EDITORIAL ADVISORY BOARD G. S. AGARWAL,
Ahmedabad, India
T. ASAKURA,
Sapporo, Japan
M.V BERRY,
Bristol England
C. COHEN-TANNOUDJI,
Paris, France
V. L. GINZBURG,
Moscow, Russia
E GORI,
Rome, Italy
A. KUJAWSKI,
Warsaw, Poland
J. PE~NA,
Olomouc, Czech Republic
R. M. SILLITTO,
Edinburgh, Scotland
H. WALTHER,
Garching, Germany
PREFACE With the publication of the fortieth volume of Progress in Optics, a significant milestone has been reached. The first volume was published in 1961, a year after the invention of the laser, an event which triggered a wealth of new and exciting developments. Many of them have been reported in the 228 review articles published in this series since its inception. The present volume contains six review articles on a variety of subjects of current research interests. The first, by T.R. Wolifiski, is concerned with polarimetric optical fibers and sensors. These devices have created a novel generation of powerful sensory-oriented techniques. The article reviews the main efforts and achievements in this field within the last two decades. It discusses the physical origin of polarization phenomena in birefringent fibers, both at the fundamental and the applied levels, and various deformation effects due to pressure, strain, twist and temperature on propagation of the lowest-order mode in fibers. The second article, by J. Tanida and Y. Ichioka, presents a review of recent researches on digital optical computing. After introducing the basic concepts needed for understanding the developments in this field, some feasibility experiments as well as software studies are discussed. The article by V. Pefinov~ and A. Luk~ which follows, deals largely with photodetection from the standpoint of the theory of open systems, bordering on novel techniques for testing irreversibility via quantum trajectories. Both destructive and non-destructive models of the process of photodetection are discussed. The fourth article, by Z. Zalevsky, D. Mendlovic and A.W. Lohmann, presents an account of modern theories of resolution in optical systems, based on the concepts of communication theory. The next article, by J. Turunen, M. Kuittinen and E Wyrowski, is concerned with the design of microstructured optical elements by the use of electromagnetic diffraction theory. Such an approach is required when the paraxial approximation is inadequate to describe their performance, or when it becomes necessary to take into account the state of polarization of the light. Diffractive elements based on linear or modulated gratings which operate in zero-order, first-order and multiorder modes are discussed.
vi
PREFACE
The concluding article by Z. Ficek and H.S. Freedhoff deals with the theory underlying the interaction of an atom with an intense polychromatic driving field, with particular reference to certain experiments. Several different systems which have been studied to date are discussed, including subharmonic resonances in the absorption spectrum of a strong probe, the fluorescence, near-resonance absorption and the Autler-Townes absorption by the entangled driven systems. In publishing this fortieth volume it is appropriate to acknowledge the substantial help which I have received over the years. There are too many persons to acknowledge individually. Three of them, however, deserve special mention: Mr. Jeroen Soutberg, director of ISYS Prepress Services in the Netherlands, is largely responsible for the production of these volumes. He must be credited for consistently maintaining the highest possible standards. I wish to thank Dr. M. Suhail Zubairy, one of my former students and now Professor at a University in Islamabad, Pakistan for preparing, for many years, the subject indexes for these volumes. I also wish to express my appreciation to Dr. Joost Kircz, a former publisher of Elsevier, who provided much help and advice with the publication of earlier volumes in this series. Finally, I wish to thank members of the Editorial Advisory Board of Progress in Optics for their part in having made this series such a successful enterprise. Emil Wolf
Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA October 1999
E. WOLF, PROGRESS 1N OPTICS XL 9 2000 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
POLARIMETRIC OPTICAL FIBERS AND SENSORS
BY
TOMASZ R. W O L I l q S K I
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland phone: (+48 22) 660-8212,-7262, fax: (+48 22) 628-2171; e-mail:
[email protected], http://www.if.pw.edu.pl./-wolinski
CONTENTS
PAGE w 1.
INTRODUCTION
w 2.
POLARIZATION
w 3.
BIREFRINGENCE
w 4.
DEFORMATION FIBERS
. . . . . . . . . . . . . . . . . . . PHENOMENA
IN O P T I C A L F I B E R S
1N O P T I C A L F I B E R S
.
. . . . . . . . .
3 .
.
4 22
E F F E C T S IN H I G H L Y B I R E F R I N G E N T
. . . . . . . . . . . . . . . . . . . . . . .
28
w 5.
APPLICATION OF POLARIZATION EFFECTS . . . . . . .
52
w 6.
CONCLUSIONS
69
AND FUTURE PERSPECTIVES
ACKNOWLEDGEMENTS REFERENCES
. . . . . .
. . . . . . . . . . . . . . . . . .
70
. . . . . . . . . . . . . . . . . . . . . . .
70
w 1. Introduction Over the last two decades significant progress has occurred in optical fiber technologies from the use of intensity (amplitude) modulation to that of modulation of the optical polarization of the electromagnetic wave propagating along a fiber. At the same time new possibilities have opened up for both optical communication and also optical fiber sensors and systems. The key to successful construction of these new sensing devices and coherent communication systems is in high-performance polarimetric optical fibers and sensors. They are mostly based on highly birefringent (HB), polarization-maintaining (PM) fibers, which have aroused great interest from both theoretical and practical points of view. Although polarization effects in optical fibers initially played a minor role in the development of lightwave systems, their importance is still growing, due to an enormous increase in optical path lengths that can be achieved with singlemode fibers and also to an increase in bit rates in digital systems, as reviewed by Poole and Nagel [ 1997]. These two events recently precipitated a rediscovery of polarization phenomena in lightwave systems. Before 1980 it was impossible to exploit the polarization modulation in a fiber for sensing applications, since the conventional single-mode fibers manufactured for telecommunication use do not hold the optical wave amplitude in a particular polarization state. The appearance of HB fibers created a new generation of fiber-optic sensors known as polarimetric fiber sensors, which use polarization (phase) modulation within these fibers or at their output due to various external perturbations describing the physical environment. The aim of this chapter is to review the foremost achievements and efforts in research activities related to the development of a new generation of polarimetric optical fibers and sensors at both fundamental and applied levels during the past twenty years. The review underlines the physical origin of the perturbations (e.g., those induced by pressure, strain, bend, twist, temperature) on the lowestorder mode propagation in HB polarization-maintaining fibers together with their impact on applications in optical fiber sensors and systems. Several papers and chapters in textbooks have been published on polarizationmaintaining fibers and polarization effects in fibers, for example, by Kaminow [ 1981 ], Payne, Barlow and Ramskov-Hansen [ 1982], Rashleigh [ 1983a], Noda,
4
POLARIMETRIC OPTICAL FIBERS AND SENSORS
[I, w 2
Okamoto and Sasaki [1986], Tsao [1992], and Huard [1997]. However, they have not addressed all aspects of the rapidly growing part of fiber optics that is analyzed in this review.
w 2. Polarization Phenomena in Optical Fibers The phenomenon of polarization was discovered by Huyghens in 1690 by passing light through two calcite crystals, and only in 1808 Malus passed partially reflected light through a calcite crystal and found that it was polarized (Born and Wolf [1993]). In analogy to magnetic bodies, Malus called oriented light "polarized light" (Kliger, Lewis and Randall [1990]). Optical fibers exhibit particular polarization properties, (e.g., Kaminow and Ramaswamy [1979], Cancellieri, Fantini and Tilio [1985], Tsao [1986], Shafir, Hardy and Tur [1987]). Contrary to ordinary plane waves in bulk media, of which the amplitudes are constant in the wave plane, guided electromagnetic fields in optical-fiber waveguides are called inhomogeneous plane waves, since their amplitudes are no longer stable within the plane wave (Huard [ 1997]) and the fields are generally characterized by non-transverse components. Two approaches are generally used in the description of polarization phenomena in optical fibers (Tsao [1992]). The first approach treats an optical fiber as an optical waveguide, in which light as a kind of electromagnetic wave of optical frequencies can be guided in the form of waveguide modes. This approach identifies basic polarization eigenmodes of a fiber and relates them to the polarization state of the guided light. Changes in output polarization are described in terms of polarization-mode coupling due to birefringence changes acting as perturbations along the fiber. The polarization coupling, except for some simple cases, is described by the coupled mode theory (Rashleigh [ 1983a]). The second approach treats an optical fiber like any other optical device that transmits light, and the fiber can be divided into separate sections that behave like polarization state shifters. Here, polarization evolution in a fiber can be described by one of the three general formalisms: by the Jones vectors and matrices formalism, by the Stokes vectors and Mueller matrices formalism, or by the Poincar6 sphere representation. Since optical fibers allow large propagation distances, even extremely small birefringence effects can accumulate along the fiber, and their random distribution over such distances generally make it difficult to determine the polarization properties of guided light; this applies both to the state and the degree of polarization.
I, w 2]
POLARIZATION PHENOMENA IN OPTICAL FIBERS
5
2.1. M O D E S OF O P T I C A L F I B E R
An optical fiber consists of a core of dielectric material with refractive index nco and a cladding of another dielectric material with a refractive index ncl less than nco. The exact description of the modes propagating in a fiber is complicated, since they are six hybrid-field components of great mathematical complexity. Detailed analyses of guided-wave modes in cylindrical optical fibers have been conducted in many review papers and textbooks by, among others, Snitzer [1961], Marcuse [1974], Clarricoats [1976], Kaminow and Ramaswamy [1979], Snyder and Love [1983], Cheo [1985], Cancellieri, Fantini and Tilio [1985], Tsao [1986], Yeh [1987], Shafir, Hardy and Tur [1987, 1988], and Huard [1997]. The modes with a strong electric E- field compared with the magnetic Hz field along the direction of propagation (z-axis) are designated as EH modes. Similarly, those with a stronger ~ field are called HE modes. These modes are hybrid since they consist of all six field components (3 electric and 3 magnetic) and possess no circular symmetry. The propagating modes are discrete and require identification by two indexes (/,p): HE0, and EHI,, both of which are integer indexes. The first, 1=0,1,2,3,..., separates the variables in the scalar wave equation, whereas the second, p = 1,2,3,..., indicates the pth roots of the Bessel function of the first kind Jl and the modified Bessel function KI. F o r / = 0 , the hybrid modes are analogous to the transverse-electric (TE) and the transverse-magnetic (TM) modes of planar waveguides, and two linearly polarized sets of modes exist that are circularly symmetric with vanishing either the E or H longitudinal field components" TE0t~ (E_- = 0) and TM0p (/-/: = 0). The lowest-order transverse modes TE01 and TM01 have cutoff frequencies V = Vc = 2.405, where V is the normalized frequency, defined as 2 = v/u 2 + w 2 V = -2~J a V n/ c o2 - ncl
(2.1)
with a the core radius, X the free space wavelength, n~o (nd) the refractive index of the fiber core (cladding), V/ nc2o- nd2 = NA the numerical aperture of the fiber used in optics to express the ability of the system to gather light, and u and w are parameters defined as follows: u=a
ncok0 -
,
w=a
-n:lk~,
where/3 is the propagation constant and k0 = 2x/2.
(2.2)
6
POLARIMETRIC OPTICAL FIBERS AND SENSORS
[I, w 2
The lowest-order mode of cylindrical waveguide is the HEll mode, which has zero cutoff frequency. This is the fundamental mode of an optical fiber and is also the only mode propagating in the region of frequencies 0 < V < 2.405. Hence, in this region a fiber is considered to be single-mode. The field distribution E(r, t) corresponding to the HEll mode, has three non-zero components Ex, E y and Ez (in Cartesian coordinates), among which either Ex or E y dominates. Even a single-mode fiber is not truly single-mode however, since the electric field of the HE l~ mode has two polarizations orthogonal to each other that constitute two polarization modes of a single-spatial-mode fiber. A significant simplification in the description of these modes is based on the fact that most fibers for practical applications use core materials in which the refractive index is only slightly higher than that of the surrounding cladding; that is, nco - ncl 0) are fourfold degenerate: twofold orientational degeneracy (even and odd) and twofold polarization degeneracy (x and y). In this case four polarization modes, x, e, v, e x, o, ,. o, can be guided along the fiber. namely LPtp LP~p , LPlp and L p ip In the isotropic case single-mode fibers (normalized frequency parameter V ~a.
The normalization constant amplitude E0 can be determined from the power relation (Yeh [ 1987]):
wJo+( zo)u +( zo)
E o - V Jl(U)
;ra2ncl
V Kl(w)
.Tra2ncl '
where zo = ~u/ko is the plane-wave impedance in a vacuum, Jl and Kl (1= 0, 1) are the Bessel functions of the first kind and the modified Bessel functions, respectively, and ~0 is the angular frequency corresponding to the free space wavelength ]l. The next four higher-order modes TE01, TM01, l_.llTeven =-~21 and H E ~ d (with 2.405 < V < 3.832), have slightly different propagation velocities and almost the same cross-sectional optical intensity distributions. A new method for measuring cutoff frequencies of TE01, TM01, and HE21 modes was proposed by Kato and Miyauchi [ 1985]. In the weakly guiding approximation these four second-order modes become fourfold degenerate and are denoted as LP~ modes. The field distributions of four independent linear combinations of the waveguide modes, TM01-HE21, TE01-HE21, TE0~ +HE21 and TM0~ +HE21, shown in fig. 2, constitute the
I, w 2]
POLARIZATIONPHENOMENAIN OPTICAL FIBERS
9
Table 1 Polarization modes of an isotropic weakly guiding two-mode fiber
Fundamental LPol mode:
Normalized frequency:
V~< Vc=2.405
Waveguide modes:
HE i"l' HE;'I
Linearly polarized modes:
LPoVl, LPo 1
Propagation constants:
ridx = riO' (twofold degeneracy)
Second-order LPll model:
Normalized frequency:
2.405 < V < 3.832
Waveguide modes:
TM01 -HE21, TE01-HE21, TE01 +HE21, TM01 +HE21
Linearly polarized modes:
LP~,
Propagation constants:
fi~x = fi~y= fi~x= fi~-" (fourfold degeneracy) ~ even = )odd cut "~cut
Cutoff wavelengths:
~1'
LPll,
OX
LPll,
01'
LP(l
linearly polarized second-order (LPll) modes as a single linear electric field vector. Fibers operating in this regime (2.405 < V < 3.832) are two-mode (or bimodal) fibers. In fact, the two-mode fiber supports six modes: two polarizations of the fundamental LP0~ mode and two polarizations of each of two lobe orientations (even and odd) of the second-order LPI1 mode" LP~l and LP~'1 (table 1). Consequently, the LP02 mode is the sum of TE02, TM02, and the HE22 modes, and the LP21 mode is the sum of HE31 and EHI1 modes, and so on. 2.3. OPTICAL FIBERS SENSITIVE TO POLARIZATION EFFECTS
An ideal isotropic fiber propagates any state of polarization launched into the fiber unchanged. However, the realization of the perfectly isotropic singlemode fiber demands huge manufacturing requirements with respect to the ideal circularity of the core and lack of mechanical stress. Since in the ideal cylindrical fiber the fundamental LP01 mode contains two degenerated orthogonally polarized modes they are propagating at the same phase velocity. In real single-mode fibers that possess non-zero internal birefringence, both orthogonally polarized modes have randomly different phase velocities, causing fluctuations of the polarization state of the light guided in the fiber. The absolute magnitude of birefringence in such fibers is typically of the order
10
POLARIMETRIC OPTICAL FIBERS AND SENSORS
[I, w 2
l[3y- fix] ,~ 1 m -1 (Ulrich [1994]). This means that after propagation through a length of about 1 m the polarization will be modified in an unpredictable way and consequently these fibers cannot preserve any state of polarization launched into the fiber. Over the past twenty years numerous authors have developed and analyzed different types of optical fibers sensitive to polarization effects (e.g., Ramaswamy, French and Standley [1978], Okamoto, Edahiro and Shibata [1982], Simpson, Stolen, Sears, Pleibel, Macchesney and Howard [1983], Snyder and Riihl [1983], Burns, Moeller and Chen [1983], Okamoto, Varnham and Payne [1983], Noda, Shibata, Edahiro and Sasaki [1983], Okamoto [1984], Snyder and Riihl [1984], Marrone, Rashleigh and Btaszczyk [1984], Chen [1987], Alphones and Sanyal [ 1987], Hayata and Koshiba [ 1988]).
2.3.1. Low-birefringence polarization-maintaining fibers The concept of polarization preservation in single-mode fibers may be realized either by manufacturing fibers with negligible birefringence or by enhancing internal birefringence to very high values. In the first case we obtain lowbirefringence (LB) fibers fabricated by special techniques so as to exhibit particularly low birefringence of the order lily- fi~] ~< 10-2m -1. The LB polarization-maintaining fibers (PMFs) can preserve any and all polarization states (Schneider, Harms, Papp and Aulich [1978], Eickhoff and Brinkmeyer [1984]), whichever is injected, but they require the high stability of the environment since the polarization state is extremely sensitive to external perturbation effects.
2.3.2. Highly birefringent polarization-maintaining fibers In highly birefringent (HB) polarization-maintaining fibers, the difference between the phase velocities for the two orthogonally polarized modes is sufficiently high to avoid coupling between these two modes. Fibers of these class have a built-in, well-defined, high internal birefringence obtained by designing a core and/or cladding with non-circular (mostly elliptical) geometry or by using anisotropic stress applying parts built into the cross-section of the fiber. Various types of HB polarization-maintaining fibers are presented in fig. 3. These include (a) elliptical-core, (b) stress-induced elliptical internal cladding, (c) bow-tie, or (d) PANDA fibers. The magnitude of the internal birefringence is characterized by the beat length of the two polarization modes, 2Jr
Z~- IG-/~xl'
(2.6)
I, w2]
POLARIZATIONPHENOMENAIN OPTICALFIBERS
11
Fig. 3. Various types of HB polarization-maintaning fibers: (a) elliptical core; (b) elliptical internal cladding; (c) bow-tie; (d) PANDA.
Fig. 4. Definition of the beat length of the fundamental mode LP01 in a single-mode HB PM fiber.
and is responsible for phase difference changes along the longitudinal axis z of the HB fiber. The spatial period of these changes reflects the changes in the polarization states along the fiber (fig. 4). Since linearly birefringent (anisotropic) optical fibers have a pair of preferred orthogonal axes of symmetry (birefringence axes), two orthogonal quasilinear polarized field components HE{1 and HE~ 1 of the fundamental mode HEll (LP01), which propagate for all values of frequency (wavelength), have electric fields that are polarized along one of these two birefringence axes. Hence, light polarized in a plane parallel to either axis will propagate without any change in its polarization but with different velocities. However, injection of any other input polarization excites both field components HEi"1 and HEiVl, and since these two orthogonal mode components are characterized by different propagation constants fix and [3y (degeneracy of the fundamental mode is lifted), they run into and out of phase at a rate determined by the birefringence of the HB fiber. At the same time they produce a periodic variation in the transmitted polarization state from linear through elliptic to circular and back again (see fig. 4). It is apparent that transmission properties of such fibers, when propagating only the fundamental modes, are similar to those of anisotropic crystals, in that the fiber has a pair of optical axes (Snyder and Love [1983]).
12
[I, w 2
P O L A R I M E T R I C O P T I C A L FIBERS A N D S E N S O R S
Table 2 Degeneracy lifting of polarization modes in HB two-mode fiber fundamental LP01 m o d e
second-order LP l I mode
two-mode HB fiber (LP 01 + LP~I )
Operating wavelength:
aeven ~' > "~cut
'~ < aodd "~cut < ~even "~cut
~odd "~cut < ~" < aeven "~cut
Normalized frequency:
V~< Vc = 2 . 4 0 5
2.405 < V < 3 . 8 3 2
Polarization modes:
LPbVl, LP~' 1
LP~~, L P Iev I , LPIIox , LPI"ov!
ev LPbVI , LP~)'I, LP~'~, LP(1
Propagation constants:
flo'c ~ fi0v
fi~x ~ file.'" ~ fi~x ~ fl~y
riOv ~ riO'v fi~x = fi(, file.`.- fi~'
Polarization birefringences:
Aft0 = r0 r -fi~'
Afi~ = fl~x_ file.,. Afi~ = fi~x _ fi/y
Aft0 = fi0~ - f i 0 v Aft I = fiiv _ fiiv
Aft) = f i ~ - fi~x Aft I, = file``- ill"'
Aft x = fi6~- _ fii~-
Modal birefringences:
-
Cutoff wavelengths:
2.4. P O L A R I Z A T I O N BIREFRINGENT
Aft,, = rid'- fi( Aft0 - Afil = Afix - A[3y
Relation 9~even/aodd "cut -'~cut = (3e + 1)/(3 + e)
EVOLUTION
IN T H E L O W E S T - O R D E R
(e-core ellipticity)
MODES
OF A
FIBER
Two important effects are a direct consequence of birefringence properties of the fiber in the polarization evolution in the lowest-order modes of the HB fiber. The first effect is lifting the degeneracy of the modes, which means that the different polarization modes will have different propagation constants and the greater the birefringence the greater the difference. The second effect is that the even and the odd LPll modes - in the case of a two-mode fiber- will have different cutoff wavelengths that can reduce several propagating modes in the HB fiber (table 2). For wavelengths slightly shorter than a critical value (cutoff wavelength), the next higher-order mode with greater propagation velocity compared with that of the fundamental mode is guided (fig. 5). The relevant feature of HB twomode fibers is that only two second-order modes (LP~I) propagate instead of four. This means that over a large region of the optical spectrum, the two-mode HB fiber guides only four polarization modes: two orthogonal linearly polarized 1' ex fundamental LPdl and LP61 eigenmodes and the even second-order LPll and LPle~ spatial modes, the propagation constants of which are denoted by fi6", rio,
I, {} 2]
POLARIZATION PHENOMENA IN OPTICAL FIBERS
13
Fig. 5. Spectral distribution of the lowest-order linearly polarized modes in a two-mode fiber; ~.o and ~e stand for cutoff wavelengths for odd and even LPll modes
Fig. 6. Mode pattern orientation of the even LPll mode in an elliptical-core and a bow-tie fiber.
/31x and fil instead of six as in the case of isotropic fibers with perfect circular cores. This effect is clearly observed in elliptical-core (e-core) and bow-tie HB fibers in which the orientation of the even LP~ mode is shown in fig. 6. The schematic diagram of the propagation constants and the corresponding mode patterns is shown in fig. 7. The separation between the cutoff wavelengths for the LP~l and the LP~'1 modes will increase with increasing birefringence (table 2). To describe quantitatively polarization transformation due to birefringence changes (intrinsic and induced) in HB fiber, both the Jones matrix formalism and the formalism using Stokes vectors and Mueller matrices can be applied. The Jones formalism is limited to the strictly monochromatic light sources when the light propagating in the fiber is completely polarized, whereas in the quasimonochromatic case the evolution of the state and the degree of polarization along a birefringent fiber is described by the Mueller-Stokes matrix formalism. For visualization and graphic representation the Poincar6 sphere is convenient. Many authors have addressed the question of the polarization evolution in birefringent fibers (e.g., Stolen, Ramaswamy, Kaiser and Pleibel [1978], Ulrich [ 1979], Wagner, Stolen and Pleibel [ 1981 ], Sakai, Machida and Kimura [ 1982], Crosignani and Di Porto [1982], Rashleigh and Marrone [1982], Love, Hussey, Snyder and Sammut [1982], Varnham, Payne and Love [1984], Sakai [1984],
14
POLARIMETRIC OPTICAL FIBERS AND SENSORS
[I, w 2
Fig. 7. Propagation constants in a single- and two-mode HB fiber.
Takada, Okamoto, Sasaki and Noda [1986], Zheng, Henry and Snyder [1988], Jaroszewicz [ 1994], Menyuk and Wai [ 1994], Eftimov and Bock [ 1998]).
2.4.1. The Jones formalism Polarization properties of a birefringent fiber can be adequately described in the monochromatic case by a 2 x 2 unitary complex matrix, the Jones matrix (Jones [1941]): E ~
--
T(X, ~). E
in,
(2.7)
where ~, is the wavelength of propagating light, and M is the propagation matrix depending on the physical environment represented by vector V and usually expressed as a product of three terms (Culshaw and Dakin [1989]) M = Te io J,
(2.8)
where T is the scalar transmittance, 0 denotes the mean phase retardance, and J is the birefringence (Jones) matrix of the fiber. The matrix becomes the identity matrix I in the case of an isotropic fiber with a perfect cylindrical symmetry.
I, w 2]
POLARIZATION PHENOMENA IN OPTICAL FIBERS
15
For a linearly birefringent fiber, J = Jg =
e iAO/2 0 ] 0 e -iAr '
(2.9)
where Aq~ denotes linear relative phase retardance between the eigenmodes and the fiber behaves like a simple linear retarder. For a circularly birefringent fiber, B=Bc=[C~ ~] sin6O cos6q~ '
(2.10)
where 26r denotes circular relative phase retardance between the eigenmodes and the fiber behaves like a simple circular retarder. The general Jones matrix for any optical fiber with evenly distributed retardations, including linear retardance A0, circular retardance 26q~ and axial rotation, was analyzed and described by Ysao [1992]: Q p,
,
(2.11)
where P=cos~-j(Ar
sin
& ,
Q=(r+6O)
Z ~ ,
sin
/
~=~Aq~ 2+(r+6q~) 2.
(2.12) If the fundamental mode LP01 of the linearly birefringent, weakly-guiding twomode fiber is labeled with an index 0, the LPll mode is labeled with an index 1, and the electric field excitation coefficient is denoted by It, the resultant electric field in the monochromatic case can be expressed as coherent superposition of the electric fields E0 and E1 of each of the modes: (2.13)
E ( r , q~,z) = ltoEo(r,z) + ltlEl (r, r
where E o ( r , z ) : Ax(z)fo(r)ex + A.,.(z)fo(r)e ,,
(2.14)
E l ( r ' O ' z ) : Bx(z)fle(r'O)ev + Bv(z)fle(r' O)e"
(2.15)
+ Cx(Z)fl~
O)e, ~ + C.,.(z)fl~
O)e,.,
where/to and/t, are the mode excitation coefficients (It2 + lt~ = 1), j6(r), f~e (?. , q~) andfl~ r, 0) are the corresponding modal spatial distributions, and e~ and ey are
16
POLARIMETRIC OPTICAL FIBERS AND SENSORS
[I, w 2
Fig. 8. Propagation constants for the lower-order waveguide and LP modes of an HB fiber (after Eftimov [1995]).
unit vectors along the birefringence axes. As the modes propagate along the fiber (z-axis), the amplitudes of polarization modes evolve. The general case for all LP01 and LPll (even and odd) modes was analyzed by Eftimov [1995]. Figure 8 shows propagation constants for the lower-order waveguide and LP modes of a linearly birefringent HB fiber, where A / ~ = firM- fiNE, AfiE = firE- fiHE, and /3 = (]~M + 2fiHE + riTE)/4 = (fie +/30)/2, and Aft0 = fi6" - rio" The stability of the second-order mode is set by the stability parameters introduced by Snyder and Love [ 1983], defined as
fi e _
~o
= ~ . AM(E) 2AfiM(E)
(2.16)
Isotropic fibers are characterized by degenerated values of second-order birefringencies (fie = rio), hence the stability parameters turn zero and the second-order modes are unstable, changing their polarization and orientation along the fiber. In HB anisotropic fibers fie_/3o >> A/~, AfiE, which means that stability parameters tend to infinity (AM,AE~ZX~), stabilizing the LPll modes. Limiting the analysis to the practical case when only fundamental and second-
I, w 2]
POLARIZATION PHENOMENA
IN
OPTICAL FIBERS
17
order LP~I modes are excited, polarization evolution along the fiber can be described by the coupled-mode equations
A,(z) A,(z) Bx(z)
d By(z)
(2.17)
B,(~)
[Ji "
The transformation of the amplitudes A,-(z), A ,.(z) and B,-(z), B,,(z) along the fiber is given by the Jones matrices jr and jill as: Ax(z)
l
A~(O)
AI~(O)
'
B,, 150 e~O 9r~ "
9~
100
50
I
0
9
I
20
'
I
40
'
I
60
'
I
80
'
I
100
'
I
120
'
I
140
'
I
'
160
Pressure, MPa Fig. 28. Comparison of the pressure characteristics of separated HB fiber and HB fiber-based polarimetric smart structure.
structure significantly modifies the output characteristics of the HB fiber. As a consequence, the induced birefringence of the embedded HB fiber changes less markedly than for the separated HB fiber, probably because pressure-induced
68
POLARIMETRIC OPTICAL FIBERS AND SENSORS
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stresses in the structure are separated between both the epoxy coating and the embedded HB fiber. 5.5. DYNAMICPOLARIMETRICOPTICAL FIBER SENSING In dynamic stress/strain measurements, the influence of temperature may be neglected due to its long period changes compared with high-speed mechanical changes. An example of a dynamic fiber-optic polafimetric sensor is shown in fig. 29. The sensor was proposed by Domafiski, Karpierz, Sierakowski, Switto and Wolifiski [1997] as part of an early-detection system for wheel-flat of a moving train. A laser diode pigtailed to a single-mode fiber and operating at 780nm wavelength was used as a light source. The fiber was wound onto a piezoceramic cylinder, which generated polarization modulation of the light passing inside the single-mode fiber. An additional fiber-optic phase shifter controlled the polarization coupling into the measuring HB fiber. The running wheel causes stress, which is transformed into strain on the bottom part of the rail and consequently into strain on the measuring fiber. The fiber-optic polarizer then converted the polarization modulation into intensity modulation. The idea of wheel-flat test systems is based on the application of an all-fiber polarimetfic strain sensor with the measuring HB fiber glued between pillars of a rail.
Fig. 29. Configurationof a polarimetric optical fiber sensor for dynamic strain measurement.
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In field tests, a multipoint semidistributed polarimetric sensor with compensated birefringence was used. The multipoint sensor with the HB fibers measuring a few meters long was spliced 90 ~ in the middle of the sensing fiber to compensate birefringence in both parts of the HB fiber and then to resist deterioration of the DOP along the sensing fiber.
w 6. Conclusions and Future Perspectives Polarimetric optical fibers and sensors have emerged as a result of investigations of polarization fluctuation in long-distance optical fiber communication systems. They have created a new generation of powerful, mostly sensing-oriented techniques, in which polarization of the guided optical field is the important issue. The rapid development of optical fiber technology over the last two decades has led to extensive research activities and progress in the field of polarimetric optical fibers and sensors. The use of highly birefringent optical has been stimulated by a significant decrease in the costs of manufacturing polarization-sensitive fibers and polarization-preserving optical fiber elements as well as semiconductor light sources. Current research trends in the field include multiplexed and distributed sensors (Udd [1991 ], Belgnaoui, Picherit and Turpin [1994], Bock and Karpierz [1999]), few-modes polarimetric fibers for multiparameter sensing (Bock and Eftimov [ 1994], Wolifiski and Muszkowski [ 1995], Eftimov and Bock [ 1998]), combined dynamic and static polarimetric sensing (Charasse, Turpin and le Pesant [1991 ]), as well as the development of new types of polarization-sensitive fibers configured in polarimetric structures suitable for environmental monitoring (Woliliski, Konopka and Domafiski [1998]), for modern industrial civil engineering (Calero, Wu, Pope, Chuang and Murtha [1994]), and for machinery. These new structures also include side-hole, elliptical-core fibers (Bock, Urbaflczyk, Wdjcik and Beaulieu [1995], Fontaine, Wu, Tzolov, Bock and Urbaflczyk [1996]), and liquid-crystal-core fibers (Yuan, Li and Palffy-Muhoray [1991], Chen and Chen [1995], Wolifiski, Szymafiska, Nasflowski, Konopka, Karpierz, Kujawski and Dgbrowski [1998]) which are extremely sensitive to external parameters (Wolifiski, Nasitowski, Szymafiska, Konopka, Karpierz, Domafiski and Bock [1997], Woliflski [1999]). Very recently (Wolifiski, Szymaflska, Nasitowski, Nowinowski-Kruszelnicki and Dobrowski [1999]) it also became apparent that the elliptical-core liquid-crystal fiber could act as a single-polarization, fewmode fiber in which only one polarization is guided. More basic problems arise during distributed polarimetric sensing. The early work by Kurosawa and Hattori [1987] showed that high birefringence of fibers
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used in distributed polarimetric sensors did not permit finding a location of small external disturbance, such as local fluctuation of hydrostatic pressure, strain, and stress, since a total birefringence compensation in this location was required. New research indicates that white-light interferometry (Urbaficzyk, Kurzynowski, Wo~niak and Bock [ 1997]) with different fiber lengths and special kinds of splices in between might find significant use. One of the most encouraging potential applications of polarimetric optical fibers and sensors is to embed them directly inside various ceramic and composite materials, and to measure strain distribution in different structures (e.g., aircraft, bridges, highways, concrete structures), using the concept of socalled smart skins and structures. The identification of polarization phenomena existing in optical fibers opens up new perspectives on basic physical effects that occur when light is confined to optical fiber waveguides. This simultaneously creates new opportunities for applications in modern optical fiber-sensing technology that holds still greater potential for optical fiber telecommunications.
Acknowledgements I am greatly indebted to two colleagues from the Faculty of Physics at Warsaw University of Technology: Professor Adam Kujawski for his encouragement to write this review, and Dr. Andrzej Domafiski, for his inspiration to conduct research in fiber optics and his continued valuable collaboration since the late 1980s. The technical assistance of Agnieszka Szymafiska, a doctoral student, during the preparation of the chapter is gratefully acknowledged. I apologize to all distinguished individuals whose papers may have been omitted in this review due to space limitations. This research was partially supported by the Warsaw University of Technology, Warsaw, Poland.
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Ulrich, R., and S.C. Rashleigh, 1982, IEEE J. Quantum Electron. QE-18, 2032. Ulrich, R., and A. Simon, 1979, Appl. Opt. 18, 2241. Urbaficzyk, W., P. Kurzynowski, W.A. Wo2niak and W.J. Bock, 1997, Opt. Commun. 135, 1. Varnham, M.P., A.J. Barlow, D.N. Payne and K. Okamoto, 1983, Electron. Lett. 19, 699. Varnham, M.P., D.N. Payne, A.J. Barlow and R.D. Birch, 1983, J. Lightwave Technol. LT-1, 332. Varnham, M.P., D.N. Payne, A.J. Barlow and E.J. Tarbox, 1984, Opt. Lett. 9, 306. Varnham, M.P., D.N. Payne, R.D. Birch and E.J. Tarbox, 1983, Electron. Lett. 19, 246. Varnham, M.P., D.N. Payne and J.D. Love, 1984, Electron. Lett. 20, 55. Varshney, R.K., and A. Kumar, 1984, Opt. Lett. 9, 522. Vassallo, C., 1987, J. Lightwave Technol. LT-5, 24. Vengsarkar, A.M., B.R. Fogg, K.A. Murphy and R.O. Claus, 1991, Opt. Lett. 16, 464. Wagner, R.E., R.H. Stolen and W. Pleibel, 1981, Electron. Lett. 17, 177. Wang, A., G.Z. Wang, K.A. Murphy and R.O. Claus, 1991, Opt. Lett. 17, 1391. Wolifiski, T.R., 1993, Proc. SPIE 2070, 392. Wolifiski, T.R., 1994, Proc. SPIE 2341, 29. Wolifiski, T.R., 1997, Proc. SPIE 3094, 41. Wolifiski, T.R., 1999, Optica Applicata XXIX, 191. Wolifiski, T.R., and WJ. Bock, 1993, J. Lightwave Technol. LT-11,389. Wolifiski, T.R., and WJ. Bock, 1995, IEEE Trans. Instrum. Meas. 44, 708. Wolifiski, T.R., W. Konopka and A.W. Domafiski, 1998, Proc. SPIE 3475, 421. Wolifiski, T.R., and M. Muszkowski, 1995, IEEE Trans. Instrum. Meas. 44, 704. Wolifiski, T.R., T. Nasitowski, A. Szymafiska, W. Konopka, M.A. Karpierz, A.W. Domafiski and W.J. Bock, 1997, in: Proc. Int. Conf. Opt. Fiber Sensors (OFS-12, Williamsburg, USA), OSA Technical Digest Series, Vol. 16 (Optical Society of America, Washington, DC) p. 277. Wolifiski, T.R., A. Szymafiska, T. Nasitowski, W. Konopka, M.A. Karpierz, A. Kujawski and R. D0browski, 1998, Mol. Cryst. Liq. Cryst. 321, 113. Wolifiski, T.R., A. Szymafiska, T. Nasitowski, E. Nowinowski-Kruszelnicki and R. D0browski, 1999, Mol. Cryst. Liq. Cryst., in press. Wong, D., 1992, J. Lightwave Technol. LT-10, 523. Wu, C., and G.L. Yip, 1987, Opt. Lett. 12, 522. Wu, R.-B., 1992, J. Lightwave Technol. LT-10, 6. Xiaopeng, D., H. Hao and Q. Jingren, 1991, Proc. SPIE 1572, 56. Xie, H.M., Ph. Dabkiewicz and R. Ulrich, 1986, Opt. Lett. 11,333. Xie, H.M., Ph. Dabkiewicz, R. Ulrich and K. Okamoto, 1986, Opt. Lett. 11, 33. Yeh, C., 1987, IEEE Trans. Educ. E-30, 43. Yen, Y., and R. Ulrich, 1981, Appl. Opt. 20, 2721. Yuan, H.J., L. Li and P. Palffy-Muhoray, 1991, Mol. Cryst. Liq. Cryst. 199, [701]/223. Zhang, E, and J.WY. Lit, 1992, Appl. Opt. 31, 1239. Zhang, E, and J.W.Y. Lit, 1993, Appl. Opt. 32, 2213. Zhao, W., and E. Bourkhoff, 1993, IEEE J. Quantum Electron. 29, 2198. Zheng, X.-H., WM. Henry and A.W Snyder, 1988, J. Lightwave Technol. LT-6, 1300.
E. WOLE PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED
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DIGITAL O P T I C A L C O M P U T I N G
BY
JUN TANIDA AND YOSHIKI ICHIOKA Department of Material and Life Sciences, Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan
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CONTENTS
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INTRODUCTION
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FURTHER D I R E C T I O N S
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CONCLUSION
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w 1. Introduction Optical computing is a broad term indicating concepts and technologies of information processing based on optical applications. Because of large flexibilities in the representation of information, way of processing, and architectures of processing systems, many interesting methods and technologies have been presented in this research area. Digital optical computing is a sub-area of optical computing in which a digital computing scheme is utilized as the foundation. In the digital optical computing scheme, information is represented by discrete signals and is processed in the same way as used in digital electronics. As a result, digital optical computing is suitable for cooperating with the current electronic technologies. Whereas the final goal of the optical computing technique is still vague, digital optical computing has a relatively clear target because it is based on the practical evolution of electronic technologies. As a result, digital optical computing can be viewed as one of the most promising application fields in optical technologies. In this article, various interesting ideas and technologies are reviewed to clarify the whole image of digital optical computing and to identify the promising applications. In w2, basic concepts involved in digital optical computing are explained. In w3, elemental components required for implementing a digital optical computing system are described, and in w4, methods for constructing a desired circuit are presented. In w5, experimental systems which are constructed as demonstrators of the digital optical computing scheme are mentioned. In w6, studies related to software development for digital optical computing are explained. In w7, future directions of the digital optical computing technology are discussed.
w 2. Basic Concepts The basic concepts of digital optical computing are logical operation, the logic gate array, free-space optical interconnection, the methodology for logic construction, smart pixels, the optical computing system, and software for problem solving. 79
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2.1. LOGICAL OPERATION
The fundamental basis of digital optical computing is the same as the digital computing scheme widely used in current computer science. In the digital computing scheme, information is represented by a set of binary bits, i.e., O's and l's, and is processed by logical operations to convert it into another one. The general form of the operation is y = f(x),
(1)
where x and y are sets of binary data, X -" ( X 0 , x l , . . .
,xm),
(2)
y = (yO,yl,...
,y,,).
(3)
Since any complicated operation can be decomposed into a combination of simple logical operations, if a basic set of logical operations is developed, any computing system can be constructed based on the digital computing scheme. This is an important point for constructing a digital optical computing system. 2.2. LOGIC GATE ARRAY
One of the useful features of optics is the capability of processing and transferring data in parallel format. Conventional optical elements, e.g., a lens and a mirror, can transfer information as an image. Assuming such a parallel image communication, an array of logic gates placed on a plane substrate is a reasonable form of a logic element for the digital optical computing scheme. As shown in fig. 1, a logic gate array consists of multiple logic gates functioning in parallel.
Fig. 1. Concept of logic gate array.
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There are two forms of parallel processing to identify the operation. One is a single instruction stream multiple data stream (called SIMD), and the other is a multiple instruction stream multiple data stream (called MIMD). Referring to fig. 1, the case in which the function of all logic gates is identical is SIMD and the case in which the function is different is MIMD. An MIMD system can provide flexible computing capability, but is difficult to control. An SIMD system is easy to control, but processing flexibility is inferior. Due to ease of implementation, most of logic gate arrays developed for optical computing are assumed to execute the SIMD type of parallel processing. 2.3. FREE-SPACEOPTICAL 1NTERCONNECTION Once logic gate arrays are implemented, an individual logic gate on the gate array must be connected to the others to form a logical circuit for a desired operation. In digital optical computing, free-space optical interconnection is considered as a fundamental technique for this purpose. As shown in fig. 2, appropriate optical elements, e.g., a lens, a mirror, and a holographic optical element, are used to configure interconnection between the individual logic gates. As a rule of thumb, information capacity handled by an optical system is estimated by the space bandwidth product of the system. When the sizes and the maximum spatial frequency of the transmitted image are x, y, fx and f., the space-bandwidth product is calculated as SBWP =
32XyfxL.
(4)
Since the maximum spatial frequency is limited by diffraction of the optical system, the information capacity of the system can be estimated easily. For example, an imaging system with a typical camera lens has the value of more than one million. As a result, a huge amount of information can be transferred by simple imaging.
Fig. 2. Free-space optical interconnection.
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Although such an interconnection line is established in three dimensional space, the individual lines do not interfere with each other, so that the free-space interconnection scheme provides a large degree of freedom in connectivity. In addition, using specific optical elements capable of changing their characteristics, such as a spatial light modulator, a reconfigurable interconnection network is also implemented.
2.4. METHODOLOGYFOR LOGIC CONSTRUCTION To implement a desired operation by a combination of logical gates, specific procedures are required in the design process. Whereas many methodologies exist in the field of information science, those are not necessarily effective for the digital optical computing scheme. Since huge numbers of logic gates on a plane substrate are connected with more relaxed constrainsts than in electric interconnection, a different type of logic construction can be developed to utilize the potential capability of the optical digital computing scheme. The physical characteristics of logic gate arrays and free-space optical interconnection should be taken account of in the methodology. As described later, various physical phenomena are utilized in logic gates and optical interconnection. For effective use of these technological components, consideration of their operational characteristics is indispensable.
2.5. SMARTPIXELS A practical way of implementing functions in digital optical computing is the positive use of a new category of optoelectronic devices called smart pixels. The smart pixel is a device composed of optical signal emitters or modulators, optical signal receivers, and electronic signal processing circuits. Such a configuration is effective to utilize the characteristics of both optics and electronics. Optics serves in interconnection while electronics are dedicated to processing tasks. In addition, we can effectively design and optimize functional blocks for processing, detection, and modulation. Since state-of-art techniques and environments can be used for the development, high performance systems are expected to be realized. Note that this form of system construction can go along with the evolution of electronics; therefore, we can effectively utilize the fruitful results of current VLSI (very large scale integration) technologies.
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2.6. OPTICALCOMPUTINGSYSTEM One of the most sophisticated systems based on the digital scheme is the computer. Highly organized digital circuits provide extremely large capability and flexibility in computing. With the same methodology used in the current computer, a computing system can be constructed within the digital optical computing scheme. To demonstrate the capabilities of the digital optical computing scheme, various types of optical computing systems have been designed and developed experimentally. The architectures aim toward both general purpose and special purpose computing. The experimental systems are expected to provide useful information on practical system construction and the requirement for logic gate arrays and optical interconnection. 2.7. SOFTWAREFOR PROBLEM SOLVING Based on the methodologies for logic construction, a wide range of problems can be solved within the framework of optical digital computing. Such efforts are considered as software development specified for optical digital computing systems. The targets are image processing, numerical processing, emulation of parallel processes, logical processing, etc. Although the methodologies for logic construction are not necessarily the same as in current computer science, the accumulated resources, e.g., data representation schemes and algorithms, can be utilized to develop sophisticated methods for various problems.
w 3. Logic Gates To develop a logic gate array, various methods have been considered. They are categorized into optical logic devices, coded pattern processing, and other procedural techniques. In addition, data representation is relevant to each implementation, so that we also focus on the format for data representation. 3.1. OPTICALLOGIC DEVICES The most straightforward way to get functionality of logic gates is to find appropriate physical phenomena for logic operations and to apply it to device embodiment. To achieve a logic operation, a kind of nonlinear response is required. As a result, a variety of nonlinear optical phenomena have been studied and adopted for optical logic gate devices.
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In the early days, intrinsically nonlinear materials were investigated and various optical logic devices were developed. For example, the change of the refractive index was used in a Fabry-Perot etalon device to achieve optical bistability (Gibbs, McCall and Venkatesan [1980]). Although the principle is simple and interesting, the slow time response and high operation power are disadvantages of this approach. An effective way to get large nonlinearity for optical signals is to use a hybrid effect of the optical light wave and other physical properties; e.g., electric field, magnetic field, elastic field, and so on. Although Fabry-Perot types of bistable optical devices were fabricated in the early stage (Smith, Turner and Maloney [ 1978]), simpler device structures have been developed successfully. In general, optical devices capable of controlling optical signal distribution in a twodimensional format are called spatial light modulators. Spatial light modulators are not necessarily designed for logic operations, but their nonlinear response and capability of parallel operation are ideal for parallel optical logic gates. PROM (Feinleib and Oliver [ 1972]) based on the electro-optical effect, magnetooptical spatial light modulators (Farhat and Shae [1989]), liquid crystal light valves (Fatehi, Wasmundt and Collines [ 1981], Mukohzaka, Yoshida, Toyoda, Kobayashi and Hara [1994]), and deformable mirror devices (Pape and Hornbeck [ 1983]) are typical examples of spatial light modulators. In terms of speed and driving power, spatial light modulators fabricated of semiconductor materials are the most promising. The multiple quantum well structure of semiconductor materials shows interesting characteristics for optical signals. With the help of self-feedback electric field, high speed and power effective optical logic devices have been developed. SEED families (Miller, Chemla, Damen, Gossard, Wiegmann, Wood and Burrus [1984], Lentine, Hinton, Miller, Henry, Cunningham and Chirovsky [1988], Lentine, Tooley, Walker, McCormick, Morrison, Chirovsky, Focht, Freund, Guth, Leibenguth, Smith, D'Asaro and Miller [1992]), EARS (Amano, Matsuo and Kurokawa [1991]), and an optical thyristor device (Tasiro, Ogura, Sugimoto, Hamao and Kasahara [1990]) are mentioned as typical examples. The data formats of the optical logic devices presented here are rather simple. Each bit of information is expressed by optical intensity or the state of polarization of the light signal for most logic devices. Two different intensity levels or two distinguishable states of polarization, e.g., horizontal and vertical linear polarization, are assigned to a logical one and a logical zero. As a special case, the intensity ratio of a couple of light signals is used for S-SEED to increase operational tolerance (Lentine, Hinton, Miller, Henry, Cunningham and Chirovsky [ 1988]).
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Fig. 3. Schematic diagram of coded pattern processing.
3.2. CODED PATTERN PROCESSING
The effective use of optical linear processing is another way for logic operation. Although logical operations cannot be implemented only by linear processing, a spatial coding technique enables us to achieve them. Figure 3 shows a schematic diagram of the method. Information to be processed is converted into a spatial pattern which is then processed by optical linear methods. The processed pattern is retrieved to the same form as the original information, which provides the result of the logic operation. Optical shadow casting (Tanida and Ichioka [1983]) and spatial filtering logic (Bartelt, Lohmann and Sicre [1984]) are good examples of this technique. Figure 4 shows a conceptual diagram of optical shadow casting for a logic gate SLOAUNRCE~
INPUT PLANE
CODED INPUT B
SCREENs ~ "" ~Jn
DECODING MASK CODED INPUT A
Fig. 4. Optical shadow casting for logic gate array.
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Fig. 5. Conceptual diagram of symbolic substitution. array. Arbitrary logical operation between two binary images is accomplished by this technique. Each couple of pixels at the same location on the input images is converted into any one of four different spatial patterns according to the coding rule shown by the table in the figure. The coded image composed of the spatial pattems is illuminated by an array of point light sources and their shadowgrams are projected onto the screen. Then, the optical signals through the decoding mask provide a result of a logical operation. In this technique, a uniform operation is executed for all data set on the input images in parallel. The contents of the operation are specified by the pattern of the on-state sources. The optical processing used here is considered as a specific version of a cross-correlation called discrete digital correlation. Several extended versions of optical shadow casting have been developed with the introduction of a space-variant decoding mask or polarization (Yatagai [1986], Li, Eichmann and Alfano [ 1986], Karim, Awwal and Cherri [ 1987]). For the case of spatial filtering logic (Bartelt, Lohmann and Sicre [1984]), information is converted into a spatial pattern within a pre-determined set of spatial frequencies. For example, grating patterns with different spatial frequencies or random dot patterns with different grain sizes are used for the coding. Since signals with the same spatial frequency are converged into the same position on the spectrum plane, signals with a specific condition can be selected by the spatial filtering technique. Therefore, this processing is implemented by a typical coherent optical system (Goodman [1996]). As a more interesting technique for logic operation, symbolic substitution has been presented (Brenner, Huang and Streibl [1986]). Figure 5 shows the conceptual diagram of symbolic substitution. Although this technique is quite intuitive, symbolic substitution is recognized as a two-dimensional extension of Boolean logic. A logic operation takes a set of bit information as the input and puts out another set of bits according to a pre-determined transition rule. In contrast, symbolic substitution converts a spatial pattern of bit information into
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another one. The concrete procedure of symbolic substitution is to find specific spatial patterns and to substitute them with other ones according to a substitution rule describing the processing contents. As shown in fig. 5, with specific spatial patterns and substitution rules, desired logic operations can be implemented with this technique. For the coded pattern processing, various kinds of spatial patterns of optical signals, e.g., light intensity or polarization states, play an important role in data representation. Although those techniques require more space on the image plane than direct data representation, flexibility in processing is obtained with the penalty. 3.3. OPTICAL PROCEDURAL TECHNIQUES
To utilize the multi-state nature and parallelism involved in optical signals, various interesting techniques have been presented. These are residue arithmetic (Huang, Tsunoda, Goodman and Ishihara [1979]), multi-level logic (Abraham [1986]), and table lookup processing (Guest and Gaylord [1980]). The first two techniques are useful to suppress carry generation during arithmetic operations, and the last one provides a flexible method for logic generation with a simple optical setup. Residue arithmetic (Huang, Tsunoda, Goodman and Ishihara [1979]) is a number representing system using a set of residue numbers of different divisors. In this system, a non-negative integer x is represented by a set of integers, ( r l , r 2 , . . , rk). ri is the residue of x/yi, and the y;'s are arbitrary prime numbers which are different from each other. With this number representation, any x satisfying the inequality k
O Z, 0 otherwise.
(8)
By configuring the kernel function and the thresholding value Z, a variety of processing can be implemented; e.g., image enhancement, noise reduction, and simulation of parallel processes.
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Fig. 20. Processingprocedure of cellular logic.
6.1.2. Symbolic substitution As mentioned in w3.2, symbolic substitution (Brenner, Huang and Streibl [ 1986]) is a powerful technique for digital optical computing. Not only logic operations but also higher levels of processing can be achieved by the symbolic substitution scheme. The procedure is as follows: (1) Define code patterns to represent the information to be processed; (2) Under this definition, find substitution rules to accomplish the desired processing; (3) With the substitution rules, execute the procedure of symbolic substitution; and (4) Retrieve the processed patterns into the initial form of information. Symbolic substitution has large flexibility in data representation and processing algorithms, which shows the potential capability of the scheme. However, no systematic study for describing arbitrary processing has been done for symbolic substitution.
6.1.3. Optical array logic Optical array logic (Tanida and Ichioka [1988]) described in w is also a logic system suitable for the digital optical computing scheme. Complicated processing can be described and executed by the optical array logic scheme. Compared with symbolic substitution, optical array logic uses a conservative way of processing description; viz., all processing in optical array logic is described by logical operations for pixels within a neighborhood area on the two input
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p r o g r a m addition; var i, N; image imagea=../image/add.attr; image imageb=../image/add.data; kernel add; N:7;
/* N bit
add:
I 1. I I_0. I *
+I 1.1 I_o. I
*/
*
.0 .i .i .0
+1_1. I I 0.1 * t _ . l l [ .11@(o,i); for i:l to N do i m a g e b : exec( end; imout
add.result
imagea,
imageb,
add);
imageb;
end addi t ion; Fig. 21. Program example used in optical array logic.
images. A general form of the operation executed by optical array logic is represented as follows: K
L
L
k=l
m=-L
n=-L
where a i + m , j + n and b i + m , j + n are input images and c;.j is the output image, f ( . ) indicates any one of two-variable binary logic functions, and L and K are the size of the neighborhood area and the number of product terms, respectively. Arbitrary logical operations can be executed by configuring f,,.,,: k's for individual m, n and k. Unfortunately, it is difficult for us to capture the meaning of this general form, so that a specialized notation language is prepared. Using the language, processing in optical array logic can be written as the program format shown in fig. 21.
6.1.4. Binary image algebra Binary image algebra (Huang, Jenkins and Sawchuk [1988]) is an algebraic system specialized for compact description of operations on binary image data.
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Binary image algebra is constructed by an image space and a family of operations including three fundamental operations and five elementary images; viz.,
BIA - (P(W); -, U, |
(10)
where P(W) is an image space which is the power set of a predefined universal image W. The fundamental operations are complement, -, union, U, and dilation, | Each elementary image consists of just one pixel located at either the origin or one of its neighboring positions. The binary image algebra is applied to design effective logical circuits for the DOCIP (Huang, Sawchuk, Jenkins, Chavel, Wang and Weber [1993]).
6.1.5. Image logic algebra Image logic algebra (Fukui and Kitayama [1992]) is designed as a genetic language for parallel image processing. In image logic algebra, binary image processing is treated as a sequence of logical template matching. Each logical template matching is described by a kernel pattern used in a discrete correlation for the target image. A neighborhood configuration pattern is introduced for compact description of processing. For example, a transformation from A[M; N] to B[M; N] is expressed as follows:
B[M;N] = A[M;N] |
Ilrll ,,,,
(11)
where [[r[[(k,t) is the neighborhood configuration pattern. Image logic algebra comprises three operations of images, six transformations of images and three operations of the neighborhood configuration patterns. As shown in fig. 22, image casting, multiple imaging, and test transformation are unique
Fig. 22. Transformations defined in image logic algebra: (a) image casting and (b) multiple imaging (Fukui and Kitayama [1992]).
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transformations. In addition, extended erosion and dilation are defined with the help of the neighborhood configuration pattern. 6.2. APPLICATIONSOF PARALLELLOGIC SYSTEMS Various kinds of problems have been solved by the parallel logic systems explained in the previous section. Applications of the parallel logic systems cover a wide field, including image processing, numerical processing, parallel process emulation and non-numerical processing.
6.2.1. Image processing Image processing is an essential application of the digital optical computing system. Since an image is a primitive data format of the digital optical computing scheme, it can be processed effectively, and a variety of studies has been made of the subject. Due to differences in processing methods, image data are often categorized into two major classes: binary and multi-valued images. The former is composed of binary pixels, which can be processed by simple logic operations. The latter consists of an array of multi-level pixels, which requires a complicated procedure for data representation and processing. As the data format of the multivalued image, a bit expansion and a bit slice format are considered, as shown in fig. 23. Once the target image is represented by a binary form, it can be processed by logical operations in the digital optical computing scheme. For simple image processing, heuristic approaches are effective, as shown by Brenner, Huang and Streibl [1986] and Tanida and Ichioka [1988]. For flexible processing, techniques based on digital filtering are effective. Goodman and
Fig. 23. Data format for multi-valued image: (a) bit expansion and (b) bit slice.
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Rhodes [1988] present a procedure using the symbolic substitution, and Tanida and Ichioka [ 1988] show a counterpart by the optical array logic scheme. Another method is the use of morphological operations (Serra [1982]). Huang, Jenkins and Sawchuk [1988] and Fukui and Kitayama [1992] show how to implement image processing based on morphological operations within the frameworks of binary image algebra and image logic algebra, respectively.
6.2.2. Numerical processing The parallelism provided by digital optical computing is attractive for scientific computing. In scientific computing, large-size matrices often appear and their efficient calculation holds the key for high-performance computing. Therefore, various methods for matrix computation have been studied actively. The four elementary operations, i.e., addition, subtraction, multiplication, and division, are implemented by many researchers. For the case of symbolic substitution, addition was demonstrated by Brenner, Huang and Streibl [1986] and multiplication and division are described by Hwang and Louri [ 1989]. For optical array logic, addition, subtraction, and multiplication are discussed by Tanida, Fukui and Ichioka [ 1988]. Also, multiplication by image logic algebra is discussed by Fukui and Kitayama [ 1992]. Although most parallel logic systems treat data in the SIMD manner, numerical processing usually requires minute data manipulation. To overcome the problem, various sophisticated techniques have been developed. Figure 24 shows a processing procedure of addition by optical array logic (Tanida, Fukui and Ichioka [1988]). In addition to an image containing the data, an auxiliary image is prepared for storing data attributes. Use of the attribute image is a key point of the implementation. Whereas the data image holds both addends and augends, the attribute image keeps the information of location of pairs of an addend and an augend. Addition is achieved by repeated application of the following operations calculating sum and carry of each bit position: sum / = 2iy i + xiy; i, carry/+ 1
_
.
xiyi,
(12)
(13)
where x ~and y/are the ith bit position of the addend and the augend, respectively. Because the addend and the augend are placed on the data image, the actual operation for the images becomes Ci, j = ~ l i , j a i _ l , j b i , j l ) i _ l , j + ~ l i , j a i _ l , j l ) i , j b i _ l , j + a i , j a i + l , j b i , j + l b i + l , j + l ,
(14)
where a, b, and c are pixels in the attribute, data, and output images, respectively. The suffix indicates the address of the pixel on the image. Note
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Fig. 24. Processing procedure of addition by optical array logic (Tanida, Fukui and Ichioka [1988]). that a product of a's, e.g., ~li,jai_l,j, becomes 1 only if the condition is satisfied. Therefore, such a product can be used to identify the location where x and y are placed on the image. Although the operation in eq. (14) is more complicated than those in eqs. (12) and (13), we do not take care of the position and the length of each couple of the addend and augend. This feature should be emphasized as an important feature of the implementation.
6.2.3. Emulation of parallel processes Because of its inherent parallel nature, the digital optical computing scheme seems to be suitable for emulation of various parallel processes. Actually, by describing a primitive process or behavior of an individual element by logic operations, we can effectively emulate a complicated phenomenon consisting of large number of elements. Although the main purpose of the emulation is to analyze a complicated phenomenon, it can be extended as a computing technique. Based
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Transition rule
~~
(a)
(b)
l'l I'I [o1,Iol
(c)
1-1
l'~.Jl
(d)
Fig. 25. Virtual logic circuit emulated by symbolic substitution (Murdocca [1987]).
on the idea, an interesting technique was proposed with symbolic substitution (Murdocca [1987]). As shown in fig. 25, a logic circuit is constructed using spatial patterns on an image. Then an information token is placed on the image as a spatial pattern. Applying a simple substitution rule appearing in the figure, one can emulate information propagation along with the logic circuit. Since all processes are driven in parallel, large processing capability can be expected. The Turing machine is emulated by two parallel logic systems to prove their computational capability (Brenner, Huang and Streibl [1986], Tanida, Nakagawa, Yagyu, Fukui and Ichioka [1990]). Although the Turing machine itself is not an optimized system, the same technique used in the Turing machine can be used to emulate various types of virtual processing systems. Reconfiguring a processing system for a given problem, we can virtually construct an effective processing system for the problem. As an example, matrix processing was performed by emulation of a systolic processor (Fey and Brenner [1990], Fukui, Tanida and Ichioka [1990]). Emulation of a data flow machine by optical array logic was reported (Iwata, Tanida and Ichioka [1993b]). 6. 2.4. Non-numerical processing The large parallelism provided by digital optical computing seems to be ideal for tasks handling a large amount of information. Processing of huge amounts of data is one of the important applications of digital optical computing. As an example, processing for database management is considered as a promising application of digital optical computing.
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Fig. 26. Processing example of database managementby optical array logic: (a) relational database image, (b) selection rule, and (c) resultant image.
An example of optical implementation of a relational database is reported by Iwata, Tanida and Ichioka [1993a]. Figure 26 shows a processing example of database management by optical array logic. Mapping data onto an image plane as a database, one can manage the database by logical operations on the image. As an extended application of database management, inference operations are demonstrated by optical array logic (Iwata, Tanida and Ichioka [1992]). Using a specially designed database in which relations between objects are stored, objects satisfying a given condition are extracted. An expert system was constructed with the same technique.
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w 7. Further Directions Without a doubt, today's electronic computing systems are highly developed and their performance is increasing steadily. It seems to be difficult for new technologies to take the place of the steady reign of electronics. Instead, a 'newcomer' should cooperate with electronics to perform specialized functions. This is true for digital optical computing. A reasonable strategy for optics is to implement specialized functions supplementing weak points of electronics and to enlarge its application field more and more. Reflecting this situation, the research field of 'optical computing' is gradually shifted to that of 'optics in computing'. A promising architecture of the digital optical computing system is that based on smart pixel technologies. In this architecture, the state-of-art technologies for device design and fabrication can be utilized effectively. This form of application is nothing but an extension of optical interconnection technology; viz., free-space optical interconnection is employed effectively to connect elemental processors comprising a high-performance parallel information processing system. In addition, such a smart pixel system has an advantage in compactness. With appropriate packaging methods, a compact and reliable information processing system could be constructed. In terms of applications suitable for digital optical computing, a signal exchange system for optical communication and a processing system for multimedia applications are promising. The importance of an information highway like the Internet is beyond discussion. On the network various forms of information including sounds, images, and movies are transferred frequently. Because of the large amount of information and the tremendous number of potential users, the communication system is required to handle effectively extremely large quantities of information. For the demand, development of a new parallel communication network system, which has an ultra-high bandwidth capable of transmitting high-resolution images at high speed would be expected. For the new network system, the digital optical computing system would play an important role. In addition, development of a practical system will be an additional impetus to promote digital optical computing.
w 8. Conclusion Various interesting ideas and technologies related to digital optical computing have been reviewed to clarify the concept of the scheme. From basic concepts to practical applications, some of the results accumulated in this field have
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been presented in overview form. The important features of the digital optical computing scheme are parallelism and familiarity with current electronics. Due to these features, the digital optical computing scheme is a promising candidate for high-performance information processing. As a result, digital optical computing would be expected to be one of the most important and most fundamental techniques in the information processing area in the near future.
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E. WOLE PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
Ill
CONTINUOUS M E A S U R E M E N T S IN QUANTUM OPTICS
BY
V P E ~ X I O V . , i, AND A. L U K S
Laboratory of Quantum Optics, Faculty of Natural Sciences, Palaclo) University, Tr Svobody 26, 771 46 Olomouc, Czech Republic
115
CONTENTS
PAGE
w 1.
INTRODUCTION
. . . . . . . . . . . . . . . . . . .
w 2.
THE M A N D E L AND SRINIVAS-DAVIES APPROACHES TO
118
M O D E L I N G PHOTODETECTION . . . . . . . . . . . . w 3.
121
REVERSIBILITY AND IRREVERSIBILITY OF PHOTOCOUNTING . . . . . . . . . . . . . . . . . .
135
w 4.
MODELS OF CONTINUOUS M E A S U R E M E N T
148
w 5.
PROBABILITY-DENSITY FUNCTIONAL FOR QUANTUM
. . . . . .
PHOTODETECTION PROCESSES . . . . . . . . . . . .
169
w 6.
QUASICONTINUOUS SCHEMES OF P H O T O D E T E C T I O N .
w 7.
SPECIAL STATES OF AN OPTICAL MODE BY MEANS
.
173
OF CONTINUOUS AND NEARLY C O N T I N U O U S M E A S U R E M E N T S ON THIS MODE . . . . . . . . . . .
182
w 8.
PRODUCTION OF CORRELATED PHOTONS
185
w 9.
CONDITIONAL GENERATION OF SPECIAL STATES USING
. . . . . . .
IDEAL STATE R E D U C T I O N OF ENTANGLED FIELDS.
. .
189
w 10. SPECIAL STATES OF ONE OR TWO OPTICAL MODES BY MEANS OF CONTINUOUS M E A S U R E M E N T ON A DIFFERENT MODE . . . . . . . . . . . . . . . .
226
w 11. EIGENVALUE PROBLEMS FOR INTELLIGENT STATES GENERATED 1N IDEAL AND C O N T I N U O U S M E A S U R E M E N T S . . . . . 116
241
w 12. C O N C L U S I O N
.
.
.
.
ACKNOWLEDGEMENTS
. .
. .
. .
.
.
.
.
257 258
Appendix A. SUPEROPERATOR-VALUED M E A S U R E AND RELATED CONCEPTS. .
258
Appendix B. ITO'S CALCULUS WITHOUT ITO'S DIFFERENTIAL
261
REFERENCES
.
.
.
.
.
.
.
117
.
.
.
.
262
w I. Introduction The description of the photodetection process is an indispensable part of quantum optical courses, and yet many papers have been published contributing to this topic. The researchers have taken into account more and more items of detail, such as polarization, special characteristics, and inefficient detection. Both thermal and laser radiations were studied using the photodetection equation, first in the framework of the semiclassical approach to the photodetection (Mandel [1958, 1959]), and then respecting the quantum optical approach to the photon-number measurement (Mandel [1963a], Kelley and Kleiner [1964], Glauber [1965], Mollow [1968]). In accordance with this, the formally perfect model of the photon absorption by a detector (Srinivas and Davies [1981]) was not first accepted and its use in quantum optics was debated for some time (Mandel [ 1981 ], Srinivas and Davies [ 1982]). The Srinivas-Davies model can be generalized to involve a variable absorption rate which, in a particular case of its time dependence, gives a constant intensity of the photocount process. It could be interesting with respect to photon bunching and antibunching, which were first defined for a stationary field (Mandel and Wolf [1995], Torgerson and Mandel [ 1997]). The relationships of these concepts to the subPoissonian behavior have been treated (Singh [1983], Jakeman [1986], Zhou and Mandel [ 1990]), and they have been generalized to nonstationary processes (Singh [1983], Dung, Shumovsky and Bogolubov [1992]), whilst Srinivas and Davies [1981] considered these concepts only for their nonstationary process. Believing that the quantum theory is not complete without the projection postulate, we will mention the attempts to avoid this postulate (Omn+s [ 1994], Haus and K~irtner [ 1996]) as well as a scheme of successive measurements which is based on this postulate (Wigner [1963]). On the other hand, it is important that the master equation in the Lindblad form (Lindblad [ 1976]) describing the photon absorption by the detector suggests its unraveling and also the unraveling of its solutions; i.e., their expression as expectation values of the time-dependent statistical operators respecting the detection process. The master equation relates to the evolution equation for an unnormalized statistical operator between registrations and a transformation of the unnormalized statistical operator due 118
III, w 1]
INTRODUCTION
119
to a registration [Ueda, Imoto and Ogawa [1990a], Carmichael [1993a], Knight and Garraway [1994], Ban [1997a]). A general description is applied both to the case of an ordinary photon counter which does not have a logically reversible dynamics, and to the case of a quantum counter (Mandel [1966]), with a logically reversible dynamics (Ueda and Kitagawa [1992], Ueda, Imoto and Nagaoka [ 1996]). Concentrated on atoms, some papers have been concerned with the reversibility of quantum jumps (Mabuchi and Zoller [1996], Pellizzari, Beth, Grassl and Mfiller-Quade [1996], Ekert and Macchiavello [1997], Mensky [ 1996a], Nielsen and Caves [ 1997], Bennett, Brassard, Cr6peau, Jozsa, Peres and Wootters [ 1993]). The time evolutions, quantum jumps, the quantum trajectory idea, and the stochastic wave-function evolution concept have been applied in numerical simulations and formalized in the quantum stochastic calculus together with the Gaussian (or diffusive) continuous measurement, stochastic equations and Monte Carlo simulation schemes. Although based in part on formal measurement arguments, the literature also proves that the quantum trajectories arise naturally (cf. Carmichael [ 1996]). We will develop an independent stochastic formulation of the detection theory, although a standard approach exists (Parthasarathy [1992]). We will base the considerations on the damped quantum harmonic oscillator (Lax [ 1966], Louisell [ 1973], Gardiner [ 1991 ]). First we will expound the Heisenberg-Langevin approach to the description of the damped harmonic oscillator and more general open systems interacting with a thermal reservoir. Besides the It6 integral (It6 [ 1944]), we will introduce the stochastic Stratonovich integral (Stratonovich [1963, 1964]). We will then introduce the quantum stochastic Schr6dinger equation for a system interacting with a reservoir. We will outline a derivation of a stochastic nonlinear Schr6dinger equation for the state conditioned on the outcomes of photon-number measurements (Carmichael [ 1993a], Wiseman and Milburn [ 1993a], Barchielli and Belavkin [ 1991 ]). We will mention a nonlinear Schr6dinger equation for a state conditioned on the results of measurement of continuous observables (Gisin [1984], Gisin and Percival [ 1992, 1993a,b]). Even the continuous quantum nondemolition measurement of photon number can be treated as a particular case of the back-action of the continuous measurement and the photocount registration on the radiation field (Ueda, Imoto, Nagaoka and Ogawa [ 1992]). A quantum theory of feedback has been established (Wiseman and Milburn [1993b, 1994], Wiseman [ 1994a,b,c]). The developed formalism can also be applied in the case of the microscopic models of photonnumber measurements (Imoto, Ueda and Ogawa [ 1990], Ueda, Imoto, Nagaoka and Ogawa [ 1992]).
120
CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
[III, w 1
The probability-density functional for a quantum photodetection process has been invented to provide complete information on the photocount process dependent on the initial statistical operator (Ueda [1989a,b, 1990]). We will focus our attention on the generating functionals, which in comparison with the probability-density functional, present fewer difficulties and describe well the random-point process. An analogy to continuous measurement is also useful in the description of an apparatus consisting of many lossless beam splitters and photodetectors, which can be called a quasicontinuous photodetection scheme with relationship to a destructive continuous photon-number measurement (Ban [1994]). There exists an analogous device with the parametric amplifiers equivalent to the quantum counter, and one with four-wave mixers equivalent to the quantum nondemolition continuous measurement. The continuous destructive photon-number measurement can be used for generating a Schr6dinger-cat state from an ordinary squeezed state (Ogawa, Ueda and Imoto [ 1991 a]). By means of continuous photodetection, one produces novel states of a one-mode optical field conditioned on specific situations during the photodetection. Ideal and continuous measurements lead to a peculiar behavior when applied to one of two entangled fields which can be produced either on a beam splitter or in a parametric down-conversion process. Cascaded down-conversion processes can produce entanglement of three and four modes. In addition to these theoretical proposals, encouraged by experiments which were positive in clear violation of the Bell inequality (Walls and Milburn [ 1994], Pefina, Hradil and Jur6o [1994]), there are some experiments which were implemented in the continuous-wave regime. Interest in the generation of the sub-Poissonian light dates back to Yuen [1986a], concentrating on the use of the parametric amplifier and the photonnumber measurement on one of the correlated modes. One can also consider the complex amplitude measurement or quadrature measurement on one of these fields (Watanabe and Yamamoto [1988]). A general analysis also applies to a down-conversion stimulated by a two-mode input coherent state (Agarwal [1990]). Many of the quantum statistics of the special states due to the photon-number measurement have been calculated (Luke, Pefinovfi and K~epelka [ 1994]). For the beam splitter also, the photon-number measurement, along with the complex amplitude and quadrature measurements, can be analyzed (Ban [1996a]). This single device has also been proposed as a means of generating the Schr6dinger-cat states (Dakna, Anhut, Opatrn~,, Kn611 and Welsch [1997], Dakna, Kn611 and Welsch [1998]). In addition to these suggestions, we will pay attention to the states arising in the measurement of three and four correlated
III, w 2]
SOME APPROACHES TO MODELING PHOTODETECTION
121
modes after down-conversions with complete or partial alignment of the idler modes (Luis and Pefina [ 1996]). A somewhat more difficult analysis is needed in this connection for the description of the continuous destructive photon-number measurement carried out simultaneously with the nondegenerate parametric amplification (Holmes, Milburn and Walls [1989]). We will concentrate rather on a situation where the counting is done after the interaction which produced the correlated state. In the case of measurement on one of two entangled modes, the evolution of the moments of the measured and unmeasured fields depends on the variances and covariances of the amplitude of these fields (Ueda, Imoto and Ogawa [1990b]). The evolution of many conditional statistics of the unmeasured mode has been obtained in the case of destructive continuous photon-number measurement on the idler mode of a two-mode squeezed state (Pefinovfi, Luk~ and K~epelka [1996a,b]). The conditional statistics have also been obtained in the case of the nondemolition continuous photon-number measurement (Pefinovfi, Luk~ and K~epelka [ 1996b]). The continuous quadrature measurement has also been analyzed (Breslin and Milburn [1997]) and the scheme of the quantum nondemolition continuous-wave detection of the quadrature has been implemented in experiments (Schiller, Bruckmeier, Schalke, Schneider and Mlynek [1996], Bruckmeier, Schneider, Schiller and Mlynek [1997], Bruckmeier, Hansen and Schiller [1997]). Luke, Pefinovfi and K~epelka [1994] and Luis and Pefina [1996] have formulated eigenvalue problems and analyzed the ideal reduction of the correlated state of light fields. A generalization of this theory to the case of the destructive continuous state reduction has been accomplished (Pefinovfi, Luk~ and K~epelka [ 1996a]).
w 2. The Mandel and Srinivas-Davies Approaches to Modeling Photodetection With the discovery of the photoelectric effect, photometry has gained an important tool. Whereas purely classical apparatuses present no problems, photoelectric detectors should be described in a more sophisticated way. 2.1. S E M I C L A S S I C A L APPROACH TO P H O T O D E T E C T I O N
The semiclassical theory of photoelectric detection of light describes the situation where the light intensity is converted into irregular releases of
122
CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
[III, w 2
photoelectrons. In the simple case of a quasimonochromatic electromagnetic wave, the energy E of the wave, considered within the finite interval from t to t + T, is converted into several photoelectrons in dependence on the detector efficiency r/, and it should be regarded as consisting of m photons of energy hco, where h is Planck's constant divided by 2st, and underlining indicates that a variable is stochastic. Since in this situation the Poisson process is appropriate and the equation E = (hmm) may express that the wave and particle aspects are not separated, we arrive at the probability p'(m, t, T) for m photons to occur within the interval from t to t + T,
I (E)
p'(m, t, r ) = ~
~-~
m
(
E)
exp -~-~
.
(2.1)
The photocount distribution takes into account the efficiency r/, and it reads
p(n, t, T) = ~1 W" exp(-W),
(2.2)
with the parameter W = rlE/(hoo). Since the field need not be quasimonochromatic, we write
f
t+T
W = ~1
I(t') dt',
(2.3)
dt
where I(t) is the 'light intensity'. Strictly speaking, this quantity, having the dimension of time -1 , cannot be just the light intensity in the nonquasimonochromatic case. The Poisson process has the property that the numbers of counts in the nonoverlapping intervals are distributed according to the respective lengths and are statistically independent. Let us consider the numbers of counts n l , . . . , ~ and the intervals [tl, tl + T i ) , . . . , [tj, tj + Tj). The statistical independence implies a joint photocount distribution of the form d
p(nl,. . . ,nj, tl, T1, . . . , tj, Tj) = H p ( n j , tj, Tj),
(2.4)
j=l
where the photocount distributions are
p(nj, tj, Tj) = 1 Wj~jexp(-Wj), nj!
j = 1 "'
J,
(2.5)
with
Wj = rl
f
d t;
ts+~
I(t') dt'.
(2.6)
III, w 2]
SOME APPROACHES TO MODELING PHOTODETECTION
123
When the optical field fluctuates, the light intensity is a random process/(t), and the relation (2.6) takes the form t/+r/ (2.7) W_j = 7/ /(t') dt'. at/ The relation (2.4) becomes
/
p ( n l , . . . , r t j , tl, T1,. . . , tj, T j ) =
rl--~.
(2.8)
"
j=l
When the intervals [tj, tj + T/), j - 1,... ,J, do not overlap, we consider the joint probability density P ( W 1 , . . . , W j , tl, T l , . . . , tj, T j ) and we can write the averaging operation as an integral: p ( n l , . . . , n j , tl, T1, . . . , tj, T j )
-fo ~
9
~0"~
,,
ii
j:l
1 Wj+ exp(- W/)P(W1
~
, ' ' ' 9
nj!
W/t,,T1, ,
" ' ' ,
tiT/) ,
"
fi
d~.
j:l
(2.9) The case J = 1 is known as the Mandel photodetection formula (Mandel [1958, 1959]). Using the notation of the left-hand side of relation (2.4), we can introduce other notation due to Ueda [1988] (cf., a notation on the probability distributions of triggered counting (Ueda [1989b]); e.g., (l(h), n, l(t2)) =p(1,n, 1 , t l , A t l , t l + A t l , t 2 - tl - A t l , t 2 , A t 2 ) , (2.10) ( e ( t l ) , n , l(t2))=p(n, 1,tl + Atl, t2 - tl -Atl,t2, At2). We can obtain the following expansions (Ueda [1988]): lim
(l(h),n, l(t2)) _
At, -+ O, 6t2 ~ 0 lim
A h At2
,~ m 0 ill
~
=
0k
( e ( t l ) ' n ' l(t2)) = - ~
At2
At2 ----+0
=
0
0
Otl Ot2 p ( k ' tl , t2 - tl ),
~2P(k,
(2.11) t ~ , h_-
t~).
k=0
We mention the rest of the notation due to Ueda [1988]: P~,.cl(tl, t2), P c e l ( t l , t2), P s l l ( h , t 2 ) , P c l l ( t l , t 2 ) . Particularly, to quote only the simplest formulas, P s e l ( t l , t2) -"
lim
(e(tl),O,l(t2))
At2 --+ 0 O(3
Pcel (tl, t2) = Z
At2
( e ( t l ) , n , 1(t2))
lim At2 --' 0
(2.12)
At2
n=0
Applications to the fundamental statistics, that is, the Poisson and BoseEinstein statistics, have been provided. For the Poisson statistics, the probability
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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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distribution p(n, T) of n photoelectrons being registered during time T is given by (2.13)
p(n, T) = ~ ( I T ) " exp(-IT),
where I is the average number of photoelectrons per unit time. We can obtain that Psll(T)
= Psel(T) =
I exp(-IT),
Pc1 I ( T ) = P c e l ( T ) = L
(2.14)
The facts that Psll(T) and Poll(T) are equal to Psel(T) and Pcel(T), respectively, indicate that no two photoelectrons in the Poisson process are correlated. On the other hand, for the Bose-Einstein statistics two photoelectrons are correlated or bunched if their time interval is comparable to or shorter than the coherence time rc of light. The photocount distribution p ( n , T ) for the Bose-Einstein statistics is given by (Mandel [1958, 1959, 1963b]): p(n, T) =
(IT)" (1 + IT) n+l '
(2.15)
where T is required to be much shorter than re. We can obtain that (Glauber [1967]): Pse~(T) =
I (1 + IT) 2'
P~e~(T) = I.
(2.16)
For short time intervals such that IT < 1, Psll(T)-
21 > Psel(T), (1 + IT) 3
(2.17)
which shows that photons which obey the Bose-Einstein statistics tend to be registered in bunches. P~ll(T) is twice as large as P~eI(T): (2.18)
Pcll(T) = 2I = 2Peel(T).
Two probability distributions under the same initial condition, but for different time intervals, have been compared (Mehta and Wolf [1964a,b]), P~ll(r) = 2P~ll(r'),
r :,(l, t. r) P[t,t+r)(rl,..., TM-1, rMIM)= fort ~< ri < . . . < r.lt-1 < Lw < t + T, 0 otherwise. (2.62) This probability density can be interpreted as that of ordered M initially independent and identically distributed random variables. Each of them would have the probability density I(1, r)/W('~')(1, t, T). Since W('~'t(1,0, ~ ) = 1, the light intensity I(1, r) is a probability density and the renormalization takes into account the restriction to the interval [t, t + T). Considering both sides of the relations (2.58) and (2.61) right multiplied by /3(0) and taking into account the trace, we see that p(M,t,T)=
..
,It
dt
P[t,t+ll(M,
r l , . . . , r~t-l, r M ) d r l . . , drM_l drM
"
(2.63) and P[t,t+T)(M, r l , . . . , TM-1, TM)= P[t,,+ri(rl,... , r.~l_l, r;~ilM)p(M,t, T),
(2.64) where the probability density is ,I
(2.65) In a slightly different context, the importance of a formula such as the relation (2.62) has been emphasized (Shapiro [1998]).
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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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2.4. BUNCHING AND ANTIBUNCHING
The bunching of counts is usually considered in terms of the coincidence probability in the dependence on some interval To between the counts (Mandel and Wolf [ 1995]). This theory is connected to stationary fields more than it can be expected, but the definitions are also relevant to more general fields, in which we are interested. Of course, the stationary fields occur regularly (Torgerson and Mandel [1997]). We shall generalize the well-known result (Srinivas and Davies [1981]) for the conditional probability c(to + T0[to) that, given that a count occurs at to, another (not necessarily next) count is registered around to + To, expressed by:
c(to + T0]to) =
Tr{),o+roGfto,,o+r,,,)to~fO, to,[~(O)} .
(2.66)
Tr {Jto ~[o, to,/3(0)) Introducing the quasidistribution q~x(a, t),
1 to) a5X q~Ar(a, to) = A[o,
v/A[a 0, to) )
(2.67)
'
we obtain that
f ]Of4q~Af(a, t0) d2a c(to + To Ito) = R(to + To)A[ro, to+ro)f Iot 2@A/.(a, t0) d2(zEvoking the fourth-order degree of eq. (2.68) in the form
coherence
(2.68)
y~'2)(tl,t2), we can rewrite
c(to + To It0) = R(to + To)A[to, ro+ro)(h)(to) y~'z)(to, to),
(2.69)
where
(h(h- ])... [ h - ( k -
1)]])(t) = f ]al2k~N(a,t)d2a, A/-(2'2)(t,
t) = (h(h - i))(t) [ (~/) (t)]2
"
k ~> 1,
(2.70) (2.71)
This conditional probability is used for the comparison of detection of an arbitrary state with the coherent state of the same mean photon number (h)(to). Obviously, this amounts to the use of the reduced fourth factorial moment
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SOME APPROACHES TO MODELING PHOTODETECTION
133
(to, to)- 1 for the comparison with zero. Since any coherent state has the
Poisson statistics of photons, a more usual means of comparison is the Fano factor,
d(t) = ((Ah)2)(t),
Ah = h - ( h )
0, at To = 0: ~~(to
[(n)(t~ y~/3I(to to), (tmax - t0)2
+rolto)
OTo
To= 0
(2.80)
with ffN.(3,3)tt (n(n -- i)(h - 2i))(t) ~,~, t) = [(~}(t)] 3 ,
(2.81)
as a particular instance of the sixth-order degree of coherence },~;'3)(tl, t2). We cannot establish a connection with the sub-Poissonian behavior (cf. Jakeman [ 1986]), because
o. 0 > ~ 0 CFock(tO +
[
Tolto) To=0
o
> ~Cc~
+
Tolto)
(2.82) 7o=O
for the Fock state In) and the coherent state [a), with ]a[ 2 = n ~> 1, but it still results in bunching. Whereas the decrease of the waiting time probability density is faster for the coherent state than that for the equivalent Fock state, so that the detection process appropriate to the Fock state is antibunched in the sense of this comparison, the analysis for small To performed just as in the case of coincidence probability here also indicates bunching.
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REVERSIBILITY AND IRREVERSIBILITY OF PHOTOCOUNTING
135
w 3. Reversibility and Irreversibility of Photocounting The status of irreversibility was doubted at the beginning of the statistical mechanics era. A maw-particle system which is closed or isolated from the external world has a reversible evolution in classical physics. On the other hand, the maw-particle system should have an irreversible evolution of gas or a solid. This apparent paradox has been cleared up long ago. The considerations of quantum statistical physics are analogous. A simple quantum system, either mechanical or optical, which is closed or isolated, has a genuinely reversible evolution. Open quantum mechanical systems frequently exhibit irreversible behavior as a consequence of interaction with their environment. Such systems are described by master equations, which do not comprise external world variables. According to this general view, master equations can be derived by reduction of a Hamiltonian dynamics describing the interaction between the system and the environment. The reduced description would be of a less fundamental nature. The common knowledge that quantum mechanics cannot be formulated as a Markov process has been discussed recently (Gillespie [1996a,b, 1997], Hardy, Home, Squires and Whitaker [1997]). A genuine irreversibility enters the physical conditions via von Neumann's projection postulate. In the Copenhagen interpretation, an external observer 'perceives' only classical eigenvalues of a quantum observable and he or she 'causes' a collapse of the quantum state onto the appropriate eigenspace of the observable. When so markedly formulated, this interpretation can be doubted, and indeed the fundamentals of quantum theory comprise alternatives (Omn~s [1994], Haus and K~irtner [1996], Hay and Peres [1998]). In this work, we adhere to the Copenhagen interpretation. Von Neumann pointed out the two fundamentally different dynamics for describing the change of a quantum state in quantum theory: The umtary continuous evolution of a closed system and the instantaneous but unpredictable projection of its state due to a measurement causing a reduction of information on the history.
3.1. S C H E M E OF S U C C E S S I V E M E A S U R E M E N T S
Basic principles of nonrelativistic quantum mechanics are formulated for state vectors and discrete observables (Ozawa [1997]). Generalizations to statistical operators and continuous observables are possible, but the projection postulate 'generalized' to continuous observables can no longer be consistent with the statistical formula.
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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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The time evolution of the system is given by the unitary transformation
,~p(t+ T)) = exp(-~ TH) ,qJ(t)),
(3.1)
where h is the reduced Planck constant and/2/is the Hamiltonian of the system, as long as it is time independent. We let ~i denote the observable under consideration and assume that it has a discrete spectral decomposition, L
= Z a~A (ak),
(3.2)
k=l
where L is either an integer or infinity, EA(ak) are projection operators and ak, k = 1 , . . . , L , are eigenvalues of A. The probability distribution of the outcome A of the measurement of the observable A in the state ]~), (~ ~) = 1, is given by the statistical formula ^
Prob{A = a} =
(%1%)= (N %),
(3.3)
where I~pa) = L'J(a) I~p),
(3.4)
with a an eigenvalue of ~i. The function L'J( ) can be continued as a projection-valued measure; i.e., it can be defined for all parts of the set { a l , . . . ,dE} if L < oc and of {al,a2,...} if L = oc, and for all Borel subsets I of the real line R. Particularly, ~'A(R) i. Such a measure also exists for a continuous observable. One need not speak of measurement of an observable and see it as an operator, but one can apply the concept of a probability operator measure (Helstrom [1976]). The projection postulate is formulated as follows (cf. Ozawa [1997]): The state change caused by the measurement of an observable ~i with the outcome a is given by the relation: I~Pla) =
Ira> v/Prob{A_ = a}
(3.5)
Note that (~plal~Pla) = 1 and the statistical formula can be used again after the system evolves in time. For a continuous observable/} we may define the operator-valued density P#(b), /2"b(b) =
lim L'b([b'b+ Ab)) Ab --, 0+ Ab '
(3.6)
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REVERSIBILITY AND IRREVERSIBILITY OF PHOTOCOUNTING
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and we obtain the probability density pk(b) = tj if there is no further measurement. We associate with this measurement a time series with the Markovian property ((l~(t)), t ~ [O, tl]),(lW__(tl)),Al),(]~p(t)),t C [tl + 0, t2]), (]lP(t2)),A2), 9 .., (IE(tJ)),~),
I~_(tj + 0))),
(3.14) where the notation t E [t/+ 0, tj+l] means that ]~p(tj-)) is replaced by ]~(tj. + 0)), j = 1 , . . . , J - 1. We speak of time series because we disregard the intervals [0, tl ], (tl, t2],..., (tj_l, tj], where unitary evolution proceeds. On the condition that the event of which the probability is given by eq. (3.11) has occurred,
I__W(t))--IhOla,.....aj(t)),
(3.15)
where I~pla,.....aj(t)) =
]~Pa,.....,,j(t)) . V/(~Pa, .....~(t)l~Po, .....~(t))
(3.16)
When the eigenvalues aj can be summed, as for example when the operators Aj are number operators, we may consider a time series: ((]~(t)), t c [0, tl]), (]~(tl)),A_l), (A_l, ]~(t)), t ~ [tl + 0, t2]), (A__l, ]~(t2)),A_2), J-1
...,
J
A_j, j=l
+ 0)>)) j=l
(3.17) This time series also has the Markovian property. 3.2. UNRAVELING THE MASTER EQUATION
In what follows, we will generalize the Srinivas-Davies model in another direction than in w2.3. Again assuming that R is time independent, we study a composite system of a single mode and a detector described by a master equation:
Ot
-L/~,
(3.18)
with an initial condition /5(t)lt= o =/5(0),
(3.19)
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REVERSIBILITY AND IRREVERSIBILITYOF PHOTOCOUNTING
139
r
where L is a superoperator having the property
Li, =
i
R(2Of~Ot_O, Of~_DO, O),
bl + 7
(3.20)
w i t h / 2 / a Hamiltonian which does not include the interaction with the detector and with 0 the operator which affects the state of the mode whenever a count is detected. The generalization to more modes and even to a composite system comprised of an electromagnetic field and an atomic system will be obvious. In the following, we assume [ / = 6 because essentially everything is clear from the case of a free field, the free-field Hamiltonian not being explicit due to the use of the interaction representation. Since a continuous measurement is under consideration, there is a back-action of the detector also between the successive counts when no other counts are detected (Srinivas and Davies [1981]). The operation of the detector can be described by the related random process M(t), oo
M(t) = Z O(t- r__j),
(3.21)
j=l
where 0 is the Heaviside unit-step function. Formally, we may treat the detector as another degree of freedom with a Hilbert space Hdet enlarging the Hilbert space H, 7-{ ~ ~-l~m = ']"{@ '}-{det, where the subscript m stands for 'modified'. Macroscopically, the detector does not admit the superposition principle (the Schr6dinger-cat paradox excluded), and we work only with the mixtures of the number states described by ]M)det det (M]. Nevertheless, it is useful to define the shift operators, 0(2)
elx'p(iq)det)
= i @ Z [M)det det(M + 1 , M=O
eIx"p(--iq)det) = [e~p(i~det)] t ,
(3.22)
but to rely only on their pairwise use, as in the characteristic relation e~p(--i~det)lM)det det(Mle'xp(iqgdet) = M + 1)det det(M + 11.
(3.23)
Let us mention the single-mode scalar product such that: det (Mlm,Mt)rn =
~MM'In).
(3.24)
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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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Now we pass from/5(0 to the statistical operator/Sm(t) and introduce the master equation 0,, 1 1 ^ (t)~tm~m] -~pm(t) = R [~)mPm(t)()tm- -~()tm ()m[gm(t)_ SPm
,
(3.25)
where (3.26)
Om = Oetx-"p(-i~det),
and the initial condition is /~m(t) t=0 = r
10)det det(0 I.
|
(3.27)
The perturbative solution to the master equation (3.25) with the initial condition (3.27) for t = T is of the form O(3 / S m ( T ) -- Z ~/[0, T ) ( M ) / ) ( 0 ) ( ~ M=0
(3.28)
IM)det d e t ( M 9
^
Here the superoperator hi0, r)(M) is defined by the property fi[0, v)(M)/5(0)=
JO JO
•
hi0, r)(M, r l , . . . , rg-1, rg)
...
(3.29)
... drg_~ drg,
where u[o, v)(M, r l , . . . , rM),b(O)= h[o, r)(M, r l , . . . ,
rM)~(O)h~o,r)(M, r l , . . . ,
rM), (3.30)
with b/[0, T ) ( M , T I , . . . ,
TM)= R M/2 exp [ - 8 9
• Oexp[- 89
rM)Ot0]
rg_,)OtO] ...Oexp(- 89 (3.31)
To get closer to the relation (2.59), we may write ^
^ r)(M, fi[o,
T1,. 9 9 , TM) =
~ ~ ...),~ S[r~,,r)JS[r,,_,,r,,)
[o, r,),
(3.32)
where the superoperator Sir,,_., r,,), having the property S[t,t+r) = S[o, r),
(3.33)
III, w 3]
REVERSIBILITY AND IRREVERSIBILITY OF PHOTOCOUNTING
141
is such that
S[o,r)p=exp(-1RTOtD)pexp(- 89
(3.34)
"k
and the superoperator J is such that
)~ = RO~O t.
(3.35)
^
We introduce h[t,t+r)(M, q , . . . , r~t) generally. Substituting {,bm(T)}
/ 3 ( T ) = Trdet
-
u[0.rt(M, rl - t , . . . , rM - t) more
(3.36)
into the left-hand sides of eqs. (3.18) and (3.19), using the achieved property (3.25) and the cyclic property of Yrdet, we derive that eq. (3.36) is the solution of eqs. (3.18) and (3.19) for t - T. Substituting eq. (3.28) into eq. (3.36), we arrive at the form /5(T) = ~[o, r)/5(0),
(3.37)
where (cf. eq. 2.47) (x)
u[0, r)(M),
~[0, r) = Z
(3.38)
M=0
and either a coarser expansion,
/,(r)
p(M, O,
=
(3.39)
M=0
or a finer one,
~(T) = Z
..
P[o,r)(M,r , , . . . , r u)/SL.,1.r~.....~,,(T)dri ... drM_, drM,
M=0
(3.40) where p(M, 0, T) is the probability of M counts being registered during the interval [0, T),
p(M, 0, T) = Tr ( ,ha4(T) },
(3.41)
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CONTINUOUS MEASUREMENTSIN QUANTUMOPTICS
[III, w 3
with ^
(3.42)
/SM(T) = u[o, r)(M)/5(0),
and P[o, r)(M, rl,..., rM) is the probability density of M counts being registered at the times rl,..., rM in the interval [0, T), (3.43)
P[o, r ) ( M , T1,... , TM) = Tr {/3M, r, ..... r,t ( T ) } ,
with ^ RM,
1"1 . . . . . TM
(3.44)
(T) = u[o, r)(M, r , , . . . , rM)/5(0).
The normalized statistical operator filM(T) describes the resulting single-mode state on the coarser condition, 1
'OIM(T) =
p(M, O,T) DM(T)'
(3.45)
and ,blM,r,.....r,(T) describes the resulting state on the finer condition, 1
PlM,rl
...., r M ( T )
"-"
(3.46)
P ,. P[o, r)(M, rl,..., rM) -M'r'
Taking into account the decomposition (3.28), we obtain coupled master ^ equations for unnormalized statistical operators ~M(T)- u[0, r)(M)~(0), 0 ^ ~-~pM(T) =
R
[
(9/3M-l(T)O* ~l b t O p M ( y ) _ -
1^ ~pM(T) ~)t O]
(3.47)
Tracing over the mode under study in eq. (3.47), using eq. (3.41) and the fact that Tr{O t
O~M(T)} = Tr{~IM(T)Ot O} p(M, 0, T) = Tr{~M(T)O t 0},
Tr{0/SM_I(T) 0t } = Tr{/)IM_I (T) 0* O}
p(M-1,0,
(3.48) (3.49)
T),
we arrive at the rate equations:
oOp(M, O,T) = -R [Tr {/51M(T) 0 t ~)} p(M, O,T)
0}
0,
(3.50)
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REVERSIBILITY ANDIRREVERSIBILITYOF PHOTOCOUNTING
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By speaking of unraveling the master equation (3.25), or more commonly, of unraveling the master equation (3.18) with (3.20), we understand introducing a random process ~m(t) such that pm(t) = E(~m(t)),
(3.51)
where E means the expectation value (ensemble average) of the quantities whose stochastic character must be taken into account. We choose a random process with an almost obvious Markovian property, and such that /~m(t) = ~(t) Q M(t))det det(M(t)l.
(3.52)
NOW, the process has simply two components, /5(t) and M_M_(t). The initial condition of the process (3.52) can be chosen deterministic, /gm(t)] t-- 0 = ,bm (0).
(3.53)
This random process is defined explicitly with respect to the component M(t), but it also implies a definition of/~(t). Let us use the transition probabilities within the time interval [t, t + At) together with their typical asymptotic behavior for small At: Prob ( M ( t + At) = MI/5(0 ) = ,b(0), M(t) = M,_r 1 = r , , . . . ,_Lv = r~t) Tr{C3M,~, ..... r,,(t + At)} Tr{/SM,r, ..... rM(t)}
= Tr{h[t,t+at)(O)~lM,r , ..... r,,(t)}
(3.54)
= 1 -RTr{/51M,r , .....rM(t) O* 0 } At + ol(At) Prob(M___(t + A t ) = M[~_(t)=/5!.,,.r, .....r,, (t), M ( ' ) = M ) , ProblM(t + At) = M + 1 [/5_(0) = ,b(0), M(t) = M, _rI = r , , . . . , _r;,.t = r @ m
f,+at Tr{/SM+l,r, l
"'" ' TM +
1
(t + At)} dr,,+,
Tr {,bM,r, .....r,, (t)} = Tr
{it+At ,,
h[t,t+at)(1, rM+l)dr~t+l/51.~t.~ 1..... ~,,(t)
}
,It
= RTr{/)IM,r , .....r,,(t) O* O}At + o2(At) Prob ( M ( , + A t ) = M + ll/5_(t) = r
.....r,, (t), M ( ' ) = M ) ,
(3.55)
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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
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Prob = o3(At), (3.56) where oj(At), j = 1,2, 3, have the property that o j ( A t ) / A t ~ 0 for At ~ 0+. From this it follows that we define a random process with the Markovian property with respect to both of the components. To complete the definition of the Markovian process, we must define /5(t + At) on the condition that ,b(t) =/51M,~, .....r~ (t) and M ( t + At) - M(t) = 0" ^
/5(t + At) = -
h[t't+At)(O)plM'r' .....r,,(t)
,
(3.57)
Tr{u[t, t+At)(O) [~lM,r, .....r,, (t)}
and we must define /5(t + At) on the condition that /5(t) = /51M,r, .....r~,(t), M ( t + A t ) - M(t) - 1, and that _r = r, where _r is a continuous-time random variable such that M(__r + O) - M ( _ r - O) - 1" ^
/5(t + At) =
--
h[t, t+At,(1, r)/SiM,r , .....r,, (t)
(3.58)
Tr{u[t,t+At)(1, r),biM.r , .....r,,(t)}'
the fight-hand side being independent of r. We need not define /5(t + At) on the condition that M ( t + A t ) - M(t) ~ 0, 1, because this event has a negligible probability. On the condition that M ( t + T) = M and still M(rj + 0 ) M ( r j - 0 ) = 1, j = 1,... ,M, we have the deterministic value /5(t + T) =/51M,r , .....~, (t + T).
(3.59)
For the operator 0 chosen appropriately, e.g., ( ) = h (Ueda, Imoto and Ogawa [1990a]) and 0 - h t (Ueda and Kitagawa [1992]), it holds that (3.60)
/'lM.~. .....~.(T) =/,IM(T);
i.e., the result does not depend on the times at which the counts have been registered. In this case, the 'rate equation' (3.50) is a genuine one for the sodefined Markov process. The superoperator (3.32) simplifies to ^
^
h[o, r)(M, r l , . . . , rM) = P[0, r ) ( r l , . . . , rM [M) hE0' r)(M),
(3.61)
with an appropriate conditional probability density P[0, r ) ( r l , . . . , rM M); cf. eq. (2.61). The nonunitary operator (3.31) has a similar form" h[o, r)(M, r l , . . . , rM) = v/P[o, T ) ( r l , . . . , rM IM) h~o,v)(M),
(3.62)
III, w 3]
REVERSIBILITY AND IRREVERSIBILITYOF PHOTOCOUNTING
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where the nonunitary operator h[o. T)(M) is defined implicitly by this relation, and u[0, T)(M) /5 = fit0, T)(M) ~h~o' rl (M).
(3.63)
In the situation where the initial statistical operator/5(0) corresponds to a pure state I~p(0)), the random process/5(t) remains pure for all times. This means that there exists a random process I~p(t)) such that
iS(t) = I,p(t))(,p(t)l.
(3.64)
Indeed, because ,blM,r, .....r~,(t)simplifies as
I~,,,,
.....r,,(t))(~Pl.,,.,, .....r,,(t), the relation (3.57)
h[,,,+A,~(0)
Iq,(t+At))=
tpl.~,t.r , ..... r,,(t))
,
(3.65)
V/(IPlM,r, .....,,,(t)lh* [,.,+~,t(0) "I,.,+~,t(0) ~PI.~.,, ..... ,,,(t)) the relation (3.58) simplifies as
~[,.,+A,,(1, r)I~'t~l.,, ..... ,,,(t))
I q,(t + At)) = V/
{VJiM,,, ..... ,,,(t) I~ [,.,+Ati(1, t r) u[,.,A,)(1, T) q~l.~1.,,.....,,,(t)) (3.66)
and
I,p(t + At)) ~
Ol~,l~,,,,, _
V/(IPlM.r,
,,,(t)> .....
..... r,,
(3.67)
(t)10* 01,j,~,,.,, .....,,, (t))
In the foregoing exposition, we have touched on the essential concepts from work by Ueda, Imoto and Ogawa [1990a], Carmichael [1993a] and Knight and Garraway [ 1994], deviating slightly from their notation. It has been confirmed for - ~ that the total entropy of the cavity photons and the photodetector does not decrease, and the entropy change and information gain in the photon-counting measurements have been investigated in detail (Ban [1997a]). The master equation (3.18) with (3.20) is frequently referred to as the master equation in the Lindblad form (Lindblad [1976]), which is however much more general, since it contains several Lindblad operators O/. Holland [1998] has considered unraveling statistical operator evolution in the frequency domain and continuously sweeping between use of the time and frequency domains. He has presented a master equation in the Lindblad form, but he let the number of the
146
CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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Lindblad operators depend on the size of a cascaded array of filters. Jack, Collett and Walls [ 1999a] have presented a general form for non-Markovian quantum trajectories, focusing on the trajectories that simulate real-time spectral detection of the light from a localized system. 3.3. L O G I C A L R E V E R S I B I L I T Y
IN Q U A N T U M
MEASUREMENT
We will call a quantum measurement process logically reversible if there exists an operator b(0,rj(M, rM,..., r~) such that b(0,rl(M, rM,..., rl)u[0, r)(M, r l , . . . , rM)= i.
(3.68)
This new operator is the left inverse of the operator ~[0, r)(M, r l , . . . , I'M), which is nonunitary. The concept of logical reversibility was introduced by Ueda and Kitagawa [1992] and analyzed by Ueda, Imoto and Nagaoka [1996], who have also given a general condition for physical reversibility. If the left inverse exists, we can arrive at the initial value of the quantum trajectory from its actual value, ]~p(0)) =
-, ~ T r { (E(t) I_bt__bE(t)) }
(3.69)
where b_V_- b(o,rl(M(t), _rM~t),..., __r1).
(3.70)
The choice O - h corresponds to the usual photon counter (Srinivas and Davies [1981 ]), and O - h t to the quantum counter (Mandel [1966]). The case of the photon counter has been treated in w2, and formulas (2.29) and (2.34) read, for t = 0, [ W-(Ar)t~, ,0, T)]Mexp[-Wr176
p(pc)(M, O, T) = ~
0, T)] q)Ar(a) d2a, (3.71)
where ~(Ac)(Ot pc ~. , 0, T) =
lalZ[1 -
exp(-RT)].
(3.72)
For the photon counter, the conditional probability density (2.62) and the implicit one in eq. (3.61) reads: MW l 9(pc) [ ~ [ 0 , T) ~ . r l ' " 9 9 , "/'M IM )
exp -R
1 - exp(-RT)
" =
rj j= 1
for
0
0~
0, is in the ground state initially; i.e.,
0 can be determined from
III, w 6]
QUASICONTINUOUS DETE('TION SCHEMES
173
I(a, 0).
Ueda [1989b, 1990] assumes 'existence' of the functional T'[r/I(t)] with the property = fI cI)~:(a,O)A4[rlI(a, t)] d e a,
(5.28)
J
while we refer to the original papers for the functional 7)[r/I(t)].
w 6. Quasicontinuous Schemes of Photodetection Ban [1994] proposed a quasicontinuous measurement with lossless beam splitters and photodetectors. Futhermore, he considered a measurement by means of a parametric amplifier and a photodetector and a measurement in terms of a four-wave mixer and a photodetector. He discussed the relationships to the continuous destructive quantum measurement of photon number, the continuous measurement of photon number with a quantum counter, and the continuous quantum nondemolition measurement of photon number. 6.1. MEASUREMENT WITH BEAM SPLITTERS
In the Heisenberg picture, the beam splitter is characterized by the relation a(1) = ta(O) + r'b(O),
b(1) = rb(O)+ t'b(O),
(6.1)
where h(0), at(0) and b(0), bt(0) are the annihilation and creation operators of the input signal and reference modes, and a(1), at (1) and b(1), Dt (1) are those of the output modes. The coefficients t, r, r', t ' have the properties Itl 2 + Ir[ 2 = 1,
Ir'l 2 + It'] 2 = 1,
tr'*
+ rt'*
= O.
(6.2)
Here we assume that t ' = t > 0, since phase shifts are not important for our purpose. The transmittance 7- and the reflectance 7-s of the beam splitter are given by = itl2,
j~
=
ir 2.
(6.3)
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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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In the following, we assume that the input state of the reference mode is the vacuum state. In the Schr6dinger picture, the output state/3(1) of the beam splitter is given by /3(1) = V'[p(a)(0) @ [O)b b(Ol]V t,
(6.4)
~"
(6.5)
= exp(~]+ - ~*J_),
where ,b(~)(O) is the statistical operator of the signal mode input and [O)bb (0 is the statistical operator of the reference mode input. In eq. (6.5), r
- - ]-~ arccos t
(6.6)
and J + and J0 are the generators of the SU(2) Lie algebra given by J+ =
;Ttb,
hb* ,
J_ =
Jo = :'(a*a -
(6.7)
b'b),
which satisfythe commutation relations [J+,J_] = 2Jo and [fro,J+] = +J• Here we use the simplified notation h - h(0), h t - ht(0), and t) - b(0), ~)t - Dr(0). The relation (6.4) reflects the well-known fact that a lossless beam splitter is characterized by the SU(2) Lie algebra (Campos, Saleh and Teich [1989]). If the photodetector for the output of the reference mode measures photons, we obtain that (6.8)
,bm(1 + 0) =/5(~)(1 + 0) | ]m)h b(ml, where
p(ma)(1+0)-
s189189 m!
(6.9)
Now we introduce the superoperators k+, k0 as ~+ll~/= ~t)l~/~,
~ _ ~ / = ~/t~ t,
~0)1~/=~ t ~ / + ll~/~t,
(6.10)
for an arbitrary operator l~/ (Pefinovfi, Luk~ and K~epelka [1996b]). The superoperators (6.10) obey the commutation relations [k_,k+] [~0,~+] = +2k+. Then r
=
k0 and
+ 0)is expressed as follows"
/5~)(I+0)= u(T, m)/5(m(0),
(6.11)
where u(T, m) = (1 - T)m T89 m! The normalized output state of the signal mode is given by
~)(a) O) Im (1 + =
u(T' m) p(a)(o)
Tra{U(T, m) f)(a)(O)}
(6.12)
(6.13)
where Tra is the trace operator over the Hilbert space of the signal mode.
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175
It is interesting to remark that the output state (6.9) (or 6.11) is equal to the state of the light beam, after a photon counter registers M - m photons, in the continuous measurement of photon number described by the quantum Markov process. When we do not refer to the result of the measurement on the output of the reference mode, the output state of the signal mode becomes 0(3
/5(a)(1 + O) = Z/5(ma)(o)'
(6.14)
m----O
= T 89(i'~ i )-Z75(")(0),
(6.15)
where Tra/~(a)(1 + 0 ) = Tr,/5(")(0)= 1 is satisfied. It is easily seen that this state is merely attenuated (Pe~inovfi, Lukg and K~epelka [1996b]). Using the statistical operator (6.13) of the signal mode output, we can calculate the mean (hth)lm, the second-order moment of the photon number ((hth)2)l,,,, and the Fano factor decreased by unity (Mandel [1979]):
fm
((hth)2>lm- m), we get (Ban [1994])
Up
(hth)l o = ( N - m) _---L-~p, 1
((h*h)2)l m = (N - m) 1 - ~ p
(N - m - 1) 1 - ~ p +1
(6.20)
~rp fro--
1--~p"
We see sub-Poissonian photon statistics from f,,, < O. (iii) The coherent state. When the input state is a coherent state, such that /5(~ = la)(a[ and n0 = ]a 2, we obtain
(hth)l., = (~t~)to = T-~o, (6.21)
((hth)2)lm = 7-~o(7"ffo + 1), fm = 0.
(iv) The thermal state. When the input signal mode is in the thermal state, oo
,o(a)(o) = Z p ( n )
n)(nl,
(6.22)
n=0
where --//
p(n) =
no
(1 + n0)n+l
(6.23) '
with if0 the mean photon number, we get the results (hth)l, , = ( m + l ) ( h t h ) l 0 = (m + 1)T~o
((hth)2)lm =
1 + 7~-no
(m + 1)Tff0 l+Rff0 '
(m + 2)'T~o
1
1 + 7~-no + lj ,
(6.24)
T-~o fm = 1 + ~-~0 We have super-Poissonian photon statistics (f,,, > 0). We find bunching correlation of photon numbers in the thermal state, since the mean photon number in the output state of the signal mode increases if the photodetector for the output of the reference mode registers the photons, (htgt)l m >1 (hth)l o. It is a possible definition of this property. We find
III,w 6]
QUASICONTINUOUS DETECTION SCHEMES
177
antibunching correlation of photon numbers in the binomial and number states, since the mean photon number in the output state decreases if the photodetector registers the photons, (gttgt)l,, , 0. The quasidistribution related to the antinormal ordering reads 4~A(a, T)lr=o
=
exp(-[a]2) { [~ exp Re(a 2) - 2fi Re a :r~t g
(~t - v) fi2 g
]} ' (7.2)
where kt = cosh r,
v = sinh r.
(7.3)
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183
S P E C I A L S T A T E S OF AN O P T I C A L M O D E
They introduced the normalization constants
ZM(T)=2MOM[411 /exp ( -
(1
+y)~tv
(7.4)
'
particularly 1
Zo(T) =
V/1 _y2 exp (1
(7.5)
+y)Itv '
where
1/
y = ~ exp(-RT).
(7.6)
While no photons are detected (M = 0; no-count process), the q~A representation evolves as exp(-la[ 2) 4'A(a, rio) = JrZo(r)
x exp {- [Y~exp(-RT) Re( a 2) 2fi ~
exp(- 51RT) Re a]
}
(7.7) However, as soon as one photon is detected (M = 1), the q~A representation changes abruptly into v z0(r) q~a(a, T[1) = 2 - ~
12Zl(T)
• {Iexp(-~1RT)R e a - f iv
a.
+ [exp(- 89
a] 2 q~A(a, T 0).
(7.8) The subsequent one-count event further emphasizes the two-peaked character of the q~A representation, which is given by q'A(a, TI2) Zo(r) [ v2 /32 = 4Z2(T) / 1 + 1/314~ exp(-2RT) + 4--1t: exp(-RT)
(
+ 2 fi_2_2 /32 /*v ~ - 1 + 4/3 exp(-1RT)
la
) ( +2
E 1
fi2 ~-1
Itv
~exp(-RT)Re(a 2) v exp(-RT)
~t
a2
Re a
}
~A(a, T 0).
(7.9) If the quantum counter is used for measuring an initial vacuum state, conditional states are reminiscent of photon-added thermal states. Although
184
CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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such states need exhibit neither squeezing nor sub-Poissonian statistics, their nonclassical character can be tested using a quantitative criterion developed by Agarwal and Tara [1992]. Agarwal, Puri and Singh [1997] studied the properties of wave-packet states with vortex structure. The quantum system can be a two-dimensional harmonic oscillator or a two-mode radiation field. They demonstrated the generation of such states by the interaction of A systems with squeezed radiation in a twomode cavity. The state of each mode is mixed, even though Agarwal, Puri and Singh [ 1997] define the state of the two-mode field as a pure state. With 0 = 1 (h + e-iqb), we have an essentially one-mode measurement, which is interesting since it influences two modes. Agarwal, Graf, Orszag, Scully and Walther [1994] examined the cases of an initial two-mode coherent state la)al/3)b or the product of a coherent state with a number state ]a)a]N)b. They paid attention to the mode b and assumed ]al 2 = ]/3]2 or ]a] 2 = N initially. They found that as the counting interval increases, bimodal quasidistributions evolve for the mode b starting from an initial number state. The phase distribution (Pegg and Barnett [1988, 1989a,b]) first narrows and then bifurcates. The state of the separate mode described by the statistical operator /~(b) IM = Tra {/51M} (7.1 0) is mixed, and a numerical calculation has shown that /~(b)
1
IM = 5(IA+)(A+[ +
A_)(A_ ),
(7.11)
where IA+) are states which have bimodal (/)A quasidistribution. Ban [ 1995] investigated photon statistics and quadrature squeezing in the even and odd coherent states (Klauder and Skagerstam [ 1985]) and the Yurke-Stoler state (Yurke and Stoler [ 1986]) under the influence of a continuous measurement of photon number. He found that the photon statistics of the cavity mode oscillate between sub-Poissonian and super-Poissonian distributions, and that the quadrature fluctuation oscillates between squeezing and nonsqueezing at each time when one photon of a cavity mode is registered by a counter. He has also treated a 'phase cat' (Schleich, Pernigo and Le Kien [1991]): [~p) = - ~1 ([ae i~c/2) + [ae-'q~/2 )),
(7.12)
where a is a positive real parameter and the normalization constant Z becomes
{1 + exp
cos, o 2 sin
},
to show the statistical oscillation in the continuous measurement of photon number. These oscillatory behaviors have been related to the oscillation in the
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PRODUCTIONOFCORRELATEDPHOTONS
185
photon-number distribution, (cf. Ogawa, Ueda and Imoto [1991 b]). A possibility of taking into account an imperfection of the photon counting is provided by the hybrid model of photon-counting measurement proposed by Lee [1994]. In this model, there are some photons taken away from the cavity without being registered by the counter. When we use the hybrid model, the results are to be modified. Cirac, Gardiner, Naraschewski and Zoller [1996] used continuous measurement theory to describe the evolution of two Bose condensates in an interference experiment. They have shown how the system evolves into a state with a fixed relative phase. The quasicontinuous schemes of photodetection should be completed by a connection to the generation of special states. Ban [1997b] derived the equivalence relation between a lossless beam splitter and a nondegenerate parametric amplifier with respect to the conditional output state. He found that, passing from the beam splitter to the nondegenerate parametric amplifier, the roles of the input state and the postmeasurement state of the reference mode are exchanged. In more detail, we assume that the input state is I~p)~,~,(~p[ and the vacuum state is found in the output port of the reference mode. The equivalent input state of the reference mode of the nondegenerate parametric amplifier is the vacuum state, and the equivalent postmeasurement state of this mode is with oo
I~p*)b = Z(h(nl~P)b)* [n)h.
(7.14)
n=O
Photon-counting measurement, heterodyne detection, and homodyne detection are considered as the conditional quantum measurements. Although a mismatch factor e -i'~-~2f in the effective interaction Hamiltonian (6.34), with g replaced by ge -i'~21, leads to the inhibition of emission, it cancels with its complex conjugate in the limit of continuous measurement with O(t) = e-i'~th t. It indicates an enhancement of the emission in the scheme analyzed by Luis and Sfinchez-Soto [1998a]. This scheme comprises an instantaneous, yet destructive, measurement.
w 8. Production of Correlated Photons We proceed with correlated photons in nonlinear optical processes. The correlated photons are produced in a variety of optical devices such as parametric
186
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
[III, w8
amplifiers, four-wave mixers, and down-converters. The photons produced in a nonlinear optical process such as four-wave mixing or parametric downconversion are known to have unusual correlation properties which result in many nonclassical aspects of the radiation field. Agarwal [ 1990] noted that the amount of correlation depends on the nature of the nonlinear process, the strength of the pumping field, the losses in the medium, and the transmission from the mirror if nonlinear processes in cavities are considered. The correlated photons have been used in a number of studies on nonclassical aspects of light, including questions like the Bell inequalities. Correlation between the signal and readout modes is present in the optical back-action evasion (La Porta, Slusher and Yurke [1989], Song, Caves and Yurke [1990]) using a single-pass parametric amplifier (parametric down-conversion process). Smith, Collett and Walls [1993] have considered quantum nondemolition measurements with a degenerate parametric amplifier driven by a classical pump field, which is able to provide good quantum nondemolition correlations. A beam splitter is one of the key optical devices in quantum optical experiments and quantum communication systems, which is mathematically equivalent to the Mach-Zehnder interferometer and linear directional coupler. The input-output relationships of these devices are characterized conveniently in the language of the SU(2) Lie group when dissipation can be ignored (Yurke, McCall and Klauder [1986], Prasad, Scully and Martienssen [1987], Ou, Hong and Mandel [1987], Fearn and Loudon [1987], Huttner and Ben-Aryeh [1988b], Campos, Saleh and Teich [1989], Janszky, Sibilia and Bertolotti [1991], Lai, Bu~ek and Knight [1991], Leonhardt [1993], Luis and Sfinchez-Soto [1995]). In optical signal detections such as the homodyne and heterodyne detections, a beam splitter is used to mix a signal mode with a local oscillator mode. It is also indispensable for interferometric experiments. In some cases the signal mode is divided into two parts by the beam splitter, and the two output modes of the beam splitter are correlated to each other (Ban [ 1996a]). Some authors speak of an 'anticorrelation' since an increase of the photon number in one output mode is equivalent to a decrease of the photon number in the other. Ban [1996b] has explained the model of a degenerate four-wave mixer. The photon number of the signal mode is conserved in this model, and only the phase of this mode changes. The rate of phase shift is given by the positionlike quadrature of the reference mode. The momentum-like quadrature of the reference mode shifts and the rate of quantum shift is given by the photon number of the signal mode. The initial state of the reference mode is vacuum so that the change of the phase of the signal mode is slow at the beginning. The process produces correlation between the output modes.
III, w8]
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187
Joobeur, Saleh and Teich [1994] developed a theory for the second- and fourthorder spatiotemporal coherence properties of spontaneous parametrically downconverted light, assuming a crystal of finite length and a pump of finite spectral width. Joobeur, Saleh, Larchuk and Teich [1996] extended this theory to include the effect of pump beam waist (or equivalently, of the pump transverse width). Kono, Koashi, Hirano and Matsuoka [1996] performed correlation measurements of the outputs of a nondegenerate optical parametric amplifier with a weak coherent input field. By varying the relative phase between the input and pump fields, they observed positive and negative correlations between the signal and idler photons. The cross-correlation function between the signal and idler photons is g~2) = (as,
ailb~bsb~bilas, ai>
(8.~)
n,
cbs(a, ln) (ml) =
[q~s(a, In)(nl)]*
for m < n. (9.96)
In case (c), the eigenstate is given by the finite expansion M
Po
=
M
Z
(9.97)
Z c,c,*, n)(m],
n=0 m=0
where c, - c,,(-~o,M) are expressed as follows:
Cn --
I
1 O" q72 VLM(-'~
M v (7o)u-'' (M-n)!
0
f o r n ~
M.
We obtain the Wigner function M
9 so(a) = Z
M
Z c,c~,-~s(a, In)(ml).
(9.99)
n=0 m=0
The general form of the Wigner function is
9 s(ot) = qbso(e-ir(a- y)) M
=Z
M
Z
exp(inr)c,[exp(imr) c,,,]* ~s(a - y, n)(ml).
(9.100)
n=0 m=0
Here exp(ir) is given in eq. (9.84). It is well known that the distributions of the generalized position and momentum operators are given as marginal integrals of the Wigner function (Beck, Smithey and Raymer [1993]), and the distributions of quadrature operators Q(0),/3(0) are linked to them by a simple rescaling. Nevertheless, Luke, Pefinovfi and I~epelka [1994] derived the formulas for the distributions
206
CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
[III, w 9
of these quadratures using the technique to be described in w11, with the choice r = 0, r/ > 0 (which corresponds to )' and 7 real and such that-7 < Y). The distribution of quadrature operator Q(0) is
~Q(Q)= (QIPIQ)=I(Q W) I2 , where I~p)= Y'~'~n~oCnln).Here, IQ) have the properties
(9.101)
Q(O)IQ) = QIQ),
(9.102)
(Q'IQ)=6(Q'-Q).
(9.103)
and
Using the formula
(Q]M)=HM(---~2) [2MMI!x/~exp (-Q-~2) ] ~/2
(9.104)
with the Hermite polynomials [~]
gM(x) = ~
M~
k ! ( M - 2k)! (--llk(2x)M-2k'
k=O
(9.105)
M forM even, M5---2for M odd, we derive the quadrature representation (Q]~p): (Q]V) = exp(-r/2) [L~
•
-1/2 exp[r/(Q - 2~)]
~
x/2 J [2MM 1,-x( Q / ~-e2x~p)(2 .
2
)] 1/2 (9.106)
and q~Q(Q) = exp(-2r/2) [L~ •
x/~] -1 exp[2r/(Q - 2~)]
(Q-2-~)21 [HM(Q-2-~2 ~
..)] ,2
(9.107)
III, w 9]
CONDITIONAL GENERATION OF SPECIAL STATES
207
with -
l yl z- lYl z
Ix z + 2 y ~ + 171 z
rt = 21y + Y*I'
rl =
2ly +-Y*I
(9.108)
'
the parameter ~ is real and rl is a complex number generally. Similarly, the distribution of the quadrature operator/5(0) is 05~(P) :
(PIPlP):
(9.109)
I(PIW)I z ,
where [P) are the quadrature eigenstates, P(O)IP) =
PIP),
(9.11 O)
normalized as (P'IP) = 6(P'-
n).
(9.111)
With the aid of the formula p2
(P[M)=(--i,MHM(~2) ( 2MMIv/~_~ exp (--~-) ]
1/2 ,
(9.112,
we get the quadrature representation (PIW)" (P ~p) = exp(-r/2)[L~
-1/2 exp(-i~P)
x (--i)MHM( P +i2r/X/~) [2MM}x/~-~ exp( (P +i2r/)22
)] 1/2
p
(9.113) and q~(P) = [L~
exp (-~-~2) (9.114)
• HM
X/~
Let us note that the distribution (9.114) does not depend on the parameter r/(r/ real), which can be easily related to the same property of the integral expression q~(P) = ~1 / ~
qss ( 2x )+ i
dx.
(9.115)
O(3
A shifting property of the parameter ~ is obvious from eq. (9.107). Purely imaginary displacement from eq. (9.114) has its counterpart in eq. (9.107) as
208
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
[III, w9
an exponential factor exp[2r/(Q- 2~)] amplifying on the right (r/ > 0) and attenuating on the left. In general, the formulas (9.107) and (9.114) can be applied to quadrature components Q(r),/5(r) ((9.83)), which are, as can be seen easily, the principal quadrature operators (LukL Pe?inovfi and Pe?ina [1988]). The condition r = 0 can be reached by using a suitable phase shifter in the local oscillator beam. Nevertheless, r/is complex generally, and we thus use the formula (9.114), with the modification r / + Re r/and with a displacement by 2Im r/. Let us note that the quadrature variances used to measure the amount of squeezing are invariant in displacement. The squeezing of vacuum fluctuations occurs in the principal component with the variance ((AQ(r)) 2) for suitable values of the parameters 7, ~ and each M > 0. No squeezing is possible in the quadrature ~b(r) (Pe?inovfi, Luk~ and K~epelka [1994]). We will investigate the unitary dynamical time evolution followed by the measurement process on the idler mode. For the description of this evolution, three quantities are used and are important; namely, --9*(t), al(t), a2(t), given by the relations (9.81) and (9.95). For chosen initial complex amplitudes and the outcome M of the photon-number measurement, we see the motion of these points in fig. 1. The maximum al(t) moves from ~1(0) over an arc to the point
(9.116)
The saddle point a2(t) moves from infinity (and from low probability) to a2(oo) = - a l (cx~).
(9.117)
The minimum o f - 7 * (t) moves from infinity to the origin over a ray. At an equal time the three points are collinear. Straight-line segments are plotted to express the time dependence of connected points. The line segment linking the points al (t), a2(t) may play the role of the diameter of a three-dimensional plot of the q).a quasidistribution. In the unitary dynamics, the time evolution is reflected in the increasing correlation between the signal and idler modes. Whenever the measurement process closes the evolution, the amount of correlation causes leakage of the properties of the measurement outcome (the resulting Fock state of the idler mode) to the signal mode. However, we assume the Fock state ]M) to be the outcome, so that for suitable times its phase uncertainty manifests itself in
III, w 9]
CONDITIONAL GENERATION OF SPECIAL STATES
209
10-
Ins
8 (o) -2-4
I
i
I
'
I
~
~
'
I
-4
Re Fig. 1. The evolution o f the quantities - y * ( t ) , al(t) and a2!t) for t E [10 -2, 10] and the sweep o f the crescent axes for times 9 = exp{ln 10 -2 + ( l n 1 0 - I n 1 0 - 2 ) ~ } , j = 13 . . . . . 20, for ~1(0) = 6, qJ1 = 0, [_~2(0) = 6, g[ = 1, lp = 0, and M = 10.
the signal mode, and the parametric down-conversion can evolve to very strong correlations. Thus, for very large times the signal mode possesses an identical copy of the number state IM). To this extent the photon number of the signal mode is modulated for the purposes of quantum communication. On the contrary, for t = 0, when the unitary dynamics is not present, the measurement process does not affect the coherent state of the signal mode. For suitable times crescent states result and the straight lines represent their axes. In figs. 2-4 we present the quasidistributions q~.a(a, t) describing the reduced state. Figure 2 demonstrates a more-or-less coherent state for t = 0.001. At time t-- 3.5, a weak correlation causes an annulus to arise. An increasing correlation in fig. 3 makes the annulus more pronounced, with a well-defined saddle in the superstructure graph for t = 4. Figure 4 (t = 10) corresponds to a Fock state with undefined saddle point. For t = 0.001, it holds t h a t - ~ * ( t ) ,~ 6000i, a l ( t ) ~ 6.0, and a2(t) ~ 6000i, and the points al(t), a2(t) fall outside the figure, although already for t = 3.5 they are contained in the figure. The appropriate Wigner functions easily demonstrate the time reversal. For t = 10, we almost obtain the familiar picture of the number state with two
210
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
[III, w 9
Fig. 3. As fig. 2, but for t = 4.
local maxima presented in fig. 5, one corresponding to the flat m a x i m u m of q ~ t ( a , t ) in fig. 4 and the other situated at the origin. In fig. 6 (t = 4), the local m a x i m u m at the origin becomes weaker and the m a x i m u m preserved in the quasidistribution q~A(a,t) increases. The oscillations connected with the
III, w 9]
CONDITIONALGENERATIONOF SPECIALSTATES
Fig. 5. The quasidistribution ~s(a,t) for I_~r 6, ~1 and t = 10.
= 0,
211
1_~2(0)1= 6, Igl = 1, ~p - 0, M = 10,
occurrence of the negative quasiprobabilities survive persistently, and only for t = 0.001 does the Gaussian form prevail over nonclassical properties (fig. 7). In the following, we will demonstrate the distributions of quadrature operators by applying eqs. (9.107) and (9.114). With the special choice o f initial phases of qg~ = 0, ~p = - ~1 Jr, we achieve that the principal quadratures coincide with the basic ones for all interaction times. Similarly, as in the previous considerations,
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Fig. 7. As fig. 5, but for t = 0.001. we obtain the result that the variance of the position-like quadrature can be squeezed, whereas that of the momentum-like quadrature cannot. In fig. 8, the squeeze of the position-like quadrature variance manifests itself with the crest followed by branching into oscillations of the near-number states. The Gaussian shape applies for near-coherent states. The oscillations in the Q-quadrature distribution correspond to the bimodality in the P-quadrature distribution and they replace the squeezing when the promontories in the q~s quasidistribution begin to manifest themselves and, eventually, the kettle
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CONDITIONALGENERATIONOF SPECIALSTATES
213
Fig. 9. Another view of the Q-quadrature distribution from fig. 8. closes. This bifurcation is analogous to the situation with number and quantum phase in the ordinary squeezed state (Schleich and Wheeler [1987]). From another viewpoint, we can distinguish in fig. 9 the effect of a symmetry related to the quasidistribution q~A(a, t) of crescent contour lines. Here we do not emphasize the marginal property of the Wigner function, because for moderate times the oscillations are not present, although the crest has disappeared and the peak has fallen. Of course, the study of squeezing based on using all of the
214
CONTINUOUSMEASUREMENTS|N QUANTUMOPTICS
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Fig. 10. The P-quadrature distribution for ~l(0)[ = 20, q~l = 0, ~2(0) = 6, g = 1, ~p= -~, and M=10. quadrature distribution is not so abstract and simple as a plot of the variance only. The momentum-like quadrature squeezing is not present, as can be seen in fig. 10, where the Gaussian shape evolves into the Hermitian oscillations without going through a stage of crest formation. Moreover, the distribution is symmetric regardless of special initial conditions. The quasidistributions of the complex amplitudes provide not only distributions of the quadratures, but also those of the optical phase. Along with such phase distributions, the canonical phase distribution ranks, which is derived more straightforwardly from the number-state representation (Pefinovfi, Luk~ and ~(1) Pefina [1998]). The reduced statistical operator PlM = ]~PlM)(~PlM , with the vector [~plM) --[lPlM(t)), has the number-state representation
C.IM -- C.IM(t)= (nl~PlM(t)).
(9.118)
The optical phase is multivalued, in principle, which is reflected in the 2:r periodicity of the following 'raw' phase distribution:
1 oc Praw(q~'tlM't) = ~
Z
[2 exp(-inq~)c"lM(t)
"
(9.119)
n=0
One of the phase characteristics is the time-dependent preferred phase ~,
- ~ - -~(t[M,t)- arg [(etx"p(iq~))(t)],
(9.120)
where argz = Im(lnz) and e~p(iq~) is an exponential phase operator of Susskind and Glogower (Susskind and Glogower [1964], Lukg and Pe~inovfi [ 1991 ], Luke,
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CONDITIONALGENERATIONOFSPECIALSTATES
215
Pefinovfi and Kfepelka [1992]). The expectation value ( ) has been computed according to the formula o~
(etx"p(iq~)) = Z
C~'IM(t) cn+'IM(t)"
(9.121)
n=O
This 'raw' preferred phase is multivalued, in principle, as an example shows, in which the comb of the peaks in the raw phase distribution determines a multivalued preferred phase. The peaks have a period of 2r they do not provide a single value. Therefore, we confine the preferred phase ~ to the interval [-r r Having a single-valued preferred phase ~ - ~_~ at our disposal, we generate nonperiodic phase distributions: P(99, tiM, t) = ~ Praw(qg' tiM, t) for ~ C [~ - Jr, ~ + Jr), otherwise. / 0
(9 122) "
Because the preferred phase is time dependent, we are led to the continuous multivaluedness of functions. The graph of a continuous multivalued function consists of many continuous component curves preserving a vertical spacing. Here the graph preserves 2Jr vertical spacing. Of course, the restriction to the interval [-Jr, zr) should be required only for t = 0, and everything else follows from the continuity principle. We name the function defined by this connected component of the graph ~cont- The phase distribution (9.122) has not been produced only formally, because this distribution is possessed by a physically important random phase-valued variable confined to the interval [ ~ - Jz, 99 + ~). The phase distribution (9.122) may be interpreted as^ the distribution of eigenvalues of an orthogonally generated phase operator ~ _ ~ (Lukg and Pefinovfi [1994]). But the preferred phase can be subtracted from these eigenvalues and the appropriate distribution becomes
P(cp,-~, tiM, t) = P(q) + -~, t M, t).
(9.123)
Quantum-mechanically, the phase distribution (9.123) should be understood as that of eigenvalues of the rotated phase operator ~_~(~), ~-:r(~) = exp(i~h) ~_~ exp(-i~h) = q)-r - ~ i ;
(9.124)
the second relation was established by Luk~ and Pefinovfi [ 1993] when studying s-phase formalisms.
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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
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Fig. 11. Dependence of the preferred phase on the measured photon number for ~i (0) = 6, ~2(0) = 6, Igl- 1, and arg g = 0.
The dynamics of the preferred phase is obvious from fig. 11, which comprises the arguments of mean values of the exponential phase operator. For t = O, the preferred phase independent of M corresponds only to that of the signal mode. It emerges that, for the mode entanglement increasing with t, the dependence on the measured photon number manifests the initial phase of the idler mode, which is not measured, but enters the result via the optical parametric process. Particularly for very large t, when the determination of the preferred phase is mostly the matter of an exact computation, it is demonstrated that in the large-M limit the preferred phase tends to arg(-~* (e~)). In fig. 12, the conditional phase distribution P(qJ, t[M, t) is plotted, which is located at q0 = 0 (mod 2:0 for t = 0 and which for increasing t moves at first very quickly towards negative values of q~. (The t discretization manifests itself in the cock crest.) The diffusion of the phase also continues successively, which seems to moderate the decrease of the preferred phase, so that it does not reach the value arg (~*(~)). The phase distribution flattens ultimately. The transformation, which makes the preferred phase zero, leads to the centered distributions P(cp,-~, t]M, t) plotted in fig. 13. This picture can be compared with fig. 12. The similarity is enhanced by the phase diffusion, which dominates over
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CONDITIONALGENERATIONOF SPECIALSTATES
217
Fig. 13. Evolution of the phase distribution P(q~, ~,t M,t), q~ E [-:r, Jr), related to the rotated phase operator ~9-~_~(~) for _~1(0) = 6, ~2(0) = 6, g = 1, arg g - 0, and M = 10.
the shift o f the p r e f e r r e d phase. T h e effect o f the shift o f the p h a s e distribution is e l i m i n a t e d a n d the p h a s e diffusion m a y be m o r e p e r s p i c u o u s .
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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
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9.2. ENTANGLEMENTVIA OTHER TECHNIQUES Ban [ 1997c] has shown that a nondegenerate parametric amplifier can produce a single-mode squeezed state with any squeeze parameter and coherent amplitude. He assumed that a homodyne detection as a conditional measurement on the idler mode and controlled beam splitting for the signal mode are combined. He obtained the statistical operator of the output state when the idler mode is in the vacuum state at the initial time. The unnormalized quantum state of the signal mode after the measurement on the idler mode, when the initial state is pure, is given as I~P(q)) = fih~
(9.125)
where ]~p) is the initial state vector of the signal mode and t~h~
~1 r
exp( ~/-T
q2~exp
,, -~
~^t2 - ~-~ a
/2~
^t
V --T--q a
1 2 -~
.
(9.126) If in addition the initial state vector of the signal mode is a vacuum, I~p)s = [0)s, we obtain by comparison of the scalar product (a]~p(q)) with the well-known formula for (alfi) (Yuen [1976]) that fi = v / 2 - R r q / v / r + R and [~p(q)) is an unnormalized squeezed state,
]~P(q))= ~/Jr(T+74) exp
- @ ~ ( T + 7 4 ) exp
I 2(7-+7"4)q2 '
~, 2 ' 0
2(T+7"4)q2 b
b
T+7-4 q
v/T+7.4 q
~ 2'0
10)
]0),
(9.127) where 0 = ln(T + 7"4), and S (r, rl) is the single-mode squeezing operator, 1 (h2e-i2~ - ht2e i2~)], S(r, r/)= exp [~r
(9.128)
where r is a nonnegative squeeze parameter and r/is the phase of squeezing. Next the single-mode displacement operator has been used, /3(/3) = exp(/3h* -/3" h),
(9.129)
where/3 is a complex displacement. The squeeze parameter of this state can obviously be controlled, but its coherent amplitude comprises a random factor q.
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CONDITIONAL GENERATION OF SPECIAL STATES
219
Since even this random factor is known after the homodyne measurement, we can set the coherent amplitude to any value. To achieve this, the controlled displacement of the complex amplitude of the signal mode is performed. Ban [ 1996a] assumed that an observation is carried out at one of the two output ports of a lossless beam splitter, and that the input state of the reference mode is the vacuum state. He investigated photon statistics of a conditional output state at the unobserved output port for input states of two kinds; viz., for the Fock state input and the thermal state input. He considered photon counting, heterodyne detection, and homodyne detection as the means of observations at the output port. The unnormalized output state at the unobserved output port when the input state is pure is given as
lip(m) ) = USU(2)(m)] ,,pc IP), I~,(a)>
= "het
(9.130)
USU(2)(a)] IP), I~P(q)) = Usu(2)(q)l*P), ^hom
where I~p) is the input state of the signal mode, and ~Pu(2)(m)
1
-~m'. ,,het
1
,,hom
1
Usu(2)(a ) = ~
Usu(2)(q) = ~
7-4 ~ h,,T~
(9.131)
~
(~_~) exp -
(q2)
exp --~-
exp
exp
(
-~--~a +
7 -89
qa 7-~
(9.132)
(9.133)
Pegg, Phillips and Barnett [1998] consider even a generalized measurement. The authors are concerned with the operator ]event)b b(event l, which is not a projector, since levent)b is an unnormalized vector, [event)b =c (CIR* ]1)b [0)~ 1
- v~(r~lO)~ + iy~[1)b),
(9.134)
where k = exp [88
+ b*~)]
(9.135)
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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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is the unitary operator describing the operation of a second beam splitter (in terms of the annihilation operators/~ and ~ for modes b and c), and O(3
[C)e - ~-~ ?',,[n)c.
(9.136)
n=0
Their first beam splitter is described similarly as given in relation (9.13 5) (but in terms of the annihilation operators h and/~ for modes a and b). They assumed that a single-photon field is incident on the input port a and have shown that, after the detection of the event [event)b, the output a is in the state K'(Y010)a+ Y~ll)a), where K" is a normalization constant. In other words, the field in output mode a is the same as that obtained by truncating the state [C) after its first two photon-number components and then renormalizing the state. In their analysis, the authors also take account of detector inefficiency, r/< 1. Dakna, Anhut, Opatm~,, Kn611 and Welsch [1997] proposed a simple beamsplitter scheme for generating a Schr6dinger-cat-like state of a single-mode optical field. They assumed that a squeezed vacuum is injected in one of the input channels (the second input channel being unused) and photons are counted in one of the output channels. They have shown that the conditional states in the other output channel exhibit properties of superpositions of two coherent states with opposite phases. Further, they discussed the effect of realistic photocounting on the states. Dakna, Kn611 and Welsch [1998] have shown that conditional output measurement on a beam splitter may be used to generate photon-added states for a large class of single-mode quantum states, such as thermal states, coherent states, squeezed states, and displaced photon-number states. They assumed that the beam splitter combines a mode prepared in such a state and a mode prepared in a photon-number state, and that no photons are detected in one of the output channels. Photon-added states can be highly nonclassical ones. Photonadded coherent states are non-Gaussian squeezed states. Photon-added squeezed vacuum states exhibit all of the typical properties of Schr6dinger-cat-like states. Photon adding to a squeezed vacuum can therefore be regarded as a method for producing Schr6dinger cats. Dakna, Clausen, Kn611 and Welsch [ 1998] have compared single-detector photocounting with N-fold photon chopping. The photon-added states were introduced by Agarwal and Tara [ 1991 ]. They studied the mathematical and physical properties of such states and discussed how such states can be produced in practice. They considered the passage of initially excited atoms through a cavity, and modeled atoms as two-level atoms. The interaction Hamiltonian has the form (cf. eq. 4.58) /t~/int
=
h ( g h ?r+ + g'hi(r_).
(9.137)
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CONDITIONALGENERATIONOF SPECIALSTATES
221
The initial state of the atom-field system is [a)]e), where [a) is the coherent state of the field and [e) is the excited state; the combined system is (approximately) equivalent to a beam splitter with a low reflectivity and the input state [a)[ 1). If the atom is detected to be in the ground state [g), then the state of the field is reduced to a photon-added coherent state. Back-action evasion is achieved by a quantum noise evader (Yurke [1985]). The optical back-action-evading apparatus can be described by an effective interaction Hamiltonian /-)in, = hKDsglR,
(9.138)
where K is an effective coupling constant, the subscripts S and R stand for signal and readout, respectively, and ^
glj = --~ Qj = v/-2Re(hj), j = S,R, = 1 _ ~ , . = x/2Im(hj),
j = S,R.
(9.139) (9.140)
The coupling between the photon number and a quadrature has been described in eqs. (6.46)-(6.49), but here we expound that between two quadratures. In the Heisenberg picture, this coupling is described by the following transformation of the signal and readout modes (Song, Caves and Yurke [1990]):
(qs) OR
=(1 r) (qs) out
01
OR
(9.141) in'
(/Ss) =( 1 0) (/Ss) /3R out -r 1 /3R
(9.142)
in'
where r = Kt = 2 sinh r, and r is the squeeze parameter. Let the input state be described by a Wigner function of the form q~,~(qs,Ps, qR,PR) = q~s(qS,PS)
exp(-q~ _p2).
(9.143)
In the Schr6dinger picture the 'back-action evasion' is described by the relation for the output Wigner function q~,~Ut(qs,Ps, qR,PR) = qs~n(qs -- rqR,PS, qR, rps +PR),
(9.144)
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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
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or, according to eq. (9.143): z
1
(•)souttqs,Ps, qR,PR) = q~s(qs -- rqR,PS)~
exp[-q 2 - ( - r p s +pR)2].
(9.145) The momentum-like quadrature operator fiR is measured. When it has a realization PR, we encounter the signal mode in an unnormalized state: pout
--
s (qS,PS,PR) =
f
OG
pout z
-
S tqS,PS, qR,PR)dqR
OG
-~r.out z
(9.146) 1
- 'vS tqs,PSl~R)--~_ exp[--qSR + rps)2] 9 V'Jr Here the conditional state is described by the Wigner function
q~,~Ut(qs,Ps]fiR) =
(y2)
~ CI)s(qs --Y, Ps) V/~ r exp --~Z
dy.
(9.147)
The readout quadrature f i R can, in principle, measure the quadrature operator/3s with an arbitrary large signal-to-noise ratio since r can be large. The back-action noise is 'evaded' and appears in the position-like quadrature component of the transmitted signal (see the convolution in the relation (9.147)). Song, Caves and Yurke [ 1990] described a method for generating superpositions of classically distinguishable quantum states using the optical back-actionevading apparatus shown by La Porta, Slusher and Yurke [ 1989]. They assumed that two modes of the electromagnetic field, the signal and readout modes, are correlated through a back-action-evading device consisting of a nondegenerate parametric amplifier and polarization rotators. They took advantage of the correlation between signal and readout to generate a Schr6dinger cat on the signal mode. They assumed that both modes are injected in the vacuum state and the number of photons in the readout is measured at the output. To separate the superposed states more distinctly, they proposed to process the signal mode through a degenerate parametric amplifier. Yurke, Schleich and Walls [ 1990] proposed to inject a squeezed vacuum at the signal frequency instead of amplifying the signal after processing by the backaction-evading apparatus and the measurement. Ban [1996b] proposed a scheme to generate the Fock state via a degenerate four-wave mixing and partial measurement on the reference mode whose initial state he assumed to be the vacuum. As an example of the initial state of the signal mode, he considered the Fock state, coherent state, and superposition of
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223
two Fock states. He considered the direct, heterodyne, and homodyne detection measurements on the reference mode as conditioning for the signal mode. The unnormalized output state of the signal mode after the measurement on the reference mode, when the initial state is pure, is given as:
I~P(r, m)) = ~PC(r, re)lip(O) ), I*P(r, a)) = cthet(1", a)l,p(0)), Iq,(r, q)) = ~h~ q)l~P(O)),
(9.148)
where I~p(O)) is the input state of the signal mode, and ~pc(T,m ) -- ( - i~v Y ) m exp(- 89 ~/het(g, a )
= - ~1
~h~
= ~
1
m,
(9.149)
, ^2), exp(- 89lal 2) exp (_ix/~ra,h - ~grn
(
)
e x p ( - l q 2) exp i 2v/~grrqh ,
(9.150) (9.151)
where gr = (/./1-)2, and r is the interaction time of the four-wave mixing. In the cases of photon counting and heterodyne detection, the conditional signal state of the degenerate four-wave mixer reduces to a Fock state in the limit gr --+ cx). The same is valid for the homodyne detection, with the exception of the case treated by Ban [1996b]. The phase of the local oscillator has been chosen such that we cannot obtain any information of the signal mode. If we get some information of the signal mode, the scheme generates a Fock state; otherwise, the homodyne detection of the reference mode does not induce the reduction of the conditional signal state to a Fock state. Luis and Pefina [ 1996] studied the states generated from the vacuum in two parametric down-conversion crystals with aligned idler beams when the idler beams are completely or partially connected. First they assumed that the idler modes are perfectly superimposed and aligned. The parametric interaction in the two nonlinear media is described by the effective interaction Hamiltonians/~/lint and/-~/2int, /2/jint =
h(gj~l~a[+gf&jai),
j = 1,2,
(9.152)
where hsj and ai are the annihilation operators for the corresponding signal and idler beams and gj are parameters depending on the pump and the
224
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
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nonlinear characteristics of the media. Luis and Pefina introduced the unitary transformations (9.153) i^
Tj-= exp ( - ~/-/jint Tj) , j = 1,2,
(9.154)
where rj. are the corresponding interaction times. In the Heisenberg picture, the operator N = hsl + hs2 - h i , with hsj = h~hsj, hi = h~hi, is conserved. It can be derived that
/Wout--/Qin,
(9.155)
where the subscripts in and out mean that the operator N is calculated using input and output photon-number operators, respectively. The operator N behaves similarly to the operator 7",
~/'out-- ~'in,
(9.156)
the operator ]~[in commutes with 7" and/Vout- 7't]Qinff'- The study is realized in the Schrfdinger picture. The relation between the input lip) and output ]~p) field states of the whole device is provided by the unitary operator T, I~) = 7"I~P)9
(9.157)
Luis and Pefina [1996] first assumed that a photon-number measurement is performed on the idler beam. They derived that the conditional states are the SU(2) coherent states, while the photon-number statistics of the idler beam are Bose-Einstein with the mean ~i,
ni = ~12~/~- 1,
/tj = cosh([gjl~-),
j - 1,2.
(9.158)
They further considered a measurement on the second signal beam. The conditional states are the SU(1,1) coherent states. The photon-number statistics of the second signal beam are Bose-Einstein with the mean ~s2,
-ns2 =/t~lv2] 2,
gJ sinh(gjlrj), vj =-i[--~j[
j = 1 2.
(9.159)
They commented on a result of the same quality in the case of the first signal beam. Here, nsl = / t ~ - 1 = [Vl [2.
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CONDITIONAL GENERATION OF SPECIAL STATES
225
To show that other special states can be produced, they considered partial coupling of the idler modes. One way to accomplish this is by inserting a beam splitter in the idler beam between the two crystals. The beam splitter couples the output idler mode of the first crystal with an 'escape' mode they described by the annihilation operator h0. The beam splitter was described by real reflection, r, and transmission, t, coefficients with a Jr phase change in one of the reflections. Its action on the field state is given by the unitary operator I"BS : exp[y(a~ai- a~a0)],
(9.160)
where ~' = arctan(-~). The relation between the states entering, I~p), and leaving, I~p), the whole device is now given by the unitary operator 7" = 1"2TBsl"l. An example with decoupled idler modes indicates that the conditional states are products of number states in the limit. First they studied the case corresponding to a measurement of the photon number on the h0 and ai modes. They did not present the joint photon-number distribution of the two modes (it is more complicated than that of the four modes), but they described the states as solutions to eigenvalue problems. This will be expounded in w 11. They obtained 'photon-added' SU(2) coherent states (with respect to the first signal mode, the photons are subtracted in the second one). They demonstrated a connection to the ix-J,, intelligent states. They also considered measurements on the other two output modes. First, they again treated the two output signal beams, but one obtains similar special states as above. A measurement on the h0 and as2 modes leads to 'photon-added' SU(1,1) coherent states. As many photons are added as escape. The authors related these states to the Ky-Kz intelligent states. A measurement on the hsl and ai modes leads to SU(1,1) special states spanned by a finite number of photon-number states. Here also, a relation to the K,,-k- or the k~-kv intelligent states can be found. In the Heisenberg picture, the operator N' = hsl + hs2 - h0 - hi, where ho = h~ho, is conserved; i.e., ^I __ j~qT./ Nout m"
(9.161)
From such conservation laws, one can simply discern which case leads to the SU(2) Lie group and which one leads to the SU(1,1) Lie group. De Martini [ 1998] dealt with a situation which resembles somewhat the case of complete alignment and detection of one photon on the first signal beam. The second crystal is, due to polarization, described as two equal and independent
226
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
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optical parametric amplifiers. The author proposed a novel optical device, the quantum-injected, entangled optical parametric amplifier, which produces fourmode SU(1,1) Schr6dinger-cat states. Steuernagel [1997] proposed a quantum optical selection scheme to generate Fock states 12q) on pulsed light beam. His idea uses an array of (2 q - 1) beam splitters with 2 q input ports which are fed with single-photon Fock states. As a source, spontaneous down-conversion crystals are suggested, since this allows one to apply a time gating technique to all of the detectors involved. The detectors survey (2 q - 1) output ports; the free port is the one to produce the sought Fock state. It has been shown that the use of balanced beam splitters and symmetrical input states should give best results. Luis and Sfinchez-Soto [1998b] analyzed a conditional-generation scheme consisting of a parametric down-converter and a generalized measurement on some of its output modes. They found that this scheme generates approximately the state associated with the generalized measurement. In more detail, the scheme generates the desired state up to the action of a nonunitary operator and the normalization. This enables one to associate the desired state with a suitable generalized measurement in many situations.
w 10. Special States of One or Two Optical Modes by Means of Continuous Measurement on a Different Mode The destructive and nondemolition versions of the process of continuous measurement have the property in common that they converge to the ideal photon number and even to the collapsed wave function in the long-time limit when some assumptions are observed. As a consequence, the continuous measurement on one of two or three quantum-correlated modes presents a natural generalization of the conditional generation of special states using the ideal measurement. 10.1. CONTINUOUS STATE REDUCTION Yuen [1986a] proposed generation of high-intensity photon-number eigenstate fields from parametric processes with measurement feedback. According to that paper, strong correlation and ideal photodetection can entail that, after one counts n photons on the idler mode, the corresponding signal mode is just in state In) (see w9). One may thus consider the generation of the number state In) on the signal mode by stopping the signal after n counts on the idler. In addition to a 'continuous generation' of a near-number state, which we do not analyze in detail
III, w 10]
SPECIAL STATES
227
in this chapter, Yuen [ 1986a] obviously assumed a continuous measurement of the photon number, even though this is not the same as the one we have focused on here. Let us remember that the theory in w3 has been restricted to the single-mode fields only for the sake of concreteness, but it is valid in general for an arbitrary number of modes provided that one defines on which mode the measurement is being carried out. Holmes, Milburn and Walls [1989] assumed the measurement was on the signal, not on the idler mode. Nevertheless, in their notation, the measurement was still carried out on the mode b. Their model includes the interaction in the photodetection process; i.e., they performed the unraveling of the master equation (3.18), (3.20), where H is the interaction Hamiltonian of a nondegenerate parametric amplification. They introduced coupled master equations of the kind of eq. (3.47). They solved them in the representation by q~A quasidistributions (Q functions) for initial coherent states. They determined the conditional state of both modes and the conditional state of separate a (idler) mode. For simplicity, they assumed the initial vacuum state in a sequel. Similarly as in a somewhat different case (Agarwal [1990]; w9 here), one can obtain only an amplified number state on this condition on the idler mode. They compared the uncompleted measurement using a perfect detector with the completed measurement limited by a low quantum efficiency. As is usual in the analysis of optical parametric amplifiers including losses on the idler mode (Pefina [ 1991 ], p. 146), they discerned two cases dependent on the constants of the model, namely, whether the amplification dominates the damping or the damping dominates the amplification. Obviously, the latter case would be closer to a situation in which the counting is done, after the interaction which produces the correlated state is turned off (Walls and Milburn [1994]). They focused on this case, although they could only approximate it when the losses on the idler mode did not occur. Ueda, Imoto and Ogawa [1990b] developed a general theory that describes continuous state reduction of an arbitrary two-mode state by continuous photodetection on one of the modes. It is not very difficult to generalize again, and we present such an attempt here. We assume that the operator 0 acts only on the idler mode. The average photon number of the idler field for a postmeasurement state just after the one-count event is obtained from the relation (3.58) as lira (hz)(s)= s---~ t +
(Oth20)(t) (0*0)(t)
=
(10.1)
(Oth2b)(t)- (~2)(t)(OtO)(t) (n2)(t) + (t)
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CONTINUOUSMEASUREMENTSINQUANTUMOPTICS
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Keeping in mind the special ordering of the operators 0 t, n2, 0, we can write eq. (10.1) in the form lim (a2)(s) = (n2)(t) + s --+ t +
[ (Art2A(b i O)) (/)]ord (t)
(10.2)
where the subscript ord indicates the considered ordering. Particularly, for 0 = h2 (Ueda, Imoto and Ogawa [ 1990b]), the ordering ord is the normal ordering of the annihilation and creation operators h2, h~ and the second term on the right-hand side in eq. (10.2) is the Fano factor decreased by unity. In a similar way, it can be shown that the kth moment of the photon number of the idler field for a postmeasurement state has the form lim (hw
[(Anw
= (hzk)(t)+
S ----+ t +
tO)) (t)lord (0t0>(t)
(10.3)
where the subscript ord means a special ordering of the operators 0 t, n2, O. The time development of the average photon number of the idler field in a no-count process satisfies the following differential equation: d dt (h2) = -R cov(h2, 0 t O), (10.4) where we have introduced a 'covariance' (Merzbacher [1970])
cov(&, bt O) = ' [(Ah2A(O*O)) + (A(0*O)A&)].
(10.5)
The average photon number of the signal mode just after the one-count event is given by
lim (h,)(s) =
s-~,+
(bt;~,O)(t)
(b*b)(t)
= (~)(t) +
cov(a,, OtO)(t)
(10.6)
(OtO)(t)
Here there are no ordering problems in the definition of the covariance. The kth moment of the postmeasurement signal photon number is given by lim (h~)(s)= (h~)(t)+
s~,+
coy(a{ O*O)(t) ' (OtO)(t)
(10.7)
For the average photon number of the signal mode in a no-count process, we have d (hi) = - R c o v ( ~ , O* b). (10.8) dt The conditioning on the number M of photocount registrations in the interval [t,t + T) is simply written (or reasonably complicated) only if
III, w 10]
SPECIAL STATES
229
the postmeasurement state is not dependent on the times of registration rl, ~'2,..., rM, U[0, T)(M, Tl, 1"2,..., TM) = v/P(T1, T 2 , . . . , TM IM)h[o, r)(M).
(10.9)
In analogy to eq. (10.7),
(h*,)wM(T) = z ~_zz : XI~>,
(11.17)
with ~ and r/given in eq. (9.108). The general eigenvalue problems (11.12) or (11.17) can be reduced to any of the following three normal (canonical) forms:
(a) where
[h + ir/o/3(0)] I~Po)= Al~o), I~Po) = 0tl~p),
~r - 0(~', r ) = b(~')exp(irh),
(11.18) (11.19)
with r/o and ~' given in eq. (9.86);
(b) where
(h-),oht)l~po>= ~1~Oo>,
(11.20)
[~Po) = Ertl~P),
(11.21)
~r -- ~(_~*, r) = b(-~*)exp(irh),
with Yo given in eq. (9.88); and
(c) where
(h + ~oa)[tOo) = XltOo), ]~Po) = ~r*l~p),
~ - ~r(y, r ) = D(y)exp(irh),
(11.22) (11.23)
with Yo given in eq. (9.90). In all the cases, the unitary displacement operator D(fl) is defined in eq. (9.129). It can be demonstrated that by using an
244
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
[III,w 11
appropriate rotation followed by a suitable displacement, any normal state can be transformed into any solution of the more general eigenvalue problem. Indeed, the normal forms rest on a displacement and a rotation of the points -~*, ), situating them on the real axis as the points -r/0, r/0 (case a), 0, )'0 (case b), and -Y0, 0 (case c). For small ),, ~, the near-number state is understood, which occurs for t going to infinity in the evolution. A consideration of normal eigenvalue problems is not very convincing with respect to the quasidistributions qsa, since we obtain only special forms of the formula (9.79). Nevertheless, it is important that problem a) (eq. 11.18) is the definition of the number-quadrature minimum uncertainty state (Pefinovdt, Luk~ and I~epelka [1994]). These states have been studied in connection with the quantum phase problem, when the problematic quantum phase operator has been replaced by the quadrature operator. For further detail, see Pe[inov~, Luk~ and Pe~ina [ 1998]. The representation of the state I~p) in the number state basis for case (a) is m~ u
exp(-~)
n-M n-M
v/G(_4,~,) r/(j
-..
LM (-r/02) for n ~> M,
c. = (n[ ~p) =
(11.24) n~ exp(-'~
v'G(-4,~)
r/y-nLM-n(--r/2)for n ~< M.
Applying the formula (11.14) to the definition ~ 1,
,,
(11.26) where
am,=
I(m+M)! r xn-mrn-m ~" ,4~,,2x for m ~< n, M ! ~/" rIO1 l~m+M ~--"e l]O]
(11.27) (n+M)! t o n ~m-nrm-n i A~2~ M! ~./-ulO) ~n+M~,--'.ttlo)
for m ~> n.
Particularly, 1
(h)
-_
r/o LO(_4r/2
(h2) = L~
)
~0(_l)l_..~,,r. = ,. ~M(--4r/2),
(11.28)
L (2n)(-1)2-"2"L'/w(-4r/2)"
(11.29)
III, w 11]
EIGENVALUEPROBLEMSFOR INTELLIGENTSTATES
245
The coherent state lr/0) solves the problem for M = 0. In case (b) (eq. 11.20), the role of the quadrature is taken over by the photon creation operator, and the number state representation of the appropriate eigenstate is (Yamamoto, Machida, Imoto, Kitagawa and BjBrk [1987])
( )
n<M,
,/2
Cn =
exp -~-
n!
(11.30)
1 y(~-M
v/LOM(_yg) (n--M)!
,
n >~ M.
The expectation values of the antinormally ordered field operators read 1
=
(k+M)! . l - k r l - k
Lo(_ro~) M~ Yo ~k+M(-Y2),
k ~< 1,
,,
k > t.
(11.31)
For M = 0, the coherent state ]Y0) results. In case (c) (eq. 11.22), the eigenvalue problem is related to the number operator and the photon annihilation operator, which results in the finite number state representation (9.98). The moments are expressed as follows:
k
(~k~t/)=
Z~
1
2)
l
m~~ (km) (--frO )k-m ~ =
(In)(--ffo)l-namn,
(11.32)
B=0
where (m+M)!q-in-mln-m M!
IO
L'm+M
(__y2)
for m ~< n, (11.33)
amn = (n+M)! [q-i ~ m - n l m - n M! ~.1"01 L'n+M
(__y2) for m ~> n.
For M = 0, the vacuum state results for all values of the parameter Y0. Luis and Pefina [1996] presented a number of eigenvalue problems, and outlined ingenious derivations. Since the states are to be generated from the vacuum, the roles of relations (11.1) are played by
Tasl~'tl~> = (lUlasl q- tv1V;as2 - tVl~2t~ [ - Z~vla;)[~> = 0 ,
(11.34)
Tas2ftl~> = (/t2hs2- v2h[)lip) = 0,
(11.35)
7'aiT't[-~>
= (t2~l~2ai-k- r[21a 0 - Vla~l- t/A 1v2a~2 ) 11/9>= 0,
(11.36)
taofrtl~>
= ( - ~ 2 a i + tao + ~2a~:)I~> - o,
(11.37)
when one considers partial coupling of the idler modes. They started by first studying the case corresponding to a measurement of the photon number on the
246
CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS
[III, w 11
'escape' and idler modes by means of photodetectors placed at the output of the beam splitter and at the output idler beam. If the results of such a measurement are no and ni, respectively, the state of the signal modes ]~Pln~,n0) is given by the projection of ]~p) on ni)iln0)0 (and a normalization). Equations (11.36) and (11.37) can be solved for hi]~P> and h0]~P). Equations (11.34) and (11.35) are solved for h~ ]~) and h~l~), but solutions are presented somewhat implicitly: =
-
,
(11.38)
^tl~)
rvl
(as, - ~asz)lq,> = tt-Tao
,
where ~ = tV1/(V2gl) , while (explicit solutions) it2
(11.39)
r v1
aol'lp> = --~1 h~l]~).
The recursive relations (11.7) can be avoided here. On multiplying the first and second equations in (11.39) by hi and h~, respectively, and on appropriate substitutions, we obtain that
~ i ] ~ ) = ( ~ , + ~2)~s21~), tl~ao]~> -- a ~ l ( a s l -
(11.40)
~:as2)l~>-
We can project these two equations over photon-number states of the idler and 'escape' modes, respectively. We obtain easily that
dz -
Nl~Plni,no) = (no + ni) q~ln~,,o)' ~J+)l~P,n,,no) - ~(no - ni)l-~ln,,no>,
(11.41)
1
where /~r = ~:ll~S 1 -t- l~2as2 ,
) z = 1 ( a : l a s , - t~2tffs2 ) ,
) + = asll~s2 9
(11.42)
Taking into account the commutation relations
Dz,J+] = J + ,
[/V,J+] : O,
(11.43)
we have ,Jz - ~,J+ = exp(~J+) ,Jz exp(-~J +),
(11.44)
III, w 11]
EIGENVALUEPROBLEMSFOR INTELLIGENTSTATES
247
which gives the solution as = jVexp(
j+)l
o)s.
ni- k
k=0
where N" is a normalization constant, ni ni-k
i
N'=
~ k=0
no + k k i_~lzk
(11.46)
By means of an SU(2) transformation (i.e., the action of a beam splitter), an SU(2) intelligent state is obtained from I~Pln~,,,0) (the state 11.45). Luis and Pefina [1996] discussed the kinds of states which arise when the photon-number measurement is performed on other pairs of modes. They assumed that the two detectors were placed at the output signal modes. When the outcomes of such photon-number measurement are n l and n2 for the first and second signal modes, respectively, the state of the idler and 'escape' modes [~Pln,,~2) is given by the solution of the equations
_
~
(11.47)
d z + ~ff-)IlPlnl,n z) = ~ ( r t l - rt2)[~ln~,n2),
where ~ - t/(/~2 r), and
= a[Zti +a~Zto, Jz
= 1 (a~a0- a[ai) ,
J+ = a~ai.
(11.48)
The solution of eqs. (11.47) reads 1~1,,,,:) = N" exp(~9-)ln2)ilnl )i,
"(c )(;,)
= ./V. Z k=0
\
n2 + k n2
~k n2 + k)iIn, - k)o,
(11.49)
where N" is a normalization constant, 1
N =
_n2 \
(11.50) n2
Note that the notation is light, and its meaning is somewhat different from that in relation (11.41). To the contrary, the properties of the states I~pl,,,,2) are the same as those of I~pln,,n0).
248
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
[III, w 11
If the measurement is performed on the modes as2 and h0, giving n2 and no, respectively, the state of the idler and first signal modes ]lpln2,n0) is to be found. Equations (11.36) and (11.37)can be solved for h0]~) and h~21-~). Equations (11.34) and (11.35) are solved for hs2]-~), h~]-~), but solutions h~2 ~), h~ ]-~) are presented somewhat implicitly:
~2~ ~/,
(11.51) as21~> = V ~01~,>-
-~
I~>,
where r/= tVl/(ttltt2). On multiplying the third and fourth equations in (11.51) by h~2 and h~, respectively, and on appropriate substitutions, we obtain that h~ho]~p) = a~,(as, - r/a[)l~ ),
(11.52)
aZ=as21~>= a~(a~- ~as,)l~>. ^t can project these two equations over photon-number states of the second and 'escape' signal modes, respectively. We obtain easily that
We
Rl~,:,no>
=
(no - n: - 1)l~ln2,.,,>, 1
(k:-
(11.53)
~k+)l~Pln:,n,,> = ~(no + n:
+
1)l~ln:..,,),
where
"
_
_ ,
'(a~ asl-k-a[a,-k-i)
K+ = asia ^t "ti 9
(11 .54)
From the commutation relations [k~,K+] = K+,
[K,K+] = 0,
(11.55)
we have K~ - r/K+ = exp(r/K+) K~ exp(-r/K+),
(11.56)
which leads to the solution ]~lnz,no> = .A/"exp(r/K+)lno)sl ]n2>i, ~ ~(n2+k )( )no+k = ./V'Z ~kln0 + k>sl n2 + k>i, k=o \ n2 k
(11.57,
III, w11]
EIGENVALUEPROBLEMSFORINTELLIGENTSTATES
249
where the normalization constant is given by
.IV.= [ L (n2ff-k/(no k k=0 \ n2 +
k)
[?][2k]
-89
(11.58)
By means of an SU(1,1) transformation, an SU(1,1) intelligent state is obtained from Ilpl,2,n0) (the state 11.57). Finally, when we measure the photon number on the first signal and idler modes, with outcomes n l and n2, respectively, we search for the state of the second signal and 'escape' modes. Equations (11.36) and (11.37) can be solved for hi[~) and h~l [~). Equations (11.34) and (11.35) are solved for hs, I~P), h~ [~), but solutions h~, ~), hi [~p) are presented somewhat implicitly:
=
' --
~s211~) ---- "V2 ~.l-Ilp)
,
'
//2
Iv,)
=
(11.59)
--~1("as2 --t--
,
V2
where r/= t:/(rv2). On multiplying the third and fourth equations in (11.59) by ^ t and h~ respectively, and on appropriate substitutions, we obtain that asl (11.60) We can project these two equations over photon-number states of the first signal and idler modes, respectively. We obtain that ^
m
Xl~lnl,n,>
(Kz
-
(nl
-
ni -
1)l~lnl,,,.>,
(11.61)
+ r/K-)l~l,2,,0 ) = ~(n0 1 + n2 + 1)IIPL,,1,,,,),
where
R : a~ao- a~2as2- i,
k , -,-.=.
l (a~ao-+-a~2~/s2q-i)
~
K- = aohs2. (11.62)
250
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
[III, w 11
Again, the notation does not mean the same as that in relation (1 1.53). Taking into account suitable commutation relations, we arrive at the solution [~ln,,n,) = A/" exp(rlk_)lni>s2ln~ )0,
(()(:1) ni
= "A/" Z
ni - k
k=0
n ~ l n i - k>s21n, - k)0,
1163.
where the normalization constant is given by
.1.64. After distinct SU(1,1) transformations for [rll > 1 and I~1 < 1, SU(1,1) intelligent states are obtained from I~l,l,~i) (the state 11.63). The case corresponding to a measurement of the photon number on the first signal and 'escape' modes yields (only) an SU(1,1) coherent state,
~22as2ai
I~pl,,,,no) =.A/'exp =.A/"'~.
m + nl - no
~0=
]O)s2ln~-n0)i "v2
]m)s2]m + nl - n0)i,
(11.65)
m
where the normalization constant is given by 1
.A/. = [~-~ (m+nl-no)
v2 2ml ~
m
m=0
~22
(1)n,-n0+l = ~22 "
(1 1.66)
When we measure the photon number on the second signal and idler modes, the state of the first signal and escape modes is an SU(1,1) coherent state, -
I~]n2,ni)
frvl
"--
^,
.A/'exp ~ - ~ a s , h ~/ ]ni- n2)i]O)0 N
I
*
(11.67)
= .A/" Z(x:)~l ni
-
n n2
[n)sl In -- ni-+- n2)0,
n2
where the normalization constant is given by
.A/" =
m + n i - n2 m
r 'v1
-~l
=
1-
r gl
71
(11.68)
m=O Luis and Pefina [1996] also considered input coherent states, mainly on the assumption of the complete coupling of the idler modes.
III, w 11]
EIGENVALUE PROBLEMS FOR INTELLIGENT STATES
251
11.2. C O N T I N U O U S STATE R E D U C T I O N
Agarwal and Tara [ 1992] investigated a 'necessary condition' of the positiveness or a classical character of the P function (@• quasidistribution). Its violation is sufficient for the nonclassical character of this function. Such a condition can be applied to the states under study. They provided the example of the photonadded thermal state and that of the superposition of two coherent states similar to an amplitude cat (Schaufler, Freyberger and Schleich [1994]). Pef-inovfi, Lukg and I~epelka [1996a] derived a generalization of the problem (11.12). They observed that it is equivalent to an eigenvalue-eigenoperator problem: (11.69) We will derive a generalization of the problem (11.69) using the property of rescaling in the @.a quasidistribution. Starting from a modification of the problem (11.69),
where
~' = M + A y ~ , ,
(11.71)
introducing the A-dependent statistical operator (11.72) where the superoperator S means amplification by the factor ~A' and using the rescaling property in the form (11.73) we obtain the problem +
-
/5(A, y, ~,)= ~'/5(A, y, ~,).
(11.74)
Here the superoperators h+, ht+ are given in eq. (2.40) and
h+ = ht+h+.
(11.75)
252
CONTINUOUS MEASUREMENTSIN QUANTUMOPTICS
[III, w 11
Since ,~-1
Sh+S
= Aa*+a*_ + a*+(a+ - a*),
~-l 1 sa+s = v~a*_ + ~ ( a + ,,
(11.76)
a*),
9,-1
~a*+s = ~a*+, where the superoperators h_, ht_ are given in eq. (2.40), we arrive at the eigenvalue-eigenoperator problem Aa*~a + a*[a,/,] +Ap~a + ~,[h,[~]-Ayh*[~ = 2,'/5,
/5 -/5(A, y, p). (11.77) Let us remark that the amplification superoperator can be expressed as 1
) 2a+ta-t-ala-t-a+a+t
= v/Aexp [(1-A)ht+ht] (x/A) a+ta+(v/A) a-a~ CO
= x / ~ Z (1-A)Jht+j j!
(11.78)
(V/--~)ht+h+~ltJ (V/~)a-ht-
j=0
From the relation min(n, m)
PnmIM(A , Y, }')= Z
/'+) "-" '+) (A)P,,-j,m-jlM(1 ' x/-Ay, x/A~) , VP,,In-jta)Pmlm-/
j=0
(11.79) where the (displaced) negative binomial distribution pnlk(A) =
(1 -
,
n=k,k+l,k+2,...,
(11.8o)
and PnmlM(1, x/My, v/-A~) -- CnlM(1, x/Ay, v/A~)C,*,IM (1, X/~y, V ~ ' ) , (11.81) with C,IM(1, Y, Y)--
C ~.. n! [t~ • exp(-5 lyl
~. 12)] -l
L~-n(-y~),
(11.82)
III, w 11]
EIGENVALUE PROBLEMS FOR INTELLIGENT STATES
253
we derive the formula
[Lo (-AIy + P*lZ)] -' PnmlM($, 7, ~)- v~!m! M!
A M+I exp(-AlYl2)
min(n,m)
x
Z k=0
(1 - A) k ~Ml.+k rMl.+k ( - A yr/) k! ~.-k
(11.83)
x (~,)M-m+k ~,,,-krM-m+k( - A y* ~*).
Applying displacement and rotation operators to the density operator r a situation can be achieved with real parameters 70, ~'0 such that-~'0 < 70. Substituting particular values of 7, )' into the eq. (11.83), we obtain formulas pertinent to the following three cases: (i) For 7' = ~*, denoting )70 = 7' = ~'*, the matrix elements can be written as
P, mIM (At = ~
1
x/m!n!A M+I
exp(-A ]r/012 ) L~ ]r/0 [2)
min(m, n)
•
Z
(1 - A ) k rl~t_,,,+k rM_m+k(_AlrlOi2 ) k!
k=0
(11.84)
~,,-k
x ()7~) M-"+k "-'.-k'M-"+k(--A]~o]2)"
For A = 1 the appropriate form of the coefficients (11.82) generalizes the formula (11.24), removing the constraint r/0 > 0. (ii) For ~, = 0, denoting 70 = Y, the matrix elements simplify dramatically; i.e., 1
P,,mIM (A) = --~.. x/~m!n!A M+'
•
min(m-M,n-M)~ Z...,
k=0
exp(-A ]y0 ]2) L~ ]y0]2)
(1 -
A)k(Ay~)"'-M-k(Ayo) "-M-l
o,
we obtain the tautology A[/l~/(t)N(t)] = [A/~(t)] N(t) +/l~/(t)[A/V(t)] + A/~(t) AdV(t),
(B.2)
where ~/(t) and N(t) are adapted operator-valued processes. Dividing both sides of eq. (B.2) by At and passing to a limit, we obtain the Leibniz formula with the It6 correction:
+ lim At ~ o+
(B.3)
AM(t)AN(t). At
Using the Stratonovich product where necessary, we persevere on the Leibniz formula d d--~[~Vl(t) N(t)] = ( [d ]f/l(t)lN(t)} s
+
}s
262
CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS
[III
where d
^
1
AM(t) AIV(t) At
{ /~/(t) [~t N(t) ]}
s
lim AI~(t)AIV(t) = { ~l(t) [d~-~N(t) } + 21 At---,0+ At
, (B.5) (B.6)
I
The indicated limits do not vanish, as for instance
lim 1 f,+A, L(r) dr atf,+A, Lt(r') dr'
At---~0+ At at
= Ri
'
(B.7)
although lim h1 ft+a, Lt(r,) dr,
At ~ 0+ At
1 atlim --. o+ A-t
t
[ftt+At
it+AtL(r) dr = 0, Jt
L(r) dr j 2 = Ate0+ lim At1
[/
t+At Lt(r)dr ]2 = ~'
(B.8)
(B.9)
where L(t) is quantum noise according to eq. (4.8), with the properties given in eq. (4.3).
References
Agarwal, G.S., 1990, Quantum Opt. 2, 1. Agarwal, G.S., M. Graf, M. Orszag, M.O. Scully and H. Walther, 1994, Phys. Rev. A 49, 4077. Agarwal, G.S., R.R. Puri and R.P. Singh, 1997, Phys. Rev. A 56, 4207. Agarwal, G.S., and K. Tara, 1991, Phys. Rev. A 43, 492. Agarwal, G.S., and K. Tara, 1992, Phys. Rev. A 46, 485. Aliskenderov, E.I., H.T. Dung and L. Kn611, 1993, Phys. Rev. A 48, 1604. Araki, M., and E. Lieb, 1970, Commun. Math. Phys. 18, 160. Audretsch, J., and M.B. Mensky, 1997, Phys. Rev. A 56, 44. Ban, M., 1994, Phys. Rev. A 49, 5078. Ban, M., 1995, Phys. Rev. A 51, 1604. Ban, M., 1996a, J. Mod. Opt. 43, 1281. Ban, M., 1996b, Opt. Commun. 130, 365. Ban, M., 1997a, Phys. Lett. A 235, 209. Ban, M., 1997b, Opt. Commun. 143, 225. Ban, M., 1997c, Phys. Lett. A 233, 284. Ban, M., 1998, Phys. Lett. A 249, 167. Barchielli, A., 1986, Phys. Rev. A 34, 1642. Barchielli, A., 1990, Quantum Opt. 2, 423.
III]
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E. WOLF, PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED
IV
OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
BY
ZEEV Z A L E V S K Y , DAVID M E N D L O V I C
Tel-Aviv University, Faculty of Engineering, 69978 Tel-Aviv, Israel
AND
ADOLF W. L O H M A N N
University of Erlangen, Lab. far Nachrichtentechnik, Cauer Str. 7, 91058 Erlangen, Germany
271
CONTENTS
PAGE w 1.
INTRODUCTION
. . . . . . . . . . . . . . . . . . .
w 2.
T H E S P A C E - B A N D W I D T H P R O D U C T (SW) AS A T O O L F O R SUPER RESOLUTION STUDIES
273
. . . . . . . . . . . .
275
w 3.
S U P E R R E S O L U T I O N AS S W A D A P T A T I O N
w 4.
S U P E R R E S O L U T I O N B A S E D ON T E M P O R A L C O N S T R A I N T S 293
w 5.
THE GENERALIZED WIGNER FUNCTION FOR THE
. . . . . .
A N A L Y S I S OF S U P E R - R E S O L U T I O N S Y S T E M S
282
. . . . .
w 6.
S U P E R R E S O L U T I O N F O R O B J E C T S W I T H F I N I T E SIZE
w 7.
WAVELENGTH-MULTIPLEXING SUPER RESOLUTION
w 8.
CONCLUSIONS . . . . . . . . . . . . . . . . . . .
312 .
320
. .
329 338
L I S T OF A B B R E V I A T I O N S A N D S Y M B O L S . . . . . . . . . .
339
ACKNOWLEDGEMENTS
. . . . . . . . . . . . . . . . .
339
. . . . . . . . . . . . . . . . . . . . .
340
REFERENCES
272
w 1. Introduction
Speaking in general terms, 'resolution' means that information about small details, 6x, of an object is available as output of an optical system. 'Being available' usually means that the object is displayed as an image. But in a broader sense, 'resolution' means the possibility to somehow infer object information, such as a small size 6x, from the observed output data. In this study the final output will be an image, but on its way from input to output, the information about the object may be totally unrecognizable due to some coding schemes. The common goal of all projects described in this chapter is to achieve 'super resolution'. The attribute 'super' refers to the capability of obtaining more information about the object than could be expected when considering only the lenses and the apertures of a standard optical instrument, such as a microscope or a telescope. Additional components, such as moving gratings, spectral prisms, polarization components, etc., are used to improve the performance of the system; i.e., to create 'super resolution' (SR). The field of super resolution can be divided into 'classical SR' and 'modern SR'. The latter is also called 'near-field microscopy' (a recent paper on that subject by Blattner, Herzig and Dandliker [ 1998] may serve as an entrance into the literature on near-field microscopy). Another categorization considers the various causes, which limit the resolution. Foremost in this review are the limits caused by diffraction. We refer to this category as 'diffraction resolution' (DR). Another category, called 'geometrical resolution' (GR), is concerned with the discrete structure of certain detectors, such as CCD arrays. The third category, called 'noise equivalent resolution' (NER), is the most fundamental limitation of all information gathering systems, and deals with the ability of each detector cell to distinguish a signal out of a noise. However, in most of the existing image forming systems, it is the DR that one wants to improve. This has been accomplished with some success, as the following sections will show. 273
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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
[IV, w 1
1.1. DIFFRACTIONRESOLUTION (DR) According to Abbe (Lummer and Reiche [1910]), the diffraction limit of spatial resolution, in the camera plane, may be expressed as 6XD~F~ 1.22~.F#,
(1)
where/l is the wavelength and F# is the F number of the imaging system. This spatial resolution is related to the size of the aperture, since
f
F#- D'
(2)
where D is the diameter of the imaging lens and f is its focal length. Toraldo Di Francia [1955] said that if one only wants to know the lateral distance Ax between two stars, there is no diffraction limit on the accuracy of Ax. It could very well be Ax < 6x = 2f/D. However, if Ax is smaller than 6x, the recorded image is very similar to the single star case (Ax = 0). So, the signal-to-noise ratio will set the limit. What this case teaches us is that 'image formation' (here seeing two separate bright points) is only a very luxurious case of 'information gathering'. The fewer questions we ask, the more accurate will be the answer, if the system is properly tailored to these particular questions. Many techniques have been suggested to improve the diffraction limit. As will be further specified, all of those techniques were based on an a priori knowledge about the object. This knowledge was used to synthesize in effect an extra large aperture of the imaging system. One can characterize the a priori information types according to the following groups: object shape, temporally restricted object, wavelength-restricted object, one-dimensional object, and polarizationrestricted object. 1.2. GEOMETRICALRESOLUTION (GR) This second type of resolution is related to the finite size of the detector pixels. Assuming that this size is Ax and the focal length of the lens is f , then if such a system is aimed at a scene located a distance R from the camera, the spatial resolution in the scene's plane is Ax
8x = -~R.
(3)
For instance, assuming that Ax = 30 gm, R = 10 km a n d f = 300 mm, one obtains 8x = 1 m. According to the Johnson Criterion (Waldman and Wootton [1993]),
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275
for a probability of 50% one needs 1.5 pixels in order to detect an object, 6 pixels to recognize it, and 12 pixels to identify it. Thus, having an object with a size of AL = 3 m, the above mentioned distances will be
Rdet
-
Rrec
-
Riden
-
ALf 3m .0.3m = 20 km, 1.5Ax 1.5 930 gm ALf 3m.0.3m = 5 km, 6Ax 6 . 3 0 gm ALl 3m .0.3m = 2.5 km. 12Ax 1.5 930 ~tm
(4) (5)
(6)
The algorithm optimal for achieving this type of super resolution is coined a sub pixeling algorithm and it is related to the Gabor transform (Gabor [1946]). Briefly, the procedure for obtaining the improvement is to record N images. Between two recordings the camera is shifted by a sub pixel distance of Ax/N. Then the images are properly merged, a Fourier transform is performed, the result is divided by G ( - v ) (which is a Fourier transform of the pixel's shape), and eventually an inverse Fourier transform is calculated. 1.3. N O I S E
EQUIVALENT
RESOLUTION
(NER)
This third type of resolution limit is related to the noise developed in each one of the detector's cells. As had been mentioned before, noise may be the limiting factor, for example if one wants to measure the distance of two stars (Toraldo Di Francia [1955, 1969]). The cause of noise might be stray light, or temperature dependent detector noise, or quantum noise if the light level is very low, or even quantization noise which is caused due to the fact that the camera has a finite number of sampling bits. We assume in our study that these types of resolution impairments are negligible compared with the influence of diffraction upon the resolution. As an overall rule, an averaging operation, whether it is temporal or spatial, is helpful for improving the NER.
w 2. The Space-Bandwidth Product (SW) As a Tool for Super Resolution Studies As previously mentioned, the improvement of resolution requires a priori knowledge about the signal. The term 'signal' will be used instead of the narrower
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[IV, w 2
term 'object'. One can characterize the a priori information types according to the following classes: object shape (Toraldo Di Francia [1955, 1969]), temporally restricted signal (Francon [1952], Lukosz [1966]), wavelength-restricted signal (Kartashev [1960]), one-dimensional signal (Bartelt and Lohmann [1982]) and polarization-restricted signal (Gartner and Lohmann [1963]). Mendlovic and Lohmann [1997] described super resolution as a rearrangement of the signal's degrees of freedom and the transmitting system's degrees of freedom such that these two sets of freedom are well matched. A trivial example may serve to illustrate the concept. Suppose that a set of objects is 1 mm wide, and that the finest details are 1 ~tm wide. Suppose also that a detector (e.g., photographic film) can resolve details of 100 ~tm in size. However, the size of the detector is 100 mm. In this case, an ordinary magnification is enough to adapt the shape of the signal SW to the shape of the detector SW. Or, for instance, let us assume that the spatial aperture of a system is small and that some of the signal's information is lost due to this fact. If it is also known a priori that the signal's information is the same for all wavelengths, one may convert part of the spatial information into wavelength information, in a way that the aperture of the system is expanded synthetically. Based on the distinction between the signal information and system's capabilities, Lohmann, Dorsch, Mendlovic, Zalevsky and Ferreira [1996] and Mendlovic and Lohmann [1997] proposed a way to adapt the signal to the system. The Wigner chart is often a useful conceptual tool. A brief introduction can be found as an appendix of an earlier article in Progress in Optics (Lohmann, Mendlovic and Zalevsky [1998]). Super resolution apparently had an early phase in the 1960s, and a recent phase in the 1990s. In between these two phases the field was almost dormant, apart from a few isolated attempts (Cox and Sheppard [1986]). 2.1. THE W I G N E R D I S T R I B U T I O N F U N C T I O N
The Wigner distribution function (WDF), introduced by Wigner [1932] in the context of quantum mechanics, is a space-frequency representation of a signal which was applied to optical signals by Bastiaans [1979a,b]. A recent review was presented by Dragoman [ 1997]. The Wigner chart is a wave-optical generalization of the 'Delano diagram' (ray optics Y~" diagram). Several wellknown optical transformations can be performed in the Wigner domain simply by changing coordinates. For example, the Fourier transform (FOU) is represented by a 90 ~ rotation of the Wigner distribution function (WDF). A Fresnel transform or free space propagation (FSP) corresponds to an x-shearing of the WDE Passage through a lens (LENS) means v-shearing. Therefore the FSP and the
IV, w 2] THE SPACE-BANDWIDTHPRODUCT (SW)AS A TOOL FOR SUPER RESOLUTION STUDIES
277
(a)
(f) l MAG
I"
FT
FRT
FSP
~
,=x
4. 1/
(e)
V
.,'-
/%
(d)
V
,,
,x
(d)
1"
~x
Fig. 1. Basic transformation applied over the WDE (a) Input, (b) lens, (c) free space, (d) Fourier, (e) fractional Fourier, (f) magnification.
LENS operation are Fourier conjugates. A fractional Fourier transform (FRT) corresponds to a rotation of the WDF by an arbitrary angle (Lohmann [ 1993]). These basic transforms are illustrated in fig. 1. The mathematical definition of the WDF is
W(x, v) = f cx:~u(x + X-~t )u *(x - X-~t ) exp(-i2a'vx') dx'.
(7)
oo
Apparently the Wigner chart simultaneously presents spatial and spectral information. It doubles the number of dimensions; thus, a 1D object has a 2D Wigner chart. The WDF is closely related to the intensity lu(x)] 2 and to the power spectrum Ifi(v) l2 as
W(x, v)dv = lu(x)l 2
f
~ m(x, v)dx = I~(v)l 2,
(8)
oc
where (x)
u(x) e x p ( - 2 m vx) dx.
fi(v) =
(9)
O(3
The double integral yields the total energy
f o ~ / ~ 1 7 W(x, 6 v) dx dv = Etot" oo
oo
(10)
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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
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The way back from the WDF to the signal domain is possible by
f_
~ W ('-~x, v) exp(2:rivx) d v = u(x)u* (0)
(11)
2.2. THE S P A C E - B A N D W I D T H PRODUCT OF SIGNALS: SWI
2.2.1. The SWI-number The letter 'I' in SWI refers to the second letter of 'signal', in contrast to the 'Y' in SWY, which alludes to the space-bandwidth product of systems. Suppose that a signal is bounded in the space domain and also in the frequency domain:
Ix[
u(x) ~ 0
only within
< Ax/2,
(12)
fi(v) ;~ 0
only within Iv[ < Av/2.
(13)
From eq. (13) it follows that u(x) may be represented as a sampling series: u(x)= Z
u(m6x) sinc ( X - 6xm6X
"
(14)
The sampling step, 6x, which is the inverse of the bandwidth, determines the finest detail within u(x): 6x = 1/Av.
(15)
Strictly speaking, only one of the two bounds (12) and (13) can be valid rigorously. However, if the product of the two bounds AxAv is large, say 100, both equations may be valid in good approximation (Marks [ 1993]). In that case, the series (14) contains only a finite number N of samples: Iml ~ N/2;
Ax N - 6x - AxAv.
(16)
The number N counts the degrees of freedom of the signal u(x), as expressed in a physicist's language (VanderLugt [1992]). In communications theory, AxAv is called the 'space-bandwidth product'. We will refer to this product as the SWI-number. Such a number makes sense not only in the context of a particular signal but also for a set of signals with common bounds (eqs. 12 and 13).
IV, w 2] THE SPACE-BANDWIDTH PRODUCT (SW) AS A TOOL FOR SUPER RESOLUTION STUDIES
279
One brief comment about the case where N is not large: a more careful definition of signal size Ax and bandwidth A v is required; for example, as
x21u(x)l 2 dx f_~ lu(x ) 2 dx '
(17)
f _ ~ ]~(V)]2 dv
(18)
The product AxAv has a lower bound, which is actually reached if u(x) is a Gaussian function. This is the essence of the uncertainty principle (Marks [1993]).
2.2.2. The SWI area In w2.1 on the Wigner distribution function WDF, we illustrated how the shape of W(x, v) is modified if the signal u(x) itself experiences a process like free space propagation (FSP), transmission through a lens, Fraunhofer-Fourier diffraction (FOU), self imaging, magnification, or fractional Fourier transformation (FRT). The size of the area remains the same, but the shape may change drastically (FSP, lens, magnification). A rectangle, which encloses the WDE and whose boundaries are parallel to the two axes (x, v), may now have a larger area than the WDF area (FSP, lens, FRT). For instance, a detector whose SWY is rectangular, has to be wider in x if the signal has traveled through free space, as shown in fig. l c. In the case of magnification, the shape has been changed, but not the size, because the boundaries remain parallel to the axes (x, v). These considerations lead us to define the space-bandwidth product SWI of a set of signals u(x) as the area occupied by the set of associated WDFs (Lohmann [1967]). Notice that this (x, v) definition of the SWI has in common with the SWI-number definition that the amount of SWI area remains unchanged if the SWI number also remains the same. However, the shape of SWI(x, v) may change even if the SWI number remains constant. This tells us that the SWI shape is more informative than the SWI number. This fact will turn out to be crucial for the various super-resolution methods, where the shape of the SWI is manipulated, but not the SWI number. 2.3. T H E S P A C E - B A N D W I D T H P R O D U C T OF A S Y S T E M - S W Y
2.3.1. The SWY number Suppose a photographic camera has an image size of 25 mm (for simplicity, in one dimension only), and the resolution may be 25/tm. Such a system can handle
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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
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= 25 mm/25 ~tm - 1000 pixels. This number 1000 is in this case the SWY number. This concept can be extended to higher dimensions: imagine a TV set which is switched on for one hour (At = 3600 s). The frame rate may be 25 per second (6t = ~ s). 400 lines with 500 pixels per line mean a SWY number of N~v = 400 x 500 = 2 9 105. Together with the temporal 'SWY number' Nt = 3 6 0 0 / ~ = 9 9 106, the overall SWY number is N,~,,Nt = Ntotal = 1.8 * 1012. This number may be called 'space-time-bandwidth product', to be consistent. Another generalization from spectroscopy: the usable wavelength range may be A~. = 100nm and the spectral resolution 6/l = 0.01 nm. This leads to NA = 104. Ax/6x
2.3.2.
The S W Y area
The three examples mentioned in w2.3.1 have in common that they correspond to rectangular domains in phase space, with the boundaries parallel to the axes. There exists a good reason for these rectangular SWY areas, which will now be illustrated by a counter example. Suppose a cheap lens is used as a wide-angle photographic objective. The image resolution will decrease from the center of the image field towards the edges. Expressed in another way: the local bandwidth capability will vary as A v ( x ) = A - B x 2,
Ix] ~< Ax/2.
(19)
The problem with such a system is that two identical object details, one located near the center (x = 0), the other close to the edge (x = Ax/2), will yield different images. Or, the other way around, if two image details look alike, the two objects may actually be quite different. Hence, an unbiased image interpretation is not possible. 2.4. THE SW AS A 2D F U N C T I O N
A certain signal can only pass through a certain system if the following relation holds: SWI c SWY.
(20)
Note that this is a graphical relation. That is, the SWI area must be included in the SWY area. In other words, not only the pure number but also the shapes of SWI and of SWY are important. Otherwise some information of the signal is
IV, w 2] THE SPACE-BANDWIDTH PRODUCT (SW) AS A TOOL FOR SUPER RESOLUTION STUDIES
281
lost in passing through the system. The SW of the system may cut the SW of the signal. Again, this statement can be visualized easily in the Wigner domain. For a signal having a rectangular shape, the SWY plot of the system could be the area of the conventional space-bandwidth product SW. In conclusion, no information is lost if a necessary and also a sufficient condition are satisfied. It is necessary that the SWI and SWY numbers satisfy SWI(#) ~< SWY(#).
(21)
A sufficient condition is that the SWI and SWY plots obey SWI(plot) c SWY(plot).
(22)
The definition of the SW can be generalized in two ways. First, so far we dealt with a Wigner chart of a single signal. Now, let us present the ensemble average of Wigner charts due to a set of signals that may enter the optical system (Mendlovic and Lohmann [1997]). So far the SW has been a pure number. Now, as a second generalization, SW(x, v) is a binary function of two variables, for a 1D object, with the following definition:
SWB(x, v ) =
1 (W(x, v)) > WTR, 0
otherwise.
(23)
The symbol (.) means an ensemble average operation, and WTR is a certain threshold value. Now, the SW is a binary function that is suitable for the estimation of Ax and A v. To be more precise, one should consider that the total area of W(x, v) is related to the total energy (eq. 10). For the following discussion it is advantageous to keep this property also for the SW chart definition. Thus, we define
SW(x, v)= sT swB(x, v),
(24)
when ST is selected in such a way that
(25) As a result, we get:
sT = f f swB(x, v) W(x, v)dx dv f f SWB(X, v)dx d v
(26)
2.4.1. The SW in a hyper space So far, our SWB(x, v) has been binary. In other words, the average properties of a set of signals, or the dynamic range capabilities of the system (which includes
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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
[IV, w 3
a detector) are the same, wherever the SW is non-zero. In other words, the set of signals, or the capabilities of the system, are space invariant. That may not always be the case. For example, the signals (images) may be brighter than average around the center, or scattered light may affect predominantly the corners of the image plane. Having signals with wider dynamic range and a detector that can sense it, influences the number of degrees of freedom and the shape of the SW function. For example, consider having a priori information that a certain point source is a binary point source that can be located at 100 possible locations. This means that the input signal has 100 degrees of freedom. We may assume that this point source is imaged by a CCD camera. For a binary detector (pixel of the CCD) one needs 100 detector cells in order to know the exact location of the point source. Now assume that the detector has an infinite dynamic range. Then, based on the a priori information about the image (a point source), by only one pixel one may find the exact location of the point source since the intensity readout of the detector is proportional to the position of the point source. Thus, the dynamic range also affects the number of degrees of freedom. Therefore, instead of binarizing the Wigner chart, one should leave it as it is and define an SW hyper space (having a non-binary value per each x, v): SW(x, v ) =
(W(x, 0
v))
(W(x, v)) > WVR otherwise.
(27)
Now, the volume of the shape defined by SW is the number of degrees of freedom. The WTR is a suitable threshold value.
w 3. Super Resolution as SW Adaptation 3.1. LOSSLESS TRANSMISSION THROUGH A SYSTEM
For some optical system given by its SWY(x, v) and a given input signal described by its SWI(x, v), a necessary condition for transmitting the whole signal without losing information is SWI(x, v) C SWY(x, v).
(28)
Note that in this context C is a graphical relation comparing two shapes. As a result of this graphical relation we obtain a numerical condition: (Volume{SWI} -)
Nsigna I ~ Nsystem
(-- Volume{SWY}).
(29)
If the last two conditions are not fulfilled, some information of the input signal may be lost while passing through the system.
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283
In many cases the condition (29) about the SW numbers N may be satisfied, whereas the two SW shapes do not obey eq. (28), or SWI(x, v) r SWY(x, v),
Nsigna 1 ~ Nsystem.
(30)
That is the situation where super-resolution methods can be useful. 3.2. SUPER-RESOLUTION STRATEGY
We assume that the number of degrees of freedom of the system is larger (or at least not smaller) than the degrees of freedom of the input signal. Thus, from the information capacity point of view, the system should be able to handle the signal. However, let us assume that the SWI shape is not included in the SWY shape. For such a case we propose the SW-adaptation strategy, which adapts SWI to be included in SWY (Mendlovic and Lohmann [1997]). The adaptation of the SWI shape can be accomplished by using one of the basic optical processes: 9 x-sheafing of the SWI using free space propagation; v-sheafing using a lens; 9 rotation caused by the FRT; 9 x shift, or v shift due to a prism or grating; 9 changing of the aspect ratio (x scaled by a, v by 1/a) by means of image magnification; 9 any combination of these processes. Figure 1 portrayed those shape distortions. Based on this list of processes, the possibilities of obtaining the SW adaptation vary widely as the following examples show. SW adaptation may be accomplished by several of those processes in cascade, as illustrated in fig. 2. We start with a given SWI and SWY. We notice that SWI and SWY satisfy the condition of eq. (29) since they have the same area but different locations and orientations. First, we should adapt SWI to be enclosed by SWY. Now the adapted signal can be transmitted by the system. Since we transformed the original signal, in some cases there is a need for performing an inverse adaptation process after passing through the system (using the same list of processes mentioned above, but in opposite sequence). The final result is the output. In the example of fig. 2, the adaptation process contains three steps. First, a prism shifts the signal along the v direction. Then a FOU rotates the SW chart by 90 ~ Another prism then shifts the SWI chart to be included in the SWY chart. This example is trivial, of course, but it serves to illustrate our concept in general. 9
284
[IV, w 3
OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
I swi(x, v)
Ifneeded Inverse
SW ~ adaptation
...
. pu[
V
SWY
SWl I X
2rism y
;X
ET.
;x
Prism r
Fig. 2. Schematic illustration of the SW adaptation process.
3.3. HYPER SPACEADAPTATION Having information with wide dynamic range and a detector with low dynamic range means that the system will not be able to reconstruct the complete information of the signal. It will show perhaps all of the frequencies existing in the signal, but not in their correct intensities. The tool used for the dynamic range adaptation is a grating, since in the Wigner domain a multiplication of a signal by a grating causes its Wigner chart to be replicated while the height of each replica is decreased. Thus, a grating conserves the volume (the energy) but it enlarges the area of the function occupied in the (x, v) plane. In other words, the SW hyper space adaptation process actually consists of two stages. In the first stage the dynamic range is adapted using a grating. Then, a 2D (area) SW adaptation process is performed as illustrated in fig. 2 in order to feed the frequency distribution in the (x, v) plane of the signal to the acceptance SW area of the system. The Fourier coefficients of the grating regulate the magnitudes of the v-shifted replicas of the original W(x, v). The geometrical super resolution in terms of SW adaptation is illustrated schematically by fig. 3. This is the case where the spatial resolution of the viewed background is much finer than the spatial resolution to be viewed by the sensing device. Figure 3a is the SW function of the signal (SWI), and fig. 3b is the accepted SW of the system (SWY). In order to increase spatial resolution, the following SW adaptation process is applied. First, some of the dynamic range degrees of freedom are converted to spatial degrees of freedom, for example, by a grating. The SWI after this stage is shown in fig. 3c. Then, based on time
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285
Fig. 4. SW hyper space adaptation process illustrating sub-pixeling operation. (a) SWI. (b) SWY. (c) SWY after dividing each pixel into three regions.
multiplexing (using the temporal degrees of freedom), each time slot, a part of the SWI shown in fig. 3c, is transferred. Figure 4 demonstrates another example where SW hyper space adaptation
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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
[IV, w 3
is performed. In this case the adaptation was performed with the help of the detector which is a part of the system. The adaptation was done by dividing each detector's pixel into three regions. This operation allowed the detector to sense spatial frequencies which are three times higher, but it decreased its dynamic range. This operation adapted the SWY to the SWI charts and allowed a full transfer of the signal's information. 3.4. G E N E R A L I Z A T I O N S
So far, the SW chart is a function of the spatial parameters x and v. This can be generalized by taking into account all other parameters: temporal information, wavelength, polarization, etc. This leads to the following SW chart: S W ( x , Vx, y, Vy, t, ~,, POL, . . .).
(31)
This generalized definition is useful for the application of the SW-adaptation process not only with spatial parameters but also with all other proposals for achieving super resolution. Now the adaptation process is done on the M-dimensional SW function where M is the number of parameters involved; for example, M = 7 in eq. (31). For example, if it is known a priori that the signal is temporally constant, then the SW space can be divided into many time slots. Each time slot handles different spatial windows. As a result, the total spatial resolution is increased. This approach is equivalent to the 'time multiplexing' approach for super resolution that was suggested by Francon [1952] and by Lukosz [1966]. 3.5. SURVEY OF T H E E X P L O I T E D SIGNAL C O N S T R A I N T S
The classification and the demonstration of the SW-adaptation process has been discussed by Mendlovic, Lohmann and Zalevsky [1997]. Below is a brief summary. Francon [1952] as well as Lukosz [1966] previously considered this issue. 3.5.1. Restricted object shape
The first family of examples is concerned with the spatial information of the object. One example is the trade-off between the finest detail of the object and its extent Ax. For instance, assume an object with finest detail of 6x that should be captured by a CCD camera whose pixel size is 6XCCD = M 6 x , where M is
IV, w 3]
SUPER RESOLUTION AS SW ADAPTATION
287
Fig. 6. Adaptation using a lens.
a magnification factor. We assume that the signal and the camera (system) have the same number of degrees of freedom (number of pixels). Figure 5 illustrates the example. The ratio A x / 6 x = Ns is not changed by the magnification process. In that sense the process of magnification is a simple case of SW adaptation. Another example of this type of adaptation is a human eye looking at a distant bird. If the bird contains too fine details, once again an adaptation can be executed by a magnification device. Here it is common to use a telescope (Kepler or Galilei). Figure 5 illustrates this type of adaptation. In both examples the object had to be magnified in order to adapt the image resolution to the resolution capability of the detector. The price to be paid is a smaller object field. A third example is connected with coupling an optical signal into a GRIN fiber. Conceptually, the acceptance shape of the fiber in the Wigner plane is a distorted rectangle, sheared along the frequency axis (fig. 6). This phenomenon can be explained in the following manner: Input locations which are close to the upper outer part of the fiber can contribute only negative ray directions (spatial frequencies) which are inserted into the fiber. Locations which are close to the lower entrance of the fiber can contribute only positive ray directions. Input points which are located in the center of the entrance plane (the core) contribute
288
OPTICAL SYSTEMS WITH IMPROVED RESOLVINGPOWER
[IV, w 3
Fig. 8. Adaptation using a fractional Fourier transform.
a certain range of positive and negative directions. On the other hand, a shearing of the Wigner chart of the input signal along the frequency direction can be done using a lens. The focal power of the lens determines the amount of the shearing and should be matched to the GRIN acceptance Wigner shape (fig. 6). The fourth example concerning adaptation of restricted object shapes is related to the fact that the SW of many objects have a higher bandwidth around the center and lower bandwidth toward the edges. For example, a portrait photo with a neutral background belongs to that category. On the other hand, a common SW shape of a system is rectangular. Figure 7 illustrates this phenomenon. In order to transmit all of the signal's information through the system, a relatively big rectangular system SW should be used, which is expensive. Using the adaptation process illustrated in fig. 8, one can reduce the requirements and the price of the system. The input signal is minified so that its Wigner shape will be a rotated square. Then this square should be rotated by 45 ~ using the fractional Fourier transform
IV, w 3]
SUPER RESOLUTION AS SW ADAPTATION
289
Fig. 9. Adaptation using the moir6 effect.
(Lohmann [1993]) which can be easily implemented optically. Eventually, another magnification is performed for full adaptation to the SW shape of the system. These three adaptation steps are implemented by simple optical elements such as lenses and free space propagation distances. Thus, the fact that three adaptation steps are used barely affects the total price of the system. If needed, after transmission, inverse steps of this process might be performed in order to return to the original representation of the signal. Note that the final remark is relevant to most of the SW-adaptation examples. The fifth example is related to the moir6 effect, which was essential for a project called 'spatial pulse modulation' (Lohmann and Werlich [1971]). A spatial high-frequency signal was down-modulated by superposing it with a high-frequency grating. Then, a low-frequency filter eliminated unwanted terms, which were caused by unsuitable Fourier components of the demodulating grating (only two unwanted terms are shown in fig. 9). The output (far fight) is adapted to the low-frequency capabilities of the detector.
3.5.2. Temporally restricted signals A second type of a priori information is related to the time coordinate. In many cases, it is known a priori that the signal changes only slowly as a function of time, or not at all. This allows one to achieve super resolution using time multiplexing. We denote this action as 'temporal adaptation' of the generalized SW function. A different part of the SW chart is transmitted in each time slot. We must assume that the signal is constant over the scan duration. Figure 10 illustrates the first method for performing such a temporal adaptation that is based on synchronized moving pinholes. This method was introduced by Francon [ 1952]. Notice that the quality of the lens influences the light efficiency, but not the resolution. The resolution depends only on the size of the scanning
290
OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER
[IV, w 3
Fig. 11. Time multiplexing by using a pair of scanning gratings.
pinhole. Francon's idea is the root of today's scanning confocal microscopes (Osterberg and Smith [ 1964], Lewis [ 1994]). The second approach was introduced by Lukosz and Marchand [ 1963] and by Lukosz [ 1967]. The method is similar to that of Francon, but instead of scanning pinholes, scanning gratings were used. The system is illustrated in fig. 11. In this suggested system the light efficiency is better and the integration time is shorter. The temporal adaptation of the generalized SW is done as follows: Each diffraction order carries a different section of the spatial frequency spectrum of the object. Each section has its own temporal carrier frequency. The analysis of the encoded information is done with the second synchronously moving grating. A detector with temporal integration is needed at the output. The spatial resolution of the transmitting system is low. Hence, most of the spatial information of the object must be converted into temporal information. w4 contains the details of temporal super resolution.
3.5.3. Waoelength-restricted signals Another type of signal adaptation is connected with wavelength multiplexing (codification) (Kartashev [1960], Bartelt [ 1979a,b]). If the object is color neutral
IV, w3]
White
SUPER RESOLUTION AS SW ADAPTATION
1 Prism
Io(x)
-
Spectrum
291
transparency
Io(a2)
Fig. 12. ~, multiplexing adaptation.
(black-gray-white), one may use the wavelength ~, as a parameter which replaces the x coordinate. An optional system along this direction is shown in fig. 12. Using dispersive prisms, the temporal spectrum of the white light is spread spatially over the spatial input information Io(x). Each slot of the spatial information is encoded with a different wavelength and then sent into a fiber for transmission. The ~, multiplexed signal is transmitted through the fiber and reconstructed (decoded) again in the output using the same prism type: /o(X)
,/o(aA) =r TRANSMISSION :=~/B(a,~)
,/B(X).
(32)
3.5.4. One-dimensional signals A one-dimensional signal with high resolution may be represented as a twodimensional signal with less resolution along each direction (Lohmann and Werlich [1971 ], Bartelt and Lohmann [ 1982]). We start with an SW function that contains two spatial axes, of which one is unused (a 6 function). After adaptation, the SW function contains information in both spatial axes. Then, one performs the transmission through the system, and if needed, there should be an inverse adaptation step that returns the information into a single spatial axis. An example of such an implementation is a moir6 pattern of a one-dimensional highresolution object and a slightly rotated grating which produces a low-resolution two-dimensional raster representation (Grimm and Lohmann [ 1966]).
3.5.5. Polarization-restricted signals The last type of a priori information is connected with the polarization state of the signal. We assume the object to be unpolarized. But the optical system is able to transmit two sets of data in two orthogonal states of information. Here
292
OPTICAL SYSTEMSWITH IMPROVEDRESOLVINGPOWER
%(v)
[IV, w 3
P(v)
h, separated by an inhomogeneous slab in 0 < z < h. Region z < 0, which contains the incident field Ei(r) and the reflected field Er(r), has a real refractive index nr. The index distribution in 0 < z < h, where the field is denoted by Eg(r), may be complex: h(r) = n ( r ) + ire(r). Finally, the refractive index in region z > h, which contains the transmitted field Et(r), is real (nt) for transmission-type elements and complex (ht = nt + iK't) for reflectiontype elements (other possible geometries exist, such as a reflective dielectric multilayer-coated surface profile). Our task is to solve the unknown fields Er(r), Eg(r), and Et(r) from known Ei(r) and the refractive-index distribution. Maxwell's equations reduce to two
350
DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH
[V, w 3
x
n=nt
n =n r
n=n(x)
_ ~ r m Rm
O,n
z
0
h
Fig. 5. Diffraction of a plane wave by a grating located in the region 0 < z < h between two uniform media in half-spaces z < 0 and z > h.
independent sets in the y-invariant geometry of fig. 5. One set describes TE polarization, in which E(x,z) points in the y direction and all other nonvanishing electromagnetic field components (H~ and/-/~) can be expressed in terms of Ey. The other set describes TM polarization, in which E(x,z) lies in the xz plane and the non-vanishing components Ex and E~ can be expressed in terms of Hr. Consequently, the full solution of the electromagnetic diffraction problem can be obtained by solving two independent scalar differential equations instead of the vectorial Maxwell equations. 3.2. RAYLEIGH EXPANSIONS AND GRATING EQUATIONS
Let us assume a periodic two-dimensional index distribution h(x + d, z) = h(x, z) for all z, noting that this condition applies both inside the grating and (trivially) in the uniform regions around it. Then every scalar component U(x,z) of the twodimensional electromagnetic field satisfies the Floquet-Bloch pseudoperiodicity condition, U(x + d,z) = U(x,z) exp(iad),
(3.1)
where a = k n r sin 0, k = 2:r/~., and 0 is the angle of incidence. This condition discretizes the angular spectrum representations of all electromagnetic field components in the homogeneous regions z < 0 and z > h. So-called Rayleigh expansions, Ur(x,z < O) = Z m=--oo
Rm
exp[i(amx- rmZ)],
(3.2)
V, w 3]
ELECTROMAGNETICTHEORYOFGRATINGS
351
O0
Ut(x,z > h) = Z
Tm exp{i[amx + tm(Z-- h)]},
(3.3)
m=--oo
are obtained, where
am = a + 2arm~d,
(3.4)
= / [ ( k n r ) 2 - ~ ] 1/2 rm
if laml
knr,
(3.5)
and, if we assume for simplicity that region z > h is dielectric,
tm
=
[(knt)2-a2] I/2
if [aml ~knt,
i [ a 2 _ (knt)2]l/2
if lam[ > knt.
(3.6)
If the region z > h is metallic, the square root is taken such that the sum of the real and imaginary parts of tm is positive. Expansions (3.2) and (3.3) contain several types of plane waves with complex amplitudes Rm and Tm. All waves with real-valued rm and tm are homogeneous backward- and forward-propagating plane waves, respectively. Waves with imaginary-valued rm or tm are evanescent, with the surfaces of constant amplitude and phase perpendicular to each other. If region z > h is metallic, all plane waves in it are inhomogeneous: the surfaces of constant amplitude and phase are neither parallel nor perpendicular. Inhomogeneous waves (including evanescent waves) decay exponentially with the distance from the reference plane z - 0 or z - h. For homogeneous waves the conventional plane-wave interpretation gives am = knr sin 0 m in region z < 0 and am = knt sin Om in region z > h, where Om are the propagation directions in the appropriate half-spaces. Equation (3.4) gives the grating equation m~,
nr sin Om = nr sin 0 + d
(3.7)
for reflected diffraction orders, and m~
nt sin Om = nr sin 0 + --d-
(3.8)
for transmitted orders. The law of reflection and Snell's law follow from eqs. (3.7) and (3.8) when m = 0.
352
DIFFRACTIVEOPTICS: ELECTROMAGNETICAPPROACH
[V, w 3
3.3. DIFFRACTION EFFICIENCIES
The diffraction efficiency r/m of order m is the fraction of incident flux directed in that order. Its measure is the z-component of the time-averaged Poynting vector, calculated for the mth plane wave in the appropriate Rayleigh expansion. For reflected propagating orders we obtain COS 0m Om -- COS 19
iRm[2
(3.9)
while for transmitted orders, ~m-
Om
nt c o s
nr cos 0
ITml
2
(3 10)
in TE polarization, and ~m -
nr COS Om nt c o s 0
ITml
2
(3 11)
in TM polarization. Evanescent waves do not carry energy in the z direction; i.e., r/m = 0. Hence, for a dielectric grating, energy conservation requires that the sum of the efficiencies of the reflected and transmitted propagating orders is one. For metallic gratings the sum of the efficiencies of reflected orders is less than one and the difference is a measure of absorption. 3.4. EXACT EIGENMODE MODEL FOR BINARY GRATINGS
Let us assume a dielectric lamellar grating profile of fill factor f = c/d in region 0 < z > ~, but the convergence is rather slow, in particular i f f = 1/4 or f - 3/4. A comprehensive analysis of the limits of the thin-element approximation is outside the scope of this review because of the large number of parameters to be considered: d, c, h, n, 0 in both TE and TM polarization (see Pommet, Moharam and Grann [1994]). We saw in w2.3 that binary dielectric transmission gratings can be efficient beam deflectors at Bragg incidence if the period is chosen such that only orders m = 0 and m = -1 can propagate. Early discussions of this concept are due to Loewen, Nevibre and Maystre [ 1979] and Moharam and Gaylord [1982]; further details can be found in papers by Gupta and Peng [ 1993], Noponen and Turunen [1994a], and Gerritsen and Jepsen [1998]. The Bragg-type 3 solution presented in w2.3 is nearly polarization-independent. However, its aspect ratio (ratio of groove depth and minimum feature size) a ~ 5.0 is rather high. The aspect ratio can be reduced if the grating is designed for either TE or TM polarization. If f = 1/2 and h = 1.6342, (a ,~ 3.27), we obtain r/-i = 97.7% in TE polarization. Similarly, i f f = 1/2 and h = 2.1542, (a ,~ 4.3), we have r/_l = 97.9% in TM polarization. Enger and Case [1983] obtained experimentally r/_l > 85% in SiO2 using interference lithography and reactive ion etching. More recently, Glaser, Schr6ter, P6hlmann, Fuchs and Bartelt [1998] obtained r/_l - 96%. Nguyen, Shore, Bryan, Britten, Boyd and Perry [1997] measured r/_l = 94% for a grating with d = 350nm and demonstrated that corrugated SiO2 surfaces are as resistant to laser-induced damage as flat SiOe surfaces, which confirms the applicability of transmission-mode diffractive optics in high-power visible and UV laser technology. The Bragg-angle selectivity as well as the dependence of r/_l o n f and h/2, are analyzed in fig. 7 for the polarization-insensitive solution presented in w2.3. The angular selectivity is much weaker than in conventional volume gratings because of smaller relief depth and higher index modulation (small value of the KleinCook parameter). Reasonable fabrication errors in the fill factor f are tolerated, especially in TM polarization, and the design is not exceedingly sensitive to depth errors. Consider a metallic binary reflection grating with real refractive indices nr = n l = n and complex n2 = nt = h in fig. 2. The thin-element approximation still gives eqs. (4.1) and (4.2), but now with A p - 2nh. Some limits of
3 Note that these gratings are not Bragg gratings in the classical sense: the Klein-Cook parameter Q = 2;r(Md)(h/d), which should satisfy Q >> 1, is only ~ 10.
V, w 4]
LINEAR GRATINGS OPERATING IN FIRST-ORDER MODE
(a)
(b)
1[~,~~'~ 0.8 J
1 /
(c)
~
1
0.8
06~fi~
0.8
~ 06
0.4
~
0.2 0
~ 06
0.4
~
0.2 5
10
15
20
361
25
30
0 0.3
35
o.,
0.2 0.4
0.5
0.6
0 [deg]
0.7
0.8
0.9
0 1.2
1.5
2
f
2.5
3
h/~,
Fig. 7. Dielectric binary Bragg gratings. Dependence of diffraction efficiency on (a) angle of incidence, (b) fill factor, and (c) groove depth. Solid lines: TE polarization. Dashed lines: TM polarization.
Table 1 Efficiencies r/_ 1 of binary reflection gratings made of real metals but optimized by assuming perfect conductivity (Bragg incidence) Metal
A = 488nm
A = 633 nm
A = 1064nm
TE
TM
TE
TM
TE
TM
Ag
0.874
0.634
0.919
0.184
0.972
0.956
A1
0.892
0.736
0.877
0.805
0.936
0.914
Au
0.228
0.137
0.940
0.695
0.983
0.966
Cr
0.596
0.423
0.535
0.423
0.519
0.434
Cu
0.484
0.300
0.899
0.600
0.962
0.923
Mo
0.473
0.367
0.436
0.365
0.600
0.424
Ti
0.352
0.239
0.419
0.294
0.434
0.342
this approximation have been evaluated for perfectly conducting gratings by Gremaux and Gallagher [1993]. Hessel, Schmoys and Tseng [1975], Jull, Heath and Ebbeson [1977], and Cheo, Schmoys and Hessel [1977] have shown that r/_l = 100% can be achieved for many combinations of d/~, f and h, and simultaneously for TE and TM polarization. If only one polarization state is of interest, one possible solution in TM mode is d = ~, n = 1, f = 0.5 and h = 0.234~.. In TE mode, deeper and wider grooves are needed; for example, d = ~, n = 1, f = 0.75, and h = 0.407~. Little attention has been paid to finitely conducting gratings even though it is known that the assumption of perfect conductivity fails at near-infrared and visible regions. As illustrated in table 1, where the two structures presented above are considered, the efficiency depends strongly on the choice of metal
362
DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH
(a)
(c)
(b)
1
1
0.8
0.8
1 0.8
~ ""
i"
r
~
[V, w 4
0.6
0.6
0.4
0.4
0.2
0.2
0 10
20
30
40
50
0 [deg]
0 0.8
~ ~
0.6 0,4 0.2 0
0.9
1
f/,/i~
11
2
0.8
0.9
1
1.1
1.2
h/h o
Fig. 8. Binary AI Bragg gratings at A = 633 nm. Dependence of r/_ l on (a) angle of incidence, (b) normalized fill factor and (c) normalized groove depth. Solid lines: TE polarization. Dashed lines: TM polarization.
and the operating wavelength. At ~. - 488 nm the efficiency is poor for all metals, especially in TM polarization. At 2. = 633 nm, A1 is reasonably good also for TM polarization, and at 2. = 1064 nm several metals give acceptable values of r/_l. The efficiencies can be improved by readjusting f and h: for A1 gratings we obtain 17-1 = 0.898, 0.882, and 0.940 for 2 = 488 nm, 633 nm, and 1064 nm, respectively, in TE polarization. In TM polarization the corresponding efficiencies are 0.865, 0.854, and 0.926; i.e., the improvement is substantial in visible light. Figure 8 illustrates more closely the properties of the reflection-mode A1 gratings at 2. = 633 nm. Here j~ and h0 represent the (different) ideal values of fill factor and groove depth for each state of polarization. The near absence of Bragg selectivity, which is a characteristic of reflection-mode solutions in TM polarization, is evident from fig. 8a. In TE mode the solution is nearly independent o f f over a wide range. Yokomori [ 1984] showed that many profile shapes in addition to the binary profile can give high efficiencies at Bragg incidence if only two orders can propagate, and Miller, de Beaucoudrey, Chavel, Turunen and Cambril [ 1997] demonstrated slanted binary gratings, which display the Bragg effect at normal incidence and give high first-order efficiencies. 4.2. MULTILEVEL GRATINGS
High-efficiency deflection of an on-axis light beam is accomplished traditionally using triangular (blazed) grating profiles or, for convenience of fabrication by certain lithographic methods, their Q-level staircase approximations. Considering transmission gratings with step height ~.(Q- 1)/Q(n- 1), the thin-element approximation gives r/_l = TsincZ(1/Q),
(4.3)
V, w 4]
LINEAR GRATINGS OPERATING IN FIRST-ORDER MODE (a)
(b)
0.8 =
(D . ,...,
363
0.8
~=
0.6
o.6
04
0.4
0.2
0.2 0 2
4
6
8
0
1
2
4
d/A,
6
8
10
d/A,
Fig. 9. Efficiencies of Q-level stair-step SiO2 gratings. (a) TE polarization. (b) TM polarization. Solid lines: Q = 16. Dashed lines: Q = 8. Dotted lines: Q = 4.
(a)
(b) ,
0.8
1
0.8
0.6
.
.
.
.
.
.
.
.
.
.
.
.
.
0.6
-
t~
~ o.4
0.4
0.2
0.2 '
2
4
6
dA
8
10
0
1
2
4
6
8
10
d/k
Fig. 10. Same as fig. 9, but for a silicon grating.
where T is the Fresnel transmission coefficient. In fig. 9 we assume normal incidence from substrate (n = 1.46) to air: the results of eq. (9) significantly exceed the rigorous results when d is comparable to ~. A sharp drop occurs when d ~ 2~, followed by a peak at d ~ 1.5+l; an explanation in terms of multiple scattering was offered by Noponen, Turunen and Vasara [1993] (the peak disappears in the case of normal incidence from air to substrate). Some further comparisons with approximate methods were presented by Pommet, Moharam and Grann [1994]. Gratings made of high-index semiconductor materials for infrared applications suffer a far less dramatic reduction of efficiency than SiO2 gratings when the period is reduced, because the profile is considerably shallower. Figure 10 shows the efficiency curves for ideal multilevel Si gratings for +l = 1550nm (n = 3.474). In TM polarization, r/_l actually reaches its maximum at the 2Z Wood anomaly. Because of high n, the asymptotic values of ~7-1 in the limit d/~ ~ ~ are relatively low, but use of antireflection coatings significantly
364
DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH (a)
[V, w 5
(b)
.
0.8
0.8
0.6
~ 0.6
0.4
0.4
0.2
0.2 0 d/~
10 d/X
Fig. 11. Efficiencies of four-level A1 gratings. (a) TE polarization. (b) TM polarization. Solid lines: A = 1064nm. Dashed lines: ~. = 633 nm. Dotted lines: A = 488nm.
improves the efficiencies (see Pawlowski, Engel, Ferstl, Ffirst and Kuhlow [1994]). For example, a single-layer ~,/4 coating (TazOs, n = 2.054, thickness 0.122~,) improves the efficiency of a four-level grating with d = 5Z from 51.9% to 75.7% in TE polarization, and from 57.2% to 79.2% in TM polarization. Additional results for coated Q-level gratings were presented by Kleemann and Gfither [ 1998]. For metallic reflection gratings the groove depth is also rather shallow, ~,(Q- 1)/2Q, and the reduction of the efficiency is consequently modest, as shown by Shiono, Kitagawa, Setsune and Mitsuyu [1989] in TE polarization. Figure 11 illustrates the efficiency curves of four-level metallic gratings for three different wavelengths in both TE and TM polarization. In TE polarization, the efficiency is nearly constant down to d ~ 3~ and then drops rapidly. In TM polarization, strong Wood anomalies occur at integer values of d/Z, but efficiencies substantially higher than the thin-element value are observed for small d/~,. Shiono and Ogawa [ 1991 ] have used depth compensation to improve the efficiency in case of off-axis incidence.
w 5. Linear Gratings Operating in Zeroth-Order Mode According to the grating equations (3.7) and (3.8), only the zeroth transmitted and reflected orders can propagate if the period d is sufficiently smaller than Z. One might think that little can be done with gratings in this domain, but in fact such subwavelength-period gratings can introduce substantial wavelengthdependent effects in the division of power between the two zeroth orders, much like thin-film stacks (c.f. w2.4). Additionally, the transmittance and reflectance properties of subwavelength-period corrugated surfaces may depend strongly on
V, w 5]
LINEAR GRATINGS OPERATING IN ZEROTH-ORDER MODE
(a)
(b)
9. TM 0.8
1
"". . . . . . . . . . . . . . . . "'.
0.8
0.6 ;.7..1
365
............ '..
~ o.6
0.4
~
0.2
~
0
~
0.4 0.2 0
0
0.5
0
d/Z,
0.5
1
d/'X
Fig. 12. Efficiencies of the zeroth transmitted orders of metal stripe gratings made of (a) A1 and (b) Au. Solid lines: h = 50nm. Dotted lines: h = 200 nm. polarization, facilitating the construction of polarization components as pointed out in w2.2. 5.1. M E T A L L I C S U B W A V E L E N G T H T R A N S M I S S I O N GRATINGS
Perhaps the best known subwavelength-period grating is the conducting wire-grid polarizer, which transmits TM-polarized light but reflects TE-polarized light. These components are employed extensively at long wavelengths, such as in radio engineering. Subwavelength-period wire grids are difficult to fabricate for visible light, but the same effect is achieved with an array of metal stripes (thickness h, width d/2, rectangular cross section) on a dielectric substrate. These structures can be made using, e.g., electron-beam or phase-mask lithography followed by lift-off or electroplating. Figure 12 illustrates rigorously calculated TE and TM mode transmission curves for A1 and Au metal-stripe gratings for two values of h. When d = 200 nm, good polarizers are obtained when d ~ 0.5~. or smaller. This is indeed the conventional domain of operation of metal-stripe polarizers. With smaller values of h, such as 50nm, the TE transmission increases in the range d 3 matrices Ae, Br and Dr, and P is an infinite-dimensional vector with the non-zero c o m p o n e n t - ~ 1F6~,0I.
VI, w 3]
THEORETICALMETHODS
403
The matrix equation (19) is a simple differential equation with timeindependent coefficients, and is solved by direct integration. For an arbitrary initial time to, the integration of eq. (19) leads to the following formal solution for Y(t): Y(t) = Y(to)e ~ t - (1 - eKt) K-1p.
(20)
In order to proceed further, we must truncate the dimension of the vector Y(t). The validity of the truncation is ensured by requiring that the solution (20) does not change as the dimension of Y(t) increases or decreases by one. Because the determinant of the finite-dimensional (truncated) matrix K is different from zero, there exists a complex invertible matrix T which diagonalizes K, and/~ = T-1K T is the diagonal matrix of complex eigenvalues. By introducing L = T-IY and R = T-1p, we can rewrite eq. (20) as L(t) = L(to)e )~t- ( 1 - e '~t) ~,-IR,
(21)
or, in component form, q
Zi(t) = Zi(to)e )~it- Z
(/~-l)ij ( 1 - e zjt) Rj,
(22)
j=l
where q is the dimension of the truncated matrix. To obtain solutions for X/.~0(t), we determine the eigenvalues ~.; and eigenvectors Li(t) by a numerical diagonalization of the matrix K. The steady-state values of the components X/It)(t) can be found from eq. (22) by taking t ~ ~ , or more directly by setting the left-hand side of eq. (19) equal to zero. Thus, q Y/(oo) = - Z (K-1)/j j=l
PJ"
(23)
In subsequent sections, we will use the solution (23) to calculate the stationary fluorescence intensity, Is = [89 +X3~~
,
(24)
and absorption and dispersion coefficients of the probe fields, W(~o) = Z'(O)) = -Im [X~-')(~)],
(25)
D(o)) = Z " ( o ) ) = - R e [X~-')(oo)],
(26)
where 2'~(0)) and 2'"(09) are the real and imaginary parts, respectively, of the field susceptibility. We will use eq. (22), together with the quantum-regression
404
SPECTROSCOPY IN POLYCHROMATIC FIELDS
[VI, w 3
theorem (Lax [1968]), to calculate the incoherent and coherent parts of the fluorescence spectrum:
F(m) = ReX(~
(27)
oo
=
6 (m- ms + g6),
(28)
~ ---- - - ( X )
where v = ( m - ms)/F and X( 0 = limt ~ ~
X~)(t).
3.2. D R E S S E D - A T O M M E T H O D
In this section, we present the dressed-atom method of calculating the fluorescence and absorption spectra of a two-level atom driven by a polychromatic field. The method was first introduced by Cohen-Tannoudji and Reynaud [1977] for a monochromatic driving field. In this picture, every field mode whose Rabi frequency 292 is sufficiently large that g2 > F is considered to 'dress' the atom, and to form along with it a single, entangled quantum system. This reflects the fact that photons are exchanged between the atom and driving field modes via absorption and stimulated-emission processes many times between successive spontaneous emissions by the atom into the vacuum field.
3.2.1. Dressed states The Hamiltonian of the entangled system is written as H = HA +HE + W = H 0 + W,
(29)
where HA is the Hamiltonian of the atom, given by eq. (2), P
HL=hZoi(a~ai+ 89
(30)
i=l
is the Hamiltonian of the driving field, P
W=hZgi(a~S-+S+ai )
(31)
i=1
is the interaction (in the RWA) between the atom and the driving field, and the coupling constant between the atom and the ith field mode.
gi is
VI, w3]
THEORETICALMETHODS
405
The 'undressed' states of the system are the eigenstates of H0. They can be written as direct products of the form la) | ]Nl) | | ]Np), where ]a) is an atomic state (a = e, g), and Ni is the number of photons in driving mode i. We diagonalize the total Hamiltonian (29) in the basis formed by these undressed states. The interaction W has non-vanishing matrix elements only between those undressed states between which the atom has a non-vanishing dipole moment and the number of photons in one of the field modes changes by one; the matrix elements are thus proportional to v/Ni + 1 or x/~/. However, for strong driving fields (Ni) >> 1, we can approximate
gi v/Ni + 1 ~ gi ~
~ . . . . 2-Qi,
(32)
where 2g2i is the (on resonance) Rabi frequency of mode i. The eigenstates of H, denoted by In, N), are the dressed states of the system, and must be calculated individually for every system considered. In general, they are grouped into manifolds, which are labelled s where N is the total number of excitations of the states in oe(N). The dressed-atom method is most useful when the energy differences between the dressed states In, N) and Im, N) are large compared with the damping rate F: ](_Dnm] > 1". It is then possible to make the secular approximation, in which we neglect coupling between diagonal and off-diagonal elements of the density matrix. The calculations performed within this approximation are valid only at the lowest order in F~ [(_Dnm].
3.2.2. Fluorescence Next, the dressed atom is allowed to interact with the vacuum field. This results in spontaneous emission by the dressed atom down its energy manifold l a d d e r - equivalent to fluorescence in the bare-atom picture. The temporal and spectral properties of this emission/fluorescence are a signature of the entangled system, its energy levels and their populations. Additionally, the system may be probed by measuring the absorption or dispersion of a weak applied field, nearly-resonant either with the driven transition (Mollow absorption) or with a transition of the atom between one of the driven levels and a third level ]d) (Autler-Townes absorption). The fluorescence and Mollow absorption processes involve transitions between dressed states of the system, and depend on nonvanishing matrix elements of the atomic dipole moment operator between states in which the numbers of driving field photons remain constant. If the atom has no permanent dipole moment (see w4.3.3), these occur only between neighboring manifolds, and we denote them by/t,,, = l/t[ (n,N IS+[ m , N - 1).
406
SPECTROSCOPYIN POLYCHROMATICFIELDS
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The probability of a transition between the dressed states [n,N) and m, N - 1) is Fnm = F [/tnml2, and the total spontaneous emission decay rate from In, N) into the manifold below is given by
Fn=ZFnm.
(33)
m
3.2.3. Populations The populations of the dressed states P n , N = (n,N [p[ n,N) are the diagonal elements of the density matrix of the system, and are found by projecting the master equation (1) onto In, N) on both the left and the fight. In the secular approximation, the reduced populations P, = ~-~X P n , N (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 1992]) satisfy the following set of coupled equations of motion: d -~Pn(t) :-FnPn(t) + ~ F,mPm(t). (34) m
The first term on the right-hand side of eq. (34) is due to transitions out of In, N) into the manifolds below, and the second term to transitions into In, N) from the states Im, N + 1) of the manifolds above. The intensity of every spectral line which corresponds to a transition originating in In, N), whether in fluorescence or in absorption, is proportional to Pn. The populations P~ can be used as well to calculate Pc, the population of the excited state le) of the strongly driven atom. The total intensity of fluorescence by the atom is then FPe.
3.2.4. Coherences The projection of the master equation (1) onto ]re,N - 1 ) on the right and (n,N on the left gives the equations of motion for the coherences flnm,N -- (n,N [p[ re, N - 1 ) between the dressed states of the neighboring manifolds. If there is only one transition which has (Bohr) frequency r the reduced coherence Pnm = ~--~X Pnm,U obeys the equation -~ Dnm
[iOOnm+ F~"m)]Dnm,
(35)
where F (nm) is the coherence damping rate, F(nm) _. -21(F n q_ Fm ) q_Dnm,
(36)
and D n m takes into account the 'transfer of coherence' between manifolds (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 1992]). The transition frequen-
vI, w3]
THEORETICALMETHODS
407
cies O)~m give the frequencies of the lines in both the fluorescence and nearresonance absorption spectra. If more than one transition has the same Bohr frequency, however, the evolution of their corresponding coherences may be coupled. A simple general expression cannot then be given for the evolution; each case must be studied individually.
3.2.5. Fluorescence spectrum The steady-state fluorescence spectrum is given by the real part of the Fourier transform of the two-time correlation function of the dipole moment operators, F(co) = Re f o o d r e i'~ lim (/t + (t + r)/t-(t)) t --+ OO
(37) d r e i~~ lim Ztl,,m(a~m(t+r)ll-(t)), = Re f0 ~176 t ----+ OO nm
where (Ynm = ] n , N ) ( m , N - 1 [ . From the quantum regression theorem (Lax [1968]), it is well known that for r > 0 the two-time average (a.m (t + r ) g - ( t ) ) satisfies the same equation of motion as the one-time average (a.m(t)) with the initial condition
(38)
(Onm(t)ll-(t)) : FnmPn,
where rnm is the probability of the transition from n to m and Pn is the reduced steady-state population of level n. The one-time average (a,m(t)) satisfies the same equation of motion as the coherence pmn(t). When the spectral lines do not overlap, the fluorescence line corresponding to the transition n =~ m is simply (39)
Fnm((-1)) = F n m P n L n m ( ~ ) ,
where 1
r (nm)
L.m(OO) -
(40) \
/
The line is centered at the Bohr frequency O~,m, has a width determined by the damping rate of the coherence F~~"'), and an intensity (in the steady state)
408
SPECTROSCOPY IN POLYCHROMATIC FIELDS
[VI, w 3
proportional to the product of the steady-state population of state n and the transition rate I " n m from n to m. The integrated fluorescence intensity Is = f F(~o)d~o is proportional to the population Pe of the excited state le) of the bare atom, found using the dressed populations it',.
3.2.6. Near-resonance absorption spectrum The absorption spectrum of a weak probe beam monitoring the driven system is given by the real part of the Fourier transform of the two-time commutator ([W(t),W(t + r)]). The term (W(t)W(t')) is associated with absorption, and the term (/t+(t~)/t-(t)) with stimulated emission of the probe beam: the net absorption between the levels is equal to the difference between absorption and stimulated emission by the atom. Using the same procedure as for the fluorescence spectrum, we find that the absorption line corresponding to a transition from n to m is (41)
J/Vnm(O))- - I-'nm(Pm - Pn) Lnm(fO),
where Lnm(O)) is given by eq. (40). The components of the absorption spectrum have the same positions and widths as their counterparts in the fluorescence spectrum, but widely different intensities. The net absorption at any frequency is proportional to the transition rate F,,m and the difference between the populations of the lower and upper levels in the transition. If the population difference is positive, the probe is absorbed by the system, whereas it is amplified if the difference is negative. The central component of the absorption spectrum is not given correctly by the dressed-atom method within the secular approximation. This component is reproduced correctly only in higher order (Grynberg and Cohen-Tannoudji [1993]). We can also calculate the refractive index of the probe beam, which is proportional to the imaginary part of the two-time commutator. The index is given by a sum of dispersive profiles, where the profile corresponding to the transition n :=~ m is D,m(~O) = 1 F~m (Pm -- P,)
6O- 09,,,,
2"
(42)
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BICHROMATIC FIELDS
409
3.2.7. Autler-Townes absorption spectrum The structure and the population distribution of the dressed states can also be studied by monitoring the system with a weak probe beam coupled to a third atomic level Id), connected to either Ig) or ]e), which has a transition frequency tod different from tOo (Autler and Townes [1955]). In the transition from Ig, N) to Id, N), for example, the number of drivingfield photons remains constant. The atomic feature is split by the presence of the driving fields into components, whose intensity is the product of Pn and a weight factor I(g,N n,N)l 2. Thus the number of components is equal to the number of dressed states 'contaminated' by Ig, N). The shift of each component is determined by the shift from NhtOo of the energy of In, N), and the width by 1_ 2 (Fd + Fn), where Fd is the natural width of level Id)
w 4. Bichromatic Fields The subharmonic resonances appear in the absorption and dispersion spectra of an intense field used to probe a monochromatically strongly driven atom; i.e., the so-called 'strong probe'. However, a 'strong probe' is a contradiction in terms. In reality, the system consists of an atom that is being driven by two strong fields, and which should be probed by a (third) weak field. If this weak field is simply the vacuum, we observe fluorescence by the system. Therefore, in this case one could expect significantly different spectral properties of the fluorescence from those observed with monochromatic driving. Zhu, Lezama, Wu and Mossberg [1989] measured the fluorescence spectrum of a two-level atom driven by a bichromatic field consisting of two components of equal amplitude displaced symmetrically by frequency 6 about the atomic resonance. The observed spectrum was composed of a central component at the atomic resonance frequency together with a series of sidebands separated equally by 6. The most striking features of the spectrum were that the positions of the sidebands were independent of the Rabi frequency, which, however, determined their number and relative intensities (in a complicated manner). This was in complete contrast with the situation for the Mollow triplet observed for monochromatic driving (Schuda, Stroud and Hercher [1974], Wu, Grove and Ezekiel [1975], Hartig, Rasmussen, Schieder and Walther [1976]), in which the displacements of the sidebands depended (linearly) on the Rabi frequency, while their number (2) and intensities did not. These observations were explained by a number of theoretical analyses of the fluorescence spectrum (Kryuchkov
410
SPECTROSCOPYINPOLYCHROMATICFIELDS
[VI, w4
[ 1985], Tewari and Kumari [ 1990], Freedhoff and Chen [ 1990], Agarwal, Zhu, Gauthier and Mossberg [1991], Agarwal and Zhu [1992], Ficek and Freedhoff [ 1993]). The specific structure of the dressed states was also confirmed in another experiment by Papademetriou, Van Leeuwen and Stroud [ 1996], who observed the Autler-Townes spectrum of a bichromatically driven, two-level transition in a three-level cascade atom. There have now been several experimental investigations as well which examined the response of a two-level atom to a bichromatic field composed of one strong and one weaker component. This special case of bichromatic excitation has led to the discovery of many interesting effects, such as phasedependent atomic dynamics (Wu, Gauthier and Mossberg [1994a, 1994b]), the double Stark effect (Yu, Bochinski, Kordich, Mossberg and Ficek [1997]), and the multi-photon ac Stark effect (Greentree, Wei, Holmstrom, Martin, Manson, Catchpole and Savage [ 1999]). If the time evolution of the fluorescence intensity is observed (Wu, Gauthier and Mossberg [1994a, 1994b]), it exhibits a modulation, and the atomic response depends on the phase even if the interaction begins with the atoms in their ground states (Chien, Wahiddin and Ficek [1998]). In the double Stark effect, the components of the Mollow triplet are each split into a triplet, and additional triplets appear at harmonics of the Rabi frequency (Ficek and Freedhoff [1996]). The multiphoton ac Stark effect appears whenever the frequency of the weaker component is near the multiphoton resonances induced by the strong component, and is manifested by the appearance of additional triplets at subharmonics of the Rabi frequency. A theoretical interpretation of these results has been given in terms of Floquet states and with a doubly-dressed atom model that provides a simple physical explanation of the observed features (Rudolph, Ficek and Freedhoff [1998], Rudolph, Freedhoff and Ficek [1998a,b]). 4.1. SYMMETRICEXCITATION We consider first a symmetric excitation in which the components of the bichromatic field have equal Rabi frequencies and are displaced equally about the atomic resonance. We calculate the dressed states of the entangled atom-field system, and use these to explain the observed spectra. 4.1.1. Dressed states
The Hamiltonian of the non-interacting atom and bichromatic field is given by Ho = h~ooS z + h (090 + 6) at+a+ + h (O9o - 6) at_a_,
(43)
vI, w4]
BICHROMATICFIELDS
411
and has the eigenvalue equations Ho IZn, 2 N ) = [EzN + 2nh6] IZn, 2 N ) , Ho IZn - 1 , 2 N ) - [E2N + (2n - 1) h6] IZn - 1,2N),
(44)
where a• ( a ~ ] are the annihilation (creation)operators for the driving modes \
/
of frequencies (-Oo + 6, E 2 N = 2Nhooo, and Ig, N + n, N - n) = 12n,2N) and ]e, N + n - 1, N - n) - 12n - 1,2N) are the 'undressed' states of the noninteracting system corresponding to energy manifold E(2N). When we include the interaction between the atom and the bichromatic field, the states recombine to form a new ladder of states (dressed states) (Freedhoff and Chen [1990]): 2N, ~ , m
=
J,-m - - - g
In, ZN),
(45)
n=--OO
corresponding to energies E~N) = E2N + mh6,
m = 0, 4-1, + 2 , . . . .
(46)
In a similar way, the undressed states ]g,N + n + 1 , N - n) and le, N + n , N - n) of manifold E (2N + 1) recombine to form the eigenstates of that manifold. Each manifold contains an infinite number of states. Neighboring manifolds are separated by frequency co0, while neighboring states within each manifold are separated by 6. We now introduce the interaction between the atom and the vacuum field, leading to a spontaneous emission cascade by the dressed system down its energy manifold ladder. It is straightforward to show that there are two distinct transition rates between manifolds of even and odd numbers of excitations. The transition rate from [2N + 1,-~, m) to IZN, ~, ~ m' ) is Fmm' = -~F 6ram' + (-1)mJm'-m
-~
(47)
,
Q m t ) is while the rate from IZN, -8,m) to 12N - 1, ~, I'm,,, : ~F
6mm' -- (-1)'J~,_,,,
-~
(48)
The total transition rates out of 12N + 1,-~,m) and 12N, ~, ~2 m) are then given by J0 - ~
(49)
where (+) and (-) stand for the rates from the dressed states ]2N + 1 -g,m) e and 12N, ~e , m ), respectively. ,
412
SPECTROSCOPYIN POLYCHROMATICFIELDS
[VI, w 4
d
(5
(5 . i
o4
d 1--
d
0
0
1o
20
3o
2fl/r
(b) O
,
,
i
i
1 /0
I
i
i
0
,
.
20
,
,
.
l
30
n/r Fig. 6. Stationary total fluorescence intensity for a symmetric bichromatic field plotted (a) as a function of 2#2 and constant 6 = 5F, (b) as a function of 6 and constant 2g2 = 20F. The p o p u l a t i o n s o f the dressed states are then found f r o m the rate equations. F r o m them, the steady-state p o p u l a t i o n s o f the atomic levels are f o u n d to be
ee,g((X)) =
1(
~
4Jo(-4#2/6))
1 :t: 3 + Jo ( - 8 . Q / O )
"
(50)
The steady-state total intensity o f fluorescence by the b i c h r o m a t i c a l l y driven a t o m is Is =
FPe(oO). This
expression was also derived by K r y u c h k o v [1985],
vI, w4]
BICHROMATICFIELDS
413
using a method based on quasi-energy states, and is plotted in fig. 6a for constant 6 = 5F as a function of 2s and in fig. 6b for constant 2f2 = 20F as a function of 6. These figures display the 'subharmonic-resonance' behavior observed experimentally by Chakmakjian, Koch and Stroud [ 1988], with maxima appearing in Is or, equivalently, minima in the absorption by the 'strong probe', at the zeros of the function J0(-4f2/6). 4.1.2. F l u o r e s c e n c e
spectrum
We obtain an analytical expression for the fluorescence spectrum by studying the evolution of the coherences between dressed states in neighboring manifolds. There are four principal (reduced) coherences, O0s and o0, both of which correspond to polarization of the system at frequency COo, and azk and azk-l, corresponding to polarizations at even and odd sideband frequencies, respectively. These can be shown to satisfy the equations of motion
-
d ~ I F +) ~o~ = - (~i0)o dt -
d
+
d dtO2k =~do 2 k - ~ td
= -
[ [
OOs, (3 _ jo ( _ 8 ~ )
i (0)o + 2 k 6 ) + l F i (~o + (2k -
(( 8J))l ((82))1 3 - Jo - - -
1) 6) + 1F
(51)
o2k,
3+Jo
-~
O2k-1.
At frequency 0)0 there are two lines, with widths ~l F a n d G = ~1 F [ 3 - J o ( - ~ )] 9 The even sidebands are located at frequencies 0)o + 2 k 6 and also have widths Fe. The odd sidebands are located at 0)o + ( 2 k - 1)6, and their widths are Fo = ~1 F [ 3 + J o ( - 8 ~-2- ) ] . The full spectrum is given by the expression Fe [Q + J0 (-4.(2/6)] J0 (-4f2/6) F { F 1 + QJ0 (-4s + F(0)) = ~-~ ~ (0)_ 0)0)2 + (F/2)z 4 (~0- ~00)2 + r 2
/;
O(9
+ Z k=-oG
J2k (-4f2/6)
(o) - ~o - 2k 6) 2 + F 2
OO
+ Z k=-o~
J2k-I(-4.(2/6)
(0)-0)0-(2k-1)6)2+Fo 2
' (52)
414
SPECTROSCOPY IN POLYCHROMATICFIELDS
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(*) 3 8~
v I~
O
-10
0
10
0
10
(b)
3 v i,
tl~ O
6
-10
(:~
-
%)/r
Fig. 7. Fluorescence spectrum for a symmetric bichromatic field with constant 6 = 2 . 5 F and two different Rabi frequencies: (a) 2s = 10F; (b) 2s = 6/'. The parameter values correspond to the experimentally observed spectra, presented in figs. 2(a) and 2(b) of Zhu, Wu, Lezama, Gauthier and Mossberg [ 1990].
where 4J0 (-4g2/6) O = 3 + J0 ( - 8 Q / O ) "
(53)
This expression is plotted in fig. 7 for 6 = 2 . 5F and two different Rabi frequencies: 2g2 = 10/" (fig. 7a) and 2g2 = 6/" (fig. 7b). The theoretical curves are in excellent agreement with the experimental observations, presented in figs. 2a and 2b of Zhu, Wu, Lezama, Gauthier and Mossberg [1990]. The intensities of the sidebands at +k6 are proportional to J~ (-492/6), so the number of sidebands observed increases with (2/6. The widths of the odd sidebands in the experimental spectra are somewhat less than those in their theoretical counterparts. This narrowing results from the contribution to the odd sidebands of the coherent scattering (Agarwal, Zhu, Gauthier and Mossberg [ 1991 ]), which is not included in the theoretical expression (52).
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BICHROMATIC FIELDS
415
Other techniques have also been used to study the interaction of a twolevel atom with a symmetric bichromatic field. Wilkens and Rza~ewski [1989] proposed a purely numerical technique based on the It6 prescription of the integration of the time-dependent differential equations. Tewari and Kumari [1990] and Kryuchkov [1985] have derived analytical formulae for the spectrum using an iterative procedure on the damping parameter and the separated lines approximation, respectively. This latter technique has also been used to calculate photon statistics and squeezing under bichromatic excitation (Kryuchkyan [1991 ], Kryuchkyan and Kheruntsyan [1992], Kryuchkyan, Jakob and Sargsian [1998]), and to explain the experiments on the two-photon decay of Rydberg atoms in a bichromatically driven cavity (Lange and Walther [1994], Lange, Walther and Agarwal [ 1994]).
4.1.3. Near-resonance absorption spectrum A nearly-resonant probe field can also be used to monitor the energy structure and population distribution of the system. We calculate the spectrum with the Floquet method, however, because the dressed-atom approach will reproduce the structure at the central frequency only in higher order (Grynberg and CohenTannoudji [1993]). In fig. 8, we plot the absorption spectrum for a symmetric bichromatic field with 292 = 14F and 6 = 5F. The spectrum shows a series of dispersive features located at n6, (n = -t-1, + 2 , . . . ) with no structure at the central frequency too. As in the fluorescence spectrum, the separations of the sidebands are independent of the Rabi frequency of the driving field, but their number increases with g2. However, depending on g2 the spectrum can also exhibit a large absorption peak at the central frequency. This is shown in fig. 9, where we plot the absorption spectrum for 6 - 5F and 292 = 15F. The dispersive structure of the sidebands and their constant separation can be explained quantitatively by the dressed-atom model. To explain the oscillation of the central peak amplitude with the Rabi frequency, we refer to the optical Bloch equations (8) and the oscillatory properties of the steady-state fluorescence intensity, shown in fig. 6a. For A = 0 and a = 1, we can write the absorption spectrum as 1
W(to) = ~F
(i s _
1
/-'
2) !F2 + ( t o - too) 2 + 2Re Ul~
=
-i(,o)-~oo)/r'
(54)
4
where Is is the steady-state fluorescence intensity and U~~ is the Laplace transform of the zeroth-order harmonic component of U(t)= ~1 [(5'-(t)) + (S+(t))].
416
SPECTROSCOPY IN POLYCHROMATIC FIELDS
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? o x If)
I
v o 3=
? o x ut) I
I
I
I
I
o
-20
I
2o
(o - % ) / r Fig. 8. Near-resonance absorption spectrum for a symmetric bichromatic field with 2f2 = 14F and di = 5F.
v
d
-
20
.
.
0
.
.
20
(~ -~o)/r Fig. 9. Near-resonance absorption spectrum for a symmetric bichromatic field with 2~2 = 15F and 6 = 5F.
We see from eq. (54) that the central component of the absorption spectrum is distinct from the remaining features, depending only on Is. The oscillations with ~2 of its amplitude reflect the oscillations of the fluorescence intensity, shown in fig. 6a. Whenever Is ~ ~, 1 there is no central line in the absorption spectrum. [There is also a contribution (dispersive-type) to the central component
VI, w 4]
BICHROMATIC FIELDS
417
from U(~ its amplitude, however, is very small and it does not affect the spectrum.] 4.1.4. Autler-Townes absorption spectrum
In fig. 10, we plot the spectrum of the Autler-Townes absorption to a third atomic level Id) when a symmetric bichromatic field drives the Ig) --* Ie) transition. The spectrum shows a series of peaks separated by 6, revealing the constant b-dependent separation of the dressed states.
3'
O_4o
.
. - 2 0.
.
!
0
20
4O
Fig. 10. Autler-Townes absorption spectrum for a symmetric bichromatic field with 2s 6=5FandF d= IF.
= 20F,
This multi-peaked Autler-Townes spectrum was observed experimentally by Papademetriou, Van Leeuwen and Stroud [1996] in atomic sodium. In the experiment the absorption was measured on the 3P3/2 --* 4D5/2 transition of a three-level cascade atom in which the 3S~/2 ---* 4P1/2 transition was driven by a 100% amplitude-modulated field. 4.2. NEARLY SYMMETRIC EXCITATION
4.2.1. Fluorescence spectrum
When the average frequency of the field components is detuned from the atomic resonance (,4 ~ O) and/or the Rabi frequencies of the two fields are unbalanced (a ~ 1), symmetry is destroyed and the positions of the sidebands in the
418
SPECTROSCOPY IN POLYCHROMATICFIELDS
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tN
ci
ci
ci
g ci
0
-20
0
20
(,~ - ,~,)/r Fig. 11. Fluorescence spectrum for a detuned bichromatic field with 2s = 8F, a = 1, A = 3F and 6 = 5F.
V
!
-20
0
20
(~ - ~,0)/r Fig. 12. Fluorescence spectrum for an asymmetric bichromatic field with 2g2 = 8F, A = 0, 6 = 5F and a = 0.75. f l u o r e s c e n c e s p e c t r u m d e p e n d o n the R a b i f r e q u e n c i e s a n d d e t u n i n g s . In fig. 11, w e p l o t the s p e c t r u m for 292 = 8 F , a = 1, 6 = 5 F a n d A = 3 F ; i.e., for e q u a l R a b i f r e q u e n c i e s b u t a s y m m e t r i c d e t u n i n g . T h e s p e c t r u m s h o w s an i n t e r e s t i n g d e v i a t i o n f r o m that for A = 0: It c o n t a i n s m o r e p e a k s , w h i c h arise f r o m the splitting o f the c e n t r a l line a n d the even s i d e b a n d s into doublets. T h e p o s i t i o n s
VI, w 4]
BICHROMATIC FIELDS
419
of the odd sidebands are unaffected by A. A similar modification to the spectrum appears for A = 0 and a , 1. This is shown in fig. 12, where we plot the spectrum for A = 0, 6 = 5F, 2g2 = 8F and a = 0.75. As in the case of A , 0, the central line and the even sidebands split into doublets, whereas the odd sidebands remain unchanged. The splitting of the spectral lines into doublets might suggest that non-zero detunings or unequal Rabi frequencies have the effect of splitting some of the dressed states into doublets. This, however, is not the case. The splitting in fact arises from shifts of the dressed-state energies. We show this by considering the asymmetry to be a perturbation on the symmetric (a - 1, A = 0) Hamiltonian, and using first-order perturbation theory to find the corrected energies of the system (Ficek and Freedhoff [1993]). Because the eigenstates of the even manifold ,5' (2N) are different from those of the odd ,5 ( 2 N - 1), the shifts of their dressed states occur in different directions. For example, for a = 1 and A ;~ 0, the energies are E~N) = E2N
+
1
m6 + -~A [1 + (-1)mJ0 (-4g2/6)],
E(m) 1 2N-1 = E z N - 1 + mt~ + ~A[1
(55)
- (-1)mJ0 (-492/6)].
The central line and the even sidebands result from 2m =~ 2m' and from 2m + 1 =~ 2m' + 1 transitions within adjacent manifolds. Because the shifts of these states occur in opposite directions, the transition frequencies are to = too + 2 (m - m') 6 + AJ0 (-4g2/6),
(56)
and the lines split into doublets. The odd sidebands, however, are due to transitions 2m =~ 2m' + 1 and 2m + 1 =~ 2m' within neighboring manifolds. These states are shifted in the same direction, so the transition frequencies are unshifted. Similarly, the perturbed energies for a , 1, A - - 0 are given by E~N)
= E2N +
mb + (-1)rag2 ( a - 1) J1 (4Y2/b),
(m) = E2N-1 + 2N-1
mb-(-1)mg2(a
(57)
- 1) J1 (4g2/6)
As a result, the central line and even sidebands are split by an amount 4g2 ( a - 1) J1 (4g2/6), while the odd sidebands are unaffected. The total fluorescence intensity and the fluorescence spectrum have been calculated by Agarwal and Zhu [ 1992] for both symmetric and nearly symmetric bichromatic driving for a range of experimental parameters using the Floquet method.
420
SPECTROSCOPYIN POLYCHROMATICFIELDS
[VI, w4
d
I
-20
I
,,
i
0
20
(o -~,)/r Fig. 13. Near-resonance absorption spectrum for a bichromatic field with 292 = 8F, a = 1, 6 = 5F, Fd = l F, and A = 5F.
4.2.2. Near-resonance absorption spectrum The dispersive shape of the sidebands of the absorption spectrum and the oscillatory properties of the central line are characteristic of symmetric bichromatic driving. As with the fluorescence spectrum, the absorption spectrum changes drastically when A ~ 0 or a ~ 1. This is shown in fig. 13, where we plot the absorption spectrum for a = 1,292 = 8F, 6 = 5F and A = 2F. In this case, the central component and the even sidebands of the absorption spectrum are composed of absorption-emission doublets, whereas the odd sidebands remain dispersive for all values of A and a. With a nonsymmetric driving field there are more regions of frequency where the probe field can be amplified instead of being absorbed by the atom. The properties of the absorption spectrum are explained qualitatively by the dressed-atom model. The splitting of the spectral lines is related directly to the shifted dressed-atom frequencies. The absorptive and emissive behavior is related to an unequal population of the dressed states. As in the case of a monochromatic driving field detuned from atomic resonance, the dressed states are populated unequally, giving a net absorption or amplification of the probe field on transitions between states of unequal population.
4.2.3. Autler-Townes absorption spectrum Figure 14 shows the Autler-Townes spectrum for a driving field with a - 1, A - 5F. In this case, the spectrum is asymmetric and composed of pairs of
VI, w 4]
BICHROMATIC FIELDS
421
[
,(N+M)O+> I(N+M)O->
E(N)
i i
IN->
o
99
o" 9
9
9
9
(a)
(b)
E(N-1)
(c)
Fig. 15. Energy levels of a doubly driven atom: (a) singly dressed states driven by a strong field of frequency to2 = too + g2; (b) energy levels of the non-interacting singly dressed atom plus the second mode; (c) doubly dressed states of the interacting singly dressed atom and the second mode.
for n ~> 2, at the subharmonic frequencies m 2•17 as well. The higher a, the greater the number and the intensity of the subsidiary features, whose splittings and spectral distributions display an intricate dependence on n and on a. As examples, we present the fluorescence spectrum for n - 1, the dispersive profile of the probe for n - 2, and the Autler-Townes absorption spectrum for n - 1, 2 and 3. The correlation between successive photons scattered by the atom into individual lines of its fluorescence spectrum has also been studied theoretically, by Ben-Aryeh, Freedhoff and Rudolph [ 1999].
4.3.1.1. Fluorescence spectrum. In fig. 16, we plot the theoretical spectrum corresponding to parameters 292 = 220 MHz and several values of 2s along with the fluorescence spectrum observed experimentally (Yu, Bochinski, Kordich, Mossberg and Ficek [1997]) for the same parameters. In this case, to
424
SPECTROSCOPYIN POLYCHROMATICFIELDS (a) ~ 2 = 0 M H z
E
[VI, w4
(c) D- 2 = 100 M H z
ii
c~ (b) D- 2 = 60 M H z
(d) D- 2 = 140 M H z
g
-I ~
ii
u_
-440-220
o
220 4 4 0
V c -- V a
-440-220
o
220 440
(MHz)
Fig. 16. Experimentallyobserved spectra (lines i) togetherwith theoretical spectra (lines ii) of doubly driven Ba atoms with a strong resonant field of Rabi frequency 2(2 = 220 MHz and a weaker field of several Rabi frequencies 2f22 and detuned from the resonance by 292 (from Yu, Bochinski, Kordich, Mossberg and Ficek [1997]). obtain agreement it was necessary to include as well in the theoretical curve both the elastic scattering and the effect of the different isotopes of the Ba atoms.
4.3.1.2. Near-resonance index of refraction. The modifications of the absorption spectrum, especially the vanishing absorption at the central frequency, may prove useful in the preparation of optical materials having a large index of refraction accompanied by vanishing absorption. In fig. 17 we plot the dispersive profile of the probe field for n = 2. At the lower frequency sideband (where the probe is amplified strongly), the refractive index varies rapidly with frequency and vanishes at the point of maximum amplification. The situation is different at the central frequency (where the absorption vanishes). Here, the refractive index is different from zero with a strong negative dispersion. Moreover, near the central frequency both the absorption and dispersion change slowly with frequency. This property makes the system a convenient candidate for practical applications, since it does not require a precise matching of the probe field frequency to the point of vanishing absorption. The central structure is remarkably stable against variation in frequency.
VI, w 4]
BICHROMATIC FIELDS
425
q 0
o
d
v
,
-50
,
,
.
,
,
,
0
.
.
.
,
,
50
(= -=0)/r Fig. 17. Dispersive profile of the probe beam monitoring a doubly driven atom with 2g2 = 50F, n = 2 and a = 0.4.
4.3.1.3. Autler-Townes absorption spectrum. In fig. 18 (overleaf), we plot the theoretical Autler-Townes absorption spectra for n = 1, 2 and 3, 2g2 = 40F, and a = 0.45. Autler-Townes absorption and dispersion have been measured by Greentree, Wei, Holmstrom, Martin, Manson, Catchpole and Savage [1999] in a nitrogen-vacancy center in diamond. The resolution of these experiments is so high that the splitting of the doublets is measured as a function of a and of the detuning of the weaker field from exact (subharmonic) resonance. It is found that the minimum splitting occurs for o92 shifted slightly from ~o0 + ~-, a feature we discuss in the next section. 4.3.1.4. Shifts of the subharmonic resonances. The minima in the splittings of the Autler-Townes doublet features appear as well, of course, in the corresponding features of the fluorescence and near-resonance absorption spectra, and can be explained fully by the dressed-atom picture in terms of dynamic Stark shifts (Rudolph, Freedhoff and Ficek [1998b]). Furthermore, at those frequencies O,)(mni)n = (D O q- 2~2 + An , the central component of the n fluorescence spectrum F(to0) vanishes identically, a fact which allows one to pinpoint sharply the positions of tOmi ~')n 9 Lastly, those same frequencies t~mi "(")n are the frequencies which correspond to the shifts of the subharmonic resonances shown in figs. 4 and 5. All these phenomena can be unified and explained in the dressed-atom picture.
426
SPECTROSCOPY IN POLYCHROMATIC FIELDS
[VI, w 4
n--1
!
9
,
I
....
,
,
,
n-2
:3 vt.~
Jt~
A * A
n---3
-40
-20
0
20
40
(~ - ~n)/r Fig. 18. Autler-Townes absorption spectra for 2~2 = 4 0 F , a = 0.45, F d = 89F and different n.
The shifts of the subharmonic resonances were first predicted theoretically by Berman and Ziegler [1977] and Ruyten [1989, 1992a,b] in a numerical calculation which involved the solution of the optical Bloch equations for the system. Because the shifts appear in the same positions as do the Bloch-Siegert shifts in the Bloch equations for monochromatic driving without the RWA (Allen and Eberly [ 1975]), Ruyten termed them the 'generalized Bloch-Siegert shifts'. For bichromatic driving, however, they occur within the RWA, and are much
VI, w4]
BICHROMATICFIELDS
427
larger than the original Bloch-Siegert shifts (Bloch and Siegert [1940]), which makes it possible to experimentally observe them. The physical origin of the shifts can be explained with the energy-level diagram of fig. 15b (above). Before calculating the splitting of the doublet states [a~m ")) and ]b~ )) using (higher-order degenerate)perturbation theory, we first take into account the shifts of the two levels caused by all doublets within ,5 (N + M) having m' ;~ m. These (second-order) shifts are of order of magnitude g22/g2, represent a dynamic Stark shift of the levels of doublet m due to the second field mode, and shift the levels in opposite directions. The result is a shift of their anticrossing point, given (to lowest order in a) by An =
!a2g2 8 ,a 2 2(~,2_1)(2
for n = 1 ' for n/> 2.
(61)
At the shifted frequencies tOmi n'(n) = (DO + -n--2f2+ An, the splittings are easily shown to be a minimum, and equal to the matrix element of the effective n-photon coupling interaction between are then simply given by
IN + M,m+)=
~
1
t,) ) [am
and ,Ib~,~')) - Furthermore, the dressed states
(62)
([a~m"') • Ibm,7')),
causing a vanishing of the fluorescence at ~o0 due to destructive interference between the transition amplitudes from la~,~')) and ]b~,~')). Finally, these shifted I
l
i
t
energies determine the true subharmonic resonances of the doubly driven system, at the shifted points of anticrossing.
4.3.2. Twofields with equalfrequencies The phenomena described in the previous sections arise from the excitation of a two-level atom by two fields having different frequencies. In this section, we consider driving the atom with two fields having equal frequencies. For a fixed relative phase between the fields, this is equivalent to interaction with a monochromatic field whose electric field is the vector sum of those of the two fields. However, for an arbitrary relative phase between the fields the problem is more complicated and, in fact, is similar to the phase diffusion effect (Zoller [ 1979], Agarwal and Lakshmi [ 1987], Lobodzinski [ 1997]). A complementary view of this phase diffusion is obtained by studying this system as an example of 'double dressing'. The two-level atom is driven by two
428
SPECTROSCOPYIN POLYCHROMATICFIELDS
[VI, w4
field modes which have equal frequencies but are otherwise distinguishable (e.g., by different k,), with 292 > 2922. For simplicity, we assume that the frequencies are equal to the atomic transition frequency o~0. The interaction of the atom with the strong field results in the singly dressed states ]N-+-) of eq. (59). The Hamiltonian H0 of the noninteracting dressed atom + weaker field satisfies the eigenvalue equation (Freedhoff and Ficek [ 1997]) So I(N - n) • n) = h [No)o :t: .Q] I(N - n) +, n),
(63)
where I ( N - n ) + , n ) = ] ( N - n ) + ) | In), and n is the number of photons in the weaker field. We see from eq. (63) that the eigenstates form an infinite ladder of doublets, each composed of an (effectively) infinite number of degenerate states. When we include the interaction between the singly dressed atom and the weaker field, the degeneracy is lifted. The resulting doubly dressed states are found by diagonalizing the matrix of the interaction W in the basis of the undressed states (assuming negligible interaction between + a n d - manifolds, true for g2 > g22). The matrix elements of W are W(~ ) = ((N - n) :i:, n IWI (N - m) ::k, m)
(
~hg2 v / n + 1 6 . + l , m + ~ 6 . _ l , m
)9
(64)
This matrix has the same form as that which represents the position operator in the basis of the energy eigenstates of the one-dimensional harmonic oscillator (Cohen-Tannoudji, Dupont-Roc and LaloE [1977]). The eigenvalues of manifold g (N) are -cx~ < A < cx~,
(+) A = h [N~0 • (g2 + 89
(65)
and the corresponding eigenvectors are
IN • ~.) = n~O
-q- - ~
I( N - n ) + , n ) ,
(66)
where q~,(x) =
(VS- 2"n! )-,J2 H,(x) e-~, x-,
(67)
and H,(x) is the Hermite polynomial of order n. The energy levels of the doubly dressed atom thus consist of an infinite ladder of doublet continua.
VI, w4]
BICHROMATICFIELDS
429
Transition rates between the continua in neighboring manifolds are given by
G+--FI(N+~,IS+I(N-1)•
2 =a16(/~,
- ~,) ,
(68) Is+I( N - 1) m A') 2 =a16( ,~, q- ,~,,) 9
GT--FI(N+'a.
Thus transitions between pairs of +(-) doublets can occur only between dressed states corresponding to the same eigenvalue ~' = ~,, resulting in the central component of the fluorescence spectrum being unchanged, while transitions corresponding to the sidebands of the Mollow triplet (4- =~ :F) occur to 2,'=-~,, producing a broad continuum9 The total populations of the + and - continua are equal for resonant driving9 Within each continuum, however, the population distribution depends on ~,: If there are M photons in field mode 2, the populations P (~,) are given by
P(~) = [~M (]~/V/2) [2, where
OM(~,/V/2) is
given by eq. (67). The population
distribution has maxima for ~,/x/~ in the vicinity of the classical turning points of the harmonic oscillator eigenfunction CM ( V A// 2 )
1 ; i.e., for AM ~ +2 C M + ~,
or for energies 1 I~Mlhg2 ~ h g 2 v ~ = 89 For I~1 < I~MI the populations P (zl) are smaller, but nonzero, while for I~1 > I'~MI ,P (~) goes rapidly to zero. Following the standard dressed-atom procedure, we find that the incoherent part of the fluorescence spectrum is given by iF
F
s~o~)- ~
~o-~o0/+ (89 2 1
+ 8
/+
( )[
dA q~M
~
+
A
~
2
3_4F
(to - co0 - 2s _ ~,g2)2 + (3 F) 2
4 (co- ~Oo + 2 s
Xg2)2 + ( 3/-') 2
l}
(69) This spectrum is plotted in fig. 19 for 2f2 = 20F and 2f22 = 7F. The graph shows a central component which is the same as that in the Mollow triplet, together with sidebands consisting of a convolution of Lorentzian functions, centered at r 4- 2g2- ~g2 and having width 3F/4, multiplied by a weight factor 9]q~M --(~/X/2)]2. Therefore, the overall width of the sidebands is determined by I
I
the population distribution P(~,) 9 Because q~M ( ~ / V ~ ) ] 2 has maxima near the \
/
i
classical turning points, the sidebands display peaks near to0 4- 2~2 4- s weaker continua centered at COo 4- 2s of width ~ 2922.
with
430
SPECTROSCOPYIN POLYCHROMATICFIELDS ~,,,
,
,
,
,
,
[VI, w4 ,,,
r
c~
c~
c~
0
-20
0
20
(~, - ,%)/r
Fig. 19. Fluorescence spectrum of a two-level atom driven by two fields of equal frequencies, 292 = 20F and 2922= 7F. The Autler-Townes spectrum can also be calculated. It is found theoretically that each line of that spectrum is replaced by a continuum of width Y22. These continua were observed by Wei [1994] and by Greentree, Wei, Holmstrom, Martin, Manson, Catchpole and Savage [1999]. They studied the response of a two-level atom which is driven by two lasers with the same frequency, but with significantly different intensities and without phase locking. The response was monitored by recording the (Autler-Townes) absorption spectrum corresponding to a transition to a third atomic level.
4.3.3. Driving at the Rabi frequency If the atom has a permanent dipole moment, transitions can occur at the Rabi frequency between the states iN+) of the same energy manifold (Freedhoff and Smithers [1975], Freedhoff [1978], Dalton and Gagen [1985]). These transitions have been observed by measuring the absorption of a (weak) probe field tuned near the Rabi frequency 292 of the strong resonant field (Barrett, Woodard and Lafyatis [1992], Holmstrom, Wei, Windsor, Manson, Martin and Glasbeek [1997]). When a strong field is applied at the Rabi frequency, the dressed atom is dressed again. The doubly dressed states of manifold E (N) are IM (N) ~) = ~
1
( N+, M - 1) • i N - , M ) ) ,
(70)
VI, w5]
POLYCHROMATIC FIELDS, p/> 3
431
corresponding to energies E M ( N ) • -- h [Nw0 + (2M - 1) g2 + G]. Here we have [N+,M) - I N + ) | IM), where IN+) are the singly dressed states (59) and M is the number of photons in the field at Rabi frequency 2s 2G is the 'Rabi of the Rabi frequency' g2 v/M, where the atomic dipole moment element in g2 is "~1 (,Ue e -- ~gg). Thus each manifold contains an infinite number of doublets, with interdoublet splitting 2g2, and intradoublet splitting 2G, very similar to the energy level spectrum in w4.3.1. Subharmonic resonances in the absorption of a 'strong probe' tuned near the Rabi frequency 2g2 of the resonant driving field have been observed by Brunel, Lounis, Tamarat and Orrit [1998], who monitored the integrated intensity of fluorescence by their driven molecule as a function of the probe frequency. The near-resonance absorption spectrum of this doubly driven system has been measured by Windsor, Wei, Holmstrom, Martin and Manson [1998], and interpreted in terms of the doubly dressed states (70).
w 5. Polychromatic Fields, p t> 3 5.1. AMPLITUDE-MODULATEDDRIVING FIELDS The earliest theoretical studies of an atom driven by a polychromatic field involved semi-classical amplitude-modulated (am) fields of the form (Armstrong and Feneuille [1975], Feneuille, Schweighofer and Oliver [1976], Thomann [1976, 1980], Blind, Fontana and Thomann [1980], Agarwal and Nayak [1984, 1986], Friedmann and Wilson-Gordon [ 1987], Ruyten [ 1990], Smirnov [ 1994])
E(+)(t) - E 0~+)e --i~OLt ( l + a c o s b t ) ,
(71)
where a is the modulation amplitude and 6 the modulation frequency. We can rewrite the field (71) in the form
E~+)(t) = E~+) [e-i~~
1 (e-i(wL-~)t+ + ~a
e-i(~oL+0)t)] ,
(72)
showing that the am field is in general equivalent to a trichromatic field. For a >> 1, the carrier frequency is effectively suppressed, so that a 100% modulated field corresponds to bichromatic driving with a = 1, considered in w4. In this section, we consider a to be finite.
5.1.1. Floquet treatment We consider a trichromatic driving field (p = 3) with 61 = 0, 02 = - 6 3 = 6, Rabi frequencies of the detuned fields 2s - 2s and Rabi frequency of the resonant
432
SPECTROSCOPYIN POLYCHROMATICFIELDS
[VI, w5
central component 2s = 2g2. For this situation, the optical Bloch equations (8) reduce to the (Laplace transform) recurrence relation
AeX 3
435
A v
3
'
'
I
437
'
'
'
I
'
'
'
Rabi of the Rab~
r ..O x__
t.__O .t-, C~ O