Cavity-Enhanced Spectroscopies, Volume 40

Experimental Methods in the Physical Sciences VOLUME 40 CAVITY-ENHANCED SPECTROSCOPIES

EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES Robert Celotta and Thomas Lucatorto, Editors in Chief FoundingEditors L. MARTON C. MARTON

Volume40

Cavity-Enhanced Spectroscopies Edited by Roger D. van Zee J. Patrick Looney

National Institute of Standards and Technology Gaithersburg, Maryland, USA

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1

To the one-hundred-year-long tradition of accuracy and precision i n physical and chemical measurements at the National Bureau of Standards/ National Institute of Standards and Technology. " . . . when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind," it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state o f science . . . . " Lord Kelvin

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CONTENTS CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

VOLUMES IN SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

.

Stabilizing Diode Lasers to High-Finesse Cavities by RICHARD W. FOX, CHRIS W. OATES, and LEO W. HOLLBERG 1.1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2. I n t r o d u c t i o n to Diode Laser Locking . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3. G e n e r a t i n g the E r r o r Signal: The P o u n d - D r e v e r - H a l l Method .....................................................

5

1.4. The L o o p Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.5. Locking the Laser and L o o p Optimization . . . . . . . . . . . . . . . . . .

29

1.6. Repetitive Locking for Cavity R i n g - D o w n Spectroscopy . . . .

37

1.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

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

44

2. Spectroscopic Applications Using Ring-Down Cavities by GIEL BERDEN, GERARD MEIJER, and WIM UBACHS 2.1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.2. 02: A B e n c h m a r k System for R i n g - D o w n Techniques . . . . . . .

47

2.3. Spectroscopic Applications of C R D

63

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

2.4. C R D and Alternative Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Acknowledgment ...............................................

74

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

74

3. Absorption Detection at the Quantum Limit: Probing High-Finesse Cavities with Modulation Techniques by JUN YE and JOHN L. HALL 3.1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.2. Cavity E n h a n c e m e n t : A Simple Physics Picture . . . . . . . . . . . . .

86

vii

viii

CONTENTS 3.3. Signal E n c o d i n g a n d Extracting: Modulation-plus-Heterodyne Detection . . . . . . . . . . . . . . . . . . . . .

90

3.4. W e a k A b s o r p t i o n M e a s u r e d by Field D e c a y (Time-Domain) ............................................

95

3.5. W e a k A b s o r p t i o n M e a s u r e d by Field Phase (Frequency-Domain) ......................................

105

3.6. C o n c l u d i n g R e m a r k s on Signal E x t r a c t i o n a n d Cavity E n h a n c e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

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

122

Cavity-Enhanced Absorption: Intracavity Laser Spectroscopy

,

by GEORGE H. ATKINSON 4.1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

4.2. Principles of ILS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

4.3. D e v e l o p m e n t of ILS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

4.4. C o n c l u d i n g R e m a r k s . . . . . . . . . . . . . . . . . . . . . .

145

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

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

146

The Continuous-Wave Hydrogen Raman Laser

.

by PETER A. ROOS, LEI S. MENG, a n d JOHN L. CARLSTEN

.

5.1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157

5.2. Semiclassical T h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162

5.3. N d : Y A G - P u m p e d System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

166

5.4. D i o d e - P u m p e d Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

170

5.5. Practical C o n s i d e r a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

5.6. S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

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

181

2D Microcavities: T h e o r y a n d E x p e r i m e n t s by JENS U. NOCKEL a n d RICHARD K. CHANG 6.1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

CONTENTS

.

ix

6.2. Dielectric Microcavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186

6.3. Whispering-Gallery M o d e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

6.4. Scattering Resonances a n d Q u a s i b o n d States . . . . . . . . . . . . .

192

6.5. Cavity R i n g - D o w n and Light Emission . . . . . . . . . . . . . . . . . . .

195

6.6. Wigner Delay Time and the Density of States . . . . . . . . . . . .

197

6.7. Lifetime Versus Linewidth in Experiments . . . . . . . . . . . . . . . .

200

6.8. H o w M a n y M o d e s Does a Cavity Support? . . . . . . . . . . . . . .

201

6.9. Cavities W i t h o u t C h a o s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

204

6.10. Chaotic Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207

6.11. Phase Space R e p r e s e n t a t i o n with Poincar6 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

208

6.12. U n c e r t a i n t y Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

210

6.13. Husimi Projection . . . . . . . . . . . . . . . . .

211

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

6.14. Constructive Interference with Choatic Rays . . . . . . . . . . . . . .

213

6.15. Chaotic Whispering-Gallery M o d e s . . . . . . . . . . . . . . . . . . . . . . .

216

6.16. D y n a m i c a l Eclipsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

218

6.17. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219

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

220

Microparticle Photophysics: Fluorescence Microscopy and Spectroscopy of a Photonic Atom by STEPHEN ARNOLD and STEPHEN HOLLER 7.1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

7.2. P h o t o n i c A t o m M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229

7.3. S p o n t a n e o u s Emission in a Microsphere: Cavity Q E D . . . . .

232

7.4. S p o n t a n e o u s Emission f r o m Molecules at a Microsphere Surface: A Semiclassical A p p r o a c h . . . . . . . . . . .

236

7.5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

244

Acknowledgment ..............................................

250

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

251

x

CONTENTS .

Single Atoms Moving in a High-Finesse Cavity

by P. W. H. PINKSE and G. REMPE

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255

8.2. Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259

8.3. Cavity-Mediated Light Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

270

8.4. Experimenta! Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

282

8.5. O u t l o o k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

288

Acknowledgments

289

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

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

.

289

The Cavity-Enhanced Optical-Frequency Comb Generator and

its Applications

by MOTONOBU KOUROGI, BAMBANG WIDIYATMOKO, a n d MOTOICHI OHTSU 9. l. I n t r o d u c t i o n

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

297

9.2. G e n e r a t i n g a n O p t i c a l - F r e q u e n c y C o m b . . . . . . . . . . . . . . . . . . .

298

9.3. U s i n g O p t i c a l - F r e q u e n c y C o m b s

305

9.4. S u m m a r y

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

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

References ..................................................... INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

314 315 321

CONTRIBUTORS Numbers in parentheses indicate the pages on which authors' contributions begin.

STEPHEN ARNOLD (227), Polytechnic University, Brooklyn, New York, USA GEORGE H. ATKINSON (129), University of Arizona, Tucson, Arizona, USA GILL BERDEN (47), FOM Institute for Plasma Physics, Nieuwegein, The

Netherlands JOHN L. CARLSTEN (157), Montana State University, Bozeman, Montana,

USA RICHARD K. CHANG (185), Yale University, New Haven, Connecticut, USA RICHARD W. F o x (1), National Institute of Standards and Technology,

Boulder, Colorado, USA JOHN L. HALL (83), National Institute of Standards and Technology and

University of Colorado, Boulder, Colorado, USA LEO W. HOLLBERG (1), National Institute of Standards and Technology,

Boulder, Colorado, USA STEPHEN HOLLER (227), Los Gatos Research, Mountain View, California,

USA MOTONOBU KOUROGI (297), Tokyo Institute of Technology, Yokohama,

Japan GERARD MEIJER (47), FOM Institute for Plasma Physics, Nieuwegein, and

University of Nijmegen, Nijmegen, The Netherlands LEI S. MENG (157), Montana State University, Bozeman, Montana, USA JENS U. NOCKEL (185), University of Oregon, Eugene, Oregon, USA CHRIS W. OATES (1), National Institute of Standards and Technology,

Boulder, Colorado, USA MOTOICHI OHTSU (297), Tokyo Institute of Technology, Yokohama, Japan P. W. H. PINKSE (255), Max Planck Institute of Quantum Optics, Garching

bei Mfinchen, Germany G. REMPE (255), Max Planck Institute of Quantum Optics, Garching bei

Mfinchen, Germany PETER, A. ROOS (157), Montana State University, Bozeman, Montana, USA WIM UBACHS (47), Free University, Amsterdam, The Netherlands BAMBANG WIDIYATMOKO (297), Tokyo Institute of Technology, Yokohama,

Japan JUN YE (83), National Institute of Standards and Technology and University

of Colorado, Boulder, Colorado, USA

xi

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VOLUMES IN SERIES EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES (formerly Methods of Experimental Physics)

Editors-in-Chief Robert Celotta and Thomas Lucatorto Volume 1. Classical Methods Editedby Immanuel Estermann Volume 2. Electronic Methods, Second Edition (in two parts) Editedby E. Bleuler and R. O. Haxby Volume 3. Molecular Physics, Second Edition (in two parts) Edited by Dudley Williams Volume 4. Atomic and Electron Physics~Part A: Atomic Sources and Detectors; Part B: Free Atoms Editedby Vernon W. Hughes and Howard L. Schultz Volume 5. Nuclear Physics (in two parts) Editedby Luke C. L. Yuan and Chien-Shiung Wu Volume 6. Solid State Physics-Part A: Preparation, Structure, Mechanical and Thermal Properties; Part B: Electrical, Magnetic and Optical Properties Editedby K. Lark-Horovitz and Vivian A. Johnson Volume 7. Atomic and Electron Physics--Atomic Interactions (in two parts) Editedby Benjamin Bederson and Wade L. Fite Volume 8. Problems and Solutions for Students Editedby L. Marton and W. F. Hornyak xiii

xiv

VOLUMES IN SERIES

Volume 9. Plasma Physics (in two parts) Edited by Hans R. Griem and Ralph H. Lovberg Volume 10. Physical Principles of Far-Infrared Radiation Edited by L. C. Robinson Volume 11. Solid State Physics Edited by R. V. Coleman Volume 12. Astrophysics~Part A: Optical and Infrared Astronomy Edited by N. Carleton Part B: Radio Telescopes; Part C: Radio Observations Edited by M. L. Meeks Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams Volume 14. Vacuum Physics and Technology Editedby G. L. Weissler and R. W. Carlson Volume 15. Quantum Electronics (in two parts) Edited by C. L. Tang Volume 16. Polymers~Part A: Molecular Structure and Dynamics; Part B: Crystal Structure and Morphology; Part C: Physical Properties Edited by R. A. Fava Volume 17. Accelerators in Atomic Physics Edited by P. Richard Volume 18. Fluid Dynamics (in two parts) Edited by R. J. Emrich Volume 19. Ultrasonics Edited by Peter D. Edmonds Volume 20. Biophysics Editedby Gerald Ehrenstein and Harold Lecar

V O L U M E S IN SERIES

Volume 21. Solid State Physics: Nuclear Methods EditedbyJ. N. Mundy, S. J. Rothman, M. J. Fluss, and L. C. Smedskjaer Volume 22. Solid State Physics: Surfaces Editedby Robert L. Park and Max G. Lagally Volume 23. Neutron Scattering (in three parts) Editedby K. Skold and D. L. Price Volume 24. Geophysics--Part A: Laboratory Measurements; Part B: Field Measurements Editedby C. G. Sammis and T. L. Henyey Volume 25. Geometrical and Instrumental Optics Editedby Daniel Malacara Volume 26. Physical Optics and Light Measurements Editedby Daniel Malacara Volume 27. Scanning Tunneling Microscopy Editedby Joseph Stroscio and William Kaiser Volume 28. Statistical Methods for Physical Science Editedby John L. Stanford and Stephen B. Vardaman Volume 29. Atomic, Molecular, and Optical Physics--Part A: Charged Particles; Part B: Atoms and Molecules; Part C: Electromagnetic Radiation Editedby F. B. Dunning and Randall G. Hulet Volume 30. Laser Ablation and Desorption

Editedby John C. Miller and Richard F. Haglund, Jr. Volume 31. Vacuum Ultraviolet Spectroscopy I

Editedby J. A. R. Samson and D. L. Ederer

Volume 32. Vacuum Ultraviolet Spectroscopy II

EditedbyJ. A. R. Samson and D. L. Ederer

Volume 33. Cumulative Author Index and Tables of Contents, Volumes 1-32

XV

xvi

VOLUMES IN SERIES

Volume 34. Cumulative Subject Index Volume 35. Methods in the Physics of Porous Media Edited by Po-zen Wong Volume 36. Magnetic Imaging and its Applications to Materials Editedby Marc De Graef and Yimei Zhu Volume 37. Characterization of Amorphous and Crystalline Rough Surface: Principles and Applications Editedby Yi Ping Zhao, Gwo-Ching Wang, and Toh-Ming Lu Volume 38. Advances in Surface Science Edited by Hari Singh Nalwa Volume 39. Modern Acoustical Techniques for the Measurement of Mechanical Properties Editedby Moises Levy, Henry E. Bass, and Richard Stern Volume 40. Cavity-Enhanced Spectroscopies Edited by Roger D. van Zee and J. Patrick Looney

PREFACE

While the proposition that lasers have revolutionized spectroscopy is unlikely to generate controversy, the claim that cavity-enhanced methods have effected comparable change might provoke protest. But we believe that this assertion can be successfully argued too. Witness the ubiquity of build-up cavities in atomic spectroscopy, the incredible sensitivity of cavity-enhanced, frequency-modulation spectroscopy, or the explosive growth in the laboratory use of the ring-down techniques. This volume is aimed at practising spectroscopists who are interested in cavity-enhanced methods but may lack the photonics expertise of an electrical engineer. In this vein, and because this volume is dedicated to cavity-enhanced methods, the authors were encouraged to highlight the enabling role the resonator plays. This volume is not, however, a monograph on optoelectronic theory, and any such theoretical ruminations are counterbalanced with plenty of practical, experimental discussions. Thus, the introductory chapter is a practical review of locking lasers to optical resonators. This volume then presents an eclectic sampling of spectroscopic methods that use resonators for signal enhancement or otherwise exploit light/matter interactions within a resonator. The second and third chapters discuss the simplest cavity-enhanced method: the use of a passive resonator to do spectroscopy on atoms and molecules in a cavity. The next chapter deals with a similar concept, absorption measurement in a cavity, but an active one. The book closes with chapters that discuss microresonators and phenomena unique to or enhanced by the resonator environment, such as the hydrogen Raman laser, cavity QED, and optical-frequency comb generators. Other subjects could certainly have been included in this book, and the ones we selected are based partly on our fancy. Equally important in our selection, however, was the criterion that a method should have matured beyond conceptualization and proven its mettle, but not be in common use. In this regard, these chapters should prove a timely aid, for much of what is described in this volume is known mostly to expert practitioners. Hopefully, this volume will become a standard and authoritative reference and a useful guide to more widespread implementation of these important techniques. Finally, we would also like to take this opportunity to acknowledge the contributions of the many who helped us during the course of our work, in

xvii

xviii

PREFACE

particular Dr. Joseph T. Hodges, Dr. James R. Whetstone, and Dr. Hratch G. Semerjian.

National Institute of Standards and Technology Gaithersburg, Maryland, USA July 2002

xviii

R. D. van Zee J. P. Looney

1. STABILIZING DIODE LASERS TO HIGH-FINESSE CAVITIES Richard W. Fox, Chris W. Oates, Leo W. Hollberg National Institute of Standards and Technology, Boulder, Colorado, USA

1.1 Introduction This chapter is written for the chemist, physicist, or engineer who is interested in locking a diode laser to an optical cavity. There already exist many good references on the locking of lasers [1, 2], including some on diode lasers in particular [3, 4]. In this chapter we aim to provide a practical hands-on guide, with most or all of the theory left to cited references. The motivations for locking lasers to high-finesse optical cavities include applications as varied as laser cooling, length metrology, and analytical and precision spectroscopy. In some cases it may be the stabilization of the laser's frequency to a narrow cavity resonance that is of interest, while in others it may be the huge build up of intracavity power or the long effective path length between the mirrors that is important. The case of locking diode lasers is of particular interest because of their unique characteristics, relatively low cost and widespread use in many applications. Before proceeding further, we should offer a note of caution that the term "diode laser" now encompasses many devices whose electrical and optical characteristics differ greatly. Included in this category are the Fabry-P6rot laser, vertical-cavity surface-emitting lasers, and distributed feedback lasers. While the basic approach for locking these different types of lasers is the same, the performance of the lock and the details in the electronic feedback loops will be quite different. Rather than overwhelm the reader with details for all these different types, we instead focus on one type in particular, the widely used extended-cavity Fabry-P6rot laser. For those interested in locking other types of diodes to cavities, this chapter should still be a good place to start. This chapter consists of the following. We begin with a general discussion of the issues involved in diode laser locking and introduce the reader to some of the terminology. We then describe in detail the various steps needed to lock the laser to a cavity resonance: (1) derivation of the error (locking) signal, (2) design of the electronic feedback circuitry, (3) initial locking of the laser, (4) adjustment of the feedback design, and (5) evaluation of the lock performance. We illustrate this discussion by frequency locking an 1 EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES Vol. 40 ISBN 0-12-475987-4

Published by Elsevier Science (USA) All rights of reproduction in any form reserved. ISSN 1079-4042/01 $35.00

2

STABILIZING DIODE LASERS

extended-cavity diode laser, reducing the linewidth to a few hertz relative to the cavity. We conclude with an example in which we modified the locking apparatus for a cavity ring-down demonstration. Included are results showing the laser repetitively locking and unlocking to the cavity.

1.2 Introduction to Diode Laser Locking Regardless of the application, the basic goal of locking the frequency of a laser to a cavity is to reduce the frequency fluctuations between the laser and cavity. The noise spectrum of the laser's frequency fluctuations leads to an effective "linewidth" of the laser, which conceptually describes the broadening of the laser's spectrum around its central frequency. The concept of laser linewidth can be a little confusing but is valuable, so it is worth a brief discussion here. The basic confusion arises from the fact that the laser linewidth is dependent on the timescale over which it is evaluated, as a laser's noise spectrum typically contains very different fast and slow components. A useful intuitive picture is that the fast fluctuations give rise to what is called the "fast linewidth" (or sometimes just "linewidth"); by "fast" we usually mean in a time less than the effective spectroscopic interaction time (typically tens or hundreds of microseconds). With this definition, the linewidth defines the narrowest feature that the laser is capable of resolving. The low-frequency fluctuations (usually of larger magnitude) then cause jitter of this narrow spectral line in frequency space. Even slower changes often occur because of thermal effects, which can cause the laser's central frequency to drift. In order to reduce the laser linewidth, one needs a stable frequency reference suitable for measuring the laser's frequency fluctuations. One can then construct a feedback loop that attempts to compensate these fluctuations. A Fabry-P6rot cavity, which usually consists of two highreflectivity mirrors separated by a spacer, is a convenient choice for a frequency reference. With mirror reflectivities > 99% over a given range of wavelengths, a cavity has a series of evenly spaced, sharp resonances (typically with linewidths < 10 MHz). The resonance or "fringe" separation is c/2L, where c is the speed of light and L the length of the optical cavity (i.e. the distance between the mirrors). Radio-frequency modulation techniques can be used to derive an electronic error signal that represents the deviations of the laser frequency from a given cavity reference fringe. One then uses electronic feedback (primarily to the laser current in the case of diode lasers) to control the laser frequency and minimize its deviations relative to the cavity fringe.

I N T R O D U C T I O N TO DIODE LASER L O C K I N G

3

If one succeeds in making the electronic feedback bandwidth (i.e. range of frequencies over which corrections are effectively imposed) wider than the dominant noise spectrum of the laser, the laser's fluctuations can be controlled such that the laser appears tightly locked to the peak of the cavity mode. In this case, the linewidth relative to the cavity's center frequency can be a small fraction (~ 10-4) of the cavity's linewidth. (In this case, the actual laser linewidth will almost certainly be determined by the mechanical stability of the cavity itself.) Likewise, even though the power is not actively stabilized, the power transmitted from the cavity can be stable to better than 1%. On the other hand, if the electronic feedback stabilizes only the lowfrequency portion of the laser's fluctuations, the center frequency will be fixed, but no linewidth narrowing will occur and relatively little of the laser power will be coupled into the mode. Thus we see that there are two basic determining factors in the final performance of the lock: the initial noise spectrum of the laser and the electronic bandwidth achievable in the feedback or "servo" system. The short cavity lengths and low facet reflectivity found in most diode structures result in "low-Q" cavities, and generally the optical output will have a minimum spectral linewidth of 10 MHz, and often much larger. Thus, if we expect to narrow the linewidths of these lasers significantly, we will need feedback loops with sufficient gain up to 10 MHz and beyond. However in most cases (for most commercial diodes), a feedback bandwidth of 10 MHz is quite difficult to achieve because of the characteristics of the diode laser itself. At low modulation frequencies, the frequency shift with injection current is primarily caused by thermal effects. That is, because the injection current causes heating and the index of refraction is temperature dependent, we find that the subsequent frequency modulation (df/d/) is negative. This effect diminishes with modulation frequency, and at times on the order of 1 gs, another smaller effect, caused by the electronic charge carriers, starts to dominate. For this case we have the opposite dependence, i.e., df/dI is positive. The resulting net phase response of the diode is troublesome for the servo designer, making electronic feedback bandwidths beyond a few megahertz increasingly difficult to achieve. To achieve tight locking, it is therefore often necessary to reduce the frequency fluctuations of the laser by other means before locking it to the cavity. A common technique is to increase the optical cavity's Q by adding optics external to the laser chip in order to form a longer cavity. A laser cavity built in this manner is often called an ECDL, "extended-cavity diode laser," or "external-cavity diode laser." The use of frequency-dependence reflection for the external cavity (e.g. with a diffraction grating) can offer the additional benefits of frequency pulling and reduced mode competition. There are many extended-cavity lasers available commercially, and several

4

STABILIZING DIODE LASERS

AR Coating

l

--

PZT

Mirror

Output Beam

Fio. 1.1. A diode laser arranged in a Littman-Metcalf configuration. The beam is incident at grazing incidence on the diffraction grating, and the first order is reflected back by a mirror. The zero order serves as the laser output. Coarse wavelength tuning can be done by adjusting the mirror angle, and fine-tuning by using a PZT to adjust the cavity length. However, true single-mode scanning over a large range requires that the laser cavity length be changed synchronously with the mirror angle.

good references for their design and construction [5, 6]. The laser used as an example here is an off-the-shelf commercial diode that was antireflection coated and built into a Littman-type extended cavity about 8cm long (see Fig. i.1). The extended cavity is formed with a diffraction grating (for wavelength-selective optical feedback) and a high-reflectivity mirror mounted on a piezo-electric transducer (PZT). The laser operates around 830nm and yields an output power of 6 m W at an injection current of 70 mA. We can tune the laser frequency quickly over small excursions with the laser current, or slowly over larger excursions with the voltage on the PZT. The resulting fast linewidth for this laser is roughly 50 kHz, while there are low-frequency vibrations of the mechanical structure (caused by room acoustics), which cause the laser output to jitter in frequency on the order of a megahertz. We note that in many cases, one must use an off-the-shelf commercial diode (without an additional antireflection coating), which usually requires that a shorter extended cavity be used to reduce laser mode competition. Unfortunately, shorter extended cavities yield less frequency noise reduction, leaving more work to be done by the electronic feedback system. Nonetheless, the techniques described in this chapter should still be directly applicable to this case as well and provide good locking performance [7]. We should also add that while the use of electronic feedback is convenient, it is not the only option for stabilizing the diode laser's frequency to a cavity. The use of resonant optical feedback directly from a cavity is a wellestablished technique that can be used to stabilize even a simple Fabry-P6rot diode to certain cavities [8-10]. A requirement is that the

G E N E R A T I N G THE ERROR SIGNAL

5

(a)

(b)

FIG. 1.2. Two configurations to optically lock a diode laser to an optical cavity. The cavity must return light to the laser when on resonance, while reflecting away the off-resonance beam. Diagram (a) is a three-mirror V-shaped cavity, while diagram (b) is a confocal optical cavity aligned in a V-configuration. For stable operation, the path length from the laser to the cavity must be servo controlled. Not shown are mirrors and a coupling lens to align the laser to the cavity.

optical alignment be such that light is reflected or returned to the laser diode only when the cavity is in resonance. Two possible configurations include a three-mirror V-shaped cavity, and a confocal cavity aligned in the c/4L configuration (see Fig. 1.2). Additionally, it is necessary to electronically control the phase of the light returning to the diode laser, which can be accomplished with a mirror mounted on a PZT [11].

1.3 Generating the Error Signal:

The Pound-Drever-Hall Method

The first step in the locking procedure is to generate an electronic error signal that can be further processed for locking the laser to the cavity. Perhaps the two most convenient approaches for generating this signal are side-locking and the Pound-Drever-Hall (PDH) method [1]. With the simpler technique, side-locking, the laser is stabilized to the side of the cavity fringe without the use of modulation techniques. While this technique is straightforward to implement and has some advantages (e.g. simplicity, modulation-free), it suffers from several serious disadvantages. First, because it is modulation-free, one necessarily detects the error signal

6

STABILIZING D I O D E LASERS

at DC, where there can be significant amplitude noise. Second, side-locking has a much smaller acquisition range, which means the laser and cavity frequencies need to be nearly coincident before the system will lock. Third, the lock is less robust because perturbations (e.g. vibrations) that drive the laser to the opposite side of the resonance will cause the feedback loop to push the laser further from resonance. Fourth, because one locks to the side of the resonance rather than the top, there is reduced buildup of optical power in the cavity, and there may be increased noise on the transmitted intensity. The PDH technique circumvents most of these drawbacks by modulating the frequency of the light, enabling detection of the error signal at a high frequency where the technical noise is near the shot-noise limit. The resulting demodulated error signal has a high signal-to-noise ratio and a large acquisition range, which can produce robust locks. Furthermore, this error signal has odd symmetry about the line center that enables locking to the top of a cavity fringe. For these reasons, we prefer the PDH method and focus on its implementation throughout the remainder of this chapter. The PDH technique is described in detail in many references [1-4], but we will review the general approach here. Consider a purely frequency modulated (FM) laser beam impinging on the input mirror of an optical cavity and reflecting back to a detector. For low modulation index (as we typically use), one can view the frequency spectrum of the modulated light as consisting of a carrier with two sidebands: one at higher frequency with a phase relative to the carrier that is in phase with the modulation, and one at lower frequency that is out of phase by 180 ~. As long as there is no absorption or phase shift of the laser carrier or modulation sidebands with respect to one another, the detector photocurrent will not have a signal at the modulation frequency. (Only the optical phase is modulated, not the optical power.) A simple view of this fact is that the beating between the carrier and the upper frequency sidebands creates a photocurrent modulation that is exactly canceled by the out-of-phase modulation from the lower frequency side. If a sideband is attenuated or phase shifted, or the carrier's phase is shifted, the photocurrents will not cancel and RF power at the modulation frequency will appear on the detector signal. Near a cavity resonance, the resultant optical reflection of the carrier from the cavity is phase shifted with respect to the sideband components that are further away from the cavity resonance. Consequently, the detector photocurrent will show power at the modulation frequency. The laser's frequency noise will then appear as noise sidebands centered around the modulation frequency. When this signal is mixed to base-band (using phasesensitive detection with the appropriately chosen phase), the result is

GENERATING THE ERROR SIGNAL

7

Fro. 1.3. A schematic representation of the Pound-Drever-Hall error signal as if the laser was scanned over a cavity resonance. The cavity reflection is detected and demodulated by a mixer using a phase-delayed local oscillator. Proper adjustment of the phase delay results in a sharp central frequency discriminator, with secondary peaks spaced by the modulation frequency.

a frequency discriminator with odd symmetry that may be used to correct the frequency of the laser as shown schematically in Figure 1.3. Here we note that the light seen by the detector actually consists of two components: the fraction of the input beam that is reflected, plus the fraction of the internal cavity wave that is transmitted back out of the input coupler. The detected photocurrent represents the interference of these two components. For Fourier components of the laser's frequency noise below the cavity linewidth, this system acts like a frequency discriminator as described above. At frequencies above the cavity linewidth, the input field is essentially heterodyned with the cavity wave. Thus, for these Fourier components, the system acts as a phase discriminator, which causes the system response to faster frequency fluctuations to decrease as 1If. In electronic terms, the P D H technique gives us a frequency error signal with a sensitivity that can be measured in volts per hertz of optical frequency. At the Fourier frequency corresponding to the cavity linewidth, the sensitivity starts to decrease, and continues to decrease as 1/f. At some higher frequency, the error signal will cease to be useful as the magnitude decreases to the level of the background noise, although this limit is usually well above the attainable servo bandwidth. For those who are perhaps puzzled by some of terminology above, recall that phase-sensitive detection is the process that a simple lock-in amplifier uses, although most commercial lock-in devices do not operate at the

8

STABILIZING DIODE LASERS

frequencies required here (10-50 MHz). In this chapter we show how to build the necessary circuitry using RF amplifiers and filters, a simple mixer, and subsequent operational amplifier (op-amp) filtering and amplification. Our experimental layout is shown in Figure 1.4. The light goes from our ECDL through an optical isolator and electro-optic modulator to the optics used for coupling to the optical cavity. While this setup would be suitable for most types of lasers, let us emphasize a couple of aspects particular to diode lasers. Because diode lasers are extremely sensitive to optical feedback, good optical isolation (> 50dB) is necessary to attenuate the light reflected from the cavity. Additionally, the laser's spatial mode is not round, but is instead oval because the laser's output aperture is asymmetric. Although a spherical collimating lens is often used at the diode output, correcting the beam's aspect ratio with anamorphic prisms or cylindrical lenses would result in better spatial matching to the cavity mode. This would in turn increase the power coupled to the cavity. In the examples here, a single lens and an adjustable aperture are used for spatial mode-matching to the cavity, and two alignment mirrors are provided between the lens and the cavity. The cavity reflection is returned to the photodetector using a quarter-wave plate along with a polarizing beam splitter. The detector signal is amplified and filtered to pass the modulation frequency, and phasesensitively demodulated using a balanced mixer. The resulting signal is then

Polarizing Beamsplitter

X/4

i LaserI "'--" I Phase-Modulator

IP'FSynthezerI I

Cavity

I Delay I Mixer

~

r Band-pass Detection Amplification

LoopFilter [~ Error signal

Fla. 1.4. A basic Pound-Drever-Hall locking scheme. A modulation frequency in the range 10-40 MHz is typically used. The phase of the RF local oscillator input to the mixer is adjusted to give a symmetrical frequency discriminator at the mixer output. In the absence of residual amplitude modulation or RF pickup, the center of the cavity resonance will correspond to zero at the output. The error signal must be properly filtered and amplified to control the laser.

GENERATING THE ERROR SIGNAL

9

amplified and filtered as desired for the feedback loop, and finally sent to the diode laser frequency control elements. Let us now see how to design and construct these various pieces, starting first with the optics before moving on to the electronics.

1.3.1 Coupling Light into the Cavity In this section, we describe how to choose and align the optics used for coupling the light from the diode laser into the optical cavity. The optimization of this procedure requires consideration of two distinct and important concepts, mode-matching and optical impedance-matching. Modematching refers to adjustment of the input beam's size, shape, and wavefront curvature to match the cavity mode. Impedance-matching refers to the adjustment of the cavity parameters (e.g. mirror reflectivities) to maximize coupling into the cavity. This occurs when the transmission through the input mirror equals all the other losses of the resonator. For instance, a two-mirror cavity would be impedance-matched if the loss from the input mirror equaled the sum of the second mirror's loss plus the loss resulting from scatter and absorption on the mirrors and anything else in the cavity. If this ideal condition is met, the reflection of a perfectly spatially matched input beam will destructively interfere completely with the cavity wave transmitted back through the input coupler. Thus, no net power will be reflected off the input mirror on resonance in steady state. Usually this condition is not met, and even a perfectly spatially matched beam will not couple all the power into the cavity. However, coupling efficiencies of 80% are not uncommon. Instead, the reflected signal from the cavity on resonance will show a dip to a minimum, but not a dip to zero. In some applications, however, such as ring-down spectroscopy, impedancematching may be less important than some other consideration such as maximizing the ring-down time. In such a case, one might minimize the loss at each mirror, for example, even if it means that less power is coupled. Matching the spatial mode of a diode laser's beam to that of a spherical mirror cavity is a challenge because the laser's output beam is neither round nor "diffraction limited." Conceptually, think of a beam emanating from a waist in the cavity towards the laser, and let us call this the "cavity wave." The beam waist and radius of curvature of the cavity wave may be calculated from the mirror geometry [12]. The mode-matching task is to shape and focus the diode beam such that it approaches the conjugate of this cavity wave, i.e., a wave that is exactly like it except moving towards the cavity. Typically we start with a beam that is collimated with an additional lens, if necessary, after it emerges from the optical isolator or electro-optic

10

STABILIZING DIODE LASERS

modulator. A first step is often then to make the collimated diode beam's cross section somewhat round by using a pair of anamorphic prisms. Next, at the fictitious point where the diode beam and the cavity wave are the same size, a piano-convex lens of the proper power will focus the diode beam to a waist at the same position as the cavity's waist. If the collimated diode beam is larger than the cavity wave, use either a collimating lens of shorter focal length on the diode, or a multiple lens solution to reduce the beam's cross section. For instance, a positive lens can be used to create a converging diode beam that at some point will have the same cross-sectional size as the cavity wave. A negative lens of the proper focal length at this point will reduce the wavefront curvature, and cause the diode beam to reach a waist at the same position as the cavity waist. See Reference [12] for more details on mode-matching. A brief description on exactly how to align the diode beam to a highfinesse, two-mirror Fabry-P6rot cavity is warranted here. Much of this discussion applies to ring cavities as well. An adjustable diaphragm prior to the lens is useful, both for alignment and for reduction of the power coupled to higher order modes. It is easiest to align the diode beam to the cavity if there are two adjustable turning mirrors of good quality between the cavity and the focusing optics. Adjust the two mirrors such that the input beam is incident at the center of the cavity mirror, and also such that the cavity reflection hits the turning mirrors at the same points as the input diode beam. Observe the cavity reflection on the diaphragm; if the beam size is approximately the same size as the input beam at the diaphragm, then input wavefront curvature is well chosen. If not, either the input diode beam's radius of curvature is incorrect, or the position of the waist is not near the cavity wave's waist. Careful calculation or some experimentation should enable one to get close. Now, with a detector placed just after the cavity, one can monitor the cavity transmission on an oscilloscope, while sweeping the laser frequency slowly (~ 30Hz) with a triangle drive signal fed to the PZT-controlled mirror. Alternatively, the cavity length can be swept if one of the cavity mirrors is mounted on a PZT. Once modes are observed, the laser's wavelength sweep should be adjusted so that about two cavity flee-spectralranges are covered. Fine adjustment of the alignment (and mode-matching) should then lead to good coupling to the fundamental TEM00 mode. One common problem that arises is to figure out which peak actually corresponds to the TEM00 mode. To this end we offer several suggestions. Closing the adjustable aperture to a small size will attenuate the coupling to other modes more than the TEM00 mode, because it is spatially symmetric and the smallest mode. Also, all the mode peaks except for the TEM00 mode are actually two or more modes with approximately the same resonance

G E N E R A T I N G THE E R R O R SIGNAL

11

frequencies. This degeneracy is broken in high-finesse cavities by any departure of the mirror surfaces from a spherical shape [13]. This leads to a fine splitting of all the higher order transverse modes that is easily seen by zooming in on an unknown mode by reducing the laser sweep and increasing the oscilloscope gain. The fundamental mode does not have this degeneracy and will appear as a single peak. Observation of the transmitted spatial mode profile by eye or with a video camera after the cavity can help to optimize alignment into the TEM00 mode as well. With some patience, one can usually get to the point where the TEM00 peak is a factor of five or ten times larger than any of the competing modes. When working with very high-finesse cavities, one may notice that the peak heights vary drastically from sweep to sweep. Sweeping more slowly may increase the heights of the peaks, but may or may not reduce the variation. These effects result from the laser frequency not staying in resonance long enough (i.e. the cavity decay time) to reach equilibrium (recall that the cavity buildup time is long for high-finesse cavities). Alignment under these conditions requires patience, although we will see later on that when the servo system is operating, one can easily optimize the alignment by working with the laser locked to the cavity fringe. We also note that mirror contamination can affect the loss of the fundamental mode relative to higher order modes, especially the first odd transverse mode. It is well known that the mirrors of high-power dye laser systems will become contaminated by material deposited on the mirror surface just where the beam power is maximum. With good low-loss mirrors, the ring-down cavity power can be 10 W continuous-wave, even for milliwatt input powers, so one can expect this problem occasionally if the cavity is operated with high power. A characteristic sign of this occurring is a change in the spatial coupling such that the TEM00 mode power is degraded in relation to the transverse modes.

1.3.2 Modulating the Laser Frequency There are several competing factors to consider when choosing the modulation frequency for the PDH lock. The frequency needs to be sufficiently high so that the process of filtering and demodulating to base-band does not yield a significant phase shift within the desired feedback bandwidth. We find that choosing a modulation frequency an order of magnitude higher than the required bandwidth is more than sufficient. Although modulating at even higher frequencies (hundreds of megahertz) is possible, this requires more skill to avoid distorting the DC baseline because of RF pickup. It also makes the local-oscillator phase more

12

STABILIZING DIODE LASERS

susceptible to variations through changes in temperature of the cables, and more susceptible to residual amplitude modulation (AM) offset errors associated with variations in the optical path length. In practice, we find that modulation frequencies ranging from 15 MHz to 40 MHz work well. To modulate the laser frequency, we either modulate the laser's injection current directly or use an electro-optic modulator (EOM) placed between the laser and the cavity. Of these two schemes, using an EOM is preferable because it yields much less residual AM. Residual AM can be quite troublesome for the locking process, as it leads to DC offsets after demodulation, which in turn shift the lock point away from the peak of the cavity transmission. In principle such offsets can be compensated, but often the factors responsible for the residual AM are unstable, and lead to a drifting offset that is more difficult to null. Thus, we use an EOM when frequency drift relative to the cavity is important, but employ direct-current modulation for less critical locks. Commercial EOM units are generally configured either as resonant circuits or as broadband modulators. In order to generate sidebands of sufficient amplitude (5-10% of the carrier peak intensity), a resonant configuration makes sense because it reduces the required RF power by about a factor of ten. The addition of a low loss inductor and capacitor external to a broadband EOM (as shown in Fig. 1.5) works well and Q's > 10 are easily achieved. In order to know what fraction of the light is coupled into the sidebands by the modulator, one can couple light into the cavity and monitor the transmission spectrum. The sidebands can then be observed by sweeping the laser's frequency over a cavity resonance, or alternatively by sweeping the cavity if its length is adjustable with a PZT. This signal can then be used to adjust the RF power sent to the EOM in order to generate sidebands of the desired amplitude. Of course, an EOM adds considerable cost to the system, and it is often possible to lock by modulating the laser directly. The success of this approach depends on the magnitude of the amplitude modulation that accompanies the desired frequency modulation. This in turn depends on the laser design, specifically on parameters such as the length of the extended cavity and the output coupling of the laser. Unfortunately the same process that serves to reduce the laser's frequency noise and emission linewidth , relative to the bare diode laser also reduces the laser's frequency deviation for a given modulation current. Consequently, modulating the injection current of a laser with a long extended cavity and a high reflectivity grating will produce large AM sidebands, with very little associated frequency modulation. One can check the modulation characteristics by aligning the laser to an optical cavity and observing the amplitudes of the induced sidebands. If direct modulation produces obviously asymmetric sidebands,

GENERATINGTHE ERRORSIGNAL

13

(b)

(a)

lOpF T

?

1 uH

-li

C (crystal)@

Fla. 1.5. A broadband nonresonant electro-optic modulator can produce sidebands adequate for locking if it is driven resonantly. With an inductor mounted externally but close to the modulator, the tank circuit shown will resonate at 2rcf~ (LC) -1/2. In configuration (a), an additional external capacitor is necessary for coupling (an airspaced adjustable capacitor works well). Tuning the inductor will affect primarily the resonance frequency, while tuning the coupling capacitor will change the strength of coupling and hence the input impedance. Stronger sidebands will be obtained with configuration (b), however, this requires that neither side of the modulator crystal be grounded. The highest Q will be obtained by using nonadjustable air-core inductors and tuning the modulation frequency into resonance. this is an indication that the relative amounts of amplitude and frequency modulation are comparable, and that the degree of frequency modulation is sufficient to create an error signal for locking. The asymmetry of the sidebands is caused by the AM and F M components being in-phase on one side of the carrier and out-of-phase on the other side. For an ECDL, if the direct modulation sidebands are symmetrical, it is possible that they are primarily AM sidebands, and the mixer output will have a DC component that is large with respect to the error signal. The amount of AM is also easily detected by a high-speed photodiode. Direct modulation can be accomplished by coupling an R F signal to the injection current with a small capacitor (_< 50pF), as shown in Figure 1.6. The standard electrostatic precautions and warnings apply to this current input terminal. For instance, one needs to ensure that no current pulse is delivered to the laser when connecting a cable or turning on the RF function generator. When in doubt, the safe practice would be to connect the function generator while it is on, but with the output at zero. Regardless of the modulation technique, the amplitude of the sidebands imposed on the laser is not critical, and it is possible to trade some modulation depth for gain elsewhere in the loop. A good starting point is to set the sideband amplitude at approximately 5-10% of the carrier. If the modulation sidebands are too weak, then the noise floor of the resulting system will be higher than necessary, and the resulting laser linewidth relative to the cavity will be higher than necessary. Also, the acquisition range will be poor, as the wings of the P D H error signal will be degraded to the noise level. If the modulation sidebands are too large, the power in the carrier and the subsequent cavity power buildup and transmittance will suffer.

14

S T A B I L I Z I N G D I O D E LASERS 50 pF 1[

r",'-'c--ra__ Current Source Laser Diode

Fro. 1.6. A simple circuit to apply RF modulation to the laser through a coupling capacitor.

1.3.3 Detecting the Reflected Light Light reflected off the input mirror of the high-finesse cavity can be picked off with a simple beam splitter, or a polarizing beam splitter and either a quarter-wave plate or a Faraday rotator. The latter choices use the light more efficiently, while the former is cheaper and reduces the number of optics in the beam path (which can cause residual AM). Attention must be paid to multiple reflections in the path between the cavity and the detector, because they can lead to ~talon effects that are all too familiar to spectroscopists, l~talon effects usually manifest themselves as ripples in the mixer output as the laser is swept in wavelength. These baseline ripples can be quite large and seriously compromise the locking performance. Thus, it is important to take some care with the optics to avoid normal reflections that can lead to unintentional 6talons. Slight tilting of the detector will avoid sending the reflection from the detector surface back towards the cavity. Likewise, defocusing or using a longer focal length lens to capture the light on the detector will result in less scattered light being reflected back towards the cavity. The detector monitoring the cavity reflection should be a lownoise detector with a reasonable bandwidth and also have a low-frequency DC output. Although a simple photodiode and a load resistor may be used to capture the reflected signal, the signal size will be limited because the load resistor needs to be relatively small in order for the detector to have sufficient bandwidth. The signal then must be routed through an RF amplifier to reach levels optimum for the mixer input (see Fig. 1.4). Serious consideration should be given to using a transimpedance amplifier to convert the photocurrent output to a voltage. There are wide-band transimpedance amplifiers available with built-in feedback resistors and impressively low input noise. Although not necessary for locking purposes, the DC output is useful for monitoring the laser power and to diagnose laser-cavity coupling problems. The low-noise aspect of the detector is important if the reflected light power from the cavity is expected to be low. To have optimum signal-tonoise for the error signal, the photocurrent fluctuations must be larger than the detector noise. The amount of light power needed for this to be the case

GENERATING THE ERROR SIGNAL

15

can be calculated assuming the photocurrent fluctuations are attributable to shot noise. Of course if the photocurrent noise is actually higher than the shot level, the detector noise level will be exceeded with even less light than this calculation indicates. With no light incident, the detector-amplifier combination will exhibit some output noise density at the modulation frequency, measured in V/~/Hz which can easily be measured by an RF spectrum analyzer. One can calculate the power P0 that would be necessary for the photocurrent noise to exceed the detector noise by setting this background noise level equal to the shot noise,

Gv/2qPoR In this expression G is the transimpedance gain in V/A, q is the electronic charge, 1.6x10 -19 C, and R is the detector's responsivity in A/W. We note that there are commercial photoreceivers sold as "low noise" that will require more than 1 mW of power on the detector before the photocurrent shot level is larger than the detector noise. However, there are also transimpedance amplifiers with more than 100 MHz of bandwidth that require less than 50 gW of power before the light noise is larger than the detector's noise.

1.3.4 Demodulation Once a satisfactory signal is obtained from the detector, it is time to demodulate the error signal so that it will be ready to be used in the servo system. The effect of frequency modulation was to encode the laser frequency fluctuations around the modulation frequency (rather than DC); now it is time to "decode" this error information by mixing it down to DC. This demodulation process is easily accomplished by combining our error signal from the detector with a phase-shifted version of the local-oscillator signal used to generate the modulation in a doubly balanced mixer. We note that it can be beneficial to add an RF band-pass filter between the detector and the mixer, as signals outside of the anticipated servo bandwidth only add excess noise at the mixer input. However, care needs to be taken that the band-pass filter is not too narrow, as this will lead to phase shifts that will limit the servo bandwidth. Unless there is significant noise outside the servo bandwidth, a good starting point is to use no filter prior to the mixer. For good mixer performance, the local-oscillator (LO) signal power should be enough to fully turn on the mixer diodes. (While mixers are available for a number of different power levels, we find that those requiring an LO power of + 7 d B m to be a convenient choice.) The size of the resonance signal from the detector into the mixer's R F port should be within

16

STABILIZING DIODE LASERS

a few decibels of the maximum given in the manufacturer's specifications (typically the RF port level will be a maximum of +1 dBm for a +7 dBm LO). If the output of the detector is much less than this, a low-noise RF amplifier can be inserted before the mixer's RF port. The RF signal, which is centered around the modulation frequency, is down-converted by the mixer to base-band (i.e. DC), with the resultant output accessible at the mixer IF port. This is done in a coherent manner, and with some conversion efficiency, typically about - 6 dB. The IF port output also contains the same information at twice the modulation frequency, as a doubly balanced mixer converts the input signal at fRF to fLO--fRV and fLO+fRF. In addition, if the mixer IF port output is observed with a spectrum analyzer, a smaller but often significant narrowband component at lEo will be found. These high-frequency outputs of the mixer can be sufficiently attenuated by the loop filter, which as we shall see is actually a low-pass filter. However, it is possible that these residual RF signals can generate a small DC signal through rectification by any nonlinearity in the circuit. This can wreak havoc with the frequency stability of a system that was designed to closely track the laser to the center of the cavity. It thus makes sense to attenuate the strong high-frequency components with resonant LC notch filters at the mixer output. The resonances should be of sufficient Q to avoid unnecessary phase shifts within the desired servo bandwidth [14]. In Figure 1.7, we show circuit components to reduce f and 2f (30 MHz and 60 MHz) while introducing only I~ of phase shift in the region from 1MHz to 10 MHz. The lineshapes one detects at the output of the mixer IF port depend on the relative phase between the RF signal and the LO. In order to generate a frequency discriminator with odd symmetry and maximum amplitude, it is essential to set this relative phase to be dispersion-sensitive (i.e. 90 ~ outof-phase). The phase of the LO fed to the mixer is easily adjusted with phase-shifting IC, a delay box, or simple lengths of cable. One could instead adjust the phase of the RF signal, but this may lead to undesired delay o~ phase shifts in the servo loop. To set the LO phase to the ideal value, it is easiest to start by using an oscilloscope to monitor the signal from the IF port while sweeping the laser or cavity over a cavity resonance. A low-pas~, filter (e.g. with a corner at ~ 100 kHz) will attenuate the 2fsignal and help tc see the features more clearly. One then adjusts the LO phase and looks foJ the appropriate discriminator line shape (see Fig. 1.3). There will in fact b( two such "ideal" settings; these yield mirror images of the desirec discriminator signal (switching between these two settings gives an eas3 way to change the overall sign of the feedback signal). If the laser frequenc) jumps around too much to find the phase setting that optimizes th( discriminator amplitude, an alternative approach is to first adjust the phas~ to minimize the central feature. Then shift the phase by 90 ~ by adding o:

17

G E N E R A T I N G THE E R R O R SIGNAL m

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n

Frequency(Hz) Fla. 1.7. Series LC traps at the mixer IF port can reduce the high-frequency components while adding very little phase shift within the servo bandwidth. The curves show measured data for a two-notch filter that adds only I~ of phase shift in the region from 1-10 MHz. Once the filter is installed in the circuit, tunable elements are used to set the frequency.

subtracting the appropriate time delay, which equals one-quarter of the modulation period. The signal will appear larger as the laser is swept past the cavity resonance more slowly, because there is more time for the light to build up in the cavity mode. The error signals for a system using EOM modulation and one using injection-current modulation are shown in Figure 1.8. Note that the error signal observed when current modulation is employed contains a large DC component. This is attributable to the strong intensity modulation (residual AM) that must be compensated at the integrator input. Subsequent changes of the optical system (alignment for instance) will change this DC level, and require some adjustment of the offset compensation. The optimum locking will be obtained when the highest possible signalto-noise (S/N) exists on the error signal. The error signal's discriminator will have a linear slope measured in units of volts per laser frequency, for instance V/kHz. The factors that determine this slope include the cavity's linewidth, the sideband amplitude, the power on the detector, detector gain (V/A), postdetector amplification, and the mixing process. The noise on this discriminator should be attributable to the laser frequency noise and ultimately at a much lower level by shot noise and detector noise. The best performance comes when the system is shot-noise-limited, i.e., the S/N is

18

STABILIZING

DIODE

LASERS

0.100 30 MHz

Sweep turn around

modulation

L -0.000

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.

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.

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Time (ms) (From trigger point) FIG. 1.8. The low-pass filtered output from the mixer is the error signal. The local oscillator phase is set to maximize the amplitude, slope, and symmetry of the central frequency discriminator. Intensity modulation in the lower example results in a large DC offset, which must be compensated by adding a DC current to the integrator input.

limited by the intrinsic noise on the photocurrent. However this depends on whether the laser intensity fluctuations at the modulation frequency are at the shot-noise limit. If the laser exhibits technical noise in excess of the shot-noise level, noise reduction is possible by using a subtracting detector configuration. In this case, a fraction of the laser beam is sampled by a photodiode prior to the cavity, and the photocurrent is subtracted from the reflection signal. With proper attention to the phase of the two signals, much of the excess common-mode noise can be canceled. We will not dwell further on this approach, because ECDLs are usually within 10dB of the shot level at practical modulation frequencies. Furthermore, even though a little tighter frequency control is to be gained, there is less light available for the cavity ring-down signal. A good approach is to ensure that the noise on the signal reaching the mixer is indeed caused by the light. This is most easily checked by sending the RF input to a spectrum analyzer and monitoring the baseline noise levels

THE LOOP FILTER

19

near the modulation frequency. With the laser off-resonance (and the light level set to the approximate value that exists when the system is locked), one simply compares the noise levels with the light blocked and unblocked. If the noise does not increase when the light is unblocked, then the system is likely to be limited by the detector-amplifier noise and either the load resistance or the light level should be increased.

1.4 The Loop Filter The loop filter connects the error signal to the laser, thus completing the feedback loop. The goal of the feedback electronics is to supply enough gain to drive the laser's frequency fluctuations to the noise floor over as much of the servo bandwidth as possible. With proper design, the electronics should neither limit the residual frequency noise level, when the laser is locked, nor limit the maximum servo bandwidth. The residual frequency noise should be limited by the light's amplitude fluctuations (technical or shot noise), and the correction speed of the loop should be limited primarily by the laser chip itself. In this section we describe the construction of a loop filter that attempts to achieve these objectives. We start with a very simple introduction to servo system theory and then proceed to a concrete example, namely, a loop-filter design suitable for locking the frequency of a diode laser to a high-finesse cavity.

1.4.1 Introduction to Feedback Systems A short introduction to the stability of negative feedback systems may be useful a t this point. Our objective here is not to treat general control systems [15], but simply to provide some basic understanding and a few rules-of-thumb to aid the construction of a feedback loop to lock a diode laser. With that in mind, we note that a feedback loop supplies corrections to some device (e.g. the laser) within some electronic bandwidth. By "loop" we refer to a signal path that goes from the laser to the cavity, to a detector, to an electronics gain stage, and back to the laser. At low frequencies, the loop will have very high gain, so that small deviations of the error signal are greatly amplified and the device is forced to move in the proper,direction, effectively suppressing the deviation. In general, the perturbations will be reduced by the factor of [l+A(f)] -1, where A(f) refers to the system gain at the Fourier frequency f. However, consider a small sine-wave perturbation of the laser frequency occurring at a somewhat higher rate, say 100 kHz. As this signal traverses the loop,

20

STABILIZING DIODE LASERS

various components will shift its phase. For instance, a simple low-pass filter made from a resistor and capacitor will begin to introduce a noticeable phase shift at one-tenth of the filter - 3 d B frequency, (2rcRC) -1. As the signal frequency increases, the filter starts attenuating the signal, and the resultant phase shift increases. This phase shift stops accumulating at a frequency about ten times (2TcRC)-1, with the filter output lagging the input by 90 ~ Primarily as a consequence of the various loop components that act as low-pass filters, the correction signal will be phase shifted from the original perturbation. (Any additional time delay, such as lengths of cable, will further increase the phase lag.) We can safely predict that at a high enough frequency, the correction signal will in fact be 180 ~ away from the negative feedback; in other words, positive feedback! In this situation, if the correction signal causes the laser to move more than the original perturbation, an oscillation will of course quickly develop. To avoid this scenario, one must ensure that the loop gain is less than 0 dB, or unity gain, in the frequency range where the phase shift of the loop is approaching 180 ~. The phase margin of the loop refers to exactly how far (in degrees) from 180 ~ the phase shift is when the loop gain falls below unity. If the phase margin is less than about 45 ~, the system will tend to oscillate. In the time domain, insufficient phase margin is associated with an under-damped system, and a ringing response to transients will be observed. The task of the servo design is to ensure that there is sufficient gain at low Fourier frequencies while keeping the unity-gain frequency low enough such that the loop is stable. This naturally leads to a gain versus frequency dependence with negative slope. In fact, a plot of this dependence is a standard part of the servo designer's toolbox. These plots, called Bode plots, are drawn with log-log axes to simplify the interpretation of the various shapes. An example of such a plot is shown in Figure 1.9. A simple picture of the open loop transfer function is the gain as a function of frequency that a signal would experience if it were injected into the diode laser and detected after it travels through the whole system. The bandwidth of the loop refers to the frequency at which the gain falls to 1 (or 0 dB). This is commonly called the unity-gain point, and is often in the range of several megahertz for a servo controlling a diode laser via the injection current. A rule-of-thumb is that at the unity-gain point, the slope of the loop transfer function may not be decreasing by much more than 20dB per decade of frequency, which corresponds to a factor of 10 per decade or an f - 1 slope on the log-log plot. At lower frequencies the slope may be steeper, as shown in Figure 1.9, but the transition from a steeper slope to the f - 1 slope may not occur near the unity-gain frequency. This model

21

THE LOOP FILTER 120 110

Cavity Av

100 S

90 rn

80 Integrator switch closed

"o

60

Integrator RC roll-off

70

40 dB/decade slope l.-- 40

20 10

Unity gain -10 102

r

r

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2 3 4 567 103

t

q

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,

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2 3 4 567 104

................ 2 3 4 567 105

2 3 4 567 106

2 3 4 567 107

Frequency (Hz) FIG. 1.9. The loop gain transfer function, or Bode Plot, shows loop gain as a function of frequency. Transitions to a steeper slope are caused by RC roll-off elements and the optical cavity. The transition from -40dB/decade to -20dB/ decade near 500 kHz is caused by a differentiating stage. Note that the loop crosses the 0 dB axis with a slope of 20 dB/decade.

transfer function fulfills the requirements discussed earlier, namely, large gain at low frequencies while maintaining sufficient phase margin at unity gain to protect against loop oscillation.

1.4.2 Characterizing the Loop Transfer Function We attempt to achieve this stable transfer function by compensating the response inherent in the actual components such as the laser, cavity, and detector. First, the transfer function of the elements in the loop (except for the loop filter itself) needs to be estimated and/or measured. Then a loop filter is designed using op-amps and filters to provide gain and adjust the shape of the actual loop to approach the desired form. In this section we will investigate the loop transfer function before the addition of the loop filter, while the next section focuses on the loop filter itself.

22

STABILIZING DIODE LASERS

The two elements that should have the greatest effect upon the shape of the transfer function are the cavity and the diode laser. Other elements such as the detector have been designed to have minimal effect on the loop transfer function. (We also assume that the cables in the loop path are as short as possible to minimize loop time delay.) As mentioned previously, the reference cavity acts as a frequency discriminator at frequencies below the cavity linewidth and as an integrator ( f - 1 slope) for frequencies above the cavity linewidth. The transfer function of the diode laser itself is considerably more complicated and varies significantly among devices. However, most common low-cost diode lasers are similar in that the phase response rolls off sharply near 1 MHz. As we will see, to compensate this phase roll-off we use phase-lead circuits. While we often determine the component values for the compensation of the laser empirically, it is certainly possible (and probably preferable when time and equipment are available) to measure the complete phase and amplitude transfer function of the laser's frequency modulation response. This can be accomplished with a frequency discriminator such as that provided by the side of an optical cavity resonance. A plot of the laser's transfer function can be obtained by modulating the current of the laser and recording the amplitude and relative phase of the reflected or transmitted intensity modulation. A number of issues must be addressed in order to make sure the measured phase shift actually represents the laser's phase response. The detector's response must be known, along with that of the frequency discriminator [16]. The laser frequency must loosely be controlled such that it remains on a linear portion of the fringe during the measurement. Also, the laser's amplitude modulation must be removed from the measurement, which can be done with a subtracting-detector arrangement. Careful adjustment and matching of the DC intensity of the beams can be avoided by using negative feedback to match the photocurrents at low frequencies [17]. Although we have mentioned the shape of the servo loop's response curve, another quantity that we need to determine is the amount of gain required from the loop filter. Insufficient gain could easily bring the 0 dB point (the unity-gain point) up to the steep portion of the gain curve (i.e., the f - 2 portion of the curve), rendering the system unstable. Conversely, too much gain could push the unity-gain point up to a frequency with insufficient phase margin, again rendering the system unstable. The gain of the loop depends on the slope of the discriminator and on the modulation characteristics of the laser Rself. For Fourier frequencies less than the cavity linewidth, the discriminator slope is independent of frequency, and can be calculated by slowly sweeping over the cavity line while observing the error signal. If you know the cavity linewidth, simply

THE LOOP FILTER

23

divide the peak-to-peak error-signal magnitude by the cavity linewidth to obtain an estimate of the slope in volts per hertz. 1 The modulation function of the laser can be determined at a single frequency (we suggest 10 kHz) by driving the laser current with a sine-wave function generator applied through a coupling resistor. (Our circuit, shown later, is coupled to the laser via a 2 kS2 resistor.) A measurement of the deviation of cavity resonance will then result in a hertz per volt measurement of the laser modulation characteristic at the 10 kHz rate. 2 When multiplied together, these two measurements yield a dimensionless number, call it A', that indicates the loop gain at 10 kHz without the loop filter. Expressed in decibels, this would be 20 log[A]. One can then use this to scale the overall gain needed to achieve the desired transfer function shown in Figure 1.9. As we will see in the next section, there are many ways to adjust the overall gain in the circuit.

1.4.3 Loop-Filter Electronics The loop transfer function described in the previous section is realized by building the proper electronic loop-filter circuit. Fortunately, a fairly simple circuit based on op-amp electronics is sufficient to provide the gain and shaping required for locking a diode laser. The circuit diagram for the loop filter we have implemented for our test laser is shown schematically in Figure 1.10. The basic feedback circuit consists of a first stage op-amp that filters and amplifies the error signal coming from the mixer. The output from the first stage is split into two channels, a fast channel that goes to the laser's injection current, and a slower channel that goes to the PZT controlling the length of the extended cavity. The circuit also contains resistors and capacitors used to shape the transfer function and control the overall gain, as well as six switches (these can be controlled manually or by TTL signals) that are used to enhance the performance of the servo system.

lIf the approximate cavity linewidth is unknown, an oscilloscope with cursors can be used to measure the time between peaks of the central discriminator. Then the measurement can be calibrated using the known frequency offset of the sidebands as a frequency scale. 2Care is necessary to avoid electrically damaging the laser. It is safe to connect the function generator while the power is on, but the output should be adjusted to its lowest setting. With the laser frequency sweeping slowly so that the cavity resonance is visible, one can turn up the drive voltage until a measurement of the deviation can be made with calibrated oscilloscope cursors.

Laser Current Source

39 p~

82 Q

Laser

To PZT Amplifier

FIG.1.10. The loop-filter circuit. The switches across the first amplifier and the PZT amplifier are used for increasing the DC gain, and the input and output switches can be used to rapidly turn the servo off and on. The offset on the second-stage amplifier is used to bias the output such that the unlocked laser is not perturbed by the circuit. See text for circuit details, most of the switches shown are optional.

THE LOOP FILTER

25

Let us consider this circuit in more detail, with the initial assumption that all the switches are closed. The first stage consists of an op-amp with parallel R C feedback channels. At DC, the effective op-amp gain is 220K/1K or 46.8 dB, but at the frequency (2TcRC)-1 or 8 kHz, the gain starts decreasing and falls as f -a. This stage is responsible for one half of the - 4 0 dB/decade slope shown in our model transfer function in Figure 1.9. The remaining half is attributable to the cavity. (As mentioned earlier, the cavity roll-off that occurs beyond the [5 kHz] cavity linewidth contributes a f -1 or - 2 0 dB/ decade response.) Additional shaping of the transfer function with the first stage is achieved by including a capacitor in parallel with the input resistor. The result is a decreasing input impedance starting at the frequency (2~RC) -1 or 480 kHz for the values shown. The purpose is to bring the system gain slope from a f - 2 or - 4 0 dB/decade slope to a f - 1 or - 2 0 d B / decade slope, prior to the unity-gain frequency. Note this inflection point is shown in Figure 1.9. A much smaller resistor is included in series with the input capacitor in order to limit the very high frequency gain. Depending on the op-amp used, this resistor may not be necessary, and in any event should be as small as possible. An extra boost in low-frequency gain (below Fourier frequencies of 8 kHz) can be achieved by opening switch $3, because then the op-amp functions as an integrator. In fact, the gain at lower frequencies increases as f - a until the amplifier "open-loop" gain is achieved. This level will depend on the particular op-amp and supply voltage, but could easily be 100 dB. Thus, along with speed and noise performance, the open-loop gain should be considered when selecting an op-amp. Because of the increased gain at low frequencies, opening the integrator switch $3 forces the error signal ever closer to zero. However, with $3 open, if the laser is not locked to a cavity mode then amplifier "a" will integrate to a voltage supply rail and relocking will not necessarily occur. With $3 closed, the gain is moderate and the system can respond if the laser frequency is tuned near a cavity mode. Therefore, a good strategy is to acquire the locked condition with $3 closed; one can then open it after the lock is established. In Figure 1.9 we show (with a dashed line), the system gain through the injection current channel with $3 closed. We emphasize however, that this transfer function does not include the low-frequency gain supplied by the PZT channel. As we will describe, for Fourier frequencies under 1 kHz the PZT channel gain becomes significant and is in fact dominant near DC. Another feature of the first stage is the external offset current that is fed in via switch $2. The op-amp itself should be balanced using the standard offset compensation circuit that can be found in the amplifier datasheet. We have chosen to include an additional offset current channel because the circuit may be used with injection current modulation instead of an

26

STABILIZING DIODE LASERS

electro-optic crystal. In this case, substantial current offsets are necessary to compensate for the laser's amplitude modulation. We stress once again that current modulation is not recommended if drift of the locked laser relative to the exact center of the cavity is critical, because the amount of AM from the laser typically changes with time. The voltages that the offset current is derived from (shown in Fig. 1.10 as 4-Vo) should be well filtered and no larger than necessary to compensate for the mixer offset. The output from the first-stage amplifier is then sent through separate paths to the two correction elements. First let us consider the fast path, which passes through a variable attenuator and goes to op-amp "b" that feeds current to the laser diode. If gain beyond that provided by the first stage is needed, one can appropriately set the feedback resistance on op-amp "b." Otherwise, one can simply omit op-amp "b," which will enable a slight increase in the attainable servo bandwidth. Regardless, the resulting "fast" correction signal is sent to the diode laser through switch $6 and a 2 kS2 resistor. This resistance together with any capacitance inherent in the laser will form a low-pass filter, possibly contributing phase lag that would affect the system bandwidth. Another possible coupling configuration would be to use a smaller coupling resistor and limit the voltage of the amplifier output, but this is not recommended as it could expose the laser to dangerously large current spikes. We emphasize here that caution is required to avoid damaging the laser, because the feedback is coupled directly through the resistor to the laser anode or cathode (depending on the diode's polarity). Note that, although Figure 1.10 is drawn with a coupling to the anode of a positively biased laser, no changes are necessary to couple to the cathode of a negatively biased laser. At the end of this section, we describe how to use $6 to protect the laser against potentially fatal current spikes from the loop-filter electronics. In order to provide phase-lead to compensate the diode laser, we have added parallel capacitance on the second stage input resistance and on the coupling resistor to the laser (see Fig. 1.10). While the optimum values for these components will depend on the particular laser, the values shown are probably good starting points. In addition, in both of these places and on the first-stage input there is also a second resistor in series with the capacitor. These resistors will limit the circuit's highfrequency gain in the region above 30 MHz. The purpose is to prevent parasitic loop oscillations that can develop at very high frequencies. Again the values shown are probably good starting points, although decreasing the resistances may allow a wider bandwidth as long as the loop remains stable. Also included on the input to op-amp "b" is an offset control. Its purpose is to bias the second amplifier's output such that under steady-state

T H E LOOP F I L T E R

27

conditions (or for the unlocked state) there is very little current through the coupling resistor. In other words, we adjust the pot so that the amplifier's output is close to the laser's junction voltage. This gives the circuit the advantage that it can be connected to the laser by closing switch $6 without appreciably perturbing the laser frequency. The second path, or "slow" feedback, goes to op-amp "c" that controls the voltage sent to the PZT driver for the laser cavity end mirror (see Figure 1.1). The advantage of this configuration is that the low-frequency corrections are dominated by the large gain in the PZT channel so that the laser current DC level remains constant (large changes in this current can lead to laser mode hops). In fact, under the locked condition the DC level after the first stage should be equal to zero when this stage is configured as an integrator. This highlights one of the aspects of using integrators in these feedback loops--a nonzero correction can be applied (after amplifier "c" in our circuit) while a zero-error signal can be maintained. We also benefit from the increased low-frequency gain in the overall loop transfer function. For added flexibility in this stage we have included a switch, $5, across the feedback capacitor, so we can switch between flat gain and a full integrator (as we did with $3 on the first stage). We have assumed that the PZT driver has a polarity switch, so one can find the right sign for the feedback. If the PZT driver does not have a polarity switch, a simple inverting amplifier following the integrator may be required. We note also that for more accurate locking (a smaller offset from the center of the cavity resonance), amplifier "c" should have its inherent offset adjusted prior to any input. This can be done with a trim pot wired as shown in the amplifier datasheet. This adjustment is not important for the second-stage amplifier "b," because the gain of this stage is small relative to that of the first stage. We control the overall loop gain with a potentiometer that varies the attenuation between the first and second amplifier stages. A one-turn pot is used to minimize stray capacitance that might limit the feedback bandwidth. Care should be taken to minimize the length of the signal path by positioning the pot close to the amplifiers, preferably by mounting it on the board. The choice of amplifiers for the various stages is based on bandwidth, noise density, and offset performance. For the two amplifiers in the current correction path ("a" and "b"), we want to have amplifiers with little time delay (recall that time delay leads to unwanted phase shifts) and good noise performance. Thus we usually work with op-amps with gainbandwidth products > 30 MHz and a noise level substantially less than that of the input signal. Of course, the level of the input noise most likely depends upon how much light is incident on the detector (recall the

28

STABILIZING DIODE LASERS

discussion in Section 1.3.3). For light power above 100 gW, the noise requirement on the op-amp is not severe and should not present a problem. The op-amp in the PZT channel ("c") does not need to be fast, so a standard field-effect transistor op-amp with good DC characteristics should suffice. As we have already seen, there are several analog switches in the circuit, which we use to increase the low-frequency gain, and to allow quick, remote unlocking and relocking via TTL signals. Most of these switches are optional, with the exceptions being the PZT integrator switch $5 and the output protection switch $6. However, these may be simple manual toggle switches. The TTL-controlled switches are available from many manufacturers with many different features. Because speed is relatively important, we implement switches that can respond in about lOOns. Low-leakage current is also important in the integrator configurations; the switches we chose are specified at 100pA. Minimizing switching transients is another consideration, with the pertinent specification showing up on many datasheets as "charge injection." In order to maximize performance and flexibility, it is important to consider the layout of the electronics. The electronics board should be located close to the laser to avoid unnecessary time delays and cable capacitance. To achieve higher loop bandwidths we typically place the loop filter in a small box adjacent to the laser on the optical table. We have also chosen to place the coupling resistor and capacitor to the laser on the electronics board instead of physically placing them next to the laser. This allows an additional TTL-controlled switch to be placed on the laser-side of the coupling resistor, for the purpose of rapidly shunting a few milliampere of current away from the laser to quickly shift the laser frequency. Such switching may be useful in cavity ring-down spectroscopy, and is discussed in the following sections. We conclude this section with some words of caution. The output switch $6 is necessary in the present implementation in order to protect the laser against current spikes from the loop-filter electronics during cable connect/disconnect episodes or power up/down. When connecting or disconnecting the cable between the circuit and the laser, one first opens $6. Similarly, before turning the circuit power off, it is wise first to open this switch and disconnect the laser. While there are certainly more elegant solutions that might be included to protect the laser without contributing unwanted phase shift in the signal channel, we have found that this approach works fine in the laboratory setting. A more robust solution may be necessary, however, if the circuit is to be handled by less-experienced users.

LOCKING THE LASER AND LOOP OPTIMIZATION

29

1.5 Locking the Laser and Loop Optimization 1.5.1 Initial Locking With the loop filter designed, all the pieces are in place to lock the laser to a cavity fringe. It is easiest to start by sending a triangle wave to the laser's or cavity's PZT driver to sweep slowly over a cavity free-spectral range and view the transmitted signal on an oscilloscope. A sweep rate of ~ 3 0 H z is a good compromise between sweeping too quickly which reduces the power buildup in each mode, and too slowly which makes alignment difficult. First we need to compensate any DC offsets in the signal coming from the mixer. With the laser nominally aligned as described in Section 1.3 and the output switch $6 open, the signal at the output of op-amp "a" should yield a low-pass filtered version of the error signal, centered at zero volts DC. Any offset from zero can be adjusted with the offset current through switch $2. We usually check for locking action by closing switch $6 and turning up the gain knob (with the PZT feedback gain turned off for the moment). An increase in transmission, widening of the resonance, or even a flat section on top of the fringe are all indications that the polarity of the system is correct and the feedback loop is attempting to keep the laser locked to the fringe (see Fig. 1.11). If the overall sign of the servo is wrong, the fringe will be suppressed and the system will try to lock to the sidebands instead. This situation can be corrected by choosing the opposite slope for the discriminator (e.g. by changing the demodulation phase by 180~ Adjusting the overall gain by turning the potentiometer will give some clues as to whether more gain is required or not. If the servo seems to be trying to lock, one can then reduce the sweep to zero and the laser should stay in lock for increasing fractions of the sweep. If this is indeed the case, now is a good time to optimize the alignment to the optical cavity, because the sweep-to-sweep amplitude variations will be greatly reduced. If the servo clearly perturbs the laser but no locking action is observed, even on the sidebands, it is possible that the gain is too low and the servo is unstable. Remember that a portion of the servo loop's gain curve is decreasing as f - 2 , or - 4 0 dB per decade. If the overall loop gain is such that the servo's unity-gain point occurs on this steep slope then no locking will occur. Determining the system gain as detailed in Section 1.4.2 will offer a clue as to whether this is the problem. When the laser is locked to the cavity, increasing the gain past the optimum point will cause a servo oscillation that will be evident on the error signal. Another indication that this is occurring is a reduction of

30

STABILIZING DIODE LASERS 0.5

--

Sweep Up

-------------

Sweep Down

-

1

O --h

"~

0.4

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V

E

......

~

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.5 O

v

t_

0.3

Q_ "o 0.2

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O

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,

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6

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I

8 Time (ms)

I

I

10

i

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12

14

Fla. 1.11. Traces of the laser power transmitted through, and reflected from a highfinesse cavity as the laser frequency is swept up and down. The injection-current portion of the electronic servo is on, and briefly locks the laser to the TEM00 mode as the laser passes the resonance from each direction. Here the asymmetry is attributable to the offset adjustment of the high-gain stage in the feedback loop, but similar behavior is possible from other causes (see text). the power transmitted through the cavity as power is taken out of the carrier and put into oscillation sidebands. Reducing the gain slightly (just below oscillation) provides a good initial setting. In a similar way, the PZT gain can be increased until instability ensues and then reduced slightly. If turning on the PZT gain drives the laser out of lock, the sign of the PZT feedback may need to be reversed. With the PZT gain turned up, the signal observed after the first amplifier stage ("a" in Fig. 1.10) should be driven to zero. As an aside, we note that with high-finesse cavities there is often an asymmetry in the locking performance in regard to the direction of the frequency sweep (see Fig. 1.11). This usually is the result of a nonzero offset in the locking electronics. Such asymmetries can also be caused by localized heating of the high-finesse cavity mirrors, caused by absorbed light as the mode power builds up. In this case, the thin films will expand and the cavity resonance will shift slightly toward higher frequencies. This effect is similar to the nonlinearity observed with gas-filled cavities [18]. Regardless of the cause, the net result is that the effective sweep rate of the laser relative to the cavity is effectively higher in one direction than the other and/or the locking range is reduced. Consequently, the laser spends less time locked to the cavity, possibly to the extent that maximum transmission is never achieved.

LOCKING THE LASER AND LOOP OPTIMIZATION

31

1.5.2 Adjusting the Loop Parameters In the previous section we identified the lock condition simply by observing the light level transmitted through the cavity. In order to optimize the feedback system, it is necessary to have a better diagnostic. Fortunately, much of this information can be obtained from the same error signal one uses for the locking. A directional coupler placed just before the mixer can be used to extract a small fraction of the error signal without significantly changing the overall loop gain. One can then use an RF spectrum analyzer to view the noise sidebands, which are centered at the modulation frequency. The spectrum analyzer display should resemble the trace shown in Figure 1.12, which consists of a center peak at the modulation frequency with noise sidebands on either side. If the gain is increased, the sidebands will increase in amplitude as the servo starts to oscillate; if the gain is then reduced slightly so that the system is well behaved, the width of the noise spectrum is an indication of the servo bandwidth. One can now attempt to increase the bandwidth and reduce the level of the low-frequency noise by changing component values in the loop-filter circuit. We recall that there are three resistors in the circuit of Figure 1.10 that serve as convenient points to place a parallel capacitor, thus creating a high-pass element with its corresponding phase advance. These resistors include the first- and second-stage amplifiers' input resistors, and the coupling resistor to the laser diode. One can use the spectrum analyzer display to monitor the error signal's noise spectrum and adjust these component values in an iterative fashion to obtain the best performance. It is worth spending some time on the process by changing components and values to arrive empirically at a wider system bandwidth. A simple technique that often helps is to place a finger on one of the phase-lead capacitors, thereby increasing the actual capacitance slightly. This may offer a clue as to whether more or less capacitance is desired. Not only can the capacitor values be adjusted, but different op-amps can also be inserted to determine whether amplifier speed is playing a role in the system bandwidth. Similarly, the length of the cable between the detector and mixer can be adjusted to determine whether time delay is an issue. A spectrum analyzer monitoring the error signal can be useful in improving the performance of the servo loop as illustrated using our demonstration system. Figures 1.12a and 1.12b show a noise spectrum for the locked ECDL using EOM modulation. These data are referenced to the mixer input, and are measured by using a - 1 0 dB coupler prior to the mixer and subsequently correcting the data plots by 10dB. The lowest level depicted in Figure 1.12a is the laser's amplitude noise level. This represents the lowest level to which we can reduce the frequency noise--further gain

32

STABILIZING DIODE LASERS

only writes amplitude noise onto the frequency of the laser. The AM noise was measured by unlocking the system and attenuating the beam reflected from the cavity such that the power incident on the detector was the same as when the system was locked. The signal when no light was incident on the detector is not shown, but is approximately 10 dB below the light level. A number of technical noise spikes are evident in the laser's AM spectrum, along with a peak at the modulation frequency, 29 MHz. The top trace shows the error signal, which contains several important features. One can see at 4-4 MHz from the center, large "servo bumps," which are characteristic of the servo system. Here the phase for the correction signal is approaching 180 ~ (positive feedback), so the noise is increased rather than suppressed. If one turns up the loop gain, the system will oscillate (displaying tall, sharp peaks) in this region. The oscillation frequency indicates roughly the bandwidth of the servo--if the bandwidth is much less than anticipated, then further evaluation of the loop filter and laser/detection system is needed. Not surprisingly, we have observed that the bandwidth can change with the laser's operating parameters. This is to be expected, because we know that the frequency noise spectrum of the laser -50 ,~,

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33

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FIG. 1.12b. The noise spectrum at the mixer input, under the same conditions as Figure 1.12a, but measured with a 3 kHz resolution bandwidth. The lower trace, generated by a flashlight, shows the noise fluctuations of a shot-noise-limited photocurrent of the same DC level as when the system is locked. The laser's amplitude noise shown in the preceding figure is essentially at the same level. Note that the error signal approaches the intensity noise at low frequencies, indicating that the system is suppressing frequency fluctuations to the shot-noise limit in the kilohertz range. changes as the laser sees optical feedback, or approaches a mode hop, for instance. An increase in the magnitude of the F M noise spectrum would be accompanied by an increase of the laser's F M modulation coefficient, which in turn will affect the servo loop's gain. Similarly, optical feedback characteristics (e.g. alignment of the feedback mirror) of the laser cavity will change the operating parameters, so tweaking up the alignment can sometimes restore the locking performance. The error-signal noise at low frequencies appears to be well above the amplitude noise in Figure 1.12a, however this is a measurement artifact caused by the wide spectrum analyzer bandwidth. Taking a closer look, as in Figure 1.12b, the error-signal noise when locked is approaching the amplitude noise at approximately 20 kHz from the carrier. At even lower frequencies where the servo has higher gain, the error-signal noise will appear to be less than the amplitude noise level, however this is simply an indication that the laser is being frequency modulated by the amplitude noise. Note that we have calibrated our noise measurements to the shot-noise-limit with the aid of a flashlight to supply a shot-noise-limited photocurrent. As in the case

34

STABILIZING DIODE LASERS

of the laser amplitude noise measurement in Figure 1.12a, the shot-noise measurement was accomplished by adjusting the DC photocurrent to the same level as exists when the system is locked. The data traces show that the laser's AM noise is essentially at the shot-noise level in the region near 30 MHz. We noted the power peak at the modulation frequency in Figure 1.12a. There is almost always some residual power detected at the modulation frequency because of RF pickup by the laser cables or housing. Another possible source of AM power at the modulation frequency that often occurs is attributable to improper alignment of the modulator crystal with respect to the laser's polarization axis. However, the residual level attributable to these sources is indicated by the lower trace. The coherent spike increases by 20 dB when locked, and is most likely caused by one or more transverse cavity modes near the modulation frequency. The affected sideband would be slightly phase shifted if any power was coupled into a transverse mode. Consequently, the RF modulation detected on the cavity reflection would not cancel at the exact center of the cavity. For applications in which the small offset from the center of the cavity resonance is important, attention to the proper choice of modulation frequency, cavity geometry, and spatial coupling is required. Note that the coherent peak is still a very small fraction of the power at the mixer input when the laser is unlocked but near the cavity mode. The power when the system is unlocked may be measured with an oscilloscope as the laser is swept over a cavity fringe, and for this system it is on the order of 0dBm, or - 5 0 dBm/Hz when referenced to the 100 kHz spectrum analyzer bandwidth of Figure 1.12a. At this point the performance of the laser lock should be adequate for most applications that require narrow linewidths and high spectral resolution. It is now appropriate to expand the capabilities of the lock and start to take advantage of the switches to make the lock more robust and more flexible.

1.5.3 Evaluating the Locked System With the servo working reasonably well, one might be curious as to the linewidth of the locked laser and how to measure it. This problem is actually trickier than one might guess. The best way is to beat it against a more stable laser on a photodiode and evaluate the resulting fluctuations. The strength of this technique is that it reveals both of the major contributors to the laser's frequency-noise spectrum: the residual noise of the laser lock, and the fluctuations of the cavity length. However, this requires access to a more stable laser of the same color. One can of course build a replica of the first locked laser system and make a comparison. An independent Fabry-P6rot

LOCKING THE LASER AND LOOP OPTIMIZATION ,

35

cavity can also be used as a frequency discriminator. However, both of these approaches will suppress any noise that is common-mode to the two systems under comparison, so the resulting beat-note noise spectrum may be artificially clean. This is a serious problem because a frequency fluctuation of kilohertz magnitude corresponds to a vibration of the cavity length below the picometer level (25cm cavity length, 800nm wavelength). The selfheterodyne techniques involving long optical fibers are impractical, given the long coherence length of a laser locked to a high-finesse cavity, and the additional phase noise caused by the fiber. For many applications including evaluation of the lock itself, it is useful to measure the noise within the servo loop relative to the cavity. It is very important to emphasize that measurements of the noise within servo loops are not good indications of the actual laser linewidth or lineshape. One reason is that the servo gain acts to minimize the noise at the integrator input, even if the noise is not caused by frequency fluctuations. For instance if some of the signal fluctuations at the integrator input are actually amplitude fluctuations, the servo will suppress this signal by increasing the frequency fluctuations. Nonetheless, the noise measurements are useful in determining the performance of the electronics, transducers, and servo components. In the absence of excess noise of the electronics or cavity (usually not the case), one can make an estimate of the performance of the servo system. In particular, consider the case in which the noise spectrum is flat out to some bandwidth, and whose noise level is small compared to that bandwidth (typically the case for a well-stabilized laser). Then the contribution of this noise to the laser line shape is a Lorentzian with a linewidth equal to A v = rc(Av~ms)2/B, where A vrms is the root-mean-squared fluctuation derived from the spectrum analyzer noise level divided by the discriminator slope, and B isthe bandwidth for the spectrum analyzer measurement. Many spectrum analyzers give the noise level as spectral power, in dBm/Hz; one needs only to.convert dBm to volts 2 and then divide by the square of the discriminator slope to convert the measurement into hertz. While our noise spectrum is not usually completely flat, one can nonetheless use this formula to make estimates that are reasonable enough, especially in light of the fact that this approach neglects fluctuations of the cavity itself, which for most good locks, probably are the dominant noise source. There are also mathematical techniques for mapping spectra that are more complicated into laser line shapes [19, 20].

1.5.4. Making the Lock More Robust While the method described thus far should produce a good laser lock, there are several minor modifications we can add to make the locking

36

STABILIZING DIODE LASERS

system more robust and reliable. In particular, the system will be easier to work with if it can keep the laser locked in the face of moderate disturbances and quickly recover from more severe events. Increased servo gain is the answer to acoustic perturbations, as the loop will reduce the error-signal fluctuations by nearly the inverse of the gain. For examPle , at acoustic frequencies the system gain may be 100 dB, or 105. The frequency noise of the laser caused by voices will be reduced to 10.5 of what it would have been in the unlocked case. More challenging problems come with sharp raps on the mechanical system caused, for instance by dropping a tool on the optical table, as the system has much less gain at a few hundred kilohertz. As a first step toward reducing the sensitivity to such effects, it is advisable to use solid mechanical structures for the optical components in the system. Good vibration isolation of the reference cavity and the laser box will reduce the instability of the system and hence the servo requirements. Other effective passive approaches include the use of robust mounts (i.e. no laboratory posts) for the mirrors and beam splitter. Even with good mechanical mounting, the system will still be susceptible to out of lock events caused by sharp mechanical perturbations. Here the inclusion of two electronic switches in the control circuit, one manual, and one electronic, serves to improve the lock and make it more robust. The first of these, $5 in the circuit diagram, simply turns the op-amp "c" from a proportional gain stage to a pure integrator. This increases the lowfrequency gain and drives the signal after the first stage to zero as desired. During acquisition it is preferable to have this switch closed (i.e. integrator off); otherwise, this op-amp will sit on the rail (because of even tiny offsets, an integrator will drift to rail quickly when the laser is out of lock) and make acquisition unidirectional at best. When lock is achieved, one can simply flip the switch to increase the low-frequency gain. A second switch, $4 in the circuit diagram, can be inserted to make the lock considerably more robust. This switch may be configured so that it simply turns off the input to the PZT integrator when the laser momentarily goes out of lock for any reason. The TTL signal that controls the switch is derived from a comparator that monitors whether the transmission through the cavity (i.e. the voltage from a detector placed after the cavity) is greater than a reference voltage (set to be roughly half of the detector voltage when the laser is locked to the cavity). When the laser is pulled into resonance by the current, the switch is closed and the PZT channel is turned on. If a perturbation momentarily knocks the laser out of lock, the switch automatically opens and the value on the capacitor is held, so that when the perturbation is over the laser frequency should be close to the previous value and relocking to the cavity can occur easily. We have found that the addition of this switch will keep the laser locked in the presence of fairly

REPETITIVE LOCKING FOR CAVITY RING-DOWN SPECTROSCOPY

37

significant noise. This makes it possible to work on the table without knocking the laser out of lock. Note that if the laser stays out of lock for more than a short time, the integrator will start to drift away from the "lock" value toward a rail and relocking is less likely. To reacquire the lock, usually one needs to close $5 to turn off the integrator. In principle, this can pull the laser frequency quite far from that of the desired cavity mode, especially if the previous "lock" value was large. We have found that with occasional monitoring, this correction voltage from the amplifier can be kept near zero simply by tuning the DC knob on the PZT driver. Then relocking to the same cavity mode is usually straightforward. During momentary unlock events the input switches (S1 and $2) to op-amp "a" could also be opened in a manner analogous to using $4 as discussed above. This would allow the amplifier feedback switch $3 to remain open, in integrator mode. However, the RC time constant of this amplifier is much shorter than amplifier "c," and our experience has been that relocking is much less likely if switch $3 remains open. There are other approaches to building "smart" servos that automatically relock the laser. For a processor-controlled system, it is possible to control the locking electronics and the laser in such a manner as to mimic manually relocking of the laser. Alternatively, this can be achieved using analog electronics. For instance, one approach upon an accidental unlock is to sweep the laser back and forth to once again "find" the cavity mode. Our implementation uses a single-pole, double-throw switch on the input to the first op-amp, in place of S1. The normally closed switch setting would connect the mixer output to the op-amp input. The other switch position connects the op-amp to a small oscillating current, generated by a low-pass filtered 555-timer circuit. When the cavity transmission falls below threshold, S1 is set to sweep the laser. Once the laser relocks, S1 is switched and the sweep is no longer sent to the circuit. This works quite well, as pounding on the optical table will cause the laser to unlock, reacquire, and lock within a fraction of a second. The circuit is shown in Figure 1.13.

1.6 Repetitive Locking for Cavity Ring-Down Spectroscopy We conclude this chapter with some modifications to the locking circuits that make it well suited for cavity ring-down spectroscopy (CRDS) experiments. Many CRDS measurements are realized with pulsed lasers

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FIG. 1.13. An analog solution to automatically relocking the laser after it becomes unlocked because of some perturbation. The cavity transmission is monitored with a threshold circuit, and when the laser is off-resonance this circuit injects a small, slowly sweeping current generated by a 555 timer into the feedback electronics. The three-way switch can be set such that the input stage always receives the error signal, or always receives the sweep current, or receives the error signal only when some light is transmitted by the cavity.

REPETITIVE L O C K I N G FOR CAVITY R I N G - D O W N SPECTROSCOPY

39

or continuous-wave (CW) lasers that are swept through the cavity fringe. CW locking of the laser to the cavity has advantages that have been previously recognized and discussed in the literature [21-24]. These advantages come from the ability to excite a single-cavity mode, and with more power than is possible by simply sweeping past the mode. Such experiments use repeated locking and intentional unlocking to generate ringdown decays from the cavity, improving the signal-to-noise through averaging. While we briefly discussed relocking issues in the previous section, the process of rapid unlocking and relocking of a laser to a cavity requires a more detailed understanding of the reacquisition process. In this final section, we will discuss repetitive locking and show how the addition of a few more switches to the loop filter can accommodate its requirements. A limitation during reacquisition of the lock is the cavity ring-up time, which of course is the same as the ring-down time. In other words, one expects that the system cannot lock instantly, and that the time period required for the cavity transmission to reach a stable intensity will be no shorter than the cavity-decay time. For a cavity with a high Q, we expect that the contribution of the servo to the time required to reach an equilibrium state is relatively minor, because the servo bandwidth is approximately 1 MHz, while the timescale to equilibrium for a typical highQ cavity for ring-down spectroscopy is on the order of hundreds of microseconds. As expected, we find that during the relocking process the error signal returns to zero well before the cavity transmission stabilizes. There are two caveats to this; the first being that switching the servo "on" does not mean that the laser will immediately be driven with high gain to the resonance center. In fact if the laser frequency is off resonance by more than a few cavity linewidths (but less than the modulation frequency), the laser could still be driven in the proper direction, but relatively slowly, as the error signal is near zero. Thus, one expects that acquisition will take longer than the microsecond timescale that corresponds to the servo bandwidth. The second caveat is that for a system with a frequency control "knob" other than the laser injection current (such as the piezoelectric length of an extended cavity), the laser frequency and the transmitted power through the cavity may not be perfectly correlated. For instance, in response to a transient that shifts the cavity resonance towards the blue, the servo may quickly return the laser frequency to the center of the cavity resonance by decreasing the injection current and hence also the laser power. With the laser frequency fixed once again on the cavity resonance, the circuit in Figure 1.10 will slowly bring the injection current back up to the original value by acting on the PZT, decreasing the laser cavity's mechanical length. Consequently, we expect the "ring-up" of power in the cavity to the previous stable value may take slightly longer than the ring-down, because

40

STABILIZING DIODE LASERS

the laser power was lowered to relock the system. This process occurs in the reverse sense also, causing a slightly faster return to equilibrium if the laser power is increased to relock. In order to switch the laser in and out of lock repetitively, we need some fast way to turn the servo off and back on. Simply switching the error signal off (with S 1) at the first amplifier's input serves to unlock the laser, because the servo loses access to the error (i.e. correction) signal. Switching the output switch $6 has the same effect. We keep the PZT integrator switch ($5 in Fig. 1.10) open during the unlocking and locking cycle so that the laser's PZT drive voltage remains appreciably unchanged. Furthermore, for the same reason the integrator input switch $4 remains open during "unlock" and closed during "lock." If the input switch $4 is not utilized, the output of the integrator may be dependent on the duty-cycle of the unlocking and locking. This is a function of the current through the integrator's input resistor during the "unlock" portion of the cycle. If it is nonzero, the integrator output will attempt to steer the laser away from the resonance, which will result in the system breaking lock after only a few cycles of the relocking sequence. To allow the cavity ring-down time constant to be measured, CRDS experiments have most often used acousto-optic modulators to switch the input beam away from the cavity mirror. Switching the laser to a subthreshold value has also been employed [25]. Rapid frequency shifting of the laser to a stable off-resonance frequency has been used to allow heterodyne ring-down measurements in reflection [26]. Observing the ring-down decay in transmission by frequency shifting is also possible for some cavity configurations. The shift must be accomplished in a manner such that the cavity is a high-attenuation filter. This requires attention to the cavity's transverse-mode structure in the vicinity of the fundamental mode, and the cavity's vibrational stability. Here we have used a half-symmetric resonator constructed from a plane mirror and a 30cm radius mirror, separated by about 25 cm. There is a 15 MHz wide-frequency region to the blue side of each fundamental mode of this cavity that appears free from transverse modes. A modal analysis indicates that in this region there are no modes with combined transverse-mode orders (n+m) of less than 100. Thus by shifting the laser frequency so that the laser sits in this "dark" frequency region during cavity ring-down, we should have minimal contamination of the ring-down signal attributable to leakage of the carrier into the cavity. This approach may require neglecting some data points at the beginning of the ring-down decay if the laser does not shift frequency quickly enough. To ensure the locking sidebands at 4-29 MHz do not leak into the cavity, they are turned off with another CMOS analog switch at the start of each ringdown event. The switch was arranged in the center of a "pi"-shaped 50

REPETITIVE LOCKING FOR CAVITY RING-DOWN SPECTROSCOPY

41

configuration, which provided 30 dB of "off" attenuation and about 1 dB of "on" attenuation [14]. We note that more attenuation can be accomplished by using three switches, one in series with each resistor of the "pi" attenuator. The laser-frequency shift is accomplished with another analog switch, which is not shown in Figure 1.10. The switch is positioned on the diode laser side of the 2 kS2 coupling resistor, and, when closed, connects the diode anode to ground through a variable resistor. The switch is closed by the same TTL signal that unlocks the laser. Severallhundred microseconds later, after the ring-down measurement is completed, the switch is opened and the laser frequency returns to the vicinity of the cavity fringe (i.e. within the P D H capture range). A single cycle of this unlock, relock sequence is illustrated by Figure 1.14. During this cycle the laser frequency is monitored by using an auxiliary reference cavity, positioning the resonance such that

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FIO. 1.14. Data from a single ring-down and relocking cycle. The exponential decay from the high-finesse measurement cavity corresponds to the left scale, while the laser's optical frequency is indicated by using the transmission from an auxiliary reference cavity. Prior to the ring-down event (triggered at t = 0), the laser is locked to the high-finesse cavity and the reference cavity transmission is approximately 75%. The laser is shifted to the blue approximately 3-4MHz by shunting 0.3 mA of current away from the diode when it is unlocked (see text). The relock signal happens at 355 ps, and reacquisition and locking occurs in the next 100 gs.

42

STABILIZING DIODE LASERS

the laser frequency is on the side of a fringe. The sample rates of both the ring-down data and the reference cavity data are 0.4 gs per point. Within three data points (~1.2 gs), the laser frequency has shifted about 3-4 MHz away from the high-finesse resonance. We find that the repetitive locking works better with only one full integrator in the system, so we keep switch $3 closed for both locking and unlocking. A system undergoing this repetitive locking is shown in Figure 1.15. Although only six cycles are shown, the system is capable of indefinitely unlocking and relocking. Using software to set the TTL "unlock" and "lock" delays, we have cycled through 106 ring-down sequences. However, some caution must be exercised before adding a large number of decays together to measure the exponential decay time. The limitation on the measurement time and subsequent number of averages will be given by any nonrandom process that may change the cavity loss, for instance a subtle change in gas pressure or temperature. In such a situation, although the random noise will appear to decrease with averaging, the

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Time (ms) Fla. 1.15. Repetitive locking of the laser is possible by opening and closing the input switches or the output switch. At t = 0, the first ring-down cycle is triggered, and the laser is simultaneously unlocked and frequency shifted to the blue by decreasing the current to the diode. The laser is shifted back near the cavity resonance and the input switch closed 215 gs later. The transmitted power through the cavity returns to an equilibrium value, and the process is triggered again at a 820 Hz repetition rate.

REPETITIVE LOCKING FOR CAVITY RING-DOWN SPECTROSCOPY

43

measurement repeatability will not improve. Shown in Figure 1.15 is an arbitrary 1000 gs delay between the end of each "unlock" period and the start of the next "unlock" period. Higher throughputs (faster averaging times) are possible by reducing this dead time. As may be observed in Figure 1.14, this system returns to the steady-state transmission in about 350 gs from the TTL "lock" signal. Even though the system will relock to the cavity center in a shorter amount of time, this does not necessarily mean that the cavity transmittance has stabilized at its previous value. A set of twenty-five sequential decays is averaged together and is shown in Figure 1.16. The exponential decay appears linear on a log scale, until the trace reaches the noise floor set by the detector noise level. Fitting an exponential curve to this data results in a 1/e time constant of 24.2 gs. The exponential fit was weighted for shot noise (square root of the signal level), and includes a constant term for the offsets that account for the analog-todigital converter and detector noise. 10.0

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44

STABILIZING DIODE LASERS

1.7 Conclusions In this chapter we have described in some detail a fairly simple approach for locking an ECDL to a high-finesse Fabry-P6rot cavity. In our lab, we have used this approach to lock diode lasers for many different applications including precision metrology of atom-trapping experiments, high-resolution atomic spectroscopy, cavity ring-down spectroscopy, and length metrology. While the different lasers required different values for the electronic components, we have found the overall approach to be quite flexible and yield good results on a fairly short time scale. Our experience has shown that laser linewidths of well under 1 kHz can be readily achieved with this technique. We note that many other laboratories have used a similar approach to achieve excellent results as well. We have attempted to make this chapter "user-friendly" in the sense that it includes details not normally included in the standard research papers. We even added descriptions of a few "bells and whistles" in the form of switches that might make the locking system more reliable and easy to work with. Of particular interest and importance in the present context is the demonstration of rapid locking and unlocking of an extended-cavity diode laser to a high-finesse optical cavity. This technology can greatly enhance the signal-to-noise ratio and data rate in cavity ring-down spectroscopy.

References I. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Laser phase and frequency stabilization using an optical resonator, Appl. Phys. B 31, 97-105 (1983). 2. M. Zhu and J. L. Hall, Stabilization of optical-phase frequency of a laser system--application to a commercial dye-laser with an external stabilizer, J. Opt. Soc. Am. B 10, 802-816 (1993). 3. H. R. Telle, Narrow linewidth laser diodes with broad, continuous tuning range, Appl. Phys. B 49, 217-226 (1989). 4. H. R. Telle, Stabilization and modulation schemes of laser diodes for applied spectroscopy, Spectrochim. Acta Rev. 15, 301-327 (1993). 5. A. S. Arnold, J. S. Wilson, and M. G. Boshier, A simple extended-cavity diode laser, Rev. Sci. Instrum. 69, 1236-1239 (1998). 6. P. Zorabedian and W. R. Trutna, Jr., Alignment-stabilized gratingtuned external-cavity semiconductor laser, Opt. Lett. 15, 483-485 (1990).

REFERENCES

o

o

10.

11.

12. 13. 14. 15. 16.

17. 18.

19. 20. 21. 22.

45

A. Schoof, J. Grunert, S. Ritter, and A. Hemmerich, Reducing the linewidth of a diode laser below 30Hz by stabilization to a reference cavity with a finesse above l0 s, Opt. Lett. 26, 1562-1564 (2001). B. Dahmani, L. Hollberg, and R. Drullinger, Frequency stabilization of semiconductor lasers by resonant optical feedback, Opt. Lett. 12, S76-S78 (1987). H. Li and H. R. Telle, Efficient frequency noise reduction of GaA1As semiconductor lasers by optical feedback from an external high-finesse resonator, IEEE J. Quantum. Electron. 25, 257-263 (1989). H. Patrick and C. E. Wieman, Frequency stabilization of a diode laser using simultaneous optical feedback from a diffraction grating and a narrow-band Fabry-P6rot cavity, Rev. Sci. Instr. 62, 2593-2595 (1991). S. Ohshima and H. Schnatz, Optimization of injection current and feedback phase of an optically self-locked laser diode, J. Appl. Phys. 71, 3114-3117 (1992). A. E. Siegman, "Lasers." University Science Books, Mill Valley, California, 1986. C. Fabre, R. G. Devoe, and Brewer, R. G. Ultrahigh-finesse optical cavities, Opt. Lett. 11, 365-367 (1986). C. Hutchinson, Ed., "The ARRL Handbook for Radio Amateurs." The American Radio Relay League, Newington, Connecticut, 2002. R. C. Dorf, "Modern Control Systems." Addison-Wesley Publishing, Reading, Massachusetts, 1989. P. Tremblay and R. Ouellet, Frequency response of a Fabry-P~rot interferometer used as a frequency discriminator, IEEE Trans. Instrum. Meas. 40, 204-207 (1991). P. C. D. Hobbs, Ultrasensitive laser measurements without tears, Appl. Opt. 36, 903-920 (1997). P. Dub6, L.-S. Ma, J. Ye, P. Jungner, and J. L. Hall, Thermally induced self-locking of an optical cavity by overtone absorption in acetylene gas, J. Opt. Soc. Am. B 13, 2041-2053 (1996). D. S. Elliott, R. Roy, and S. J. Smith, Extracavity laser band-shape and bandwidth modification, Phys. Rev. A 26, 12 (1982). R. W. Fox, Trace Detection with Diode Lasers. Ph.D. Thesis, University of Colorado, Boulder, 1995. D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, CW cavity ring-down spectroscopy, Chem. Phys. Lett. 264, 316-322 (1997). B. A. Paldus, C. C. Harb, T. G. Spence, B. Wilke, J. Xie, J. S. Harris, and R. N. Zare, Cavity-locked ring-down spectroscopy, J. Appl. Phys. 83, 3991-3997 (1998).

46

STABILIZING DIODE LASERS

23. C. R. Bucher, K. K. Lehmann, D. F. Plusquellic, and G. T. Fraser, Doppler-free nonlinear absorption in ethylene by use of continuouswave cavity ring-down spectroscopy, Appl. Opt. 39, 3154-3164 (2000). 24. T. G. Spence, C. C. Harb, B. A. Paldus, R. N. Zare, B. Willke, and R. L. Byer, A laser-locked cavity ring-down spectrometer employing an analog detection scheme, Rev. Sci. Instrum. 71,347-353 (2000). 25. A. A. Kosterev, A. L. Malinovsky, F. K. Tittel, C. Gmachl, F. Capasso, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, Cavity ring-down spectroscopic detection of nitric oxide with a continuouswave quantum-cascade laser, Appl. Opt. 40, 5522-5529 (2001). 26. M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, R. N. Zare, M. J. Lawrence, and R. L. Byer, Frequency-switched heterodyne cavity ring-down spectroscopy, Opt. Lett. 25, 920-922 (2000).

2. SPECTROSCOPIC APPLICATIONS USING RING-DOWN CAVITIES Giel Berden

FOM Institute for Plasma Physics, Nieuwegein, The Netherlands

Gerard Meijer

FOM Institute for Plasma Physics, Nieuwegein, and University of Nijmegen, Nijmegen, The Netherlands

Wim Ubachs

Free University, Amsterdam, The Netherlands

2.1 Introduction The present chapter deals with the applications of cavity ring-down spectroscopy and its variants. In the first section, a rather large number of studies involving the weak transitions in the oxygen molecule, including the effects of magnetic fields and collisions, are described. This molecule has served as a benchmark system to demonstrate the opportunities of ringdown techniques. In a subsequent section, we give an overview of the studies in which ring-down techniques have been applied to produce new spectroscopic data on molecules. In the last section, we have listed some pros and cons of ring-down techniques and the alternative techniques commonly used in molecular spectroscopy: laser-induced fluorescence, resonantly enhanced multi-photon ionization, coherent anti-Stokes Raman spectroscopy, degenerate four-wave-mixing and the non-laser Fourier transform spectroscopic technique.

2.2 02: A Benchmark System for Ring-down Techniques In many studies performed to demonstrate aspects of ring-down techniques the oxygen molecule is chosen as the target; this choice is connected to the properties of a somewhat peculiar set of low-lying electronic states in this molecule. Its lowest molecular orbital configuration ,2 ,2 ,2 (lSO'g)2 (ls%) (2SO'g)2 (2Set.) (2pCrg)2 (2pZru)4 (2pZrg) gives rise to the three wellknown lowest electronic states of the molecule: the electronic ground state X 3Eg and the excited states a 1Ag and b 1Eg.+ Transitions between the ground 47 EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES Vol. 40 ISBN 0-12-475987-4

Copyright 9 2002 by Academic Press All rights of reproduction in any form reserved. ISSN 1079-4042/01 $35.00

48

APPLICATIONS USING RING-DOWN

SPECTROSCOPY

state and these excited states are well studied: the b 1Ng-X + 3~2 transition, in the visible range is called the Atmospheric system, while the a 1Ag-X3Ng transition is called the Infrared atmospheric system. Also known is the Noxon (b 1Ng-a + 1Ag) system connecting the two electronically excited states. As shown in Figure 2.1 besides these states there are more bound states correlating with two ground state oxygen atoms O(3p)4-O(3p): the c 1~C~-, A t3Au, and A3~C+ state. The three systems connecting the electronic ground state to these states, observed in the ultraviolet, A 3I ]+, - X 3~g, At3 A,--X 3leg, and c 1]C~--X3~C2, are referred to as the Herzberg I, II, and III systems. The lowest excited state, a lag, is connected to these three excited states as well, but only the Chamberlain bands (A'3A,-a 1Ag) and the Richard-Johnson system (cl~C-2-alAg) were experimentally observed. As follows from an ab initio study of the low-lying states of 02, apart from the six observed

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02: A BENCHMARK SYSTEM FOR RING-DOWN TECHNIQUES

49

bound states there are a number of additional states, 3FI,, l I-[g, 51-[u, 5~u, and 5rig all correlating with O(3p)+o(3p); these states are basically repulsive at the first few Angstroms of internuclear separation and are all very weakly bound at large internuclear separation [1]. The 5rig state, with the largest well-depth of the quintet states (0.16 eV), is included in Figure 2.1. Between the lowest six of these states fifteen radiative transitions are conceivable, but the peculiarity lies in the fact that they are all forbidden in the electric dipole approximation. Hence the electronic transitions are very weak and only eight of them have been observed. The excited state lifetime of the b 112g+ state is about 11 s; the transition, connecting to the ground state, is threefold forbidden in view of I~+-Y:-, triplet-singlet, and g-g transitions. The b-X system, allowed for magnetic dipole transitions, is a factor of 109 weaker than typical electric dipole allowed transitions. The higher lying v > 0 vibrational levels in the b 1YZg + state suffer from much lower Franck-Condon overlap with the xayZg, v - 0 ground state level, resulting in oscillator strengths offl0 - 1.6 x 10-11 and f 2 0 - 6.3 x 10-13. Nevertheless it was these two weak transitions, the Atmospheric B and y-bands, that was recorded in the first experiment demonstrating cavityring-down (CRD) spectroscopy [2]. Particularly because of the weakness of the bands, and in view of the easy availability of oxygen gas, and the atmospheric transitions being in the range of commercially available tunable lasers, these transitions have served as the prototypical test systems in many subsequent studies, demonstrating the power and sensitivity of ring-down techniques. In fact before the invention of the powerful CRD method, some physically related techniques using the enhancement properties of cavities, also focused on the forbidden optical transitions of the oxygen molecule to demonstrate their power and feasibility [3, 4].

2.2.1 The bl ~]g-FX3~g Atmospheric System Naus and Ubachs and co-workers have measured the CRD absorption spectra of the b 112g+-X312g (v, 0) bands (v = 0-3) of 1602, 160180, 160170, 1802, 170180, and 1702 isotopomers of oxygen [5-8]. In these works the focus was on spectroscopy and the generic pulsed CRD technique, with a fitting of the decay transients. As a result new or improved molecular constants for the b 1Y;+, v states were determined for all isotopomers, while for 170 2 the constants in the X ly;+, v - 0 ground state were determined. Note that the (3, 0) band near 590 nm, also referred to as the 3-band is 104 times weaker than the (0, 0) band; part of a recorded spectrum for the ~-band of 1602 at a pressure of 183 torr of natural oxygen is shown in

50

APPLICATIONS USING RING-DOWN SPECTROSCOPY I

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Figure 2.2. The same 3-band was also investigated by means of intracavity laser absorption spectroscopy (ICLAS) [9] and this allows for a comparison of this method with the pulsed laser CRD technique; the ICLAS spectrum yielded a similar signal-to-noise ratio (SNR), but was recorded at 3.5 times higher pressure, resulting also in somewhat collisionally broadened lines. The SNR in the generic CRD experiment depends of course strongly on the reflectivity of the mirrors used, and this was R > 99.99% in Reference [8]. The CRD studies of References [5-8] focused on an improvement of the spectroscopy of the b - X system of 02, in particular for the less abundant isotopomers. In view of the inherent Doppler broadening of the CRD technique, the accuracy in the old work of Babcock and Herzberg [10] could not be significantly improved. In fact also the sensitivity, previously obtained from long atmospheric path lengths, is comparable to that of the recent CRD studies; some lines of the weakest 3-band were already recorded back in 1948. In Reference [5], using pulsed CRD near 760 nm with mirrors of R ~ 99.998%, a quadrupole transition in the b--X(0, 0) band, with an intensity of 2 x 10 .9 cm -1 could be observed, demonstrating the high sensitivity. It is noted, however, that such transitions were observed, even with higher SNR, from atmospheric absorption measurements over 43 atm km absorption path lengths [11].

02:A BENCHMARKSYSTEMFORRING-DOWNTECHNIQUES

51

In the pulsed C R D experiments some of the isotopic species were observed in natural abundance, as previously in atmospheric absorption, but also isotopically enriched samples could be used. This demonstrates a notable advantage of the ring-down setup: long effective path lengths, and therewith high spectroscopic sensitivity can be obtained from small cavities for which only minute gas samples are required. Here the C R D technique is favored over the alternative application of FT spectrometers attached to White-multipass-cells, where much higher gas inputs are required. Particularly when applied to expensive isotopically enriched gas samples this advantage is of importance. With the use of a cell of 40 cm length, a diameter of 1 cm and a pressure of 180 torr, a spectrum of the very weak g-band of 1802 could be recorded [8] from a sample of ~10 atm cm 3 or 4 x 10 -4 M. Similarly, in an application of phase-shift C R D the v-band of 1802 was measured using a minute sample cell [12]. Also the use of the cavity-enhanced-absorption (CEA) variant of ringdown techniques (discussed below) in application to an absorption measurement of oxygen in a slit jet demonstrates a specific advantage over FT-techniques. The ~P](1) line of the b 1Eg-X+3Eg(0, 0) band of 1602 could be measured with a cavity of only 10 cm length [13]. The slit-jet configuration resulted in a linewidth of less than 0.01 cm -1 on this resonance shown in Figure 2.3, which is due to residual Doppler-broadening

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52

APPLICATIONS USING RING-DOWN SPECTROSCOPY

in the planar jet-geometry. This example constitutes the first measurement of an optical transition in 02, which is not limited by the full Dopplerbroadening encountered when probing static gas samples. In this example only the lowest rotational energy level could be probed, due to the jet conditions, where a mixture with strongly cooling argon gas was used; this technique might be used in future to improve upon the electronic spectroscopy of the electronic transitions in 02. Xu et al. [14] performed quantitative CRD studies on three atmospheric bands b-X(v', 0) for v ' = 0-2, with a laser of 0.09 cm -1 bandwidth, just somewhat broader than the molecular absorption feature. In the analysis of the decay transients the signal was assumed to originate from two contributions, giving rise to a biexponential transient: one part was assumed to be exactly on resonance and absorbing proportional to the strength of the spectral line, while the remaining part was assumed to be off-resonant. Within this model-assumption, pressure dependent rotational line strengths were analyzed as a function of pressure for a number of lines in the three bands. Resulting values for the line strengths confirm the quoted values in the HITRAN database. Pressure broadening, involving self-broadening effects as well as collisions with N2, was investigated by CRD on the b--X(1,0) band at 688 nm [15]; there is a vast literature on this subject coming from FT studies.

2.2.2 The a 1Ag--X3~g Infrared Atmospheric System The a 1Ag-X3]Eg (0, 0) band at 1.27 gm was investigated independently by two groups using the generic pulsed CRD method [16, 17]. This nearinfrared atmospheric system is, with an upper state lifetime exceeding 1 h, even weaker than the b--X system. In the study of Newman et al. [16] CRD was compared with long path Fourier-transform spectroscopic investigations involving 60 atmm effective path lengths. As for the sensitivity, the CRD and FT methods are found to be equally suited as spectroscopic tools. In this region of the near-infrared, where the combined Doppler and collision broadened width (for atmospheric pressure) on oxygen lines is typically 0.1 cm -1, most commercially available pulsed laser systems have bandwidths beyond this value. As in the case of Newman et al., the bandwidth of 0.25 cm -1 gives rise to problems in deducing quantitative information on the line intensities. This bandwidth problem has been recognized and treated formally by various groups [18-20] and is in a sense similar to the slit-width problem in classical spectroscopic investigations. In CRD problems arise because the measured transients are no longer monoexponential if the bandwidth of the laser exceeds the widths of the

02: A B E N C H M A R K SYSTEM FOR R I N G - D O W N TECHNIQUES

53

molecular resonance. Note that Xu et al. [14] assessed this problem slightly differently with the assumption of a two-component decay, as discussed above. Newman et al. followed the estimates and the analysis of Zalicki and Zare [19], who showed that the deviation is reasonably small and quantifiable if the absorbance per cavity round-trip remains small. In such case the integrated intensity, i.e. across a spectral line, deviates only by an amount of less than 10%. However, to ensure the smallness of the single round-trip absorption only a 0.5 gs portion from the decay transient could be used. We note that in the experiment, the cavity decay time was 7 gs (mirror reflectivity R = 9 9 . 9 3 % and 1.5 m cavity length) and hence the majority of data points, usually giving rise to improved signal-to-noise, were omitted. As a final result the total integrated band intensity on the a 1Ag-X3Ng (0, 0) band could be brought into agreement with the Fouriertransform spectroscopic measurements to within 2%, yielding a value for the Einstein coefficient of A = 2.19(7) x 10 .4 s -1 [16]. In the study of Miller et al. [17] a more state-of-the-art laser system was used with a nearly Fourier-transform limited bandwidth of 0.017 cm -1 at 1.27 gm, therewith avoiding the aforementioned bandwidth problems in the quantitative analysis of the CRD-transients. However, as was demonstrated previously by Martin et al. [21], when the coherence time exceeds the roundtrip time in the cavity, disturbing oscillations on the ring-down transients will occur, that prohibit a straightforward interpretation of the decay transient in terms of an absorbance. Miller et al. overcame this problem by lowering the detector response to 600 ns, therewith washing out the oscillations. It is noted that such a procedure will decrease the obtainable sensitivity in the experiment, but here the focuswas on a determination of the band integrated intensity and the Einstein A-coefficient. Measurements were performed over a range of pressures, varying from 15 to 557 torr, resulting in a value of A = 2.3(3) x 10 -4 s -1, which is in good agreement with the average from a number of classical experiments, and confirming the study of Newman et al. [16].

2.2.3 The Herzberg Bands in the Ultraviolet The extension of ring-down techniques to the ultraviolet wavelength domain is relatively straightforward, and again the oxygen molecule provides a benchmark test system for C R D studies. Frequency doubling techniques, converting the output of commercially available tunable lasers into the ultraviolet (UV), usually reduce the laser output by an order of magnitude, but since only millijoules of laser intensity are required in

54

APPLICATIONS USING R I N G - D O W N SPECTROSCOPY

a generic C R D experiment, this has no adverse effect. A major point of concern is the reduced reflectivity of avaliable mirrors to a maximum of typically 99.8%; this severely limits the sensitivity of C R D in the UV. Huestis et al. [22] reinvestigated the spectra of the Herzberg bands close to the dissociation threshold by C R D and by an alternative technique of probing the oxygen atoms, produced by collisional dissociation involving an excited oxygen molecule. In an initial study, a laser with a bandwidth of 0.5 cm -1 was applied, and it was demonstrated that C R D had a superior sensitivity and new features were observed. Also the first direct information on oscillator strengths of the Herzberg II and III systems was generated in these experiments. In a subsequent study [23] focusing on the 242-244 nm region, where new features had been identified, a laser of improved bandwidth (0.2 cm -1) was used for the purpose of better resolving the features and for addressing the bandwidth issue in quantitative CRD. It should be noted that this narrower bandwidth is still a factor of three above t h e Doppler width on the molecular oxygen resonances. In this region of the ultraviolet again a comparison can be made between the C R D technique and the concomitant developments in Fourier transform spectroscopy. A UV Fourier transform spectrometer was developed [24, 25] that could be set to an effective resolution of 0.06 cm -1 yielding an accuracy of 0.005 cm -1 in the line positions. In terms of resolution, this F T setup is indeed superior to the C R D setup used in References [22, 23] although it should be noted that there do exist UV lasers that would match this resolution. In terms of sensitivity the C R D experiment, here limited b y the mirror reflectivity of 99.6%, made possible the observation of five new excited vibrational levels A3~ +, v - 12, A'3A,, v - 12, 13 and clI~2, v 17, 18; moreover of the A - X ( l l , 0) band 12 branches were observed via CRD, and only 8 with FT spectroscopy. Apart from the newly identified, features, the C R D experiment yielded also some 100 additional--yet unidentified--spectral lines just below the dissociation threshold in O2. As for the oscillator strengths, an attempt was made in the C R D studies to correct for the effect of the laser bandwidth, but a comparison with t h e higher resolution FT study [25] indicates that there is still an underestimate of 30% on the line strengths. The developments in C R D and F T techniques have progressed simultaneously. After the work of Slanger et al. [23], Jenouvrier et al. [26] performed FT studies in the UV, again focusing on the energy region below the first dissociation threshold in O2. With an improved sensitivity some lines reported in the C R D experiments were not observed in F T and vice versa. Nevertheless, based on the resolution of 0.12 cm -1 many hitherto unidentified lines could be given an assignment. F r o m a~ deperturbation analysis, some transitions could be assigned to a weakly

02: A BENCHMARK SYSTEM FOR RING-DOWN TECHNIQUES

55

bound state of 3rI, symmetry of which vibrational levels v = 0 and v = 1 were identified and for which a dissociation energy of De -- 140 cm -1 was determined.

2.2.4 The (02)2 Complex Since the end of the 19th century it was known that high-density oxygen gas features some additional absorption phenomena. Broad resonances, investigated in the condensed phase and in the high-pressure regime, are understood as caused by 0 2 - 0 2 collisional complexes. During a collision an electronic transition is induced in the (02)2 system. This is an effect of symmetry breaking by the collision partner, giving rise to a change of the selection rules. As a result both 02 partners may leave in an electronically excited state after absorption of a single photon. At the 630 nm resonance both molecules are excited to the a lag, v = 0 state. Naus and Ubachs [27] employed the sensitivity of pulsed CRD to investigate the (02)2 features at 630 and 580 nm in the pressure regime below 1 atm. Density-dependent cross sections were determined at discrete wavelength settings from an analysis of measured C R D extinction curves as a function of pressure as shown in Figure 2.4. The derived cavity-decay times have a quadratic dependence on gas pressure because of the collisional phenomenon involving two molecules; however the extinction is so weak that a linear contribution, related to Rayleigh scattering had to be included in the analysis; also a constant term related to the mirror reflectivity was included in the fits to the data. The C R D method appears to be ideally suited for such measurements: (i) for the broad resonances there is no bandwidth problem and the extinction can be straightforwardly interpreted in terms of cross sections; (ii) the contributions of Rayleigh scattering and collision-induced absorption can be unraveled; (iii) by choosing discrete frequency positions of the laser the features of the (O2)2-complex at around 630 nm can be disentangled from the linearly absorbing ?,-band in the O2-monomer, also at 630 nm. The latter entails a marked advantage over the FT technique, which was also applied to investigate (02)2 [28]. While the (02)2 collisional complex is associated with the repulsive part of the O2-O2 intermolecular potential, van der Waals bound states do exist below the dissociation threshold within this potential. Sharp and wellresolved absorption spectra of the O2-dimer have been recorded in a supersonic slit-jet expansion using the continuous wave (CW)-CRD technique [29]. Here the slit-jet configuration provides sub-Doppler resolution on the resonances of below 0.01 cm -1 as shown in Figure 2.5.

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3.4.3 Experimental Results of AC Chop Ring-Down Spectroscopy The concept discussed in the preceding section has been demonstrated in the measurement of rovibrational transitions of acetylene. The experiment

WEAK ABSORPTION MEASURED BY FIELD DECAY (TIME-DOMAIN)

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used a Yb:YAG laser. The transition involved was the 3v3 R(29) overtone of C2H2, located at 1031.6528 nm, with an absorption coefficient of 4 x 10 .6 (torrcm) -1 In the 46.9 cm long cavity, a gas pressure of a few millitorr (1 t o r r = 133 Pa) was typically used, giving 1 x 10 .6 level of absorption. The beam chopping frequency was 1.4 kHz, corresponding to At = 714 Its. The cavity transmission was received by an avalanche photodiode (APD), and the beat signal was sent to an rf spectrum analyzer for demodulation. The frequency reference of the rf spectrum analyzer was tied to the rf signals used to drive the AOMs. In the linear amplitude display mode, the video output of the spectrum analyzer in zero-span-mode provides a phase-insensitive demodulation for the heterodyne beat. To measure the empty cavity finesse, both modes 1 and 2 were tuned out of the molecular resonance. Figure 3.5 shows a representative trace of the demodulated heterodyne beat ring-down waveform with an overlaid theoretical fit. As can be seen, the model presented above produces an excellent fit, giving the empty cavity field ring-down (1/e) time of 90 Its. This leads to a cavity linewidth (fullwidth at half-maximum) of 3.5 kHz and a finesse of 90,000. Within the detection bandwidth of 173 kHz, the recovered SIN was 150, about half the expected value. This was partly attributable to "ringing" of the spectrum analyzer's rf filter function, which is optimized for the frequency-domain analysis. It was not possible to use a smaller bandwidth on the instrument without distorting the signal.

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102

ABSORPTION DETECTION TECHNIQUES

Mode 2 is then tuned to the center of the acetylene resonance, and the ring-down waveform becomes clearly asymmetric in the neighboring halfcycles. Figure 3.6 shows a set of experimental data where intracavity gas pressure was varied to generate four different intracavity absorption levels (expressed in terms of single-pass in the graph). The respective ring-down beat waveforms are shown in the left column of the figure. The absorption data (shown in the right column) were produced in the following way. First, a copy of the original data was shifted by a half-chopping cycle along the time axis. The differences between the original data and the shifted data gave the absorption signals. With a single-pass absorption of 1.7 x 10-6, the acquired SIN was 10 with a bandwidth of 173 kHz. The absorption sensitivity normalized to 1 s averaging time was then 1.6 x 10-1~ In steady state (no chopping), each mode has 3 gW (P0) in the cavity transmission. Given an r/for the APD of 0.3 A/W, the shot-noise limited sensitivity should have been ~ 1.2 x 10-11 at 1 s averaging. However, because the APD has an excess noise factor of about three, the expected minimum absorption sensitivity was ~ 4 x 10-11, which is within a factor of four of the experimental result. One can notice in Figure 3.6 that the recovered signal amplitude does not increase linearly with respect to the additional intracavity absorption. Of course one expects that if the gas absorption greatly exceeds the level of the empty cavity loss, signal saturation will occur. In Figure 3.7(a), the signal contrast is shown against the intracavity absorption normalized to the empty cavity loss. The dotted curve is calculated assuming that the coupling power to mode 2 (the absorbing mode) remains a constant. However, we know that with added loss inside, the power-coupling efficiency to the cavity changes, and the available power for mode 2 will decrease. Therefore for a fixed incident power, the signal saturation occurs sooner. This is the scenario plotted in the solid curve of Figure 3.7(a). Figure 3.7(b) illustrates saturation of the experimental data of Figure 3.6. Model from Figure 3.7(a) is used to fit the data. Remember the fraction round-trip losses of this empty cavity were 7 x 10-5. One solution to this problem is to increase the input power as the intracavity absorption level rises. A faster chopping cycle could also be used. But in any case this affects only the signal size scale; the recorded decay information is unaffected. Further improvement of the system includes the use of faster chopping cycles and the replacement of the APD with a sensitive PIN diode in a resonant matching circuit. At present the chopping frequency is limited by the locking loop between the laser and the cavity. After all, the sideband locking system also uses the heterodyne principle; in this case the beat is between the field of the direct reflection off the cavity-input

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50.0

I 50.0

~: i~

40

20

30

0

,

I 51.0

!

| l i 52.0 53.0 SIN = 33.3

|

20 10

-20

0 50.0

51.0

52.0

50.0

51.0

52.0

40 30

20

2O

0

10

-20

50.0

51.0

52.0

time (ms)

53.0

53.0 SIN = 37

53.0

time (ms)

FIG. 3.6. Demodulated heterodyne beat amplitudes between the two chopped cavity modes (left column) in the presence of intracavity absorption (labeled in the graph). Shown in the right column is the absorption signal obtained by differencing the beat amplitudes in the adjacent half-cycles. Reproduced with" permission from Reference [44].

104

ABSORPTION DETECTION TECHNIQUES .~_---- 0 . 2 5 c-

=

...........

(a)

......

~' 0 . 2 0 -

.--

.

,~ 0.15"~ (" o t,j

o.-~176

0.10 -

~9 0 . 0 5 r ._~ oo 0.00

,

,

.

.." ~

-"

..'"

Solid line: Cavity input coupling efficiency included

. ~

~ I 1

0

I 2

I 3

4

Intracavity gas absorption (normalized to empty cavity loss) 60x10 .3 50I--

Empty cavity loss (single pass) 35

xlO -6

40

O o

30-

c

20-

._~

(b)

f 10 - ~,

0-

--

I

10

t

20

9

signal contrast from data theoretical signal contrast I

30

Intracavity gas absorption (single pass)

I

40xl 06

FI~. 3.7. (a) Signal contrast vs. intracavity molecular absorption. The change

of cavity-input coupling efficiency is (not) taken into account in the solid (dotted) curve. (b) Experimental data from Figure 3.4, showing signal saturation when the molecular absorption approaches the empty cavity loss (35 x 10.6 single-pass). Reproduced with permission from Reference [44].

mirror and the field leaked out from the cavity storage. If the mode is being switched too fast, the cavity field does not have sufficient time to establish itself as the frequency/phase reference for the incident instantaneous laser field to compare against. The lock will then be based on equal contributions from two weaker cavity fields. Experimentally, there is a transient associated with the switching, and this leads to some problems with the lock's robustness. An alternative is to stabilize the laser on the cavity with a third mode, completely off from the molecular resonance and independent of the other two modes. The third mode can be left on all time to maintain lock while the switching can go on as before between the first two modes. The heterodyne detection rf system can conveniently filter out the contribution from the third mode. The only penalty in this arrangement is a somewhat increased level of shot noise, resulting from the added contribution of the third mode to the photocurrent. A hybrid of the on-resonance/off-resonance switch with transmission heterodyne detection against the third mode is another clear avenue for high-sensitivity detection.

WEAK ABSORPTION MEASURED BY FIELD PHASE

105

3.5 Weak Absorption Measured by Field Phase (Frequency Domain) In Section 3.4 the advantages of an enhancement cavity were discussed in a time-domain application. The signal exiting the cavity can also be measured using phase-sensitive detection methods, i.e. in frequency-domain applications. The phase of the light, along with the cavity-resonance structure, is perturbed by the molecular radiation, which leads to additional phase shifts. The objective of this section is to discuss phase-sensitive optical-heterodyne spectroscopy, using an enhancement cavity, as a tool for signal acquisition. The advantage of this approach lies in the characteristic property of FM spectroscopy: the simultaneous and continuous observation and subtraction of the signal and background optical phases. As one considers how to probe an external cavity signal with the FM technique, the first approach that comes to mind is to lock the laser frequency to a cavity resonance and then modulate that cavity mode around the desired molecular resonance while monitoring the cavity transmission. This approach is a simple lock-in derivative line shape recovery process. In order for this method to be successful, it is important to have a very tight frequency lock loop between the laser and the cavity because any laser frequency noise relative to the cavity will be converted to amplitude noise. To implement this scheme, a piezoelectric transducer (PZT) is mounted on one of the cavity mirrors. This assembly is used to modulate the cavity length, and the laser tracks the modulation. The modulation frequency is usually limited to the audio range on account of mechanical resonances and roll-off of the PZT and mirror assembly frequency response. Depending upon the laser's amplitude noise spectral distribution, the attainable modulation frequency may be too low to reach shot-noise limited detection. Moreover, the apparent line shape would also be broadened by this modulation process. However, promising results have been obtained. Using a solid-state Nd:YAG laser, locked to a high-finesse (100,000) cavity (corresponding to ~ 1 mHz relative linewidth), a cavity-dither (at 500 Hz), and lock-in detection, a detection sensitivity of 3 x 10-11(6.4 x 10-13/cm) has been measured at 1 s averaging [47].

3.5.1 Principle of NICE-OHMS To benefit from the full noise-reduction advantages of FM spectroscopy, one needs to increase the phase-modulation frequency of the probe field, usually to be much larger than the resonance linewidth under study. In addition to the laser-cavity locking issue, another obstacle remains; namely,

106

ABSORPTION DETECTION TECHNIQUES

the cavity-bandwidth limit. Specifically, F M sidebands at a high frequency are needed to eliminate low-amplitude noise, and at the same time the cavity must respond to the sidebands in exactly the same manner as it responds to the carrier. This will reduce the frequency-to-amplitude noise conversion process. This goal can be realized by frequency modulating the input laser beam at exactly the FSR of the cavity. We then detect and demodulate the cavity-transmitted light at the modulation frequency. Any small residualfrequency variations of the laser will still lead to some amplitude fluctuations and phase shifts of the transmitted carrier, but these will also lead to exactly the same amplitude fluctuations and phase shifts of the sidebands, which are transmitted on nearby cavity-axial orders. So the transmitted light still accurately represents an FM spectral triplet, with minimal AM conversion caused by the relative laser-cavity-frequency jitter. Thus the noise level can approach the intrinsic AM noise level of the laser at the FSR frequency. Figure 3.8 shows the case where the central component is used to detect the intracavity molecular resonance, illustrating how the saturated molecular dispersion causes an unbalance of the laser FM spectrum by a phase shift on the carrier component. Initially, all the FM components are lined up on their respective cavity modes. The central cavity mode will then be pulled because of the additional phase shift attributable to the molecular dispersion. After the phase-sensitive demodulation, the detector viewing the transmitted light will generate a dispersion signal in the rf beat. We refer

FIG. 3.8. The optical spectrum and the detection principle for NICE-OHMS: col denotes the laser frequency and A is the phase-modulation frequency that matches the cavity FSR. The original FM symmetry is upset when the molecular dispersion shifts a cavity resonance by COm.

WEAK ABSORPTION MEASURED BY FIELD PHASE

107

to this technique as (laser frequency-) noise-immune, cavity-enhanced, optical-heterodyne molecular spectroscopy (NICE-OHMS) [45, 29, 47, 48]. This modulation and detection scheme makes it possible to use a highfinesse cavity without introducing additional noise.

3.5.2 The Sensitivity of the NICE-OHMS Technique To estimate the sensitivity associated with NICE-OHMS, we notice that the cavity-enhancement effect applies only to the signal; no additional noise source has been introduced. Therefore the noise-equivalent absorption signal is that of Eq. (3.7) for ordinary FM spectroscopy, divided by the cavity-enhancement factor (2F/re). The power in the denominator should be that of the cavity-transmitted light, Pt. This argument can be supplemented with a more rigorous proof. Suppose the molecular dispersion changes the intracavity refractive index by An, with the shift of cavity resonance given by (An 9coo). Light going through the cavity will thus acquire an extra phase shift of r = arctan(An, coo/?'), where )i is the cavity HWHM (measured in radians/s). Following the previous treatment and under the assumption of a small 4~, we derive 2 coo cad ~ A n . coo/?' - ~-~a . . . . 7/ 2?' d

.

FSR (ad) . . . y

2F(~_d) rc

(3.16)

The signal amplitude is ~/r2rIPtJoJ 1 9r and the shot-noise limit is ~/2eBoPt. The minimum detectable absorption at S / N = 1 is thus,

(~

zc ~

~/2

2F V rlPtJo(fl)Jl(fl)

(317)

When/3 ~ 1, this sensitivity result is the same as that expressed in Eq. (3.15) for the AC chop ring-down method, except for a small numerical factor (~ 1.4), an expected result inasmuch as both methods are shot-noise limited. The difference arises from the fact that in the cavity-enhanced FM spectroscopy, some portion of the carrier power is converted to the sidebands, leading to a slight loss of sensitivity for a fixed total optical power. A numerical example helps demonstrate the potential of this technique. Suppose the modulation index,/3, is 0.5, and the photodiode responsivity, r/, is 0.85 A/W. Also, take the optical power, Pt, to be 5 mW and the detection bandwidth, B, to be 1/2n Hz, which corresponds to a 1 s time constant. Then, for a single-pass cell, the noise-equivalent integrated absorption,

108

ABSORPTION DETECTION TECHNIQUES

(o/d)min, is 2.2 x 10-8. Under the same conditions, a cavity with a finesse of 100,000 improves the sensitivity to 3.5 x 10-13. As discussed below, a noiseequivalent sensitivity of 5.2 x 10-13 of an integrated absorption at 1 s averaging has been achieved. This corresponds to an absorption of 1 x 10-14/cm for a cavity length of 50 cm [47].

3.5.3 Implementing a NICE-OHMS Experiment: Signal Line Shape, Size, and Sensitivity An experimental schematic is shown in Figure 3.9. Two electro-optic phase modulators are used to impose two sets of FM sidebands on the laser beam. Modulation at a low frequency, 3, is detected in the reflected cavity signal and is used to produce an error signal for locking the laser to the cavity. Sidebands at a higher modulation frequency, A, are set to the cavity's FSR frequency and are used to probe the intracavity molecular resonances. This signal is detected in transmission, with adequate optical isolation between the cavity and the photodiode. To study the resonance signal line shape and width, a precise scanning capability is important. A frequencyoffset locking loop is implemented to permit sweeps with rf resolution of the laser frequency relative to an independent stable reference. During the scan, the cavity FSR changes slightly. To maintain the noise-immune property, the sideband frequency must track this changing FSR.

FIG. 3.9. General schematic of the NICE-OHMS spectrometer, showing the major components of the laser-cavity locking, the transmitted sideband detection, and the precision tuning control. Reproduced with permission from Reference [47].

WEAK

ABSORPTION

MEASURED

BY FIELD

PHASE

109

As shown in Figure 3.8, the laser spectrum has three major components: the carrier at col and two phase-modulation sidebands at wi~ 4-A. Each of these three components has its own two sidebands located at 4-g away from itself, namely at col -t- g, col + A -t- ~ and col - A -t- ~ which are not shown in Figure 3.8. The three corresponding cavity-resonant peaks are denoted as coo and coo + F S R , with FSR = A. The beam reflected from the cavity carries all these frequency components and is detected by a photoreceiver. Demodulation at frequency ~ produces the servo error signal used to lock the laser frequency to the narrow linewidth cavity. Because of the additional modulation, this locking error signal now has three contributions. These are the two sideband resonances of strength J l ( f l ) 2 n e a r coo 4-A, and the carrier contribution J0(fl) 2 near coo. Together, these additively define the lock point. When none of the laser frequency components is affected by a resonance of the intracavity molecules, the servo error signal keeps the carrier, coL, and its two sidebands, coL 4-A, on the cavity resonances, coo and coo 4-A. Thus, the transmitted beam has the original, perfectly balanced FM spectra because the carrier (coL) and the sidebands (coL =k A) experience the same phase shifts and amplitude attenuations. However, when a molecular resonance affects any of these three components, the interaction converts part of the FM into AM, which is then detected by the photoreceiver viewing the transmitted light. For example, as shown in Figure 3.8, when a molecular resonance is n e a r the cavity resonance coo, the carrier will experience a modified intracavity refractive index, which has been changed by An because of the molecular absorption. As a result, coo is shifted by COm - - - - co0 /k // . If we neglect for the moment the servo contributions from the two sidebands, then the carrier coL could be kept at the new cavity resonance center (coo+com) and the sidebands, after being shifted to the new positions of coL+A +corn, will no longer lineup with the cavity resonances. The resulting phase shifts then lead to the AM signal recovered by the detector viewing the transmitted light. In practice, the two sidebands contribute certain servo error components that will partially offset the servo error signal provided by the carrier. The net result is that the laser frequency will be shifted back by laser servo in the amount of x - 2comJl(fl) 2, where x is derived from the requirement that xJo(fl) 2 - 2(corn- X)Jl(fl) 2 - O. Here we have used the approximation that J0(fi)2 § 2Jl(fi) 2 - 1, valid for the interesting range fi_< 1. However this shift of locking point causes little effect in the signal detection because it is very small (< 100 Hz) compared with the cavity linewidth, which is typically a few tens of kilohertz. Taking into account the nonlinear molecular phase shift q~ and the additional phase shift ~bx due to the small change in the locking offset, we

110

ABSORPTION DETECTION TECHNIQUES

can express the field of the cavity-transmitted light in the following form: Et(t) = Et(0)[J0 exp[- i(cot - 4~x)] + J1 exp[- i((co + A)t + 4) - ~bx)] - Ja exp[-i((co - A)t + ~ - 4~x)]]

(3.18)

We can see that the locking offset phase shift (exp[ i4~x]) is a common factor for all three contributions and can be factored out. The signal current at the frequency A can be readily derived as is cx Jo(fi)Jl(fi) sin(dp) s i n ( A t ) ,~ Jo(fi)Jl(fi)dpsin(At)

(3.19)

The signal has a pure dispersion line shape and is independent of the lasercavity locking point. An important aspect of this line shape is that it contains only the odd-symmetric response ~ 4~sin(At), and so the line shape and the apparent line center position are independent of any less than optimal setting of the rf detection phase at A. This property will allow precise locking to these resonances, even though the rather high modulation frequency generally causes problems in the stability of the detection phase. We should note that although the requirement of the laser-cavity locking is much more relaxed for N I C E - O H M S than in the direct cavity transmission detection, the laser linewidth still needs to be narrowed so that a stable optical power is effectively coupled into the cavity. For metrology purposes, this laser-cavity locking loop also serves as the shortterm frequency stabilizer. For a cavity linewidth of, say, 10 kHz, it is straightforward to lock a commercially available external-cavity diode laser below 100 Hz relative to the cavity, using a feedback control of the laser diode's current and the PZT of its external grating. Using a solid-state Nd:YAG laser, the combined servo actions of the laser's internal PZT and an external acousto-optic modulator, the laser-cavity relative frequencynoise spectral density was as low as 20 mHz/x/-H~. This indicates that the laser's linewidth relative to the cavity is a mere 1.3 mHz. Another important technical issue is that the recovered line shape is influenced by a residual amplitude modulation (RAM) associated and synchronous with FM at the cavity-FSR frequency. With an active control loop, RAM at A can be eliminated, and the resulting line shape matches the model line shape. To reject further noise and minimize baseline drift, a small dither can be applied on the cavity resonance (with the modulation amplitude matching the width of molecular resonance) at a low audio frequency. This allows a lock-in detector to process the demodulated rf signal from the output of the double-balanced mixer that is driven at frequency A. Hence, the line shape from the rf channel, resulting from modulation detection of an isolated dispersion resonance, approximates the derivative of a dispersion profile [40]. Indeed the experimental data are fit by a theoretical line shape rather well,

WEAK ABSORPTION MEASURED BY FIELD PHASE

6~

4-

5.3 m T o r r s a t u r a t i o n ~ 1.3 6 4 0 k H z p-p d i t h e r Fit F W H M = 0 . 7 0 5 M H z

111

dither Fit FWHM

I~~--~1 Observed FWHM

I=

2-

t

O-

-2-

I

I

I

I

-2

-1

0

1

=1

MHz

FIG. 3.10. Frequency scan of the C2HD (v2 + 3v3) P(5) transition line shape and overlaid theoretical fit, with fit residuals magnified by ten times. Reproduced with permission from Reference [44]. as shown in Figure 3.10, with the fit residual magnified by ten times. Although the intrinsic transition width associated with the (v2 + 3v3) P(5) line of CzHD is on the order of kilohertz, the observed linewidth (FWHM) is 705 kHz (after removal of the modulation broadening by the fit) which includes contributions from the power-saturation (~ 1.3 times) and pressurebroadenings (~ 35 kHz/mtorr) of the 270 kHz transit-time linewidth. As explained earlier, one way to maintain lock between a laser and cavity, is to dither some component of the system and lock-in on the transmitted light. We refer to this low-frequency operation as DC detection, to differentiate it from the high-frequency rf approach of NICE-OHMS. The line shape measured by using DC detection follows the original Wahlquist formula for a modulation-broadened, derivative line shape [39, 40]. Comparing NICE-OHMS and DC signals gives us an appreciation for the noise-immune nature of the NICE-OHMS detection, as shown in Figure 3.11. This figure shows signals collected using the two techniques and two different laser-cavity lock conditions. Under one set of experiments, the laser and cavity were tightly locked, and in the second, the laser-cavity lock was deliberately set to be loose and even oscillating. We can now compare the recovered signal-to-noise ratios before and after the lock was sabotaged. The DC detection of the intracavity molecular absorption (upper row) is shown to be critically dependent upon the performance of the laser-cavity lock. (A fast laser-cavity frequency-lock servo was used for the graphs obtained in the left column while a slow and noisy servo was used for those in the right column.) However, increased laser frequency noise relative to the cavity yields little effect in FM detection (bottom row).

112

ABSORPTION DETECTION TECHNIQUES Tight Cavity Lock DC detection

V

Loose Cavity Lock DC d

] 7.5E-9 saturated

absorption

Fro. 3.11. Demonstration of the noise-immune property of NICE-OHMS. The C2HD (v2 + 3v3) P(5) resonance signal is recovered by both cavity-dither lock-in (DC) detection and NICE-OHMS technique, under the conditions of a tight lasercavity lock (left column) and a poor lock (right column). Reproduced with permission from Reference [47].

Figure 3.12 shows the experimental sensitivity that was achieved using 1.8 mtorr of gaseous C2HD. The transition under study is the (v2 + 3v3) P(5) overtone line of C2HD. The cavity finesse was 100,000 and the intracavity power ~ 300 W, giving a saturation parameter of ~ 1.75 and a saturation peak contrast of 13.2%. The single-pass (46.9 cm long cavity) linear absorption was about 3 x 10 -8. Therefore the absolute level of saturated absorption by the intracavity molecules was 4 x 10 -9. This is verified by the DC detection of the cavity transmission, shown in the top graph of the figure. The calibration process involves measurement of the cavity finesse, on-resonance transmission, and reflection dip contrast, from which the residual round-trip cavity losses are calculated. With the laser locked to the cavity with a relative linewidth of ~ 1 mHz, the simple cavity-dither and lock-in detection of the transmission yields a S/N (amplitude/rms noise) of 130 at 1 s averaging. This corresponds to a detection sensitivity of 3 x 10 -11 at 1 s. The corresponding SIN from the N I C E - O H M S detection is 7700 with a 1 s time constant, as shown in the bottom graph of the figure. This translates into a noise-equivalent detection sensitivity of 5.2 x 10 -13 at 1 s averaging, ~ 1.5 times worse than the calculated shot-noise limit. The N I C E - O H M S result is about sixty times better than the straightforward

113

W E A K ABSORPTION M E A S U R E D BY F I E L D PHASE

0.5--

SIN = 130 at ls, Sensitivity: 3 x 10"11

.

0.0-

-0.5 E

I

E _l

2.0--

1.51.0-

0.5- ~

~

j

/

DC detection (1.8 mTorr) 4E-9 saturated absorption I

-1

L_

. .,..,

]

V

I

0

/

I

1

~

2

MHz

5.2 x 10"13 ~ v i t y

at ls: 5E-13

0.0I

-1

I

0

I

1

I

2

MHz

FIG. 3.12. Sensitivity measurement of the NICE-OHMS technique. The upper graph shows the level of the saturated absorption while the lower graph shows the corresponding SIN obtained via NICE-OHMS. The noise-equivalent detection sensitivities (normalized to 1 s time constant) are 3 x 10-11 for cavity-dither detection and 5.2 x 10-13 for NICE-OHMS. Reproduced with permission from Reference [47].

dither detection, because of its higher modulation frequency and its insensitivity towards the laser frequency noise relative to the cavity. And we emphasize that the gain in sensitivity by N I C E - O H M S over simple dither detection will be even more impressive if we had not done a good job in locking the laser to the cavity. There is an optimum value of the intracavity sample pressure for the maximum signal size. An increase of pressure raises the size of linear absorption, but at the same time reduces the level of saturation because of the pressure-broadening of the homogeneous transition width. For fixed cavity parameters, change of pressure also influences the input power coupling. Therefore the signal size and its "discriminator" slope vary depending upon the pressure. A useful model of this behavior is based on an axially averaged field picture [49, 50] in which the moving molecules interact with the average standing intracavity field. At line center, the counterrunning waves interact with the same molecules, thus causing the absorption coefficient to be reduced from the off-resonance value of ~0/~/1 + S to ~0/~/1 + 2S. Here d0 is the linear absorption coefficient at the center of the

114

ABSORPTION DETECTION TECHNIQUES

Doppler profile and S is the saturation parameter. The observed nonlinear signal is thus proportional to A~ -

or0 _ or0 ~/I+S ~/1 + 2S

(3.20)

To perform detailed calculations on signal size, we use Eqs. (3.2) and (3.3) to determine the cavity transmission, along with the following useful relations: An - Ac~2/4~, oto - e p . P , S - I / I s a t , and/sat - I0" (FT + F0" p)2. Here An is the refractive index change caused by the saturated molecular resonance; Otp is the molecular absorption coefficient per unit length and unit pressure; P is the gas pressure; /sat is the required saturation intensity (W/ cm 2 ); FT is the residual linewidth, at zero pressure and zero power attributable to transit-time broadening; F0 is the pressure-broadening coefficient; and I0 is a power-scaling constant that can be determined experimentally. The natural linewidth (~ kHz) of a vibrational overtone transition is negligible in a saturation calculation because transit-time broadening at room temperature is typically a few hundred kilohertz. We can calculate in steps, the relative depth of the saturated absorption, the cavity frequency shift caused by the molecular resonance, and the F M signal size and discrimination slope as functions of gas pressure. With the parameters associated with a typical cavity and gas sample, we find that a pressure around 10 mtorr gives the maximum saturation signal. The model fits to the experimental data rather well, as shown in Figure 3.13, which displays the experimental signal slope (signal amplitude/linewidth) versus pressure. As will be explained later, this slope is used as a frequency discriminator for locking the laser frequency to the molecular transition, so the maximum slope leads to the best locking precision.

"T"

3.0--

2~ d

2.5-

> 9

El. 0 03 _

2.0o

1.5-

tE~

03 1 . 0 -

I

I

I

I

I

I

0

5

10

15

20

25

30

Pressure (mTorr)

FIG. 3.13. Signal slope recovered from the NICE-OHMS spectrometer with respect to the pressure of the intracavity gas sample. Reproduced with permission from Reference [48].

WEAK ABSORPTION MEASURED BY FIELD PHASE

115

The high detection sensitivity of the NICE-OHMS method, opens up many possible spectroscopic applications. For example, some weak transitions have been measured within the tuning range of a Nd:YAG laser at 1.064 gm. Two such lines were measured, 12C2H2 (2Vl + v2 + vs) R(12) [51] and 12C1602 (2v1 + 3v3) R(6) [52]. These have transition dipole moments of 50 and 6 gD, respectively ( 1 D - 3.33564 x 10-3~ m). These absorption lines are weaker than the CzHD (v2 + 3v3) P(5) transition, which has a transition dipole moment of ~ 7 0 gD [53]. Using the same gas pressure, optical power, and cavity-dither amplitude, the saturated absorption signals of CO2, CzHD and C2H2 are compared in Figure 3.14. The C2H2 transition is recovered with an excellent signal-to-noise ratio, as shown in Figure 3.14(c). The signal size is about 0.23 of that for CzHD, at the same gas pressure and optical power. It provides another frequency reference for the Nd:YAG laser. The 12C1602(2Vl + 3v3) R(6) transition shown in Figure 3.14 is quite different than either of the acetylene lines. For one thing, the saturated absorption signal is much weaker than the C2HD line, by more than a factor of 350. More strikingly, the recovered line shape associated with the transition is vastly different. Nevertheless, this spectrum can be fit using two separate resonances that have different line centers and linewidths. Interestingly, the narrower negative-going peak indicates a physical process reversed from the normal saturated absorption. The intracavity optical power was 410 W, where the saturation of this CO2 transition is estimated to be only about 2% in the free-flight regime. As the input power is reduced by a factor of four, the resonance line shape remained relatively unchanged. The signal size, however, decreased roughly by the square of the power change. The line shape did not depend upon the laser dither frequency either. The change of the dither amplitude, on the other hand, had a larger impact on the signal size of the relatively wider peak, causing an apparent change on the signal line shape. But this change is well accounted for by the two-resonance model, and the fit was able to produce consistent linewidths corresponding to a fixed gas pressure. As the intracavity gas pressure was changed, linewidth broadenings and line center shifts were observed for both peaks. Extrapolated to zero pressure, I/he-inverted peak has a width of 100 kHz. This CO2 resonance involves two quanta of symmetric stretch and three quanta of antisymmetric stretch in the C-O bond. The bending mode (v2) is not excited. The lifetime of the excited vibrational state is estimated to be ~ 2 ms, with relaxation dominated by IR fluorescence through the vibrational transition (2,0o, 3) --+ (2,0o, 2). The relevant molecular constants are given in Reference [54]. One likely explanation for the abnormal (sign-reversed) part of the line shape in the CO2 spectrum

116

ABSORPTION DETECTION TECHNIQUES 5 mTorr 0 0 2 , 4 1 0 W 50 kHz p-p dither

(a) 0.20.1-

.~_

Saturated on

0.0-0.1-0.2-

.,_._..._.-Absorption

-0.3-

residual

-0.4-

-1.5

(b)

I

I

I

I

I

I

-1.0

-0.5

0.0

0.5

1.0

1.5

5 mTorr C2HD, 410 W

15-

/~ i

Saturated

i

Absorption

(1)

m

10-

O

0O 5i..

(~

0-

t,m

_J

.5

(C)

I

I

I

I

I

I

-1.0

-0.5

0.0

0.5

1.0

1.5

4-

/~

5 mTorr 02H 2,

410W,

3-

.

p-p d~ther

210I

I

I

I

I

I

-1.0

-0.5

0.0

0.5

1.0

1.5

MHz

FIG. 3.14. Line shape comparison among the resonances of (a) CO2, (b) C2HD, and (c) C2H2, under the same experimental conditions. Reproduced with permission from Reference [47]. is that it arises from a near-resonant two-photon absorption transition in the neighborhood of the one-photon resonance, which we see as a saturated absorption resonance. This idea is supported by the fact that the zero-pressure linewidth of the inverted peak, only about 100 kHz, is just half of that dictated by the transit-time broadening (210 kHz). It is as if during the transit, the molecule experiences twice as many radians of signal phase, for example by resonating with a two-photon response. Alternatively, there may be a quantum interference resulting from the state mixing of the excited vibrational state.

117

WEAK ABSORPTION MEASURED BY FIELD PHASE

3.5.4 A NICE-OHMS Application" Laser Frequency Stabilization The NICE-OHMS technique can provide us with the line centers for weak molecular lines in the visible wavelength region with metrological precision. The narrow linewidths associated with saturated line shapes are especially useful, as the line centers are narrowly defined, which improves the longterm stability of a light source locked to such a transition. Moreover, the high signal-to-noise ratio improves short-term stability, permitting more effective intercomparisons among various frequency standards. With the narrower linewidth--but lower S/N--of the C2HD overtone transition, a frequency stability comparable to that of the I2-stablized system has already been achieved [55, 56]. The NICE-OHMS spectrometer provides laser frequency discrimination information relative to both the cavity resonance and the molecular transition. It is thus an ideal system with which to achieve simultaneously good short- and long-term frequency stabilizations. As discussed earlier, the laser tracks the cavity resonance to within a few millihertz. The vibrational noise and long-term drift in the cavity can be suppressed by stabilizing the cavity to an intracavity molecular resonance. The NICE-OHMS signal is intrinsically dispersive when the molecular resonance is probed by the carrier of the FM triplet. Used for locking, this could basically eliminate the influence of the local oscillator frequency drift on the recovered line center. In practice it is necessary also to dither the cavity length and make a secondharmonic signal recovery of the rf mixer output. This is partly to suppress the baseline offset problem associated with the imperfect FM modulation at the FSR frequency. Figure 3.15 shows such a discrimination curve associated 10 mTorr

4-

(50 ms)

S/N=8700@ls 2-

>0 "0

0-

E
1000 s), a promising indicator for an ultrastable frequency reference. This frequency stability, achieved by locking to an extremely weak reference transition, is a direct result of the spectrometer's high detection sensitivity. Notice that the C2HD-stabilized system shows only three times more frequency noise than the I2 system, which is notable because the I2 transition strength is almost a million times stronger than the P(5) line of the C2HD (v2 d- 3v3) overtone band. The short-term frequency stability of the optical sources can be comparable to or better than the state-of-the-art microwave standards. However, the reproducibility and accuracy of optical-frequency measurements are not yet comparable to microwave standards. One approach to reducing some of the systematic shifts of the reference frequency is to slow down the motion of the target quantum absorber, thereby reducing the second-order Doppler effect. Slower speeds also imply longer interaction times. Slower molecules give a narrower linewidth because the natural lifetime of vibrational transitions usually far exceeds (more than 300 times, for the acetylenes) the transit-controlled interaction time, limited by the laser field dimensions. While effective schemes for molecular cooling and trapping are emerging, optical selection of slow molecules based on interaction time has been actively pursued since the late 1980s [60-62]. The idea is this: the homogeneous linewidth originating from collisional broadening can be reduced by lowering the gas pressure until the mean-free-path of molecules becomes much larger than the transverse field dimension. This is referred to as the transit-time regime. To optically select the slow molecules, a low optical power is necessary so that the low RaN frequency leads to appreciable saturation only for the slowest molecules. The observed signal is thus dominated by interaction with the slowest molecules and consequently the effective interaction time is increased, limited finally by saturation and residual pressure broadening. Unfortunately, this avenue toward linewidth reduction is expensive in S/N, scaling approximately according to the fifth power of the instrumental resolution increase. In the free-flight regime, molecules with the mean thermal velocity cross the laser beam without suffering any collision. In other words, if we define

120

ABSORPTION DETECTION TECHNIQUES

the collision-broadened homogeneous linewidth (the half-width at half maximum) a s / ' p and the transit-time linewidth as FT, then Fp c~, while the Stokes field decays exponentially at its empty cavity rate as defined by the resonator losses at the Stokes wavelength. By including the effect of the gas, the Raman gain can be seen to deplete the pump field through conversion to Stokes. The coupled rate Eqs. (5.1) and (5.2) can be used to numerically analyze the dynamic response of the system including its relaxation oscillations and intensity noise transfer functions as performed in Reference [25]. Taking the steady-state limit of Eqs. (5.1) and (5.2) under the assumption that R ~ 1, and converting the intracavity optical fields to optical powers external to the resonator, the following analytic expressions can be obtained for the Stokes optical power exiting the Raman resonator as well as the pump optical powers reflected from (front) and transmitted through (back) the resonator above threshold [35]:

PSf(b)

--

21 /~p 2s rSf(b)

[(

Tpf

1-

)1/2

jsfesb "eOep

1 -- e 0 ( l -- CRpfePb)

ePf--[r162 ePb -- 1poTPb(1 -- CRsfRsb)

2

1

(5.6)

(5.7)

(5.8)

PP is the input pump optical power (note that this is the pump power coupled into the resonator) and we have defined the following constant

164

THE CONTINUOUS-WAVEHYDROGEN RAMAN LASER

with units of power: P0 ----

'~P -]-/~S

o/-4-,, 2~ csc-1 ~/L

(5.9)

In this expression, r is the radius of curvature of the resonator mirrors. Physically, Po/2 represents the pump power that gives an exponential gain of unity per pass, as defined by amplifier theory [40]. From Eq. (5.7), one can obtain the power necessary to achieve impedance matching by equating two terms within the brackets. Also note that the transmitted pump optical power given in Eq. (5.8) is independent of the incident pump power. Its value is clamped at that of the empty cavity just before the onset of lasing. This characteristic is also exhibited in the transmitted pump power of a nondegenerate optical parametric oscillator. The Stokes optical power grows as the square root of the input pump power (linear with the input pump field), in direct analogy with the down-converted power of the nondegenerate optical parametric oscillator. Reference [35] gives a more complete analysis of these equations and uses them to find the optimal construction and operation parameters for the Raman resonator with respect to mirror reflectivities, cavity geometry, and detuning from the Raman resonance line center. The Raman laser threshold occurs when the two terms within the square brackets of Eq. (5.6) are equal. Explicitly, the threshold input pump power is

eThresh _ ~1 P0 (1 - v/Rsf RSb]/ (1 - v/RPfTpfRPb)2 ~ 71;24Fp.P~

(5.10)

where Fp and Fs are the cavity finesses at the pump and Stokes wavelengths, respectively, and where the approximate result holds for the case where the front and back mirrors are identical. From Eq. (5.6) it can also readily be shown that the photon conversion efficiency reaches a maximum value of

PSb /~S Pp/~p

Tabrpf

1

4 (1

-

v/RsfRSb)(1

-

(5.11) v/RpfRPb)

As long as the transmitted pump power of Eq. (5.8) remains clamped, this maximum occurs at a pump rate of four times the threshold value regardless of mirror reflectivities, resonator geometry, or Raman plane-wave gain coefficient. When the front and back resonator mirrors are identical, the maximum photon conversion efficiency from Eq. (5.11) is 50% in the absence of mirror absorptions and including Stokes emission from both

SEMICLASSICAL THEORY

165

ends of the resonator. However, improved impedance matching can be achieved by reducing the reflectivity of the front mirror at the pump wavelength. In fact, photon conversion efficiencies can approach 100%, limited only by nonproductive losses such as transmitted pump power, mirror absorptions, and scattering [32, 35]. To this point, we have assumed that the pump and Stokes fields are simultaneously resonant within the HFC and that the frequency difference between the two resonator modes corresponds precisely to the linecenter of the two-photon Raman resonance. By relaxing these requirements, the continuous tuning properties of the device can be analyzed. A more complete treatment of the continuous tuning behavior is given in Reference [41]. To assess the Stokes power dependence on detuning from the Raman resonance linecenter in the high-pressure limit [19, 20], we assume that the Raman resonance is described by a Lorentzian profile. In this case, all of the above equations are valid with the modification Po --~ (Po/~), where (F/2) 2 -

A 2 nt- ( f ' / 2 ) 2

(5.12)

Here, F is the full-width at half-maximum (FWHM) of the Raman resonance and A is the detuning from the two-photon Raman resonance linecenter. The fact that the pump and Stokes resonator modes tune together when the cavity length is changed, albeit at different rates, modifies the Stokes frequency tuning range relative to the Raman resonance width. Specifically, tuning of the Stokes resonator mode, and therefore the Stokes output frequency, is related to tuning across the Raman resonance in the following manner: -

A.

1

(5.13)

Plots of the resulting tuning behavior compared to data can be found in Reference [41]. In particular, it is shown that for a pump power of six times the threshold value, nearly constant Stokes output power can be obtained for detuning from one half maximum of the Raman resonance to the other. Therefore, to estimate the constant power continuous tunability of the Raman system, we can replace A in Eq. (5.13) with the F W H M of the Raman resonance. This estimate assumes that other resonator modes (both longitudinal and transverse) do not acquire sufficient gain to dominate the lasing process during tuning [41]. Note that when the wavelength ratio in Eq. (5.13) is less than two, the tuning range is larger than the Raman

166

THE CONTINUOUS-WAVE HYDROGEN RAMAN LASER

resonance width, whereas when the ratio is larger than two the range is smaller. When operating the system below the four times threshold value, the peak conversion efficiency occurs on linecenter (A =0). However, for pump powers greater than this, detuning from linecenter actually increases the Stokes generation efficiency. The conversion efficiency reaches a maximum when detuned from the linecenter by: /-" / PP A -- -4-~ ~ 4PThres----------~

1

(5.14)

where PThresh is given by Eq. (5.10). Physically, this effect can be viewed quite simply. When operating above four times threshold, detuning effectively decreases the Raman gain and therefore raises the laser threshold until the four times threshold condition is again realized for the higher pump power value. In this section we have treated the doubly resonant cw Raman system mathematically. Equations (5.1) through (5.5) provide the time dependent dynamics of the two intracavity field amplitudes while Eqs. (5.6) through (5.9) give the steady-state emitted optical powers expected from the resonator for both the pump and Stokes wavelengths. In the following section we will compare these theoretical results to experimental data for a cw Raman laser that was pumped by a frequency-doubled Nd:YAG laser.

5.3 Nd:YAG-Pumped System In order to establish the feasibility of the cw Raman laser concept, the first system was designed to accommodate a large margin for error. A 200 mW, highly stabilized, cw frequency-doubled Nd:YAG laser with an emission linewidth < 10 kHz was used as the pump source. Resonator mirrors were chosen to give a finesse of greater than 13,000 at both the pump (532 nm) and Stokes (683 nm) wavelengths, yielding a projected laser threshold below 1 mW. The experimental setup is shown in Figure 5.4. PDH locking stabilized the laser to the HFC. A pair of Faraday isolators was used to attenuate optical feedback by > 60 dB. The beam was weakly focused through an electrooptic modulator (EOM) to place 12 MHz phase modulation sidebands on the carrier for locking purposes. The beam was then double-passed through an acousto-optic modulator (AOM), which served as an actuator for frequency corrections. Two lenses were used to match the beam to the fundamental spatial mode of the Raman laser resonator. A half-wave plate

ND:YAG-PUMPED SYSTEM

167

Fro. 5.4. Nd:YAG-pumped cw Raman laser setup. The frequency actuators are PZT tubes (slow) and a double-passed AOM (fast). The EOM places 12 MHz phase sidebands on the carrier for locking purposes. PBS: polarizing beam splitter. From Reference [25], reproduced by permission of the Optical Society of America.

and polarizer were used in tandem as a variable attenuator for the input pump light. The same polarizer, combined with a subsequent quarter-wave plate served as a beam separator allowing for detection of the reflected light from the HFC, which was used for locking. The Raman laser resonator consisted of two identical mirrors with radii of curvature (25 cm) greater than the separation (7.7 cm). The mirrors were spaced by a set of three piezoelectric tubes, which served as the low bandwidth ( < 1 kHz) frequency actuator in the feedback control loop. This resonator was placed within a hermetically sealed cell containing 10 atm of diatomic hydrogen gas. The output Stokes light was spatially separated from the transmitted pump light using a Pellin-Broca prism. The steady-state output power versus input power was the first characteristic of the system to be studied. Results for the o u t p u t Stokes optical power and photon conversion efficiency as functions of the input pump power coupled into the resonator are given in Figure 5.5. The solid lines represent best fits to the experimental data using Eq. (5.6) from Section 5.2 (see Reference [25] for fitting parameters). The Stokes laser threshold was measured to be 640 gW. The transmitted pump power remained constant to within a few percent after threshold. A maximum output power of 2.5 m W and a maximum photon conversion efficiency of

168

THE C O N T I N U O U S - W A V E H Y D R O G E N R A M A N L A S E R 30

_

3

25 .~ 20 o

o

r/l

15

O

9 , ,...~ r.z'3

~> 10

Conversion Efficiency Stokes Power

9

r 9

5

Theory

0 I

0

,

I

5

,

I

10

,

I

15

,

I

20

,

I

25

,

30

Pump Power (mW)

Fla. 5.5. Output Stokes power and photon conversion efficiency as functions of coupled input pump optical power. This system demonstrated a threshold of 640 btW, a maximum Stokes power of 2.5 mW and a maximum conversion efficiency of 27% for a pump rate of four times threshold. 27% for an input power of four times the threshold value (2.6 mW) were measured, confirming predictions. The mirror reflection coefficients were measured by cavity ring-down while the transmission coefficients were measured directly from single-mirror throughput. The dynamic behavior of the system was also experimentally investigated [25]. The relative intensity noises (RIN) (i.e. the ratio of the root-meansquare of the time-averaged power) of the input pump, transmitted pump, and emitted Stokes light were measured as functions of Fourier frequency and input pump power. Figure 5.6 shows the measured R I N of the transmitted pump and emitted Stokes as functions of input pump power. The data was accumulated for a Fourier frequency of 30 kHz and have been normalized to the transmitted pump R I N at threshold (i.e. 0 dB/Hz in the figure) to isolate the dynamic effects of the lasing process from that of the pump laser itself. The lines represent numerical solutions using Eqs. (5.1) and (5.2) from Section 5.2 as described in Reference [25]. Quantum noise is not included in this analysis. The RINs of the pump and Stokes both drop as the pump rate is increased. Note that because the time-averaged transmitted pump intensity is fixed above the laser threshold, the decrease in its R I N also indicates a decrease in its absolute noise power. The Raman process therefore serves as a "noise eater" for the transmitted pump light, which becomes more effective with increased pump rate. The R I N of the Stokes, on the other hand, is initially well above the threshold transmitted

ND:YAG-PUMPED SYSTEM 40

. . . .

30 ~ "r-

9 Stokes

~m

20

-

~

......

~

z

.

0 N -10

-"i

-20

9

9

~

Experimental Data

Stokes Fit

Pump Fit

PumpExperimentalData

HI '' ~

,

i1

-

.

,

i ='"

if

9

]

"'~ 9 e'-9

-3o -40

,

. . . .

ta

10

Z

I

169

0

"~",;"':--~.,~-e~-~... .

.

.

.

i

.

.

.

.

.

.

.

9 .

"- :, ,, ,o,. 9 .

.

;

.

.

Pump Power (roW) FIG. 5.6. Transmitted pump and emitted Stokes RINs, normalized to the threshold transmitted pump RIN at 30 kHz, as functions of coupled input pump power. The symbols are data while the lines represent theoretical predictions from Section 5.2. The RINs of both emitted beams decrease by as much as 35 dB/Hz with increasing pump rate. From Reference [23], reproduced by permission of the Optical Society of America. pump value (0 dB/Hz) because of its small time-averaged amplitude. However, for larger pump rates (and hence larger Stokes time-averaged amplitudes), the Stokes R I N falls below threshold transmitted pump value. To understand the origin of the observation that R I N decreases with increasing pump power, it is necessary to also analyze the R I N as a function of noise frequency. Figure 5.7 shows the RIN of the transmitted pump and emitted Stokes for a pump rate of four times threshold. Again, the data has been normalized to the transmitted pump RIN at threshold and the dashed curves represent numerical solutions. The arrows denote the H F C linewidth for the pump (125 kHz) and the Stokes (144 kHz) wavelengths. For frequencies below the cavity linewidths, the transmitted pump noise is suppressed as a result of efficient conversion to Stokes. This is particularly evident at dc where the transmitted pump power is clamped (see Eq. (5.8) in Section 5.2) and any change in the low-frequency input power affects only the Stokes output. Above the cavity linewidths, the transfer of input pump fluctuations to the Stokes output is suppressed because the fluctuations exceed the response bandwidth of the cavity. To examine the linewidth of the Stokes output, two identical Raman resonators were constructed, and the heterodyne beat signal between the Stokes outputs was detected using a fast detector and a spectrum analyzer. The beat note F W H M was measured to be 8 kHz for a spectrum analyzer sweep time of 10 ms [25]. This is substantially greater than the fundamental

170

THE CONTINUOUS-WAVE HYDROGEN RAMAN LASER 40

'

I

'

30

- - - ' - - Pump RIN Stokes RIN

20

....

I

'

Stokes cavity linewidth

Theory

,,j~_~

,-cJ

.~

_~,~ p,t,~

-10

~l

Pump cavity linewidth

-20 -30 -40

-

/

0

,

I

,

50

I

100

,

150

Frequency (kHz) FIG. 5.7. Transmitted pump and emitted Stokes RINs, normalized to the threshold transmitted pump RIN, as functions of noise frequency. The symbols are data while the dashed curves represent theoretical predictions from Section 5.2. The arrows denote the cavity linewidths for the pump (125 kHz) and Stokes (144 kHz) wavelengths.

width predicted by the Schawlow-Townes limit [42]. A possible cause of the implied broadening could be Raman resonator cavity length fluctuations. Another indicator of the Stokes laser phase noise is the root-Allan variance as shown in Figure 5.8. This phase noise measure shows a minimum of 1 kHz, occurring for a 1 s integration time. Finally, a small amount of on-axis coherent anti-Stokes emission was observed at 436 nm from this system [26]. Parametric anti-Stokes emission typically exhibits an extended ring spatial output pattern resulting from phase matching requirements, but has also been observed on-axis in other systems using pulsed laser sources [43, 44]. For the Nd:YAG-pumped system, the anti-Stokes power dependence on Stokes power was measured to be linear, with a slope of 26 n W / m W when tuned to the Raman resonance line center and slightly greater when detuned from line center.

5.4 Diode-Pumped Systems The Nd:YAG-pumped system proved to be a useful tool for studying the cw Raman laser, but diode lasers, for reasons of size, cost, and tunability,

DIODE-PUMPED SYSTEMS 100

- ,,j

........

i

........

J

........

i

........

i

171 ........

i

........

i

t 10 G) .,..,

I

> ?;

,_..,




0.25

0.00

.

1

.

.

.

I

I

2

3

k

J

,

I 4

._

I

5

Time (ms)

FIG. 5.14. Mode pulling and dynamic behavior as observed by scanning across the Raman cavity resonance above the laser threshold. The dotted line denotes the change in cavity length while the solid line gives the cavity pump transmission as functions of scan time. Note the thermally induced asymmetry between the forward and backward scans. The resulting cavity instabilities are important considerations for frequency locking. Both the mode pulling and the dynamic behavior associated with the heat generation can be observed by scanning the HFC length when operating above the Stokes laser threshold as shown in Figure 5.14. First, note the asymmetry in the cavity pump transmissions for the forward and backward as a result of the mode pulling. In the region of increasing cavity mirror separation (> 3 ms), the transmission exhibits multiple peaks over a broad range of cavity lengths. This behavior is not observed for in the region of decreasing cavity length ( < 3 ms). Similar asymmetries have been previously observed in other nonlinear systems. For this case, as Stokes light is generated the optical path length decreases because of heat deposition. This effectively increases the cavity scan rate when the physical cavity length is decreased, but counteracts the cavity scan when increasing the physical cavity length. As one might expect, this also affects the discriminator slope for frequency locking. For the Nd:YAG system discussed in Section 5.3, mode pulling towards the blue has been measured to be 35 MHz per mW generated Stokes power (compared to the typical HFC linewidth of < 1 MHz) [27]. Also of interest in Figure 5.14 are oscillations in the transmitted pump power for the forward scan. These can be interpreted as interplay between

PRACTICAL CONSIDERATIONS

179

the mode pulling and the Lorentzian bandpass of the cavity resonance. Temporal oscillations caused by heat deposition have been observed previously in pulsed Raman experiments that did not use a resonator [50, 51]. Observation of these phenomena have prompted modeling of the dynamic behavior of these systems [52]. Consideration of such dynamics is crucial for stable cw Raman laser operation. First, note that heat deposition (and therefore refractive index change) is directly associated with the generation of Stokes photons, so the resulting instabilities can occur no faster than the buildup and decay of the Stokes power. This means that it is advantageous to design a servo locking bandwidth which at least exceeds the HFC linewidth at the Stokes wavelength. Because the pump laser phase/frequency locking bandwidth can be related to the cavity decay rate at the pump wavelength, it is best to choose mirror reflectivities that give a pump cavity bandwidth larger than the Stokes cavity bandwidth. In this way, any dynamics associated with the production of Stokes will fall well within the locking bandwidth for the pump laser. To help emphasize this point, Figure 5.15 shows the intracavity pump and Stokes field amplitudes, generated using Eqs. (5.1) through (5.5) of Section 5.2, during an abrupt startup for three different regimes of cavity finesses. Figure 5.15(a) depicts the cavity dynamics when the finesses for the pump and Stokes wavelengths are equal. The Nd:YAG-pumped system discussed in Section 5.3 is close to this regime. Figure 5.15(b) shows the amplitudes when the finesse at the pump wavelength is much lower than that of the Stokes so that the pump field can build up and decay quickly compared to the Stokes. The asymmetric-mirror diode-pumped systems discussed in Section 5.4 fall into this category. Note that the relaxation oscillations are highly damped in this regime. Figure 5.15(c) shows the amplitudes when the finesse of the pump is much greater than that at the Stokes wavelength. In this regime, the relaxation oscillations are accentuated to the point of "spiking" (as observed in traditional laser systems) [53], and the Stokes power fluctuations can exhibit Fourier frequencies well above the pump cavity linewidth. The final concern with respect to heat deposition is thermal lensing. This effect has been observed and addressed in other nonlinear systems. We have not yet experimentally verified the effects of thermal lensing in low-power systems. However, for higher power systems above the Stokes laser threshold, we expect a drop in the pump coupling efficiency into the fundamental mode of the HFC as well as a rise in the transmitted pump power caused by changes in the resonator mode waists [30]. The first of these effects can be compensated by adjusting mode-matching lenses to thermally load the cavity in preparation for higher power operation as with solid-state systems.

180

THE CONTINUOUS-WAVE I

I

I

HYDROGEN 1

I

I

RAMAN I

LASER

I

I

ump -- ~Stokes pump

yp.~,,.p= 5ps

Stokes

/ ............................................................................

"

I

I

5

1;

1;

2'0

2'5

I

i

i

i

i

3;

I

35

i

40

1

4'5

i

50

i

.,"'" ...................................

C'3


peak structure in scattering or emission spectra and are caused by constructive interference in the resonator. Nonintegrable cavities introduce added freedom into the design of novel optical components, especially when we apply results from the field of quantum chaos, in which modern quasiclassical methods are of central importance [3-9]. The term "quantum" in "quantum chaos" relates to the fact that cavity modes are discrete as a consequence of the constructive interference requirement mentioned above; the term "chaos" refers to a degree of complexity in the ray picture which renders inapplicable all simple quantization schemes such as the paraxial method. The optics problem we are addressing is not one of "quantum optics," but of the quantized (i.e. discrete) states of classical electrodynamics in spatially confined media. By quasiclassics, we therefore mean the short-wavelength treatment of the classical electromagnetic field. Confusion should be avoided between quasiclassics as defined here, and the semiclassical treatment of light in the matter-field interaction, which couples quantum particles to the electromagnetic field via the classical (vector) potentials: how we obtain the modes of a cavity (e.g. quasiclassically), should be distinguished from how we use them (e.g. as a basis in quantum optical calculations). Our main emphasis in this chapter will be on the "how" part of the problem, but the wealth of physics contained in these modes themselves points to novel applications as well. Having made this clarification, we henceforth use the term "semiclassics" synonymosly with "quasiclassics" as defined above.

6.2 Dielectric Microcavities Many of the microcavities which are the subject of this chapter employ "mirrors" of a simple but efficient type: totally reflecting abrupt dielectric interfaces between a dielectric body and a surrounding lower index medium.

D I E L E C T R I C MICROCAVITIES

187

Resonators occurring in nature, such as droplets or microcrystals, use this mechanism to trap light. This allows us to draw parallels between nature and a variety of technologically relevant resonator designs that are based on the same confinement principle. A recurring theme in numerous systems is the combination of total internal reflection (TIR) with a special type of internal trajectory that skips along the boundary close to grazing incidence: the "whispering-gallery" (WG) phenomenon, named after an acoustic analogue in which sound propagates close to the curved walls of a circular hall without being audible in its center [10, 11]. An example for the use of the WG effect in cavity ring-down spectroscopy is reported in Reference [12]. A WG cavity can provide the low loss needed to reduce noise and improve resolution in the detection of trace species; the trace chemicals are located outside the cavity and couple to the internal field by frustrated total internal reflection, or evanescent fields. In this section, we review Fresnel's formulas before introducing the WG modes. This provides the basis for Sections 6.9-6.16. In the ray picture, the Fresnel reflectivity, RFresnel, o f a dielectric interface depends on the angle of incidence X (which we measure with respect to the surface normal) and relative refractive index n. In particular, RFresne 1 drops significantly below the critical angle for total internal reflection, 1

XTIR ~

arcsinn

(6.1)

In the limit of normal incidence on a plane dielectric interface, the reflectivity becomes independent of polarization, RFresnel --

n+ 1

(6.2)

If the electric field is polarized perpendicular to the plane of incidence (we shall denote this as transverse magnetic, TM, polarization), then Eq. (6.2) constitutes the lower bound for the reflectivity. Hence, the limit of a closed cavity is approached with increasing refractive index of the cavity. If, on the other hand, the electric field lies in the plane of incidence (TE polarization), then the reflectivity drops to zero at the

Brewster angle, 1

XB - arcsin ~/1 +n e

(6.3)

and thus the closed-cavity limit is never fully reached. Because XB < XTIR, the polarization does not affect the trapping of light at incident angles

188

2D MICROCAVITIES: T H E O R Y A N D EXPERIMENTS

X>XTIR, (to leading order in frequency). In addition, because the wavelength 2 does not enter the Fresnel formulas, internal reflection can be classified as a "classical" phenomenon which can be understood based on Fermat's principle. This should be contrasted with the intrinsic frequency dependence of Bragg reflection--the other widespread mechanism for confining light in dielectrics. From an engineering point of view, Bragg reflectors are challenging to realize for lateral confinement. In particular for low-index materials, high-reflectivity windows (stop bands) are direction- and frequency-dependent with narrow bandwidth. TIR, on the other hand, is broadband and technologically simple. At higher orders of 2, wavelength-dependent corrections to Fresnel's formulas do arise because TIR is truly total only for a plane wave incident on an infinite and flat interface. The latter does not hold for boundaries with finite curvature or even sharp corners, and similarly in cases where the incident beam has curved wave fronts, as in a Gaussian beam [13-16]. The physical reason for radiation leakage in these situations is that light penetrates the dielectric interface to some distance which depends exponentially on 2, as in quantum-mechanical tunneling. This allows coupling to the radiation field outside the cavity [13, 14]. The analogy to quantum mechanics rests on the similarity between the Schr6dinger and the Helmholtz equation for the field 7t,

V2~-k-n2(r)k2~ = 0

(6.4)

to which Maxwell's equations often reduce. Here, n(r) is the index profile and k the free-space wavenumber. The WG modes of a circular resonator are straightforward to calculate exactly, because the Helmholtz equation separates in cylinder coordinates r, 4~,z. The circular cylinder and sphere are the two main representatives of this small class [7, 17] of integrable dielectric scattering problems. The analytic solution, due to Lorentz and Mie, of Maxwell's vector wave equations for a dielectric sphere, has a long history [13, 14, 17-20]. It finds application in a wide range of different optical processes, ranging from elastic scattering by droplets to nonlinear optics [25] and to cavity quantum electrodynamics [21-23]. Similarly, dielectric cylinders are used, e.g. as models for atmospheric ice particles [24] or edge-emitting microdisk and micropillar lasers [26-28]. Because of rotational symmetry, the ray motion in a sphere is always confined to a fixed plane. By contrast, rays in a cylinder can spiral along the axis [29, 30], but the propagation becomes planar if the incident wave is aligned to be perpendicular to the cylinder axis. Similar symmetry arguments make it

WHISPERING-GALLERY MODES

189

possible to find analytic solutions in concentrically layered dielectrics or ring resonators [31-33]. Each time a degree of freedom can be separated due to symmetry, the effective dimensionality of the remaining wave equation is reduced by one-in the above examples one finally arrives at an ordinary (one dimensional) equation for the radial coordinate. Even without symmetry properties such as in the sphere or the cylinder, one often finds approximate treatments by which such a reduction from three to fewer dimensions can be justified: in fact, in integrated optics, most functions are performed by planar optical devices, for which the mode profile in the vertical direction can be approximately separated from the wave equation in the horizontal plane, leaving a two-dimensional problem. For a discussion of methods exploiting this assumption (e.g. the effective-index method), the reader is referred to the literature [34]. One of the advantages of reducing the cavity problem to two degrees of freedom is that polarizations can approximately be decoupled into TE and TM. The resulting wave equations are then scalar, with the polarization information residing in the continuity conditions imposed on the fields at dielectric interfaces. Therefore, the fields are formally obtained as solutions of Eq. (6.4) in two dimensions. This scalar problem is the starting point for our analysis. In the absence of absorption or amplification, the index n(r) which defines our microcavity is real-valued, and approaches a constant (taken here as n = 1 for air) outside some finite threedimensional domain corresponding to the cavity.

6.3 Whispering-Gallery Modes Figure 6.1 illustrates the formation of whispering-gallery modes (WGMs) for a dielectric ellipse illuminated by a TM polarized plane wave. The wavenumber k is made dimensionless by multiplying with the mean radius R of the two-dimensional shape. The sharp features in the spectra of Figure 6.1 are caused by modes which rely on Fresnel reflection. This can be deduced from the fact that the spectrum becomes more crowded as the refractive index is doubled from n = 1.5 (a) to n = 3 (b); the intuitive reasoning is that a larger set of ray paths acquires high reflectivity as n increases. The more regularly spaced peaks of Figure 6.1(a) can in fact be identified as WGMs: light circumnavigates the perimeter of the cavity in such a way as to insure TIR during a complete roundtrip, i.e. X> XTIR at all encounters with the boundary. Consequently, long-lived cavity modes can form by constructive interference. In the stability diagram of linear paraxial resonator theory [2], a circular

190

2D MICROCAVITIES:

(a) ~'m Index:

~4.

n=

THEORY

/g'~

1.5

EXPERIMENTS

'~'3[I Index: tu!

k,,

~

_..c 2

30

AND

=

c

31

32

1

33

kR

34

35

30

31

32

33

kR

34

35

FIG. 6.1. Light scattering spectra for a resonator in the shape of an ellipse with eccentricity e = 0.8, defined in terms of the major and minor axes a, b as e = v / 1 - (b/a) 2L. The length scale R - q/-ab is used in our calculations to convert the wavenumber into the dimensionless "size parameter" kR. The index of refraction is n = 1.5 in (a) and n = 3 in (b), showing how the openness of a dielectric cavity increases when n is reduced. The incoming plane wave in (a) and (b) travels parallel to the major axis, and the scattered intensity is detected at 90 ~ from incidence.

dielectric W G cavity could be classified as a confocal resonator with closed but leaky mirrors. Gaussian beam paraxial optics fails in circular cavities because the beam parameters become undefined in the confocal limit. The Lorentz-Mie treatment of circular and spherical micro-objects m a y serve as starting points for perturbative treatments of resonators whose shape deviates only slightly from rotational symmetry [35], or when objects in the vicinity of the cavity exert a weak influence [36]. Perturbation methods are among the most powerful tools of wave physics, but one must be aware that there are phenomena outside their reach (most apparent when energy denominators diverge). As a simple example, we return to the elliptical resonator [37]. If we assume impenetrable boundaries (of Dirichlet type), it is a classic textbook problem [38] to obtain the W G M s of the ellipse by applying perturbation theory to a circular cavity. The weakness of this approach is revealed when we recall that the modes can be found by an exact separation of variables and fall into two classes: W G M s and beam-like "bouncing-ball" states; in Figure 6.2 we illustrate these two classes with slightly more complicated dielectric boundary conditions. Since bouncing-ball modes do not exist in the circle, perturbative expansions for this type of modes can be expected to become problematic, especially when attempting to describe how some modes of the circle lose their W G character under a continuous shape deformation.

WHISPERING-GALLERY MODES

191

FIG. 6.2. Numerical solution of Maxwell's equations for the internal and external intensity of TM modes in an elliptic cylinder with a refractive index of n = 1.5 (a) and n = 3 (b), respectively (assuming n = 1 on the outside). The eccentricity is e = 0.709 in both plots. The dimensionless wavenumber in (a) is kR = 21.27636 at a resonance width of KR = 0.0085; the state in (b) is found at kR = 20.69345 and has a width of tcR - 0.05. The greyscale shows high intensity as white.

The wave solutions shown in Figure 6.2 show field intensity extending to the exterior of the cavity because the dielectric interface is "leaky." In general, such a calculation must be performed numerically; we shall discuss the definition and emission properties of these leaky modes further below, in Section 6.4. For now, we are only concerned with the internal intensity patterns. The reader will recognize a strong similarity between the "bouncing-ball" mode and a higher order transverse Gauss-Hermite beam; this arises because the ray motion corresponding to this mode is a stable oscillation between the flat sides of the cavity. The analytic transverse form of the bouncing-ball beam in the ellipse is however not a Gaussian, but a Matthieu function. The caption of Figure 6.2 gives the linewidth K of the two types of modes, indicating that the W G resonance is almost an order of magnitude narrower than the bouncing-ball mode, despite the twofold higher refractive index used in Figure 6.2(b). The ellipse has been chosen as an example because it is integrable in the limit of no leakage. Other oval deformations of the circle do not have this simplifying property. However, families of W G ray trajectories have been proven [39] to exist in any sufficiently smooth and oval enclosure, provided that its curvature is nowhere zero. This makes W G M s a very robust phenomenon of general convex oval cavities. Before we discuss in more detail the relation between the ray picture and the internal structure of the resonator modes, we now turn to some general considerations on what allows us to define the modes of an open cavity.

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AND EXPERIMENTS

6.4 Scattering Resonances and Quasibound States W G M s are not infinitely long lived even in an ideal dielectric cavity, as we saw in the finite linewidths of Figure 6.1. In fact, for a finite, threedimensionally confined dielectric body, all solutions to Eq. (6.4) are extended to infinity, forming a continuous spectrum. A basis of eigenfunctions is given by the scattering states, consisting of an incoming wave grin that is elastically scattered by the dielectric microstructure of index n(r) into an outgoing wave grout. In the asymptotic region where n = 1, the relation between incoming and outgoing waves is mediated by the scattering operator S, known from quantum scattering theory [40], to which the electromagnetic resonator problem is conceptually analogous. The S-matrix formalism has long been in use in microwave technology as well as optics [41]: a matrix representation is obtained by defining basis states Iv)(I/~)) in which the asymptotic incoming (respectively, outgoing) fields can be expanded; one has 1/rscat - - l/tin -t- l~rou t M

(6.5) v=l

Here, S is a unitary matrix if the incoming and outgoing waves are normalized to carry unit flux, and if n(r) is real. The dimension M is the number of open scattering channels; in the Fabry-Phrot cavity there are two channels: both, the incident field In) and the outgoing states Ira) can be a plane wave either traveling on the left or on the right of the cavity. In 2D and 3D, it is often useful to use angular momentum eigenstates as the basis defining the channels. Only the properties of the index profile, not of the particular incoming wave, enter S. For a review of scattering theory see, e.g. Reference [42]. As was seen in Figure 6.1, the actual cavity modes in this continuum of scattering states are revealed if we measure the scattering of light as a function of wavenumber k. The amplitude of the scattered field shows resonant structure at discrete values of k which do not depend on the detailed spatial form of the exciting field grin. These resonances are caused by poles of S in the complex k plane, and are a characteristic of the microcavity itself. In Eq. (6.5), we note that a pole of S admits nonvanishing grout in the absence of any incoming waves, grin. In these solutions at complex k, also known as the quasibound states, we have finally found a proper definition of what we simply called "cavity modes" earlier.

SCATTERING

RESONANCES AND QUASIBOUND

STATES

193

A well-known example is the linear two-mirror (Fabry-P6rot) cavity [2]. Its resonances are easily obtained within physical optics by writing the transmission of an incident ray as a geometric series over multiple round-trips. In each round-trip, the amplitude of a ray in the cavity accrues a factor rl r2 exp(inkg + i4~1 + i~b2), where s is the round-trip path length. We have split the amplitude reflectivities at the individual reflections v = 1,2 into modulus rv and phase Cv. Attenuation comes from r~ < 1(v = 1,2 in the two-mirror case). The refractive index n in the cavity is real, as stated below Eq. (6.4), and the wave number k is measured in free space. The transmitted amplitude Ttotal is obtained by summing this over all repetitions, (x)

Ttotal - - I Z

(rlr2einkg+ir

r

(6.6)

o'=1

where the prefactor I is determined by the incident wave and the mirror transmittivities. This geometric series can be summed, leading to a denominator 1 - g r t ( ( ~ ) , with the round-trip "gain" grt((~)

--

e inke/c I-I

rvei~v

(6.7)

1)

In our example, v runs from 1 to 2. Whenever grt c o m e s close to 1, the resonator exhibits a transmission peak. The equality ~rt(OS)= 1 can be satisfied only if we admit complex frequencies, (5 _= co + iy, and choose for the imaginary part C

7' - - ~ ~

In r,

(6.8)

1)

On the real frequency axis, y determines the resonance linewidth. Were we to look at the complex-frequency solution directly and reinstate the time dependent factor exp(icSt) that accompanied the original wave equation, the field at every point in space would decay with a factor exp(-yt). An interpretation of this can be given in the ray picture: for a ray launched inside the cavity, the field is attenuated by a factor rl r2 in each round-trip; after time t, the number of round trips is ct/g., leading to the exponential law

E(t) - e ctln(nr0/e = e -•

(6.9)

Hence, each transmission peak is uniquely associated with a metastable, or quasibound state, and the decay rate is determined by the reflectivities encountered during a round-trip.

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Equation (6.9) straightforwardly justifies the concept of a metastable state within the ray picture. We introduced resonances as peaks in the transmitted field, arising as near-divergences of a geometric series over ray paths undergoing multiple reflections inside the cavity. This is in fact just a special case of a very general quasiclassical approach which has found widespread use in chemical physics [43] and other fields [44] since its original introduction by Miller [6, 45]. It relates the S-matrix element between arbitrary channels tt and v of a multichannel scattering process to a sum over all possible ray paths starting in channel v and ending in tt: S,~ - ~

V/p~) exp(ik cI)(~) - ia (~))

(6.10)

o-

where a parameterizes the family of ray trajectories leading from incoming channel v to outgoing channel tt, just as in Eq. (6.6). ~(~/ is a phase shift acquired by the rays as they encounter caustics and interface reflections along their path. The phase shift k~b(~) is the generalization of the dynamical phase ke in the linear example, and xn- /(~) is the transition zv probability, corresponding to the product of reflection and transmission coefficients in our two-mirror example. For a cavity defined by Fresnel reflection, rtrt t v~) can to lowest approximation be determined purely within ray optics. The virtue of Eq. (6.10) is that it points the way from geometric optics to wave optics even in systems where the ray paths are not as easily enumerated as in the Fabry-P6rot cavity. The Fabry-P6rot cavity discussed above is an example where Eq. (6.10) in fact yields exact results, because only plane-wave propagation is involved. Although corrections to this quasiclassical formula are necessary in more complicated cavity geometries, Eq. (6.10) makes it plausible that ray considerations are a powerful tool for understanding quasibound states in many open systems. In this spirit of ray-based scattering theory, we can ask how to extract the quasibound states as poles of the physical-optics expression Eq. (6.10). This means we want to generalize the logical transition (illustrated for the Fabry-P6rot cavity) from a transmission amplitude determined by Eq. (6.7) to an internal ray loop with attenuation given by Eq. (6.9). Thus, the original scattering problem should be replaced by a Monte Carlo simulation of a suitable ensemble of rays inside the resonator, and the internal ray dynamics suffers dissipation owing to the openness of the cavity. Several technical problems make it difficult to carry out this idea in a general cavity: the first question is what would be the proper choice of initial conditions for a ray ensemble in a two-dimensional cavity such as

CAVITY R I N G - D O W N AND LIGHT EMISSION

195

the ellipse of Figure 6.2. The answer is provided by quasiclassical quantization conditions that put some constraint on the ray paths to be used in the ray calculation. Employing a strategy along these lines, it is possible to predict not only the decay rate of a quasibound state as in Eq. (6.9), but also the directionality of the emitted radiation [51]. This will be expounded in Sections 6.11 and 6.14.

6.5 Cavity Ring-Down and Light Emission We now discuss the significance of complex frequencies in Eqs. (6.4) and (6.5). This will highlight the relation between quasibound states and the observable cavity response to an external field or pump signal, which is of particular interest in spectroscopy. In Eq. (6.4), the wavenumber only appears in the form of a product nk, so that an imaginary part in k = k + itc = &/c can immediately be reinterpreted as part of a complex refractive index t7 at real k, setting n/~ - n (k + ix) = k (n + inK~k) -- fik

(6.11)

The resulting Helmholtz equation has a real wavenumber, and thus describes the steady-state wave solutions in an amplifying medium, because/c > 0: a plane wave would have the form exp(icot- i~kx), which grows in the propagation direction. From Eq. (6.11) it follows that quasibound states appear naturally as approximate solutions for the lasing modes of a microlaser with a homogeneous gain medium [52]. For a physical understanding of lasers [53], the openness of the system as contained in the quasibound state description is essential. When a cavity is excited with a pulse, on the other hand, we are not looking for steady-state solutions but for transients. As we noted below Eq. (6.9), quasibound states describe such a decay process. These nonstationary states are exploited in many fields, e.g. nuclear physics (where they are called "Gamov states"), and their properties are well-known [54, 55]. One peculiar property that may cause confusion is that they formally diverge in the far field, as can be seen by noting that the outgoing wave in Eq. (6.5) obeys the radiation boundary condition, which in the continuation to complex frequency reads 1/rou t (X

ei(~t-b)

(r --+ ~ )

(6.12)

This holds for any dimensionality if r is understood to be the radial coordinate, and algebraic prefactors (which depend on the dimension of

196

2D MICROCAVITIES: T H E O R Y A N D E X P E R I M E N T S

the solid angle element) are not considered. At fixed t, the imaginary part tc of k causes exponential growth with r. However, grouping together the exponential dependences on position and time, the amplitude of the quasibound state is controlled by the factor e -•215

(6.13)

Clearly, this remains constant along the space-time trajectory r = ct. The complex-valuedness of oSk therefore has nothing unphysical, provided causality is taken into account: within the ring-down time 1/co of the cavity, the maximum distance to which we can extend measurements of the radiated field is of order R ~ c / y = 1/K. In this range, r < R, the decaying pre-exponential factors dominate over the exponential r dependence. Equation (6.11) represents a level of approximation which does not take microscopic properties of the light-emitting medium into account. However, it forms a starting point which emphasizes the effect of the resonator boundaries on the mode structure from the outset; an effect that becomes essential as the cavity size decreases. If we allow the polarization P, which in Maxwell's wave equation effects the coupling between matter and field, to become a function of position, and possibly acquire a nonlinear dependence on the electric field, then it turns out that the quasibound states of the homogeneous medium, discussed above, are nevertheless a convenient basis in which to describe the emission and mode coupling caused by P. To make this plausible, recall that any given electromagnetic cavity field can be expanded in an integral over the scattering states ~scat of Eq. (6.5) as a function of k; these are the "modes of the universe" in the presence of the cavity. To evaluate such an integral, one can extend the integration contour into the complex plane and apply the residue theorem. The integral is then converted to a sum over those k at which the integrand has a pole. But the poles of the scattering states are just the poles of the S-matrix, i.e. the quasibound states. For details of these arguments, the reader should see Reference [54]. The radiation from a source distribution d(r) is obtained directly by summing up those quasibound states which lie in the spectral interval of interest, weighted according to their overlap with the distribution d(r), and with an energy denominator that makes sharp resonances contribute more strongly than broad ones. In particular, for small cavities where the free spectral range is large and the linewidths are small, the emission properties are determined by the spatial form and temporal behavior of individual quasibound states.

W I G N E R DELAY TIME AND THE DENSITY OF STATES

197

6.6 Wigner Delay Time and the Density of States In spectroscopic applications, the quality Q and finesse F of a cavity are important figures of merit. Conventionally, one defines Q=

kr K

,

F=

FSR K

(6.14)

where FSR is the free spectral range and kr is the wavenumber of the mode. To decide whether the linewidth of an individual resonance, as given by Eq. (6.8), is narrow or broad, we can choose as our yardstick the separation between neighboring modes. However, as seen in Figure 6.1(b), this can become rather ambiguous if the spectrum is crowded. A free spectral range is then hard to define, and should be replaced by a continuous function of frequency: the spectral density, or density of states, p(k). If Q and F were independent of k, then the total number of cavity modes with wavenumber less than or equal to k would be N(k) = k / F S R Q/F. The density of states is the k-dependent generalization of the inverse FSR:

N(k) --

p(k')dk'

(6.15)

The structure of p(k) contains all the information about both Q(k) and F(k). For a dielectric scatterer with sharp boundaries, there is in principle a clear distinction between "inside" and "outside." An incoming wave propagates freely outside the dielectric, but can be trapped inside for some time. Therefore, an incident wave pulse emerges from the scatterer with a time delay, TD, compared to the time it takes in the absence of the obstacle ( S - 1). The time delay is a continuous function of k, whereas the resonance decay time 1/y labels discrete poles of the S-matrix. To clarify the relation between delay time and decay time, consider the simplest case of a one-channel scattering system, in which ~in O( exp(icot + ikx) and Oout cx exp[icot- ikx + i0(k)]. Here, 0 is the scattering phase shift, which is related to the (unitary) S-matrix of Eq. (6.5) by

S(k) - exp[i 0(k)]

(6.16)

The outgoing wavepacket then has the Fourier decomposition

7t(x, t) - f dk ~(k) eik(ct-x)+i~ d

(6.1 7)

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2D M I C R O C A V I T I E S : T H E O R Y A N D E X P E R I M E N T S

We can look at this general expression in two instructive limiting cases: the magnitude of of the Fourier spectrum, I~(k)l, could be assumed to be either sharply peaked or slowly varying. Consider first the case where [~(k)[ has a narrow peak at some central wavenumber k0; the opposite limit will be treated in Eq. (6.27). Then the variation of 0 with k need only be retained to linear order, yielding

d0

O(k) ~ O(ko) + (k - ko) -~o =~ ~(x, t) ,~ const, x

f dk ~ ( k ) e ik(ct-x+dO/dk~ (6.18)

If we denote by fr0(x, t) the corresponding pulse for 0 - - 0 , i.e., without scattering, then the last equation means O ( x , t) -

~o

( x, t + - 1 d~0)

(6.19)

c

This shift in t is the Wigner-Smith delay time, TD(k) [56, 57]; it is a continuous function of k. For a general scattering system with M R 1 channels, rD(k) is obtained by adding the k-derivatives of all the phases 0, entering the M eigenvalues exp(i0,) of the S-matrix: TD(k) --

lw ~ 30.Ok= ci T r {\ddSk* S )

1

---clmTr

( d ) lnS(k)

(6.20)

/x=l

The last equality is a matrix identity; it contains a trace over all open channels/z - 1,..., M. The logarithmic derivative can be executed immediately if one channel is resonant: near a resonance frequency (-Or - - ckr one has S o( 1 / ( k -

kr - itc)

(6.21)

with x describing the linewidth, or the decay rate in Eq. (6.13) via y = cK. Then "CD(kr) ~ -

1 ctc

=-

1 y

(6.22)

which means that the resonant time delay equals the resonance lifetime. The spectral density of the open system is closely related to the delay time rD, as we can understand from the following (nonrigorous) argument (for more stringent derivations, see, e.g. Reference [60]): rD defines a length scale L ~ - CrD/n (n is the refractive index) which is the characteristic distance over which the wave will propagate inside the scatterer. If we interpret this as the effective "cavity length," then a mode

W I G N E R DELAY TIME AND THE DENSITY OF STATES

199

should naively be expected when an integer number of wavelengths fits into this length, i.e. nkL~ = 2rcv + const

(6.23)

where v is an integer, and the constant takes into account phase shifts at interface reflections or caustics. The number of modes that are contained in a small interval Ak around k is then given by the corresponding change A v in the above equation, which to lowest order in k is 1

A v -- ~

(6.24)

L~nAk

Therefore, the spectral density is simply Ap

/7

c

p(k) - A k = 2--~ L~ -- ~

TD(k)

(6.25)

This, combined with Eq. (6.20), is the K r e i n - F r i e d e l - L o y d f o r m u l a for the density of states in an open system, which reappears almost invariably whenever linear response or time-dependent perturbation theory in the presence of a continuous spectrum are considered [58-65]. An important example is Fermi's golden rule which relates electronic transition rates W to the squared matrix element ]M] 2 of the interaction and the density of states, 27~

W(k) - ~ IM[2p(k)

(6.26)

Based on this and Eq. (6.25), we can state that microcavities are able to enhance optical transition rates, because at certain wavenumbers the light is trapped (i.e. delayed) in the cavity for long time. The density of states is one of the fundamental quantities that make small cavities interesting, as was recognized long ago by Purcell [66-70, 85] who observed that Eq. (6.26), when applied to the probability for spontaneous emission, can lead to an enhancement of this atomic decay process compared to its rate in free space, by many orders of magnitude. What we have seen here is that peaks in the spectral density are associated with rapid variations in the S-matrix, i.e. with the quasibound states discussed in Section 6.4. Small cavities also enable the opposite effect, a suppression of spontaneous emission when no cavity modes fall within the emission spectrum [68, 71, 72]. The limit of a well-defined frequency in the wavepacket of Eq. (6.17) allowed us to assume that the radiation interacts only with a single quasibound state, leading to the resonant delay time Eq. (6.22). On the other hand, if we take I~(k)]- C to be constant, the whole spectrum enters with equal weight. One could still make the assumption that there is only a single isolated resonance in the spectrum; then one immediately arrives at

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2D M I C R O C A V I T I E S : T H E O R Y A N D E X P E R I M E N T S

the well-known relation between Lorentzian line shape and exponential resonance decay: take the resonance to be at wavenumber kr (it is in fact always accompanied by a partner state [35] at - k r , but we can ignore it at large enough frequencies); replacing exp(i0)= S in Eq. (6.17) by Eq. (6.21), what remains is the Fourier transform of a Lorentzian, ~(x, t) -- C

f

1 eik(ct-x) (x e-K(ct-x)eikr(ct-x) dk k - kr - iK

(6.27)

This is a damped oscillation as a function of c t - x at frequency Or - - C kr with decay constant Yr = CKr. Comparing with Eq. (6.13), we have recovered the decay law of an individual quasibound state in the field envelope. According to Eq. (6.27), if we could make a time-resolved measurement of the oscillating electric field in the outgoing wave, the resonance lifetime would show up in the decaying envelope of the successive peaks of the rapidly oscillating field. It is impractical to make such a measurement at optical frequencies. However, recent advances have been able to approach this limiting case of Eq. (6.17) by making the wavepackets into ultrashort pulses, where correspondingly ]c(k)[ becomes very broad. This will be discussed further in the next section. The results of a real experiment are, however, modified from this simple Lorentzian approximation because the scatterer in general supports more than one distinct quasibound state. The Fourier transform in Eq. (6.17) consequently does not yield a purely exponential envelope, except at long times the decay when it is dominated by the narrowest resonance [73].

6.7 Lifetime Versus Linewidth in Experiments The two limiting cases of ~(k) in the wavepacket Eq. (6.17) are just extremes of the time-frequency uncertainty relation. Experiments on microcavities have been performed both in the spectral and time domain. However, there has been only one report of the temporal nature of an ultrashort optical wavepacket (100 fs pulse, 30 m short in air) incident on a pendant-shaped, hanging droplet with an equatorial radius of 520 m [74]. Such pulses are spectrally broad but have the intriguing property that their spatial length is much shorter than the cavity size. Conceptually, the analogy to a well-defined particle trajectory suggests itself, and hence one looks for ballistic propagation in the cavity. In the experiment, wavepackets were observed to circulate inside the droplet in the region where the WGMs reside; such paths are characteristic of high-order rainbows in that entry and exit of the light are separated by a large number of internal reflections [19, 47].

HOW MANY MODES DOES A CAVITY SUPPORT?

201

The time-resolved measurement showed that coherent excitation of a large number of WGMs allows propagation of a short wave packet along the sphere's equator for several round-trips without significant decoherence, except for decrease in the wave packet intensity because of leakage of the WGMs. The novel aspect of the experiment that enabled the time-resolved measurements was the use of two-color, two-photon-excited Coumarin 510 dye molecules imbedded in the liquid (ethylene glycol) that formed the pendant droplet. The Coumarin fluorescence (near 510 nm) appears when one wave packet of 21 and another wave packet of 22 spatially overlap. Thus, the Coumarin fluorescence acts as a correlator between wave packets. This time-resolved experiment was designed to answer the following questions: (1) does the excitation wave packet remain intact and ballistic after evanescent coupling with WGMs? (2) after a few round-trips, would dispersion cause the wave packet to broaden? and (3) can the cavity ringdown time be observed for those wave packets that make several roundtrips? The answers were reached that the shape of excitation wave packet remained intact, the wave packet was not broadened after a few round-trips, and that cavity ring-down was observed for each round-trip. This is an extension of the single-mode ring-down determined by Eq. (6.27). In particular in the context of cavity ring-down spectroscopy, the usefulness of time-domain measurements is recognized [75]. There, one deals with very high Q factors and their modification by the sample to be studied. Another application of temporal observation is encountered in microdroplets, where the optical feedback provided by WGMs makes it possible to reach the threshold for stimulated Raman scattering (SRS). The SRS spectrum consists of sharp peaks, commensurate with the higher Q WGMs located within the Raman gain profile. The highest Q value of the WGMs can be determined either by resolving the narrowest linewidth A2 of the SRS peak or by measuring the longest exponential decay of the SRS signal after the pump laser pulse is off. When Q > 105, the decay time r = Q/co (where co ~ 3 x 1015 Hz for 2 = 620 nm) becomes a much easier quantity to measure directly because it is longer than 100 ps. Otherwise, the spectral linewidth (A2 = 2/Q) needs to be resolved better than 0.006 nm (for 2 = 600 nm). Cavity decay lifetimes as long as 6.5 ns have been observed [76].

6.8 How Many Modes Does a Cavity Support? Having defined the density of modes for the open system in Section 6.6, we can go one step further and perform the the average ~(k) over some finite spectral interval Ak. The suitable choice for such an averaging interval

202

2D MICROCAVITIES: THEORY AND EXPERIMENTS

should of course contain many spectral peaks. Two relevant examples where an average spectral density is of use are the Thomas-Fermi model of the atom [77] and Planck's radiation law. The latter shall serve as a motivation for some further discussion of the average iS(k) in the next section. For a presentation of the subtleties involved in this procedure, cf. References [78, 79]. For our purposes, we simply remark that using Eq. (6.25) as a starting point, one way of arriving at an averaged spectral density is to make the formal substitution k -~ k + iK. This amounts to an artificial broadening of all resonances as a function of k to make them overlap into a smoothed function; K then plays the role of the averaging interval [80]. In the limit of the closed cavity, the spectral density becomes a series of Dirac delta functions as the resonance poles move onto the real k axis and become truly bound states. The resonator then defines a Hermitian eigenvalue problem of the type we encounter in all electromagnetics textbooks, and many fundamental properties of realistic cavities can be understood within this loss less approximation. As a point in case, it is worth recalling the problem of black-body radiation. From a historical point of view, this thermodynamic question was the nemesis of classical mechanics as the foundation of physics, because it led to the postulate of discrete atomic energy levels. From a practical point of view, the black-body background can be a source of noise in spectroscopic measurements, and its spectrum is modified by the presence or absence of a cavity. From an electrodynamic point of view, the central nontrivial aspect of Planck's problem is that the average spectrum of the black body can be observed to be independent of the cavity shape. The explanation of this universality rests on the average spectral density of cavity modes, which is found to be independent of the resonator geometry to leading order in frequency. Although it may be intuitively convincing that the shape of the enclosure should become unimportant when its dimensions are large compared to the wavelength [81], the actual proof requires a large measure of ingenuity. In a series of works beginning in 1913 [82], Weyl showed that for a closed, three-dimensional electromagnetic resonator of volume V, the average spectral density as a function of wavenumber k is (including polarization) k2 -3D (k) ~ ~ V PWeyl

(in 3D),

-2D 1(k) ~,~ /gwey

k

A

(in 2D)

(6.28)

The second equation applies when, as discussed above, the wave equation can be reduced to scalar form and two degrees of freedom for a cavity

HOW MANY MODES DOES A CAVITY SUPPORT?

203

of area A. The average number of modes in an interval Ak then is [)Weyl(k) Ak for large k. Geometric features other than the volume enter in this quantity only as corrections with lower powers of the wavenumber. These terms depend on the boundary conditions, surface area (or circumference) and curvature, as well as on the topology of the cavity [83, 841. As mentioned in the previous section, Eq. (6.26), microcavities can lead to enhanced spontaneous emission because of their highly peaked density of electromagnetic modes. Although Weyl's formula is strictly valid only for short wavelengths, it nevertheless allows us to estimate the limiting size for ultrasmall cavities that should be approached if we want to observe such density-of-states effects: note that Eq. (6.28) approaches a "quantum limit" as the volume V approaches (2/2) 3 at fixed wavelength 2: the number of modes with wavenumber below k = 2rc/2 is then N-w3D eyl(k)

--

L k PWeyl(k) -3D dk ~ ~7"I; ~ 1

(6.29)

which means that only a single mode remains in the cavity. In reality, the ultrasmall size approached here is not necessarily the optimal choice, because it is hard to maintain good quality for small cavities. One of the best ways to achieve both small size and long lifetimes is by making use of W G M s in dielectric cavities. The universality of Weyl's spectral density is surprising if we look at it using an analysis for stable resonators [86]. This shows that the number of modes which can be obtained within the paraxial approximation depends strongly on details of the cavity geometry. Recall the nature of the paraxial approximation; it is in fact a short-wavelength approximation to Maxwell's wave equations, in the sense that the characteristic length scale (of the cavity or variations in the refractive-index profile) along the propagation direction must be large compared to the wavelength. In that framework, the geometric-optics picture is therefore the backbone on which the mode structure of a cavity is built. However, paraxial modes appear only if stable, closed ray patterns exist in the cavity. What makes Weyl's formula nontrivial is that it does not distinguish between a cavity in which such stable orbits are readily available (as was the main subject of the previous chapter), and the extreme yet realizable case of a chaotic cavity in which no stable paths can be found at all, see Section 6.10. Stability is a property of particular rays, not of a cavity as a whole. To understand all the modes of a generic cavity, one has to go beyond paraxiality. It also must be realized that paraxial optics is itself a special case of a more general approximation scheme, known as the

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parabolic-equation method [85, 87]. This name refers to the fact that the resulting wave equation of Schr6dinger type is mathematically classified as a parabolic differential equation. This is mentioned here because at a more abstract level, one can build approximations of paraxial type not only around rectilinear rays: e.g. for WGMs, an envelope function ansatz in polar coordinates, 7~(r, ~b)= x(r,~b)exp(-i/3~b) allows ~b to become the "propagation direction." The question of whether a mode can be called paraxial or not then becomes dependent on the coordinate system one uses (e.g. cartesian vs. cylindrical). A more fundamental distinction by which the cavity as a whole can be classified is that of integrability. Next, we discuss some examples for this concept.

6.9 Cavities Without Chaos One feature that can be observed in both Figure 6.2(a) and (b) is that the modes exhibit caustics inside the dielectric, i.e. well-defined curves of high intensity which in the ray picture correspond to envelopes at which the rays are tangent. In the W G M , the caustic is an ellipse confocal with the boundary; it can be parameterized by its eccentricity, ec. In Figure 6.2(b), the caustic consists of two confocal hyperbola segments. Caustics separate the classically allowed from the forbidden regions, in the sense of the WKB approximation. In this section, we discuss some examples of how quasiclassical and exact solutions can be obtained in nonchaotic but nontrivial cavities, if coupling to the exterior region is neglected. In this closed limit, all fields can be written as real-valued functions obeying standard boundary conditions (Dirichlet or Neumann). Figure 6.3(a) illustrates the W G caustics of an ellipse and their relation to the quasiclassical quantization method of Einstein, Brillouin, and Keller (EBK, the multidimensional generalization of the WKB approximation) [88]: given a particular caustic of eccentricity ec, we examine the ray segments that are tangent to it. For a fixed sense of rotation, there are exactly two distinct rays going through any given point r between the caustic and the boundary--one traveling from the caustic toward the boundary and the other vice versa. These two unique ray directions as a function of r define two vector fields, which are called "ray congruences" [88] and are shown in the figure as a flywheel of rays. They are furthermore normal to the phase fronts shown as curved grey lines. These are the eikonals ~l,2(r) appearing in the quasiclassical ansatz for the wave, O(r) = A l(r) exp[-i k ~ba(r)] §

A2(r) exp[-i k ~2(r)]

(6.30)

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205

Fla. 6.3 (a) In a WGM of the ellipse, the point r is visited by two wavefronts: one carrying rays from the caustic to the boundary (top), and the other vice versa (bottom); note that both have the same sense of rotation (indicated by the arrow). A complete ray trajectory alternates between these two wave fronts as it encounters boundary and caustic repeatedly. (b) A paraboloid (surface of revolution generated by a parabola), closed by a mirror intersecting its focal point. The focal length equals 3.25 2. The azimuthal mode number is m = 1; high intensity is shown in black. The corresponding rays (bottom) come in families and generally do not close on themselves (bottom). (c) A 2D equilateral triangle cavity. The side length is 13.5 2. The mode can be constructed rigorously from a family of periodic ray orbits, one example of which is shown.

We have suppressed the monochromatic time dependence exp(icot). At least two terms are necessary when there are two degrees of freedom; additional, symmetry-related terms may be needed to make the wave field real-valued. In the standard EBK quantization, the amplitude functions A1,2 are assumed to be slowly varying, and one can achieve the single-valuedness of the wave function only if the phase advance in the exponentials is an integer multiple of 2re for any closed loop in the planar cavity. This occurs only for certain discrete combinations of the unknown parameters k (the wavenumber), and ec (the eccentricity of the caustic). Hence, the semiclassical method quantizes not only the wave parameter k of the modes, but also the classical parameter ec defining the corresponding ray trajectories. A simple example is the circular resonator, a limiting case of Figure 6.3(a). The internal caustic in that case is a concentric circle with radius Ri. By geometry, the rays corresponding to that caustic have a fixed angle of incidence given by sin X = R i / R , if R is the cavity radius. Single-valuedness of the wave field then requires that the phase advance along a loop encircling the caustic (= circumference x wavenumber) equals 2rcm, where

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m is an integer: 2nRink - 2nm

=:~ sin X

m

nkR

(6.31)

where n is the index of refraction. The meaning of m follows if we interpret p =_ hnk as the linear photon momentum, and recall the definition of

classical angular momentum, L = r • p. Then if r lies on the surface, the z component of L is Lz = R p sin X, which identifies Lz = hm by comparing with Eq. (6.31). One can thus call m an "angular momentum quantum number." In addition to this orbital angular momentum quantization, there is a radial single-valuedness condition which forces k to become discrete. This yields a complete set of two quantum numbers for two degrees of freedom, and it implies that X in Eq. (6.31) becomes discretized as wellanother way of understanding the quantization of the caustic ec. For a basic discussion of the EBK method in rotationally invariant, separable cavity geometries, cf. Reference [89]. One result of the EBK quantization is that the WGMs do not in general correspond to closed ray orbits except in the limit when the internal caustic approaches the cavity surface. The quantized caustics instead belong to a family of rays which encircle the perimeter quasiperiodically, coming arbitrarily close to any given point on the boundary after a sufficiently long path length. This generalizes to most other resonator problems: What counts for the formation o f modes is not that the associated rays close on themselves, but only that the wave fronts, to which these rays are normal, interfere constructively.

Even in three-dimensional integrable cavities, the EBK method can yield highly accurate results down to the lowest-frequency modes. As an example, we mention recent work on a microlaser cavity with strong internal focusing properties, which consists of a dome in the shape of a paraboloid, on top of a layered semiconductor [90], cf. Figure 6.3(b). As in the ellipse, the exact solution for the mode shown for this piano-parabolic mirror geometry bears some resemblance to the more familiar paraxial optics, in this case a Gauss-Laguerre beam. However, just like the circular resonator, the parabolic dome has no stable ray orbits, owing to the confocal condition. The modes can be found exactly because the geometry allows separation of variables in parabolic cylinder coordinates. The short-wavelength approximation can be made highly accurate (as in the parabolic dome) or even exact (as in the Fabry-P6rot cavity of Section 6.4.) As a nontrivial generalization of exact quantization based on rays, Figure 6.3(c) shows the equilateral triangle. All its modes can be obtained by superimposing a finite number of suitably chosen plane waves [10, 91, 85]. This is achieved by "unfolding" the cavity into an infinite lattice created by its

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mirror images. These examples show that ray optics, as the skeleton which carries the wave fields, remains a useful tool far beyond the paraxial limit. However, the reader may ask: what is the use for semiclassical methods in exactly solvable problems such as the above systems? After all, semiclassics is simple in separable systems! Beyond quantitative estimates, the value of quasiclassics is that the connection between modes and rays can be carried over to deformations of the cavity shape where the separability is destroyed. When this happens, eigen states cannot be labeled uniquely by global quantum numbers anymore. However, as Weyl's formula teaches us, the absence of good quantum numbers does not imply the absence of good modes. In order to classify the latter, we shall attempt to label them according to the ray trajectories to which they quasiclassically correspond.

6.10 Chaotic Cavities Taken together, the discussions of stable resonators in the previous chapter [86] and of integrable systems in the foregoing section provide the essentials of "conventional" resonator physics. However, in the infinite space of possible cavity shapes, most geometries are nonintegrable and hence display chaos in their ray dynamics, as mentioned in the introduction to this chapter. To illustrate the transition to chaos, Figure 6.4 shows the continuous evolution of an individual quasibound state as the shape of an oval resonator is deformed. The cavity shape is a two-dimensional quadrupole of mean radius R. Now the distinction between the mathematical ellipse and other oval shapes becomes important. Although the difference between the shapes in Figures 6.4 and 6.2 is barely discernible at small e, the quadrupole is not an integrable cavity and displays a far more intricate internal mode structure. In particular, caustics become frayed, and nodal lines form ever more complicated patterns as e increases. Given this complicated scenario, it is not immediately clear that ray considerations can help at all in understanding the properties of states such as those in Figure 6.4. However, it turns out that quite the opposite is true: it is the added complexity of the internal ray dynamics that can be identified as the cause of the more complex wave fields. We make this somewhat provocative statement because the previous examples have proven the success of using ray considerations as a scaffolding for constructing the cavity modes. Unfortunately, there is so far no complete theoretical framework for the quasiclassical quantization of partially chaotic systems; the EBK method, which in quantum mechanics gives rise to the corrected Bohr-Sommerfeld quantization rules [88], fails when the system does not exhibit well-defined caustics, as was already noted by Einstein [92].

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Fro. 6.4. The refractive index in (a-d) is n = 2.65. (a) Position /< = k + i~c of a quasibound state, parametrized by the quadrupolar shape deformation r(0)= R(1 + e cos20)/v/1 + E2/2. Here, E is the fractional deformation from the circle. Arrows show the location of the quasibound states shown in (b-d), corresponding to e=0.114 (b), e=0.126 (c), and e =0.135 (d). The resonance widths are tcR = 0.087, 0.069, and 0.047 in (b), (c), and (d) respectively. Thus, increasing deformation leads to linewidth narrowing and more focused emission.

However, it is worth following the quasiclassical route because there exists a vast amount of knowledge about the classical part of the problem: chaotic ray dynamics can help gain insights into the wave solutions that cannot be gleaned from numerical computations alone. Ray optics can be formally mapped onto the classical mechanics of a point particle; this allows us to leverage a rich body of work on chaotic classical m e c h a n i c s - - a mature, though by no means complete field [1, 5, 93, 94]. Hence, quasiclassics in the presence of chaos is a challenging undertaking, but also a useful one because the numerical costs for obtaining exact wave solutions in nonintegrable systems are so high. As observed in Section 6.7, pulses that excite many cavity modes can behave ballistically, i.e. seem to follow well-defined trajectories. The idea of the quasiclassical approach is complementary to this: from the behavior of whole families of ray trajectories, we want to extract the properties of individual cavity modes.

6.11 Phase Space Representation with Poincar~ Sections As the ellipse already taught us, different types of ray motion can coexist in a single resonator--e.g. W G and bouncing-ball trajectories. In such cases, it is desirable to know what combination of initial conditions for a ray will result in which type of motion. Initial conditions can be specified by giving the position on the boundary at which a ray is launched, and the angle X it forms with the surface normal. The proper choice of initial conditions was identified at the end of Section 6.4 as a prerequisite in the

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209

FIG. 6.5. Poincar6 surfaces of section: at each reflection, record the position of impact (parametrized by the polar angle ~b) and angle of incidence X, in a plot of sin X versus 4~; repeat for several trajectories. (a) ellipse of eccenticity e = 0.8 (same shape as in Figure 6.1) (b) quadrupole with E = 0.135, as in Figure 6.4(d). In (a), oscillatory paths between the flat sides appear as closed loops, whereas WG circulation creates curves that span all polar angles ~b. While all rays in (a) form caustics, the chaotic paths shown in (b) do not. Chaotic rays can undergo WG circulation (bottom left) for long times before covering the whole cavity in a quasirandom way (bottom right). quasi-classical modeling of resonance decay. The present section introduces the tools necessary for solving this problem. Information on the possible types of ray motion, and how they are grouped in families, is contained in the Poincar6 surface o f section (SOS), cf. Figure 6.5. It is a representation of the classical phase space in the 4r-sin X plane, spanning all possible angles of incidence and reflection positions along the boundary. The definition of X in the center of Figure 6.5 includes a sign, measured positive in the direction shown by the arrow from the outward normal. This reflects the observation made below Eq. (6.31) that sin X is proportional to the z component of the instantaneous angular momentum at the reflection, and the sign thus distinguishes the sense of rotation. By combining the two variables q~ and sin X, dynamical structure can be revealed which would remain hidden in a collection of real-space ray traces. Nonchaotic trajectories are confined by local or global conservation laws to one-dimensional lines, called invariant curves. Almost all rays in the ellipse follow such curves, as seen in Figure 6.5(a), which shows a clear-cut division into oscillatory and rotational (WG) motion, cf. also Figure 6.2. This is analogous to the phase space of a physical pendulum [95]. The separatrix between the two types of motion corresponds to a diametral ray orbit connecting the points q~ = 0, rc on the boundary.

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Rays launched on an invariant curve must remain on it for all subsequent reflections. Such unbroken curves persist for ]sin Xl--+ 1 even in Figure 6.5(b); this is just the WG limit. A lesser degree of robustness is observed for the oscillatory trajectories: they are surrounded by elliptical "stable islands," but a chaotic sea forms in-between, owing to the fact that in Figure 6.5(b) the diametral separatrix orbit mentioned above is unstable, developing the sensitivity to small deviations in initial conditions which is typical for chaotic behavior. Chaos develops preferentially around such separatrix orbits. The analogy to a pendulum makes this plausible: there, the separatrix is the unstable equilibrium point at which the pendulum balances upside down. Note that the different types of motion described here are mutually exclusive, i.e. chaotic orbits never cross over into the islands of stability. This has the important effect that chaotic motion is indirectly affected by the presence of stable structure in the SOS. The central reason why the SOS is introduced in this chapter is that it allows us to form a bridge between the discrete electomagnetic modes of the cavity and their measurable emission characteristics. Emission means coupling to the environment and hence appears as a dissipation mechanism in the internal cavity dynamics, as illustrated in Eq. (6.9). Recalling Section 6.2, the emission from a dielectric cavity is governed foremost by Fresnel's formulas, which give a wavelength-independent relation for the reflectivities rv along any ray path, determined only by the local angle of incidence Xv at reflection v; in the example of the Fabry-P6rot cavity, nothing else is needed to determine the decay rate y of a cavity mode, cf. Eq. (6.8). Now we note that the SOS shows the angle of incidence on the vertical axis, and hence the Fresnel coefficient of any given reflection can be read off with ease. In particular, the critical angle for TIR, Eq. (6.1), is represented in the SOS as a horizontal line. When this line is crossed from above (in absolute value), refractive escape becomes possible. The SOS then tells us which combinations of initial conditions give rise to ray trajectories that eventually reach this classical escape window in phase space.

6.12 Uncertainty Principle To complete the bridge between a cavity mode and its corresponding set of initial conditions in the ray dynamics, we have to find a way of projecting quasibound fields onto the SOS. To this end, let us briefly discuss some aspects of how mode fields can be measured. Individual quasibound states can be studied in great detail in microlaser experiments [16, 96-98], because one can make spatially and spectrally resolved images

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of the emitter under various observation angles. As is evident from Figure 6.4, the far-field intensity depends on the polar angle 0 of the detector relative to some fixed cross-sectional axis of the object (say, the horizontal axis). Instead of simply measuring this far-field intensity, however, one can also ask from which points on the cavity surface the collected light originated--i.e, an image of the emitter can be recreated with the help of a lens. The dielectric will then exhibit bright spots at surface locations 4~whose distribution in the image may change as a function of observation angle 0. The two variables 4~ and 0 are conjugate to each other, because 0 measures a propagation direction whereas ~b is a position coordinate of the cavity. The conjugacy between ~b and 0 means that they are incompatible, in the sense that they obey an uncertainty relation: the field distribution as a function of 4~ in the image can be deduced from the far-field distribution in 0 by a Fourier transformation, and that is the function performed by the imaging lens. But quantities related by Fourier transformation cannot simultaneously have arbitrarily sharp distributions. Physically, in an imagefield measurement a large lens is needed to get good spatial (qS) resolution, but this leaves a larger uncertainty about the direction 0 in which the collected light was traveling [98]. If we could plot the measured intensity as a function of both q5 and 0, we could generate a two-dimensional distribution similar to the SOS of the preceding ray analysis. In fact, knowing the surface shape, it is only a matter of trigonometry and the law of refraction to transform the pair (4~, 0) measured on the outside to (~b, X) inside the cavity and hence make the analogy complete. But if one has already calculated the wave field of a quasibound state, what is the advantage of representing it in this phase space rather than in real space? The answer is that the real-space wave patterns often obscure information about the correlations between the two conjugate variables (q~, sin X). As mentioned above, the SOS which plots these variables would allow us to understand whether a given mode is allowed to emit refractively; and if so, we can furthermore determine at what positions on the surface and in which directions the escape will preferentially occur. This is determined by the joint distribution of (4~v, sin Xv) over all reflections v encountered by the rays corresponding to the given mode [51].

6.13 Husimi Projection To begin a phase-space analysis, we now return to the question of how to extract correlations between conjugate variables from a wave field. This can be illustrated by analogy with a musical score: any piece of music can be recorded by graphing the sound amplitude as a function of time.

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On the other hand, musical notation instead plots a sequence of sounds by simultaneously specifying their pitch and duration. These are conjugate variables because monochromatic sounds require infinite duration [99], but their joint distribution is what makes the melody. The reason why musical scores can be written unambiguously is that they apply to a shortwavelength regime in which the frequency smearing by finite "pulse" duration goes to zero (in this sense, all music is "classical"...). In the same way that musical notes are adapted to our perception of music, phase-space representations of optical wave fields project complex spatial patterns onto classical variables relevant in measurements. This reasoning is familiar from quantum optics as well: there, amplitude and phase of the electromagnetic field are conjugate variables, and their joint distribution is probed in correlation experiments [100, 101]. A possible way of obtaining a phase-space representation is the Husimifunction, obtained by forming an overlap integral between the relevant quantum state and a coherent state corresponding to a minimum-uncertainty wave packet in the space of photon number versus phase. Here, we want to extract the same type of information about the joint distribution of the conjugate variables 4~, sin X relevant to an optical imaging measurement on a cavity mode, as described above. By projecting the electromagnetic field onto the SOS, measurable correlations are revealed which can be compared to the classical phase-space structure. This is another realization of a Husimi function; the examples mentioned above differ essentially only in the actual definition of the coherent state basis onto which the wave field is projected [102]. Motivated by the above remarks on measurement of emission locations versus detector direction, we now note that the coherent states with which the cavity modes should be overlapped are in fact the Gaussian beams [2]. The fundamental Gaussian beam 7tG has the property of being a minimum-uncertainty wave packet in the coordinate x transverse to its propagation direction z, evolving according to the paraxial (Fresnel) approximation in complete analogy with a Schr6dinger wave packet, 7tG(X,Z)_

1 1 [ X2 1 (2~) 1/4 v/a + (iz)/(2kcr) exp - 4 a 2 -~-2iz/k exp(ikz)

(6.32)

The "waist" of the beam at z -- 0 is ~, and the angular beam spread is A 0 - arctan

~

kcr

(6.33)

In the imaging setup, ~ determines the spatial resolution. Only beams with angular spread smaller than a maximum A0 are admitted by the aperture,

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213

and hence features smaller than the corresponding a are unresolved. Clearly, the uncertainty product a A0 vanishes for high wavenumbers. We are concerned with the internal cavity, and thus do not want the definition to contain the particular leakage mechanism (e.g. the law of refraction). Therefore, imagine that our detector could be placed inside the cavity, close to its boundary. Given the internal field ~int of the mode, we then make a hypothetical measurement by forming the overlap with a minimum-uncertainty wavepacket OG of width a, centered around a certain value of 4) and sin X. The form of 7*0 is analogous to Eq. (6.32), but transformed to polar coordinates because our position variable is an angle, 4). The radial coordinate can be eliminated because we constrained our detector to lie on the cavity surface. Although individual Gaussian wave packets are not good solutions of the wave problem inside the cavity, they are always an allowed (though over complete) basis in which the field can be expanded. This is all we require in order to obtain the desired, smoothed phase-space representation of ~int: The value of the overlap integral, H(qS, sin X) -= I(~rintl~a)12

(6.34)

as a function of 4) and sin X can (with some caveats) be interpreted as a phase-space density, and is the desired Husimi projection; the parameter a in Eq. (6.32) should be chosen so as to optimize the desired resolution in 4) and sin X, i.e. the "squeezing" of the minimum-uncertainty wave packets which probe ~int. Details on the definition, properties and applications of Husimi functions on the ray phase space are found in References [8, 97, 103-105, 107]. The price we pay for obtaining a quasi-probability distribution is that Eq. (6.34) discards all phase information. Figure 6.6 shows how the prescription Eq. (6.34) maps the mode of Figure 6.4(c) and (d) onto the classical ray dynamics. Although the chaotic trajectories corresponding to (d) show some accumulation near certain paths in real space, only the Husimi plot reveals what combinations of 05 and sin X occur in the internal field. This in turn determines at what positions refractive ray escape is expected, and in which directions the light will radiate.

6.14 Constructive Interference with Chaotic Rays In this section, we return to the "mystery of the missing modes" with which Section 6.8 confronted us: Weyl's density of states is asymptotically the same for all cavities of the same volume, even if we cannot classify their

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Fla. 6.6. The Husimi projections of states (b) and (d) in Figure 6.4 are shown in (a) and (b), resp., as grayscale images (dark for high intensity) superimposed onto the SOS. In (a), the field is localized in the chaotic boundary layer of the bouncing-ball island, with strong maxima on an unstable periodic orbit. Its star shape is shown in the bottom center inset, displaying ray escape according to Snell's law is as thin arrows. In (b), the cavity is the same as in Figure 6.5(b), but the SOS shows a different set of ray trajectories, forming a homoclinic tangle (interweaving black lines). The rightmost inset shows the unstable periodic orbit from which the tangle emanates. Its invariant manifolds guide the transport of phase-space density to the refractive escape window, I sin XI < 1/n ~ 0.377 (dashed horizontal lines).

modes according to some numbering that follows from integrability, or at least paraxiality. Figure 6.4 shows that a given mode does not cease to exist as integrability is destroyed with increasing e, but evolves smoothly into a new and complex field pattern with a well-defined spectral peak that can even become narrower with increasing chaos in the ray dynamics. As noted in Section 6.10, the "mystery" is how at large shape deformation the increasingly numerous chaotic rays, which crisscross the cavity in a pseudor a n d o m way and even diverge from each other, can give rise to constructive interference, and hence to a resonant mode. In Section 6.9, the quasiclassical quantization approach led from the ray picture to the wave fields. Now, we follow the opposite direction and use the Husimi projection of a given cavity mode to perform a "dequantization" which leads to the ray picture. In this way, the novel physics of chaotic modes can be described without much technical detail. By superimposing the Husimi projection of a mode onto the SOS for the same deformation, we can identify classical structures on which the mode is built. We have so far encountered two types of structure, cf. the caption of Figure 6.5: (I) invariant curves such as those formed by W G rays, permitting E B K quantization as discussed in Section 6.9; (II) stable islands around periodic orbits, for which the paraxial approximation of the previous chapter [86] can be used. In both cases, a finite (and usually small) number of wavefronts suffices to achieve the constructive interference without which no cavity mode can form. The mode shown in Figure 6.6, on

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the other hand, overlaps with neither of these nonchaotic phase-space components; it is a chaotic mode. One possible way to understand that chaotic modes are possible at all, is to realize that even the chaotic sea is not structureless. The classical objects of crucial importance for an understanding of chaotic modes are the unstable periodic orbits of the cavity. They give rise to a third type of invariant sets in the ray dynamics: (III) stable and unstable manifolds. These are one-dimensional curves which, like the structures (I) and (II), have the property that rays launched anywhere on the manifold will always remain there. Along these manifolds, trajectories rapidly approach or depart from a given periodic orbit, such as the one shown in Figure 6.6(b, right inset). A nonperiodic ray trajectory which moves along one of the corresponding manifolds is displayed in the top center inset. The stable and unstable manifolds form an intricate web, the "homoclinic tangle," which was already recognized by Poincar6 as the cause of severe difficulties in calculating the dynamics [5]. Part of this tangle is shown in Figure 6.6(b), forming interweaving lobes which play an important role in controlling the transport of phase-space density. A discussion of this "turnstile" action and the relation to the stability of the orbits from which the tangle originates is found, e.g. in References [1, 108]. As the Husimi projection in Figure 6.6(b) shows, the wave intensity is guided along the invariant manifolds of the unstable periodic orbit [110], imparting a high degree of anisotropy onto the internal intensity and on the emission directions. Particularly strong Husimi intensity is found at the reflection points of the periodic orbit shown in the inset to Figure 6.6(b); this corresponds to the high-intensity ridges in the real-space intensity of Figure 6.4(d). Such wave function scarring [109] is a surprising feature, because a wave field concentrated near the origin of the homoclinic tangle, i.e. at the unstable periodic orbit itself, spatially appears similar to a sequence of Gaussian beams, despite the fact that paraxial optics requires the underlying modes to be stable. Even if we do not start from this paraxial point of view, but follow the historical developments in quantum chaos, scars seem to run counter to the early conventional wisdom that asserted chaotic modes should posses a random spatial distribution [108]. In a recent study on lasing in a large, planar laser cavity with no stable ray orbits [111], laser operation with focused emission was observed and explained by wave function scarring. This phenomenon is of interest not only from the applied point of view, but also because a complete understanding of the quasiclassical theory for modes corresponding to the invariant manifolds of type (III) is as yet missing [112, 113].

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6.15 Chaotic Whispering-Gallery Modes The lifetime of the quasibound states in Figure 6.4(d) yields a Q factor of Q - k/K = 700, which is of the same order as that of a high-power microlaser [97] that was demonstrated recently as an application of cavity design using the concepts introduced in Section 6.11. The bow tie-shaped mode encountered there was of paraxial type, centered around a stable periodic ray orbit. If low lasing thresholds and hence higher Q factors are desired, we recall from Figure 6.7 that W G M s are preferred. As shown in Figure 6.2, W G M s in partially chaotic cavities display high Q and strongly anisotropic emission patterns. Comparing the nonchaotic case of Figure 6.2(a) and the chaotic cavity of Figure 6.7, we indeed find stronger focusing in the latter case. Chaotic W G M s are among the few types of chaotic modes for which an approximate quasiclassical description along the lines of Section 6.9 is possible. They correspond to ray trajectories which are part of the chaotic sea but circulate along the cavity perimeter many times before exploring other phase-space regions, cf. Figure 6.5(b), bottom. There is thus a separation of time scales between the fast W G circulation and a slow deviation from W G behavior which inevitably ends in chaotic motion. This makes it possible to use an adiabatic approximation [114] in which the "diffusion" away from W G motion is neglected for times long enough to contain many round-trips of the ray.

FIG. 6.7. A WGM in the quadrupole cavity at e =0.104, at dimensionless wavenumber kR = 32.695 and width KR = 0.0025. The focused near-tangential emission from the points of highest curvature is explained in the ray picture (b). The ray-wave connection is illustrated in the Husimi plot (c), superimposed onto the SOS (only sin X> 0 is plotted to magnify details). The stable and unstable manifolds of the rectangular periodic orbit funnel the rays downwards to the escape condition along two narrow pathways. The resulting emission therefore is not attributable to a single periodic orbit, but to a family of (generally aperiodic) trajectories following the invariant manifolds shown here.

CHAOTIC WHISPERING-GALLERY MODES

217

As a consequence, one can formulate approximate quantization conditions for chaotic WG modes by ignoring the chaotic dynamics [51], leading to equations of the EBK type, Eq. (6.30). The approximate nature of this quantization can be recognized in Figure 6.7(a), which differs from the ellipse-shaped cavity of Figure 6.2(a) in the fact that a small but nonvanishing field persists in the cavity center. This indicates that there no longer is a well-defined caustic, as we required in Section 6.9. Nevertheless, the Husimi distribution (grayscale) in Figure 6.7(c) condenses approximately onto a one-dimensional curve (the shape of which can be given analytically within the adiabatic approximation [51]), following the homoclinic tangle (the SOS in Figure 6.7(c) is vertically expanded compared to Figure 6.6(b)). The location of this adiabatic invariant curve is determined by the EBK quantization; its minima are seen to approach the TIR condition. As indicated in Section 6.14, chaos does become important when the emission properties of a chaotic WG mode are concerned. It is the deviation of the rays from the above adiabatic assumption which allows an initially confined ray to violate the total internal reflection condition after some time, and hence escape refractively, cf. Figure 6.7(b). In Figure 6.7(c), this escape results from wave intensity leaking across the TIR condition near 4~= 0, re. This can be reproduced within a ray simulation by launching an ensemble of rays on the adiabatic invariant curve, and recording the distribution of classically escaping rays. This implements the program outlined at the end of Section 6.4. The stochastic branching of a ray trajectory into transmitted and reflected parts, also known as ray-splitting [48], could be summed up as a geometric series in the Fabry-P~rot example, Eq. (6.6). In the oval cavity, Monte Carlo simulations for large ensembles of incident rays are required [51]. This ray approach has been applied in scattering problems along the lines of Eq. (6.10) [49]; the value and applicability of ray considerations in scattering from chaotic optical cavities was demonstrated in Reference [50]. The difference to the quasibound state approach is that our incident rays are launched inside the cavity, not outside of it. Because chaotic diffusion makes classical escape from WGMs possible in the first place, the ray approach in fact becomes more useful at large deformations where chaos increases. That is just the regime where exact scattering calculations encounter difficulties. Starting from the ray approach to emission directionality and decay rates, one can then successively introduce corrections at higher orders in 2. In particular, in Reference [51], the opennesss of the system was taken into account in the semiclassical quantization step by introducing additional phase shifts into the EBK equations. The reason is that penetration into the outside region, as mentioned in Section 6.2, creates an enlarged, effective cavity boundary

218

2D MICROCAVITIES: T H E O R Y A N D E X P E R I M E N T S

which lengthens the ray path lengths. This "Goos-Haenchen effect" [15] is well known from plane interfaces as well. The effective cavity is actually of slightly different shape than the physical geometry. This has the consequence that even the elliptic cavity of Figure 6.2, which for impenetrable boundaries is integrable, becomes nonintegrable as a dielectric body [46, 47].

6.16 Dynamical Eclipsing In the 2 -+ 0 limit, the internal dynamics of the ellipse displays no chaos, cf. Figure 6.5(a). This makes it an important test case with which to compare whenever we want to identify fingerprints of chaos. The power of such comparisons has been illustrated in an experiment on prolate, dyedoped lasing microdroplets [98]. The geometry of the droplets was rotationally symmetric around the polar axis, which is also the major axis. If one analyzes only the light propagating in a plane containing the two poles, then the cavity is effectively formed by the 2D axial cross section whose shape is an oval of the type we have been discussing above, cf. Figure 6.5. The experimental technique of recording with a CCD camera an image of the micro-object at various observation angles 0 (measured with respect to the polar axis of a spheroidal droplet) reveals prominent bright and dark regions along the cavity rim, from which the position and exit angle of the laser emission can be extracted simultaneously. As discussed in Section 6.3, the lasing modes in low-index materials such as droplets should be of WG-type. The experiment illustrates both the focusing property, as well as the robustness of WGMs. When the lasing droplets are imaged from the side perpendicular to the polar axis (0 = 90~ one thus expects emission directionality of the type shown in Figures 6.2(a) or 6.7(a), showing emission from the points of highest curvature (i.e. the poles). However, neither case describes the observed emission correctly. The droplets appeared dim at the poles instead of being brightest there. This counter-intuitive phenomenon can be explained by the phase space structure shown in Figure 6.5(b). An essential role is played by the small islands belonging to the stable four-bounce diamond-shaped orbit depicted in the figure. Its effect is that WG rays of the type shown below the SOS are prevented from reaching the high-curvature points (~ = 0, re) at angle of incidence sin X = 0.65, because chaotic and stable motions are mutually exclusive (cf. Section 6.11). Now XTm = arcsin0.65 just happens to be the critical angle for the droplet-air interface; hence, near-critical escape from the highest curvature points is ruled out for chaotically moving WG rays in the droplet. In Figure 6.5(b), it is impossible to see from the ray

CONCLUSIONS

219

trace in real space that the combination 4~= 0, X = X T I R will never occur. Likewise, this information is by no means obvious from the wave equation itself. Only the phase-space analysis using the SOS reveals the small islands that cause this effect, which has been called dynamical eclipsing. The emission profile is determined by the phase-space structure in the vicinity of the emission line sin XTIR, and therefore we do not detect dynamical eclipsing in the higher index example of Figure 6.7. A glance at Figure 6.5(a) makes it clear that the effect will never occur in the ellipse, because the relevant islands do not exist there.

6.17 Conclusions The subtle difference in shapes between ellipsoid and quadrupole raises the question: How short does the wavelength have to be compared to the cavity dimensions, in order to be able to resolve such differences in geometry? Clearly, the droplets are so large compared to the optical wavelength (35 gm diameter for an equivalent-volume sphere) that we are deep in the short-wavelength limit. What the above example teaches us is that it is not the structure in real space that has to be resolved by the waves, but the structure in phase space. The uncertainty principle will limit the size of the islands that can have a noticeable effect on the cavity mode structure, but the important thing to realize is that small differences in cavity shape can lead to large differences in phase-space structure. As a result, the phase-space effects introduced in this chapter turn out to be observable even in cavities that are not much larger than the wavelength. This is an extension of the claim made for integrable systems in Section 6.9, that quasiclassical methods often extend all the way to the longestwavelength modes. Future work on nonintegrable dielectric cavites can proceed in various directions: our understanding of the classical ray transport in partially chaotic systems has to be explored further, and the systematic wave corrections to these ray ideas must be investigated. Two such corrections which are of particular importance in chaotic systems are dynamical localization [51, 114, 115] and chaos-assisted tunneling [107]. Dynamical localization is an interference effect that suppresses decay rates below the value expected from ray simulations as outlined in this chapter. It becomes more pronounced in smaller cavities, and in fact helps maintain high Q-factors in the presence of chaotic ray dynamics. Localization arises in many areas of physics as a name for wave corrections to a classical diffusion picture. In our case, the diffusion happens in the chaotic phase space of the ray dynamics. Chaos-assisted tunneling, on the other hand,

220

2D MICROCAVITIES: THEORY AND EXPERIMENTS

acts to enhance radiation losses of nonchaotic WGMs at small deformations when chaos pervades only the low sin X regions of the SOS. It also leads to a coupling between WG wavefronts encircling the cavity in counterclockwise and clockwise sense of rotation, e.g. between the extreme bottom and top of the SOS in Figure 6.4(a). In the ray picture, spontaneous reversal of rotation direction is forbidden for WG trajectories on invariant curves; the mechanism at work here is tunneling in phase space. The phase-space approach to microcavity electrodynamics provides a means of classifying the wide variety of modes in cavities where "good quantum numbers" do not uniquely enumerate the spectrum. Semiclassical methods provide the connection between individual modes and manifolds in phase space, such as the caustics in Section 6.9, belonging to invariant curves in the SOS. Open questions remain about how to establish this connection in the general case when caustics are broken up by ray chaos. The particularly important case of WGMs, however, is amenable to approximate treatments which permit quantitative predictions. Many systems in which strong matter-field coupling enables quantumelectrodynamic studies, are actually large enough to be in the quasiclassical regime: examples are semiconductor domes [90] (where the effective wavelength is shortened by a high refractive index), or silica microspheres (cf. References [21, 23] and citations therein) where atoms close to the dielectric interface interact evanescently with WGMs whose angular momentum according to Eq. (6.31) is m--~ 103. At all levels of microcavity optics--quantum or classical, linear or nonlinear--the fundamental task is to establish a modal basis, and the methods we sketched in this chapter accomplish this in a way that provides physical insight. References

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7. MICROPARTICLE PHOTOPHYSICS: FLUORESCENCE MICROSCOPY AND SPECTROSCOPY OF A PHOTONIC ATOM Stephen Arnold

Polytechnic University, Brooklyn, New York, USA

Stephen Holler Los Gatos Research, Mountain View, California, USA

7.1 Introduction Orbital motion is one of the most familiar physical models in nature. Newton's gravitational interaction describes planetary and intragalactic motions in terms of orbits. The Coulomb interaction allowed Bohr to construct a simple description of an atom in which the electron orbits the nucleus, with a quantized angular momentum. In the world of photonics, a photon orbiting within a microsphere [1] (Fig. 7.1) has many of the mathematical properties of an electronic atom. Although others have described this state as a Mie resonance, or whispering gallery mode, we will alternatively call it a photonic atom mode [2]. Our simplistic picture (Fig. 7.1) neglects diffraction and polarization, but it shows how total internal reflection can trap a photon. Surprisingly this sort of trapping works on a small scale (a few wavelengths) and produces a microcavity with a long confinement time. Equivalently, in the frequency domain, this leads to an extremely compact high-Q filter, with a myriad of applications: integrated optical filters for wavelength division multiplexing applications [3, 4], sensitive sensors for chemical and biological agents,

FIG. 7.1. Photonic atom model. 227 EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES

Copyright 9 2002 by Academic Press

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MICROPARTICLE PHOTOPHYSICS

as well as a variety of optical microelectromechanical machining ap tions (e.g. microspectrum analyzer [5]). Although some of these applic~ will be discussed at the end of this paper, our goal is to understanc spontaneous emission occurs from a molecule within such a structure Our interest in spontaneous emission in microparticles stems from th that the photon within a meso-optic domain (size of about 1 t~ wavelengths) plays a profound role in contemporary physics and techn [6-8]. It is a subject that has come to be known as cavity qu~ electrodynamics. The name is appropriate. Both fluorescence and R are spontaneous processes, and are therefore the result of zero. fluctuations. In our earliest courses in quantum mechanics the ide~ never discussed. We were taught to calculate "stationary excited stat, hydrogen, etc. But of course, they are really not stationary. In a sens~ of the problem was left out. We never allowed a quantized electroma field to interact with the atom. The effect of photonic atom modes on spontaneous emission , microspheres is best demonstrated by observing the fluorescence impurity fluorophores within a suspended liquid droplet. Witne, fluorescence spectrum of rhodamine 6G in a glycerol droplet ~10 diameter at room temperature (Fig. 7.2) [9]. The familiar smooth rhod 6G spectrum bristles with new peaks. They are associated molecular e states coupling their energy into high-Q modes of the sphere. The outline for the rest of this paper is as follows. We will look at th angular momentum modes of a microsphere by using analogies from

lOO 80

and excited state ]e), connected by an electric dipole transition with a transition energy of hCOA and transition moment /x. The atomic Hamiltonian is J'~A--h(-~ Cr+Cr-, with ~r+= le><e[ the atomic rising and lowering operators, respectively. The Hamiltonian has eigenvalues 0 and hCOA. The decay rate of the atomic dipole is given by ?/. Assuming that the transition is purely radiatively broadened, the natural linewidth (FWHM) is 2?/, as given by the Weisskopf-Wigner theory [8]: 23/-

1 4/xzco3 4roe0 3 hc 3

(8.12)

The parameter ?/ can be interpreted as the coupling constant of the atomic dipole to the continuum of empty free-space modes. If a cavity is placed around the atom, the mode supported by the cavity should be treated separately. As long as the solid angle over which this mode extends is small, one can assume that the coupling to the continuum of free-space modes is almost unchanged. The coupling to the near-resonant cavity mode is given by S2(n,r)- 2v/-ng(r), where n > 0 now determines the number of energy quanta in the combined atom-cavity system and /3,2e)c g(r) -- V2h~ ~ V 7t(r) -= go O(r)

(8.13)

is the single-photon coupling constant [8]. Apart from a factor two, it is the coupling constant S2(n,r) for n = 1, i.e. when the cavity contains one photon and the atom is in its ground state or when the atom is excited and the cavity has no photon. Note that g is a coherent coupling constant and not an incoherent decay rate like the spontaneous emission rate. The atom-cavity coupling introduces a term "]-[CA = hg(r)( ~+a + ate-) in the Hamiltonian, where the rotating wave approximation is made. Because of the interaction, new eigenvalues and eigenvectors of the total Hamiltonian, ~JC = ~ c + ~-~A nt- "]-{CA, must be found. The system is named after Jaynes and Cummings, who first studied it in detail, shortly after the invention of the laser [42]. Note that the Jaynes-Cummings Hamiltonian does not take into account losses in the cavity and caused by

BASIC THEORY

265

atomic decay. After diagonalization, the new eigenvalues are given by 7-{jclg, 0) - 1 hcoclg, 0)

(n - 0)

7-/jc]n, + , r ) - h[coc(n + 8 9 21-AcA:klaR(n,

ACA, r)][n, + , r )

(n _> 1) (8.14)

with f2R(n, Aca, r ) - v/f22(n,r)+ ACA2 and A C A - COG- OA, the detuning between the cavity and the atomic resonance frequency. The new eigenenergies form a ladder of doublets. The distance between each consecutive doublet of the ladder is ha)c, and each doublet has a splitting of hS2R(n, ACA, r). The eigenenergies of the two bare systems, the atom and the cavity, and of the coupled system are plotted in Figure 8.3. For n _> 1, the new eigenstates or dressed states are a superposition of the unperturbed (bare) atomic and cavity product states as follows: (In,+,r))_(sin0n) In, - , r ) cos0n

[g,n) +

(cos0n) -sin0n

le, n - 1)

(8.15)

where the mixing angle, On, is defined by tan 20n - -S2(n' r) ACA

0 _< 20n < n

(8.16)

FIG. 8.3. Energy levels of the uncoupled (left) and strongly-coupled (right) atomcavity system for the case of coc > COA.The splitting of the states In, 4-, r) depends on the position of the atom and is large in the center of the cavity.

266

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

For an atom initially in the ground state and n photons in the cavity, the probability of finding the system in one of the bare states, Ig, n) or le, n 1), can be found by solving the Schr6dinger equation, and is cos2(ls2Rt) or sin2( 89 respectively, showing characteristic Rabi oscillations at frequency S2R. The physical meaning of the oscillations is quite clear: the cavity field stimulates an excited atom to emit its excitation as a photon into the cavity, where the photon is stored until the atom reabsorbs it. Then, of course, the atom can emit it again and so forth. This oscillatory exchange of energy between the atom and the cavity can already occur for a single excited atom in a cavity containing initially only a vacuum field. Hence, the single-photon coupling constant, g = 82(1)/2 for ACA--0, is sometimes called half the vacuum R a b i frequency [43]. As already mentioned in the introduction, a consequence of the oscillatory energy exchange is that the notion of spontaneous emission as an irreversible decay process looses its meaning. The Jaynes Cummings system as described above provides a starting point for further studies of the atom cavity system. For a realistic description of an experiment in the optical domain, the losses of both the cavity and the atom must be taken into account. We assume that the dissipative coupling to the environment is only a weak perturbation and that the environment has no memory, i.e., we apply the Born and the Markov approximations, respectively [44]. In this case, the environment serves as a bath with constant temperature. The temperature is assumed to be zero, which is a good approximation for the optical regime, where the thermal excitation of the modes is negligible. To compensate for the dissipation, the cavity field is driven by an external laser, as has already been discussed above for the case of an empty cavity. The time dependence of the coupled atom-cavity system can be found by solving its quantum master equation. If the excitation of the atom, Pe - (o'+o'-), is small, an analytic solution can be found. As derivations can be found elsewhere (see, e.g., Reference [16]), we will o n l y state the results. For example, the mean number of photons in the cavity is (n(r))

(at a)

772 A2 + V2 IA(r)l 2

(8.17)

where AA = COL COA is the difference between the laser frequency and the frequency of the atomic transition and A(r) -- Vtc + g2(r) - AAAC nt- i(Acy + AAtc)

(8.18)

The output power of the cavity is Pout(r) = ( n ( r ) ) h c o c T c / ( 2 Q

(8.19)

267

BASIC THEORY 1.0

0.8 c-

.9

u~ 0.6

._~ E E

0.4

s,_

0.2

0.0 -30

-20

-10

0

10

20

laser detuning A [2~ MHz]

30

FIG. 8.4. Theoretical transmission Pout/Pmax of a high-finesse cavity as a function of laser detuning A = AA = A c f r o m the common resonances of the free-space atom and the empty cavity, in the limit of weak excitation. The transmission curves are normalized to the maximum transmission of the empty cavity. The solid line is the transmission for an empty cavity with g = 0. The width (FWHM) of the transmission peak is given by 21(. The dashed line shows the transmission for a single maximally coupled atom with g = go. The parameters are as in Reference [54]: (go, V, ~c) = 2~x (16, 3, 1.5) MHz. In this case, the transmission on resonance (A = 0) is more than 3 orders of magnitude smaller than in the empty cavity case. At A = +go, however, the transmission is increased and two peaks have appeared.

As an illustration, Figure 8.4 shows the normalized output power of a cavity without and with an atom at an antinode as a function of the laser frequeny for (go, V,K)= 2re x (16,3, 1.5)MHz. Unless otherwise stated, these parameters are u s e d in most of the experiments discussed in this chapter. F o r the parameters of Figure 8.4, the empty cavity shows a resonance with a width of 2K, as is indicated by the solid line in the figure. Obviously, the presence of a single atom strongly coupled to the light field dramatically changes the transmission spectrum, as is shown by the dashed line in the figure. Because of the presence of the atom, two new peaks appear at the normal-mode frequencies of the combined atom-cavity system [13, 30, 43, 45-50]. The two peaks can be interpreted as an Autler-Townes splitting caused by the strong single-photon cavity field and probed by the weak pump laser. The distance between the two peaks for a maximally coupled atom is 2g0. The width of the new peaks is a weighted average of the widths of the cavity resonance and the atomic resonance [51]. Obviously, for the two peaks to be separated completely, the vacuum Rabi frequency, 2g0, must be larger than the dissipative decay rates, 2 ? / a n d 2to. The condition go > (?/,K) defines the strong-coupling regime of cavity QED.

268

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

Its realization requires a long cavity lifetime and a small cavity volume. In fact, Eq. (8.13) shows that the atom-photon coupling constant continuously increases with decreasing cavity volume. Equation (8.19) together with Eqs. (8.17) and (8.18) is the basis on which the presence of an atom can be detected. Also, information on the position of the atom can be obtained, as the transmission of the cavity depends on the coupling of the atom which, in turn, depends on the atomic position [52, 53]. Note that (n) as defined in Eq. (8.17) is linear in the pump intensity, as long as the assumption of low atomic excitation is fulfilled. We emphasize that Eqs. (8.17 8.19) are only valid for small excitation of the atom. Within that approximation, the excitation probability of the atom is Pe(r)

r/2 g2 (r) g2 (r) IA(r) 2 -= (n(r)) A2 + ?/----------5"

(8.20)

A practical validity check is that Pe 1/2, by inserting the expressions for go, g and K from Eqs. (8.13), (8.12) and (8.6), respectively. Assuming O)A ~--O9C, one finds the result 2C1 - 3Q~.3/(4rc2V) > 1. This expression is exactly the one found in the introduction on the basis of some intuitive arguments. Note that, apart from the wavelength, the single-atom co-operativity parameter does not depend on any atomic property. In fact, the ratio (n)/(n0) depends only on the mirror reflectivity and the cavity-mode waist. Both are design parameters and, hence, adjustable by the experimentalist to optimize atom detection. When the laser-cavity detuning is not zero, the transmission of the empty cavity is lower than the maximally possible transmission. According to Eq. (8.17), it depends now on the various detunings whether the presence of an atom causes a decrease or an increase of the cavity transmission. For the special case A AA c ,-~ g2(r), the laser is near resonant with the transition from Ig, 0) to either I1, ,r) or I1, + ,r), and the transmission increases. Classically, this can be understood as the effect of the additional phase shift that the atom causes, shifting the cavity into resonance with the pump laser. Whether an atom can be observed or not, also depends on the available signal divided by the noise of the detector observing the transmitted light. This noise is usually dominated by shot noise caused by the finite photon flux emerging from the cavity. For a given cavity, the photon flux cannot be increased indefinitely, because the intracavity intensity must be kept below the saturation limit of the atom. Transit signals of single slow atoms passing a high-finesse resonator have been seen in two experiments [17, 54] and [55, 56]. As an example, Figure 8.5 shows the transmission of a small cavity with a waist w0 = 29 pm and finesse F - 4 . 3 x 105 as a function of time for a wavelength of ) ~ - 780 nm. In the experiment [54], rubidium atoms are captured in a magneto-optical trap and launched towards the cavity in an atomic fountain. This allows to inject single atoms with an adjustable speed into the volume of the high-finesse cavity. Depending on the operating parameters, the passage of an atom leads to a dip or a peak in the transmission. The average duration of these transit signals is given by the velocity of the atoms and the waist of the cavity mode.

270

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

a)

1

A4 iv

0

0.0

0.5

11o

11~

time [ms]

21o

b)

'

I l

2 v

1 0

o.o

o!5

11o

115

21o

time [ms]

FIG. 8.5. Intracavity photon number while several 85Rb atoms pass the cavity mode one after the other. (a) With (AA, Ac) = 2re x (-26.5, 0) MHz, resulting in a decrease of the transmitted light. (b) With (AA, Ac) = 2re x (-26.5, - 4) MHz, resulting in an increase of the transmitted light on passage of an atom. The transmission decrease in (a) as well as the increase in (b) are the result of dispersion, not absorption. Other parameters in this experiment are (go, y, to) = 2re x (16, 3.0, 1.4) MHz, and the atoms are injected into the cavity with a speed of 0.8 m/s. The cavity that was used is shown in Figure 8.13.

An interesting observation in an experiment like the one in Figure 8.5(a) was that the number of dips depends in an unexpected way on the c o m m o n detuning of the laser and the cavity from the atom. It was found that the number of atoms detected close to an antinode, where the atoms produce a deep dip in the cavity transmission, is larger if the laser is red detuned from the atomic resonance than for blue detuning [17]. This observation could be explained by taking into account the effect of light forces, which we have left out of the discussion, so far. However, the motion of slow atoms is considerably influenced by light forces caused by the cavity field, as the energy associated with the single-photon Rabi frequency can exceed the kinetic energy of a slow atom. In this case, an atom can quickly move into or out of the region of strong coupling close to an antinode, depending on the laser detuning. To obtain a full understanding of the transit signals and the atom's dynamics in a high-finesse cavity, we now turn to a discussion of light forces in this fundamental system.

8.3 Cavity-Mediated

Light Forces

Our present understanding of light forces is based on the notion of Einstein that a quantum of light carries a m o m e n t u m of hk. M o m e n t u m conservation

CAVITY-MEDIATED LIGHT FORCES

271

in the absorption and emission of the photon by a particle is the basic mechanism behind radiation pressure. It is a small effect for thermal light sources. Nevertheless, it could already be demonstrated around 1900 for macroscopic objects [57-59] and in the 1930s for atoms [60]. It was, however, not until the availability of narrow-band lasers in the 1980s that light forces could be used efficiently for manipulating atomic samples. This triggered the development of a variety of very powerful laser cooling and trapping methods. These methods rely on many subtle effects, depending on the properties of the light and the atom (for a review, see, e.g. References [61, 62]). With the new techniques, atomic gases could be cooled to record low temperatures, paving the way to quantum degeneracy of dilute gases. For an atom at rest, light forces can be divided into two classes [61]. One class contains the dissipative forces relying on spontaneous emission. For atoms, radiation pressure is an example of such a force. The acceleration can be as large as 105 m/s 2, but it vanishes rapidly with the squared detuning of the laser from the atomic resonance, 1/A 2. Hence, it is arbitrarily small when the laser is detuned far enough. However, an intense light field can still induce a dipole in a polarizable particle with a magnitude proportional to 1/AA. As the potential energy of a particle with polarizability o~ in an electric field E is --olE2/2, it experiences a force in a light field exhibiting an intensity gradient. This force is called the dipole force. It is a reactive or conservative force and belongs to the second class. It results from stimulated exchange of photons between light beams propagating in different directions. The theory describing light forces in free space is a good starting point for the discussion of light forces in a cavity. There are important differences, though. The most obvious is that caused by the power buildup in a cavity, the light forces in a cavity are enhanced together with the intracavity intensity. However, there is more to the influence of a small cavity than just power buildup. As discussed in Section 8.2 above, an atom can change the amplitude of the intracavity field dramatically, depending on its position. As a consequence, the light forces change and qualitatively new effects appear, leading to an intricate dynamical interplay of atomic motion and field dynamics. Here we want to give an overview of the light forces on a single atom in a high-finesse cavity. We will discuss the physical mechanisms and give the most important results. For the derivation we refer to the literature. We mainly follow the analytical results of References [15, 16, 22], although other approaches also exist [21, 63, 64]. Forces on a fixed atom will be considered first, and then the effect of atomic motion will be taken into account,

272

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

8.3.1 Light Forces on an Atom at Rest

The light force on an atom at rest can be derived from the expectation value of the force operator -h(Vg(r))(cr+a + atcr-), the negative gradient of the Jaynes-Cummings interaction Hamiltonian discussed above. This can be calculated once the steady-state solutions of the master equation are known [16]. With A(r) taken from Eq. (8.18), the conservative dipole force is Fdip(r) -- -hr/2AA Vg2(r) IA(r)l 2

(8.22)

In the dressed-state picture of Figure 8.3, the dipole force is the (negative) gradient of the position-dependent energy of the dressed state times the probability of finding the system in this state. The dipole force can be integrated to a potential, -r/2hAA ["Re[A(r)]'~ Udip(r) -- Im [A(r)] arctan ~i--mm~ - ~ ] ) ,

(8.23)

for all Im[A(r)] r 0, where Re[A(r)] and Im[A(r)] denote the real and imaginary part of A(r), respectively. For Im[A(r)] = 0, the potential is Udip(r)- -2rlZhAa/(Re[A(r)]) 3. I n t h e limit of low atomic saturation, the dipole potential cannot exceed hg. But as g is large in the strong-coupling regime, the potential created by a single intracavity excitation can be deep enough to trap an atom [25, 26]. This will be discussed further in Section 8.3.4. If an atom absorbs a photon from the cavity field, the atom experiences a momentum kick in the propagation direction of the photon. The opposite situation occurs when the atom emits a photon. In a standing wave, the net radiation pressure is zero, because the probability to absorb a photon from either of the two beams forming the standing wave is identical. However, the repeated absorption and emission of photons leads to a random walk in momentum space. This is characterized by a momentum diffusion constant, which is defined by D = d/dt((p 2) -(p)2), with p the momentum operator. It is convenient to split the diffusion into two contributions, D = Dse+ Odip. The first one is attributable to spontaneous emission events and is given by Ose(r) -- Pe(r)h2k2y-

~2/s

g2(r) iA(r)l 2

(8.24)

where Pe is defined in Eq. (8.20). The second contribution comes from fluctuations of the dipole force around its mean value defined in Eq. (8.22). It can be calculated by taking into account correlations of the force operator

CAVITY-MEDIATEDLIGHT FORCES

273

at different times. For weak fields, this leads to

Acy_~A__~)[ ~2. t-AAK' n ~,]

Ddip(r) -- h2(Vg(r)) 2 IA(r)l 2/']2Y (1 +4 AAg2(r~)g

(8.25)

The first term between the brackets of Eq. (8.25) coincides with the result obtained in the standard (free-space) theory of atomic motion in laser light (see, e.g., Reference [61]) in the limit of small atomic saturation, provided the mean intracavity intensity derived from Eq. (8.17) is substituted in the free-space theory. The second term between the brackets of Eq. (8.25) is a genuine cavity term [15], which vanishes in the limit of a bad cavity with z --+ co. Depending on its sign, it can enhance or reduce the dipole fluctuations compared to those Of a free-space standing wave. Note that the gradients of the standing wave pattern, Vg(r), are to a good approximation directed along the cavity axis. Hence, Odip acts predominantly along the cavity axis, whereas the spontaneous emission is directed in all directions, with a characteristic dipole emission pattern. For circularly polarized light, the total diffusion along the cavity axis is given by Dax(r) = 0Dse(r) + Ddip(r). In each perpendicular direction it is D x ( r ) - D y ( r ) Dse(r).

8.3.2 Velocity-Dependent Light Force So far we have assumed the atom to be at rest. If an atom moves, corrections depending on the velocity of the atom will appear. In free space, for example, the spatially modulated light intensity in a standing wave can lead to a nonconservative force, which can be used to cool the atom by means of a combination of the conservative dipole force and the dissipative spontaneous emission [61]. In the standing wave of a high-finesse cavity, there is the additional effect that the cavity field cannot follow fast changes of the atom's position. In fact, the field needs a time of the order of K-1 to respond and settle to the new equilibrium photon number corresponding to the new position of the atom. But the equilibrium might never be reached if the atom moves. This has an interesting consequence. Consider, for example, an atom that moves along the cavity axis. The pump laser is assumed to be red detuned with respect to the atom and the cavity. As a consequence, the intracavity field is small for an atom close to a node (g ~ 0). On its way from a node to an antinode, the coupling of the atom to the field increases, leading to an increase of the intracavity field. Because of the finite response time, the field intensity is smaller than for an atom at rest. Therefore, the atom is pulled towards the antinode less strongly than expected from the steady-state dipole force. Leaving the antinode, the

274

S I N G L E A T O M S M O V I N G I N A H I G H - F I N E S S E CAVITY

atom will be pulled backwards stronger than expected from the steady-state dipole force, as the field cannot dim fast enough. The net effect is that on top of the conservative dipole force a correction occurs. This correction is larger if the cavity field is further away from its equilibrium level. Therefore, it depends on the cavity decay time K and the velocity of the atom. The result is a dissipative force that cools in this example. There are other parameters where this force heats. The explanation is completely analogous to the cooling case described above, with the correction resulting from the finite response time of the cavity field now increasing the atoms' momentum. Note that this description makes use only of the timedependent dipole force. It is an optical cooling mechanism that does not rely on spontaneous emission by the atom! The cooling force can also be understood in the dressed state picture introduced in Figure 8.3. Consider the situation depicted in Figure 8.6(a). For these detunings, the coupled atom-cavity system is excited preferentially near the nodes where the energy of the upper dressed state of the first manifold with n = 1 is lowest. Because of the small but finite velocity of the atom, the system adiabatically follows the dressed state-energy curves and decays back to the ground state at a position where the energy of this dressed-state has increased. Hence, the kinetic energy of the atom decreases. The mechanism also works if the laser excites the lower dressed state when the atom is close to an antinode, as depicted in Figure 8.6(b). This mechanism has similarities with Sisyphus cooling in a free-space standing wave [61]. There are, however, important differences. The main difference is that the amplitude of the light field is influenced by the atom. Moreover, the cavity resonance provides an extra degree of freedom, enabling to choose optimum cooling parameters. In addition, the decay back to the ground state can be caused by a spontaneous emission of the atom, but also by the emission of a photon from the cavity. For strong coupling and large detuning from the atomic resonance, decay of the cavity field can dominate atomic losses in the form of spontaneous emission. It is in this limit that the mechanism becomes independent of spontaneous emission by the atom. Here, the cooling can also be understood as coherent scattering of photons, as is explained in Figure 8.6(c). Note that the laws of thermodynamics are not violated, as the cavity-field decay balances the entropy loss within the cavity with an increase in entropy of the transmitted light. The derivation of the velocity-dependent force, F~, is rather complex. It involves the calculation of expectation values of cavity field and atomic operators to higher order in the velocity of the atom. Again, an analytic solution can be obtained in the limit of low atomic saturation and to first order in the velocity [15, 16, 22]. We quote the somewhat lengthy result

CAVITY-MEDIATEDLIGHT FORCES

275

FIG. 8.6. Two different explanations of the velocity-dependent force for parameters that cool. (a) and (b) Time domain picture for kv < (z, y) after Reference [15]. Plotted are the dressed energy levels as a function of the position along the cavity axis for a cavity which is near resonant with an atomic transition. For cooling, the pump laser should excite the system to the bottom of either the upper or the lower dressed state, as indicated by the vertical arrows in (a) and (b), respectively. Cooling originates from the difference between the energy at which the system is preferentially excited and where it decays from. (c) Frequency domain picture for the limit of ]AcA[ --+ cx~ after Reference [18]. Plotted is the coherent spectrum of a polarizable particle, far away from its optical resonances, in a standing-wave light field. The four ways in which a photon can be scattered are indicated by the thin arrows. The atom moves up through the standing wave (indicated by the small arcs), which modulates the field the atom sees at rates +2kv, leading to two side bands in the spectrum. By choosing the cavity frequency resonant with the blue side band, the coherent scattering of photons in the forward direction is enhanced, reducing the atom's velocity. without further derivation: F~(r) --

h4r/2(v 9Vg)VgiA(r)[ -6

[(-A3AA4 V

AZAA3g2 V + AAA~g4V + ACg6V

+ A cg 3 2~/3 -- 2AA~cg 3 ~2 2tC + 2AA g 6K nt- 2Ac

_ A2AAcg2 gtc2 + 3AA g4gz2

AAA4y 3

g4y2tc 2A3AA2Vz2

2AAA2V 3tc2 + Ac g2y3tc2 - 2A3g2tc 3

A3AYK4- AAy3K4) -t-AAg2( 2A2A2cY@2g4y 2A~y 3 4A3AAcZ + 4A2g2z _ 4AAAcV2tc + 4g2g2tc + 2A2gz 2 + 2V3tc2)] (8.26) where g = g(r). This force is linear in the velocity of the atom, v, and points in the direction of the gradient of the coupling strength. As this force has an overall factor c< ]Vg] 2, it will only be i m p o r t a n t in cavities with large coupling gradients in the direction of the atomic propagation. To get an idea of the p a r a m e t e r dependence of this force, the one-dimensional friction coefficient, fi = IFv, axl/lvaxl, averaged over a wavelength along the cavity

276

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

FIG. 8.7. Contour plot of the coefficient/3 of the velocity-dependent force, averaged over a line of length k/2 along the cavity axis, as a function of the detunings for constant atomic excitation, Pe = 0.1. The unit is 1000 Ks -l/K, expressing the relative change of the kinetic energy (in Kelvin) per second for a 85Rb atom with mass m = 1.42 x 10.25 kg. The other parameters are (go, V, z) = 2re x (16, 3, 1.4) MHz.

axis, is plotted in Figure 8.7 for constant atomic excitation. Close to AA -- 0 and for large IAcI, the well-known Doppler cooling is most important. However, close to Ac = 0 and for large IAAI, the cavity-mediated velocitydependent mechanism dominates. It leads to heating for pump light that is blue detuned with respect to the cavity resonance, and leads to cooling for pump light that is red detuned with respect to the cavity resonance. Note that for very large IAAI, the sign of the atomic detuning becomes unimportant. For parameters where the velocity-dependent force cools, the system eventually equilibrates at a temperature given by the ratio fl/Dax. In theory, this temperature can be much smaller than the Doppler temperature, even for a two-level atom and can approach the recoil limit in the case of small z [15]. In a real experiment, the conditions of low atomic excitation and low velocity might not be met. In that case, numerical methods must be applied to get valid results. There is a variety of possible methods, of which we want to mention a few. Most simulations treat the motional degrees of freedom in a classical way in order to keep the required calculational resources acceptable. One method which stays close to the analytical results above, is to solve the master equation numerically. This is not

CAVITY-MEDIATED LIGHT FORCES

277

difficult to implement, but is relatively slow for higher photon (or atom) numbers, as the required processor time scales quadratically in the number of basis vectors needed to describe the system. One can also solve the master equation in a stochastic way by means of a quantum jump approach [65, 66]. This scales only linearly in the number of basis vectors, but needs additional averaging to get rid of the "noise". Another method is solving the Focker-Planck equation for the Wigner function describing the system [19, 21, 67]. The velocity-dependent force can also cool ensembles of particles [18, 20]. For this purpose, analytic expressions for the force in the limit of low atomic saturation can be derived [22]. The results indicate that the cooling force per atom can be kept constant by rescaling detuning and pump power as the atom number increases, provided the atoms move independently and do not lock in collective motion in which the individual contributions to the effective coupling of the atoms to the cavity field tend to cancel.

8.3.3 Experimental Studies of Light Forces To find out how detailed information about the light forces acting on an atom in a cavity can be obtained from experimental data, it is instructive to discuss first the results of a simulation. Consider an atom moving on a trajectory perpendicular to the cavity axis, with only a small velocity component in the direction of the cavity axis. For an atom entering the standing wave from a direction perpendicular to the cavity axis, the antinodes form a row of slits with the shape of a stack of pancakes viewed from the side. For a nonresonant light field, the dipole force acting on the atom leads to an oscillatory motion in the standing wave. Depending on the detuning of the light field with respect to the atomic transition, the atom will be confined to the vicinity of an antinode of the standing wave for red-detuned light, or to a node for blue-detuned light. A simulation of such a channelling motion in a red- and in a blue-detuned light field is depicted in Figures 8.8(a) and (b), respectively. The simulation includes all reactive and dissipative forces discussed above. The expectation value of the intracavity photon number (n) is a function of the atom-light coupling and therefore of the position of the atom. The channeling motion leads to a modulation of (n) with twice the oscillation frequency. Although the absolute value of the detunings ( A A - +g0) in the simulations of Figures 8.8(a) and (b) are identical, there are interesting differences between this redand blue-detuned case due to the nonlinearity of the system: in the red-detuned case, the a t o m moves through a relatively wide channel around an antinode, with a slow- and large-amplitude oscillation. Nevertheless, the

278

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

FIc. 8.8. (a) Motion of an atom with initial vertical velocity v0= 1.4 m/s in a reddetuned cavity field (AA = - g o , Ac =0, pump rate r/Z= 2to2 and mode waist w0 = 29 gm). The mode function [~r(r)[2 is indicated as gray scale on the left. Below that, the potential energy is indicated; the potential depth is approximately 0.2 mK. On the right, the expectation value of the intracavity photon number is plotted against time. (b) Idem in a blue-detuned field (AA -- +go, Ac --0).

modulation of the transmitted intensity originating from this oscillation is hardly visible, due to the nonlinearity of (n) as a function of g2. In the blue-detuned case, however, the atom oscillates fast through a narrow channel around a node. Although the amplitude of this oscillation is small, it leads to a deep modulation of the transmitted light. Note that in both Figures 8.8(a) and (b), the motion of the atom is not completely periodic, indicating that r a n d o m forces leading to m o m e n t u m diffusion are important. The channeling motion of an atom passing through a cavity field was studied experimentally in Reference [17]. As discussed above, the light transmitted through the cavity contains information on the motion of a passing atom. For a single transit, however, the typically small number of detected photons allows one to obtain little information about the axial motion. By taking the autocorrelation function of the transmitted intensity, g ( 2 ) ( r ) = (I(t)I(t+ r))/(I(t)) 2, where I(t) is the intensity, and appropriately correcting for the finite measuring time, the information of many individual transits can be combined. In particular for blue detuning, the characteristic oscillation of the channeling atoms appears, as can be seen from the dip of g(2)(r) at r = 1 ItS in Figure 8.9. Here, the parameters of the experiment were chosen such that each atom passing through the cavity increases the transmission. In contrast to Figure 8.8, most photons are counted when an atom is coupled to the cavity mode. This leads to g(2)(r) > 1 for r < 30 its, the transit time of an atom through the cavity mode. When comparing the experimental data with the result of a Monte Carlo simulation, satisfactory agreement could only be obtained if all optical forces acting on the atoms were taken into account. In particular, when leaving out the velocity-dependent force discussed in Section 8.3.2,

CAVITY-MEDIATED LIGHT FORCES ........

'

........

'

'

9

1 .25

a)

9 o....

1.2 o

o ~

o

~176 ~

.

~

279

, - . . . . . --, [ "", dipoleforce b) o k , "', + diffusion

.'.

"....,.:

1.1

i

.

.

.

.

.

.

9 .~

1.65 ~ . . . ,~L^,~ "dipoleior'ce C) ] " 9~ , ,

1.0

.......... Simulated: Dipoleforce Simulated: All forces . . . . . . . .

0.1

I

1

. . . . . . . .

I

10 time ~ [gs]

1 . . . . . . . .

100

.

1.55 0.1

6

.

0

.

~

0.2 0.5 1 time -c [gs]

2

FIG. 8.9. Intensity autocorrelation function g(2)(r) of the light transmitted through the cavity, while single atoms pass through the cavity field [17]. The circles and triangles are experimental data points. The dotted and solid lines are results of a Monte-Carlo simulation, each one rescaled to fit the data at r = 100 gs and r = 2 gs. (a) For parameters where the velocity-dependent force heats. The dashed line is the result of a simulation including only conservative dipole forces; the solid lines includes all conservative and dissipative forces. Best agreement between theory and experiment is obtained when all light forces, namely the dipole force, diffusion and the velocity-dependent force, are taken into account. The contribution from the velocity-dependent force is especially apparent in (b), a zoom in to short time scales of (a). The dashed line now includes the dipole force and diffusion. (c) Idem, for parameters that cool. In this red-detuned case, the experimental data does not discriminate between the simulation with and without velocity-dependent force. The cooling effect cannot be observed as the axial velocity of the atom is small throughout the whole interaction.

the calculated and measured autocorrelation function disagree for parameters where the velocity-dependent force heats the motion of the atom along the axial direction.

8.3.4 T r a p p i n g a Single A t o m

Amazingly, the dipole force in a high-finesse cavity can be large enough to hold an atom against gravity, even for an intracavity photon number of order unity. However, the dipole f o r c e is a conservative force and one needs either fast cooling or active feedback to trap an atom that enters the cavity. Active feedback works as follows: when an atom is detected in the vicinity of an antinode, the laser light pumping the cavity is suddenly switched to a higher value. This increases the depth of the optical dipole potential and traps the atom in the cavity field. Two groups have used this technique to trap individual atoms [23, 24].

280

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY 4 b

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FI~. 8.10. (a) Experimental transmission signal from Reference [24]. The data points are averaged over 10 ~ts and then smoothed by nearest-neighbor averaging over 5 data points. The dashed line shows the expected output intensity of an empty cavity. It is increased by a factor of 8 for a preset time of 3 ms after an atom caused a trigger at t = 0, (b) relative coupling strength inferred from (a) as a function of time. Experimental parameters are (go, y,K)= 2~ x (16,3, 1.5)MHz for Rb atoms at - 780 nm. In Figure 8.10(a), a transmission signal of a successful trapping event is shown from Reference [24]. In this particular trace, the atom enters the cavity mode at t = 0, where it triggers the feedback mechanism that increases the pump intensity for a duration of 3 ms. Of course, this increases the intensity of the transmitted light even for a cavity without an atom (dashed line). The signal from a trapped atom resides on top of that. Obviously, the atom remains in t h e c a v i t y for approximately 1.7 ms. The fluctuations of the transmitted power reflect the motion of the trapped atom and, of course, shot noise. This can be estimated from the fact that 1 p W of transmitted power corresponds to only ~ 20 detector clicks in each observation interval of 10 ps duration. To understand the motion of the trapped atom better, we plot in Figure 8.11 the calculated transmission signal from a q u a n t u m - j u m p Monte Carlo simulation o f an atomic trajectory. It can be seen that there is a correlation between the changes of the transmitted power and the radial excursion of the atom, especially on longer time scales. As the experimental signal shows similar structure in the transmitted intensity, we interpret this as evidence for the bound motion of the atom, moving back and forth along the radial direction of the cavity mode. The long time (of the order of a few ms) some atoms spend in the cavity in References [24, 27] is consistent with the simulated trajectories. The simulations also indicate that the atom sometimes escapes from an antinode, flies over several nodes before being cooled down by the

281

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Fla. 8.11. Result of a quantum-jump Monte Carlo simulation where a point like atom is trapped for 1.4 ms. Parameters are identical to those of Figure 8.10. (a) Solid line: expectation value of the intracavity photon number, (n), versus time. The noiselike variation of (n) on short time scales reflects both the fast motion of the atom along the cavity axis and the fact that the state of the cavity field differs from that of a coherent state in the presence of an atom. After the atom has left the cavity, the coherent field is time independent. Dashed line: radial coordinate, p, of the atom. The pump rate is increased by a factor of 8 after the atom caused a trigger at t = 0, (b) Projection of the atom's trajectory on a plane perpendicular to the cavity axis, marked by the cross.

friction force and recaptured in another antinode. Eventually the atom leaves the cavity in the transverse (radial) direction because of heating due to spontaneous emission. When leaving out the velocity-dependent force in the simulation, but keeping the m o m e n t u m diffusion caused by the dipole fluctuations, the atom unavoidably hits one of the t w o mirrors and long trapping times do not occur. This provides first indirect evidence for the existence of the cavity-mediated cooling force in a high-Q cavity. As compared to Reference [24], the experiment of Reference [23] employed a cavity with about the same finesse, but smaller Q, as the length was much smaller. This results in a larger atom-cavity coupling constant, so that the atom-laser detuning could be increased. Moreover, a heavier atom was observed with light at a longer wavelength. As a consequence, the m o m e n t u m diffusion caused by spontaneous emission was smaller. This made the motion of the atom more regular, and it could be assumed that the angular m o m e n t u m of the particle changed only slowly. With this assumption, it was possible to estimate the angular m o m e n t u m of the atom each half round-trip around the cavity axis from the experimental transmission data. Then, assuming an entrance point and angle, one could reconstruct atomic trajectories ("orbits") from the transmitted light.

282

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

FIG. 8.12. (a)and (b): Reconstructed trajectories from experimental transmission signals (not shown). (c) and (d): Simulated trajectories (white lines) and trajectories that are reconstructed from transmission signals derived from the simulated trajectories in the same plot (gray lines). Experimental parameters are (g0, y,K) = 2re x (110,2.6, 14.2)MHz for Cs atoms at )~ = 852 nm and a cavity length ~ = 10.9 gm. Reprinted with permission from Hood et al. [23]. Copyright 2000 American Association for the Advancement of Science.

In Figures 8.12(a) and (b) two examples are shown that are reconstructed from data. To estimate the quality of the reconstruction, the inversion a l g o r i t h m was also applied on simulated trajectories, as shown in Figures 8.12(c) and (d). It was found that this procedure only fails for highly elliptical orbits, and that it works well for the more circular orbits.

8.4 Experimental Considerations The experiments discussed in this chapter require a high-finesse cavity consisting of two highly reflective mirrors with losses caused by transmission, absorption and scattering of only a few parts per million (ppm). The resonance frequency of the cavity must both be stabilized and tunable relative to the resonance frequencies of the atom and the laser. These issues will be discussed in the following. The problem of how to create and

E X P E R I M E N T A L CONSIDERATIONS

283

manipulate ultracold atoms that can be placed inside the cavity by means of laser cooling and trapping techniques is beyond the scope of this book, and the reader is referred to the laser cooling and trapping literature [62].

8.4.1 Mirrors

Since the early 1990s, the best mirrors available for the near infrared have a transmission and loss of less than a few ppm [14]. It should be realized that there is a trade-off between low loss,/2, and high reflectivity, ~ . The reason is that high reflectivity is reached by placing a large number of dielectric layers on a super-polished substrate. With each new layer small imperfections or contaminations might be added, causing absorption and scattering losses to increase with the number of layers. For strong coupling, only a high reflectivity is important. However, absorption and scatter losses in the mirror can dramatically reduce the number of detected photons, thereby imparting the detection efficiency significantly. Ultrahigh reflectivity mirrors are not only difficult to make, it is also nontrivial to measure their properties or check their specifications [68]. An ordinary transmission and reflection measurement can establish T, but cannot measure losses of a few ppm, and hence, fails to measure /2 + T - 1 ~ with a useful accuracy. Therefore, it is best to set up a cavity to measure the properties of the mirrors. The cavity linewidth, 2to, can easily be measured by scanning either the laser or the cavity length and using a modulation sideband as frequency marker. Of course, this requires a laser whose linewidth is much narrower than the bandwidth of the cavity resonance. As the linewidth is inversely proportional to the length of the cavity, it is favorable to construct a small cavity in order not to be too restrictive on the laser linewidth. An alternative technique, where the laser linewidth is not an important parameter, is cavity ring-down spectroscopy. In its simplest realization, this works by storing a certain amount of light in the cavity and subsequently measuring the decay time r - 1/(2K) of the cavity intensity. An elegant realization is to sweep the cavity resonance quickly across the laser line by scanning the length of the cavity. This will effectively simulate a delta-form excitation [69]. Another possibility is to quickly turn off the input light at the moment when the laser frequency coincides with the cavity frequency during the sweep. This measurement scheme does not require fast frequency scanning but fast intensity switching, which can be realized by, e.g., an acousto-optic modulator.

284

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

By scanning the laser or the cavity length one can determine the distance between the longitudinal modes, the free spectral range (FSR). Note, however, that for a small cavity, the F S R can be of the order of 2re x 1012 Hz or, equivalently, several nanometers. From the linewidth and the free spectral range follows the finesse, F = F S R / ( 2 K ) = r c ~ / (1 7~), and, hence, 7~. The length of the cavity can, in principle, be measured directly with, e.g. a micrometer caliper, but this is difficult for short cavities. It can be determined more accurately from the free spectral range, F S R = rcc/g~.More elegant is to map out the distance between the transversal modes, Acot, and calculate the length from the measured frequency ratio [39] -

FSR

= - arccos 1 -

(8.27)

and the known mirror curvature R. This method has the advantage that it does not depend on an absolute frequency calibration. Finally, the absorption and scatter losses can be obtained from a measurement ofthe maximum transmission of the cavity, Tmax = T2/(1 - 7~)2, and/2 = 1 - ~ T. However, this requires perfect mode matching, which is difficult to achieve i n practice. A way around this is to measure the reflected and transmitted power simultaneously [68], allowing to calculate T from Pout

T2F2

Prefl - Pin

/22F 2 _ g2

(8.28)

where Pout, Prefl and Pin, are the transmitted, reflected and impinging light powers, respectively. Handling ultrahigh reflectivity mirrors requires stringent precautions. They must be kept in a clean environment a t all times, as the smallest dust particle can degrade performance. A clean-airflow box is the minimum equipment required, Another issue to consider is birefringence caused by strain. Shrinking glue or clamping the mirrors too firmly in their mounts can cause strain on the mirror substrate that c a n be large enough to cause birefringence in the coating, resulting in a cavity with nondegenerate polarization eigenmodes. The mirrors should also remain clean during the experiment. Consequently, directing a high-intensity atomic beam onto a mirror surface is probably not a good idea. With all these precautions, however, a high-finesse cavity can be a reliable part of the experiment. For example, our system is operated in a moderately good vacuum of about 0.1 gPa, with no direct line of sight

EXPERIMENTAL CONSIDERATIONS

28 5

between the cold atom source and the mirror surfaces. We have operated the system for more than a year without noticeable degradation of the cavity finesse.

8.4.2 Making a Small Cavity The simplest cavity geometry that one can think of, namely the F a b r y P6rot cavity, is also well suited for making a small cavity. Of course, the mirrors should be curved to make a stable mode. In addition, a strong mirror curvature confines the mode to a small waist w0. For a given mirror curvature, the volume can be reduced by placing the mirrors closer together, limited only by the condition that the atoms should be able to get into the cavity volume. Thus, the volume is limited by the sag height (the depth of the concave surface) of the curved mirrors. Pushing the limits of high coupling, the mirror diameter should be reduced. However, it should be realized that it is difficult to make good mirror coatings on too strongly curved substrates, at least on mirrors with a large diameter. Moreover, machining the mirrors can introduce birefringence or cause contamination of the high-finesse coating. Also, as a rule of thumb, good coating should be available within a circle with a diameter of at least 6w0 around the center of the mode, to ensure that the effect of truncating the tail of the Gaussian TEM00 mode is less than 1 ppm. As an example, in the experiment described in Reference [54], the cavity consists of two dielectric mirrors with a radius of curvature of 20 cm and a diameter of 7.75 mm. It is depicted in Figure 8.13. A piezo-ceramic tube encloses the cavity mirrors and controls the length of the cavity. The tube has four holes allowing atoms to enter the volume between the mirrors and residual gas to be removed. Both mirrors are assumed to be identical.

Fro. 8.13. Left: Construction drawing of the assembled cavity consisting of the two mirrors, their mounts, and the piezoelectric tube. Right: Photograph of the cavity before assembling.

286

SINGLE ATOMS MOVING

IN A HIGH-FINESSE

CAVITY

The cavity length is 116 gm. The finesse of the cavity was estimated from the measured cavity linewidth, t c - 2re x 1.5 M H z (HWHM), to be better than 4.3 x 10 s. This procedure of obtaining the cavity finesse was checked by cavity ring-down measurements. F r o m this, the mirror reflection losses were calculated to be 7.3 x 10 -6, including the transmission loss of 2.8 x 10 -6. Birefringence was small as the cavity resonances for left and right circularly polarized light overlap within the cavity linewidth. The TEM00 mode of the cavity has a waist w0 - 29 gm, resulting in a maximum atom-field coupling of go -- 2rt x 1 6 MHz for t h e 5281/2 F -- 3, m F - - 3 +-~ 52p3/2 F -- 4, m F - - 4 transition in 8SRb The cavity described above is not the smallest one can make. A cavity with a length of only 10.9 gm was reported in Reference [23], using coned mirrors to get a small sag height. Even smaller cavities seem feasible, and although experimental difficulties will hamper further development, optical cavities with a length of a few wavelengths do not seem impossible.

8.4.3 Stabilization and Vibrations In many experiments with high-finesse cavities, it is crucial that the length of the cavity is fixed in order to have a stable resonance frequency. Only in experiments where the cavity can be far detuned from a resonance in the particle, one can in principle lock the laser on or near the cavity resonance, which now slowly drifts. Here we discuss the first case, in which the relative accuracy in the length setting must be much better than Ag/g = x/COc, which is of the order of a few times 10 9. This corresponds to an absolute accuracy of much better than A g - ()~/2)(K/FSR)= )~/(4F), which is of the order of 10 -12 m. Hence, a length stability better than 100 fm is essential for the experiment described in this chapter. This requires a very good vibration isolation, temperature stability and precise active length stabilization. In Reference [54], for instance, the cavity was mounted on a vibration isolation stage inside the vacuum system, to suppress acoustical noise. This is depicted in Figure 8.14. Moreover, care was taken to remove sources of vibration (like water cooling) from the optical table, which was also pneumatically isolated from the lab floor. The light used for the stabilization should be of low power, as a significant fraction of its power is absorbed in the high-finesse mirror coatings. In our experiments, for example, it was not possible to stabilize the cavity well enough if more than a few microwatts of power was used, in particular when the light intensity had to be changed rapidly during the experiment. This is probably due to thermal expansion of the mirrors because of absorption in the mirror coatings, an effect that can even be used to measure the absorption losses of these mirrors [70]. The complete

EXPERIMENTAL CONSIDERATIONS

287

Fro. 8.14. Photograph of the high-finesse cavity mounted on a vibration isolation stage. The stage, which is mounted in an ultrahigh-vacuum chamber, consists of three copper plates connected by viton cylinders. The cavity mount with its mirrors and piezo is clamped in a block of Teflon | for extra damping.

stabilization therefore had to operate at extremely low light levels. This was accomplished in the following way: After passing through an attenuator, narrow-band (< 100 kHz) laser light, frequency stabilized to a saturated absorption line in a gas of Rb atoms, was coupled into the ultrahigh-finesse cavity. The cavity length was actively stabilized by means of a frequencymodulation technique (Pound-Drever-Hall method [71]), which required less than 100 pW of incident laser power. To this end, laser light was frequency modulated with an EOM creating two side bands at 4-10.7 MHz. The beat signal of the carrier with the sidebands in the light reflected from the cavity was measured with an avalanche photodiode and demodulated with the original fm drive frequency. The resulting error signal was used as input for a servo loop driving a low-noise high-voltage amplifier, controlling the piezo and, hence, the cavity length. As the passive stability of the cavity is good on time scales below 1 ms, the bandwidth of the stabilization feedback loop was limited by an electrical low-pass filter (3 dB point at 65 Hz). In this way, the residual rms-fluctuations in the cavity length were reduced to less than 10 -13 m. During the measurement interval, when atoms were injected into the cavity, the length stabilization was frozen, the radio frequency side bands were switched off, and the laser-beam power and frequency were switched to the desired value for the actual experiment. The passive short-term stability of the cavity was better than 100 kHz/ms, which guaranteed a measurement interval of a few microseconds.

288

SINGLE ATOMS MOVING IN A HIGH-FINESSE CAVITY

This stabilization scheme has the advantage that it is independent of what happens behind the cavity, where the transmitted beam can be manipulated at will without affecting the lock loop. The disadvantage is the high background of reflected light that is either not well mode matched or reflected back from the input mirror because of impedance mismatch. Alternatively, the transmitted light can be used to lock the cavity on maximum transmissi0nl; This has the advantage of being a zerobackground method. For this scheme, the laser light impinging on the cavity can be frequency modulated and this modulation can be detected in the transmitted light. With standard lock-in techniques, this can then be converted to an error signal for the length adjustment. A disadvantage of both described methods is, that the light used for the length stabilization is also influenced by atoms. As a result, during the actual experiment the active stabilization must be frozen and the maximum duration of the experiment is limited by the passive stability of the cavity. This can be overcome by stabilizing the cavity with light that is faroffresonant with the atomic transition, but resonant with another longitudinal cavity mode. As the free spectral range of a small cavity is rather large, this second laser beam will have a wavelength which differs by at least a few nanometers [72].

8.5 Outlook Future research will certainly try to construct and operate smaller cavities in the search for stronger coherent coupling. As technical difficulties will grow, alternative routes to strong coupling should be pursued as well. For example, a carefully designed hollow defect in a photonic band gap material could act as an ultimate strong-coupling cavity. The single-photon atom trap of Section 8.3.4 already hints at the prospects of high-finesse optical cavities for fundamental studies and their applications: first, one or a few atoms interacting with one or a few light quanta form the textbook examples of light-matter interaction, and are therefore a beautiful playground to test existing theories. In this context, the mechanical forces on an atom in a high-finesse cavity still has unsolved questions, as for instance the relation between the quantum fluctuations of the cavity light field and the momentum diffusion of the 'atom. Stated in more general terms, radiation physics in confined space could benefit from high-finesse cavity experiments, and experiments testing subtle Lamb-shifts and Casimir-Polder forces in high-finesse cavities can be imagined. Moreover, a single particle trapped in a high-finesse cavity has applications in the field of quantum communications and computing [73-77].

REFERENCES

289

For example, it should allow the transmission of quantum bits ("qubits") between distant cavities [78, 79], to build arbitrary states of the electromagnetic field [80], or to generate a bit stream of single-mode photons [81-83]. For this, it will be necessary to confine the atom to a very stable circular orbit, or better, cool the atom down to lower temperatures, perhaps even to the motional ground state of the strongly coupled atomcavity system. The quantization of the atomic motion in a high-Q cavity has been studied theoretically [84, 85], but remains unexplored experimentally, so far. As an alternative to cooling atoms in the cavity field, one could try to start with an ultracold and dense sample in the first place, by loading the cavity with a Bose-Einstein condensate [86]. Here the cavity field could serve as a sensitive probe for properties of the Bose-Einstein condensate. Cavity-mediated cooling has a more down-to-earth application in the optical cooling of particles that do not have closed transitions, like the overwhelming majority of molecules. Finally, high-finesse optical cavities might be used to probe single complex (biological) molecules with high temporal and spatial resolution close to the standard quantum limit [87]. In this context, a novel scheme to reconstruct an atomic trajectory from the measured intensity pattern at the output of a multimode cavity has recently been proposed [88].

Acknowledgments This text is based on work done at the University of Konstanz and the Max-Planck-Institut ffir Quantenoptik in Garching, Germany, performed by Thomas Fischer, Peter Maunz, Peter Mtinstermann, Thomas Puppe, and the authors. The help of our colleagues in the preparation of the manuscript is gratefully acknowledged.

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290 o

~

o

o

10. 11. 12.

13.

14.

15.

16. 17.

18. 19. 20. 21.

SINGLE ATOMS M O V I N G IN A H I G H - F I N E S S E CAVITY

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292

S I N G L E ATOMS M O V I N G IN A H I G H - F I N E S S E CAVITY

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2, 3.

4.

5. 6~

77.

8,

9.

80. 81. 82.

83.

84.

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dynamics of Bose-Einstein condensates in high-finesse optical cavities, Phys. Rev. A 61, 033609 (2000). 87. Vladimir B, Braginsky and Farid Ya. Khalili, "Quantum Measurement." Cambridge University Press, Cambridge, 1992. 88. P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, An optical kaleidoscope using a single atom, Phys. Rev. Lett. 88, 043601, 1-4 (2002).

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9. THE CAVITY-ENHANCED OPTICAL-FREQUENCY COMB GENERATOR AND ITS APPLICATIONS Motonobu Kourogi, Bambang Widiyatmoko, Motoichi Ohtsu Tokyo Institute of Technology, Yokohama, Japan

9.1 Introduction While continuous-wave lasers with narrow power spectra are widely used for highly accurate optical frequency and interference measurements [1], the use of pulsed lasers in such applications remains uncommon [2]. This situation may be a result of the common notion that pulsed lasers cannot be used for frequency measurements because of the broad linewidth of the power spectrum inherent to these light sources. This notion is perhaps odd, inasmuch as the frequency spectrum of a pulse train that is obtained by coherent modulation of a continuous-wave laser is a series of equally spaced, discrete frequencies distributed over a large frequency range. Radiation with this type of a power spectrum is referred to as an opticalfrequency comb (OFC) and is widely used in precision frequency metrology. Furthermore, as a consequence of the Fourier transform relationships, the time dependence of a field with an OFC spectrum is nothing other than a series of pulses, separated in time by the reciprocal of the frequency spacing between each comb "tooth." The duration of each individual pulse is in turn related to the number of "teeth" within the comb. An OFC is useful for applications requiring precise measurement of optical frequencies distributed over a wide frequency range. This type of measurement is required for applications as varied as dense frequencydivision-multiplexed communication networking, high-resolution spectroscopy, and the measurement of fundamental physical constant [3, 4]. An OFC simplifies the accurate optical-frequency measurements needed in these applications. For example, by using an OFC with a tunable laser, a stable and accurately tunable coherent light source can be realized, which is nothing other than an optical-frequency synthesizer [5]. Furthermore, OFCs have not only frequency-domain applications, but also novel applications in time domain. For example, an optical pulse of the desired wave shape can be synthesized by controlling the amplitude and the phase of each frequency component within an OFC [6].

297 EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES Vol. 40 ISBN 0-12-475987-4

Copyright 9 2002 by Academic Press All rights of reproduction in any form reserved. ISSN 1079-4042/01 $35.00

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OPTICAL-FREQUENCY COMB GENERATOR

This chapter discusses an OFC generator based on an electro-optic modulator within a Fabry-P6rot cavity [7-16]. Section 9.2 describes the basics of OFC generation, including expansion of the comb by self-phase modulation [15]. This section also discusses the methods used for optical difference frequency measurements using an OFC generator. Section 9.3 describes how OFCs are used, focusing on applications in precision spectroscopy and a novel implementation of optical coherent tomography.

9.2 Generating an Optical-Frequency Comb 9.2.1 Optical-Frequency Comb Basics The frequency at which an electromagnetic wave oscillates is typically determined by heterodyne measurements. In a heterodyne measurement, the frequency difference between a known, standard frequency and the frequency under test is measured by counting the heterodyne beat signal between them, by which the frequency under test is determined [17]. Frequently, however, the frequency difference between the standard and the test frequency will be so large that the heterodyne beat note frequency exceeds the electronic bandwidth of the photoreceiver (GHz), making a straightforward-heterodyne measurement impossible. In such situations, a comb generator can be used advantageously. In order to understand why an optical-frequency comb is useful for measurements at optical frequencies, let us first consider electronic signals at microwave frequencies. A microwave frequency pulse train at a constant repetition rate can be generated by locking the generating electronics to some standard microwave frequency reference. The resulting series of pulses is shown in Figure 9.1(a). The power spectrum of this pulse train, obtained by Fourier transformation, is the comb-like spectrum shown in Figure 9.1(b). The frequency difference between adjacent sidebands is constant. The absolute frequency of each sideband component is known, because the generating electronics were fixed to a standard frequency. In the opticalfrequency region, however, generating a pulse train in this direct fashion is not possible because the field oscillates at hundreds of terahertz. However, a pulse train can be generated at optical frequencies by modulating the amplitude of a monochromatic optical field at a fixed frequency. The resulting field is shown in Figure 9.1(c). On Fourier transformation of this field, a comb-shaped spectrum with constant frequency difference between adjacent sideband components is obtained, as shown in Figure 9.1(d). If the modulation frequency is known, then the frequency of each sideband component can be calculated relative to that of

GENERATING AN OPTICAL-FREQUENCY COMB




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the original field. Furthermore, if the wavelength of the field being modulated is known, say by locking the laser to some well-established atomic transition, then the absolute frequency of each sideband will also be known. Of course, not only must the modulation frequency and the carrier frequency be stabilized, but they must also be sufficiently narrow to detect a heterodyne signal. Because there is but a single "active" component to the system (the laser), an OFC generator is a simple and comparatively robust way to generate radiation over a wide range of frequencies. The efficiency with which sidebands are generated can be greatly enhanced by producing an OFC inside a Fabry-P&ot (FP) cavity [18]. Figure 9.2(a) shows an OFC generator based on a Fabry-P6rot electro-optic (EO) modulator. A Version of this modulator was proposed some thirty years ago for generating short optical pulses [7]. It consists of an efficient EO phase modulator installed in an optical FP cavity with the modulation frequency being a harmonic of the cavity's free spectral range (FSR). This gives a deep modulation of the light that resonates in the FP cavity. The output of this cavity is a pulse train with a repetition rate twice the modulation frequency. Each pulse in this train has a Lorentzian time profile, and the pulse width AT is expressed as A r -- (2Fflfm)-1

(9.1)

where F is the finesse of the FP cavity [18],/3 is the modulation index for a single pass in the EO modulator, and fm is the modulation frequency.

300

OPTICAL-FREQUENCY COMB GENERATOR

FIG. 9.2. Optical-frequency comb generators. (A) Conventional, and (B) monolithic with coupling cavity. M: Mirror, EOM: Electro-optic modulator, R: Reflectorized surface, CC: Coupling cavity.

This means that a high modulation index must be applied to the EO modulator to generate a short pulse, or wide span, OFC. This can be achieved by matching the group velocity of the light within the modulator to the phase velocity of the microwave signal, or the modulation frequency should be integer times the free spectral range of the resonator. The power transferred into high-order sidebands is increased when a highfinesse FP cavity is used. One way to realize this design is to construct a monolithic cavity structure from the EO modulator itself by-applying reflective coatings on both ends of the crystal [10]. This approach can give a cavity with a the finesse between 400 and 600. Combining the monolithic cavity with a coupling cavity [13] has the added advantage of increasing the overall transmission of the OFC generator. The coupling cavity length is judiciously selected, so that the incident light is resonant in this cavity, while the sidebands generated by the EO modulator are not. Thus, this second cavity acts as a window for the incident light and a high reflection mirror for the sidebands. Figure 9.2(b) illustrated such a monolithic type modulator with a coupling cavity. The transmission of the OFC generator like the one shown in this figure can be 25%. An example of the spectral envelope of OFC is shown in Figure 9.3. In this example, the laser wavelength was 1.5 gm, the finesse of the monolithic cavity was about 600, the modulation index was 0.7 radian, and the modulation frequency was 6 GHz, which is twice the FSR of the monolithic

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cavity. The spectral span for this type of OCF generator, using LiNbO3 as the electro-optic crystal, is about 10 THz. Thus, this O F C can measure the frequency difference between two lasers as high as several terahertz. The accuracy of this measurement has been compared with more conventional methods, such as using an optical parametric oscillator [19] or optical frequency interval dividers [20, 21]. It has been shown that the accuracy of measurements using an O F C and an optical frequency interval divider can be as high as 6.8 x 10 -15 [21]. A closer examination of the power spectrum shown in Figure 9.3 reveals two features. First, the power spectrum decays exponentially away from the center wavelength. Second, there is a stepwise discontinuity on both sides of the spectrum. The first of these simply reflects that decreasing amounts of power can be shifted into increasing higher order sidebands, even when using a high-finesse cavity [8]. The second observation, however, is a nonideality introduced by dispersion in the EO crystal. Specifically, when the finesse of the monolithic cavity is sufficiently high, dispersion of the EO crystal limits the O F C span. What happens in essence is this: The resonance condition within the cavity includes not only the usual geometry-dependent terms, but must also terms from the frequency-dependent speed of light within the EO crystal. The result is that the resonant frequencies of the oscillator are not exactly

302

OPTICAL-FREQUENCY COMB GENERATOR

evenly spaced. The modulator, however, operates at a fixed frequency, generating sidebands at fixed frequency intervals. At some point, the frequency of the sideband is outside of the comparatively narrow resonance of the resonator, and this results in the sharp cutoff. The dispersion-limited span of a cavity-enhanced optical comb is given by the simplified equation, Spandispersion_limited -- VTcD)2Lc

(9.2)

where D is the material dispersion of the EO crystal, 2 is the wavelength of incident laser, c is the speed of light in vacuum, and Lc is the length of the crystal [11]. Using this equation, the dispersion limit of the span in the optical-comb generation can be calculated as a function of wavelength of laser. This prediction is plotted in Figure 9.4 along with some experimental data. It is seen that the experimental data are consistent with the predicted span interval. As shown in this figure, a flat top appears near 2 lam, which is attributable to higher order dispersion in the EO crystal.

9.2.2 Expanding the Span of a Cavity-Enhanced OFC Although the span of a cavity-enhanced OFC can reach > 10 THz, this interval is still only about 5% of the carrier frequency at optical

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GENERATING AN OPTICAL-FREQUENCYCOMB

303

wavelengths. To achieve an increased span, one would have to develop dielectric coatings that are reflective over an extended wavelength range, discover electro-optic crystals than can dissipate large amounts of microwave power without being damaged, and eliminate group-velocity dispersion within the electro-optic crystal. A simpler approach is to exploit the self-phase modulation (SPM) that occurs in an optical fiber [22]. Self-phase modulation takes place in a nonlinear medium in which the refractive index depends on the intensity. Because the output of an opticalfrequency comb generator is a series of pulses, SPM will take place as the pulse train passes through an appropriate medium. By analogy to supercontinuum generation [23-25], one would expect to be able to generate an OFC with a span of more than 100 THz. The 1.5 gm wavelength region is ideal for this sort of application, because optical fibers in this wavelength (used primarily by telecommunications industry) are manufactured with very low losses, are designed so that the material dispersion is compensated for by the structural dispersion, and are readily available and comparatively inexpensive. It is also important to note that the power-dependent modulation index is readily increased by the use of a small fiber core. This all suggests that it should be possible to expand the span of the OFC using optical fibers. Figure 9.5 shows the experimental configuration of an expanded-span OFC based on self-phase modulation in an optical fiber. The light from an external-cavity diode laser is coupled into a cavity-enhanced OFC generator. Its output is introduced into a long dispersion-flattened fiber (DFF) [26, 27], after being amplified in an erbium-doped optical fiber amplifier (EDFA) [28]. A DFF typically has an anomalous dispersion coefficient of 0.12 ps/km per nm. The EDFA increases the average power of the pulse train by a factor of ten to sixty, giving peak pulse powers of watts. Because SPM under anomalous-dispersion conditions supports soliton formation, efficient spectral broadening is expected inasmuch as there is pulse compression of the higher order solitons [29].

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FIG. 9.6. Power spectra at various stages with in the expanded-span opticalfrequency comb generator. (A) Before the erbium-doped fiber amplifier, (B) after the erbium-doped fiber amplifier, (C) after passing through the dispersion-flattened fiber, with 80 mW of input power, and (D) with 250 mW of input power. Because the resolving power of the spectrograph (~ 125 GHz) was greater than the modulation frequency (6 GHz), an overall envelope is observed, rather than the detailed structure of the comb.

Figure 9.6 shows the power spectrum of the light from an expanded-span OFC at various stages. Curve A shows the spectrum of the output from the monolithic cavity OFC generator, and Curve B shows the spectrum after passing through the fiber amplifier. As it turns out, the spectral envelope of the light from the OFC monolithic cavity generator exceeded the gain profile of EDFA, so it is unnecessary to use the full generating power of the monolithic cavity OFC generator when seeding this system. Curves C and D show the power spectrum after amplification and passing through the dispersion-flattened fiber. In the case of curve D, the spectrum is extended to a span of 50 THz. This span corresponds to 8000 sidebands. It is approximately five times larger than the span of OFC obtained without fiber, and it is a full 25% of the carrier frequency of the original laser. Higher order nonlinear effects do introduce certain nonidealities into the system. The most obvious of these can be seen in, Figure 9.6, Curves C and D. Both of these spectra have side peaks around 9 THz away from central frequency. This is caused by a higher order group-velocity dispersion in the optical fiber. Another effect is found when measuring the pulse duration after the fiber amplifier. Given the spectrum of Figure 9.6, curve C, Eq. (9.1) predicts the pulse duration of 200 fs for the light exiting the fiber amplifier.

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The pulse duration can easily be measured using an autocorrelator, and the pulse durations are found to be about 2 ps. The presumed cause of this increased pulse duration is the phase retardation, introduced by the dispersion in the electro-optic crystal, on the various spectral components of the frequency comb. To demonstrate that the expanded-span O F C is in fact useful for measuring frequency difference at widely differing frequencies, the signal-tonoise of the beat note was measured as function of frequency separation. This measurement is shown in Figure 9.7. The measurement was carried out for differing output powers. In all cases, the signal-to-noise fell off at a rate of ,~ 6 dB per octave. A better signal-to-noise can be expected by using a larger optical power in the O F C and improving the characteristics of the D F F . The measurement demonstrates, nevertheless, that reliable frequencydifference measurements can be made over a wide span using the output of this generator.

9.3

Using

Optical-Frequency

Combs

As discussed in the introduction, optical-frequency comb filters have a variety of time-domain and frequency-domain applications. Here, two of

306

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these will be discussed, the accurate determination of the absolute frequencies of spectroscopic transitions and optical coherence tomography.

9.3.1 Precision Spectroscopy: A Frequency-Domain Application of Optical-Frequency Combs Knowing the exact frequency at which atomic and molecular spectral transitions occur provides insight into the fundamental physics, internal motions, and structure of the atom or molecule under study. In the case of atoms, linewidths, under favorable environmental circumstances, are very narrow (Hz). This attribute is the basis for atomic clocks and interferometric measurement of quantities such as the local gravitational acceleration. However, while atomic transitions are narrow, they are also sparse. Molecules, by contrast, have many transitions, corresponding to rotations and vibrations, over a wide frequency range. As such, molecular spectra can provide reference frequencies over a range of wavelengths and at closely spaced frequency intervals. Optical-frequency comb generators are useful tools in making precision measurements of spectroscopic transitions and have been widely used in atomic spectroscopy [4, 30, 31]. The idea behind these experiments is to build a frequency chain, with the OFC serving as a link between a light source for which the absolute frequency is known and the light source which is used to make the spectroscopic measurement (cf. Fig. 9.8). A laser, such as a methane-stabilized He:Ne operating at 3.392 gm, serves as the absolute reference frequency of the system. Ideally, the frequency of this reference source will be calibrated against a primary standard maintained by a national metrology institute. From this reference light, another color is generated using nonlinear optical techniques, such as second-harmonic generation, and a second laser is reference to this new frequency. Depending on the target frequency, multiple lasers and nonlinear light generation processes may be needed. Next, it is time for the OFC generator to work its magic. The light from the secondary laser is fed into a comb generator, from which a whole set of sidebands is created. The laser to be used in the actual experiment is locked to one of these sidebands, and the light generated by this laser is used in the actual spectroscopic measurement. In this manner, the OFC has transferred the absolute frequency of the reference laser onto the measurement of the absolute frequency of the spectroscopic transition, albeit with a slightly increased uncertainty. A number of precise and accurate measurements spectral-line positions have been made or validated using the approach describing the preceding

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paragraph. These have included the shift in the 1S-2S two-photon resonance between hydrogen and deuterium [4]. It was found to be 670.99433464 GHz 4- 150 Hz. In addition to the inherent beauty of some many decimals of precision, this number was used to calculate the difference in the radius of the proton and deuteron, which was reported to be 1.97535 fm. In another set of experiment, the absolute frequency of the cesium D1 and D2 lines were measured [30, 31]. The precision in these measurements exceeded previous measurements by decades. Measurement of the D2 line allowed the fine-structure constant to be determined with unprecedented precision. Such precision in frequency measurement naturally finds application in the time measurements [32], and experiments are being planned and undertaken to construct an optical-frequency synthesis system that can determine optical frequencies with a precision limited by primary frequency standards. This system will rely on precisely determined atomic transition frequencies, measured using OFCs, and an OFC to fill in the gaps between these atomic transitions [33]. Optical-frequency combs have also been used to determine the absolute frequencies of molecular transitions. For example, some ninety rovibrational line positions were measured for acetylene near 1.54 ktm [34]

308

OPTICAL-FREQUENCY COMB GENERATOR

FIG. 9.9. Frequency chain for the determination of absolute transition frequencies. ES-OFCG: Expanded-span optical-frequency comb generator, G: Grating, PH: Pinhole, PD: Photodiode, AE: Analyzer electronics, 1)C2H2/CH4/h" frequency for acetylene transition (known), methane transition (to be measured), and heterodyne signal, respectively, fro: modulation frequency, n: sideband number.

and few lines of methane near 1.66 gm [35]. The approach to these measurements is somewhat different than described earlier, and to illustrate this technique the measurement of the methane line positions will be discussed. In these experiments, shown schematically in Figure 9.9, the lasers were locked to the molecular transitions by standard methods [36], one laser to the methane line to be measured and the other to an acetylene transition for which the absolute frequency position was known from standard OFC measurements [34]. The frequency difference between these two lines was just too large to be spanned by a monolithic OFC generator, and thus the expanded-span generator discussed in the previous section was used. The output from the laser locked to the acetylene transition passed through an expanded-span OFC generator. The output from the laser locked to the acetylene transition acted, in essence, as the local oscillator for the system. The two beams were combined, sent to a photodiode, and the resulting heterodyne signal was measured. Using the known frequency of the acetylene transition and the beat-note frequency, the absolute frequency at which the methane transition occurred could be calculated. One trick proved critical to extracting a good beat note. Because the comb power was so much higher than that of reference laser, a grating and a pinhole were used as a filter to eliminate the amplitude noise of the comb. In summary, optical-frequency comb filters can be used to determine the absolute frequency of spectroscopic transitions, in many cases with decades greater precision than possible before. Furthermore, by using an expandedspan OFC, the absolute frequency of spectral transitions can be measured over a wide range of frequency. Using optical-frequency combs and

USING OPTICAL-FREQUENCY COMBS

309

frequency chains, it is now possible to measure at optical frequencies far removed from standardized microwave frequencies.

9.3.2 Optical Coherence Tomography: A Time-Domain Application of Optical-Frequency Combs Optical coherence tomography (OCT) is used to study the geometry and internal structure of objects. OCT imaging has its roots in ultrafast optics and draws upon many technologies including white-light interferometery, fiber optics, and Fourier transform spectrometry. OCT creates highresolution, cross-sectional images of microstructures within highly scattering media and biological systems and is used in broad range of research and clinical applications. Conventional OCTs, shown schematically in Figure 9.10(a), are based on a Michelson interferometer with a broadband, short coherence time light source [37, 38]. The technique uses coherent cross correlation of light reflected or backscattered from the object being studied. The object reflects or scatters light back to beam splitter from many internal locations. Of course, the maximum interference occurs when the optical path length difference is smaller than coherence length of the optical source. As the translating mirror is scanned, signal appears whenever the time delay to the translating mirror matches that of a reflection or backscattering site in the object. The delay information is then used to determine the

Fla. 9.10. Optical coherence tomographs. (A) A conventional OCT, and (B) an optical-frequency comb-based OCT. fm: modulation frequency, Afm: frequency difference.

310

OPTICAL-FREQUENCY COMB GENERATOR

longitudinal location of the reflection site. A cross-sectional image is produced when a set of longitudinal profiles are recorded across the object being studied. In this arrangement, the mirror in the reference arm of the interferometer must be continuously moved, and this mechanical motion limits the speed at which a complete image can be collected. Measurements must be faster than the movements of a living object in order to minimize image blur and other motion artifacts [39, 40]. The reported maximum speed for longitudinal scans is 1.6 m/s for the conventional method [41-45]. Variants on the conventional approach have achieved scan speeds of 3 m/s [46] and 21 m/s [49], but these methods still change the optical path length of the reference arm to match that of the sample arm. Another variant, call optical frequency-domain reflectometry, uses fixed path lengths of both arms and sweeps the wavelength of a narrow-band light source over a broad range. However, the rate at which the light source can be scanned still limits the speed with which images can be collected [48, 49]. An OCT based on optical-frequency comb generators can eliminate the problems associated with conventional approaches. This system eliminates moving parts, which greatly increases the longitudinal scan speed, and thus minimizes image blur caused by motion. The result is a high-definition coherence tomograph image acquired in a short time interval. How is it possible to obtain such rapid scanning speed without moving parts? With two OFC generators operated at slightly different modulation frequencies used as follows: one OFC generator is placed in the object arm, and a second is placed in the reference arm of a Michelson interferometer instead of a mirror-moving actuator, as shown in Figure 9.10(b). The two generators are modulated with two different modulation frequencies, having a frequency difference of Af. The time- and frequency-domain outputs of these generators are illustrated in Figure 9.11. The frequency interval in between a sideband and its adjacent sidebands is equal to its modulation frequency. The center frequency, v0, injected into both generators is the same, but the sideband frequencies are different by multiples of Afm. Thus, the time intervals between output pulses also differ, by the amount ~ Afm/f2. A time-domain analogy outlines the basic principles of a conventional OCT and a OFC-based OCT (cf. Fig. 9.12). Imagine twins capable of running at the same speed. If the twins run the same distance, they arrive at their destination simultaneously. In OCT, the twins are light pulses that interfere. They only interfere constructively when the optical path lengths are equal. In a conventional optical coherent tomograph, the moving reference mirror is what makes the optical path lengths equal. Now imagine what happens when our sprinting twins run different distances; they will arrive at their destination at different times. But the twins can arrive at their

USING OPTICAL-FREQUENCY COMBS

V0

(fm)-1
i i


> time

(A)

,,,llll

>

f

time

(c)

,,llll

fm

Ill ,,.

frequency

(g)

(fro + Afro)-~
l ~fm + Arm II,-

frequency

(g)

FIo. 9.11. Output from the optical-frequency comb generators in an OCT. (A) Time dependence of the field for generator operated at modulation frequency fro, and (B) field spectrum for generator operated at fro, (C) time dependence of the field for generator operating at fm + Afm,and (D) field spectrum for generator operating at

fm+ Afm.

9

9

,,~ @ ,,~

4" 9 4,

Same path lengths

(A)

tl

t 2

@ 9

9

54" | 4,

4" 9 4"

Different path lengths

(g)

to t~

t 2'

t 1'

(c)

t o'

t 1

cat

t o,

t 1'

(D)

FIG. 9.12. Heuristic explain of optical-frequency comb-based OCT. Consult text for details. goal simultaneously if one is delayed. This delayed-start principle is behind the optical-frequency comb OCT. To understand the interferometric equivalent, let us consider two pulse trains that have a slightly different time interval, At, between the pulses. The pulses from one train arrive at

312

OPTICAL-FREQUENCY COMB GENERATOR

times to, tl, and t2, and those from the other at t~, t], and t~. The delay between the arrival time of the two pulses decreased by At with each set of pulses, until the pulses arriving at time to (which equals t~) arrive simultaneously and interfere (Fig. 9.12(c)). Of course, had the different optical path lengths been different, say if a delay length c a t had been introduced on the path travel by the primed pulses, then pulses arriving at tl (equal to t~) would have interfered (Fig. 9.12(d)). That's the idea behind an optical-frequency comb optical coherence tomograph. The opticalfrequency comb generator launches a series of pulse into the reference and object arms of an interferometer. Using phase-sensitive detection, the delay between the launch time of two interfering pulses can be determined. This delay, in turn, gives the difference in the optical path for the two pulses and from this information, the depth at which reflection and backscattering occurs can be determined. A whole series of these allows an image to be reconstructed. We now have an OCT without any moving parts. How fast can an optical-comb filter OCT scan? It depends on the modulation frequency,fro, and the frequency difference, Afm [50]. Figure 9.13 shows this dependence for several modulation frequencies. As can be seen, longitudinal scan speed increases linearly with Afm, and that there is a strong dependence of the scan speed on the modulation frequency, fm. Thus, one would like to operate with some modest modulation frequency, but with a fairly large difference. To determine what longitudinal scan speed could be realized, a series of signals were recorded, using a mirror as the "object," at varying frequency differences. These are shown in Figure 9.14. The measured scan speed does increase proportionally with increasing

10

frn = 3 GHz ~,8

6G H z /

E

v"O 6

(D 00 4

ttO

03 2

100 200 300 400 Frequency Difference (kHz)

FIG. 9.13. Optical-frequency comb-based OCT longitudinal scan speed as a function of frequency difference (Afro) for various modulation frequencies (fm).

USING OPTICAL-FREQUENCY COMBS

Scan Speed 2 5 m s -~

313

Siqnal

I

--4--

Af m = 1 kHz

-JF-

250 rn s -~ Af m = 10 kHz 2.5 km s -~ Af m = 1 0 0 k H z 12.5 km s-~ Af m = 500 kHz

~

.

_

~

j ~ , .

FIG. 9.14. Measured OCT signals at various scan speed (i.e. frequency difference (Afro)).

frequency difference, as predicted. However, it can be seen that resolution and signal-to-noise ratio degrade as the scanning speed increases, because the frequency of detected beat signal was beyond the bandwidth of the photodetector. This problem can, of course, be solved by improving the circuit of the photodetector. Another important practical consideration is the depth resolution of an OCT. The depth resolution, Az, is given by Arc

Az -- - -

n

=

c 2nFfifm

(9.3)

where as before c is speed of light, n is refractive index, At is time fraction of pulse width (Eq. (9.1)), F is finesse, /3 is modulation index, and fm is modulation frequency. Figure 9.15 is a plot of Eq. (9.3) for three different modulation frequencies, where F = 4 0 0 , and n = 1.0. This figure shows that optimal depth resolution is achieved at modest modulation frequencies and indices. To test the depth resolution of this OCT, an experiment was been performed using a glass plate as the object, as shown in Figure 9.16. The light source was a monolithic Nd:YAG laser (1319 nm, 206 mW). OFCG1 and OFCG2 were driven with modulation frequencies of 6.20233 and 6.202319 GHz, respectively (i.e. Afm= 11 kHz). This corresponds to 275 m/s longitudinal scan speed. The one-way optical path length from the front surface of a slide glass to the back surface was 1.32 mm at an incidence angle of 45 ~ as shown in Figure 9.17. It agrees well with the range of resolution since the depth resolution is approximately 100 gm.

314

OPTICAL-FREQUENCY COMB GENERATOR

FIG. 9.16. Experimental arrangement for measuring object thickness. OI: Optical Isolator, BS: Beam Splitter, M: Mirror, O: Object (glass plate), OFCG: Opticalfrequency comb generator, DBM: Dual-balanced mixer, PD: Photodiode, S: Signal processing electronics, fro: modulation frequency, Arm: frequency difference.

These proof-of-principle measurements showed that it should be possible to make a useful optical coherence tomograph using an optical-frequency comb.

9.4 Summary This chapter has reviewed the cavity-enhanced optical-frequency comb generator, which can generate combs with frequency span of up to 50 THz.

REFERENCES

315

Fla. 9.17. Optical-frequency comb-based OCT signals, cf. text for details.

This technology has made possible a new generation of measurements that are testing fundamental physical principles in unprecedented detail and are determining physical constants with unprecedented precision and accuracy. At the same time, this technology may be the key to a new, noninvasive imaging technology for use in situations (ambulances, emergency helicopters) where conventional optical coherence tomographs cannot be used. As optical-frequency comb generation technology advances in the future, we can reasonably expect to find increased applications in scientific and real-world situations. References

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16. 17.

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INDEX

A Acetylene, 307 Acousto-optic modulators, 97 Active devices, 129, 185 Allan variance, 118, 171

E

Electro-optic modulators, 8, 12, 13, 97, 172, 287, 299 Erbium-doped optical fiber amplifier, 303 l~talon, 14

B

Ballistic propagation, 200 Beer-Lambert law, 130 Benzene, 64 Bistability, 260, 268 Bode plots, 20 Brewster angle, 187 Brilluoin scattering, 140 Buildup power, 87

F

Fabry-P6rot, 1, 2, 10, 34, 44, 84, 159, 192, 260, 285, 298 Feedback, 19, 185 Fermi golden rule, 233 Fiber optic, 249 Fine-structure constant, 307 Finesse, 84, 87, 88, 101, 166, 180, 257, 258, 299 Fluorescence emission, 232, 256 Fourier components, 7 Fourier frequency, 19 Fourier spectrum, 198 Fourier transform, 52, 53, 54 Free spectral range, 234, 261,299 Frequency modulation, 141 Fresnel reflectivity, 187

C

C2H2, 115, 307 C2HD, 111, 112, 115, 119 CARS, 73 Cavity Q, 3, 39, 201,230, 255, 281 Cavity-enhanced-absorption, 51, 60 Cesium, 307 CH4, 66, 308 Chaos, 204, 207 CHC13, 64 Circular resonator, 205 Circulating power, 261 CO2, 115, 135 Conversion efficiency, 164, 166

H

HCN, 64 Heisenberg equation, 263 Herzberg bands, 53 Heterodyne, 35, 91, 169 HITEMP, 72 HITRAN, 52, 72 Hollberg, L., 171 Husimi projection, 211 Hydrogen, 161,307 Hydrogen peroxide, 64

D

Density of states, 197, 232 Depth resolution, 313 Deuterium, 307 Diffuse interstellar bands, 69 DiI(3), 238 Diode lasers, 1, 64, 74, 158, 170, 173, 303 Doppler cooling, 276 Dressed states, 265 Dye lasers, 11, 145

I

I2-stablized, 117 Impedance mismatch, 176 Intensity noise, 92 Intracavity laser spectroscopy, 129 321

322 ,l Jaynes-Cummings, 264 K

Kastler, A., 84 Krein-Friedel-Loyd formula, 199 L Lamb, W.E. Jr., 162, 288 Langmuir-Blodgett film, 239 Lehmann, K.K., 64 Local oscillator, 15, 89, 92, 96 Loop transfer function, 21, 23 Lorentz-Mie treatment, 190 M

Magnetic fields, 56 Magnetic rotation, 61 Methane, 65, 308 Metrology, 1, 142, 306 Mie resonance, 227 Minimum detectable absorption, 91, 94, 100, 107, 134 Mirrors, 2, 10, 50, 63, 105, 132, 158, 172, 258, 283 Mode competition, 138, 140 Mode-matching, 9 Modes, 186 Molecular beams, 67 N N20, 64 Nd:YAG, 115, 161, 166 NICE-OHMS, 105 Nitric acid, 64 Noise spectrum, 32, 34 Nonintegrable cavities, 186 O

(02)2, 55 Optical coherence tomography, 309 Optical feedback, 8 Optical fiber, 158, 303 Optical-frequency comb, 297 Optical impedance matching, 9, 176 Optical parametric oscillator, 137 Output power, 266 Overtone, 64, 83, 122, 143 Oxygen, 47, 48, 49, 56, 60, 64

INDEX P

Parasitic effects, 138 Paraxial approximation, 203, 212, 262 Particle microscope, 237 Phase margin, 20 Phase shift, 109, 257 Photocurrent, 15, 100, 110 Photonic atoms, 231 Poincar6 sections, 208 Polarimetry, 72 Pound-Drever-Hall, 5, 7, 11, 97, 160, 172, 287 Predissociation studies, 66 Propyne, 64 Pulse width, 299 Pump field, 162 Purcell, E.M., 199

Q Quantum electrodynamics, 89, 228, 232, 260 Quantum-noise limit, 83 R

Rabi frequency, 86, 266 Radiation pressure, 271 Raman, 73, 157 Raman scattering, 157 Rate equations, 136, 162 Rayleigh length, 262 Rayleigh scattering, 70, 140 Reflected power, 87, 163 Relative intensity noises, 168 Ring-down, 9, 28, 37, 39, 41, 43, 47, 49, 84, 96, 99, 100, 187, 195, 247, 283 S Saturation parameter, 120 Saykally, R.J., 67 Scattering phase shift, 197 Schawlow-Townes limit, 170 Self-phase modulation, 303 Short-wavelength approximation, 185, 203 Shot-noise, 15, 18, 19, 90 Shot-noise limit, 94 Sidebands, 6, 300, 310 Signal-to-noise, 6, 14, 17, 50, 90, 137, 142, 305

INDEX Single-mode, 63 Spectrum analyzer, 31, 32, 169 Spontaneous emission, 255, 272 Stokes field, 162 T Technical noise, 18, 19, 83, 90, 92 Thermal lensing, 179 Threshold power, 164 Total internal reflection, 187 Transient species, 64, 144, 145 Transimpedance amplifier, 14 Transmitted power, 87, 163 Transverse mode, 11, 40, 240, 284

323

W

Water vapor, 64 Weisskopf-Wigner theory, 264 Weyl, H., 202 Whispering-gallery modes, 189, 216, 227, 247 Wigner-Smith delay time, 198 Z Zalicki, P., 53, 65 Zare, R.N., 53

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