e.g., aperture) and the sample surface did not occur to him. This concept had to wait fifty years and was a key ingredient in the development of scanning tunneling microscopy (STM) (Binnig, Rohrer, Gerber and Weibel [1982]). In the same 1932 paper, Synge also suggested for the first time to use image processing to highlight certain features of an image before displaying it. According to McMullen (McMullan [1990]), Synge had many other visionary ideas, some of which became of importance in other scientific fields. In 1936 Synge had a mental breakdown and had to spend the rest of his life in a Dublin nursing home. Although Einstein did not believe in the practical realization of Synge’s original idea the concept of using the scattered light from a tiny particle as a light source is today well-established and experimentally verified (Fischer and Pohl [1989], Malmqvist and Hertz [1994], Kawata, Inouye and Sugiura [1994], Anger, Bharadwaj and Novotny [2006], Kühn, Hakanson, Rogobete and Sandoghdar [2006]). In 1985 John Wessel proposed a method very similar to Synge’s original idea (Wessel [1985]). His proposed arrangement is depicted in fig. 3(b). Wessel suggests to excite the particle resonantly thereby creating a locally enhanced field which serves as a light source. He is the first to mention the analogy to classical antenna theory. He writes: “The particle serves as an antenna that receives an incoming electromagnetic field.” Wessel did not know of Synge’s ideas and his proposal was likely inspired by the invention of STM (Binnig, Rohrer, Gerber and Weibel [1982]) and the discovery of surface enhanced Raman scattering (SERS) (Fleischmann, Hendra and McQuillan [1974], Jeanmaire and Van Duyne [1977], Albrecht and Creighton [1977]). The quest for an understanding of SERS

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gave rise to many theoretical studies aimed at predicting the electromagnetic field enhancement near laser-irradiated metal particles and clusters thereof (Gersten and Nitzan [1980], Wokaun, Bergman, Heritage, Glass, Liao and Olson [1981], Boardman [1982], Metiu [1984], Meier, Wokaun and Liao [1985]). This era can be considered as the first phase of what is called today nanoplasmonics (Xia and Halas [2005]). In 1988 Ulrich Ch. Fischer and Dieter W. Pohl carried out an experiment that is very similar to Synge’s and Wessel’s proposal (Fischer and Pohl [1989]). Instead of using a laser-irradiated solid metal particle as a local light source, they used a gold-covered polystyrene particle [cf. fig. 3(c)], a structure that was later extensively developed and named gold nanoshell (Jackson, Westcott, Hirsch, West and Halas [2003]). Fischer and Pohl imaged a thin metal film with 320 nm holes and demonstrated a spatial resolution of ∼50 nm. Their results provide the first experimental evidence that near-field scanning optical microscopy as originally proposed by Synge is feasible. As an illustration of the experimental feasibility of Synge’s original idea (and Wessel’s proposal) fig. 4 shows a fluorescence image of single molecules dispersed on a quartz substrate and imaged with a single laser-irradiated gold particle (Anger, Bharadwaj and Novotny [2006], Kühn, Hakanson, Rogobete and Sandoghdar [2006]). It turns out that the polarization conditions of the exciting laser radiation are important – an ingredient that is not discussed in either Synge’s or Wessel’s proposal. In many experiments Synge’s particle is replaced by a bare metal tip and these methods became to be known as ‘apertureless’ or

Fig. 4. Single-molecule fluorescence imaging using the scattered light from a laser-irradiated gold particle as light source (Anger, Bharadwaj and Novotny [2006]). (a) Experimental arrangement. An 80 nm gold particle is attached to the end of a etched glass tip and irradiated by a radially polarized laser beam. A sample with dispersed fluorescent molecules is raster-scanned underneath the gold particle. (b) Corresponding near-field fluorescence image.

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‘tip-enhanced’ near-field optical microscopy (Zenhausern, Martin and Wickramasinghe [1995], Novotny, Sanchez and Xie [1998]). Ironically, the very idea of Synge, communicated in his first letter of 22 April 1928 to Einstein, has been patented in 2001 by Fuji Xerox Co. (Ueyanagi [2001]). The inventor, Kiichi Ueyanagi, describes different applications making use of an antenna structure (such as a particle) fabricated into the end face of a dielectric medium to localize optical radiation. One of the figures is reproduced in fig. 3(d).

§ 4. First developments Before the first experimental developments were undertaken, Einstein’s and Synge’s idea of using an irradiated aperture in a flat metal screen (Synge [1928]) resurfaced in several other proposals. Not knowing of Synge’s prophetic work, John A. O’Keefe publishes in 1956 a short proposal starting with “The following is presented as a concept illustrating a method by which it might conceivably be possible to go beyond the resolving power of visible light” (O’Keefe [1956]). Similar to Synge, he proposes to employ an irradiated small hole in an aluminum coating as a light source. He writes “By scanning, an image can be built up whose detail will correspond to the size of the hole, even if it is smaller than the wavelength of the light used.” O’Keefe indicates that he believes the realization of his proposal is remote because of the difficulty of providing for relative motion between the pinhole and the object. Only a few months later, Albert V. Baez publishes an article in which he discusses the results of a near-field acoustical experiment (Baez [1956]). For his demonstration, Baez used acoustic waves with a wavelength of 14 cm (2.4 kHz) and confirmed a resolution not limited by diffraction. Baez used his own fingers as test objects and his results are mentioned only very qualitatively in the article. More than ten years later Charles W. McCutchen discusses the possibility of overcoming the diffraction limit of optical imaging by convolving the spatial frequencies of the sample with the spatial frequencies of a probe object (McCutchen [1967]). His consideration is based on the assumption that the plane containing the probe object constitutes a spatial filter function for the field emanating from the sample plane. The probe object shifts the spatial spectrum of the sample field thereby making high spatial frequencies propagate. This principle is analogous to a radio receiver where a local oscillator (probe particle) shifts the signal frequencies (sample field) into the pass band (propagating radiation). According to McCutchen, it does not matter whether a particle or an aperture is scanned over the surface of a sample. What matters is that the probe object is very small so

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it provides the necessary high spatial frequencies. Of course, the application of simple linear filter theory is an oversimplification and it cannot account for strong interactions between sample and probe, e.g., for field enhancement effects and resonant interactions. Nevertheless, it provides an intuitive picture which is important for the understanding of image formation in near-field optical microscopy. McCutchen believes that it is not practical to place the probe object right next to the sample and he considers the effect of moving the probe object behind the condenser lens. He argues that the bandwidth of spatial frequencies can be doubled – a principle employed today in structured illumination microscopy (Gustafsson [2005]). McCutchen writes “In an extreme form, the technique would consist of scanning the specimen past a minute aperture, much smaller than the Abbe limit. . . . Obviously, it would be hard to use this method on any but the flattest of surfaces, and I wonder if there are many jobs it could do that reflection electron microscopy would not do better.” McCutchen’s scepticism is justified, but it is the scientific curiosity and the potential benefit of combining the chemical specificity of optical spectroscopy with high-resolution optical microscopy which drives the progress in near-field optics forward. In 1984, Gail A. Massey uses Fourier optics to formulate the image-forming process described by McCutchen (Massey [1984]), however without knowing of McCutchen’s work. Massey concludes that for an aperture of size d a lateral resolution of ∼d can be achieved with a depth of focus of ∼d/3 (Massey [1984]). In 1970, H. Nassenstein proposes to illuminate a sample with evanescent waves, thereby generating scattered waves which contain information on spatial frequencies of the sample spectrum beyond the classical resolution limit (Nassenstein [1970]). Similar experiments were reported before in the context of holography (Stetson [1967]). Since evanescent waves are bound to their source (primary or secondary), Nassenstein’s proposal implies that a secondary object (the probe) be brought close to the sample. Today, total internal reflection fluorescence (TIRF) microscopy is a powerful tool in biological research because it effectively reduces background fluorescence. There are commercially available TIRF microscope objectives with NA’s of 1.5–1.65. The first experimental validation of near-field microscopy using electromagnetic radiation has been undertaken by Eric A. Ash and G. Nicholls at University College, London. In a paper published in 1972 they use 10 GHz microwaves (λ = 3 cm) and an aperture of 1.5 mm to image an aluminum test pattern deposited on a glass slide (Ash and Nicholls [1972]). At a separation between aperture and sample plane of 0.5 mm they are able to achieve a resolution of better than λ/60, clearly beyond the diffraction limit of standard microscopy. Ash and Nicholls state “. . . one might hope to build a super-resolution optical micro-

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scope”, and they add “Although the optical microscope may not be beyond reach, in our view a more immediately hopeful application is the construction of an infrared microscope with a resolution comparable with that normally attainable in the visible optics spectrum.” However, the history of near-field optics proceeded in the other sequence. In 1982 the first optical scan images were recorded by Dieter W. Pohl and co-workers at the IBM Research Laboratories (cf. fig. 2) and the infrared analogue has been demonstrated later in 1986 by Gail A. Massey using 100 µm radiation (Massey, Davis, Katnik and Omon [1985]).

§ 5. Surface plasmons and surface enhanced Raman scattering The field of near-field optics has been greatly established by the quest for diffraction-unlimited microscopy and spectroscopy. But there are at least two other important developments which had a significant influence on the shaping of the field: (1) research in the field of metal optics and (2) the discovery of surface enhanced Raman scattering (SERS). In 1968 two different experiments, one by Andreas Otto and the other by Erwin Kretschmann and Heinz Räther, have demonstrated that plasma oscillations on thin metal films can be excited with an optical beam undergoing total internal reflection (Otto [1968], Kretschmann and Räther [1968]). It was shown that these plasma oscillations are polaritons, that is, their existence is coupled to an electromagnetic field, and today these excitations are referred to as surface plasmon polaritons. It is interesting to note that already in 1959 T. Turbadar used the Kretschmann configuration for reflectance measurements on aluminum films and he noticed a characteristic reflection dip beyond the critical angle of total internal reflection. The reflection dip was shown to be consistent with thin film theory and it was noted that it must be associated with an evanescent wave because there is no transmitted light (Turbadar [1959]). The existence of optical surface waves on metal surfaces was also noticed in 1941 by Ugo Fano in his theoretical analysis of anomalous diffraction gratings (Fano [1941]). The later experiments of Otto, Kretschmann and Raether gave rise to various dedicated experiments aimed at understanding the confinement of optical fields near the surface of metals. On planar metal surfaces, plasmons are nonradiative. They propagate along the surface and their wavelength at a given frequency ω is shorter than the one in free space. At the surface plasmon frequency, and in the ideal case of no damping, the wavelength of the plasmon polariton goes to zero and the associated field becomes localized to the very surface of the metal, that is, the decay length of the evanescent wave goes to zero. Besides the interest in surface plasmons propagating along extended interfaces, experimental and theoretical work

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had also been undertaken to understand the optical response of finite-sized metal particles. The work of Uwe Kreibig provided an understanding of the transition of macroscopic electromagnetic theories applicable to particles larger than the mean-free path of electrons in the metal to quantum theories applicable to clusters of a few metal atoms (Kreibig [1974]). Surface plasmons associated with finite metal particles and clusters thereof gained a lot of importance for the understanding of the SERS effect which was discovered in 1974 (Fleischmann, Hendra and McQuillan [1974]). Shortly after, it was realized that the phenomenon is due to the increased surface of roughened metal surfaces and the associated electromagnetic field enhancement (Jeanmaire and Van Duyne [1977], Albrecht and Creighton [1977]). In the early 1980s much theoretical work was done with the aim of understanding the SERS effect (Chang and Furtak [1981]). The nearfields of various metal nanoparticle arrangements were calculated and it was concluded that the effect is mediated by so-called ‘hot spots’, i.e., localized regions between metal-particle aggregates that exhibit particularly strong electromagnetic field enhancement. A recent retrospective was published by Martin Moskovits (Moskovits [2005]). The enhanced near-fields around metal nanoparticles have also been shown to lead to several interesting photochemical phenomena, including near-field driven plasmon-resonant particle growth (Chen and Osgood [1983]). The quest for an understanding of the SERS effect marked the beginning of the first wave of nanoplasmonics and gave rise to new schemes of near-field optical microscopy, such as Wessel’s proposal (Wessel [1985]). The electromagnetic theories and calculations developed in the early phase of SERS proved very beneficial for the understanding of light localization and near-field interactions. On the other hand, the development of near field optical microscopy provided for the first time direct access to the fields associated with surface plasmon polaritons (Specht, Pedarning, Heckl and Hänsch [1992], Marti, Bielefeldt, Hecht, Herminghaus, Leiderer and Mlynek [1993], Dawson, de Fornel and Goudonnet [1994]) and gave rise to the recent revival of nanoplasmonics.

§ 6. Studies and applications of energy transfer Independent of the activities in SERS, plasmonics, or near-field microscopy, short-range optical interactions were also investigated in the context of energy transfer between molecules (Förster [1946, 1948]) and fluorescence quenching near metal surfaces (Kuhn [1968], Drexhage [1970], Zingsheim [1976]). Already in 1970 Hans Kuhn suggested to make use of optical near-fields for contact imaging (Kuhn [1968]). He envisioned the use of short-range energy transfer from

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Fig. 5. Near-field optical contact printing originally proposed by Hans Kuhn. A nanoscale pattern is transfered to a dye layer by placing a laser-irradiated nanostructured metal film on top of the dye layer. Short-range bleaching the fluorescence of the dye layer gives rise to spatial patterning of the fluorescence intensity. (a) Fluorescence image of the dye layer during contact with a metal pattern consisting of platinum disks (diameter 8 µm). Close to the disks the fluorescence is diminished because of fluorescence quenching. (b) After release of the metal mask regions of the dye layer which were close to individual metal disks exhibit stronger fluorescence (quenching lowers the bleaching rate). Parts (c,d) show the corresponding arrangements of layer and mask. From Fischer and Zingsheim [1982].

electronically excited dye molecules as a method for the duplication of nanostructures. The scheme was later implemented by Hans P. Zingsheim and Ulrich Ch. Fischer (Fischer and Zingsheim [1982]). In this experiment a monomolecular film of dye molecules serves as a light-sensitive film, and a very thin, only partially absorbing planar metal pattern embedded in the surface of a pliable polymer film serves as a conformal mask (cf. fig. 5). After the film has been brought into contact with the mask and has been irradiated with light, the structure is transferred from the mask to the monomolecular film as a pattern of areas where the dye is bleached and areas where it is not bleached. The resolution of this pattern transfer is not limited by the wavelength of light but by the range of the energy-transfer mechanism and the distance between the mask and the dye layer (Fischer [1998]). In principle, this process of pattern transfer makes use of the short range of the near-field of single electronically excited molecules in order to obtain diffraction-unlimited resolution down to the molecular scale. Studies of

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energy-transfer processes established that many types of near-field interactions can be understood on a classical phenomenological basis (Chance, Prock and Silbey [1978], Kerker, Wang and Chew [1980]). The phenomenological theory treats the quantum-mechanical transition probability between two states as a classical dipole oscillating at the transition frequency. The theory turns out to be equivalent to a rigorous quantum-electromagnetic theory in the weak coupling regime (Novotny and Hecht [2006]) but is much easier to deal with. Extensive reviews of surface enhanced spectroscopy and energy-transfer processes between molecules and surfaces of various shapes has been provided in 1984 by Horia Metiu (Metiu [1984]) and by George W. Ford and Willes H. Weber (Ford and Weber [1984]). The latter review also discusses the nonlocal dielectric response near material boundaries (spatial dispersion). In later work, Förster energy transfer has been proposed as an interaction mechanism for near-field optical imaging (Kopelman and Tan [1994], Fujihira [1996], Sekatskii and Letokhov [1996], Vickery and Dunn [1999]). In 1999 the group of Robert C. Dunn provided the first experimental demonstration of near-field energy-transfer microscopy with sub-wavelength resolution (Vickery and Dunn [1999]) and the following year the emission from a single molecule was used for the first time as a local light source (Michaelis, Hettich, Mlynek and Sandoghdar [2000]).

§ 7. First developments of near-field optical microscopy The first dedicated efforts to demonstrate near-field microscopy at optical frequencies started in the early 1980s. The first developments proceeded without the knowledge of previous proposals and the prophetic papers by Synge. On 27 December 1982 Dieter W. Pohl, then working at the IBM Research Laboratory in Switzerland, filed a patent titled optical near-field scanning microscope (Pohl [1984]) which describes an aperture-based method very similar to that conceived by Synge more than fifty years earlier. The patent states “The term ‘near-field’ is intended to express the fact that the aperture is located near the object at a distance smaller than the wavelength. The term ‘aperture’ is used here to describe the pointed end of a light waveguide which forms an entrance pupil with a diameter of less than 1 µm.” In the same year, Pohl and co-workers managed to overcome the remaining experimental hurdles and recorded the first optical scan trace, shown in fig. 2. Their first scientific publication appeared with some delay in 1984 (Pohl, Denk and Lanz [1984]). In this publication near-field optical microscopy is referred to as optical stethoscopy in analogy to the acoustic stethoscope used in medical diagnosis. Pohl, Denk and Lanz write “The familiar

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medical doctor’s stethoscope, for instance, allows localization of the position of the heart to within less than 10 cm by moving the stethoscope over the patient’s chest, and listening to the sound of the heart beat. Assuming a sound frequency of 30–100 Hz, corresponding to a wavelength of almost 100 m, the stethoscope provides a resolving power of roughly λ/1000!” Their aperture light-source was formed by pressing an aluminum-coated, electrochemically etched quartz crystal towards a transparent sample. Laser light is coupled into the crystal and as soon as light is transmitted through the sample a tiny aperture is formed. This procedure for forming apertures has been later named pounding or punching. During the same period other groups have started similar efforts. Results by Ulrich Ch. Fischer were presented in a talk by Hans Kuhn at the second Meeting of Molecular Electronic Devices in 1983. The proceedings of this meeting were published much later, in 1987, because the editor passed away in the meantime (Kuhn [1987]). One of the scan traces was also reproduced in the 1984 yearbook of the Max-Planck-Society (Kuhn [1984]) (summary of research activities of 1983) and is shown in fig. 6 along with the outline of the experimental setup. In their experiments, Fischer and Kuhn used a glass hemisphere coated with a tantalum/tungsten layer in which a 100 nm hole was fabricated. A similar metal layer (20 nm thickness) with ten times larger holes (1 µm) was fabricated onto a glass slide and served as a test sample. The hemisphere was brought close to the sample and then scanned (without feedback control) along the surface. In fig. 6(b) two different scan traces are shown, one through the center of the sample hole (top) and one closer to its edge (bottom). The slope of the upper trace indicates a resolution bet-

Fig. 6. Experiment by Ulrich Ch. Fischer performed in 1983 while working with Hans Kuhn. (a) Schematic of the experiment. A metal-coated glass hemisphere with a 100 nm hole is used to irradiate a test sample made of a metal layer with 1 µm holes deposited onto a glass slide. From Kuhn [1987]. (b) Optical scan traces recorded along the solid line (top trace) and along the dashed line (bottom trace). From Kuhn [1984].

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ter than the diffraction limit. At the same time, Fischer and Zingsheim pioneered what is today called nanosphere lithography (Fischer and Zingsheim [1982]). In this technique, a suspension of colloidal particles with a diameter of >100 nm is deposited onto a plane surface. Subsequent evaporation of the solvent arranges the particles in a hexagonal two-dimensional lattice. Vacuum deposition of metals and other materials through the voids between neighboring spheres leads to arrays of triangular structures on the surface. The size and periodicity of these patterns can be varied by the size of the original particles. Double exposure from different angles of incidence allows more complex pattern to be created (Haynes and Van Duyne [2001]). The resulting Fischer patterns are used as test samples for microscopy or for Raman-active substrates for the detection of target agents. At the time of the experiments of Fischer and the experiments performed at IBM by Pohl and co-workers another development was underway at Cornell University. Aaron Lewis and co-workers studied light transmission through arrays of holes in planar metal films and were working towards a “scanning nanometer optical spectral microscope”. The first document mentioning their effort is an abstract from the 1983 Biophysics Meeting (Lewis, Isaacson, Muray and Harootunian [1983]). One year later they published an article presenting light transmission through 30 nm apertures in metal films (Lewis, Isaacson, Harootunian and Muray [1984]). In the same article they discuss “the possibility of constructing a scanning optical microscope based on near field imaging which could potentially have spatial resolutions as small as one-tenth (of) the wavelength of the incident light” (Lewis, Isaacson, Harootunian and Muray [1984]). Their first experimental results made use of thermally pulled glass capillaries, as used in the patch-clamp technique, and were published in 1986 (Harootunian, Betzig, Isaacson and Lewis [1986]). These experiments used near-field excited fluorescence as contrast mechanism and demonstrated a resolution on the order of the aperture diameter of 100 nm. In the same publication, Lewis and co-workers introduced the acronym NSOM, standing for “near-field scanning optical microscopy”. Earlier in the same year Urs Dürig, Dieter W. Pohl and Flavio Rohner published transmission near-field images recorded with aperture probes and by using active distance control via electron tunneling between probe and sample (Dürig, Pohl and Rohner [1986]). They referred to the technique as NFOS microscopy. The acronym SNOM was created later in 1988 to emphasize the analogy to SEM, STM, and other scanning microscopies. The optical near-fields near irradiated apertures were not only considered for microscopy but also for sensing applications. In 1986 Ulrich Ch. Fischer publishes a study on the transmission of tiny apertures in metal films (Fischer [1985]). The following year he demonstrates that the aperture acts as probe of its microenvironment through enhanced light scatter-

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ing and fluorescence (Fischer [1986]). This approach has later been revisited and applied to studies of single-molecule dynamics (Levene, Korlach, Turner, Foquet, Craighead and Webb [2003]). A significant breakthrough in the application of near-field optical microscopy came in 1991 when Eric Betzig and co-workers introduced aperture probes formed at the end of metal-coated, thermally pulled quartz fibers (Betzig, Trautman, Harris, Weiner and Kostelar [1991]). This method became widely used and was also adopted by the first companies that pursued a commercialization of nearfield optical microscopes. Another important technical improvement came in the following year, in 1992, when two independent groups introduced the shear-force technique to control the distance between probe and sample (Toledo-Crow, Yang, Chen and Vaez-Iravani [1992], Betzig, Finn and Weiner [1992]). In 1995, the shear-force method was combined with the sensing capability of a tuning-fork crystal (Karrai and Grober [1995]), which today is the most widely used method for probe-sample distance control. The group of Betzig at Bell Laboratories pioneered many applications of near-field optical microscopy, published many highlights, and contributed to the general acceptance of the method (Betzig and Trautman [1992]). In 1994, near-field microscopy was applied for the very first imaging of single fluorescent molecules (Trautman, Macklin, Brus and Betzig [1994]). While single molecules had been detected before by use of spectral identification at cryogenic temperatures (Moerner and Kador [1989], Orrit and Bernard [1990]), it was the spatial mapping which triggered the birth of single-molecule spectroscopy (Xie and Trautman [1998]). As an illustration, fig. 7 shows a fluo-

Fig. 7. (a) SEM image of smooth aperture probe fabricated by slicing away the end of a metal-coated, tapered optical fiber with a focused ion beam (FIB). From Veerman, Otter, Kuipers and van Hulst [1998]. (b) Single molecule fluorescence image acquired with a FIB milled aperture probe. From Moerland, van Hulst, Gersen and Kuipers [2005].

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rescence image of a sample with single dye molecules recorded with an aperture probe that was created by focused ion beam milling (Veerman, Otter, Kuipers and van Hulst [1998]). Over the years, many variations of near-field optical microscopy have been explored and developed. Probably the most notable one is photon scanning tunneling microscopy (PSTM), also called scanning tunneling optical microscopy (STOM), developed in 1989 by three different groups (Courjon, Sarayeddine and Spajer [1989], Reddik, Warmack and Ferrell [1989], deFornel, Goudonnet, Salomon and Lesniewska [1989]). This method is the optical analog of STM as it probes the tunneling photons between a surface and a local probe. Typically, an evanescent field is created by total internal reflection at a dielectric–air interface and the tip of a pointed optical fiber is used to locally convert the evanescent field into a propagating waveguide mode, similar to frustrated total internal reflection (Novotny and Hecht [2006]). PSTM is a very attractive method because of its simple physical picture, but the recorded images were not easy to interpret, mainly because of multiple scattering between separated objects on the sample. The reconstruction of the sample features based on the measured optical information, the inverse scattering problem, can be facilitated by tuning the angle of incidence of the total internally reflected wave (Garcia and NietoVesperinas [1995]). Several groups have theoretically established that PSTM measurements are inherently holographic and that they provide enough information to determine the two-dimensional structure of a thin sample (Greffet, Sentenac and Carminati [1995], Bozhevolnyi and Vohnsen [1996], Carney, Frazin, Bozhevolnyi, Volkov, Boltasseva and Schotland [2004]). A series of related studies on near-field phase conjugation have been performed by Sergey I. Bozhevolnyi and co-workers (Bozhevolnyi, Keller and Smolyaninov [1994], Bozhevolnyi, Keller and Smolyaninov [1995]). In modified form, the PSTM found interesting applications in optoelectronics and optical waveguides. Here, the evanescent tail of a waveguide mode is directly measured with a near-field probe rendering spatial maps of the electromagnetic field distribution. In a landmark experiment, Niek van Hulst’s group combined this method with heterodyne detection and generated phase and amplitude maps of femtosecond pulses propagating along ridge waveguides (Balistreri, Gersen, Korterik, Kuipers and van Hulst [2001]). These experiments, shown in fig. 8, directly visualized the electromagnetic field associated with an optical pulse and demonstrated that optical processes and phenomena can be probed by sampling their evanescent fields. Recently it was demonstrated that waveguide modes can also be probed by light scattering from a metal tip (Stefanon, Blaize, Bruyant, Aubert, Lerondel, Bachelot and Royer [2005]). PSTM was also applied to var-

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Fig. 8. Measurement of the electromagnetic field of a light pulse propagating along a ridge waveguide. Left: Schematic of a heterodyne PSTM with variable time delay in the reference arm. The instrument probes the evanescent tail of a waveguide mode and renders a temporally and spatially resolved map of the phase and amplitude distribution of the electromagnetic field associated with the pulse. Right: Map of the measured amplitude multiplied with the cosine of the measured phase for a single tracked pulse. The area depicted is an enlargement of a small part of the actual scan. From Balistreri, Gersen, Korterik, Kuipers and van Hulst [2001].

ious other phenomena, such as light localization in random media (Gresillon, Aigouy, Boccara, Rivoal, Quelin, Desmarest, Gadenne, Shubin, Sarychev and Shalaev [1999], Bozhevolnyi, Volkov and Leosson [2002]) or propagation of surface plasmon polaritons (Marti, Bielefeldt, Hecht, Herminghaus, Leiderer and Mlynek [1993], Dawson, de Fornel and Goudonnet [1994], Krenn and Weeber [2004]). Figure 9 shows an example of a recent measurement by the group of Joachim R. Krenn. The figure depicts a map of the measured light intensity of a surface plasmon propagating along a silver nanowire. In the 1990s near-field optical microscopy was applied to various problems, and the progress is best summarized by referring to more detailed review papers (Girard and Dereux [1996], Fischer [1998], Dunn [1999], Pohl [2004]). Very high spatial resolutions were demonstrated, but the nature of the optical contrast in the recorded images was often not understood. In fact, some optical images exhib-

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Fig. 9. Cylindrical surface plasmon propagating towards the end of a silver nanowire. The figure shows a map of the light intensity measured with a PSTM. From Ditlbacher [2005].

ited suspiciously close resemblance to the simultaneously recorded shear-force images. It was soon realized that two distinct properties contribute to the recorded optical signal: (1) the local material-specific response due to the probe–sample interaction, and (2) the vertical motion of the probe due to probe–sample distance control. While the former is the true optical contrast, the latter is an artifact due to the dependence of the strength of the near-field interaction on the proximity of the probe. A variation of the optical signal is generated even if the probe is located at a fixed lateral position and its vertical position over the sample surface is varied. In 1997, Bert Hecht and collaborators published an article titled Facts and artifacts in near-field optical microscopy in which they concluded that many published images represent the path of the probe rather than the true optical properties of the sample (Hecht, Bielefeldt, Novotny, Inouye and Pohl [1997]). Following this paper, previous results had to be re-examined and new results had to be tested critically before being published.

§ 8. Theoretical near-field optics Initial theoretical work in the field of near-field optics aimed at developing an understanding of image formation in near-field optical imaging. Approximate methods such as scalar diffraction theory break down in the near-field, making it necessary to solve the full vectorial wave equation for a given problem. In a 1931 publication, Synge writes “This theory [diffraction theory] is, in most cases, a very good approximation, and forms the basis of the theory of resolution of optical instruments as usually presented, but it is by no means an absolute theory, such a theory requiring the solution of the electromagnetic equations, subject

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to boundary conditions which have been a bar to their solution except in one very simple case. In general we may say that when we come down to magnitudes of the order of a wavelength the approximate theory ceases to be a good approximation.” (Synge [1931]). The theory for understanding the field distributions near tiny apertures in metal screens had been developed long before the interest in near-field optics started. It was Hans Bethe in 1944 who provided the first rigorous description (Bethe [1944]). His theory was corrected and generalized by C.J. Bouwkamp in 1950 (Bouwkamp [1950a, 1950b]). For real metals and for screens of finite thickness the Bethe–Bouwkamp theory still agrees qualitatively with the true field distribution (Novotny and Hecht [2006]). The first theoretical studies of near-field optical microscopy made use of the formulas of Bethe and Bouwkamp and used Fourier optics to propagate the fields (Massey [1984], Betzig, Harootunian, Lewis and Isaacson [1986], Roberts [1987], Roberts [1989], Roberts [1991], Leviatan [1986], Leviatan [1988]). Also, intuitive models were developed in which the near-field probe was treated as an elementary dipole (Van Labeke and Barchiesi [1993], Labani, Girard, Courjon and Van Labeke [1990], Girard and Courjon [1990]). In the 1990s different methods were introduced to solve the full vectorial wave equation for a given near-field configuration. Among the most widely used methods were plane-wave expansion techniques (Van Labeke and Barchiesi [1992]), Green’s function techniques (Dereux and Pohl [1993], Girard and Dereux [1994], Martin, Dereux and Girard [1994]), and the multiple multipole (MMP) technique (Novotny, Pohl and Regli [1994], Novotny and Hafner [1994], Novotny and Pohl [1995]). These methods have an analytical foundation and are summarized in the book by Novotny and Hecht [2006]. A review of early theoretical activities in near-field optics has been written by Christian Girard and Alain Dereux (Girard and Dereux [1996]). The goal of these early studies was to understand how interactions in the near-field are mapped to the far-field. Most commonly, the inverse scattering problem cannot be solved in a unique way and calculations of field distributions are needed to provide prior knowledge about source and scattering objects and to restrict the set of possible solutions. For example, experimental near-field images revealed contrast reversal if the collection angle of the near-field scattered light was modified (Hecht, Pohl, Heinzelmann and Novotny [1995]). It was shown that this contrast reversal originates from evanescent wave scattering giving rise to supercritical light propagation (forbidden light) in a dielectric sample (Novotny [1997a]). In 1995 it was shown that in some cases it is possible to assume that the interaction between probe and object can be neglected (weak coupling) which results in a simple linear detection process (Carminati and Greffet [1995]). In this regime, the reciprocity theorem requires that near-field optical images recorded in the

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PSTM mode and images recorded with aperture-based microscopy are equivalent (Mendez, Greffet and Carminati [1997]). The theory underlying image formation in near-field optics has been reviewed in 1997 by Jean-Jacques Greffet and Remi Carminati (Greffet and Carminati [1997]). Theoretical studies gave input to improved instrument design and predicted new detection strategies for optimizing the signal-to-noise ratio (SNR) in a given measurement (Girard, Dereux, Martin and Devel [1994], Greffet and Carminati [1997], Novotny [1997b], Hecht, Bielefeldt, Pohl, Novotny and Heinzelmann [1998]). Ultimately, it is the SNR which determines the best achievable resolution because, according to the principle of analytical continuation, a signal with finite support can be reconstructed exactly from a noise-free measurement in an arbitrarily small spatial domain (Devaney and Wolf [1974], Wolf and Nieto-Vesperinas [1985]). More recent theoretical studies aimed at understanding the physical properties of optical near-fields. Among the topics studied were coherence properties (Carminati and Greffet [1999], Roychowdhury and Wolf [2003]), the polarization state of optical near-fields (Setälä, Kaivola and Friberg [2002], Ellis, Dogariu, Ponomarenko and Wolf [2005]), spontaneous emission (Girard, Martin and Dereux [1995], Novotny [1996]) and local density of states near nanoscale structures (cf. fig. 10) (Colas des Francs, Girard, Weeber, Chicanne, David, Dereux and Peyrade [2001], Joulain, Carminati, Mulet and Greffet [2003], Novotny and Hecht [2006]), reciprocity relations in the optical near-field (Carminati, NietoVesperinas and Greffet [1998]), and fluctuation-induced friction (Zurita-Sanchez, Greffet and Novotny [2004]). The fluctuational properties of optical near-fields have been discussed in two recent review papers (Henkel [2005], Joulain, Mulet, Marquier, Carminati and Greffet [2005]). Studies of near-field inverse scattering have been pursued by different groups, mainly using the PSTM geometry (Garcia and Nieto-Vesperinas [1995], Greffet, Sentenac and Carminati [1995], Bozhevolnyi and Vohnsen [1996], Greffet and Carminati [1997], Carney and Schotland [2003]). It was shown that samples can be uniquely reconstructed from near-field optical measurements by use of near-field tomography (Carney, Frazin, Bozhevolnyi, Volkov, Boltasseva and Schotland [2004]). While the first studies on the quantum nature of evanescent fields have been performed already in the 1970s by Girish S. Agarwal (Agarwal [1975]) and Chuck Carniglia and Leonard Mandel (Carniglia and Mandel [1971]), a quantized theory of spatially confined light has been put forth by Ole Keller in 1998 (Keller [1998, 2000a, 2000b, 2005]). Keller also described the birth process of the photon wavefunction and pointed out that in a near-field interaction the photon is destroyed before it is fully born (Keller [2000b]). Another problem is the fact that it is not strictly possible to separate the source of radiation from the sink of radi-

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Fig. 10. Calculations of the electromagnetic local density of states (LDOS). (a) LDOS above a circular ‘optical corral’ built from dielectric particles deposited on a quartz substrate (λ = 440 nm). From Colas des Francs, Girard, Weeber, Chicanne, David, Dereux and Peyrade [2001]. (b) LDOS over a semi-infinite sample of aluminum as a function of frequency and evaluated at different heights. From Joulain, Carminati, Mulet and Greffet [2003].

ation. Instead, source and sink appear as a coupled object or, more formally, it is not possible to independently define the states of source and detector (Power and Thirunamachandran [1997]). A remarkable result of Keller’s work is the finding that only at an infinite time after its birth the energy of a photon is h¯ ωo , ωo being the transition frequency. At shorter times, the photon energy is larger than hω ¯ o. Part of the problem of defining a near-field photon is associated with the fact that the near-field is not purely transverse, which can be easily verified for an evanescent wave and its excitation. Standard quantum electrodynamics (QED) proceeds by invoking the Coulomb gauge and quantizing the retarded transverse field. It gives little attention to the ‘attached’ field. However, from single-molecule experiments it is known that a molecule close to an interface interacts with the total

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field and not only with its transverse part (Drexhage [1974]). Future theories and experiments will shed more light on the existence of near-field photons. Because of their localized nature, optical near-fields can vary substantially over the length scale defined by a quantum system, such as a quantum dot or a molecule. Hence, the light–matter interaction can no longer be restricted to the dipole selection rules, and higher-order multipolar transitions need to be considered (Zurita-Sanchez and Novotny [2002]). In the extreme case, the multipolar expansion does not converge and the optical near-field becomes a probe for the local orbital overlap between the system’s ground and excited states. It is also interesting to note that in the light–matter interaction the momentum of a photon (p = 2π h/λ) is typically neglected because it is much smaller than the electron ¯ momentum in matter (p = 2meff E). Consequently, photoinduced band-to-band transitions in an electronic dispersion diagram happen vertically. However, the momentum of a photon associated with the optical near-field is no longer defined by the wavelength but by a characteristic length d associated with the optical confinement (p = 2π h¯ /d). This makes the near-field momentum comparable with the electron momentum and hence intraband transitions become possible (Beversluis, Bouhelier and Novotny [2003]). The high momenta associated with the optical near-field as well as the possibility of accessing dipole-forbidden transitions will enrich optical spectroscopy and open up new and exciting frontiers.

§ 9. Near-field scattering and field enhancement Let us now go back in time to revisit Synge’s original proposal of using the light scattered by a small particle as a light source. When brought close to a sample surface, the particle not only scatters the incident field but also the field that is scattered from the surface. In fact, there is an infinite number of scattering iterations between particle and sample. Depending on the properties of particle and sample and on the excitation conditions only one or a few terms in this series are relevant. For example, in Wessel’s proposal (Wessel [1985]), the particle’s response is resonant with the incident field and hence the enhanced field generated by the particle can be regarded as an independent light source exciting the sample at short distance. On the other hand, one could consider a situation in which the interaction between the exciting field and the sample is more dominant than the interaction of the external field and the particle. In this case, the particle acts as a passive probe that scatters away the near-field of the irradiated sample. Both approaches have been implemented in near-field optics using pointed probes such as dielectric or metal tips as local scatterers. In weak scattering the probe acts as

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a local perturbation (Zenhausern, O’Boyle and Wickramasinghe [1994], Gleyzes, Boccara and Bachelot [1995], Inouye and Kawata [1994]) whereas in strong scattering the interaction between the probe and the exciting field dominates and the probe acts as an optical antenna (cf. Section 10), a device that efficiently converts the energy of free propagating radiation into localized energy, and vice versa (Keilmann [1995], Novotny, Sanchez and Xie [1998]). Whether a probe acts as a local perturbation or as an optical antenna depends on the particular experimental implementation. A recent review of tip-based near-field optical microscopy is by Novotny and Stranick [2006]. As mentioned before, the first experimental demonstration of Synge’s particlebased idea has been presented by the pioneers of near-field optical microscopy, Ulrich Ch. Fischer and Dieter W. Pohl (Fischer and Pohl [1989]). In 1992, scattering from a metal tip was applied for the detection and imaging of surface plasmon polaritons (Specht, Pedarning, Heckl and Hänsch [1992], de Hollander, van Hulst and Kooyman [1995]) and in 1994 experiments by the groups of Satoshi Kawata (Inouye and Kawata [1994]), A. Claude Boccara (Gleyzes, Boccara and Bachelot [1995]), and H. Kumar Wickramasinghe (Zenhausern, O’Boyle and Wickramasinghe [1994]) established that scattering-based near-field microscopy is a viable alternative to standard aperture-based approaches. While the original role of the aperture was to confine an optical field beyond the limits of diffraction, the role of the tip was to establish a local interaction and to scatter away the local field. To express this different viewpoint, scattering-based approaches are also referred to as apertureless near-field optical microscopy. Wickramasinghe’s experiments were already proposed in a patent filed in 1989 in which the method was named “apertureless near-field optical microscopy” (Wickramasinghe and Williams [1990]). Figure 11 reproduces the first illustration from this patent. They write: “For example, an ideal conical tip having a single atom or group of atoms at the very end which is illuminated by a focused light source, will result in optical evanescent fields diverging from the tip. The divergent fields will interact with the sample surface on a local scale. These fields will be scattered by the surface and a portion will propagate into the far field, where the fields may be detected, providing a useful signal for measuring the local optical and topographical properties of the surface with high resolution.” This sentence assumes that the interaction between the tip and the excitation field is stronger than the interaction between the sample and the excitation field, similar to Wessel’s proposal. In their patent, Wickramasinghe and Williams propose to use a double modulation technique combined with heterodyne detection in order to extract the near-field signal from the detected scattered light. In their later experiments, Wickramasinghe and co-workers implemented a slightly different interferometric detection

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Fig. 11. ‘Apertureless’ near-field optical microscope using heterodyne detection. Original drawing from the patent of H. Kumar Wickramasinghe and Clayton C. Williams (Wickramasinghe and Williams [1990]). (12) end of a tip, (14) tip, (40) optical source, (42) acousto-optic modulator, (44,46) lens, (48) beam splitter, (50) pin photodiode.

scheme and demonstrated extremely high spatial resolutions (Zenhausern, Martin and Wickramasinghe [1995]). However, as discussed before, the origin of the contrast in the recorded images has been debated. A Japanese patent with very similar ideas was filed in 1992 by Satoshi Kawata and co-workers (Kawata, Inouye and Sugiura [1992]). The patent describes a near-field optical microscope using a combination of total-internal reflection illumination and light scattering from a vertically modulated tip. A parallel effort in scattering-type near-field microscopy was also pursued by Fritz Keilmann with experiments performed at radio and microwave frequencies and later in the infrared (Keilmann, van der Weide, Eickelkamp, Merz and Stockle [1996], Knoll, Keilmann, Kramer and Guckenberger [1997], Knoll and Keilmann [1998]). In 1989, a few months after Wickramasinghe’s patent submission, Keilmann filed a patent titled scanning tip for optical radiation in which he proposes the fabrication of a coaxial tip, i.e., an aperture with a center conductor (Keilmann [1991]), an endeavor first undertaken by Fee, Chu, and Hänsch at microwave frequencies (Fee, Chu and Hänsch [1989]) and later by Fischer and Zapletal in the optical frequency regime (Fischer and Zapletal [1992]). Coaxial waveguides have no cut-off and therefore provide near-unity power transmission. Later, Fritz Keilmann and Bernhard Knoll implemented a method to extract the near-field signal from the scattered light (Wurtz, Bachelot and Royer [1998], Knoll and Keilmann [2000]). The method makes use of vertical probe modulation with frequency Ω and demodulation of the scattered light at higher harmonics nΩ. A combination with heterodyne detection allowed Fritz Keilmann and Rainer Hillenbrand to separately measure amplitude and phase of

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Fig. 12. Scattering-based near-field optical microscopy of a gold Fischer pattern with polystyrene residues deposited on a silicon surface. The method detects the backscattered optical signal at a higher harmonic of the modulation frequency Ω. (a) Schematic of the experiment, (b) topographical map, (c) optical map. Based on the different optical contrast, gold and polystyrene can be distinguished with a spatial resolution of ∼10 nm. From Hillenbrand and Keilmann [2002].

the scattered signal and to extract material specific optical parameters (Hillenbrand and Keilmann [2000, 2002], Keilmann and Hillenbrand [2004]). As an example, fig. 12 shows an image of a gold Fischer pattern with polystyrene residues deposited on a silicon surface. Polystyrene and gold yield clearly different optical contrast. In 1997 it was proposed to use the enhanced field at a laser-irradiated metal tip as an excitation source, similar to Wessel’s idea (Novotny, Sanchez and Xie [1998], Novotny, Bian and Xie [1997]). The interaction of the locally enhanced field with the sample surface generates an optical response which is then coupled out by the same tip and detected in the far-field. This strong-scattering scheme makes it possible to detect a spectroscopic response at frequencies different from the excitation frequency, thereby making it possible to explore the full range of linear and nonlinear optical spectroscopy. The first experimental

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demonstration employed two-photon excited fluorescence and was published in 1999 (Sanchez, Novotny and Xie [1999]). Similar experiments have also been performed by other groups (Hamann, Gallagher and Nesbitt [2000]), and today the method is most widely referred to as tip-enhanced near-field optical microscopy (Novotny and Stranick [2006], Bouhelier [2006]). Following these experiments, the method was extended to other spectroscopic interactions such as Raman scattering (Stöckle, Suh, Deckert and Zenobi [2000], Anderson [2002], Hayazawa, Inouye, Sekkat and Kawata [2002], Hartschuh, Sanchez, Xie and Novotny [2003]) and coherent anti-Stokes Raman scattering (Ichimura, Hayazawa, Hashimoto, Inouye, and Kawata [2004]). As an illustration, fig. 13 shows a near-field Raman

Fig. 13. Near-field Raman imaging of a single-walled carbon nanotube sample. (a) Topography showing a network of carbon nanotubes overgrown with water droplets. (b) Raman scattering image of the same sample area recorded by integrating, for each image pixel, the photon counts that fall into a narrow spectral bandwidth centered around ν = 2615 cm2 (indicated by the shaded stripe in (c)). (c) Raman scattering spectrum recorded on top of the nanotube. Adapted from Hartschuh, Sanchez, Xie and Novotny [2003].

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scattering image of a single-walled carbon nanotube sample along with a Raman scattering spectrum recorded when the tip is placed on top of the nanotube. The field enhancement at a laser-irradiated metal tip has been the subject of many theoretical studies. Winfried Denk and Dieter W. Pohl demonstrated that the quasi-static fields in the gap between a tip and a substrate can be extremely strong and that this effect can be instrumental for inelastic tunneling and light emission during scanning tunneling microscopy (STM) (Denk and Pohl [1991]). Later, it was theoretically established that the field-enhancement effect must be driven by an external field polarized along the axis of the pointed probe (Novotny, Bian and Xie [1997], Martin and Girard [1997], Furukawa and Kawata [1998]). Interestingly, the field enhancement was also studied in the context of STM as it was believed to mediate the transfer of atoms from the tip to the sample (Jersch and Dickman [1996], Gorbunov and Pompe [1994], Bragas, Landi and Martinez [1998]).

§ 10. Near-field optics and antenna theory In this last section I intend to outline similarities between near-field optics and antenna theory (Pohl [2000]). These similarities project the roots of near-field optics more than 100 years back to the time of Guglielmo Marconi’s experiments on radiowave transmission. The quality of an antenna is characterized by quantities such as radiation efficiency and antenna gain, and it is plausible that similar quantities are applicable to optical near-field probes. Therefore, established antenna-concepts can provide inspiration for novel near-field probes. The primary function of a near-field probe is the concentration of electromagnetic energy on a sample surface, similar to a standard electromagnetic antenna that concentrates propagating radiation into a confined zone called the feedgap. In the feedgap, electric circuitry either releases or receives the power associated with the electromagnetic field. The challenge in the design of an antenna is to efficiently couple the power flow between the near-zone and the far-zone of the source (or receiver). This criterion holds also for a quantum source, such as a single molecule, which emits a single photon at a time. The most efficient antenna designs that have been implemented at optical frequencies are the half-wave antenna (Mühlschlegel, Eisler, Martin, Hecht and Pohl [2005]) and the bow-tie antenna (Grober, Schoelkopf and Prober [1997], Crozier, Sundaramurthy, Kino and Quate [2003], Schuck, Fromm, Sundaramurthy, Kino and Moerner [2005], Farahani, Pohl, Eisler and Hecht [2005]). However, by making use of electromagnetic resonances associated with surface plasmons and phonons any nano-

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structure can be viewed as an optical antenna. Of course, the efficiency depends on the material properties and the geometry of the nanostructure. As mentioned earlier, the association of near-field optical microscopy with antenna theory has already been made by Wessel (Wessel [1985]). However, the notion of ‘optical antennas’ can be traced back even earlier. For example, an antenna attached to a metal-to-metal point contact was used in 1968 by Ali Javan and co-workers for frequency mixing of infrared radiation (Hocker, Sokoloff, Daneu, Szoke and Javan [1968]). The rectification efficiency of these whisker diodes could be increased by suitably kinking or bending the wire antenna (Matarrese and Evenson [1970]). The length L (tip to kink) had to be adjusted in relation to the wavelength and angle of incidence, and the strongest response was obtained for the fundamental resonance of L ≈ λ/2.7 (see, for example, Kirschke and Rothammel [2001]). These experiments were performed at infrared wavelengths of λ = 10 . . . 337 µm where metals are good conductors. Whisker wires are routinely employed for delivering electromagnetic energy to miniature semiconductor circuits such as diodes or field-effect transistors. Similar frequency-mixing experiments have later been performed by Wolfgang Krieger et al. at the tunnel junction of an STM (Krieger, Suzuki, Völcker and Walther [1990]). In essence, an antenna efficiently converts the energy of free-propagating radiation to localized energy, and vice versa. For example, the antenna of a cell phone is used to concentrate the energy of incoming radiation onto a receiver chip with dimensions much smaller than the wavelength of the incoming radiation. In the context of microscopy, an optical antenna basically replaces a conventional focusing lens (objective), thereby concentrating external laser radiation to dimensions much smaller than the diffraction limit. The controlled and reproducible development of efficient optical antenna designs will provide new applications and opportunities not only in near-field optical microscopy but also in sensing applications and in optical device architectures. Due to reciprocity a good transmitting antenna is also a good receiving antenna. However, most radiowave or microwave antennas are employed only in one mode or the other. In near-field optical microscopy, on the other hand, the near-field probe can act as both a receiving antenna for localizing optical energy and a transmitting antenna for emitting the optical response. Classical antenna design assumes that there is no time lag between the conduction electrons and the exterior field. Consequently, electromagnetic penetration into the metal (skin effect) is negligible. At optical frequencies, on the other hand, the ‘inertia’ of conduction electrons causes electromagnetic fields to penetrate into the metals, and antenna design is no longer a pure function of the ‘exterior’ wavelength of the fields. Therefore, antenna designs cannot be directly downscaled from the

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microwave to the optical regime (cf. fig. 14) and a scaled (effective) wavelength has to be invoked instead (Novotny [2007]). Existing antenna concepts can provide the necessary inspiration for their optical counterparts. For example, the selfsimilar antenna proposed by Mark Stockman and co-workers (Li, Stockman and Bergman [2003]) has similarities with the well-established Yagi–Uda antenna developed in the 1920s in Japan and today widely used as receiving antenna in the UHF/VHF region. As an illustration, fig. 14 shows a calculation of the resonance of a λ/2 antenna. Because of surface plasmon resonances, the response at optical frequencies is very strong but the resonance is no longer at λ/2. Instead, the resonance wavelength is determined both by the external irradiation and the dielectric properties of the material. Pointed metal structures were used for the localization of electromagnetic energy already in the 18th century, long before the era of antenna design began. The most prominent invention making use of this property is Benjamin Franklin’s lightning rod (Krider [2006]). It can be viewed as a sharply pointed electrode with a strongly enhanced static field at its apex. This field originates from the potential difference between the tip’s support (ground) and a counter electrode (atmosphere). For strong enough fields the tip initiates a plasma channel (discharge) and conducts electrons into ground thereby lowering the initial potential difference. In a letter sent to Peter Collinson, a fellow of the Royal Society in London, on 25 May 1747, Benjamin Franklin writes “. . . In pursuing our Electrical Enquiries, we had observ’d some particular Phenomena, . . . ” and then adds “The first is the wonderful Effect of Points both in drawing off and throwing off the Electrical Fire” (Labaree et al. [1961a]). If the ‘Fire’ to which Franklin is referring were associated with ‘Radiation’ we could accept his phrase as the definition of an antenna, namely a device that attracts or sends off electrical radiation. The sketch shown in fig. 15 has been made by Franklin in a letter to Collinson, dated 29 July 1750 (Labaree et al. [1961b]). The contour defined by A, B, C, D, E represents an electrified body and the dashed lines around it indicate an “Atmosphere of Electrical Particles”, i.e., the electric field. Franklin argues that the Atmosphere to the right of L,C,M has the least surface to rest on and hence the attraction at the point C is weakest. As a consequence, the particles expand easiest at point C. He concludes “On these Accounts, we suppose Electrified Bodies discharge their Atmospheres upon unelectrified Bodies more easily, and at greater Distance, from their Angles and Points, than from their smoothe Sides.” Franklin adds “These Points will also discharge into the Air, when the Body has too great an electrical Atmosphere . . . ” In the same letter, Franklin speculates that the Fires of Electricity and Lightning are the same and he proposes the construction of grounded lightning rods to be fixed on buildings for the protection against lightning. He also

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Fig. 14. Adaptation of the λ/2 antenna to the optical regime. (a) Resonant field distribution near a 220 nm gold rod (λ = 1250 nm, thickness 20 nm). The curve under the figure depicts the charge distribution at an instant of time. (b) Spectral dependence of the intensity enhancement at the extremities of a 220 nm gold rod. (c) Resonant field distribution near a 220 nm rod made of an ideal conductor (λ = 520 nm, thickness 20 nm). (d) Spectral dependence of the intensity enhancement at the extremities of a 220 nm ideally conducting rod. The resonance deviates from λ/2 because of the finite thickness of the rod. (a,c) The contour lines are displayed with a logarithmic scaling (factor of 2 between successive lines).

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Fig. 15. Sketch from Benjamin Franklin’s letter sent to Peter Collinson on 29 July 1750. The contour defined by A,B,C,D,E represents an electrified body and the dashed lines around it indicate an “Atmosphere of Electrical Particles”. See text for details. From Labaree et al. [1961b].

writes “To determine the Question, Whether the Clouds that contain Lightning are electrified or not, I would propose an Experiment to be try’d where it may be done conveniently.” His following proposal inspired the famous ‘sentry-box’ experiment performed at Mary-la-Ville, France, in 1752 by Dalibard and Delors (Krider [2006]). It might appear to be a far stretch from Franklin’s lightning rod to near-field optical microscopy but, similar to an antenna or a near-field probe, the efficiency of a lightning rod is characterized by its ability to concentrate and localize electric fields (Denk and Pohl [1991]). However, despite this apparent similarity a lightning rod is not an antenna. The lightning rod is a static device whose properties are dictated by the Laplace equation. Consequently, the surface of the lightning rod is an equipotential surface and the geometrical singularity at the tip apex gives rise to a singular field, irrespective of the shape of the shaft or wire on which the tip is mounted. On the other hand, an antenna interacts with electromagnetic radiation and hence its behavior is dictated by the wave equation. Because of the finite penetration of optical radiation into metals the quasi-static approximation is only valid for structures smaller than the skin depth, but such small structures are not the most efficient antennas. Therefore, inspiration for efficient optical nearfield probes has to be drawn from antenna theory despite the fact that a direct downscaling of traditional antenna designs to optical wavelengths is not possible.

§ 11. Concluding remarks This article provides an overview of discoveries and achievements which were influential to the shaping of near-field optics. Important achievements cannot always be accredited to a single person or a single group because different developments

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progressed in parallel. I would like to emphasize that the history presented in this article cannot be complete because all resources are finite. I hope no important details have been left out. Any omissions are solely due to my personal ignorance. As is evident from this article the roots of near-field optics are not well defined, and the shaping of the field depended on many developments. The modern interest has its origins in the 1982 experiments by Pohl and co-workers and in parallel developments by other groups. Near-field optics received important inspiration from the early work in nanoplasmonics which was mainly aimed at answering open questions in SERS, and from studies of energy transfer. The prophetic papers and letters of Synge emerged at a later stage. From today’s perspective, near-field optics has a lot in common with classical antenna theory extended into the optical regime, and this is where it connects to the current second wave of nanoplasmonics. Interestingly, the first experimental developments in near-field optical microscopy were concentrated on the idea of using an aperture as a means to further confine a diffraction-limited focus, and this mindset delayed experimental work on antenna-inspired concepts. Today, Synge’s original particle (tip) idea finds applications in various studies and, despite Einstein’s initial skepticism, it is likely to be adopted in future commercial instruments.

Acknowledgements This article is the result of the valuable input and help of many friends and colleagues. I am very grateful to Barbara Schirmer who patiently helped to research and translate the resources forming the basis of this article, and I like to thank my parents in law, Grazia and Hans Schicht, for ‘full board’ during the writing period in Switzerland. I appreciate the input and advice from various friends and colleagues, particularly Niek F. van Hulst, Fritz Keilmann, Ulrich Ch. Fischer, Jean-Jacques Greffet, Alexandre Bouhelier, Ole Keller, and Bert Hecht. I would like to acknowledge the Einstein Archives in Jerusalem, Israel, for making the Einstein–Synge letters available and for the permission to cite them. I am grateful for financial support by the Department of Energy (grant DE-FG02-01ER15204) and the Air Force Office of Scientific Research (grant F-49620-03-1-0379).

References Abbe, E., 1873, Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung, Archiv f. Miroskop. Anat. 9, 413.

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Chapter 6

Light tunneling by

H.M. Nussenzveig Instituto de Física, Universidade Federal do Rio de Janeiro Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(07)50006-5 185

Contents

Page § 1. Introduction: Newton and contemporaries . . . . . . . . . . . . . . .

187

§ 2. Classical diffraction theory . . . . . . . . . . . . . . . . . . . . . . .

195

§ 3. The optomechanical analogy . . . . . . . . . . . . . . . . . . . . . .

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§ 4. Modern developments in diffraction theory . . . . . . . . . . . . . .

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§ 5. Exactly soluble models . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 6. Watson’s transformation . . . . . . . . . . . . . . . . . . . . . . . .

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§ 7. CAM theory of Mie scattering . . . . . . . . . . . . . . . . . . . . .

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§ 8. Impenetrable sphere . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 9. Near-critical scattering . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 10. The rainbow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 11. Mie resonances and ripple fluctuations . . . . . . . . . . . . . . . .

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§ 12. Light tunneling in clouds . . . . . . . . . . . . . . . . . . . . . . . .

227

§ 13. The glory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 14. Further applications and conclusions . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction: Newton and contemporaries The concept of tunneling was introduced in the early days of quantum mechanics (Merzbacher [2002]). It was first applied by Hund [1927], to account for hopping between wells in a double-well molecular bound state. For continuum states, the earliest application is Nordheim’s [1927] treatment of thermionic emission and electron reflection by metals, modeled as electron penetration of rectangular potential barriers. Pioneering applications to nuclear physics were independently proposed by Gamow [1928] and by Gurney and Condon [1928] to explain alpha radioactivity in terms of Coulomb barrier penetration. The huge range of observed lifetimes follows from the exponential sensitivity of the barrier penetration factor to the energy of the emitted alpha particle. An early application to condensed matter physics was Fowler and Nordheim’s [1928] theory of field emission (Eckart [1930]). For a uniform applied electric field, this amounts to penetration of a triangular potential barrier. The name “wave-mechanical tunnel effect” was first employed by Schottky [1931] in connection with photoemission from a metal–semiconductor barrier layer. In all these instances, “tunneling” refers to the transmission through a potential barrier of a particle with insufficient energy to surmount it according to classical mechanics. It is a consequence of the wave–particle duality, one of the wave properties of matter. There is a well-known wave-mechanical analogy (Gamow and Critchfield [1949]) with the phenomenon of frustrated total reflection in optics, also treated as light tunneling (Sommerfeld [1954]). As will be seen below, light tunneling and barrier penetration are closely related. The earliest known observation of frustrated total reflection, reported in Query 29 of his “Opticks”, is due to Newton [1952], who may therefore be regarded as the discoverer of tunneling (cf. Hall [1902]). It is mentioned in his letter to Oldenburg of 9 December 1675 (Newton [1757]). He was observing the phenomenon known as Newton’s rings. As described in Query 29, this was done “. . . by laying together two Prisms of Glass, or two Object-glasses of very long Telescopes, the one plane, the other a little convex, and so compressing them that they do not fully touch, nor are too far asunder.” 187

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He had observed that “The Rays of Light in going out of Glass into a Vacuum, are bent towards the Glass, and if they fall too obliquely on the Vacuum, they are bent backwards into the Glass, and totally reflected”; however, by compressing them as he described, “. . . the Light which falls upon the farther Surface of the first Glass where the Interval between the Glasses is not above the ten hundred thousandth Part of an Inch, will go through that Surface, and through the Air or Vacuum between the Glasses, and enter into the second Glass”. Newton was a remarkable experimenter: he made very accurate measurements of the radii of the rings, and thereby determined the corresponding thicknesses of the “Interval between the Glasses”. How did he interpret these results, given that the visual appearance of the rings immediately suggested periodicity and waves? Newton’s views on the nature of light were as ambivalent as that nature itself. One of the most remarkable expositions of those views is his “Hypothesis explaining the Properties of Light” in the letter to Oldenburg (Newton [1757]). The dispute (Hall [1995]) following the publication of his first paper, “New Theory about Light and Colors” (Newton [1672]) rendered him extremely averse to publishing his conjectures (“Hypotheses non fingo”), so that his design in the Opticks “. . . is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments” (Newton [1952]). “Were I to assume an hypothesis”, Newton writes to Oldenburg, it should be that light “is something or other capable of exciting vibrations in the aether”. These vibrations “succeed one another . . . at a less distance than the hundred thousandth part of an inch” – a correct estimate for half a wavelength of visible light. Newton assumes that the aether density varies in different media (playing a role similar to the refractive index), with no discontinuity at an interface between two media, which he views as a continuous variation through a transition layer “of some depth”. He proceeds: “. . . refraction I conceive to proceed from the continual incurvation of the ray all the while it is passing” through this layer. However, beyond the critical angle, the ray “must turn back and be reflected”. How does this take place? To model total reflection, in the “Principia” (Newton [1946]) he invokes the corpuscular model of light, though not committing himself to this model. In the Scholium to Proposition XCVI of Book I, he states: “Therefore because of the analogy there is between the propagation of the rays of light and the motion of bodies, I thought it not amiss to add the following Propositions for optical uses; not at all considering the nature of the rays of light,

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Fig. 1. Newton’s model for total reflection of a light ray (Newton [1946], Book I, Proposition XCVI). The reflected ray is laterally displaced.

or inquiring whether they are bodies or not; but only determining the curves of bodies which are extremely like the curves of rays.” This statement is one of the earliest formulations of the optomechanical analogy, which plays a crucial role in the history of light tunneling. There are prior hints of this analogy in the work of the great medieval scholar known as Alhazen (Ronchi [1952]), and Descartes had derived the laws of reflection and refraction by comparison with the trajectory of a ball (Descartes [1637]): “. . .l’action de la lumière suit en ceci les mêmes lois que le mouvement de cette balle”. Newton’s model for the transition layer is mechanically equivalent to interpolation by a linear potential field, like gravity near the earth, so that the light ray describes an arc of a parabola (fig. 1), implying, by the way, that the reflected ray also undergoes a lateral displacement – an early version of the Goos–Hänchen shift (Goos and Hänchen [1947]) which, as we will discuss later, is another manifestation of light tunneling. A tough challenge to the corpuscular model was how to explain that light is partially reflected and partially transmitted at an interface. How does a light particle choose which path to take? To deal with this conundrum, Newton formulated the first dual model for light, endowing it with both particle and wave properties (Newton [1952], Book Two, Proposition XII): “Every Ray of Light in its passage through any refracting Surface is put into a certain transient Constitution or State, which in the progress of the Ray returns at equal Intervals, and disposes the Ray at every return to be easily transmitted through the next refracting Surface, and between the returns to be easily reflected by it. D EFINITION. The return of the disposition of any Ray to be reflected I will call its Fit of Easy Reflexion, and those of its disposition to be transmitted its Fits of easy Transmission, and the space it passes between every return and the next return, the Interval of its Fits.”

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Fig. 2. Newton’s explanation of the rainbow (Newton [1952], Book I, fig. 15).

In Proposition XIII, Newton states that “. . . light is in Fits of easy Reflexion and easy Transmission, before its Incidence on Transparent Bodies . . . For these Fits are of a lasting nature” – an explicit assertion of the dual nature of light. Newton’s value (“ 89 1000 of an Inch”) for the “Interval of Fits” (half-wavelength) of light “in the Confine of yellow and orange”, derived from his measurements of Newton’s rings, is quite acceptable by present standards. As an application of his theory of dispersion, Newton explained the colors of the rainbow (for contributions from his predecessors, see Boyer [1987]). His diagram (fig. 2) shows that the primary bow is formed by rays undergoing a single internal reflection within water droplets, and the secondary bow is formed by rays that undergo two internal reflections. The dark band between primary and secondary bows was first reported by the Greek philosopher Alexander of Aphrodisias circa 200 AD. Newton computed the angular width of the rainbow arcs, taking into account the angular diameter of the sun, and verified the results by his own measurements. In Book III of his “Opticks”, Newton discusses the diffraction of light (that he calls “Inflexion”). The discoverer of diffraction (who also gave it that name) was the Jesuit father Francesco Maria Grimaldi, who described his observations in a posthumously published book (Grimaldi [1665]). In one of them (top illustration in fig. 3(a)) he had a small opaque body lit by a pencil of sun rays and noticed the

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(a) Fig. 3. (a) Grimaldi’s experiments on diffraction (Grimaldi [1665]). (b) Title-page of his book and reproduction of Proposition I (next page).

presence of three parallel fringes beyond the penumbra (regions CM and ND). In the other one (bottom illustration in fig. 3(a)), a cone of light going through two diaphragms spreads beyond the domain ON predicted by geometrical optics. His Proposition I (fig. 3(b), bottom) states that “Light propagates and spreads not only directly, through refraction, and reflection, but also by a fourth mode, diffraction”. Newton referred to Grimaldi’s experiments at the beginning of Book III and then reported his own observations of the shadows of hairs, pins, knife-edges and other very thin or sharp objects. The high accuracy of these experiments has been verified by recent tests (Nauenberg [2000]). Fresnel blamed Newton for not reporting diffraction fringes inside the shadow, but only those outside, as stated in “Opticks”, Query 28:

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(b) Fig. 3. (Continued.)

“The Rays which pass very near to the edges of any Body, are bent a little by the action of the Body, as we shew’d above; but this bending is not towards but from the Shadow.” However, by reproducing Newton’s experimental conditions, it has been verified (Stuewer [2006]) that the intensity of the diffraction lines inside the geometrical shadow was far too low for Newton to have observed them visually. What about the explanation of diffraction? As Newton states in Book III, “I was then interrupted, and cannot now think of taking these things into farther Consideration. And since I have not finish’d this part of my Design, I shall con-

6, § 1]


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clude, with proposing only some Queries in order to a farther search to be made by others: Query 1. Do not Bodies act upon Light at a distance, and by their action bend its Rays, and is not this action (cæteris paribus) strongest at the least distance? Qu. 2. Do not the Rays which differ in Refrangibility differ also in Flexibility, and are they not by their different Inflexions separated from one another, so as after separation to make the Colours in the three Fringes above described? And after what manner are they inflected to make those Fringes? Qu. 3. Are not the Rays of Light in passing by the edges and sides of Bodies, bent several times backwards and forwards, with a motion like that of an Eel? And do not the three Fringes of colour’d Light above mention’d, arise from three such bendings?” An illustration of Newton’s proposed explanation appears in the abovementioned Scholium in the “Principia” (fig. 4), representing diffraction by a knife-edge. According to Query 1, light rays passing outside the edge are bent into the shadow, the amount of their “Inflexion” increasing as they come closer to the edge. In Query 2, Newton proposes that “Inflexion”, like refractive index, depends on colour. As will be seen below, contemporary diffraction theory provides qualitative support for all three of these queries. Newton also conjectured that an increasing optical density of the “Aetherial Medium” might be responsible for the postulated action at a distance: “Qu. 20. . . . And doth not the gradual condensation of this Medium extend to some distance from the Bodies, and thereby cause the Inflexions of the Rays

Fig. 4. Newton’s interpretation of diffraction by a knife edge (Newton [1946], Book I, Proposition XCVI, Scholium). He states that “. . . the rays which fall upon the knife are first inflected in the air before they touch the knife”.

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of Light, which pass by the edges of dense Bodies, at some distance from the Bodies?” In 1657, Pierre de Fermat formulated his Principle of Least Time (Fermat [1891]): “ The path really followed by light from a point A to a point B corresponds to the shortest possible time”, from which he derived the law of refraction (for reflection, this had already been done by Heron of Alexandria). Taking into account the inverse relation between velocity of light and refractive index n, and that travel time in general is stationary, not necessarily a minimum (Born and Wolf [1999]), this amounts to stationarity of the optical path between A and B, B n ds = 0,

δ

(1.1)

A

where ds is the element of arc length along the path. Newton’s contemporary Christian Huygens explicitly proposed a wave theory of light (Huygens [1690]). This was actually a theory of the propagation of light pulses, excluding any idea of periodicity. Employing a billiard-ball model of the ether as scaffolding, he asserts that, in propagation, each ether particle collides with all those that surround it, so that “. . . around each particle there is made a wave of which that particle is the centre”. This leads him to the celebrated “Huygens’ Principle”, illustrated in fig. 5, taken from his treatise. Each point of a pulse front, according to Huygens, gives rise to secondary spherical pulses, the envelope of which at a given time is the propagated pulse. Secondary pulses are only effective at their point of contact with the envelope, which explains rectilinear propagation (fig. 5): “For if, for example, there were an opening BG, limited by opaque bodies BH, GI, the wave of light which issues from the point A will always be terminated by the straight lines AC, AE, as has just been shown; the parts of the partial waves which spread outside the space ACE being too feeble to produce light there.” In contrast with Huygens’ pulse theory of light, a theory of light based on oscillatory wave propagation in an elastic ether was apparently first proposed by Leonhard Euler (Euler [1746]), who emphasized the analogy with sound propagation and employed it as well to account for diffraction. He was also responsible for a fundamental contribution to the mathematical description of wave tunneling, Euler’s formula (Euler [1748]), eix = cos x + i sin x, which connects the exponential and trigonometric functions.

(1.2)

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Fig. 5. Propagation of a spherical light pulse according to Huygens’ Principle (Huygens [1690], fig. 6).

Its special case eiπ + 1 = 0

(1.3)

has been called “the most beautiful formula of mathematics”, as it combines some of the most basic elements and operations of arithmetic, algebra, geometry and analysis.

§ 2. Classical diffraction theory Light propagation in terms of oscillatory waves, pioneered by Euler, was taken up by Thomas Young, who formulated the associated basic principle of interference. He states it explicitly (Young [1802]): “Radiant Light consists in Undulations of the luminiferous Ether”. . .“When two Undulations, from different origins, coincide either perfectly or very nearly in Direction, their joint effect is a Combination of the Motions belonging to each”. Young made several applications of his principle. Employing Newton’s measurements of Newton’s rings, he derived very accurate values for the wavelengths associated with different colors. He also applied it (Young [1804]) to explain the supernumerary rainbow arcs that appear just below the primary bow when relatively small and uniformly sized water droplets are present (Lee and Fraser

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[2001]). However, as is discussed below, Young’s interference theory of the rainbow must be amended to account for diffraction effects. The celebrated two-slit interference experiment, illustrated with water waves, appears in fig. 6, taken from Young [1807]. An excellent modern survey of Young’s contributions is given by Berry [2002]. Young’s theory of diffraction differed from Newton’s. He interpreted Newton’s diffraction experiments with knife-edges in terms of “interference of the light reflected from the edges of the knives” (Young [1804]), so that in his view the diffracted light arises from sharp edges and object boundaries. Incident rays that just graze the edge of an aperture or obstacle would undergo “a kind of reflection”. The resulting boundary diffracted wave would penetrate into the geometrical shadow and would interfere with incident and geometrically reflected waves in the illuminated region, accounting for the diffraction fringes observed. The classical wave theory of light diffraction was formulated by Augustin-Jean Fresnel (Fresnel [1816]), who combined Huygens’ principle with Young’s principle of interference. According to Fresnel, “. . . elementary waves arise at every point along the arc of the wave front passing the diffracter and mutually interfere. The problem was to determine the resultant vibration produced by all the wavelets reaching any point behind the diffracter.” Thus, what in Huygens’ formulation was just a geometrical construction for wavefronts becomes a dynamical principle for wave propagation. Figure 7 illustrates the application of Fresnel’s idea to diffraction of a monochromatic plane wave by an opaque half-plane. Each point of the unblocked portion of the incident plane wavefront becomes the source of a spherical secondary wave, as in Huygens’ principle, giving rise to the geometrical shadow boundary (thin vertical line in fig. 7). However, while Huygens just applied his envelope construction to get a transmitted cut-off pulse front (envelope), Fresnel lets all spherical wavelets interfere. Their resultant yields the diffracted wave, which propagates into the shadow and produces diffraction fringes around the shadow boundary. We will call Fresnel’s explanation “diffraction as blocking”. Fresnel submitted his proposal (Fresnel [1816]) to the French Academy of Sciences as an entry in the competition for the Grand Prize offered for the explanation of diffraction. Most members of the award committee, chaired by Arago, favored the corpuscular theory, but the prize was awarded to Fresnel following a remarkable experimental confirmation, as reported by Arago: “One of your commissioners, M. Poisson, had deduced from the integrals reported by [Fresnel] the singular result that the centre of the shadow of an opaque

6, § 2] Classical diffraction theory Fig. 6. Young’s illustration of two-slit interference in terms of the pattern “obtained by throwing two stones of equal size into a pond at the same instant” (Young [1807]). He mentions the analogy with acoustics and optics. 197

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Fig. 7. According to Fresnel, diffraction by a half-plane arises from blocking of the incident plane wavefront. Secondary spherical waves propagate and interfere beyond the shadow boundary (thin vertical line).

circular screen must, when the rays penetrate there at incidences which are only a little more oblique, be just as illuminated as if the screen did not exist. The consequence has been submitted to the test of direct experiment, and observation has perfectly confirmed the calculation.” This bright spot at the center of the shadow of a circular disc became known as Poisson spot (a prior observation in 1723 by Giacomo Maraldi remained unrecognized). A more precise mathematical formulation of the Huygens–Fresnel principle followed from the work of Gustav Kirchhoff (Kirchhoff [1882]) on scalar monochromatic wave propagation. Kirchhoff derived an exact integral representation of the wave function in terms of its boundary values over a surface (Born and Wolf [1999], Baker and Copson [1950]). For an aperture in an opaque plane screen, this surface may be taken as the plane of the screen. However, the exact boundary values are unknown. The classical Fresnel–Kirchhoff theory of diffraction is based upon Kirchhoff’s approximation, a perturbative assumption on the unknown boundary values, expected to hold when all relevant dimensions of the obstacle (aperture) as well as the wavefront are much larger than the wavelength. It is assumed that one can re-

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place the unknown boundary values by the incident wave over unobstructed parts of the wavefront, and that one can take them to vanish over obstructed (shadow) parts – as would be prescribed by geometrical optics. An important application of classical diffraction theory was Airy’s treatment of the rainbow problem (Airy [1838]). The primary rainbow direction is an extremal deflection angle, i.e., a caustic direction, for rays incident on a water droplet that undergo just one internal reflection (cf. Nussenzveig [1977, 1992]). Smaller scattering angles represent a shadow region for this class of rays (Alexander’s dark band); at larger angles, two such rays with different impact parameters interfere, leading to Young’s supernumerary arcs. The title of Airy’s paper is “On the intensity of light in the neighbourhood of a caustic”. Airy applied Huygens–Fresnel theory to an S-shaped cubic wavefront within a droplet, approximating the unknown wave amplitude along it by a constant. The result was his “rainbow integral”, now known as the Airy function Ai(z), which satisfies the differential equation Ai (z) = zAi(z).

(2.1)

Figure 8 is a plot of Ai(x). For negative x, it has a slowly damped oscillatory behavior, with peaks related to the supernumerary arcs; for positive x, it undergoes faster than exponential damping, associated with penetration by diffraction on the shadow side of the rainbow.

Fig. 8. Plot of Ai(x) for −15 x 5.

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For diffraction by a (large) aperture, classical diffraction theory turns out to yield acceptable approximations. Why is that so? For scalar diffraction by an aperture in a plane screen, the diffraction amplitude (asymptotic amplitude of the outgoing spherical wave) in the direction of unit vector sˆ is given by Rayleigh’s exact formula (Bouwkamp [1954]) sˆ · sˆ0 f sˆ, sˆ0 = (2.2) exp −ik sˆ − sˆ0 · x u(x) d2 x, iλ where sˆ0 is the direction of the incident wave, λ = 2π/k is the wavelength, the integral is extended over the plane of the screen, and u(x) is the exact (unknown) wave function on this plane. The short-wavelength assumption implies that dominant contributions to the amplitude arise from the near-forward domain, in which k sˆ − sˆ0 · x kd sin θ ∼ k⊥ d ≡ 2πd/λ⊥ 1, (2.3) where θ is the diffraction angle, d is the diameter of the aperture and λ⊥ is the transverse wavelength. Thus, the main diffraction pattern depends only on spatial Fourier components of u(x) on the aperture plane with λ⊥ d, which are insensitive to fine details of the aperture boundary values: they feel mainly the gross blocking effect captured by Kirchhoff’s approximation (Nussenzveig [1959]). Finer details affect large diffraction angles, where the intensity is weaker. Equivalently, one may extend the two-dimensional Fourier expansion of the exact wave function over the aperture plane (x, y) to the half-space z > 0 beyond it by the wave equation propagation factor exp(ikz z) (kx2 + ky2 + kz2 = k 2 ), but this introduces evanescent waves, with kz = i kx2 + ky2 − k 2 , that decay exponentially with z. In this angular spectrum of plane waves (Nussenzveig [1959], Born and Wolf [1999]), finer details of the aperture distribution give rise to evanescent waves (for applications to microscopy, see Section 14.1.6). This already gives us a foretaste of light tunneling effects in diffraction. Young’s interpretation of diffraction is actually consistent with classical diffraction theory, in spite of the apparent differences. Indeed, the Fresnel–Kirchhoff representation of the diffracted wave as a surface integral over the unobstructed part of a wavefront can be converted into a line integral over the edge and interpreted in terms of Young’s boundary diffraction wave (Rubinowicz [1917], Sommerfeld [1954]). For other important contributions to the wave theory of light in the 19th century, culminating in Maxwell’s electromagnetic theory, we refer to Born and Wolf [1999].

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§ 3. The optomechanical analogy Still in the first half of the 19th century, an important conceptual reformulation of optics was the development of the Hamiltonian analogy between geometrical optics and classical mechanics (Hamilton [1828]). The basis for Hamilton’s reformulation is the optical path function between two points along a ray, that appears in (1.1), B [AB] =

n ds ≡ S(B) − S(A).

(3.1)

A

As stated by Hamilton, “The mathematical novelty of my method consists in considering this quantity as a function of the co-ordinates of these extremities, which varies when they vary, according to a law which I have called the law of varying action; and in reducing all researches respecting optical systems of rays to the study of this single function.” The function [AB] is Hamilton’s point characteristic, and S is the eikonal function, which satisfies the relation ˆ ∇S = nu,

(3.2)

where uˆ is the ray direction (unit tangent vector). From (3.2) follows the eikonal equation (∇S)2 = n2 .

(3.3)

If one now considers the path of a point particle with mass m and energy E in a potential field V (r), one finds that it can be obtained from the Hamilton–Jacobi equation (Goldstein [1957]) (∇W )2 = 2m E − V (r) , (3.4) provided that the initial position and the initial direction are specified, the same conditions required for (3.3). Remembering that the refractive index is dimensionless, the analogy leads to V (r) n(r) = 1 − (3.5) , E which is a real quantity in the domain accessible to the motion, V (r) < E. Thus, the paths of the above-considered particle according to classical mechanics are identical to geometrical optic light rays in an inhomogeneous medium with refractive index (3.5).

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Fermat’s principle (1.1) is the analogue of Maupertuis’ principle of least action. Further discussion of variational principles and of the application to electron optics can be found in Born and Wolf [1999]. The optomechanical analogy played an important role in the formulation of quantum mechanics. In one of his earliest communications on wave properties of matter, Louis de Broglie states (De Broglie [1923]): “The new dynamics of the free point particle is to the old dynamics . . . as wave optics is to geometrical optics”. Erwin Schrödinger made the same point in his second paper on wave mechanics (Schrödinger [1926]): “. . .our classical mechanics is the complete analogy of geometrical optics. . . Then it becomes a question of searching for an undulatory mechanics. . . working out. . . the Hamiltonian analogy on the lines of undulatory optics.” He stated it succinctly as “Ordinary mechanics : Wave mechanics = Geometrical optics : Undulatory optics”. The analogy can be employed in the reverse direction, by associating an effective potential field with a refractive index distribution. Light tunneling is thereby identified as the analogue of quantum tunneling through a potential barrier (Section 1).

§ 4. Modern developments in diffraction theory 4.1. The geometrical theory of diffraction A beautiful heuristic extension of geometrical optics for treating diffraction problems, known as the geometrical theory of diffraction, was proposed by Keller [1962]. It is based upon the following main postulates: (i) The diffracted field propagates along diffracted rays, that generalize the concepts of reflected and refracted rays, as well as Young’s ideas on edge diffraction. They are determined by extending Fermat’s Principle (1.1) of the stationary optical path, allowing the paths to include points or arcs on discontinuous boundaries, such as corners, edges or vertices. For diffraction by a smooth obstacle, an example is illustrated in fig. 9: an incident ray PD, tangential to the surface at D, travels along a geodetic arc DT on the surface and leaves it again tangentially at T, propagating along a straight line to Q, a point in the shadow region. (ii) The transport of amplitude and phase along a diffracted ray obeys the laws of geometrical optics. (iii) The analogue of a reflection coefficient, relating the initial field on a diffracted ray to the incident field, is a diffraction coefficient D, that is supposed to

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Fig. 9. The tangentially incident ray PD excites the diffracted ray DT, that leaves the surface tangentially at T to reach a point Q in the shadow.

be determined by the local geometry and nature of the boundary around the point of diffraction D. An account of the history of the geometrical theory of diffraction is given by Keller [1979, 1985], who also discusses the shortcomings of his theory. In common with geometrical optics, it leads to singularities at focal points and caustics, including the surface of a smooth obstacle, which is a caustic of diffracted rays. As a consequence, the diffraction coefficients cannot be determined self-consistently within the theory. To determine them, in accordance with (iii), the theory makes use of comparisons with the asymptotic short-wavelength behavior of solutions to canonical problems with locally similar geometry, such as diffraction by a sphere or cylinder. A survey of applications is given by Borovikov and Kinber [1994]. Transitions between domains covered by different numbers of rays are also not treated. This includes, in particular, the penumbra region around the light/shadow boundary.

4.2. Fock’s theory of diffraction Fock’s treatment of the penumbra region, originating from his work on the propagation of radio waves around the Earth’s surface (Fock [1965]) employs the conceptual picture of diffraction as transverse diffusion, developed by Leontovich and Fock [1946]. To explain this concept we note that, in geometrical optics, amplitude and phase are transported longitudinally along the rays, with no constraints in transverse directions (Sommerfeld [1954]), so that discontinuities at shadow boundaries are allowed. Diffraction is associated with smoothing of these discontinuities by transverse diffusion. To see how this arises, consider monochromatic wave propagation along the z-direction and substitute the ‘Ansatz’

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u(x, y, z) = A(x, y, z) exp(ikz)

[6, § 4

(4.1)

into the Helmholtz monochromatic wave equation. One finds

∂A i ∂ 2A ∂ 2A i ∂ 2A (4.2) = + − , ∂z 2k ∂z2 2k ∂x 2 ∂y 2 where the second term on the left may be neglected for slow amplitude variation per wavelength. Traveling along with a wavefront, z = ct, we get

2 ∂ A ∂ 2A ic ∂A + =D , , D≡ (4.3) 2 2 ∂t 2k ∂x ∂y which is Leontovich and Fock’s “diffusion equation” in this simplest situation. However, the fact that D is imaginary reveals that the proper analogy is with Schrödinger’s equation, rather than the diffusion equation. Physically, what happens along the traveling wavefront is akin to the oscillatory spreading behavior of a free Schrödinger wave packet, rather than to stochastic diffusion. In view of the analogy between physical optics and wave mechanics discussed in Section 3, this is hardly surprising. As will be seen below, the Schrödinger analogy is indeed the correct one. In Fock’s theory of diffraction by a smooth body, in order to treat the penumbra, the body surface near the glancing ray is approximated by a parabolic surface, and an ‘Ansatz’ generalizing the above one is substituted into the Helmholtz equation in suitable coordinates, describing transverse behavior in terms of the “diffusion” analogy. An integral representation for the solution to this local approximation leads to a new set of special functions known as Fock functions, related to complex Fourier transforms of the inverse of the Airy function. They are taken to represent the behavior in the penumbra region, interpolating between the illuminated and shadow domains. An exposition of Fock’s theory and its applications is given by Babich and Kirpichnikova [1979]. However, as is discussed below, Fock’s theory yields only a transitional asymptotic approximation, that cannot be extended very far on either side of the geometrical light/shadow boundary. What is needed is a uniform asymptotic approximation. A very interesting independent treatment of the penumbra, equivalent to Fock’s, was formulated by Pekeris [1947], in connection with microwave propagation around the Earth’s horizon. He introduced an “Earth flattening transformation”, representing the Earth’s surface as flat, and simulating the curvature of the rays by an “atmosphere” in which the refractive index increases linearly with height above the surface. In view of (2.1), the connection with the Airy functions arises from this assumed linear dependence.

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§ 5. Exactly soluble models 5.1. Exact solutions The earliest exact solution of a diffraction problem was Sommerfeld’s celebrated treatment of diffraction by a half-plane (Sommerfeld [1896, 1954]). Though physically unrealistic, because the half-plane is assumed to be infinitely thin yet opaque, it represents diffraction, for the first time, through analytic continuation. The solution is a uniform analytic function over a two-sheeted Riemann surface, with the diffracted wave penetrating into the second Riemann sheet. The diffracted field behaves asymptotically like a cylindrical wave emanating from the edge, consistent with Young’s picture of boundary diffraction. It is worth noting that Newton’s Query 3 (Section 1) is also vindicated by plotting the energy flow close to the edge (Braunbek and Laukien [1952], Berry [2002]): there are indeed eel-like undulations. Analytic continuation is also important in the solution of the integral-equation formulation of other diffraction problems by the Wiener–Hopf technique (Noble [1958]) as well as in more general scattering problems (Nussenzveig [1972], Wolf and Nieto-Vesperinas [1985]). For a small number of separable geometries, exact solutions in the form of infinite series of eigenfunctions have been found (Bouwkamp [1954], Hönl, Maue and Westphal [1961]). However, as pointed out by Sommerfeld [1954], “. . .a mathematical difficulty develops which quite generally is a drawback of this ‘method of series development’: for fairly large particles (ka > 1, a = radius, k = 2π/λ) the series converge so slowly that they become practically useless”. The reason for this is discussed below.

5.2. Mie scattering The keystone to a deeper understanding of the dynamics of diffraction, to which the rest of this work is mainly devoted, is a reformulation of a problem that has an exact solution in terms of an infinite series of eigenfunctions, allied to the unique feature of being physically realistic: the scattering of light by spherical particles. A historical survey of the scattering of plane waves by a sphere has been prepared by Logan [1965]. An early treatment was that of Clebsch [1863], who dealt with elastic waves incident on a rigid sphere. Lord Rayleigh [1872] found the exact series solution for the scattering of sound by a sphere. The corresponding solution for scattering of a plane electromagnetic wave by a transparent sphere was given by Lorenz [1890] (cf. Keller [2002]) and rediscov-

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ered by Mie [1908], who was concerned with explaining the colors of colloidal suspensions of metallic particles. It was also treated by Debye [1909a] in his doctoral thesis about the radiation pressure of an electromagnetic wave on a metallic sphere. Historically, referring to Lorenz–Mie scattering is better justified, but we stick with the more usual designation Mie scattering. The Mie solution is a partial-wave series Sj (β, θ ) =

∞ 1 (j ) (i) 1 − Sl (β) tl (θ ) + 1 − Sl (β) pl (θ ) 2 l=1

(j = 1, 2; i = j ),

(5.1)

where Sj are the polarized scattering amplitudes in the direction θ for perpendicular (j = 1) and parallel (j = 2) polarizations (with respect to the scattering plane), β ≡ ka is the size parameter, Sl(i) (β) are the S-matrix elements for magnetic (i = 1) and electric (i = 2) multipoles of order l, and tl and pl are angular Legendre-type functions. The S-matrix elements are rational combinations of spherical Bessel and Hankel functions and their derivatives (for their expressions and the notation, see Nussenzveig [1992]).

5.3. The localization principle In quantum mechanics, the index l in the partial-wave series also labels eigenvalues of the orbital angular momentum, that are given by

1 1 2 1 (5.2) l(l + 1)h¯ = − h¯ ≈ l + h¯ for l 1. l+ 2 4 2 In semiclassical (short-wavelength) scattering (Berry and Mount [1972]), large l prevails, and the above orbital angular momentum can be associated with a corresponding impact parameter bl by bl =

(l + 12 )h¯ (l + 12 ) = , p k

(5.3)

where p = h¯ k is the linear momentum. This association between a partial-wave term and the impact parameter of an incident path, already implicit in Debye’s thesis, was emphasized by Van De Hulst [1957] (cf. also Roll, Kaiser, Lange and Schweiger [1998]), and it is known as the localization principle. A closely related wave-mechanical picture is a description of the incident beam on a transverse plane in terms of annular rings centered on the scatterer, with areas h¯ consistent with the uncertainty relation, bl being the

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average radius of the ring between the values l and l + 1 (Blatt and Weisskopf [1952]). In geometrical optics, only incident rays with impact parameters a are scattered by the sphere, so that the partial-wave series (5.1) may be cut off at a maximum value of l given by lmax ka = β. Partial waves with l > lmax are associated with above-edge incident rays. For solar radiation and cloud water and aerosol droplets, β reaches ∼10 000, and numerical convergence of the Mie series does not set in until values actually exceeding the above estimate for the cut off are reached, because of above-edge effects to be discussed below. This justifies Sommerfeld’s comment on the convergence problem.

§ 6. Watson’s transformation An important historical incentive for trying to overcome the convergence problem of the Mie series was the treatment of radiowave propagation around the Earth, beyond the horizon. As noted by Love [1915], the value of β in this situation is of order 104 . During the first two decades of the 20th century, rival proposals to account for the propagation were the diffraction theories and those invoking ionosphere reflection (for a detailed account, see Yeang [2003]). Diffraction theories modeled the Earth as a perfectly conducting sphere. Henri Poincaré worked on the diffraction model from 1909 until his death in 1912, treating it in nine papers and a monograph (Poincaré [1910]). His basic idea was to convert the partial-wave series into a contour integral using the residue theorem and to obtain an asymptotic approximation of this integral. An improved version of Poincaré’s approach was developed by Watson [1918]. His transformation of the partial-wave series is equivalent to the formula ∞

1 exp(−iπλ) 1 φ l + ,x = φ(λ, x) dλ, 2 2 cos(πλ) l=0

(6.1)

C

where C is a contour encircling the positive real axis and φ(λ, x) is a holomorphic function of λ within C that takes on the values φ(l + 12 , x) at the half-integers. For the Mie series, analyticity of φ(λ, x) follows from the properties of cylindrical and Legendre functions. Watson deformed the path C onto a symmetric neighbourhood of the imaginary λ axis, over which the integral vanishes, because the integrand is odd. The deformation is allowed in the shadow region. In this process, the only singularities met are poles (the integrand is meromorphic), so that the transformation results

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in a residue series over the poles. The residues represent surface waves, traveling around the sphere and getting damped by tangential shedding of radiation, like Keller’s diffracted rays in fig. 9. The angular damping is associated with the imaginary part of the poles. Watson found that this damping was too large to account for radiowave propagation into the shadow, and in a subsequent paper (Watson [1919]) he extended his treatment to include reflection from the ionosphere, modeled as a concentric spherical surface. He showed that the results were consistent with experimental observations. Watson was concerned with radiation from a dipole antenna on the Earth’s surface. The extension of his method to the scattering of a plane electromagnetic wave by a perfectly conducting sphere was undertaken by White [1922]. Watson’s original path deformation does not converge in the illuminated region. White showed that, in this region, the contour can be deformed to yield, besides the residue series, a path through a saddle point. For the asymptotic evaluation of the remaining integral, he used the method of steepest descents, that had been introduced by Debye [1909b] to deal with the analogue of the Mie series in scattering by a circular cylinder (Debye [1908]). The saddle-point contribution yields the Wentzel–Kramers–Brillouin (WKB) approximation to the reflected wave, previously obtained by Nicholson [1910]. The WKB approximation is often improperly referred to as “geometrical optics”: strictly speaking, geometrical optics describes only the propagation of intensity, not phase. However, it is customary to refer to the WKB dominant term as geometrical optics (or ray optics) approximation. For the development of the WKB method, see Berry and Mount [1972] and Keller [1979]. The application to electromagnetic theory was systematized in 1944 by Luneburg (Luneburg [1964]). The application of Watson’s transformation to radio wave propagation was further elaborated by Van Der Pol and Bremmer [1937], and Bremmer [1949]. Watson’s surface waves, referred to as “creeping waves”, as well as White’s extension to the illuminated region, were rederived by Franz and Depperman [1952], and also extended to transparent spheres (Beckmann and Franz [1957], Franz [1957]). Applications of complex angular momentum to quantum potential scattering and high-energy physics followed from the work of Regge (1959, 1960) (see Newton [1964] and De Alfaro and Regge [1965]). Poles of the scattering amplitude in the λ plane, like those found by Watson, became known as Regge poles (cf. Nussenzveig [1970, 1972]).

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§ 7. CAM theory of Mie scattering 7.1. The Poisson sum formula An improved version of complex angular momentum theory, hereafter referred to as CAM, was developed by Nussenzveig [1965]. Its applications to Mie scattering, with special reference to light tunneling, will be reviewed in the remainder of this work. For more detailed expositions, see Nussenzveig [1992] (and the original references therein), Grandy [2000] and Adam [2002]. The starting point, instead of (6.1), is the Poisson sum formula (Titchmarsh [1937]), apparently first employed in this connection by Bremmer [1949],

∞

∞ ∞ 1 1 φ l + ,x = exp 2imπ λ + φ(λ, x) dλ. 2 2 m=−∞ l=0

(7.1)

0

A beautiful physical interpretation of this Poisson representation, applied to the partial-wave expansion of the scattering amplitude in semiclassical scattering by a central potential, has been proposed by Berry and Mount [1972]. Substituting the partial-wave amplitudes by their WKB approximations, and Legendre polynomials by their asymptotic expansions (near-forward and near-backward directions excluded), and applying the stationary phase approximation to the integrals, they reinterpret (7.1) as a sum over paths. Stationary phase points are roots of Θ λ¯ = −(2mπ ± θ ), (7.2) where Θ is the classical deflection angle associated with a classical path that takes m turns about the origin. The corresponding phase of the integrand becomes the classical action along this path. The Poisson representation then represents a sum over a family of classical paths with continuous angular momenta. The envelope of all such paths is a caustic sphere with radius r0 (λ), the classical distance of closest approach (outermost radial turning point) for angular momentum λ. The caustic is a singular solution of the equations of motion. A particle with any λ may undergo any deflection Θ by following a pseudoclassical path, formed by joining an incoming classical path from infinity to an outgoing one, through a piece that “coasts” along the caustic sphere and takes any number of turns about the origin. Keller’s diffracted rays represent a special case of pseudoclassical paths, in which the caustic sphere is the surface of the scatterer. The Poisson representation stands halfway between classical and wave mechanics: summing over sta-

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tionary phase points yields classical path contributions, whereas summing over the topological number m leads back to “quantized” λ = l + 12 . 7.2. Basic tools of CAM theory The CAM theory of Mie scattering employs the following basic tools: (i) The optomechanical analogy – We denote by N the refractive index of the sphere. According to (3.5), the corresponding potential function is given by −(N 2 − 1)k 2 (0 r < a), V (r) = (7.3) 0, (r > a) so that it represents a rectangular well for N > 1 and a rectangular barrier for N < 1. The partial-wave S-function for magnetic multipoles is in fact identical to that for quantum scattering by the potential (7.3), so that for perpendicular polarization the optical and mechanical problems are essentially the same. (ii) The effective potential – The radial equation for the lth multipole wave yields an effective radial potential λ2 , (7.4) r2 where V (r) is given by (7.3) and the last term represents the centrifugal barrier. (iii) The localization principle – According to (5.3), for β 1, we associate angular momentum λ with incident rays having impact parameter Uλ (r) = V (r) +

bλ = λ/k.

(7.5)

(iv) Path deformations – Like the Poincaré–Watson method, the CAM treatment works with path deformations of the Fourier integrals of (7.1) in the λ plane. In contrast with the partial-wave series, in which a large number of terms contribute, the idea is to employ the path deformations so as to collect dominant asymptotic contributions from a small number of critical points in the λ plane. As happened with White’s extension of the Watson transformation (Section 6), different path deformations are required in different spatial regions (e.g., shadow and illuminated regions). (v) Critical points – The following are the main types of critical points: (a) Real saddle points. These are also stationary-phase points, so that they are associated with classical paths, i.e., geometrical-optic rays. The evaluation of their asymptotic contribution by the saddle point method yields the WKB approximation (including higher-order correction terms).

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(b) Complex saddle points. These are associated with complex rays (Chapman, Lawry, Ockendon and Tew [1997], Kravtsov, Forbes and Asatryan [1999]), that represent dynamical tunneling effects usually disregarded by classical diffraction theory. (c) Complex poles. For the polarized amplitudes, these are Regge poles, associated with resonances. They are also related to tunneling, as will be seen. (d) Uniformity. The need (iv) to employ different path deformations in different regions brings out the problem of transitions between them – where diffraction effects are found (e.g., light/shadow). To obtain smooth joining, it is essential to employ uniform asymptotic expansions (Berry [1969], Ludwig [1970], Olver [1974]) for the special functions that appear in the Poisson representation.

§ 8. Impenetrable sphere 8.1. Structure of the wave function All relevant features are already apparent for a scalar field, which may represent scattering by a quantum hard sphere (Dirichlet boundary condition). It is useful to describe first the CAM results for the structure of the wave function at finite distances (Nussenzveig [1965]). The subdivision into spatial regions is represented in fig. 10. The deep shadow region is the domain (shaded in fig. 10) r/a β 1/3 , θ0 −θ −1/3 β , where θ0 = sin−1 (a/r) is the shadow boundary angle. This is the only domain where the original Watson transformation is rapidly convergent, yielding exponentially damped Regge pole contributions associated with surface waves and diffracted rays. Much weaker damping is found in the shadow of a circular disc (Jones [1964]), signaling the failure of classical diffraction theory, which predicts identical behavior in both cases. In the deep illumination (lit region), θ − θ0 β −1/3 , the dominant term is the WKB approximation, that includes the incident and geometrically reflected waves. Near the shadow boundary, |θ − θ0 | β −1/3 , β −1/3 z/a β 1/3 (Fresnel region in fig. 10), we find the classical Fresnel pattern of a single straight edge, rather than that of a slit (as classical diffraction theory would predict). In the Fresnel–Lommel region, θ θ0 , β 1/3 r/a β, classical diffraction theory does work: we get the classical Fresnel pattern of a circular disc (Lommel [1885]). In particular, along the axis, one finds the Poisson spot, reproducing the

212 Light tunneling [6, § 8

Fig. 10. Subdivision into spatial regions of the field scattered by an impenetrable sphere.

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213

incident intensity. However, in contrast with the circular disc, it only develops at large axial distances, of order β 1/3 a. It grows into a cone with angular opening of order β −1 , surrounded by diffraction rings. In the Fraunhofer region, r/a β, the field may be described in terms of the total scattering amplitude f (k, θ ). Up to θ ∼ β −1 , the region of the forward diffraction peak, the amplitude is dominated by the classical Airy pattern, f (k, θ ) ≈ ia

J1 (β sin θ ) . β sin θ

(8.1)

For θ β −1 , the geometrical reflection region (fig. 10), the WKB approximation holds, yielding the reflected wave. The region between the forward peak and the geometrical reflection region is the penumbra. A transitional asymptotic approximation in this domain yields Fock-type functions. However, it does not quite bridge the gap between the adjacent regions. For this purpose, a uniform approximation is required.

8.2. Diffraction as tunneling The effective potential Uλ (r) associated with an impenetrable sphere by (7.4) is an infinitely high wall surrounded by the centrifugal barrier. It is shown in fig. 11(i) for three values of λ, together with the associated incident rays, shown in fig. 11(ii), employing the localization principle to determine the corresponding

Fig. 11. (i) Effective potential for an impenetrable sphere and three different angular momenta λ: () above-edge, with turning point b. (ii) The corresponding incident rays.

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impact parameter bλ . The edge ray (bλ = a) is associated with λ = β, for which the “energy” k 2 lies exactly at the barrier top. For below-edge rays (bλ < a), k 2 is above the barrier. For above-edge rays (bλ > a), k 2 meets the barrier at the turning point r = bλ . Such rays do not interact with the sphere according to geometrical optics. However, in the above-edge domain λ − β = O(β 1/3 ), tunneling through the centrifugal barrier to the sphere surface leads to appreciable interaction and yields an important contribution to the scattering amplitude. Similarly, in the below-edge domain β − λ = O(β 1/3 ), geometrical-optic reflection from the surface is strongly distorted by interaction with the centrifugal barrier (anomalous reflection). The associated “peculiar interference effect” is discussed by Van De Hulst [1957], Section 17.21. In the penumbra angular region θ = O(β −1/3 ), which is much broader than the forward peak angular width θ = O(β −1 ), both effects are important. Fock’s theory of diffraction amounts to approximating the region near the top of the centrifugal barrier by a linear potential (equivalent to Pekeris’s “Earth flattening transformation”). In view of eq. (2.1), this leads to the Airy functions that appear in Fock-type diffraction integrals. The Leontovich–Fock diffusion picture is misleading: the natural analogy is with the Schrödinger equation. Fock’s theory yields a transitional approximation because it neglects the curvature of the centrifugal barrier. CAM theory (Nussenzveig and Wiscombe (1987, 1991), Nussenzveig [1988]) leads to uniform asymptotic approximations, that match smoothly with WKB results in the geometrical reflection region. It is possible to separate the effects of the “diffraction as blocking” picture of classical diffraction theory (Section 2) from the new contributions of tunneling and anomalous reflection by employing the localization principle to isolate aboveedge and below-edge effects. To illustrate how blocking is connected with partial-wave contributions only from rays that hit the sphere, we note that their contribution, according to Babinet’s principle (Sommerfeld [1954]), is the same as that from a circular aperture, for which rays go through without interaction, yielding the forward diffraction Airy pattern [β]

l=0

J1 (βθ ) 1 Pl (cos θ ) = β + O β −1 , l+ 2 θ

β 1,

(8.2)

where [β] denotes the largest integer contained in β and we have approximated the sum by an integral. Thus, the blocking effect arises from the cut off in the partial-wave series at [β].

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Fig. 12. Relative contributions from classical diffraction (blocking), tunneling (above-edge) and anomalous reflection (below-edge) in scattering by an impenetrable sphere; γ ≡ (2/β)1/3 . Reprinted with permission from Nussenzveig and Wiscombe [1991]. © 1991 by the American Physical Society.

The absolute values of the relative contributions from blocking, tunneling and anomalous reflection are plotted in fig. 12, where γ ≡ (2/β)1/3 , for β = 10. Classical diffraction (blocking) is dominant within the central part of the forward diffraction peak, but the three effects become comparable already at θ ∼ β −1/3 . At larger scattering angles, tunneling contributes as much as classical diffraction. The tunneling range of impact parameters above the edge that contributes significantly is given by 1/3 b − a = O λ20 a ,

(8.3)

where λ0 ≡ 2π/k is the wavelength. This weighted geometric average between wavelength and size may be thought of as the range, for an impenetrable sphere, of the effective “action at a distance” conjectured by Newton in his Query 1 (Section 1). How much improvement is brought by the uniform CAM approximation is illustrated by the error plots in fig. 13, which represent percent errors in the extinction efficiency (ratio of the total cross-section to the geometrical cross-section) in acoustic scattering by a rigid sphere (Nussenzveig and Wiscombe [1987]), with size parameters ranging from 1 to 10. The CAM percent error is three to four

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Fig. 13. Comparison between percent errors in the uniform CAM approximation and in the Fock approximation to the extinction cross-section for acoustic scattering by a rigid sphere, in the range 1 < β < 10. Reprinted with permission from Nussenzveig and Wiscombe [1987]. © 1987 by the American Physical Society.

orders of magnitude smaller than that of the Fock theory, which was the best previous approximation. It is of order 10% even at β = 1 and only a few ppm at β = 10. For β = 100, it is found that the uniform CAM approximation is more accurate, for all scattering angles, than typical “exact” results from numerical partial-wave summations (1 ppm).

§ 9. Near-critical scattering For transparent spheres with relative refractive index N < 1, a new diffraction effect occurs. It is manifested, for instance, in clouds of air bubbles in seawater, seen near the critical angle for total reflection: colored bands in the scattered light were observed under these circumstances by Pulfrich [1888], who compared them with a rainbow. A physical optics approximation to explain this near-critical scat-

6, § 9]

Near-critical scattering

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tering effect, analogous to Airy’s classical diffraction theory of the rainbow, was proposed by Marston [1979]. Because the Fresnel reflectivities approach total reflection with vertical slope (Born and Wolf [1999]), the scattered intensity according to geometrical optics undergoes a sharp break in slope at the critical scattering angle, a singularity which may be called a weak caustic. The effective potential (7.3), (7.4) for this problem (rectangular plus centrifugal barrier for a given λ) is shown in fig. 14(a), together with four “energy levels” k 2 related with different impact parameters b and with the angle of incidence θ1 through the localization principle b = λ/k = a sin θ1 . The corresponding situations are shown in fig. 14(b). In situation 1, θ1 is below the critical angle and the incident ray gets inside the sphere, where it undergoes multiple reflections. The radial turning point r = b/N gets closer to the surface as critical incidence (situation 2) is approached. Situations 3 and 4 have turning points at the surface r = a, corresponding to total reflection; for the grazing ray 4, effects similar to those found for an impenetrable sphere take place. The new effect occurs in situation 2. Critical incidence on a plane interface gives rise to a lateral displacement of the reflected beam, the Goos–Hänchen shift (Goos and Hänchen [1947], Bryngdahl [1973]). Its origin is directly connected with Newton’s discovery of tunneling (Section 1): an evanescent wave penetrates into the rarer medium, running along the interface. A similar effect takes place in situation 2, but it is modified by the interface curvature (centrifugal barrier), leading to a spherical Goos–Hänchen angular displacement of the totally reflected beam, another light-tunneling effect. The CAM theory of this effect (Fiedler-Ferrari, Nussenzveig and Wiscombe [1991]) yields a new type of diffraction integral, the Pearcey–Fock integral, that describes near-critical scattering. Results for the perpendicular polarization gain factor (ratio of polarized intensity to that for an ideal isotropic scatterer), near the critical angle for an air bubble in water with β = 104 are shown in fig. 15. Contributions from non-critical paths that produce rapid interference oscillations (fine structure) have been subtracted out. The WKB curve shows the slope break at the critical angle. The physical optics approximation (POA) yields a Fresnel-like diffraction pattern, sharply disagreeing with the Mie results, while CAM produces a very good approximation in the near-critical region. The angular width of the critical region is δ = O(β −1/2 ). The spherical Goos–Hänchen angular displacement is different for electric and magnetic polarizations.

218 Light tunneling Fig. 14. (a) The effective potential for scattering by a sphere with N < 1, showing four “energy levels”. (b) Corresponding incident rays – 1: subcritical incidence; 2: critical incidence; 3: supracritical incidence; 4: edge incidence; α ≡ Nβ. Reprinted with permission from Fiedler-Ferrari, Nussenzveig and Wiscombe [1991]. © 1991 by the American Physical Society. [6, § 9

6, § 10]

The rainbow

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Fig. 15. Perpendicular polarization gain for N = 0.75, β = 10 000, in the near-critical region: Mie result (with fine structure subtracted out), CAM (open circles), physical optics approximation POA, and WKB approximation (dashed line); θt = critical scattering angle; angular width of near critical region δ = O(β −1/2 ). Reprinted with permission from Fiedler-Ferrari, Nussenzveig and Wiscombe [1991]. © 1991 by the American Physical Society.

The Goos–Hänchen shift on a plane interface is of the order of the wavelength, so that measurements in the visible require amplification by multiple passages (Goos and Hänchen [1947]). In contrast, the spherical Goos–Hänchen angular displacement is a macroscopic effect, observable with the naked eye. Experimental observations (Tran, Dutriaux, Balcou, Le Floch and Bretenaker [1995]) for large size parameters are in very good agreement with CAM theory predictions, including the polarization dependence.

§ 10. The rainbow 10.1. The Debye expansion Going over to N > 1, the chief applications are to light scattering by water droplets in the atmosphere, that gives rise to several striking meteorological optics effects (Greenler [1980], Lynch and Livingston [2001], Minnaert [1993], Tricker [1970]). The CAM treatment is based on the Debye expansion (Debye [1909a]),

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Light tunneling

[6, § 10

Fig. 16. Interpretation of the Debye expansion. (i) An incoming spherical multipole wave is partially reflected at the surface (reflection amplitude R22 ) and partially transmitted (transmission amplitude T21 ); (ii) The transmitted spherical wave goes through the center and back to the surface, where it is partially transmitted (amplitude T12 ) and partially reflected (amplitude R11 ).

an exact representation of the S function that parallels the ray optics description in terms of multiple internal reflections within a droplet. We denote by 1 and 2 the interior of the sphere and the external region, respectively. The interaction of an incoming spherical multipole wave with the sphere is broken up into an infinite series of partial transmissions through the surface, going through the center, which plays the role of a perfect reflector, converting incoming into outgoing waves that are again partially reflected at the surface and partially transmitted to the outside region (fig. 16). Each interaction with the surface gives rise to spherical reflection and transmis(p) (p) sion coefficients Rij (λ, β), Tij (λ, β), (i, j ) = (1, 2), p = (1, 2), where p is the polarization index (Van Der Pol and Bremmer [1937], Nussenzveig [1969a]). They are related with the Fresnel reflection and transmission amplitudes. Each term in the Debye expansion has a Poisson representation that is treated by the CAM methods discussed in Section 7.2: all integrands are meromorphic functions. CAM yields rapidly convergent asymptotic approximations for β 1. Background integrals are dominated by saddle-point contributions that yield the WKB approximation for ray paths with corresponding numbers of internal reflections. In different angular regions, the number of real saddle points (geometricaloptic rays) can vary, giving rise to transition regions described by penumbra Focktype effects, as well as to rainbow-type transitions, that are discussed below. The complex poles, named Regge–Debye poles, give rise to surface waves generated by glancing incident rays, as was found for an impenetrable sphere, again

6, § 10]

The rainbow

221

associated with tunneling. However, after describing any arc along the surface, they can now take “shortcuts” across the sphere (cf. fig. 25(b), below), with amplitudes determined by surface wave transmission coefficients, and the contributions from all possible such paths must be summed over. One must also consider how fast the Debye series itself converges. Successive terms are damped roughly by the average internal reflection amplitude within the domain of angles of incidence that contributes. For below-edge rays, the internal reflection amplitude tends to be very small, implying fast convergence, except for near-glancing incidence. Thus, for water droplets, it usually suffices to consider only the first three terms of the Debye series (direct reflection, direct transmission and transmission after one internal reflection), which, in geometrical optics, contribute over 98.5% of the total intensity (Van De Hulst [1957]). This does not prevent the intensity from becoming highly concentrated in narrow angular regions, as happens in the glory (Section 12), or in narrow sizeparameter ranges (Mie resonances, Section 11). To survive internal reflection damping, such contributions must arise from near-total internal reflection, i.e., incidence near the edge. Thus, for the glory and for Mie resonances, slow convergence of the Debye expansion is expected, and one must either consider many terms or go back to the Mie series. 10.2. The primary rainbow For rainbow history, we refer to Boyer [1987], Jackson [1999], and Lee and Fraser [2001]. We have already commented above on the contributions from Descartes, Newton, Young and Airy. The primary rainbow is contained in the third Debye term, associated with a single internal reflection. The rainbow scattering angle θR separates the shadow side of the primary rainbow (Alexander’s dark band: no real rays with a single internal reflection) from the bright side, covered by two geometric rays with different paths (in a domain θL > θ > θR ); the interference between them gives rise to the supernumeraries. The two real rays on the bright side are associated with two real saddle points in the λ plane, shown as black and white circles in fig. 17(a). As θ decreases from θL to θR , the two saddle points move towards each other, coalescing at the rainbow angle θ = θR . On the shadow side, for θ < θR , the two saddle points become complex, moving apart along complex conjugate paths. The contribution from the saddle point in the lower half-plane is dominant. It represents a complex ray, describing light tunneling into the shadow side. Each saddle point has a range, its neighborhood in the λ plane that gives a significant contribution to the integral. When the two ranges overlap, as happens

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Fig. 17. (a) Saddle point trajectories in the λ plane for scattering angles θ around the rainbow scattering angle θR ; ! " – saddle points. (b) Saddle points with overlapping ranges.

around the rainbow angle (fig. 17(b)), the ordinary saddle-point method must be modified, in order to yield a uniform asymptotic approximation. The uniform procedure was developed by Chester, Friedman and Ursell [1957]. It was applied by Berry [1966] to quantum potential scattering, by Nussenzveig [1969a] to scalar wave rainbow scattering, and by Khare and Nussenzveig [1974] and Nussenzveig [1979] to electromagnetic rainbow scattering. The Airy theory is a transitional lowest-order approximation (in all respects) to the uniform theory. Interference with the Debye direct reflection contribution produces rapidly oscillatory “fine structure”. Figure 18 shows a comparison, in the primary rainbow region, between parallel polarized intensities from Mie, CAM and Airy theories, after subtracting out fine structure, for water droplets with size parameter 1500 (requiring more than 1500 terms in the Mie summation). The uniform CAM approximation agrees very well with the exact theory: the small oscillatory differences arise from higher-order Debye non-rainbow contributions. However, Airy theory fails completely for this polarization, reversing the locations of supernumerary maxima and minima. For the dominant perpendicular polarization, agreement within the main rainbow peak (though not outside it) is much better (Khare and Nussenzveig [1974]). The reason for the poor performance of Airy theory for parallel polarization is that the two interfering contributions to the supernumeraries have incidence angles on opposite sides of the Brewster angle, leading to an additional 180◦ phase shift not accounted for in Airy’s classical diffraction theory. Experimental imaging polarimetry of the rainbow supports these results (Können and De Boer [1979], Barta, Horváth, Bernáth and Meyer-Rochow [2003]).

6, § 10]

The rainbow

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Fig. 18. Mie, CAM and Airy parallel polarized intensity near the rainbow angle for N = 1.33, β = 1500 (direct reflection has been subtracted out). Reprinted with permission from Khare and Nussenzveig [1974]. © 1974 by the American Physical Society.

Extended numerical comparisons between the Mie and Airy theories have been made (Wang and Van De Hulst [1991], Lee [1998]). Taking into account the size dispersion of water droplets in clouds tends to smooth out rapid interference oscillations and decrease discrepancies. The dark band, where tunneling occurs, becomes dominated by other Debye contributions, such as externally reflected light and higher-order rainbows. In terms of visual perception, rather than physical variables, differences are hard to detect. On the other hand, applications to rainbow refractometry require more precise quantitative data. A comparative analysis of Mie, CAM and Airy theories (Saengkaew, Charinpatnikul, Vanisri, Tanthapanichakoon, Mees, Gouesbet and Grehan [2006]) shows that, for an extended angular domain which includes the secondary rainbow, and taking into account droplet size dispersion, Airy theory overestimates intensities and yields incorrect supernumerary positions at larger angles, whereas CAM agrees closely with Mie theory and requires dramatically smaller computation times. Shape distortion of larger raindrops by air-drag forces changes rainbow features only slightly, an indication of their structural stability. Structurally stable

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[6, § 11

caustics, of which the rainbow is the simplest one, are associated with “diffraction catastrophes” (Berry and Upstill [1980]). § 11. Mie resonances and ripple fluctuations 11.1. Mie resonances The effective potential (7.4) for N > 1 is shown in fig. 19(a). It superposes a rectangular well onto the centrifugal barrier, recalling the nuclear well surrounded by the Coulomb barrier models employed in the early treatments of quantum tunneling (Section 1). This is a typical shape that gives rise to resonances. We consider impact parameters b in the range a b Na

(11.1)

corresponding to incident rays that pass outside the sphere (fig. 19(b)), so that the “energy” k 2 lies between the top T of the external barrier at (λ/a)2 and its bottom B at the well depth (7.3) below (fig. 19(a)). Besides the outer turning point at r = b, there are two inner turning points, at r = a and at r = b/N . Correspondingly (fig. 19(b)), the incident ray tunnels through the centrifugal barrier to the surface and takes a shortcut inside, so that it undergoes multiple internal reflections between the two radial turning points, the inner sphere representing a caustic for the transmitted rays.

Fig. 19. (a) The effective potential for a sphere of radius a with N > 1, with a sketch of resonance wave functions for n = 0 and n = 1. (b) A corresponding incident ray with impact parameter b > a.

6, § 11]

Mie resonances and ripple fluctuations

225

Since internal incidence is beyond the critical angle, reflection would be total in geometrical optics, producing a bound state of light. However, tunneling back to the outside through the centrifugal barrier allows radiation to escape. Within narrow neighborhoods of “resonance energies”, analogous to quantum energy levels within the well, boundary condition matching leads to very large ratios of internal to external wave amplitudes, as illustrated in fig. 19(a). This provides the physical interpretation of Mie resonances (Nussenzveig [1989], Guimarães and Nussenzveig [1992, 1994], Johnson [1993]). Mie resonances have also been referred to as “whispering gallery modes”, by analogy with acoustics (Lord Rayleigh [1910]), and as “morphology-dependent resonances” (Chang and Campillo [1996]). They are also associated with the natural modes of oscillation of a dielectric sphere (Debye [1909a], Nussenzveig [1972]). For a given angular momentum, several resonances may arise. They are characterized by a “family number” n (= 0, 1, 2, . . .), the number of nodes of the radial wave function within the well in the limit of zero leakage: wave functions for n = 0 and n = 1 are sketched in fig. 19(a). The lower the value of n, the deeper the quasibound state lies within the well and the longer is the resonance lifetime, since it must tunnel across an increasing barrier width. In CAM theory, the resonances are described by the complex Regge poles (Sections 6, 7) λnj (β), where j = 1, 2 is the polarization index. As β varies, the poles describe Regge trajectories, that give unified descriptions of all resonances with the same family number n. A resonance with polarization j in partial wave l arises when Re λnj crosses the physical value λ = l + 12 . The resonance width follows from Im λnj , which is determined by the transmissivity of the potential barrier; as above, the lowest n yield the narrowest resonances. 11.2. Ripple fluctuations Resonance contributions to the polarized amplitudes correspond to residues at the Regge poles. The contributions from Debye terms that were discussed above play the role of background amplitudes, and their interference with resonance contributions must be taken into account in the total polarized intensities and crosssections. In the behavior of cross-sections as functions of size parameter, Mie resonances contribute to the “ripple”, a very complex structure showing rapid quasiperiodic fluctuations and peaks with a variety of heights and widths (Shipley and Weinman [1968]), that appears at all scattering angles and is highly sensitive to refractive index changes, representing a serious nuisance in numerical computation. An exam-

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[6, § 11

Fig. 20. Extinction efficiency for water droplets with size parameters from 4 to 50, displaying ripple fluctuations.

ple is the extinction efficiency Qext (β) (ratio of total to geometrical cross-section) of water droplets for β between 4 and 50, shown in fig. 20. A typical water droplet in the atmosphere, with radius of the order of 10 µm, is highly transparent in the visible and is very close to spherical shape because of surface tension, so that the Mie model is very realistic and the sharp ripple fluctuations are present. For an oil droplet of this size, they are so narrow that a fractional change in average radius of the order of 0.1 Å could be experimentally detected by monitoring radiation pressure (Ashkin [1980])! In view of the extreme narrowness of the resonances for the large values of β that are required in many applications, it is important to determine resonance positions with accuracy better than the resonance width. This has been done employing uniform asymptotic approximations (Guimarães and Nussenzveig [1992]). The quasiperiodic features of the resonances arise from the period of Regge recurrences, successive passages of a Regge trajectory near physical values of the angular momentum. The quasiperiod is approximately given by (Chylek [1990]) δβ ∼ M −1 tan−1 M, M ≡ N 2 − 1, (11.2) which is 0.821 for N = 1.33. A CAM fit to Qext within the quasiperiod 58.2 β 60.0 (Guimarães and Nussenzveig [1992]) is shown in fig. 21. All resonances that contribute are la-

6, § 12]

Light tunneling in clouds

227

Fig. 21. CAM fit to the Mie extinction efficiency for N = 1.33 within a quasiperiod. The background contribution, including and excluding the p = 4 forward glory term, is shown. The contribution offset 2 includes negative (antiresonant) contributions. by 2 from the broad resonance M61

beled by the notation (E, M)nl (E or M polarization, lower index angular momentum, upper index family number). The CAM approximation is evaluated to order O(β −2 ), and it fits the Mie result to this order ( 0. Let Ex (r, t) and Ey (r, t) be the components of the complex electric vector (derived from components of the real electric vector, via complex analytic signals (see Born and Wolf [1999], p. 158), at point r and time t, along two mutually orthogonal directions perpendicular to the z direction. The polarization matrix is the matrix ∗ Jxx (r) Jxy (r) Ex (r, t)Ex (r, t) Ex∗ (r, t)Ey (r, t) J(r) ≡ = . Jyx (r) Jyy (r) Ey∗ (r, t)Ex (r, t) Ey∗ (r, t)Ey (r, t) (3.13) In terms of this matrix the degree of polarization is given by the expression (Born and Wolf [1999], p. 628) 4 Det J(r) P(r) ≡ 1 − (3.14) , [Tr J(r)]2 where Det and Tr denote the determinant and the trace of the matrix, respectively.

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The influence of Young’s interference experiment

[7, § 4

Fig. 7. The role of Young’s interference experiment in the formulation of the theory of coherence – scalar theory.

§ 4. Unification of the theories of polarization and coherence It is clear from the preceding account that both polarization and coherence are manifestations of the same phenomenon, namely of correlations between fluctuations in optical fields. Recent research, which we will now briefly describe, has revealed that coherence properties and polarization properties are, in fact, intimately related and that it is possible to study them within the framework of a unified theory (Wolf [2003a]). The theory has already provided a deeper understanding of these two basic aspects of statistical optics and has predicted some

7, § 4]

Unification of the theories of polarization and coherence

265

interesting new effects. As we will soon see, Young’s interference experiment has played a central role in this development. The basic mathematical tool of this new theory is the so-called cross-spectral density matrix of a stochastic, statistically stationary, electromagnetic beam. It is a generalization of the scalar cross-spectral density function which we encountered earlier [eqs. (3.8) and (3.10)]. The matrix may be defined as follows: Consider again a statistically stationary electromagnetic beam which propagates close to the z-direction into the half-space z > 0. Let {Ex (r, ω)} and {Ey (r, ω)} be the statistical ensembles, in the sense of coherence theory in the space–frequency domain, of the components of the complex electric field vector in two mutually orthogonal directions perpendicular to the beam axis. The crossspectral density matrix of the beam is the 2 × 2 matrix W(r1 , r2 , ω) ≡ Wij (r1 , r2 , ω) = Ei∗ (r1 , ω)Ej (r2 , ω) , i = x, y; j = x, y. (4.1) The four elements of this matrix may be determined from measurements employing Young’s interference experiment, with filters, polarizers and a rotator placed behind the pinholes (Roychowdhury and Wolf [2003]8 ), as indicated schematically in fig. 8(b) and (c). (In the special case when r2 = r1 , the elements of the matrix have to be determined by a somewhat different scheme, with the help of a beamsplitter.) It can be shown by simple calculation (Wolf [2003a]) that, apart from a trivial proportionality factor, the spectral visibility V(r, ω) =

Smax (r, ω) − Smin (r, ω) , Smax (r, ω) + Smin (r, ω)

(4.2)

in the vicinity of a point r in the interference pattern is proportional to the absolute value of the quantity Tr W(r1 , r2 , ω) . η(r1 , r2 , ω) = √ √ Tr W(r1 , r1 , ω) Tr W(r2 , r2 , ω)

(4.3)

Here r1 and r2 are again the position vectors of the two pinholes. The quantities Tr W(rj , rj , ω), j = 1, 2, are just the spectral densities S(rj , ω) of the electric field at the pinholes [fig. 8(a)]. The phase of η can be determined experimentally from measurements of the location of the maxima of the spectral density in the

8 In this reference, the sentence after eq. (12) is somewhat misleading and should be replaced by “The other off-diagonal element may be determined in an analogous manner.”

266


[7, § 4

Fig. 8. The role of Young’s interference experiment in the formulation of the theories of coherence and polarization – electromagnetic theory.

interference pattern. Because of the relation between η and V which we just mentioned, it is clear that η(r1 , r2 , ω) may be identified with the (generally complex) spectral degree of coherence of the electromagnetic beam.9 It is of interest to note that only the diagonal elements Wxx and Wyy of W appear in the expression (4.3) for the degree of coherence. This fact is a consequence of 9 A degree of coherence of an electromagnetic field in the space–time domain was introduced much earlier by Karczewski [1963a, 1963b]. It has, however, not found applications, probably because it is not easy to measure.

7, § 4]


267

the orthogonality between the components Ex and Ey of the electric vector and is in agreement with one of the Fresnel–Arago laws mentioned earlier. This, of course, does not mean that the two components are necessarily uncorrelated. The lack of appreciation of this fact has led to some confusion in recent publications regarding the definition of the degree of coherence on an electromagnetic beam. In terms of the elements of the cross-spectral density matrix one can also define the spectral degree of polarization by a formula analogous to eq. (3.14) as 4 Det W(r, r, ω) . P(r, ω) = 1 − (4.4) [Tr W(r, r, ω)]2 The cross-spectral density matrix W(r1 , r2 , ω) obeys precise propagation laws. In free space they are the two Helmholtz equations ∇j2 W(r1 , r2 , ω) + k 2 W(r1 , r2 , ω) = 0,

j = 1, 2.

(4.5)

The solution of these two equations for paraxial (beam) propagation from a plane z = 0 (usually the source plane) into the half-space z > 0 is W(r1 , r2 , ω) = W(0) (ρ1 , ρ2 , ω)K(ρ1 − ρ1 , ρ2 − ρ2 , z, ω) d2 ρ1 d2 ρ2 , (z=0)

(4.6)

where K(ρ1 − ρ1 , ρ2 − ρ2 , z; ω) = G∗ (ρ1 − ρ1 , z; ω)G(ρ2 − ρ2 , z; ω),

(4.7)

G being the free-space Green function for paraxial propagation from a point ρ in the “source plane” z = 0 to a point ρ in a transverse plane at distance z from that plane (see fig. 9), W(0) denoting the cross-spectral density matrix of the electric

Fig. 9. Notation relating to paraxial propagation.

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[7, § 4

Fig. 10. Changes in the spectral degree of polarization along the axis of a certain partially coherent electromagnetic beam propagating in free space. (After James [1994].)

field in the plane z = 0. Explicitly (see Mandel and Wolf [1995], Eq. (5.6-17))

2

ik G(ρ − ρ , z; ω) = − (4.8) exp ik ρ − ρ /2z eikz 2πz (k = ω/c, c being the speed of light in vacuum). The formula (4.6) may readily be generalized to propagation in any linear medium, whether deterministic or random (Wolf [2003b]). Because the cross-spectral density matrix propagates according to precise laws it is clear that, in general, the spectral density S(r, ω), the spectral degree of polarization P (r, ω) and the spectral degree of coherence η(r1 , r2 , ω) will all change as the beam propagates. That the degree of polarization of a beam may change on propagation was first predicted by James [1994]. Examples of changes in the whole state of polarization were given by Korotkova and Wolf [2005]. Such changes are illustrated in figs. 10 and 11. To determine the changes, one only needs to make use of the propagation law for the cross-spectral density matrix which we just mentioned and substitute the elements of the “propagated” crossspectral density matrix into the formulas defining these quantities. It is clear that because the degree of polarization and the degree of coherence are expressed in terms of the elements of the same correlation matrix, they will generally be related. A relation between coherence and polarization was first predicted theoretically by Roychowdhury and Wolf [2005] (see also Li, Lee and Wolf [2006]) and

7, § 4]


269

Fig. 11. Changes in the polarization ellipse of the polarized portion of a typical partially coherent electromagnetic beam propagating in free space. (After Korotkova and Wolf [2005].)

confirmed experimentally by Gori, Santarsiero, Borghi and Wolf [2006] by means of Young’s interference experiment (see fig. 12). The unified theory of coherence and polarization which we just outlined, although formulated only very recently, has already found useful applications,

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[7, § 4

Fig. 12. Degree of polarization P at the center of the fringe pattern, as a function of the spectral degree of coherence μ(0) of light at the pinholes in Young’s interference experiment with a typical, partially coherent stochastic electromagnetic beam. The light at the pinholes is assumed to be unpolarized. The different curves pertain to beams characterized by different values of their parameters. (After Gori, Santarsiero, Borghi and Wolf [2006].)

for example, in elucidating propagation of stochastic electromagnetic beams in the turbulent atmosphere (Salem, Korotkova, Dogariu and Wolf [2004], Roychowdhury, Ponomarenko and Wolf [2005]), in fibers (Roychowdhury, Agarwal and Wolf [2005]) and through human tissues (Gao [2006], and Gao and Korotkova [2007]). The results will undoubtedly prove useful in fields such as optical communications and medical diagnostics. It should be clear from the preceding discussion that Young’s interference experiment, with straightforward extensions which use filters, polarizers and rotators, has become of basic importance in the formulation and applications of the theories of polarization and coherence, and has pointed a way towards the unification of the two disciplines. In spite of the tremendous impact the Young interference experiment had on the development of physics, it was received with suspicion and hostility. Here are two examples: “From such a dull invention, nothing can be expected. It only removes all the difficulties under which the theory of light laboured, to theory of this new medium, which assumes its place. It is a change of name; it teaches no truth, reconciles no contradictions, arranges no anomalous facts, suggests no new experiments, and leads to no new enquiries. It has not even the

7]

References

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pitiful merit of affording an agreeable play to the fancy. It is infinitely more useless and less ingenious, than the Indian theory of the Elephant and Tortoise.” [From Edinburgh Review, vol. I (1803), p. 456]. “. . . we exposed the absurdity of the law of interference, it pleases him to call one of the most incomprehensible supposition that we remember to have met with in the history of human hypotheses.” [From Edinburgh Review, vol. V (1805), p. 97].

Because of attacks such as these, Young’s principle of interference was at first largely ignored. Young replied to some of the criticisms and actually published a pamphlet responding to them. Apparently only a single copy of it was sold! There is a lesson in this for young scientists; namely that they should not become discouraged when a referee criticizes their manuscripts. They should just remember that, in spite of the very harsh criticism that his work initially received, Thomas Young eventually made it!

Acknowledgement I am very grateful Ms. Patricia Sulouff, the Head Librarian and to Ms. Sandra Cherin, a member of the staff of the Physics-Optics-Astronomy Library at the University of Rochester, for very considerable help that she has given me, by locating most of the old publications of historical interest cited in this article. I am also obliged to Proffesor Taco Visser for helpful comments.

References Born, M., Wolf, E., 1999, Principles of Optics, 7th edition, Cambridge University Press, Cambridge, UK. Brosseau, C., 1998, Fundamentals of Polarized Light, Wiley, New York. Collett, E., 1971, Mathematical foundation of the interference laws of Fresnel and Arago, Amer. J. Phys. 39, 1483–1495. Collett, E., 1993, Polarized Light, Marcel Dekker, New York. Crease, R.P., 2003, Prism and the Pendulum, Random House, New York. Gao, W., 2006, Changes of polarization of light beams on propagation through tissue, Opt. Commun. 260, 749–754. Gao, W., Korotkova, O., 2007, Changes in the state of a random electromagnetic beam propagating through tissue, Opt. Commun. 270, 474–478. Gori, F., Santarsiero, M., Borghi, R., Wolf, E., 2006, Effects of coherence on the degree of polarization in Young interference pattern, Opt. Lett. 31, 688–690. Gurney, H., 1831, Memories of the Life of Thomas Young, John and Arther, Arch., London. Hills, V.L., 1978, Thomas Young’s autobiographical sketch, Proc. Amer. Phill. Soc. 122, 248–260. James, D.F.V., 1994, Change of polarization of light beams on propagation in free space, J. Opt. Soc. Amer. A 11, 1641–1649.

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Jönsson, C., 1961, Elektroneninterferenzen am mehreren künstlich hergestellet Feinspalten, Z. f. Physik 161, 454–474. English translation: Electron diffraction and multiple slits, Am. J. Phys. 42 (1974) 4–11. Karczewski, B., 1963a, Degree of coherence of the electromagnetic field, Phys. Lett. 5, 191–192. Karczewski, B., 1963b, Coherence theory of the electromagnetic field, Nuovo Cimento 30, 909–915. Kipnis, N., 1991, History of the Principle of Interference of Light, Birkhäuser, Germany. Korotkova, O., Wolf, E., 2005, Changes in the state of polarization of a random electromagnetic beam of propagation, Opt. Commun. 246, 35–43. Li, Y., Lee, H., Wolf, E., 2006, Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams, Opt. Commun. 265, 63–72. Mach, E., 1926, The Principles of Physical Optics, B.F. Dutton & Company, Dover, New York, reprinted by. Mandel, L., Wolf, E., 1970, Selected Papers on Coherence and Fluctuations of Light, Dover, New York. Reprinted 1990 by SPIE Optical Engineering Press, Belington, Washington, USA. Mandel, L., Wolf, E., 1995, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, UK. Mollow, J.D., 2002, The origin of the concept of interference, Phil. Trans. Roy. Soc. London 360, 807–819. Mujat, A., Dogariu, A., Wolf, E., 2004, A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel–Arago interference laws, J. Opt. Soc. Amer. A 21, 2414–2417. Peacock, G., 1855, Life of Thomas Young, J. Murray, London. Robinson, A., 2006, The Last Man Who Knew Everything, Pi Press, New York. Roychowdhury, H., Agarwal, G., Wolf, E., 2005, Changes in the spectrum and in the spectral degree of coherence of a partially coherent beam propagating through a gradient index fiber, J. Opt. Soc. Amer. A 23, 940–948. Roychowdhury, H., Ponomarenko, S., Wolf, E., 2005, Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere, J. Mod. Phys. 52, 1611– 1618. Roychowdhury, H., Wolf, E., 2003, Determination of the electric cross-spectral density matrix of a random electromagnetic beam, Opt. Commun. 226, 57–60. Roychowdhury, H., Wolf, E., 2005, Young’s interference experiment with light of any state of coherence, Opt. Commun. 252, 268–274. Salem, M., Korotkova, O., Dogariu, A., Wolf, E., 2004, Spectral degree of coherence of a random three-dimensional electromagnetic field, J. Opt. Soc. Amer. A 21, 2382–2385. Stokes, G.G., 1852, On the composition and resolution of streams of polarized light from different sources, Trans. Cambridge Phil. Soc. 9, 399–416. Stokes, G.G., 1901, Mathematical and Physical Papers, Cambridge University Press, Cambridge, UK. Swindell, W., 1975, Polarized Light, Hutchinson & Ross, Dowden, UK. Taylor, G.I., 1909, Proc. Cambridge Phil. Soc. 15, 114–115. Verdet, E., 1865, Etude sur la constitution de la lumière non polarisée et de la lumière partiellement polarisée, Ann. Scientifiques de l’Ecole Normale Supérieure 2, 291–316. Whewell, W., 1875, History of the Inductive Science, vol. II, D. Appleton and Company, New York. Whittaker, E.T., 1951, A History of the Theories of Ether and Electricity – the Classical Theories, T. Nelson and Sons, London, England. Wiener, N., 1927–1928, Coherence matrices and quantum theory, J. Math. Phys. 7, 109–125. Wiener, N., 1930, Generalized harmonic analysis, Acta Math. 55, 117–258. Wolf, E., 1954, Optics in terms of observable quantities, Nuovo Cimento 12, 884–888. Wolf, E., 1955, A macroscopic theory of interference and diffraction of light from finite sources II: fields with a spectral range of arbitrary width, Proc. Roy. Soc. A 230, 246–265.

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Chapter 8

Planck, photon statistics, and Bose–Einstein condensation by

Daniel M. Greenberger City College of New York, New York, NY 10031, USA

Noam Erez Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 77843, USA

Marlan O. Scully Applied Physics and Materials Science Group, Engineering Quadrangle, Princeton University, Princeton, NJ 08544, USA Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 77843, USA

Anatoly A. Svidzinsky Applied Physics and Materials Science Group, Engineering Quadrangle, Princeton University, Princeton, NJ 08544, USA Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 77843, USA

M. Suhail Zubairy Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 77843, USA ISSN: 0079-6638

DOI: 10.1016/S0079-6638(07)50008-9 275

Contents


277

§ 2. Planck’s blackbody radiation law . . . . . . . . . . . . . . . . . . . .

279

§ 3. Bose–Einstein condensation . . . . . . . . . . . . . . . . . . . . . . .

299

§ 4. The quantum theory of the laser . . . . . . . . . . . . . . . . . . . . .

304

§ 5. Bose–Einstein condensation: laser phase-transition analogy . . . . . .

310

§ 6. Hybrid approach to condensate fluctuations . . . . . . . . . . . . . .

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Appendix A: Mean condensate particle number and its variance for weakly interacting BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

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§ 1. Introduction Optics was the original handle by which classical physicists learned to pry their way into the mysteries of quantum physics. It was optics that led to the realization that light possesses a dual character: in one limit the purely classical wave theory, and in the other the purely quantum-mechanical particle limit. In 1900, when the spectrum of blackbody radiation was being studied in detail, only the classical side was known. This was used to connect it with thermodynamics, from which many of its properties could be derived. But the Wien spectral law, which characterized the general, but not specific form of the spectral law, was as far as thermodynamics could take one. In order to get a specific law, Planck had to also draw on the probabilistic considerations of Boltzmann, a real departure for Planck, and he inadvertently drew into focus the particle aspect of the problem, without at that time understanding just how radical his innovation was. But this added statistics and fluctuations into the mix. A main point in this chapter will be to show the role that fluctuations played in Planck’s and Einstein’s thinking in the early days of quantum theory, the important role it played in the development of the quantum theory of the laser, and finally, how the laser theory allows one to treat the fluctuations in a Bose– Einstein gas above, below, and at the critical temperature. A second major theme in the chapter will be to pursue the historical thread running through Planck’s work. In their desire to present a coherent story, leading from classical physics to quantum physics, most textbooks leave out or distort the history of the subject, which is consequently not well known. But in this case, the courage of Planck in abandoning his lifelong distrust of probability, coupled with his total reluctance to abandon the principles of classical physics, led to a series of fascinating ironies that strongly affected the history of the subject, and they deserve to be known better. A further hidden element guiding the development of the early quantum theory, the laser, and Bose–Einstein condensation, was the connection between advancing technology and experimental technique. The effects of technology are apparent in the laser and Bose–Einstein condensation, although they are usually not appreciated as an input into early quantum theory, but the accurate mea277

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[8, § 1

surements of the blackbody spectra were made possible by the invention of the bolometer by Langley, who became the first director of the Smithsonian Institution in the USA. But also, the funding for the improvements of the bolometer so that measurements could be extended into the infrared, which became the most relevant measurements, leading to the breakdown of Wien’s specific radiation law, was provided by the power company of Berlin, which city had recently been electrified. Blackbody radiation is the least efficient means of illumination (one wants to be far from equilibrium) and it set a standard against which to measure efficiencies. So it turns out that the interest in funding such an abstruse subject as blackbody radiation was actually driven by the technology of the day. The interconnections between all these threads constitute an interesting subject in itself, but here we shall only go so far as to follow a few of them. We shall emphasize some of the interesting historical details that surround Planck’s work, which seem to be almost unknown to physicists. We have drawn heavily on Planck’s original papers (Planck [1900a, 1900b]), reproduced with comments by Ter Haar [1967], and Planck’s [1913] book on heat radiation. We have also extensively used publications by Kuhn [1978], Hermann [1971], Jammer [1966] and Heilbron [1996]. Some other good references are Klein [1975], Mehra and Rechenberg [1982], Rosenfeld [1936], Varro [2006] and Kangro [1976]. The anthology by Brush and Hall [2003] contains reprints of some papers we refer to, with comments. We go on to describe in some detail exactly what Planck did, and did not do, and the importance of fluctuations in his work. Their true meaning and importance was established by Einstein [1909]. We then proceed in the second half of this paper to explain how fluctuations enter into the theory of the laser, and how this theory has been used to treat fluctuations in Bose–Einstein condensation (BEC). Historically, Einstein [1924, 1925] was the first to demonstrate the existence of the “Bose” condensate. After Bose [1924] had created his “photon as a particle” path to the Planck distribution, Einstein [1925] went on to show that there is a critical temperature below which a macroscopic number of atoms would occupy the lowest energy state of the potential holding the atoms. One might think that all problems concerning the ideal Bose gas would have been solved in the 20s or 30s. Not so! As late as the 90s, it was noted that “the grand canonical fluctuation catastrophe” has been discussed by generations of physicists who have not solved the problem. Motivated by the above, we found a new approach to the N body boson problem that treats the fluctuations very accurately, by extending the laser-phase transition analogy to include BEC.

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Planck’s blackbody radiation law

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§ 2. Planck’s blackbody radiation law 2.1. Some ironical historical details concerning Planck It is rather universally assumed that when Planck introduced the quantum in 1900 (Planck [1900a, 1900b]), he quantized the energy levels of an oscillator. But in fact, what he did was very ambiguous (Kuhn [1978]), and we shall produce some strong evidence that at that time he was thinking more along the lines of quantizing the size of cells in phase space. Furthermore, he fought the idea of quantizing both radiation and the oscillator. In fact, as late as 1913, when he published the second edition of his book on heat radiation (Planck [1913]), he did not believe that the energy levels of either the oscillator or the radiation were quantized, even though Einstein had quantized both of them: the photon in 1905 (Einstein [1905]) and the oscillator in 1907 (Einstein [1907]). We shall introduce a number of quotes from Planck’s original theoretical paper on quantum theory (Planck [1900b]) which is usually taken as the birth of quantum theory, and from his heat radiation book, to prove his aversion to quantizing anything physical. It was not until Bohr had quantized the levels of the hydrogen atom (Bohr [1913]), and the discussions that followed this, that Planck and most of his colleagues accepted quantization as a fact of nature. It is well known that Planck rejected the idea of photons until quite late, but here is a quote from the introduction of his 1913 book that not only proves that, but that also outlines his philosophy on the subject, which we think not only explains his opinions, but also made it possible for him to discover the law of blackbody radiation long before either he or anyone else understood its consequences. He says, “While many physicists, through conservatism, reject the ideas developed by me, or at any rate maintain an expectant attitude, a few authors have attacked them for the opposite reason, namely, as being inadequate, and have felt compelled to supplement them by assumptions of a still more radical nature, for example, by the assumption that any radiant energy whatever, even though it travel freely in a vacuum, consists of indivisible quanta, or cells. Since nothing probably is a greater drawback to the successful development of a new hypothesis than overstepping its boundaries, I have always stood for making as close a connection between the hypothesis of quanta and the classical dynamics as possible, and for not stepping outside of the boundaries of the latter until the experimental facts leave no other course open. I have attempted to keep to this standpoint in the revision of this treatise necessary for a new edition.” Obviously he is referring disapprovingly to Einstein’s photons in the quote. However we shall see that the last part of the quote is also very relevant in deciding what he actually did.

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What then led to the radically new form of the radiation law? It was the breaking up of the energy cells into finite units, so that statistics could be applied, which as he said he took directly from Boltzmann [1877a] (who ultimately let the cell size go to zero), and his introduction of a new way of counting microstates. We point out that the introduction of the quantum alone was not enough to produce his formula. This is because when one uses Boltzmann statistics, if one changes the cell size one merely changes the thermodynamic probability by an exponential multiplicative factor, which in turn leads to an additive constant in the entropy. Since in classical entropy an additive constant has no physical significance, this is why the cell size does not matter in classical physics. Planck’s argument, which led to a counting scheme that looks very like Bose statistics, introduced a cell size that is unique, and in fact is a fundamental constant of nature, and this was caused by his taking the entities that occupied these quantized units as indistinguishable. He was silent on this matter, and it took a long time for people to realize it. The first inkling of what was happening came from two papers in 1911, one by Natanson [1911], and the other by Ehrenfest [1911]. They both singled out Einstein’s [1905] derivation of the localized photon-like properties of electromagnetic waves in the limit where Wien’s radiation formula worked (Einstein took Wien’s formula as his starting point). They pointed out that his argument would not work with Planck’s formula instead of Wien’s and that Einstein’s argument presumes Boltzmann statistics. They then point out that Planck’s argument assumes that the energy units are indistinguishable, which they each find very puzzling. [Of course with hindsight, we realize that Wien’s formula holds in the particle-like domain where Einstein was operating, while the Rayleigh–Jeans formula holds in the wave-like regime. Einstein’s later 1909 paper on fluctuations (Einstein [1909]) sets out the particle–wave dichotomy for photons for the first time.] Before we begin, we would like to point out that there were many historical ironies in Planck’s development. His thesis advisor, Phillip von Jolly, in 1879 told him that the development of the first and second laws of thermodynamics had completed the structure of theoretical physics and that a bright young man should think twice about entering the field (quoted by Heilbron [1996]). (This in spite of the fact that Maxwell’s equations had been developed only ten years earlier. But it is known that Einstein in 1900 couldn’t find a course on electrodynamics at Zurich, and had to teach himself. New advances percolated at a slow rate in those days.) Planck nonetheless thought that there was a lot left to do regarding entropy, and spent most of his early career developing the consequences of the second law, namely in chemical thermodynamics. This led to a call to Berlin in 1889 for him

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to replace Kirchhoff, who had retired (but Planck was not initially appointed as a full professor). He soon complained in a letter that “nobody in Berlin is interested in entropy” (Heilbron [1996]). But when he started working on blackbody radiation, he immediately looked for the connection between entropy and energy, while he said everyone else was looking for the connection between frequency and temperature. Although he was initially under the influence of Ostwald and Mach (the great disbelievers in atomic theory, since atoms were then considered unobservable), Planck had slowly come to believe in atoms, as he thought it was the only way to treat certain problems, such as heat conduction and osmotic pressure, but he was sure that they were to be treated by mechanics. He was bitterly against the probability arguments of Boltzmann, whom he otherwise respected, because he thought the second law had to be exact. In fact he set an assistant, Zermelo (later of axiomatic set theory fame), to develop one of the two main arguments against Boltzmann, the “ergodic” argument (Zermelo [1896a, 1896b], Boltzmann [1896, 1897]), that a system in phase space will ultimately return to a point arbitrarily close to where it is now, even if far from equilibrium. [The other argument, due to Loschmidt, was the “time reversal” argument (Boltzmann [1877b]), that for every state heading toward equilibrium, there is another time-reversed state heading away from it.] He started to work on the blackbody radiation problem in 1896, and he thought (Planck [1900c]) he had proven Wien’s empirical radiation law (an exponential form, which actually holds only at relatively low temperatures, or high ν/T ). By 1900, experiments were being carried out at higher temperatures and lower, infrared frequencies, and the experimentalists, Lummer and Pringsheim, and Planck’s colleagues, Rubens and Kurlbaum, were finding out that Wien’s law did not work. The energy at a given frequency at higher temperatures was becoming linear in the temperature (in accord with the not-yet-stated Rayleigh–Jeans law). Planck developed his radiation law in a somewhat ad-hoc manner; the law worked very well, and he then set about to develop a theoretical explanation. At that time, there were a number of proposed hypothetical laws to deal with the discrepancy being discovered in Wien’s law. Planck’s worked almost perfectly, and was quickly accepted by the experimentalists. But it was clear that for any law to be taken seriously, it had to be theoretically motivated. Planck had become convinced that one could never discover the universal energy function by purely thermodynamic means, and he reluctantly decided to switch to Boltzmann’s methods. In a later famous letter that Planck [1931] wrote about the success of that effort (letter to R.W. Wood, quoted by Hermann [1971]), he said that switching to probability arguments was “an act of desperation”, but that breaking the cell into units

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of hν was “purely a formal assumption and I really did not give it much thought” (since the dependence on ν is actually required by Wien’s spectral law, a direct consequence of the second law). In order to apply the probability theory, Planck wrote S = kB ln W , to connect the thermodynamic probability W with the entropy S. In doing so, he wrote this equation for the first time, as Boltzmann, on whose tombstone the equation appears, never actually wrote it. Boltzmann always used the H-theorem, or something equivalent. Another related irony is that Boltzmann never wrote kB as a separate constant, but always used (R/N0 ), the gas-constant per molecule. Planck’s radiation law allowed one to calculate h, kB and N0 accurately for the first time, as well as the electrical charge e, from the Faraday constant. As a result, Planck thought it only simple justice that kB should be called Planck’s constant, or at least the Planck–Boltzmann constant, but it never happened. The poor fellow was stuck with h! In 1908, Arrhenius (who wielded tremendous influence) tried to convince the Swedish Academy (quoted in Heilbron [1996]) to give Planck the Nobel Prize because “it has been made extremely plausible that the view that matter consists of molecules and atoms is essentially correct . . . No doubt this is the most important offspring of Planck’s magnificent work.” No mention of the quantum of action. But Planck had to wait another 10 years, because Lorentz [1908] had come up with an argument that the Rayleigh–Jeans law had to be the correct classical law, and that the reason it failed at high temperature was that the system could not come to equilibrium at high temperatures. He withdrew this opinion after the experimentalists convinced him that if the Rayleigh–Jeans law were correct, with its ultraviolet catastrophe, many substances would glow in the dark at room temperature. But the Academy decided that the jury was still out on Planck’s work. An even further irony is that Planck was ultimately convinced of the truth of the new quantum theory by Nernst’s Heat Theorem, that CV → 0 as T → 0. This implies that W → 1, and S → 0, with no additive constant, so that an absolute minimum entropy is reached at absolute zero. This is a purely probabilistic argument, so far had Planck gone in changing his view. The ultimate irony is that when Boltzmann committed suicide, from sickness and frustration with all his critics, the University of Vienna offered Planck his chair. (Planck, who loved Vienna and was a professional-quality pianist, was tempted. But his colleagues at Berlin managed to make it worthwhile for him to stay.) Planck’s [1913] book praises Einstein’s derivation of CV , but he never mentions his quantization of the energy levels of the oscillator to En = nhν, or uses it. He merely says it is beyond the scope of his book, but it strikes us as rather strange

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that he chose not to further comment on it, as it seems to disagree with Planck’s interpretation.

2.2. Thermodynamic background leading to the radiation law The concept of a black body was introduced by Kirchhoff [1860]. In what follows, including Planck’s Law, we are going to give a rather self-contained argument that will not always be historically complete, although we will indicate certain occasions where historic remarks are relevant, because they determine motivations, and provide a context for what people did. We note that the history is often fairly complicated, controversies arose and sometimes took many years to get resolved. Sometimes it is even true that no one knew precisely what had been accomplished until much later. [As an example, we note that the “ultraviolet catastrophe” was not even named as such (Ehrenfest [1911]) until 11 years after Planck had solved the problem!] We are not professional historians, but at least in the case of Planck, there are many “smoking guns” within his work to justify what we say. Kirchhoff knew from looking at spectral lines from the sun that there was heat energy in empty space, and postulated equilibrium radiation. But the knowledge of what it consisted of was primitive. Maxwell’s equations had not yet been postulated, and the identity of heat rays and light rays had not yet been established. Nor had the existence of atoms in the walls of a cavity, nor the fact that an oscillator radiates and absorbs electromagnetic energy, or that such energy carries momentum. Thus it is rather amazing that Kirchhoff should have established on the basis of relatively simple arguments that within a cavity at equilibrium, this radiation should be independent of the substance of the walls of the cavity, and that at a fixed temperature a good emitter of radiation should be a good absorber. A perfect absorber should then radiate an energy equivalent to everything that falls upon it within the cavity at equilibrium, independently at each frequency. The radiation emitted by such a perfect absorber he called black radiation, and there should then be a universal function u(ν, T ) that describes the radiation energy density in equilibrium with the walls, that on average gets both absorbed and re-emitted, at any particular frequency and temperature. Because of the unknown nature of what happened within the cavity, Kirchhoff was attacked for each of the assumptions he made leading to this conclusion, and the existence of this universal function was dismissed by many. Meanwhile others tried to change the assumptions and re-derive the results. Even after the turn of the century this argument went on (well after Planck’s work). Although Planck does not explicitly mention these controversies in the 1913 edition of his book on

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heat radiation, he was nonetheless clearly affected by them, as he takes over 20 pages to discuss and justify Kirchhoff’s law. However, after Maxwell, Boltzmann tried to find a thermodynamic “equation of state” for the radiation in 1884 (similar to P V = N RT for particles) (Boltzmann [1884]), and after Hertz had produced electromagnetic waves in 1888, Wien in 1893 tried to find the spectral function of Kirchhoff (Wien [1893]). He succeeded to the extent of reducing the problem to a single function of ν/T , which is as far as one can go thermodynamically, and for which he ultimately won the Nobel Prize. Since the radiation hitting an area A of the wall of a cavity carries both momentum density and energy density, Boltzmann was able to treat it similarly to a particle flux hitting the wall, and showed that 1 (2.1) u(T ), 3 where Pν and uν refer to the pressure and energy density between frequencies ν and (ν + dν). The difference between this formula and the non-relativistic one is the factor 1/3, rather than the 2/3 for particles, which comes from the nonrelativistic form for the energy (E = mv 2 /2 = pv/2) rather than the extreme relativistic form for light (E = pc). Boltzmann [1884] then used this in connection with the second law of thermodynamics P (T ) =

dU = T dS − P dV , ∂U ∂S ∂P =T −P =T − P, ∂V T ∂V T ∂T V together with U = V u, to get 1 du 1 u=T − u, 3 dT 3 du 4u = T , dT ln u = 4 ln T + const, u = σ T 4 ,

(2.2)

(2.3)

the Stefan–Boltzmann law. Before Boltzmann derived it theoretically, Stefan [1879] had correctly guessed its form by examining some data that were not only inadequate, but that we now know were inaccurate as well. For the entropy, we again use eq. (2.2). If we define the entropy density s by S = V s, then 1 u = T s − u, 3

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4u 4 (2.4) , S = V s = σ T 3V . 3T 3 Equation (2.4) implies that during an adiabatic expansion of the cavity, so that the entropy is constant, we will have V T 3 = const. s=

2.2.1. Wien’s spectral law In 1888 Hertz showed the reality of Maxwell waves. In 1893 Wien applied the laws of thermodynamics and electromagnetism to the problem of blackbody radiation (Wien [1893]) and succeeded in reducing Kirchhoff’s universal function to a function of one variable. That is as far as one can go in classical physics. Wien tackled the problem of including the frequency in the blackbody law by considering an adiabatic motion of a wall of the cavity. This induced a Doppler shift on the radiation, while at the same time the wall did work on the radiation. Born’s Atomic Physics book (Born [1949]) has a simplified treatment in an appendix. But we will consider a much simpler technique based on adiabatic invariance, that was not available to Wien, but was first introduced by Ehrenfest in 1913. Ehrenfest [1913] was looking for some quantity that does not change while the external parameters of the system undergo a slow adiabatic change. He reasoned that such a quantity would be a good candidate for quantization, since it would not undergo a gradual change during the process, but could only change abruptly. This became the theoretical underpinning for the Bohr–Sommerfeld–Wilson quantization rule. First we have to find the normal modes of the radiation. We assume the cavity is a cube, of side L, since for all but the lowest normal modes the shape does not matter. We also assume that the walls are fully reflecting and use standing-wave boundary conditions. Then the modes for a Fourier expansion of the field satisfy kx L = nx π,

ky L = ny π,

kz L = nz π,

k 2 = ω2 /c2 .

(2.5)

The last of these equations comes from the wave equation for the fields. Here the n’s are positive integers. The number of modes in a region is given by L3 kx ky kz π3 L3 L3 8π 3 2 2 → 4πk dk = 4π ν dν. (2.6) 8π 3 8π 3 c3 The 8 in the denominator in the second line is due to the fact that the n’s are positive, so only the first octant is important, but we are integrating the k’s over all of k space. Finally we must introduce another factor of 2 because there are two degrees of polarization for each direction in k space. So nx ny nz ≡ Σn =

Σn = V

8π 2 ν dν. c3

(2.7)

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Rayleigh [1900] introduced the counting of modes of the field. He did it only qualitatively, following an earlier procedure he had used for sound waves. He then said the total energy density should be 8π 2 (2.8) ν kB T dν. c3 Here ε¯ represents the average energy of a mode, which by the equipartition theorem should be kB T . In 1905 he added the numerical factors in the above equation (Rayleigh [1905]), but made a minor mistake which was corrected by Jeans [1905], who emphasized how important and inescapable the above formula is. It has since been known as the Rayleigh–Jeans law. Later Lorentz [1908] also gave a very general derivation, and for a while he and Jeans believed that the reason the equation did not work experimentally was because it was difficult to establish equilibrium at high frequencies, and the experiments were therefore not correct. But the equation blows up at high frequencies and so cannot be correct, a problem labeled by Ehrenfest [1911] as the “ultraviolet catastrophe”. To establish Wien’s law, one need only note that in eq. (2.5), if one slowly changes L, then ki will slowly change, but ni cannot and will stay fixed (Ter Haar [1967]). This leads to uν dν = dU/V = Σn ε¯ n =

ki L = const,

νL = const,

ν 3 V = const,

T 3 V = const (adiabatic change),

(2.9)

ν/T = const. The second line above is just eq. (2.4), and so since the entropy of each mode, sn , remains constant during an adiabatic change, one must have 8π S/V = Σn sn = 3 (2.10) ν 2 sn(ν) (ν/T ) dν ≡ sν dν, c and therefore 3 3 8π (2.11) T sν = ν 3 (T /ν) 3 sn (ν/T ) = ν 3 f (ν/T ). 4 4 c The Rayleigh–Jeans law obviously takes this form, and so does an empirical radiation law proposed by Wien [1896]: uν =

uν = aν 3 e−bν/T .

(2.12)

[We call this Wien’s radiation law, to differentiate it from eq. (2.11), Wien’s spectral law, firmly embedded in the laws of thermodynamics. Equation (2.11)

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is sometimes called Wien’s displacement law, but we reserve this for the statement concerning the frequency at which the energy distribution uν is a maximum, νmax /T = const, a consequence of eq. (2.11).] Prior to 1900, all measurements were taken in the relatively high frequency domain, and Wien’s empirical law held pretty well. In fact, Planck had convinced himself that it must be the universal law. But the situation started to change after improvements were made to the experimental equipment. Then Rubens reported to Planck that at higher temperature for a given frequency the results were becoming linear in T , and Planck realized he had to rethink his ideas.

2.3. Planck’s introduction of the quantum of action In his first theoretical paper in 1900, Planck [1900b] makes two very confusing statements about the quantization of energy. He gives two successive sentences that are totally contradictory. After telling us that he will use Boltzmann’s method, he says, “If E [the energy of the N resonators of energy ν] is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways. We consider, however – this is the most essential point of the whole calculation – E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55×10−27 erg·s. This constant multiplied by the common frequency ν of the resonators gives us the energy element ε in erg, and dividing E by ε we get the number P of energy elements which must be divided over the N resonators.” This statement is often quoted in history of quantum theory books and articles, and it certainly looks like Planck is talking about quantized energy levels. However the very next sentence reads, “If the ratio is not an integer, we take for P an integer in the neighborhood.” Now if he really meant for the energy units to be quantized, P would naturally be an integer. Instead, we believe that he meant that the resonators could have any energy between nε and (n + 1)ε, and one just lumped them all together as nε. In other words, he was quantizing in phase space, as Boltzmann had done, because as he said, one could not count states otherwise. He went on to count and characterize the energy elements, ε, but he never said that an oscillator’s total energy must be nε, as Einstein later did. This is because, as we shall show below, he never believed it to be so. In 1906–7, Planck gave a series of lectures in Berlin, which were published as a rather comprehensive book on “heat radiation”. He put out a second edition in 1913 (Planck [1913]). So the statements in the book should be indicative of how Planck thought about the subject as late as 1913.

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There is no doubt that he introduced a quantum of action. He says as much in opening the preface to the second edition, “Recent advances in physical research have, on the whole been favorable to the special theory outlined in this book, in particular to the hypothesis of an elementary quantity of action”. But exactly what was quantized? He says on p. 125, “By the preceding developments the calculation of the entropy of a system of N molecules in a given thermodynamic state is, in general, reduced to the single problem of finding the magnitude G of the region elements in the state space. That such a definite finite quantity really exists is a characteristic feature of the theory we are developing, as contrasted with that due to Boltzmann, and forms the content of the so-called hypothesis of quanta.” It would seem fairly certain from this statement that his interest was in quantizing phase space. Shortly thereafter, in Part III, Chapter III, p. 135, he introduces a model of the linear harmonic oscillator, specifically in phase space. He talks about the energy as an ellipse, and makes the transition from the coordinates p and x to E and ϕ. He introduces the unit of action and takes the ellipses to have the average energy (n + 1/2)ε. He then makes an argument defending the appearance of what we now call “zero-point energy” (although his interpretation of it is totally different, having nothing to do with the uncertainty principle). It is hard to see why he would do that unless he thought the actual energies were distributed throughout the ellipse. In Part IV, Chapter III, he shows in more detail his ideas about the emission of radiation. To modern eyes, this new theory of Planck’s looks very strange, as it makes absorption and emission totally different processes. But it was used by a number of people for a while, and it could explain the photoelectric effect, and a few other things, but it was forgotten relatively soon after Bohr quantized the hydrogen atom later that year. (Bohr’s theory itself took some time to become accepted.) But it shows how Planck’s thinking was totally in flux, and how even then he was unwilling to believe in the quantization of energy levels. On p. 161 he says, “Whereas the absorption of radiation by an oscillator takes place in a perfectly continuous way, so that the energy of the oscillator increases continuously and at a constant rate, for the emission we have, in accordance with sec. 147, the following law: The oscillator emits in irregular intervals, subject to the laws of chance; it emits, however only at a moment when its energy of vibration is just equal to an integral multiple n of the elementary quantum ε = hν, and then it always emits its whole energy of vibration nε.” He then describes how the oscillator absorbs energy at a constant rate, so that its energy increases linearly in time, and as it passes a given energy nε, it may or may not radiate. If not, it continues on toward (n + 1)ε. So the oscillator energy

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is not quantized, but it emits in quantized units, of multiples of the quanta. On the basis of this model, he then goes on to calculate, p. 166, “Hence in the state of stationary equilibrium the number of oscillators whose energy lies between nhν and (n + 1)hν is . . . ” and proceeds to give a complicated formula. But it is clear that the energy levels of the oscillator are not quantized, nor is the absorption of radiation. Only the emission of radiation is. Presumably after emission, the radiation got thermalized. So by this time in his thinking, something was quantized, but it did not stay quantized. He even draws a diagram giving the saw-toothed form described above for the energy of a single oscillator as a function of time. We would like to say something about Planck’s intellectual attitude, which was summarized in the quote we gave at the beginning. He was an insider, an intellectual leader of the German community, and a man of total integrity. He had not the slightest desire to overthrow, or to see the overthrow of, the hard-won victories of classical science. And yet in times of crisis, he had the moral courage both to introduce a notion that he knew was radical, and whose implications no one could comprehend at the time, and also to suddenly abandon a strong belief that had sustained him throughout his career until then, namely that statistical considerations could not play a fundamental role in the understanding of physics at a profound level. The quote shows clearly that he would willingly go as far as he thought he had to go, but absolutely no further, and he lived up to this conviction. For this reason, we believe that nobody but Planck could have made the advance that he made, when he did. His first paper was a purely phenomenological gimmick, which he made by performing his analysis in terms of entropy. As he said, he had devoted his life to examining entropy, which few people at the time took seriously. In his second paper he realized that the gimmick of the first paper had to correspond to a fundamental finite unit of action. But what that meant, nobody was prepared to say at that time. His own explanations were fuzzy, arbitrary, and had many loopholes. We think his revelation took the subject as far as it could have gone without a deeper analysis, which after all would consume many years of work by many people. In the total state of ignorance at that time, we think he did exactly what he was mentally inclined to do. He took the subject as far as it could go at that time, and no further. He introduced the quantum of action, and it worked, but its significance was very obscure. However it is important to realize that quantized energy levels, for both radiation and matter, are features of nature. Quantized cells in phase space are artifacts of theory. It is interesting that he was willing to accept the latter, which he could hope to fix, but was not willing for a long time to accept the former, which would invalidate most of classical physics. His conservatism led him for many years to try to find a close-to-classical explanation for what he had done, and he was strongly inclined against the radical

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advances of others, which is why it was left for Einstein to quantize both the oscillator levels and the electromagnetic field. On the other hand, in subjects where radical ideas could immediately lead to clear conceptual advances, he was quick to approve, and he was one of the earliest supporters of relativity theory, and in fact spent most of his research time between 1905 and 1908 trying to advance the theory, and convince his peers of its validity. Planck’s position in the German Physical Society made his voice the primary one in deciding what should be published in Annalen der Physik, the leading physics journal of the time, and his openness to radical new ideas, such as Einstein’s, is almost without parallel (one wonders whether a paper such as Einstein’s special relativity paper would get published in Physical Review today?) He even allowed Einstein to publish his photon paper, with which he strongly disagreed.

2.4. Planck’s derivation of the blackbody radiation law When Planck attacked the problem of blackbody radiation, he realized that since the results were independent of the nature of the material in the cavity, one could use a simple model for the cavity. So he chose to consider a damped harmonic oscillator as a model for the material in the walls. His results are arrived at simply in Born’s [1949] book. For absorption of radiation, if one has an oscillator of natural frequency ω0 , and weak damping, γ , which is being driven at frequency ω, the equation of motion will be the real part of mx¨ + mγ x˙ + mω02 x = E0x eiωt .

(2.13)

Then when one compensates for the three-dimensionality of the problem, and assumes that E0 represents the equilibrium radiation present at temperature T , one finds that the power absorbed is πκe2 dEabs = u(ν0 ), dt 3m where κ = 1/4π 0 in mks units, while the power radiated is given by

(2.14)

2κe2 ω02 dErad 2κe2 a¯2 (2.15) = ε¯ , = 3 dt 3mc 3mc3 where the term with a is the average of the acceleration-squared, and ε¯ represents the average energy of the oscillator. Combining eqs. (2.14) and (2.15) gives uν =

8πν 2 ε¯ . c3

(2.16)

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Planck had this result well before Rayleigh had published his node-counting argument. All Planck had to do was insert the equipartition result kB T for ε¯ , and he would have had the Rayleigh–Jeans formula considerably before Rayleigh. But he never did, and there has been considerable debate as to why. Could he have not known about equipartition, since at this time he was an avid attacker of the entire statistical mechanics enterprise? This would seem very unlikely, as he was interested in specific heats, and would have known about the Dulong–Petit law controversy (Dulong and Petit [1819]; some solids did and some did not have U = 3RT V ). Or was he aware of it but already had no confidence in it, as Wien’s empirical law, eq. (2.12), seemed to be holding up nicely? We are unlikely to ever know. At any rate in 1900 Planck found out that Wien’s law was not holding up, and he had to make a report to the Berlin physical society. From his long experience in thermodynamics, he later said that he immediately started searching for the solution in the relation between entropy and energy, while everyone else was worried about the relation between ν and T . Planck had derived a formula for the approach to equilibrium by an oscillator in a blackbody cavity that had a small excess energy U over its equilibrium value (Planck [1900d]). Then if its energy changed by dU , the change in entropy of the entire system (oscillator plus field) would be dStot =

3 d2 S dU U. 5 dU 2

(2.17)

So the function d2 S/dU 2 is clearly connected to fluctuations about equilibrium, although at the time Planck was not thinking statistically. It was Einstein in 1909 (Einstein [1909]) who clearly brought out the direct meaning of this function as a statistical measure of fluctuations. He inverted the formula S = kB ln W to the form W = eS/kB . Then one can connect the entropy of an arbitrary state to its probability. If W is a maximum for S = S0 = S(E0 ), the state with maximum entropy and minimum energy, then very close to equilibrium we can write 1 ∂ 2S S = S0 − α(E − E0 )2 , −α = , 2 ∂E 2 0 W = eS0 /kB e−α(E−E0 )

2 /k B

.

(2.18)

There is no linear term since S0 is a maximum. If we then ask for the energy fluctuations about equilibrium, we get (E − E0 )2 W (E) dE kB 2 2 E = (E − E0 ) = (2.19) . = 2α W (E) dE So 1/α = −2/(∂ 2 S/∂E 2 )0 is a measure of the energy fluctuation.

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Planck’s first derivation of his radiation formula was a purely numerical manipulation. Nonetheless, it is very interesting because it is profoundly and directly connected to fluctuations, in a way that Planck could not have foreseen. He knew that entropy was the key to the problem, and he thought the answer was directly related to the quantity 1/α of eqs. (2.18) and (2.19), which governed the return to equilibrium, via eq. (2.17). Until a few days earlier, when Rubens had come to him, he had thought that Wien’s empirical law, eq. (2.12), was the correct solution to the problem. Using his own eq. (2.16), together with eq. (2.12), he wrote c3 caν −bν/kB T (2.20) uν = . e 2 8π 8πν One could use this to express S directly in terms of E by eliminating T , since at constant V , 1/T = (∂S/∂E)V = ∂s/∂ ε¯ , where s is the entropy per oscillator. Therefore from eq. (2.20), ε¯ =

∂s 1 1 8π ε¯ = = − ln 3 , T bν c aν ∂ ε¯ 1 ∂ 2s =− . (2.21) 2 bν ε¯ ∂ ε¯ This is the expression which Planck had previously thought to be exact, and even that he could derive it with some plausible assumptions. The new knowledge given to him by Rubens that, at low frequencies in the newly accessible infra-red region, uν ≈ ν 2 T as had just been predicted by Rayleigh, he wrote as (using uν = Aν 2 T ) c3 c3 u = AT = kB T , ν 8π 8πν 2 kB ∂s kB ∂ 2s 1 =− 2. = = , (2.22) 2 T ∂ ε¯ ε¯ ∂ ε¯ ε¯ (The last equation of the first line is just the equipartition theorem, which was used by Rayleigh, although not by Planck, to give the value of A.) Planck says he spent the next few days looking for an extrapolation between these two extremes that gave plausible behavior, and finally came up with ε¯ =

∂ 2s kB =− , 2 ε¯ (Δ + ε¯ ) ∂ ε¯

(2.23)

where Δ is independent of the temperature. In the limit ε¯ Δ, ∂ 2s kB →− , ε¯ Δ ∂ ε¯ 2

Δ = bkB ν ≡ hν,

(2.24)

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where h is a new physical constant. The fact that Δ must depend linearly on ν comes from Wien’s spectral theorem, a thermodynamic necessity. In the other limit, ε¯ Δ, we have eq. (2.22), the Rayleigh–Jeans law. We can of course integrate eq. (2.23), to get Planck’s formula, which is still valid today, ∂ 2s kB kB 1 1 = − = − − , ε¯ (Δ + ε¯ ) Δ ε¯ Δ + ε¯ ∂ ε¯ 2 ∂s 1 kB ε¯ Δ + ε¯ = = − ln , = eΔ/kB T = ehν/kB T , ∂ ε¯ T Δ Δ + ε¯ ε¯ hν . ε¯ = hν/k T (2.25) B −1 e 2.4.1. Planck’s theoretical derivation As we have said, Planck imagined that there were a series of oscillators in the walls, in equilibrium with the radiation. Since each oscillator reaches equilibrium with the same frequency of radiation as the oscillator itself, and those of different frequencies all behave independently, we can consider each frequency independently. This had previously given rise to much controversy, the problem being how independently behaving oscillators could ever come to equilibrium, especially if one considered the walls of the cavity to be perfectly reflecting. The prevailing opinion was that this was an abstraction, and if one thought of a small lump of coal (that absorbed all frequencies) as also being inside the cavity, it would force all frequencies to come to equilibrium together. Next he considered that for each frequency, if there were N oscillators, the total energy was divided into P discrete units of size ε = hν. As we have said, it does not matter whether one considers this to be because the energy is quantized or because one considers all the energy E between frequencies ν and ν + dν to be lumped together and considered as E/ε = P discrete units. Planck in any case was psychologically not disposed to seeing the energy as quantized, and as we have emphasized, long resisted it. If Planck were to continue following Boltzmann, he could further divide this into nk oscillators with energy kε so that nk = N, (2.26) kεnk = E, and then find the distribution of nk ’s which has maximum probability. But Planck stated that one didn’t even have to go this far. He merely said that most of the time the system will be very close to equilibrium, and the rest constitute rare events that will hardly contribute, so he just took the total number of possible ways to distribute the P units of energy over the N oscillators. How many such ways are there?

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A simple way to see this (due to Ehrenfest) is just to draw two vertical bars, and randomly distribute P circles, and N − 1 other bars between them. For example, the arrangement |oo||o|ooo|||oo| . . . . . . |oo|o| would represent 2 units of energy in the first box (oscillator), none in the second, 1 in the third, 3 in the fourth, etc., altogether taking up N boxes. How many possible such arrangements are there? There are N − 1 + P objects, which we can distribute in (N − 1 + P )! ways, and since the order of the circles and bars does not matter, the total becomes W =

(N − 1 + P )! . (N − 1)!P !

(2.27)

This is the total number of ways of distributing the energy amongst the oscillators, and the overwhelming majority of such arrangements lie close to equilibrium. Planck next assumed that N 1, P 1, and ln N ! ≈ N ln N − N . Thus S = kB ln W ≈ kB [ln(N + P )! − ln N ! − ln P !] = kB (N + P ) ln(N + P ) − (N + P ) − (N ln N − N )

− (P ln P − P ) P P P P ln 1 + − ln . + kB N 1 + N N N N

(2.28)

So the entropy only depends on the average number of energy units per oscillator, P /N. Then since ε¯ = E/N = P ε/N,

P /N = ε¯ /ε.

(2.29)

(We have been using ε¯ to represent the average energy per oscillator, while ε is just hν, the energy unit.) So finally, ε¯ ε¯ ε¯ ε¯ s = S/N = kB 1 + (2.30) ln 1 + − ln . ε ε ε ε Then, as before 1 ∂S ∂s kB ε¯ + ε = = = ln , T ∂E ∂ ε¯ ε ε ε ε¯ = ε/k T , e B −1 ε 8πν 2 8πν 2 uν = 3 ε¯ = 3 ε/k T . c c e B −1

(2.31)

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This is Planck’s derivation of his formula. If we take an extra derivative of the first line of eq. (2.31), we get kB ∂ 2s =− . ε¯ (ε + ε¯ ) ∂ ε¯ 2

(2.32)

This reduces to Planck’s previous numerical formula, where we see that ε plays the role of his constant Δ, which was necessary to make the formula work. If ε → 0, we lose the behavior of Wien’s empirical formula at high energies, which is the limit in which Einstein introduced the particle behavior of photons. Rosenfeld, in writing a history of early quantum theory (Rosenfeld [1936]), claimed that Planck probably worked backward from eq. (2.31) to get the entropy (2.30), from which he could guess the right combinatorial law for W , eq. (2.27), which appears in Boltzmann’s [1877a] original article.

2.5. Some comments on the Planck derivation There are a number of things to notice about Planck’s derivation, some of which we have noted earlier. First, what does it mean to keep ε finite, since for the case of particles using classical statistics, cell size doesn’t matter? We pointed out earlier that it was noticed independently by Natanson and Ehrenfest in 1911 that the Planck derivation treats all the energy elements as equivalent, so that it is clearly different from Boltzmann’s statistics, and in fact makes them indistinguishable. Ehrenfest also showed in 1906 (Ehrenfest [1906]) that the Planck derivation puts an extra constraint on the system that he said could be satisfied in several ways, but that the most natural was to strictly quantize the energy levels of the oscillators. Einstein actually did this in 1907, in his famous specific heat paper (Einstein [1907]). A number of people noticed that since ε = hν, one of the basic assumptions of the theory cannot work. Equations (2.29) and (2.31) assume that N and P 1. But for high enough frequencies at a given temperature, ε becomes quite large, and most of the oscillators will be in their ground state. This is why equipartition breaks down, since classical physics scales the frequency so that all frequencies are equally important, and they all have the same average energy, kB T . The Planck formula correctly identifies the parameter ε/kB T as the important dividing line, but the assumptions of the derivation also break down at high frequency. The Einstein derivation of 1907 (where the energy levels are quantized, and the probability that a state En = nε is occupied is Pn = A exp(−nε/kB T )), does not suffer from this defect.

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Once he had shown that the energy levels of the oscillator are quantized, Einstein also realized that eq. (2.16), connecting uν with the average energy of an oscillator, ε¯ , is inconsistent, since it was derived using a classical oscillator that absorbs and emits energy continuously. But he thought the equation must be true on the average. So it is clear why Planck’s derivation left the situation in a state of great confusion for a long time. 2.6. Einstein’s fluctuation argument In 1909, Einstein looked at the fluctuations in the Planck formula and noticed a simple, but very deep relation (Einstein [1909]). It was in this paper that Einstein introduced eq. (2.19) for the fluctuations. We can see the result already from Planck’s early ad-hoc derivation of his result, eq. (2.23). and if we insert eq. (2.23) into eq. (2.19), we get

kB kB E 2 = = − 2 = ε¯ Δ + ε¯ 2 . ∂ s 2α 2

(2.33)

∂ ε¯

If one believes that the energy levels of the oscillator are quantized, so that En = nΔ = nhν, as Einstein did, and ε¯ = nΔ, ¯ where n¯ represents the average level of the oscillator, one can also put this into the form E 2 (2.34) = (n)2 = n¯ + n¯ 2 . Δ2 This also holds true for the field excitations if one considers the field modes to be oscillators. Einstein then pointed out that in his paper on photons in 1905 [they were not explicitly called “photons” until 1926 (Lewis [1926])], he had used the Wien radiation formula when he discussed the radiation as resembling individualized excitations, and had shown how it resembled the independent particles of a perfect gas. He then identified the first term with the fluctuations of a group of independent particles, while the second term must correspond to the fluctuations in a cavity of classical waves. [This result was expressed in terms of the energy density of a small finite volume of the cavity, via eq. (2.16), but the justification was essentially a dimensional argument, which said that in the classical limit where Δ does not contribute, and one has nothing else with the dimensions of energy, one needs E 2 ∝ ε¯ 2 . An explicit later calculation by Lorentz [1912] proved the result.] But eqs. (2.33) and (2.34) are exact and hold even when one is not in either of the two classical limits represented by particles or waves. And so this paper is generally taken as the birth of the wave–particle duality that has perplexed physicists up to the present time.

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2.7. Einstein’s A and B coefficients In 1917, Einstein [1917] published his famous A and B coefficients paper. The paper was in two parts, the first of which discussed energy transformations and rates of absorption and emission for the various processes that go on in an atom or molecule in equilibrium with the radiation in a cavity. The second part discusses momentum transfer during these processes. This paper was very seminal in that it taught us how to think about radiation. It is not only the starting point for laser physics, but it also pretty much made the existence of energy levels essential, showing how they lead naturally to Planck’s radiation law. Einstein assumed that the molecule could occupy only a discrete set of allowed states {Zn } which had energies {εn }, and whose relative probability of occupation at temperature T is Wn = pn e−εn /kB T ,

(2.35)

where the pn represent statistical weights. He then assumes that a molecule can decay spontaneously from a state Zm to Zn (such that εm > εn ), and emit energy εm − εn . The probability per molecule for this to occur in time dt he takes as dW = Anm dt.

(2.36)

As analogies, he quotes radioactive γ decay and Hertzian oscillators. He then assumes that there are induced (stimulated) emission and absorption processes, which he calls a quantum-theoretical hypothesis, that he assumes take place with a probability dW = Bnm uν dt

(2.37)

for absorption from the lower level to the higher level, and n dW = Bm uν dt

(2.38)

for emission from the higher level to the lower level. These are for transitions induced by the external field. Even without introducing the quantized states, in the classical picture for absorption and emission used by Planck, eqs. (2.14) and (2.15), the rates were proportional to the density of the surrounding radiation. If we then equate emission and absorption at equilibrium (detailed balance), we get

n pn e−εn /kB T Bnm uν = pm e−εm /kB T Bm uν + Anm .

(2.39)

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Then if we take the limit T → ∞, for which also uν (T ) → ∞, then pn Bnm = n , and pm Bm uν =

n Anm /Bm (ε −ε )/k e m n BT

. (2.40) −1 This formula immediately leads to the Bohr rule εm − εn = hν, and in the high-temperature limit, where the Rayleigh–Jeans law holds, we can evaluate A/B, which leads to the Planck radiation law. Even after the development of non-relativistic quantum mechanics, until the advent of field theory, Einstein’s derivation was needed to calculate the spontaneous emission of radiation. Like in much of the rest of this story, there is an irony in Einstein’s introduction of his A and B coefficients. To Einstein himself, the most important part of the paper was the second part. The derivation in this part is more difficult, and is usually ignored today, but the point of the calculation was the consideration of momentum conservation in the radiation process, rather than merely energy conservation. By methods reminiscent of his derivation of Brownian motion, he proved that to preserve thermal equilibrium in a gas of atoms, or molecules, during the decay process one must consider that in the individual decays, the atom recoils, acquiring the appropriate momentum. In his words, “If the molecule undergoes a loss of energy of magnitude hν without external influence, by emitting this energy in the form of radiation (spontaneous emission), this process is also a directed one. There is no emission in spherical waves. The molecule suffers in the spontaneous elementary process a recoil of magnitude hν/c in a direction which is in the present state of the theory determined only by ‘chance’.” The irony implicit in this derivation is brought out in his subsequent statement, “These properties of the elementary processes required by Eq. (12) [an equilibrium equation of the momentum fluctuations] make it seem practically unavoidable that one must construct an essentially quantum theoretical theory of radiation. The weakness of the theory lies, on the one hand, in the fact that it does not bring any nearer the connection with the wave theory and, on the other hand, in the fact that it leaves moment and direction of the elementary processes to ‘chance’; all the same, I have complete confidence in the reliability of the method used here.” This is the paper that introduced chance into the radiation process in an essential way. After this, it was an inevitable and inescapable part of the quantum landscape. He had to introduce it in order to make it clear that the photons were emitted in individual quantum processes, and carried both energy and momentum. This was very important to him, because even at this late date, which was already 1917, the existence of the photon was not yet generally accepted. But even as he introduced the element of chance in an essential way, he lamented it.

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There is a strong parallel here between Einstein and Planck, who both introduced revolutionary thoughts brought about by necessity after a long intellectual odyssey. Yet no sooner had Planck let the genie of the quantum out of the bottle, than he devoted many years of effort to unsuccessfully trying to force it back in, without destroying the revolution it had brought about. And Einstein had the same experience. Once he had let the genie of chance out of the bottle, he unsuccessfully spent the rest of his life trying to stuff it back in. Not that this diminishes by one iota the accomplishments of these two great men, but it does point up the ironies that life has in store for the best of us. § 3. Bose–Einstein condensation In 1924, Bose made the seminal observation that it is possible to derive Planck’s radiation law from purely corpuscular arguments without invoking at all the wave properties of light resulting from Maxwell’s field equations. The main ingredient in Bose’s argument was the indistinguishability of the particles in question and a new way of counting them – now universally known as “Bose–Einstein statistics” – which pays careful attention to what is implied by their being indistinguishable. In the case of light quanta, an additional feature is that their number is not conserved, because light is easily emitted and absorbed. Massive particles (atoms, molecules, . . . ), by contrast, are conserved and therefore, as Einstein [1924, 1925] emphasized, their indistinguishability has further consequences, of which the phenomenon of Bose–Einstein condensation (BEC) is the most striking one. Bose–Einstein condensation has long been a fascinating subject and has attracted renewed interest in light of successful experimental demonstrations of BEC in dilute He4 (Crooker, Hebral, Smith, Takano and Reppy [1983], Chan, Blum, Murphy, Wong and Reppy [1988], Crowell, Van Keuls and Reppy [1995]) and ultracold atomic gases (Anderson, Ensher, Matthews, Wieman and Cornell [1995], Bradley, Sackett, Tollett and Hulet [1995], Davis, Mewes, Andrews, van Druten, Durfee, Kurn and Ketterle [1995], Fried, Killian, Willmann, Landhuis, Moss, Kleppner and Greytak [1998], Miesner, Stamper-Kurn, Andrews, Durfee, Inouye and Ketterle [1998]). Furthermore the production of “coherent atomic beams”, the so called atom laser (Mewes, Andrews, Kurn, Durfee, Townsend and Ketterle [1997], Andrews, Townsend, Miesner, Durfee, Kurn and Ketterle [1997], Anderson and Kasevich [1998], Bloch, Hänsch and Esslinger [1999]), and its relation to the conventional laser is intriguing; as is the relation between the BEC phase transition and the quantum theory of the laser (Scully and Lamb [1966], Scully and Zubairy [1997]).

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The physics of BEC is subtle with many pitfalls and surprises. For example, Uhlenbeck [1927] criticized Einstein’s arguments concerning the implied singularity in the equation of state at the critical temperature Tc . Einstein’s results require that the thermodynamic limit be taken, i.e., the number of particles N and the volume V are taken to be infinite with the density N/V being finite. This however leaves the question of how best to think about and define Tc for finite mesoscopic systems. A canonical ensemble, in which N particles inside a trap can interact and exchange energy with a thermal reservoir at temperature T , provides a natural approach to BEC. This canonical-ensemble approach is a useful tool in studying BEC properties in the current experiments on cold dilute gases (Anderson, Ensher, Matthews, Wieman and Cornell [1995], Bradley, Sackett, Tollett and Hulet [1995], Davis, Mewes, Andrews, van Druten, Durfee, Kurn and Ketterle [1995], Han, Wynar, Courteille and Heinzen [1998], Ernst, Marte, Schreck, Schuster and Rempe [1998], Hau, Busch, Liu, Dutton, Burns and Golovchenko [1998], Esslinger, Bloch and Hänsch [1998], Anderson and Kasevich [1999], Miesner, Stamper-Kurn, Andrews, Durfee, Inouye and Ketterle [1998], Fried, Killian, Willmann, Landhuis, Moss, Kleppner and Greytak [1998], Mewes, Andrews, Kurn, Durfee, Townsend and Ketterle [1997], Andrews, Townsend, Miesner, Durfee, Kurn and Ketterle [1997], Anderson and Kasevich [1998], Bloch, Hänsch and Esslinger [1999]). It is also directly relevant to the He-in-vycor BEC experiments (Crooker, Hebral, Smith, Takano and Reppy [1983], Chan, Blum, Murphy, Wong and Reppy [1988], Crowell, Van Keuls and Reppy [1995]). The dynamics and statistics of the condensate are then obtained from the canonical partition function. However, the N-particle constraint associated with the canonical ensemble is rather cumbersome, and no simple analytic expressions for the canonical partition function are known to exist for three-dimensional traps. Even numerical calculations for large N may become impractical. A way out is to calculate the grand canonical properties for the ideal Bose gas where the constraint of fixed particle number is relaxed. This was how Einstein derived the characteristics of the condensate and obtained the expression for the critical temperature. In general, we would expect that the macroscopic properties of the condensate for both canonical and grand canonical ensembles should be equivalent. However, as we discuss below, only properties related to mean number of condensed particles are almost identical in the two ensembles, and the mean-square fluctuations are remarkably different. Even as the temperature T approaches zero when all N particles condense in the ground state, the fluctuations in the grand canonical ensemble becomes huge, of the order of N 2 , as discussed below. This is clearly unacceptable.

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Recently, realizing the inherent similarity in the phase-transition behavior between laser and the Bose–Einstein condensation, a new approach was developed to study the nonequilibrium approach to BEC in the canonical ensemble using the methods employed in the quantum theory of the laser (Scully [1999], Kocharovsky, Scully, Zhu and Zubairy [2000]). The advantage of this approach is that analytic, though approximate, expressions are obtained for the canonical partition function for the Bose–Einstein condensate for arbitrary traps. The differences between the various moments for the condensate based on these analytic expressions and the exact numerical results are in most cases negligible. This approach also allows us to extend the critical-temperature concept to mesoscopic systems, involving, say, 103 atoms, in a natural fashion. However, before proceeding to give the details of the laser-theory based analysis of BEC, we recall Einstein’s arguments based on grand canonical ensemble and see whether we can extend these arguments in a natural way to describe a mesoscopic system. We also present the salient features of the quantum theory of the laser that become relevant in seeing the close connection between the nonequilibrium approach to the dynamics and statistics of the condensate of N -atom Bose gas and the photons inside a laser.

3.1. Average condensate particle number Here we present a derivation of the average condensate particle number following the original derivation of Einstein. We recall that Einstein considered particles inside a box in the thermodynamic limit. We consider particles in a harmonic trap and first discuss the thermodynamic limit. The difference between a box and the harmonic trap is in the density of states. We then go on to consider the mesoscopic number of particles. Following Einstein we work with the grand canonical ensemble in which the average condensate particle number n¯ 0 is determined as follows (Ketterle and Druten [1996]). The total number of atoms N in the trap is given by N=

∞ k=0

n¯ k =

∞ k=0

1 , exp[β(εk − μ)] − 1

(3.1)

where for the three-dimensional (3D) isotropic harmonic trap we have εk = h¯ Ω(kx + ky + kz ), with Ω the trap frequency, β = 1/kB T and μ the chemical potential. In the following we demonstrate how to calculate the mean number of condensed particles n¯ 0 for a 3D isotropic harmonic trap. Using n¯ 0 = 1/(exp(−βμ)−

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[8, § 3

1), we can relate the chemical potential μ to n¯ 0 as 1 + 1/n¯ 0 = exp(−βμ) and rewrite eq. (3.1) as N=

∞ ∞ nk = k=0

k=0

1 . (1 + 1/n¯ 0 ) exp(βεk ) − 1

(3.2)

For large n¯ 0 we neglect 1/n¯ 0 in comparison with 1. Following Einstein, we proceed to separate off the ground state so that eq. (3.2) can be written as N − n¯ 0 = H,

(3.3)

where H=

k>0

1 . −1

(3.4)

eβ k

For an isotropic harmonic trap with frequency Ω the degeneracy of the nth energy level is (n + 2)(n + 1)/2, and we obtain ∞

H=

1 1 (n + 2)(n + 1) ≈ 2 exp(βnΩ) − 1 2 n=1

∞ 1

(x + 2)(x + 1) dx. exp(xβΩ) − 1

(3.5)

In the limit kB T Ω we find 1 H≈ 2

∞ 0

x2 dx = exp(xβΩ) − 1

kB T Ω

3 ζ (3),

(3.6)

where ζ (x) is the Riemann zeta-function. We define the critical temperature Tc such that when T = Tc we have n¯ 0 = 0. This yields Ω N 1/3 Tc = kB ζ (3)

(3.7)

as the temperature of BEC transition in the thermodynamic limit. The resulting expression for the mean number of particles in the condensate is 3 T n¯ 0 (T ) = N 1 − , Tc

(3.8)

which shows a cusp at T = Tc . For a mesoscopic number of particles (e.g., a few hundred) eq. (3.8) becomes inaccurate as the thermodynamic limit is not reached. To improve the accuracy we first rewrite eq. (3.2) in the following way (Kocharovsky, Kocharovsky, Holthaus, Ooi, Svidzinsky, Ketterle and Scully

8, § 3]


[2006], Jordan, Ooi and Svidzinsky [2006]): 1 1 N − n¯ 0 = 1 . β k − n¯ 0 n¯ 0 + 1 k>0 e n¯ 0 +1

303

(3.9)

For n¯ 0 1, the term n¯ 0 /(n¯ 0 + 1) inside the summation may be approximated by 1. Then we obtain a quadratic equation for the mean number of particles in the ground state: N − n¯ 0 =

H 1 n¯ 0

+1

⇒ n¯ 20 + n¯ 0 (1 + H − N ) − N = 0

(3.10)

whose solution is 1 n¯ 0 = N − H − 1 + (N − H − 1)2 + 4N . (3.11) 2 The analytical expression (3.11) shows a smooth crossover near Tc for a mesoscopic number of particles N as shown in fig. 1. Here we compare the mean condensate number as given by eq. (3.11) obtained in the grand canonical ensemble (solid line) for N = 200 and the solution (3.8) (dashed line) that is valid only for a large number of particles N with the numerical calculation of n¯ 0 (T ) from the exact recursion relations in the canonical ensemble (dots) (Wilkens and Weiss [1997]). In the canonical ensemble the total number of particles N is fixed, rather than the chemical potential. We see that, for the average particle number, both ensembles (grand canonical and canonical) yield very close answers. The interesting

Fig. 1. The average condensate particle number versus temperature for N = 200 particles in an isotropic harmonic trap. The solid line is eq. (3.11), while the dashed line shows the thermodynamic limit formula (3.8). “Exact” dots are obtained numerically in the canonical ensemble (Wilkens and Weiss [1997]).

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[8, § 4

observation is that the approximate expression (3.8) obtained in a suitable limit within the grand canonical ensemble yields results that are indistinguishable from the exact results from the canonical ensemble. However, as we discuss below, this is not the case for the BEC fluctuations.

3.2. Fluctuations in the number of particles in the condensate Condensate fluctuations are characterized by the central moments μm = (n0 − n¯ 0 )m . The first of them is the squared variance (n0 − n¯ 0 )2 = n20 − n0 2 . (3.12) When the temperature T approaches zero, all N particles are forced into the system’s ground state, so that the mean square (n0 )2 of the fluctuation of the ground-state occupation number has to vanish for T → 0. However the grand canonical description gives (n0 )2 → N (N +1), clearly indicating that with respect to these fluctuations the different statistical ensembles are no longer equivalent. What, then, would be the correct expression for the fluctuation of the groundstate occupation number within the canonical ensemble, which excludes any exchange of particles with the environment, but still allows for the exchange of energy? Various aspects of this riddle have appeared in the literature over the years (Ziff, Uhlenbeck and Kac [1977], Ter Haar [1970], Fierz [1956]), mainly inspired by academic curiosity, before it resurfaced in 1996 (Grossmann and Holthaus [1996], Politzer [1996], Gajda and Rz¸az˙ ewski [1997], Wilkens and Weiss [1997], Weiss and Wilkens [1997]), this time triggered by the experimental realization of mesoscopic Bose–Einstein condensates in isolated microtraps. Condensate fluctuations can be measured by means of scattering of a series of short laser pulses (Idziaszek, Rz¸az˙ ewski and Lewenstein [2000], see also Chuu, Schreck, Meyrath, Hanssen, Price and Raizen [2005]). Since then, much insight into this surprisingly rich problem has been gained. Much of this insight follows directly from the quantum theory of the laser, to which we now turn.

§ 4. The quantum theory of the laser The quantum (photon) picture of maser/laser operation is a difficult problem in the interaction of radiation with matter. Even several years after the development of the maser and the laser there was not a fully quantized theory of laser action. The difficulties inherent in this problem were most succinctly stated by Roy Glauber [1964] in his Les Houches lectures:

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305

“The only reliable method we have of constructing density operators, in general, is to devise theoretical models of the system under study and to integrate [the] corresponding Schrödinger equation, or equivalently to solve the equation of motion for the density operator. These assignments are formidable ones for the case of the laser oscillator and have not been carried out to date in quantum mechanical terms. The greatest part of the difficulty lies in the mathematical complications associated with the nonlinearity of the device. The nonlinearity physics plays an essential role in stabilizing the field generated by the laser. It seems unlikely, therefore, that we shall have a quantum mechanically consistent picture of the frequency bandwidth of the laser or of the fluctuations of its output until further progress is made with these problems.”

Following the Les Houches meeting, Marlan Scully and Willis Lamb took up the challenge and developed a fully quantum-mechanical theory of laser that yielded the photon statistics above, at, and below threshold (diagonal elements of the laser field density matrix), showed that the laser linewidth was contained in the time decay of the off-diagonal elements of the density matrix, and made the physics clearer by comparing the laser threshold to a ferromagnetic phase transition (Scully and Lamb [1966]). They presented their theory of “optical maser” at the famous Puerto Rico Conference on the “Physics of Quantum Electronics” in the summer of 1965. The treatment of the laser near threshold must include a nonlinear active medium consisting of atoms that are pumped in their excited states and a damping mechanism to account for the loss of photons from the cavity through end mirrors. To obtain laser pumping action we introduce atoms in their upper level |a at random times ti , decaying to a far-removed ground state |b. Cavity field damping is included by coupling the field to an ensemble of atoms in their ground state (γ subsystem in fig. 2). Here we concentrate on the study of the photon distribution function for the laser field, which is given by the diagonal matrix elements of the reduced density operator of the field. The photon statistical distribution for the laser is of interest for several reasons. Historically, it was initially thought by some that the statistical photon distribution should be a Bose–Einstein distribution. A little reflection shows that this cannot be, since the laser is operating far from thermodynamic equilibrium. However, a different paradigm recognizes that many atoms oscillating in phase produce what is essentially a classical current, and this would generate a coherent state, the statistics of which are Poissonian. But, for example, the photon statistics of a typical helium–neon laser are substantially different from a Poissonian distribution. Of course, well above threshold, the steady-state laser photon statistical distribution is Poisson which is the characteristic of a coherent state. In order to see these interesting features we consider the master equation of the laser in various regimes of operation. Here we omit details of the full theory and point out that the diagonal elements ρnn ≡ p(n), which represent the probability of n photons in the field, satisfy the

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[8, § 4

Fig. 2. Model.

equation of motion (n + 1)A nA p(n) ˙ =− p(n) + p(n − 1) 1 + (n + 1)B/A) 1 + nB/A − Cnp(n) + C(n + 1)p(n + 1).

(4.1)

where A is the linear gain coefficient, B is the self-saturation coefficient and C is the decay rate. The first two terms on the right-hand side of eq. (4.1) describe pumping and the last two terms come from damping (decay). It is interesting to note that the diagonal elements are coupled only to diagonal elements. More generally, only off-diagonal elements ρnn with the same difference (n − n ) are coupled. Before we begin the solution of the above equation we want to give a simple intuitive physical picture of the processes it describes in terms of a probability flow diagram, fig. 3. The left-hand side is the rate of change of the probability of finding n photons in the cavity. The right-hand side contains the physical processes that contribute to the change. Each process is represented by an arrow in the diagram. The processes are proportional to the probability of the state they are starting from and this will be the starting point of the arrow. The tip of the arrow points to the state the process is leading to.

8, § 4]


307

Fig. 3. Detailed balance and the corresponding probability flow diagram.

A simple physical meaning can be given to eq. (4.1) for the photon distribution function in terms of a probability flow diagram (fig. 3) by expanding the terms in the denominator of eq. (4.1). There we see the ‘flow’ of probability in and out of the |n state from and to the neighboring |n + 1 and |n − 1 states. For example, the A(n + 1)p(n) term represents the flow of probability from the |n state to the |n + 1 state due to the emission of photons by lasing atoms initially in the upper states. Here An is the rate of stimulated emission, A is the spontaneous emission rate, and these rates are multiplied by p(n) to yield the total probability flow rate. Since the probability flows out of p(n), this term is negative. The first term in the expansion of the square-bracketed term in eq. (4.1), namely B(n + 1)2 p(n) = A(n + 1)(B/A)(n + 1)p(n), corresponds to the process in which photons are emitted and then reabsorbed, the reabsorption rate being (B/A)(n + 1). Similar explanations exist for the other terms, including the loss terms. After this brief discussion of the meaning of the individual terms we now turn our attention to the solution of the laser master equation (4.1). Although it is possible to obtain a rather general time-dependent solution to eq. (4.1), our main interest here is in the steady-state properties of the field. To obtain the steadystate photon statistics, we replace the time derivative with zero. Notice that the right-hand side of the equation is of the form F (n + 1) − F (n), where F (n) = Cnp(n) −

nA p(n − 1), 1 + nB/A

(4.2)

simply meaning that in steady state, F (n + 1) = F (n). In other words F (n) is independent of n and is, therefore, a constant c. Furthermore, the equation F (n) = c has a normalizable solution only for c = 0. From eq. (4.2) we then immediately obtain A/C p(n) = (4.3) p(n − 1), 1 + nB/A which is a very simple two-term recurrence relation to determine the photonnumber distribution. Before we present the solution a remark is called for here. The fact that F (n) = 0 and F (n+1) = 0 hold separately is called the condition of detailed balance. As a consequence we do not need to deal with all four processes

308


[8, § 4

affecting p(n). It is sufficient to balance the processes connecting a pair of adjacent levels in fig. 3, and instead of solving the general three-term recurrence relation, resulting from the steady-state version of eq. (4.1), it is enough to solve the much simpler two-term recursion (4.3). It is instructive to investigate the photon statistics in some limiting cases before discussing the general solution. Below threshold the linear approximation holds. Since only states with very small n are populated appreciably, the denominator on the right-hand side of (4.3) can be replaced by unity in view of nB/A 1. Then n A . p(n) = p(0) (4.4) C The normalization condition, ∞ n=0 p(n) = 1, determines the constant p(0), yielding p(0) = (1 − A/C). Finally n A A p(n) = 1 − (4.5) . C C Clearly, the condition of existence for this type of solution is A < C. Therefore, A = C is the threshold condition for the laser. At threshold, the photon statistics change qualitatively and very rapidly in a narrow region of the pumping parameter. It should also be noted that below threshold the distribution function (4.5) is essentially of thermal character. If we introduce an effective temperature T defined by e−h¯ ω0 /kT = A/C,

(4.6)

we can cast (4.5) in the form

p(n) = 1 − eh¯ ω0 /kT e−nh¯ ω0 /kT .

(4.7)

This is just the photon number distribution of a single mode in thermal equilibrium with a thermal reservoir at temperature T . The inclusion of a finite temperature loss reservoir to represent cavity losses will not alter this conclusion about the region below threshold. There is no real good analytical approximation for the region around threshold, although the lowest-order expansion of the denominator in (4.3) yields some insight. The solution with this condition is given by n n A (1 − kB/A). p(n) = p(0) (4.8) C k=0

This equation clearly breaks down for n > A/B = nmax , where p(n) becomes negative. The resulting distribution is quite broad, exhibiting a long plateau and a

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rapid cut-off at nmax . The broad plateau means that many values of n are approximately equally likely and, therefore, the intensity fluctuations are large around threshold. The most likely value of n = nopt can be obtained from the condition p(nopt − 1) = p(nopt ) since p(n) is increasing before n = nopt and decreasing afterward. This condition yields nopt = (A − C)/B which is smaller by the factor C/A than the value obtained from the full nonlinear equation. The third region of special interest is the one far above threshold. In this region A/C 1 and the n values contributing the most to the distribution function are those for which n A/B. We can then neglect 1 in the denominator of (4.3), yielding p(n) = e−n¯

n¯ n , n!

(4.9)

with n¯ = A2 /(CB). Thus the photon statistics far above threshold are Poissonian, the same as for a coherent state. This, however, does not mean that far above threshold the laser is in a coherent state. As we shall see later, the off-diagonal elements of the density matrix remain different from those of a coherent state for all regimes of operation. Figure 4 shows the photon number distribution in different limits.

Fig. 4. Photon number distributions for (a) thermal photons plotted from eq. (4.7) (dashed line), (b) coherent state (Poissonian) (thin solid line), and (c) He–Ne laser plotted using eq. (4.8) (thick solid line). Insert shows an atom making a radiation transition.

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[8, § 5

§ 5. Bose–Einstein condensation: laser phase-transition analogy Bose–Einstein condensation in a trap has intriguing similarities with the threshold behavior of a laser which also can be viewed as a kind of a phase transition (DeGiorgio and Scully [1970], Graham and Haken [1970], Kocharovsky, Kocharovsky, Holthaus, Ooi, Svidzinsky, Ketterle and Scully [2006]). In both cases stimulated processes are responsible for the appearance of the macroscopic order parameter. The main difference is that for the Bose gas in a trap there is also interaction between the atoms which, in particular, yields stimulated effects in BEC. On the other hand there are two subsystems for the laser, namely the laser field and the active atomic medium. The crucial point for lasing is the interaction between the field and the atomic medium. Thus, the effects of different interactions in the laser are easy to trace and relate to the observable characteristics of the system. This is not the case in BEC, and it is important to separate different effects. As we discussed in the previous section, the laser light is conveniently described by a master equation obtained by treating the atomic (gain) media and cavity dissipation (loss) as reservoirs which when “traced over” yield the coarsegrained equation of motion for the reduced density matrix for laser radiation. We thus arrive at the equation of motion for the probability of having n photons in the cavity given by eq. (4.1). From eq. (4.8) we have that partially coherent laser light has a sharp photon distribution (with width several times Poissonian for a typical He–Ne laser) due to the presence of the saturation nonlinearity, B, in the laser master equation. Thus, we see that the saturation nonlinearity in the radiation– matter interaction is essential for laser coherence. Next we turn to an ideal Bose gas and derive a master equation for the particles in the condensate. The steady-state description of the condensate arises from the inherent nonlinearities in the system. One naturally asks: Is the corresponding nonlinearity in BEC due to atom–atom scattering, or is there a nonlinearity present even in an ideal Bose gas? In the following we show that the latter is the case.

5.1. Condensate master equation Here we consider a model of a dilute Bose gas of atoms wherein interatomic scattering is neglected. This ideal Bose gas of N atoms is confined inside a trap and the atoms exchange energy with a reservoir at a fixed temperature T (Scully [1999], Kocharovsky, Scully, Zhu and Zubairy [2000], Kapale and Zubairy [2001]). The “ideal gas + reservoir” model corresponds to a canonical ensemble, and it allows

8, § 5]

Bose–Einstein condensation: laser phase-transition analogy

311

Fig. 5. Simple harmonic oscillators as a thermal reservoir for the ideal Bose gas in a trap.

us to demonstrate most clearly the master-equation approach to the analysis of dynamics and statistics of BEC. It provides the simplest description of many qualitative and, in some cases, quantitative characteristics of the experimental BEC. In particular, it explains many features of the condensate dynamics and fluctuations and allows us to obtain the particle number statistics of the BEC. An extension of the present approach to the case of an interacting gas which includes usual manybody effects due to interatomic scattering will be discussed in the next section. For many problems a concrete realization of the reservoir system is not very important if its energy spectrum is dense and flat enough. For example, one expects (and we find) that the equilibrium (steady-state) properties of the BEC are largely independent of the details of the reservoir. For the sake of simplicity, we assume that the reservoir is an ensemble of simple harmonic oscillators whose spectrum is dense and smooth, see fig. 5. The interaction between the gas and the reservoir is described by the interaction picture Hamiltonian gj,kl bj† ak al† e−i(ωj −νk +νl )t + h.c., V = (5.1) j

k>l

where bj† is the creation operator for the reservoir j oscillator (“phonon”), and ak† and ak (k = 0) are the creation and annihilation operators for the Bose gas atoms in the kth level. Here h¯ νk is the energy of the kth level of the trap, and gj,kl is the coupling strength. Following along the lines of the quantum theory of the laser we can derive an equation of motion for the distribution function of the condensed bosons (pn0 ≡ ρn0 ,n0 ) (Kocharovsky, Scully, Zhu and Zubairy [2000]) p˙ n0 = −κ Kn0 (n0 + 1)pn0 − Kn0 −1 n0 pn0 −1 + Hn0 n0 pn0 − Hn0 +1 (n0 + 1)pn0 +1 , (5.2)

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[8, § 5

where κ embodies the spectral density of the bath and the coupling strength of the bath oscillators to the gas particles and Kn0 = (5.3) (ηk + 1)nk n0 , Hn0 = ηk nk n0 + 1 , k>0

k>0

with ηk =

1 . eh¯ νk /T − 1

(5.4)

The particle number constraint comes in since k>0 nk n0 = N − n¯ 0 . The steady-state distribution of the number of atoms condensed in the ground level of the trap can be determined from eq. (5.2), and the various moments, including the mean value and the variance, can then be determined. It is clear that there are two processes: cooling and heating. The cooling process is represented by the first two terms with the cooling coefficient Kn0 , and the heating by the third and fourth terms with heating coefficient Hn0 . In the cooling process the atoms in the excited atomic levels in the trap jump to the condensate level and transfer energy to the thermal reservoir whereas, in the heating process, the atoms in the condensate absorb energy from the reservoir and get excited. The cooling and heating coefficients have an analogy with the saturated gain and cavity loss in the laser master equation (4.1). According to eq. (5.3), these coefficients depend upon trap parameters such as the shape of the trap, the total number of bosons in the trap, N , and the temperature T . In general, the cooling and the heating coefficients are complicated and depend upon the condensate probability distribution pn0 . In this sense, eq. (5.2) is a transcendental equation for pn0 . This equation can however be simplified in certain approximations, and we obtain analytic results for the condensate distribution that are close to the exact numerical results.

5.2. Low-temperature approximation At low enough temperatures, the average occupations in the reservoir are small and ηk + 1 1 in eq. (5.3). This suggests the simplest approximation for the cooling coefficient: Kn0 (5.5) nk n0 = N − n¯ 0 . k>0

In addition, at very low temperatures the number of non-condensed atoms is also very small. We can therefore approximate nk n0 + 1 by 1 in eq. (5.3). Then

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the heating coefficient is a constant equal to the total average number of thermal excitations in the reservoir at all energies corresponding to the energy levels of the trap, Hn0 H,

H≡

k>0

ηk =

k>0

1 (eh¯ νk /T

− 1)

.

(5.6)

Under these approximations, the condensate master equation (5.2) simplifies considerably and contains only one non-trivial parameter H. We obtain p˙ n0 = −κ (N − n0 )(n0 + 1)pn0 − (N − n0 + 1)n0 pn0 −1

+ H n0 pn0 − (n0 + 1)pn0 +1 . (5.7) It may be noted that eq. (5.7) has the same form as the equation of motion for the photon distribution function in a laser operating not too far above threshold (Bn/A 1)). The identification is complete if we define the gain, saturation, and loss parameters in laser master equation by A = κ(N + 1), B = κ, and C = κH, respectively. The mechanisms for gain, saturation, and loss are however different in the present case. The resulting steady-state distribution for the number of condensed atoms is pn0 =

1 HN−n0 , ZN (N − n0 )!

(5.8)

where ZN = 1/pN is the partition function. It follows from the normalization condition n0 pn0 = 1 that ZN = eH (N + 1, H)/N!, (5.9) ∞ α−1 −t where (α, x) = x t e dt is an incomplete gamma-function. The mean value and the variance can be calculated from the distribution (5.8) for an arbitrary finite number of atoms in the Bose gas, n0 = N − H + HN+1 /ZN N!,

n20 ≡ n20 − n0 2 = H 1 − n0 + 1 HN /ZN N ! .

(5.10) (5.11)

As we shall see from the extended treatment in the next section, the approximations (5.5), (5.6) and, therefore, the results (5.10), (5.11) are clearly valid at low temperatures, i.e., in the weak trap limit, T ε1 , where ε1 is the energy gap between the first excited and the ground levels of a single-particle spectrum in the trap. However, in the case of a harmonic trap the results (5.10), (5.11) show qualitatively correct behavior for all temperatures, including T ε1 and T ∼ Tc (Scully [1999]).

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[8, § 5

In particular, for a harmonic trap we have from eq. (5.6) that the heating rate is 3 1 T kB T 3 ζ (3) = N . ≈ H= exp[β h¯ Ω(l + m + n) − 1] Tc hΩ ¯ l,m,n (5.12) Thus, in the low temperature region, the master equation (5.7) for the condensate in the harmonic trap becomes (Scully [1999])

1 p˙ n = − (N + 1)(n0 + 1) − (n0 + 1)2 pn0 + (N + 1)n0 − n20 pn0 −1 κ 0 3

T n0 pn0 − (n0 + 1)pn0 +1 . −N (5.13) Tc The resulting plots for n¯ 0 , the variance and the third and fourth central moments are given in fig. 6 (dashed line). These analytical results give a qualitatively correct description of the ideal Bose gas when compared with the exact solution for

Fig. 6. The first four central moments for the ideal Bose gas in an isotropic harmonic trap with N = 200 atoms as calculated via the solution of the condensate master equation (solid lines: quasithermal approximation, eq. (5.18); dashed lines: low temperature approximation, eq. (5.8)). Dots are “exact” numerical result obtained in the canonical ensemble.

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315

the moments as derived in the canonical ensemble. This quantitative agreement with the exact numerical results can be considerably improved in the quasithermal approximation to which we turn next.

5.3. Quasithermal approximation for non-condensate occupations At arbitrary temperatures, a very reasonable approximation for the average noncondensate occupation numbers in the cooling and heating coefficients in eq. (5.3) is given by nk n0 = ηk

nk n0 / ηk = k>0

k >0

(N − n¯ 0 ) . (eεk /T − 1)H

(5.14)

Equation (5.14) satisfies the canonical-ensemble constraint, N = n0 + k>0 nk , independently of the resulting distribution pn0 . This important property is based on the fact that a quasithermal distribution (5.14) provides the same relative average occupations in excited levels of the trap as in the thermal reservoir. The cooling and heating coefficients (5.3) in the quasithermal approximation of eq. (5.14) are Kn0 = (N − n0 )(1 + η),

Hn0 = H + (N − n0 )η.

(5.15)

Compared with the low-temperature approximation (5.5) and (5.6), these coefficients acquire an additional contribution (N − n0 )η due to the cross-excitation parameter, i.e., η=

1 1 1 ηk nk n0 = . ε /T k N − n0 H (e − 1)2 k>0

(5.16)

k>0

At arbitrary temperatures, the condensate master equation (5.2) contains two nontrivial parameters, H and η,

p˙ n0 = −κ (1 + η) (N − n0 )(n0 + 1)pn0 − (N − n0 + 1)n0 pn0 −1

+ H + (N − n0 )η n0 pn0

− H + (N − n0 − 1)η (n0 + 1)pn0 +1 . (5.17) The steady-state solution of eq. (5.17) is given by pn0

N−n0 1 (N − n0 + H/η − 1)! η = , ZN (H/η − 1)!(N − n0 )! 1 + η

(5.18)

316


[8, § 5

Fig. 7. Probability distribution of the ground-state occupation, pn0 , at the temperature T = 0.2, 0.5, 0.8 and 1.0Tc in an isotropic harmonic trap with N = 200 atoms as calculated from the solution of the condensate master equation (5.2) in the quasithermal approximation, eq. (5.18), (solid line) and the “exact” numerical dots obtained in the canonical ensemble.

where the canonical partition function ZN = 1/pN is N−n0 N η N − n0 + H/η − 1 ZN = . N − n0 1+η

(5.19)

n0 =0

The master equation (5.17) for pn0 , and the analytic approximate expressions (5.18) and (5.19) for the condensate distribution function pn0 and the partition function ZN , respectively, are among the main results of the condensate masterequation approach. Now we are able to present the key picture of the theory of BEC fluctuations, that is, the probability distribution pn0 , fig. 7. Analogy with the evolution of the photon number distribution in a laser mode (from thermal to coherent, lasing) is obvious from a comparison of fig. 7 and fig. 4. With an increase of the number of atoms in the trap, N , the picture of the ground-state occupation distribution remains qualitatively the same, just the relative width of all peaks becomes narrower. The average number of atoms condensed in the ground state of the trap is n0 = N − H + p0 η(N + H/η).

(5.20)

The squared variance and higher central moments can be also calculated analytically, e.g., n20 = (1 + η)H − p0 (ηN + H)(N − H + 1 + η) − p02 (ηN + H)2 , (5.21)

8, § 5]


where p0 =

N 1 (N + H/η − 1)! η ZN N!(H/η − 1)! 1 + η

317

(5.22)

is the probability that there are no atoms in the condensate. The first four central moments for the Bose gas in a harmonic trap with N = 200 atoms are presented in fig. 6 as functions of temperature in different approximations. It is clearly seen that the analytic results based on quasithermal distribution are indistinguishable from the exact numerical results for the mean, and the second and the fourth moments. The results for the third moment are however quantitatively somewhat different. The success of the master-equation approach is that the analytic expressions are available for the partition function as well as the condensate distribution function that mimic the exact solution to a remarkable degree for a mesoscopic ideal Bose gas. As a final point, we mention that a laser phase-transition analogy exists via the P -representation of the density matrix (DeGiorgio and Scully [1970], Graham and Haken [1970]) 2 d α P (α, α ∗ )|αα|, ρ= (5.23) π where |α is a coherent state. The steady-state solution of the Fokker–Planck equation for a laser near threshold is (Scully and Zubairy [1997]) 1 B A−C ∗ 2 4 P (α, α ) = (5.24) exp |α| − |α| N A 2A which clearly indicates a formal similarity between

ln P (α, α ∗ ) = − ln N + 1 − H/(N + 1) n0 − 1/2(N + 1) n20

(5.25)

for the laser equation and the Ginzburg–Landau type free energy (Scully and Zubairy [1997], DeGiorgio and Scully [1970], Graham and Haken [1970]) G(n0 ) = ln pn0 ≈ const. + a(T )n0 + b(T )n20 , where

|α|2

(5.26)

= n0 , a(T ) = −(N − H)/N and b(T ) = 1/2N for large N near Tc .

5.4. Squeezing, noise reduction and BEC fluctuations The term “squeezing” originates from the studies of noise reduction in quantum optics. In the present BEC context this aspect of (quantum) optical physics is relevant to the characteristic function for the total number of atoms in the two,

318


[8, § 5

k and −k, modes squeezed by Bogoliubov coupling. A detailed derivation of the characteristic function for the fluctuations of the number of atoms in the two excited modes squeezed by the Bogoliubov coupling was presented by Kocharovsky, Kocharovsky and Scully [2000]; it utilizes known results for the squeezed states of the radiation field and is given by

2 ˆ+ ˆ ˆ+ ˆ ˆ+ ˆ ˆ+ ˆ Θ±k (u) ≡ Tr eiu(βk βk +β−k β−k ) e−εk (bk bk +b−k b−k )/T 1 − e−εk /T (z(Ak ) − 1)(z(−Ak ) − 1) , = (5.27) (z(Ak ) − eiu )(z(−Ak ) − eiu ) where Ak =

V h¯ 2 k2 n¯ 0 Uk − , εk − n¯ 0 Uk 2M V

z(Ak ) =

εk is the energy of Bogoliubov quasiparticles: 2 2 n¯ 0 Uk 2 n¯ 0 Uk 2 h¯ k − , + εk = 2M V V

Ak − eεk /T , Ak eεk /T − 1

(5.28)

(5.29)

M is the atomic mass, V is the condensate volume, and Uk is the atom–atom scattering energy. In eq. (5.27) βˆk are bare canonical ensemble quasiparticles which are related to Bogoliubov quasiparticles bˆk by the canonical transformation + , βˆk = uk bˆk + vk bˆ−k

βˆk+ = uk bˆk+ + vk bˆ−k ,

(5.30)

where uk and vk are Bogoliubov amplitudes uk =

1 1 − A2k

,

Ak vk = . 1 − A2k

(5.31)

The characteristic function for the distribution of the total number of the excited atoms is equal to the product of the coupled-mode characteristic functions, Θn (u) = k=0,mod{±k} Θ±k (u), since different pairs of (k, −k)-modes are inde pendent to the first approximation. The product runs over all different pairs of (k, −k)-modes. By doing all calculations via the canonical-ensemble quasiparticles we automatically take into account all correlations introduced by the canonical-ensemble constraint. The important conclusion is that for a square-well trap the ground-state occupation fluctuations are not Gaussian even in the thermodynamic limit. It is more convenient, in particular, for the analysis of the non-Gaussian properties, to solve for the cumulants κm which are defined as coefficients in the Taylor expan m sion ln Θn (u) = ∞ m=1 κm (iu) /m!, where Θn (u) is the characteristic function iu n ˆ Θn (u) = Tr{e ρ}. ˆ There are simple relations between κm and central moments

8, § 5]


319

μm , in particular, ¯ κ1 = n,

κ2 = μ2 ,

κ3 = μ3 ,

κ4 = μ4 − 3μ22 .

(5.32)

The “generating cumulants” κ˜ m are simply related to the cumulants κm by κ1 = κ˜ 1 ,

κ2 = κ˜ 2 + κ˜ 1 ,

κ3 = κ˜ 3 + 3κ˜ 2 + κ˜ 1 ,

κ4 = κ˜ 4 + 6κ˜ 3 + 7κ˜ 2 + κ˜ 1 .

(5.33)

For Gaussian distribution κm = 0 for m = 3, 4, . . . . The explicit formula for all cumulants in the dilute weakly interacting Bose gas was obtained by Kocharovsky, Kocharovsky and Scully [2000] as 1 1 1 + . κ˜ m = (m − 1)! (5.34) 2 (z(Ak ) − 1)m (z(−Ak ) − 1)m k=0

For the ideal gas the answer is 1 κ˜ m = (m − 1)! . ε /T (e k − 1)m

(5.35)

k=0

In comparison with the ideal Bose gas, eq. (5.35), for the interacting particles we have effectively a mixture of two species of atom pairs with z(±Ak ) instead of exp(εk /T ). It is important to emphasize that the first equation in (5.34), m = 1, is a nonlinear self-consistency equation, N − n¯ 0 = κ1 (n¯ 0 ) ≡

1 + A2k eεk /T

k=0

(1 − A2k )(eεk /T − 1)

,

(5.36)

to be solved for the mean number of ground-state atoms n¯ 0 (T ), since the Bogoliubov coupling coefficient Ak , and the energy spectrum εk , are themselves functions of the mean value n¯ 0 . Then, all the other equations in (5.34), m 2, are nothing else but explicit expressions for all cumulants, m 2, if one substitutes the solution of the self-consistency equation (5.36) for the mean value n¯ 0 . Equation (5.36), obtained for the interacting Bose gas in the canonical-ensemble quasiparticle approach, coincides precisely with the self-consistency equation for the grand-canonical dilute gas in the so-called first-order Popov approximation (see a review by Shi and Griffin [1998]). The latter is well established as a reasonable first approximation for the analysis of the finite-temperature properties of the dilute Bose gas and is not valid only in a very small interval near Tc given by Tc − T < a(N/V )1/3 Tc Tc , where a = MU0 /4π h¯ 2 is a usual s-wave scattering length. The analysis of eq. (5.36) shows that in the dilute gas

320


[8, § 5

Fig. 8. Temperature scaling of the first four cumulants, the mean value n¯ 0 /N = N −κ1 /N , the variance √ 1/3 κ2 /N = (n0 − n¯ 0 )2 1/2 /N 1/2 , the third central moment −κ3 /N 1/2 = (n0 − n¯ 0 )3 1/3 /N 1/2 , the fourth cumulant |κ4 |1/4 /N 1/2 = |(n0 − n¯ 0 )4 − 3κ22 |/N 2 , of the ground-state occupation fluctuations for the dilute weakly interacting Bose gas with U0 N 1/3 /ε1 V = 0.05 (thick solid lines), as compared with the ideal gas (thin solid lines) and with the “exact” numerical result in the canonical ensemble (dot-dashed lines) for the ideal gas in the box; N = 1000. For the ideal gas the thin solid lines are almost indistinguishable from the “exact” dot-dashed lines in the condensed region, T < Tc (N ). Temperature is normalized by the standard thermodynamic-limit critical value Tc (N = ∞) that differs from the finite-size value Tc (N ), as is clearly seen in the graphs.

the self-consistent value n¯ 0 (T ) is close to that given by the ideal gas model, and for very low temperatures goes smoothly to the value given by the standard Bogoliubov theory (Lifshitz and Pitaevskii [1981], Abrikosov, Gorkov and Dzyaloshinskii [1963], Fetter and Walecka [1971]) for a small condensate depletion, N − n¯ 0 N. This is illustrated by fig. 8 in which we show the first four cumulants. Near the critical temperature Tc the number of excited quasiparticles is relatively large, so that along with the Bogoliubov coupling other, higher-order

8, § 6]

Hybrid approach to condensate fluctuations

321

effects of interaction should be taken into account to get a complete theory. Note that the effect of a weak interaction on the condensate fluctuations is very significant, see fig. 8, even if the mean number of condensed atoms changes by a relatively small amount.

§ 6. Hybrid approach to condensate fluctuations We now show how to combine ideas from the canonical-ensemble quasiparticle formalism of (Kocharovsky, Kocharovsky and Scully [2000]) (which works well √ for an interacting gas at temperature not too close to Tc when μ2 n¯ 0 ) with the physics of the master-equation approach (in the spirit of the quantum theory of the laser) (Kocharovsky, Scully, Zhu and Zubairy [2000]), in order to obtain essentially perfect quantitative agreement with the exact numerical solution of the canonical partition function at all temperatures for the fluctuation statistics of the Bose gas. Such a hybrid technique was proposed by Svidzinsky and Scully [2006]. We recall the master equation for the condensate probability distribution for a non-interacting Bose gas (5.2): 1 p˙ n = −Kn0 (n0 + 1)pn0 + Kn0 −1 n0 pn0 −1 κ 0 − Hn0 n0 pn0 + Hn0 +1 (n0 + 1)pn0 +1 .

(6.1)

The detailed balance condition yields Kn0 pn0 +1 = . pn0 Hn0 +1

(6.2)

Since the occupation number of the ground state cannot be larger than N there is a canonical-ensemble constraint pN+1 = 0 and, hence, KN = 0. In contrast to pn0 , the ratio pn0 +1 /pn0 as a function of n0 shows simple monotonic behavior. We approximate Kn0 and Hn0 by a few terms of the Taylor expansion near the point n0 = N : Kn0 = (N − n0 )(1 + η) + α(N − n0 )2 ,

(6.3)

Hn0 = H + (N − n0 )η + α(N − n0 ) .

(6.4)

2

The parameters H, η and α are independent of n0 ; they are functions of the occupation of the excited levels. We derive them below by matching the first three central moments in the low-temperature limit with the result of Kocharovsky, Kocharovsky and Scully [2000]. We note that the detailed balance equation (6.2) is the Padé approximation (Baker and Graves-Morris [1996]) of the function

322


[8, § 6

pn0 +1 /pn0 . Padé summation has proven to be useful in many applications, including condensed-matter problems and quantum field theory. Equations (6.2)–(6.4) yield an analytical expression for the condensate distribution function: 1 (N − n0 − 1 + x1 )!(N − n0 − 1 + x2 )! , pn0 = (6.5) ZN (N − n0 )!(N − n0 + (1 + η)/α)! where x1,2 = (η ± η2 − 4αH )/2α and ZN is the normalization constant deter mined by N n0 =0 pn0 = 1. In the particular case η = α = 0 eq. (6.5) reduces to eq. (5.8) obtained in the low temperature approximation. Using the distribution function (6.5) we find that, in the validity range of Kocharovsky, Kocharovsky and Scully [2000] (at low enough T ), the first three central moments μm ≡ (n0 − n¯ 0 )m are n¯ 0 = N − H,

μ2 = (1 + η)H + αH2 ,

μ3 = −H(1 + η + αH)(1 + 2η + 4αH). Equations (6.6) and (6.7) thus yield 1 μ3 4μ2 η= −3+ , H = N − n¯ 0 , 2 μ2 H μ3 1 1 μ2 − − . α= H 2 H 2μ2

(6.6) (6.7)

(6.8) (6.9)

On the other hand, the result of Kocharovsky, Kocharovsky and Scully [2000] for an interacting Bogoliubov gas is (see Appendix A for derivation of n¯ 0 and μ2 )

u2k + vk2 fk + vk2 , n¯ 0 = N − (6.10) k=0

μ2 =

1 + 8u2k vk2 fk2 + fk + 2u2k vk2 ,

k=0

μ3 = −

u2k + vk2

(6.11)

1 + 16u2k vk2 2fk3 + 3fk2 + fk

k=0

+ 4u2k vk2 (1 + 2fk ) ,

(6.12)

where fk = 1/[exp(εk /kB T ) − 1] is the number of elementary excitations with energy εk present in the system at thermal equilibrium, and uk and vk are Bogoliubov amplitudes. Substituting for n¯ 0 , μ2 and μ3 in eqs. (6.8) and (6.9) their expressions (Kocharovsky, Kocharovsky and Scully [2000]) (6.10)–(6.12) yields the unknown parameters H, η and α. The beauty of the present “matched asymptote” derivation is that the formulas for H, η and α are applicable at all temperatures, i.e. not only in the validity range of Kocharovsky, Kocharovsky and Scully

8, § 6]

Hybrid approach to condensate fluctuations

323

Fig. 9. Average condensate particle number n0 , its variance n0 = (n0 − n¯ 0 )2 , third and fourth central moments (n0 − n¯ 0 )m (m = 3, 4), and fourth cumulant κ4 , as a function of temperature for an ideal gas of N = 200 particles in a harmonic trap. The solid lines (CNB5) show the result of the hybrid approach while Kocharovsky, Kocharovsky and Scully [2000] yields the dashed lines (CNB3). Dots are “exact” numerical simulation in the canonical ensemble. The temperature is normalized by 1/3 /k ζ (3)1/3 , where ω is the trap the thermodynamic critical temperature for the trap Tc = hωN ¯ B frequency.

[2000]. The distribution function (6.5) together with eqs. (6.8) and (6.9) provides complete knowledge of the condensate statistics at all T . Taking vk = 0 and uk = 1 in (6.10)–(6.12) we obtain the ideal gas limit. Figure 9 shows the average condensate particle number n¯ 0 , its variance, third and fourth central moments μm and fourth cumulant κ4 as a function of T for an ideal gas of N = 200 particles in a harmonic trap. The solid lines result from the present approach, which is in remarkable agreement with the “exact” dots at all temperatures both for μm and κ4 . Central moments and cumulants higher than fourth order are not shown here, but they are also remarkably accurate at all temperatures. Results of Kocharovsky, Kocharovsky and Scully [2000] are given by the dashed lines which are accurate only at sufficiently low T . Deviation of

324


[8, § 6

higher order cumulants (m = 3, 4, . . .) from zero indicates that the fluctuations are not Gaussian. Clearly the present hybrid method passes the ideal gas test with flying colors. We note the excellent agreement with the exact analysis for the third central moment and fourth cumulant κ4 given in fig. 9. Next we apply this technique to N interacting Bogoliubov particles confined in a box of volume V . The interactions are characterized by the gas parameter an1/3 , where a is the s-wave scattering length and n = N/V is the particle density. The energy of Bogoliubov quasiparticles εk depends on n¯ 0 , hence, the equation n¯ 0 = N ¯ 0 must be solved self-consistently. In fig. 10 we plot n¯ 0 , the n0 =0 n0 pn0 for n variance n0 , third and fourth central moments as a function of T for an ideal and an interacting (an1/3 = 0.1) gas in the box. The solid lines show the results of the present approach, while those of Kocharovsky, Kocharovsky and Scully [2000]

Fig. 10. Average condensate particle number, its variance, third and fourth central moments as a function of temperature for an ideal (an1/3 = 0) and interacting (an1/3 = 0.1) Bose gas of N = 200 particles in a box. Solid lines are the result of the hybrid approach. Kocharovsky, Kocharovsky and Scully [2000] yields dashed lines (CNB3). The temperature is normalized by the thermodynamic critical temperature for the box Tc = 2π h¯ 2 n2/3 /kB Mζ (3/2)2/3 , where M is the particle mass.

8, Appendix A]

Mean condensate particle number and its variance

325

are represented by dashed lines. The present results agree well for all μm with the result by Kocharovsky, Kocharovsky and Scully [2000] in the range of its validity. Near and above Tc the result by Kocharovsky, Kocharovsky and Scully [2000] becomes inaccurate. However, the results of the present method are expected to be accurate at all T . Indeed, in the limit T Tc the present results (unlike the result by Kocharovsky, Kocharovsky and Scully [2000]) merge with those for the ideal gas. This is physically appealing since at high T the kinetic energy becomes much larger than the interaction energy and the gas behaves ideally. Similar to the ideal gas, the interacting mesoscopic BEC n¯ 0 (T ) exhibits a smooth transition when passing through Tc . One can see from fig. 10 that the repulsive interaction stimulates BEC, and yields an increase in n¯ 0 at intermediate temperatures, as compared to the ideal gas. This effect is known as “attraction in momentum space” and occurs for energetic reasons (Leggett [2001]). Bosons in different states interact more strongly than when they are in the same state, and this favors multiple occupation of a single one-particle state.

Acknowledgements We gratefully acknowledge the support of the Office of Naval Research (Award No. N00014-03-1-0385) and the Robert A. Welch Foundation (Grant No. A-1261).

Appendix A: Mean condensate particle number and its variance for weakly interacting BEC In the framework of Bogoliubov theory the particle operator can be expressed in terms of the quasiparticle creation and annihilation operators as + , βˆk = uk bˆk + vk bˆ−k

βˆk+ = uk bˆk+ + vk bˆ−k ,

(A.1)

where uk and vk are Bogoliubov amplitudes. The total number of particles out of the condensate is given by the expectation value of the operator

+

u2k bˆk+ bˆk + vk2 bˆk bˆk+ + uk vk bˆk bˆ−k + bˆk+ bˆ−k . βˆk+ βˆk = Nˆ out = k=0

k=0

(A.2)

Using the particle number constraint nˆ 0 + Nˆ out = N we obtain

n¯ 0 = N − Nˆ out = N − u2k + vk2 fk + vk2 , k=0

(A.3)

326


[8, Appendix A

where fk = bˆk+ bˆk = 1/[exp(εk /kB T ) − 1] is the number of elementary excitations with energy εk present in the system at thermal equilibrium. Particle fluctuations can be calculated in a similar way (Giorgini, Pitaevskii and Stringari [1998]). Using eq. (A.2) we have 2 Nˆ out = βˆ + βˆk βˆq+ βˆq k

k=0

q=0

u2k u2q bˆk+ bˆk bˆq+ bˆq + vk2 vq2 bˆk bˆk+ bˆq bˆq+

=

k,q=0

+ u2k vq2 bˆk+ bˆk bˆq bˆq+ + bˆq bˆq+ bˆk+ bˆk

+ + u2k uq vq bˆk+ bˆk bˆq bˆ−q + bˆq+ bˆ−q

+ + vk2 uq vq bˆk bˆk+ bˆq bˆ−q + bˆq+ bˆ−q

+ ˆ+ ˆ + uk vk u2q bˆk bˆ−k + bˆk+ bˆ−k bq b q

+ ˆ ˆ+ b q bq + uk vk vq2 bˆk bˆ−k + bˆk+ bˆ−k

+ + +

bˆq bˆ−q + bˆq+ bˆ−q . + uk vk uq vq bˆk bˆ−k + bˆ bˆ k −k

(A.4)

To calculate the expectation value of the terms with four quasiparticle operators 2 we use Wick’s theorem which holds for the operators of staappearing in Nˆ out tistically independent excitations. In particular, non-zero averages come from the terms with q = ±k. Using Wick’s theorem we obtain + + + ˆ bˆk bˆk bˆ−k b−k = fk2 , bˆk bˆk bˆk+ bˆk = 2fk2 + fk , + + + bˆk bˆk bˆk bˆk+ = 2fk2 + 3fk + 1, = (fk + 1)2 , bˆk bˆk bˆ−k bˆ−k + bˆk bˆk bˆk bˆk+ = bˆk bˆk+ bˆk+ bˆk = 2fk2 + 2fk , + + + ˆ+ ˆ = bˆ−k bˆ−k bˆk bˆk bˆ−k bˆ−k bk bk = fk2 + fk , + + ˆ+ bˆk bˆ−k bˆk+ bˆ−k = bˆk bˆ−k bˆ−k bk = (fk + 1)2 , + + + + bˆk bˆ−k bˆk bˆ−k = bˆk bˆ−k bˆ−k bˆk = fk2 , and therefore 2 Nˆ out =

u2k u2q fk fq + vk2 vq2 (fk + 1)(fq + 1) + 2u2k vq2 fk (fq + 1)

k=±q=0

+

k=0

+ ˆ u4k bˆk+ bˆk bˆk+ bˆk + bˆk+ bˆk bˆ−k b−k

+ + vk4 bˆk bˆk+ bˆk bˆk+ + bˆk bˆk+ bˆ−k bˆ−k

+ u2 v 2 bˆ + bˆk bˆk bˆ + + bˆk bˆ + bˆ + bˆk + bˆk bˆ−k bˆ + bˆ + k k

k

k

k k

k −k

8]

References

327

+ ˆ ˆ + + ˆ+ ˆ + bˆ−k bˆ−k + bˆk+ bˆ−k bk b−k + bˆk+ bˆk bˆ−k bˆ−k b k bk

+ ˆ+ + ˆ + bˆk bˆ−k bˆ−k bk + bˆk+ bˆ−k b−k bˆk

u2k u2q fk fq + vk2 vq2 (fk + 1)(fq + 1) + 2u2k vq2 fk (fq + 1) = k=±q=0

+

u4k 3fk2 + fk + vk4 3fk2 + 5fk + 2

k=0

+ u2k vk2 10fk2 + 10fk + 2 . On the other hand

Nˆ out 2 = u2k + vk2 fk + vk2 u2q + vq2 fq + vq2 .

(A.5)

(A.6)

k=0 q=0

Using the particle number constraint together with eqs. (A.5) and (A.6) and u2k − vk2 = 1 we find the following answer for the squared variance of condensate fluctuations (Giorgini, Pitaevskii and Stringari [1998]): 2 2 2 μ2 ≡ nˆ 20 − nˆ 0 = Nˆ out − Nˆ out

1 + 8u2k vk2 fk2 + fk + 2u2k vk2 . = (A.7) k=0

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328


[8

Boltzmann, L., 1896, Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo, Annalen der Physik 57, 773–784. This and Boltzmann [1897] are Boltzmann’s answers to Zermelo [1896a] and Zermelo [1896b]. All four papers are reprinted in Brush and Hall [2003]. Boltzmann, L., 1897, Zu Hrn. Zermelo’s Abhandlung über die mechanische Erklärung irreversibler Vorgange, Annalen der Physik 60, 392–398. Born, M., 1949, Atomic Physics, 5th edition, Hafner, New York. Reprinted by Dover, New York, 1989. Bose, S.N., 1924, Z. Phys. 26, 176. Bradley, C., Sackett, C., Tollett, J., Hulet, R., 1995, Phys. Rev. Lett. 75, 1687. Brush, S.G., Hall, N.S. (Eds.), 2003, The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, Imperial College Press, London. This contains reprints and commentaries on the papers Boltzmann [1877a, 1877b 1896, 1897], Zermelo [1896a, 1896b]. Chan, M.H.W., Blum, K.I., Murphy, S.Q., Wong, G.K.S., Reppy, J.D., 1988, Phys. Rev. Lett. 61, 1950. Chuu, C.S., Schreck, F., Meyrath, T.P., Hanssen, J.L., Price, G.N., Raizen, M.G., 2005, Phys. Rev. Lett. 95, 260403. This paper reports the recent experiment of M. Raizen and coworkers on observation of number statistics in a degenerate Bose gas. Crooker, B.C., Hebral, B., Smith, E.N., Takano, Y., Reppy, J.D., 1983, Phys. Rev. Lett. 51, 666. Crowell, P.A., Van Keuls, F.W., Reppy, J.D., 1995, Phys. Rev. Lett. 75, 1106. Davis, K., Mewes, M., Andrews, M., van Druten, N., Durfee, D., Kurn, D., Ketterle, W., 1995, Phys. Rev. Lett. 75, 3969. DeGiorgio, V., Scully, M.O., 1970, Phys. Rev. A 2, 1170. Dulong, P.L., Petit, A.T., 1819, Sur quelques points importants de la théorie de la chaleur, Annales de Chimie et de Physique 10, 395–413. Ehrenfest, P., 1906, Planckschen Strahlungstheorie, Physikalische Zeitschrift 7, 528–532. Reprinted in his collected papers, p. 123 (see Klein [1959]). Ehrenfest, P., 1911, Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle, Annalen der Physik 36, 91–118. Ehrenfest, P., 1913. This article was published in Dutch. It is reprinted in English translation as “A mechanical theorem of Boltzmann and its relation to the theory of energy quanta”, appears in Proceedings of the Amsterdam Academy 16, 591–597 (1914), and was reprinted in his collected papers, pp. 340–346 (see Klein [1959]). Einstein, A., 1905, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt, Annalen der Physik 17, 132–148. Einstein, A., 1907, Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme, Annalen der Physik 22, 180–190. Einstein, A., 1909, On the present status of the radiation problem, Physikalische Zeitschrift 10, 185– 193. Einstein, A., 1917, Zur Quantentheorie der Strahlung, Physicalische Zeitschrift 18, 121–128. Reprinted in English translation in Ter Haar [1967]. Einstein, A., 1924. Sitzungsberichte der Preussischen Akademie der Wissenschaften, XXII. Gesamtsitzung, p. 261. Einstein, A., 1925. Sitzungsberichte der Preussischen Akademie der Wissenschaften, I. Sitzung der physikalisch-mathematischen Klasse, p. 3. Ernst, U., Marte, A., Schreck, F., Schuster, J., Rempe, G., 1998, Europhys. Lett. 41, 1. Esslinger, T., Bloch, I., Hänsch, T.W., 1998, Phys. Rev. A 58, R2664. Fetter, A.L., Walecka, J.D., 1971, Quantum Theory of Many-Particle Systems, McGraw-Hill, San Francisco. Fierz, M., 1956, Helv. Phys. Acta 29, 47. Fried, D.G., Killian, T.C., Willmann, L., Landhuis, D., Moss, S.C., Kleppner, D., Greytak, T.J., 1998, Phys. Rev. Lett. 81, 3811.

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Author index for Volume 50 A

Balcou, Ph., 9 Balistreri, M.L.M., 159, 160 Baltuska, A., 10 Barchiesi, D., 162 Barnett, S.M., 132 Barta, A., 222 Barton, G., 103 Barty, C.P.J., 7 Barut, A.O., 67 Bechtel, J.H., 7 Becker, P.C., 5 Beckmann, P., 208 Bell, J.S., 122 Belsky, A.M., 17, 32, 37, 45 Bergman, D.J., 172 Bergman, J.G., 149 Berman, P.R., 103 Bernard, J., 158 Bernáth, B., 222 Berry, M.V., 17, 22, 24, 25, 34, 37–39, 42–46, 196, 205, 206, 208, 209, 211, 222, 224 Bethe, H.A., 162 Betzig, E., 157, 158, 162 Beversluis, M.R., 165 Bharadwaj, P., 148, 149 Bialynicki-Birula, I., 82, 89, 91 Bian, R.X., 168, 170 Bielefeldt, H., 153, 160, 161, 163 Binnig, G., 148 Blaire, B., 67 Blaize, S., 159 Blatt, J.M., 207 Bloch, I., 299, 300 Bloembergen, N., 4, 7, 24, 32, 42, 45 Blum, K.I., 299, 300 Boardman, A.D., 149 Boccara, A.C., 160, 166 Bohr, N., 53, 66, 68, 279 Boltasseva, A., 159, 163

Abbe, E., 142 Abrikosov, A.A., 320 Acín, A., 126 Ackerhalt, J.R., 103 Adam, J.A., 209 Agarwal, G., 270 Agarwal, G.S., 139, 163 Aigouy, L., 160 Airy, G.B., 199 Albrecht, M.G., 148, 153 Alfano, R.R., 10 Allen, L., 63 Anderson, B.P., 299, 300 Anderson, M., 299, 300 Anderson, M.S., 169 Andrews, M.R., 299, 300 Anger, P., 148, 149 Antonoyiannakis, M.I., 242 Apostol, A., 139 Armstrong, J.A., 6 Arnold, D.V., 37 Arons, A.B., 56 Asatryan, A.A., 211 Ash, E.A., 151 Ashkin, A., 226 Aspect, A., 122, 123 Aubert, S., 159 B Babich, V.M., 204 Bachelot, R., 159, 166, 167 Baez, A.V., 150 Bainier, C., 243 Baker, B., 198 Baker, G.A., 321 Balachandran, R.M., 243 Balcou, P., 219 331

332

Author index for Volume 50

Boltzmann, L., 59, 280, 281, 284, 295, 328, 330 Borghi, R., 269, 270 Born, M., 15, 18, 19, 21, 26, 46, 55, 65, 77, 79, 99, 106, 194, 198, 200, 202, 217, 260–263, 285, 290 Borovikov, V.A., 203 Bose, S.N., 55, 59, 278 Bothe, W., 69 Bouhelier, A., 141, 165, 169 Boulanger, B., 24, 42 Bouwkamp, C.J., 162, 200, 205 Boyd, R.W., 123, 131 Boyer, C.B., 190, 221 Bozhevolnyi, S.I., 159, 160, 163 Brabec, T., 4, 9 Bradley, C., 299, 300 Bragas, A.V., 170 Braginsky, V.B., 241 Braunbek, W., 205 Bremmer, H., 208, 209, 220 Bretenaker, F., 219 Bricard, J., 232 Brillouin, M., 72 Britto Cruz, C.H., 5 Brosseau, C., 257 Brown, R.H., 101 Brownrigg, W., 230 Brus, L.E., 158 Brush, S.G., 278, 327, 328, 330 Bruyant, A., 159 Bryant, H.C., 231 Bryngdahl, O., 217, 243 Bucerius, H., 232 Bucksbaum, P.H., 10 Burin, A.L., 243 Burnett, N.H., 9 Burnham, R.D., 131 Burns, M.M., 300 Busch, B.D., 300 C Campillo, A.J., 225, 241 Canto, L.F., 21 Cao, H., 243 Carminati, R., 139, 159, 162–164 Carney, P.S., 159, 163 Carniglia, C.K., 139, 140, 163 Cederbaum, L.S., 17

Chabanov, A.A., 242 Chan, M.H.W., 299, 300 Chance, R.R., 155 Chang, R.K., 153, 225, 241, 242 Chang, R.P.H., 243 Chapman, S.J., 211 Charinpatnikul, T., 223 Chen, C.J., 153 Chen, G., 242 Chen, Y., 158 Chester, C., 222 Cheville, R.A., 239 Chew, H., 155 Chiao, R.Y., 122, 124, 243, 244 Chicanne, C., 163, 164 Christov, I.P., 8 Chu, S., 167 Chu, S.Y., 21 Chuang, I.L., 126 Chuu, C.S., 304 Chylek, P., 226, 230 Clauser, J.F., 115, 122, 123 Clebsch, A., 205 Cohen-Tannoudji, C., 89 Colas des Francs, G., 163, 164 Collett, E., 257 Collins, R.J., 4 Compton, A.H., 53, 66, 69 Condon, E.U., 187 Copson, E.T., 198 Corkum, P.B., 8–10 Cornell, E., 299, 300 Courjon, D., 141, 142, 159, 162, 243 Courteille, Ph., 300 Craighead, H.G., 139, 158 Crease, R.P., 253 Creighton, J.A., 148, 153 Critchfield, C.L., 187 Crooker, B.C., 299, 300 Crowell, P.A., 299, 300 Crozier, K.B., 170 Cummings, F.W., 116 D Dagenais, M., 118 Daneu, V., 171 Dave, J.V., 228 David, T., 163, 164 Davies, R., 229

Author index for Volume 50 Davis, J.A., 152 Davis, K., 299, 300 Davisson, C.J., 73 Dawson, P., 153, 160 De Alfaro, V., 208 De Boer, J.H., 222 De Broglie, L., 56, 67, 68, 70–72, 75, 78, 202 De Broglie, M., 65 De Carvalho, C.A., 228, 243, 244 de Fornel, F., 153, 160 de Hollander, R.B.G., 166 De Maria, A.J., 4 Debye, P., 53, 55, 66, 206, 208, 219, 225 Deckert, V., 169 deFornel, F., 159 DeGiorgio, V., 310, 317 Denk, W., 155, 170, 174 Dennis, M.R., 17, 45 Depperman, K., 208 Dereux, A., 140, 160, 162–164 Descartes, R., 189 Deschamps, G.A., 40 Desmarest, C., 160 Devaney, A.J., 163 Devel, M., 163 Dickman, K., 170 Dietrich, P., 9 Dirac, P.M., 79, 86 Ditlbacher, H., 161 Dogariu, A., 139, 163, 257, 270 Donangelo, R., 21 Doron, E., 242 Doumont, J.M.P.L., 131 Dowling, J.P., 67 Drescher, M., 9 Dresden, M., 66 Drexhage, K.H., 153, 165 Drude, P., 140 Dulong, P.L., 291 Dunn, B., 141, 160 Dunn, R.C., 155 Dupont-Roc, J., 89 Durfee, D.S., 299, 300 Dürig, U., 157 Dutriaux, L., 219 Dutton, Z., 300 Dyba, M., 144 Dzyaloshinskii, I.E., 320

333

E Eberly, J.H., 117 Eckart, C., 187 Economou, E.N., 242 Egner, A., 144 Ehrenfest, P., 55, 280, 283, 285, 286, 295 Eickelkamp, T., 167 Einstein, A., 53–56, 59–61, 64, 65, 67, 68, 76, 278–280, 291, 295–297, 299, 330 Eisler, H.-J., 170 Ellis, J., 139, 163 Ensher, J., 299, 300 Ernst, U., 300 Erramilli, S., 141 Esslinger, T., 299, 300 Euler, L., 194 Evenson, K.M., 171 Exner, F.M., 231 F Fano, U., 152 Farahani, J.N., 170 Fearn, H., 106 Fee, M., 167 Fermat, P., 194 Fermi, E., 86 Ferrell, T.L., 159 Ferretti, A., 20, 21 Fetter, A.L., 320 Fève, J.P., 24, 42 Feynman, R.P., 108, 122 Fiedler-Ferrari, N., 217–219 Fields, M.H., 241 Fierz, M., 304 Fillard, J.P., 141 Finn, P.L., 158 Firth, W.J., 131 Fischer, U.Ch., 139, 141, 146, 148, 149, 154, 157, 158, 160, 166, 167 Fleischmann, M., 148, 153 Fock, V.A., 203, 244 Foldy, L.L., 87 Foquet, M., 139, 158 Forbes, G.W., 211 Ford, G.W., 155 Fork, R.L., 4, 5 Forrester, A.T., 108 Förster, Th., 153

334


Fowler, R.H., 187 Franz, W., 208 Fraser, A.B., 195, 196, 221 Frazin, R.A., 159, 163 Fresnel, A., 196 Friberg, A.T., 139, 163 Fried, D.G., 299, 300 Friedman, B., 222 Friedman, R.S., 17 Frischat, S.D., 242 Fromm, D.P., 170 Fry, E.S., 122 Fujihira, M., 155 Furtak, T.E., 153 Furukawa, H., 170 Futterman, S.N., 230 G Gadenne, P., 160 Gaeta, A.L., 131 Gajda, M., 304 Gallagher, A., 169 Gamow, G., 187 Gao, W., 270 Garcia, N., 159, 163 Gardiner, C.W., 131 Geiger, H., 69 Gerber, C., 148 Germer, L., 73 Gersen, H., 158–160 Gersten, J., 149 Giorgini, S., 326, 327 Girard, C., 140, 160, 162–164, 170, 243 Gisin, N., 126 Glass, A.M., 149 Glauber, R.J., 105, 110, 304 Gleyzes, P., 166 Gohle, Ch., 10 Goldberg, P., 129–131 Goldstein, H., 73, 76, 77, 201 Golovchenko, J.A., 300 Gomes, A.S.L., 243 Goos, F., 189, 217, 219 Gorbunov, A.A., 170 Gori, F., 269, 270 Gorkov, L.P., 320 Gorodetsky, M.L., 241 Goudonnet, J.-P., 153, 159, 160 Gouesbet, G., 223

Goultelmakis, E., 10 Graham, R., 310, 317 Grandy, W.T., 209 Graves, R.P., 15, 23, 24, 26 Graves-Morris, P., 321 Greenler, R., 219 Greffet, J.-J., 139, 159, 162–164 Grehan, G., 223 Gresillon, S., 160 Greytak, T.J., 299, 300 Griffin, A., 319 Grimaldi, F.M., 190, 191 Grischkowsky, D., 239 Grober, R.D., 158, 170 Grossmann, S., 304 Grynberg, G., 89 Guckenberger, R., 167 Gudmundsen, R.A., 108 Guerra, J.M., 243 Guimarães, L.G., 225, 226, 229, 237 Gurney, H., 253 Gurney, R.W., 187 Gustafsson, M.G.L., 144, 151 H Hafner, C., 162 Hakanson, U., 148, 149 Haken, H., 310, 317 Halas, N.J., 149 Hall, A.R., 188 Hall, E.E., 187 Hall, N.S., 278, 327, 328, 330 Hamann, H.F., 169 Hamel, W.A., 130 Hamilton, W.R., 15, 23, 201 Han, D.J., 300 Hänchen, H., 189, 217, 219 Hänsch, T.D., 167 Hänsch, T.W., 10, 153, 166, 299, 300 Hanssen, J.L., 304 Hare, J., 241, 242 Hargrove, L.E., 4 Haroche, S., 103 Harootunian, A., 157, 162 Harris, S.E., 6 Harris, T.D., 158 Hartschuh, A., 169 Hasbani, R., 10 Hashimoto, M., 169 Hau, L.V., 300

Author index for Volume 50 Hayazawa, N., 169 Haynes, C.L., 157 Hayter, A., 231 Hebral, B., 299, 300 Hecht, B., 140–142, 144, 153, 155, 159–163, 170 Heckl, W.M., 153, 166 Heilbron, J.L., 278, 280–282 Heinan, H., 4 Heinzelmann, H., 162, 163 Heinzen, D.J., 300 Heinzmann, U., 9 Heisenberg, W., 55, 65, 79, 80, 86 Hell, S.W., 142, 144 Heller, E.J., 242 Hellwarth, R.W., 3, 10 Hendra, P.J., 148, 153 Henkel, C., 139, 163 Henry, C.H., 128, 131 Hentschel, M., 9, 10 Heritage, J.P., 149 Hermann, A., 278, 281, 330 Herminghaus, S., 153, 160 Hertz, H.M., 148 Herzberg, G., 17 Hettich, C., 155 Hillenbrand, R., 168 Hills, V.L., 253 Hirsch, L.R., 149 Hocker, L.O., 171 Hogervorst, W., 45 Holt, R.A., 123 Holthaus, M., 302–304, 310 Hondros, D., 140 Hong, M.K., 141 Hönl, H., 205 Hori, M., 141 Horne, M.A., 123 Horváth, G., 222 Hu, X.H., 243 Hulet, R., 299, 300 Hund, F., 187 Huygens, C., 194, 195 I Iblisdir, S., 126 Ichimura, T., 169 Idziaszek, Z., 304 Ilchenko, V.S., 241, 242

Indik, R.A., 17, 45 Inouye, S., 299, 300 Inouye, Y., 148, 161, 166, 167, 169 Ippen, E.P., 4, 5 Irie, M., 141 Isaacson, M., 157, 162 Itatani, J., 9 Ivanov, M.Yu., 9, 10 J Jackson, J.B., 149 Jackson, J.D., 221 Jahn, R., 144 Jakobs, S., 144 James, D.F.V., 106, 268 Jammer, M., 278, 330 Jarmie, N., 231 Javan, A., 171 Jaynes, E.T., 116 Jeanmaire, D.L., 148, 153 Jeans, J.H., 60, 286 Jeffrey, M.R., 17, 24, 25, 34, 42–46 Jersch, J., 170 Joannopoulos, J.D., 242 John, S., 242 Johnson, B.R., 225 Johnson, P.O., 108 Jones, D.S., 211 Jönsson, C., 253 Jordan, A.N., 303 Jordan, P., 55, 65, 79, 87 Joulain, K., 139, 163, 164 Juan, J., 231 K Kac, M., 304 Kador, L., 158 Kaen, P.N., 5 Kaiser, T., 206 Kaivola, M., 139, 163 Kane, D.J., 6 Kangro, H., 278 Kapale, K.T., 310 Kapteyn, H.C., 8 Karczewski, B., 266 Karrai, K., 158 Kasevich, M.A., 299, 300 Katnik, S.M., 152 Kawata, M.S., 141

335

336


Kawata, S., 148, 166, 167, 169, 170 Kazak, N.S., 45 Kazarinov, R.F., 128, 131 Keilmann, F., 166–168 Keldysh, L.V., 7 Keller, J.B., 202, 203, 208 Keller, O., 82, 87, 89, 91, 139–141, 159, 163, 205 Kelley, P.L., 110 Kerker, M., 155 Ketterle, W., 299–303, 310 Khapalyuk, A.P., 32 Khare, V., 222, 223, 234, 236 Khilo, N.A., 45 Killian, T.C., 299, 300 Kim, K.E., 229 Kimble, H.J., 118 Kinber, V.Y., 203 King, T.A., 45 Kino, G.S., 170 Kipnis, N., 253 Kirchhoff, G., 198, 283 Kirpichnikova, N.Y., 204 Kirschke, A., 171 Kirsten, G., 65 Klar, T.A., 144 Klein, M.J., 59, 278, 328 Klein, N., 117 Kleiner, W.H., 110 Klenberger, R., 9 Kleppner, D., 103, 299, 300 Klett, J.D., 230 Knight, P.L., 63, 116, 118, 132 Knoll, B., 167 Knox, R.S., 153 Kocharovsky, V.V., 301–303, 310, 311, 318, 319, 321–325 Kocharovsky, Vl.V., 302, 303, 310, 318, 319, 321–325 Können, G.P., 222 Kooyman, R.P.H., 166 Kopelman, R., 155 Körber, H., 65 Korlach, J., 139, 158 Korotkova, O., 268–270 Korterik, J.P., 159, 160 Kostelar, R.L., 158 Kramer, A., 167 Kramers, H.A., 53, 66, 68

Krausz, F., 4, 9, 10 Kravtsov, Y.A., 211 Kreibig, U., 153 Krenn, J.R., 160 Kretschmann, E., 152 Krider, P., 172, 174 Krieger, W., 171 Kuhn, H., 153, 156 Kühn, S., 148, 149 Kuhn, T.S., 278, 279 Kuipers, L., 158–160 Kurlbaum, F., 60 Kurn, D.M., 299, 300 Kwiat, P.G., 122, 124 L Labani, B., 162 Labaree, W., 172, 174 Lagendijk, A., 243 Lalor, E., 32 Lamb, W.E., 299, 305 Lamb Jr, W.E., 110, 113 Lami, A., 20, 21 Lamoreaux, S., 104 Landau, L.D., 15, 19, 20, 56, 78–80 Landhuis, D., 299, 300 Landi, S.M., 170 Lange, S., 206 Lanz, M., 155 Laport, O., 90 Laukien, G., 205 Laven, P., 236 Lawandy, N.M., 243 Lawry, J.M.H., 211 Lazutkin, V.F., 242 Le Floch, A., 219 Lee, B.J., 244 Lee, H., 268 Lee, R.L.J., 17 Lee Jr., R.L., 195, 196, 221, 223 Lefèvre-Seguin, V., 241, 242 Légaré, F., 10 Leggett, A., 325 Leiderer, P., 153, 160 Leontovich, M.A., 203 Leosson, K., 160 Lerondel, G., 159 Lesins, G.B., 230 Lesniewska, E., 159

Author index for Volume 50 Letokhov, V.S., 155, 243 Levene, M.J., 139, 158 Levesque, J., 10 Leviatan, Y., 162 Lewenstein, M., 9, 304 Lewis, A., 157, 162 Lewis, G.N., 53, 70, 296 L’Huiller, A., 9 Li, K., 172 Li, Y., 268 Liao, P.F., 149 Lidorikis, E., 242 Liebisch, T., 45 Lifshitz, E.M., 15, 19, 20, 320 Liu, C., 300 Livingston, W., 219 Lloyd, H., 15, 24, 25, 42, 46 Lobastov, V.A., 6 Logan, N.A., 205 Lommel, E., 211 Long, M.B., 242 Long, R., 241, 242 Longuet-Higgins, H.C., 17 Lord Rayleigh, 142, 205, 225, 286 Lorentz, H.A., 65, 282, 286, 296 Lorenz, L., 205 Lotsch, H.V.K., 244 Loudon, R., 103, 105, 110, 111, 118, 132 Love, A.E.H., 207 Ludwig, D., 27, 211 Lummer, O., 60 Luneburg, R.K., 208 Lunney, J.G., 15, 17, 24, 25, 42–44, 46 Lynch, D.K., 219, 230 M Mach, E., 258 Macklin, J.J., 158 Magyar, G., 108 Maiman, T.H., 3 Mainfray, G., 7 Maleki, L., 242 Malmqvist, L., 148 Mandel, L., 82, 101–103, 105–108, 110, 111, 113, 118, 124, 132, 139, 140, 163, 262, 263, 268 Mansuripur, M., 34 Marnier, G., 24, 42 Marquier, F., 163

337

Marston, P.L., 217 Marte, A., 300 Marti, O., 153, 160 Martin, O.J.F., 162, 163, 170 Martin, Y., 150, 167 Martinez, O.E., 170 Massey, G.A., 151, 152, 162 Matarrese, L.M., 171 Matsko, A.B., 242 Matthews, M., 299, 300 Maue, A.W., 205 McCutchen, C.W., 150 Mcgowan, R.W., 239 McMullan, D., 145, 148 McQuillan, A.J., 148, 153 Mead, C.A., 17 Meade, R.D., 242 Mees, L., 223 Mehra, J., 278 Meier, M., 149 Mekis, A., 242 Mendez, E.R., 163 Merz, R., 167 Merzbacher, E., 187 Metiu, H., 149, 155 Mewes, M.O., 299, 300 Meyer-Rochow, V.B., 222 Meyrath, T.P., 304 Michaelis, J., 155 Mie, G., 206 Miesner, H.J., 299, 300 Mikhailychenko, Y.P., 24, 42 Millikan, R.A., 65 Milonni, P.W., 103, 104, 106, 113, 115–118, 126, 129–131 Milosevich, N., 9 Minnaert, M.G.J., 219 Mitchell, D.L., 227 Mlynek, J., 153, 155, 160 Mocker, H.W., 4 Moerland, R.J., 158 Moerner, W.E., 158, 170 Moiseyev, N., 17 Mollow, J.D., 253 Moroz, A., 242 Morse, P.M., 57 Moses, H.E., 89 Moskovits, M., 153 Moss, S.C., 299, 300

338


Mount, K.E., 206, 208, 209 Mourou, G.A., 7 Moyer, P.J., 141, 243 Mühlschlegel, P., 170 Mujat, A., 257 Mulet, J.-P., 163, 164 Muray, A., 157 Murnane, M.M., 8 Murphy, S.Q., 299, 300

Olson, D.H., 149 Olver, F.W.J., 211 Omon, E., 152 Ooi, C.H.R., 302, 303, 310 Oppenheimer, J.R., 54, 56, 78–80, 86 Orrit, M., 158 Osgood, R., 153 Otter, A.M., 158, 159 Otto, A., 152

N

P

Naida, O.N., 44 Narozhny, N.B., 117 Nassenstein, H., 151 Natanson, L., 280 Nauenberg, M., 191 Nedungadi, T.M.K., 24, 27, 30, 42, 46 Nernst, W., 140 Nesbitt, D.J., 169 New, G.H.C., 4 Newell, A.C., 17, 45 Newton, I., 187–190, 193 Newton, R.G., 208 Ngo, D., 230 Nicholls, G., 151 Nicholson, J.W., 208 Nielsen, M.A., 126 Nieto-Vesperinas, M., 159, 163, 205 Niikura, H., 10 Nitzan, A., 149 Noble, B., 205 Nöckel, J.U., 242 Noh, H., 243 Nordheim, L., 187 Novotny, L., 139–142, 144, 148–150, 155, 159, 161–163, 165, 166, 168–170, 172 Nussenzveig, H.M., 30, 199, 200, 205, 206, 208, 209, 211, 214–220, 222, 223, 225–229, 234, 236–241, 243, 244 Nye, J.F., 17

Paesler, M.A., 141, 243 Paschen, W., 60 Passante, R., 67 Pauli, W., 79, 80, 86 Peacock, G., 253 Pedarning, J.D., 153, 166 Peierls, R., 56, 78–80 Pekeris, C.L., 204 Pendry, J.B., 242 Peppard, M.B., 56 Perkal’skis, B.S., 42 Pernter, J.M., 231 Perry, M.D., 7 Petermann, K., 128, 131 Petit, A.T., 291 Peyrade, D., 163, 164 Pfleegor, R.L., 108 Pinnick, R.G., 230 Pitaevskii, L.P., 15, 19, 20, 320, 326, 327 Planck, M., 54, 58–60, 278, 279, 281, 282, 287, 291, 330 Poggendorff, J.C., 26, 42 Pohl, D.W., 140, 146, 148, 149, 155, 157, 160–163, 166, 170, 174 Poincaré, H., 207 Politzer, H.D., 304 Pollack, M.A., 4 Pompe, W., 170 Ponomarenko, S., 139, 163, 270 Popp, J., 241 Potter, R., 24, 26, 27, 29, 30, 42 Power, E.A., 67, 140, 164 Prasad, P.N., 141 Price, G.N., 304 Primack, H., 242 Pringsheim, E., 60 Prober, D.E., 170 Prock, A., 155

O O’Boyle, M.P., 166 Ockendon, J.R., 211 O’Hara, J.G., 15 Ohmura, T., 89 Ohtsu, M., 141 O’Keefe, J.A., 150

Author index for Volume 50 Pulfrich, C., 216 Purcell, E.M., 108 Q Qian, S.-X., 242 Quate, C.F., 170 Quelin, X., 160 Quéré, F., 9 R Radmore, P.M., 132 Raizen, M.G., 304 Rajagopalan, V.S., 24, 27, 30, 42, 46 Raman, C.V., 24, 27, 30, 42, 46 Rasmussen, J.O., 21 Räther, H., 152 Ray, B., 232 Rayleigh, J.W.S., 60 Rechenberg, H., 278 Reddik, R.C., 159 Regge, T., 208 Regli, P., 162 Reider, G.A., 9 Rempe, G., 117, 300 Reppy, J.D., 299, 300 Ribordy, G., 126 Richards, D., 243 Ridgway, W.L., 229 Ring, P., 21 Rivoal, J.C., 160 Rizzoli, S.O., 144 Roberts, A., 162 Robinson, A., 253 Rogobete, L., 148, 149 Rohner, F., 157 Rohrer, H., 148 Roll, G., 206 Ronchi, V., 189 Rosenfeld, L., 278, 295 Rothammel, K., 171 Roychowdhury, H., 139, 163, 265, 268, 270 Royer, P., 159, 167 Rubens, H., 60 Rubinowicz, A., 200 Ryaboy, V.M., 17 Ryzhevich, A.A., 45 Rz¸az˙ ewski, K., 304

339

S Sackett, C., 299, 300 Saengkaew, S., 223 Salem, M., 270 Salomon, L., 159 Sanchez, E.J., 150, 166, 168, 169 Sanchez-Mondragon, J.J., 117 Sandoghdar, V., 148, 149, 155 Santarsiero, M., 269, 270 Sarayeddine, K., 159 Sargent III, M., 110, 111, 113, 121 Sarychev, A.K., 160 Sauvain, E., 243 Savchenkov, A.A., 241, 242 Scarani, V., 126 Schanz, H., 242 Schell, A.J., 24, 32, 42, 45 Schoelkopf, R.J., 170 Schoer, J.M., 7 Schotland, J.C., 159, 163 Schottky, W., 187 Schrader, M., 142 Schreck, F., 300, 304 Schrödinger, E., 56, 73, 76, 77 Schrödinger, W., 202 Schuck, P.J., 170 Schuster, J., 300 Schweiger, G., 206 Scifres, D.R., 131 Scrinzi, A., 10 Scully, M.O., 103, 110, 111, 113, 121, 299, 301–303, 305, 310, 311, 313, 314, 317–319, 321–325 Seelig, E.W., 243 Sekatskii, S.K., 155 Sekkat, Z., 169 Sentenac, A., 159, 163 Seres, E., 10 Seres, J., 10 Setälä, T., 139, 163 Shalaev, V.M., 160 Shank, C.V., 4, 5 Shapiro, S.L., 4 Shchegrov, A.V., 139 Sheng, P., 243 Shi, H., 319 Shih, H., 45 Shimony, A., 122, 123 Shipley, S.T., 225, 237

340


Shrinivas, R., 6 Shubin, V.A., 160 Shverdin, M.Y., 6 Sibbett, W., 5 Siegman, A.E., 130, 131 Sigalas, M.M., 242 Silbey, R., 155 Silverman, M., 22 Simon, F., 69 Singh, S., 117 Slater, J.C., 53, 55, 66–68 Smilansky, U., 242 Smith, E.N., 299, 300 Smith, W.L., 7 Smolyaninov, I.I., 159 Snow, J.B., 242 Sokoloff, D.R., 171 Sommerfeld, A., 140, 187, 200, 203, 205, 214 Soskin, M.S., 17 Soukoulis, C.M., 242 Spajer, M., 159 Specht, M., 153, 166 Spence, D.E., 5 Spielmann, C., 9, 10 Stamper-Kurn, D., 299, 300 Stefan, J., 284 Stefanon, I., 159 Steinberg, A.M., 122, 124, 243, 244 Stepanov, M.A., 17, 37, 45 Stetser, D.A., 4 Stetson, K.A., 151 Stockle, D., 167 Stöckle, R.M., 169 Stockman, M.I., 172 Stokes, G.G., 259 Stone, A.D., 242 Stoyer, M.A., 21 Stranick, S.J., 139, 141, 166, 169 Streifer, W., 131 Strekalov, D., 242 Strelill, C., 10 Stringari, S., 326, 327 Stuewer, R.H., 192 Sudarshan, E.C.G., 82, 110 Sugiura, T., 148, 167 Suh, Y.D., 169 Sundaram, B., 129–131 Sundaramurthy, A., 170 Suzuki, T., 171

Svidzinsky, A.A., 302, 303, 310, 321 Swan, A., 141 Swindell, W., 259, 263 Synge, E.H., 145, 147, 148, 150, 162 Szoke, A., 171 T Takano, Y., 299, 300 Talamantes, J., 230 Tan, W.H., 155 Tanthapanichakoon, W., 223 Taylor, G.I., 253 Teich, M.C., 7 Tempea, G., 10 Ter Haar, D., 278, 286, 304, 328, 330 Tew, R.H., 211 Thirunamachandran, T., 140, 164 Tip, A., 242, 243 Titchmarsh, E.C., 209 Tittle, W., 126 Toledo-Crow, R., 158 Tollett, J., 299, 300 Tomsovic, S., 242 Toraldo di Francia, G., 144 Townes, C.H., 108, 127 Townsend, C.G., 299, 300 Tran, N.H., 219 Trautman, J.K., 140, 158 Trebino, R., 6 Tricker, R.A.R., 219 Truhlar, D.G., 17 Turbadar, T., 152 Turner, S.W., 139, 158 Twiss, R.Q., 101 Tzeng, H.-M., 242 Tzeng, H.M., 242 U Uberacker, M., 10 Udem, Th., 10 Ueyanagi, K., 146, 150 Uhlenbeck, G.E., 90, 304 Uhlenbeck, K., 17 Uhlmann, A., 27 Ulloa, A., 231 Upstill, C., 224 Ursell, F., 222 Ussishkin, I., 242

Author index for Volume 50 V Vaez-Iravani, M., 158 Vahala, K.J., 242 Van Albada, M.P., 243 Van De Hulst, H.C., 30, 206, 214, 221, 223, 233, 234, 237, 239, 241 Van Der Pol, B., 208, 220 van der Waerden, B., 90 van der Weide, D.W., 167 van Druten, N.J., 299–301 Van Duyne, R.P., 148, 153, 157 Van Huele, J.F., 67 van Hulst, N.F., 158–160, 166 Van Keuls, F.W., 299, 300 Van Labeke, D., 162 Van Tiggelen, B.A., 243 Vandenbem, C., 242 Vanisri, H., 223 Varro, S., 278 Vasnetsov, M.V., 17 VCT, 43 Veerman, J.A., 158, 159 Verdet, E., 259 Verhoeft, A.J., 10 Vickery, S.A., 155 Videen, G., 230 Vigneron, J.P., 242 Vigoureux, J.M., 142, 243 Villani, G., 20, 21 Villeneuve, D.M., 10 Vohnsen, B., 159, 163 Voigt, W., 26, 45 Völcker, M., 171 Volkov, V.S., 159, 160, 163 Von Klitzing, W., 241, 242 Von Neumann, J., 21 W Walecka, J.D., 320 Walker, D.H., 6 Wall, K.F., 242 Waller, I., 80, 86 Walther, H., 111, 117, 171 Walther, T., 122 Wang, D.-S., 155 Wang, J., 242 Wang, R.T., 223 Warmack, R.J., 159 Warnick, K.F., 37

341

Watson, G.N., 207, 208 Watson, W., 230 Weaire, D., 15 Webb, W.W., 139, 158 Weber, W.H., 155 Weeber, J.-C., 160, 163, 164 Weibel, E., 148 Weinberg, S., 113 Weiner, J.S., 158 Weinman, J.A., 225, 237 Weiss, C., 303, 304 Weisskopf, V., 67, 207 Wessel, J., 146, 148, 153, 165, 171 West, J.L., 149 Westcott, S.L., 149 Westphal, K., 205 Westphal, V., 144 Wheaton, B.R., 65, 66 Whewell, W., 258 White, F.P., 208 Whittaker, E.T., 253, 258 Wickramasinghe, H.K., 150, 166, 167 Wiederrecht, G.P., 141 Wieman, C., 299, 300 Wien, W., 60, 284–286 Wiener, N., 263 Wigner, E.P., 21, 67 Wilkens, M., 303, 304 Wilkinson, M., 17 Williams, C.C., 166, 167 Willig, K.I., 144 Willmann, L., 299, 300 Wilson, C.T.R., 231 Winn, J.N., 242 Wiscombe, W.J., 214–219, 227, 241 Wobranskii, P., 10 Woerdman, J.P., 130 Wokaun, A., 149 Wolf, E., 15, 18, 19, 21, 26, 46, 77, 82, 99–103, 105–108, 110, 111, 118, 139, 163, 194, 198, 200, 202, 205, 217, 257, 260–265, 268–270 Wolfram, S., 36 Wolga, G.J., 7 Wong, G.K.S., 299, 300 Wong, J.G.D., 230 Wootters, W.K., 126 Wouthuysen, S.A., 87 Wurtz, G., 167 Wynar, R.J., 300

342


X Xie, X.S., 140, 150, 158, 166, 168–170 Y Yakovlev, V.S., 10 Yamilov, A., 243 Yang, P.C., 158 Yao, A.M., 131 Yavuz, D.D., 6 Yeang, C.P., 207 Yin, G.Y., 6 Young, T., 195–197, 253, 255, 256 Yudin, G.I., 9 Z Zapletal, M., 167

Zbinden, H., 126 Zenack, A.Z., 242 Zender, C.S., 230 Zenhausern, F., 150, 166, 167 Zenneck, J., 140 Zenobi, R., 169 Zermelo, E., 281, 328 Zernike, F., 260 Zewail, A.H., 6 Zhang, Z.M., 244 Zhu, S.Y., 301, 310, 311, 321 Ziff, R.M., 304 Zingsheim, H.P., 153, 154, 157 Zoller, P., 131 Zubairy, M.S., 103, 299, 301, 310, 311, 317, 321 Zurek, W.H., 126 Zurita-Sanchez, J.R., 139, 163, 165

Subject index for Volume 50 – partition function 316 Casimir force 104 Casimir–Polder effect 103 Cauchy–Riemann conditions 47 chirality 17 coherence, complex degree of 261 – , degree of 260, 262, 264, 266 – theory, classical 99, 104, 264, 269 – – in space-frequency domain 262 – – , quantum 104 coherency matrix 263 coherent state 102, 109 Compton effect 66, 68, 69 – wavelength 72, 79 conical diffraction 17, 34, 42–44 – – , paraxial theory of 31 – refraction 15, 17, 18, 25, 47 – – , external 23 – – , internal 23 cooling coefficient 312 correlation function 106 – , intensity 107 Coulomb gauge 85 cross-spectral density matrix 265, 268 crystal optics 43

Abbe’s limit 144 Airy function 142, 199, 204 – pattern 231 – theory 223 angular spectrum representation 139 antenna theory 170 atom laser 299 attosecond regime 8–10 – streak camera 9 Bell inequality 122–124 Bell’s theorem 122 Belsky–Khapalyuk theory 33 biaxial crystal 15 – dielectric, nonchiral 18 birefringence 3 blackbody radiation 54, 57, 59, 64, 277, 278, 281 Bogoliubov coupling 318 – gas, interacting 322, 324 Bohr–Kramers–Slater theory 54, 67, 68 Bohr–Sommerfeld stability condition 76 Bohr–Sommerfeld–Wilson quantization rule 285 Boltzmann’s constant 59 – principle 57, 63 – statistics 280 Bose gas 300, 317 – – , ideal 310, 314 – – , weakly interacting 319 Bose–Einstein condensation 277, 278, 299– 305, 310 – statistics 55, 78, 299 Bragg reflection 5 Brownian motion 298

de Broglie wave 71, 86 de Broglie’s phase wave 71 de Broglie’s world vector relation 73 Debye expansion 219, 220 diabolical point 30, 44, 45 – singularity 20 diffraction as tunneling 213 – by a half plane 205 – , Fock’s theory of 203 – , geometrical theory of 202 – limit 139, 142–144, 148 – – , resolution limit beyond 144 – theory, classical 195–200, 222 Dirac electron 80

canonical ensemble 300, 304, 318 – – , grand 301 – momentum 75 343

344

Subject index for Volume 50

– equation 85, 86 Dirac–Waller–Oppenheimer light quantum theory 80 Dirichlet boundary condition 211 Doppler effect 69 double refraction 254 Duling–Petit law 291

Heisenberg uncertainty principle 142 Helmholtz equation 139, 204, 262, 267 Heron of Alexandria 194 hole burning 113 Huygens principle 244 Huygens–Fresnel principle 198 – theory 199

eikonal equation 73 – function 201 Einstein’s A coefficient 113, 114, 297–299 – B coefficient 113, 297–299 – fluctuation formula 64, 65, 100, 108, 115 – light-quanta 54, 56–65 electron interferometry 10 – microscopy 6 energy-momentum four vector 75 entangled state 126 entropy of radiation 61 Euler’s formula 194 evanescent wave 139, 140, 163

index of refraction, intensity dependent 3

femtochemistry 6 femtosecond pulse 4, 5 Fermat’s principle 71, 73, 75, 202 Fermi–Dirac statistics 78 Fermi’s golden rule 112 field-effect transistor 171 Fischer pattern 157 Fock state 109, 110, 111 Fresnel–Arago laws 257 Fresnel–Kirchhoff representation 200 – theory of diffraction 198 Gaussian beam 30, 37, 40, 45 geometrical-optics 39, 56, 73 Goos–Hanschen angular displacement 217 – shift 189, 217, 219 group velocity dispersion 5 Hamilton equation 19 Hamilton–Jacobi equation 56, 73, 77, 91, 92 – – , quantum mechanical 78 Hamilton’s principle 27, 29 – ray cone 20 Hanbury Brown and Twiss experiment 107 harmonic generation 4 heating coefficient 312 ,313 Heisenberg equation 104, 119, 120, 262 Heisenberg’s matrix mechanics 79

Jaynes–Cummings model 116 Kerr effect 3 Kirchhoff’s approximation 198, 199 Lamb dip 113 – shift 103 Landau–Peierls operator 89 – theory 56, 80, 82, 86 laser cooling 115 – linewidth 126–129 – , neodymium glass 4 – phase transition analogy 310 – , quantum theory of 299, 304–309, 311 – , semiconductor 128 – , Ti-sapphire 5, 7,9 light tunneling 202 – – in clouds 227–230 local hidden-variable theory 122, 123, 132 Mandel’s Q-parameter 102, 118 matter waves 70 Maupertuis action 76, 202 Maxwell’s equation 18, 20, 21, 80, 81, 85, 90, 91, 280 Michelson interferometer 105, 107 microscopy, dark field 145 – , near-field 153, 155, 243 – , total internal reflection fluorescence 151 Mie efficiency factors 241 – resonance 224, 228, 229, 237, 240 – scattering 205 – – , CAM theory of 209 – theory 223, 232 mode-locking 113 – , Kerr-lens 7 multiple multipole technique 162 mutual coherence function 261 – intensity 260

Subject index for Volume 50 nanoplasmonics 149, 153 nanosecond pulse 4 natural linewidth 127 near-field optics 140, 142, 161–165, 170 Nerst’s heat theorem 282 non-classical light 117 numerical aperture 142 Oppenheimer’s equation for light quantum 87 optical activity 17 – equivalence theorem 110 – imaging 243 – maser 305 – microscopy 150 – – , aperture-based near-field 147 – – , high-resolution 139 – – , near field 158, 159, 162, 166 Pade approximation 321 paraxial approximation 28 – differential equation 47 – propagation 267 Pauli spin matrices 32 Petermann factor 128, 130, 131 phase matching 124 photoelectric effect 57, 68, 100–112, 113 – emission, two-photon induced 7 photoelectron counting 110 photon 70 – antibunching 108, 118 – bunching 101, 102 – distribution function 305, 308, 309 – , eikonal equation for 91–93 – statistics 308 – wave function 80, 82, 84 – – mechanics 79–91 photonic crystal 242 Planck formula 100 – spectrum 114 – radiation law 54, 55, 57–60, 62, 64, 299 Planck’s black-body radiation law 279–299 Planck’s constant 56 Poggendorff’s dark ring 26, 27, 30, 31, 35, 38, 40, 42 Poincaré–Watson method 210 Poisson distribution 109 – sum formula 209 polarization, degree of 259, 267 – matrix 263 – , theory of 264, 269

345

ponderomotiv energy 7 Popov approximation 319 population inversion 3 Poynting vector 21 principle of least action 74 Q-switching 3, 113 quantum computation 126 – cryptography 111, 126, 132 – electrodynamics 67, 99, 113 – – , cavity 103, 241 – information 132 – noise 99, 126, 131, 133 – of action 287, 289 – optics 99, 103, 110 – recurrence theorem 117 – theory of radiation 99 – tunneling 202 Rabi frequency 117 Raman’s bright spot 26, 38–40 – scattering, coherent anti-Stokes 169 – – , stimulated 4 Rayleigh spectrum 114 Rayleigh–Jeans formula 100, 291 – law 60, 281, 282, 286, 293, 298 Regge pole 225 – trajectory 225 Regge–Debye pole 220, 234 resonance fluorescence 117, 118 saturable absorption 3 scanning nanometer optical spectral microscope 157 – tunneling microscopy 148 Schawlow–Townes laser linewidth (see laser linewidth) Schrodinger equation 9, 79 second harmonic generation 6 – law of thermodynamics 284 self-focusing 3, 5 self-Kerr-lens 5 self-phase modulation 3–5, 10 semiclassical radiation theory 111–117 Slater’s virtual radiation field 66 spontaneous emission 113, 114 – – noise 127 – – , stochastic amplified 5 squeezed state 132, 318 squeezing 317 stationary-phase approximation 34

346

Subject index for Volume 50

– , principle of 75 Stefan–Boltzmann law 284 Stirling’s formula 58 stimulated emission 3, 113 Sudarshan’s equation 82 surface enhanced Raman scattering 152 – plasmon 152 – – polariton 152 thermionic emission 187 time-resolved spectroscopy 6 total internal reflection, frustrated 147 tunneling ionization 7 ultraviolet catastrophe 61, 283 van Cittert–Zernike theorem 262 Van De Hulst’s theory 233 Van der Waals attraction 140

Watson’s transformation 207, 208 wave mechanics 71 wave-particle duality 54, 64, 65, 78, 253, 296 – – , covariant 73 Weisskopf–Wigner approximation 116 whispering gallery modes 225 Wien’s radiation law 58, 60, 61, 63, 100, 101, 280, 286, 291 – spectral law 282, 285, 286 WKB approximation 208, 210, 213, 220, 228 Yagi–Uda antenna 172 Young interference 196, 253, 265, 270 – interferometer 105 Young’s two-slit experiment 102 zero-point energy 55, 104 – fluctuations 67

Contents of previous volumes*

VOLUME 1 (1961) 1 2 3 4 5 6 7 8

The modern development of Hamiltonian optics, R.J. Pegis Wave optics and geometrical optics in optical design, K. Miyamoto The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat Light and information, D. Gabor On basic analogies and principal differences between optical and electronic information, H. Wolter Interference color, H. Kubota Dynamic characteristics of visual processes, A. Fiorentini Modern alignment devices, A.C.S. Van Heel

1– 29 31– 66 67–108 109–153 155–210 211–251 253–288 289–329

VOLUME 2 (1963) 1 2 3 4 5 6

Ruling, testing and use of optical gratings for high-resolution spectroscopy, G.W. Stroke The metrological applications of diffraction gratings, J.M. Burch Diffusion through non-uniform media, R.G. Giovanelli Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi Fluctuations of light beams, L. Mandel Methods for determining optical parameters of thin films, F. Abelès

1– 72 73–108 109–129 131–180 181–248 249–288

VOLUME 3 (1964) 1 2 3

The elements of radiative transfer, F. Kottler Apodisation, P. Jacquinot, B. Roizen-Dossier Matrix treatment of partial coherence, H. Gamo

1 2 3 4

Higher order aberration theory, J. Focke Applications of shearing interferometry, O. Bryngdahl Surface deterioration of optical glasses, K. Kinosita Optical constants of thin films, P. Rouard, P. Bousquet

1– 28 29–186 187–332

VOLUME 4 (1965)

* Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

347

1– 36 37– 83 85–143 145–197

348 5 6 7

Contents of previous volumes The Miyamoto–Wolf diffraction wave, A. Rubinowicz Aberration theory of gratings and grating mountings, W.T. Welford Diffraction at a black screen, Part I: Kirchhoff’s theory, F. Kottler

199–240 241–280 281–314

VOLUME 5 (1966) 1 2 3 4 5 6

Optical pumping, C. Cohen-Tannoudji, A. Kastler Non-linear optics, P.S. Pershan Two-beam interferometry, W.H. Steel Instruments for the measuring of optical transfer functions, K. Murata Light reflection from films of continuously varying refractive index, R. Jacobsson X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor 7 The wave of a moving classical electron, J. Picht

1– 81 83–144 145–197 199–245 247–286 287–350 351–370

VOLUME 6 (1967) 1 2 3 4 5 6 7 8

Recent advances in holography, E.N. Leith, J. Upatnieks Scattering of light by rough surfaces, P. Beckmann Measurement of the second order degree of coherence, M. Françon, S. Mallick Design of zoom lenses, K. Yamaji Some applications of lasers to interferometry, D.R. Herriot Experimental studies of intensity fluctuations in lasers, J.A. Armstrong, A.W. Smith Fourier spectroscopy, G.A. Vanasse, H. Sakai Diffraction at a black screen, Part II: electromagnetic theory, F. Kottler

1– 52 53– 69 71–104 105–170 171–209 211–257 259–330 331–377

VOLUME 7 (1969) 1 2 3 4 5 6 7

Multiple-beam interference and natural modes in open resonators, G. Koppelman Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Pegis Echoes at optical frequencies, I.D. Abella Image formation with partially coherent light, B.J. Thompson Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian The photographic image, S. Ooue Interaction of very intense light with free electrons, J.H. Eberly

1– 66 67–137 139–168 169–230 231–297 299–358 359–415

VOLUME 8 (1970) 1 2 3 4 5 6

Synthetic-aperture optics, J.W. Goodman The optical performance of the human eye, G.A. Fry Light beating spectroscopy, H.Z. Cummins, H.L. Swinney Multilayer antireflection coatings, A. Musset, A. Thelen Statistical properties of laser light, H. Risken Coherence theory of source-size compensation in interference microscopy, T. Yamamoto 7 Vision in communication, L. Levi 8 Theory of photoelectron counting, C.L. Mehta

1– 50 51–131 133–200 201–237 239–294 295–341 343–372 373–440

Contents of previous volumes

349

VOLUME 9 (1971) 1 2 3 4 5 6 7

Gas lasers and their application to precise length measurements, A.L. Bloom Picosecond laser pulses, A.J. Demaria Optical propagation through the turbulent atmosphere, J.W. Strohbehn Synthesis of optical birefringent networks, E.O. Ammann Mode locking in gas lasers, L. Allen, D.G.C. Jones Crystal optics with spatial dispersion, V.M. Agranovich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J. Petykiewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden

1– 30 31– 71 73–122 123–177 179–234 235–280 281–310 311–407

VOLUME 10 (1972) 1 2 3 4 5 6 7

Bandwidth compression of optical images, T.S. Huang The use of image tubes as shutters, R.W. Smith Tools of theoretical quantum optics, M.O. Scully, K.G. Whitney Field correctors for astronomical telescopes, C.G. Wynne Optical absorption strength of defects in insulators, D.Y. Smith, D.L. Dexter Elastooptic light modulation and deflection, E.K. Sittig Quantum detection theory, C.W. Helstrom

1– 44 45– 87 89–135 137–164 165–228 229–288 289–369

VOLUME 11 (1973) 1 2 3 4 5 6 7

Master equation methods in quantum optics, G.S. Agarwal Recent developments in far infrared spectroscopic techniques, H. Yoshinaga Interaction of light and acoustic surface waves, E.G. Lean Evanescent waves in optical imaging, O. Bryngdahl Production of electron probes using a field emission source, A.V. Crewe Hamiltonian theory of beam mode propagation, J.A. Arnaud Gradient index lenses, E.W. Marchand

1– 76 77–122 123–166 167–221 223–246 247–304 305–337

VOLUME 12 (1974) 1 2 3 4 5 6

Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. Harwit, J.A. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin

1– 51 53–100 101–162 163–232 233–286 287–344

VOLUME 13 (1976) 1

On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment, H.P. Baltes 2 The case for and against semiclassical radiation theory, L. Mandel 3 Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G. Schulz, J. Schwider

1– 25 27– 68 69– 91 93–167

350


5

Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi 6 Aplanatism and isoplanatism, W.T. Welford

169–265 267–292

VOLUME 14 (1976) 1 2 3 4 5 6 7

The statistics of speckle patterns, J.C. Dainty High-resolution techniques in optical astronomy, A. Labeyrie Relaxation phenomena in rare-earth luminescence, L.A. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, P.J. Vernier Optical fibre waveguides – a review, P.J.B. Clarricoats

1– 46 47– 87 89–159 161–193 195–244 245–325 327–402

VOLUME 15 (1977) 1 2 3 4 5

Theory of optical parametric amplification and oscillation, W. Brunner, H. Paul Optical properties of thin metal films, P. Rouard, A. Meessen Projection-type holography, T. Okoshi Quasi-optical techniques of radio astronomy, T.W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe

1– 75 77–137 139–185 187–244 245–350

VOLUME 16 (1978) 1 2 3 4 5

Laser selective photophysics and photochemistry, V.S. Letokhov Recent advances in phase profiles generation, J.J. Clair, C.I. Abitbol Computer-generated holograms: techniques and applications, W.-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission from high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky

1– 69 71–117 119–232 233–288 289–356 357–411 413–448

VOLUME 17 (1980) 1 2 3

Heterodyne holographic interferometry, R. Dändliker Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi 4 Michelson stellar interferometry, W.J. Tango, R.Q. Twiss 5 Self-focusing media with variable index of refraction, A.L. Mikaelian

1– 84 85–161 163–238 239–277 279–345

VOLUME 18 (1980) 1 2

Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan Photocount statistics of radiation propagating through random and nonlinear media, J. Pe˘rina

1–126 127–203

Contents of previous volumes Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U. Zavorotnyi 4 Catastrophe optics: morphologies of caustics and their diffraction patterns, M.V. Berry, C. Upstill

351

3

204–256 257–346

VOLUME 19 (1981) 1 2 3 4 5

Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy Light scattering spectroscopy of surface electromagnetic waves in solids, S. Ushioda Principles of optical data-processing, H.J. Butterweck The effects of atmospheric turbulence in optical astronomy, F. Roddier

1– 43 45–137 139–210 211–280 281–376

VOLUME 20 (1983) 1 2 3 4 5

Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtès, P. Cruvellier, M. Detaille Shaping and analysis of picosecond light pulses, C. Froehly, B. Colombeau, M. Vampouille Multi-photon scattering molecular spectroscopy, S. Kielich Colour holography, P. Hariharan Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff

1– 61 63–153 155–261 263–324 325–380

VOLUME 21 (1984) 1 2 3 4 5

Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, L.A. Lugiato The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, W.C. Schieve

1– 67 69–216 217–286 287–354 355–428

VOLUME 22 (1985) 1 2 3 4 5

Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema Spectral and temporal fluctuations of broad-band laser radiation, A.V. Masalov Holographic methods of plasma diagnostics, G.V. Ostrovskaya, Yu.I. Ostrovsky Fringe formations in deformation and vibration measurements using laser light, I. Yamaguchi 6 Wave propagation in random media: a systems approach, R.L. Fante

1– 76 77–144 145–196 197–270 271–340 341–398

VOLUME 23 (1986) 1

Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown 2 Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka 3 Optical films produced by ion-based techniques, P.J. Martin, R.P. Netterfield

1– 62 63–111 113–182

352 4 5

Contents of previous volumes Electron holography, A. Tonomura Principles of optical processing with partially coherent light, F.T.S. Yu

183–220 221–275

VOLUME 24 (1987) 1 2 3 4 5

Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L. Rothberg Interferometry with lasers, P. Hariharan Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, I. Glaser

1– 37 39–101 103–164 165–387 389–509

VOLUME 25 (1988) 1

Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci 2 Coherence in semiconductor lasers, M. Ohtsu, T. Tako 3 Principles and design of optical arrays, Wang Shaomin, L. Ronchi 4 Aspheric surfaces, G. Schulz

1–190 191–278 279–348 349–415

VOLUME 26 (1988) 1 2 3 4 5

Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, I.C. Khoo Single-longitudinal-mode semiconductor lasers, G.P. Agrawal Rays and caustics as physical objects, Yu.A. Kravtsov Phase-measurement interferometry techniques, K. Creath

1–104 105–161 163–225 227–348 349–393

VOLUME 27 (1989) 1 2 3 4

The self-imaging phenomenon and its applications, K. Patorski Axicons and meso-optical imaging devices, L.M. Soroko Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P. Porter

1–108 109–160 161–226 227–313 315–397

VOLUME 28 (1990) 1 2

Digital holography – computer-generated holograms, O. Bryngdahl, F. Wyrowski Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, I.A. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, R.J. Cook

1– 86 87–179 181–270 271–359 361–416


353

VOLUME 29 (1991) 1 2

Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev 3 Generation and propagation of ultrashort optical pulses, I.P. Christov 4 Triple-correlation imaging in optical astronomy, G. Weigelt 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol

1– 63 65–197 199–291 293–319 321–411

VOLUME 30 (1992) 1 2 3 4 5

Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Fabre Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P. Shchepinov Localization of waves in media with one-dimensional disorder, V.D. Freilikher, S.A. Gredeskul Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa Cavity quantum optics and the quantum measurement process, P. Meystre

1– 85 87–135 137–203 205–259 261–355

VOLUME 31 (1993) 1 2 3 4 5 6

Atoms in strong fields: photoionization and chaos, P.W. Milonni, B. Sundaram Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, J.R. Simpson Adaptive multilayer optical networks, D. Psaltis, Y. Qiao Optical atoms, R.J.C. Spreeuw, J.P. Woerdman Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre

1–137 139–187 189–226 227–261 263–319 321–412

VOLUME 32 (1993) 1 2 3 4

Guided-wave optics on silicon: physics, technology and status, B.P. Pal Optical neural networks: architecture, design and models, F.T.S. Yu The theory of optimal methods for localization of objects in pictures, L.P. Yaroslavsky Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, V.U. Zavorotny 5 Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus

1– 59 61–144 145–201 203–266 267–312 313–361

VOLUME 33 (1994) 1 2 3 4

The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin Quantum statistics of dissipative nonlinear oscillators, V. Peˇrinová, A. Lukš Gap solitons, C.M. De Sterke, J.E. Sipe Direct spatial reconstruction of optical phase from phase-modulated images, V.I. Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, F. Wyrowski

1–127 129–202 203–260 261–317 319–388 389–463

354

Contents of previous volumes VOLUME 34 (1995)

1 2 3 4 5

Quantum interference, superposition states of light, and nonclassical effects, V. Bužek, P.L. Knight Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov The statistics of dynamic speckles, T. Okamoto, T. Asakura Scattering of light from multilayer systems with rough boundaries, I. Ohlídal, K. Navrátil, M. Ohlídal Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss

1–158 159–181 183–248 249–331 333–402

VOLUME 35 (1996) 1 2 3 4 5 6

Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis Interferometric multispectral imaging, K. Itoh Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo Coherent population trapping in laser spectroscopy, E. Arimondo Quantum phase properties of nonlinear optical phenomena, R. Tana´s, A. Miranowicz, Ts. Gantsog

1– 60 61–144 145–196 197–255 257–354 355–446

VOLUME 36 (1996) 1 2 3 4 5

Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders Super-resolution by data inversion, M. Bertero, C. De Mol Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan Photon wave function, I. Bialynicki-Birula

1– 47 49–128 129–178 179–244 245–294

VOLUME 37 (1997) 1 2 3 4 5 6

The Wigner distribution function in optics and optoelectronics, D. Dragoman Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura Spectra of molecular scattering of light, I.L. Fabelinskii Soliton communication systems, R.-J. Essiambre, G.P. Agrawal Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg

1– 56 57– 94 95–184 185–256 257–343 345–405

VOLUME 38 (1998) 1 2 3

Nonlinear optics of stratified media, S. Dutta Gupta Optical aspects of interferometric gravitational-wave detectors, P. Hello Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W. Nakwaski, M. Osiński 4 Fractional transformations in optics, A.W. Lohmann, D. Mendlovic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Horner 6 Free-space optical digital computing and interconnection, J. Jahns

1– 84 85–164 165–262 263–342 343–418 419–513


355

VOLUME 39 (1999) 1 2

Theory and applications of complex rays, Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrný 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs

1– 62 63–211 213–290 291–372 373–469

VOLUME 40 (2000) 1 2 3 4

Polarimetric optical fibers and sensors, T.R. Woliński Digital optical computing, J. Tanida, Y. Ichioka Continuous measurements in quantum optics, V. Peˇrinová, A. Lukš Optical systems with improved resolving power, Z. Zalevsky, D. Mendlovic, A.W. Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski 6 Spectroscopy in polychromatic fields, Z. Ficek, H.S. Freedhoff

1– 75 77–114 115–269 271–341 343–388 389–441

VOLUME 41 (2000) 1 2 3 4 5 6 7

Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur Ellipsometry of thin film systems, I. Ohlídal, D. Franta Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu Quantum statistics of nonlinear optical couplers, J. Peˇrina Jr, J. Peˇrina Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sánchez-Soto Optical solitons in media with a quadratic nonlinearity, C. Etrich, F. Lederer, B.A. Malomed, T. Peschel, U. Peschel

1– 95 97–179 181–282 283–358 359–417 419–479 483–567

VOLUME 42 (2001) 1 2 3 4 5 6

Quanta and information, S.Ya. Kilin Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrný, B.A. Malomed Quantum Zeno and inverse quantum Zeno effects, P. Facchi, S. Pascazio Singular optics, M.S. Soskin, M.V. Vasnetsov Multi-photon quantum interferometry, G. Jaeger, A.V. Sergienko Transverse mode shaping and selection in laser resonators, R. Oron, N. Davidson, A.A. Friesem, E. Hasman

1– 91 93–146 147–217 219–276 277–324 325–386

VOLUME 43 (2002) 1 2 3

Active optics in modern large optical telescopes, L. Noethe Variational methods in nonlinear fiber optics and related fields, B.A. Malomed Optical works of L.V. Lorenz, O. Keller

1– 69 71–193 195–294

356


4

Canonical quantum description of light propagation in dielectric media, A. Lukš, V. Peˇrinová 5 Phase space correspondence between classical optics and quantum mechanics, D. Dragoman 6 “Slow” and “fast” light, R.W. Boyd, D.J. Gauthier 7 The fractional Fourier transform and some of its applications to optics, A. Torre

295–431 433–496 497–530 531–596

VOLUME 44 (2002) 1 2 3

Chaotic dynamics in semiconductor lasers with optical feedback, J. Ohtsubo Femtosecond pulses in optical fibers, F.G. Omenetto Instantaneous optics of ultrashort broadband pulses and rapidly varying media, A.B. Shvartsburg, G. Petite 4 Optical coherence tomography, A.F. Fercher, C.K. Hitzenberger 5 Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, F.Kh. Abdullaev, S.A. Darmanyan, J. Garnier

1– 84 85–141 143–214 215–301 303–366

VOLUME 45 (2003) 1 2 3 4 5 6

Anamorphic beam shaping for laser and diffuse light, N. Davidson, N. Bokor Ultra-fast all-optical switching in optical networks, I. Glesk, B.C. Wang, L. Xu, V. Baby, P.R. Prucnal Generation of dark hollow beams and their applications, J. Yin, W. Gao, Y. Zhu Two-photon lasers, D.J. Gauthier Nonradiating sources and other “invisible” objects, G. Gbur Lasing in disordered media, H. Cao

1– 51 53–117 119–204 205–272 273–315 317–370

VOLUME 46 (2004) 1 2

Ultrafast solid-state lasers, U. Keller Multiple scattering of light from randomly rough surfaces, A.V. Shchegrov, A.A. Maradudin, E.R. Méndez 3 Laser-diode interferometry, Y. Ishii 4 Optical realizations of quantum teleportation, J. Gea-Banacloche 5 Intensity-field correlations of non-classical light, H.J. Carmichael, G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice

1–115 117–241 243–309 311–353 355–404

VOLUME 47 (2005) 1 2 3 4 5 6

Multistep parametric processes in nonlinear optics, S.M. Saltiel, A.A. Sukhorukov, Y.S. Kivshar Modes of wave-chaotic dielectric resonators, H.E. Türeci, H.G.L. Schwefel, Ph. Jacquod, A.D. Stone Nonlinear and quantum optics of atomic and molecular fields, C.P. Search, P. Meystre Space-variant polarization manipulation, E. Hasman, G. Biener, A. Niv, V. Kleiner Optical vortices and vortex solitons, A.S. Desyatnikov, Y.S. Kivshar, L.L. Torner Phase imaging and refractive index tomography for X-rays and visible rays, K. Iwata

1– 73 75–137 139–214 215–289 291–391 393–432


357

VOLUME 48 (2005) 1 2 3 4 5

Laboratory post-engineering of microstructured optical fibers, B.J. Eggleton, P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel, M.J. Steel Optical solitons in random media, F. Abdullaev, J. Garnier Curved diffractive optical elements: Design and applications, N. Bokor, N. Davidson The geometric phase, P. Hariharan Synchronization and communication with chaotic laser systems, A. Uchida, F. Rogister, J. García-Ojalvo, R. Roy

1– 34 35–106 107–148 149–201 203–341

VOLUME 49 (2006) 1 2 3 4 5 6

Gaussian apodization and beam propagation, V.N. Mahajan Controlling nonlinear optical processes in multi-level atomic systems, A. Joshi, M. Xiao Photonic crystals, H. Benisty, C. Weisbuch Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield, C. Brosseau, A. Dogariu Quantum cryptography, M. Dušek, N. Lütkenhaus, M. Hendrych Optical quantum cloning, N.J. Cerf, J. Fiurášek

1– 96 97–175 177–313 315–380 381–454 455–545

Cumulative index – Volumes 1–50* Abdullaev, F.Kh., S.A. Darmanyan, J. Garnier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media Abdullaev, F.Kh., J. Garnier: Optical solitons in random media Abelès, F.: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical frequencies Abitbol, C.I., see Clair, J.J. Abraham, N.B., P. Mandel, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, V.M., V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in lasers Arnaud, J.A.: Hamiltonian theory of beam mode propagation Asakura, T., see Okamoto, T. Asakura, T., see Peiponen, K.-E. Asatryan, A.A., see Kravtsov, Yu.A. Babiker, M., see Allen, L. Baby, V., see Glesk, I. Baltes, H.P.: On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-free diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy * Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

359

44, 303 48, 35 2, 249 7, 139 16, 71 25, 1 11, 1 9, 235 26, 163 37, 185 9, 179 39, 291 9, 123 41, 97 36, 179 35, 257 6, 211 11, 247 34, 183 37, 57 39, 1 39, 291 45, 53 13,

1

29, 65 1, 67 21, 217 12, 287

360

Cumulative index – Volumes 1–50

Bassett, I.M., W.T. Welford, R. Winston: Nonimaging optics for flux concentration Beckmann, P.: Scattering of light by rough surfaces Benisty, H., C. Weisbuch: Photonic crystals Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M. Berry, M.V., M.R. Jeffrey: Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics Berry, M.V., C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., see Chumash, V. Bertolotti, M., see Mihalache, D. Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Biener, G.: see Hasman, E. Bloembergen, N.: From millisecond to attosecond laser pulses Bloom, A.L.: Gas lasers and their application to precise length measurements Bokor, N., N. Davidson: Curved diffractive optical elements: Design and applications Bokor, N., see Davidson, N. Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Boyd, R.W., D.J. Gauthier: “Slow” and “fast” light Brosseau, C., A. Dogariu: Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., T. Scheermesser, F. Wyrowski: Digital halftoning: synthesis of binary images Bryngdahl, O., F. Wyrowski: Digital holography – computer-generated holograms Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Bužek, V., P.L. Knight: Quantum interference, superposition states of light, and nonclassical effects Cagnac, B., see Giacobino, E. Cao, H.: Lasing in disordered media Carmichael, H.J., G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice: Intensity-field correlations of non-classical light Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur: Principles of optical disk data storage Casasent, D., D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications

27, 161 6, 53 49, 177 33, 319 35, 61 50, 13 18, 257 36, 129 36, 1 27, 227 16, 357 36, 245 47, 215 50, 1 9, 1 48, 107 45, 1 22, 77 4, 145 43, 497 49, 315 23, 1 35, 61 15, 1 4, 37 11, 167 33, 389 28, 1 2, 73 19, 211 34,

1

17, 85 45, 317 46, 355 41, 97 16, 289 21, 287

Cumulative index – Volumes 1–50 Cerf, N.J., J. Fiurášek: Optical quantum cloning Chang, R.K., see Fields, M.H. Charnotskii, M.I., J. Gozani, V.I. Tatarskii, V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T., Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y., A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, W.M. Christov, I.P.: Generation and propagation of ultrashort optical pulses Chumash, V., I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.J., C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.J.B.: Optical fibre waveguides – a review Cohen-Tannoudji, C., A. Kastler: Optical pumping Cojocaru, I., see Chumash, V. Cole, T.W.: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtès, G., P. Cruvellier, M. Detaille: Some new optical designs for ultra-violet bidimensional detection of astronomical objects Creath, K.: Phase-measurement interferometry techniques Crewe, A.V.: Production of electron probes using a field emission source Cruvellier, P., see Courtès, G. Cummins, H.Z., H.L. Swinney: Light beating spectroscopy Dainty, J.C.: The statistics of speckle patterns Dändliker, R.: Heterodyne holographic interferometry Darmanyan, S.A., see Abdullaev, F.Kh. Dattoli, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., N. Bokor: Anamorphic beam shaping for laser and diffuse light Davidson, N., see Bokor, N. Davidson, N., see Oron, R. Decker Jr, J.A., see Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses De Mol, C., see Bertero, M. DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Desyatnikov, A.S., Y.S. Kivshar, L. Torner: Optical vortices and vortex solitons De Sterke, C.M., J.E. Sipe: Gap solitons Detaille, M., see Courtès, G. Dexter, D.L., see Smith, D.Y. Dogariu, A., see Brosseau, C. Domachuk, P., see Eggleton, B.J.

361 49, 455 41, 1 32, 203 41, 283 37, 345 41, 97 13, 69 29, 199 36, 1 16, 71 14, 327 5, 1 36, 1 15, 187 20, 63 28, 361 20, 1 26, 349 11, 223 20, 1 8, 133 14, 1 17, 1 44, 303 31, 321 45, 1 48, 107 42, 325 12, 101 7, 67 9, 31 36, 129 23, 1 47, 291 33, 203 20, 1 10, 165 49, 315 48, 1

362


Dragoman, D.: The Wigner distribution function in optics and optoelectronics Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dušek, M., N. Lütkenhaus, M. Hendrych: Quantum cryptography Dutta, N.K., J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media Eberly, J.H.: Interaction of very intense light with free electrons Eggleton, B.J., P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel, M.J. Steel: Laboratory post-engineering of microstructured optical fibers Englund, J.C., R.R. Snapp, W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry Erez, N., see Greenberger, D.M. Essiambre, R.-J., G.P. Agrawal: Soliton communication systems Etrich, C., F. Lederer, B.A. Malomed, T. Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C., see Reynaud, S. Facchi, P., S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., see Chumash, V. Fercher, A.F., C.K. Hitzenberger: Optical coherence tomography Ficek, Z., H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Fiurášek, J., see Cerf, N.J. Flytzanis, C., F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. Foster, G.A., see Carmichael, H.J. Françon, M., S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlídal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, V.D., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions Friesem, A.A., see Oron, R. Froehly, C., B. Colombeau, M. Vampouille: Shaping and analysis of picosecond light pulses

37,

1

43, 433 12, 163 14, 161 49, 381 31, 189 38, 1 7, 359 48,

1

21, 355 16, 233 50, 275 37, 185 41, 483 37, 95 30, 1 42, 147 22, 341 36, 1 44, 215 40, 389 41, 1 1, 253 49, 455 29, 321 4, 1 39, 1 46, 355 6, 71 41, 181 40, 389 30, 137 9, 311 42, 325 20, 63

Cumulative index – Volumes 1–50 Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T. Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H., G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tana´s, R. Gao, W., see Yin, J. García-Ojalvo, J., see Uchida, A. Garnier, J., see Abdullaev, F.Kh. Garnier, J., see Abdullaev F.Kh. Gauthier, D.J.: Two-photon lasers Gauthier, D.J., see Boyd, R.W. Gbur, G.: Nonradiating sources and other “invisible” objects Gea-Banacloche, J.: Optical realizations of quantum teleportation Ghatak, A., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, V.L.: Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Ginzburg, V.L., see Agranovich, V.M. Giovanelli, R.G.: Diffusion through non-uniform media Glaser, I.: Information processing with spatially incoherent light Glesk, I., B.C. Wang, L. Xu, V. Baby, P.R. Prucnal: Ultra-fast all-optical switching in optical networks Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., see Charnotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, V.D. Greenberger, D.M., N. Erez, M.O. Scully, A.A. Svidzinsky, M.S. Zubairy: Planck, photon statistics, and Bose–Einstein condensation Grillet, C., see Eggleton, B.J. Hache, F., see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: Interferometry with lasers Hariharan, P.: The geometric phase Hariharan, P., B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M., J.A. Decker Jr: Modulation techniques in spectrometry

363 8, 51 41, 283 1, 109 3, 187 34, 333 35, 355 45, 119 48, 203 44, 303 48, 35 45, 205 43, 497 45, 273 46, 311 18, 1 13, 169 17, 85 30, 1 31, 321 32, 267 9, 235 2, 109 24, 389 45, 53 9, 281 8, 1 32, 203 12, 233 30, 137 50, 275 48, 1 29, 321 29, 1 20, 263 24, 103 48, 149 36, 49 12, 101

364


Hasegawa, A., see Kodama, Y. Hasman, E., G. Biener, A. Niv, V. Kleiner: Space-variant polarization manipulation Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Hendrych, M., see Dušek, M. Herriot, D.R.: Some applications of lasers to interferometry Hitzenberger, C.K., see Fercher, A.F. Horner, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images

30, 205 47, 215 42, 325 30, 1 38, 85 10, 289 49, 381 6, 171 44, 215 38, 343 10, 1

Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y. Ishii, Y.: Laser-diode interferometry Itoh, K.: Interferometric multispectral imaging Iwata, K.: Phase imaging and refractive index tomography for X-rays and visible rays

40, 77 28, 87 46, 243 35, 145 47, 393

Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation Jacquod, Ph., see Türeci, H.E. Jaeger, G., A.V. Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W., B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation Javidi, B., J.L. Horner: Pattern recognition with nonlinear techniques in the Fourier domain Jeffrey, M.R., see Berry, M.V. Jones, D.G.C., see Allen, L. Joshi, A., M. Xiao: Controlling nonlinear optical processes in multi-level atomic systems

5, 247 3, 29 47, 75 42, 277 38, 419 20, 325 38, 343 50, 13 9, 179 49, 97

Kastler, A., see Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V. Lorenz Keller, O.: Historical papers on the particle concept of light Keller, U.: Ultrafast solid-state lasers Khoo, I.C.: Nonlinear optics of liquid crystals Khulbe, P., see Carriere, J. Kielich, S.: Multi-photon scattering molecular spectroscopy Kilin, S.Ya.: Quanta and information Kinosita, K.: Surface deterioration of optical glasses Kitagawa, M., see Yamamoto, Y. Kivshar, Y.S., see Desyatnikov, A.S. Kivshar, Y.S., see Saltiel, S.M. Klein, M.C., see Flytzanis, C.

5, 1 37, 257 43, 195 50, 51 46, 1 26, 105 41, 97 20, 155 42, 1 4, 85 28, 87 47, 291 47, 1 29, 321


365

Kleiner, V., see Hasman, E. Klyatskin, V.I.: The imbedding method in statistical boundary-value wave problems Knight, P.L., see Bužek, V. Kodama, Y., A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, F.: The elements of radiative transfer Kottler, F.: Diffraction at a black screen, Part I: Kirchhoff’s theory Kottler, F.: Diffraction at a black screen, Part II: electromagnetic theory Kozhekin, A.E., see Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes, A.A. Asatryan: Theory and applications of complex rays Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrný, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities

47, 215 33, 1 34, 1 30, 205 7, 1 3, 1 4, 281 6, 331 42, 93 26, 227 36, 179 39, 1 29, 65 1, 211 40, 343

Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, F., see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, V.S.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Fractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sánchez-Soto: Quantum phase difference, phase measurements and Stokes operators Lukš, A., V. Peˇrinová: Canonical quantum description of light propagation in dielectric media Lukš, A., see Peˇrinová, V. Lukš, A., see Peˇrinová, V. Lütkenhaus, N., see Dušek, M.

14, 47 11, 123 41, 483 16, 119 6, 1 16, 1 39, 373 8, 343 41, 97

43, 295 33, 129 40, 117 49, 381

Machida, S., see Yamamoto, Y. Mägi, E.C., see Eggleton, B.J. Mahajan, V.N.: Gaussian apodization and beam propagation Mainfray, G., C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas

28, 87 48, 1 49, 1 32, 313

42, 93

5, 287 38, 263 40, 271 35, 61 21, 69 41, 419

366


Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, V.I. Mallick, S., see Françon, M. Malomed, B.A.: Variational methods in nonlinear fiber optics and related fields Malomed, B.A., see Etrich, C. Malomed, B.A., see Kurizki, G. Mandel, L.: Fluctuations of light beams Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., see Abraham, N.B. Mansuripur, M., see Carriere, J. Manus, C., see Mainfray, G. Maradudin, A.A., see Shchegrov, A.V. Marchand, E.W.: Gradient index lenses Martin, P.J., R.P. Netterfield: Optical films produced by ion-based techniques Masalov, A.V.: Spectral and temporal fluctuations of broad-band laser radiation Maystre, D.: Rigorous vector theories of diffraction gratings Meessen, A., see Rouard, P. Mehta, C.L.: Theory of photoelectron counting Mendez, E.R., see Shchegrov, A.V. Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Meystre, P., see Search, C.P. Michelotti, F., see Chumash, V. Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L.: Self-focusing media with variable index of refraction Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, P.W.: Field quantization in optics Milonni, P.W., B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., see Tana´s, R. Miyamoto, K.: Wave optics and geometrical optics in optical design Mollow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A., A. Thelen: Multilayer antireflection coatings Nakwaski, W., M. Osiński: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navrátil, K., see Ohlídal, I. Netterfield, R.P., see Martin, P.J.

22, 1 33, 261 6, 71 43, 71 41, 483 42, 93 2, 181 13, 27 25, 1 41, 97 32, 313 46, 117 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 46, 117 38, 263 40, 271 30, 261 47, 139 36, 1 27, 227 17, 279 7, 231 19, 45 50, 97 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201

38, 165 41, 97 25, 1 34, 249 23, 113


367

Nguyen, H.C., see Eggleton, B.J. Nishihara, H., T. Suhara: Micro Fresnel lenses Niv, A., see Hasman, E. Noethe, L.: Active optics in modern large optical telescopes Novotny, L.: The history of near-field optics Nussenzveig, H.M.: Light tunneling

48, 1 24, 1 47, 215 43, 1 50, 137 50, 185

Ohlídal, I., D. Franta: Ellipsometry of thin film systems Ohlídal, I., K. Navrátil, M. Ohlídal: Scattering of light from multilayer systems with rough boundaries Ohlídal, M., see Ohlídal, I. Ohtsu, M., T. Tako: Coherence in semiconductor lasers Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T., T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Omenetto, F.G.: Femtosecond pulses in optical fibers Ooue, S.: The photographic image Opatrný, T., see Kurizki, G. Opatrný, T., see Welsch, D.-G. Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orozco, L.A., see Carmichael, H.J. Orrit, M., J. Bernard, R. Brown, B. Lounis: Optical spectroscopy of single molecules in solids Osiński, M., see Nakwaski, W. Ostrovskaya, G.V., Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.I., V.P. Shchepinov: Correlation holographic and speckle interferometry Ostrovsky, Yu.I., see Ostrovskaya, G.V. Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, V.D., see Barabanenkov, Yu.N.

41, 181

Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The modern development of Hamiltonian optics Pegis, R.J., see Delano, E. Peiponen, K.-E., E.M. Vartiainen, T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C., see Carriere, J. Pe˘rina, J.: Photocount statistics of radiation propagating through random and nonlinear media

34, 249 34, 249 25, 191 44, 1 34, 183 15, 139 44, 85 7, 299 42, 93 39, 63 42, 325 46, 355 35, 61 38, 165 22, 197 30, 87 22, 197 24, 165 33, 319 29, 65 39, 291 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 57 41, 97 18, 127

368


Peˇrina, J., see Peˇrina Jr, J. Peˇrina Jr, J., J. Peˇrina: Quantum statistics of nonlinear optical couplers Peˇrinová, V., A. Lukš: Quantum statistics of dissipative nonlinear oscillators Peˇrinová, V., A. Lukš: Continuous measurements in quantum optics Peˇrinová, V., see Lukš, A. Pershan, P.S.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U., see Etrich, C. Petite, G., see Shvartsburg, A.B. Petykiewicz, J., see Gniadek, K. Picht, J.: The wave of a moving classical electron Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., see Fields, M.H. Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems Presnyakov, L.P.: Wave propagation in inhomogeneous media: phase-shift approach Prucnal, P.R., see Glesk, I. Psaltis, D., Y. Qiao: Adaptive multilayer optical networks Psaltis, D., see Casasent, D.

27, 315 34, 159 45, 53 31, 227 16, 289

Qiao, Y., see Psaltis, D.

31, 227

Raymer, M.G., I.A. Walmsley: The quantum coherence properties of stimulated Raman scattering Reiner, J.E., see Carmichael, H.J. Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino, C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Rice, P.R., see Carmichael, H.J. Riseberg, L.A., M.J. Weber: Relaxation phenomena in rare-earth luminescence Risken, H.: Statistical properties of laser light Roddier, F.: The effects of atmospheric turbulence in optical astronomy Rogister, F., see Uchida, A. Roizen-Dossier, B., see Jacquinot, P. Ronchi, L., see Wang Shaomin Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M., J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Roy, R., see Uchida, A.

41, 359 41, 359 33, 129 40, 115 43, 295 5, 83 41, 483 41, 483 44, 143 9, 281 5, 351 31, 139 41, 1

28, 181 46, 355 31, 321 30, 1 29, 321 46, 355 14, 89 8, 239 19, 281 48, 203 3, 29 25, 279 35, 1 13, 69 24, 39 4, 145 15, 77 29, 321 48, 203


369

Rubinowicz, A.: The Miyamoto–Wolf diffraction wave Rudolph, D., see Schmahl, G.

4, 199 14, 195

Saichev, A.I., see Barabanenkov, Yu.N. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Saltiel, S.M., A.A. Sukhorukov, Y.S. Kivshar: Multistep parametric processes in nonlinear optics Sánchez-Soto, L.L., see Luis, A. Sanders, B.C., see Hariharan, P. Scheermesser, T., see Bryngdahl, O. Schieve, W.C., see Englund, J.C. Schirripa Spagnolo, G., see Paoletti, D. Schmahl, G., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G.: Aspheric surfaces Schulz, G., J. Schwider: Interferometric testing of smooth surfaces Schwefel, H.G.L., see Türeci, H.E. Schwider, J.: Advanced evaluation techniques in interferometry Schwider, J., see Schulz, G. Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics Scully, M.O., see Greenberger, D.M. Search, C.P., P. Meystre: Nonlinear and quantum optics of atomic and molecular fields Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V., see Jaeger, G. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchegrov, A.V., A.A. Maradudin, E.R. Méndez: Multiple scattering of light from randomly rough surfaces Shchepinov, V.P., see Ostrovsky, Yu.I. Shvartsburg, A.B., G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media Sibilia, C., see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see De Sterke, C.M. Sipe, J.E., see Van Kranendonk, J. Sittig, E.K.: Elastooptic light modulation and deflection Sizmann, A., G. Leuchs: The optical Kerr effect and quantum optics in fibers Slusher, R.E.: Self-induced transparency Smith, A.W., see Armstrong, J.A. Smith, D.Y., D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W.: The use of image tubes as shutters Snapp, R.R., see Englund, J.C.

29, 65 28, 87 6, 259 26, 1 47, 1 41, 419 36, 49 33, 389 21, 355 35, 197 14, 195 17, 163 25, 349 13, 93 47, 75 28, 271 13, 93 10, 89 50, 275 47, 139 16, 413 42, 277 39, 213 46, 117 30, 87 44, 143 27, 227 31, 189 33, 203 15, 245 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355

370


Sodha, M.S., A.K. Ghatak, V.K. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.J., see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S., M.V. Vasnetsov: Singular optics Spreeuw, R.J.C., J.P. Woerdman: Optical atoms Steel, M.J., see Eggleton, B.J. Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Steinvurzel, P., see Eggleton, B.J. Stoicheff, B.P., see Jamroz, W. Stone, A.D., see Türeci, H.E. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., see Mills, D.L. Suhara, T., see Nishihara, H. Sukhorukov, A.A., see Saltiel, S.M. Sundaram, B., see Milonni, P.W. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Svidzinsky, A.A., see Greenberger, D.M. Sweeney, D.W., see Ceglio, N.M. Swinney, H.L., see Cummins, H.Z. Tako, T., see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets Tana´s, R., A. Miranowicz, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, V.I., V.U. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, V.I., see Charnotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C., B.E.A. Saleh: Photon bunching and antibunching Ter-Mikaelian, M.L., see Mikaelian, A.L. Thelen, A., see Musset, A. Thompson, B.J.: Image formation with partially coherent light Thyagarajan, K., see Ghatak, A. Tonomura, A.: Electron holography Torner, L., see Desyatnikov, A.S. Torre, A.: The fractional Fourier transform and some of its applications to optics Torre, A., see Dattoli, G. Tripathi, V.K., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering

13, 169 39, 213 27, 109 42, 219 31, 263 48, 1 5, 145 37, 345 48, 1 20, 325 47, 75 9, 73 2, 1 19, 45 24, 1 47, 1 31, 1 12, 1 50, 275 21, 287 8, 133 25, 191 23, 63 35, 355 17, 239 40, 77 18, 204 32, 203 5, 287 26, 1 7, 231 8, 201 7, 169 18, 1 23, 183 47, 291 43, 531 31, 321 13, 169 2, 131


371

Türeci, H.E., H.G.L. Schwefel, Ph. Jacquod, A.D. Stone: Modes of wave-chaotic dielectric resonators Turunen, J., M. Kuittinen, F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.

47, 75 40, 343 17, 239

Uchida, A., F. Rogister, J. García-Ojalvo, R. Roy: Synchronization and communication with chaotic laser systems Upatnieks, J., see Leith, E.N. Upstill, C., see Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids

48, 203 6, 1 18, 257 19, 139

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