Optical Whispering Gallery Modes for Biosensing: From Physical Principles to Applications [1st ed.] 9783030602345, 9783030602352

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Biological and Medical Physics, Biomedical Engineering

Frank Vollmer Deshui Yu

Optical Whispering Gallery Modes for Biosensing From Physical Principles to Applications

Biological and Medical Physics, Biomedical Engineering Editor-in-Chief Bernard S. Gerstman, Department of Physics, Florida International University, Miami, FL, USA Series Editors Masuo Aizawa, Tokyo Institute Technology, Tokyo, Japan Robert H. Austin, Princeton, NJ, USA James Barber, Wolfson Laboratories, Imperial College of Science Technology, London, UK Howard C. Berg, Cambridge, MA, USA Robert Callender, Department of Biochemistry, Albert Einstein College of Medicine, Bronx, NY, USA George Feher, Department of Physics, University of California, San Diego, La Jolla, CA, USA Hans Frauenfelder, Los Alamos, NM, USA Ivar Giaever, Rensselaer Polytechnic Institute, Troy, NY, USA Pierre Joliot, Institute de Biologie Physico-Chimique, Fondation Edmond de Rothschild, Paris, France Lajos Keszthelyi, Biological Research Center, Hungarian Academy of Sciences, Szeged, Hungary Paul W. King, Biosciences Center and Photobiology, National Renewable Energy Laboratory, Lakewood, CO, USA Gianluca Lazzi, University of Utah, Salt Lake City, UT, USA Aaron Lewis, Department of Applied Physics, Hebrew University, Jerusalem, Israel Stuart M. Lindsay, Department of Physics and Astronomy, Arizona State University, Tempe, AZ, USA Xiang Yang Liu, Department of Physics, Faculty of Sciences, National University of Singapore, Singapore, Singapore David Mauzerall, Rockefeller University, New York, NY, USA Eugenie V. Mielczarek, Department of Physics and Astronomy, George Mason University, Fairfax, USA

Markolf Niemz, Medical Faculty Mannheim, University of Heidelberg, Mannheim, Germany V. Adrian Parsegian, Physical Science Laboratory, National Institutes of Health, Bethesda, MD, USA Linda S. Powers, University of Arizona, Tucson, AZ, USA Earl W. Prohofsky, Department of Physics, Purdue University, West Lafayette, IN, USA Tatiana K. Rostovtseva, NICHD, National Institutes of Health, Bethesda, MD, USA Andrew Rubin, Department of Biophysics, Moscow State University, Moscow, Russia Michael Seibert, National Renewable Energy Laboratory, Golden, CO, USA Nongjian Tao, Biodesign Center for Bioelectronics, Arizona State University, Tempe, AZ, USA David Thomas, Department of Biochemistry, University of Minnesota Medical School, Minneapolis, MN, USA

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Frank Vollmer Deshui Yu •

Optical Whispering Gallery Modes for Biosensing From Physical Principles to Applications

123

Frank Vollmer Physics and Astronomy University of Exeter Exeter, UK

Deshui Yu Physics and Astronomy University of Exeter Exeter, UK

ISSN 1618-7210 ISSN 2197-5647 (electronic) Biological and Medical Physics, Biomedical Engineering ISBN 978-3-030-60234-5 ISBN 978-3-030-60235-2 (eBook) https://doi.org/10.1007/978-3-030-60235-2 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

I first learned about the whispering-gallery modes (WGM) as a Ph.D. student in Prof. Albert Libchaber’s laboratory, which was part of the Center for the studies in Physics and Biology at the Rockefeller University in NYC, around the year 2000. I attended a physics lecture that was held as part of the Center’s seminar. The lecturer, Prof. Stephen Arnold from Brooklyn Polytechnic, introduced the fascinating experiments with WGMs to the audience. He described the experiments that analyzed the WGMs in the fluorescence spectra of liquid microdroplets. Professor Richard K. Chang at Yale University was a pioneer of such studies. The WGMs were investigated through the light emission from the fluorescent droplets that were excited by laser light. The droplets were generated, for example, by a piezoelectric droplet generator. Some researchers also levitated the charged microdroplets in a Paul trap. Then, they could carefully study the morphology-dependent resonances in well-controlled and nearly perfect spherical microdroplet shapes. Could one use the enhanced light–matter interactions of WGMs for the ultra-sensitive biodetection? I found this question very intriguing. The challenges to build such a biosensor were formidable. One would have to find a way to replace the liquid microdroplet with one made of glass. Furthermore, one had to apply the laser interferometry to read out the slight changes in the WGM spectra. The glass-made WGM sensors would have to provide a very high finesse and quality factor. The WGM sensors also had to operate in an aqueous environment, without the use of fluorescently labeled molecules in order to find broad applications in biology. I decided to take on these challenges and embarked on my own research toward developing a WGM single-molecule biosensor. Little did I know that it would take much more than a Ph.D. thesis to achieve this goal. Demonstrating single-molecule sensing with WGMs in my own laboratory would require the contributions from many excellent Ph.D. students, postdocs, undergraduate researchers, and collaborators that I had the pleasure of working with over the next 20 years. After initial demonstrations of WGM biosensing with glass microspheres at Rockefeller University, the first milestone for single nanoparticle detection was achieved at Harvard. In my laboratory at the Rowland Institute at Harvard University, we set up an experiment to demonstrate single nanoparticle detection. This was around the v

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year 2008. We demonstrated the detection of single Influenza A virus particles, which are virions with about 100 nm in diameter. These studies were a great step forward. They also showed the limitations of the technique. A novel idea was needed in order to dramatically boost the WGM detection sensitivity. A factor of more than 1000 was needed to boost the WGM sensor signal to enable the detection of a 5 nm protein. In 2010, a powerful idea for boosting the WGM frequency shift signal in response to a single protein emerged. The idea was to use plasmonic nanoparticles to enhance the interaction of light with a single protein. This idea prepared the ground for Optoplasmonic WGM sensing. Optoplasmonic sensing led to dramatic advances in single-molecule detection. Around the year 2016, my laboratory at Max Planck Institute in Germany demonstrated the detection of single atomic zinc and mercury ions in aqueous solution. The characterization of single-molecule reactions of small 77 Da cysteamine molecules with reactant gold as well as other disulfide moleules was demonstrated in 2020 at University of Exeter. This study also demonstrated the detection of single molecules at attomolar concentrations. The research on WGM biosensing has always been very exciting. It combines interesting physics with emerging applications in biophysics and biology. The idea for writing a book about the physics of WGMs and their applications in sensing and biosensing emerged around the year 2011. I started teaching an interdisciplinary course on the physics of biosensing with WGMs. A more careful review of the subject showed that since 2000, many important advances in physics were made with WGMs. The field of cavity optomechanics had emerged; nonlinear optical effects were studied with WGMs including multi-wave mixing, parameteric downconversion, and frequency comb generation. The demonstration of strong coupling between atoms and molecules with WGMs sparked the interest for using WGMs in cavity QED. A number of world-leading laboratories joined the effort and made seminal contributions in the research area of optical microcavities. The WGMs provide a particularly versatile experimentation platform for those researchers to study the fundamentals of light–matter interactions, from the classical to the quantum regimes. Many of the groundbreaking physics advances were made with WGMs, and almost always they point to an interesting application in biosensing. Examples for this are the exceptional points and the quantum light generated or collected with WGMs that might be able to further enhance detection sensitivities of single molecules. A comprehensive book on WGM biosensing would have to cover all of these exciting fundamental physics of WGMs. Book chapters on the fundamental understanding of the sensing with light, the physics of WGM in optical microcavities, the application of WGM microcavities in physics, and the fundamentals of quantum optics and molecular cavity QED would be needed. To begin with, let alone completing such a comprehensive book on the physics of WGMs and their biosensing applications did not come about easily. It required the right co-author. After I moved my laboratory to University of Exeter, UK, in 2016, the right co-author joined my group: Dr. Deshui Yu. Deshui is an exceptionally talented theorist trained at Peking University. Deshui joined my group at University of Exeter in 2018 to work on the theory of WGMs. Deshui was very excited when I mentioned the book project to him. Although the idea for the book was still in its very early stages

Preface

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even at that time, Deshui was very enthusiastic to join the project. Deshui is not only one of the finest theorists I met, but also an exceptional lecturer of physics. Deshui can find simple explanations for complicated physics. He is able to explain complicated physics without taking shortcuts. Effortlessly, he is able to bridge theory and experiment. He engages with the researchers from other fields and disciplines. A little more than a year later after Deshui joined, the first draft for our book was ready. I am deeply indebted to Deshui for his hard work and commitment to completing a high-quality book. Deshui’s immense contributions resulted in the inclusion of advanced theoretical chapters that are essential for a deep understanding of the WGM physics. The book now fully conveys the excitement and appreciations for this exciting research topic. The book covers most of the fundamental WGM physics and combines them with the chapters on sensing with light, surface plasmon resonance, and single molecule sensing. The book touches not only on core subjects in physics, but also on some fundamental aspects of biophysics and chemistry. It is, we hope, an inspiring and comprehensive read for an interdisciplinary readership. I quote Deshui: “Science without theory is lame, theory without experimentation is blind”. Frank Vollmer We hope this book will be a valuable resource for teaching the physics of optical microcavities. We also hope for it to become a valuable resource for the active users of WGMs, in their various applications including biosensing. The future for WGM sensors is bright and just emerging. WGM sensors will help us to explore the biophysics and biochemistry at the very limits of what humans are capable of investigating. They are a platform to develop various nano- and quantum sensing methodologies. They can allow us to explore Nature’s smallest entities such as single molecules, single photons, and femto-Newton forces, as well as intricate molecular optical properties such as single-molecule chirality. Researchers may find this book useful for developing novel WGM sensing techniques, such as those that may further enable the detection and visualization of processes at the nanoscale, at ultrahigh sensitivity, and in a specific and sensitive manner, down to the level of molecules, atoms, and bonds. The authors believe that WGM sensors and spectrometers will have many important applications in health, nanotechnology, metrology, environment, biology, defense and security, and astronomy. We hope this book may aide in the development of these next-generation biosensor applications, and that the book may help extend these applications to include chip-scale single-molecule laboratories for clinical diagnostics that provide maximum information by analyzing a biological sample molecule-by-molecule. In summary, we would like to thank Sequoia Alba Webster and Mithil Parekh for contributing to the very early stages of the book project. The authors thank all who have directly and indirectly contributed to the successful completion of the book. Thank you too, Mayumi Noto and Dongjie Wang. Exeter, UK

Frank Vollmer Deshui Yu

Contents

1 Sensing with Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Light Propagation in Space . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Vector and Scalar Potentials . . . . . . . . . . . . . . . . 1.2.2 Wave Equations for Electric and Magnetic Fields 1.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 1.3 Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Electric Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Local Field Correction . . . . . . . . . . . . . . . . . . . . 1.3.3 Lorentz Oscillator Model . . . . . . . . . . . . . . . . . . 1.4 Frequency and Intensity Fluctuations of Light . . . . . . . . . 1.4.1 Autocorrelation Function and Spectrum . . . . . . . . 1.4.2 Lorentzian and Gaussian Broadening . . . . . . . . . . 1.4.3 Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Hanbury Brown–Twiss Effect . . . . . . . . . . . . . . . 1.5 Frequency Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Fabry–Pérot Resonator . . . . . . . . . . . . . . . . . . . . 1.5.2 LC Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Laser Frequency Locking to Transmission Line . . 1.5.4 Pound–Drever–Hall Technique . . . . . . . . . . . . . . 1.5.5 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.6 Optical Heterodyne and Homodyne Detection . . . 1.5.7 Measuring the Photon Number . . . . . . . . . . . . . . 1.6 Optical Sensing Technologies . . . . . . . . . . . . . . . . . . . . . 1.6.1 Michelson–Morley Interferometer . . . . . . . . . . . . 1.6.2 Optical Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Single-Atom Detection . . . . . . . . . . . . . . . . . . . . 1.6.4 Optical Tweezers . . . . . . . . . . . . . . . . . . . . . . . .

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Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Whispering Gallery Modes in Optical Microcavities . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Microspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Strict Analytical Eigenmodes . . . . . . . . . . . 3.2.2 Analytical WGM Frequencies . . . . . . . . . . . 3.2.3 Effective Mode Volume . . . . . . . . . . . . . . . 3.2.4 Approximate Solutions . . . . . . . . . . . . . . . . 3.2.5 Intrinsic Quality Factor . . . . . . . . . . . . . . . . 3.2.6 Excitation of WGMs . . . . . . . . . . . . . . . . . 3.3 Microbottles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Microdisks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Microtoroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Thermorefractive-Noise-Limited Frequency Stability . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Applications of WGM Microcavities in Physics 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 Laser Generation . . . . . . . . . . . . . . . . . . . 4.3 Nonlinear Frequency Conversion . . . . . . . .

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2 Surface Plasmon Resonance . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Plasmonics: Interaction of Light with Metals . . . . 2.2.1 Drude–Sommerfeld Model . . . . . . . . . . . 2.2.2 Interband Transitions of Bound Electrons 2.2.3 Drude–Lorentz Model . . . . . . . . . . . . . . 2.3 Surface Plasmon Polaritons—SPPs . . . . . . . . . . . 2.3.1 SPPs at Plane Interfaces . . . . . . . . . . . . . 2.3.2 Excitation of SPPs . . . . . . . . . . . . . . . . . 2.3.3 SPR Sensors . . . . . . . . . . . . . . . . . . . . . 2.3.4 Surface Plasmon Optics . . . . . . . . . . . . . 2.4 Localized Surface Plasmon Resonances—LSPRs . 2.4.1 Spherical Nanoparticles . . . . . . . . . . . . . 2.4.2 Energy Stored in Metal Nanoparticles . . . 2.4.3 Nanoellipsoids . . . . . . . . . . . . . . . . . . . . 2.4.4 Mie Theory of Light Scattering . . . . . . . . 2.4.5 Numerical Methods for LSPRs . . . . . . . . 2.5 Plasmonic Coupling of Nanoparticles . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.1 Pockels Effect . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Multiple-Wave Mixing . . . . . . . . . . . . . . . . . . . 4.3.3 Second-Order Nonlinearity . . . . . . . . . . . . . . . . 4.3.4 Frequency Doubling . . . . . . . . . . . . . . . . . . . . . 4.3.5 Parametric Down-Conversion . . . . . . . . . . . . . . 4.3.6 Third-Order Nonlinearity . . . . . . . . . . . . . . . . . 4.4 Parity-Time Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Parity and Time-Reversal Transformations . . . . . 4.4.2 Spectral Reality of Non-Hermitian Hamiltonians 4.4.3 Parity-Time Symmetry in Optics . . . . . . . . . . . . 4.4.4 Observation in Experiments . . . . . . . . . . . . . . . 4.5 Electromagnetically Induced Transparency . . . . . . . . . . . 4.6 Optical Frequency Combs . . . . . . . . . . . . . . . . . . . . . . . 4.7 Optomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Classical Description . . . . . . . . . . . . . . . . . . . . 4.7.2 Acoustic Modes in Microcavities . . . . . . . . . . . 4.7.3 Optical Measurement of Mechanical Motion . . . 4.7.4 Optomechanical Cooling . . . . . . . . . . . . . . . . . . 4.8 Nanoparticle Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Single-Molecule Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sensing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 WGM Microcavity Perturbed by Single Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 WGM Microcavity Perturbed by a Nanolayer . . . . 5.3 Chemical Reaction Analysis . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Kinetics of Association and Dissociation Reactions 5.3.2 Determination of Kinetic Constants . . . . . . . . . . . . 5.3.3 Thermodynamic Analysis . . . . . . . . . . . . . . . . . . . 5.4 Sensing Based on Passive WGM Microcavities . . . . . . . . . 5.4.1 Experimental Demonstrations . . . . . . . . . . . . . . . . 5.4.2 Mode Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Sensing Based on Active Microcavities . . . . . . . . . . . . . . . 5.6 Other Schemes of Single Particle Detection . . . . . . . . . . . . 5.6.1 Optomechanical Sensing of Single Molecules . . . . 5.6.2 Exceptional Point-Based Sensor . . . . . . . . . . . . . . 5.6.3 Self-referenced Photonic Molecule Biosensor . . . . . 5.7 Practical Limits of Sensing Detection . . . . . . . . . . . . . . . . . 5.8 Optoplasmonic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 LSPR Enhanced Local Electric Field . . . . . . . . . . . 5.8.2 Detection of Microsphere–Nanorod Interaction . . . .

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5.8.3 5.8.4 5.8.5 5.8.6 5.8.7 5.8.8

Optoplasmonic Sensing Mechanism . . . . . . . . Detection of Nucleic Acid Hybridization . . . . Detection of Ion–Nanorod Interactions . . . . . Detection of Protein Conformational Changes and Chemical Reactions . . . . . . . . . . . . . . . . Attomolar Detection of Single Molecules . . . Characterizing Thiol Gold and Amine Gold Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . 293 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

6 Fundamentals of Quantum Optics . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Light–Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Rabi Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Rabi and Ramsey Measurements . . . . . . . . . . . . . . . . . . . 6.5 Quantization of Electromagnetic Field . . . . . . . . . . . . . . . 6.6 Fock States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Photon Correlation Functions . . . . . . . . . . . . . . . . . . . . . 6.8 Interaction Between Single Emitter and Photons . . . . . . . . 6.9 Multiple Emitters Interacting with Photons . . . . . . . . . . . . 6.10 Spontaneous Emission of Emitter in Free Space . . . . . . . . 6.11 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.1 Two-Level Emitter . . . . . . . . . . . . . . . . . . . . . . . 6.11.2 Dynamical Polarizability of an Emitter . . . . . . . . 6.11.3 EIT in K-Type Emitters . . . . . . . . . . . . . . . . . . . 6.11.4 Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.5 Interaction-Induced Spectral Shift and Broadening 6.11.6 Diagonalization Method . . . . . . . . . . . . . . . . . . . 6.11.7 Two-Time Correlation Functions . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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299 299 300 302 308 310 314 317 319 322 324 326 327 332 333 335 337 339 340 343 344

7 Molecular Cavity QED . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Purcell Effects . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Molecular Emitters . . . . . . . . . . . . . . . . . . . . . 7.4.1 Dipole Moment . . . . . . . . . . . . . . . . . 7.4.2 Jablonski Diagram . . . . . . . . . . . . . . . 7.4.3 Franck–Condon Principle . . . . . . . . . . 7.4.4 Absorption and Emission Spectroscopy 7.4.5 Beer–Lambert Law . . . . . . . . . . . . . . . 7.4.6 Einstein Coefficients . . . . . . . . . . . . . .

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Contents

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7.5

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Molecular Cavity QED in Experiment . . . . . . . . . . . . . 7.5.1 Single Molecules Interacting with Plasmonic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Strong Coupling Between Molecules and Microcavities . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Molecular Microlasers . . . . . . . . . . . . . . . . . . 7.5.4 Single Molecule Interacting with a Microcavity 7.6 Heisenberg-Limited Quantum Metrology . . . . . . . . . . . 7.6.1 Heisenberg Uncertainty and Squeezed States . . 7.6.2 Heisenberg-Limited Phase Measurement . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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365 365 367 369 370 375 378 381

Appendix A: Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Appendix B: Cylindrical and Spherical Coordinate Systems . . . . . . . . . . 389 Appendix C: Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Appendix D: Diagonalization of the Two-Dimensional Matrix . . . . . . . . 401 Appendix E: Eigenmodes of Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . 403 Appendix F: Solving the First-Order Linear Matrix Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Chapter 1

Sensing with Light

Abstract The interaction of highly coherent light and matter is the key in the science of measurement. In the classical electromagnetic theory, the absorption and dispersion properties of the light propagation in a medium are described by the classical Lorentz oscillator model. The unavoidable phase and intensity fluctuations of a light field, which are commonly characterized by the first- and second-order correlation functions, respectively, broaden the light spectrum and restrict its practical applications in metrology. In order to suppress the spectral linewidth, the light’s frequency is usually stabilized to an optical resonator. Several light-based sensing applications, such as the gravity-wave detection and optical clocks, are briefly introduced.

1.1 Introduction Owing to the unique features of extremely large frequency (1014 Hz), high coherence (monochromaticity), and light quanta (photons), laser light has been extensively applied in modern physics, chemistry, biology, and information engineering. The well-established techniques for laser cooling and trapping of neutral atoms lead to the realization of the fifth state of matter, i.e., Bose–Einstein condensate [1, 2], which is currently one of the most active fields in condensed matter physics. The optical frequency standards (i.e., a well-stabilized laser oscillation) have much surpassed the microwave-frequency clocks, based on which the second is defined, in both stability and accuracy [3, 4]. The giant optical interferometer LIGO enables the observation of the gravitational waves from the merger of black holes [5, 6], which were first predicted by Einstein’s general theory of relativity in 1916. Nowadays, the optical tweezers, formed by a tightly focused laser beam, have become a common tool in biology to ‘pin down’ or immobilize the live cells [7]. The sensing with the single-biomolecule resolution has been achieved based on the platform of the light propagating inside ultrahigh-quality-factor microresonators [8]. The rapid chemical reactions may be monitored and controlled by means of using the ultrashort light

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2_1

1

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1 Sensing with Light

pulses, giving the birth of a new branch of chemistry, i.e., femtochemistry [9]. Moreover, the quantum teleportation [10] and quantum key distribution [11] between satellites and ground stations with a distance over one thousand kilometers have been demonstrated by using the flying entangled photons, paving the way toward a global-scale quantum network [12]. All these applications are attributed to the intricate light–matter interaction. Different mathematical formalisms have been developed to interpret the various interaction processes. Light can behave as either an electromagnetic wave or a flux of particles (photons), i.e., the so-called wave–particle duality. Most macroscopic optical phenomena, such as the reflection, refraction, diffraction, and absorption of light, can be understood through the classical theory of electromagnetism that focuses on the wave nature of light. In this picture, the microscopic particles (i.e., atoms and molecules) comprising the medium are modeled by the classical Lorentzian oscillators. The response of the medium to the light field is determined by its relative permittivity. In contrast, in some cases the detailed internal electronic states of the particles must be considered and the Lorentzian model fails. Thus, the semiclassical theory, where the particles are described in a quantum mechanical way while the light field is still treated as an electromagnetic wave, should be applied. This method has been widely adopted in nonlinear optics and laser physics. In particular situations, such as squeezing of light, single photon emission and one-atom lasing, the particle nature of light plays a more important role than its wave nature. Consequently, the full quantum theory, where both light and particles are considered by using the quantum mechanics, is essential to understand the nonclassical behavior of the interaction between light and matter. This theoretical framework is fundamental to quantum optics, which has been vigorously developed over past several decades. In this chapter, we introduce the classical theory of the light–matter interaction. The semiclassical and full quantum theory will be discussed in the following chapters. The starting section is given to the introduction of Maxwell’s equations, which are the fundamental basis of the light traveling in the medium. We then consider the polarization of the medium in an external light field by using the Lorentz oscillator model, where the electric-field component of the light periodically stretches and compresses a virtual spring that links the electron to the heavy nucleus. As we will see, the macroscopic optical properties of matter are attributed to the microscopic polarizability of the particles. Next, we focus on how to measure the quality of light, e.g., phase and intensity fluctuations, by using the correlation functions and the common approaches of stabilizing the light’s frequency. In the last section, we briefly introduce the sensing applications of the optical waves.

1.2 Light Propagation in Space

3

1.2 Light Propagation in Space All the phenomena associated with the classical electricity and magnetism are governed by a set of coupled differential equations, i.e., Maxwell’s equations: ∂ B, ∂t ∂ Ampere’s circuital law: ∇ × H = J f + D, ∂t Gauss’s law for electricity: ∇ · D = ρ f , Gauss’s law for magnetism: ∇ · B = 0, Faraday’s law of induction: ∇ × E = −

(1.1a) (1.1b) (1.1c) (1.1d)

with the electric field strength E(r, t) (measured in volts per meter, V/m), the electric displacement field (in units of coulomb per square meter, C/m2 ) D(r, t) = ε0 E(r, t) + P(r, t),

(1.2)

the medium’s polarization density P(r, t) (defined as the dipole moment per unit volume), the magnetic field strength H(r, t) (measured in amperes per meter, A/m), the magnetic flux density/magnetic induction B(r, t) (measured in teslas, symbol: T), the free charge density ρ f , the free current density J f , the electric permittivity of free space ε0 = 8.854 × 10−12 F/m (farads per meter), and the magnetic permeability of free space μ0 = 4π × 10−7 H/m (henries per meter). The macroscopic polarization P(r, t) depends proportionally on the electric field E(r, t) via P = ε0 χe · E,

(1.3)

with the electric susceptibility (rank-2) tensor χe of the medium. The relation between B(r, t) and H(r, t) is written as B = μ0 (H + M) = μr μ0 H,

(1.4)

with the magnetization vector M and the dimensionless relative permeability μr . The speed of light in vacuum c is directly linked to the constants ε0 and μ0 via c= √

1 . ε0 μ0

(1.5)

Equations (1.1a)–(1.1d) are the formulation in International System of Units (SI), which uses the MKS (meter-kilogram-second) system. In some literatures, the Gaussian units convention is applied, where the CGS (centimeter-gram-second) system is used. In Gaussian units, the expression of the Gauss’s law for magnetism (1.1d) does not change while the other three Maxwell’s equations take the form

4

1 Sensing with Light

∇ ×E = −

1 ∂ B, c ∂t 

(1.6a) 

1 ∂ 4πJ f + D , c ∂t ∇ · D = 4πρ f .

∇ ×H =

(1.6b) (1.6c)

The SI and Gaussian units of energy are joule (symbol: J) and erg, respectively, with 1 erg = 10−7 J. In the rest of this book, the SI units are applied. Maxwell’s equations are the foundation of the classical optics and the electric circuits. In this book, we only focus on the nonmagnetic, homogeneous, linear, and isotropic media, for which μr = 1 and the linear polarization P(r, t) are linked to E(r, t) via (1.7) P = χe ε0 E, with the scalar susceptibility χe . (Note: the ‘homogeneous’ denotes the material has a uniform composition and properties throughout the sample while the ‘isotropic’ means the material’s properties are the same in all directions.) Thus, the electric displacement D(r, t) is rewritten as D = ε0 E,

(1.8)

 = 1 + χe .

(1.9)

with the relative permittivity The dielectric constant  is real for a non-dissipative medium, whose refractive index √ is given by n = . Moreover, one may derive the continuity equation ∂ ρ f + ∇ · J f = 0, ∂t

(1.10)

which denotes the conservation of the local charge, from Maxwell’s equations. The operations of the differential operator ∇ on the scalar and vector fields have been summarized in Appendix A.

1.2.1 Vector and Scalar Potentials We consider the electromagnetic waves propagating in free space,  = 1. Since the magnetic field B(r, t) is a divergence-free vector [see (1.1d)], one can relate B(r, t) to a vector potential A(r, t) B = ∇ × A, (1.11) due to the identity ∇ · (∇ × A) = 0. Inserting (1.11) into Faraday’s law (1.1a) leads to

1.2 Light Propagation in Space

5

  ∂ ∇ × E + A = 0, ∂t

(1.12)

for which a scalar potential φ(r, t) may be introduced to re-express the electric field E(r, t) as ∂ E = −∇φ − A, (1.13) ∂t because of the identity ∇ × (∇φ) = 0. We then rewrite the rest two Maxwell’s equations in terms of A(r, t) and φ(r, t) as ρf ∂ ∇ 2 φ + (∇ · A) = − , ∂t ε0     2 ∂ 1 1 ∂ 2 ∇ A − 2 2 A − ∇ ∇ · A + 2 φ = −μ0 J f , c ∂t c ∂t

(1.14a) (1.14b)

by using the identities ∇ × (∇ × A) = ∇(∇ · A) − ∇ 2 A, ∇ · ∇φ = ∇ φ. 2

(1.15a) (1.15b)

The definitions (1.11) and (1.13) are not unique. Indeed, the electric E(r, t) and magnetic B(r, t) fields are unaffected under the transformation ˜ = A + ∇ψ, φ˜ = φ − ∂ ψ, A ∂t

(1.16)

with an arbitrary function ψ(r, t), i.e., the so-called gauge invariance [13]. Nonetheless, such a freedom enables us to choose a certain gauge to simplify the mathematical formalism. One of commonly-adopted gauges in the classical electromagnetic theory is the Lorenz gauge, 1 ∂ (1.17) ∇ · A + 2 φ = 0. c ∂t Substituting the above equation into (1.14a) and (1.14b) leads to the wave equations for the scalar and vector potentials ρf 1 ∂2 φ= , 2 2 c ∂t ε0 1 ∂2 −∇ 2 A + 2 2 A = μ0 J f . c ∂t − ∇2φ +

(1.18a) (1.18b)

It is seen that the free charge ρ f is the source of the scalar potential φ while the free current J f is the source of the vector potential A.

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1 Sensing with Light

Another gauge that is also often applied is the Coulomb gauge (also known as the transverse gauge) ∇ · A = 0, (1.19) under which (1.14a) and (1.14b) are reduced to ρf , ε0

(1.20a)

1 ∂2 1 ∂ A = −μ0 J f + 2 ∇φ. c2 ∂t 2 c ∂t

(1.20b)

∇2φ = − ∇2A −

Equation (1.20a) takes the form of the Poisson equation, which leads to the solution of the instantaneous Coulomb potential 1 φ(r, t) = 4πε0



ρ f (r , t)  dr . |r − r |

(1.21)

We then decompose the current density J f into the transverse (rotational) J f,⊥ and longitudinal (irrotational) J f, components, J f = J f,⊥ + J f, , with ∇ · J f,⊥ = 0, ∇ × J f, = 0.

(1.22)

Using the vector identity ∇ × (∇ × J f ) = ∇(∇ · J f ) − ∇ 2 J f , one finds J f,⊥ and J f, satisfy the Poisson’s equations ∇ 2 J f,⊥ = −∇ × ∇ × J f , ∇ 2 J f, = ∇(∇ · J f ),

(1.23)

whose solutions are  J f (r , t)  1 ∇ ×∇ × dr , J f,⊥ (r, t) = 4π |r − r |  ∇  · J f (r , t)  1 dr . J f, (r, t) = − ∇ 4π |r − r |

(1.24) (1.25)

In deriving the above solutions, we have used (see Problem 1.3) ∇2

1 = −4πδ(r − r ) |r − r |

(1.26)

with the Dirac delta function1 δ(r). In addition, the continuity equation (1.10) leads to 1 The

one-dimensional Dirac delta function is defined as  ∞ x = x0 δ(x − x0 ) = , 0 x = x0

1.2 Light Propagation in Space

7

1 ∂ ∇φ = μ0 J f, . c2 ∂t Consequently, the vector potential A(r, t) only owns the transverse (rotational) component A⊥ (r, t) and (1.20b) reduces to ∇ 2 A⊥ −

1 ∂2 A⊥ = −μ0 J f,⊥ . c2 ∂t 2

(1.27)

It is convenient to use the Coulomb gauge when no sources are present, i.e., φ = 0.

1.2.2 Wave Equations for Electric and Magnetic Fields We restrict ourselves to the situation of no free charge and current J f = ρ f = 0. According to Maxwell’s equations in free space ( = 1), both E(r, t) and B(r, t) satisfy the same three-dimensional wave equation 1 ∂2 E(r, t) = 0, c2 ∂t 2 1 ∂2 ∇ 2 B(r, t) − 2 2 B(r, t) = 0. c ∂t ∇ 2 E(r, t) −

(1.28a) (1.28b)

The plane-wave solution of the electric field to (1.28) may be written as E(r, t) = E(+) (r, t) + E(−) (r, t) E∗ E0 = e eik·r−iωt + e∗ 0 e−ik·r+iωt , 2 2 with the integral

 a

b

(1.29)

 1 a < x0 < b . δ(x − x0 )d x = 0 x0 < a or x0 > b

The delta function has the scaling and symmetry properties, δ(αx) = δ(x)/|α| and δ(−x) = δ(x). In the three-dimensional case, the delta function takes the form δ(r − r0 ) = δ(x − x0 )δ(y − y0 )δ(z − z 0 ). The volume integral of δ(r − r0 ) against a continuous function f (r) reads as   f (r0 ) r0 ∈ V f (r)δ(r − r0 )d 3 r = . /V 0 r0 ∈ V .

8

1 Sensing with Light

Fig. 1.1 Transverse optical wave. The electric field E, magnetic field B, and wavevector k form a mutually orthogonal set. The Poynting vector S is parallel to k

with the unit polarization vector e, wavevector k, amplitude E 0 , frequency ω = 2πc/λ, and wavelength λ. We see that E(r, t) consists of two parts, where E(+) (r, t) denotes the positive-frequency (e−iωt ) part while E(−) (r, t) corresponds to the negative-frequency part (eiωt ). For the light (optical wave), which is a transverse electromagnetic wave, its polarization is perpendicular to the direction of propagation (along k), e · k = 0. The associated magnetic field component is orthogonal to but in phase with E(r, t), k × E(r, t) . (1.30) B(r, t) = ω Actually, the directions of E, B, and k obey the right-hand rule as displayed in Fig. 1.1. The light propagation represents an directional energy flux. The Poynting vector is usually used to describe the energy transfer per unit area perpendicular to the propagation direction of the light within unit time, S(r, t) = E(r, t) × H(r, t) =

1 E(r, t) × B(r, t). μ0

(1.31)

The light intensity, which refers to the time-averaged power transferred per unit area, is then calculated as I =

cε0 k 1 · S(r, t) = |E 0 |2 = |E 0 |2 , |k| 2 2cμ0

(1.32)

where we have used the identity a × b × c = b(a · c) − c(a · b) and the symbol  f (t) denotes the time average of a periodic function f (t) within one period. The intensity may be alternatively expressed as I = cε0 E(r, t)2 =

1 E(r, t)2 . cμ0

(1.33)

1.2 Light Propagation in Space

9

The above expression is more often used in practice. It should be pointed out that although I is written in terms of the electric field E, the intensity actually includes the contributions from both E and B. In addition, from Maxwell’s equations we have H · ∇ × E = −μ0 H · E · ∇ × H = ε0 E ·

∂ H, ∂t

∂ E. ∂t

(1.34a) (1.34b)

Equation (1.34a) minus (1.34b) leads to ∂ W + ∇ · S = 0, ∂t

(1.35)

with the electromagnetic energy density defined as W =

  1 1 1 (ε0 E · E + μ0 H · H) = ε0 E · E + B · B . 2 2 μ0

(1.36)

In deriving the above equation, we have used the cross-product rule for two arbitrary vectors X and Y ∇ · (X × Y) = (∇ × X) · Y − X · (∇ × Y).

(1.37)

Equation (1.35) is called Poynting’s theorem [14], which indeed represents the energy conservation law. The total energy density W consists of both electric and magnetic components. Using (1.29) and (1.30), one finds that the electric- and magnetic-field energy densities are equal 1 B · B. (1.38) ε0 E · E = μ0 This conclusion is important and useful to simplify the calculation. In practice, only the light intensity/power is measurable in experiment, and one may only have to focus on either electric E or magnetic B field to compute the light intensity/power. The average value of W over one oscillation period is related to the light intensity I via W¯ ≡ W (t) = I /c. (1.39) When the light travels in a homogeneous dielectric (an electrical insulating material), the wave equation of the electric field can be derived from Maxwell’s equations as ∂2 1 ∂2 (1.40) ∇ 2 E(r, t) − 2 2 E(r, t) = μ0 2 P(r, t). c ∂t ∂t

10

1 Sensing with Light

The nonzero polarization density of the medium P(r, t) affects the light propagation, such as absorption/amplification, dispersion, confinement, and scattering, depending on the specific light–matter interaction process. For an isotropic medium, P(r, t) is linearly proportional to the local electric field E(r, t), and in the absorptionless limit (1.40) is rewritten as 1 ∂2 ∇ 2 E(r, t) − 2 2 E(r, t) = 0, (1.41) v ∂t √ with the light velocity in dielectric v = c/n. Here, n =  is the refractive index of the medium. The plane-wave solution of (1.41) takes the same form of (1.29) except the revised dispersion relation ω = v|k|. We have the wavevector |k| = n|k0 | and the wavelength λ = λ0 /n of the light traveling in dielectric, where k0 denotes the wavevector of the light in vacuum with the corresponding wavelength λ0 . In addition, the associated magnetic component has the expression same as (1.30). Similarly, one may derive the light intensity I =

1 vε0 |E 0 |2 = |E 0 |2 . 2 2vμ0

(1.42)

The Poynting’s theorem (1.35) is still valid for the optical wave in dielectric, and the energy density reads as 1 W = (E · D + H · B), (1.43) 2 whose time-averaged value is related to the intensity I via W¯ = I /v.

(1.44)

Again, the electric and magnetic fields share the energy equally. We should point out that (1.43) is only applicable to the materials with Re() > 0 and Im() = 0 (nondissipative). For the light traveling in a medium with a negative Re() or nonzero Im(), Poynting’s theorem must be re-derived from Maxwell’s equations by taking into account the specific expressions of E(r, t), B(r, t), and P(r, t) with the complex electric susceptibility χe (see Sect. 2.4.2).

1.2.3 Boundary Conditions When a light propagation encounters the interface between two dielectrics with different relative dielectric constants, the reflection and refraction (transmission) processes occur. The relevant mechanism is embedded in the boundary conditions for the electric and magnetic fields. We consider the interface between two homogeneous media with i=1,2 . The unit normal vector n12 is located at the interface and points from the medium 1 to the medium 2 . The specific boundary conditions may be summarized as follows [15]:

1.3 Polarizability

11

(1) The tangential component of the electric field E is continuous across the interface, (1.45) n12 × (E2 − E1 ) = 0. Here, Ei=1,2 denotes the electric field at the interface on the i -medium side. The condition (1.45) can be directly derived from Faraday’s law of induction (1.1a) via integrating over a surface bounded by a closed loop across the interface and applying Stokes’ theorem. (2) The normal component of the electric displacement field D owns a step value of the surface charge density ρ f at the interface between two media, n12 · (D2 − D1 ) = ρ f .

(1.46)

This condition can be obtained from Gauss’s law for electricity (1.1c) by integrating over a small volume across the interface and using Gauss’s (divergence) theorem. In particular, the normal component of D varies continuously across the interface when no charge exists at the surface. (3) The normal component of the magnetic flux density B is continuous across the interface, (1.47) n12 · (B2 − B1 ) = 0. This condition comes from Gauss’s law for magnetism (1.1d) via integrating over a small volume across the interface and applying Gauss’s theorem. (4) The tangential component of the magnetic field strength H varies discontinuously at the interface due to the surface current density J f , n12 × (H2 − H1 ) = J f .

(1.48)

Integrating over a closed loop across the interface and using Stokes’ theorem, Ampère’s law (1.1b) leads to the condition (1.48). In the absence of J f , the tangential component of H is continuous across the boundary. The above four boundary conditions are the direct results by applying Maxwell’s equations at the interface [see Problem 1.2]. We should point out that they are also valid to the metal–dielectric interface. In the following of this book, we set both free charge and current to zero, J f = ρ f = 0.

1.3 Polarizability As we have pointed out, the optical property of nonmagnetic matter is mainly determined by its polarization P [see (1.40)], more precisely speaking, the dynamical response of the particles that make up the matter to an external electric field. We consider a dielectric insulator composed of atoms. In each atom, the negatively charged electrons are tightly bound to the positively charged nucleus as illustrated in

12

1 Sensing with Light

Fig. 1.2 Mechanism of the dielectric polarization. a Unpolarized atom with a spherically symmetric distribution of the electron cloud. b Polarized atom with separated positive and negative charge centers. c Polarized medium with positive and negative charges on surface

Fig. 1.2a. The net charge density cancels out, leading to no permanent dipole moment of the dielectric medium. When an external (static or time-varying) electric field E is applied, the center of the electron cloud is slightly shifted away from its corresponding nucleus [see Fig. 1.2b]. Such a deviation between negative and positive charge centers results in an induced dipole moment d (i.e., the electric dipole approximation). Consequently, the net dipole moment of the dielectric bulk is nonzero and the material becomes polarized, P = 0 [see Fig. 1.2c]. The polarization P may be parallel or nonparallel to the direction of E, depending on the linear or nonlinear material. In this section, we briefly introduce the concept of the electric-field-induced polarization by using the classical electromagnetic theory.

1.3.1 Electric Dipole We start with the simplest electric dipole consisting of a negative charge −q separated from a positive charge q by a vector l. The direction of l points from the negative to positive charge. We assume that two charges have the same mass and set the origin of the coordinate system at the center of mass. Thus, the q and −q charges are located at l/2 and −l/2, respectively. The electric potential produced by two charges takes the form   q q 1 − φ(r) = 4πε0 |r + l/2| |r − l/2|   d · er ≈ + O |l/r|2 . (1.49) 2 4πε0 |r| The last expression is obtained by using the Taylor expansion in the limit of |l|  |r|. The unit vector er is along the radial direction in spherical coordinates. The electric dipole moment is defined as d = ql, (1.50)

1.3 Polarizability

13

whose SI unit is Coulomb meter (C · m). In atomic physics, d is usually measured in atomic units (a.u.), 1 ea0 = 8.478 × 10−30 C · m with the elementary charge e = 1.602 × 10−19 C and Bohr radius2 a0 = 0.053 nm. In addition, the CGS unit, debye (symbol: D) with 1 D = 3.33564 × 10−30 C · m, is also often used in many literatures. As suggested by (1.49), a pair of closely spaced opposite charges do not form an ideal electric dipole in the near-field region |l| ∼ |r|. The dipole approximation is only valid in a spatial scale much larger than the separation |l|, i.e., the point-dipole limit. The negative gradient of the potential gives the electric field generated by the dipole moment, 1 3(d · er )er − d . (1.51) E(r) = −∇φ(r) = 4πε0 |r|3 The amplitude of E(r) drops off as |r|3 . The concept of the electric dipole moment may be generalized to a system composed of multiple charges {qi , i = 1, 2, ..., n} with their corresponding locations {ri , i = 1, 2, ..., n}. The total dipole moment is then defined as d=

n i=1

qi ri .

(1.52)

It is worth noting that the above definition depends on the choice of the origin of coordinates. Shifting the origin of coordinates to r0 leads to a new dipole moment d =

n i=1

qi (ri + r0 ) = d + r0

n i=1

qi .

(1.53)

As one can see, the dipole moment nis invariant upon the choice of the coordinate qi is equal to zero. The dipole moment may system only when the net charge i=1 be further generalized to a continuous-charge-distribution system q(r) 1 d= v

 q(r)rdv,

(1.54)

v

with the system’s volume v. Again, the dipole approximation requires that the spatial scale of interest should be much larger than v. The electric dipole moment d is a microscopic physical quantity associated with a single point-like particle (e.g., atom and molecule). Combining all the dipole moments i di within a volume element ΔV leads to the macroscopic polarization density 1  di . (1.55) P= i ΔV

2 The Bohr radius is defined as a

0

=

4πε0 2 m e e2

with the reduced Planck’s constant  = 1.054 × 10−34

J · s and the mass of electron m e = 9.11 × 10−31 kg.

14

1 Sensing with Light

In a homogeneous and isotropic medium, the dipole moment is distributed uniformly within any small volume. The polarization density vector is then reduced to P = N d,

(1.56)

with the number of dipoles per unit volume N .

1.3.2 Local Field Correction The atoms and molecules with a symmetric structure do not own a permanent dipole moment, but they can acquire an induced dipole moment if placed in an external electric field Eext . In contrast, for the nonsymmetrical atoms and polar molecules the skewed electron distribution leads to a permanent dipole moment for such particles. We focus on the dielectrics made up of the former kind of particles (specifically, atoms) and discuss the relation between the induced polarization P and the external electric field Eext . For a dielectric bulk polarized by Eext , the electric field inside the dielectric E is a superposition of Eext and the depolarization field Edep produced by the polarization P, (1.57) E = Eext + Edep . As illustrated in Fig. 1.2c, the electric dipole moment vectors are in the tail-to-head alignment inside the dielectric. Thus, the net negative and positive charges are only distributed at the dielectric surface and Edep is opposite to Eext . Actually, E is a field averaged over a large number of atoms. We now consider the local electric field Eloc acting on a certain site of the atom (target) in the dielectric medium. The field Eloc denotes the real value that induces the target dipole moment. Due to the effect coming from the target’s surroundings, Eloc differs from the macroscopic field E. Here, we follow the Lorentz’s approach to evaluate Eloc . As shown in Fig. 1.3, the target atom is assumed to be surrounded by a spherical cavity. The cavity’s radius R is large enough that the sphere includes multiple dipoles around the target atom. The local field Eloc is thus composed of four parts [16] (1.58) Eloc = Eext + Edep + Elor + Edip , with the applied field Eext , the depolarization field Edep induced by the polarization charges on the sample’s surfaces, the Lorentz field Elor generated by the polarization charges on the surface of the Lorentz spherical cavity, and the near field Edip from other dipoles lying inside the Lorentz sphere. The Lorentz field Elor may be derived by assuming a continuous surface-charge distribution |P| cos θ, Elor =

1 4πε0

 0

π

(2π R 2 )

P cos θ P cos θ sin θdθ = L · , 2 R ε0

(1.59)

1.3 Polarizability

15

Fig. 1.3 Lorentz local-field correction. A bulk of dielectric is polarized in an external electric field Eext . The macroscopic field Edep is generated by the surface polarization charge. The local electric field Eloc denotes the actual field acting on the target atom. An imaginary sphere divides the induced dipole moments into two groups. The dipoles inside the sphere produce an electric field Edip at the target site while the electric field produced by the dipoles in the other group is Elor

with the so-called depolarization factor L = 1/3 for the spherical cavity. The factor L relies on the geometrical structure of the cavity. For a thin sheet, L is equal to 1 while L = 0 for a long needle [17]. The near field Edip depends on the specific arrangement of dipoles, i.e., the crystal structure of the medium. For an isotropic material, one has Edip = 0. To illustrate this, we take the simple cubic lattice as an example and assume that all dipoles are along, for instance, the z-axis. The electric field at the origin (the location of the target atom) produced by other dipoles is calculated to be Edip =

 3(d · ri )ri − r 2 d  3z 2 − r 2 i i i =d = 0. 5 i i ri ri5

(1.60)

In the last step of deriving (1.60), we have used the fact that the x-, y-, and z- directions are equivalent for the cubic lattice. Eventually, the local electric field reads Eloc = E + L ·

P . ε0

(1.61)

This local field induces the dipole moment d of the target atom, d = αEloc ,

(1.62)

where the proportionality constant α is the so-called atomic polarizability‘. The SI unit of α is C · m2 · V−1 . In atomic physics, the unit of α is given by3 1 a.u. = 1.649 × 10−41 C · m2 · J−1 . Using (1.61), the polarization (1.56) is then expressed as   P , (1.63) P = Nα E + L · ε0 31

a.u. =

e2 a02 Eh

with the Hartree energy E h =

2 m e a02

= 4.35974 × 10−18 J.

16

1 Sensing with Light

from which we find the electric susceptibility χe =

Nα P = . ε0 E ε0 − N αL

(1.64)

The above equation links the microscopic quantity α (related to the local electric field Eloc ) to the macroscopic quantity χe (related to the in-medium electric field E). It is seen that χe is proportional to α in the low-density limit ε0  N αL. Further, one arrives at the relation between the relative dielectric constant  = 1 + χe and the polarizability Nα −1 = . (1.65) L + (1 − L) ε0 In particular, the depolarization factor L =

1 3

for the spherical structure gives

−1 Nα , = +2 3ε0

(1.66)

which is referred to as the Lorentz–Lorenz equation, equivalent to the Clausius– Mossotti relation. One may define the Lorentz local-field factor as the ratio of the local electric field Eloc to the macroscopic field E L=

+2 Eloc = . E 3

(1.67)

The Lorentz local-field correction is usually applied to the solids. There is another model, known as the Onsager local-field correction [18], which has been widely used for the polar liquids because the dipolar reaction field from other particles is taken into account in this method.

1.3.3 Lorentz Oscillator Model After obtaining the relation between the microscopic quantity α and the macroscopic quantity χe , we now look for the specific expression of α. In principle, the polarizability α can be derived rigorously from the quantum mechanical theory, which will be introduced in Chap. 7. But here we give a classical description of the polarizability based on the well-known Lorentz oscillator model [19]. In this model, the light–atom interaction is approximated by a driven mass-spring system. Although the Lorentz classical theory has severe limitations (e.g., unable to explain the spontaneous emission of atoms), it is still commonly used even today as a trial tool to obtain an intuitive understanding of a complex physical system. We focus on an atom in a homogeneous dielectric medium and for simplicity, assume that the atom is composed of an electron bound to the nucleus [see Fig. 1.4a].

1.3 Polarizability

17

Fig. 1.4 Lorentz oscillator. a A virtual spring links the electron to the nucleus. The electron is vibrating around its equilibrium position and is perturbed by an external light. b Harmonic oscillation of the electron in the absence of the damping and driving sources. c Damped harmonic oscillation of the electron without the driving source. d Driven harmonic oscillation of the electron. For (b– d), γ = 2π × ν, ω0 = 2π × 10 ν, Fdri (t)/m e x0 = 0.025 × ω02 cos(ω0 t), and the initial conditions x(0)/x0 = 1 and x(0) ˙ = 0.0. Here, ν is an arbitrary frequency in units of Hz

Since the mass of the nucleus is much larger than the electron’s mass, the nucleus may be viewed as motionless and the external perturbation (light driving) primarily influences the motion of electron. For a free (unperturbed) atom, the relative position between the nucleus and the electron is in equilibrium, and the origin of the coordinate system may be set at the equilibrium position of the electron. We restrict ourselves to the one-dimensional model. Following Lorentz’s postulation, the binding force between the nucleus and the electron may be described by Hooke’s law k Fres (t) = − x(t) = −ω02 x(t), me me

(1.68)

where k is the spring constant and x is the displacement of the electron from its equilibrium position. Thus, the electron is linked to the nucleus via a virtual spring and the force Fres acts as a restoring force. In the last step of the derivation, we have defined the oscillation frequency

18

1 Sensing with Light

ω0 =

k . me

(1.69)

Additionally, in order to emulate the damping processes, such as the spontaneous emission and collisions, we artificially introduce a viscous (damping) force exerted on the electron, d (1.70) Fdam (t) = −m e γ x(t), dt where γ denotes the energy dissipation rate. Normally, γ is much smaller than ω0 by several orders of magnitude. When the atom is placed in an external light field, the electron bears an extra driving force exerted by the local electric field E loc (t) Fdri (t) = −eE loc (t).

(1.71)

Here, we have used the facts that (1) the light field hardly affects the motion of nucleus due to its heavy mass and (2) the magnetic force exerted on the electron is negligible in comparison to the electric force. According to Newton’s second law, one may write the equation of motion of the electron as d2 (1.72) m e 2 x(t) = Fres (t) + Fdam (t) + Fdri (t). dt In the non-damping limit γ = 0 and in the absence of the external field E loc = 0, the general solution of the above equation is x(t) = x0 cos ω0 t + C,

(1.73)

where the amplitude x0 and constant C are related to the initial conditions of the electron. Equation (1.73) corresponds to a harmonic oscillation at the frequency ω0 [see Fig. 1.4b]. In contrast, when γ = 0 but E loc = 0, (1.72) describes a damped harmonic oscillator. As shown in Fig. 1.4c, the electron dissipates its acquired energy to the environment. The oscillator’s lifetime is determined by the characteristic time τ = γ −1 . We rewrite (1.72) as d2 d Fdri (t) x(t) + γ x(t) + ω02 x(t) = . 2 dt dt me

(1.74)

The left side of the above equation is analogous to the spontaneous emission process of the atom prepared in an excited state. The frequency ω0 determines the energy separation between the ground and excited states of the atom. In contrast, the rightside term corresponds to a driving source, i.e., pumping the energy from the light to the atom. One may define the quality factor for this damped harmonic oscillator as Q=

ω0 . γ

(1.75)

1.3 Polarizability

19

As we will see later, (1.74) has a form same as the intracavity field of a light-driven optical resonator as well as an electric-field-driven LC circuit. We are interested in the response of the atomic dipole moment d = −ex to the external light field at different frequencies. It is convenient to map (1.74) into the frequency domain. For an arbitrary time-dependent signal F(t), its Fourier transform is defined as  ∞ F(t)e−iωt dt. (1.76) F(ω) = F [F(t)] = −∞

The inverse Fourier transform can convert the frequency-domain function F(ω) back to the signal  ∞ 1 −1 F(ω)eiωt dω. (1.77) F(t) = F [F(t)] = 2π −∞ Performing the Fourier transform on (1.74), one obtains the electric dipole moment d = α(ω)Eloc (ω),

(1.78)

with the atomic polarizability α(ω) = −

1 e2 , m e ω 2 − ω02 − iγω

(1.79)

and the Fourier transform of the local electric field Eloc (ω) = F [E loc (t)]. The polarizability α(ω) is complex because of the energy dissipation (γ = 0). The relative dielectric constant of a homogeneous medium with a density of atoms N is related to α(ω) via (ω) = 1 + N α(ω)/ε0 . Here we have assumed ε0  N αL in (1.64). Within the near-resonance region of ω ∼ ω0 , (ω 2 − ω02 ) approximates to 2ω0 (ω − ω0 ) and (ω) is then simplified as (ω) ≈ 1 −

1 N e2 . 2m e ε0 ω0 ω − ω0 − iγ/2

(1.80)

We further express the complex dielectric (ω) as  = (n + iκ)2 , with the refractive index n and the attenuation coefficient κ of the medium   || + Re() || − Re() , κ= . n= 2 2

(1.81)

(1.82)

For an ex -polarized light traveling in the medium along the z-direction, the wavevector of the light takes the form k = [2π(n + iκ)/λ0 ]ez with the free-space wavelength λ0 . The light propagation term eik·ez z can be re-expressed as e−2πκz/λ0 ei2πnz/λ0 , indi-

20

1 Sensing with Light

Fig. 1.5 Atomic polarizability. a Refractive index n(ω) and attenuation coefficient κ(ω) as a function of the light frequency ω with N e2 /m e ε0 ω0 = 0.1γ. The decay rate γ is chosen as a frequency unit. b Dispersion part of the polarizabilities α S,P (ω) for 88 Sr atom in the ground (5s5 p)1 S0 (dashed) and metastable state (5s5 p)3 P0 (solid)

cating the light is attenuated along its propagation direction. Figure 1.5a plots the dependence of n(ω) and κ(ω) on the light’s frequency ω. For the resonant light–matter interaction, the refractive index n(ω0 ) is equal to unity, indicating that the medium does not change the light’s velocity and wavelength. In contrast, κ(ω) reaches its maximum value, i.e., the light suffers from a strong attenuation, at ω = ω0 . When ω departs from the resonant frequency ω0 , n(ω) raises (goes down) on the red-detuned (blue-detuned) side of ω0 and arrives at the maximum (minimum) at ω − ω0 = −γ/2 (ω − ω0 = γ/2). The coefficient κ(ω) decreases as ω moves away from ω0 and falls to half of its maximum height at ω − ω0 = ±γ/2. As we have pointed out, the resonant frequency ω0 corresponds to the atomictransition frequency between the ground and excited states and γ gives the decay rate of the excited state. The Lorentz oscillator model may be extended to multiple equilibrium positions (corresponding to multiple atomic transitions at frequencies ωi=0,1,2,··· ), for which the generalized polarizability of the atom is written as α(ω) = −

 e2 fi . 2 m e ω − ωi2 − iγω i

(1.83)

The dimensionless quantity f i is the so-called oscillator strength and measures the difference of the i-th atomic transition rate calculated by classical and quantummechanical theories [20]. Indeed, the quantum mechanics gives fi =

2m e ωi |di |2 , 3e2

(1.84)

1.4 Frequency and Intensity Fluctuations of Light

21

with the i-th transition dipole moment di , and f i is related to the Einstein coefficient for spontaneous emission Ai (corresponding to the i-th electric dipole transition) via Ai =

e2 ωi3 fi . 2πε0 m e c3

(1.85)

One has γi ≈ Ai when the spontaneous emission dominates the decay. Figure 1.5b shows an example of the dispersion part of the light-induced atomic polarizability Re(α). The dispersion curve depends on the internal electronic state of the atom. For two specific electronic states, there exist a group of intersections, at which the dispersion values of two electronic states are equal. The wavelengths at these intersections are called the magic wavelengths, which are of particular importance for trapping the atoms in an optical potential without distorting the overlap of wavefunctions of two electronic states [21].

1.4 Frequency and Intensity Fluctuations of Light The light sources commonly applied in physics and chemical and biological sensing are the lasers because of their unique properties, namely, high monochromaticity, directionality, and (temporal and spatial) coherence. Nonetheless, for a free running laser its frequency (or phase) is inevitably influenced by the environment. The laser cavity length suffers from the short-term fluctuations (induced by the acoustic vibration of the mirrors) and the long-term drifts (mainly caused by the temperature drift and slow pressure change), which are mapped onto the laser frequency. Also, the light’s intensity does not stay constant over time for manifold reasons, such as intermode switching, polarization instability, atmospheric turbulence, and temperature and power-supply (driven-current) fluctuations. One needs to find a general way to qualify the optical waves. Such an analysis can be done in either time or frequency domain. These two representations are equivalent and linked through the Fourier transform.

1.4.1 Autocorrelation Function and Spectrum In the time domain, we usually employ the autocorrelation function, which is defined as R(τ ) = E (−) (t + τ )E (+) (t)  1 T /2 (−) = lim E (t + τ )E (+) (t)dt, T →∞ T −T /2

(1.86)

22

1 Sensing with Light

to analyze the temporal coherence of the light field. Here, E (+) (E (−) ) is the positivefrequency (negative-frequency) part of the light field and τ denotes a time delay. The above equation actually measures the similarity between the light beam and itself shifted by τ . The function R(τ ) is an experimentally measurable quantity [22]. For example, a light beam is split into two sub-beams via a 50/50 beam splitter. One beam passes through a long optical fiber, which induces a time delay, and then is superimposed with the other beam whose optical path is shorter. The combined beams eventually enter a photodetector and the resultant photocurrent is proportional to R(τ ). For an ideal plane wave, the value of |R(τ )| is independent of τ . However, the unavoidable environmental fluctuations influence both phase and amplitude of the light wave, for which |R(τ )| decays as τ is increased. One may imagine that |R(τ )| approaches zero as τ → ∞. According to the Wiener–Khintchine theorem, the autocorrelation function R(τ ) is directly linked to the power spectral density (PSD) of the light  S(ω) = 2cε0 A · F [R(τ )] = 2cε0 A ·

∞ −∞

R(τ )e−iωτ dτ ,

(1.87)

with the inverse Fourier transform 2cε0 A · R(τ ) = F −1 [S(ω)] =

1 2π



∞ −∞

S(ω)eiωτ dω.

(1.88)

Here, A is the cross-sectional area of the laser beam. The PSD, whose SI unit is W/Hz (watts per hertz), describes how the power carried by the light is distributed with respect to the frequency ω. This may be illustrated by the zero-time-delay autocorrelation function  ∞ 1 S(ω)dω. (1.89) 2cε0 A · R(0) = 2π −∞ The left side of the above equation corresponds to the light power while the right side gives the area covered by the PSD√ curve in the frequency domain. Substituting the Fourier transform of the light field T E(ω) = 2F [E (+) (t)] into (1.86) yields  R(τ ) =

∞



∞





T /2

lim

−∞

−∞

T →∞ −T /2



ei(ω −ω)t dt 2π

E ∗ (ω)E(ω  ) e−iωτ dωdω  . 4 2π

(1.90)

Further, using the Dirac delta function δ(ω − ω  ) = Equation (1.90) can be rewritten as

1 2π



∞

−∞



e−i(ω−ω )t dt,

(1.91)

1.4 Frequency and Intensity Fluctuations of Light

1 R(τ ) = 2π



∞

−∞

|E(ω)|2 −iωτ e dω. 4

23

(1.92)

Comparing (1.88) and (1.92), one finds S(ω) =

cε0 A cε0 A |E(−ω)|2 = |E(ω)|2 . 2 2

(1.93)

The above equation provides us an alternative way to derive the PSD directly from the Fourier transform of the light field. This dramatically simplifies the mathematical calculation. In many cases, the specific value of R(τ ) is not important and we are mainly interested in its relative value. Thus, it is convenient to introduce the normalized first-order correlation function g (1) (τ ) =

E (−) (t + τ )E (+) (t) , E (−) (t)E (+) (t)

(1.94)

to study the temporal coherence of the light field. The zero time delay τ = 0 gives g (1) (0) = 1 and one has g (1) (0) ≥ g (1) (τ ). The ideal (perfectly coherent) light has g (1) (τ ) = e−iωτ . However, in many practical cases |g (1) (τ )| may go to zero monochromatically as |τ | is increased. The coherence time of the light τc is generally measured by (1.95) |g (1) (τc )| = 1/e. In particular, when |g (1) (τ )| follows the exponential function exp(−γτ ) or Gaussian profile exp[−(γτ )2 ] with a constant γ, τc is directly given by γ −1 . The coherent time τc is of great importance since the phases of an optical wave at two time points with an interval shorter than τc may be, on average, considered to be synchronized.

1.4.2 Lorentzian and Gaussian Broadening The PSD of light has two fundamental line shapes, whose broadening mechanisms can be categorized into either homogeneous or inhomogeneous effects. The homogeneous broadening follows the Lorentzian function L(ω) =

ΔωFWHM /2 1 , π (ω − ω0 )2 + (ΔωFWHM /2)2

(1.96)

which is maximized at the central frequency ω0 with L(ω0 ) =

2 0.637 = . πΔωFWHM ΔωFWHM

(1.97)

24

1 Sensing with Light

Fig. 1.6 Lorentzian and Gaussian broadening. a Spectral line shapes. The corresponding first-order correlation functions are shown in (b)

Figure 1.6a displays the profile of L(ω). The integral of L(ω) is given by 

ω+ ω−

L(ω)dω =

     ω + − ω0 ω− − ω0 1 arctan − arctan . π ΔωFWHM /2 ΔωFWHM /2

(1.98)

In the limit of ω± → ±∞, one obtains the normalization 

∞ −∞

L(ω)dω = 1.

(1.99)

The frequency ΔωFWHM is recognized as the full width at half maximum (FWHM) of the line shape,4 L(ω0 ± ΔωFWHM /2) = L(ω0 )/2. The ratio of the light power dropped in the spectral linewidth to the total light power reaches  ΔωFWHM /2

−ΔωFWHM /2 L(ω)dω

  

∞ −∞

 L(ω)dω = 0.5.

(1.100)

The Lorentzian line shape is generally caused by the homogeneous dissipative processes, for instance, the spontaneous emission and collisions of the photon emitters in the light-source medium. When two homogeneous broadening mechanisms, which have the same central frequency ω0 but different linewidths ΔωFWHM,i=1,2 , take 4 In

1958, Schawlow and Townes derived the fundamental (quantum-noise-limited) limit for the spectral linewidth ΔωFWHM of a good-cavity laser with the Lorentzian broadening, i.e., the 2 /P , where Δω Schawlow–Townes linewidth [23], ΔωFWHM = ω0 Δωcav out cav is the linewidth of the cold cavity and Pout is the laser output power. In 1967, Lax proved that the correct linewidth 2 /2P should be ΔωFWHM = ω0 Δωcav out [24]. The Schawlow–Townes linewidth can be generalized 2 /2P to the form ΔωFWHM = n sp ω0 Δωcav out [25] with the inversion factor n sp = N2 /(N2 − N1 ). The parameters N1,2 are the populations of the active particles in two lasing states 1 (lower) and 2 (upper), respectively. One has n sp = 1 for an ideal four-level laser.

1.4 Frequency and Intensity Fluctuations of Light

25

effects simultaneously, the overall line shape is given by the convolution  L(ω) =

∞

−∞

L 1 (ω  )L 2 (ω − ω  )dω  .

(1.101)

It can be proven that L(ω) has the same expression as (1.96) with the total linewidth ΔωFWHM = ΔωFWHM,1 + ΔωFWHM,2 .

(1.102)

The above equation further leads to the following quality factor:  1  ΔωFWHM,i 1 ΔωFWHM = = = . i i Q ω0 Qi ω0

(1.103)

The first-order correlation function corresponding to the Lorentzian broadening is derived as   ΔωFWHM |τ | (1) g (τ ) = exp iω0 τ − , (1.104) 2 with the Fourier transform F [g (1) (τ )] = 2πL(ω).

(1.105)

It is seen that the coherent time for the light field with the Lorentzian broadening reads τc = 2/ΔωFWHM [see Fig. 1.6b]. Another typical line shape takes the form of the Gaussian function  √  √ 2 ln 2 ω − ω0 2 G(ω) = √ exp − 2 ln 2 , ΔωFWHM πΔωFWHM which also follows the normalization condition  ∞ G(ω)dω = 1.

(1.106)

(1.107)

−∞

The maximum of G(ω) is √ 2 ln 2 0.939 G(ω0 ) = √ = . ΔωFWHM πΔωFWHM

(1.108)

Thus, for the same ΔωFWHM , G(ω) is higher than L(ω) [see Fig. 1.6a]. The FWHM of the line shape is given by ΔωFWHM , i.e., G(ω0 ± ΔωFWHM /2) = G(ω0 )/2. The power of the light within the linewidth ΔωFWHM occupies  ΔωFWHM /2

−ΔωFWHM /2 G(ω)dω

  

∞ −∞

 G(ω)dω ≈ 0.761,

(1.109)

26

1 Sensing with Light

of the total power. Generally, the Gaussian broadening results from the inhomogeneous mechanisms (e.g., Doppler effect) and usually happens in the gas and diode lasers, where the velocity of the photon emitters (atoms, molecules, and electrons) follows the Maxwell–Boltzmann distribution. The corresponding first-order correlation function is  2 Δω τ FWHM , (1.110) g (1) (τ ) = exp iω0 τ − √ 4 ln 2 with the Fourier transform F [g (1) (τ )] = 2πG(ω).

(1.111)

The coherence time is then given by √ 4 ln 2 . τc = ΔωFWHM

(1.112)

The convolution of two Gaussian line shapes with the central frequencies ω0(i=1,2) and the linewidths ΔωFWHM,i=1,2 still results in a Gaussian line shape, whose central frequency and linewidth are ω0 = ω0(1) + ω0(2) , 2 2 ΔωFWHM = (ΔωFWHM,1 + ΔωFWHM,2 )1/2 .

(1.113a) (1.113b)

In a practical situation, the spectral profile is neither pure Lorentzian nor pure Gaussian broadening. A more general line shape is given by the so-called Voigt integral  V (ω) =

∞

−∞

L(ω  )G(ω − ω  )dω  .

(1.114)

The central frequency of L(ω) may not coincide with that of G(ω). When the linewidth of one function component much exceeds that of the other, the latter function can be approximated by a δ-function. As a result, V (ω) is reduced to the former line shape but its central frequency may be shifted.

1.4.3 Phase Noise In principle, the spectral broadening is caused by the fluctuations in both amplitude and phase of the light field. Here, we first consider the relation of the light’s PSD to the phase fluctuation. We write the positive-frequency part of a monochromatic optical wave in the following form:

1.4 Frequency and Intensity Fluctuations of Light

E (+) (t) =

27

E0 exp[−iω0 t − iϕ(t)], 2

(1.115)

where E 0 is the constant amplitude, ω0 is the central frequency, and the noiseperturbed phase ϕ(t) varies slowly in comparison to ω0 . The instantaneous angular frequency reads as d (1.116) ω(t) = ω0 + ϕ(t). dt To derive the PSD, one needs to calculate the autocorrelation function R E (τ ) =

|E 0 |2 iω0 τ e exp[iϕ(t + τ ) − iϕ(t)] , 4

(1.117)

where the subscript ‘E’ denotes that R(τ ) is associated with E (+) (t). Generally, the phase fluctuation fulfills a Gaussian process [26], which leads to exp[iϕ(t + τ ) − iϕ(t)] = exp[Rϕ (τ ) − Rϕ (0)],

(1.118)

with the autocorrelation function for the phase noise ϕ(t) Rϕ (τ ) = ϕ(t + τ )ϕ(t) .

(1.119)

The Wiener–Khintchine theorem gives us  Rϕ (τ ) − Rϕ (0) =

∞ −∞

Sϕ ( f )(ei2π f τ − 1)d f,

(1.120)

with the PSD of the phase fluctuation Sϕ ( f ). One should note that the noise PSD is usually expressed in the f = ω/2π domain. Thus, the PSD of the light field S E (ω) is linked to Sϕ ( f ) via S E (ω) =

cε0 A |E 0 |2 · 2



∞

e−i(ω−ω0 )τ exp

−∞



∞

−∞

 Sϕ ( f )(ei2π f τ − 1)d f dτ .

(1.121) The general form of Sϕ ( f ) may be decomposed into a set of independent noise processes (or colors), for instance, the flicker-frequency, white-frequency, flicker-phase, and white-phase noises with the corresponding power laws f 02 h −1 f −3 , f 02 h 0 f −2 , f 02 h 1 f −1 and f 02 h 2 f 0 , and the constants h i=−1,0,1,2 . Here, we have used f 0 = ω0 /2π. We focus on the white frequency noise [see Fig. 1.7a and b] that is usually caused by the environmental influences, such as temperature and vibrations. In practice, Sϕ (ω) has a finite bandwidth  Sϕ ( f ) =

h 0 f 02 / f 2 | f | ≤ B . 0 |f| > B

(1.122)

28

1 Sensing with Light

Fig. 1.7 White-frequency-noise-induced spectral broadening. a Single-sided noise spectrum of the numerically generated white frequency noise with Sϕ ( f ) ∝ f −2 . The dashed line denotes the linear curve fitting. The sampling frequency is f s . b Correlation functions exp[iϕ(t + τ ) − iϕ(t)] and exp[Rϕ (τ ) − Rϕ (0)]. c Lorentzian line shape with h 0 f 02 / f s = 0.1. The solid line gives the Lorentzian curve fitting. d Gaussian line shape, where the phase noise is filtered by a low-pass filter with a bandwidth of B = 0.025h 0 f 02 . The solid line gives the Gaussian curve fitting

Substituting the above equation into (1.121) leads to [27]    π B|τ | sin2 x h 0 ω02 |τ | e−i(ω−ω0 )τ exp − d x dτ . π x2 −∞ 0 (1.123) For a wide bandwidth with B  h 0 f 02 , one may extend the upper limit of the integral in round brackets to infinity and obtain S E (ω) =

cε0 A |E 0 |2 · 2



∞

 0

∞

sin2 x π dx = . 2 x 2

As a result, the PSD of the light is given by

(1.124)

1.4 Frequency and Intensity Fluctuations of Light

29

 ∞ cε0 A 2 e−i(ω−ω0 )τ −h 0 ω0 |τ |/2 dτ |E 0 |2 · 2 −∞ ΔωFWHM /2 1 cε0 A |E 0 |2 · 2π · , = 2 π (ω − ω0 )2 + (ΔωFWHM /2)2

S E (ω) =

(1.125)

with the linewidth ΔωFWHM = 2π(2πh 0 f 02 ).

(1.126)

(Note: In some literatures, the single-sided spectrum Sϕ ( f ) is employed and ΔωFWHM = 2π(πh 0 f 02 ).) It is seen that S E (ω) presents the Lonrentzian line shape as shown in Fig. 1.7c. In contrast, using the Taylor expansion in the narrow-band limit with B  h 0 f 02 , we find 

π B|τ | 0

sin2 x dx ≈ x2



π B|τ |

d x = π B|τ |.

(1.127)

0

The corresponding PSD is further derived as S E (ω) =

 ∞ cε0 A cε0 A 2 2 |E 0 |2 |E 0 |2 · 2π e−i(ω−ω0 )τ −h 0 Bω0 τ dτ = 2 2 −∞ ⎡  √ 2 ⎤ √ 2 ln 2 ln 2(ω − ω ) 2 0 ⎦, ×√ exp ⎣− (1.128) ΔωFWHM πΔωFWHM

with the spectral linewidth √  ΔωFWHM = 4 ln 2 h 0 Bω02 .

(1.129)

(Note: When the single-sided spectrum Sϕ ( f ) is employed, ΔωFWHM = √ 8 ln 2 h 0 Bω02 .) Thus, S E (ω) follows the Gaussian broadening [see Fig. 1.7d]. The spectral linewidths for the phase noise in other colors have been studied in [28, 29].

1.4.4 Hanbury Brown–Twiss Effect In the above, we have discussed the spectral broadening of the light-field PSD S E (ω) caused solely by the fluctuations in the light’s phase ϕ(t). However, the intensity of a practical light beam does not stay constantly either. We now introduce how to measure the light’s intensity fluctuations, which is equivalent to deriving the correlation function of the light’s intensity. Figure 1.8a shows the general measurement scheme.

30

1 Sensing with Light

Fig. 1.8 Second-order correlation function g (2) (τ ). a Hanbury Brown–Twiss (HBT) interferometry. The beam output from the light source is split into two sub-beams, whose intensities are measured by two photodetectors, respectively. The intensity correlation is obtained through mixing the measured photocurrents. b g (2) (τ ) versus the time delay τ for the Lorenztian- (dashed) and Gaussian-broadened (solid) light fields as well as the light in the coherent state (dotted)

A light beam is split into two sub-beams, which respectively enter two photodetectors, via a 50/50 beam splitter. One photodetector converts the corresponding light signal into the optocurrent i(t) ∝ 2cε0 A · E (−) (t)E (+) (t) = AI (t),

(1.130)

with the light intensity I (t). In contrast, the current signal generated by the other photodetector has a relative time delay τ , i.e., i(t + τ ). Two current signals are then mixed by a multiplier. Analogous to g (1) (τ ) that actually is the first-order amplitudecorrelation function, one can define the normalized second-order correlation function to characterize the intensity correlation g (2) (τ ) =

I (t + τ )I (t) i(t + τ )i(t) = . i(t + τ ) i(t) I (t + τ ) I (t)

(1.131)

For a classical optical wave, the Cauchy–Schwarz inequality gives g (2) (τ ) ≤ g (2) (0).

(1.132)

In the limit of the zero time delay τ ∼ 0, we have g (2) (0) = 1 +

I 2 (t) − I (t) 2 [I (t) − I (t) ]2 = 1 + ≥ 1. I (t) 2 I (t) 2

(1.133)

The intensity correlation was firstly measured in the well-known Hanbury Brown– Twiss experiment [30]. Now this method has been widely applied to verify the singlephoton generation [31].

1.5 Frequency Stabilization

31

For the chaotic light emitted from an ensemble of independent photon emitters, it can be proven that g (2) (τ ) is related to g (1) (τ ) via [32] chaotic light : g (2) (τ ) = 1 + |g (1) (τ )|2 .

(1.134)

Thus, one obtains g (2) (0) = 2 and g (2) (τ ) approaches the unity as τ → ∞. The normalized intensity correlation function takes the form g (2) (τ ) = 1 + exp (−ΔωFWHM |τ |) ,

(1.135)

for the Lorentzian line shape while the Doppler-broadened light source gives



ΔωFWHM τ g (τ ) = 1 + exp −2 √ 4 ln 2 (2)

2 ,

(1.136)

as shown in Fig. 1.8b. For the light field in the coherent state, also referred to as the most classical state of light, g (2) (τ ) always stays at unity coherent state : g (2) (τ ) = 1.

(1.137)

One may convert the light’s intensity to the number of light quanta (photons), i.e., n(t) = AI (t)Δt/ω0 , within a short duration Δt. Thus, the second-order correlation function g (2) (τ ) actually measures the fluctuations in n(t). For the coherent state, the photon number fulfills the Poissonian distribution, where the probability of detecting k (∈ Z) photons within Δt reads p(k) = e−n¯

n¯ k . k!

The number n¯ represents both the average photon number n¯ = variance n¯ = ( k p(k)k 2 ) − n¯ 2 as well.

(1.138) k

p(k)k and the

1.5 Frequency Stabilization The spectral linewidth of a free running laser is typically of the order of MHz (megahertz) and GHz (gigahertz). For many practical applications, the lasers with such a large linewidth are inapplicable. Thus, the laser’s frequency must be stabilized to a frequency reference so as to narrow its linewidth. The most common reference is the optical resonators (or cavities), whose frequency-discrimination mechanisms include the multiple-beam interferometry (e.g., macroscale [33, 34] and micropillar [35] Fabry–Pérot resonators and whispering-gallery-mode microcavities in the disk [36], spherical [37], and toroidal [38] structures) and the photonic band gap

32

1 Sensing with Light

(e.g., photonic crystal microcavities [39]). Here, we focus on the simplest optical resonators, i.e., the Fabry–Pérot resonators, which have been widely used in modern optics. The state-of-the-art laser system, whose frequency is locked to a meticulously designed and protected Fabry–Pérot resonator, possesses a mHz (millihertz) spectral linewidth. A laser system with such an ultra-narrow linewidth is particularly desired for the high-resolution spectroscopy, gravitational wave detection, and optical clocks.

1.5.1 Fabry–Pérot Resonator The Fabry–Pérot resonator is formed by a pair of parallel mirrors (M1 and M2 ) separated with a distance L as shown in Fig. 1.9a. We use ri (ti ) with i = 1, 2 to denote the amplitude reflection (transmission) coefficient of the i-th mirror. For simplicity, we assume that the resonator is placed in free space and r1,2 and t1,2 are (+) real. An incident ray E inc = (E 0 /2)e−iωt enters the resonator from the mirror M1 and then bounces back and forth between two mirrors. At each time of the ray hitting the mirrors, a fraction of the ray leaks out of the resonator. The summation of the leaking components at the mirror M1 (M2 ) gives the amplitude of the light reflected by (+) (+) and E tra . Thus, the amplitude reflection (transmitted through) the resonator, i.e., E ref (+) (+) and transmission coefficients of the resonator are defined as rFP = E ref /E inc and (+) (+) tFP = E tra /E inc , respectively, and derived as

Fig. 1.9 Fabry–Pérot interferometer composed of a pair of parallel mirrors M1 and M2 . a Schematics of a light interacting with a Fabry–Pérot resonator. The light is perpendicularly incident on the mirror M1 . To explicitly illustrate the multiple-beam interferometry, the beams are artificially tilted with respect to the normal of the mirrors. b Reflection (dashed) and transmission (solid) spectra of the Fabry–Pérot resonator with r1 = r2 = 0.8

1.5 Frequency Stabilization

33

rFP = r1 − t12 r2 e−i2ϕ − t12 r1r22 e−i4ϕ − t12 r12 r23 e−i6ϕ + . . . t12 r2 e−i2ϕ , 1 − r1r2 e−i2ϕ = t1 t2 e−iϕ + t1 t2 r1r2 e−i3ϕ + t1 t2 r12 r22 e−i5ϕ + . . . = r1 −

tFP

=

t1 t2 e−iϕ , 1 − r1r2 e−i2ϕ

(1.139a)

(1.139b)

where ϕ = ωL/c is the extra phase acquired by the light propagating from one mirror to the other. In writing the above equations, we have used the fact that the optical waves undergo a phase shift π when they travel in a lower-refractive-index medium and are reflected by a higher-refractive-index barrier. The intensity (power) reflection and transmission coefficients RFP = |rFP |2 and TFP = |tFP |2 are then given by (r1 − r2 )2 + 4r1 r2 sin2 ϕ , (1 − r1r2 )2 + 4r1r2 sin2 ϕ t12 t22 . = (1 − r1r2 )2 + 4r1r2 sin2 ϕ

RFP =

(1.140a)

TFP

(1.140b)

Obviously, the energy conservation leads to RFP + TFP = 1.

(1.141)

It is seen that TFP reaches the maximum Tmax = t12 t22 /(1 − r1r2 )2 when ϕ = mπ with m ∈ Z, meaning that the round-trip phase shift 2ϕ equals an integer multiple of 2π. In particular, Tmax reaches the unity when r1 = r2 and t1 = t2 . Figure 1.9b illustrates the reflection and transmission spectra of the Fabry–Pérot resonator as a function of the incident frequency ω. It is seen that a group of equally separated dips and spikes are presented. At each dip or spike, the light transmits through the resonator maximally, i.e., the light is resonant with the Fabry–Pérot resonator. The resonance occurs only when the laser frequency ω fulfills the condition ω = ωm ≡ mωFSR . Each resonant frequency ωm denotes a certain longitudinal mode that can exist inside the resonator. Here, ωFSR = 2πνFSR with νFSR =

c 2L

(1.142)

is the so-called the free spectral range (FSR) and corresponds to the frequency interval between two neighboring resonant modes. Due to r1,2 = 1 (i.e., the partial reflection), the dips and spikes have a FWHM ΔωFWHM c 1 − r1 r2 νFSR = , = √ 2π 2L π r1r2 F

(1.143)

34

1 Sensing with Light

determined by TFP = Tmax /2. In (1.143), we have defined the finesse of the resonator √ π r1 r2 F= , 1 − r1 r2

(1.144)

which depends solely on the reflectivity of the mirrors. Actually, F can be interpreted as the average number of times that a photon (the quantum of light) inside the resonator is reflected back and forth until it finally transmits through one of the mirrors. As r1,2 approach unity, the finesse F goes to infinity and ΔωFWHM becomes narrow. This linewidth narrowing mechanism entirely arises from the interference among F sub-beams. The nonzero linewidth ΔωFWHM indicates that the light energy cannot be stored inside the resonator forever. The effective lifetime of the intraresonator photons τ is generally defined as the time scale at which the energy of the intra-resonator light field decays to 1/e of its initial value. It may be proven τ = 1/ΔωFWHM . We focus on a certain resonant frequency of the Fabry–Pérot resonator ωFP . The quality factor ωFP Q= , (1.145) ΔωFWHM i.e., the storage time τ = 1/ΔωFWHM in units of the oscillation period 1/ωFP , is commonly employed to characterize the decay behavior of the resonant mode. The Q factor can be enhanced by choosing a high resonant frequency ωFP and a narrow linewidth ΔωFWHM . Table 1.1 lists the quality factor Q for different types of resonators. The spherical and toroidal microcavities own a quality factor up to Q ∼ 109 . When the detuning between the incident frequency ω and the resonant frequency ωFP is much smaller than the free spectral range ωFSR , |ω − ωFP |  ωFSR , the transmission spectrum may be approximated as tFP (ω) ≈

ΔωFWHM /2 . ΔωFWHM /2 + i(ω − ωFP )

(1.146)

As illustrated in Fig. 1.10a, the dependence of the real (imaginary) part of tFP on the detuning (ω − ωFP ) exhibits an absorption (dispersive) behavior similar to that of the atomic polarizability α(ω) [see Fig. 1.5a]. Additionally, Fig. 1.10a shows that the transmission |tFP (ω)| is maximized at the resonance ω = ωFP and decreases as ω moves away from ωFP . The transmitted light acquires a positive (negative) phase arctan[Im(tFP )/Re(tFP )] compared to the incident light when ω < ωFP (ω > ωFP ). In the limit of a large detuning |ω − ωFP |  ΔωFWHM , the acquired phase approaches ±π/2. In the above, we have assumed the amplitude of the incident light is a constant E 0 . In a more general situation, we assume the Fourier transform of the incident (+) (+) (ω) and the corresponding transmitted amplitude has a form Etra (ω). It light is Einc (+) (+) is straightforward to obtain Etra (ω) = tFP (ω)Einc (ω). Mapping this relation into the time domain results in

1.5 Frequency Stabilization

35

Table 1.1 Quality factor Q for different types of resonators with the central frequency ν0 = ω0 /2π and the ambient temperature T Type ν0 T Q Fabry–Pérot resonator [33, 34] Micropillar [35] Microsphere [37] Microtoroid [38] Photonic band-gap cavity [39] Coplanar waveguide resonator [40] LC resonator [41]

194 THz 321 THz 437 THz 441 THz 200 THz 1.8 GHz 8.2 GHz

124 K 14 K 300 K 300 K 20 mK 30 mK

>107 >2.5 × 105 0.8 × 1010 2.3 × 108 7.5 × 105 1.1 × 106 >103

Fig. 1.10 Near-resonance transmission coefficient tFP (ω). a Real (dashed) and imaginary (solid) parts of tFP (ω). b Amplitude (dashed) and phase (solid) of tFP (ω)

  d (+) ΔωFWHM ΔωFWHM (+) (+) E tra (t) = − − iωFP E tra E inc (t). (t) + dt 2 2

(1.147)

The term ΔωFWHM /2 in parentheses gives the decay rate of the transmitted amplitude while ωFP denotes its central frequency. The incident light plays the role of a driving source, whose injection rate is also ΔωFWHM /2.

1.5.2 LC Oscillators The optical resonators are capable of confining the electromagnetic waves at certain frequencies (longitudinal modes) for a time scale determined by the quality factor Q. This reminds us of another type of the electromagnetic-energy storage devices, i.e., LC circuits whose fundamental elements include an inductor L and a capacitor C. The electrical energy is alternatively stored into the magnetic field of L and the electric field of C and also suffers from an unavoidable dissipation due to the nonzero resistance R.

36

1 Sensing with Light

Fig. 1.11 LC harmonic oscillators. The parallel (series) circuit is driven by a time-varying current source Idri (voltage source Vdri ). The extra resistance R is introduced to describe the energy loss

The L and C components can be connected in either parallel or series way as shown in Fig. 1.11. The parallel (series) loop is driven by an external time-varying current Idri (voltage Vdri ) source. The extra R converts the electrical energy into the heat. According to Kirchhoff’s voltage law, the voltages (VR , VL , VC ) across R, L , C are related to the currents flowing through them (I R , I L , IC ) via I R (t) =

VR (t) d d , VL (t) = L I L (t), IC (t) = C VC (t), R dt dt

(1.148)

respectively. We assume the capacitor C is composed of a pair of parallel plates with an area A and an interplate separation l, i.e., C = ε0 A/l, and focus on the intracapacitor electric field E(t) = VC (t)/l. The equation of motion of E(t) for both types of circuits may be written in the form d d2 E(t) + κ E(t) + ω 2LC E(t) = iκω LC E dri (t), dt 2 dt

(1.149)

where κ denotes the loss rate of the electrical energy, ω LC is the characteristic oscillation frequency, and E dri (t) corresponds to the external driving source. The specific expressions of κ, ω LC , and E dri (t) for different circuit types are listed Table 1.2. Table 1.2 Specific expressions of the damping rate κ, oscillation frequency ω LC , and driving source E dri (t) in (1.149) for the parallel and series LC circuits Parallel Series Summation κ ω LC E dri (t)

Idri = I R + I L + IC 1 RC 1 √ LC √ R LC d −i Idri (t) l dt

Vdri = V R + VL + VC R L 1 √ LC  1 L −i Vdri (t) Rl C

1.5 Frequency Stabilization

37

Using the Fourier transforms, E(ω) = F [E(t)] and Edri (ω) = F [E dri (t)], we map (1.149) into the frequency domain with ω close to ω LC E(ω) κ/2  . Edri (ω) κ/2 + i(ω − ω LC )

(1.150)

Comparing (1.150) with (1.146), one finds the linewidth ΔωFWHM (mode frequency ωFP ) of the FP resonator is equivalent to the energy loss rate κ (LC oscillation frequency ω LC ). It should be noted that unlike the fixed ω LC , ωFP has multiple values (different longitudinal modes). One may also see that (1.147) is a firstorder time-differential equation while (1.149) takes the form of second-order time derivative. To interpret this difference, we re-express the electric field inside C iω LC t ˜ ˜ . Generally, the envelope E(t) varies slowly compared to the as E(t) = E(t)e iω LC t ˜ ˜ , i.e., |d E(t)/dt|  ω LC | E(t)|. In this slowly varying envelope fast variation e approximation (see Problem 1.1), (1.149) is reduced to κ  κ d E(t) = − − iω LC E(t) + E dri (t), dt 2 2

(1.151)

which is equivalent to (1.147).

1.5.3 Laser Frequency Locking to Transmission Line Since the longitudinal modes of a Fabry–Pérot resonator have the fixed frequencies ωm = mωFSP , they can be used as a frequency reference and a laser frequency may be locked to its nearest resonator mode. Figure 1.12a displays the intensity transmission line (also called the absorption line) corresponding to the ωFP mode. The curve has a Lorentzian profile L(ω) in the near-resonance zone ω ∼ ωFP . A laser light with its frequency ω close to ωFP passes through the resonator. We assume ω is weakly modulated by a sinusoidal local oscillator at a small frequency Ω. In general, either the laser frequency ω or the resonator’s frequency ωFP can be modulated by the local oscillator, depending on which one is more convenient for the specific application. These two approaches are similar in principle. Here, we apply the former one. Varying ω can be implemented through either directly modulating the transition frequency of the lasing active medium or adding an external modulator to modulate the laser’s output beam. A fraction of the laser light is incident on the Fabry–Pérot resonator. We write the incident field as (+) (t) = E inc

E0 exp [−i(ω + δω sin Ωt)t] , 2

(1.152)

with the modulation amplitude δω much smaller than the FWHM of the resonator’s mode, δω  ΔωFWHM . The power of the transmitted light is then written as

38

1 Sensing with Light

Fig. 1.12 Laser frequency stabilization based on the transmission line of a Fabry–Pérot resonator. a In the upper graphic, the laser frequency is modulated by a harmonic local oscillator (LO) at the frequency Ω. The resonator’s transmission spectrum presents a Lorentzian profile L(ω). The transmitted power of the laser power Itra (t) oscillates at either Ω or 2Ω. The lower graphic shows the weight of the Ω-frequency component in Itra (t) as a function of the laser-resonator detuning Δ. The line shape is equivalent to the derivative curve of L(ω). b Schematic diagram of locking laser frequency to the transmission line of the resonator, with the symbols: BS (beam splitter), PD (photodetector), LO (local oscillator), and PID (proportional-integral-derivative) controller

Itra (t) ∝

(ΔωFWHM /2)2 E 02 · , 4 (ΔωFWHM /2)2 + (Δ + δω sin Ωt)2

(1.153)

with the detuning Δ = ω − ωFP . When the laser frequency is resonant with the optical resonator Δ = 0, Itra (t) oscillates at a frequency of 2Ω. In contrast, the transmitted intensity with Δ = 0 contains the component oscillating at Ω. As |Δ| is increased, the weight of the Ωfrequency component becomes large. For |Δ  δω|, Itra (t) oscillates approximately at Ω, where Itra (t) with Δ < 0 (Δ > 0) is in-phase (out-of-phase) with the local oscillator. A photodetector is used to convert the transmitted photon flux into a photocurrent. The photocurrent further enters a typical lock-in amplifier, which is capable of extracting a small signal (i.e., the Ω-frequency component) buried in a strong background. The dependence of the lock-in amplifier’s output on the detuning Δ is also shown in Fig. 1.12a. As one can see, the frequency-dependence curve crosses zero at the resonance ω = ωFP , which is referred to as the locking point, and exhibits a derivative behavior of the Lorentzian line shape. Within the near-resonance region around ωFP , the lock-in amplifier’s response is almost linearly proportional to the detuning Δ. This region is called the locking range and is narrower than ΔωFWHM . The output from the lock-in amplifier is finally fed back into the laser system via a proportional-integral-derivative circuit, thereby locking the laser frequency ω to the

1.5 Frequency Stabilization

39

frequency of the resonator’s mode ωFP . The schematic diagram of the laser-frequency stabilization is summarized in Fig. 1.12b. This technique is also applicable to lock the laser frequency to the absorption line of medium, e.g., the saturated absorption spectrum of rubidium or cesium atoms [42].

1.5.4 Pound–Drever–Hall Technique Besides the transmission line, the frequency locking of the light field can be also implemented by using the reflection spectrum of a Fabry–Pérot resonator, which is known as the Pound–Drever–Hall (PDH) method [43]. Nowadays, the PDH technique has become a standard tool for obtaining a narrow-linewidth laser. The best record of laser’s linewidth achieved so far reaches as small as 5 mHz [33, 34]. The PDH approach is schematically depicted in Fig. 1.13. A portion of the light output from the laser system passes through an electro-optic modulator. The modulator is driven by a local oscillator at a microwave frequency Ω. Thus, the light’s phase is modulated with an amplitude β and the frequency Ω, resulting in E (+) (t) =

E 0 −iωt−iβ sin Ωt e . 2

(1.154)

Here, E 0 and ω are the light’s amplitude and frequency, respectively. Using the Jacobi–Anger identity, we decompose the above expression as

Fig. 1.13 Schematic diagram of Pound–Drever–Hall method with the symbols: EOM (electro-optic modulator) and PBS (polarization beam splitter)

40

1 Sensing with Light

∞ E (+) (t) = J0 (β)e−iωt + 2e−iωt J2m (β) cos(2mΩt) m=1 E 0 /2 ∞ −2ie−iωt J2m+1 (β) sin[(2m + 1)Ωt], m=0

(1.155)

with the n-th (n ∈ Z) Bessel function of the first kind Jn (β). Note that J−n (β) = (−1)n Jn (β). In the limit of β  1, we only retain the first three terms and obtain E (+) (t) ≈ J0 (β)e−iωt − J1 (β)e−i(ω−Ω)t + J1 (β)e−i(ω+Ω)t . E 0 /2

(1.156)

The light is then reflected by a Fabry–Pérot resonator, whose amplitude reflection coefficient is rFP (ω) with a central frequency ωFP close to ω. The reflected light reads as E (+) (t) = rFP (ω)J0 (β)e−iωt − rFP (ω − Ω)J1 (β)e−i(ω−Ω)t E 0 /2 +rFP (ω + Ω)J1 (β)e−i(ω+Ω)t .

(1.157)

The resonator changes the relative weights of different frequency components as the coefficient rFP is frequency-dependent. The intensity of the reflected light is measured by a photodetector, Iref (t) ∝ E (+) (t)E (−) (t). To the first order of J1 (β), Iref (t) mainly contains three components Iref (t) ≈ Icon + IΩ,c (ω) cos Ωt + IΩ,s (ω) sin Ωt,

(1.158)

with the constant term Icon ∝ |rFP (ω)|2 J02 (β) + |rFP (ω − Ω)|2 J12 (β) + |rFP (ω + Ω)|2 J12 (β),

(1.159)

and the cosine and sine terms at the frequency Ω having the amplitudes IΩ,c ∝ 2J0 (β)J1 (β)Re[H (ω)], IΩ,s ∝ 2J0 (β)J1 (β)Im[H (ω)].

(1.160)

We have defined the frequency-discrimination function ∗ ∗ (ω)rFP (ω + Ω) − rFP (ω)rFP (ω − Ω). H (ω) = rFP

(1.161)

Figure 1.14 illustrates the dependence of IΩ,c (ω) and IΩ,s (ω) on the laser frequency ω. We see that IΩ,c (ω) follows a dispersion-like curve while IΩ,s (ω) exhibits an absorption behavior. The dispersion IΩ,c (ω) presents a steep slope around the central frequency ωFP , providing an excellent frequency-discrimination ability. The width of this slope region is called linear dynamic range, within which the frequency fluctuation may be strongly suppressed via the feedback control system. Indeed, this range depends on the Q factor of the optical resonator and characterizes the

1.5 Frequency Stabilization

41

Fig. 1.14 Pound–Drever–Hall frequency discrimination. Dispersion IΩ,c and absorption IΩ,s versus the frequency ω of a probe beam. The cavity length is L = 10 cm and the modulation frequency is Ω = 2π × 20 MHz

sensitivity of the frequency locking system. In order to pick up the IΩ,c (ω) component in photocurrent, Iref (t) is mixed with the local-oscillation signal (after the phase shifter) and further passes though a low-pass filter. The output is eventually fed back into the laser system via a proportional-integral-derivative circuit, stabilizing the frequency ω. Locking the laser frequency to a Fabry–Pérot interferometer may narrow its linewidth by several orders of magnitude. However, such a narrowing cannot go down infinitely but is ultimately limited by the inevitable Brownian motion of the resonator, including the mirror coatings, substrates, and spacer [44]. The thermal vibration of the mirrors, coming from the temperature fluctuation, seismic and acoustic vibration, and the change of the atmospheric pressure can cause the fluctuation δL of the resonator length L. This fluctuation is further mapped into the laser frequency, influencing the stability of the laser frequency. A lot of efforts have been paid to reduce the Brownian thermomechanical noise in the resonator. For example, the Fabry–Pérot resonator is protected by a vacuum chamber and the in-chamber temperature is stabilized. The chamber is further placed on a vibration isolation platform. In addition, the optical cavity should be made of the low-expansion materials, such as ultralow expansion glass (ULE), fused silica, and single-crystal silicon. It is worth noting that locking the laser frequency to a Fabry–Pérot resonator only improves its short-term stability. This is because the slow frequency drift of the resonator cannot be eliminated completely. A good long-term frequency reference is essentially reproducible and hardly affected by the environmental perturbations. The narrowline optical transitions of atoms and ions are determined by nature and, therefore, are suitable to serve as the long-term frequency references.

42

1 Sensing with Light

Besides the PDH technique, the Hansch–Couillaud method [45] also utilizes the reflection property of the Fabry–Pérot resonator to stabilize the laser frequency. However, this method relies on the feature of the frequency-dependent polarization of the light [see Problem 1.4], rather than the phase modulation.

1.5.5 Shot Noise In experiments, only the light power is measurable. Practically, the detectable light signal cannot be infinitely small. For example, an optical sinusoidal wave at a frequency ωsig is incident on a photodetector. The light power is given by 2 /2 with the amplitude E sig and the beam area A. The photodetector Psig = Acε0 E sig converts the light into a dc current i dc = geη

Psig , ωsig

(1.162)

with a gain g and a quantum efficiency η ≤ 1. The mean power of the electric signal 2 is proportional to the root mean square (rms) of the photocurrent i dc , i.e., square-law photodetector. We now consider the background fluctuations in current, among which two dominant components are the shot and thermal noises, in the photodetector. The shot noise arises from the discrete nature of the electrons. We assume that within a characteristic time scale Δt, the number of the electrons flowing from the cathode to the anode of the photodiode is n. The fluctuation in n obeys the Poisson statistics, for which the ¯ Thus, the rms fluctuation in the variance of n is equal to its mean value, Δn 2 = n. current is given by the Schottky formula [46] 2 = g 2 e2 i sn

Δn 2 n¯ i dc = 2gei dc Δf, = g 2 e2 = ge (Δt)2 (Δt)2 Δt

(1.163)

where Δf = 1/2Δt is the detection bandwidth. The thermal noise (also known as Johnson noise) results from the random motion of the electrons in a resistor [47] and happens regardless of applied voltage. The rms thermal fluctuation in the current takes the form of the Nyquist formula [48] i tn2 =

4k B T Δf , R

(1.164)

with Boltzmann’s constant k B = 1.380649 × 10−23 J/K, the absolute temperature T , and the input resistance of the detection circuit R. Consequently, the signal-to-noise (S/N) ratio reads as [49]

1.5 Frequency Stabilization

43



 Psig 2 g e η ωsig . = 4k B T Δf 2gei dc Δf + R 2 2 2

2 S i dc = 2 + i2 N i sn tn

(1.165)

In the limit that the shot noise power much exceeds the thermal noise power, we have S η Psig = . N 2Δf ωsig

(1.166)

Setting S/N = 1 leads to the minimum detectable optical power min(Psig ) =

ωsig . ηΔt

(1.167)

It is seen that increasing the incident power can always enhance the detection sensitivity. In addition, narrowing the bandwidth Δf reduces the minimum detectable signal but extends the detection (integration) time Δt.

1.5.6 Optical Heterodyne and Homodyne Detection Various techniques have been developed to measure a weak-power optical signal. One approach is the so-called heterodyne technique (also known as coherent detection), whose basic idea is to mix the weak signal E sig cos ωsig t with a strong local-oscillation beam E LO cos ωLO t, as illustrated in Fig. 1.15a. The frequency ωsig is close to ωLO . We assume that the signal and local-oscillator beams have the same transverse profile, i.e., perfectly overlapping each other. The typical signal and local-oscillator 2 2 /2 ∼ fW (femtowatt) and PLO = Acε0 E LO /2 ∼ mW powers are Psig = Acε0 E sig (milliwatt), respectively, with the area of the beam spot A. The photodetector converts the superimposed light into the electric current with a dc component

Fig. 1.15 Optical heterodyne (a), balanced heterodyne (b), and homodyne (c) detection with the symbols sig (signal), LO (local oscillator), PD (photodetector), and PBS (polarization beam splitter)

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1 Sensing with Light

i dc =

geη (PLO + Psig ), ωLO

(1.168)

and an oscillating component (i.e., beat note) i beat (t) =

2geη  PLO Psig cos(ωsig − ωLO )t. ωLO

(1.169)

The frequency difference |ωsig − ωLO | is typically within the radio or microwave band. Thus, one advantage of the heterodyne detection is to shift the optical frequency of the signal down into the electronically tractable band. In addition, due to  PLO Psig ∼ nW (nanowatt), the amplitude of the electrical signal is several orders of magnitude larger than that of the direct detection. The SNR of the square-law detector is then derived as   geη 2 2 PLO Psig S ωLO =  2 2  . N g e η 4k B T 2 (PLO + Psig )Δf + Δf ωLO R

(1.170)

In the shot-noise limit, the above equation is simplified as η Psig S ≈ , N Δf ωsig

(1.171)

which is twice that of the direct detection [see (1.166)]. The eventual SNR cannot be improved by the heterodyne technique but is still limited by the shot noise of the signal only. This is understandable, because the powers of the electrical signal and the shot noise are both increased as the power of the local oscillator PLO is raised. Nevertheless, the heterodyne detection enables the sensitive detection of the weak signal. For the direction detection, (1.166) is only valid for a large enough signal power Psig , i.e., the shot noise much exceeds the thermal fluctuation and the shotnoise limit is reached. In practice, for a low-power optical signal the thermal noise dominates the background fluctuations in the direct detection, rather than the shot noise. However, by using the heterodyne technique, the short-noise-limited SNR can be simply accessed through increasing the local oscillator’s power PLO . One disadvantage of the heterodyne scheme is that the local oscillator directly introduces the excess noise (i.e., the light intensity fluctuations), mainly coming from the term PLO into the detection. This issue may be avoided in the balanced heterodyne method [50, 51] as shown in Fig. 1.15b. The optical signal and local-oscillation beams are mixed by a polarization-dependent 50/50 beam splitter, which produces two subbeams. The signal and local-oscillation components are in-phase in one sub-beam while out-of-phase in the other. We assume that the powers of the signal components in two sub-beams are the same and so are two local-oscillation components. Two

1.5 Frequency Stabilization

45

sub-beams then respectively enter two photodetectors and the measured currents are calculated to be   geη  PLO + Psig + 2 PLO Psig cos(ωsig − ωLO )t , (1.172) i 1 (t) = ωLO    geη i 2 (t) = PLO + Psig − 2 PLO Psig cos(ωsig − ωLO )t . (1.173) ωLO The difference between i 1,2 cancels the terms of PLO and Psig , i 1 (t) − i 2 (t) =

4geη  PLO Psig cos(ωsig − ωLO )t. ωLO

(1.174)

The shot-noise power in either photodetector is about 2(g 2 e2 η/ωLO )PLO Δf . Thus, the resultant SNR is given by S 4η Psig , (1.175) ≈ N Δf ωsig enhanced by four times over the common heterodyne detection [see (1.171)]. A variant of the balanced heterodyne detection is the balanced homodyne detection with ωLO = ωsig . Generally, both signal and local-oscillator beams come from the same light source. This technique is commonly used to extract the phase information of the light field. Figure 1.15c shows that a medium placed in the optical path of the signal sub-beam introduces an extra phase ϕ(t) to the signal. The current produced by the corresponding photodetector is then derived as i(t) =

  geη  PLO + Psig + 2 PLO Psig cos ϕ(t) , ωLO

(1.176)

extracting the phase information ϕ(t) carried by the signal light.

1.5.7 Measuring the Photon Number As an important optical device in experiments, the photodetectors need to precisely count the number of photons. In general, the photodetectors are classified into the semiconductor-based and photoemissive (photoelectric effect-based) types. The semiconductor photodiodes, which are commonly used in experiments, belong to the former type. The incident photons hit on the p-n (or p-i-n) junctions of the photodiodes and create the hole-electron pairs. Two kinds of charge carriers, holes, and electrons drift in the opposite directions under the reverse-biased voltage, giving rise to the electric current. In contrast, the photodetectors of the second type (e.g., phototransistors) are similar to the photodiodes but incorporate the internal gain that amplifies the photocurrent within the detectors. Such a mechanism much enhances the detector’s sensitivity.

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1 Sensing with Light

Some photodetectors, such as the photomultiplier tubes (PMTs), avalanche photodiodes (APDs), and visible light photon counters (VLPCs) [52, 53], are even capable of measuring the weak light power at the single-photon level [54]. In a PMT, a photon incident on the photocathode scatters a single electron. This electron is multiplied at the successive dynode stages, leading to a detectable current. In an APD, an external voltage is applied to accelerate the charge carriers to a high kinetic energy, i.e., Geiger-mode operation. Further carriers can be liberated when the charge carriers collide with the lattice. As a result, the number of carriers increases exponentially, forming a huge avalanche current. In a VLPC, the intrinsic absorber and gain regions are separated, where a hole–electron pair is generated in the absorber region and only the hole is accelerated toward the gain region, triggering an avalanche. Several factors are usually used to assess a photodetector: (i) Quantum efficiency η, which is defined as the ratio of the number of the photoelectrons collected by the detector to the number of the incident photons. The photocurrent I produced by the incident light with a power P is given by I = eη P/ω = R P, where R = (ηλ/1.24) A/W is the photodetector’s responsivity5 and the incident wavelength λ is in units of nm. The practical photocathodes have a quantum efficiency less that 30%. In comparison, the antireflective coating on the surface of the APDs enhances the quantum efficiency up to 60 ∼ 80% at the peak wavelength around 500−600 nm [55]. (ii) Dead time (recovery time), i.e., the time interval during which the detector is unable to absorb a second photon after the previous photon-detection event. For example, after a single photon triggering an avalanche of the charge carriers, the APD must be prepared for the next detection event, i.e., removing the excess carriers. This quenching process leads to an undesired dead time, limiting the detector’s counting rate. The typical dead time of the APDs ranges tens of ns to 1 µs. In contrast, the dead time of the PMTs is about 1 ∼ 10 ns. (iii) Dark count rate is associated with the false detection events caused by the dark current in the detectors. (iv) Resolving photon number, i.e., the capability of distinguishing the number of photons. The APDs can only distinguish zero and ‘one or more’ photons. In contrast, the VLPCs are capable of distinguishing the higher photon numbers.

1.6 Optical Sensing Technologies The laser light has been applied in numerous scientific research. Its functions include restricting the movement of the microscopic particles, changing the internal and external state of quantum emitters, imaging, measuring, and remotely transferring both classical and quantum information. In this section, we briefly introduce several relevant applications in sensing.

5R

=

eηλ hc

with the Planck’s constant h = 6.626 × 10−34 J · s.

1.6 Optical Sensing Technologies

47

1.6.1 Michelson–Morley Interferometer One of widely applied optical sensors is the Michelson–Morley interferometer (MMI) [56], whose structure is illustrated in Fig. 1.16a. A laser beam at the wavelength λ is split into two branches via a beam splitter. Two sub-beams travel along different paths (arms) and are reflected back toward the beam splitter by the mirrors 1 and 2, respectively. The length difference between two optical paths is d. Two subbeams are recombined together and enter a photodetector, measuring their interference. From the recorded pattern, one may evaluate the phase difference Δφ = 2πd/λ between two optical paths. The displacement of either mirror changes Δφ, thereby varying the interference pattern. This feature can be utilized to measure a tiny displacement of an object. The MMI was originally proposed to probe the luminiferous aether in 1887. However, its null result disproved the aether’s existence. The most well-known MMI is perhaps the LIGO (Laser Interferometer Gravitational-Wave Observatory), whose arm length reaches 4 km. The Fabry–Pérot resonator is applied in each arm to further extend its length up to 1,600 km. LIGO was built for detecting the cosmic gravitational waves, which are one prediction of Einstein’s general relativity theory. When a gravitational wave passes the LIGO, it causes the distortion of two arms in an opposite fashion. In 2015, the LIGO firstly recorded a transient gravitational-wave signal with a peak strain of one part in 1021 [5].

Fig. 1.16 Optical sensors. a Schematic diagram of the Michelson–Morley interferometer. b Principle of atomic clocks. The laser light from a local oscillator (LO) interacts with an ensemble of lattice-confined atoms whose central transition frequency is ω0 . The measured error signal ds/dω is fed back into the oscillator, stabilizing the oscillator’s frequency ω to ω0 . The optical frequency can be converted down to the radio (RF) or microwave (MC) frequency via a frequency comb

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1 Sensing with Light

1.6.2 Optical Clocks The entire SI is constructed from seven base units: kilogram for mass, second for time, kelvin for temperature, ampere for electric current, mole for the amount of a substance, candela for luminous intensity, and meter for distance. However, only the second is the most precisely measurable quantity owing to the continuous development in ultrahigh-resolution spectroscopy and atomic physics. The current SI second is defined as the duration of 9, 192, 631, 770 periods of the unperturbed ground-state hyperfine transition of 133 Cs. The most accurate timekeepers are the so-called atomic clocks, whose working principle is illustrated in Fig. 1.16b. A typical atomic clock mainly consists of a local oscillator (producing a periodic-time/frequency signal), a frequency reference (whose frequency is reproducible and uninfluenced by the external perturbations), a detector (measuring the frequency difference between the oscillator and the reference), and a servo loop (feedback controlling the frequency of the local oscillator) [57]. The local oscillator, whose frequency ω is usually prestabilized to a microwave or optical frequency resonator, outputs an electromagnetic signal to interact with the frequency reference at ω0 . The transmitted electromagnetic field is measured by the detector. The error signal drives the servo to compensate the oscillator’s frequency. The heart of the atomic clocks is the frequency reference, whose frequencydiscrimination curve is required to be stable and accurate. It is natural to choose the quantum absorbers (or emitters), such as atoms, ions, and molecules, as the frequency discriminator since the transition frequency ω0 between two specific states of the quantum absorbers is completely determined by nature. According to the range of the operating frequency ω0 , the atomic clocks may be divided into microwave (e.g., Cs and Rb) and optical (e.g., Sr, Yb, and Hg) groups. The stability (or instability) and accuracy (or uncertainty) are commonly employed to evaluate the performance of an atomic clock. So far, the optical clocks have already surpassed the microwave clocks in terms of both stability and accuracy. The stability σ y (τ ) measured by the Allan deviation [58] refers to how consistent the clock’s certain-time-interval-separated ticks are. Here, τ is the measurement time of each tick. The ultimate frequency stability in short term, limited by the quantum projection noise (QPN) [59], may be expressed in an analytical form σ QPN (τ ) y

Δω 1 = √ ω0 N



Tc , τ

(1.177)

where Δω is the linewidth of the frequency-discrimination curve, N is the number of quantum absorbers, and Tc is the clock cycle time. The smaller σ y (τ ) is, the more stable the clock is. It is seen that a large ω0 can strongly reduce σ y (τ ). For this reason, the optical clocks stay in the frontier of the modern measurement science. In addition, σ y (τ ) goes down as N is increased. Thus, in comparison to the single-ion optical clocks, the neutral atoms are the most promising candidates for implementing an ultra-stable clock.

1.6 Optical Sensing Technologies

49

The motion of atoms needs to be maximally frozen so as to minimize the Doppler effect and interatomic collisions. To this end, the ensemble of neutral atoms are tightly confined in a standing-wave optical potential that is formed by a pair of counter-propagating laser beams. Setting the lattice-laser wavelength λm at the socalled magic wavelength enables the clock-transition frequency ω0 immune to the fluctuations in lattice potential [21]. In addition, the atoms follow the harmonic oscillation within the Lamb–Dicke regime [60], where the photon recoil energy of the probe light (i.e., clock laser) E rec =

h2 2Mλ2p

(1.178)

is much smaller than the quantized energy spacing of the harmonic oscillation E osc

h = λm



2U , M

(1.179)

√ i.e., the Lamb–Dicke parameter η = E rec /E osc  1 (see Problem 1.5). Here, M is the atomic mass, λ p is the wavelength of the clock laser, and U is the lattice potential depth. In the Lamb–Dicke regime, the effect of the internal state transition of the atom on the external (motional) state of the atom is negligible. Moreover, prestabilizing the local optical oscillator to a high-Q cryogenic Fabry–Pérot resonator, which is well isolated from the external seismic and acoustic vibrations, significantly narrows the spectral linewidth of the interrogation signal down to a value less than 10 mHz [34]. All these techniques, i.e., the lattice-confined atoms and the narrowlinewidth local oscillator, eventually lead to the birth of optical lattice clocks [61], which represent the state of the art in modern measurement. To date, the record for √ the best stability held by optical lattice clocks is σ y (τ ) = (4.8 × 10−17 )/ τ [62], which is two orders of magnitude lower than that of single-ion optical clocks [4] but (τ ) yet because of the Dick effect [63]. still have not hit σ QPN y The outstanding short-term stability of optical lattice clocks enables the fast evaluation of the systematic uncertainties. Careful elimination of the environmental perturbations on the quantum absorbers significantly improves the clock accuracy Δωshift /ω0 to the order of 10−18 . Here, Δωshift is the shift of the clock-transition frequency caused by the environment. Currently, the accuracy of optical lattice clocks is primarily limited by the Stark shift of the reference frequency induced by various ambient electric fields, among which the blackbody radiation plays the main role [64]. Addressing this issue relies on a highly accurate blackbody-radiation thermometry (∼mK) [65] or directly placing the quantum absorbers in a cryogenic environment [3]. The best accuracy achieved in optical lattice clocks so far is Δωshift /ω0 = 1.4 × 10−18 [66], which denotes that the clock won’t gain or lose a second in 23 billion years, longer than the age of the universe (13.8 billion years).

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1 Sensing with Light

The exceptional stability and accuracy of such optical time-keeping devices enable the wide sensing applications in (1) Constraining the coupling of fundamental constants, i.e., the fine-structure constant α=

1 e2 , 4πε0 c

(1.180)

and the electron–proton mass ratio μ = m e /m p , to the gravitational field [67]. Such a measurement is motivated by testing the unified theories, which predict that the coupling of these constants to the gravity may cause their variation in time and space. The latest constraints on their relative uncertainties [68] are α/α ˙ = −0.7 × 10−17 yr−1 −17 −1 (i.e., per year) and μ/μ ˙ = 0.2 × 10 yr , respectively. (2) Observing gravitational waves in new frequency ranges. The LIGO and Virgo optical interferometers have detected the gravitational waves with a frequency from tens to hundreds of Hz [5, 6]. Recently, there is a large interest in extending the detection range toward the low-frequency band of 10−4 ∼ 1 Hz because of a large number of gravitational wave sources near the earth. The gravitational wave detection can be implemented through the Doppler-tracking method [69]. In this method, two synchronized optical lattice clocks are carried by the spatially separated satellites, respectively, which fly in space, avoiding the seismic and gravity gradient noises. (3) Relativistic geodesy (height measurement). According to Einstein’s relativity, the clock situated at a smaller gravitational potential (lower height) evolves slower than that positioned in a larger gravitational field (higher height), i.e., the time dilation. When a photon flies from a lower to a higher height, its frequency becomes smaller, i.e., the gravitational red shift. Comparing two optical lattice clocks has shown a fractional frequency shift of 10−18 per height difference of 1 cm [70]. (4) Hunting for dark matters, which have long been a scientific mystery. It has been pointed out that the huge dark-matter objects can cause a transient-in-time change of the fundamental constants [71], which may be detected by using a network of optical atomic clocks [72].

1.6.3 Single-Atom Detection At room temperature and standard atmospheric pressure (1 atm), the number of oxygen molecules within one liter of air is about 5 × 1021 and the root-mean-square velocity of the gas molecules approximates 500 m/s. It was inconceivable to capture, control and manipulate a single individual among such numerous moving particles four decades ago. Thanks to the techniques of laser cooling and trapping of neutral atoms [73], the detection and manipulation of single atoms become implementable in the laboratory now. As demonstrated in [74], an ensemble of ultracold atoms are confined in a twodimensional square optical lattice (lattice constant of a = 532 nm). The lattice potential U is slowly raised from a lower to a higher value (much higher than the recoil energy E rec of the lattice laser). The quantum gas undergoes a phase transition to the

1.6 Optical Sensing Technologies

51

Fig. 1.17 Fluorescence images of single atoms. Reprinted by permission from Springer Nature Customer Service Center GmbH: Springer Nature, Nature, [74], © 2011

so-called Mott insulating state [75] with only one atom per lattice site. The internal state of a single atom at a certain site may be manipulated by using an off-resonant laser beam, which is strongly focused by a high-resolution imaging system with a resultant beam diameter of 600 nm, and microwave pulses. The laser beam can be moved and positioned with an accuracy of 0.1a. The Landau–Zener method is applied to induce the transition of the target atom between two ground-state hyperfine states. The final state of the atom is detected via the fluorescence imaging. Figure 1.17 displays the result of single-atom control and detection. The resulting fidelity reaches as high as 95% and the influence of the manipulation of the target atom on its neighboring atoms is negligible.

1.6.4 Optical Tweezers Optical tweezers utilize the laser light, which is strongly focused via a high numerical aperture lens, to hold and manipulate the microscopic dielectric objects. The laser beam directly passes through the object, whose size ranges from micrometer (e.g., living cells) to nanometer (e.g., atoms and molecules). In general, the light force exerted on an object includes the gradient and scattering components. The gradient force is also referred to as the trapping force and has a typical value of the order of 1 ∼ 102 piconewtons. The region close to the beam waist presents a strong electric field gradient. When the object is displaced from the beam center, for example, in a transverse direction, the object’s part located in the intensive light field refracts more light power than the part located in the relatively weak light field [see Fig. 1.18a]. This results in a lateral net force toward the beam center exerting on the object. The gradient force always points to the beam center, i.e., the three-dimensional confinement, and the dielectric object can be trapped at the beam center in principle. In contrast, the scattering force arises from the object absorbing the photons, acquiring the excess momentum along the direction of the beam propagation and scattering the photons in all directions. The conservation of momentum leads to a net force along the beam propagation [see Fig. 1.18b]. As a result, the object is trapped at the position displaced

52

1 Sensing with Light

Fig. 1.18 Principle of optical tweezers. a Net force Fnet in the transverse direction. b Axial force Fnet

slightly downstream from the beam center since a portion of the gradient force needs to compensate the scattering force. In common cases, an optical tweezer holds one particle in the trap [76]. However, an optical-trap array with a number up to 102 may be implemented by means of diffracting the laser beam into multiple spots [77, 78]. Unlike an optical lattice with a fixed nearest-neighbor separation, the distance between two optical tweezers in an array can be tuned within 1 ∼ 10 µm. This provides a way to control the interparticle interactions. In addition, the position of a tweezer in an array is movable, for which the array pattern may be artificially designed. The applications of optical tweezers include deforming the living cells, single molecule force measurement, and nanotechnology, all of which make the optical tweezers an indispensable tool for studying a variety of physical, chemical, and biological systems.

Problems 1.1 A light field is polarized in the x-direction and propagates along the z-axis, E(r, t) = ex E (+) (z, t) + ex E (−) (z, t), in a medium. The electric polarization density of the medium P may be written in a similar way, P(r, t) = ex P (+) (z, t) + ex P (−) (z, t). Inserting the above expressions into (1.40), one obtains 

∂ 1 ∂ + ∂z c ∂t

  ∂ 1 ∂ ∂2 − + E (+) (z, t) = −μ0 2 P (+) (z, t). ∂z c ∂t ∂t

The positive-frequency parts E (+) (z, t) and P (+) (z, t) may be written in the following way: 1 1 E (+) (z, t) = E(z, t)eik0 z−iω0 t , P (+) (z, t) = P (z, t)eik0 z−iω0 t , 2 2

1.6 Optical Sensing Technologies

53

Fig. 1.19 Boundary conditions. Stokes’ theorem is applied on (a) while Gauss’s theorem is used on (b)

where E(z, t) and P (z, t) are the slowly varying amplitudes. The wavenumber k0 is related to the light’s central frequency ω0 via k0 = ω0 /c. Derive the following equation:   k0 ∂ 1 ∂ E(z, t) = i P (z, t), (1.181) + ∂z c ∂t 2ε0 in the slowly varying envelope approximation (SVEA)    ∂E(z, t)       k0 |E(z, t)|,  ∂E(z, t)   ω0 |E(z, t)|,  ∂P (z, t) |  ω0 P (z, t)|. ∂z ∂t ∂t 1.2 Derive the interface conditions for the electromagnetic fields (1.45) and (1.48) by integrating Faraday’s law (1.1a) and Ampère’s law (1.1d) over an oriented smooth surface S bounded by a closed boundary curve C across the interface [see Fig. 1.19a]. Stokes’ theorem   (∇ × F) · dS = F · dl (1.182) S

C

should be applied. The curve C is positively oriented, meaning that the surface S is always on your left side when you are walking along the curve C. The direction of the element vector dS is parallel to the unit normal vector of the curve C that follows the right-hand rule. The vector field F may be E and H. Derive the boundary conditions (1.46) and (1.47) by integrating (1.1b) and (1.1c) over a volume V surrounded by a surface S across the interface [see Fig. 1.19b]. Gauss’s theorem   (∇ · F)d V = F · dS (1.183) V

S

should be used. Here, the vector field F may be D and B. The outward-pointing element vector dS is normal to the locally planar surface. 1.3 We consider the solution to Poisson’s equation within a volume V . The associated Green’s function G(r, r0 ) is defined as ∇ 2 G(r, r0 ) = δ(r − r0 ),

(1.184)

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1 Sensing with Light

Fig. 1.20 Hansch–Couillaud frequency stabilization. a Schematic diagram with the symbols: LP (linear polarizer), PBS (polarization beam splitter), and PD (photodetector). b Polarization spectroscopy as a function of the detuning (ω − ωFP ). The amplitude reflection coefficients of mirrors of the FP resonator are both r1,2 = 0.9. The free spectral range of the Fabry–Pérot resonator is ωFSR

where G(r, r0 ) may be viewed as a potential generated by a point source located at is the solution r0 ∈ V . As |r − r0 | → ∞, G(r, r0 ) approaches zero. Equation (1.26)  of the above Poisson’s equation. Derive (1.26) by using the fact V δ(r)d V = 1 and Gauss’s theorem. 1.4 The frequency stabilization scheme based on the Hansch–Couillaud method [45] is illustrated in Fig. 1.20a. The incident light (frequency ω) output from a laser system is vertically polarized and enters a Fabry–Pérot resonator (central frequency ωFP ) via a reflection mirror. A linear polarizer, whose transmission direction forms a 45◦ angle with respect to the incident beam’s polarization, is inserted into the Fabry– Pérot resonator. The reflected wave contains two components with the respective

1.6 Optical Sensing Technologies

55

√ √ field amplitude E  = (E 0 / 2) · rFP (ω) and E ⊥ = (E 0 / 2) · r1 , where E 0 is the amplitude of the incident beam, rFP (ω) is the amplitude reflection coefficient of the Fabry–Pérot resonator [see (1.139a)] , and r1 is the amplitude reflection coefficient of the resonator’s mirror M1. The reflected light passes through a λ/4 waveplate and enters a polarization beam splitter (PBS). We assume that the fast axis of the λ/4 waveplate is parallel to the transmission direction of the intracavity linear polarizer. Using the Jones calculus, the light fields output from the PBS are given by 

E1 E2



    1 1 −1 10 E = . E⊥ 0i 2 −1 1

(1.185)

The intensities of the outputs from the PBS I1 ∝ |E 1 |2 and I2 ∝ |E 2 |2 are measured by two photodetectors, respectively. The difference (I1 − I2 ) gives the error signal [see Fig. 1.20b]. This error signal is finally fed back into the laser, stabilizing the frequency ω. Derive the error signal (I1 − I2 ). 1.5 In a 87 Sr optical lattice clock, the 87 Sr atoms are confined in a one-dimensional optical lattice with a potential depth of 20 µK. The lattice lasers operate at the red-detuned magic wavelength 813 nm. The clock transition wavelength is 698 nm. Estimate the harmonic-oscillation frequency of the atoms moving inside a lattice site and the Lamb–Dicke parameter.

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Chapter 2

Surface Plasmon Resonance

Abstract The sensitivity of an optical measurement is closely related to the light– matter interaction strength. Enhancing the local-field intensity raises the sensor’s sensitivity. The surface plasmon resonance of metal films and the localized surface plasmon resonance of metal nanoparticles enable the subwavelength confinement of the light field, overcoming the classical diffraction limit. The resulting enhancement factor of the local-field intensity can exceed 103 . The interaction of light and metal films/nanoparticles is studied by means of the classical electromagnetic theory. The analytical analysis is applicable to the simple metal structures while the finite-difference time-domain and finite element methods are used to investigate the complex structures.

2.1 Introduction In the previous chapter, we have briefly introduced the interaction between light and the dielectric medium based on the Lorentz classical theory. We focus in this chapter on the interaction of metals (here we mainly consider the noble metals) with the optical-frequency radiation. As we will see, the light–metal interaction gives rise to a combined oscillation of the conduction electrons in metals and the optical waves. The repulsive interaction among the electrons makes the collective oscillation occur on the metal surface, i.e., the so-called surface plasmon resonance [1–3]. Basically, such collective oscillations are entirely caused by the negative permittivity of metals and can be divided into two groups. The surface plasmon polaritons (SPPs) exist at the dielectric–metal interface. This kind of plasmons travel along the planar surface waveguides but suffer the strong confinement in the vertical direction. To date, a great deal of studies have been carried out on the SPP-based nanophotonic integrated circuits [4–6], where a wide variety of plasmonic devices and components are developed to perform different optical functions. The other kind of surface plasmons, known as the localized surface plasmon resonances (LSPRs), are non-propagatable but highly localized around the metal particles, whose sizes are much smaller than the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2_2

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radiation wavelengths. The curved surface of these nanostructures enables the direct excitation of the surface plasmons via the light illumination. The strong confinement of the light overcomes the classical diffraction limit, leading to a huge enhancement of the near-field radiation. This feature has been widely applied in biosensing and single-molecule detection [7–9]. The first section is dedicated to a brief introduction of surface plasmons. We take the dielectric function of metals as a starting point since it is the basis of all plasmonic properties. Although a rigorous treatment relies on quantum mechanics, here we choose the classical electromagnetic theory so as to deliver a simple and intuitive description. The SPPs and LSPRs are respectively discussed in the next two sections, where the plasmonic oscillations are derived from Maxwell’s equations plus the specific boundary conditions. The resonant plasmons can be identified through the light scattering measurement. The last section is focused on the interplasmon coupling in assembled nanoparticles, which is a common way to implement the near-field intensity enhancement.

2.2 Plasmonics: Interaction of Light with Metals Noble metals have been widely applied in the fields of plasmonics and nanophotonics. There is, however, actually no strict definition of the membership in the noble-metal group. In chemistry, the noble metals are the metals that are resistant to the corrosion and oxidation in moist air, for instance, gold (Au), silver (Ag), and platinum (Pt). In atomic physics, a noble metal has the fully occupied d-like bands of the electronic structure, for which copper (Cu) is counted in the list. The optical properties of noble metals are normally ascribed to the complex relative permittivity , which is related to the index of refraction n and the extinction coefficient κ via  = n 2 − κ2 + i2nκ.√Writing  as a summation of the real and imaginary parts,  = 1 + i2 , yields n = (1 + ||)/2 and κ = 2 /(2n). The refractive index n denotes the ratio of the phase velocity of the optical waves in vacuum to that in the material. The extinction coefficient κ accounts for the exponential attenuation of the optical waves when they travel in the material. The complex dielectric function  of the noble metals depends on the radiation frequency ω, i.e., (ω), and is primarily determined by the response of the electrons in materials to the external light field. Studying (ω) may be within the framework of either classical or quantum theory. Here we choose the former approach. When a light field E(ω) at the frequency ω is applied on the material, a net displacement r is imposed on the motion of electrons. Such a light-induced displacement further leads to a macroscopic polarization P(ω). The dielectric function (ω) can be then derived from D(ω) = (ω)ε0 E(ω) = ε0 E(ω) + P(ω). In the following, we consider the contribution from the actions of both conduction and bound electrons to the material’s relative permittivity (ω).

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63

2.2.1 Drude–Sommerfeld Model We start with the optical response of the free (conduction-band) electrons in the noble metals driven by an optical wave E0 e−iωt . Following Newton’s second law, the equation of motion for the instantaneous deviation of an electron from its equilibrium position takes the form F(t) , (2.1) r¨ + γ f r˙ = me with the mass of a conduction electron m e , the force F(t) = −eE0 e−iωt , and the elementary charge e. The second term on the left-hand side refers to the damping force caused by the inter-electron collisions. The energy loss rate γ f approximates γ f = vF /lmfp with the Fermi velocity vF and the mean free path lmfp of the electrons. The ansatz r(t) = r0 e−iωt leads to the solution r0 =

eE0 . ω(ω + iγ f )

(2.2)

The macroscopic polarization of the electron gas is calculated to be P = n e d with the density of conduction electrons n e and the field-induced electric dipole moment d = −er. The relative dielectric constant is then given by =1−

ωp2 ω(ω + iγ f )

,

(2.3)

where the plasma frequency is defined as  ωp =

n e e2 . m e ε0

(2.4)

Equation (2.3) is usually referred to as the Drude–Sommerfeld model. For most metals and noble metals, ωp is of the order of 1016 s−1 (∼0.1 μm or ∼10 electronvolts1 ), i.e., in the near-ultraviolet regime. We express the complex  as  = 1 + i2 . The imaginary part 2 originates from the nonzero damping rate γ f , corresponding to the inelastic scattering of the light. In the limit γ f ∼ 0, the real part 1 is negative when ω < ωp , whereas it becomes positive when ω > ωp [see Fig. 2.1a]. The meaning of the plasma frequency may be interpreted as follows: When the light’s frequency ω is below ωp , the electrons can respond to the oscillation of the external electric field in a timely fashion, thereby canceling the electric field inside a metal at any time. That explains why the light cannot enter a metal but gets totally reflected (i.e., the shine of metals). When ω exceeds ωp , the metal’s reflectivity drops drastically. It is because the electrons can no longer follow the relatively rapid oscillation of the 1 One

electronvolt (eV) is equal to 1.602 × 10−19 J.

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Fig. 2.1 Relative dielectric constant  = 1 + i2 for Au as a function of the driving frequency ω. a Drude–Sommerfeld model for the conduction electrons with ωp = 8.89 eV and γ f = 0.07088 eV [10]. b Lorentz oscillator model for the bound electrons with ω˜ = 2.97 eV, γ˜ = 0.87 eV, and w = 0.071 [11]. The inset illustrates the schematic diagram of the interband transition Table 2.1 Drude-model parameters for Au, Ag, Cu, and Al (aluminum)a with the free carrier concentration n e from [12], the effective mass m ∗ from [13], the plasma frequency ωp from [14], the Fermi velocity vF and the mean free path lmfp from [15], and the decay constant γ f Au Ag Cu Al a Al

n e (1028 m−3 )

m ∗ /m e

ωp (eV)

vF (106 m/s)

lmfp (nm)

γ f (eV)

5.91 5.86 8.43 18.07

0.99 0.96 1.49 1.45

9.01 8.98 10.77 15.77

1.38 1.45 1.11 1.60

37.7 53.3 39.9 18.9

0.024 0.018 0.018 0.056

is not in the common noble metal group, but it supports the surface plasmon polaritons

external electric field and hence lose the ability of screening the light waves. This accounts for the so-called UV transparency of metals. The plasma frequency ωp and the damping rate γ f for the free-electron gas of several metals are listed in Table 2.1.

2.2.2 Interband Transitions of Bound Electrons The Drude–Sommerfeld model fails in predicting the optical properties of noble metals in the terahertz and visible regime due to the existence of high dissipative loss. This is because the light quanta (photons) can excite the bound electrons from the lower-lying bands to the conduction band. As illustrated in the inset of Fig. 2.1b, one bound electron e− acquires the energy of a single photon ω and then transits to an unoccupied state in the conduction band above the Fermi level E F . The

2.2 Plasmonics: Interaction of Light with Metals

65

excitation of an electron inevitably leaves a hole h+ , which carries a positive charge e, at the electron’s previous position and gives rise to a transient dipole moment. This process is reversible, i.e., the excited electron can return to the bound band via the so-called stimulated emission, filling a hole in the bound bands. Creating and eliminating an electron–hole pair are analogous to the mass–spring system in classical mechanics. In addition, the conduction electrons can also transit back to the bound bands via spontaneously emitting the photons. Consequently, the interband transition of a bound electron may be described by the Lorentz oscillator model r¨ + γ˜ r˙ + ω˜ 2 r =

F(t) . m∗

(2.5)

Here, m ∗ denotes the effective mass of a bound electron (see Table 2.1), γ˜ accounts for the radiative decay rate of the electrons in the conduction band, and ω˜ corresponds to the frequency of the interband transition. Using the same ansatz as in the previous section, the electric dipole moment is then derived as d(t) = −

1 e2 E0 e−iωt , ∗ 2 m ω − ω˜ 2 + i γω ˜

(2.6)

and one further obtains the relative dielectric constant =1− with the weight factor

 w=

wωp2 ω 2 − ω˜ 2 + i γω ˜ n˜ ne



m∗ me

,

(2.7)

 ,

(2.8)

and the density of the bound electrons n. ˜ As shown in Fig. 2.1b, the real part of  exhibits a dispersion behavior around ω˜ while the imaginary part of  is peaked at ω˜ with a width of about γ, ˜ denoting a strong inelastic radiation scattering at ω. ˜ According to the classical electrodynamics, an oscillating electric dipole emits the electromagnetic radiation, Erad and Brad . We assume the light-induced dipole moment d is located at the origin of the spherical coordinate system and directed along the z-axis. In the far-field limit |r|  λ = 2πc/ω, the electric field Erad is given by [16]  1  1 ¨ − r/c) , r × r × d(t (2.9) Erad (t) = 2 3 4πε0 c r while the associated magnetic field Brad reads Brad (t) =

1 [r × Erad (t)] . cr

(2.10)

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2 Surface Plasmon Resonance

The power radiated per unit solid angle is expressed as dP = S · (rr ), d

(2.11)

 r S = 2c2 ε0 Erad × B∗rad = 2cε0 |Erad |2 r

(2.12)

where the Poynting vector

represents the directional energy flux, that is, the electromagnetic energy transfer per unit area per unit time along r. The total radiated power is computed to be

π

P=



0

 =

2π

sin θdθ

dϕ 0

2

1 e 4πε0 m ∗

2

dP d

16πε0 ω4 |E0 |2 . 3c3 (ω 2 − ω˜ 2 )2 + (γω) ˜ 2

(2.13)

The incident light is assumed to propagate along the x-direction and the corresponding energy flux is Sinc = 2cε0 |E0 |2 . Thus, one obtains the interaction cross-section2 of the dipole moment σ(ω) =

ω4 P = σT 2 , 2 Sinc (ω − ω˜ )2 + (γω) ˜ 2

(2.14)

with the so-called Thomson cross-section 8π 2 r , 3 0

(2.15)

e2 1 . 4πε0 m ∗ c2

(2.16)

σT = and the electron’s classical radius r0 =

For an electron m ∗ = m e , the Thomson cross-section is calculated to be σT = 6.65246 × 10−29 m2 .

2 The

scattering cross-section denotes an area of a plane oriented perpendicular to the propagation direction of the incident wave. This area would intercept the total incident power similar to the total power radiated by the scatterer.

2.2 Plasmonics: Interaction of Light with Metals

67

Table 2.2 Drude–Lorentz parameters for Au and Ag [11] Au Ag j 0 1 2 3 4 5

wj 0.760 0.024 0.010 0.071 0.601 4.384

ω˜ j (eV) 0 0.415 0.830 2.969 4.304 13.320

γ˜ j (eV) 0.053 0.241 0.345 0.870 2.494 2.214

wj 0.845 0.065 0.124 0.011 0.840 5.646

j 0 1 2 3 4 5

ω˜ j (eV) 0 3.886 4.481 8.185 9.083 20.29

γ˜ j (eV) 0.048 0.816 0.452 0.065 0.916 2.419

2.2.3 Drude–Lorentz Model Combining the contributions of both free and bound electrons, the final relative dielectric function may be written as [17] (ω) = (∞) −

w0 ωp2 ω 2 + i γ˜ 0 ω

−

J 

w j ωp2

j=1

ω 2 − ω˜ 2j + i γ˜ j ω

,

(2.17)

where J is the number of the included Lorentz oscillators with the transition frequencies ω˜ j and decay rates γ˜ j . The constant (∞) is the high-frequency offset of the relative permittivity. In most cases, (∞) is equal to the unity but it can be greater than the unity if J is not large enough. The weight w j accounts for the ratio of ω˜ 2j to the square of the plasma frequency ωp . Here, we have also involved the extra effects of the intraband transitions3 on the Drude–Sommerfeld model of free electrons into w0 and γ˜ 0 . Equation (2.17) is known as the Drude–Lorentz model. The relevant parameters for Au and Ag are listed in Table 2.2.

2.3 Surface Plasmon Polaritons—SPPs After discussing the relative permittivity of noble metals, we consider the electromagnetic waves propagating at a metal–dielectric interface. The strong resonant coupling between the light field and the collective oscillation of the conduction electrons gives rise to the so-called surface plasmon polaritons (SPPs). These quasi-particles were first predicted and demonstrated in the pioneering works of Richie, Powell, and Swan in the 1950s [2, 18]. Due to their appealing features of the electric-field enhancement and channeling the optical waves beyond the diffraction limit of light, the SPPs 3 An intraband transition denotes a transition of electrons between electronic states within the same

band. In contrast, an interband transition corresponds to an electronic transition between different bands.

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2 Surface Plasmon Resonance

have recently experienced an explosion of interest in various fields, ranging from the boosted light–matter interaction and the miniaturized photonic circuits to the sensitive chemical and biological detection.

2.3.1 SPPs at Plane Interfaces The SPPs propagating at the metal–dielectric interface hybridize the collective motion of free electrons and the optical waves, which may be well described by the classical electromagnetic theory. The SPP wavelength is shorter than that of the incident (driving) light. The electromagnetic field at the interface strongly decays along the direction perpendicular to the interface. For an incident light within the infrared and visible regime, the penetration depth of the electromagnetic field in the dielectric region is shorter than the light wavelength, leading to the subwavelengthscale confinement. The propagation length of the plasmon waves at the interface reaches only a few to tens of micrometers due to the large ohmic loss of metals.

2.3.1.1

Semi-infinite Metal–Dielectric Interface

We start with considering the simplest semi-infinite structure as shown in Fig. 2.2. The light incidents on the metal–dielectric interface that is in the x − y plane. The region of z > 0 is made of the metal (symbol M) with a complex relative permittivity M . The dielectric medium (symbol D) stays in the region of z < 0 and has a relative permittivity D > 0. The light field at the frequency ω (vacuum wavelength λ = 2πc/ω) propagates along the x-direction and is polarized in the z-axis. Our aim is to find the propagating modes E(r, t) = 21 E(r)e−iωt + c.c. and H(r, t) = 21 H(r)e−iωt + c.c. that can exist at the metal–dielectric interface under

Fig. 2.2 Semi-infinite metal–dielectric interface. The metal in the region of z > 0 has a complex dielectric function M . The relative permittivity of the dielectric medium in the region of z < 0 is D

2.3 Surface Plasmon Polaritons—SPPs

69

certain boundary conditions. Here, c.c. denotes the complex conjugate. Substituting E(r, t) and H(r, t) into Maxwell’s equations (1.1a) and (1.1b), we obtain ∇ × E(r) = iωμ0 H(r),

(2.18a)

∇ × H(r) = −iωε0 E(r).

(2.18b)

The frequency-dependent (ω) denotes the relative-permittivity distribution in the whole space, i.e., (ω) = M for z > 0 while (ω) = D for z < 0. We first consider the p-polarized (transverse magnetic or TM) solutions, where no magnetic field exists in the propagation direction, because the s-polarized (transverse electric or TE) modes are not allowed to propagate at the interface (see below). Equation (2.18) gives ∂ ∂ E x (x, z) − E z (x, z) = iωμ0 Hy (x, z), ∂z ∂x ∂ Hy (x, z) = −iωε0 E z (x, z), ∂x ∂ Hy (x, z) = iωε0 E x (x, z), ∂z

(2.19a) (2.19b) (2.19c)

and Hx = Hz = 0 and E y = 0. The nonzero electric and magnetic components E x,z and Hy are the functions of x and y. We look for the solutions in the following form: δ j,M

[Hy( j) (x, z), E x( j) (x, z), E z( j) (x, z)] = e(−1)

κ j z ik j x

e

[Hy( j) , E x( j) , E z( j) ].

(2.20)

The parameter j = M or D denotes different spatial regions. The positive decay coefficient κ j characterizes the exponential attenuation of the electric and magnetic fields in the z-axis. The delta function δ j,M controls the direction of increasing the distance from the interface. The wavenumber k j is the spatial frequency of the electromagnetic wave varying along the x-direction. Inserting (2.20) into (2.19), it is easy to obtain the following relation: k 2j − κ2j =  j

ω2 . c2

(2.21)

At the metal–dielectric interface, we have the boundary conditions for the electricfield components E x(D) (x, z = 0) = E x(M) (x, z = 0), D E z(D) (x, z = 0) = M E z(M) (x, z = 0),

(2.22a) (2.22b)

Hy(D) (x, z = 0) = Hy(M) (x, z = 0),

(2.22c)

which lead to the conclusion kD = kM ≡ k, i.e., the electromagnetic waves in the metal and dielectric regions have the same wavenumber. This is understandable

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2 Surface Plasmon Resonance

because the resonant light–matter interaction requires the phase matching between the electromagnetic waves and the collective oscillation of the conduction electrons. In addition, one may derive the following solutions: ω k= c



  D M ω 2D ω 2M , κD = − , κM = − . D + M c D + M c D + M

(2.23)

It should be noted that the above solutions are also valid for the complex M . For the sake of simplicity we assume M is real in the following discussion. Spectral Dispersion. Applying the Drude–Sommerfeld model in the nondissipative limit (γ f ∼ 0), M = 1 − ωp2 /ω 2 , the relation between ω and k is governed by

 (ck)2 1 + D 4 2 2 2 (ck) + ωp2 = 0. (2.24) ω − ω ωp + D D Equation (2.24) gives two dispersion k − ω branches, in which the lower one corresponds to the SPP branch. As displayed in Fig. 2.3a, an anti-crossing occurs between two branch curves, indicating the formation of polaritons (i.e., the quanta of the electromagnetic field coupled to the collective excitation of the conduction electrons). The k − ω plane is divided into three regions: In region (1), the value of ω is smaller than ωp , (2.25) ωsp = √ 1 + D for which both M and D + M are negative, resulting in a real wavenumber k. The dispersion curve approaches the light line ck ω=√ , D

(2.26)

when k is decreased to zero and is saturated to ωsp at a large k. Since the entire SPP branch is on the right side of the light line, the SPP wavelength λSPP = 2π/k is always longer than that of the light propagating in the dielectric. The law of conservation of energy prevents the energy transformation from the SPP into the light. Therefore, the SPP is nonradiative. The region (2) is the gap between the lines ω = ωp and ω = ωsp , where k is an imaginary number. The electromagnetic waves suffer from a strong decay along the propagation direction. As a result, none plasmon modes exist in this region. In region (3), ω starts with the plasma frequency ωp and grows strongly with an asymptote  ω = ck

1 + D . D

(2.27)

2.3 Surface Plasmon Polaritons—SPPs

71

Fig. 2.3 Propagation modes at the semi-infinite interface between gold metal and silica dielectric. In the Drude–Sommerfeld model, the plasma frequency of the gold metal is ωp = 8.89 eV. The relative permittivity of the silica dielectric is D = 2.25. a Spectral k − ω dispersion relation. b Skin depths. The inset shows a sketch of the exponential decay of the electric component E z in the metal and dielectric media

Decay. In the z-axis, the electromagnetic field at the interface can only penetrate into the metal/dielectric medium within a finite distance, i.e., skin depth. One may use the 1/e decay lengths (2.28) δM,D = κ−1 M,D to characterize the electromagnetic-field penetration. In the limit of k ∼ 0, we have ω ωp and |M |  D , for which the electromagnetic-field penetration into the metal is much shorter that into the dielectric, δM δD [Fig. 2.3b]. In addition, the skin depth δM is less than the SPP wavelength λSPP , which means the plasmon waves can only propagate at the metal–dielectric interface. The typical values of the penetration lengths are δM ∼ 10 nm in metal and δD ∼ 100 nm in dielectric, overcoming the diffraction limit of the light. As k is increased, both δM and δD decrease and approach each other.

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2 Surface Plasmon Resonance

Typical noble metals that support SPPs are gold and silver, but the metals such as copper, titanium, nickel, and chromium have also been used in experiments. For a practical metal, the imaginary part of the relative permittivity is nonzero, Im(M ) = 0, which results in the decay of the plasmon waves along the propagation direction. Indeed, this decay is caused by the movement of electrons heating the metals, i.e., ohmic loss. The solutions (2.23) are still valid for the complex M . Similarly, we define the 1/e decay length L = 1/Im(k) as the effective propagation length of the plasmon waves. Using γ f = 0.071 eV for gold, L is about 20 μm at ω = 1.6 eV (corresponding to λ = 500 nm). In general, L varies within the range 10–100 μm for the light frequency ω in the infrared and visible ranges. The nonzero Im(M ) also strongly affects the dispersion k − ω relation (see Problem 2.1). So far, we have only discussed the p-polarized solutions. Now we consider the s-polarized waves propagating at the metal–dielectric interface along the x-direction. Maxwell’s equations lead to ∂ ∂ Hx − Hz = −iωε0 E y , ∂z ∂x ∂ E y = iωμ0 Hz , ∂x ∂ E y = −iωμ0 Hx , ∂z

(2.29a) (2.29b) (2.29c)

and E x = E z = Hy = 0. The nonzero electric and magnetic components E y and Hx,z depend on x and z. We look for the solutions of the following form: δ j,M

[E y( j) (x, z), Hx( j) (x, z), Hz( j) (x, z)] = e(−1)

κ j z ik j x

e

[E y( j) , Hx( j) , Hz( j) ].

(2.30)

Substituting (2.30) into (2.29) and using the boundary condition Hx(D) (x, z = 0) = Hx(M) (x, z = 0),

(2.31)

one arrives at κD + κM = 0, which is apparently impossible since both κD and κM are positive. Thus, the s-polarized SPPs cannot exist at the metal–dielectric interface.

2.3.1.2

Thin Film

The limited propagation length L of the plasmon waves at the semi-infinite metal– dielectric interface restricts their applications in channeling the light. However, the situation becomes very different in the case of metal films. As we will see below, the decay length L can be strongly extended for a metal film whose thickness d is much smaller than the light wavelength. We consider a thin metal film placed between two dielectric substrates [see Fig. 2.4a]. For simplicity, two substrates are assumed to have the same dielectric

2.3 Surface Plasmon Polaritons—SPPs

73

Fig. 2.4 SPPs at the surfaces of a thin metal film. a The film has a thickness of d and a dielectric function M . The relative permittivity of two identical dielectric substrates is D . Two groups of SPP (2+) (2−) (2+) (2−) = Hy (even) and Hy = −Hy (odd) exist at the metal–dielectric intermodes with Hy faces. b Dispersion relation for a gold metal thin film with silica substrates. The plasma frequency of the gold metal is ωp = 8.89 eV. The relative permittivity of the silica dielectric is D = 2.25

constant D . The semi-infinite-interface model concludes that the wavenumer k of a plasmon wave does not change across the metal–dielectric interface and only ppolarized SPPs exist. Thus, we write three nonzero electromagnetic components existing at the interface as [Hy (x, z), E x (x, z), E z (x, z)] = eikx [Hy (z), E x (z), E z (z)].

(2.32)

Substituting (2.32) into (2.18), one obtains iω

∂ E x (z) = (c2 k 2 − ω 2 )μ0 Hy (z), ∂z ∂ Hy (z) = iωε0 E x (z). ∂z

(2.33a) (2.33b)

The metal film is between the z = ±d/2 planes. We look for the solution of Hy in the following form: ⎧ (1) −κ1 z ⎪ , z > d/2, ⎨ Hy e Hy (z) = Hy(2+) eκ2 z + Hy(2−) e−κ2 z , |z| < d/2, ⎪ ⎩ (3) κ1 z z < −d/2, Hy e ,

(2.34)

where κ1,2 > 0 are the decay constants of the electromagnetic waves in metal and dielectrics along the z-direction. The continuity condition of the electric and magnetic components E x (z) and Hy (z) at two metal–dielectric interfaces leads to the following matrix equation:

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2 Surface Plasmon Resonance

      ⎤ κ2 d −κ2 d κ1 κ1 κ2 κ2  (2+)    exp exp + − ⎥ ⎢  M M  ⎢ D  2  D  2  ⎥ Hy(2−) = 0 . ⎣ κ1 −κ2 d κ2 d ⎦ H y κ1 κ2 κ2 0 exp exp − + D M 2 D M 2 (2.35) The existence of nontrivial solutions of Hy(2+) and Hy(2−) requires that the determinant of the square matrix of coefficients is equal to zero. This results in two propagation modes: the even mode Hy(2+) = Hy(2−) with ⎡

M κ1 κ2 d = − tanh , D κ2 2

(2.36)

and the odd mode Hy(2+) = −Hy(2−) with M κ1 κ2 d . = − coth D κ2 2

(2.37)

Here tanh(·) and coth(·) are the hyperbolic functions. Similarly, we assume the electric component E x (z) takes the form ⎧ (1) −κ1 z ⎪ , z > d/2, ⎨Ex e E x (z) = E x(2+) eκ2 z + E x(2−) e−κ2 z , |z| < d/2, ⎪ ⎩ (3) κ1 z z < −d/2. Ex e ,

(2.38)

For the even mode Hy(2+) = Hy(2−) , (2.33) gives E x(2+) = −E x(2−) and κ21 = k 2 − D

ω2 ω2 , κ22 = k 2 − M 2 . 2 c c

(2.39)

Combining (2.36) and 2.39), one may obtain the spectral dispersion relation between the wavenumber k and the frequency ω. Following the similar way, we have E x(2+) = E x(2−) for the odd mode Hy(2+) = −Hy(2−) and (2.39) is still valid. The corresponding k − ω dispersion for the odd mode can be derived by combining (2.37) and (2.39). Figure 2.4b displays the spectral dispersion of the even and odd SPP modes by using the Drude–Sommerfeld model in the damplingless limit (γ f ≈ 0). It is seen that both even and odd modes are under the dispersion line of the light in the √ dielectric medium ω = ck/ D . As the film thickness d is increased, the difference between two modes is reduced. One may expect that even and odd modes will become degenerate as d approaches infinity. For a practical metal, the imaginary part of the dielectric function Im(M ) is nonzero. The inevitable ohmic loss limits the propagation of the plasmon modes to a 1/e decay length L = 1/Im(k). Equations (2.36), (2.37), and (2.39) are still valid for the complex M . For instance, a gold film with d = 200 nm is positioned between two silica substrates with D = 2.25. We apply the Drude–Sommerfeld model with

2.3 Surface Plasmon Polaritons—SPPs

75

γ f = 0.071 eV. For the SPP with ω = 0.62 eV, the propagation length of the odd mode is estimated to be L o = 32.6 μm while that of the even mode reaches L e = 1.3 cm. Thus, we call the odd and even modes as the short-range and long-range SPPs, respectively. Typically, the propagation length of the long-range SPPs is of the order of centimeters in the infrared regime [19], for which they can be applied for the signal transmission. In Fig. 2.4a, the dielectrics in the regions of z > d/2 and z < −d/2 are same, corresponding to a symmetric layer structure. In the asymmetric case, where the dielectrics in two regions are different, it is found that the propagation length of the long-range SPPs may be further extended [20].

2.3.2 Excitation of SPPs Now we come to the question of how to excite the SPPs. As illustrated in Fig. 2.3a, for a metal–dielectric interface the k − ω dispersion curve of the SPPs is always under √ that of the light propagating in the dielectric ω = ck/ D . This is understandable because a portion of the light energy has to be consumed to drive the collective oscillation of the surface charges. As a result, the momentum of the incident beam does not match that of the plasmon modes. This forbids the light field in the dielectric to directly excite the SPPs at the metal–dielectric interface. The simplest way to increase the wavenumber of the exciting light relatively to that of the plasmon modes is to let the in-dielectric light beam excite the metal–air-interface SPPs. As shown in Fig. 2.5a, for a metal–air interface the SPP branch is below the line ω = ck, but it has an intersection with the line ω = ck/n. Here, n is the refractive index of a dielectric that does not directly contact the metal. Consequently, the light propagating in the dielectric medium may excite the SPPs at the metal–air interface. In Otto configuration displayed in Fig. 2.5b, a glass prism is positioned close enough to a metal surface. An external p-polarized light beam illuminates the prism wall and is totally reflected internally. The evanescent field at the glass–air interface interacts with the free electrons at the metal–air interface and hence excites the SPPs. The wavenumber matching is fulfilled through tuning the angle of incidence θinc with respect to the normal to the glass–air interface. The resonant excitation of SPPs gives rise to a dramatic drop of the reflected beam’s intensity since the resonant driving of the plasmon modes consumes a large energy of the light. Although the Otto configuration is valid in principle, precisely controlling the tiny air gap between the prism and the metal challenges its feasibility in experiments. A more ubiquitous way to excite the SPPs is the Kretschmann configuration [21], where a thin metal film directly contacts the intermediary prism wall [see Fig. 2.3b]. The evanescent light penetrates through the metal film and the SPPs are excited at the outer side of the metal film. The light resonantly driving the plasmon modes occurs when the SPP wavelength at the metal–air interface coincides with that of the normal component of the incident wavevector,

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2 Surface Plasmon Resonance

Fig. 2.5 Excitation of SPPs. a Dispersion relation of SPPs at the interface between gold metal and air. There exists an intersection between the SPP curve and the dispersion line ω = ck/n of the light traveling in a dielectric medium with a refractive index n. b Otto and Kretschmann configurations of the excitation of SPPs

λSPP =

1 2πc , √ ω P sin θinc

(2.40)

where P is the prism’s dielectric constant and ω is the frequency of the incident light. The Kretschmann configuration can be further extended to the dielectric-metal-prism structure as long as the relative permittivity of the dielectric medium is smaller than P . There are also other clever methods to excite the SPPs. (i) A diffraction grating structure is formed on a bulk metal and the exciting light beam propagates in the dielectric [22]. The discrepancy between the SPP wavenumber 2π/λSPP and the wavenumber component of the incidence light along the same direction of the SPP waves is compensated by multiple reciprocal grating vectors. (ii) The SPP excitation can be achieved straightforwardly on a rough metallic surface, which may be viewed as a superposition of a large number of different gratings. However, such an approach has a low light-to-SPP transformation efficiency [22]. (iii) Using an SNOM (scanning near-field optical microscope) fiber tip does not only excite the circular plasmon waves but also allows to directly probe the local SPP fields [23, 24].

2.3.3 SPR Sensors The sharp angle-dependent surface plasmon resonance, (2.40), has been tremendously employed in the detection of chemical and biological substances. The sensing setup is normally based on the Kretschmann configuration [see Fig. 2.6a], where a thin metal film (M) is in contact with a prism (P) and an analyte layer (A) is

2.3 Surface Plasmon Polaritons—SPPs

77

Fig. 2.6 Schematic diagram of a surface-plasmon-resonance sensor based on the Kretschmann configuration. a The sensor consists of a prism, a metal film, and an analyte with the corresponding relative permittivities P , M , and A , respectively. The incident angle of the probe light is θinc . The reflected beam enters a photodetector. The light-intensity reflectivity |r |2 is sensitive to the change √ in the refractive index A of the analyte. b Dependence of the reflectivity |r |2 on the incident angle θinc . The dielectric function of Au is given by the Drude–Sommerfeld model with the plasma frequency ωp = 8.89 eV and the decay constant γ f = 0.071 eV. The prism dielectric is P = 2.30. Varying the analyte concentration tunes A from 1.77 to 1.80

placed above the film. An incident light beam at the frequency ω is reflected at the metal–prism interface. The metal film has a thickness d and a dielectric function M . Both dielectric constants of prism and analyte P and A are positive. In general, √ the refractive index A depends proportionally on the analyte concentration in the aqueous solution. A photodetector is applied to measure the power of the reflected beam. Again, we only consider the p-polarized light because of the absence of the s-polarized SPPs. According to the Fresnel theorem, one may derive the ratio of the amplitude of the reflected ray to that of the incident ray [25] r=

rP−M + rM−A e2ikM d , 1 + rP−MrM−A e2ikM d

(2.41)

where the reflection coefficient rα−β with (α, β = P, M, A) between α and β media is given by kβ α − kα β (2.42) rα−β = kβ α + kα β with the parameter kα defined as  ω α − P sin2 θinc . kα = c

(2.43)

As plotted in Fig. 2.6b, a sharp dip is presented in the reflection spectrum |r |2 since the surface plasmon resonance consumes a large amount of the light energy. The position of the spectral dip responds sensitively to the change in the refractive index √ A of the analyte layer. Such an angle-dependent-dip shifting may be utilized to

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2 Surface Plasmon Resonance

precisely measure a small perturbation of the analyte concentration. The simplicity and sensitivity of this optical technique make it particularly useful in chemical and biological reaction analysis.

2.3.4 Surface Plasmon Optics The behavior of the plasmon waves in a surface plane is similar to the twodimensional light propagation. It is natural to consider if the SPPs can be manipulated via the elements analogous to the common optical devices, like lenses, mirrors, resonators, and waveguides, in conventional optics. In addition, the well-recognized capability of the SPPs to overcome the diffraction limit of the light is beneficial to building a compact photonic integrated circuit, where the nanoscale plasmon lasers work as the light sources and the optical information is transferred via the plasmonic waveguides [26]. Currently, the modern electronic devices are approaching their fundamental speed and bandwidth limitations. It is believed that using the light signals instead of the electronic signals is a promising solution to boost the speed and bandwidth of the information processing. However, the major problem is how to minimize the sizes of the conventional optical devices beyond the classical diffraction limit. The SPPs provide a potential way to address this issue.

2.3.4.1

Plasmonic Focusing

One essential process in the surface plasmon optics is to concentrate the plasmon waves on a nanoscale spot. Such a function may be simply accomplished through a group of properly arranged surface defects [27]. It is also possible to implement both SPP excitation and focusing in the same surface structure. For instance, the SPP waves are launched in one focal point of an elliptical Bragg reflector and focused on the other focal point [28]. The plasmonic focusing based on a circular/elliptical structure that is milled into a metal film has been profoundly studied in [29]. The constructive interference of the SPPs from an array of nanoscale holes, which are arranged in a circular pattern, in a metal film can focus the plasmon waves onto a spot with a subwavelength width (∼380 nm) [30]. A similar structure but replacing the holes with the nanoparticles has been performed in [31]. Other surface structures have also been proposed and demonstrated, including the circular gratings [32, 33], the holographic arrays [34, 35], the plasmofluidics [36], the symmetry broken nanocorrals [37], and the Luneburg and Eaton lens [38].

2.3.4.2

Plasmonic Waveguides

One of the basic components in the integrated plasmonic circuits is the plasmonic waveguides, which are responsible for transferring the optical information among

2.3 Surface Plasmon Polaritons—SPPs

79

different on-chip devices. The mode confinement of the plasmonic waveguides differs fundamentally from that of the conventional optical waveguides. For an optical dielectric waveguide (fiber), the cross-sectional size of the guided mode is suppressed as the waveguide’s diameter (thickness) d is decreased from a value much larger than the light wavelength λ to a value similar to λ. When d is further reduced below λ, the effects of the light diffraction become predominant and the evanescent field outside the waveguide is strongly enhanced (i.e., the transverse size of the guided mode is extended). The situation for the plasmonic waveguides is totally different, where the size of the SPP modes can be monotonically suppressed as d is decreased to a value smaller than λ, exceeding the diffraction limit. However, as illustrated in Fig. 2.3b the strong confinement of the plasmon waves is accompanied by a shortened propagation (attenuation) length that results from the ohmic loss. Actually, the trade-off between the mode confinement and the limited propagation length is the key factor in designing the waveguide structure, although the relevant mechanism has not been fully understood yet. The long-range SPPs can propagate over a centimeter-scale length but have a loose confinement. In contrast, the metal–insulator–metal structure confines the plasmon modes below the diffraction limit of the light but the SPP propagation can only sustain a few wavelengths [39, 40]. A wide variety of channel plasmon-polariton waveguides have been proposed in theory and demonstrated in experiments: (1) The simplest SPP waveguide structure is the metal stripe [41], whose typical width, thickness, and length are about 1 ∼ 10 mircometers, tens of nanometers and tens of mircometers, respectively. The observed 1/e propagation length reaches a few tens of micrometers. The experiment has confirmed that narrowing the stripe width drastically increases the SPP propagation loss [42]. (2) A stripe-shaped waveguide becomes a nanowire when its width is reduced down noticeably below the light wavelength. The excitation of the SPPs in a directional nanowire may be accomplished by linking the nanowire to a stripe that is terminated in a triangular taper [43, 44] as shown in Fig. 2.7. (3) Another guiding structure has a wedge shape and the plasmon waves propagate along the wedge’s apex [45, 46]. The propagation length obtained in experiments is only a few mircometers. (4) In comparison, the groove-shaped plasmonic waveguide exhibits a relatively low-loss feature with a propagation length ∼100 μm at the telecommunication wavelength of 1550 nm while the mode confinement does not reach the subwavelength scale [47]. (5) A new type of plasmonic waveguide, i.e., gap plasmon waveguides [48, 49], has attracted much attention for the reasons of the high transmission of the plasmon waves through the sharp bends, the high tolerance to

Fig. 2.7 Channel plasmon-polariton guiding with the stripe, wedge, groove, and gap structures

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Fig. 2.8 Plasmonic lasers. a A CdS semiconductor nanowire with a diameter d sits on the top of a silver substrate. A MgF2 layer with a thickness h exists between nanowire and substrate. Reprinted by permission from Springer Nature Customer Service Center GmbH: Springer Nature, Nature, [54], © 2009. b Cavity formed by a rectangular semiconductor pillar encapsulated in a silver layer. Reprinted with permission from [55] ©The Optical Society. c Gold–silica-dye core-shell nanoparticle

the structural imperfections, and the relatively simple fabrication. In this structure, a dielectric slot with a subwavelength width is situated in a thin metal film and the SPPs are strongly localized within this gap. (6) The electromagnetic energy flux can also be guided by a one-dimensional chain of closely separated nanoparticles, where the guiding mechanism is due to the near-field coupling among the nanoantennas [50, 51], but the achieved propagation length reaches only a few hundreds of nanometers. Combining the conventional optical waveguides with the SPPs leads to the socalled hybrid plasmonic waveguide [52], where a higher-index dielectric waveguide is placed above a metal surface by a lower-index gap whose thickness is much smaller than the light wavelength. The optical mode is coupled to the plasmon mode across the nanoscale gap, resulting in a strong confinement of the electromagnetic field within this gap. Such a hybrid (conductor-gap-dielectric) structure has gained a considerable interest since it does not only achieve the strong mode confinement but also extends the propagation length of the plasmon waves. The dielectric waveguide can be a cylinder or has a rectangular-shaped cross-section. However, the fabrication of the cylindrical structure is not straightforward in comparison to the rectangular one. The hybrid plasmonic waveguide has various transformations (such as a crosssection with a metal cap on a silicon-on-insulator rib [53]) and can also be used to implement the nanoscale plasmonic lasers with a mode area well beyond the diffraction limit of the light [54].

2.3 Surface Plasmon Polaritons—SPPs

2.3.4.3

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Plasmonic Lasers

Analogous to the conventional lasers, the SPPs may be amplified via an external gain medium, which overlaps spectrally and spatially with the surface plasmons. The miniaturization of such devices, i.e., the plasmonic lasers, holds the potential to be used as the light sources for the on-chip optical communication. One challenge in the implementation of the surface plasmon amplification in the visible regime is to sufficiently compensate the large propagation loss of the surface plasmon waves through the stimulated emission. However, the extremely small mode volume of the surface plasmons leads to a Purcell factor comparable to or even larger than that of the photonic structures, which facilitates the lasing action. Lasing with a nanowire on a metal film. The lasing action can be simply achieved by using a single dielectric nanowire, whose diameter and length are tens to hundreds of nanometers and tens of micrometers, respectively. The single nanowire plays the role of both gain medium and an optical Fabry–Pérot resonator, where two ends of the dielectric nanowire function as the reflecting mirrors. The lasing oscillation may be sustained through electrically/optically injecting the external energy into the nanowire. The conventional laser behaviors, such as the threshold and spectral linewidth narrowing, have been experimentally identified in the nanowire lasers [56]. Such a simple structure is usually referred to as the photonic laser. The nanowire’s diameter has a critical value, around which the lasing threshold increases dramatically. This is ascribed to the large mismatch between the transverse mode of the resonator and the cross-section of the gain medium. Reducing the nanowire’s diameter enhances the evanescent field outside the nanowire, enlarging the mode volume and weakening the light–matter interaction. Consequently, the photonic lasers cannot overcome the diffraction limit of the light. Placing the dielectric nanowire above a metal film with a subwavelength separation forms a nanometer-scale laser [see Fig. 2.8a]. Such a structure is similar to the hybrid plasmonic waveguide and is called the plasmonic laser. The optical resonator is still formed by two ends of the nanowire. It is known that the plasmon waves propagating at the metal–dielectric interface strongly enhance the light field near the surface. One effect of the SPPs on the laser dynamics is the mode-volume suppression. The electromagnetic energy is distributed in the inside-nanowire and inside-metal-film regions as well as the gap between the nanowire and the metal film. Although the volume of the gap region is tiny, the highly intense light field within this gap may account for the predominant amount of the electromagnetic energy. As a result, the effective mode volume is primarily determined by the gap region, overcoming the diffraction limit. The mode-volume suppression enables the lasing action even when the nanowire’s diameter is well below the critical value for the photonic-laser counterpart. In addition, a large amount of the gain is consumed to compensate the ohmic loss of the SPP propagation. Thus, the threshold of the plasmonic lasers is higher than that of the photonic lasers when the nanowire’s diameter is above the critical value. Besides the nanowire-type cavity, the plasmonic laser based on a pillar cavity has also been demonstrated [55]. As shown in Fig. 2.8b, a rectangular cross-section

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2 Surface Plasmon Resonance

InP/InGaAs/InP pillar is surrounded by an insulating silicon nitride (SiN) layer and then encapsulated in silver. Two ends of the pillar forms a Fabry–Pérot cavity, whose Q factor is estimated over 102 at room temperature. The plasmonic-laser cavity can also be in the coaxial structure, where a metallic rod is enclosed by a ring of metal-coated gain medium [57]. Spasers. In [58], a new type of SPPs, i.e., the so-called spasers (surface plasmon amplification by stimulated emission of radiation), was proposed. In this scheme, the pumped quantum emitters (i.e., molecules) transfer their internal energy to the quantized surface plasmon mode in a radiationless manner. Such a spaser has been claimed to be demonstrated in [59]. In the experiment, a metal core was surrounded by a gain-medium shell [see Fig. 2.8c] and a narrow-linewidth emission, whose central frequency matches the surface plasmon resonance frequency, was observed. Nonetheless, as pointed out in [60], the quantization of the electromagnetic field within a low-Q (∼10) cavity is ambiguous since the cavity is not strictly (or even quasi-) closed. That is, the environment (reservoir) significantly influences the intracavity field. One needs to be careful when applying the conventional electromagneticfield quantization approach. In addition, the concept of the plasmonic cavity for a single metal nanoparticle is problematic. Due to the negative permittivity at an optical frequency, the single metal nanoparticle presents an optical potential barrier, rather than a potential well. Thus, no bound modes of the light exist.

2.4 Localized Surface Plasmon Resonances—LSPRs In the above we have discussed the SPPs that propagate at the planar metal–dielectric interfaces. Indeed, the SPPs are the longitudinal waves, where the electrons’ displacement relative to the uniform background of the positive charges is parallel to the plasmon wave propagation. The plasmon waves suffer the tight subwavelengthscale confinement (overcoming the diffraction limit) in the direction perpendicular to the metal–dielectric interface while they can be guided in the interface with a distance of the order of micrometers to millimeters. This opens a new research field in nanophotonics, i.e., the highly integrated planar plasmonic circuits. In this section, we consider another type of the surface plasmons that, in contrast to the SPPs, are non-propagating excitations and exist on the surfaces of the metal nanoparticles. The particles with their geometric sizes in the range of 1 ∼ 100 nm are called nanoparticles. Since the strongly curved surface exerts an effective restoring force on the conduction electrons [61], the surface plasmons can be excited through the direct light illumination. These nanoparticles may be distributed in free space or in gaseous, liquid, or solid embedding substances. They can also be covered by the dielectric shells.

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83

2.4.1 Spherical Nanoparticles As shown in Fig. 2.9, we first consider a monochromatic light Einc (r, t) = ez

E 0 ik·r−iωt E0 + ez e−ik·r+iωt , e 2 2

(2.44)

interacting with the metal nanospheres that are located in a dielectric substrate. The √ light polarization is parallel to the z-axis while the wavevector k = ex D ω/c is along the x-direction. Here, ω = 2πc/λ is the light frequency with the wavelength in free space λ and D is the relative dielectric permittivity of the substrate. We focus on the nanoparticles with a size much smaller than the in-substrate wavelength of √ the incident light λ/ D . Under this circumstance, the electromagnetic field distributes uniformly across the nanoparticle at any fixed instant of time and can be approximated as an electrostatic field E0 = E 0 ez . This is known as the quasi-static approximation [62]. In addition, we assume the distance between nanospheres is much longer than the sphere radius a, for which the intersphere coupling can be neglected. As a result, the physical system is reduced to the interaction between E0 and a single metal nanosphere. It is convenient to study the system in spherical coordinates, r = (r, θ, ϕ). The origin of coordinates is set at the nanosphere’s center. The electric field E0 is rewritten as (2.45) E0 = E 0 cos θ er − E 0 sin θ eθ , where er and eθ are the unit vectors in the direction of increasing the radial distance r and the polar angle θ. We use E(r) to denote the electric field at the point r. Since the electric field is a conservative field, ∇ × E(r) = 0, one may define an electrostatic potential φ(r) whose gradient gives the electric field, E(r) = −∇φ(r). In the charge-free medium, the potential φ(r) follows the Laplace’s equation: ∇ 2 φ(r) = 0,

Fig. 2.9 LSPR of nanospheres driven by an incident light in the quasi-static approximation

(2.46)

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2 Surface Plasmon Resonance

which takes the form     1 ∂ 1 ∂ 1 ∂2 ∂ 2 ∂ φ + φ + 2 2 φ = 0, r sin θ 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂ϕ2

(2.47)

in spherical coordinates (see Appendix B). Following the method of separation of variables, one obtains the formal solution of φ(r) as φD (r) = φM (r) =

∞  l=0

∞

l=0

Al r + l

Bl

 Pl (cos θ), r > a;

r l+1

Cl r l Pl (cos θ),

r < a,

(2.48a) (2.48b)

by using the Legendre polynomials Pl (cos θ). The subscripts D and M are used to distinguish φ in the outside-nanosphere and inside-nanosphere regions. It is seen that φD (r) and φM (r) are ϕ-independent due to the azimuthal symmetry of the system. In writing (2.48), we have used the fact that φM (r) is non-divergent at the nanosphere’s center. The electric fields in the dielectric and metal regions are then derived as ED (r) = ErD (r)er + E θD (r)eθ = −∇φD (r), EM (r) = ErM (r)er + E θM (r)eθ = −∇φM (r),

(2.49a) (2.49b)

respectively. The arbitrary parameters Al , Bl , and Cl are be determined by imposing the specific boundary conditions. In the limit r → ∞, ED (r) must approach the incident field E0 . We find φD (r, θ) = A0 − E 0 r cos θ +

∞ l=0

Bl r l+1

Pl (cos θ).

(2.50)

Additionally, the boundary conditions at the nanosphere’s surface require that the tangential components, E θD and E θM , and normal components, D ErD and M ErM , are continuous across the metal–dielectric interface. Expressing in terms of the potential functions, we have 



∂ ∂ = , (2.51a) φD (r, θ) φM (r, θ) ∂θ ∂θ r =a r =a



 ∂ ∂ D φD (r, θ) = M φM (r, θ) . (2.51b) ∂r ∂r r =a r =a From the above conditions, one may easily obtain φD (r, θ) = −E 0 r cos θ + φM (r, θ) = −

a3 M − D E 0 2 cos θ, M + 2D r

3D E 0 r cos θ. M + 2D

(2.52a) (2.52b)

2.4 Localized Surface Plasmon Resonances—LSPRs

85

Here, we have set the constant potential components to be zero, A0 = C0 = 0. Finally, the electric fields ED and EM are derived as ED = E0 + EM =

1 3er (er · P) − P , r > a; 4πD ε0 r3

3D E0 , M + 2D

r < a,

(2.53a) (2.53b)

where the total polarization of the nanosphere is given by P = V p with the volume V = 4πa 3 /3 and the dipole moment density p = D αEloc .

(2.54)

α M − D = . 3ε0 M + 2D

(2.55)

The polarizability α is defined as

The local field Eloc is equal to the applied electric field E0 . Equation (2.55) is known as the Lorentz–Lorenz equation, which is also equivalent to the Clausius–Mossotti relation. The factor 1/3 arises from the Lorentz local-field correction and is the result specific to the spherical particle. Replacing EM (r) with the symbol E(r) that represents the real electric field inside the nanosphere, the dipole moment reads  p = D α E +

p 3D ε0

 = D ε0 χe E,

(2.56)

with the susceptibility of the nanoparticle in the dielectric medium χe =

M − 1. D

(2.57)

Further, we obtain the the electric displacement D = D ε0 E + p within the nanosphere (2.58) D = D ε0 (1 + χe )E = M ε0 E. As one can see, the metal nanosphere is polarized by the applied light field. √ Due to a (λ/ D ), only the electric dipole moment is induced while the highorder dipole moments are suppressed. It is also seen that |EM | is maximized when Re(M ) = −2D , i.e., Fröhlich condition, where the incident wave resonantly drives the surface plasmon. In the free environment with D = 1, the Drude–Sommerfeld model leads to the resonant frequency of the nanoparticle plasmon excitation ωp ω=√ , 3

(2.59)

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Fig. 2.10 Distribution of the electric-field intensity |ED (x, y = 0, z)|2 around the gold nanosphere in the x − z plane. The nanosphere owns a radius a = 10 nm and is embedded in the silica medium with D = 2.25. The wavelength of the incident light is λ = 536 nm. The relative permittivity of the gold sphere M is computed based on the Drude–Lorentz model with the relevant parameters listed in Table 2.1

which is independent on the nanosphere’s radius a. The presence of the nanosphere strongly affects the distribution of the outside-sphere electric field within a distance of ∼a to the nanosphere’s surface. As illustrated in Fig. 2.10, the electric field is enhanced in the direction same as the light polarization, which, actually, results from the light wave constructively interfering with the collective oscillation of the free electrons. However, unlike the SPPs, such a collective motion of the free electrons cannot propagate. The enhancement of the electric field near the metal surface is in favor of the coupling of the light field to an extra quantum emitter. It has been utilized in biosensing and chemical science. In the experiment, one of measurable physical quantities is the extinction crosssection σext , which represents the total loss of the energy from the incident wave due to both the absorption σabs and the scattering σsca , σext = σabs + σsca .

(2.60)

The absorption cross-section of the nanosphere σabs is related to the polarizability α via   Vα , (2.61) σabs = kIm ε0 √ with the wavenumber in dielectric k = D (2π/λ). The zero Im[M ] cancels the absorption of the nanospherical particle, σabs = 0. The elastic scattering of the incident wave is dominated by the Rayleigh scattering in the far-field zone since the

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87

wavelength of the incident wave is much long compared to the nanosphere’s size. The corresponding differential scattering cross-section along the esca direction is given by esca · [(er × P∗ ) × (er × er × P)] dσsca k4 . (2.62) = d (4πD ε0 )2 |E0 |2 The differential element of the solid angle is d = sin θdθdϕ. The unit vector esca = er denotes the scattering direction. Integrating (2.62) over the full solid angle, one obtains k 4  V α 2 (2.63) σsca =   . 6π ε0 It is seen that σsca ∝ λ−4 , which is one character of the Rayleigh scattering. The ratio σabs /σsca scales as (λ/a)3 , suggesting that the absorption dominates the light–nanoparticle interaction. Figure 2.11a plots the dependence of the extinction cross-section σext on the wavelength of the incidence light λ. A localized surface plasmon resonance (LSPR) peak, corresponding to the maximum of σext , is located at the position predicted by the Fröhlich condition. As the sphere radius a is increased, the peak’s height raises strongly while the peak’s position does not shift. Indeed, this nanosphere-size independence is not true because the quasi-static approximation fails for a large a. As we will see below, when a approaches λ, the nanosphere’s multipole moments become comparable to or even larger than that of the electric dipole moment, strongly affecting the profile (width and height) and location of the

Fig. 2.11 Extinction cross-section σext of the gold nanoparticles versus the incident wavelength λ. (a) Nanospheres with radii a = 1 nm (dotted), 10 nm (dashed), and 15 nm (solid). (b) Cross-section σext of the nanoellipsoids with the light polarization along ex (transverse) and ez (longitudinal). The nanoellipsoids have a half-length of the major axis c = 10 nm (dashed) and 13 nm (solid). The transverse radius is set at a = b = 5 nm. For all curves, the nanoparticles are embedded in the silica medium with D = 2.25. The relative permittivity of the gold nanoparticles M is computed based on the Drude–Lorentz model with the relevant parameters listed in Table 2.2

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LSPR peak. Finally, we should point out that (2.61) and (2.63) are valid for the nanoparticles with an arbitrary shape in the quasi-static limit.

2.4.2 Energy Stored in Metal Nanoparticles When a metal object is placed into an electrostatic field, the electric field exerts an extra force on the free electrons inside the object. Under this force, the free electrons move in a direction opposite to the external electric field. The electrons pile up on one side of the object, leaving a net positive charge on the other side of the object. Such a separation between negative and positive charges produces an electric field opposite to the external electric field. The free electrons cease their movement until the inner-object electric field is completely canceled out. All the negative and positive charges eventually reside on the object’s surface, shielding its interior from the external electric field. As a result, no electric energy is stored inside the metal object. However, the situation becomes different when the metal object is immersed in an electromagnetic field. As illustrated by (2.53a), the electric field in the interior of the metal nanoparticle is nonzero, indicating that a portion of the electromagnetic energy is deposited inside the nanoparticle. In general, Re(M ) for a metal material is negative within the visible region of the electromagnetic spectrum [see Fig. 2.1a] and Im(M ) is also non-negligible. Thus, (1.43), which is derived for an absorptionless dielectric medium with a positive relative permittivity, fails in calculating the electromagnetic energy in the metal materials. Here, we use another way to derive the electromagnetic energy inside a metal nanoparticle. From Maxwell’s equations, one may obtain

 −

σ

S · dσ = V

∂ 1 (ε0 E · E + μ0 H · H) d V + ∂t 2



 ∂ E · P d V, ∂t V

(2.64)

where the area element dσ on the nanoparticle’s surface is along the outward-pointing (pointing toward the exterior of the nanoparticle) normal direction. The nanoparticle’s volume is V with the corresponding volume element d V . The Poynting vector is S = E × H with the inner-nanoparticle fields E and H and the nanoparticle’s polarization density is P(t). In the quasi-static approximation, the electric field is distributed homogeneously inside the nanoparticle. One may express E(t) and P(t) as E∗ E −iωt e + e eiωt , 2 2 E −iωt E ∗ iωt P(t) = eε0 χe e e , + eε0 χ∗e 2 2

E(t) = e

(2.65a) (2.65b)

with the nanoparticle’s susceptibility χe . It is convenient to rewrite P(t) as P(t) = ε0 Re(χe )E(t) + ε0 Im(χe )E(t − T /4),

(2.66)

2.4 Localized Surface Plasmon Resonances—LSPRs

89

with the period T = 2π/ω. Substituting the above equation into (2.64), we obtain the generalized Poynting theorem



 ∂ E(t) · E(t − T /4) d V, S(t) · dσ − ε0 Im(χe ) ∂t V σ V (2.67) where the electromagnetic energy density W (t) is related to the real part of the susceptibility χe through

∂ W (t)d V = − ∂t

W (t) =



ε0 μ0 [1 + Re(χe )] E(t) · E(t) + H(t) · H(t). 2 2

(2.68)

The two terms on the right side of (2.67), respectively, correspond to the electromagnetic energy injection/leakage across the nanoparticle’s surface and the dissipation (ohmic loss) due to the nonzero Im(χe ). We now focus on a metal nanosphere (relative permittivity M ) placed inside a dielectric medium (relative permittivity D ). The vacuum permittivity ε0 in (2.64)– (2.68) should be replaced with D ε0 . An external light field Einc (t) (frequency ω and amplitude E 0 ) is applied to interact with the nanoparticle. As discussed in the previous section, the complex susceptibility of the nanosphere is given by χe = (M − D E . D )/D and the electric field amplitude inside the nanosphere reads E = M3 +2D 0 √ The amplitude of the magnetic field H is related to E via H = − |M |ε0 /μ0 E. Thus, using (2.68), the time-averaged electromagnetic energy stored in the metal nanosphere is calculated to be W¯ sto =

W (t)d V = V

V ε0 |E|2 |M | + Re(M ) , 2 2

(2.69)

where · · ·  denotes the time average over one period. Similarly, one may find the time-averaged energy dissipation D ε0 Im(χe ) W¯ dis = 



E(t) ·

V

 V ε0 |E|2 ω ∂ E(t − T /4) d V = Im(M ). (2.70) ∂t 2 

Here, we have artificially introduced the characteristic√energy-dissipation rate  of the nanoparticle. Recalling√the refractive index n = [|M | + Re(M )]/2 and the extinction coefficient κ = [|M | − Re(M )]/2, one arrives at V ε0 |E|2 2 V ε0 |E|2 2ωnκ · n , W¯ dis = · . W¯ sto = 2 2 

(2.71)

Both W¯ sto and W¯ dis are positive and W¯ dis is completely induced by the nonzero κ. Equation (2.71) is consistent with the results obtained in [63, 64]. Moreover, one may obtain the time-averaged electric dipole interaction

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2 Surface Plasmon Resonance

−2

 V

 V D ε0 E 02 3(M − D ) (−) · P(+) (t) · Einc (t) d V = − , 2 M + 2D

(2.72)

where P(+) (t) denotes the positive-frequency part of P(t) while E(−) inc (t) corresponds to the negative-frequency part of Einc (t). Indeed, this dipole interaction gives rise to the storage of a portion of the light energy inside the metal nanoparticle as well as the dissipation of the light energy. It can be proved that the imaginary part of the dipole interaction (2.72) is equal to −W¯ dis with  = ω, denoting the energy dissipation. However, the real part of (2.72) is unequal to −W¯ sto . This is because the above dipole interaction is the result obtained by neglecting the details inside the nanosphere (e.g., Lorentz local-field correction). That is to say the nanosphere is viewed as a point-like dipole.

2.4.3 Nanoellipsoids The isotropic nanosphere cannot illustrate the dependence of the LSPR on the nanoparticle’s geometry. A more general shape of the nanoparticle is the ellipsoid whose surface is defined by y2 z2 x2 + + = 1. (2.73) a2 b2 c2 Three principal semi-axes have the lengths of a ≤ b ≤ c and the nanoellipsoid’s volume is V = 4πabc/3. The incident light travels along the x-axis and is polarized in the e j=x,y,z direction. Again, we assume the size of the nanoellipsoid is much smaller √ than the in-medium wavelength of the incident light, c λ/ D . Consequently, the physical system is simplified to an external electric field E0 = E 0 e j interacting with an ellipsoidal metal particle (relative permittivity M ). The LSPR may be studied by analytically solving Laplace’s equation (2.46) in ellipsoidal coordinates [65]. For the light polarization e j , the nanoellipsoid’s polarizability in the dipole approximation is derived as M − D , (2.74) α j = ε0 D + L j (M − D ) where the geometrical factors are defined as

Lj = 0

∞

(a 2 δ

j,x

+

b2 δ

j,y

+

c2 δ

(abc/2) dq  . 2 2 2 j,z + q) (q + a )(q + b )(q + c )

(2.75)

In the special case a = b = c, we have L x = L y = L z = 1/3 and the expression of α returns to (2.55). In the limit c  a = b, the nanoparticle approaches a rod shape with L x = L y ≈ 1/2 and L z ≈ 0. Such a geometric anisotropy must lead to a direction-dependent collective motion of the free electrons on the surface of the metal nanorod [66]. The

2.4 Localized Surface Plasmon Resonances—LSPRs

91

scattering spectrum may be computed by using (2.61) and (2.63). As illustrated in Fig. 2.11b, two distinct plasmonic modes are presented. The high LSPR peak results from the surface electrons oscillating in the long axis (the longitudinal direction). As the length of the semi-major axis c is increased, the peak’s height strongly grows. Also, the center of the peak moves toward the long wavelength side (i.e., the red shift), which is unlike the LSPRs of the isotropic nanospheres. The position of the longitudinal plasmon resonance occurs at Re(M ) = D (L z − 1)/L z . Due to L z ∼ 0, the plasmon resonance wavelength (frequency) is sensitive to the change of the environmental dielectric constant D . This highly sensitive peak shift may be utilized as a fundamental basis of the sensing applications. The other LSPR peak, whose height is much lower than the longitudinal one, corresponds to the transverse electron oscillation in the x − y plane. In comparison, the height and center of the transverse peak hardly depend on the nanorod’s length 2c. This transverse plasmon mode happens when Re(M ) = D (L x − 1)/L x , which is located at the shorter-wavelength side of the longitudinal plasmon resonance. Actually, the transverse LSPR peak is coincident with the plasmon band of the spherical nanoparticles (see Fig. 2.11).

2.4.4 Mie Theory of Light Scattering In the above, we have only considered the LSPRs in the Rayleigh scattering limit, where the size of the scattering particles is much smaller than the radiation wavelength λ in free space. The occurrence of the LSPRs is identified by the resonance Rayleigh scattering peak. However, such a treatment fails when the particle’s size becomes comparable to or larger than λ. A more general light scattering theory of a spherical particle with an arbitrary radius was developed by Gustav Mie in 1908 [67]. Indeed, the Mie solutions are based on the multipole expansions of the incident, scattered and inside-sphere electric and magnetic fields enforced by the appropriate boundary conditions [68, 69]. To date, the Mie theory has been extended to various structures including the infinite cylinder, ellipsoid, spherical shell, two spheres, and a sphere on a plane. Here, we only focus on the simplest case, i.e., the optical scattering by a spherical particle, so as to convey the general idea of the Mie theory. As shown in Fig. 2.12, an incident light Einc at the frequency ω = 2πc/λ travels along the z-axis and is scattered by a sphere with a radius a and dielectric function M . The light polarization is in √ the x-direction and the wavevector is given by k = kez = D (2π/λ)ez , where D is the dielectric constant of the outside-sphere medium. In spherical polar coordinates (0 ≤ r, 0 ≤ θ < π, 0 ≤ ϕ < 2π) (see Appendix B), we decompose the vectors of the electric and magnetic fields, 21 Ee−iωt + 21 E∗ eiωt and 1 He−iωt + 21 H∗ eiωt , into the radial, polar, and azimuthal components, E = Er er + 2 E θ eθ + E ϕ eϕ and H = Hr er + Hθ eθ + Hϕ eϕ . Here, the complex amplitudes E and H represent the electromagnetic field at an arbitrary spot. Then, one may obtain

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Fig. 2.12 Light scattering by a metal sphere. The incident light propagates in the z-direction and is polarized along the x-axis. The metal sphere has a radius a and is immersed in the medium with the dielectric constant D . The dielectric function of the metal particle is M



∂ 1 ∂ (E ϕ sin θ) − Eθ , iωμ0 Hr = r sin θ ∂θ ∂ϕ

 1 ∂ 1 ∂ Er − (r E ϕ ) , iωμ0 Hθ = r sin θ ∂ϕ ∂r 

1 ∂ ∂ (r E θ ) − Er , iωμ0 Hϕ = r ∂r ∂θ

(2.76a) (2.76b) (2.76c)

from (2.18a) and 

∂ 1 ∂ (Hϕ sin θ) − Hθ , r sin θ ∂θ ∂ϕ

 1 ∂ 1 ∂ Hr − (r Hϕ ) , −iωε0 E θ = r sin θ ∂ϕ ∂r 

1 ∂ ∂ (r Hθ ) − Hr , −iωε0 E ϕ = r ∂r ∂θ

− iωε0 Er =

(2.77a) (2.77b) (2.77c)

from (2.18b). The above equations may be solved by separating the electromagnetic fields into the linearly independent TM and TE components.

2.4.4.1

TM-wave Solutions

We first study the TM-wave solutions, i.e., the radial component of the magnetic field vanishes, Hr = 0. The whole space can be divided into two subregions (inside and outside sphere) according to the dielectric parameter . In either subregion,  is distributed homogeneously. Since we will later separately consider the electric and magnetic fields in different subregions,  here may be treated as a constant for now, ∂/∂r = 0. Combining (2.76b), (2.76c), (2.77b), and (2.77c), we have

2.4 Localized Surface Plasmon Resonances—LSPRs

93



 ∂2 1 ∂ Er ω2 , +  2 (r Hθ ) = −iωε0 ∂r 2 c sin θ ∂ϕ  2  ∂ ∂ Er ω2 (r Hϕ ) = iωε0 +  , ∂r 2 c2 ∂θ and (2.76a) leads to

∂ ∂ (E ϕ sin θ) = Eθ . ∂θ ∂ϕ

(2.78a) (2.78b)

(2.79)

Equation (2.79) suggests a potential U, which is related to E ϕ and E θ via 1 ∂U 1 ∂U , Eϕ = . r ∂θ r sin θ ∂ϕ

(2.80)

∂ (r W), ∂r

(2.81)

∂2 1 ∂2 1 (r W), E ϕ = (r W). r ∂r ∂θ r sin θ ∂r ∂ϕ

(2.82)

Eθ = We may further replace U with

U= and arrive at Eθ =

Substituting (2.82) into (2.77b) and (2.77c) yields Hθ = −iωε0

∂ 1 1 ∂ (r W), Hϕ = iωε0 (r W), r sin θ ∂ϕ r ∂θ

(2.83)

in the condition of TM waves. We combine (2.83) with (2.77a) and arrive at

   ∂ ∂ 1 ∂2 1 Er = − sin θ W + W . r sin θ ∂θ ∂θ sin θ ∂ϕ2

(2.84)

Finally, inserting (2.83) and (2.84) into (2.78a), one obtains Helmholtz’s equation   ω2 2 ∇ +  2 (r W) = 0. c

(2.85)

Using (2.85), Er is rewritten as  Er =

∂2 ω2 +  ∂r 2 c2

 (r W).

(2.86)

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2 Surface Plasmon Resonance

Thus, the electric- and magnetic-field components Er , E θ , E ϕ , Hθ , and Hϕ are all linked to the function W(r, θ, ϕ). The problem of solving six equations (2.77)–(2.76) is now reduced to finding the solutions of (2.85). Applying the separation variable method, the general form of W is derived as r W(r, θ, ϕ) =

 √ ω ! √ ω !" cl ψl  r + dl χl  r l=0 m=−l c c ×Pl(m) (cos θ) [am cos(mϕ) + bm sin(mϕ)] , ∞ l

(2.87)

where am , bm , cl , and dl are the arbitrary constants, ψl (ρ) =

πρ πρ Jl+ 21 (ρ), χl (ρ) = − Nl+1/2 (ρ) 2 2

(2.88)

are the so-called Riccati–Bessel functions (see Appendix C) with the Bessel functions Jl+1/2 (ρ) (i.e., Bessel functions of the first kind) and Neumann functions Nl+1/2 (ρ) (i.e., Bessel functions of the second kind), and Pl(m) (cos θ) are the associated Legendre functions. As a result, W is expressed in the form of a superposition of multipole expansions, where, for examples, l = 1 and l = 2 respectively correspond to the dipole and quadrupole oscillations. The above solution is applicable to either subregion.

2.4.4.2

TE-wave Solutions

Next we consider the TE-wave solutions, i.e., the radial component of the electric field vanishes, Er = 0. In a similar way, we find 

 ∂2 1 ∂ Hr ω2 , +  2 (r E θ ) = iωμ0 2 ∂r c sin θ ∂ϕ  2  ∂ ∂ Hr ω2 (r E ϕ ) = −iωμ0 , +  2 2 ∂r c ∂θ

(2.89a) (2.89b)

from (2.77b), (2.77c), (2.76b), and (2.76c). Equation (2.77b) leads to ∂ ∂ (Hϕ sin θ) = Hθ , ∂θ ∂ϕ

(2.90)

based on which we define a potential function V related to Hθ and Hϕ via Hθ = and then replace V with

1 ∂V 1 ∂V , Hϕ = , r ∂θ r sin θ ∂ϕ

(2.91)

2.4 Localized Surface Plasmon Resonances—LSPRs

V=

95

∂ (r M). ∂r

(2.92)

Using the function M, we express the spherical components of the electric field E as ∂ 1 (r M), r sin θ ∂ϕ 1 ∂ E ϕ = −iωμ0 (r M), r ∂θ E θ = iωμ0

(2.93a) (2.93b)

and the components of the magnetic field H as

   ∂ ∂ 1 ∂2 1 Hr = − 2 sin θ (r M) + (r M) , r sin θ ∂θ ∂θ sin θ ∂ϕ2 1 ∂2 (r M), Hθ = r ∂r ∂θ ∂2 1 (r M). Hϕ = r sin θ ∂r ∂ϕ

(2.94a) (2.94b) (2.94c)

Finally, (2.89a) leads to Helmholtz’s equation   ω2 ∇ 2 +  2 (r M) = 0, c

(2.95)

and Hr may be rewritten as  Hr =

∂2 ω2 +  ∂r 2 c2

 (r M).

(2.96)

Again, using the multipole expansions, the general form solution of (2.95) is given by r M(r, θ, ϕ) =

 √ ω ! √ ω !" gl ψl  r + h l χl  r l=0 m=−l c c ×Pl(m) (cos θ) [em cos(mϕ) + f m sin(mϕ)] ,

∞ l

(2.97)

with the arbitrary constants em , f m , gl , and h l .

2.4.4.3

General Forms of Electric and Magnetic Fields

Adding the TM and TE waves, one obtains the complete solutions of the electric field E

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2 Surface Plasmon Resonance

 ∂2 ω2 Er = +  2 (r W), ∂r 2 c 2 ∂ 1 1 ∂ Eθ = (r W) + iωμ0 (r M), r ∂r ∂θ r sin θ ∂ϕ ∂2 1 ∂ 1 (r W) − iωμ0 (r M), Eϕ = r sin θ ∂r ∂ϕ r ∂θ 

(2.98a) (2.98b) (2.98c)

and the magnetic field H  ∂2 ω2 (r M), +  ∂r 2 c2 ∂ 1 ∂2 1 (r W) + (r M), Hθ = −iωε0 r sin θ ∂ϕ r ∂r ∂θ 1 ∂2 1 ∂ (r W) + (r M), Hϕ = iωε0 r ∂θ r sin θ ∂r ∂ϕ 

Hr =

(2.99a) (2.99b) (2.99c)

in terms of the auxiliary functions W and M. The final physically accepted solutions shall be determined by the specific boundary conditions.

2.4.4.4

Mie Scattering

We now look for the Mie solution of the light scattering by a spherical particle. The electric and magnetic fields ED and HD in the dielectric medium (outside the sphere) may be written in a superposition of the incident Einc and Hinc and the scattered Esca and Hsca components, ED = Einc + Esca , HD = Hinc + Hsca .

(2.100a) (2.100b)

The incident electromagnetic field propagates along the z-axis and is polarized in the x-direction, k Einc = E 0 eikz ex , Hinc = E 0 eikz e y . (2.101) ωμ0 Here E 0 is the electric field amplitude. The incident wavenumber k is equal to √ D (ω/c). In spherical polar coordinates, we re-express the electric field Einc = Erinc er + E θinc eθ + E ϕinc eϕ as Erinc = E 0 eikr cos θ sin θ cos ϕ, E θinc E ϕinc

ikr cos θ

= E0 e cos θ cos ϕ, ikr cos θ = −E 0 e sin ϕ,

(2.102a) (2.102b) (2.102c)

2.4 Localized Surface Plasmon Resonances—LSPRs

97

and the magnetic field Hinc = Hrinc er + Hθinc eθ + Hϕinc eϕ as k E 0 eikr cos θ sin θ sin ϕ, ωμ0 k = E 0 eikr cos θ cos θ sin ϕ, ωμ0 k = E 0 eikr cos θ cos ϕ. ωμ0

Hrinc =

(2.103a)

Hθinc

(2.103b)

Hϕinc

(2.103c)

The symbols EM = ErM er + E θM eθ + E ϕM eϕ and HM = HrM er + HθM eθ + HϕM eϕ are used to represent the electric and magnetic fields inside the sphere, respectively. We define the W (M) functions associated with the incident, scattered, and insidesphere electromagnetic fields as Winc (Minc ), Wsca (Msca ), and WM (MM ), respectively. Using the fact that4 eikr cos θ sin θ = −

1 ∞ l−1 i (2l + 1)ψl (kr )Pl(1) (cos θ), l=1 (kr )2

(2.104)

one may calculate the specific expressions of W inc and M inc , 1 ∞ inc c ψl (kr )Pl(1) (cos θ) cos ϕ, l=1 l k2 1 ∞ inc k =− E0 2 g ψl (kr )Pl(1) (cos θ) sin ϕ, l=1 l ωμ0 k

r Winc = −E 0

(2.105a)

r Minc

(2.105b)

from (2.86) and (2.96). The parameters clinc and glinc are the same, clinc = glinc = i l−1

(2l + 1) . l(l + 1)

(2.106)

Additionally, in the far-field region (r → ∞), the scattered field must behave as eikr /r , i.e., the spherical wave, for which the appropriate form of W sca and Msca should be 1 ∞ sca c ξl (kr )Pl(1) (cos θ) cos ϕ, l=1 l k2 1 ∞ sca k =− E0 2 g ξl (kr )Pl(1) (cos θ) sin ϕ, l=1 l ωμ0 k

r Wsca = −E 0

(2.107a)

r Msca

(2.107b)

√ (1) (ρ) with the Hankel functions of the where ξl (ρ) = ψl (ρ) − iχl (ρ) = πρ/2Hl+1/2 (1) (ρ) = Jl+1/2 (ρ) + i Nl+1/2 (ρ). Furthermore, since the Neumann first kind Hl+1/2 4 This

(1)

d equation is obtained by using (C.32) and Pl (cos θ) = − sin θ d cos θ Pl (cos θ).

98

2 Surface Plasmon Resonance

functions possess a singularity at the origin r = 0, the WM and MM functions for the inside-sphere electromagnetic field only have the ψl components 1 ∞ M c ψl (mkr )Pl(1) (cos θ) cos ϕ, (2.108a) l=1 l (mk)2 1 ∞ M mk r MM = − E0 g ψl (mkr )Pl(1) (cos θ) sin ϕ, (2.108b) l=1 l ωμ0 (mk)2 r WM = −E 0

√ where m is defined as m = M /D . Determining the parameters clsca , clM , glsca , and glM relies on the specific boundary conditions, which require that the tangential components of E and H must be continuous across the sphere’s surface, E θD (r = a) = E θM (r = a),

E ϕD (r = a) = E ϕM (r = a),

(2.109a)

HθD (r = a) = HθM (r = a),

HϕD (r = a) = HϕM (r = a).

(2.109b)

The above boundary conditions may be further simplified in terms of the W and M functions as



 ∂ ∂ r (Winc + Wsca ) (r WM ) = , (2.110a) ∂r ∂r r =a r =a (2.110b) D (Winc + Wsca )r =a = M (WM )r =a ,



 ∂ ∂ r (Minc + Msca ) (r MM ) = , (2.110c) ∂r ∂r r =a r =a (2.110d) (Minc + Msca )r =a = (MM )r =a , which lead to the amplitudes for the scattered waves (2l + 1) mψl (ka)ψl (mka) − ψl (ka)ψl (mka) , l(l + 1) mξl (ka)ψl (mka) − ξl (ka)ψl (mka) (2l + 1) mψl (ka)ψl (mka) − ψl (ka)ψl (mka) , = i l+1 l(l + 1) mξl (ka)ψl (mka) − ξl (ka)ψl (mka)

clsca = i l+1

(2.111a)

glsca

(2.111b)

∂ with, for example, ψl (mka) = [ ∂ρ ψl (ρ)]ρ=mka . We are only interested in the scattered waves, where the electric-field components are calculated as

1 ∞ l(l + 1)clsca ξl (kr )Pl(1) (cos θ) cos ϕ, l=1 (kr )2  cos ϕ ∞  sca  cl ξl (kr )u(θ) − iglsca ξl (kr )v(θ) , = E0 l=1 kr  sin ϕ ∞  sca  cl ξl (kr )v(θ) − iglsca ξl (kr )u(θ) , = E0 l=1 kr

Ersca = −E 0

(2.112a)

E θsca

(2.112b)

E ϕsca

(2.112c)

2.4 Localized Surface Plasmon Resonances—LSPRs

99

and the magnetic-field components are given by 1 ∞ k E0 l(l + 1)glsca ξl (kr )Pl(1) (cos θ) sin ϕ, (2.113a) l=1 ωμ0 (kr )2  sin ϕ ∞  sca k = E0 cl ξl (kr )v(θ) + iglsca ξl (kr )u(θ) , (2.113b) l=1 iωμ0 kr  cos ϕ ∞  sca k c ξl (kr )u(θ) + iglsca ξl (kr )v(θ) , (2.113c) =− E0 l=1 l iωμ0 kr

Hrsca = − Hθsca Hϕsca

with the shorthands u(θ) = sin θ

∂ 1 P (1) (cos θ), v(θ) = P (1) (cos θ). ∂(cos θ) l sin θ l

(2.114)

In the far-field limit, r → ∞, we have the asymptotic behaviors ξl (kr ) ∼ (−i)l+1 eikr , ξl (kr ) ∼ (−i)l eikr , and the scattered fields Esca ∼

eikr eikr , Hsca ∼ . r r

(2.115)

(2.116)

We focus on the electric field of the forward-scattered wave (θ = 0) along the same polarization direction of the incident wave (ϕ = 0), (E θsca )θ=ϕ=0 = E 0

eikr ∞ (−i)l l(l + 1)(clsca + glsca ), l=1 2kr

(2.117)

where we have used5 u(0) = −v(0) = l(l + 1)/2. According to the optical theorem [70] (a general law of wave scattering), the extinction cross-section of the spherical particle is given by σext

sca  (E θ )θ=ϕ=0 4π Im . = k E 0 eikr /r

(2.118)

Thus, the total cross-section, including both scattering and absorption components, depends only on the imaginary part of the scattering amplitude evaluated in the forward direction. Consequently, one obtains σext =

" 2π ∞ l+1 sca sca Re (−i) l(l + 1)(c + g ) , l l l=1 k2

(2.119)

result may be obtained from the expression of the Legendre polynomials Pl (x) =  2 l (x − 1)l−k (x + 1)k with x = cos θ. k=0 k

5 This

#l

100

2 Surface Plasmon Resonance

where the scattering component accounts for [62] σsca =

2π ∞ [l(l + 1)]2 sca 2 (|cl | + |glsca |2 ), l=1 (2l + 1) k2

(2.120)

while the rest part σabs = σext − σsca

(2.121)

corresponds to the absorption component. The effect of the sphere size has been involved in the amplitudes clsca and glsca . Figure 2.13 illustrates the dependence of σext on the dimensionless parameter q = √ D (2πa/λ) in the limit of Im (M )  0 with Re (M ) > 0, i.e., the elastic scattering by a dielectric sphere. It is seen that when the sphere size is increased from zero, the cross-section is strongly enhanced. After that, multiple maxima and minima are presented. Different values of M give a similar behavior. As q goes to ∞, i.e., the sphere’s size is much larger than the wavelength, σext approaches a value twice as large as the area of the particle’s cross-section. Since no loss sources are introduced, the scattering cross-section σsca is equal to σext .

Fig. 2.13 Mie scattering cross-section. The extinction cross-section σext of a dielectric (water √ with M = 1.332 ) sphere as a function of the parameter q = D (2πa/λ) with the free-space wavelength λ = 2πc/ω and the dielectric constant D = 1. Inset: Dependence of the extinction σext (solid), scattering σsca (dash), and absorption σabs (dotted) cross-sections of a gold metal sphere on the wavelength of the incident light λ. The sphere has a radius a = 50 nm and is placed in the silica medium with D = 2.25. The dielectric function M follows the Drude–Lorentz model (2.17)

2.4 Localized Surface Plasmon Resonances—LSPRs

101

We are interested in the light scattering by a metal sphere with a nonzero Im (M ). In the small-particle limit a λ, the electric dipole term6 sca cl=1 ≈ i(ka)3

M − D M + 2D

(2.122)

sca primarily contributes to σext while the effects of cl>1 and glsca are negligible [71]. Consequently, the extinction cross-section approximates

σext ≈ 9

ω c



4π 3 a 3



3/2

D Im (M ) , [Re (M ) + 2D ]2 + [Im (M )]2

(2.123)

and the scattering cross-section reads σsca ≈

3 ω !4 2π c



4π 3 a 3

2 2D

[Re (M ) − D ]2 + [Im (M )]2 . [Re (M ) + 2D ]2 + [Im (M )]2

(2.124)

The Fröhlich resonance condition, at which σext reaches its maximum, corresponding to the LSPR extinction peak, is then given by Re (M ) = −2D . Although the LSPRs originate from the imaginary part of M [see (2.123), σext = 0 when Im(M ) = 0], the nonzero Im (M ) also broadens the plasmon resonance. Equation (2.124) suggests that the scattering cross-section σsca scales as (ω/c)4 , i.e., the property of the Rayleigh scattering. Due to (σsca /σext ) ∝ (ka)3 , σsca is much smaller than σext for the small particles. Hence, the absorption accounts for the main source of the cross-section. As the particle’s radius a grows, the role of the higher-order multipole terms in (2.119) and (2.120) becomes important, indicating that the collective oscillation of the free electrons can no longer fully follow the varying of the light wave. This results in the retardation effects of the electromagnetic field across the particle, broadening the LSPRs (compare Fig. 2.11a with the inset of Fig. 2.13). In addition, the position of the LSPR peak moves toward the long-wavelength side when a is increased, i.e., the red shift. Meanwhile, the scattering cross-section σsca rises strongly while the absorption σsca is suppressed. Generally, the term of the Mie scattering refers to the light scattering by the particles with a comparable to λ, though the Mie theory has no upper limit on the particle’s size. Although the Mie theory has shown a great success in the studies of the light scattering by the spheroidal geometries, its applications to the irregularly shaped and randomly oriented particles are severely limited. The main reason lies in the fact that the whole theory is established on the multipole expansion in spherical coordinates. In practice, the resonance peak presented in the Mie scattering can be utilized to estimate the particle size. Forward scattering is commonly applied, rather than backscattering. This is because backscattering is more sensitive to the particle’s shape.

6 This

result may be obtained by using (C.2) and (C.13).

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2 Surface Plasmon Resonance

2.4.5 Numerical Methods for LSPRs So far, we have discussed the LSPRs for the regularly shaped nanoparticles, which may be solved in an analytical way by using the appropriate approximation (i.e., quasi-static analysis). However, this approach does not possess the versatility and also fails for the nanoparticles with the intricate shapes. The study on the irregularly shaped nanoparticles relies on the numerical tools. Here, we introduce two popular methods, namely, the finite-difference time-domain (FDTD) and finite element method (FEM) methods. They have been widely used in the plasmonics and nanophotonics to find the optical properties of the systems with intricate structures.

2.4.5.1

FDTD Method

The FDTD method was originally proposed by K. S. Yee in 1966 [72], where the continuous time-dependent Maxwell’s equations are approximately represented by a set of finite difference equations. The recent advances in the computer technology have significantly prompted the application of this algorithm. For one thing, the FDTD method may simulate an up to 103 × 103 × 103 three-dimensional grid with a fast speed, and for another, its remarkable simplicity and versatility enable one to solve complex engineering problems in a wide spectrum of fields. From Maxwell curl equations (1.1a) and (1.1b), we obtain ∂ Ey ∂ Ez ∂ Hx = − , ∂t ∂z ∂y ∂ Hy ∂ Ez ∂ Ex μ0 = − , ∂t ∂x ∂z ∂ Ey ∂ Ex ∂ Hz = − , μ0 ∂t ∂y ∂x

μ0

∂ Hy ∂ Dx ∂ Hz = − , ∂t ∂y ∂z ∂ Dy ∂ Hx ∂ Hz = − , ∂t ∂z ∂x ∂ Hy ∂ Dz ∂ Hx = − , ∂t ∂x ∂y

where the free current density J f is assumed to be zero. We discretize the threedimensional region into a mesh of cuboidal cells (see Fig. 2.14). The x-, y-, and z-direction lengths of a cell are Δx, Δy, and Δz, respectively. The location of the (i, j, k)-th grid point is given by (iΔx)ex + ( jΔy)e y + (kΔz)ez , where i, j, k ∈ Z. The electric-field components are located in the middle of the edges while the magnetic-field components are situated in the center of the faces. We also discretize the time line with an interval Δt, leading to the time series of the electricn = E x,y,z (nΔt) [or the electric-displacement and magnetic-field components E x,y,z n+1/2 n component Dx,y,z = Dx,y,z (nΔt)] and Hx,y,z = Hx,y,z [(n + 1/2)Δt], respectively. Then, the spatial and temporal derivatives of the electric- and magnetic-field components are approximated by their central differences. For example, the Taylor series of the functions f (x ± Δx/2) around x are written as

2.4 Localized Surface Plasmon Resonances—LSPRs

103

Fig. 2.14 Threedimensional Yee cell with the grad spacings of Δx, Δy, and Δz in three orthogonal directions

 f

x±

Δx 2



    ∂ f (x) Δx 1 ∂ 2 f (x) Δx 2 + ∂x 2 2! ∂x 2 2   3 3 1 ∂ f (x) Δx ± + ··· . 3! ∂x 3 2

= f (x) ±

(2.126)

Subtracting two expansions, one arrives at 1 ∂ f (x) = ∂x Δx

     Δx Δx f x+ − f x− + O Δx 2 , 2 2

(2.127)

achieving a second-order accuracy. We finally obtain the central-difference equations for the magnetic-field components, for example,

 1 1 1 1 μ0 Hxn+1/2 (i, j + , k + ) − Hxn−1/2 (i, j + , k + ) Δt 2 2 2 2

 1 1 1 = E yn (i, j + , k + 1) − E yn (i, j + , k) Δz 2 2

 1 1 1 n n E z (i, j + 1, k + ) − E z (i, j, k + ) , − Δy 2 2

(2.128)

and the electric-displacement components, for example,

 1 1 1 n n−1 Dx (i + , j, k) − Dx (i + , j, k) Δt 2 2

 1 1 1 1 1 n−1/2 n−1/2 Hz = (i + , j + , k) − Hz (i + , j − , k) Δy 2 2 2 2

 1 1 1 1 1 n−1/2 n−1/2 Hy − (i + , j, k + ) − Hy (i + , j, k − ) . (2.129) Δz 2 2 2 2

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2 Surface Plasmon Resonance

Fig. 2.15 FDTD simulation for a gold nanorod by using Lumerical. a Scanning electron microscopy (SEM) image of the gold nanorods. Reprinted by permission from Springer Nature Customer Service Center GmbH: Springer Nature, Journal of Nanoparticle Research, [73], © 2017. b Nanorod consisting of a cylinder with two hemispherical ends. The cylinder length is l = 55 nm and the hemispherical radius is a = 6 nm. c Extinction cross-section σext versus the incident wavelength λ performed by Lumerical FDTD with the material data of ‘Au (Gold)—CRC’. The nanorod is in free space and oriented along the z-axis. The incident light propagates in the x-direction. The light polarization is along cos φ e y + sin φ ez with φ = 7◦ . Two resonant modes, longitudinal (LM) and transverse (TM), are identified. d Electric field distribution in the x − z plane for LM and TM, respectively. E 0 is the amplitude of the incident light

The central-difference equations for other components can be derived in a similar way. The electromagnetic field propagates forwards in the time domain, where the instant values of the electric-field (electric-displacement) components are updated prior to the magnetic-field components. The computational stability requires that the change of the electromagnetic field over one cell must be small enough, for which √ (λ/ D )  Δx, Δy, Δz. Once the cell size has been set, the time increment can be chosen accordingly. To involve the inter-field-component coupling between two √ adjacent cells, the electromagnetic field  propagation distance (cΔt/ D ) within the duration Δt needs to be smaller than Δx 2 + Δy 2 + Δz 2 . Owing to its broad applications, the FDTD algorithm has been incorporated in several commercial simulation softwares, such as Lumerical. As an example, Fig. 2.15 shows an FDTD simulation result of the extinction cross-section σext of a nanorod

2.4 Localized Surface Plasmon Resonances—LSPRs

105

by using Lumerical. It is seen that the transverse and longitudinal LSPR modes have been well presented. A typical simulation scheme involves a light source (generating the incidence light), an object (whose optical properties are under investigation), and a monitor (measuring the light scattered from the object). Generally, the light source injects a short pulse, rather than a continuous wave, which enables the capture of the system’s response over a broad spectral band by a single simulation. Actually, such an efficient method is based on the fact that the frequency-domain monitor automatically normalizes the response at each single frequency by the Fourier transform of the light source. Moreover, due to the limited computer memory, the simulation has to be performed within a bounded space. That is to say the light field must be truncated on a chosen outer boundary, i.e., the so-called absorbing boundary condition. Ideally, none of the outgoing waves should be reflected. However, the finite difference approximation inevitably leads to a small reflection at the outer boundary, or even causes a numerical instability.

2.4.5.2

FEM Method

In comparison, the FEM can handle the more challenging and complicated geometries (e.g., mechanical parts and industrial equipment) and a wider variety of mathematical and engineering problems (e.g., solid mechanics, fluid dynamics, heat transfer, and electromagnetic analyses). The basic principle underlying the FEM is cutting a large system under study into an ensemble of smaller and simpler pieces that are called the finite elements. Each element is modeled by a set of simple algebraic equations, and the entire system is computed by assembling these equations. Such a subdivision is beneficial for accurately catching the details of a complex object as well as the relevant local effects. The FEM solver has also been merged into many commercial or open-source software packages, for instance, COMSOL Multiphysics. Figure 2.16a illustrates an example of the two-dimensional mesh of a square panel with a disk hole located at the panel’s central. The whole domain is divided into a number of triangular elements, which are joined by multiple nodes. The mesh becomes denser in the region close to the boundary, i.e., a higher gradient of a physical quantity (e.g., density and dielectric constant) demands the smaller elements to capture the local details efficiently. Increasing the global number of finite elements may enhance the simulation accuracy, but it also consumes plenty of computer resources. Generally, the element types for the two-dimensional surface meshing include 3-node triangle, 4-node quadrilateral, 6-node triangle, and 8-node quadrilateral as shown in Fig. 2.16b. For the three-dimensional solid meshing, the common element types have 4-node tetrahedron , 8-node brick, 10-node tetrahedron, and 20-node brick. We take the two-dimensional triangular mesh as an example. The value of a variable ϕ(x, y) defined within an element may be approximated by ϕ(x, y) = N1 (x, y)ϕ1 + N2 (x, y)ϕ2 + N3 (x, y)ϕ3 ,

(2.130)

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Fig. 2.16 FEM elements. a Two-dimensional triangular mesh. b Examples of two- and threedimensional element types

where ϕi=1,2,3 = ϕ(xi , yi ) are the values of the variable at the nodes (xi , yi ) and the functions Ni=1,2,3 (x, y) interpolate the solutions among the discrete nodes. The nodal values ϕi are treated as the unknown constants to be determined. The interpolation (also known as shape or blending) functions are equal to unity at the corresponding node while becoming null at other nodes. Normally, the interpolation functions take the low-order polynomial forms. For instance, the linear interpolation functions are expressed as Ni (x, y) = ai + bi x + ci y. Imposing the condition Ni (x j , y j ) = δi, j , one has ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ N1 1 1 1 1 A ⎝ N2 ⎠ = ⎝x ⎠ with A = ⎝x1 x2 x3 ⎠ . (2.131) y N3 y1 y2 y3 The above linear matrix equation may be solved via Cramer’s rule, Ni (x, y) =

detAi , detA

(2.132)

where Ai is the matrix formed by replacing the i-th column of A with (1 x y)T . The superscript symbol T denotes the transpose operation on a matrix. It is more complex to calculate the higher-order (e.g., quadratic and cubic) interpolation functions for the elements with more nodes. The relevant discussion can be found in [74]. Equation (2.130) may be written in a more general form ϕ(r) =

n i=1

Ni (r)ϕi

(2.133)

2.4 Localized Surface Plasmon Resonances—LSPRs

107

with the number n of nodes in one element. We assume the variable ϕ(r) fulfills a partial differential equation Pϕ(r) = f (r). (2.134) where P denotes the differential operator and f (r) is called the source function. Two methods are commonly used to formulate the FEM. (i) Variational approach. One may define an energy/potential function E[ϕ(r)], whose variance returns to (2.134). Generally, E[ϕ(r)] has an integral form. Substituting (2.133) into E[ϕ(r)] and equating the partial derivatives of E[ϕ(r)] with respect to ϕi to zero, one obtains a set of algebraic equations associated with ϕi . Solving these equations produces ϕ(r). It is seen that the prerequisite for the variational approach is the existence of E[ϕ(r)]. However, for a nonlinear or dissipative system, E[ϕ(x, y)] cannot be found straightforwardly or even does not exist. One has to turn to other approaches. (ii) Weighted residual method. Inserting the approximation (2.133) into (2.134) leads to a residual n (2.135) R(r) = [P Ni (r)] ϕi − f (r). i=1

In general, R cannot be zero within the element’s domain . In the so-called weak ( sense, we require the weighted residual Riw =  wi (r)R(r)dr over  to be zero. The test (or weighting) functions wi (r) are usually chosen to be same as the interpolation functions Ni (r), i.e., Galerkin’s method. Solving the set of equations Riw = 0 gives us the approximation solution of ϕ(r).

2.5 Plasmonic Coupling of Nanoparticles As we have pointed out, the LSPRs of single nanoparticles may enhance the outerparticle electromagnetic field within a short distance near the nanoparticle’s surface (see Fig. 2.10). The typical enhancement factor reaches 10 ∼ 103 . However, the experimental observation of the single-molecule sensitivity in surface-enhanced Raman scattering [75–77] shows an enhancement of the electromagnetic intensity up to 106 , larger than that of single nanoparticles by several orders of magnitude. This suggests the strong plasmonic coupling of a group of nanoparticles. The simplest composite nanostructure is composed of a pair of metal nanoparticles spaced closely with a distance d. These two nanoparticles are not necessary to be identical. The electromagnetic field generated by one particle inevitably imposes an extra force on the oscillating surface charges on the other particle. Consequently, the combined electron behavior forms a new oscillation mode. When the interparticle separation d is shorter than the optical wavelength, the resonant frequency of this new plasmon mode differs much from that of either single nanoparticle. A simple way to interpret the LSPRs of nanoparticle dimers is the phenomenological picture of the coupled spring-mass system [78]. In the dipole approximation, the collective oscillation of the surface electrons on one nanoparticle can be

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Fig. 2.17 Nanoparticle dimers. a Coupled spring-mass model. b Plasmon hybridization model

viewed as a mechanical spring-mass oscillator [79]. The interaction between two nanoparticles is modeled by an additional spring that links two masses as shown in Fig. 2.17a. When two nanoparticles are far apart, the effect of this third spring is weak. One may intuitively expect the appearance of two synchronized-oscillation modes: In one mode, the oscillations of two masses are completely in phase. In particular, the coupling spring maintains a fixed length for two identical nanoparticles. That is, no additional force is exerted on either mass by the coupling spring and the synchronized-oscillation frequency is same as that of individual spring-mass system. In the second mode, two mechanical oscillators are antiphase, where the deformation of the coupling spring is always opposite to the other two springs. This is equivalent to shorten the spring length of either spring-mass oscillator, resulting in an oscillation frequency higher than that of either oscillator. The coupling between nanoparticles can also be modeled via an alternative way named the plasmon hybridization model [80] as shown in Fig. 2.17b. In this picture, the plasmonic properties of a composite nanostructure may be intuitively understood through hybridizing the plasmons of elementary subnanostructures. In more detail, the deformation of the surface charges on a nanoparticle is decomposed into a set of spherical harmonics, i.e., elementary plasmons [81]. These plasmonic modes are

2.5 Plasmonic Coupling of Nanoparticles

109

analogous to the atomic/molecular orbitals characterized by the quantum number l. The instantaneous Coulomb interaction between surface charges on two nanoparticles accounts for the interparticle coupling that is controlled by the interparticle separation d. In the quantum mechanical language, this means that the overlap of wave functions (i.e., hybridization of elementary plasmons) gives rise to the energylevel shift (i.e., the frequency shift of LSPR). The red-shifted LSPR (compared with the original LSPR frequency) corresponds to the bonding mode (symmetric coupling) while the antibonding mode (antisymmetric coupling) has a frequency higher than either of the original LSPR frequencies. For a relatively large separation d, the dipole–dipole interaction with l = 1 dominating the interparticle coupling and the LSPR frequency shift scales as 1/d 3 . As the separation d is narrowed, the overlap of higher-order orbitals with l > 1 becomes larger, resulting in the strong effects of multipole interactions. Thus, the LSPR frequency moves stronger and faster. This hybridization picture is principally applicable to the complex nanostructures with an arbitrary shape, like nanoparticle aggregates, and its validity has been verified through the fully quantum mechanical calculation [82]. In the hybridization model, the plasmon response of a composite nanostructure can be simply obtained by hybridizing the plasmons of individual nanoparticles. In the special case with two identical subnanostructures, the interparticle interaction leads to the splitting of the original degenerate plasmon resonances into two new resonant modes as illustrated in Fig. 2.18a. This contradicts the prediction of the coupled-spring–mass picture because of the absence of the quantum effect in the latter. Figure 2.18b displays the electric-field distribution of, as an example, a gold nanoparticle dimer. It is seen that the electric field is significantly enhanced within the junction region between two nanoparticles, i.e., the hot spot. The enhanced electric field decays rapidly away from the nanoparticle’s surface, which provides a huge field gradient and thereby an enhanced optical force [83]. This feature has been utilized to trap and manipulate the nanostructures [84]. We now meet a fundamental question: Can the nanoparticles be viewed as an optical resonator/cavity? One may answer this question from the following aspects. (i) In quantum electronics and photonics, the longitudinal/transverse eigenmodes of an optical resonator are generally determined by the general eigenvalue equation 

ω2 − ∇ 2 u(r) + β 2 − (r) 2 u(r) = 0 c

(2.136)

where β is recognized as the eigenvalue and the dielectric distribution (r) provides an effective three-dimensional potential well. A deep enough well supports multiple discrete bound states u(r) of the light field, achieving the light localization. In contrast, for a nanoparticle like nanosphere or nanorod, the negative value of Re(M ) results in a potential barrier, rather than a well. As a result, no optical eigenmodes exist for a single nanoparticle. For the side-to-side and end-to-end coupled metal nanoparticle dimers, the light may be confined between two potential barriers in one dimension. (ii) Although the electromagnetic energy distributes primarily within the tiny hot spot region of the nanoparticle dimers, the oscillating surface charges on the

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Fig. 2.18 Coupled nanoparticles. Two identical gold nanospheres with a radius 40 nm and the minimum distance between two spherical surfaces d. a Extinction cross-section σext for the coupled system with d = 1 nm (solid) and 10 nm (dashed) and single nanoparticle (dash-dotted). The simulation is performed by using Lumerical FDTD with the material data of ‘Au (Gold)—CRC’. b Distribution of the electric field in the x − z plane for the dimer with d = 1 nm at the higherwavelength plasmonic resonance. The centers of two spheres are aligned in the z-axis and the origin of coordinates is located at the middle spot between them. E 0 is the amplitude of the incident light

nanoparticles radiate the electromagnetic waves into the space. This is the reason why they are called the nanoantennas, making the concept of a closed cavity problematic. (iii) The ohmic loss leads to an LSPR spectral linewidth of the order of tens of nm, indicating a quality factor Q ∼ 10. Such a low Q factor makes the application of the common quantization approach of an optical cavity ambiguous. Especially, the common perturbation method, where the dissipation caused by the system–reservoir interaction is modeled through the second-order perturbation theory, is obviously invalid. Indeed, the LSPRs denote the plasmon modes, more precisely speaking, the collective oscillation modes of the surface charges, rather than the photon oscillation within a closed (quasi-closed) region.

2.5 Plasmonic Coupling of Nanoparticles

111

Fig. 2.19 Dispersion relation for a semi-infinite gold film calculated based on Drude–Sommerfeld (solid) and Drude–Lorentz (dashed) models

Problems 2.1 The dispersion relation between the wavenumber k of an SPP to the frequency of an incident light ω shown in Fig. 2.3a is the result based on the Drude–Sommerfeld model in the non-dissipative limit. In practice, the effect of the energy dissipation is non-negligible, especially in the visible range. Plot the k − ω relation for a semiinfinite gold film (see Fig. 2.19) based on (1) the Drude–Sommerfeld model with ωp = 8.89 eV and γ f = 0.07088 eV, and (2) the Drude–Lorentz model with the data listed in Table 2.2.

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73. B. Törngren, S. Sandén, J.O. Nyman, A. Tiihonen, H. Jiang, J. Ruokolainen, J. Halme, R. Österbacka, J.H. Smått, Minimizing structural deformation of gold nanorods in plasmon-enhanced dye-sensitized solar cells. J. Nanoparticle Res. 19, 365 (2017). https://doi.org/10.1007/s11051017-4062-9 74. K.H. Yang, Basic Finite Element Method as Applied to Injury Biomechanics (Academic, 2017), Chapter II 75. K. Kneipp, Y. Wang, H. Kneipp, I. Itzkan, R.R. Dasari, M.S. Feld, Population pumping of excited vibrational states by spontaneous surface-enhanced Raman scattering. Phys. Rev. Lett. 76, 2444–2447 (1996). https://doi.org/10.1103/PhysRevLett.76.2444 76. S. Nie, S.R. Emory, Probing single molecules and single nanoparticles by surface-enhanced Raman scattering. Science 275, 1102–1106 (1997). https://doi.org/10.1126/science.275.5303. 1102 77. H. Xu, E.J. Bjerneld, M. Käll, L. Börjesson, Spectroscopy of single hemoglobin molecules by surface enhanced Raman scattering. Phys. Rev. Lett. 83, 4357–4360 (1999). https://doi.org/10. 1103/PhysRevLett.83.4357 78. L. Novotny, Strong coupling, energy splitting, and level crossings: A classical perspective. Am. J. Phys. 78, 1199–1202 (2010). https://doi.org/10.1119/1.3471177 79. P. Biagioni, J.S. Huang, B. Hecht, Nanoantennas for visible and infrared radiation. Rep. Prog. Phys. 75, 024402 (2012). https://doi.org/10.1088/0034-4885/75/2/024402 80. E. Prodan, C. Radloff, N.J. Halas, P. Nordlander, A hybridization model for the plasmon response of complex nanostructures. Science 302, 419–422 (2003). https://doi.org/10.1126/ science.1089171 81. P. Nordlander, C. Oubre, E. Prodan, K. Li, M.I. Stockman, Plasmon hybridization in nanoparticle dimers. Nano Lett. 4, 899–903 (2004). https://doi.org/10.1021/nl049681c 82. E. Prodan, P. Nordlander, Structural tunability of the plasmon resonances in metallic nanoshells. Nano Lett. 3, 543–547 (2003). https://doi.org/10.1021/nl034030m 83. H. Xu, M. Käll, Surface-plasmon-enhanced optical forces in silver nanoaggregates. Phys. Rev. Lett. 89, 246802 (2002). https://doi.org/10.1103/PhysRevLett.89.246802 84. O.M. Maragò, P.H. Jones, P.G. Gucciardi, G. Volpe, A.C. Ferrari, Optical trapping and manipulation of nanostructures. Nat. Nanotechnol. 8, 807–819 (2013). https://doi.org/10.1038/nnano. 2013.208

Chapter 3

Whispering Gallery Modes in Optical Microcavities

Abstract The whispering-gallery-mode microcavities have been widely applied in chemical and biological sensing due to their tiny mode volume and high-quality factor. In this chapter, the exact and approximate solutions of the whispering gallery modes of microspheres, microbottles, microdisks, and microtoroids are studied in an analytical manner. In addition, the microcavity’s quality factor and the evanescentwave excitation are also discussed in detail.

3.1 Introduction The whispering gallery modes (WGMs) were first introduced by Lord Rayleigh in the nineteenth century to describe the curvilinear propagation of the sound waves under the dome of St. Paul’s Cathedral. Now, the relevant concept has been generalized to the light waves creeping tangentially at the closed concave surface of an optical cavity. The mechanism of optical WGMs originates from the continuous total internal reflection. In the wave picture, a WGM acquires an extra phase of an integer times 2π after one round-trip, constructively interfering with itself. So far, the best record for the Q factor of the optical WGMs has exceeded 1011 , which is achieved based on the crystalline CaF2 microcavities [1, 2]. The miniaturized optical cavities that support WGMs may be divided into two groups [3]: (i) microsphere, microbubble, and microbottle structures made of melting glassy fibers, polymer materials, and liquids; and (ii) on-chip micro-ring, microdisk, and microtoroid structures microfabricated from the transparent dielectric materials. Owing to the unique features of tiny mode volume (∼ 103 λ3 ) and high Q factor, the WGM microcavities possess wide potential applications in the integrated optics and biosensing [4]. The analytical treatment on WGMs can be launched to the microcavities possessing spherical or cylindrical symmetries. This chapter is started with the fundamental principle and the detailed description of the WGMs in microspheres, involving the exact and approximate solutions, the quality factor, and the excitation approaches. The subsequent sections are dedicated © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2_3

117

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3 Whispering Gallery Modes in Optical Microcavities

to other popular microstructures, e.g., microbottles, microdisks, and microtoroids, which support the WGM propagation. The relevant optical properties are also discussed in detail.

3.2 Microspheres The simplest WGM microstructure is in the spherical shape with a radius R of tens of µm [see Fig. 3.1a]. The isotropic dielectric microsphere owns a relative permittivity i higher than that of the outside-sphere region o . The wave nature of the light tells us that only the light at certain wavelengths can be sustained within the cavity without suffering a large propagation loss. From the geometric perspective, the light beam is trapped inside the microsphere by means of the multiple reflections under the

Fig. 3.1 Schematic diagram of WGMs in a microsphere. a The coupling between a microsphere and a tapered fiber. The WGM exists in the equatorial plane. b and c The microsphere’s capability of confining the light for a long period of time arises from the continuous total internal reflection at the cavity’s surface

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119

resonance condition, i.e., the optical paths of different round-trips are completely overlapped as shown in Fig. 3.1b. However, this is not the whole story. Since the beam is partially reflected when hitting on the cavity’s wall, the confinement has a finite lifetime. Due to i > o , the angle of the refracted ray θout is larger than that of the incident ray θinc . One may expect that as θinc is increased the refraction will disappear eventually, i.e., the occurrence of the total internal reflection [see Fig. 3.1c]. Hence, the combination of the resonance condition and the total internal reflection gives rise to the long-lived bound states of the intracavity light.

3.2.1 Strict Analytical Eigenmodes The high symmetry of the microsphere enables one to find the exact analytical solutions of the WGMs. We write the electric and magnetic fields as 21 E(r)e−iωt + c.c. and 21 B(r)e−iωt + c.c. with a complex frequency ω. From Maxwell’s equations, one obtains ∇ · E(r) = 0, ∇ × E(r) = iωB(r), (3.1) k2 ∇ · B(r) = 0, ∇ × B(r) = −i E(r), ω √ where the wavenumber of the light field in the medium is given by k = ω/c with the relative permittivity of the medium . Equation (3.1) suggests that both amplitudes E(r) and B(r) follow the same equation of motion: ∇ 2 C(r) + k 2 C(r) = 0.

(3.2)

In spherical coordinates (0 ≤ r , 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π), three independent vector solutions of (3.2) may be constructed in the following way: Llm (r) = ∇Ψlm (r), Mlm (r) = Llm (r) × r, Nlm (r) =

1 ∇ × Mlm (r), k

(3.3)

where Ψlm (r) with l ∈ Z and m = −l, −l + 1, ..., l − 1, l are the solutions of the scalar equation (3.4) ∇ 2 Ψ (r) + k 2 Ψ (r) = 0. Using the method of separation of variables, Ψlm (r) takes the form Ψlm (r) = zl (kr )Ylm (θ, ϕ).

(3.5)

The radial function zl (kr ) fulfills the equation   2 ∂ l(l + 1) ∂2 2 zl (kr ) = 0. zl (kr ) + k − zl (kr ) + ∂r 2 r ∂r r2

(3.6)

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3 Whispering Gallery Modes in Optical Microcavities

Indeed, zl (kr ) may be written in terms of the spherical Bessel jl (kr ) and n l (kr ) functions or the Ricatti–Bessel ψl (kr )/kr and χl (kr )/kr functions. The spherical harmonic function of the degree l and the order m  Ylm (θ, ϕ) =

(2l + 1) (l − m)! m P (cos θ)eimϕ 4π (l + m)! l

(3.7)

is related to the associated Legendre polynomial Plm (cos θ). Actually, the spherical harmonics are the solutions of the eigenvalue problem 

∂ 1 ∂2 1 ∂ sin θ + sin θ ∂θ ∂θ sin2 θ ∂ϕ2 m = −l(l + 1)Yl (θ, ϕ).

r 2 ∇ 2 Ylm (θ, ϕ) =

 Ylm (θ, ϕ) (3.8)

From (3.3), it is easy to prove ∇ × Llm (r) = 0, ∇ · Llm (r) = ∇ 2 Ψlm (r) = −k 2 Ψlm (r), 1 ∇ · Mlm (r) = 0, ∇ · Nlm (r) = 0, Mlm (r) = ∇ × Nlm (r). k

(3.9)

For the reader’s convenience, the explicit expressions of Llm (r), Mlm (r) and Nlm (r) are listed below  ⎛  ⎞ ∂ m z (kr ) Y (θ, ϕ) l ⎜ ∂r l ⎟ ⎜ ⎟   z ∂ (kr ) ⎜ ⎟ l (3.10a) Llm (r) = er eθ eϕ ⎜ Ylm (θ, ϕ) ⎟ , ⎜ ⎟ ⎝ zl (kr )r 1∂θ ∂ ⎠ Y m (θ, ϕ) r sin θ ∂ϕ l ⎛ ⎞ 0 1 ∂ m ⎟ ⎜  ⎜ ⎟ Mlm (r) = er eθ eϕ ⎜zl (kr ) sin θ ∂ϕ Yl (θ, ϕ)⎟ , (3.10b) ⎝ ⎠ ∂ m −zl (kr ) Yl (θ, ϕ) ∂θ ⎛ ⎞ l(l + 1) zl (kr )Ylm (θ, ϕ) ⎜ ⎟  ⎜ 1  ∂kr ⎟ ∂   ⎜ m r zl (kr ) Yl (θ, ϕ) ⎟ Nlm (r) = er eθ eϕ ⎜ ⎟ . (3.10c) ∂r ⎜ kr ⎟  ∂θ ⎝1 ∂ ⎠ 1 ∂ m r zl (kr ) Yl (θ, ϕ) kr ∂r sin θ ∂ϕ Introducing the vector spherical harmonics

3.2 Microspheres

121

Xl,m (θ, ϕ) = [∇Ylm (θ, ϕ)] × r, Yl,m (θ, ϕ) = Zl,m (θ, ϕ) =

r ∇Ylm (θ, ϕ), Ylm (θ, ϕ)er ,

(3.11a) (3.11b) (3.11c)

the solutions Llm (r), Mlm (r), and Nlm (r) can be rewritten as 

 ∂ 1 zl (kr ) Zl,m (θ, ϕ) + zl (kr )Yl,m (θ, ϕ), (3.12a) ∂r r (3.12b) Mlm (r) = zl (kr )Xl,m (θ, ϕ),   ∂ l(l + 1) 1 zl (kr )Zl,m (θ, ϕ) + r zl (kr ) Yl,m (θ, ϕ). (3.12c) Nlm (r) = kr kr ∂r Llm (r) =

The radial distance r and the vector r = r er are introduced in (3.11) to ensure that the vector spherical harmonics are independent of the radial coordinate r . In addition, it can be found that Xl,m (θ, ϕ) ⊥ er , Yl,m (θ, ϕ) ⊥ er . Zl,m (θ, ϕ)  er . Finally, the general solution of (3.2) is given by the superposition of Llm (r), Mlm (r), and Nlm (r), Clm (r) =

∞ l l=0

m=−l

 m m  αl Ll (r) + βlm Mlm (r) + γlm Nlm (r) ,

(3.13)

with the arbitrary constants αlm , βlm , and γlm determined by the appropriate boundary conditions. We now consider the specific expressions of E(r) and B(r). In either r > R or r < R region, both electric and magnetic fields are non-divergent, ∇ · E(r) = 0 and ∇ · B(r) = 0. Thus, E(r) and B(r) do not contain the Llm (r) components. In addition, the vector Mlm (r) is perpendicular to r, i.e., r · Mlm (r) = 0. One may find that ETM (r) ∝ Nlm (r) for the TM (r · B(r) = 0) modes and ETE (r) ∝ Mlm (r) for the TE (r · E(r) = 0) modes. Using B(r) = − ωi ∇ × E(r), we obtain the corresponding magnetic fields BTM (r) ∝ − ikω Mlm (r) and BTE (r) ∝ − ikω Nlm (r), respectively. 3.2.1.1

TM Modes

center, The intracavity electric field ETM i (r) must have a finite value at the spherical √ thereby zl (kir ) ∼ ψl (kir )/kir with the intracavity wavenumber ki = i ω/c. Addiiko r for a large tionally, the outside-cavity electric field should follow ETM o (r) ∼ e r . Thus, zl (kor ) takes the form of ξl (kor )/kor with ξl (ρ) = ψl (ρ) − iχl (ρ) and the √ outside-cavity wavenumber ko = o ω/c. The functions ψl (ρ) and χl (ρ) are the Riccati–Bessel functions (see Appendix C). The subscripts ‘i’ and ‘o’ denote the

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3 Whispering Gallery Modes in Optical Microcavities

inside- (r < R) and outside-cavity (r > R) regions, respectively. The TM-polarized electric field is then expressed as   ψl (kir ) ψl (kir ) l(l + 1) Yl,m (θ, ϕ) , (3.14a) = Zl,m (θ, ϕ) + (kir )2 ki r   ξl (kor ) ξl (kor ) TM TM Yl,m (θ, ϕ) , (3.14b) Zl,m (θ, ϕ) + Eo (r) = Ao l(l + 1) (kor )2 ko r

ETM i (r)

ATM i

with the amplitudes ATM and ATM o . One should note that, for example, i ψl (kir )



∂ψl (ρ) = ∂ρ

 ρ=ki r

.

(3.15)

According to the boundary condition   TM er × ETM i (R, θ, ϕ) − Eo (R, θ, ϕ) = 0,

(3.16)

we obtain the following relation: ATM i

ψl (ki R) ξl (ko R) = ATM . o ki ko

(3.17)

In addition, the boundary condition   TM er · i ETM i (R, θ, ϕ) − o Eo (R, θ, ϕ) = 0

(3.18)

leads to the other relation TM ATM i ψl (ki R) = Ao ξl (ko R).

(3.19)

Combining (3.17) and (3.18), the complex frequency ω is determined by the following modal equation: 1 ξ  (ko R) 1 ψl (ki R) =√ l . (3.20) √ i ψl (ki R) o ξl (ko R) Further, one may derive the inside- and outside-cavity magnetic fields as iki ψl (kir ) Xl,m (θ, ϕ), ω ki r TM iko ξl (ko r ) Xl,m (θ, ϕ). BTM o (r) = −Ao ω ko r

TM BTM i (r) = −Ai

(3.21a) (3.21b)

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123

Once ω is computed from (3.20), the distributions of the corresponding electric and magnetic field vectors can be obtained straightforwardly. Generally, the total electromagnetic energy of a WGM is normalized to the one-photon energy ω.

3.2.1.2

TE Modes

Similarly, one may derive the electric field for the TE modes as ψl (kir ) Xl,m (θ, ϕ), ki r TE ξl (ko r ) Xl,m (θ, ϕ) ETE o (r) = Ao ko r

TE ETE i (r) = Ai

(3.22a) (3.22b)

and the corresponding magnetic field   ψl (kir ) ψl (kir ) iki TE Ai l(l + 1) Y Z (θ, ϕ) + (θ, ϕ) , (3.23a) l,m l,m ω (kir )2 ki r   ξl (kor ) ξl (kor ) iko TE l(l + 1) A Y (r) = − Z (θ, ϕ) + (θ, ϕ) . (3.23b) BTE l,m l,m o ω o (kor )2 ko r BTE i (r) = −

Using the boundary condition at the surface of the microsphere   TE er × BTE i (R, θ, ϕ) − Bo (R, θ, ϕ) = 0,

(3.24)

TE we find the relation between the amplitude constants ATE i and Ao  TE  ATE i ψl (ki R) = Ao ξl (ko R).

(3.25)

The other boundary condition   TE er · BTE i (R, θ, ϕ) − Bo (R, θ, ϕ) = 0

(3.26)

gives us the following relation: ATE i

ψl (ki R) ξl (ko R) = ATE . o ki ko

(3.27)

Combining (3.25) and (3.27), the modal equation for the TE-polarized WGMs reads √ ψl (ki R) √ ξl (ko R) = o , i ψl (ki R) ξl (ko R) from which the light frequency ω is determined.

(3.28)

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3.2.1.3

3 Whispering Gallery Modes in Optical Microcavities

Mode Numbers

A microsphere can only sustain the WGMs whose frequency ω fulfills (3.20) or (3.28). More specifically, the polar number l and the polarization p = TM/TE can be used to characterize a WGM. Actually, l approximates the ratio of the optical length of the microsphere’s equator to the mode wavelength (see below). However, using l and p is not enough to identify a certain WGM. The WGMs are also related to the number m that is associated with the azimuthal angle. The positive value m > 0 denotes the optical wave is rotated around ez in the clockwise direction while the negative value m < 0 corresponds the counterclockwise circulation. Indeed, the azimuthal number m refers to the projection of the orbital angular momentum of the light on the ez axis. Since the modal (3.20) and (3.28) are independent of m, the WGMs having the same l and p but different m are degenerate. Up to now, the properties of the WGMs along the radial direction have not been considered. As shown in Fig. 3.2, for a certain l the radial function zl (kr ) of some WGMs presents only one maximum while the function zl (kr ) of the other WGMs exhibits multiple maxima. We use the radial number n to represent the number of intensity maxima in the radial direction. Eventually, a specific WGM is described by a set of integers (n, l, m) along with p = TM/TE. The WGM frequency ω is related to (n, l), i.e., ωn,l . Figure 3.3 displays the intensity distributions of several WGMs, allowing one to better interpret l and m. It is seen that the WGM exhibits 2m intensity maxima as the azimuthal angle ϕ is increased from 0 to 2π. In addition, the WGM has (l − |m| + 1) intensity maxima along the polar direction. The fundamental mode corresponds to n = 1 and m = l, i.e., only one intensity maximum is presented in both radial and polar directions. Figure 3.4 illustrates the wavelength difference Δλn,l = λ˜ l − λn,l between the √ WGM wavelength λn,l = 2πc/ωn,l and the approximation λ˜ l = 2π R i /(l + 1/2) versus the mode numbers (n, l). The ratio (Δλn,l /λ˜ l ) is less than ten percent for n = 1. As n is increased, (Δλn,l /λ˜ l ) becomes larger. Nonetheless, λ˜ l still provides a good approximate value for a small n. For a given radial number n, λn,l always goes down as l grows. The wavelength in the visible regime has a polar number l ranging roughly from 400 to 1000. In addition, the TE wavelength is slightly higher than the TM wavelength. All the above features are obtained from Fig. 3.4. Below, we will derive an approximate analytical expression of the WGM frequency ωn,l (or wavelength λn,l ), which well matches the numerical solutions and, in particular, links to the number n of the intensity maxima in the radial direction.

3.2.2 Analytical WGM Frequencies In practice, the microsphere’s radius R (typically, tens or hundreds of µm) much exceeds the optical wavelength of interest and the typical value of the polar index l is greater than 100. Thus, the radiative leakage of the mode energy outside the cavity

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125

Fig. 3.2 Radial function zl (kr ) (solid) of the TE modes and the corresponding optical potential V (r ) = l(l+1) − 1 (dashed). The number of the intensity maxima corresponds to the mode number k2r 2 n. The microsphere is made of silica (i = 1.44) and stays in vacuum (o = 1). The spherical radius is R = 50 µm

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3 Whispering Gallery Modes in Optical Microcavities

Fig. 3.3 Intensity distributions of the TE-polarized modes of a silica (i = 1.44) microsphere in vacuum (o = 1). The spherical radius is R = 50 µm

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127

Fig. 3.4 Dependence of the WGM wavelength λ = 2πc/ω on the mode numbers (n, l). The wave√ length difference Δλn,l is defined as Δλn,l = λ˜ l − λn,l with λ˜ l = 2π R i /(l + 1/2). The microsphere is same for Fig. 3.3

is negligible. One may make the approximation ξl (kor ) ≈ −iχl (kor ). Defining x = √ ko R and N = ki /ko = i /o , we write two modal (3.20) and (3.28) in a compact form ψ  (N x) χ (x) P l = l , (3.29) ψl (N x) χl (x) with P = 1/N (N ) for the TM (TE) modes. As pointed out in [5], the total internal reflection leads to x ≤ ν ≤ N x, (3.30) with the definition ν ≡ l + 1/2. For a given mode wavelength λ = 2πc/ω in free space, the above equation gives the range of λ √ 2π R √ 2π R ≤ λ ≤ i . o ν ν

(3.31)

The difference |N x − ν| for the tightly confined WGMs scales as ν 1/3 , N x = ν + zν 1/3 ,

(3.32)

with a variable z. In order to derive the analytical expression for the variable z, we use the asymptotic expansion for the Bessel functions in the transition region [6]

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3 Whispering Gallery Modes in Optical Microcavities

   ∞ Pk (z) 21/3  Jν ν + zν 1/3 = 1/3 Ai −21/3 z k=0 ν 2k/3 ν 22/3   1/3  ∞ Q k (z) Ai −2 z + , k=0 ν 2k/3 ν     ∞ Rk (z) 22/3 Jν ν + zν 1/3 = − 2/3 Ai −21/3 z k=0 ν 2k/3 ν 21/3  1/3  ∞ Sk (z) + 4/3 Ai −2 z , k=0 ν 2k/3 ν

(3.33a)

(3.33b)

where Ai(−21/3 z) is the Airy function and Pk (z), Q k (z), Rk (z), and Sk (z) are the polynomials in z of degree growing with k. Below we list the first few polynomials P0 (z) = 1,

1 P1 (z) = − z, 5

P2 (z) = −

9 5 3 z + z2, 100 35

3 2 17 1 z , Q 1 (z) = − z 3 + , 10 70 70 4 3 1 R0 (z) = 1, R1 (z) = − z, S0 (z) = z 3 − . 5 5 5

Q 0 (z) =

In addition, replacing x with (ν sechα), we have Debye’s expansions for the Neumann functions [6] √

2eν(α−tanh α) ∞ Uk (coth α) Nν (νsechα) = − √ (−1)k , k=0 νk πν tanh α  sinh 2α ν(α−tanh α) ∞ Vk (coth α)  Nν (νsechα) = e (−1)k , k=0 πν νk

(3.34a) (3.34b)

with the first few polynomials U0 ( p) = V0 ( p) = 1, U1 ( p) =

1 1 (3 p − 5 p 3 ), V1 ( p) = (−9 p + 7 p 3 ). 24 24

Substituting (3.33) and (3.34) to (3.29), we arrive at 

 21/3 4 21/3 z − 1/3 + Ai (−21/3 z) P ν 5 ν √     1 z 3 21/3 z 2 N2 − 1 1/3 1− Ai(−2 z) + Ai (−21/3 z) 5 ν 2/3 10 ν 2/3 N2 z ≈ −1 + 2 , (3.35) 2/3 N −1ν to the order of ν −2/3 . The above equation implies that the argument (21/3 z) must be close to a root αn of the Airy function Ai(−α), i.e., αn = 21/3 z n , where the number n denotes the n-th

3.2 Microspheres

129

Table 3.1 First 15 roots of Ai(−α) = 0 n αn n 1 2 3 4 5

2.338 4.088 5.521 6.787 7.944

6 7 8 9 10

αn

n

αn

9.023 10.040 11.090 11.936 12.829

11 12 13 14 15

13.692 14.528 15.341 16.133 16.906

root. Indeed, n corresponds to the number of the intensity maxima along the radial direction. Several roots αn have been listed in Table 3.1. We then expand Ai(−21/3 z) and Ai (−21/3 z) around z n 1 Ai(−21/3 z) = Ai (−21/3 z n ) · (−21/3 Δz) + Ai (−21/3 z n ) · (−21/3 Δz)2 2 1 (3) 1/3 1/3 + Ai (−2 z n ) · (−2 Δz)3 + · · · , (3.36a) 6 Ai (−21/3 z) = Ai (−21/3 z n ) + Ai (−21/3 z n ) · (−21/3 Δz) 1 + Ai(3) (−21/3 z n ) · (−21/3 Δz)2 + · · · , (3.36b) 2 with a small displacement Δz. Further, Δz is expanded in power of ν −1/3 , Δz = z − z n = a1 ν −1/3 + a2 ν −2/3 + a3 ν −1 + · · · .

(3.37)

As pointed out in [7], the higher derivative of the Airy function Ai(k) (−α) is related to Ai(−α) and Ai (−α) via Ai(k) (−α) = G k (−α)Ai(−α) + Hk (−α)Ai (−α),

(3.38)

with the polynomials G k (−α) and Hk (−α). At αn we have Ai(k) (−αn ) = Hk (−αn ) Ai (−αn ). The first few G k (−α) and Hk (−α) are listed below G 1 (−α) = 0, G 2 (−α) = −α, G 3 (−α) = 1, G 4 (−α) = α2 , H1 (−α) = 1, H2 (−α) = 0, H3 (−α) = −α, H4 (−α) = 2. Inserting (3.36)–(3.38) into (3.35), one finds the parameters in (3.37) a1 = − √

P N2 − 1

, a2 =

3z n2 P(N 2 − 2P 2 /3) , a3 = − zn , 10 (N 2 − 1)3/2

Finally, (3.32) yields the WGM wavelength

··· .

(3.39)

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3 Whispering Gallery Modes in Optical Microcavities

λn,l

 αn 3 αn2 −1/3 P √ + = 2π R i ν + 1/3 ν 1/3 − √ ν 2 N 2 − 1 10 22/3 −1 P(N 2 − 2P 2 /3) αn −2/3 −1 − ν + O(ν ) . (N 2 − 1)3/2 21/3

(3.40)

The above equation implies that for a large polar number l the resonant wavelength √ roughly occurs at λ˜ l = 2π R i /(l + 1/2), which is mainly determined by the microsphere’s size and the polar number l. The free spectral range of the microsphere (i.e., the frequency difference between two adjacent modes l and l + 1) is approximated as ΔωFSR = √

c . i R

(3.41)

It is seen that ΔωFSR is independent of the relative permittivity of the outside-sphere medium o . Further, the cavity’s finesse is given by F=

ΔωFSR λ Q. = √ ΔωFWHM 2π i R

(3.42)

A quality factor ∼ 109 leads to F > 106 . One may prove that the difference between the WGM wavelength estimated by (3.40) and the numerical value obtained from (3.29) is lower than 1 percent.

3.2.3 Effective Mode Volume As we have mentioned, one unique feature of the microcavities is their tiny mode volume. Actually, the mode volume measures the coupling strength between the cavity mode and the extra emitters (e.g., atoms and molecules). A small (large) mode volume results in a strong (weak) cavity–emitter interaction. In additional, the tiny mode volume can enhance the sensitivity of the microcavity-based biosensors. We assume the complex amplitudes of the electric and magnetic fields of a cavity mode are E(r) and B(r), respectively, and have E(r) ∼ B(r) ∼ 0 as |r| → ∞. That is, the light field is confined within a quasi-closed region. The stored light energy is given by    (r)ε0 1 (3.43) |E(r)|2 + |B(r)|2 dr. W = 2 2μ0 Normally, we choose the maximum energy density 

Imax

 (r)ε0 1 2 2 |E(r)| + = max |B(r)| , 2 2μ0 r∈R3

(3.44)

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131

to assess the effective volume of the quasi-closed region, Veff = W/Imax .

(3.45)

However, this way to choose I , i.e., (3.44), is inappropriate in practice. Equation (3.44) implies that a volume Veff is independent of r. That is to say, the cavity– emitter interaction for an emitter located at r = Rer is same as the interaction for an emitter located at |r| → ∞. This is obviously incorrect. Moreover, as shown in Figs. 3.2 and 3.3, Imax is always located inside the microsphere. However, the extra emitter cannot enter the microsphere. Therefore, the effective volume of a cavity mode should be evaluated based on the local energy density where the emitter is located,  (r)|E(r)|2 dr . (3.46) Veff = (r)|E(r)|2 In the above equation, we have used the fact that the electric and magnetic fields have the same energy density. Equation (3.46) indicates that Veff is minimized at Imax while it approaches ∞ as |E(r)| → 0. As depicted in Fig. 3.5, Veff goes down as the polar number l is increased, i.e., a large l facilitates the light confinement. Within the visible regime, the typical value of Veff evaluated based on Imax is ∼ 103 µm3 . In contrast, the energy density at the microsphere’s surface around the equator is about ten times smaller than Imax , resulting in an extended Veff . The ratio of Veff to the cubic of the mode wavelength

Fig. 3.5 Dependence of the fundamental WGM volume Veff on l. The solid curves correspond to Veff evaluated based on Imax while the dashed curves are the results obtained based on the light field at the microsphere’s surface around the equator E(r = R, θ = π/2, ϕ = 0). The microsphere is same for Fig. 3.4

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3 Whispering Gallery Modes in Optical Microcavities

λ3 is related to the Purcell factor. It is seen that (Veff /λ3 ) evaluated based on Imax ranges 102 –103 . Thus, the light confinement cannot exceed the diffraction limit.

3.2.4 Approximate Solutions The strict solutions discussed in Sect. 3.2.1 do not facilitate to conduct a preliminary analysis in many practical applications. Here, we present an approximate analysis for the microsphere’s WGMs with a polar number l of the order of 102 (i.e., small leakage) and |m| close to l (i.e., the light field is mainly distributed around the sphere’s equator). The derived analytic solutions are useful for estimating the coupling strength and the quality factor for a general configuration of a microsphere interacting with other optical devices (e.g., tapered fiber) [8].

3.2.4.1

TE Modes

We assume the polarization of the WGMs is dominantly along the polar direction eθ and the electric field vector is approximated as ETE (r) ≈ E θTE eθ = Ψ (r, θ, ϕ)eθ ,

(3.47)

with ErTE = E ϕTE = 0. The scalar function Ψ (r, θ, ϕ) follows the Helmholtz equation, (∇ 2 + k 2 )Ψ (r, θ, ϕ) = 0 with the in-medium wavenumber k. Using the method of separation of variables, Ψ (r, θ, ϕ) may be further written as Ψ (r, θ, ϕ) = z(r )(θ) eimϕ , where the radial z(r ) and polar (θ) functions are solved from 

 ∂2 2 ∂ l(l + 1) 2 z(r ) = 0, +k − + ∂r 2 r ∂r r2  2  ∂ m2 cos θ ∂ + l(l + 1) − (θ) = 0 + ∂θ2 sin θ ∂θ sin2 θ

(3.48a) (3.48b)

with the positive integer l ∈ Z and the azimuthal number |m| ∼ l. Polar dependence. We first consider (3.48b) and restrict ourselves to the WGMs that are primarily distributed near the equatorial plane, i.e., θ = π/2 + δ with a small variable δ. Thus, one can make the approximations cos θ ∼ −δ and sin θ ∼ 1 − δ 2 /2 and replace (θ) with (π/2 + δ). Equation (3.48b) is then re-expressed as 

   ∂2 ∂ π 2 2 + l(l + 1) − m + δ = 0. − δ (1 + δ )  ∂δ 2 ∂δ 2

We further make the transformation

(3.49)

3.2 Microspheres

133



π 2

 2 + δ = (δ)eδ /4 ,

(3.50)

where the function (δ) fulfills the equation 

   ∂2 1 1 2 2 2 δ (δ) = 0. + l(l + 1) − m + − m + ∂δ 2 2 4

(3.51)

Scaling the small variable δ by δ=

(m 2

x x ∼√ , 1/4 + 1/4) |m|

(3.52)

Equation (3.51) is simplified as 

   ∂2 x 2 + 2q + 1 − x  √ = 0, ∂x 2 |m|

(3.53)

with the definition q = l − |m| ≥ 0. Deriving the above equation demands the conditions |m|  1 and q  l, |m|. The solutions of (3.53) are the Hermite–Gaussian functions1   x 1 2  √ (3.54) = q √ 1/2 Hq (x)e−x /2 . (2 q! π) |m| Finally, one arrives at the approximate expression of (π/2 + δ) l,m

     π −1/2 2 + δ = 2(l−|m|) (l − |m|)! Hl−|m| ( |m|δ)e−|m|δ /2 . 2 |m|

π

(3.55)

The above function has been normalized. Radial dependence. We now consider (3.48a). Inside the microsphere r < R, the exact radial solutions are the spherical Bessel functions (see Appendix C) z i (r ) = ATE i jl (ki r ).

(3.56)

In the outside-sphere region of r > R, the light field rapidly decays to zero as (r − R) is increased. We approximate the outside-sphere field as −α(r −R) , z o (r ) = ATE o e

(3.57)

with a spatial decay constant α. For a strongly confined WGM, the characteristic length α−1 is of the order of the mode wavelength but is much smaller than the may use ddx Hq (x) = 2q Hq−1 (x) and Hq+1 (x) = 2x Hq (x) − 2q Hq−1 (x) to prove (3.53) is the solution of (3.54). The orthogonality of the Hermite–Gaussian functions is given by ∞ −x 2 d x = √π2q1 q !δ 1 q1 ,q2 . −∞ Hq1 (x)Hq2 (x)e

1 One

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3 Whispering Gallery Modes in Optical Microcavities

sphere’s radius R. Thus, one has αR  1. Substituting (3.57) into (3.48a), we obtain  α≈

l(l + 1) − ko2 . R2

(3.58)

So far, we have derived the expression of the electric field ETE . The corresponding magnetic component can be obtained straightforwardly, HTE = −

i ∇ × ETE = HrTE er + HϕTE eϕ . ωμ0

(3.59)

The amplitudes ATE and ATE o are determined by the boundary conditions. At the i TE TE interface r = R, the condition er · (HTE i − Ho ) = 0 demands the continuity of Hr , resulting in TE (3.60) ATE i jl (ki R) = Ao . TE In addition, the boundary condition er × (HTE i − Ho ) = 0 leads to  TE ATE i ki jl (ki R) = −α Ao .

(3.61)

Consequently, the characteristic equation for the TE modes is given by jl (ki R) = −α/ki , jl (ki R)

(3.62)

from which the WGM frequency ω can be determined. Finally, the electric field ETE reads (3.63) ETE (r) = Nnor z n,l (r )l,m (θ)eimϕ eθ , with the radial function ⎧ ⎨ jl (kir ) r ≤ R z n,l (r ) = jl (ki R) , ⎩ −α(r −R) e r>R

(3.64)

the normalization constant Nnor , and the small θ varying around π/2. Still, n ∈ N denotes the number of the intensity maxima as r goes from 0 to ∞.

3.2.4.2

TM Modes

We then consider the TM modes that take the form HTM (r) ≈ HθTM eθ = Ψ (r, θ, ϕ)eθ ,

(3.65)

3.2 Microspheres

135

with HrTM = HϕTM = 0. Again, solving the Helmholtz equation (∇ 2 + k 2 )Ψ (r, θ, ϕ) = 0 leads to  TM jl (kir ) r≤R imϕ Ai Ψ (r, θ, ϕ) = l,m (θ)e , (3.66) TM −α(r −R) r>R Ao e with the inside- and outside-sphere amplitudes ATM and ATM o . The corresponding i electric field vector is given by ETM =

i ∇ × HTM = ErTM er + E ϕTM eϕ , ωε0

(3.67)

with E θTM = 0. At the interface r = R, the boundary conditions er · (i ETM − i TM TM ) = 0 and e × (E − E ) = 0 yield o ETM r o i o jl (ki R) = ATM ATM i o ,

ATM i

ki  α j (ki R) = −ATM , o i l o

(3.68)

where we have used the assumption (∂ HθTM /∂r )  (HθTM /r ). Hence, one obtains the characteristic equation for the TM modes α ki jl (ki R) =− . i jl (ki R) o

(3.69)

The final expression of HTM (r) is the same for (3.63). The approximate solutions derived above are in good agreement with the exact solutions discussed in Sect. 3.2.1. For example, one may test that the difference between the WGM wavelengths estimated by (3.20) and (3.28) and the wavelengths derived from (3.62) and (3.69) is of the order of or even less than 1 nm.

3.2.5 Intrinsic Quality Factor Due to the presence of optical losses, the microcavity’s eigenmodes are leaky and usually referred to as the quasi-modes. Generally, the quality factor Q of a WGM consists of four components, 1 1 1 1 1 = + + + , Q Q rad Q mat Q ss Q sa

(3.70)

where Q rad , Q mat , Q ss , and Q sa quantify the optical-loss mechanisms (channels) of radiation, material attenuation, surface scattering, and surface absorption, respectively. The relative dominance of different components depends on the material properties, fabrication quality, cavity size, and surrounding circumstance.

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3 Whispering Gallery Modes in Optical Microcavities

Radiation loss. When traveling inside the microsphere, the resonant optical wave undergoes the finite times of the total internal reflection during a round-trip. For a small number of reflection times, the light approaches the microsphere’s surface at a steep angle, reducing the efficiency of the total internal reflection [see Figs. 3.1b and 3.1c]. As a result, a portion of light is lost at each reflection, i.e., the radiation loss (also called the curvature loss). For the fundamental WGMs with l = m of a microsphere in air, the radiation-loss-limited quality factor may be approximated as [9] Q rad

(2l + 1) p  = 4



  −1 exp (2l + 1)(βn,l − tanh βn,l ) , 

(3.71)

where p = 0 (1) for the TE (TM) modes,  is the relative permittivity of the microsphere, and we have defined the shorthand βn,l

⎧    −1 ⎫  1/3 ⎨√ ⎬ 2l + 1  2 1 = cosh−1  1+ − p , αn ⎩ ⎭ 2l + 1 4  −1

with the n-th root αn of the Airy function Ai(−α). The factor Q rad grows exponen14 tially with the polar number l. It appears √ that when l > 100, Q rad exceeds 10 for the silica microspheres. Due to l ∼ (2π R)/λ, increasing the spherical radius R suppresses the radiation loss for a certain wavelength λ. Thus, the influence of the radiation loss on the quality factor may be safely neglected for a large enough cavity size. However, a large R also increases the WGM volume, degrading the sensitivity in the sensing applications. Material loss. The vacuum wavelength λ = ω/c of a WGM should be located within the transparency window of the cavity material. Otherwise, the light field suffers from a strong attenuation when traveling inside the microcavity. Most microspheres are made of glass materials. The intrinsic loss is characterized by the attenuation coefficient Pout 1 , (3.72) αmat (λ) = − × 10 × log10 L Pin with the input Pin and output Pout powers and the propagation length L. The unit of αmat (λ) defined by (3.72) is dB/m (or dB/km). Generally, αmat (λ) of the silica is comprised of the Rayleigh scattering (≈ α0 (λ0 /λ)4 with α0 = 1.7 dB/km and λ0 = 850 nm) and the material absorption (mainly in the infrared regime). Theoretically, αmat (λ) is minimized at the fiber-optic-telecommunication wavelength of 1.55 µm, i.e., αmat (λ = 1.55 µm) ≈ 0.2 dB/km. The fundamental material attenuation limits the ultimate quality factor of the WGMs, Q mat =

2π n , λ αmat

(3.73)

3.2 Microspheres

137

with the refractive index n of the cavity at λ. One should note that αmat in (3.73) has × αmat (dB/m) = 0.23 × αmat (dB/m). For the a unit of m−1 and αmat (m−1 ) = ln10 10 pure silica, n may be estimated by the well-known Sellmeier formula  n(λ) =

1+

i

A i λ2 , λ2 − λi2

(3.74)

with the parameters i Ai λi

1 0.696750 0.069066

2 0.408218 0.115662

3 0.890815 9.900559

and the wavelength λ in units of µm. For the mode wavelength at λ = 1.55 µm, Q mat exceeds 1.2 × 1011 and the corresponding storage time of the intracavity photons reaches about 0.1 ms. In contrast, the attenuation coefficient at 633 nm, i.e., the classical wavelength of the Helium–Neon laser, is 7 dB/km, resulting in Q mat = 0.9 × 1010 as shown in Fig. 3.6a. Besides the silica, the microspheres can also be made of the polymers [11] and crystalline materials [12], but their Q mat is reduced because of the relative high αmat . Surface scattering. The imperfect fabrication techniques inevitably lead to a surface roughness of the microsphere, for which the light traveling at the microsphere’s surface experiences a discontinuity in refractive index. Figure 3.7 illustrates an example of the fused-silica-surface image recorded by the atomic force microscopy. We use h(x, y) to denote the height distribution of the microsphere’s surface. The standard deviation of the surface roughness is given by

Fig. 3.6 Quality factor of a microsphere with 2R = 750 µm. a Different components of the intrinsic quality factor of a silica microsphere in air. b Effect of the atmospheric-water adsorption on the decay time τ of a microsphere’s WGM. Reprinted with permission from [10] ©The Optical Society

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3 Whispering Gallery Modes in Optical Microcavities

Fig. 3.7 Surface roughness of a silica microsphere. Surface data obtained by the atomic force microscopy on a grid of 256 × 256 points. Reprinted with permission from [13] ©The Optical Society

 σ=

1 L2

  h 2 (x, y)d xd y,

(3.75)

 with the image area L 2 and the zero mean value L −2 h(x, y)d xd y = 0. One may also evaluate the spatial correlation length B via fitting the correlation function   R(u) =

h(x, y)[h(x + u, y) + h(x, y + u)]d xd y,

(3.76)

to R(u) ∼ e−(u/B) . Two points with a distance larger than B may be considered as independent scatterers. Thus, the size of one scatterer is given by V = (4π/3)σ B 2 . In the limit of the weak surface roughness, V  λ3 , the optical loss caused by the residual surface inhomogeneity may be modeled by the Rayleigh scattering [17]. We assume the microsphere is situated in air. Using the local-field correction factor to the disk-shaped scatterers (2B  σ), we express the corresponding scattering coefficient as     Nd V 2 2π 4  − 1 2 , (3.77) α= 6π λ  2

with the relative permittivity √  of the microsphere. The density √ Nd of the scatterers is estimated to be Nd = 2R/λ(2R B 2 )−1 , where the factor 2R/λ is the ratio of the external to the internal mode volume of the microsphere [13]. Similar to (3.73), the surface-roughness-limited quality factor is given by √  2 3 √  λ 2Rλ 3  . Q ss = 4 4π −1 σ2 B 2

(3.78)

Using σ = 1.7 nm, B = 5 nm, and λ = 800 nm, we estimate Q ss = 7.2 × 109 for the glass microsphere with a radius R = 400 µm. Surface absorption. Figure 3.6b illustrates a typical time dependence of the damping time τ = Q/ω of a microsphere in air. It is seen that τ rapidly decays at the beginning and then approaches the saturation. Such a time-dependent degradation of

3.2 Microspheres

139

Table 3.2 Selected experimental Q factors of microspheres Material Environment λ (nm) 2R (µm) Fused silica

Air

670, 780, 850

750, 800, 680

Fused silica Silica Liquid Silicon Dye doped polymer

Air Water Air Air Air

633 780 1560 1427 Visible

750 79 500∼1500 1000 10

Q

Ref

(8, 7.5, 7.2) × 109 8 × 109 5 × 106 5 × 105 8 × 104 103 ∼ 104

[13] [10] [14] [15] [16] [11]

the quality factor Q is presumably attributed to the extra optical loss caused by the adsorbed water upon the microsphere’s surface. After the fabrication, the hemosorption of the atmospheric water can form a layer of OH groups, which is chemically bound to the microsphere’s surface. The relevant quality factor component is estimated by [13]  πR 1 Q sa = , (3.79) 16n 3 λ δβ with the microsphere’s radius R and the thickness δ and absorption coefficient β (at the wavelength λ) of the water layer. The data of β at different wavelengths is listed in [18]. As suggested in Fig. 3.6a, Q sa plays a fundamental role in the near-infrared regime. Several selected Q factors of the microspheres achieved in experiment are listed in Table 3.2. It is seen that the Q factor mainly depends on the cavity material. The record of the highest Q factor approximates 1010 . Additionally, the circumstance also strongly affects Q. Figure 3.8 presents an example of a glass microsphere fabricated at the University of Exeter and the measured transmission spectrum. This microsphere has been used for the single-molecule sensing in the local lab.

3.2.6 Excitation of WGMs For a high-Q microsphere with a negligible radiation loss, accessing the WGMs by means of the free-space laser beams is generally inefficient. The method of the near-field evanescent-wave coupling is normally utilized to enhance the excitation of the WGMs. Three typical schemes are employed in the experiment [see Fig. 3.9a]: (1) A high-index glass prism is located below the microsphere with a separation of (0.1∼1)λ. The light beam is focused  onto the prism’s inner surface at the optimum angle of incidence θinc = arcsin s / p , where s and  p are the relative permittivities of microsphere and prism, respectively, and  p < s . The resonant coupling

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3 Whispering Gallery Modes in Optical Microcavities

Fig. 3.8 Microsphere fabricated in the local lab at University of Exeter. a The glass microsphere is approximately 2R = 81.55 µm in diameter. b Transmission spectrum around 785 nm. The free spectral range approximates ΔωFSR ≈ 2π × 10.0 GHz. c Lorentzian curve fitting (solid curve) to the spectral dip at λ = 785.020806 nm. The Q factor is estimated to be Q = λ/Δλ ≈ 4.6 × 106

efficiency may reach up to 80% [19]; (2) An equivalent coupler can be executed by using an angle-polished optical fiber [20]. The light beam is guided to the fiber tip, which is polished with a steep angle, and undergoes the total internal reflection; and (3) Coupling the WMGs to an optical fiber taper (diameter of 1 ∼ 4 µm) has the competitive advantages of the easy excitation/measurement and the facilitated inter-device transfer of the light information. We consider the microsphere–fiber coupling as shown in Fig. 3.9b. A step-index fiber along the ex -axis is placed below a microsphere cavity. The cross-section of the fiber is in a circular shape with a radius of R f while the microsphere’s radius

3.2 Microspheres

141

Fig. 3.9 Excitation of WGMs. a Different couplers for exciting the WGMs in a microsphere. b Coupling between a microsphere and an optical fiber. The fundamental mode HE11 of the fiber evanescently interacts with the equatorial WGMs of the microsphere

is Rs . For the sake of simplicity, let us restrict ourselves to the fiber’s fundamental HE11 mode with a linear polarization along ez (see Appendix E) interacting with a WGM of the microsphere. The distance between the central axis of the fiber and the microsphere’s center is equal to Rs + R f + D with a surface-to-surface separation D. The relative permittivities of the fiber and microsphere are  f and s , respectively, and the ambient environment has a relative permittivity of o . The fundamental mode of the fiber E f (r) fulfills the Helmholtz equation ∇ 2 E f (r) + ˜ f (r)

ω 2f c2

E f (r) = 0,

(3.80)

where ˜ f (r) denotes the corresponding relative-permittivity distribution without the microsphere (i.e., replacing s in Fig. 3.9b with o ) and ω f is the mode frequency. Similarly, the WGM of the microsphere Es (r) satisfies ∇ 2 Es (r) + ˜s (r)

ωs2 Es (r) = 0, c2

(3.81)

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3 Whispering Gallery Modes in Optical Microcavities

with the corresponding relative-permittivity distribution ˜s (r) without the fiber (i.e., replacing  f in Fig. 3.9b with o ) and the WGM frequency ωs . We assume that E f (r) and Es (r) follow the normalization conditions    1 2 1 , (3.82) ˜ f (r)|E f (r)|2 dydz = ˜s (r)|Es (r)|2 dr = 1, 2 L eff 2 with an effective length L eff of the fiber mode along the propagation direction. An arbitrary light field takes the superposition form E(r, t) = a f (x)E f (r)e−iω f t + as (t)Es (r)e−iωs t ,

(3.83)

with the x-dependent a f (x) and t-dependent as (t) amplitudes. The propagation of E(r, t) is governed by the wave equation ∇ 2 E(r) −

˜(r) ∂ 2 E(r) = 0, c2 ∂t 2

(3.84)

where the relative-permittivity distribution ˜(r) involves both microsphere and fiber. Using the slowly varying envelope approximation [see (1.151)] # ∂ #∂ # # # ∂2 # # # ∂2 # # # # # # # # # 2 a f (x)#  2β # a f (x)#, # 2 as (t)#  2ωs # as (t)#, ∂x ∂x ∂t ∂t

(3.85)

one may derive the relation between a f (x) and as (t)     ωs2 ˜(r) ∂ 2iωs 2 as (t) + as (t) (r) − ˜s (r) 2 Es (r)e−iωs t = c ∂t c   2 ω   ∂ f − 2iβ a f (x) + a f (x) ˜(r) − ˜ f (r) 2 E f (r)e−iω f t , ∂x c

(3.86)

with the propagation constant β of the fiber mode. Multiplying both sides of the above equation by E∗s (r) and integrating over the whole space, one obtains ω 2f −i(ω −ω )t ∂ f s as (t) ≈ iΔωs as (t) + i e ∂t 4ωs     × a f (x) ˜(r) − ˜ f (r) E∗s (r) · E f (r) dr,

(3.87)

where the coupling-induced frequency shift Δωs is recognized as Δωs =

ωs 4



  ˜(r) − ˜s (r) |Es (r)|2 dr.

(3.88)

3.2 Microspheres

143

Due to ˜(r) − ˜s (r) =  f − o with r inside the fiber, we re-express Δωs as ωs Δωs = 4

 ( f − o )|Es (r)|2 dr,

(3.89)

Vf

with the integration region inside the fiber denoted by V f . Similarly, multiplying both sides of (3.86) by E∗f (r) and integrating over y − z plane and (−∞, x] in the x-axis, we have ωs2 L eff i(ω f −ωs )t a f (x) = a f (−∞) + ias (t) e 8β(c2 / f )  x  ∞ ∞   × [˜(r) − ˜s (r)] E∗f (r) · Es (r) d xd ydz. −∞

−∞

(3.90)

−∞

Substituting (3.90) into (3.87) yields   ∂ 1 1 − + iΔωs as (t) + ia f (−∞)e−i(ω f −ωs )t as (t) ≈ − ∂t 2τ0 2τc 2  ωf   × (s − o ) E∗s (r) · E f (r) dr, 4ωs Vs

(3.91)

where the integration is carried out inside the microsphere (denoted by Vs ). In (3.91), the term  ω 2f ωs L eff   1 = (s − o ) E∗s (r) · E f (r) dr 2τc 32β(c2 / f ) Vs    × ( f − o ) E∗f (r) · Es (r) dr

(3.92)

Vf

corresponds to the rate of the light energy transferring from the microsphere into the fiber. We have also artificially inserted the intrinsic loss rate 1/2τ0 of the microsphere (i.e., the loss rate of the WGM in the absence of the fiber) in (3.91). The efficient microsphere–fiber coupling requires the phase synchronization and a significant overlap of the microsphere and fiber modes [21], leading to the equality of the modules of two integrals in (3.92). In addition, according to [21] the effective length L eff is determined by  L eff = 2π Rs s / f .

(3.93)

The circulation time of the light traveling a round-trip distance inside the microsphere √ approximates τr = 2π Rs s /c. This duration actually measures the minimum time scale of the amplitude as (t) varying, shorter than which (3.91) becomes invalid.

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3 Whispering Gallery Modes in Optical Microcavities

√ During τr , the distance of the light traveling in the fiber reaches τr c/  f , i.e., L eff . As a result, one may write (3.91) in a compact form [22]   1 1 is ∂ a˜ s (t) + √ a˜ s (t) = −i(ωs − ω f − Δωs ) − − ∂t 2τ0 2τc τc

(3.94)

where we have applied the substitution as (t) = a˜ s (t)ei(ωs −ω f )t and the shorthand √ s = a f (−∞)eiφ / τr with the phase φ of the integral in (3.91). The light energy inside the microsphere is given by |a˜ s (t)|2 while |s|2 denotes the power carried by the incident light propagating in the fiber. Setting t → ∞, we obtain the steady-state power transmission coefficient for the light wave in the fiber passing the microsphere– fiber coupling region T =1−

τ0−1 τc−1 |a˜ s (∞)|2 . = 1 − |s|2 τ0 (ωs − ω f − Δωs )2 + (τ0−1 + τc−1 )2 /4

(3.95)

At the resonance, ω f = ωs − Δωs , T is simplified as  T =

τ0−1 − τc−1 τ0−1 + τc−1

2 .

(3.96)

It is seen that T is minimized to zero when τ0−1 = τc−1 , i.e., the entire input power is lost due to the microsphere’s intrinsic loss, which is referred to as the critical coupling. Finally, the microsphere’s quality factor Q is given by 1 1 1 1 1 = + = + , Q ω s τ0 ω s τc Q0 Qc

AQ1

(3.97)

where Q 0 denotes the intrinsic quality factor while Q c is caused by the microsphere– fiber coupling. In experiment, the microsphere–fiber coupling efficiency (1 − T ) can reach as high as 72% with Q c = 2 × 106 [23]. In the above, we have only discussed the coupling between one WGM and one fiber mode. In practice, the microspheres are not the single-mode optical devices and the tapered fibers, in general, also support the multimode propagation. Although the fundamental HE11 mode of the fiber can be efficiently launched and the singleWGM excitation is highly accessible, the optical power is inevitably transferred to the higher-order fiber modes via the excited WGM. One may study the multimode microsphere–fiber coupling in a similar way. Nonetheless, in many cases we prefer that the optical power is mainly transferred between the HE11 fiber mode and a certain WGM. The parasitic coupling between two desired modes and others limits the inter-desired-mode coupling efficiency. The coupling ideality, defined as the ratio of the power coupled into a desired (fiber or WGM) mode to the power coupled into

3.2 Microspheres

145

all microsphere and fiber modes, can be employed to quantify the coupling of a single fiber mode to a single WGM. It has been verified that under the appropriate conditions the fiber–taper coupler offers a near-unity ideality [24].

3.3 Microbottles In microspheres, the WGM light is confined within a thin equatorial ring near the sphere’s surface. Such a two-dimensional confinement challenges a controllable interaction between quantum emitters and microcavities because the prism or fiber coupler restricts the mechanical and optical access [26]. Using a microbottle structure may be a way to address this issue. Figure 3.10a shows a typical microbottle, i.e., a cylindrically symmetric deformation of a glass tube. Besides the equatorial character of the WGMs in the transverse plane, the optical waves in a microbottle also exhibit the bouncing-ball modes [see Fig. 3.10b], like the modes in a standard Fabry–Pérot cavity, along the axial direction. Thus, unlike the microspheres, the microbottles provide a three-dimensional confinement of the light [27]. We choose the cylindrical coordinate system (see Appendix B), where the location of a point r is described by the radial distance ρ ∈ [0, ∞), the azimuth angle ϕ ∈ [0, 2π), and the axial coordinate z ∈ [0, ∞), for the analysis. The bouncing-ball

Fig. 3.10 WGMs of microbottles. a Microscope image of a glass bottle resonator. Reprinted from [25] under the Creative Commons Attribution License. b Microbottle geometry in cylindrical coordinates. The cavity modes exhibit the WGM characters in the ρ − ϕ plane and act like a bouncing ball along the axial direction with two caustics located at ±z c . The intensity distributions of a typical cavity mode along the axial and radial directions are displayed in c and d, respectively

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3 Whispering Gallery Modes in Optical Microcavities

modes exist between ±z c and the profile of the prolate-shaped bottle is approximated as R0 with − z c ≤ z ≤ z c , (3.98) R(z) =  1 + (Δkz)2 i.e., a quasi-parabolic spatial profile, with the maximal radius R0 and the curvature of the microbottle Δk. The symbol ‘i’ (‘o’) is used to denote the region inside ρ < R(z) (outside ρ > R(z)) the microbottle. The relative permittivity of the dielectric microbottle is i while o denotes the relative permittivity of the outside-cavity environment. The cavity’s eigenmodes with the full polarization information may be obtained from solving (3.1). Here, we focus on the experimentally relevant modes with the maximum angular momentum so as to simplify the analysis. The wavevector of a cavity mode is written as k = kρ eρ + kϕ eϕ + k z ez . Since the mode is located close to the microbottle’s surface, the radial component kρ is negligible, kρ ≈ 0. In addition, the confinement along the axial direction requires that k z (z) must vanish at ±z c , i.e., k z (±z c ) = 0 and kϕ (±z c ) = k with k = |k|. From the conservation of orbital angular momentum L = r × (k), i.e., dL/dt = 0, we conclude that the product R(z)kϕ (z) is independent of the z coordinate. Then, it is easy to derive the azimuthal kϕ (z) and axial k z (z) components of the wavevector  R2 Rc kϕ (z) = k , k z (z) = ±k 1 − 2 c , R(z) R (z)

(3.99)

with the definition Rc = R(±z c ) < R0 . We restrict ourselves to the limit (Δkz c )2  1, for which k z  kϕ , and focus on the quasi-TE-polarized modes. According to the cylindrical symmetry of the microbottle, we look for the electric field with the following form: E(r) = Ψ (ρ, z)eimϕ ez ,

(3.100)

with m ∈ Z. Inserting (3.100) into (3.2), one obtains     ∂ ∂2 m2 1 ∂ ρ Ψ (ρ, z) + 2 Ψ (ρ, z) + k 2 − 2 Ψ (ρ, z) = 0. ρ ∂ρ ∂ρ ∂z ρ

(3.101)

We then write Ψ (ρ, z) as Ψ (ρ, z) = Φ(ρ, z)Z (z) and assume that Φ(ρ, z) varies much slower with respect to z than Z (z). In the adiabatic approximation, ∂2 ∂2 Φ(ρ, z)Z (z) ≈ Φ(ρ, z) 2 Z (z), 2 ∂z ∂z Equation (3.101) leads to two eigenvalue equations

(3.102)

3.3 Microbottles

147

  ∂2 1 ∂ m2 2 Φ(ρ, z) + kϕ − 2 Φ(ρ, z) = 0, Φ(ρ, z) + ∂ρ2 ρ ∂ρ ρ 2 ∂ Z (ρ, z) + k z2 Z (ρ, z) = 0, ∂z 2

(3.103a) (3.103b)

√ √ with k 2 ≈ kϕ2 + k z2 . In the following, we use ki = i ω/c and ko = o ω/c to denote the wavenumber of the mode inside and outside the cavity, respectively, with the mode’s frequency ω. Equation (3.103a) has the form of Bessel’s differential equation. We write its solutions as Φi (ρ, z) = A

Jm (kϕ,i ρ) H (1) (kϕ,o ρ) , Φo (ρ, z) = A (1)m , Jm (kϕ,i R(z)) Hm (kϕ,o R(z))

(3.104)

√ with the azimuthal wavenumbers kϕ,i = ki Rc /R(z) and kϕ,o = o /i kϕ,i and the amplitude A. In the limit of ρ → ∞, Hm(1) (kϕ,o ρ) ∼ eikϕ,o ρ represents the outgoing wave. We then express the electric and magnetic fields inside and outside the cavity as   Ei,o (r) = Φi,o (ρ, z)ez Z (z)eimϕ ,   Φi,o (ρ, z) 1 ∂Φi,o (ρ, z) m eρ + i eϕ Z (z)eimϕ . Bi,o (r) = ω ρ ∂ρ

(3.105a) (3.105b)

Equation (3.105a) is obtained from the relation ∇ × E = iωB. Equation (3.104) ensures that Φi,o (ρ, z) fulfills the continuity condition n × (Ei − Eo ) = 0 across the microbottle’s surface ρ = R(z) with the normal unity vector n. In addition, the boundary condition n × (Bi − Bo ) = 0 leads to √ [H (1) (kϕ,o R(z))] √ [Jm (kϕ,i R(z))] = o m(1) , i Jm (kϕ,i R(z)) Hm (kϕ,o R(z))

(3.106)

from which one may evaluate kϕ,i R(z). By setting ki Rc = m, ˜ (3.103b) is rewritten as   m˜ 2 m˜ 2 ∂2 (3.107) − 2 Z (z) + 2 (Δkz)2 Z (z) = ki2 − 2 Z (z). ∂z R0 R0 Mapping (3.107) onto the eigenvalue equation for the quantum harmonic oscillator, we obtain the wavenumber of the cavity mode ki2 =

m˜ 2 (2q + 1) + , R02 z m2˜

(3.108)

and the eigenfunctions for the motion of light (i.e., the bouncing ball) in the axial direction

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3 Whispering Gallery Modes in Optical Microcavities

π −1/4 2 Z m,q = √ q Hq (z/z m˜ )e−(z/zm˜ ) /2 , 2 q!z m˜

AQ2

(3.109)

 ˜ and the Hermite polynomials Hq (z). The integer q ∈ Z correwith z m˜ = R0 /mΔk sponds to the quantum number of the harmonic oscillator. Unlike the microspheres, ki depends on the azimuthal number m and this strongly reduces the degeneracy of the WGMs. As an example, we assume that a microbottle is made of a material with i = 2.25 and stays in vacuum o = 1. The maximum radius of the microbottle is R0 = 10 µm and the curvature is Δk = 0.03 µm−1 . The z-independent number m˜ = kϕ,i (z)R(z) may be evaluated from (3.106) by choosing a certain value m and setting z = 0. The number m˜ is complex with its real part Re(m) ˜ close to m. Once obtaining m, ˜ one may further derive the parameter z m˜ and the complex wavenumber inside the cavity ki . The cavity Q factor is given by |Re(ki )/Im(ki )| and the vacuum wavelength of the √ cavity mode is λ = ω/c = 2π i /Re(ki ). For m = 108, one finds λ = 716.5 nm and 11 Q = 2.3 × 10 . Figure 3.10c shows the intensity distribution |Z (z)|2 of the cavity mode in the axial direction. There exist two maximum-intensity locations close to −z c and z c , respectively. The bouncing-ball length 2z c reaches 144.4 µm. These two well-separated regions of the enhanced field have various potential applications. For instance, a single quantum emitter is placed at one enhanced-field region, achieving the strong emitter–cavity coupling, while a thin optical fiber is coupled to the cavity at the other enhanced-field region, performing the input and output operations. The intensity distribution of the cavity mode in the radial direction is displayed in Fig. 3.10d, where multiple maxima are presented. We use n to denote the number of intensity maxima along the radial direction. Thus, a cavity mode is specified by (n, m, q). The common fabrication materials of the microbottles include silica and polymers. In general, the silica microbottles have a Q factor much higher than the polymer ones (see Table 3.3), but still smaller than that of the microspheres. The fiber-based microbottles can be fabricated by a two-step heat-and-pull process [28]. A scanned CO2 laser beam is used to heat up an optical fiber to its softening point. The fiber is then mechanically pulled, decreasing its diameter to a desired starting value. Next, the fiber is locally heated up by a focused CO2 -laser beam and is stretched to further reduce the diameter at the corresponding location. Repeating the same step at another

Table 3.3 Selected experimental Q factors of microbottles. The profile of a microbottle is characterized by R(z)/R0 ≈ 1 − (Δkz)2 /2 with the radius R0 and the curvature Δk Material Environment λ (nm) 2R0 (µm) Δk (µm−1 ) Q Refs. Glass fiber Silica Silica Polymer

Air Air Air Air

850 1550 ∼1550 1541

35 47 265 17.5

0.012 0.0133 0.0027 0.01

3.6 × 108 2.7 × 108 ∼ 107 1.8 × 106

[29] [30] [31] [32]

3.3 Microbottles

149

location results in the bottle shape. The recent improved technique may produce a microbottle with an intrinsic quality factor over 108 [29]. An alternative popular method is the soften-and-compress technique, which is a single-step process. A standard fusion splicer is used to soften a small region of a continuous piece of fiber. Simultaneously, the fiber is slightly compressed, resulting in a double-neck bottle. The quality factor of the microbottles fabricated via this process can reach about 107 [31]. The excitation of the microbottle’s modes can be implemented via the the evanescent coupling between a tapered fiber and the microbottle. The coupling strength depends on the overlap between the microbottle and taper modes. Due to the strong asphericity of the microbottle, the mode excitation relies upon the specific microbottle–taper-coupling position [33]. As shown in Fig. 3.11, the excitation spectrum becomes sparser, i.e., less modes are excited, when the coupling position moves from the microbottle’s center to its neck. This may be roughly interpreted from the caustic axial length z c . The tapered fiber placed at the center of the microbottle potentially excites all microbottle’s modes. In contrast, when the tapered fiber is away from the microbottle’s center with a distance d, the modes, whose z c are shorter than d, become no longer accessible.

Fig. 3.11 Mode excitation of a microbottle. a–c Images of the microbottle that is coupled through a tapered fiber located at the microbottle’s center and the positions 100 and 200 µm away from the microbottle’s center, respectively. d Excitation of a normal fiber. e Transmission spectra for the microbottle excited at different positions. Reprinted with permission from [33] ©The Optical Society

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3 Whispering Gallery Modes in Optical Microcavities

3.4 Microdisks As discussed in the last section, the curvature Δk of a microbottle is small and the light field moves along the axial direction. Increasing Δk shortens the inter-caustic separation. One can imagine that as the inter-caustic separation is shorter than the light wavelength λ (e.g., in the extreme case that the microbottle becomes a microdisk with a thickness smaller than λ), the light stops moving in the axial direction. As a result, the bouncing-ball modes disappear but the WGMs still remain. In this section, we study the WGMs in a disk-shaped microcavity as shown in Fig 3.12a. The microdisk’s radius R is larger than λ while its thickness d is smaller than λ. The disk wedge angle is θ. The relative permittivities of the microdisk and the outside-disk dielectric medium are i and o , respectively, and i > o . Again, the cylindrical coordinate system (ρ, ϕ, z) is applied, where the origin is set at the microdisk’s center and the unit vector ez is along the central axis of the microdisk. The WGMs of a microdisk can be investigated by using the finite element method. In contrast, the analytic solutions for the disk geometry are hardly to be derived. However, the microdisks with θ = 90◦ may be simplified by a two-dimensional effective-index model with a reasonable accuracy. In this model, the refractive index contrast, i versus o , is assumed to be high enough that the motion of the light in

Fig. 3.12 WGMs in microdisks. a SEM image of a microdisk. b Schematic diagram of the microdisk cavity. Reprinted from [34] by permission of John Wiley & Sons, Inc. c Two lowest Z (z) solutions with m = 100. The radius and thickness of the microdisk are R = 10 µm and d = 500 nm, respectively. The microdisk with i = 5 is situated in vacuum o = 1. d The radial function Φ(ρ) corresponding to the symmetric Z (z) in c

3.4 Microdisks

151

the z-direction is analogous to the light traveling in a slab waveguide. We start with considering the quasi-TE-polarized modes (E z , Bρ , Bϕ ), i.e., the light polarization is perpendicular to the disk plane [35, 36]. According to the microcavity’s symmetry, the electric field takes the form E(r) = E z ez = Φ(ρ)Z (z)eimϕ ez ,

(3.110)

which follows the equation 1 Φ(ρ)



   ∂2 1 ∂2 1 ∂ m2 2 Φ(ρ) + Φ(ρ) + Z (z) + k − 2 = 0, (3.111) ∂ρ2 ρ ∂ρ Z (z) ∂z 2 ρ

with the angular momentum number m ∈ Z, the in-medium wavenumber k = √ ω/c, and the relative-permittivity distribution (ρ, z). The solutions with m > 0 (m < 0) describe the clockwise (counterclockwise) traveling waves. In the region of ρ < R, the relative permittivity is i for |z| < d/2 while o for |z| > d/2. Introducing an effective wavenumber keff into (3.111), one obtains the following two equations:   1 ∂ m2 ∂2 2 Φ(ρ) + keff − 2 Φ(ρ) = 0, Φ(ρ) + ∂ρ2 ρ ∂ρ ρ 2 ∂ 2 Z (z) + (k 2 − keff )Z (z) = 0. ∂z 2

(3.112a) (3.112b)

We look for the self-consistent solutions of the above equations. Equation (3.112b) is equivalent to the time-independent Schrödinger equation for a particle moving in a one-dimensional finite square potential well. The corresponding bound states can be divided into the symmetric and antisymmetric groups. The symmetric solutions are expressed as ⎧ %  ⎪ 2 2 ⎪ A exp keff − ko z , z < −(d/2), ⎪ ⎪ ⎪ ⎪  % ⎨ 2 z , |z| < (d/2), ki2 − keff Z (z) = B cos ⎪ ⎪   % ⎪ ⎪ ⎪ 2 ⎪ − ko2 z , z > (d/2), ⎩ A exp − keff

(3.113)

√ √ with ki = i ω/c and ko = o ω/c. The amplitudes A and B are determined by the appropriate boundary conditions. Imposing the continuity of Z (z) and the continuity of ∂ Z (z)/∂z at z = ±d/2, one may find the relation between keff and (ki , ko ) % 2 keff

−

ko2

  % % d 2 2 2 2 = ki − keff tan k − keff . 2 i

Similarly, the antisymmetric solutions of Z (z) are given by

(3.114)

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3 Whispering Gallery Modes in Optical Microcavities

⎧ %  ⎪ 2 2z , ⎪ A exp k − k z < −(d/2), ⎪ o eff ⎪ ⎪ ⎪  % ⎨ 2 Z (z) = B sin ki2 − keff z , |z| < (d/2), ⎪ ⎪   % ⎪ ⎪ ⎪ 2 ⎪ − ko2 z , z > (d/2), ⎩−A exp − keff

(3.115)

with the relation %

  % % d 2 2 2 . keff − ko2 = − ki2 − keff cot ki2 − keff 2

(3.116)

In addition, since the radial function Φ(ρ) must be finite as ρ approaches zero, the solution of Eq (3.112a) is written as Φ(ρ) =

Jm (keff ρ) . Jm (keff R)

(3.117)

In the ρ > R region, the wavenumber k(ρ, z) is equal to ko and the term keff in (3.112a) and (3.112b) should be set at ko . The radial component takes the form Φ(ρ) =

Hm(1) (ko ρ) Hm(1) (ko R)

,

(3.118)

with the Hankel functions of the first kind Hm(1) . We have obtained the radial and axial functions, but the WGM frequency ω and the corresponding effective wavenumber keff are still unknown. To evaluate them, one has to derive the magnetic field, B(r) = −(i/ω)∇ × E(r). The continuity of B(r) at the interface ρ = R leads to keff

[Jm (keff R)] [H (1) (ko R)] = ko m(1) , Jm (keff R) Hm (ko R)

(3.119)

with, for example, [Jm (keff R)] = [∂ Jm (x)/∂x]x=keff R . Concequently, ω and keff can be numerically derived by combining (3.119) with (3.114) or (3.116). For a certain m, one may obtain multiple values of the mode frequency ω, which can be arranged in an ascending order according to Re(ω). The quantum number q is used to denote the q-th value of ω. Thus, a cavity mode is specified by (n, m, q), where n denotes the number of intensity maxima in the radial distribution. As an example, we consider that a microdisk with R = 10 µm, d = 500 nm, and i = 5 is placed in vacuum (o = 1). Figure 3.12c illustrates the first two solutions of Z (z) with m = 100, where the symmetric solution has a vacuum wavelength λ = 1.2 µm while the other with λ = 988 nm is antisymmetric. The radial function Φ(ρ) corresponding to the symmetric Z (z) is displayed in Fig. 3.12d. We see that only one field maximum is presented

3.4 Microdisks

153

and the radial number is equal to n = 1. The quality factor Q = |Re(ω)/Im(ω)| is estimated to be over 1015 . So far, we have only discussed the quasi-TM-polarized modes. To derive the quasi-TE-polarized modes (Bz , E ρ , E ϕ ), we write the magnetic field as B(r) = Bz ez = Φ(ρ)Z (z)eimϕ ez .

(3.120)

Equations (3.111)–(3.118) are still valid for the radial and axial functions Φ(ρ) and Z (z). In addition, using the relation i∇ × B(r) = (k 2 /ω)E(r) and the continuity of the tangent component E ϕ at the ρ = R interface, one finds 1 [Hm(1) (ko R)] 1 [Jm (keff R)] =√ , √ eff Jm (keff R) o Hm(1) (ko R)

(3.121)

√ with the effective relative permittivity eff = ckeff /ω. The mode frequency ω and the effective wavenumber keff can then be evaluated from (3.114), (3.116), and (3.121). For the practical uses, the WGMs with an edge inclination angle θ < 90◦ are of more interest than the WGMs with θ = 90◦ . The calculation of these WGMs relies on the finite element method. As illustrated in Fig. 3.13, when θ goes up the radial positions of the TE and TM modes move closer to the outer rims of the microdisk wedge. One direct consequence is the red shift of the resonance wavelength [37], i.e., toward the long wavelength. In addition, the Q factor falls as θ is increased. The reason is believed that when the modes are shifted away from the disk perimeter, the radiation losses of the WGMs are suppressed [39]. In Table 3.4, we list several microdisk’s Q factors achieved in the recent experiments. As one can see, the values are one order of magnitude lower than that of the microspheres [10]. Nevertheless, the microdisks can be integrated on-chip for various detection and sensing applications. The traditional semiconductors, due to their compatibility with the complementary metal oxide semiconductor (CMOS) approach, are the suitable material platform for fabricating the on-chip microdisks. The relevant fabrication process will be introduced in next section. Recently, the lithium niobate on an insulator (LNOI) has become another promising candidate for implementing the on-chip integration of microcavities. In particular, its excellent nonlinear optical properties have attracted a great deal of interest in performing the nonlinear processes on the high-Q microdisks [43]. As shown in Fig. (3.14), the specific fabrication process can be summarized in four steps [40, 42, 44]: (1) Chromium (Cr) deposition. A thin Cr layer with a thickness of hundreds of nm is coated on the surface of the LNOI; (2) Formation of the disk-shaped mask. The femtosecond laser ablation is employed to tailor the Cr film into a microdisk; (3) Polishing. The LNOI region uncovered by the Cr mask is removed by means of the chemo-mechanical polishing; and (4) Formation of the microdisk. The top Cr mask and the silica buffer layer are eliminated via two

AQ3

154

3 Whispering Gallery Modes in Optical Microcavities

Fig. 3.13 Dependence of the electric-field distributions of the fundamental TE and TM modes on the edge inclination angle θ. Reprinted with permission from [37] ©The Optical Society Table 3.4 Selected experimental Q factors of microdisks Material Environment λ (nm) 2R (µm) d (µm) Silica Silica Crystalline Polymer

Air Air Air Air

1500 1550 773 600

7500 60 140 120

10 1 0.9 1

Silica

Air

1550

140

2

Wedge angle

Q

Refs.

27◦ 22◦ 9.5◦ 8.4◦ , 13.1◦ Invertedwedge

8.75 × 108 6 × 107 1.46 × 107 ∼ 107

[38] [39] [40] [37]

∼ 106

[41]

consecutive chemical wet etching processes. The Q factor of the microdisks fabricated in this way reaches the order of 107 . The femtosecond laser micromachining can cause a high roughness of the disk’s surface. The technique of the focused ion beam milling may be used to create the ultrasmooth sidewalls of the microdisks [46].

3.4 Microdisks

155

Fig. 3.14 Fabrication process of the LNOI microdisks via the femtosecond laser ablation. Reprinted from [42] under the Creative Common CC BY license

3.5 Microtoroids The on-chip WGM microcavities can also be in the toroidal structure [see Fig. 3.15a], where a toroidal-shaped cavity is supported by a pillar [45]. The microtoroids with a dumbbell-like cross-section have a lower symmetry compared to the microspheres. This provides an extra confinement in both transverse and vertical degrees of freedom, simplifying the modal spectrum. In addition, the evanescent interaction between the relatively large-diameter microtoroid and its relatively small-diameter pedestal

Fig. 3.15 a SEM image of a microtoroid. Reprinted by permission from Springer Nature Customer Service Center GmbH: Springer Nature, Nature, [45], © 2003. b A small section of microtoroid. The microtoroid is in the x − y plane. The origin o of the cylindrical coordinate system is located at the center of the toroid. The origin o of the local toroidal coordinates is set at the central point of the microtoroid’s cross-section

156

3 Whispering Gallery Modes in Optical Microcavities

is strongly weakened, suppressing the influence of the pedestal on the local sensing ambience. As shown in Fig. 3.15b, a microtoroid is characterized by two diameters, D = 2R (major) and d = 2a (minor). The value (D + d) is called the principal diameter. So far, investigating the microtoroid’s WGMs mainly relies on the full-vectorial FEM (finite element method) solver. This is because the common approach of separation of variables is inapplicable to the electromagnetic wave equation in toroidal coordinates, making the analytical derivation complicated. Nevertheless, the approximate solutions may be obtained in the limit of the small inverse aspect ratio (a/R)  1 [47]. The perturbation approach introduced below was firstly applied to study the light propagation in a bent optical waveguide [48]. We first consider the quasi-TE-polarized modes. In the cylindrical coordinate system (ρ, ϕ, z) with ρ ∈ [0, ∞), ϕ ∈ [0, 2π), and z ∈ (−∞, ∞), the electric field is approximately written as E(r) ≈ E(ρ, z)eiβ Rϕ ez ,

(3.122)

with an effective propagation constant β. The WGMs require that the round-trip phase shift must be multiples of 2π, β = m/R with m ∈ Z.

(3.123)

The scalar electric field E(ρ, z) follows the following wave equation: 

2 ∂2 ∂2 1 ∂ ω2 2R + + +  − β ∂ρ2 ρ ∂ρ ∂z 2 c2 ρ2

 E(ρ, z) = 0,

(3.124)

with the unknown mode frequency ω. We make the coordinate transformation ρ = R + r cos φ, z = r sin φ,

(3.125)

with the local toroidal coordinates r ∈ [0, ∞) and φ ∈ [0, 2π) [see Fig. 3.15b]. From the expressions  z , (3.126) r = (ρ − R)2 + z 2 , φ = tan−1 ρ− R one may derive

∂r = cos φ, ∂ρ

∂φ sin φ =− . ∂ρ r

(3.127)

Thus, (3.124) in the local toroidal coordinates take the form 

  ∂2 1 ∂2 ∂ sin φ ∂ 1 ∂ 1 + cos φ − + + ∂r 2 r ∂r r 2 ∂φ2 R + r cos φ ∂r r ∂φ    2 −2 ω r (3.128) + 2 − 1 + cos φ β 2 E(r, φ) = 0. c R

3.5 Microtoroids

157

To simplify (3.128), we rewrite E(r, φ) as  E(r, φ) =

R Ψ (r, φ), R + r cos φ

(3.129)

and the wave equation for Ψ (r, φ) is given by 

1 ∂2 ∂2 1 ∂ ω2 + + +  ∂r 2 r ∂r r 2 ∂φ2 c2     −2 1 r 2 Ψ (r, φ) = 0. + 1 + cos φ − β R 4R 2

(3.130)

It is seen that two variables r and φ cannot be separated, for which we resort to the perturbation theory. In the limit of (a/R)  1, the scalar field Ψ (r, φ) and the propagation constant β 2 can be expanded in an ascending series of (a/R) as [48] Ψ (r, φ) =

 a j

 a j Ψ ( j) (r, φ), β 2 = β 2j . j R j R

(3.131)

Substituting (3.131) into (3.130), one may obtain the scalar wave equation for the zero-order component DΨ (0) (r, φ) = 0, (3.132) where the differential operator D is defined as  D f (r, φ) =

1 ∂2 1 ∂ ∂2 + + + q2 ∂r 2 r ∂r r 2 ∂φ2

 f (r, φ),

(3.133)

% with the propagation constant q = ω 2 /c2 − β02 . We focus on the φ-independent mode. It is straightforward to derive the solution of (3.132) ⎧ J0 (qir ) (0) ⎪ r a K 0 (qo a/i)

(3.134)

% with qi,o = i,o ω 2 /c2 − β02 . Here, i (o ) is the inside-cavity (outside-cavity) relative permittivity constant and Jν (K ν ) denote the Bessel functions of the first kind (the modified Bessel functions of the second kind). One should note that qo is an imaginary constant. The boundary condition ∂ (0) ∂ (0) Ψi (r = a, φ) = Ψ (r = a, φ) ∂r ∂r o

(3.135)

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3 Whispering Gallery Modes in Optical Microcavities

leads to the characteristic equation qi

J1 (qi a) K 1 (qo a/i) = (qo /i) . J0 (qi a) K 0 (qo a/i)

(3.136)

Similarly, one derives the differential equation for the first-order component as ' r & DΨ (1) (r, φ) = β12 − 2β02 cos φ Ψ (0) (r, φ). a

(3.137)

The first-order correction β12 may be obtained by multiplying (3.132) by Ψ (1)  (r, φ) and (3.137) by Ψ (0) (r, φ), substracting and integrating over the cross-section ( S d S), i.e.,    2 ( r cos φ Ψ (0) (r, φ) d S I, (3.138) β12 = 2β02 S a 

where we have defined

 (0) 2 Ψ (r, φ) d S.

I =

(3.139)

S

It can prove that β12 = 0. Thus, the next order correction should be considered. Equation (3.137) can be solved by writing Ψ (1) (r, φ) in the form ⎧ ) J (q r ) ⎪ ⎪Ψi(1) (r, φ) = 1 i cos φ l al (qir )l r a ⎩ o l K 0 (qo a/i)

(3.140)

Taking into account the boundary conditions Ψi(1) (r = a, φ) = Ψo(1) (r = a, φ),

∂ (1) ∂ (1) Ψi (r = a, φ) = Ψ (r = a, φ), ∂r ∂r o

we have the expression ⎧   2 K 1 (qo a/i) r 2 J1 (qir ) (β0 a)2 ⎪ (1) ⎪ +1− 2 cos φ r < a ⎨Ψi (r, φ) = 2(qi a) (q a J0 (qi a) o a/i) K 0 (qo a/i)   . 2 J1 (qi a) r 2 K 1 (qor/i) (β0 a)2 ⎪ ⎪ ⎩Ψo(1) (r, φ) = −1+ 2 cos φ r > a 2(qo a/i) (qi a) J0 (qi a) a K 0 (qo a/i) (3.141) To obtain the correction to β02 , we further consider the wave equation for the secondorder component

3.5 Microtoroids

159

  2 r 1 cos φ − 2 Ψ (0) (r, φ) DΨ (2) (r, φ) = β22 + 3β02 a 4a r  −2β02 cos φ Ψ (1) (r, φ), a

(3.142)

where we have used β12 = 0. Again, we multiply (3.132) by Ψ (2) (r, φ) and (3.142) by Ψ (0) (r, φ), integrate their subtraction over the cross-section, and obtain β22 =

    r 1 2 2 cos φ Ψ (0) (r, φ)Ψ (1) (r, φ)d S + β 0 4a 2 S a (   2 r I. −3 cos φ Ψ (0) (r, φ)Ψ (0) (r, φ)d S S a

(3.143)

Thus, the electric field to the first-order approximation and the propagation constant to the second-order approximation are expressed as 

& a ' R Ψ (0) (r, φ) + Ψ (1) (r, φ) eiβ Rϕ , (3.144a) R + r cos φ R   a 2 β22 , (3.144b) β ≈ β02 + R

E(r) ≈ ez

respectively. Combining (3.123) and (3.136), one can evaluate the WGM frequency ω. After deriving the electric field, one may further calculate the corresponding mag−1 iβ Rϕ with H (r, φ) = (iωμ0 )−1 netic field H = (iωμ0 ) ∇ × E ≈ eϕ H e 1 ∂ + ∂r E(r, φ). r Figure 3.16 displays the examples of the quasi-TE WGMs of a microtorid calculated from the perturbation theory. The intensity maximum of the fundamental (n = 1, l = 0) mode is deviated from the origin o of local toroidal coordinates in Fig. 3.15b. In a similar way, one can calculate the quasi-TM-polarized modes H ≈ H (ρ, z)eiβ Rϕ ez and the corresponding electric field is given by E = (−iε0 ω)−1 ∇ × H.

Fig. 3.16 Quasi-TE-polarized WGMs of a silica microtorid with 2a = 3 µm, 2R = 60 µm, and λ = 1.55 µm

160

3 Whispering Gallery Modes in Optical Microcavities

Fig. 3.17 Schematic diagram of the microtoroid fabrication process

Fabrication. As illustrated in Fig. 3.17, the microtoroids can be fabricated by using the conventional microelectronic fabrication techniques through four basic steps [45]: (1) A few micrometer thick layer of silica grows on the top of a silicon (Si) wafer. A uniform layer of positive photoresist is then deposited on the silica-onsilicon wafer. After that, the photolithography is performed, where the photoresist is exposed to the external radiation through a mask containing arrays of circular disks. Typically, the diameter of the finally produced microtoroids will be 25∼30% smaller than that of the disks [49]; (2) Since the unexposed-photoresist disks act as the etch masks, the wafer is selectively etched by immersing it in the buffered hydrofluoric acid (HF) solution, thereby creating the disk-shaped pads. The residual photoresist is removed by using acetone; (3) Dry etching silicon. The silicon substrate is etched by exposing the wafer to the XeF2 gas, where the remaining silica disks act as the etch masks. After equally undercuting the edges, the SiO2 disks are supported by the silicon pillars. Normally, these undercut microdisks have a Q factor of about 105 , limited by the roughness of the disk periphery; and (4) Laser reflow. A CO2 laser at 10.6 µm illuminates the undercut silica disks. The thermal conductivity of silicon is about one hundred times higher than that of silica. In addition, the optical absorption cross-section of silicon at 10.6 µm is far smaller than that of silica. Due to these two reasons, the silicon pillars are much cooler than the silica disks and function as the circular heat sinks. Hence, the silica-disk peripheries are melted and the disk diameter shrinks, forming a toroid structure. The surface tension smooths the surface. Several selected Q factors of microtoroids are listed in Table 3.5. Generally, the silica microtoroids have a Q factor much higher than the polymer ones. In comparison, the microtoroid’s Q factor is one order of magnitude lower than that of the

3.5 Microtoroids

161

Table 3.5 Selected experimental Q factors of microtoroids Material Environment λ (nm) D (µm) d (µm) Silica Silica Silica Polymer

Air Air Polymer Air

1550 1550 982.5 680, 1300, 1550

28 80 ∼ 120 120 30 ∼ 150

3.5 5∼7 10 2.5 ∼ 6

Q

Refs.

4 × 108

[50] [45] [51] [52]

∼ 1 × 108 1 × 107 ∼ 1 × 105

microspheres. Recently, the microtoroids have become one of the main platforms for the microcavity-based biosensing and fundamental studies for the nonlinear and non-Hermitian optics.

3.6 Thermorefractive-Noise-Limited Frequency Stability The central frequency ω of a WGM inevitably suffers from various environmental noises. One of them is the temperature fluctuations, giving rise to the thermorefractive noise in microstructures. Such a thermodynamic noise sets a fundamental limit to the frequency stability of the miniaturized cavities (i.e., microspheres [53], microtoroids [54], and micro-ring resonators [55]). Simple analysis. We consider that a microstructure is in contact with a heat bath whose temperature is Tc . This microstructure can exchange the energy with the heat bath. In statistical mechanics, the investigation on such a system may be performed by using a canonical ensemble (i.e., multiple copies of this system). We use E to denote the total energy of one copy, and the variance of E is given by (ΔE)2  = E 2  − E2 = k B Tc2

∂E , ∂Tc

(3.145)

where O denotes the mean value of a physical quantity O over the canonical ensemble and k B is the Boltzmann constant. Using the specific heat capacity C=

1 ∂E , ρVth ∂Tc

(3.146)

with the material density ρ and the thermal volume Vth (see below), one obtains the variance of the microstructure’s temperature fluctuations (ΔT )2  =

(ΔE)2  k B Tc2 = . 2 (CρVth ) CρVth

(3.147)

AQ4

162

3 Whispering Gallery Modes in Optical Microcavities

Table 3.6 Material constants for the fused silica at room temperature ρ (kg/m3 ) C (J/kg·K) dn/dT (K−1 ) D (m2 /s) 2200

1.45×10−5

670

Tc (K)

9.5×10−7

300

Owing to the tiny volume of the microstructure, the influence of the temperature fluctuations becomes particularly important. For example, using the data listed in Table 3.6, the relative frequency uncertainty of a WGM2 1 dn  δω ∼− (ΔT )2  ω n dTc

(3.148)

is estimated to be ∼ 10−10 at room temperature. Here, n is the refractive index of the microcavity and δn denotes the corresponding fluctuation. Frequency fluctuations. In a more detailed analysis, we use δT (r, t) = T (r, t) − Tc to denote the temperature difference in the location r at the time t. This temperature difference induces the heat transfer described by the heat diffusion equation ∂ δT (r, t) = D∇ 2 δT (r, t) + F(r, t), ∂t

(3.149)

with the thermal diffusivity D = k/ρC and the thermal conductivity k of the material. In the Markovian limit, the driving force F(r, t) satisfies the following autocorrelation function: F(r, t)F(r , t  ) = (ΔF)2 [∇r · ∇r δ(r − r )]δ(t − t  ),

(3.150)

with the variance (ΔF)2 . We introduce the four-dimensional Fourier transform of an arbitrary function O(r, t),   O(q, f ) = F [O(r, t)] =

O(r, t)eiq·r−i2π f t drdt,

(3.151)

and the corresponding inverse Fourier transform O(r, t) = F −1 [O(q, f )] =

 

O(q, f )e−iq·r+i2π f t

dq d f. (2π)3

(3.152)

2 A cavity mode E(r) fulfills the Helmholtz equation (∇ 2

+ (r)ω 2 /c2 )E(r) = 0. A small fluctuation δ(r) of (r) induces a small frequency shift δω of ω. Inserting (r) + δ(r) and ω + δω into  2  δ(r)|E(r)| dr (r)|E(r)|2 dr

δi ≈ − 2 ≈ − δn n , where i is the relative i √ dielectric constant of the microcavity, δi denotes the fluctuation of i , n = i , and δn is the  2 corresponding  fluctuation. Assuming δn is caused by the temperature fluctuation (ΔT ) , i.e., dn 2 δn = dTc (ΔT ) , one arrives at (3.148).

the Helmholtz equation leads to

δω ω

≈ − 21

3.6 Thermorefractive-Noise-Limited Frequency Stability

163

Thus, the driving term F(q, f ) = F [F(r, t)] has the autocorrelation function in the (q, f ) domain3 F(q, f )F(q , f  ) = (2π)3 (ΔF)2 q 2 δ(q + q )δ( f + f  ).

(3.153)

Applying the Fourier transform to (3.149), one obtains δT (q, f ) = F [δT (r, t)] =

F(q, f ) , Dq 2 + i2π f

(3.154)

and the temperature difference is re-expressed as   δT (r, t) =

F(q, f ) dq e−iq·r+i2π f t d f. 2 Dq + i2π f (2π)3

(3.155)

It is easy to derive the autocorrelation function of the temperature difference 

 

q2 D 2 q 4 + (2π f )2 dq   ×e−iq·(r−r )+i2π f (t−t ) d f. (2π)3



δT (r, t)δT (r , t ) = (ΔF)  2

(3.156)

We then define the time-dependent average temperature difference as δ T¯ (t) =



2 ˜ δT (r, t)|D(r)| dr,

(3.157)

where the normalized electric displacement intensity takes the form 2 ˜ |D(r)| =

(r)|E(r)|2 . (r)|E(r)|2 dr

(3.158)

Using (3.156), the autocorrelation function of δ T¯ (t) reads Rδ T¯ (τ = t − t  ) = δ T¯ (t)δ T¯ (t  )    (ΔF)2  = 

where we have defined G(q) =

 dq q 2 |G(q)|2 ei2π f τ d f, (3.159) D 2 q 4 + (2π f )2 (2π)3 2 −iq·r ˜ e dr. |D(r)|

deriving F (q, f )F (q , f  ), the identity (2π)qx2 δ(qx + qx ) is used.

3 In

∞ ∞

−∞ −∞ e

i(qx x−qx x  ) ∂ ∂ δ(x ∂x ∂x 

(3.160)

− x  )d xd x  =

164

3 Whispering Gallery Modes in Optical Microcavities

Equation (3.159) implies that the power spectral density of the temperature fluctuations has a form  dq q 2 |G(q)|2 . (3.161) Sδ T¯ ( f ) = (ΔF)2  2 4 2 D q + (2π f ) (2π)3 In addition, comparing Rδ T¯ (τ = 0) to (3.147), one finds the expressions of the effective thermal volume Vth and the variance (ΔF)2  Vth−1 =

 |G(q)|2

dq 2k B Tc2 D 2 . , (ΔF)  = (2π)3 ρC

(3.162)

The temperature fluctuations δ T¯ (t) cause the change of the refractive index of a dielectric optical cavity, resulting in a frequency shift δω(t) of the cavity modes δω(t) 1 dn ¯ =− δ T (t). ω n dTc

(3.163)

Finally, one arrives at the power spectral density of the relative frequency fluctuations  Sδω/ω ( f ) =

1 dn n dTc

2 Sδ T¯ ( f ).

(3.164)

As suggested by (3.162), Sδω/ω ( f ) depends on the specific geometry of the microcavities. The Allan deviation σ y (τ ), which is related to the spectrum Sδω/ω ( f ) via [57]  σ 2y (τ ) = 2

0

∞

Sδω/ω ( f )

sin4 (πτ f ) d f, (πτ f )2

(3.165)

with the sampling time τ , is usually employed to measure the frequency stability. As shown in Fig. 3.18, the thermorefractive-noise-limited σ y scales approximately as ∝ τ −1/2 , indicating that the dominant component in Sδω/ω ( f ) follows the f −1/2 law. However, in practice the thermorefractive noise mainly affects the relatively shortterm (10−5 ∼ 10−3 s) stability while the relatively long-term (> 10−1 s) stability is limited by the linear frequency drift of the cavity modes.

Problems 3.1 A simple analysis of the microsphere–fiber coupling is given by [58]. As shown in Fig. 3.19a, a tapered fiber is near resonantly coupled to a microsphere (radius Rs and relative permittivity s ). The light components in the fiber and the microsphere before (after) the coupling region are E f,i (E f,o ) and E s,i (E s,o ), respectively. The amplitude transmission coefficient for the light in fiber (microsphere) is defined as

3.6 Thermorefractive-Noise-Limited Frequency Stability

165

Fig. 3.18 Thermorefractive noise. Two solid curves correspond to the power spectral density Sδω/ω ( f ) versus the noise frequency f for the TE-polarized WGM (n = 1, l = 203, m = 203) of a fused-silica microsphere with R = 50 µm in air and the Allan deviation σ y (τ ) versus the sampling time τ , respectively. The effective thermal volume Vth is evaluated to be ∼ 5 × 10−15 µm3 . Two marker lines denote the experimentally measured Allan deviations (i.e., toroid 1 and 2), which are extracted from [56]

t (t  ). Let κ (κ ) be the coupling coefficient from the fiber (microsphere) to the microsphere (fiber). The relation among different light components can be described by       E f,i t κ E f,o =U with U = . (3.166) E s,o E s,i κ t • Prove the following results: t = t  , κ = κ , t 2 − κ2 = 1,

(3.167)

i.e., t is a real number while κ is an imaginary number, under the conditions of time-reversal symmetry and energy conservation. In addition, E s,i is related to E s,o via E s,i = αE s,o eiϕ , where the real α denotes the amplitude loss factor (representing the cavity round-trip loss) and ϕ is the extra phase acquired in a round-trip. • Prove that the power transmission T = |E f,o /E f,i |2 of the light passing the coupling region takes the form

166

3 Whispering Gallery Modes in Optical Microcavities

Fig. 3.19 Microsphere–fiber coupling. a Schematic diagram. b Power transmission T as a function of the frequency ω of the probing beam propagating inside the fiber, where m ∈ Z. The microsphere’s amplitude loss factor is α = 0.9. The amplitude-transmission coefficient √ of the fiber is t = 0.9 for the solid curve and t = 0.5 for the dashed line. (c) T versus κ = i 1 − t 2 at the resonance ω = mΔωFRS

C 

T =1− 1 + F sin

2

πω ΔωFSR

,

(3.168)

with the coupling parameter C  C =1− the finesse F F=

t −α 1 − αt

2

4αt , (1 − αt)2

,

(3.169)

(3.170)

3.6 Thermorefractive-Noise-Limited Frequency Stability

167

the light frequency ω, and the free spectral range of the microsphere ΔωFSR = √ c/Rs s . At the resonance, ω = mΔωFSR with m ∈ Z, the transmission reaches its minimum  Tmin =

α−t 1 − αt

2 .

(3.171)

In particular, the zero transmission Tmin = 0 is obtained when t = α, i.e., the power of the light in the fiber is completely transferred to the microsphere’s WGM [see Fig. 3.19b]. Indeed, this transmission vanishing, which is conventionally referred to as the critical coupling, results from the destructive interference between the light leaking from the microsphere and the transmitted light in the fiber. The nonzero transmission for t < α (t > α) is called the under-coupling (over-coupling). The coupling coefficient κ is related to the overlap between the fiber mode and the microsphere WGM. A larger overlap (i.e., a smaller fiber-microsphere separation) leads to a higher |κ|. However, as shown in Fig. 3.19c the minimal transmission does not occur at the maximal coupling |κ| = 1. When the straight fiber and the microsphere are far apart, the light power inside the fiber is hardly coupled into the microsphere. As the fiber moves to the microsphere, more power is transferred to the WGM and T becomes smaller until the coupling coefficient reaches the intrinsic attenuation of the light in the microsphere [see (3.96)], i.e., critical coupling. When the fiber-microsphere distance is further reduced, the large coupling coefficient leads to an increased decay of the light in the microsphere and T grows up again. • Further prove the expression of the microsphere’s quality factor √ ω π αt Q= ΔωFSR 1 − αt

(3.172)

by using (3.168).

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3 Whispering Gallery Modes in Optical Microcavities monocrystalline lithium niobate microdisk resonator. Phys. Rev. Lett. 122, 173903 (2019). https://doi.org/10.1103/PhysRevLett.122.173903 R. Wu, M. Wang, J. Xu, J. Qi, W. Chu, Z. Fang, J. Zhang, J. Zhou, L. Qiao, Z. Chai, J. Lin, Y. Cheng, Long low-loss-lithium niobate on insulator waveguides with sub-nanometer surface roughness. Nanomaterials 8, 910 (2018). https://doi.org/10.3390/nano8110910 D.K. Armani, T.J. Kippenberg, S.M. Spillane, K.J. Vahala, Ultra-high-Q toroid microcavity on a chip. Nature 421, 925–928 (2003). https://doi.org/10.1038/nature01371 J. Lin, Y. Xu, J. Tang, N. Wang, J. Song, F. He, W. Fang, Y. Cheng, Fabrication of threedimensional microdisk resonators in calcium fluoride by femtosecond laser micromachining. Appl. Phys. A 116, 2019–2023 (2014) B. Min, L. Yang, K. Vahala, Perturbative analytic theory of an ultrahigh-Q toroidal microcavity. Phys. Rev. A 76, 013823 (2007). https://doi.org/10.1103/PhysRevA.76.013823 S.J. Garth, Modes on a bent optical waveguide. IEEE Proc. J.: Optoelectron. 134, 221–229 (1987). https://doi.org/10.1049/ip-j.1987.0039 A.J. Maker, A.M. Armani, Fabrication of silica ultra high quality factor microresonators. J. Vis. Exp. 65, e4164 (2012). https://doi.org/10.3791/4164 T.J. Kippenberg, S.M. Spillane, K.J. Vahala, Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip. Appl. Phys. Lett. 85, 6113–6115 (2004) F. Monif, S.K. Özdemir, J. Friedlein, L. Yang, Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix. IEEE Photon. Technol. Lett. 25, 1458–1461 (2013). https://doi. org/10.1109/LPT.2013.2266573 A.M. Armani, A. Srinivasan, K.J. Vahala, Soft lithographic fabrication of high Q polymer microcavity arrays. Nano Lett. 7, 1823–1826 (2007). https://doi.org/10.1021/nl0708359 M.L. Gorodetsky, I.S. Grudinin, Fundamental thermal fluctuations in microspheres. J. Opt. Soc. Am. B 21, 697–705 (2004). https://doi.org/10.1364/JOSAB.21.000697 O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, T.J. Kippenberg, Cryogenic properties of optomechanical silica microcavities. Phys. Rev. A 80, 021803 (2009). https://doi.org/10.1103/ PhysRevA.80.021803 G. Huang, E. Lucas, J. Liu, A.S. Raja, G. Lihachev, M.L. Gorodetsky, N.J. Engelsen, T.J. Kippenberg, Thermorefractive noise in silicon-nitride microresonators. Phys. Rev. A 99, 061801 (2019). https://doi.org/10.1103/PhysRevA.99.061801 J.M. Dobrindt, Bio-sensing using toroidal microresonators and theoretical cavity optomechanics. Ph.D. thesis, Ludwig Maximilians Universität, Munchen (2012) F. Riehle, Frequency Standards: Basics and Applications (Wiley-VCH, Weinheim, 2006) M. Cai, O. Painter, K.J. Vahala, Observation of critical coupling in a fiber taper to a silicamicrosphere whispering-gallery mode system. Phys. Rev. Lett. 85, 74–77 (2000). https://doi. org/10.1103/PhysRevLett.85.74

Chapter 4

Applications of WGM Microcavities in Physics

Abstract Various applications have been explored for the whispering-gallery-mode microcavities. The microcavity-based laser generation and nonlinear optics potentially allow a micron-sized light source that may be used in a photonic integrated circuit. The miniaturized optical frequency combs serve as a bridge between the radio and optical domains. Besides, the microcavities can be utilized to study the fundamental problems in physics, for instance, the parity and time-reversal symmetric non-Hermitian quantum mechanics. The applications of microcavities in optomechanics and nanoparticle trapping are also introduced.

4.1 Introduction Owing to their high-quality factors and tiny mode volumes, the WGM microcavities have been widely applied to study the fundamental researches in physics. The laser emission and nonlinear optics based on the WGM microcavities exhibit the low thresholds and narrow spectral linewidths, possessing the potential uses in the photonic integrated circuits as the light sources. The interacting WGM microcavities have become a common platform for investigating the quantum mechanical problems of the PT -invariant systems. The size of the optical frequency combs can be significantly shrunk by using the WGM microcavities. Such transportable devices much facilitate the comparison between microwave and optical frequency standards. The WGM microcavities have also excited a new research field, cavity optomechanics. In this chapter, we deliberately focus on the classical mathematical treatment of the optical processes so as to deliver an easily understandable and accessible description of the relevant applications of the WGM microcavities.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2_4

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4.2 Laser Generation The unique features of the WGM microcavites, like the long lifetime of the intracavity photons and the tiny volume of the cavity modes, make them the excellent candidates for constructing a microlaser with a low threshold and a narrow spectral linewidth. The microcavities can be made from water droplets, silica, semiconductors, and polymers due to the tremendous progress in microfluidic and micro- and nanofabrication technologies. The gain materials include organic dyes, quantum dots, and rare-earth ions with the achieved laser wavelengths ranging from ultraviolet to infrared. Lasing droplets. The laser emission from the individual liquid droplets was firstly reported in [1]. The confinement of the light waves within the droplets results from the total internal reflection at the liquid–air interface. The strong surface tension shapes the droplets into the spheres with a highly smooth boundary. Figure 4.1a depicts the typical scheme of the microfluidic droplet lasers. The pump light irradiates a linear stream of the highly monodisperse dye-doped droplets that fall freely under the gravity. The lasing occurs when the round-trip gain of the light exceeds its roundtrip loss. In addition, the robust laser emission depends less on the sizes and shapes of the droplets [2]. Generally, the emitted coherent photons are accompanied by the fluorescence. Figure 4.1b illustrates a typical fluorescence spectrum of the dye molecules. Since the fluorescence peak is well separated from that of the absorption, the dye molecules are usually modeled by the four-level lasing configuration. Based on the properties of the laser emission, the fluorescence spectrum may be divided into three regions [1]. Region A contains the fluorescence peak and the absorption tail. In this region, the absorption loss is still (relatively) high, inhibiting the laser emission [see Fig. 4.1c]. By contrast, the absorption becomes low enough in the intermediate Region B that

Fig. 4.1 Droplet lasers. a Schematic diagram of the laser emission from a droplet stream. b Comparison between the absorption and fluorescence spectra of the dye molecules. c Spectra of the laser emission in different regions. Reprinted with permission from [1] ©The Optical Society

4.2 Laser Generation

173

Fig. 4.2 Energy-level diagrams of the laser transitions of erbium (Er3+ ), ytterbium (Yb3+ ), neodymium (Nd3+ ), and thulium (Tm3+ ) ions

several emission spikes, which are much higher than the fluorescence background, appear, denoting the onset of laser oscillation. In Region C, the absorption decays to a substantially low value and the emission spikes are distributed throughout the entire region. Besides vertically falling, the droplets can be also suspended by the tip of a thin wire [3] or move in a microchannel by means of the microfluidic technology [4]. In addition, a quasi-droplet laser may be achieved by using a microbubble filled with the liquid gain medium [5]. These approaches offer more flexibility in manipulating the droplets and light coupling. Silica glass. One of the common materials used in fabricating the solid-state microcavities is the silica glass. The laser emission can be implemented through doping the rare-earth ions into the microcavities or coating the microcavities with the gain media. Figure 4.2 displays the simplified energy-level diagrams of several ions that are usually employed as the active emitters in the silica microcavities. In [6], a neodymium-doped silica microsphere (diameter of 50 ∼ 80 µm), pumped by a light beam at 807 nm, supports the multimode (60 ∼ 80 modes) laser emission around 1080 nm. The lasing modes are equally spaced by the free spectral range. The high cold-cavity quality factor Q = 2 × 108 suppresses the lasing threshold as low as 200 nW, which is over three orders of magnitude smaller than that of the conventional neodymium-doped fiber lasers. Instead of shaping the active-ion-doped bulk samples into the microcavities, it is much easier to coat the ready-made undoped microcavities with an active layer. In [7], the lasing action around 1535 nm occurs in a silica microsphere coated with an erbium-doped sol-gel film (thickness of 1 µm). The microsphere is pumped by a light beam at 980 nm and the threshold is about 28 µW. Varying the doping concentration

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4 Applications of WGM Microcavities in Physics

of the active ions and the thickness of the sol-gel layer, the laser emission can operate in either continuous-wave or pulse mode. In addition, the microlasers doped with the ytterbium (pumped at 970 nm and lasing at 1040 nm), thulium (pumped at 1554 nm and lasing at 1975 nm; pumped at 1064 nm and lasing at 450, 461, 784, 802, and 816 nm), and HgTe quantum dots (pumped at 830 nm and lasing within the range from 1240 to 1780 nm) have been demonstrated in [8–11]. Semiconductors. The WGM microcavities made from the semiconductors are characterized by the large refractive index contrast between the cavity materials and their surroundings. For instance, the refractive index of InP/InGaAsP is as high as n = 3.5 in comparison to SiO2 with n = 1.5. This results in a strong suppression of the cavity size down to a few μm. Generally, the microcavities are in the disk shape with a diameter ∼1 µm and thickness ∼0.1 µm. The surface and sidewall roughness of the microdisks limits the cavity Q to 103 ∼ 104 , resulting in a relatively high threshold. In [12], the single-mode lasing, where the gain is provided by the optically pumped quantum wells, was achieved at 1.3 and 1.5 µm with a threshold below 100 µW. A quantum-dot-embedded microdisk laser was reported in [13]. The pump beam at 633 nm comes from a He–Ne laser and the lasing emission is centered at 962.7, 970.2, 992.1, and 1002.7 nm. A laser emission at 413.9 nm was demonstrated in [14], where the microdisk was fabricated via the band-gap-selective photoelectrochemical etching. In addition, [15] reported a room-temperature lasing in semiconductor microdisks. An electrically driven microlaser integrated on silicon was realized in [16]. Such a compact light source may be a potential key component for the large-scale integration of the electronic and photonic circuits. Polymers. The polymer microcavities have the advantages of low fabrication cost, fairly easy processing, and structural flexibility. The active organic dyes and quantum dots can be directly dissolved in the polymers or coated on the polymer microcavities. The laser emission in the dye-doped polymer microdisks (fabricated by the standard photolithographic techniques) and microrings (formed by the surface tension of the droplets encircling an optical fiber) was reported in [17]. The lasing based on the conical microcavities was demonstrated in [18], where the pump beam is at 532 nm and the lasing wavelength is around 600 nm.

4.3 Nonlinear Frequency Conversion In addition to the laser generation, the WGM microcavities have also been extensively used in the nonlinear optics. We discuss several selected multiwave-mixing processes in this section. As we will see below, the nonlinear frequency conversion benefits from the high Q factor and small mode volume of the WGM microcavities. Let us consider an intense light beam traveling in a nonlinear material. The electric field E (typically ∼108 V/m) can induce a non-negligible change in the refractive index of the material. This in turn changes the properties of the light, such as frequency/wavelength, polarization, intensity, and propagation direction. The material’s dielectric polarization density may be generally expressed as a power series in E [19]

4.3 Nonlinear Frequency Conversion

P=

 n

175

  . P(n) = ε0 χ (1) · E + χ (2) : EE + χ (3) .. EEE + · · · ,

(4.1)

with the n-th order electric susceptibility χ (n) . Indeed, χ (n) is a rank-(n + 1) tensor describing the material’s symmetry (e.g., χ (2) = 0 for a material with the inversion symmetry) and the polarization-dependent nonlinear interaction. The n-th order polarization, for instance, P(2) is given by the inner (double-dot “:”) product operation between χ (2) and the dyadic EE. The i-th (i = x, y, z) component of the dielectric polarization P reads 

Pi(n)    (2) = ε0 χi(1) E + χ E E + j j k j i jk

Pi =

n

j

jk

jkl

 χi(3) E E E + · · · . (4.2) j k l jkl

We write the electric field E j along the e j direction as the superposition of a number of discrete frequency components Ej =

1 E j ( p j ωm j )ei p j ωm j t , p j =±,m j 2

(4.3)

where we have used the fact E j (−ωm j ) = E ∗j (ωm j ). Similarly, we write the n-th order polarization Pi(n) in the e j axis as Pi(n) =

1 P (n) ( pi ωm i )ei pi ωmi t . pi ,m i i 2

(4.4)

Inserting (4.3) and (4.4) into (4.2), one obtains the amplitude Pi(n) at the frequency pi ωm i Pi(n) ( pi ωm i ) =

ε0  χ (n) ( pi ωm i ; p j ωm j , pk ωm k , · · · ) ( j, p j ,m j ),(k, pk ,m k ),··· i jk··· 2n−1 ×E j ( p j ωm j )Ek ( pk ωm k ) · · · , (4.5)

where the condition of energy conservation requires pi ωm i = p j ωm j + pk ωm k + · · · .

(4.6)

In addition, the law of conservation of momentum leads to an extra elementary requirement, i.e., the phase matching, for the nonlinear coupling among optical waves. In what follows we briefly introduce several common nonlinear optical phenomena that have been observed based on the WGM microcavities. Due to their long temporal coherence and tight spatial light confinement, the WGM microcavities are an excellent platform for investigating the nonlinear optics.

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4 Applications of WGM Microcavities in Physics

4.3.1 Pockels Effect The most basic nonlinear optical process is the Pockels effect, also known as the linear electro-optic effect. It is related to the following second-order nonlinear susceptibility: Pi(2) (ω) =

 ε0   (2) χi jk (ω; ω, 0)E j (ω) E¯ k + χi(2) (ω; 0, ω) E¯ j Ek (ω) . jk jk 2

(4.7)

Here we have replaced Ek (0) with E¯ k to distinguish the static amplitudes Ek (0) from the optically oscillating amplitudes E j (ω). That is, the material’s nonlinearity is induced by exerting an external static (or slowly varying) electric field. In general, (2) one has the symmetry χi(2) jk (ω; ω, 0) = χik j (ω; 0, ω). Comparing the expressions of the electric displacement components Di (ω) = ε0

 j

i j E j (ω), Di (ω) = ε0 Ei (ω) + Pi(1) (ω) + Pi(2) (ω),

(4.8)

one can derive the material’s relative permittivity  =  L +  N L , which is a secondrank tensor represented by a 3 × 3 matrix. The linear  L and nonlinear  N L parts have the components NL iLj = δi j + χi(1) j (ω), i j =

 k

¯ χi(2) jk (ω; ω, 0) E k .

(4.9)

We assume the material is lossless and non-optically active. Thus, i j is real i∗j = i j and symmetric i j =  ji . The energy density of the in-medium light is given by W =

1  −1 ( )i j Di (ω)D j (ω), ij 2ε0

(4.10)

where the reciprocal  −1 is also real ( −1 )i∗j = ( −1 )i j and symmetric ( −1 )i j = ( −1 ) ji . Defining (x, y, z) = √

1 (Dx (ω), D y (ω), Dx (ω)), 2ε0 W

(4.11)

one obtains the so-called optical indicatrix (or index ellipsoid) ( −1 )x x x 2 + ( −1 ) yy y 2 + ( −1 )zz z 2 +2( −1 )x y x y + 2( −1 ) yz yz + 2( −1 )x z x z = 1.

(4.12)

We further assume that the reciprocal  −1 can be expanded in powers of E¯ k and have ( −1 )i j ≈ [( L )−1 ]i j +

 k

ri jk E¯ k ,

(4.13)

4.3 Nonlinear Frequency Conversion

177

with the Pockels electro-optic tensor ri jk . Due to the symmetry of  −1 , the thirdrank tensor ri jk may be reduced to a two-dimensional matrix rαk by using the Voigt’s contracted notation (i j) → α with (x x) → 1, (yy) → 2, (zz) → 3, (yz or zy) → 4, (x z or zx) → 5, and (x y or yx) → 6. In addition, the linear permittivity  L can be always transformed into a 3 × 3 diagonal matrix by choosing the appropriate coordinates. In this coordinate system, the reciprocal  −1 is simplified as −2 −2 ( −1 )α ≈ n −2 x δα,1 + n y δα,2 + n z δα,3 +

 k

rαk E¯ k ,

(4.14)

1/2

with the refractive index, for example, n x = ( L )x x for the light polarization in the x-axis. The two-dimensional matrix rαk has 18 elements. Most of them vanish due to the rotational symmetry properties of a specific material. For instance, the potassium dihydrogen phosphate (KDP) has three nonzero electro-optic coefficients r4x = r5y = 8.77 × 10−12 m/V and r6z = 10.5 × 10−12 m/V. The corresponding optical indicatrix is given by y2 z2 x2 + 2 + 2 + 2r4x E¯ x yz + 2r5y E¯ y x z + 2r6z E¯ z x y = 1, 2 no no ne

(4.15)

with the ordinary n o = 1.514 and extraordinary n e = 1.472 refractive indices at 0.5461 µm. For an applied static electric field along the z-direction, E¯ x = E¯ y = 0 and E¯ z = 0, the above equation is re-expressed as 

1 + r6z E¯ z n 2o



 X2 +

 1 Z2 2 ¯ − r E 6z z Y + 2 = 1, 2 no ne

(4.16)

by using the transformation x−y x+y X = √ , Y = √ , Z = z. 2 2

(4.17)

In the limit |r6z E¯ z |  1, the optical indicatrix is further simplified as Y2 Z2 X2 + + = 1, n 2e n 2X n 2Y

(4.18)

with the new refractive indices along the X - and Y -axes 1 1 n X = n o − n 3o r6z E¯ z , n Y = n o + n 3o r6z E¯ z , 2 2

(4.19)

respectively. As a result, the presence of a static (or low-frequency) electric field varies the refractive indices of a nonlinear material in different directions.

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Fig. 4.3 Pockels effect. a Coupling between a lithium niobate microring and a waveguide. b Cross section of a microring cavity structure. c Electro-optic shift of a TM-polarization (perpendicular to the film) cavity mode around 1555 nm after applying a voltage of 100 V. d SEM image of a fabricated racetrack. e Cross-sectional view of a radio-frequency electrical field modulating the racetrack. f Eye diagram of the racetrack cavity with a data transmission rate at 22 Gbps. Figures a and c are reprinted with permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature, [20], © 2007. Figures d–f are reprinted with permission from [21] ©The Optical Society

Another common electro-optic material is the lithium niobate (LiNbO3 ). It possesses the large Pockels coefficients r3z = 30.9 × 10−12 m/V and r1z = 9.6 × 10−12 m/V (for an applied static electric field along the z-axis) and a wide transparency range from 350 nm to 5.2 µm. The ordinary and extraordinary refractive indices of LiNbO3 at 1550 nm are n o = 2.211 and n e = 2.138, respectively. The Pockels effect based on LiNbO3 has been demonstrated in the optical microcavities. In [20], a microring is coupled to a waveguide [see Fig. 4.3a]. The whole microstructure consists of a LiNbO3 thin film covered by a SiO2 layer and bonded to a LiNbO3 wafer through a benzocyclobutene (BCB) layer as shown in Fig. 4.3b. As a substrate, the low refractive index of BCB (∼1.55) provides a suitable optical confinement. Applying a static electric voltage along the z-axis of LiNbO3 changes the refractive index of the intracavity medium, resulting in a wavelength/frequency shift of the cavity modes. As exhibited in Fig. 4.3c, an applied voltage of 100 V induces a shift of Δλ = 105 pm of the cavity resonance mode with a frequency tunability of 0.14 GHz/V. In addition, an electro-optic modulator with a data transmission rate up to 40 Gbps [see Fig. 4.3d–f] was reported in [21]. The Pockels effect can be utilized in the information transfer between individuals in networks. The information is firstly encoded in the microwave electrical signals and then up-converted to the optical carriers via the microwave-modulated cavity modes. Such an information-transmission mechanism has the advantages of low loss

4.3 Nonlinear Frequency Conversion

179

and high signal-to-noise ratio (the energies of microwave and optical photons differ by five orders of magnitude).

4.3.2 Multiple-Wave Mixing We now consider the nonlinear interaction among multiple optical waves. Splitting the polarization P into the linear P L = P(1) and nonlinear P N L parts, the wave equation (1.40) for the light traveling in a nonlinear medium is rewritten as ∇ 2 E(r, t) −

 ∂2  L 1 ∂2 P (r, t) + P N L (r, t) . E(r, t) = μ 0 c2 ∂t 2 ∂t 2

(4.20)

The linear part of the electric displacement vector is then given by D L (r, t) = ε0 E(r, t) + P L (r, t), and (4.20) is re-expressed as ∇ 2 E(r, t) −

1 ∂2 L 1 ∂2 N L D (r, t) = P (r, t). ε0 c2 ∂t 2 ε0 c2 ∂t 2

(4.21)

The above equation denotes that the nonlinear polarization P N L on the right-hand side acts as a driving source. We decompose the light field into the components with discrete frequencies {ωn ; n = 1, 2, 3, · · · }. Thus, E(r, t), D L (r, t) and P N L (r, t) can be written in the superposition forms 1 ∗ 1 En (r)e−iωn t + E (r)eiωn t , n n n 2 2 1 1   L ∗ iωn t D L (r, t) = Dn (r) e , DnL (r)e−iωn t + n n 2 2 1 1   N L ∗ iωn t P N L (r, t) = Pn (r) e . PnN L (r)e−iωn t + n n 2 2 E(r, t) =

(4.22a) (4.22b) (4.22c)

Here the amplitude vectors En (r), DnL (r), and PnN L (r) have been assumed to be time independent (or slowly varying). Substituting (4.22) into (4.21), one arrives at ∇ 2 En (r) +

ωn2 L ωn2 N L  (ω ) · E (r) = − P (r), n n c2 ε0 c2 n

(4.23)

with the linear relative permittivity  L (ωn ) = 1 + χ (1) (ωn ). We restrict ourselves to the scalar case, i.e., En → En ,  L →  L , and PnN L → NL Pn . The propagation direction of the light is in the z-axis. Thus, the ampliikn z with the in-medium wavenumber kn = tude En (r) has the form En (r) = An (z)e L  (ωn )ωn /c. Under the slowly varying envelope approximation [see (1.1)], the wave equation (4.23) is simplified as

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4 Applications of WGM Microcavities in Physics

∂ ωn2 An (z) = i P N L (z)e−ikn z . ∂z 2kn ε0 c2 n

(4.24)

Due to the nonlinear polarization PnN L , the energy interchange occurs among different frequency components as the light travels in medium. In deriving (4.24), the lossless material has been assumed.

4.3.3 Second-Order Nonlinearity In the limit where the second-order nonlinearity plays the main role, PnN L is reduced to

(2) ε0 

χ (ωn ; ωm 1 , ωm 2 )Am 1 (z)Am 2 (z) PnN L (z)e−ikn z = m ,m 1 2 2 ×ei(km1 +km2 −kn )z + χ (2) (ωn ; ωm 1 , −ωm 2 )Am 1 (z)A∗m 2 (z) ×ei(km1 −km2 −kn )z + χ (2) (ωn ; −ωm 1 , ωm 2 )A∗m 1 (z)Am 2 (z)  ×ei(−km1 +km2 −kn )z . (4.25) Here m 1 ,m 2 denotes the summation of the terms with the positive frequency ωn . It has been proven that the second-order nonlinear susceptibility χ (2) vanishes for a centrosymmetric (also known as the inversion symmetry) medium. The nonlinear polarization (4.25) drives both sum-frequency and frequency-difference generation processes. As an example, we focus on the three-wave mixing, (ω1 , ω2 , ω3 ) with ω3 = ω1 + ω2 , and assume the full permutation symmetry of the nonlinear material, χ (2) = χ (2) (ω1 ; ω3 , −ω2 ) = χ (2) (ω2 ; ω3 , −ω1 ) = χ (2) (ω3 ; ω1 , ω2 ). From (4.24), one obtains the following coupled-amplitude equations: ∂ χ (2) ω1 A1 (z) = i A3 (z)A∗2 (z)e−iΔkz , ∂z 2c  L (ω1 )

(4.26a)

∂ χ (2) ω2 A3 (z)A∗1 (z)e−iΔkz , A2 (z) = i ∂z 2c  L (ω2 )

(4.26b)

∂ χ (2) ω3 A3 (z) = i A1 (z)A2 (z)eiΔkz , ∂z 2c  L (ω3 )

(4.26c)

with the wavevector (or momentum) mismatch Δk = k1 + k2 − k3 . We further apply the assumption that the conversion of the input waves at ω1 and ω2 into the sumfrequency ω3 wave is weak. Thus, the amplitudes of the input waves can be approximated to be independent of z, A1 (z) ≈ A1 and A2 (z) ≈ A2 . We use L to denote the length of the nonlinear medium. Integrating (4.26c) leads to the amplitude of the ω3 wave at z = L

4.3 Nonlinear Frequency Conversion

181

Fig. 4.4 Three-wave mixing. a Sum-frequency generation. b Energy-level diagram. c Effect of the wavevector mismatch

A3 (L) =

eiΔk L − 1 χ (2) ω3 A1 A2 , Δk 2c  L (ω3 )

(4.27)

where the amplitude of the ω3 wave at z = 0 is set to be zero. Using the expression of the intensity of the ωn wave, In =  L (ωn )cε0 |An |2 /2, we thus arrive at I3 (L) =

I1 I2 Δk L (χ (2) )2 ω32 . L 2 sinc2 L L L 2ε0 c3 2  (ω1 ) (ω2 ) (ω3 )

(4.28)

Increasing either I1,2 enhances I3 . The effect of the wavevector mismatch Δk is reflected by the sinc function. The three-wave mixing is optimized when Δk = 0, i.e., phase matching. As Δk L is increased, the efficiency of the energy flowing from the ω1 and ω2 waves into the ω3 wave goes down with an oscillating behavior [see Fig. 4.4c]. Usually, the phase matching Δk = 0 can be achieved through exploiting the birefringence. In the type-I phase matching, the ω1 and ω2 waves have the same polarization, which is perpendicular to that of the ω3 wave. In contrast, the type-II phase matching corresponds to the case where the polarizations of the ω1 and ω2 waves are orthogonal. Cavity modes. Let us now consider the multiwave mixing of optical cavity modes {Ψn (r); n = 1, 2, · · · } with the corresponding eigenvalue equation (∇ 2 + kn2 )Ψn (r) = 0. The scalar electric field E, linear displacement D L , and nonlinear polarization P N L take the form 1 an (t)Ψn (r)e−iωn t + c.c., n 2 1 D L (r, t) = ε0  L (ωn )an (t)Ψn (r)e−iωn t + c.c., n 2 1  NL P N L (r, t) = P (r, t)e−iωn t + c.c., n n 2 E(r, t) =

(4.29a) (4.29b) (4.29c)

with the complex mode amplitudes an (t). Again, we assume the second-order nonlinearity takes the main effect. Then, PnN L (r, t) is written as

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4 Applications of WGM Microcavities in Physics

PnN L (r, t) =

(2) ε0 

χ (ωn ; ωm 1 , ωm 2 )am 1 (t)am 2 (t)Ψm 1 (r)Ψm 2 (r) m 1 ,m 2 2 ×e−i(ωm1 +ωm2 )t + χ (2) (ωn ; ωm 1 , −ωm 2 )am 1 (t)am∗ 2 (t) ×Ψm 1 (r)Ψm∗2 (r)e−i(ωm1 −ωm2 )t + χ (2) (ωn ; −ωm 1 , ωm 2 )  ×am∗ 1 (t)am 2 (t)Ψm∗1 (r)Ψm 2 (r)e−i(−ωm1 +ωm2 )t .

(4.30)

The intracavity energy of the Ψn (r) mode is given by Wn (t) =

ε0 L  (ωn )|an (t)|2 2

|Ψn (r)|2 dr.

(4.31)

Introducing the amplitudes  bn (t) =

ε0 L  (ωn ) 2

1/2

|Ψn (r)| dr 2

an (t),

(4.32)

and using (4.21) and (4.29)–(4.32), one can derive the coupled wave equations for the three-cavity-mode mixing  ω1 iω1 ∗ ∂ b1 (t) = − b1 (t) + γ b3 (t)b2∗ (t) + i κ1 P1in , ∂z 2Q 1 ω3  ∂ ω2 iω2 ∗ b2 (t) = − b2 (t) + γ b3 (t)b1∗ (t) + i κ2 P2in , ∂z 2Q 2 ω3  ∂ ω3 b3 (t) + iγ b1 (t)b2 (t) + i κ3 P3in . b3 (t) = − ∂z 2Q 3

(4.33a) (4.33b) (4.33c)

Note that we have assumed the full permutation symmetry of the nonlinear material and defined the intermode coupling strength as  Ψ1 (r)Ψ2 (r)Ψ3∗ (r)dr ω3 χ (2) γ =√  1/2 ,  ε0 /2 3 L (ω ) |Ψ (r)|2 dr  n n n=1

(4.34)

which is directly linked to the spatial overlapping of three modes. We have also inserted the cavity loss terms with the quality factors Q 1,2,3 and the input fields with in and the coupling strengths κ1,2,3 . the input powers P1,2,3 For the WGMs with large l and small (l − m), the electric fields are mainly distributed around the microcavity’s equator. One may reduce the cavity modes to the scalar eigenfunctions Ψα=1,2,3 (r) = z n α ,lα (r )Ylmα α (θ, ϕ). As illustrated by (4.34), the mode coupling strength γ is proportional to the following overlap factor of the angular parts:

4.3 Nonlinear Frequency Conversion

F=

π

183

2π

Ylm1 1 (θ, ϕ)Ylm2 2 (θ, ϕ)[Ylm3 3 (θ, ϕ)]∗ sin θ dθ dϕ 0 0  3    l1 l2 l3 α=1 (2l α + 1) l 1 l 2 l 3 m3 . = (−1) 0 0 0 m 1 m 2 −m 3 4π

(4.35)

We have employed the Wigner 3 j symbols in the above equation. The selection rules for the 3 j symbols demand the conservation of the azimuthal component of the orbital momentum, m 1 + m 2 = m 3 , the triangular inequality |l1 − l2 | ≤ l3 ≤ (l1 + l2 ), and the conservation of parity, l1 + l2 + l3 = even.

4.3.4 Frequency Doubling In the special case, the ω1 and ω2 waves have the same frequency ω p ≡ ω1 = ω2 . The input power Ppin ≡ P1in is nonzero while P3in of the ωs ≡ ω3 = 2ω p wave is zero. Thus, the nonlinear medium transfers the energy from the pump wave at ω p into the signal wave at ωs . Such a nonlinear wave-mixing process is known as the secondharmonic generation (SHG). Figure 4.5a presents the dependence of the overlap factor F of the angular parts [see (4.35)] on νs = ls − m s and ν p = l p − m p . We conclude: (1) F is maximized at νs = ν p = 0. When ls = m s and l p = m p , both signal and pump modes have a single antinode at θ = π/2, giving the best coupling; (2) F is zero when νs = odd because Ψs (r) has an odd parity in the polar direction while |Ψ p (r)|2 is an even function; and (3) F is zero when 2ν p < νs since it breaks the triangular inequality of the 3 j symbols. Setting t → ∞, (4.33) leads to the following equations for the steady-state solu≡ b p (t → ∞): tions bs(ss) ≡ bs (t → ∞) and b(ss) p 2  ωs (ss) bs = iγ b(ss) , p 2Q s  ω p (ss) iγ ∗ (ss)  (ss) ∗ bs bp bp = + i κ p Ppin . 2Q p 2

(4.36a) (4.36b)

By introducing the dimensionless variables x=

|γ |2 2



2Q p ωp

3

2Q s |bs(ss) | κ p Ppin , y = , 2Q p  ωs κ p Ppin ωp

(4.37)

Equation (4.36) is reduced to the following simple form: y(1 + x y 2 ) = 1.

(4.38)

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4 Applications of WGM Microcavities in Physics

The variable x is proportional to the input pump power Ppin while y is a function of x. We then introduce the SHG efficiency [22] η=

κ p Q p κs Q s κs |bs(ss) |2 = 4x y 4 , in 2Pp ω p ωs

(4.39)

i.e., the ratio of the signal power coupled outside the cavity (with a coupling strength κs ) to the pump power coupled into the cavity. The efficiency η satisfies η ≤ 1. The function 4x y 4 may vary independently of the rest part by tuning Ppin . As shown in Fig. 4.5b, 4x y 4 goes up as x is increased and reaches its maximum value max(4x y 4 ) = 1 at the saturation input xsat = 4, where the value of the corresponding pump power Ppin is    −1 |γ |2 2Q s 2Q p 3 (4.40) Psat = κ p 8 ωs ωp with ysat = 1/2. Then, 4x y 4 decays gradually when x > 4. Increasing Q s, p suppresses the saturation input power. The decay rate (ωs /Q s ) of the signal mode can be divided into the intrinsic loss κs0 and the coupling loss κs , i.e., ωs /Q s = κs0 + κs . Similarly, we have ω p /Q p = κ p0 + κ p . Figure 4.5c depicts the dependence of the maximum ηmax of the efficiency η with 4x y 4 = 1 on the ratios r p = κ p /κ p0 and rs = κs /κs0 . One finds that ηmax approaches the unity when r p,s  1 while ηmax → 0 as r p,s → 0. The WGM-based SHG has been demonstrated in various experiments. In [23], a disk-shaped (radius of 1.5 mm and thickness of 0.5 mm) WGM cavity is made of periodically poled z-cut LiNbO3 , where the TE modes correspond to the extraordinary waves. The periodic poling is a domain-reversal technique for obtaining a quasi-phase matching (sometimes called type-0 phase matching) of the nonlinear inter-optical-wave interaction. The TE pump beam at 1.55 µm with the corresponding  L (ω p ) = (2.138)2 is coupled into the cavity via a diamond prism. More than a half of the pump power enters the cavity. The frequency-doubled wave at 775 nm has  L (ωs ) = (2.179)2 . Note that under the quasi-phase matching, the birefringence is no longer needed for achieving the phase matching. Thus, the pump and signal waves have the equal polarization state. The pump-beam absorption and SHG emission spectra are presented in Fig. 4.5d. In [24], a SHG from 1064 to 532 nm was performed. The WGM cavity is made from a MgO-doped LiNbO3 wafer. The cavity has a disk structure whose rim can be approximated as a spheroid. Again, the diamond-prism coupler is adopted. The type-I phase matching is employed. The pump wave from a Nd:YAG laser is horizontally polarized and couples to the ordinary mode of the cavity with  L (ω p ) = (2.232)2 . In contrast, the signal wave corresponds to the extraordinary mode with  L (ωs ) = (2.234)2 and is vertically polarized. Figure 4.5e displays the measurement results of the pump-beam absorption and SHG emission spectra. Figure 4.5f exhibits a typical curve of the conversion efficiency versus the input

4.3 Nonlinear Frequency Conversion

185

Fig. 4.5 Second harmonic generation. a Overlap factor F of the angular parts. b y versus x and x y 4 versus x derived from (4.38). c Maximum coupling efficiency ηmax as a function of the parameters rs, p . d Pump transmission and signal emission spectra measured in [23]. e Transmitted pump power and the second harmonic signal. f Coversion efficiency Psout /Ppin (symbol “+”) as a function of (ss)

the input power Ppin with a saturation power Pth = 0.3 W. Here Psout = κs |bs |2 is the output power of the signal light. The solid curve is the theoretical prediction. Figures d and f are reprinted with permission from [23]. Copyright (2004) by the American Physical Society. e is reprinted with permission from [24]. Copyright (2010) by the American Physical Society

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4 Applications of WGM Microcavities in Physics

power measured in experiment [23]. It is seen that the conversion efficiency reaches its saturation at an in-coupled pump power much lower than that of the theoretical prediction. This may be attributed to a nonlinear “self-limiting” effect [24]. Another SHG from 1540 to 770 nm was reported in [25]. As we have pointed out, the second-order nonlinear inter-optical-wave interaction cannot occur in the centrosymmetric media. Hence, the inversion symmetry rules out the SHG in silicon (Si) and silica (SiO2 ) crystals. However, such an intrinsic symmetry in a bulk medium may be broken at the surface. A SHG from 1555.14 to 777.75 nm was demonstrated based on an ultrahigh-Q silica microcavity in [26]. The second-harmonic light arises from the symmetry-breaking-induced nonlinear mechanism combined with the electric multipole response in the bulk. The phase matching is implemented via utilizing the cavity-enhanced thermal and optical Kerr effects. As a result, it achieves an unprecedented conversion efficiency of 0.049% W−1 .

4.3.5 Parametric Down-Conversion Another interesting second-order optical nonlinear process is the parametric downconversion (PDC), where the nonlinear crystal converts one pump photon at ω p into one signal photon at ωs plus one idler photon at ωi . The conservation of energy requires ω p = ωs + ωi [see Fig. 4.6a], and the phase matching demands the wavevectors fulfill k p = ks + ki . By convention, we set ωs ≥ ωi . The degenerate PDC corresponds to the case of ωs = ωi . Placing the nonlinear crystal inside an optical cavity that is resonant to all three waves, the optical oscillation occurs when the parametric down-conversion gain exceeds the round-trip losses. Such a device is referred to as the optical parametric oscillator (OPO). Equations (4.29)–(4.35) are still valid to the OPO, where the ω1 , ω2 , and ω3 waves correspond to the signal, idler and pump waves, respectively. Again, the overlap factor F of the angular parts is maximized for all equatorial WGM’s coupling, i.e., lα= p,s,i = m α . Solving (4.33) with Psin = Piin = 0 and Ppin = 0, one obtains the steady-state solutions of the OPO (ss) 2 | = κp |bα=s,i

  ωα 2Q α 2Q p Pth Ppin /Pth − 1 , ω p ωα ω p

(4.41)

with the threshold input pump power  Pth =

ωs ωi κ p |γ |2 2 ωp

2Q s 2Q i ωs ωi



2Q p ωp

2 −1 .

(4.42)

Obviously, the intracavity signal (∝ |bs(ss) |2 ) and idler (∝ |bi(ss) |2 ) energies have the same threshold. In addition, the signal and idler waves do not grow linearly but

4.3 Nonlinear Frequency Conversion

187

follow the square-root law as the input pump power increases. We also find that Pth is related to the saturation input pump power Psat of the SHG [see (4.40)] via Pth =

Q p Psat . Qs 4

(4.43)

The efficiency of the signal/idler generation is defined as the ratio of the signal/idler output (with the coupling strength κα=s,i ) to the input pump power  ηα=s,i =

κα |bα(ss) |2 Ppin

=

ωα κα Q α κ p Q p ω p ωα ωp

4

 Ppin /Pth − 1 Ppin /Pth

.

(4.44)

It is easily to examine that both efficiencies reach their maxima max ηα=s,i =

ωα κ α Q α κ p Q p , ω p ωα ω p

(4.45)

when Ppin = 4Pth . Again, one may define rα=s,i, p = κα /κα0 with the intrinsic loss max on rα and r p is similar to rates κα0 of the cavity modes. The dependency of ηα=s,i Fig. 4.5c. The PDC based on the WGM microcavities with a triply resonant configuration has been demonstrated in [27]. The experimental setup, which consists of a polished LiNbO3 disk coupled with a diamond prism, is schematically shown in Fig. 4.6d. The pump beam comes from a frequency-doubled Nd:YAG laser beam at 532 nm. The coupling strength κ p is tuned by adjusting the prism-cavity distance. The type-I phase matching is adopted, where the pump WGM has an extraordinary polarization while the nondegenerate signal and idler WGMs near 1064 nm have an ordinary polarization. The achieved PDC presents a threshold as low as 6.7 µW [see Fig. 4.6e]. The spontaneous PDC can be used to produce the pairs of single photons by setting the pump power Ppin below the threshold Pth . Detecting one photon from a generated photon pair unambiguously heralds the other, i.e., knowing its state (e.g., polarization and wavelength) prior to the detection of this photon. Such a measurement based on the WGM-microcavity OPO has been reported in [28], where the single-photon generation was verified by the second-order correlation function [see Fig. 4.6f]. This heralded single-photon source possesses an enormous potential in the field of linearoptical-based quantum computation. Additionally, the subharmonic light output from the degenerate OPO owns an intensity noise lower than the photon counting noise, beating the shot-noise limit in quantum metrology and quantum communication. In [29], a 1.4 dB noise reduction below the shot-noise level has been achieved [see Fig. 4.6g].

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4 Applications of WGM Microcavities in Physics

Fig. 4.6 Parametric down-conversion. a Energy-level diagram. b Schematic diagram of the optical parametric oscillator. c Overlap factor |F| with m s = m i = 102 and the equatorial pump l p = m p . d Experimental setup. e Measurement results for a critically (inset) and weakly coupled cavity. Figures d and e are reprinted with permission from [27]. Copyright (2010) by the American Physical Society. f Second-correlation function g (2) (τ ) of the heralded single-photon generation by using the WGM-based OPO. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Communications, [28], © 2013. g Comparison between the intensity noises of the squeezed and antisqueezed lights. Reprinted with permission from [29] ©The Optical Society

4.3.6 Third-Order Nonlinearity Unlike the second-order nonlinearity, the third-order nonlinear optical interaction, described by the χ (3) term in (4.2), can occur for both centrosymmetric and noncentrosymmetric media. For example, due to the inversion symmetry, the third-order nonlinearity is the lowest-order nonlinearity in silica. The third-order nonlinearity covers a vast and diverse area, including the intensity-dependent refractive index

4.3 Nonlinear Frequency Conversion

189

(i.e., Kerr effect), third-harmonic generation (SHG), and degenerate four-wave mixing (FWM). Optical Kerr effect. A strong light beam E(ω) at ω may cause a variation in medium’s refraction index, which is proportional to the local intensity I of the light. Such a nonlinear behavior is known as AC Kerr effect. From (4.5), one obtains the corresponding nonlinear polarization P N L (ω) =

3ε0 (3) χ E(ω)|E(ω)|2 . 2

(4.46)

Note that χ (3) ≡ χ (3) (ω; ω, ω, −ω) = χ (3) (ω; ω, −ω, ω) = χ (3) (ω; −ω, ω, ω). Comparing the following two expressions of the electric displacement: D(ω) = ε0 eff E(ω), D(ω) = ε0 E(ω) + P L (ω) + P N L (ω), the total refractive index of medium n =

√

n ≈ n 0 + n 2 I with n 0 =

(4.47)

eff is given by

3χ (3) 1 + χ (1) , n 2 = . 4ε0 cn 20

(4.48)

Here n 2 in units of m2 /W is the second-order nonlinear refractive index and may be positive or negative. The typical value of n 2 is of the order of 10−20 ∼ 10−18 m2 /W, for example, n 2 ∼ 2 × 10−20 m2 /W for silica. Note that, the intensity I in (4.48) is defined as I = 21 ε0 cn 0 |E(ω)|2 . Such an intensity-dependent refractive index has been demonstrated in microtoroid [30], microbottle [31], and microsphere [32]. The direct results of the optical Kerr effect are the self-phase and cross-phase modulations, which have a wide application in all-optical signal processing. The Kerr effect may lead to another interesting phenomenon, i.e., optical bistability. Let us consider a light wave 1 1 a(t)Ψ (r)e−iωt + a ∗ (t)Ψ ∗ (r)eiωt , 2 2

E(r, t) =

(4.49)

traveling inside a cavity with the cavity mode Ψ (r) and the amplitude a(t). The positive-frequency nonlinear polarization density is simply written as P N L (r) =

3ε0 (3) χ a(t)|a(t)|2 Ψ (r)|Ψ (r)|2 e−iωt . 2

(4.50)

Making the substitution  b(t) =

1 ε0  L (ω) 2

1/2

|Ψ (r)| dr 2

a(t),

one obtains the differential equation for the mode amplitude b(t)

(4.51)

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4 Applications of WGM Microcavities in Physics

Fig. 4.7 Optical bistability. a Intracavity energy z versus the input power x with the detuning y = 5. b z versus y with x = 10. Solid curves: stable solutions. Dashed curves: unstable solutions

√ ω ∂ b(t) = − b(t) + iγ b(t)|b(t)|2 + i κ P in eiΔt , ∂t 2Q

(4.52)

from the wave equation (4.21). The nonlinear coupling strength γ in the above equation is defined as  |Ψ (r)|4 dr 3ωχ (3) γ =    . 2ε0  L (ω) |Ψ (r)|2 dr 2

(4.53)

In (4.52), we have also inserted the total cavity loss (with the corresponding cavity quality factor Q) and the input term (with the coupling strength κ of the input field into the cavity, the input power P in , and the detuning Δ of the input wave to the iΔt ˜ into (4.52) and setting t → ∞ (i.e., the cavity mode). Substituting b(t) = b(t)e system’s steady state), we arrive at an algebraic equation

 1 + (z − y)2 z = x,

(4.54)

with the definitions  x=

2Q ω

3 γ κ P in , y =

2Q ˜ (ss) 2 2Q Δ, z = γ |b | . ω ω

(4.55)

Here the variables x and y are proportional to the input-field power P in and the detuning Δ, respectively. The function z(x, y) is related to the steady-state energy of the intracavity light |b˜ (ss) |2 . As shown in Fig. 4.7, the cubic equation (4.54) may lead to three positive real roots, where two of them are stable, i.e., the real steady-state solutions, while the last one is unstable. Such a bistable behavior can be exhibited via scanning either P in or Δ and has already been observed in [31].

4.3 Nonlinear Frequency Conversion

191

Fig. 4.8 Third-harmonic generation. a Energy-level diagram. b THG in a silica microtoroid resonator. Inset: Measured output power of THG as a function of the input pump power. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Physics, [33], © 2007

Frequency Tripling. Another common third-order nonlinear interaction is the frequency tripling, where the energy of three identical pump photons at ω p = ω are converted into one signal photon at ωs = 3ω [see Fig. 4.8a]. Such a process is referred to as the third-harmonic generation (THG). For a pump beam in the visible regime, for instance, ∼800 nm, the third-harmonic beam lies in the deep ultraviolet, where the significant absorption occurs. In contrast, for a near-infrared pump beam, the signal wave can be visible. By writing the intracavity field in the form E(r, t) =

1 1 a p (t)Ψ p (r)e−iω p t + as (t)Ψs (r)e−iωs t + c.c., 2 2

(4.56)

with the pump and signal modes Ψ p (r) and Ψs (r) and the corresponding amplitudes a p (t) and as (t), the nonlinear polarization source is expressed as P N L (r, t) =

3 3ε0 (3)  a p (t)Ψ p (r) e−iωs t χ 2   2 +as (t)Ψs (r) a ∗p (t)Ψ p∗ (r) e−iω p t .

(4.57)

Here the assumption χ (3) ≡ χ (3) (3ω; ω, ω, ω) = χ (3) (ω; 3ω, −ω, −ω) has been used. Defining  bs (t) = 

1 ε0  L (ωs ) 2

1 ε0  L (ω p ) b p (t) = 2

1/2

|Ψs (r)|2 dr

as (t),

(4.58a)

1/2

|Ψ p (r)| dr 2

a p (t),

(4.58b)

and inserting (4.58) into (4.21), one obtains the wave equations for the pump and signal modes

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4 Applications of WGM Microcavities in Physics

∂ ωs bs (t) = − bs (t) + iγ b3p (t), ∂t 2Q s 

2 ωp ωp ∂ b p (t) + iγ ∗ bs (t) b∗p (t) + i κ p Ppin . b p (t) = − ∂t 2Q p ωs

(4.59a) (4.59b)

Here we have inserted the loss and input terms and defined the pump-signal coupling strength  ∗ Ψs (r)Ψ p3 (r)dr 3χ (3) ωs γ =    1/2  3/2 . 2ε0  L (ωs ) |Ψs (r)|2 dr  L (ω p ) |Ψ p (r)|2 dr

(4.60)

Setting ∂t∂ bs (t) = ∂t∂ b p (t) = 0, the steady-state solutions are determined by the following algebraic equation: (4.61) y(1 + x y 4 ) = 1, with the definitions |γ |2 2Q s x= 3 ωs



2Q p ωp

5

 2 κ p Ppin , y =

|b(ss) p | . 2Q p  κ p Ppin ωp

(4.62)

Then, the THG efficiency is given by η=

κ p Q p κs Q s κs |bs(ss) |2 = 4x y 6 , in 3Pp ω p ωs

(4.63)

with the output-coupling strength of the signal mode κs . Varying the input pump power Ppin tunes the term 4x y 6 , whose behavior is similar to Fig. 4.5b. The maximum of 4x y 6 is unity, locating at the saturation input xsat = 16. Compared to SHG, the phase matching engineering in THG is more challenging for the reasons: the large material dispersion difference between the near-infrared pump and visible signal waves, the strongly reduced mode overlap between the long- and short-wavelength cavity modes, and the tiny third-order nonlinear optical susceptibility. The WGM-based THG was firstly performed in [33], where the amorphous silica is adopted so as to suppress the second-order effects and make the third-order effects obvious. The pump beam is coupled into a toroidal cavity via a fiber taper waveguide. A typical measured emission spectrum is shown in Fig. 4.8b with the pump wave at 1553.9 nm and the THG signal at 517.4 nm. In addition, a WGM-based third-order sum-frequency generation (two 1674-nm photons plus one 1553-nm photon generating one 542-nm photon) was demonstrated in [33] Degenerate four-wave mixing. A more general third-order optical nonlinear process is the so-called four-wave mixing (FWM), where two input photons at the frequencies ω1 and ω2 are converted into two new photons at the frequencies ω3 = 2ω1 − ω2 and ω4 = 2ω2 − ω1 . In particular, when two of the four frequencies

4.3 Nonlinear Frequency Conversion

193

Fig. 4.9 Degenerate four-wave mixing. a Energy-level diagram. b Schematic diagram of degenerate FWM. c Degenerate FWM spectrum. Reprinted with permission from [34]. Copyright (2004) by the American Physical Society

coincide, the corresponding nonlinear interaction is referred to as the degenerate FWM. We consider the degenerate FWM, i.e., one signal photon at ωs and one idler photon at ωi < ωs are generated from two pump photons at ω p [see Fig. 4.9a, b]. The nonlinear polarization is given by P N L (r, t) =

3ε0 (3) 2 χ a p (t)as∗ (t)Ψ p2 (r)Ψs∗ (r)e−iωi t 2 +a 2p (t)ai∗ (t)Ψ p2 (r)Ψi∗ (r)e−iωs t

 +a ∗p (t)as (t)ai (t)Ψ p∗ (r)Ψs (r)Ψi (r)e−iω p t ,

(4.64)

with the cavity modes Ψs,i, p (r) and the amplitudes as,i, p (t). The assumption χ (3) ≡ χ (3) (ωs ; ω p , ω p , −ωi ) = χ (3) (ωi ; ω p , ω p , −ωs ) = χ (3) (ω p ; ωs , ωi , −ω p ) has been used in the above equation. Substituting  bα=s,i, p (t) =

1 ε0  L (ωα ) 2

1/2

|Ψα (r)|2 dr

aα (t),

(4.65)

into (4.21), one obtains the following wave equations for the cavity modes ∂ ωs ωs bs (t) = − bs (t) + i γ ∗ b2p (t)bi∗ (t), ∂t 2Q s ωp ∂ ωi ωi bi (t) = − bi (t) + i γ ∗ b2p (t)bs∗ (t), ∂t 2Q i ωp  ωp ∂ b p (t) = − b p (t) + iγ b∗p (t)bs (t)bi (t) + i κ p Ppin , ∂t 2Q p

(4.66a) (4.66b) (4.66c)

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4 Applications of WGM Microcavities in Physics

with the cavity quality factors Q s,i, p , the input pump power Ppin , and the rate of coupling the pump beam into the cavity κ p . The inter-wave coupling strength γ is defined as 3χ (3) ω p γ =  2ε0

 ∗ 2 Ψ p (r) Ψs (r)Ψi (r)dr .   1/2 L 2  L (ω p ) |Ψ p (r)|2 dr α=s,i  (ωα ) |Ψα (r)| dr

(4.67)

Solving (4.66) leads to the following steady-state solutions: (ss) 2 |bα=s,i | = κp

  ωα 2Q α 2Q p Pth Ppin /Pth − 1 , ω p ωα ω p

(4.68)

where the threshold Pth reads  Pth =

ωs ωi κ 2p |γ |2 2 ωp

2Q s 2Q i ωs ωi



2Q p ωp

4 −1/2 .

(4.69)

As Ppin exceeds Pth , the signal and idler waves start oscillating inside the cavity, which is known as the hyperparametric oscillations (HPO). The WGM-based HPO was firstly reported in [34], where a silica microtorid was employed. The pump beam is located at 1565 nm and the signal and idler resonances are equally apart from the pump wave by one free spectral range of the cavity [Fig. 4.9c]. In addition, the well-defined phase relationship between signal and idler waves has been verified in [35].

4.4 Parity-Time Symmetry One application of the WGM microcavities in fundamental physics is to examine the spontaneous PT symmetry breaking in non-Hermitian quantum mechanics. In this section, we briefly introduce the PT symmetry optics, where the coupled WGM microcavities interact with the environment in such a way that they experience balanced optical gain and loss.

4.4.1 Parity and Time-Reversal Transformations Parity transformation. In quantum mechanics, the parity transformation (denoted by the operator P) describes the flip in the sign of both spatial coordinates r (i.e., spatial reflection) and momentum p, P : r → −r, & P : p → −p.

(4.70)

4.4 Parity-Time Symmetry

195

The operator P acting on an arbitrary wavefunction Ψ (r) gives PΨ (r) = Ψ (−r).

(4.71)

Actually, such a transformation changes a right-handed coordinate system into a left-handed one or vice versa. Since reversing the coordinate system twice leaves the wavefunction invariant (P 2 = 1), the operator P owns two eigenvalues ±1. A wavefunction Ψe (r) is even when PΨe (r) = Ψe (r). In contrast, a wavefunction Ψo (r) is odd when PΨo (r) = −Ψo (r). It is also easy to prove that the parity operator P is Hermitian (P † = P) and unitary (P ∗ = P −1 ). The parity transformation acting on an arbitrary observable (represented by a Hermitian operator O) is expressed as POP −1 . When P commutes with the Hamiltonian H of a quantum system, P H = H P, the parity is a conserved observable (the probability of measuring P is independent of time in all states). As an example, we consider the one-dimensional quantum harmonic oscillator described by the Hamiltonian H=

1 p2 + mω2 x 2 , 2m 2

(4.72)

with the momentum operator p = −i ∂∂x , the particle’s mass m and the angular frequency ω. It is obvious that H is invariant under the parity transformation (P pP −1 = − p & P xP −1 = −x). Thus, the eigenstates of H must have either even or odd symmetry. Solving the time-independent Schrödinger equation H Ψ (x) = EΨ (x),

(4.73)

one obtains the n-th (n ∈ Z) energy level   1 , E n = ω n + 2

(4.74)

with the corresponding eigenstate Ψn(h) (x)

  1 x2 = 1/2 Hn (x/x0 ) exp − 2 . √ 2x0 2n n! π x0

(4.75)

√ Here, we have defined the characteristic length x0 = /mω and the functions Hn (z) (h) (h) (−x) = Ψ2m (x) (even) and are the Hermite polynomials. One may also prove Ψ2m (h) (h) (−x) = −Ψ2m+1 (x) (odd) with m ∈ Z. Ψ2m+1 Time reversal transformation. In the classical physics, the Newton’s second law of motion for a particle with a mass m

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4 Applications of WGM Microcavities in Physics

m

d2 r(t) = −∇V (r(t)) dt 2

(4.76)

is invariant under the time-reversal transformation T : t → −t. Here V (r) is an external potential, where the mass moves. If r(t) is a solution, r(−t) is also a solution. The particle’s momentum is computed by p(t) = m

d r(t), dt

(4.77)

which suggests that the momentum corresponding to the solution r(−t) is −p(−t). Thus, under T one has T : r → r, & T : p → −p.

(4.78)

In quantum mechanics, the relation between x and p (in the one-dimensional case) is given by the commutation [x, p] = i.

(4.79)

We impose that the classical results (4.78) are still valid here and obtain the commutation relation under the time-reversal transformation T [x, p]T −1 = −[x, p].

(4.80)

In order to keep the form of (4.79), one arrives at an important result T : i → −i.

(4.81)

According to Wigner’s theorem [36], T is an antiunitary operator1 and takes the form T = U K with a unitary operator U and the complex conjugation operator K (K −1 = K ). It is straightforwardly to verify the result (4.81) in the one- dimensional K . In addition, the invariance of the commutation rule for the case by using T = eiθ angular moment J = α=x,y,z eα Jα [Jα , Jβ ] = iαβγ Jγ ,

(4.82)

(here αβγ is the Levi–Civita symbol) under the action of T requires ordinary unitary operators are linear, U (αΨ ) = αU (Ψ ) with a complex number α and a wavefunction Ψ . In contrast, T acting on a superposition state (αΨ + βΦ) gives T (αΨ + βΦ) = α ∗ T Ψ + β ∗ T Φ, which is called the antilinear (conjugate-linear) relation. Additionally, one has the inner product T Ψ |T Φ = Ψ ∗ |Φ ∗  = Ψ |Φ∗ = Φ|Ψ . The antilinear operators that satisfy this inner product relation are called the antiunitary operators. For a spinless particle, we have T = K with U = 1, U K = K U , T † = T and T 2 = 1. For a spin-1/2 particle, one may prove T = U K with U = σ y , U K = −K U , T † = −T , and T 2 = −1. Here σ y is the y-component of Pauli’s spin matrices. 1 The

4.4 Parity-Time Symmetry

197

T JT −1 = −J,

(4.83)

which is consistent with the time-reversal transformation of the orbital angular momentum L = r × p. The wavefunction Ψ (t) of a quantum system evolves in time according to the Schrödinger equation d (4.84) i Ψ (t) = H Ψ (t), dt whose formal solution is written as Ψ (t) = exp (−i H t/) Ψ (0).

(4.85)

We exert T on the Schrödinger equation and obtain i

d Ψ (−t) = T H T −1 Ψ (−t), dt

(4.86)

with Ψ (−t) = T Ψ (t). If the motion of the system is invariant under the time-reversal transformation, i.e., Ψ (−t) is also a solution to the Schrödinger equation, T must commute with the Hamiltonian, T H T −1 = H . Indeed, this time-reversal symmetry corresponds to the conservation of entropy.

4.4.2 Spectral Reality of Non-Hermitian Hamiltonians One of the postulates in quantum mechanics says that a physical observable is associated with a Hermitian operator whose eigenvalues are all real. For example, the mean value of the system’s energy must be real and positive Ψ |H Ψ ∗ = H Ψ |Ψ  = Ψ |H Ψ  = H † Ψ |Ψ ,

(4.87)

which implies that H is equal to its Hermitian adjoint, H † = H . However, as pointed out in [37], the Hermiticity is a sufficient, but not necessary, condition for the real eigenvalues. To illustrate this interesting problem, let us examine the following modified harmonic oscillator Hamiltonian H=

1 p2 + mω2 (x 2 − 2i xd x), 2m 2

(4.88)

with a constant real displacement xd . Obviously, the modified H does not own the Hermiticity. Substituting (4.88) into (4.73), one may derive the n-th eigenvalue   1 x 2 /x 2 E n = ω n + + d 0 , 2 2

(4.89)

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4 Applications of WGM Microcavities in Physics

Fig. 4.10 Energy spectrum of the Hamiltonian (4.92) as a function of α

with the corresponding eigenstate Ψn(h) (x − i xd ). All eigenvalues {E n , n ∈ Z} are real and positive. Therefore, the Hermiticity of H is not a necessary condition for ensuring the real eigenvalues. Indeed, as pointed out in [38], a non-Hermit Hamiltonian H can possess the real spectrum (i.e., all eigenvalues are real and positive), provided H satisfies the PT symmetry (4.90) (PT )H (PT )−1 = H. By separating H into the kinetic p 2 /2m and potential V (r) parts, the PT symmetry requires (4.91) V (r) = V ∗ (−r). The Hamiltonian (4.88) is invariant neither under the parity P nor under the timereversal T transformation, but it fulfills the PT symmetry. Let us further examine the following general Hamiltonian:   ∂2 ω α with α ∈ R. − − (i x/x0 ) H= 2 ∂(x/x0 )2

(4.92)

When α = 2, H is reduced to the quantum harmonic oscillator. One can prove that the Hamiltonian (4.92) is PT symmetric. The energy spectrum of H versus α is presented in Fig. 4.10. It is seen that the spectrum exhibits three distinct regions: (1) In the region of α ≥ 2, the eigenvalues of H are infinite, discrete, and completely real and positive. Thus, the condition of Hermiticity that ensures the real spectrum

4.4 Parity-Time Symmetry

199

of a Hamiltonian may be relaxed by the PT symmetry; (2) When 1 < α < 2, H has a finite number of real positive eigenvalues and an infinite number of complex conjugate pairs of eigenvalues. It indicates that the PT symmetry breaks down spontaneously at the exceptional point (EP) α = 2. As α is reduced, the number of real eigenvalues decreases until only the ground-state energy level E 0 is remained. E 0 diverges as α approaches 1 from above; and (3) For α ≤ 1, no real eigenvalues exist.

4.4.3 Parity-Time Symmetry in Optics Although the topic of PT symmetric non-Hermitian Hamiltonians has attracted extensive studies in theory, the relevant experimental support appeared almost one decade after its discovery. Multiple PT -symmetry-related concepts may be tested experimentally on the optical platforms [39, 40] because of the reasons: (1) The classical optical wave equation is formally equivalent to the quantum mechanical Schrödinger equation [41, 42]; and (2) The PT -symmetry-demanded (one dimensional) potential V (x) = V ∗ (−x) may be produced straightforwardly by artificially manipulating the refraction index and gain/loss of optical media. We consider an optical wave E(r, t) (frequency ω, wavenumber k, and linearly polarized in the ey axis) traveling in a medium along the z-direction, 1 1 E(r, t) = ey Ψ (x, z)eikz−iωt + ey Ψ ∗ (x, z)e−ikz+iωt , 2 2

(4.93)

with the amplitude Ψ (x, z) and k = n 0 ω/c. The complex refractive index of the medium is homogeneous in the y- and z-directions but is weakly inhomogeneous along the ex axis n(r) = n 0 + Δn(x) with Δn(x) = n R (x) + in I (x),

(4.94)

Here n 0 is the real background constant, which is much larger than |Δn(x)|. Substituting (4.93) and (4.94) into the Maxwell’s equations, one obtains i

  1 ∂2 ω ∂ Ψ (x, z) = − Ψ (x, z). − Δn(x) ∂z 2k ∂ x 2 c

(4.95)

By defining the dimensionless coordinates ξ = 2kz and η = 2kx, the above equation is re-expressed as i

  ∂2 ∂ Δn(η) Ψ (η, ξ ) = − 2 + V (η) Ψ (η, ξ ) with V (η) = − . ∂ξ ∂η 2n 0

(4.96)

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4 Applications of WGM Microcavities in Physics

Equation (4.96) is formally equivalent to the Schrödinger equation, where ξ plays the role of time and V (η) represents the external potential. The symmetry condition (4.91) requires a symmetric real component n R but an antisymmetric gain/loss profile n I with respect to η [see Fig. 4.11a] n R (η) = n R (−η), n I (η) = −n I (−η).

(4.97)

Moreover, (4.96) under the condition V (η) = V ∗ (−η) leads to   ∂2 ∂ ∗ − i Ψ (−η, ξ ) = − 2 + V (η) Ψ ∗ (−η, ξ ). ∂ξ ∂η

(4.98)

Multiplying (4.96) by Ψ ∗ (−η, ξ ) and (4.98) by Ψ (η, ξ ) and subtracting , we obtain the generalized continuity equation [43] ∂ ∂ P(η, ξ ) + J (η, ξ ) = 0, ∂ξ ∂η

(4.99)

with the probability density P(η, ξ ) = Ψ ∗ (−η, ξ )Ψ (η, ξ ) and the probability current density   ∂ ∗ ∂ ∗ J (η, ξ ) = −i Ψ (−η, ξ ) Ψ (η, ξ ) − Ψ (η, ξ ) Ψ (−η, ξ ) . ∂η ∂η

(4.100)

Following a similar procedure, one gets the conservation law ∂ ∂ξ

∞ −∞

Ψ1∗ (−η, ξ )Ψ2 (η, ξ )dη = 0,

(4.101)

for two different wavefunctions Ψ1 (η, ξ ) and Ψ2 (η, ξ ). In deriving the above equation, we have used the fact Ψ1 (±∞, ξ ) = Ψ2 (±∞, ξ ) = 0, i.e., the wave are distributed within a finite region along the x direction. We assume the complex amplitude Ψ (η, ξ ) is separable with respect to η and ξ and takes the form (4.102) Ψ (η, ξ ) = Φ(η)e−iβξ . Equation (4.96) is then reduced to the time-independent Schrödinger equation  −

 ∂2 + V (η) Φu (η) = βu Φu (η), ∂η2

(4.103)

with the u-th eigenvalue βu and the corresponding eigenmode Φu (ξ ). Inserting two wavefunctions Ψ1 (η, ξ ) = Φ1 (η)e−iβ1 ξ and Ψ2 (η, ξ ) = Φ2 (η)e−iβ2 ξ into (4.101) yields

(β1∗ − β2 )

∞

−∞

Φ1∗ (−η)Φ2 (η)dη = 0.

(4.104)

4.4 Parity-Time Symmetry

201

Fig. 4.11 PT symmetry in optics. a Schematic of the refractive index n R and the gain/loss distribution n I for two-mode coupling. b, c Eigenvalues β± as a function of the ratio κ/γ I with γ R = 0. d–f Dependence of the intensities |a1,2 (η)|2 on η with κ/γ I = 2, 1, and 1/2, respectively. For all curves, the system is initialized with a1 (0) = 1 and a2 (0) = 0

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4 Applications of WGM Microcavities in Physics

In the cases: (1) both β1,2 are real and β1 = β2 ; (2) one of β1,2 is real while the other is complex; and (3) both β1,2 are complex and β1∗ = β2 , Φ1 (η) is orthogonal to Φ2 (η)

∞ −∞

Φ1∗ (−η)Φ2 (η)dη = 0.

(4.105)

In other cases, (4.104) is automatically satisfied. In addition, for a real and nondegenerate β with the corresponding Φ(η) = Φ R (η) + iΦ I (η), one may prove that Φ R (η) is an even/odd function while Φ I (η) is an odd/even function, and the following integral

∞

−∞

Φ ∗ (−η)Φ(η)dη = eiθ

∞

−∞

Φ 2 (η)dη,

(4.106)

has a real value. The phase θ is equal to 0 (π ) for the even (odd) Φ R (η). Coupled-mode optics. Next, we consider the one-dimensional configuration with an optical potential V (η) consisting of two components, V (η) = V1 (η) + V2 (η). Two subpotentials are related via V1∗ (−η) = V2 (η) and possess their own centers. The refractive index n R and gain/loss n I profiles, corresponding to V (η), are plotted in Fig. 4.11a. We assume that the real part of V1 (η) is symmetric and the imaginary part of V1 (η) is symmetric or antisymmetric with respect to its center. In the absence of V2 (η), the light traveling in V1 (η) has a real  and nondegenerate  eigenvalue β with the corresponding eigenfunction Φ1 (η), i.e., −∂ 2 /∂η2 + V1 (η) Φ1 (η) = βΦ1 (η). The symmetry V1∗ (−η) = V2 (η) tells us that in the absence of V1 (η), the eigenfunction ∗ of light traveling  in V2 (η) is Φ2 (η) = Φ1 (−η) with the same eigenvalue β, i.e.,  the 2 2 −∂ /∂η + V2 (η) Φ2 (η) = βΦ2 (η). Consequently, the wavefunction of the light traveling in V (η) may be expressed as the superposition of two local eigenfunctions Ψ (η, ξ ) = [a1 (ξ )Φ1 (η) + a2 (ξ )Φ2 (η)] e−iβξ .

(4.107)

Substituting (4.107) into (4.96), one obtains ⎛ ⎞ ∞ ∗  ∞ ∗  i ∂ a1 (ξ ) Φ1 (−η)Φ1 (η)dη −∞ Φ1 (−η)Φ2 (η)dη ⎜ ∂ξ ⎟ −∞ ∞ ∞ ⎝ ∂ ⎠ ∗ ∗ Φ (−η)Φ (η)dη Φ (−η)Φ (η)dη 1 2 −∞ 2 −∞ 2 i a2 (ξ ) ∂ξ ∞ ∗   ∞ ∗  Φ1 (−η)V2 (η)Φ1 (η)dη −∞ Φ1 (−η)V1 (η)Φ2 (η)dη a1 (ξ ) = −∞ .(4.108) ∞ ∞ ∗ ∗ a2 (ξ ) −∞ Φ2 (−η)V2 (η)Φ1 (η)dη −∞ Φ2 (−η)V1 (η)Φ2 (η)dη Due to the symmetry of V1 (η) and V2 (η), and equal, and we have

∞

−∞

∞

  2 Φ1,2 (η)dη  

−∞

∞ −∞

Φ12 (η)dη and

∞

−∞

  Φ1 (η)Φ2 (η)dη,

Φ22 (η)dη are real

(4.109)

4.4 Parity-Time Symmetry

203

i.e., Φ1 (η) overlaps weakly with Φ2 (η). Thus, (4.108) is reduced to ∞

−∞

V2 (η)Φ12 (η)dη

∞

V1 (η)Φ1 (η)Φ2 (η)dη ∞ 2 a2 (ξ ), −∞ Φ1 (η)dη (4.110a) ∞ ∞ 2 V (η)Φ (η)Φ (η)dη V (η)Φ (η)dη ∂a2 (ξ ) 2 1 2 1 ∞ 2 2 i a1 (ξ ) + −∞ a2 (ξ ). = −∞  ∞ 2 ∂ξ Φ (η)dη −∞ 2 −∞ Φ2 (η)dη (4.110b) ∂a1 (ξ ) = i ∂ξ

∞

2 −∞ Φ1 (η)dη

−∞

a1 (ξ ) +

∞ ∞ It can be proved that −∞ V1 (η)Φ1 (η)Φ2 (η)dη and −∞ V2 (η)Φ1 (η)Φ2 (η)dη are real and equal. Defining the coupling coefficient κ and the propagation shift γ = γ R + iγ I ∞ κ=

−∞

we arrive at i

V2 (η)Φ1 (η)Φ2 (η)dη ∞ 2 , γ = −∞ Φ2 (η)dη

∂ ∂ξ

∞

V2 (η)Φ 2 (η)dη ∞ 2 1 , −∞ Φ1 (η)dη

(4.111)

      a1 (ξ ) a (ξ ) γ κ . =H 1 with H = a2 (ξ ) a2 (ξ ) κ γ∗

(4.112)

−∞

Consequently, the Schrödinger equation (4.96) is converted into a 2 × 2 matrix form. Obviously, the Hamiltonian H is non-Hermitian, H = H† , but is invariant under the PT transformation with   01 P = σx = , (4.113) 10 and T = K . Diagonalizing H yields two eigenvalues β± = γ R ±



κ 2 − γ I2 .

(4.114)

For simplicity, in the following we assume both κ and γ I are positive. According to the reality of β± , the system possesses two phases with distinct behaviors: (1) Unbroken symmetry. When the coupling coefficient is larger than the gain/loss parameter, κ > γ I , both eigenvalues β± take real values [see Fig. 4.11b, c]. The corresponding eigenfunctions are derived as     1 eiθ 1 −e−iθ , Φ− = √ , Φ+ = √ 1 2 1 2

(4.115)

where the phase θ is determined via  cos θ =

κ 2 − γ I2 κ

,

sin θ =

γI . κ

(4.116)

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4 Applications of WGM Microcavities in Physics

The normalization condition for Φ± is given by Φ±† Φ± = 1.

(4.117)

However, unlike the Hermitian systems, Φ± do not satisfy the orthogonality, Φ+† Φ− = 0. One may verify that both Φ± are also the eigenfunctions of the operator (PT ). That is, the PT symmetry of the optical coupler is unbroken. Using (4.115)–(4.117), we obtain the solution of (4.112) ⎛  ⎞   2 2 2 2    cos κ − γI η − θ −i sin κ − γ I η ⎟  e−iγ R η ⎜ a1 (0) a1 (η) ⎜ ⎟   =   ⎠ a2 (0) . a2 (η) cos θ ⎝ −i sin κ 2 − γ 2 η cos κ 2 − γ I2 η + θ I (4.118) As illustrated in Fig. 4.11d, the powers |a1,2 (η)|2 versus η exhibit an oscillatory behavior, i.e., the large enough coupling strength κ leads to a periodical energy exchange between two optical modes. In addition, the nonzero γ I leads to the oscillation of the total power |a1 (η)|2 + |a2 (η)|2 . This is unlike a passive coupling system γ I = 0, where |a1 (η)|2 + |a2 (η)|2 remains constant. (2) Symmetry breaking. When κ < γ I , two eigenvalues β± become complex conjugate to each other [see Fig. 4.11b, c], signifying the onset of a spontaneous PT symmetry-breaking. The eigenfunctions of H take the form Φ+ =

    − cos θ i sin θ , Φ− = , i sin θ cos θ

(4.119)

where θ is determined via ⎛ cos θ = ⎝

γI +



γ I2 − κ 2

2γ I

⎛

⎞1/2 ⎠

,

sin θ = ⎝

γI −



γ I2 − κ 2

2γ I

⎞1/2 ⎠

.

(4.120)

Again, Φ± satisfy the normalization condition (4.117) but Φ+† Φ− = 0. In addition, it is evident that Φ± are no longer the eigenfunctions of the combined operator (PT ), though the relation PT H = HPT still holds true. The mode amplitudes a1,2 (η) are derived as     e−iγ R η cosh γ I2 − κ 2 η cos 2θ + sinh γ I2 − κ 2 η a1 (0) a1 (η) = cos 2θ   (4.121a) −i sinh γ I2 − κ 2 η sin 2θ a2 (0) ,   e−iγ R η −i sinh γ I2 − κ 2 η sin 2θ a1 (0) a2 (η) = cos 2θ

4.4 Parity-Time Symmetry

205

     + cosh γ I2 − κ 2 η cos 2θ − sinh γ I2 − κ 2 η a2 (0) . (4.121b) Figure 4.11e, f show the η-dependent intensities |a1,2 (η)|2 of the coupled system at the exceptional point κ = γ I and in the symmetry-breaking regime κ < γ I , respectively. The oscillatory behavior disappears because of the weak coupling strength. Due to the complex eigenvalues β± , the amplitude of the Φ+ mode grows exponentially along the propagation distance η while that of the Φ− mode decays exponentially. As η is further increased, both |a1,2 (η)|2 go up exponentially.

4.4.4 Observation in Experiments In above, we have proven the equivalence between the optical wave equation and the time-dependent Schrödinger equation. Thus, the predictions of the PT symmetric non-Hermitian quantum mechanics may be examined in optics. The optical potential, i.e., the complex refractive index of the medium, may be manipulated via doping the active particles and applying the external beams. The advanced micro- and nanofabrication techniques have enabled the observation of the spontaneous PT symmetry breaking on various optical platforms.

4.4.4.1

Coupled Waveguides

The first platform realized in experiment was based on the coupled optical waveguides [40]. Figure 4.12a depicts a schematic diagram of the experimental setup. A pair of waveguides with the symmetric refractive index profile n R are engraved in the Fe-doped lithium niobate (Fe:LiNbO3 ) substrate via the Ti in-diffusion. The length of each channel reaches L = 2 cm and the inter-channel coupling strength is κ = 1.9 cm−1 . The Fe:LiNbO3 is an excellent photorefractive material, and the optical gain can be provided by means of the so-called two-wave mixing. In contrast, the energy consumed for exciting the electrons from the Fe2+ centers to the conduction band gives rise to the optical loss. The antisymmetric gain/loss distribution n I is formed by using a mask to block the pump beam incidence on one of the two channels. A signal beam is coupled to either channel. To avoid any index perturbations, the low-power input (∼25 nw) is used. The light intensities I1,2 of two channels at the output facet are recorded by a CCD camera. In practice, the optical gain in the unblocked channel follows an exponential temporal build-up, γG = γmax [1 − exp(−t/τ )] with a time constant τ . In other words, the system starts from the PT -symmetric state, then passes the exceptional point, and finally stays in the broken-PT regime. Such a process is manifested by the experimental results plotted in Fig. 4.12b. At the beginning, both I1,2 exhibit the oscillatory behavior, i.e., the system is in the unbroken phase. After that, I1 experi-

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4 Applications of WGM Microcavities in Physics

Fig. 4.12 PT -symmetric integrated optics. a A signal beam is coupled into one of the two coupled channels fabricated on a photorefractive Fe:LiNbO3 substrate. Two-wave mixing and an amplitude mask are used to produce an antisymmetric gain/loss distribution n I . b Intensities I1,2 measured at the output facet. Left: the channel 1 is excited. Right: the channel 2 is excited. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Physics, [40], © 2010

ences an exponential amplification while I2 decays, i.e., the system is in the broken phase.

4.4.4.2

Coupled Microcavities

Besides the coupled waveguides, the investigation of a PT -symmetric system can be also performed on a pair of coupled WGM microcavities [44]. In comparison, the resonances play a dominant role in the system’s dynamics. As shown in Fig. 4.13a, two WGM microtoroids are directly coupled with each other via the evanescent tails. The coupling strength can be adjusted through nanopositioning stages. Each microcavity is also coupled to a different tapered fiber. One of the two microcavities is doped with Erbium (Er3+ ) ions, and the gain in this microcavity is provided by optically pumping. The other resonator is undoped and exhibits the passive loss.

4.4 Parity-Time Symmetry

207

Fig. 4.13 PT -symmetric WGM microcavities. a Two coupled microcavities, active Er3+ -doped microtoroid μR1 and passive microtoroid μR2 . b Mode splitting and change in linewidth versus the coupling strength κ. Triangle: both μR1 and μR2 are passive. Circle: μR1 is active while μR2 is passive. c Unidirectional transmission. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Physics, [44], © 2014

The pump beam in 1460 nm band and the weak probe beam in 1550 nm band are input at the port 1 in Fig. 4.13a and the transmission spectrum is monitored at the port 2. In the absence of the pump beam, both microcavities are passive. When the coupling strength κ is below a threshold κ P T , the system is in the brokensymmetry phase with the zero mode splitting and the nonzero linewidth difference [see Fig. 4.13b]. As κ/κ P T passes the unity, the linewidth difference between two modes decreases to zero and the real parts of eigenvalues bifurcate. This coupled-microcavity system also enables the observation of nonreciprocal transmission behavior. As illustrated in Fig. 4.13c, when both microcavities are passive, the forward (input at the port 1 and measure at the port 4) transmission spectrum is similar to the backward (input at the port 4 and measure at the port 1) transmission spectrum. When the pump is on and the system is in the unbroken phase, both forward and backward transmission spectra display the similar mode splitting. However, for the system in the broken phase, the forward transmission spectrum has no peaks

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4 Applications of WGM Microcavities in Physics

Fig. 4.14 Synthetic anti-PT symmetric optical microcavity. a A high-Q microsphere coupled to a tapered fiber. In the WGM microcavity, the probe and Stokes modes together with an acoustic mode join in the nonlinear Brillouin scattering. b Transmission spectra of the nonlinear gain (i.e., EIT) and loss (i.e., EIA). c Transmission spectra in the forward direction. d Measured eigenvalues β± . Reprinted with permission from [45]. Copyright (2020) by the American Physical Society

while a single resonance peak is presented in the backward transmission spectrum. Such a behavior indicates a nonreciprocal light transport between ports 1 and 4.

4.4.4.3

Anti-PT Symmetry Breaking in a Single Microcavity

Most investigations of the PT symmetry in optics are based on the spatially balanced gain/loss distribution, whose generation relies on two waveguides or two microcavities. In [45], a different (anti)-PT symmetric configuration is demonstrated, where two spectrally separated optical modes in a single WGM microcavity are coupled via a nonlinear frequency conversion, such as stimulated Brillouin scattering (SBS). As shown in Fig. 4.14a, a tapered fiber interacts with a neighboring microsphere evanescently. A weak probe beam at the frequency ω p is coupled into the fiber, propagates from left to right (forward), and excites a clockwise WGM (resonant frequency ω p0 ) in microsphere. The probe beam with a high enough power can also stimulate

4.4 Parity-Time Symmetry

209

an electrostrictive-induced SBS process, which converts the ω p wave into a Stokes wave (at a lower frequency ωs ) and a coherent acoustical wave (at the frequency Ω = ω p − ωs ). Under the conservation law of momentum, the induced Stokes wave matches a counterclockwise WGM mode (resonant frequency ωs0 ) in microsphere. Therefore, the nonlinear coupling causes a loss [electromagnetically induced transparency absorption (EIA)] to the probe beam but provides a gain [electromagnetically induced transparency (EIT)] to the Stokes wave [see Fig. 4.14b]. The extra pump and seed beams are applied to maintain a stable nonlinear coupling between the probe and Stokes waves. The physical system described above may be modeled as dEp = (−iκ p − Δ p )E p − g E s , dt d Es = (−iκs − Δs )E s + g ∗ E p , i dt

i

(4.122a) (4.122b)

with the amplitudes E p,s , the mode decay rates κ p,s , the detunings Δ p,s = ω p,s − ω p0,s0 of two WGMs, and the intermode coupling strength g. Both Δ p,s and g = g R + ig I can be tuned in experiment. Inserting the substitutions E˜ p,s = E p,s e(κ p,s −iΔ)t ,

(4.123)

with the average detuning Δ = (Δ p + Δs )/2 into (4.122), one obtains d i dt



E˜ p E˜ s



 = Hs

E˜ p E˜ s



  −δ −g with Hs = . g∗ δ

(4.124)

The detuning difference δ is given by δ = (Δ p − Δs )/2. The non-Hermitian Hamiltonian Hs does not commute with the PT operator but fulfills the anticommutation relation {Hs , PT } = Hs PT + PT Hs = 0, i.e., the anti-PT symmetry. We may define a synthetic parity operator Ps = 

1 δ 2 − g 2R



 δ gR , −g R −δ

(4.125)

and Hs satisfies a generalized PT symmetry, Ps T Hs = Hs Ps T with T = K . Two eigenvalues of Hs are derived as β± = ± δ 2 − |g|2 . When |Δ p − Δs | = 2|δ| > 2|g|, both β± are real and the system is in the unbroken phase. The probe and Stokes waves inside the microcavity are hybridized into two eigenmodes, one with the + Δ + δ 2 − |g|2 and the other with the mixed frequencies mixed frequencies ω p0,s0 ω p0,s0 + Δ − δ 2 − |g|2 . The generalized PT symmetry is broken (corresponding to the unbroken anti-PT symmetric phase) when |Δ p − Δs | < 2|g|. In experiment, the eigenvalues β± may be derived from the measurement of the frequency splitting and linewidth of the transmission spectrum of the probe beam [see Fig. 4.14c, d].

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4 Applications of WGM Microcavities in Physics

4.5 Electromagnetically Induced Transparency The electromagnetically induced transparency (EIT) denotes that a medium becomes transparent within a narrow spectral window that is located in a wide absorption spectrum background. Such a nonlinear phenomenon was firstly studied in atomic physics [46], where in general, two (coupling and probe) light beams interact with an ensemble of atoms composed of three atomic states |1, 2, 3. These states may form a Λ-, V -, or Ξ -type configuration. As illustrated in Fig. 4.15a, the coupling strength between the coupling beam and the |2 − |3 transition is strong while the interaction between the probe beam and the |1 − |3 transition is weak. The dipole transition between |1 and |2 is forbidden. In experiment, it demonstrates that the probe beam suffers a strong absorption (due to the inelastic photon scattering processes, such as the spontaneous emission of |3) in the absence of the coupling beam, whereas the presence of the coupling beam makes the probe beam pass through the atoms freely in the resonant case [see Fig. 4.15b], that is, the atoms become transparent to the probe beam. One may interpret the EIT from the destructive quantum interference. The absorption of the probe beam is accompanied by the population transfer from |1 to |2. The interference among the probability amplitudes of the transitions {|1 − |2 − (|3 − |2)n ; n = 0, 1, 2, ...} opens a narrow transparency window. The more detailed discussion can be found in Sect. 6.11.3. So far the concept of EIT has been extended to electronic circuits [47], plasmonics [48], optomechanics [49], and optical cavities [50]. It is worth noting that unlike the quantum interference in atomic physics, diminishing the absorption in these systems is caused by the destructive classical interference. We consider two evanescently coupled WGM microcavities (μR1 and μR2 ) with the frequencies ω1 and ω2 , respectively. One of the two microcavities, for example, μR1 , interacts with a tapered fiber. An external light field sin (t) at the frequency ω is coupled into the fiber at one end and its transmission is measured at the other end. Using (3.94) and (4.112), one may obtain the following equations of motion for the complex amplitudes E 1,2 (t) of the intracavity fields: d dt



⎛ ⎞ γ1    √  iδ1 − iκ E 1 (t) i γc sin (t) E 1 (t) 2 ⎝ ⎠ = + , γ2 E 2 (t) E 2 (t) 0 iκ iδ2 − 2

(4.126)

Here γ1 = γ10 + γc is the total loss rate of μR1 with the intrinsic loss γ10 and the coupling loss γc between μR1 and the tapered fiber. The loss rate of μR2 is γ2 . Two detunings δ1,2 are defined as δ1 = ω − ω1 and δ2 = ω − ω2 . The coupling strength between two microcavities is κ. Setting t → ∞, we arrive at the steady-state solutions (denoted by “ss”)

4.5 Electromagnetically Induced Transparency

211

Fig. 4.15 EIT. a The coupling and probe beams interact with an ensemble of Λ-type atoms. b Experimentally measured transmission spectrum of the probe beam in the absence/presence of the coupling beam. Reprinted with permission from [46]. Copyright (1991) by the American Physical Society. c Susceptibility χe of two coupled optical microcavities with ω1 = ω2 = 1015 , Q 1 = 105 , Q 2 = 107 , and γc = κ = 0.1γ1 . d Transmission T versus the probe beam frequency ω and the intercavity coupling strength κ. All other parameters are same to c. e T versus ω with several selected κ. f Transmission coefficient T measured in experiment. Reprinted with permission from [51]. Copyright (2005) by the American Physical Society

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4 Applications of WGM Microcavities in Physics



E 1(ss) E 2(ss)



1 =  γ1   γ2  iδ2 − + κ2 iδ1 − 2 2 ⎞ ⎛ γ2  √ (ss)  −iκ iδ2 − i γc sin 2 ⎠ ×⎝ . γ1 0 −iκ iδ1 − 2

(4.127)

(ss) is then simplified as The relation between E 1(ss) and the input sin

√ (ss) with χe = − E 1(ss) = − γc χe sin

ω − ω˜ 2 , (ω − ω˜ + )(ω − ω˜ − )

(4.128)

where two eigenfrequencies are derived as ω˜ ± =

 1 γ1,2 (ω˜ 1 + ω˜ 2 ) ± (ω˜ 1 − ω˜ 2 )2 + 4κ 2 with ω˜ 1,2 = ω1,2 − i . (4.129) 2 2

Figure 4.15c illustrates the dependence of the “susceptibility” χe on the detuning δ1 . A dip exists in the imaginary part of χe around the resonance, indicating a strong reduction of the absorption of the probe beam. In the steady state, the field at the fiber’s output end is given by √ (ss) (ss) = sin + i γc E 1(ss) , sout

(4.130)

and the intensity transmission coefficient reads  (ss) (ss) 2 /sin  = |1 − iγc χe |2 . T = sout

(4.131)

The intercavity coupling strength κ plays an important role in the transmission profile. In the absence of the interaction κ = 0, a single spectral dip with a linewidth γ1 is presented in T [see Fig. 4.15d, e]. The nonzero κ gives rise to a spike in the wide absorption background, opening a narrow transparency window. The window’s height grows as κ is increased. When κ is further enhanced, the window’s height is saturated while its width becomes wide. Such a nonlinear behavior has been manifested in experiment [see Fig. 4.15f]. The formation of EIT has a few requirements: (1) The loss rate γ2 needs to be much smaller than γ1 . Due to the intercavity interaction, a portion of the light inside μR1 is coupled into μR2 . This light field should be restored in μR2 for a long enough time (relative to μR1 ) so as to destructively interfere with the following light field in μR1 , thereby canceling out the absorption of μR1 ; (2) The frequency difference |ω1 − ω2 | between two cavity modes should be much smaller than γ1 . Otherwise, the light fields in μR1 and μR2 may be distinguished in frequency, resulting in an asymmetric lineshape of the transmission spectrum T . Actually, the EIT can be

4.5 Electromagnetically Induced Transparency

213

viewed as a Fano resonance with ω1 = ω2 ; (3) A narrow transparency spike requires that the coupling strength κ is much smaller than γ1 . A large κ (> γ1 ) results in two absorption dips with the width ∼γ1 in T .

4.6 Optical Frequency Combs The WGM microcavities have also been applied to generate the optical frequency combs (OFCs), which are a crucial tool in quantum metrology and precision measurement. According to the operation frequency, the atomic clocks may be divided into two groups: microwave and optical clocks. As a frequency/time standard, the comparison with other clocks is essential in evaluating the stability and accuracy of an atomic clock. Optical waves oscillate at a frequency over 105 times faster than that of the state-of-the-art electronics. This poses a significant challenge in directly comparing optical and microwave frequency standards. Before 2000, such a comparison was implemented through the so-called frequency chain [52]. In this chain, the nonlinear elements are used to generate the harmonics of the local oscillators whose frequencies have been known precisely. The desired optical frequency, which is close to the frequency of the optical clock to be compared, may be reached via a step-wise frequency multiplication starting with the microwave frequency standard. The microwave versus optical frequency comparison is then performed via beating the optical clock against the desired harmonics. Such a frequency chain is large in size and expensive. Only a few national laboratories may afford it. In addition, each frequency chain was basically built for one single target frequency and does not own the general application. The appearance of OFCs in the early 2000s vastly facilitate the optical frequency measurement [53]. The heart of the conversational OFCs is a mode-locked laser [for example, a titanium-doped sapphire (Ti:S) laser that is pumped by a frequencydoubled Nd:YVO4 laser operating at 532 nm], whose output is a train of identical pulses [see Fig. 4.16a] and may be written as E(t) =

 N −1 n=0

e(t − nT ).

(4.132)

Here N denotes the number of pulses and e(t) is the amplitude of individual pulses. The time interval between two consecutive pulses is T and the repetition rate of pulses is fr = 1/T (in the microwave frequency domain). Mapping E(t) into the frequency domain through the Fourier transforms E( f ) = F [E(t)] and e( f ) = F [e(t)], one obtains  N −1 1 − e−i N 2π f T e−in2π f T = e( f ) . (4.133) E( f ) = e( f ) n=0 1 − e−i2π f T The frequency-dependent output intensity I ( f ) ∝ |E( f )|2 is then given by

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4 Applications of WGM Microcavities in Physics

Fig. 4.16 Optical frequency combs. a Pulses output from a mode-locked laser. b Frequency spectra of a pulse strain with different numbers of pulses. c Frequency comb spectrum

I ( f ) = i( f )

sin2 (N π f T ) , sin2 (π f T )

(4.134)

with i( f ) ∝ |e( f )|2 . In the limit of N → ∞, the second term on the right side of the above equation may be replaced by a Dirac comb, i.e., an impulse train [see Fig. 4.16b] 1 ∞ sin2 (N π f T ) → δ( f − n fr ). (4.135) lim 2 n=0 N →∞ sin (π f T ) 2π T Thus, the spectrum I ( f ) presents multiple equally spaced teeth [see Fig. 4.16c] and is potentially used as a frequency ruler. Indeed, the train of optical pulses directly results from the phase-coherent superposition of a great number (105 ∼ 106 ) of continuous-wave longitudinal cavity modes (corresponding to the comb teeth).

4.6 Optical Frequency Combs

215

For the application as a frequency ruler, the following two essential prerequisites need to be done: (1) Control of the carrier-envelope phase. A single light pulse e(t) can be described as the product of a fast oscillating carrier wave and a slowly varying envelope. When a light pulse passes through the intracavity devices, the inevitable dispersion causes a difference between the phase (carrier) and group (envelope) velocities, leading to an extra carrier-envelope phase Δϕ [see Fig. 4.16c]. Consequently, an offset frequency f 0 is introduced into the frequency comb. The n-th comb tooth has a frequency f n = n fr + f 0 with f 0 =

Δϕ fr . 2π

(4.136)

The offset f 0 , which is also in the microwave frequency domain, affects the absolute optical frequencies of the whole comb and must be stabilized. The stabilization of f 0 can be implemented by using the self-referencing technique [54]. In this method, the double frequency 2 f n of a low-frequency comb line f n is compared with the high-frequency comb line f 2n , giving rise to a beat signal at 2 f n − f 2n = f 0 . Thus, the optical spectrum i( f ) of single pulse is required to span over an optical octave, covering both f n and f 2n . Typically, the pulses output from the Ti:S laser is centered at 830 nm (361 THz) with a width of 70 nm (corresponding a spectral width of 30 THz or a pulse length of 10 fs). One needs to find a way to significantly stretch the pulse spectrum. This may be implemented by letting the laser pulses pass through an air-silica microstructure fiber, i.e., the so-called supercontinuum generation [55], resulting in a spectrally broadened pulse spectrum from 510 to 1125 nm. Setting the f 2n comb line at 520 nm and the f n comb line at 1040 nm, one may derive the offset f 0 via a beat-note measurement with a local microwave frequency standard (e.g., hydrogen maser). The carrier-envelope phase locking can be performed by feeding the error signal to an acousto-optic modulator that controls the pump power of the Ti:S laser; (2) The repetition rate fr with a typical value ∼100 MHz also drifts due to the fluctuations in the cavity length, and it needs to be stabilized as well. Measuring fr is straightforward by counting the pulses with a fast photodiode. The frequency fr may be also locked to the local microwave frequency standard by changing the length of the laser cavity via a mirror mounted on an intracavity piezoelectric transducer. After fixing two degrees of freedom f 0 and fr , an optical frequency comb is ready to measure the absolute frequency of optical waves. Since both f 0 and fr are phase-locked to a microwave frequency standard, the stability and accuracy of the frequency comb are entirely determined by this microwave frequency standard. The frequency comparison among different OFCs has experimentally testified an uncertainty of optical frequency synthesis at a 10−19 level [56], sufficiently low for the current applications in optical frequency metrology and precision measurement. Beating an independent optical frequency standard (e.g., Sr optical lattice clock at 698 nm) with its nearest comb tooth, one can evaluate the relative stability between optical and microwave frequency standards. Thus, the OFCs play a role of a bridge between optical and microwave domains.

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4 Applications of WGM Microcavities in Physics

Fig. 4.17 WGM-based OFCs. a Typical structure. b Frequency comb spectrum spanning over 160 THz with an inter-comb-tooth spacing of 850 GHz. Reprinted with permission from [58]. Copyright (2011) by the American Physical Society

Although the conventional mode-locked OFCs have been successfully commercialized and widely used, further shrinking the device footprint and increasing the repetition rate fr over 10 GHz would be advantageous to many specific applications in the integrated photonics and telecommunications. Besides the mode locking, the OFCs can be also produced through the continuous-wave pumped nonlinear parametric frequency conversion. The nonlinear optical processes based on the WGM microcavities benefit from a long interaction length and a low threshold (∼10 µW) as well as a free spectral range f FSR of (10 ∼ 103 ) GHz [57]. Thus, the WGM microcavities are ideally suited for the miniaturized OFCs. The typical structure of WGM-based OFCs is illustrated in Fig. 4.17a, where an amplified continuous-wave pump laser drives a nonlinear microcavity. As discussed in Sect. 4.3.6, the degenerate FWM (based on SiO2 , Si3 N4 and CaF2 ) can create one signal photon at f s and one idler photon at f i via annihilating a pair of pump photons at f p with f p − f i = f s − f p . When f s and f i coincide with two microcavity’s WGMs and the pump beam exceeds the threshold, the parametric oscillation occurs, generating the signal and idler sidebands equally spaced with respect to f p . Additionally, the higher pump power (∼ GW/cm2 ) may further lead to the nondegenerate FWM among the pump and signal and idler sidebands, resulting in new sidebands, for instance, f p + f s − f i . The successive nondegenerate FWM processes finally produce a group of phase-coherent sidebands, i.e., frequency combs.

4.6 Optical Frequency Combs

217

In principle, due to the variation of f FSR caused by the dispersion of the cavity material, the generated frequency combs are not mutually equidistant and the spectral distribution of frequency combs are limited within a finite bandwidth. Nevertheless, the experimental measurement displays an octave-spanning frequency comb spectrum [58]. This is because, besides the microcavity’s dispersion, the Kerr effect also shifts the spectral positions of frequency combs, i.e., the nonlinear optical mode pulling. As a result, the frequency comb bandwidth overcomes the limits imposed by the dispersion [see Fig. 4.17b]. The applications of OFCs in metrology require the stabilization of f 0 and fr . Unlike the conventional OFCs, the microcavities do not have any movable elements that tune the frequency combs. As demonstrated in [59], a full stabilization of a WGM-based OFC can be achieved through feedback controlling both frequency and power of the pump laser. Since the generation of a WGM-based OFC begins with the degenerate FWM, the offset f 0 can be directly accessed via phase locking the frequency of the pump laser to an optical frequency reference. In contrast, the second degree of freedom fr is controlled by changing the optical path length of the microcavity. The round-trip time of an optical wave circling inside the microcavity is determined by the effective refractive index. Besides the microcavity’s dispersion, the Kerr and thermal (i.e., the absorption of laser power heats the microcavity) effects also change the refractive index of the microcavity. These two effects are pump-powerdependent. Hence, fr may be stabilized via controlling the pump power launched into the microcavity.

4.7 Optomechanics This section is dedicated to the application of WGM microcavities in the optomechanics that studies the interaction of the electromagnetic radiation with the mechanical systems. The optomechanics is a rapidly developing field of research, including the precise readout of small displacements, cooling the mechanical motion, producing the optomechanical Schrödinger cat state, etc. Below, we introduce some basic concepts in this rich field.

4.7.1 Classical Description Figure 4.18 depicts a generic optomechanical system based on a Fabry–Pérot cavity. An input signal sin at the frequency ω1 drives an optical cavity mode. A small onedimensional mechanical displacement x(t) is introduced to the cavity mirror that is free to move. The dependence of the cavity-mode frequency on x(t) is given by ω0 (t) = ω0 + ξ x(t), with ξ =

∂ω0

, ∂x

(4.137)

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4 Applications of WGM Microcavities in Physics

Fig. 4.18 Schematic diagram of a Fabry–Pérot optomechanical system

where ω0 is the resonance frequency of the cavity mode for x = 0. We have the coefficient of proportionality ξ = −ω0 /L with the cavity length L. For a WGM microcavity with a radius R, ξ approximates ξ = −ω0 /R. Using (3.94), the equation of motion for the cavity-mode amplitude a(t) is derived as   √ ω0 d a(t) = − − i (ω0 + ξ x(t)) a(t) + κsin e−iω1 t , dt 2Q

(4.138)

where Q is the cavity quality factor and κ is the coupling rate of the input field into the cavity. For simplicity, we assume the displacement x(t) follows a sinusoidal oscillation at the frequency Ω, x(t) = x0 sin Ωt.

(4.139)

Here, x0 represents the oscillation amplitude. We further assume that the maximum frequency shift ξ x0 is much smaller than the mechanical oscillation frequency Ω, i.e., β ≡ ξ x0 /Ω  1. No backaction. We first consider the simple case, where the backaction of the light on the displacement x(t) is neglected. The perturbation theory may be applied to solve (4.138). The intracavity-light amplitude a(t) is expanded in series of the modulation index β (4.140) a(t) ≈ a0 (t) + βa1 (t) + O(β 2 ). The zero- and first-order amplitudes fulfill the equations of motion   √ ω0 d a0 (t) = − − iω0 a0 (t) + κsin e−iω1 t , dt 2Q   d ω0 a1 (t) = − − iω0 a1 (t) + (−iΩ sin Ωt) a0 (t), dt 2Q respectively. Solving (4.141), one obtains the approximate solution

(4.141a) (4.141b)

4.7 Optomechanics

219

⎡ ⎤ √ −i(ω1 +Ω)t −i(ω1 −Ω)t e e κsin ⎢ −iω1 t βΩ βΩ ⎥ − a(t) ≈ ω + ⎣e ⎦, ω0 ω0 0 2 2 − iΔ − i(Δ + Ω) − i(Δ − Ω) 2Q 2Q 2Q (4.142) with the detuning Δ = ω1 − ω0 of the driving signal sin . It is seen that the extra timedependent displacement x(t) gives rise to a pair of sidebands at ω1 = ω0 ± Ω. The intensity of the intracavity field I (t) = |a(t)|2 ≈ |a0 (t)|2 + β a0∗ (t)a1 (t) + a1∗ (t) a0 (t)] reads

 κ|sin |2 1 + βΩΞ+ (Δ, Ω) sin Ωt + βΩΞ− (Δ, Ω) cos Ωt , 2 2 Δ + (ω0 /2Q) (4.143) where we have defined the functions I (t) =

( f1 + f2 ) ( f1 − f2 ) + , (4.144a) 2 2 ( f 1 + f 2 ) + (ω0 /2Q) ( f 1 − f 2 )2 + (ω0 /2Q)2 ω0 /2Q ω0 /2Q Ξ− ( f 1 , f 2 ) = − . (4.144b) 2 2 ( f 1 + f 2 ) + (ω0 /2Q) ( f 1 − f 2 )2 + (ω0 /2Q)2

Ξ+ ( f 1 , f 2 ) =

Due to the nonzero cavity decay ω0 /2Q, the oscillation of I (t) is not in-phase with the mechanical oscillation x(t). Backaction. We now take into account the backaction of the cavity mode on the mechanical degree of freedom. The displacement x(t) may be modeled by a spring– mass oscillator, whose motion is governed by d F d2 x(t) + Γ x(t) + Ω 2 x(t) = , dt 2 dt m

(4.145)

with the damping rate Γ and the effective mass m. The driving force F arises from the radiation pressure of the cavity mode, i.e., the momentum transfer of the intracavity photons to the movable cavity mirror, F = |a(t)|2

2ω0 /c = −ξ |a(t)|2 . 2L/c

(4.146)

Consequently, (4.138) and (4.145) form a coupled system. Substituting a(t) = −iω1 t a(t)e ˜ into (4.138) and setting t → ∞, one obtains the steady-state solutions

a˜

(ss)

x (ss)

√

κsin = ω , 0 − i(Δ − ξ x (ss) ) 2Q ξ =− |a˜ (ss) |2 . mΩ 2

(4.147a)

(4.147b)

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4 Applications of WGM Microcavities in Physics

For a suitable pair of (Δ, sin ), the intracavity field a˜ (ss) may have three possible values. However, only two of them are physically acceptable while the last one is actually unstable. Such a nonlinear behavior is known as the bistability (see Fig.4.7). By writing a(t) ˜ and x(t) as the steady-state solutions plus small fluctuations ˜ a(t) ˜ = a˜ (ss) + δ a(t), (ss) x(t) = x˜ + δx(t),

(4.148a) (4.148b)

Equations (4.138) and (4.145) can be simplified as     ω0 d δ a(t) ˜ = − + i Δ − ξ x (ss) δ a(t) ˜ − iξ a˜ (ss) δx(t), (4.149a) dt 2Q  2   δ F(t) d d ξ  (ss) ∗ 2 + Ω a˜ δ a˜ (t) + a˜ (ss)∗ δ a(t) , + Γ ˜ + δx(t) = − 2 dt dt m m (4.149b) where we have artificially inserted an extra weak force δ F that is exerted on the mechanical oscillator. The Fourier transformations δ a(ω) ˜ = F [δ a(t)] ˜ and δx(ω) = F [δx(t)] convert the differential equations (4.149) to the algebraic equations −iξ a˜ (ss) δ a(ω) ˜ = ω δx(ω), 0 + i(ω − Δ + ξ x (ss) ) 2Q   2 δ Fr p (ω) δ F(ω) + . −ω + iΓ ω + Ω 2 δx(ω) = m m

(4.150a)

(4.150b)

The radiation pressure δ Fr p (ω) induced by the intracavity photons reads   ˜ δ Fr p (ω) = −ξ a˜ (ss) δ a˜ ∗ (ω) + a˜ (ss)∗ δ a(ω)  = −ξ 2 |a˜ (ss) |2 Ξ+ (Δ − ξ x (ss) , ω)  +iΞ− (Δ − ξ x (ss) , ω) δx(ω).

(4.151)

∗ ˜ has been used in deriving δ Fr p (ω). Finally, we find The fact δ a˜ ∗ (ω) = [δ a(−ω)] the response of the mechanical oscillator to the external force δ F(ω)

δx(ω) = χeff (ω)δ F(ω),

(4.152)

with the effective susceptibility −1

 2 . χeff (ω) = m −ω2 + iΓeff ω + Ωeff

(4.153)

The modified damping rate and oscillation frequency are given by Γeff = Γ + γ and Ωeff = Ω 2 + k/m with the backaction-induced damping and spring constants of

4.7 Optomechanics

221

the spring mass ξ 2 |a˜ (ss) |2 Ξ− (Δ − ξ x (ss) , ω), mω k = ξ 2 |a˜ (ss) |2 Ξ+ (Δ − ξ x (ss) , ω).

γ =

(4.154a) (4.154b)

The value of γ may be positive or negative, depending on the detuning Δ. That is to say the backaction can enhance or suppress the damping rate of the harmonic oscillator. The other direct effect of the backaction is shifting the oscillation frequency. In the limit of weak intracavity light, such a frequency shift depends linearly on the cavity-mode intensity. The above classical quantitative analysis only gives us the general concepts on the backaction in optomechanics. A more detailed analysis may be performed by using the so-called quantum Langevin approach [60].

4.7.2 Acoustic Modes in Microcavities The mechanical oscillation in WGM microcavities arises from the deformation of the cavity structure, which may be described by a displacement vector field u(r, t). For a free microcavity (i.e., none external forces are exerted on the microcavity), u(r, t) follows the equation of motion [61] ρ

∂2 u(r, t) = (λ + 2μ)∇∇ · u(r, t) − μ∇ × ∇ × u(r, t), ∂t 2

(4.155)

with the density ρ and the Lamé constants λ=

σE E , μ= . (1 + σ )(1 − 2σ ) 2(1 + σ )

(4.156)

The Poisson’s ratio σ and Young’s modulus E characterize the elastic properties of the cavity material. The boundary condition of the finite-size microcavity enables us to decompose u(r, t) into multiple acoustic modes u j (r) u(r, t) =

 j

c j u j (r)e−iΩ j t ,

(4.157)

with the amplitudes c j , the intrinsic vibration frequencies Ω j , and the orthonormalization

1 u∗ (r) · u j (r)dr = δ j j . (4.158) V V j The effective mass for the j-th mode can be evaluated from

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4 Applications of WGM Microcavities in Physics

m eff, j

 2 V ρ|u j (r)| dr  = , | V w(r) · u j (r)dr|2

(4.159)

where a weighting function w(r) is introduced to map the vector field u j (r) to a scalar displacement in the WGM equator. This scalar displacement is around the major r = R and along the radial direction er . For a microsphere, (4.155) has the following analytical solution: u(r, t) = ∇Φ0 (r, t) + ∇ × Φ 1 (r, t) + ∇ × ∇ × Φ 2 (r, t).

(4.160)

Two vector potentials Φ 1,2 are expressed as Φ 1,2 = (r Φ1,2 , 0, 0) in spherical coordinates. Three scalar potentials Φq=0,1,2 (r, t) take the form Φq =



(q) Anlm jl



nlm

 Ωnlm r Ylm (θ, ϕ)e−iΩnlm t , vq

(4.161)

(q)

with the amplitudes Anlm , the spherical Bessel functions jl , and the spherical harmonics Ylm . The longitudinal v0 and transverse v1,2 sound velocities read  v0 =

λ + 2μ , v1 = v2 = ρ

%

μ , ρ

(4.162)

respectively. The radial n, polar l, and azimuthal m numbers are used to specify an acoustic mode. As an example, we consider the lowest-order mode (n, l, m) = (1, 0, 0) and have u0 (r) = u r (r)er =

A r2



 Ω Ω Ω r cos r − sin r er , v0 v0 v0

(4.163)

where Ω1,0,0 has been simplified by Ω. The boundary condition requires that the stress vanishes at r = R, which leads to   u r (r ) ∂ = 0. (4.164) (λ + 2μ) u r (r ) + 2λ ∂r r r =R Consequently, one obtains the characteristic equation for the frequency Ω   v02 Ω 2 2 tan Ω R/v0 1− 2 2 R = 1. Ω R/v0 4v1 v0

(4.165)

For a microsphere made of fused silica, we have σ = 0.17, E = 74.8 × 109 Pascals (Pa), and ρ = 2.2 × 103 kg/m3 . The Lamé constants are calculated to be λ = 1.6 × 1010 Pa and μ = 3.2 × 1010 Pa with the corresponding velocities v0 = 6.0 km/s and v1 = 3.8 km/s. Solving (4.165) yields Ω R/v0 = 2.4. For R = 25 µm, the

4.7 Optomechanics

223

Fig. 4.19 Mechanical modes in a silica toroidal cavity. ©IOP Publishing and Deutsche Physikalische Gesellschaft. Reproduced from [62] by permission of IOP Publishing. CC BY-NC-SA

lowest acoustic frequency is Ω = 2π × 92.3 MHz. The weighting function may be approximated by w(r) = R −1 δ(r − R)δ(θ − π/2)δ(ϕ)er , and the effective mass for the lowest-order mode is estimated to be m eff = 8470 × R 3 kg. For R = 25 µm, we have m eff = 1.3 × 10−10 kg. For the microcavities with complicated structures, the quantitative analysis is challenging. Thus, the numerical methods (e.g., the finite-element method [62]) are necessary. Figure 4.19 displays several acoustic modes in a silica microtoroid. The weighting function for the microtoroid takes the approximate form w(r) = (2π R)−1 δ(r − R)δ(θ − π/2)er .

4.7.3 Optical Measurement of Mechanical Motion The cavity’s mechanical motion may be detected by measuring the light output from the microcavity. Various noise sources influence such a measurement. The shot (quantum) noise of counting photons restricts the optical phase readout with a √ variance Δϕ 2  = 1/2 κτ n cav . Here, n cav is the average intracavity photon number that is proportional to the input power |sin |2 and τ is the integration time (τ −1 gives the measurement bandwidth). This optical-phase uncertainty induces an imprecision in the displacement measurement Δx 2  = (ω0 /Q) Δϕ 2 /ξ . Consequently, the quantum-noise-limited imprecision noise spectral density is expressed as  Sximp x (

f) =

ω0 Q

2

1 (ω0 /2Q)2 + (2π f )2 , 4ξ 2 κτ n cav (ω0 /2Q)2

(4.166)

where the last term originates from the response of the optical cavity to the mechanical fluctuation at the frequency f . Actually, the microcavity is a low-pass filter that suppresses the fluctuation components with 2π f > (ω0 /2Q) to the optical phase measurement. In addition, the backaction noise force (4.151) affects the intrinsic acoustic modes of the microcavity and the corresponding spectral density takes the form [63] 2 ξ 2 n cav τ (ω0 /2Q)2 SF F ( f ) = . (4.167) (ω0 /Q) (ω0 /2Q)2 + (2π f )2 Immediately, one finds the following uncertainty relation:

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4 Applications of WGM Microcavities in Physics

Fig. 4.20 Measurement of the displacement noise. a Spectra of different noise components at f = Ω/2π as a function of the intracavity photon number n cav with ω0 /κ Q = 1. b Spectrum of the displacement noise in a silica toroidal cavity. Symbols: mec (mechanical), them-ref (thermorefractive), and ele-det (electronic-detector) noises. ©IOP Publishing & Deutsche Physikalische Gesellschaft. Reproduced from [62] by permission of IOP Publishing. CC BY-NC-SA

Sximp x ( f )S F F ( f ) =

2 ω0 2 ≥ . κQ 4 4

(4.168)

As a result, the total noise spectrum reads   2 th Sx x ( f ) = Sximp x ( f ) + |χ ( f )| S F F ( f ) + S F F ( f ) ,

(4.169)

'−1 &

is the mechanical susceptibilwhere χ ( f ) = m −(2π f )2 + iΓ (2π f ) + Ω 2 ity of the microcavity. The nonzero temperature T = 0 also induces the thermal Brownian force noise with the spectral density S FthF ( f ) = mΓ (2π f ) coth

(2π f ) ≈ 2mΓ k B T, 2k B T

(4.170)

where the last approximation is valid for k B T  (2π f ). The imprecision noise predominantly restricts the measurement at a lower intracavity photon number n cav , while the backaction contributes mostly in the regime of larger n cav [see Fig. 4.20a]. imp The standard quantum limit of the displacement detection is defined by Sx x ( f ) = |χ ( f )|2 S F F ( f ) at the zero temperature T = 0 and we have % imp 2 SxSQL x ( f ) = Sx x ( f ) + |χ ( f )| S F F ( f ) =

ω0 |χ ( f )|. κQ

(4.171)

Figure 4.20b illustrates a displacement noise spectrum measured in experiment. The multiple spikes correspond to different mechanical modes. The shot noise exceeds the electronic-detector noise. The strong background (higher than the shot noise) in the low-frequency regime is caused by the thermorefractive noise.

4.7 Optomechanics

225

4.7.4 Optomechanical Cooling The radiation pressure may be used to suppress the mechanical motion of a microcavity. We introduce the temperature T for an acoustic mode via kB T mΩ 2 = Δx 2 . 2 2

(4.172)

When the coupled system is in equilibrium, the mode temperature T is given by [63] T =

Γ Ti , Γ +γ

(4.173)

with the initial temperature Ti of the acoustic mode and the backaction-induced damping rate γ =

 ξ 2 |a˜ (ss) |2  − ω0 /Q A − A+ with A± = . 2mΩ (Δ ∓ Ω)2 + (ω0 /2Q)2

(4.174)

The positive γ (resulting from the red detuning Δ = ω1 − ω0 < 0) leads to T < Ti and is often referred to as the cooling rate [see Fig. 4.21a]. In contrast, the optomechanical system undergoes a heating process in the blue-detuned regime Δ > 0. Figure 4.21b presents an example of the experimentally observed suppression of the displacement noise. The cooling process is accompanied by the reduction of the phonon (quantum of mechanical-motion  energy) number. The minimum phonon  number is given by n ph = A+ / A− − A+ , and one has  n ph =

1 ω0 4Ω Q

2  1,

(4.175)

in the limit Ω  (ω0 /Q). More detailed analysis on the optomechanical cooling relies on the quantum mechanical theory and can be found in [63].

4.8 Nanoparticle Trapping In the last section of this chapter, we briefly introduce the WGM-based nanoparticle trapping. When a small particle with a dipole moment p is placed in an inhomogeneous light field E(r, t), the optical potential the particle experiences is given by U (r) = −p(r, t) · E(r, t) = −

1 Re(α)I (r), 2ε0 c

(4.176)

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4 Applications of WGM Microcavities in Physics

Fig. 4.21 Optomechanical cooling. a Backaction-induced damping rate γ versus the detuning Δ with κ = ω0 /2Q. b Cooling of the mechanical motion of a silica microtoroid. Reprinted from [64] under the terms of the Creative Commons Attribution 3.0 License

with the linear polarizability of the particle α and the light intensity I (r). For Re(α) > 0 (Re(α) < 0), a local maximum (minimum) of I (r) leads to a potential well. A large enough optical force F = −∇U (r) may trap the particle inside this potential well. We know that the outside-microcavity field of a WGM decreases to zero within a few-wavelength distance away from the microcavity’s surface. This results in a strong gradient force in the vicinity of the surface of the microcavity. Consequently, a nanoparticle can be confined within a carousel trap around the microcavity as depicted in Fig. 4.22a. Such a WGM-based nanoparticle trapping was firstly demonstrated in [65]. In this experiment, a 1060-nm tunable laser excites a WGM of a bare silica microsphere (radius 53 µm and Q = 1.2 × 106 ) in an aqueous (heavy water) environment via a tapered fiber. The resonance wavelength is monitored through tracking the transmission dip recorded by a photodiode. The molar concentration of the nanoparticles suspended in the heavy water is about 10−15 mol/L. When a nanoparticle is trapped around the microsphere, the weak nanoparticle–microcavity interaction gives rise to a wavelength shift δλ of the WGM relative to the resonance wavelength of an unperturbed microsphere (see next chapter). Besides a deterministic propulsion coming from the WGM momentum flux, the nanoparticle within the carousel trap also undergoes a Brownian motion. This can be manifested by the elastic scattering signal from the single trapped nanoparticle [see Fig. 4.22b] as well as the trajectory of δλ [see Fig. 4.22a]. The wavelength shift δλ is a function of the separation h between the nanoparticle and the microsphere’s surface. Figure 4.22c shows a separation histogram derived from the trajectory of δλ when a nanoparticle is circumnavigating the microsphere. A pronounced maximum of δλ is located at a nonzero separation h = 0, indicating a superposition of a repulsive and an attractive interaction exerted on the nanoparticle. The repulsive interaction is independent of the light power and is caused by the repulsion between the ionized silanol groups on the bare silica surface and the neg-

4.8 Nanoparticle Trapping

227

Fig. 4.22 WGM-based nanoparticle trapping. a Schematic diagram of a WGM carousel trap. A WGM of a microsphere is excited by a tunable laser. A nanoparticle (NP) is confined inside a carousel trap, leading to a wavelength shift δλ of the WGM. A photodiode (PD) measures the transmitted light and the resonance wavelength is tracked. b Elastic scattering image of a polystyrene particle (radius a = 375 nm) trapped and circumnavigating the microsphere at a velocity of 2.6 µm/s. c Separation histogram of the single tapping event of a polystyrene particle and the corresponding potential plot Utot /k B T versus the separation h between the nanoparticle and the microsphere’s surface. All figures are reprinted with permission from [65] ©The Optical Society

atively charged nanoparticle. The attractive interaction results from the polarization potential that is similar to (4.176). The whole potential Utot the nanoparticle experiences may be evaluated by using the equilibrium statistical mechanics, U (h) = −k B T ln[ p(h)/ p(h r e f )]. The probability density p(h) is derived from the separation histogram [see Fig. 4.22c] and the reference separation h r e f is determined by U (h r e f ) = 0. The resulting Utot is also presented in Fig. 4.22c. It is seen that Utot consists of two components, Utot (h) = Ur ep (h) + Uatt (h), where the short-range repulsive potential Ur ep (h) = Ur ep,0 exp(−h/ h r ep ) is positive Ur ep,0 > 0 while the long-range potential Uatt (h) = Uatt,0 exp(−h/ h att ) is negative Uatt,0 < 0 and h att > h r ep .

Problems 4.1 A general cubic equation in one variable takes the form ax 3 + bx 2 + cx + d = 0,

(4.177)

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4 Applications of WGM Microcavities in Physics

Fig. 4.23 Bistable region

with a, b, c, d ∈ R and a = 0. Three roots of (4.177) may be found via the Cardano’s method. By defining p=

27a 2 d − 9abc + 2b3 3ac − b2 , q= , 2 3a 27a 3

(4.178)

the discriminant D of the cubic equation is defined as D=

 q 2 2

+

 p 3 3

.

(4.179)

If D < 0, all three roots are real and distinct; if D = 0, there are three real roots and at least two of them are equal; and if D > 0, the cubic equation has one real and two complex conjugate roots. Use (4.179) to identify the bistable region of (4.54) as shown in Fig. 4.23.

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Chapter 5

Single-Molecule Sensing

Abstract The dielectric nanoparties/molecules in the vicinity of a WGM microcavity evanescently perturb the WGM field. Owing to the tiny mode volume and high quality factor of the microcavity, the resulting resonance shift and spectral broadening of the microcavity’s transmission spectrum can be read out with a high enough signal-to-noise ratio. This property has been utilized to monitor the chemical/biological reaction, such as the hybridization of DNA, and the concentration change of an analyte in an aqueous solution. Such a detection is label free and can reach the single-molecule level by means of the localized surface plasmon resonance of metal nanoparticles. The WGM-microcavity sensors offer the advantages of miniaturization, simplicity, high sensitivity, and a real-time dynamic response, potentially paving a way to achieve a lab-on-a-chip sensing device.

5.1 Introduction This chapter is devoted to the applications of the WGM microcavities in biosensing. Generally, the biosensing mechanisms include the highly sensitive interferometry [1], the surface plasmon resonance (SPR) [2, 3], and the high-Q WGMs supported by the micron-sized optical cavities [4]. The chemical/biological reaction and the change of the concentration of an analyte may lead to a change in the refractive index of the sample, which further induces a SPR/(micro)cavity resonance shift in the optical scattering/transmission spectrum. In comparison, the WGM microcavities possess the advantages of simple configuration, miniature size, and multiple-time light-analyte interaction. Also, the features of the tiny mode volume and the high quality factor enable the WGM-based sensing detection at single nanoparticle/molecule level. Hence, the WGM microcavities pave a potential way to establish a compact, portable, and robust lab-on-a-chip sensing device. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2_5

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5 Single-Molecule Sensing

Fig. 5.1 Sensing mechanism based on the spectral shift and broadening

Typically, the optical wave can circulate along the inner surface of a WGM microcavity with Q ∼ 106 for over one thousand times, corresponding to an optical path length ∼0.1 m. Any perturbation that varies the optical path length may be reflected in the emission/transmission spectrum of the active/passive microcavity. This perturbation can be induced by a small change in the refractive index of the medium/solution outside the microcavity, a dielectric/metal nanoparticle touching the microcavity, and the molecules binding to the surface of the microcavity. Intuitively, the interaction between particles/molecules and optical microcavities have two consequences: (1) The effective radius of the microcavity is increased and the WGM wavelength is shifted toward the long wavelength side, i.e., red shift, as shown in Fig. 5.1; and (2) The perturbation introduces an extra decay channel to the intracavity light field, broadening the spectral linewidth of WGMs. Usually, the spectral shift of the cavity resonance is monitored in real time. Sometimes, the spectral broadening of WGMs is also utilized. In this chapter, we first introduce the basic principle of the sensing mechanism and the knowledge of the kinetic and thermodynamic analysis on the biological/chemical reactions. After that, we illustrate several selected experimental schemes of the sensing detection of nanoparticles, molecules, virus, and DNA hybridization. In the last section, we discuss the optoplasmonic sensing, where the localized surface plasmon resonance (LSPR) of metal nanoparticles is employed to enhance the evanescent field in the vicinity of the microcavity surface, allowing one to access the single particle/molecule detection level.

5.2 Sensing Mechanism The sensing mechanism based on the optical wave is to monitor the change in the spectral properties of the light, such as, intensity, polarization, central wavelength λ0 , and linewidth Δλ, so as to indirectly detect the change in the analyte. The heart of

5.2 Sensing Mechanism

235

a sensor is a high-Q microcavity that is excited evanescently by the guided wave in an optical fiber or through a prism. Due to the strong light confinement and the high quality factor of a WGM, the shift δλ in wavelength is commonly employed in the sensing measurement. Such a spectral shift, which can well exceed the linewidth Δλ of the WGM, may arise from the variations in temperature, pressure, and the concentration of the analyte in a sample as well as the nanoparticles/molecules binding to the surface of a microcavity. In some situations, the change in the spectral broadening δ(Δλ) is also utilized to measure the perturbation of nanoparticles/molecules to the microcavities. Indeed, the change in the spectrum of a microcavity can be ascribed to the variation in the refractive index of its surrounding medium. According to (3.40), the sensitivity of a WGM to the variation in the refractive index n m of the medium outside the microcavity is given by  

∂λ ∂n m

∂λ ∂n m

 

TE

TM

  αn nm n 2s λ20 −2/3 1 + 1/3 2 , = (l + 1/2) 2π R (n 2s − n 2m )3/2 2 n s − n 2m  nm λ2 αn (2n 2s − n 2m ) + 1/3 = 0 2 2 2 3/2 2π R n s (n s − n m ) 2  2n 6s + n 4s n 2m − 4n 2s n 4m + 2n 6m −2/3 , (l + 1/2) × n 2s (n 2s − n 2m )

(5.1a)

(5.1b)

with the vacuum wavelength λ0 , the radius of microcavity R, the refractive index of the cavity material n s , the polar number l, and the n-th root αn of the Airy function Ai(−α). Equation (5.1) suggests that reducing the microcavity size and the refractive index contrast between n s and n m can enhance the sensor’s sensitivity. Actually, this results from the extended evanescent tail, which, however, also raises the microcavity’s leakage. For a fused silica microsphere (n s ∼ 1.47 and R = 100 µm) in aqueous environment (n m ∼ 1.33), the refractometric sensitivity is evaluated to be ∼10 nm/RIU (refractive index units) [see Fig. 5.2a, b]. Several experimentally demonstrated sensitivities [5–10] are listed in Table 5.1. Two approaches, bulk sensing and surface sensing, are generally used. In the former method, the sensor is directly immersed in a liquid sample that contains the analyte. Basically, the analyte is distributed homogeneously in the solution. The change in the analyte concentration modifies the bulk refractive index of the solution. However, the variations in the concentrations of other substances in the solution may lead to a same change in the bulk refractive index. Thus, it is unable to identify the molecular species of interest. Addressing this issue relies on the surface sensing, where the microcavity’s surface is modified with the receptors that can selectively bind the specific analyte. The refractive index change occurs only near the cavity’s surface. The surface sensing is usually measure in units of picogram per square millimeter.

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5 Single-Molecule Sensing

Fig. 5.2 Refractometric sensitivity calculated from (5.1). a Resonance shift versus the change in refractive index n m for a microsphere with n s = 1.47 and R = 100 µm. The microcavity is immersed in a solution with n m ∼ 1.33. The wavelength of the WGM with α1 is λ0 = 1 µm. b Sensitivity ∂λ/∂n m as a function of n s . Other parameters are same to a Table 5.1 Refractometric sensitivities of different microcavities. The sensitivity is in units of nm/RIU Microcavity Cavity material Analyte Sensitivity References Capillary

Capillary Microsphere

Microsphere Microsphere

Silica (coated with dye-doped polymer) Fused silica Quantum-dotembedded polystyrene Crystalline MgF2 Fused silica

Glucose

30

[5]

Ethanol Ethanol

570 160

[6] [7]

Glycerol Ethanol

1.09 30

[8, 9] [10]

5.2.1 WGM Microcavity Perturbed by Single Nanoparticles The simplest sensing structure is composed of a WGM microcavity and a single nanoparticle located close to the microcavity’s surface as shown in Fig. 5.3a. This nanoparticle can be a dielectric bead, virus, bacteria, and DNA molecule. Generally, the nanoparticle has a size less than the penetration depth of the evanescent wave and can be polarized by the evanescent field, thereby decreasing the WGM energy. Usually, the amount of the consumed light energy accounts for only a small section of the total WGM energy. Thus, the nanoparticle’s polarization acts as a perturbation to the microcavity and the WGM wavelength suffers a red shift that is determined by the excess polarization energy of the nanoparticle. In the following, we study the perturbation of a single nanoparticle on the WGM via a general approach. We consider a microcavity stays in the space without any free electric charges or currents. The space is full of a homogeneous dielectric medium whose relative dielectric constant is m . For the free space, m is equal to unity. According to Maxwell’s

5.2 Sensing Mechanism

237

Fig. 5.3 Perturbation induced resonance shift of a WGM. a Perturbation induced by single nanoparticles/molecules. b Perturbation induced by a nanolayer

equations, one may derive the following wave equation for a microcavity with a relative permittivity s : ∇ 2 E(r, t) −

(r) ∂ 2 E(r, t) = 0, c2 ∂t 2

(5.2)

where (r) denotes the distribution of the relative permittivity, i.e., (r) = s ((r) = m ) when r is located inside (outside) the microcavity. The j-th eigenmode of the microcavity takes the form with the separated spatially and temporally dependent parts, eE j (r)e−iω j t . The complex eigenvalue ω j has Re(ω j ) > 0 and Im(ω j ) < 0 and is related to the quality factor via Q = Re(ω j )/2Im(ω j ). The eigenvalue equation for the spatial amplitude reads ∇ 2 E j (r) + (r)

ω 2j c2

E j (r) = 0.

(5.3)

For simplicity, both (r) and ω j are assumed to be real, i.e., neglecting the weak absorption from the microcavity medium and the small confinement loss of the microcavity. Similarly, we use  (r) to denote the spatially dependent relative permittivity when a nanoparticle is placed close to the microcavity’s surface. This nanostructure can be in an arbitrary shape and has a frequency-dependent dielectric function  p . One may treat this combined system as a modified microcavity and focus on a certain eigenmode E 0 (r) of this microcavity. The corresponding eigenvalue equation reads as (ω  )2 (5.4) ∇ 2 E 0 (r) +  (r) 02 E 0 (r) = 0, c with the mode’s frequency ω0 . The completeness of {E j (r); j ∈ Z} allows us to expand E 0 (r) as

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5 Single-Molecule Sensing

E 0 (r) =

 j

a j E j (r).

(5.5)

Inserting (5.5) into (5.4), one obtains  j

   a j (r) ω 2j − (ω0 )2 E j (r) = (ω0 )2 a j  (r) − (r) E j (r). j

(5.6)

In addition, the orthogonality of {E j (r); j ∈ Z} indicates that

V

(r)E ∗j1 (r)E j2 (r)dr = 0 only when j1 = j2 ,

(5.7)

for which (5.6) leads to cj

ω 2j − (ω0 )2 (ω0 )2

=



   ∗  V  (r) − (r) E j (r)E j (r)dr   . c j 2 j V (r)|E j (r)| dr

(5.8)

Here, V denotes the volume of the whole space. It is worth noting that the above formalism is applicable to an arbitrary microcavity–nanoparticle coupling strength. We now restrict ourselves to the weak-coupling limit, where the nanoparticle slightly affects the microcavity’s eigenmodes. That is to say, the overlap    ∗ 2 (r)[E (r)] E (r)dr is similar to (r)|E (r)| dr and ω0 ∼ ω0 . Thus, (5.8) 0 0 0 V V is approximated as ω0 − ω0 =− ω0

   − (r) |E 0 (r)|2 dr V  (r)  . 2 V (r)|E 0 (r)|2 dr

(5.9)

Additionally, we have the facts that  (r) = (r) outside the nanoparticle while  (r) =  p and (r) = m inside the nanoparticle. Also, the nanoparticle’s size is much smaller than the optical wavelength λ0 = 2πc/ω0 and may be viewed as an electric dipole that is located at r0 . As a result, the above equation is reduced to V p αex |E 0 (r0 )|2 ω  − ω0 δω  , = 0 =− ω0 ω0 2ε0 V (r)|E 0 (r)|2 dr

(5.10)

with the excess polarizability defined as αex =  p − m , ε0

(5.11)

and the nanoparticle’s volume V p . It is seen that the real part Re(αex ) gives rise to the frequency shift of the eigenmode, Re(δω), while the imaginary part Im(αex ) reduces the eigenmode’s quality factor, Q = ω0 /Im(δω). The fractional frequency shift can be enhanced through increasing the nanoparticle’s size and the relative permittivity

5.2 Sensing Mechanism

239

contrast between the nanoparticle and the outside-microcavity medium. Actually, this corresponds to the enhancement of the nanoparticle’s polarization energy. Using δω δλ =− , λ0 ω0

(5.12)

the fractional frequency shift is converted to the corresponding fractional wavelength shift. In fact, the spectral broadening given by (5.10) accounts only for the inelastic scattering of the nanoparticle, i.e., the absorption. The nanoparticle can also scatter the WGM light in an elastic manner, i.e., Rayleigh scattering. According to (2.63), the corresponding scattering rate is given by m |E 0 (r0 )|2  σ sca 1/2 2 m V (r)|E 0 (r)| dr 5/2 m ω04 V p αex 2 |E 0 (r0 )|2  = . 2 6πc3 ε0 V (r)|E 0 (r)| dr

Γ =

c

(5.13)

It is seen that both (5.10) and (5.13) are proportional to the local light intensity. When the nanoparticle interacts with a standing wave formed by a pair of counterpropagating waves at the same frequency, δω and Γ are maximized (minimized) at antinodes (nodes). A natural question arises: Is the sensitivity at the single nanoparticle level accessible? To approach this level, it is essential to significantly enhance the intensity |E 0 (r0 )|2 of the evanescent field in the vicinity of the microcavity’s surface, which, however, sacrifices the quality factor. Usually, the LSPRs of metal nanostructures are utilized to strengthen the local electric field at the nanoparticle/molecule’s position. So far, the nanostructures in the rod, triangular, crescent, hexagonal, and star shapes have been developed in experiment and the enhancement factor can be as high as 103 . This allows one to observe the spectral shift and broadening, whose typical values are about tens of femtometers (fm), caused by single nanoparticles/molecules.

5.2.2 WGM Microcavity Perturbed by a Nanolayer In the refractive index sensing, a microsphere with a radius R is immersed in a sample solution. Multiple particles/molecules can be adsorbed on the surface of the microsphere, forming a dielectric layer with a thickness d much smaller than R [see Fig. 5.3b]. The refractive indices of microsphere, medium, and layer are n s , n m , and n l , respectively. In many cases, the adsorbed layer has a refractive index close to that of microsphere. In the intuitive picture, an WGM at the wavelength λ0 traveling on the inner surface of a free microsphere should satisfy the in-phase condition l ≈ 2π Rn s /λ0 . Here l is the polar number of the WGM. After adsorbing

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5 Single-Molecule Sensing

a layer, the microsphere’s radius R is effectively increased and the optical wave can circumnavigate a larger circumference. Thus, the WGM wavelength λ0 needs to be shifted by a small amount Δλ, which is in proportion to the change in the microsphere’s size, d Δλ (5.14) ∼ , λ0 R so as to maintain the same mode number l. For R = 100 µm and d = 1 nm, the fractional shift in wavelength is as small as about 10−5 . The quality factor of WGMs of a microsphere in aqueous circumstance is of the order of 106 . Thus, the wavelength shift caused by a 1-nm layer is one hundred times larger than the spectral linewidth of the WGMs and can be easily observed in the transmission spectrum. Let us derive the specific relation between the fractional shift and the surface density of the layer. Inserting Z (r ) = r z(kr ) into (3.6), one finds the modified radial function Z (r ) for a microsphere fulfills the following Schrödinger-like equation: ∂2 Z (r ) + (E eff − Veff )Z (r ) = 0, ∂r 2

(5.15)

with the effective energy E eff = k02 and the vacuum wavenumber k0 of the cavity mode. The effective optical potential takes the form Veff = k02 (1 − n 2 ) +

l(l + 1) , r2

(5.16)

where n is the distribution of the refractive index. Adsorbing a nanoscopic dielectric layer to the surface of the microsphere leads to a perturbation in the effective potential, δVeff = δ[k02 (1 − n 2 )].

(5.17)

To the first-order approximation, this perturbation shifts the effective energy by an amount Z | δVeff |Z

. (5.18) δ E eff = Z |Z

Substituting δ E eff = δ(k02 ) and δ(n 2 ) = n l2 − n 2m , one obtains Z | k02 (n l2 − n 2m ) |Z + Z | 2k0 δk0 n 2s |Z = 0.

(5.19)

Thus, the fractional shift is approximated as (n 2 − n 2 )|Z (R)|2 R 2 d δk0 = − l  Rm . k0 2n 2s |Z (r )|2 r 2 dr 0

The integral in the denominator is calculated to be

(5.20)

5.2 Sensing Mechanism

241

n 2s

R

|Z (r )| r dr = 2 2

0

≈

n 2s

R 0

jl2 (n s k0 r )r 2 dr

R3 2 (n − n 2m ) jl2 (n s k0 R). 2 s

(5.21)

In deriving the above expression, we have used the result obtained in [11]. Thus, the fractional shift in wavelength reads n 2 − n 2m d δλ . = l2 λ0 n s − n 2m R

(5.22)

Using the relation between the relative permittivity  and the refractive index n,  = n 2 , we have αex σ (n l2 − n 2m )d = , (5.23) ε0 where αex = ε0 (l − m )ds is the excess polarizability and σ = 1/s denotes the surface density. Here s is an area element on the microsphere’s surface. Inserting (5.23) into (5.24), we arrive at δλ αex σ . (5.24) = 2 λ0 ε0 (n s − n 2m )R A small radius R and a pair of similar refractive indices of the outside- and insidemicrocavity materials enhance the wavelength shift δλ, thereby leading to a high sensitivity.

5.3 Chemical Reaction Analysis Around a microcavity, there may exist nanoparticles/molecules of more than one species. Different nanoparticles/molecules located at different positions can cause the same perturbation to a WGM of the microcavity despite their different relative permittivities. Thus, it is unable to distinguish the spectral shift caused by the nanoparticles/molecules of a specific specie. Addressing this issue requires the extra preparation measures before the sensor operation. Let us focus on the biosensing, where a microsphere is immersed in a sample solution containing the analyte molecules to be detected. A WGM transduces the binding of the biomolecules to the microsphere’s surface into a resonance wavelength/frequency shift signal. To achieve a shift signal that is specific for the biomolecules of a certain specie, it is necessary to modify the microsphere’s surface with the receptors (e.g., antibodies), which bind selectively to the analyte and reject (or minimize) the interaction with the molecules of other species that may be present in the sample. In addition, it is necessary to prevent any unspecific binding of biomolecules to the microsphere sensor itself.

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5 Single-Molecule Sensing

Fig. 5.4 Schematic diagram of binding process. a Sensor modified with the antibody receptors. Each antibody provides two binding sites for the analyte molecules. b Sensor surface with a fixed number of equal analyte binding sites

These demands can be met by utilizing certain functionalization strategies of immobilization of the specific receptor molecules on the microsphere’s surface. One successful approach is based on the physisorption of a biotinylated dextran layer that is coated entirely on the surface of the glass microsphere. The streptavidin is then used as a linker to immobilize the biotinylated protein G, which is further used to immobilize the antibody. The use of dextran as the first layer to coat the glass surface has the advantages of (i) preventing the interaction of unspecific biomolecules with the glass surface and (ii) introducing many biotin groups for binding the streptavidin throughout the immobilized dextran matrix. Following this functionalization strategy, one can achieve a surface density of streptavidin in excess of 1010 cm−2 . The common receptor molecules other than antibodies are single-stranded DNA strands and DNA molecules that function as the so-called aptamers by providing the binding specificity to proteins. Any receptor molecules that are used for the bulk concentration sensing should own a high affinity.1 The purpose of using the receptors is to accumulate the analyte molecules at the sensor’s surface, i.e., the association reaction, so as to transduce a detectable WGM wavelength shift. Each receptor may provide multiple binding sites [see Fig. 5.4a]. Here, we restrict ourselves to the one-on-one binding (i.e., one receptor can only bind to one analyte molecule). It should be noted that the functionalization strategy may be also implemented for other types of sensors, such as surface plasmon resonance (SPR), quartz crystal microbalance (QCM), and enzyme-linked immunosorbent assay (ELISA), is shown conceptually in Fig. 5.4b.

1 In chemistry, the term “affinity” denotes the tendency of dissimilar chemical species reacting with

each other to form a chemical compound.

5.3 Chemical Reaction Analysis

243

5.3.1 Kinetics of Association and Dissociation Reactions We now model the interaction between receptor and analyte molecules at the biosensor’s surface. The receptor molecules A are immobilized and distributed homogeneously at the sensor’s surface, which defines a fixed number of the receptor sites for binding the analyte molecules B. The square brackets [X] are used to denote the concentration of a general substance X. Note that [A] is a surface concentration of the receptors in units of mol/cm2 while [B] is a molar volume concentration in units of mol/L. Generally, the receptor sites have a finite affinity. Thus, the analyte molecules can be also unbound from the sensor’s surface, corresponding to the dissociation process. At equilibrium, the forward (association) and reverse (dissociation) processes are in a dynamic balance, i.e., there is no net change in either [A] or [B]. We assume an one-on-one interaction, where one receptor molecule interacts with one analyte molecule, i.e., the so-called Langmuir model. The equilibrium binding reaction can be formalized as kon (5.25) A + B  AB. koff

The arrows represent the reaction directions. The constant kon (koff ) quantifies the binding (unbinding) rate of the analyte molecules B to (from) the receptor sites A. The rate constants kon and koff are in units of M−1 s−1 (1 M = 1 mol/L) and s−1 , respectively. In the chemical language, (5.25) denotes that the reactants A and B are bound to form the product AB. Here, the symbol AB is used to denote the bound complex, i.e., the occupied receptor sites. The rate of change in [AB] resulting from the association processes is proportional to the product of the receptor and analyte concentrations 

d [AB] dt

 = kon [A][B],

(5.26)

association

while the rate of the dissociation-induced change in [AB] is written as 

d [AB] dt

 = −koff [AB].

(5.27)

dissociation

Combining the above two rate equations, one obtains the binding reaction equation for AB d [AB] = kon [A][B] − koff [AB]. (5.28) dt The concentrations of unoccupied and occupied receptors, [A] and [AB], satisfy the conservation law (5.29) [A] + [AB] = [A]t ,

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with the total surface concentration of the receptors [A]t . Inserting (5.29) into (5.28) yields d  [AB] = kon [B] − (kon [B] + koff ) [AB], (5.30) dt  with the modified association rate constant kon = kon [A]t . This equation links the bound complex concentration [AB] to the analyte concentration [B]. Let’s examine the response at the time point t = 0. At the initial stage of the association, the clean sensor’s surface is exposed to a sample containing the analyte molecules B, and the concentration of the occupied receptor sites is zero. Thus, (5.30) is approximated as d  [B]. (5.31) [AB] = kon dt

It is seen that d[AB]/dt depends linearly on the bulk concentration of the analyte B. Since the sensor’s response is proportional to [AB], monitoring the rate of change of the sensor signal enables the measurement of the concentration [B]. Such a method may be utilized to determine the concentration of a biomarker or analyte  can be derived from the calibration in biosensing. In practice, the rate constant kon experiments, where d[AB]/dt is measured for several well-defined analyte concentrations [B], based on the same sensor structure. The slope of the plot of d[AB]/dt  . versus [B] gives kon As t → ∞, the sensor approaches a dynamic association–dissociation equilibrium. That is, the forward binding process A + B → AB is balanced by the backward unbinding process AB → A + B. Also, the sensor signal stops varying eventually. Setting d[AB]/dt = 0, (5.28) leads to [A][B] koff ≡ KD , = kon [AB]

(5.32)

where K D is the equilibrium dissociation constant and has the units of M (or mol/L). The above expression indicates that the ratio of the product between the equilibrium receptor and analyte concentrations to the concentration of the occupied sites maintains at a constant value. Similarly, one can define the equilibrium association constant K A as [AB] , (5.33) KA ≡ [A][B] which is simply the reciprocal of K D , K A = K−1 D . In Table 5.2, we have listed the values of kon , koff and K D for several different analyte-receptor pairs measured in experiment. Usually, it is convenient to use the receptor occupancy, defined as σ=

[B] K A [B] [AB] = = [A] + [AB] K D + [B] 1 + K A [B]

(5.34)

5.3 Chemical Reaction Analysis

245

Table 5.2 Binding and unbinding rate constants kon and koff and the equilibrium association constant K D for different analyte–receptor pairs (Analyte, Receptor) kon (M−1 s−1 ) koff (s−1 ) K D (M) (Biotin, Streptavidin) [12] (LexA protein, DNA) [13] (Clozapine, Dopamine) [14]

5.5 × 108

3.1 × 10−5

0.6 × 10−13

5 × 107

3.4 × 10−3

7 × 10−11

1.4 × 106

2.8 × 10−2

2.0 × 10−8

Fig. 5.5 Dependence of the occupancy σ on the analyte concentration [B] (a) and the pH value (b)

in analysis. The last expression of the response of σ to [B] is referred to as the Langmuir isotherm. The occupancy varies between zero and unity, and σ = 1 corresponds to the fully occupied sites. The higher the analyte concentration [B] is, the closer the occupancy σ approaches the maximum sensor surface coverage σ = 1. Interestingly, when the half of the receptor sites are occupied, σ = 1/2, one has [A] = [AB] and K D = [B] as shown in Fig. 5.5a. That is to say, the concentration of the analyte molecules [B], at which the half of the maximal sensor response is obtained in equilibrium, depends entirely on K D . A high K D requires a large analyte concentration to bind 50% of the receptors. Thus, when detecting the analyte concentration [B] from the equilibrium sensor response, one has to consider the equilibrium dissociation (association) constant K D (K A ) in addition to the device sensitivity (i.e., limit of detection, LOD). In general, we have no prior knowledge of K D . The value of K D needs to be determined from the calibration measurements, where the equilibrium sensor response is measured for the samples with different known analyte concentrations [B]. In many cases, the sensor response may not follow an ideal Langmuir isotherm but exhibit a different or more complex behaviour. Also, this needs to be determined in the calibration measurements. Below, we summarize two basic approaches to determining an unknown concentration of analyte [B]:

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• Measurement at the initial time. According to (5.31), the unknown [B] can be derived from measuring the rate of change of the sensor signal at t = 0 and comparing the result with the calibration measurements. A sensitive measurement on the sensor signal requires that the receptor molecules A own a high binding rate constant kon , such as antibodies, and preferably have a diffusion-limited transport to the sensor’s surface. • Measurement in equilibrium. The Langmuir isotherm denotes that [B] = K D occurs at the half of the maximum value of the sensor signal. A small equilibrium dissociation constant K D , such as that of antibodies with K D ∼ 10−12 − 10−9 M, ensures a large sensor signal for a given low concentration of the analyte molecules [B]. We note that a similar association reaction formalism is encountered when considering the degree of protonation of an acid HA. The excess hydrogen ions H+ in solution combine with the negatively charged ions A− to form the acid molecules HA. This association reaction is described by A− + H+  HA.

(5.35)

The equilibrium dissociation constant is then given by [A− ][H+ ] , [HA]

(5.36)

[HA] . [A ] + [HA]

(5.37)

KD = and the fraction of bound A− reads σ=

−

One half of the ions A− are bound by H+ when K D = [H+ ]. The pH value (dimensionless), which is defined as the negative logarithm to the base 10 of the molar hydrogen ion concentration (5.38) pH = − log10 [H+ ] is commonly employed to measure the acidity of a solution. The pH scale ranges from 0 to 14. A pH lower (higher) than the neutral value 7 denotes the solution is acidic (basic). In the similar way, one may replace K D with the ionization constant pK D = − log10 K D .

(5.39)

In biology, pK D can range over several orders of magnitude. At σ = 1/2, we have pK D = pH. Thus, pK D is of great important because it defines the pH range [0, pK D ] within which the substance HA is less ionized [see Fig. 5.5b].

5.3 Chemical Reaction Analysis

247

5.3.2 Determination of Kinetic Constants In Sect. 2.3.3, we have introduced an SPR-based biosensor that is sensitive to the change in the refractive index (induced by the change in the concentration) close to a metal chip surface. The sensor response corresponds to a shift in the SPR angle. Based on this platform, we consider the approach to derive the rate constants kon and koff and further the equilibrium constants K A and K D . As shown in Fig. 5.6a, the receptor molecules A are immobilized onto the sensor’s surface with a total (including both occupied and unoccupied sites) surface concentration [A]t . The analyte molecules B continuously flow over the sensor surface. The analyte-receptor binding process leads to an increase of the refractive index in the vicinity of the sensor’s surface, thereby shifting the SPR angle. The sensor response is monitored in real time and one obtains the sensorgram [see Fig. 5.6b]. The kinetic parameters can be evaluated from the distinct association and dissociation phases of the sensorgram. We assume the analyte injection starts at the time ta . Again, the assumption of the one-on-one interaction model between receptor and analyte molecules is employed. In addition, the continuous supply of the analyte molecules maintains the concentration [B] at a constant value. The solution of (5.30) is then derived as  [AB] [B] 1 − e−(kon [B]+koff )(t−ta ) . = [A]t [B] + K D

(5.40)

Fig. 5.6 Determination of kinetic constants. a Schematic view of SPR sensor. b Sensorgram with association and dissociation phases

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5 Single-Molecule Sensing

The concentration of complex [AB] is mapped onto the sensor’s response (denoted by R) above the base line,  R = R0 1 − e−(kon [B]+koff )(t−ta ) .

(5.41)

The maximal value R0 corresponds to the saturation signal when the receptors are sufficiently occupied. As t is increased, the sensor response R grows from the base line, indicating that the binding process suppresses the unbinding process. This period is usually referred to as the association phase [see Fig. 5.6b]. After a long enough injection time, R becomes saturated, which corresponds to the binding–unbinding equilibrium. The curve fitting gives us the value of (kon [B] + koff ). The analyte injection is stopped at the time td , which is long enough to saturate R, and the buffer injection starts subsequently. The free analyte molecules B are carried away by the buffer flow with a high efficiency. In an ideal situation, the free analyte molecules B in the dissociation phase are removed completely at any time. Hence, the rate of change of [AB] is independent of [B], and (5.28) is reduced to d [AB] = −koff [AB]. dt

(5.42)

Solving the above equation yields the sensor’s response R = R0 e−koff (t−td ) .

(5.43)

The response exhibits an exponential decay behavior with a characteristic rate equal to koff , reflecting the kinetics of the complex dissociation. Thus, the value of koff is determined through the curve fitting to the dissociation phase of the sensorgram. Since the dissociation of the complex is independent of [B], the same koff should be obtained for different analyte concentrations. Further, the association rate constant kon is determined. After obtaining kon and koff , the equilibrium dissociation constant is evaluated straightforwardly, K D = koff /kon . The experimentally measured sensorgram may deviate from the theoretical prediction under the ideal situation because of the mass transport effect and less than 100% efficient removal of the analyte molecules.

5.3.3 Thermodynamic Analysis Chemical potential. Besides the kinetics, the chemical/biological reactions may be also analyzed in the aspect of thermodynamics. We consider a mixture containing two substances A (receptor) and B (analyte). The association reaction affects the particle number (concentration) of each substance. For X = A or B, the change of the particle number is accompanied by the change of the free energy of this substance. The corresponding rate of change with respect to the particle number is measured

5.3 Chemical Reaction Analysis

249

by the so-called (molar) chemical potential in units of J/mol μX = μoX + RT ln aX .

(5.44)

Here, μoX represents the standard chemical potential of the pure substance X in the standard state. The universal gas constant R is given by the product of the Avogadro’s number N A (= 6.022 × 1023 particles per mole) and the Boltzmann constant k B , R = N A k B = 8.314 J · K−1 mol−1 . The thermodynamic temperature T is in units of Kelvin (K). The activity aX of the substance X is recognized as aX =

γX [X] , Co

(5.45)

with the dimensionless activity coefficient γX and the molar concentration [X] of the substance in the mixture. The standard amount concentration C o = 1 mol/L ensures that both aX and γX are dimensionless. The activity coefficient γX may itself also depend on [X] and can be higher or lower than the unity. When γX ≈ 1, the behavior of the substance X in the mixture approaches that of the ideal X. Actually, the activity aX is introduced to measure the effective concentration of a substance under the non-ideal condition. Initially the chemical potentials of two substances A and B are unequal. The chemical reaction derives a net transfer of particles from the higher-chemical-potential substance to the lower-chemical-potential substance until the condition of chemical equilibrium (5.46) μA = μB is satisfied. Inserting (5.44) into (5.46), one finds another equilibrium constant, i.e., the partition coefficient P≡

  (μo − μoA ) aB . = exp − B aA RT

(5.47)

The above equation indicates that the ratio of the activities (effective concentrations) of two substances is independent of the overall concentration. Using the activities, the equilibrium dissociation and association constants are rewritten as K D = K−1 A =

aA aB , aAB

(5.48)

with the activity aAB of the bound complex AB. Gibbs free energy. The Gibbs free energy G is a potential defined for a thermodynamic system at a constant temperature T and pressure (in units of pascal), G = H − T S,

(5.49)

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5 Single-Molecule Sensing

with the enthalpy H and the entropy S. Indeed, the Gibbs function G measures the energy associated with a chemical reaction that is available to do the useful (nonexpansion) work, and it is minimized when the chemical equilibrium is established. The entropy S measures the randomness and disorder of molecules in a given thermodynamic system or chemical reaction. The definition of S is S = k B ln Ω

(5.50)

with the number Ω counting the microscopic configurations of a thermodynamic system. During the chemical reaction, the change in the Gibbs free energy ΔG = G pr oducts − G r eactants is equal to the change in the enthalpy ΔH = H pr oducts − Hr eactants minus the change in the product of the temperature T and the entropy ΔS = S pr oducts − Sr eactants , ΔG = ΔH − T ΔS.

(5.51)

In chemistry, ΔG, ΔH , and ΔS are commonly measured in units of kJ/mol, kJ/mol, and J/(K · mol), respectively. The enthalpy change ΔH corresponds to the net heat (thermal energy) transfer associated with the bond breaking (absorbing heat) and the formation of new bonds (releasing heat) in a chemical reaction at constant pressure. The enthalpy change ΔH is positive (negative) when the heat energy of the reactants is greater (lower) than that of the products. The positive (negative) change in entropy ΔS denotes the thermodynamic system becomes more (less) disordered during the reaction. For ΔG < 0, the energy of the products is lower than that of the reactants. Consequently, the chemical reaction occurs spontaneously. In the opposite case (ΔG > 0), the extra energy is required to drive the reaction. Clearly, the reaction reaches a dynamic equilibrium when ΔG = 0. For instance, we consider the decomposition of 1 mol of calcium carbonate into 1 mol of calcium oxide and 1 mol of carbon dioxide. This reaction requires the absorption of 177.8 kJ of heat and we have CaCO3 + 177.8 kJ → CaO + CO2 .

(5.52)

Since the 177.8 kJ of heat is on the reactant side, the change in enthalpy is positive and ΔH = 177.8 kJ. In addition, the gas CO2 appears in the products, making the system more disordered. Thus, the change in entropy ΔS is positive and ΔS = 160.5 J/(K · mol). At the room temperature T = 298 K, the change in the Gibbs free energy is then evaluated to be ΔG = 130.0 kJ/mol, which makes the above reaction hardly occur. The spontaneous occurrence of this decomposition reaction requires a temperature at least higher than 1108 K. The relation between the change in the Gibbs free energy ΔG and the chemical potential μ tells us

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251

ΔG = −RT ln aAB + RT ln aA + RT ln aB = RT ln K D .

(5.53)

For K D = 0.6 × 10−13 M of the biotin–streptavidin reaction, ΔG is about −75 kJ/mol at room temperature. Comparing (5.51) and (5.53), one finds ∂ ∂lnK D = ∂T ∂T



ΔG RT

 =

∂ ∂T



ΔH − T ΔS RT

 ,

(5.54)

and further obtains the van’t Hoff equation ∂lnK D ΔH = . ∂(1/T ) R

(5.55)

The above equation provides a way to compute the enthalpy change ΔH in the reaction: Fit the plot of ln K D versus 1/T measured in experiment to a straight line whose slope gives ΔH . DNA hybridization. As an example, we perform the thermodynamic analysis on the hybridization of deoxyribonucleic acid (DNA) molecules. The typical DNA owns a double-stranded (DS) helix structure, where two polynucleotides chains (strands) coil around each other and are joined via hydrogen bonds. Raising the temperature can break these hydrogen bonds and the DS molecules are decomposed into the single-stranded (SS) ones. This process is referred to as melting. In contrast, lowering of the temperature enables the reinstatement of the hydrogen bonds between single strands, known as hybridization, leading to the DNA-duplex formation [Fig. 5.7a]. The detection scheme is shown conceptually in Fig. 5.7b. For the self-complementary DNA, the equilibrium between hybridization and melting processes is described by SS + SS  DS.

(5.56)

For the convenience of discussing the melting temperature below, here we consider the equilibrium association constant KA =

[DS] . [SS][SS]

(5.57)

The molar fraction of the DS molecules is recognized as σ=

[DS] , [SS]/2 + [DS]

(5.58)

which varies between zero and unity. We assume that the system is initialized with a total SS concentration of Ct ≡ [SS]initial and [DS]initial = 0. Thus, at equilibrium the concentrations of SS and DS molecules can be, respectively, re-expressed as follows:

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5 Single-Molecule Sensing

Fig. 5.7 DNA-duplex formation. a Schematic diagram of DNA hybridization. b DNA sensor chip. Single-stranded DNA strands (oligomers) are spotted on (a glass) chip. The DNA strand hybridization is detected from the fluorescence. The best detection signal is obtained for σ = 1, where each oligomer receptor strand is bound to one complementary DNA strand (the analyte). c Plot of (1 − σ) versus T . At the melting temperature Tm , the fraction σ is equal to one half. d Example of experimental measurement of Tm−1 versus ln Ct . Reprinted from [15] with the granted permission

[SS] = Ct (1 − σ), [DS] =

Ct σ , 2

(5.59)

where we have used the relation Ct = [SS] + 2[DS]. The equilibrium constant K A is then rewritten in terms of σ and Ct as KA =

σ . 2Ct (1 − σ)2

(5.60)

When the ratio of the double strands to the single strands becomes equal, σ = 1/2, one obtains the simple expression KA =

1 . Ct

(5.61)

The temperature of the thermodynamic system, at which (5.61) is fulfilled, is defined as the melting temperature Tm . In practice, Tm may be determined from the measurement of the melting curve [see Fig. 5.7c]. The sample is initially prepared at a low temperature (e.g., 20 ◦ C), at which all single strands have been paired. Then, the DNA in solution is heated up to a high temperature (usually more than 80 ◦ C) at a slow

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253

rate (for example, one degree per minute). At the high temperature, all DS molecules are melted into single strands, whose molar concentration is Ct . The quantitation of DS molecules at different T is carried out through the absorption or fluorescence spectroscopy. As a result, one obtains the melting curve corresponding to Ct , and the melting temperature Tm can be directly read out. Actually, Tm is located at the peak of the derivative of the melting curve. The total change in the Gibbs free energy for the system undergoing the forward reaction (duplex formation) is given by ΔG =

 l

ΔG l =

 l

ΔHl − T

 l

ΔSl ,

(5.62)

which takes into account of the enthalpy and entropy contributions from each base of the DNA sequence with a length l ∈ N. Substituting (5.61) into (5.53), the melting temperature has the following expression: R ΔS 1 ln Ct + . = Tm ΔH ΔH

(5.63)

The above equation provides a way to determine the total changes in enthalpy ΔH and entropy ΔS. A group of samples are prepared at different concentrations Ct . The reciprocal of the measured Tm (the y-axis) is plotted with respect to ln Ct (the x-axis). The resulting plot is fitted to a straight line as shown in Fig. 5.7d that adapted from [15]. The enthalpy change ΔH corresponds to the slope and the entropy change ΔS is determined from the y-intercept at ln Ct = 0.

5.4 Sensing Based on Passive WGM Microcavities The sensing detection can be implemented based on either the passive microcavities, in which the WGMs are excited by the external light beams, or the active microcavities, where the intracavity lights are entirely generated by the doped active particles. Each method has unique advantages and disadvantages. For the passive microcavities, tuning the external laser source changes the intensity and wavelength of the intracavity light but the coupler introduces the extra loss, reducing the Q factor. In contrast, the active microcavities have no coupling loss but the WMG wavelengths are limited by the available doped particles. In this section, we restrict ourselves to the passive-microcavity sensing.

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5 Single-Molecule Sensing

Fig. 5.8 Unlabeled protein detection. a Experimental setup. b Transmission spectrum obtained through scanning the injection current of the DFB laser. c Current and corresponding wavelength shift of the resonant WGM after injecting the BSA into the sample cell. d Detection of the streptavidin binding to the BSA biotin immobilized on the microspheroid’s surface. Reprinted from [16], with the permission of AIP Publishing

5.4.1 Experimental Demonstrations In what follows we will briefly introduce several selected sensing experiments so as to give the readers a general idea on the structure and approach of the biosensing. • Unlabeled protein detection In [16], it demonstrates an optical biosensor with an unprecedented sensitivity for detecting label-free molecules (i.e., label-free sensing2 ). The corresponding detection structure as shown in Fig. 5.8a has become one of the popular biosensing schemes. The WGM microcavity is fabricated through melting the stripped end of a singlemode silica fiber (by using the butane/nitrous oxide microtorch flame). Rotating the fiber, the melted fiber tip forms into a spheroidal shape due to the surface tension. This microspheroid owns a diameter of 2R ≈ 300 µm and can be excited evanescently via another single-mode optical fiber [see Fig. 5.8a]. The probe light enters the 2 In the label-based biosensing, a fluorescent dye molecule is bound to a target molecule or to a tracer

molecule (e.g., antibody) that is attached to the target. The sensing detection is implemented via the optimized fluorescence techniques. However, in some cases, attaching a label may be infeasible or the label may affect the kinetics of the analyte. In contrast, the label-free sensing has the advantages of the reduced sample complexity, the simplified preparation and the efficient detection of the dynamics at the single-molecule level.

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excitation fiber from one end and the intensity of the transmitted light is measured by a photodiode located at the other end of the fiber. In order to enhance the evanescent fiber–microcavity coupling, a section (about 1 cm) of the excitation fiber, which is in contact with the microspheroid, is eroded to reduce the core diameter from 6.6 to 4 µm by using the hydrofluoric acid. For the use of biosensing, the microspheroid’s surface is chemically modified by the (3-Aminopropyl) trimethoxysilane so that a rug of amine groups is formed on the microcavity’s surface. The microspheroid is then immersed in the phosphatebuffered saline (pH 7.4) hold between two glass slides (i.e., the sample cell) via the surface tension force as shown in Fig. 5.8a. The resulting quality factor of the microcavity in the aqueous solution reaches as high as 2 × 106 . The amine groups acquire the positive charges in the buffer solution and can capture the negatively charged molecules. The microcavity is excited at λ ≈ 1340 nm by the light from a distributed feedback (DFB) laser. The DFB wavelength is repeatedly scanned via tuning the injection current with a sawtooth-shaped function. The sweep range of the wavelength covers 0.2 nm. The change in the concentration of the solution is detected by tracking the position of a resonant dip in the transmission spectrum [see Fig. 5.8b] in real time. The bovine serum albumin (BSA) is injected into the sample cell. In the buffer solution, the BSA molecules carry the negative charges and can stick to the amine groups on the microspheroid’s surface. This adsorption of BSA molecules effectively increases the microspheroid’s circumference, resulting in a shift δλ of the resonant dip toward the long wavelength side, δλ > 0. Figure 5.8c presents an experimental measurement result, where the resonance wavelength decreases firstly (due to the thermal contraction of the microspheroid) and then rapidly increases until the shift δλ becomes saturated. The surface density σ of the BSA may be estimated from δλ according to (5.24). Substituting the excess polarizability αex = 4πε0 × 3.85 × 10−21 cm3 and the refractive indices of the microspheroid n s = 1.47 and the buffer solution n b = 1.33, σ is found to be 1.7 × 1012 cm−2 . This device may be further applied as a biosensor to detect a target analyte. For instance, the biotinylated BSA works as a recognition element while the streptavidin molecules, which can be bound to the surface-immobilized BSA molecules, play the analyte role. As shown in Fig. 5.8d, a subsequent injection of streptavidin induces an additional shift of the resonance wavelength. The ratio of the wavelength shift caused by the streptavidin binding to the wavelength shift induced by the BSA adsorption approximates 0.94, implying a one-on-one stoichiometry for the binding reaction between streptavidin and BSA molecules. • Nanolayer characterization through wavelength multiplexing Introducing the resonant excitation of a second WGM, one may evaluate the thickness and effective refractive index for a thin dielectric layer formed on the microcavity’s surface. As an example, we consider a microsphere with a radius R. The symbols n s and n m are used to, respectively, denote the refractive indices of the microsphere (silica, 1.47) and its environment (water, 1.33). There exists a thin dielectric layer

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Fig. 5.9 Wavelength multiplexing. a Experimental setup. b Relative wavelength shifts of two WGMs caused by the BSA adsorption. c Relative wavelength shifts induced by two sequential injections of NaCl. d (δλ/λ)760 versus (δλ/λ)1310 for the BSA adsorption and for six incremental injection of NaCl. Points: experiment. Lines: theory. All figures are reprinted with permission from [17] ©The Optical Society

with a thickness d on the microsphere surface. This layer effectively introduces a small increment δ(n 2 ) to n 2m , resulting in a wavelength shift δλ of a WGM at λ. To the first-order approximation, the layer-perturbation-induced δλ is derived as [17]  L δ(n 2 ) δλ = 1 − e−d/L , 2 2 λ R ns − nm

(5.64)

with the evanescent field length L L=



λ

4π n 2eff − n 2m

.

(5.65)

Here, n eff is the effective refractive index for the corresponding WGM. Equation (5.64) is the basis for evaluating the thickness d and the increment δ(n 2 ) through the wavelength-multiplexing method. Figure 5.9a illustrates the specific experimental setup. The key structure is a silica microsphere coupled to a single-mode fiber. A portion of the fiber is acid eroded down to a 3-µm diameter so as to enhance the evanescent fiber–microsphere coupling. Both microsphere and fiber are immersed in the buffer solution (i.e., the phosphate-buffered saline). Two WGMs at 760 and 1310 nm are excited by the lights from two DFB lasers, respectively, and the transmission spectra are detected at the

5.4 Sensing Based on Passive WGM Microcavities

257

output end of the coupling fiber. Again, the resonance dips associated with the 760and 1310-nm region are tracked by scanning both DFB lasers with a synchronous ramp. After obtaining the wavelength shifts of two WGMs, one may calculate the ratio  L 760 1 − exp (−d/L 760 ) (δλ/λ)760  , ≈ S= (5.66) (δλ/λ)1310 L 1310 1 − exp (−d/L 1310 ) which is only related to the thickness d. The ratio S approaches the unity in the limit of d/L 760,1310  1 while the theoretical prediction gives S ≈ L 760 /L 1310 = 0.58 for d/L 760,1310  1. Equation (5.66) may be tested through two experiments: (1) BSA adsorption. The surface of the microsphere is firstly treated with the (3-aminopropyl) trimethoxysilane. When the BSA is injected into the buffer solution, both resonant WGMs are shifted in a similar way as shown in Fig. 5.9b. The saturation of the relative shifts δλ/λ ∼ 1 × 10−5 corresponds to the Langmuir-like saturation of BSA, that is, the formation of a monolayer. The thickness of the layer is estimated to be 3 nm; and (2) Injection of NaCl. The NaCl is sequentially injected into the buffer solution, increasing the salt concentration by 0.1-M increment, for twice. The wavelength shifts of the 760- and 1310-nm WGMs for these two distinct experiments are apparently separated as shown in Fig. 5.9c. The relation between (δλ/λ)760 and (δλ/λ)1310 has been summarized in Fig. 5.9d. Since the thickness of the BSA layer is much smaller than both L 760 and L 1310 , the corresponding ratio S is close to the unity. In contrast, S for the NaCl experiment is derived to be 0.54, consistent with the theoretical prediction of (5.66). Equation (5.66) can be used to evaluate the layer’s thickness d. As an example, the ratio S for the microsphere immersed in the buffer solution containing Poly-Llysine is measured to be S = 0.82. Substituting this value into (5.66), one finds the thickness d = 110 nm. Further, according to (5.64), the increment δ(n 2 ) is estimated to be 0.0033. Thus, the refractive index in the vicinity of the microsphere is effectively increased by δn ≈ δ(n 2 )/(2n m ) = 0.0012, which is very small. • Single virus detection A label-free, real-time optical detection of Influenza A virus was performed in [18]. The sensitivity reaches the single-virion level and the measured resonance shift enables one to quantitatively evaluate the virion size and mass. As shown in Fig. 5.10a, a silica microsphere with R = 50 µm is mounted on the sample cell via a stem right after the fabrication. The sample cell is enclosed within a small box so as to screen the air flow and stabilize the ambient humidity. The box is filled with the phosphate-buffered saline. An equatorial WGM of the microsphere is evanescently excited by a guided wave at λ = 1310 nm in a tapered optical fiber that passes through the sample cell. The transmission spectrum manifests the WGM linewidth of Δλ = 5 pm. The corresponding quality factor Q = 2.6 × 105 is much lower than that of a free microsphere, which is mainly ascribed to the overtone vibrational absorption of water in the near infrared.

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Fig. 5.10 Single virus detection. a Experimental setup. b Trace of δλ/λ for the polystyrene particles. c Statistics of discrete step heights. d Maximum step height (δλ/λ)max versus the microsphere curvature 1/R for the polystyrene particles. e Trace for the Influenza A virus. Reprinted from [18] with the granted permission

Experiment I. The polystyrene particles with a radius 250 nm are diluted in the buffer solution and perturbs the WGM, resulting in a wavelength shift δλ. Figure 5.10b displays a trace of δλ/λ. Multiple abrupt steps are clearly visible against the background cavity noise. They are caused by the adsorption of individual particles on the microsphere’s surface. There also exist spikes as shown in the inset of Fig. 5.10b, which actually correspond to the unsuccessful adsorption attempts. Different adsorption events induce the steps with different heights. The statistics of various steps is summarized in Fig. 5.10c, from which the maximum step height (δλ/λ)max can be read out directly. The value of (δλ/λ)max is strongly related to the microsphere’s size. From the plot of (δλ/λ)max versus the curvature 1/R displayed in Fig. 5.10d, we find (δλ/λ)max ∝ R −2.67 , which matches the theoretical prediction of the scaling R −2.5 in [11] (Problem 5.1). Thus, the heights of the discrete steps can be enhanced by reducing the microsphere size. Experiment II. The Influenza A virus has an average radius of 50 nm and a refractive index smaller than that of the polystyrene particles. Detecting such tiny virus relies on a microsphere with a small radius R. However, the reduction of microsphere’s size leads to an enhanced cavity leakage. To address the issue, R is set at

5.4 Sensing Based on Passive WGM Microcavities

259

Fig. 5.11 Detection of DNA hybridization. a Experimental setup. b Fluorescent imaging of WGMs. c Transmission spectrum. d Time trace of two WGM resonances. e Optimal monovalent salt concentration for mismatch discrimination. f Single nucleotide mismatch detection. Reprinted from [19], Copyright (2003), with permission from Elsevier

39 µm and the WGM at λ = 763 nm with Q = 6.4 × 105 in aqueous solution is used to detect the virus. The virions is directly injected into the buffer solution and a trace of δλ/λ is shown in Fig. 5.10e. The binding of single virions is observed from discrete changes in the resonance wavelength. From the measurement of (δλ/λ)max , one may evaluate the virus radius (denoted by a) by a0 R 5/6 λ1/6 with a0 = a= 1 − a0 /3L D 1/3



δλ λ

1/3 .

(5.67)

max

The evanescent field length L is given by (5.65). The dimensionless dielectric factor is defined as √ 2n 2m 2n s (n 2p − n 2m ) , (5.68) D= 2 (n s − n 2m )(n 2p + 2n 2m ) with the refractive indices of the microsphere n s = 1.45, buffer solution n m = 1.33, and target particle n p (1.5 for virus and 1.59 for polystyrene). Using (5.67) and (δλ/λ)max = 1.5 × 10−8 , the radius a of the virus is estimated to be 47 nm. • Detection of DNA hybridization The detection of the DNA hybridization was demonstrated in [19]. Figure 5.11a shows the schematic diagram of the experimental setup, where two silica microspheres (radius 200 µm) are separated by several micrometers and simultaneously coupled to a single-mode fiber through the evanescent field. The infrared light beam is produced by a DFB laser diode and the intensity of the transmitted light is recorded

260

5 Single-Molecule Sensing

by a photodetector. Tuning the injection current of the laser diode scans the light wavelength within a range of 0.14 nm. Figure 5.11b displays the fluorescent images of WGMs. Both microspheres are immersed in a buffer solution (i.e., phosphate-buffered saline) whose refractive index is n m = 1.332, resulting in a quality factor of 5 × 105 . The Lorentzian-shaped dips in the transmission spectrum [Fig. 5.11c] denote that the WGMs of two microspheres are well spectrally identified and have no crosscoupling. The surface of each microsphere is chemically modified with a different 27-mer oligonucleotide. The oligonucleotide, which is complementary to the surface-immobilized counterpart on one microsphere, is firstly injected into the sample cell. Several minutes later, the oligonucleotide, which is complementary to the DNA strand immobilized on the other microsphere, is injected. The injected oligonucleotides hybridize to their surface-bound target strands, inducing the red shifts of two WGM resonances that are measured in real time. A time trace example is presented in Fig. 5.11d. The resonance shift of a microsphere occurs right after the corresponding injection of the complementary oligonucleotide. The small spikes are attributed to the turbulence-induced temperature and refractive index fluctuations. Using (5.24) with the excess polarizability σex = 4πε0 × 4.8 × 10−22 cm3 for the 27mer oligonucleotide, the surface density of hybridized oligonucleotides is estimated to be 3.6 × 1013 oligonucleotide targets/cm2 . This sensing architecture can be utilized to distinguish the single nucleotide mismatch. The surface of one microsphere is modified with a biotinylated 11-mer oligonucleotide while an 11-mer oligonucleotide differed by a single nucleotide is immobilized on the surface of the other microsphere. The fractional wavelength shifts of two WGMs at different NaCl concentrations are recorded [see Fig. 5.11e]. The shift for the matching sequence is about 10 times larger than that of the mismatching sequence at the optimal NaCl concentration 30 mM. The time traces of the resonance wavelength shifts of two WGMs are shown in Fig. 5.11f, where the hybridization to the perfect match oligonucleotide produces a resonance wavelength shift much larger than that of the mismatching sequence [see Fig. 5.11e]. • Photoinduced transformations in bacteriorhodopsin membrane Two orthogonally polarized TE and TM WGM modes are used to study the molecular structural changes in the biological photochrome bacteriorhodopsin (bR) [20]. bR is a transmembrane protein of well-known structure. The bR membrane self-assembles easily onto a silica surface of a WGM microsphere sensor with a well-known surface concentration of the bR protein. The molecular conformation of bR can be switched optically between two stable states. The photochrome retinal, which is part of the bR protein, undergoes a reversible all-trans to 13-cis conformational change shown in Fig. 5.12a. The resonance wavelength shifts of TE- and TM-polarized WGMs excited at near-infrared wavelengths around 1311 nm are observed in real time as the retinal conformation changes with the application of green (532 nm) light, see Fig. 5.12b. The retinal 13-cis conformation relaxes back to the all-trans ground state configuration, in the dark or with the application of blue light. The WGM sensor setup used to study the photochromic transformations of retinal in bR uses a visible

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261

Fig. 5.12 a Retinal conformational changes in bR are probed with TE and TM WGMs. b The TE and TM WGM resonance wavelengths blue shift when retinal changes from the all-trans to the 13-cis conformation. c WGM experimental setup that uses TE and TM WGMs excited around 1311 nm to probe conformational changes of bR triggered with 532 nm light. Accessory photodetector PD2 is used to analyze the polarization of WGMs. Reprinted from [20], Copyright (2007), with permission from Elsevier

532 nm pump light to reversibly trigger the retinal conformational change, and nearinfrared 1311 nm light to probe the conformational changes in real time by recording the wavelength shifts of TE and TM WGMs [see Fig. 5.12c]. The retinal molecule can be considered a rod-like molecule undergoing polarizability changes δα along the major retinal axis and δα⊥ perpendicular to it. The following equation shows how one can use the wavelength shifts measured for TE- and TM-polarized WGMs to estimate the angle Θ that the retinal molecule axis makes with respect to the normal axis of the bR membrane coated on a WGM microsphere sensor:   1 δα⊥ (1 + cos2 θ ) + δα sin2 θ |E TE |2 ΔλTE ≈ . ΔλTM 2 |E TM |2 δα⊥ sin2 θ + δα cos2 θ

(5.69)

This equation can be simplified further by assuming that TE and TM WGMs have equal electric-field amplitudes:

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5 Single-Molecule Sensing

ΔλTE 1 ≈ ΔλTM 2



 1 + cos2 θ + β sin2 θ

. sin2 θ + β cos2 θ

(5.70)

Analysis of the measured values of Δλ for the TE and TM WGMs using (5.70) corresponds to an angle of 61◦ and is in excellent agreement with 3D X-ray crystallographic data for bR. Given the known surface density of bR within a monolayer of the membrane one can calculate the average polarizability change of a single bR molecule from the measured TE and TM WGM wavelength shifts as –57 Å3 . • Optical microbubble sensor The sensitivity of a sensor depends on the evanescent coupling strength between the microcavity and a particle, which relies on the intrinsic properties (e.g., polarizability and size) of the target particle. An alternative way to strengthen the microcavity– particle interaction is to directly enhance the light field at the particle’s location. When the particle is located outside the microcavity, enhancing the evanescent field of the WGM decreases its quality factor, thereby limiting the sensor’s sensitivity. In contrast, placing the particle inside the microcavity, the local light field may be enhanced by choosing a high-order WGM. One physical realization of such a sensing structure is the optical microbubble (see Fig. 5.13a) with a micrometer-order wall thickness [21]. We assume the microbubble has a hollow-sphere shape with the inner and outer radii R1 and R2 , respectively. The WGMs of the microbubble can be derived from the scalar Helmholtz equation. The analytical expression of the WGMs are separated into the radial, polar, and azimuthal components, where the radial part is expressed as ⎧ ⎪ ρ < R1 ⎨ A jm (n 1 kρ), (1) (5.71) Ψ (ρ) = B jm (n 2 kρ) + Ch m (n 2 kρ), R1 ≤ ρ ≤ R2 . ⎪ ⎩ (1) Dh m (n 3 kρ), ρ > R2 with the vacuum wavenumber k and the spherical Bessel jm and Hankel h (1) m functions. The parameters A, B, C, and D are determined by the appropriate boundary conditions, and n 1,2,3 represent the refractive indices in different regions. The microbubble’s core may be filled with either gas or liquid. Here we consider the liquid core. For the calculation of the WGMs of a complex microbubble, the use of the finite element method is necessary. As shown in Fig. 5.13b, the low-order WGMs are strongly confined inside the microcavity’s wall (R1 ≤ ρ ≤ R2 ) while the energy of the high-order WGMs is mainly distributed inside the microcavity’s core (ρ < R1 ) [23]. Besides choosing the high-order WGMs, decreasing the thickness of the microcavity’s wall (R2 − R1 ) strongly pushes the light field into the core region as illustrated in Fig. 5.13c. A more detailed dependence of the intracavity energy (i.e., the light energy within the core of the microbubble) on the thickness of the microbubble’s wall is presented in Fig. 5.13d (adapted from [24]). To reach the same fraction of the intracavity energy, a low-order WGM requires a wall thinner than that of a high-order WGM. However, the

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Fig. 5.13 Optical microbubble sensor. a Microbubble. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, The European Physical Journal Special Topics, [22], © 2014. b Normalized intensity distribution of the first- and second-order WGMs in the radial direction. Reprinted from [23], with the permission of AIP Publishing. c Field distribution of the third-order WGM derived from (5.71). Reprinted from [6], with the permission of AIP Publishing. d Percentage of the light energy inside the microbubble’s core and e quality factor of the WGMs versus the wall thickness. Here, n is the radial quantum number, denoting the mode order. Figures d and e are reprinted with permission from [24] ©The Optical Society

high-order WGMs suffer a strong inner surface tunneling loss, thereby owning a low quality factor [see Fig. 5.13e]. Thus, the optimal choice of the WGMs for the sensing detection depends on both the WGM order and the thickness of the microbubble’s wall. The microbubbles can be made from the glass capillaries. A small section of a capillary is heated (for example, by using a focused CO2 laser beam [21]) and softened. By pressurizing the air inside the capillary, the softened capillary section forms a bubble shape. The excitation of the WGMs may be implemented through a tapered optical fiber that is in contact with the microbubble.

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Since a large amount of the light resides inside the microbubble’s core, the resonance wavelength of a WGM is particularly sensitive to the change in the refractive index of the liquid core. Additionally, the liquid flows continuously and the analyte dissolved in the liquid (or the analyte concentration) can be changed straightforwardly without disturbing the sensor. A microbubble sensor based on the lasing operation was demonstrated in [25]. A gain layer of dye-doped polymer is imposed on the inner wall of a glass capillary and coupled with the WGMs via the evanescent field. A frequency-doubled Q-switched Nd:YAG pulse laser at 532 nm provides the pump energy for the laser oscillation. An ethanol solution flows inside the capillary and the quality factor of the WGMs is about 2 × 106 . Adding the methanol adjusts the refractive index of the solution. Monitoring the blue shift of a resonance peak in the lasing spectrum enables one to detect the change in the refractive index of the solution. Moreover, instead of flowing the liquid, the microbubbles can be also filled with the gases and the spectral shift of the WGMs is sensitive to the internal gas pressure, which can be utilized for the pressure sensing [26]. • Integrated multiplexed biosensor The capillary microfluidic channels with a diameter of about 100 µm and a wall thickness of a few microns do not only support the high-Q WGMs but also deliver the sample solution inside the channel. Such liquid core optical ring resonators (LCORRs) are of great interest since they facilitate a strong WGM–analyte coupling and enable an integrated label-free biosensor with an excellent sensitivity [27]. The evanescent excitation of the WGMs in a LCORR can be implemented by using an anti-reflective ridge optical waveguide (ARROW) on a chip as shown in Fig. 5.14a. The ARROW structure [see Fig. 5.14b] possesses the advantages of a low leakage of the light from the input bus waveguide into the substrate and a sufficient microcavity–waveguide coupling. The experimental measurement shows a cavity quality factor over 105 . In addition, the LCORRs own the multiplexing capability, i.e., a single LCORR may interact with multiple waveguides simultaneously and the WGMs are excited at different locations [see Fig. 5.14a]. Each LCORR-ARROW contact point potentially operates as an independent sensor. Thus, the LCORR-ARROW structure paves a way to achieve a robust lab-on-a-chip sensing device that is compact and well suited for the multiplexing use. An integrated biosensor was reported in [28]. In the refractive index sensing experiment, a solution of water and ethanol is pushed through the LCORR with the concentration of ethanol being increased gradually. The spectral shift of the guided wave transmitted from the waveguide is recorded for different concentrations of ethanol and the curve fitting shows a sensitivity of 12.5 nm/RIU [see Fig. 5.14c]. In the biosensing experiment, the BSA in phosphate buffer passes through the LCORR and the resulting sensorgram follows the first-order Langmuir model [see Fig. 5.14d], manifesting a monolayer of BSA formed on the inner wall of the capillary. • Sensing based on the spectral broadening All the experiments introduced above rely on the approach of monitoring the resonance shift of a microcavity’s mode. Here we introduce a sensing experiment [29],

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Fig. 5.14 Integrated multiplexed biosensor. a LCORR-ARROW system. b Cross-section. c Refractive index sensitivity of LCORR. d Sensorgram for BSA passing through LCORR. Reprinted from [28], with the permission of AIP Publishing

where the WGM broadening is measured in real time. The common sensing structure is slightly deformed so as to efficiently excite the WGMs of a microtoroid (major diameter 60 µm and minor diameter 50 µm) in free space and suppress the thermooptic effects. The probe beam comes from a semiconductor tunable laser and has a linewidth of about 300 kHz. The transmitted light from the microcavity is also collected in free space. The linewidth of the transmission spectrum is evaluated with a single-shot measurement time of about 10 ms. Compared to the WGM excitation through an external coupler (e.g., tapered fiber and prism), the free-space coupling hardly induces the extra spectral broadening of the WGMs. Figure 5.15a shows the dependence of the transmission spectrum on the number of spherical polystyrene nanoparticles (radius 70 nm) contacting the microcavity’s surface. The linewidth of a WGM can be derived through fitting the spectrum to the Lorentz lineshape and the results are presented in Fig. 5.15b. Interestingly, the linewidth does not always go up as the number of nanoparticles is increased. This can be interpreted from the strong dependence of the WGM–particle coupling on the specific location of a nanoparticle. The newly added nanoparticle may cause destructive interference between the pair of counter-propagating WGMs, shifting the nanoparticle’s position from an antinode to a node of a standing wave. Moreover, Fig. 5.15c displays the mode broadening induced by the lentiviruses in a diluted Dulbecco’s modified eagle medium. The measurement is performed as follows: (1)

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Fig. 5.15 Spectral broadening induced by the nanoparticles (NPs) and virus. a Transmission spectra of the WGM at 1550 nm. b Mode broadening versus the number of NPs for two WGMs (at the 1550 nm wavelength band) with intrinsic linewidths of 54.7 (square) and 143.3 (triangle) MHz. c Individual virus detection of a sample solution. Reprinted from [29] by permission of John Wiley & Sons, Inc

The microcavity is immersed in the sample solution for 120 s, then showered with pure and deionized water and finally dried with the nitrogen gas; (2) The dried microcavity is excited and the mode linewidth is measured. Such a measurement is repeated multiple times. It is seen that the binding of the viruses to the microcavity’s surface leads to a linewidth broadening ranging from 2 to 6 MHz, which is less than that of the single-virus-induced maximum signal ∼7 MHz, proving the single virus detection.

5.4.2 Mode Splitting Detection of the particles through monitoring the WGM wavelength/frequency shift is limited in two aspects: (1) The spectral shift is small due to the weak microcavity– particle coupling. It is complex to identify the sensor signal from the background fluctuations caused by the laser intensity and frequency fluctuations and the environmental noises; (2) The sensor response depends strongly on the particle’s location. Two particles with different sizes and different locations may lead to the same spectral shift, challenging the size estimation of the particles. These two limits can be overcome by introducing a second WGM. Since both WGMs experience the same noise sources, the self-referencing detection sufficiently suppresses the background fluctuations, thereby highlighting the sensor signal. In addition, the existence of the particle removes the azimuthal symmetry of WGMs. This provides a way to cancel the particle location dependence through the measurement of the mode splitting and spectral linewidths (see below). As a result, the particle size is computed with a high accuracy. Let us consider a pair of clockwise (cw) and counter-clockwise (ccw) WGMs in a microcavity. These two WGMs have the same resonance frequency ωc and spatial

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profile Ψ (r). The quality factor of the free WGMs is Q and the mode linewidth is given by κ0 = ωc /Q. A fiber taper is used to couple the pump light into and out of the microcavity at a rate κ1 . Two WGMs are excited by a light field whose frequency is ω. A photon scatterer (e.g., impurity, ion, and biomolecule) is located in the vicinity of the microcavity surface and talks to both WGMs via the evanescent light. This leads to an effective coupling between two WGMs with a strength g = −ωc

V p αex  |Ψ (r0 )|2  m , 2ε0 V (r)|Ψ (r)|2 dr

(5.72)

in the limit of Rayleigh scattering (i.e., the scatterer’s size is much smaller than the radiation wavelength, also known as the dipole approximation). Here, αex is the excess polarizability of the scatterer, V p denotes the scatterer’s volume, and r0 corresponds to the scatter’s location. The function (r) gives the spatial distribution of the relative permittivity and the ambient dielectric constant is m . The value of g may be positive or negative, depending on αex . The intermode interaction lifts the degeneracy of two WGMs, resulting in the mode splitting [30]. The wave equations of the WGMs are derived as     Γ + κ0 + κ1 Γ d Acw = − + i(ωc + g) Acw − ig + Accw dt 2 2 √ −iωt − κ1 Ain , (5.73a) cw e     d Γ + κ0 + κ1 Γ Accw = − + i(ωc + g) Accw − ig + Acw dt 2 2 √ −iωt − κ1 Ain , (5.73b) ccw e with the mode amplitudes A j ( j = cw, ccw) and the input-field amplitudes Ainj . The scatterer–WGM coupling opens an additional decay channel of the microcavity modes, i.e., the Rayleigh scattering Γ =

ωc4 √ 6π(c/ m )3



V p αex ε0

2

 V

m |Ψ (r0 )|2 . (r)|Ψ (r)|2 dr

(5.74)

As illustrated in (5.73), the presence of a scatterer causes a frequency shift g and an extra decay rate Γ to both WGMs as well as the intermode coupling with a complex strength (g − iΓ /2). Replacing the amplitudes A j with A˜ j e−iωt and setting d A˜ j /dt = 0, one finds the steady-state solutions of the coupled WGMs (denoted by “ss”) [31]   2Γ + κ0 + κ1 √  Ain i(Δ − 2g) − κ1 +, 2    κ0 + κ1 √  Ain iΔ − = κ1 −, 2

A(ss) + = A(ss) −



(5.75a) (5.75b)

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Fig. 5.16 Single-particle detection based on the mode splitting. a Transmission spectra for different numbers of deposited particles. b Normalized splitting versus the particle number for different particle radii a. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Photonics, [31], © 2010

where we have defined the pump-field detuning Δ = ω − ωc , the standing-wave amplitudes  1 ˜ Acw ± A˜ ccw , (5.76) A(ss) ± = √ 2 √ in in and the corresponding input amplitudes Ain ± = (Acw ± Accw )/ 2. Interestingly, the asymmetric amplitude A(ss) − is independent of the scatterer–WGM interaction while the symmetric amplitude A(ss) + has a frequency shift 2g and an extra spectral broadening Γ . For a free microcavity, the choice of the initial value ϕ = 0 along the azimuthal direction is arbitrary, for which the nodes and antinodes of a standing wave are not fixed. Once the scatterer is introduced, it can be viewed as a reference (or boundary). The location of the scatterer may be at an antinode of a standing wave, giving rise to the maximum scatterer–WGM coupling. The amplitude A(ss) + corresponds to this situation. In addition, the scatterer may be also located at a node of a standing wave, where the scatterer–WGM interaction is minimized, leaving the asymmetric amplitude A(ss) − unaffected. The above theoretical predictions have been verified in experiment. As shown in Fig. 5.16a, two degenerate WGMs correspond to the same spectral dip, whose linewidth is (κ0 + κ1 ), in the transmission spectrum of a free microcavity. When a particle is bound to the microcavity’s surface, the single dip is split into two. One of them is located at the position same to the original one and also its linewidth is equal to (κ0 + κ1 ). This mode corresponds to the asymmetric standing-wave mode A(ss) − while the other one denotes A(ss) + with a spectral linewidth (2Γ + κ0 + κ1 ). The separation between two dips is equal to 2g, which should be larger than (Γ + κ0 + κ1 ) so as to distinguish two dips. The depth (linewidth) of the A(ss) + dip is smaller (wider) than dip because of the extra decay rate Γ . As a second particle is bound that of the A(ss) − to the microcavity’s surface, the previously established field is redistributed to a new

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steady state and two dips are further separated apart. Thus, the indirect cw – ccw interaction is potentially enhanced as the number of particles grows. However, it is worth noting that the successive binding events do not necessarily lead to a successive increase in mode splitting as shown in Fig. 5.16b. When more particles are deposited, the position of the A(ss) − dip is approximately unmoved. But both transmission dips become broadened since more damping channels are introduced. In addition to the single-particle detection, the mode splitting can be also utilized to compute the particle size. For a single particle binding to the microcavity’s surface, the spectral linewidth difference and the separation between two transmission dips is given by Γ and |2g|, respectively. From (5.72) and (5.74), one finds 8π 2 Γ = 3 |g| 3λc



V p αex ε0

 ,

(5.77)

√ with the in-medium resonance wavelength λc = 2πc/( m ωc ). Surprisingly, the ratio Γ /|g| is independent of the particle location. We model the particle geometry as a sphere with a radius a. Using (2.55), the scatterer’s polarizability αex is expressed as  p − m V p αex = 4πa 3 , (5.78) ε0  p + 2m with the dielectric permittivity of the particle  p . The particle radius a may be then estimated by combining (5.77) and (5.78). Such an approach enables the single-shot measurement at a high accuracy.

5.5 Sensing Based on Active Microcavities So far, we have briefly introduced several selected experiments of the sensing detection. All of them operate in a passive manner, where the WGMs are excited via an external light source and the transmission spectrum is commonly employed to detect the nanoparticles/biomolecules in the vicinity of the microcavity’s surface. Besides, the lasing action can be also utilized in the sensing detection at the single particle level. The Raman lasing, also known as the stimulated Raman scattering (SRS), is applicable to most materials, avoiding doping the microcavity with extra active particles and thereby much facilitating the fabrication. Figure 5.17a illustrates the schematic energy-level diagram for the Raman transition, in which a three-level molecule absorbs one photon from the pump light at ω p and then emits a Stokes photon at ωs via a virtue molecular state. Unlike the Rayleigh scattering with ω p = ωs , two frequencies ω p and ωs are unequal and the difference ω p − ωs matches a molecular vibration. The SRS takes place when some Stokes photons have already been produced by the spontaneous Raman process or when deliberately injecting the Stokes photons. Indeed, the SRS is a third-order nonlinear optical phenomenon (i.e., four-wave mixing).

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Fig. 5.17 Raman-lasing-based sensor. a Schematic energy-level diagram of the stimulated Raman scattering. b Normalized Raman gain spectrum of the bulk silica. The Raman lasing occurs at two split modes R+ and R−. c Raman spectrum. d Raman lasing pulse generated by scanning the pump light frequency. e Detailed Raman power oscillation within the highlighted section in d. f Beat frequency change versus the number of the nanoparticles bound to the microcavity’s surface. g Beat frequency change as a function of the time. Figures b–g are reprinted from [32] with the granted permission

The features of the small mode volume and the high quality factor of the WGM microcavities significantly suppress the Raman lasing threshold. The lasing can occur simultaneously in two degenerate cw- and ccw-propagating WGMs. When a particle is located close to the microcavity’s surface, two WGMs are coupled with each other through the backscattering induced by the particle. As a result, the spectral splitting can be observed in the emission spectrum. Such a single-particle detection operates in an active manner. In addition, the beat note between two Raman lasers is inherently immune to the environmental noises, enhancing the sensitivity. The relevant experimental realization was firstly reported in [32]. A silica microcavity is driven by a pump light at 685 nm. The Raman gain is maximized at the Raman shift of 13 THz [see Fig. 5.17b]. The SRS occurs at 706 nm with a threshold

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of 350 µW as shown in Fig. 5.17c. Scanning the frequency of the pump light produces the Raman emission pulses [see Fig. 5.17d, e]. The rapid oscillation results actually from the splitting of the coupled cw and ccw modes, and the oscillation frequency (typically several tens of megahertz) corresponds to the frequency separation between two split modes, i.e., the beat frequency. As demonstrated in Fig. 5.17f, this beat frequency depends sensitively on the nanoparticle binding events and, therefore, can be applied to the real-time detection of single nanoparticles [Fig. 5.17g].

5.6 Other Schemes of Single Particle Detection The sensing detection is generally based on the scheme of either the resonance frequency/wavelength shift or the mode splitting. Both of them rely on the perturbation of single particles/molecules to the WGMs of the microcavities. Two approaches may be utilized to enhance the sensitivity of the single particle/molecule detection: (1) further mapping the resonant mode shift onto another physical quantity (e.g., the mechanical motion in optomechanics) so as to amplify the sensing response; (2) moving the operation point of the sensor from the diabolic point to, for example, the exceptional point where the mode splitting is increased. In this section, we introduce several methods that potentially enhance the sensor’s sensitivity.

5.6.1 Optomechanical Sensing of Single Molecules In Sect. 4.7, we have discussed the coupling between the optical oscillation and the mechanical motion, where the backaction of the intracavity photons enables the suppression of the mechanical fluctuations and the tiny displacement measurement. This coupling of optical and mechanical degrees of freedom can be exploited to the single-particle/molecule detection. As demonstrated in [33], when the wavelength of an external input light λ1 is blue detuned to the WGM of a microcavity λ0 , Δλ = λ1 − λ0 < 0, the mechanical motion of the microcavity may be boosted along with narrowing the mechanical spectral linewidth. This process of enhancing the coherence in mechanical motion is known as the coherent optomechanical oscillation (OMO), whose central frequency is f m and spectral linewidth is Δf m . The WGM quality factor Q needs to be maintained at a high value so that the optical backaction exceeds the threshold of the regenerative OMO. As a result, the perturbation to the WGM of the microcavity may be directly mapped onto the mechanical spectrum. The OMO frequency f m is a function of the laser-cavity detuning Δλ . When a particle/molecule is bound to the microcavity’s surface, the WGM wavelength is shifted by δλ. This shift further induces a frequency shift δ fm = −

d fm δλ, dΔλ

(5.79)

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Fig. 5.18 Optomechanical spring sensing. a Schematic diagram of the sensing mechanism. When a particle/molecule is bound to a microsphere, the optical resonance wavelength is shifted by δλ, which is transduced to a mechanical frequency shift δ f m . b Optical transmitted power as a function of the laser detuning Δλ . In the regime of Δλ ∼ 0, the transmitted power is strongly dropped and the oscillatory behavior exhibits a distorted sinusoid. The oscillation can be decomposed into multiple mechanical harmonics shown in c. For Δλ is far apart from the resonance, the transmitted power has a sinusoidal dependence on Δλ . d OMO frequency f m as a function of laser-cavity wavelength detuning. e An example of a recorded particle binding event. Reprinted from [33] under the Creative Commons CC BY license

to the OMO as illustrated in Fig. 5.18a. Replacing δ f m in (5.79) with the linewidth Δf m , one obtains the minimal detectable frequency shift of the WGM   δλ 1 d f m −1 , = Qm λ0 min f m d(Δλ /λ0 )

(5.80)

with the mechanical quality factor Q m = f m /Δf m . A large Q m and a strong dependence of f m versus Δλ enhances the sensing resolution. We consider a laser beam driving a WGM of a silica microsphere whose diameter is ∼100 µm. In the aqueous environment, the intrinsic optical quality factor of the WGM can be as high as 5 × 106 . The power of the transmitted light through the cavity is monitored in real time to measure the radial mechanical motion of the microsphere. The OMO threshold for the microcavity is a few milliwatts in power. The transmitted power exhibits an oscillatory behavior when tuning the laser detuning Δλ [see Fig. 5.18b]. As discussed in [34], in the regime of Δλ close to zero, the transmitted laser power is low, which means a large fraction of the light power is dropped into the microcavity, and the corresponding oscillation presents a distorted sinusoid due to the effects of high-order mechanical harmonics [see Fig. 5.18c]. In contrast, the dropped optical power becomes less when Δλ is far apart from the

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cavity resonance. However, the transmitted power displays a sinusoidal dependence on Δλ , corresponding to the fundamental mechanical mode. This mechanical mode has a frequency of about f m = 250 kHz and a linewidth as narrow as Δf m = 0.1 Hz. Thus, the mechanical quality factor reaches Q m = 2.5 × 106 . For the detuning Δλ in the vicinity of the resonance, the derivative of f m with respect to Δλ can be as high as d f m /dΔλ = −1.5 kHz/fm. That is, a 1-fm wavelength shift of the WGM leads to a 1.5-kHz change in the OMO frequency. As a result, the minimal sensing resolution is given by |δλ/λ0 |min ≈ 10−13 in the visible range of wavelength. Nevertheless, the experimental measurement of the Allan deviation of f m shows a minimum deviation of 10 Hz at a sampling time of around 0.3 s. This indicates that extra noise sources influence the stability of the OMO frequency f m . The estimated sensing resolution is then given by |δλ/λ0 | ≈ 6 × 10−12 , which is still 50 times higher than that of the conventional sensors. Figure 5.18e shows an example of the real-time monitoring of the fundamental OMO frequency. It clearly records a particle binding event that is reflected by a sudden change of f m . This optomechanical spring sensor enables the singleparticle/molecule detection of, for instance, silica nanobeads with a radius down to 11.6 nm and proteins with a weight of 66 kDa (1 Da = 1 g/mol) [33].

5.6.2 Exceptional Point-Based Sensor In Sect. 4.4, we have discussed the reality of the energy spectrum of a non-Hermitian Hamiltonian that possesses the PT symmetry. A spontaneous PT symmetry breaking was predicted in theory and has been experimentally manifested based on the classical optical platform. At least two eigenvalues and their corresponding eigenstates of a non-Hermitian system coalesce at the exceptional points (EPs) in the parameter space, but the eigenstates do not satisfy the orthogonality. In contrast, the degeneracy of a Hermitian Hamiltonian occurs at the so-called diabolic point (DP) and the orthogonality of the eigenstates is fulfilled. A perturbation with a strength  lifts the degeneracy of eigenstates, resulting in an energy-level splitting. The splitting around the diabolic point scales as  compared to 1/N for an EP of order N (i.e., N degenerate eigenstates). Thus, for a sufficiently small , the splitting at the EP can be (much) larger that that at the DP. Motivated by this, operating the sensor at an EP may improve the sensitivity of the single-particle detection. A more than threefold enhancement has been theoretically predicted in [35]. Let us study a more general optical system whose dynamics is governed by a Schrödinger-like equation i

d Ψ = H Ψ with H = H0 + H1 , dt

(5.81)

where  denotes the strength of the perturbation H1 to H0 . In the two-state model, the Hamiltonian H0 at a DP is recognized as

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  ω0 0 . 0 ω0

H0(DP) =

(5.82)

Here the optical loss/gain effect has been taken into account in the complex frequency ω0 . Obviously, two corresponding eigenstates     1 0 and 0 1

(5.83)

are degenerate and orthogonal to each other. Since the eigenvalue ω0 is complex, the concept of DP has been generalized beyond the Hermitian operators in quantum mechanics. In comparison, the Hamiltonian H0 at an EP is written as [36] H0(EP)

 ω0 A 0 . = 0 ω0 

(5.84)

The nonzero A0 makes H0(EP) nondiagonalizable and H0(EP) has only one linearly independent eigenstate   1 . (5.85) 0 It is seen that although H0(EP) is non-Hermitian, it does not possess the PT symmetry. Hence, we generalize the EP concept to the non-Hermitian Hamiltonian. The perturbation Hamiltonian H1 has a general form  H1 =

 ω1 A 1 , B1 ω2

(5.86)

from which the spectral splitting for the DP is derived as  Δω (DP) =  (ω1 − ω2 )2 + 4 A1 B1 .

(5.87)

As one can see, the splitting is proportional to the perturbation strength . For the EP, the splitting is given by √   (ω1 − ω2 )2 + 4 A0 B1 + 4A1 B1 √  ≈  4 A0 B1 ,

Δω (EP) =

(5.88)

which is larger than Δω (DP) for a sufficiently small . In the special case, ω1 = ω2 , one arrives at  A0 (EP) (DP) ≈ Δω . (5.89) Δω A1

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Consequently, the sensitivity of a sensor operating at the EP may exceed that of a sensor operating at the DP. Let us consider a more specific system consisting of a pair of cw- and ccwpropagating waves in a microcavity. These two optical waves have the opposite azimuthal mode numbers m and −m. The system is brought to an EP via introducing two scatterers, for instance, two nanofiber tips or two particles placed in the vicinity of the microcavity’s surface. Then, the Hamiltonian H0 is given by       ω0 + 2j=1 δω j 2j=1  j e−i2mβ j ω A0  H0 = 2 = i2mβ j B0 ω ω0 + 2j=1 δω j j=1  j e

(5.90)

where ω0 denotes the complex frequency (including the intrinsic cavity loss rate) of the unperturbed microcavity modes and β j represents the angular position of the j-th scatterer. The effects of the j-th scatterer on two modes lead to the frequency shift δω j , which is also complex due to the existence of the inelastic photon scattering, and the intermode coupling  j . The off-diagonal elements A0 and B0 quantify the backscattering-induced interaction between the cw- and ccw-traveling waves. When |A0 | = |B0 |, one obtains an asymmetric backscattering. The EP requires A0 = 0 and B0 = 0 (or A0 = 0 and B0 = 0). A third photon scatter, which is also the target particle that needs to be detected, is introduced into the system. Its role corresponds to the perturbation Hamiltonian H1 in (5.81),  δω3 3 e−i2mβ3 . H1 = 3 ei2mβ3 δω3 

(5.91)

The complex frequency splitting at the EP is given by  Δω

(EP)

≈ Δω

(DP)

A0 imβ3 e , 3

(5.92)

with Δω (DP) = 23 . The modulus |Δω (EP) | is independent of the angular position β3 of the target particle and much exceeds |Δω (DP) | when |A0 |  |3 |. The enhancement factor of the sensitivity can be defined as the ratio |Δω (EP) /Δω (DP) |, which is proportional to the square root of |A0 |. The above analysis is also valid to the perturbation caused by multiple target particles as long as N |A0 |   j e−i2mβ j . j=3

(5.93)

Such an enhancement of the sensitivity, arising from the asymmetric backscattering of counter-propagating optical waves, has been demonstrated in [37]. A microtoroid cavity operates at a DP, where the pair of cw- and ccw-traveling modes are degenerate and orthogonal. Two silica nano-tips are employed as the Rayleigh scatterers to turn the microcavity operation point to an EP [see Fig. 5.19a]. The specific

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Fig. 5.19 Exceptional-point-enhanced sensing. a Steps 1–4 of single-particle detection at an EP and the sensing at a DP. b Dependence of the complex frequency splittings Δω (EP) and Δω (DP) on the distance between the microcavity and the target scatterer. c Ratio |Δω (EP) /Δω (DP) | versus the perturbation strength. The inset displays a logarithmic plot of |Δω (EP) | and |Δω (DP) |. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature, [37], © 2017

procedures can be summarized as follows: (1) An optical mode of the microcavity is excited and no modal coupling is confirmed by the absence of mode splitting in the transmission spectrum. That is, the system is initialized at the DP; (2) A scatterer (nano-tip) is introduced to induce the modal coupling, manifested by the mode splitting and a reflection signal; (3) A second scatterer is introduced and finely tuned so that the system operates at an EP, where the light travels primarily in one direction because of the fully asymmetric backscattering between the cw- and ccw-traveling waves (i.e., A0 = 0 and B0 = 0); and (4) A third nano-tip is used to play the role of the target scatterer. As a result, the spectral splitting is observed in the transmission spectra. Figure 5.19b compares the complex frequency splittings Δω (EP) and Δω (DP) . For the splitting at the DP, the first two nano-tips are removed and only the third one is left at the same position as shown in Fig. 5.19a. As expected, the frequency shift Re[Δω (EP) ] is larger than Re[Δω (DP) ]. Also, Im[Δω (EP) ] exceeds Im[Δω (DP) ]. As illustrated in Fig. 5.19c, the ratio |Δω (EP) /Δω (DP) | grows as the perturbation √ induced by the target scatterer is reduced. The scaling relations |Δω (EP) | ∝  and (DP) |Δω | ∝  have also been manifested in experiment [see the inset of Fig. 5.19c].

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5.6.3 Self-referenced Photonic Molecule Biosensor The frequency/wavelength shift of the WGM microcavity can be caused by either the change in the bulk refractive index of the ambient medium or the binding of particles/molecules to the microcavity’s surface. Both processes may also occur simultaneously. Distinguishing these two scenarios depends on a priori knowledge of the components in the solution. In addition, the perturbation caused by the target particles/molecules binding to the microcavity’s surface may be similar to that resulting from the nonspecific molecules, making it difficult to differentiate these two events. To address these issues, a biosensing platform based on the so-called photonic molecules (i.e., coupled microcavities) has been proposed [38]. As two identical microcavities approach each other, the evanescent fields of WGMs of two microcavities overlap spatially, resulting in the inter-WGM coupling. Such an interaction can be either constructive or destructive. In the former (latter)

Fig. 5.20 Photonic molecule biosensor. a Bonding and antibonding supermodes of two identical silica microspheres with a diameter 10 µm. Two microspheres are in free space and separated by a surface-to-surface gap 300 nm. b Scattering spectra of the coupled system with different intermicrosphere gaps. The peaks correspond to the binding modes while the dips denote the antibinding modes. c Bonding and antibonding resonance wavelengths as a function of the change in the bulk ambient refractive index Δn. d Resonance wavelengths versus the thickness d of the thin dielectric layer on the surface of one of the microspheres (squares). In comparison, the results from the thin dielectric layers on the surfaces of both microspheres (circles) are also presented. e Separation between the binding and antibinding resonance wavelengths ΔλPM versus the wavelength shift of the binding supermode ΔλB . The triangle, square, and circle plots correspond to the ambient refractive index variation, the specific binding of molecules to the surface of one of microspheres, and the nonspecific binding to both microspheres, respectively. The numbers denote the corresponding slopes. All figures are reprinted with permission from [38] ©The Optical Society

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type, the WGMs in two microcavities are in-phase (out-of-phase) with respect to the axis passing through two microcavity centers [see Fig. 5.20a]. In the view of molecular physics, two microcavities form a photonic molecule, where the constructive (destructive) inter-WGM coupling is analogous to the bonding (antibonding) molecular orbitals. The bonding and antibonding supermodes may be identified in the scattering spectrum that is obtained by using the generalized multiparticle Mie theory. As shown in Fig. 5.20b, the sharp peaks (dips) correspond to the bonding (antibonding) supermodes. Basically, in a scattering spectrum, the wavelength of the bonding supermode is longer than that of the antibonding one. As the coupling strength is enhanced (i.e., the inter-microcavity gap is reduced), the peak-dip separation becomes larger. We examine the wavelength shifts of two supermodes caused by the changes in the bulk ambient refractive index and the thickness of a thin molecule layer coated on the microcavity’s surface, respectively. As illustrated in Fig. 5.20c, increasing the bulk ambient refractive index Δn shifts both binding and antibonding supermodes toward the longer wavelengths, i.e., the red shift. Similarly, increasing the thickness of the molecule layer d also induces the red shift of both supermodes [see Fig. 5.20d]. However, the wavelength shifts for a photonic molecule composed of one dielectriccoated microcavity interacting with one uncoated microcavity are weaker than that of a photonic molecule composed of both dielectric-coated microcavities. Raising Δn and increasing d both lead to a red shift of supermodes. In order to distinguish them, we consider the wavelength shift of the binding supermode ΔλB and the inter-supermode separation in wavelength ΔλPM [see Fig. 5.20c]. Figure 5.20e summarizes the ΔλPM − ΔλB relation under different situations: (i) raising the ambient refractive index variation Δn, (ii) increasing the thickness of a specific molecular layer formed on the surface of one of microspheres, and (iii) increasing the thickness of the nonspecific molecular layers binding to both microspheres. It is seen that the slope ΔλPM /ΔλB of (i) is much larger than the others while the case (iii) has the lowest slope. This provides a way to identify the wavelength shift caused by different types of events.

5.7 Practical Limits of Sensing Detection As discussed above, the single-particle/molecule sensing measurement can be implemented via monitoring the resonance frequency/wavelength shift, the mode splitting, and the spectral line broadening in real time. • The resonance shift determined by (5.10) is essential to be large enough that it can be distinguished from the environmental thermal noise-induced shift of the microcavity mode and the frequency drift of the probe laser. A 5-mK change in the ambient temperature may lead to a 10-MHz (22-MHz) resonance shift in the 1550nm (680-nm) wavelength band. The typical short-time frequency drift of the probe laser approximates 10 MHz. In addition, the shift signal is not detectable when the

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Fig. 5.21 Limits for detecting the spherical polystyrene nanoparticles in air through the resonance shift, mode splitting, and mode broadening of the WGMs. Reprinted from [29] by permission of John Wiley & Sons, Inc

resonance shift is smaller than 1% of the spectral linewidth of a cavity mode. As shown in Fig. 5.21, increasing the microcavity’s quality factor can always improve the detection limit until the thermal noise takes effect. • For the sensing detection based on the mode splitting, the splitting amount given by (5.72) must be lager than the mode linewidth ω0 /Q (including both intrinsic and coupling-induced losses) plus the extra elastic (Rayleigh) scattering loss evaluated by (5.74). Here, ω0 denotes the WGM central frequency. Again, a high Q factor improves the detection limit, which, however, cannot go down beyond the probe laser linewidth (see Fig. 5.21). • For the sensing detection based on the spectral line broadening, the Rayleigh scattering is assumed to play a predominant role and the corresponding scattering rate should exceed the uncertainty of the single-shot linewidth measurement. As shown in Fig. 5.21, the detection limit for the broadening is lower than that of the other two approaches. Basically, the detection limit of a WGM-based sensor is determined by the comparison between the magnitudes of the background fluctuation and the induced sensing signal. The noise sources can be actually divided into two groups: (1) The ones due to the technical issues, such as laser instability, Brownian motion, and flow turbulence. The detection based on the methods introduced above is currently limited by these technical noises in the experiment. In principle, such noises may be suppressed by means of improving the experimental techniques, for instance, the laser stabilization, the temperature control, and placing the sensing device in a vacuum chamber;

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(2) The fundamental noises like the quantum nature of light and the Heisenberg’s uncertainty principle. These noises are unavoidable. Since the classical light waves are mostly applied in biosensing, here we consider that the sensitivity is ultimately limited by the so-called shot noise limit (also known as the standard quantum limit, SQL)  δλ ω0 1 = , (5.94) λ0 SQL Q P0 T ητ where Q is the quality factor of the microcavity, ω0 = 2πc/λ0 is the WGM frequency, P0 is the incident power, T is the transmission efficiency of the coupler, η is the photodetector’s quantum efficiency, and τ is the integration (measurement) time. Inserting the typical sensing parameters with Q = 106 , ω0 = 2.5 × 1015 s−1 , P0 = 1 mW, T = η = 0.9, and τ = 1 s, we obtain |δλ/λ0 |SQL = 1.8 × 10−14 , much smaller than the fractional wavelength shift caused by a single virus (see Fig. 5.10). Normally, a biosensor works at room temperature. The fluctuations in the ambient temperature and the laser-heating effects induce the variations in the refractive index and the size of a microcavity, giving rise to the thermo-refractive and thermo-elastic noises, respectively. Generally, the thermal-expansion coefficient of a microcavity is sufficiently small (i.e., α = 5.5 × 10−7 K−1 for fused silica) and the effects of the thermo-elastic noise on the microcavity are negligible compared to that of the thermo-refractive noise. The thermal-refraction coefficient for the fused silica is dn/dT = 1.45 × 10−5 K−1 , leading to a fractional wavelength uncertainty δλ/λ0 ∼ 10−10 . This value is four orders of magnitude larger than that of SQL, setting a new fundamental detection limit. It should also be noted that the measurement time τ plays an important role here. The typical value of τ in the biosensing uses ranges from 1 ms to 1 s. However, the experimental results obtained so far illustrate that the sensitivity approaches the thermo-refractive detection limit only when τ < 1 ms (see Fig. 3.18). That is to say, the current sensing detection suffers a relatively low-frequency noise, although its origin is still unknown. Indeed, the measurement time τ is limited by the microcavity itself. The common sensing measurements of mode shift, broadening, and splitting are all based on the steady-state properties of a microcavity. That is to say, in order to capture the full picture of the emission/transmission spectrum, the laser frequency/wavelength scanning needs to be slow enough that the intracavity field maintains a steady state continuously. We assume the intracavity field lifetime is τ . Thus, obtaining an effective frequency point requires a time duration of at least τ f . In addition, the scanning range should cover the full spectral width of the microcavity’s mode κ = τ −1 f . We assume that the spectral width κ covers N measured points. Hence, the minimum measurement time τ is limited by N τ f and the total scan bandwidth approximates B = N κ.

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5.8 Optoplasmonic Sensors The magnitude of the sensing signal depends on the evanescent coupling strength between the particles/molecules and the microcavity. Increasing the evanescent field at the nanoparticle/molecule’s location may enhance the sensing response. Such a method has been widely applied in the microbubble sensors. In this section, we introduce an alternative way, i.e., employing the LSPRs of metal nanoparticles, to strengthen the local evanescent field. Generally, the local-field enhancement depends on the aspect ratio, orientation, and surface roughness of the nanoparticles. As we will see, the plasmon-enhanced WGM biosensors possess the single-particle/moleculelevel sensitivity in the transient regime and potentially allow one to monitor a specific chemical/biological reaction kinetics.

5.8.1 LSPR Enhanced Local Electric Field As discussed in Sect. 2.4, due to the LSPR the confinement of the light field close to the surface of a noble metal nanoparticle overcomes the classical diffraction limit, which significantly enhances the near-field intensity by a factor of 102 ∼ 103 . This can be utilized to strengthen the evanescent microcavity–nanoparticle/molecule coupling, thereby enhancing the sensing signal. The large energy loss rate of the noble metal nanoparticles (i.e., the typical Q factor of the LSPR is ∼10) may reduce the quality factor of the hybrid system. However, the resulting quality factor is still high enough for the resonance-shift and mode-broadening measurements. It is ascribed to the microcavity’s high Q factor and the weak coupling between the microcavity and the noble metal nanoparticles. Here we focus on the coupling of a fused silica microsphere (radius R and n s = 1.45367) to a rod-shaped gold nanoparticle (refractive index n p ) as shown in Fig. 5.22a. The nanorod’s length (along the z-direction) and diameter (in the x − y plane) are L and D, respectively, and the distance from the nanoparticle to the microsphere’s surface is d. The equatorial plane of the microsphere is also the nanorod’s symmetry plane. The LSPR of the gold nanorod is driven by a fundamental TE-polarized WGM with an unperturbed wavelength λ0 and a mode index l. The microsphere-nanorod hybrid system is immersed in aqueous solution, whose refractive index is n m = 1.32979 + i × 1.395 × 10−7 . The analytical analysis of the eigenvalues and eigenmodes of the hybrid system is impossible. One has to resort to the finite element method (e.g., COMSOL Multiphysics eigensolver). The extreme difference between the microsphere (∼106 µm3 ) and nanorod (∼105 nm3 ) sizes makes it impractical to simulate the whole combined system by using a lab computer. Nevertheless, the high symmetry of the hybrid system and a priori knowledge of WGM allow one to find an approximate solution of the electric-field distribution around the nanorod [39].

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Fig. 5.22 LSPR enhancement. a Electric-field amplitudes of unperturbed and perturbed TEpolarized WGMs with R = 30 µm, l = 340, λ0 = 780.911 nm, D = 12 nm, L = 42 nm, and d = 7 nm. Inset 1: schematic diagram of the microsphere-nanorod system. Inset 2: intensity distribution of the light field around the nanorod in the y − z plane. b Near-field intensity enhancement Λ versus the ratio L/D (changing L). The spherical protrusion has a radius of 1 nm. Reprinted from Baaske, Foreman, and Vollmer, Nat. Nanotechnol. 9, 933–939 (2014) [39]

In the weak coupling limit, the nanorod affects the microsphere in a perturbative manner and disturbs the electric-field distribution within a small region around the nanorod. In addition, the fundamental WGM of interest (for the sensing use) is concentrated near the microsphere’s surface along the radial axis and in the equatorial plane along the polar direction. The evanescent field penetrates into the host medium with a short distance. Thus, the numerical simulation only needs to include a small tesseroid, which contains an interior and an exterior region of the microsphere with

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a long enough distance to the microsphere’s surface. Along the azimuthal direction, this small segment of the sphere is wide enough that the effects of the nanorod at the boundaries are negligible. As illustrated in Fig. 5.22a, the light field is strongly enhanced in the vicinity of the nanorod due to the LSPR. At the nanorod’s middle point, the intensity enhancement Λ, which is defined as the ratio of the local intensity with the existence of the nanorod to the intensity at the same position in the absence of the nanorod, exceeds 60. In contrast, the near-field intensity is maximized at the ends of the nanorod with an intensity enhancement factor Λ close to 103 [see the inset of Fig. 5.22a]. The resulting quality factor of the microsphere-nanorod hybrid system can be as high as 5 × 105 , large enough for the single-molecule detection. In practice, the current imperfect fabrication techniques cannot ensure the identical size of the nanorods. Thus, the plasmonic resonance of each nanorod may lie at a different wavelength and the enhancement factor Λ depends on the aspect ratio L/D as shown in Fig. 5.22b. For an optimal ratio L/D, Λ reaches 3 × 103 . Consequently, compared to a WGM sensor, the sensitivity gain of a microcavity coupled to the nanorods can be in excess of 103 . In addition, the current imperfect fabrication techniques inevitably lead to some degree of the surface roughness of the nanorod. This may strongly modify the near-field intensity distribution [see the inset of Fig. 5.22b] and also induces the LSPR shift, thereby complicating the sensor design.

5.8.2 Detection of Microsphere–Nanorod Interaction The polarization of the gold nanorods in the vicinity of the microsphere’s surface perturbs the WGMs. This perturbation depends on the nanorod’s polarizability and is in proportion to the enhanced field strength at the nanorod’s location. Generally, the size of the gold nanorods is much smaller than the WGM wavelength λ0 , for which the dipole approximation is valid. The polarizability of the gold nanorod may be estimated by (2.74). Using (5.13) and (5.10), one can evaluate the resonance shift and the spectral broadening induced by a gold nanorod. For instance, a microsphere-nanorod structure same to Fig. 5.22a has a wavelength shift δλ ∼ 50 fm and a spectral broadening δ(Δλ) ∼ 40 fm with the nanorod’s excess polarizability αex = 16 × 10−23 + i × 7 × 10−23 m3 along the z-direction. In practice, the nanorod possesses tensorial polarizability. Since the transverse LSPR is much weaker than the longitudinal one, the nanorod is less polarized in the x − y plane and the relevant effects are negligible. In the experiment, the synthesis of gold nanorods is carried out in the presence of CTAB.3 Both CTAB-coated nanorods and microsphere are immersed in the distilled water (pH 7), and the former do not adsorb to the latter. The Brownian motion of individual nanorods changes the microsphere-nanorod distance occasionally, giving 3 CTAB

(cethyltrimetylammonium bromide) is a surfactant that can lower the surface tension of liquids. CTAB is widely used to direct the growth of gold nanorods and stabilize them after synthesis.

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Fig. 5.23 Detection of the microsphere–nanorod interaction. a Traces of the wavelength shift δλ and the change in the spectral linewidth δ(Δλ) of a WGM. The pH of the aqueous environment is 7. b Traces for pH ∼ 1.6. Inset (upper): image of a WGM excited at 780 nm with a single nanorod located at d ≈ 77 µm. Inset (lower): dimensions of the nanorod used in experiment. Reprinted from Baaske, Foreman, and Vollmer, Nat. Nanotechnol. 9, 933–939 (2014) [39]

rise to the spikes in the wavelength and linewidth traces of a WGM [see Fig. 5.23a]. Such a transient behavior is inapplicable to the sensing of the analyte molecules. Changing the pH of the aqueous environment to ∼1.6 can remove the CTAB from the surface of gold nanorods. Consequently, the nanorods can adsorb to the microsphere’s surface in an irreversible manner, manifested by the steps in the wavelength and linewidth traces [see Fig. 5.23b]. The step heights depend on the specific location of

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the adsorbed nanorod with respect to the WGM profile and the nanorod’s orientation with respect to the WGM polarization [39].

5.8.3 Optoplasmonic Sensing Mechanism The microsphere with the adsorbed nanorods is ready for the detection of single analyte molecules dissolved in the solution. We assume a gold nanorod is located at the optimal position as shown in Fig. 5.24a. The near-field intensity is maximized at the nanorod’s ends, corresponding to the most sensitive spots, i.e., hotspots. When a molecule moves towards a hotspot, the coupling strength between the molecule and the optoplasmonic sensor goes up. Meanwhile, the WGM wavelength becomes longer, i.e., the red shift, which is reflected in the transmission spectrum of the microsphere [see Fig. 5.24b]. This resonance shift δλ reaches its maximum δλmax when the molecule-hotspot distance is minimized. After a duration τ , the molecule moves away from the hotspot and the resonance shift δλ goes back to zero. Such a transient molecule–sensor interaction corresponds to a spike in the trace of the WGM wavelength as shown in Fig. 5.24c. The mean value of the spike height within τ is δλave . The statistics of the time intervals between successive molecule–sensor

Fig. 5.24 Principle of the optoplasmonic sensing. a Transient interaction between a single molecule and the optoplasmonic hybrid system. The interaction has two stages: the molecule moves toward the sensor’s hotspot and then moves away. b The WGM resonance shifts toward the long wavelength side and then shifts back. c Spike in the trace of the resonance shift. The spike has a duration τ , maximum height δλmax , and mean value δλave . Reprinted from Baaske and Vollmer, Nat. Photon. 10, 733–739 (2016) [40]

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interaction events (spikes) is expected to fulfill the Poisson distribution. Besides the transient interaction, the molecule may also be permanently bound to the hotspot, leading to a step in the wavelength trace similar to Fig. 5.23b. The relatively longterm wavelength trace suffers a slow drift caused by in the variation in ambient temperature and pressure.

5.8.4 Detection of Nucleic Acid Hybridization The optoplasmonic sensor may be utilized to monitor the single-molecule nucleic acid interaction. The experimental setup is illustrated in Fig. 5.25a. Several gold nanorods have adsorbed to a glass microsphere that is immersed in a liquid-filled polydimethylsiloxane (PDMS) sample cell containing the analyte molecules (i.e., oligonucleotide). The robust prism coupling is applied to evanescently excite a single WGM of the microsphere and measure the transmission spectrum of the probe beam. Figure 5.25b-1 shows a resonance trace measured by an optoplasmonic sensor with non-functionalized nanorods. None spikes/steps caused by the molecule–sensor interaction are observed. Thus, it is essential to modify the nanorods with the receptor molecules. Figure 5.25b-2 displays a resonance trace for a pair of unrelated receptor and analyte, where no significant resonance shifts are observed. Hence, the detection of the analyte molecules also demands that the nanorods are modified with the receptor related to the analyte. As exhibited in Fig. 5.25b-3, the spikes appear in the wavelength trace when the receptor and analyte are partially matched. However, the step shifts caused by the permanent binding of the target molecules to the receptors hardly occur. This facilitates the study of the transient interaction of the mismatched strands with the oligonucleotide receptors adsorbed on the nanorods. The histogram of the maximum shifts illustrates that δλmax ranges from 2 to 20 fm [see Fig. 5.25c]. As illustrated in Fig. 5.25b-4, when the analyte well matches the receptor, the discrete steps play the dominant role in a wavelength trace while the spike events are rarely observed, indicating the analyte-receptor hybridization. Therefore, the optoplasmonic sensor based on the specifically functionalized nanorods provides a unique tool for detecting the kinetics of single molecules with specific interactions. The analyte–receptor interaction events follow the Poisson distribution. We choose the smallest experimentally accessible time interval τm (∼20 ms) as a certain interval of time. In this time window, the expected number of the events (spikes) in a wavelength trace is n. ¯ Once n¯ is obtained, the probability of n events occurring within τm reads n¯ n p(n) = e−n¯ . (5.95) n! To evaluate n, ¯ we plot the histogram of the time intervals Δt between two successive observed events [see Fig. 5.25d] and extract the characteristic time τc by the exponential curve fitting. The expected number n¯ is then given by

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Fig. 5.25 Detection of single-molecule nucleic acid interactions. a Schematic diagram of an optoplasmonic biosensor. b Wavelength traces for different analyte–receptor pairs. c Histogram of the maximum resonance shifts in the trace (3) in b. d Statistics on the time intervals between two successive observed interaction events in the trace (3) in b. Reprinted from Baaske, Foreman, and Vollmer, Nat. Nanotechnol. 9, 933–939 (2014) [39]

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n¯ = τm /τc .

(5.96)

Using τc obtained in Fig. 5.25d, n¯ is estimated to be 0.027. Further, one finds p(1) ≈ 0.026 is over two orders of magnitude larger than p(n > 1), confirming that the spikes are primarily caused by single-molecule nucleic acid interactions. Note that this analysis is also valid for the binding interactions, i.e., the steps in the wavelength trace. So far, the hybridization of short oligonucleotides (∼2.4 kDa) and the interaction of small intercalating molecules (∼1 kDa) with double-stranded molecules have been demonstrated in experiment [39]. The corresponding average values of the observed step heights are 2.5 and 5 fm, respectively.

5.8.5 Detection of Ion–Nanorod Interactions The optoplasmonic sensor is also capable of detecting the interactions between single atomic ions with gold nanorods in the aqueous environment. When the ions access the hotspots (i.e., the nanorod’s tips), the microcavity–nanorod coupling transduces the ion–nanorod interactions into a red shift of the WGM wavelength. The characteristic inter-event interval τc measures the rate of the analyte ions interacting with the nanorods, and it depends on the analyte concentration [A]. Increasing [A] by a factor f , one may find the following scaling law under the assumption that the ion–nanorod interactions follow the Poisson process: τc ( f [A]) ≈ f −N . τc ([A])

(5.97)

Here N denotes that a spike event arises from N times of the ion–nanorod interaction. The single-ion–nanorod interactions with N = 1 are manifested by Fig. 5.26a, thereby further confirming the Poisson-process assumption. The ions can interact with the nanorods in a transient manner, corresponding to the spikes in the resonance wavelength trace, or in a permanent way, i.e., the strong covalent bonds are formed between ions and nanorods, leading to a step in the wavelength trace. The ion–nanorod interaction behavior depends on the ionic strength of the solution as illustrated in Fig. 5.26b, c. At a low ionic strength, the spike events are presented exclusively in the wavelength trace. As the ionic strength is raised, the steps appear while the observed spikes become less frequent. For high ionic strength, the number of step events goes up and the spike events are hardly recognized. Generally, the bond formation is an irreversible process. However, another type of the ion–nanorod interaction has been demonstrated in experiment, where the resonance wavelength jumps up and down between two values with different interval lengths [see Fig. 5.26d]. This discrete behavior may be interpreted through the metastable adsorption of an ion to a nanorod. At favorable ionic strength, the Hg2+ ions approach the surface of the nanorod where they can participate in the amalgamation reaction with Au. There is no reducing

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Fig. 5.26 Detection of the ion–nanorod interactions. a Dependence of the normalized spike rates on the normalized concentration of the analyte ions. b Number of spikes in successive constant intervals of four minutes and the cumulative step count at the ionic strength of 14.6 mM and [Hg2+ ] = 0.8 µM. c Resonance shift traces at different ionic strengths and [Hg2+ ] = 0.8 µM. d Resonance shift traces at different ionic strengths and [Zn2+ ] = 8 µM. Reprinted from Baaske and Vollmer, Nat. Photon. 10, 733–739 (2016) [40]

agent present in the experiments shown in Fig. 5.26c. The steps in the sensor signals might result from the reduction of Hg2+ ions by light-induced hot carriers created in the tips of the nanorods themselves, resulting in an amalgamization reaction between a Hg2+ ion and an Au atom. Similar light-induced processes have been reported previously to account for the efficient reduction of silver ions [41].

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5.8.6 Detection of Protein Conformational Changes and Chemical Reactions Gold nanorods attached to the optoplasmonic WGM sensor define detection volumes with length scales that can be on the order of few nanometers which match the diameter of a typical protein. A protein enzyme such as the pyrococcus furiosus (pfu) polymerase has a diameter of approximately 5 nm. By immobilizing a single pfu enzyme within the sensing hotspot of a gold nanorod, the optoplasmonic device becomes sensitive to the conformational (shape) changes of the protein. Conformational changes are observed from resonance wavelength shifts. Optoplasmonic sensors are sensitive to the conformational changes of the protein because the resonance wavelength depends on the overlap of the protein with the highly localized near field. Such wavelength shifts have been measured for the enzyme pfu polymerase, which has the shape of a hand (V-shaped), as shown in Fig. 5.27a. It’s opening and closing motion is associated with the enzymatic activity of DNA synthesis and has been detected from WGM resonance wavelength shift on an optoplasmonic device [42]. The magnitude and sign of the wavelength shifts Δλ caused by the conformational changes of a protein [Fig. 5.27b] are proportional to the changes in electric-field intensity integrated over the volume νm occupied by the protein molecule (with excess polarizability αe ) at times t1 and t2 : 

Δλ ∝ αe

νm (t2 )



|E(r )| d V − 2

|E(r )| d V 2

νm (t1 )

= αe (I (t2 ) − I (t1 )) = αe ΔI.

(5.98)

Fig. 5.27 a Schematic of a V-shaped protein attached at the tip of a nanorod where the near field and protein overlap. The protein changes shape by increasing or decreasing the angle θ. The pfu polymerase enzyme undergoes similar shape changes during DNA synthesis. b Example for WGM wavelength shifts that have been measured with the pfu polymerase immobilized on the nanorod. The wavelength shifts correspond to shape changes of the pfu enzyme that were recorded during catalytic activity. From [42]. © The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC)

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Fig. 5.28 a Real-time sensing of redox reactions involving individual cysteamine-L-cysteine disulfides. Disulfide bridges are formed between cysteamine linker molecules attached to the nanorod and L-cysteine thiolates/disulfides in solution, then promptly cleaved by excess reducing agent tris(2carboxyethyl) phosphine (TCEP). This reaction sequence leads to the repeated up- and down-steps observed in the resonance linewidth shifts Δκ. b Linewidth shift step pattern similar to a from individual cysteamine-TNB 5,5’-dithiobis-(2-nitrobenzoic acid) disulfides. Reprinted from [43] under the Creative Commons CC BY license

The conformational changes can potentially be observed with a high time resolution that is limited by the cavity’s photon-lifetime which can be on the order of nanoseconds for a Q in the range of 105 − 106 . Likewise, any chemical changes in the composition of the immobilized protein or the binding of a ligand molecule to an immobilized molecule can lead to measurable shifts in the optoplasmonic sensing signals. This has been demonstrated by detecting the thiol–disulfide exchange reaction at the single-molecule level using the optoplasmonic sensor [43]. Repeated reactions between sub-kDa thiolated species are detected in real time from optoplasmonic sensor signals, here resonance linewidth shifts Δκ [see Fig. 5.28a]. The observed steps in resonance linewidth shift Δκ of Fig. 5.28a and apparent absence of same steps in the wavelength shift Δλ indicate a combination of cavity lifetime variance and LSPR-WGM resonance energy invariance. This could imply

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a relationship between the LSPR resonance and molecular vibrational modes. A transition between bound vibrational states that are close in energy and reside in two continuums and may be the explanation for the observed steps in the linewidth shifts. With shifts in the electronic resonance-dependent Raman cross-section being dependent upon chemical reaction and/or charge transfer, the Raman tensor and hence the optomechanical coupling rate may be decipherable from optoplasmonic sensor measurements. Similar linewidth shift signals have been confirmed for the disulfide exchange reaction using 5,5’-dithiobis-(2-nitrobenzoic acid) (DTNB), see Fig. 5.28b.

5.8.7 Attomolar Detection of Single Molecules The reaction of ∼77 Da cysteamine molecules with the gold nanorod has been detected at concentrations down to 100’s of attomoles [43]. The cysteamine’s amine group favorably binds to our optoplasmonic sensor in a sodium carbonate/bicarbonate buffer at pH slightly above 10.75, with 1 M sodium chloride. Typical signal patterns in Fig. 5.29a for amine-gold binding [see Fig. 5.29b] are discontinuous steps in both linewidth shift Δκ and wavelength shift Δλ on the order of 1–10 fm. As time evolves and cysteamine is steadily supplied, the binding sites become occupied and binding rate decreases Fig. 5.29c. These independent shifts in Δκ and Δλ are collected in Fig. 5.29d to showcase non-monotonic linewidth narrowing and broadening once single molecules bind, and only monotonic wavelength red shifts for the singlemolecule binding events. This is an unconventional result as there are equally likely positive and negative shifts recorded for Δκ without apparent proportionality to the only positive shifts recorded for Δλ. As this work on single cysteamine molecules involves small-molecule analysis nearing extreme concentration limits, one must consider propagating experimental uncertainties in concentration. Serial dilutions will amplify the absolute concentration error according to Δci = ci



ΔVi Vi

2

 +

ΔVT,i VT,i

2 .

(5.99)

In this example, the variable pipettes for dispensing liquids are assemblies with respective maximal errors of +/−0.030 L for 1.0 µL, +/−0.11 µL for 10 µL, and +/−8.1 µL for 1000 µL. In all cases, the stock solution concentration of cysteamine is on the order of 1 mM. We estimate uncertainty in the experiment on the order of approximately +/−5aM which is less than an order of magnitude, and delineates a worst case of approximately 5% deviation from expected concentrations for optoplasmonic single-molecule detection when approaching 100 aM.

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Fig. 5.29 a Discrete steps in linewidth Δκ as well as wavelength shifts Δλ are recorded for the covalent binding of cysteamine to gold nanorods. b Cysteamine forms an amine-gold bond as indicated by the red arrow. d The binding step count decays as the sensors approaches saturation of its binding sites. d Histograms of the monotonic (positive) wavelengths and non-monotonic (positive and negative) linewidth shifts. Reprinted from [43] under the Creative Commons CC BY license

5.8.8 Characterizing Thiol Gold and Amine Gold Reactions Optoplasmonic sensors can be used to characterize single-molecule surface reactions from a low to a high affinity [44], see Fig. 5.30a. The single-molecule events recorded with optoplasmonic sensors can show characteristic signal patterns for the low-affinity (spikes) and the high-affinity interactions (steps), see Fig. 5.30b. Analysis of the step and spike signals can be used to analyze the kinetics of the chemical reaction (i.e., by analyzing count rates for the step and spike events) and the yield of the reaction (i.e., by analyzing the rate of step events divided by the rate of spike events). This analysis can be performed in real time and while varying experimental reaction conditions such as reactant concentrations, temperature, ionic strength, and pH. The real-time analysis of sensor signals can be used to optimize the chemical reaction in an approach coined “dial-a-reaction”. The reaction of thiol and amine molecules with gold have been studied in this way, by using thiol and amine-modified DNA oligonucleotides and CTAB-coated gold nanorods [see Fig. 5.31a–i]. The single-molecule signals show that the amine-

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5 Single-Molecule Sensing

Fig. 5.30 a The amine and thiol head groups that are linked to the rest of a molecule can interact with gold in different ways, either with a low affinity thereby interacting transiently, or with a high affinity thereby forming physical or chemical bonds. b Top: examples for spike signals in the wavelength shifts that identify a low-affinity interaction. Bottom: step signals in the wavelength shifts that identify a high-affinity interaction. Reprinted from [44] by permission of John Wiley & Sons, Inc

gold and thiol gold reaction kinetics and its yield depend on various experimental parameters: pH [see Fig. 5.31a–d], NaCl concentration [see Fig. 5.31e, f], number of accessible amine and thiol binding sites on the gold nanorods [see Fig. 5.31g, h] and the type of binding site which is different for amine and thiol molecules [see Fig. 5.31i]. The reaction yield is optimal near a pH=2 for the thiol gold reaction, and near a pH=10.5 for the amine-gold reaction [see Fig. 5.31a, b]. The analysis of the height of the spike signals with varying pH might indicate that the average proximity of the amine and thiol molecule with the gold surface changes with pH [see Fig. 5.31c, d]. The thiol gold reaction yield (step/spike rate) is high at 20 mM NaCl [see Fig. 5.31e], whereas the amine-gold reaction yield is high at 120 mM NaCl [see Fig. 5.31f]. Analysis of the total number of measured step counts for the thiol and amine reaction with gold can be used to estimate the number of thiol and amine binding sites by fitting a Langmuir saturation curve to the data [see Fig. 5.31g, h]. The analysis shows an average of 62 binding sites per nanorod for thiol but only an average of 4 binding sites per nanorod for amine molecules. Furthermore, the analysis of the total step counts for amine and thiol molecules [see Fig. 5.31i] confirms that amines and thiols bind to different gold atoms. It is known that amines bind to low coordinated surface gold atoms (adatoms) whereas thiols bind to (100) and (111) gold surfaces. The optoplasmonic analysis of surface reactions has already been applied to analyze other molecular interactions at the nanorod surface, such as the interaction between antigen and surface-immobilized antibody, the DNA hybridizationinteraction between oligonucleotides with one oligonucleotide species attached to the sensor, and the interaction of single molecules with a sodium dodecyl sulfate (SDS) coated CTAB nanorod [44].

Problems

295

Fig. 5.31 Measured spike rate for a thiol and b amine-gold reaction as function of pH. Analysis of the measured signal heights Δλ for the spike signals for c thiol and d amine-gold reaction for different pH values. Step and spike rates for e thiol and f amine-gold reaction for different NaCl concentrations. Total measured step counts for thiol (g) and amine (h) gold reaction. Comparison of the total step counts i for the thiol and amine reaction. Reprinted from [44] by permission of John Wiley & Sons, Inc

Problems 5.1 The fractional resonance shift given by (5.10) results from the perturbation of an individual nanoparticle/molecule with an arbitrary location. A TE-polarized WGM √ has the form E0 (r) = E 0 jl ( s k0 r )Xl,m (θ, ϕ) with the amplitude E 0 , the spherical Bessel function jl (z), the dielectric constant s of the microcavity, the vacuum wavenumber k0 = 2π/λ0 , and the vector spherical harmonics Xl,m (θ, ϕ). The dielectric constant of the outside-microcavity medium is m . The fractional resonance shift is maximized when the nanoparticle/molecule is located at the microcavity’s equator r0 = Rer + (π/2)eθ + ϕeϕ with the microcavity’s radius R. Using the following approximation

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5 Single-Molecule Sensing



R 0



2 2 R 3 s − m  √ √ jl ( s k0 r ) r 2 dr ≈ jl ( s k0 R) , 2 s

(5.100)

and (3.55), verify that the fractional resonance shift of the fundamental WGM (m = l) scales as 1 δλ ∝ 5/2 . (5.101) λ0 R

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Chapter 6

Fundamentals of Quantum Optics

Abstract Quantum optics is a field of research on the quantum mechanical behavior of the electrons in matter interacting with the light field (photons). This chapter begins with the semiclassical theory of the light–matter interaction, where the Rabi oscillations and the Ramsey fringes are studied. Then, the interaction between the matter and a quantized light is discussed via the full quantum theory. The unavoidable dissipative processes, such as the spontaneous emission of particles and the loss of intracavity photons, affect the coherent light–matter interaction. Such an open quantum system may be investigated by using the Lindblad master equation.

6.1 Introduction In Chap. 1, we have introduced the light–matter interaction based on the Lorentz theory, which directly gives us the basic physical picture of light dispersion and absorption in medium. However, this classical theory has severely limited applications, for instance, its failure to predict the blackbody radiation law and inability to explain the spontaneous emission of atoms. In this chapter, we explore the light interacting with the matter again but by using the quantum mechanics. According to the manner of describing the light field, the quantum mechanical description of the light–matter coupling can be divided into semiclassical and full quantum theories. The light field is treated as a classical electromagnetic field in the former while the energy quanta, i.e., photons, are introduced in the latter. Instead of considering the bulk of matter, we focus on the small systems composed of a microscopic number of particles (quantum emitters). The simplest physical model consists of a single twolevel emitter coupled to a single-mode optical cavity, known as the cavity quantum electrodynamics (cavity QED). Such a quantum system can be well controlled in the experiment, facilitating the fundamental studies on the light–matter interaction.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2_6

299

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In the chapter, we introduce the fundamental knowledge of the light–emitter interaction developed in quantum optics. We start with the semiclassical treatment with the emphasis on the Rabi oscillation of atoms. Next, the full quantum theory of the atoms interacting with a quantized light is briefly discussed, including the explanation of the origin of spontaneous emission. Finally, the damped light-matter interaction is considered by employing the Lindblad master equation.

6.2 Light–Matter Interaction We first establish the theoretical framework of the light–matter interface in the electric dipole approximation. Since an arbitrary light field can be decomposed into a set of monochromatic electromagnetic fields and the matter is made up of many emitters (e.g., atoms and molecules), we simplify the complex system as a monochromatic optical wave E∗ E0 (6.1) E(r, t) = e eikL ·r−iωL t + e∗ 0 e−ikL ·r+iωL t , 2 2 interacting with a single quantum emitter. The light’s polarization, amplitude, wavevector, and frequency are e, E 0 , k L and ω L , respectively. Indeed, E(r, t) mainly affects the electrons in the outer shell of the emitter. In the classical description, i.e., the Lorentz oscillator model in Sect. 1.3.3, the light field exerts an external driving force on the electron-nucleus spring system. This extra force causes a displacement of the electron from its equilibrium position, resulting in an electric dipole moment. The matter’s optical properties are primarily determined by this light-induced dipole moment. From the quantum mechanical point of view, the motion of an outer-orbit electron with the charge (−e) is governed by the Hamiltonian H=−

2 e 2  ∇ + i A(r, t) − eφ(r, t) + V (r), 2m e 

(6.2)

where the vector A(r, t) and scalar φ(r, t) potentials are related to the electric field E(r, t) via E(r, t) = −∂A(r, t)/∂t − ∇φ. The electrostatic potential V (r) corresponds to the binding potential produced by the nucleus. We replace the electron’s position vector r with r0 + Δr, where r0 is the location of the emitter’s center (nucleus) and Δr represents the electron’s relative position. Further, we assume that |Δr| varies within a range much shorter than the light wavelength, |k L · Δr|  1, that is, a point-like quantum emitter is driven by an optical wave. Thus, the vector potential can be rewritten in a Taylor series with respect to the small displacement

6.2 Light–Matter Interaction

301

A(r0 + Δr, t) = A0 (t)eikL ·(r0 +Δr) + c.c. ≈ A0 (t)eikL ·r0 (1 + ik L · Δr + ...) + c.c. ≈ A0 (t)eikL ·r0 + c.c..

(6.3)

The last step is usually referred to as the electric dipole approximation. The higherorder Taylor terms are recognized as the higher rank multipole (e.g., quadrupole and octupole) moments. For simplicity, in the following we use r instead of Δr. By setting φ(r, t) = 0 (no free charges exist), the Hamiltonian is simplified as H=−

2 e 2  ∇ + i A(r0 , t) + V (r). 2m e 

(6.4)

Within the Coulomb gauge ∇ · A(r, t) = 0, we look for the wave function of the electron in the form of   e (6.5) ψ(r, t) = Ψ (r, t) exp −i A(r0 , t) · r .  Inserting (6.5) into i

d ψ(r, t) = Hψ(r, t), dt

(6.6)

one obtains the Schrödinger equation for the motion of electron i

d Ψ (r, t) = H Ψ (r, t). dt

(6.7)

with the new Hamiltonian defined as H = H0 − d · E(r0 , t).

(6.8)

The first term on the right side of the equal sign H0 = −

2 2 ∇ + V (r), 2m e

(6.9)

corresponds to the electron moving in a free emitter while the second term gives the interaction between an electric dipole d = −er and the light field E(r0 , t). Note that r0 is located at the emitter’s center while r denotes the vector relative to r0 . The unperturbed Hamiltonian H0 accounts for the internal structure of the quantum emitter, in which the outer-shell electron can only stay in a set of discrete energy levels. When the light frequency ω L is near resonant to a pair of energy levels, we may further restrict ourselves to the simplest physical model composed of a two-level emitter interacting with a monochromatic optical wave. Generally, there are two ways to investigate the light–emitter interface described by (6.8): (1) the semiclassical theory where the emitter’s dynamics is considered in the quantum mechanical manner

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while the light field is still treated as a classical electromagnetic wave; (2) the full quantum theory where both the emitter and the light field are considered within the quantum mechanical frame. The latter is essentially applied when the nonclassical properties of the light, such as the squeezed state and the photon antibunching, take the main effect.

6.3 Rabi Oscillation We begin with the semiclassical treatment of the interaction between an emitter and a classical light E(r, t) as described in Fig. 6.1a. The quantum emitter is composed of internal |e (upper) and |g (lower) states with the completeness |e e| + |g g| = 1. The Hamiltonian (6.8) is then expressed as H = ωe |e e| + ωg |g g| − (deg |e g| + dge |g e|) · E(r0 , t),

(6.10)

where ωe (ωg ) is the energy of |e (|g) and deg = e| d |g is the matrix element of the electric dipole moment operator. The wave function of the two-level emitter can be written in a general form Ψ (t) = ce (t) |e + cg (t) |g .

(6.11)

The time-dependent coefficients ce and cg are the probability amplitudes of the emitter in states |e and |g, respectively. Substituting (6.11) into the Schrödinger equation (6.7), one may find the following equations of motion for the slowly varying amplitudes c˜e (t) = eiωe t ce (t) and c˜g (t) = eiωg t cg (t),

Fig. 6.1 Light–emitter interaction. a An optical wave is coupled to a dipole transition of a quantum emitter. b Resonant (Δ = 0, solid line) and unresonant (Δ = Ω, dashed curve) Rabi oscillations of the emitter between |e and |g states

6.3 Rabi Oscillation

303

 deg · e d c˜e (t) = i E 0 eikL ·r0 −i(ωL −ωeg )t dt 2  deg · e∗ ∗ −ikL ·r0 +i(ωL +ωeg )t c˜g (t), E0 e +i 2  dge · e d c˜g (t) = i E 0 eikL ·r0 −i(ωL +ωeg )t dt 2  dge · e∗ ∗ −ikL ·r0 +i(ωL −ωeg )t c˜e (t), E0 e +i 2

(6.12a)

(6.12b)

where ωeg = ωe − ωg is the dipole transition frequency of the emitter. Normally, the light field is nearly resonant to the emitter’s transition, ω L ∼ ωeg , and their difference (the light–emitter detuning) Δ = ω L − ωeg is much smaller than either ω L or ωeg , |Δ|  ω L , ωeg . The associated oscillating terms e±i(ωL −ωeg )t = e±iΔt in (6.12) vary much slowly in comparison to the rapidly varying terms e±i(ωL +ωeg )t . Within the time scale of interest Δ−1 , the terms e±i(ωL +ωeg )t have oscillated for multiple times. As a result, their effects on the system’s dynamics are averaged out. For this reason, one can safely omit the interaction terms associated with e±i(ωL +ωeg )t in (6.12). This is known as the rotating-wave approximation (RWA). We obtain Ω d c˜e (t) = −i c˜g (t)e−iΔt , dt 2 d Ω∗ c˜g (t) = −i c˜e (t)eiΔt . dt 2

(6.13a) (6.13b)

Here the so-called Rabi frequency Ω is defined as Ω=−

deg · e E 0 eikL ·r0 , 

(6.14)

which may be written as Ω = |Ω|e−iϕ with the phase difference ϕ between the emitter dipole moment and the light field. The exact solutions of the above differential equations are derived as   Δ Ωt t Ωt t +i c˜e (0) cos sin 2 Ωt 2    Ωt t Ω c˜g (0) e−iΔt/2 , sin − i Ωt 2    ∗ Ω Ωt t c˜e (0) sin c˜g (t) = − i Ωt 2    Δ Ωt t Ωt t −i c˜g (0) eiΔt/2 , sin + cos 2 Ωt 2 c˜e (t) =

(6.15a)

(6.15b)

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6 Fundamentals of Quantum Optics

 with is the generalized Rabi frequency Ωt = |Ω|2 + Δ2 . It is easy to verify |c˜e (t)|2 + |c˜g (t)|2 = 1, i.e., the conservation of the particle number. The population inversion W (t) = |c˜e (t)|2 − |c˜g (t)|2 is usually used to measure the probability distribution of the emitter in different states. W = 1 represents the emitter is completely in the upper |e state while the emitter in |g corresponds to W = −1. As shown in Fig. 6.1b, the quantum emitter continuously oscillates between two internal states, known as the Rabi oscillation, at a frequency equal to Ωt . Indeed, this oscillatory behavior is the direct result of the emitter alternatively absorbing and emitting the light field. The amplitude of the Rabi oscillation is maximized at the resonant interaction Δ = 0 and is strongly reduced when Δ Ω. Spin-1/2 model. The interaction between a two-level emitter and a monochromatic light is mathematically equivalent to a spin-1/2 magnetic dipole undergoing the procession in an external magnetic field [1]. It is naturally to introduce the egeinstates |e = |↑ =

  1 , 0

|g = |↓ =

  0 , 1

(6.16)

and the lowering and raising operators σ− = |g e| =

    00 01 † , σ− = |e g| = . 10 00

(6.17)

In the RWA, the Hamiltonian (6.10) may be re-expressed in the matrix form   Ωi Δ 1 −Δ Ω Ωr σx + σ y − σz = , H/ = 2 2 2 2 Ω∗ Δ

(6.18)

by using the Pauli matrices 

 01 , 10   0 −i † − σ− ) = , σ y = −i(σ− i 0   1 0 † † σ− − σ− σ− = , σ z = σ− 0 −1 † + σ− = σ x = σ−

(6.19a) (6.19b) (6.19c)

and Ω = Ωr − iΩi . The corresponding emitter’s wave function is written as Ψ (t) = c˜e (t)eiΔt/2 |e + c˜g (t)e−iΔt/2 |g .

(6.20)

According to the Schrödinger equation, the spin undergoes a Rabi flopping between |e = |↑ and |g = |↓ under the action of the driving field. One may also solve the eigenvalues and eigenstates of the 2 × 2 Hamiltonian matrix H (see Appendix D)

6.3 Rabi Oscillation

305

Ωt , 2 Ωt ω− = − , 2

ω+ =

Ψ+ = sin θ |e + cos θeiϕ |g , Ψ− = cos θ |e − sin θeiϕ |g ,

(6.21)

where we have defined cos θ = √

|Ω| , 2Ωt (Ωt − Δ)

Ωt − Δ sin θ = √ . 2Ωt (Ωt − Δ)

(6.22)

The light–emitter interaction induces the shift of the emitter’s transition frequency. In the resonant case, ω L = ωeg , two eigenstates Ψ+ and Ψ− are separated by the Rabi frequency ω+ − ω− = |Ω|, i.e., the Autler–Townes doublet [2]. Such a spectral splitting may be observed by the modified absorption spectrum of the emitter, where a strong laser beam drives the |g − |e transition while a weak probe laser sweeps across the other transition between a third internal state of the emitter and one of |g and |e. Density matrix operator. Next, we consider an alternative theoretical framework to describe the light–emitter interaction. The expectation value of an arbitrary physical quality (observable) with the associated operator O is given by the quantum mechanical average OQM = Ψ | O |Ψ . Usually, the wave function Ψ is unknown. But we know the probability PΨ of the system in Ψ , i.e., the percentage of the Ψ -state systems in an ensemble of identically prepared systems. Thus, the experimental observation O does not only include the quantum mechanical average but also the ensemble average over many systems that are prepared under the same conditions, O =



PΨ OQM .

Ψ

(6.23)

It is convenient to introduce the density operator ρ=

Ψ

PΨ |Ψ  Ψ | ,

(6.24)

and we have the measurement result O = Tr(ρO) = Tr(Oρ).

(6.25)

The conservation of probability tells us Tr(ρ) =

Ψ

PΨ = 1.

(6.26)

If the quantum system can only stay in the state Ψ0 , i.e., ρ = |Ψ0  Ψ0 | with PΨ0 = 1, this particular state is called the pure state and one has ρ = ρ2 . The opposite of a pure state is a mixed state characterized by ρ = ρ2 and Tr(ρ2 ) < 1. The equation of motion for the density matrix operator ρ can be derived from the Schrödinger equation straightforwardly,

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6 Fundamentals of Quantum Optics

d ρ= PΨ Ψ dt

 



dΨ i dΨ



Ψ | |Ψ  = − [H, ρ], +

dt dt



(6.27)

where the square brackets denote a commutator, [A, B] = AB − B A. The above equation is also known as the Liouville–von Neumann equation, which is more general than the Schrödinger equation since it involves both ensemble and quantum mechanical averages. For a two-level emitter, ρ is written as  c˜e eiΔt/2 ∗ −iΔt/2 ∗ iΔt/2  c˜e e c˜g e ρ = |Ψ (t) Ψ (t)| = c˜g e−iΔt/2 

(6.28)

which may be further simplified as 

ρee ρeg ρ= ρge ρgg

 † † † = ρee σ− σ− + ρeg σ− + ρge σ− + ρgg σ− σ− ,

(6.29)

where the density matrix elements ρee = e| ρ |e = |c˜e |2 and ρgg = g| ρ |g = |c˜g |2 denote the probabilities of the emitter in the upper and lower states, respectively, while the off-diagonal elements ρeg = e| ρ |g = c˜e c˜g∗ eiΔt and ρge = ρ∗eg are related to the emitter’s polarizability (i.e., dispersion and absorption). The Liouville equation (6.27) leads to d Ω Ω∗ ρee = −i ρge + i ρeg , dt 2 2 d Ω ρeg = iΔρeg + i (2ρee − 1). dt 2

(6.30a) (6.30b)

Bloch sphere. Due to the condition ρee + ρgg = 1, only three in four density matrix elements are independent. One may define the so-called Bloch vector R = uex + ve y + wez ,

(6.31)

u = ρeg + ρge , v = i(ρeg − ρge ), w = ρee − ρgg .

(6.32)

with three vector components

The equations of motion of the density matrix elements are then given by d u(t) = Δv(t) + Ωi w(t), dt d v(t) = −Δu(t) − Ωr w(t), dt d w(t) = Ωr v(t) − Ωi u(t). dt

(6.33a) (6.33b) (6.33c)

6.3 Rabi Oscillation

307

Fig. 6.2 Rabi oscillations represented by a Rabi sphere. a Undamped Rabi oscillation. Solid curve: Ω = − f and Δ = f . Dashed curve: Ω = f and Δ = 0. Here is f an arbitrary frequency unit. b Damped Rabi oscillation with Ω = 5γ and Δ = γ. For all curves, the emitter is initially prepared in |g

By further introducing the Rabi vector = Ωr ex + Ωi e y − Δez ,

(6.34)

the equations of motion of (u, v, w) can be included into one d R = × R. dt

(6.35)

The Bloch vector R may be represented by a Bloch sphere with a radius of unity [see Fig. 6.2a]. The points Rz,±1 = (0, 0, ±1) denote √ the emitter is in |e and |g, to the points respectively. The eigenstates |±x = (|e ± |g)/ 2 of σx corresponds √ Rx,±1 = (±1, 0, 0), while the eigenstates |± y = (|e ± i |g)/ 2 of σ y are represented by R y,±1 = (0, ±1, 0). In the absence of energy dissipation, the Rabi oscillation for arbitrary Ω and Δ can only move on the sphere surface.

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6 Fundamentals of Quantum Optics

6.4 Rabi and Ramsey Measurements The coherent light-emitter interaction provides a way to discriminate against the frequency fluctuation of an optical wave. Let us assume that the quantum emitter is initially prepared in the ground state |g. A light pulse with a duration τ is applied to near-resonantly couple to the |g − |e transition of the emitter. According to (6.15), the probability of finding the emitter in |e at the time τ is given by ρee (τ ) =

Ωt τ Ω2 sin2 . 2 2 Ωt

(6.36)

For the resonant interaction Δ = 0 with ω L = ωeg , ρee (τ ) reaches the maximum at τ = π/Ω. The corresponding light pule is called the π-pulse. Fixing τ at this π-pulse duration, a small deviation away from the resonant condition ω L = ωeg may cause a strong decrease of ρee (τ ). Figure 6.3a displays the dependence of ρee (τ ) on the light-emitter detuning Δ. The curve has a spectral linewidth of about 2Ω and may be exploited as a frequencydiscrimination curve to stabilize the light frequency ω L . A small Rabi frequency Ω narrows the spectral linewidth, which enhances the sensitivity to the frequency fluctuations in ω L but extends the measurement circle time. A long-time measurement may be restricted by the finite transit time of the emitter or the finite lifetimes of the emitter’s states. Also, a long measurement time is insensitive to the relatively rapid

Fig. 6.3 Rabi and Ramsey frequency discrimination. a The probability of the emitter in the excited state |e after interacting with a light π-pulse with its duration of τ = π/Ω. b The probability of the emitter in |e after two light π/2-pulses are applied. The interval between two pules is T = 10τ . For both curves, the emitter is initialized in the ground state |g

6.4 Rabi and Ramsey Measurements

309

frequency fluctuations. In contrast, increasing the Rabi frequency Ω (suppressing the light π-pulse duration) shortens the measurement time but broadens the spectral linewidth. In addition, a large Ω requires a high light intensity, which causes other unfavorable effects such as the huge ac Stark shifts of the emitter’s states. Thus, the optimal π-pulse duration should be determined according to the practical requirements. Moreover, the measurement of the emitter’s population distribution also introduces an uncertainty to the frequency discrimination [3], which is fundamentally limited by the so-called quantum projection noise.1 An alternative way to discriminate against the light frequency fluctuations is the Ramsey separated oscillatory field method, where the light π-pulse is divided into two π/2-pulses. Within the period between two π/2-pulses, the wave function of the emitter evolves freely for a long time T . Again, the emitter is initialized in |g at t = 0. In the first interaction zone, the emitter interacts with the first π/2-pulse. At t = τ /2, the |e amplitude of the emitter’s wave function evolves to ce (τ /2) = −i

Ω Ωt τ −iΔτ /4 −iωe τ /2 e sin e . Ωt 4

(6.37)

One should note the difference between ce,g (t) and c˜e,g (t). After the free-evolution zone, ce becomes ce (τ /2 + T ) = −i

Ω Ωt τ −iΔτ /4 −iωe (τ /2+T ) e sin e . Ωt 4

(6.38)

Passing the second interaction zone, in which the emitter interacts with the second π/2-pulse, the amplitude ce is found to be  Δ ΔT Ω Ωt τ Ωt τ − sin sin sin ce (τ + T ) = −2i Ωt 4 Ωt 4 2  ΔT Ωt τ cos e−iΔ(τ +T )/2 e−iωe (τ +T ) . + cos 4 2

(6.39)

In deriving ce (τ + T ), one needs to consider the extra phase difference between the emitter’s polarization and the light field accumulated within the period from 0 to (τ /2 + T ). 1 Deriving

the probability of an emitter in, for example, the excited state is associated with the projection operator Pe = |e e|. The measurement in experiment actually corresponds to the expectation value Pe  = Ψ (τ )| Pe |Ψ (τ ) = ρee (τ ), which is maximized at Pe  = 1/2 with Δ = 0 (the resonant interaction) and τ = π/Ω (the π-pulse). Nonetheless, the quantum mechanics imposes  √ a deviation in the projection measurement, ΔPe = Pe2  − Pe 2 = Pe (1 − Pe ). Hence, Pe  = 1/2 leads to the maximum deviation ΔPe = 1/2 and the measurement uncertainty is given by (S/N )−1 = ΔPe /Pe  = 1. For an ensemble of N√ e independent emitters, the signal is proportional to Ne Pe  while the quantum projection noise is Ne Pe (1 − Pe ). Thus, the measurement √ uncertainty reads (S/N )−1 = 1/ Ne . Increasing the number of emitters Ne suppresses the uncertainty.

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6 Fundamentals of Quantum Optics

Figure 6.3b illustrates the probability ρee (τ + T ) = |ce (τ + T )|2 as a function of the light-emitter detuning Δ. Multiple fringes, i.e., known as Ramsey fringes, are presented within the range of −Ω < Δ < Ω. The spectral linewidth of the central peak is significantly suppressed down to a value about 1/T , which is favorable to the stabilization of the light frequency. Nonetheless, the free-evolution duration T cannot be much long. Otherwise, various inevitable dissipative (relaxation and dephasing) processes, such as the spontaneous emission, may influence the cohere evolution of the emitter’s wave function. This approach of separated oscillatory fields was firstly introduced to suppress the linewidth of Doppler spectrum from an ensemble of thermal emitters [4]. Now it has been widely applied in atomic clocks and atom interferometers so as to measure the dephasing time of the polarizability of quantum emitters.

6.5 Quantization of Electromagnetic Field In above, we have briefly introduced the semiclassical treatment on the light-emitter interaction. The most important physical concept is the Rabi oscillation that describes the periodic exchange of the energy between the light and the emitter. We now turn to the full quantum mechanical description on the emitter interacting with the light quanta (i.e., photons), where we start with the second quantization of the electromagnetic field within a finite volume [5]. We write the electric E and magnetic B fields via the vector A and scalar φ potentials, E = −∂A/∂t − ∇φ and B = ∇ × A. Since no free charges exist in free space, φ can be set at zero. We consider a cubic box with an edge length l. The periodic boundary condition forces the wavevectors of the plane-wave modes that can exist in the box to be k = k x ex + k y e y + k z ez with ku=x,y,z = 2πn u /l and n u = 0, ±1, ±2, . . . . Within an elementary volume Δk x Δk y Δk z in the wavevector space, the corresponding mode number reaches  Δn =

l 2π

3 Δk x Δk y Δk z .

(6.40)

For a large enough l, one may assume a continuous distribution of wavevectors and has  3  3 l l dn = k 2 dkdΩ = k 2 dk sin θdθdϕ. (6.41) 2π 2π Here dΩ denotes the infinitesimal solid angle along the k-direction. The wavenumber k is given by k = |k| = (k x2 + k 2y + k z2 )1/2 , and θ and ϕ are the polar and azimuthal angle, respectively, in spherical coordinates. We express the vector potential A in terms of the plane waves (i.e., the combined spatial and temporal Fourier transform)

6.5 Quantization of Electromagnetic Field



A(r, t) =

k,μ

311

 ek,μ Ak,μ eik·r−iωk t + e∗k,μ A∗k,μ e−ik·r+iωk t ,

(6.42)

where ek,μ is the unit vector of the planar-mode polarization. The frequency ωk depends only on the module k, ωk = ck. The Coulomb gauge ∇ · A = 0 gives ek,μ · k = 0, indicating the polarization of an electromagnetic mode is perpendicular to its propagation direction, i.e., the transverse waves. Without losing the generality, here we assume ek,μ = e∗k,μ . In the plane perpendicular to k, these exist two independent polarization directions, ek,μ · ek,μ = δμ,μ (μ, μ = 1, 2). The electric field E(r, t) is then given by E(r, t) = i

k,μ

 ek,μ ωk Ak,μ eik·r−iωk t − A∗k,μ e−ik·r+iωk t .

(6.43)

Applying the identity ∇ × (ab) = (∇a) × b + a(∇ × b), we obtain the expression of the corresponding magnetic field B(r, t) B(r, t) = i

k,μ

  k × ek,μ Ak,μ eik·r−iωk t − A∗k,μ e−ik·r+iωk t .

(6.44)

Combining the electric- and magnetic-field energy components, the total energy of the electromagnetic field inside the cubic box reads    1 1 ε0 E · E + B · B d V H = 2 V μ0 

 ε0 = E · E + c2 B · B d V. 2 V

(6.45)

Here V = l 3 gives the quantization volume. In many cases, the boundary of the quantization volume cannot be strictly defined. One may introduce an effective volume Veff to perform the second quantization. In the following, Veff is applied instead of V , and the periodic boundary condition is enforced, which leads to 



e±i(k+k )·r d Veff = Veff δk,−k .

(6.46)

Veff

We now express H in terms of Ak,μ . Defining A˜ k,μ = Ak,μ e−iωk t , the electric-field part in H is rewritten as  (E · E)d Veff = Veff Veff

k,μ

2ωk2 A˜ k,μ A˜ ∗k,μ − Veff

k,μ,μ

ωk2 (ek,μ · e−k,μ )

×( A˜ k,μ A˜ −k,μ + A˜ ∗k,μ A˜ ∗−k,μ ). Using the identity (a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c), one has

(6.47)

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6 Fundamentals of Quantum Optics

(k × ek,μ ) · (k × ek,μ ) = |k|2 δμ,μ ,

(6.48a)

(k × ek,μ ) · (−k × e−k,μ ) = −|k| (ek,μ · e−k,μ ). 2

(6.48b)

The magnetic-field part in H is then given by  (B · B)d Veff = Veff

c2



Veff

k,μ

2ωk2 A˜ k,μ A˜ ∗k,μ + Veff

k,μ,μ

ωk2 (ek,μ · e−k,μ )

×( A˜ k,μ A˜ −k,μ + A˜ ∗k,μ A˜ ∗−k,μ ).

(6.49)

Inserting (6.47) and (6.49) into (6.45), the total energy of the electromagnetic field is simplified as ωk2 A˜ k,μ A˜ ∗k,μ . (6.50) H = 2ε0 Veff k,μ

We do the transformations −1   A˜ k,μ = 2ωk ε0 Veff (ωk Q k,μ + i Pk,μ ), −1   (ωk Q k,μ − i Pk,μ ), A˜ ∗k,μ = 2ωk ε0 Veff

(6.51a) (6.51b)

from which the real variables Q k,μ and Pk,μ are derived as   ε0 Veff A˜ k,μ + A˜ ∗k,μ ,    = −iωk ε0 Veff A˜ k,μ − A˜ ∗k,μ ,

Q k,μ = Pk,μ



(6.52a) (6.52b)

and it is easy to find the following relations d Q k,μ = Pk,μ , dt

d Pk,μ = −ωk2 Q k,μ . dt

(6.53)

Using Q k,μ and Pk,μ , the electromagnetic energy H is rewritten as H=

1 2 (Pk,μ + ωk2 Q 2k,μ ). k,μ 2

(6.54)

Equation (6.53) can be directly obtained from d ∂H Q k,μ = , dt ∂ Pk,μ

d ∂H Pk,μ = − . dt ∂ Q k,μ

(6.55)

Choosing Q k,μ and Pk,μ as the canonical position and momentum variables, respectively, we have the canonical equations of motion

6.5 Quantization of Electromagnetic Field

d Q k,μ = {Q k,μ , H }, dt

313

d Pk,μ = {Pk,μ , H }, dt

(6.56)

with the Poisson bracket {A, B} =



k,μ

∂A ∂B ∂A ∂B − ∂ Q k,μ ∂ Pk,μ ∂ Pk,μ ∂ Q k,μ

 .

(6.57)

The Poisson brackets of the canonical coordinates are calculated to be {Q k,μ , Pk ,μ } = δk,k δμ,μ , {Q k,μ , Q k ,μ } = {Pk,μ , Pk ,μ } = 0.

(6.58a) (6.58b)

Actually, (6.55) is equivalent to (6.56). To implement the quantization, we replace the Poisson brackets (6.58) with the commutators  

 Q k,μ , Pk ,μ = iδk,k δμ,μ ,    Q k,μ , Q k ,μ = Pk,μ , Pk ,μ = 0.

(6.59a) (6.59b)

Similar to the quantum harmonic oscillator, one can introduce the following annihilation and creation operators  ak,μ = ( 2ωk )−1 (ωk Q k,μ + i Pk,μ ),  † ak,μ = ( 2ωk )−1 (ωk Q k,μ − i Pk,μ ),

(6.60a) (6.60b)

which obeys the bosonic commutation relations   ak,μ , ak† ,μ = δk,k δμ,μ ,     † ak,μ , ak ,μ = ak,μ , ak† ,μ = 0.

(6.61a) (6.61b)

Thus, one has the expressions  Ak,μ =

 ak,μ , 2ωk ε0 Veff

 A†k,μ =

 a† . 2ωk ε0 Veff k,μ

(6.62)

and the electric and magnetic-field operators E(r, t) = B(r, t) =

k,μ



k,μ

ek,μ E0 ak,μ eik·r−iωk t+iπ/2 + H.c.,

(6.63a)

k × ek,μ E0 ak,μ eik·r−iωk t+iπ/2 + H.c., ωk

(6.63b)

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6 Fundamentals of Quantum Optics

with the amplitude of the electric field  E0 =

ωk . 2ε0 Veff

(6.64)

The Hamiltonian for the electromagnetic field is eventually derived as H=



  1 † . ωk ak,μ ak,μ + k,μ 2

(6.65)

6.6 Fock States After obtaining the Hamiltonian (6.65) of the quantized electromagnetic field, our next step is to find the eigenvalues and eigenstates of H . Let us first focus on a single electromagnetic mode with the creation and annihilation operators a † and a. The corresponding Hamiltonian takes the form   1 . H = ω a † a + 2

(6.66)

Assuming the ground eigenstate |0 and eigenvalue E 0 , one has the eigenvalue equation (6.67) H |0 = E 0 |0 . Applying the annihilation operator a from the left on both sides of the equality symbol leads to (6.68) H a |0 = (E 0 − ω)a |0 . The above equation implies that a |0 and (E 0 − ω) are also an eigenstate and eigenvalue of H , which, however, is contradict to the assumption that |0 is the ground state of H . Thus, it can be concluded that a |0 = 0 and E 0 = ω/2. Then, the creation operator a † is applied from the left on both sides of (6.67), which gives 3 (6.69) H a † |0 = ωa † |0 . 2 The above equation implies that 1 |1 = √ a † |0 , N1

(6.70)

with the normalization factor N1 is also an eigenstate of H with the corresponding eigenvalue

6.6 Fock States

315

E 1 = E 0 + ω =

3 ω. 2

(6.71)

Multiplying 0| a on both sides of (6.69) yields (0| a)a † a(a † |0) = (0| a)(a † |0) = N1 .

(6.72)

Using the commutation relation [a, a † ] = 1, N1 is calculated to be N1 = 1 and the normalized eigenstate |1 reads |1 = a † |0 ,

(6.73)

with the corresponding eigenvalue equation H |1 = E 1 |1 .

(6.74)

and a † a |1 = |1. Again, we apply a † from the left on both sides of (6.74) and find that 1 |2 = √ a † |1 , N2

(6.75)

is an eigenstate of H with the corresponding eigenvalue E 2 = E 1 + ω =

5 ω. 2

(6.76)

It is easy to derived the normalization constant N2 = 2. Thus, the normalized eigenstate |2 takes the form (a † )2 a† |2 = √ |1 = √ |0 , (6.77) 2 2! with the corresponding eigenvalue equation H |2 = E 2 |2 .

(6.78)

and a † a |2 = 2 |2. Repeating the above process, one obtains the n-th eigenstate of H (a † )n |n = √ |0 n!

(6.79)

which is referred to as the Fock (or number) state, and the corresponding eigenvalue   1 ω. En = n + 2

(6.80)

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6 Fundamentals of Quantum Optics

The energy spectrum of the single-mode electromagnetic field forms a ladder structure with the equal spacing of ω between two adjacent energy levels. This energy quantum ω is known as a photon. Thus, the |n state indicates that there are n photons within the effective volume Veff . The zero-photon state |0 is commonly called the vacuum state with the zero-point energy ω/2. Unlike in the classical mechanics, a quantum system still fluctuates in its ground state. That is, a vacuum cannot be viewed as an empty space. The Fock states can be used to span the Hilbert space with an infinite dimension. The photon operators a and a † acting on the Fock states gives √ a |n = n |n − 1 , √ a † |n = n + 1 |n + 1 .

(6.81a) (6.81b)

In particular, the operator a † a counts the number of photons a † a |n = n |n .

(6.82)

As an example, we consider the density operator of a thermal field in the Fockstate representation. In a canonical ensemble at the temperature T , the classical probability pn for the single-mode electromagnetic field in the |n state is proportional to the Boltzmann factor, pn ∝ e−En /k B T with the energy E n given by (6.80) and the Boltzmann constant k B . Then, the density matrix takes the form −1 e−En /k B T e−En /k B T |n n| n n  −nω/k B T

|n n| e = 1 − e−ω/k B T n 

† = 1 − e−ω/k B T e−ωa a/k B T .

ρ=



(6.83)

Using the above expression, we obtain the mean photon number n¯ = a † a = Tr(a † aρ) =

1 eω/k B T

−1

,

(6.84)

and the standard deviation of the photon number2 Δn =



−1   ω (a † a)2  − a † a2 = 2 sinh . 2k B T

(6.85)

In addition, the second-order correlation function g (2) (τ ) in the short-time-delay limit is evaluated to be g (2) (0) = 2. Taking into account multiple electromagnetic modes, ρ for a thermal field is written as 2 In

deriving Δn, the relation



n∈Z n

2 e−nx

∂ = − ∂x



n∈Z ne

−nx

is used.

6.6 Fock States

317

ρ=



k,μ

 † 1 − e−ωk /k B T e−ωk ak,μ ak,μ /k B T .

(6.86)

It is easy to derive the following expectation values † ak,μ  = ak,μ  = 0,

(6.87a)

† ak ,μ  ak,μ † ak† ,μ  ak,μ

= n¯ k,μ δk,k δμ,μ ,

(6.87b)

= ak,μ ak ,μ  = 0,

(6.87c)

† with the mean photon number n¯ k,μ = Tr(ak,μ ak,μ ρ) of the electromagnetic mode characterized by (k, μ). The above quantization procedure in principle is applicable for an arbitrary oscillation frequency ω ranging from microwave to optical regime. Now, we are ready to discuss the interaction between a quantum emitter and a quantized light field.

6.7 Photon Correlation Functions In Sect. 1.4, we have introduced the correlation functions for a classical light field. We now consider their counterparts in the quantum mechanical description. The normalized first-order correlation function is defined as g (1) (τ ) =

a † (t + τ )a(t) , a † (t)a(t)

(6.88)

where a(t) is the annihilation operator in the Heisenberg picture a(t) = ei H t/ ae−i H t/ ,

(6.89)

with the Hamiltonian of the quantum system H . In the short-time-delay limit, g (1) (τ = 0) is always equal to the unity. Generally, due to the finite photon lifetime g (1) (τ ) degrades to zero as τ is increased. The power spectral density of the light field is given by the Fourier transform of the correlation function g (1) (τ )  S(ω) =

∞ ∞

g (1) (τ )e−iωt dτ .

(6.90)

The spectral linewidth ΔωFWHM (full width half maximum) of S(ω) is the reciprocal of the photon’s lifetime. The normalized second-order (intensity-intensity) correlation function g (2) (τ ) =

a † (t)a † (t + τ )a(t + τ )a(t) , a † (t)a(t)a † (t + τ )a(t + τ )

(6.91)

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6 Fundamentals of Quantum Optics

is usually applied to measure the correlation in the photon-number fluctuation. In particular, g (2) (τ = 0) is linked to the so-called Mandel Q M parameter [6] QM =

(a † a)2  − a † a2 . a † a

(6.92)

The Q M parameter is commonly employed to measure the departure of the photon distribution from the Poissonian statistics where Q M = 1. The parameter Q M < 1 denotes a sub-Poissonian statistics while the super-Poissonian statistics corresponds to Q M > 1. It is easy to derive the relation between Q M and g (2) (0) as Q M = [g (2) (0) − 1]a † a + 1.

(6.93)

When g (2) (0) = 1, Q M is equal to the unity. For a single-mode thermal field generated  from a blackbody source, its density operator is given by ρ = (1 − e−ω/k B T ) n e−nω/k B T |n n|. Due to g (2) (0) = 2 [see Fig. 1.8b], the resulting Q M factor is higher than the unity for any ω. Thus, the photon distribution fulfills the super-Poissonian statistics. Another common state of the light field is the so-called coherent state, which is highly similar to a classical harmonic oscillator, i.e., the classical laser field. Mathematically, a coherent state |α is the eigenstate of the annihilation operator a with the associated eigenvalue α a |α = α |α . (6.94) The corresponding mean photon number is given by a † a = |α|2 . To express the coherent state in terms of the Fock states, we first introduce the displacement operator D(α) = eαa

†

−α∗ a

.

(6.95)

Using the Baker–Hausdorff formula, i.e., e A+B = e−[A,B]/2 e A e B ,

(6.96)

when two operators A and B satisfy [[A, B], A] = [[A, B], B] = 0, the displacement operator may be rewritten as D(α) = e−|α|

2

/2 αa † −α∗ a

e

e

.

(6.97)

The D(α) operator is a unitary operator D † (α) = D −1 (α) = D(−α).

(6.98)

The displacement operator acting on the photon annihilation and creation operators gives

6.7 Photon Correlation Functions

319

D † (α)a D(α) = a + α, D † (α)a † D(α) = a † + α∗ .

(6.99)

Thus, the coherent state |α can be generated from the vacuum state |0 by using the displacement operator |α = D(α) |0 . (6.100) Here we list several unique properties of the coherent states. A coherent state |α can be written in a linear superposition of Fock states as |α = e−|α|

2

/2

∞ n=0

αn √ |n . n!

(6.101)

The probability P(n) of finding n photons in |α is given by P(n) = |n|α|2 = e−|α|

2

(|α|2 )n . n!

(6.102)

Thus, the photon number distribution fulfills the Poisson distribution with the mean and variance of |α|2 . In addition, different coherent states are not orthogonal β|α = e−|α|

2

/2−|β|2 /2+β ∗ α

,

(6.103)

but the coherent states can be normalized α|α = β|β = 1. Actually, the coherent states form a complete set 1 π



∞

−∞



∞ −∞

dαdα∗ |α α| = 1.

(6.104)

For a light field in the coherent state, both second-order correlation function and Mandel Q M parameter are equal to unity, g (2) (0) = Q M = 1. For the light field in the Fock state |n, one has Q M = 0 and g (2) (0) = 1 − 1/n. In particular, when the light is at the single-photon state |1, both Q M and g (2) (0) are zero. None classical light sources can produce the Fock-state light field. Thus, we classify the light fields, whose photon distribution fulfills a sub-Poissonian statistics, as the nonclassical fields.

6.8 Interaction Between Single Emitter and Photons The coupling of a two-level emitter with a single-mode quantized light field is generally governed by the so-called Jaynes–Cummings Hamiltonian in RWA [7] H/ = ω L a † a +

ωeg † σz + gσ− a + g ∗ a † σ− . 2

(6.105)

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6 Fundamentals of Quantum Optics

The first two terms on the right side of the equal sign correspond to the energies of the free light field at the frequency ω L and the emitter with the transition frequency ωeg , respectively. The complex parameter g = |g|e−iϕ = −

(deg · e)E0 , 

(6.106)

measures the coupling strength between the emitter and one photon within an effective quantization volume Veff of the light field. Here we have set the emitter’s position at the origin of coordinates, r0 = 0. The unit vector e denotes the light’s polarization and ϕ is the phase of the emitter dipole moment. Suppressing Veff increases the amplitude E0 , thereby enhancing the emitter-photon coupling. The interaction term † a (a † σ− ) describes the process that the emitter transits from |g (|e) to |e (|g) by σ− absorbing (emitting) one photon. These two terms, varying at a frequency |ω L − ωeg | much lower than either ω L or ωeg , ensure the reversible energy exchange between the emitter and the light field. It is worth noting that in RWA we have omitted other † and σ− a. This is because they evolve at two emitter-photon interaction terms a † σ− a frequency (ω L + ωeg ), much faster than |ω L − ωeg |. The time scale of interest is † and σ− a terms of the order of |ω L − ωeg |−1 , during which the counter-rotating a † σ− have oscillated for many times. As a result, their effects on the system are averaged out and can be safely omitted. We choose the orthogonal and complete basis of the quantum system as {|u, n = |u ⊗ |n ; u = e, g; n = 0, 1, 2, ...}. The Hamiltonian matrix in this representation can be separated into a set of block submatrices, where each 2 × 2 sub-block is spanned by |e, n and |g, n + 1, except for the ground |g, 0 state. Diagonalizing the sub-blocks leads to the eigenvalues ωn,±

  1 Ωn ± , = ωL n + 2 2

(6.107)

and the corresponding eigenvectors, i.e., the so-called dressed states Ψn,+ = sin θn |e, n + cos θn eiϕ |g, n + 1 , Ψn,− = cos θn |e, n − sin θn eiϕ |g, n + 1 . Here we have defined the coefficients √ 2|g| n + 1 (Ωn − Δ) , sin θn = √ . cos θn = √ 2Ωn (Ωn − Δ) 2Ωn (Ωn − Δ)

(6.108a) (6.108b)

(6.109)

with the generalized Rabi frequency Ωn =

 Δ2 + 4|g|2 (n + 1),

(6.110)

6.8 Interaction Between Single Emitter and Photons

321

Fig. 6.4 Energy-level splittings caused by the emitter-photon coupling. a Energy spectrum ω0,± versus the detuning Δ = ω L − ωeg (solid curves) of the quantum system composed of one emitter interacting with one photon. The detuning Δ is tuned by varying ωeg . An avoided crossing occurs around the resonance. The anticrossing gap reaches 2|g| at Δ = 0. The dash lines show the energy spectrum of the non-interacting system with g = 0. b Energy-level structure of the resonantinteraction dressed states. The interaction-induced energy splitting between |e, n and |g, n + 1 states is given by mp

and the detuning Δ = ω L − ωeg . The dressed states are the direct result from the emitter-photon interaction. The energy spacing between two dressed states√is minimized at the resonant interaction Δ = 0 with Δωn = ωn,+ − ωn,− = 2|g| n + 1, corresponding to the Rabi frequency in the semiclassical theory of the light-matter interaction. In particular, the resonant coupling between the vacuum field to the emitter gives rise to the so-called vacuum Rabi splitting Δω0 = ω0,+ − ω0,− = 2|g| [see Fig. 6.4a], which is solely determined by the emitter-photon coupling √ strength g, and two lowest dressed √ states |+ ≡ Ψ0,+ = (|e, 0 + |g, 1)/ 2 and |− ≡ Ψ0,− = (|e, 0 − |g, 1)/ 2 [see Fig. 6.4b]. Besides the energy spectrum of the Hamiltonian (6.105), one may also study the time-dependent wavefunction of the quantum system Ψ (t) =

  ce,n (t) |e, n + cg,n+1 (t) |g, n + 1 . n

Applying the substitutions ce,n (t) = c˜e,n (t)e−i(nωL +ωeg /2)t and cg,n+1 (t) = c˜g,n+1 (t) e−i[(n+1)ωL −iωeg /2]t , the Schrödinger equation gives us √ d c˜e,n = −ig n + 1e−iΔt c˜g,n+1 , dt √ d c˜g,n+1 = −ig ∗ n + 1eiΔt c˜e,n . dt

(6.111a) (6.111b)

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6 Fundamentals of Quantum Optics

The exact solutions of (6.111) are derived as   Ωn t Ωn t Δ cos sin +i c˜e,n (0) 2 Ωn 2  √ Ωn t 2g n + 1 sin −i c˜g,n+1 (0) e−iΔt/2 , Ωn 2  √ 2g ∗ n + 1 Ωn t c˜e,n (0) c˜g,n+1 (t) = −i sin Ωn 2    Δ Ωn t Ωn t −i c˜g,n+1 (0) eiΔt/2 . sin + cos 2 Ωn 2 c˜e,n (t) =

(6.112a)

(6.112b)

Using the above expressions of c˜e,n (t) and c˜g,n+1 (t), one may calculate the time 2 2 evolution of the photon distribution pn (t) = |c2e,n (t)| + |c2g,n (t)| and the population inversion of the emitter W (t) = n (|ce,n (t)| − |cg,n (t)| ). Assuming the simplest case where the quantum system is initially prepared in the |e, 0 state, we have the resonant-interaction solutions pn (t) = δn,0 cos2 |g|t + δn,1 sin2 |g|t and W (t) = cos 2|g|t. Interestingly, the Rabi oscillation of the emitter still takes place even when the light field is initialized in the vacuum state. This is completely contradict to the prediction of the semiclassical theory. Indeed, such a phenomenon results from the spontaneous emission of the emitter, whose explanation relies on the full quantum mechanical treatment (see below).

6.9 Multiple Emitters Interacting with Photons In above, we have considered a single quantum emitter interacting with a quantized light field. We now study the quantum system where the photons are coupled to n e emitters. The corresponding Hamiltonian is written as ωeg ( j) ( j) ( j) σz + g j (σ− )† a + g ∗ a † σ− . (6.113) j j j j 2



The Hilbert

 space

 is spanned by the product states { u 1, u 2 , ..., u n e ; n p = |u 1  ⊗ ... ⊗ u n e ⊗ n p ; u j = e, g; n p = 0, 1, 2, ...}, where u j denotes the j-th emitter’s state and n p corresponds to the Fock state of n p photons. The sub-Hilbert-space dimension for Ne emitters is 2 Ne . The interacting system may be simplified by the assumption that the emitter-photon coupling strength are identical for all quantum emitters, i.e., the homogeneous system with g j = g. Taking into account the identity of emitters, we introduce the collective raising and lowering spin operators H/ = ω L a † a +

S+ =



( j)

j

(σ− )† , S− =



( j)

j

σ− ,

(6.114)

6.9 Multiple Emitters Interacting with Photons

323

and the z-component of the total pseudo-spin (length Ne /2) Sz =

1 ( j) σ . j z 2

(6.115)

It is easy to confirm the angular momentum commutation relations [S+ , S− ] = 2Sz , [S± , Sz ] = ∓S± .

(6.116)

The homogeneous Hamiltonian (6.113) is then simplified as H/ = ω L a † a + ωeg Sz + gS+ a + g ∗ a † S− ,

(6.117)

which clearly indicates the collective interaction between multiple emitters and photons. The Holstein–Primakoff transformation [8] is usually employed to map the spin operators S± and Sz to a pair of bosonic creation b† and annihilation b operators,   Ne . S+ = b† Ne − b† b, S− = Ne − b† bb, Sz = b† b − 2

(6.118)

The operators b† and b fulfill the bosonic commutation relation [b, b† ] = 1 and b† b counts the number The complete orthonormal basis of the Hilbert space is



of bosons.  then chosen as { n e , n p = |n e  ⊗ n p ; n e = 0, 1, 2, ..., Ne ; n p = 0, 1, 2, ...}. The |n e  state denotes that there are n e emitters in the excited state |e. As one can see, the identity of quantum emitters strongly reduces the sub-Hilbert-space dimension for the emitters from 2 Ne to Ne + 1.



We further restrict ourselves to the 2-dimension space spanned by 1e , 0 p and

0e , 1 p , i.e., the emitter-photon system has at most only one excitation. The eigenvalues of the quantum system are obtained by diagonalizing H/ ω± =

1 (ω L + ωeg − Ne ωeg ± Ω0 ), 2

(6.119)

with the corresponding eigenstates √



  2|g| Ne (Ω0 − Δ)

1e , 0 p + √ eiϕ 0e , 1 p , (6.120a) Ψ+ = √ 2Ω0 (Ω0 − Δ) 2Ω0 (Ω0 − Δ) √



  2|g| Ne

1e , 0 p − √ (Ω0 − Δ) eiϕ 0e , 1 p . (6.120b) Ψ− = √ 2Ω0 (Ω0 − Δ) 2Ω0 (Ω0 − Δ) and the generalized Rabi frequency Ω0 =



Δ2 + 4|g|2 Ne .

(6.121)

In comparison to (6.110), (n + 1) is replaced with the emitter number √ Ne . The resonant anticrossing between ω+ and ω− is calculated to be Δω0 = 2|g| Ne . Increasing

324

6 Fundamentals of Quantum Optics

Ne may enhance the vacuum Rabi splitting Δω0 . In particular, Δω0 is very sensitive to Ne for the system with a few emitters, and this has been utilized to detect the single molecular emitter [9].

6.10 Spontaneous Emission of Emitter in Free Space So far, we have not considered any effect of the decay sources on the light-emitter interaction. In practice, various dissipation mechanisms inevitably influence the coherent dynamics of the system. One of these dissipative processes is the spontaneous emission of the emitter. The emitter in the excited state |e may return back to the ground state |g through spontaneously emitting a photon with a wavevector k and polarization μ into the free space. This spontaneous process cannot be explained by the classical or semiclassical electromagnetic theory. Indeed, it is caused by the perturbation from the zero-point energy of the electromagnetic field   † ωk (−) ∗ Esp = k,μ ek,μ 2ε0 Veff ak,μ e−ik·r0 . According to the Fermi’s golden rule, the relevant transition rate of the emitter from |e to |g is given by γ=

ωk  2π

(d · e∗ )a † e−ik·r0 |e, 0 |2 δ(ck − ωeg ), (6.122) | g, 1 k,μ k,μ k,μ k,μ 2ε0 Veff 2

in the electric dipole approximation. When the quantization volume Veff approaches ∞, the emitted photon states  2 energy  spectrum, for which  1k,μ exhibit a continuous Veff k dk dΩ. Here dΩ denotes we may replace the sum k with an integral (2π) 3 a cone of solid angle centered around k. The above equation may be rewritten in a more general form 2π 2 (6.123) γ = 2 d · E(−) sp  ρ(ωeg ),  where the emitter-vacuum-field coupling intensity reads d ·

2 E(−) sp 



 1 2 dΩ = | g, 1keg ,μ d · E(−) sp |e, 0 | μ 2 4π ωeg |dge |2 , = 6ε0 Veff

(6.124)

with |keg | = ωeg /c. In the last step, we have applied the fact that the dipole moment of the emitter is randomly oriented with respect to the laboratory frame. The function ρ(ωeg ) accounts for the density of the electromagnetic modes per unit frequency around ωeg within the quantization volume Veff . In free space, ρ(ω) has the simple form ω 2 Veff (6.125) ρ(ω) = 2 3 . π c

6.10 Spontaneous Emission of Emitter in Free Space

325

Finally, the spontaneous emission rate of the emitter in free space is given by γ=

3 ωeg

3πε0 c3

|dge |2 ,

(6.126)

which actually corresponds to the Einstein’s A coefficient. Above we have derived the spontaneous emission rate based on the Fermi’s golden rule. The same result may be obtained by using the Weisskopf–Wigner theory, where the two-level emitter interacts with a reservoir † † σ− + ωk ak,μ ak,μ H/ = ωeg σ− k,μ   † † ∗ gk,μ σ− (6.127) + ak,μ + gk,μ ak,μ σ− , k,μ

with the emitter-reservoir coupling strength gk,μ

 =−

ωk deg · ek,μ . 2ε0 Veff

(6.128)

We assume the emitter is initialized in |e and none photons exist in Veff . The timedependent wavefunction of the quantum system is written as Ψ (t) = ce,0 (t) |e, 0 +

k,μ

 cg,1k,μ (t) g, 1k,μ .

(6.129)

The initial values of the probability amplitudes are ce,0 (0) = 1 and cg,1k,μ (0) = 0. Substituting Ψ (t) into the Schrödinger equation, one obtains d c˜e,0 (t) = −i gk,μ ei(ωeg −ωk )t c˜g,1k,μ (t), k,μ dt d ∗ c˜g,1k,μ (t) = −igk,μ e−i(ωeg −ωk )t c˜e,0 (t), dt

(6.130a) (6.130b)

where we have made the transforms ce,0 (t) = c˜e,0 (t)e−iωeg t and cg,1k,μ (t) = c˜g,1k,μ (t) e−iωk t . We formally integrate (6.130b) ∗ c˜g,1k,μ (t) = −igk,μ



t



e−i(ωeg −ωk )t c˜e,0 (t  )dt  ,

(6.131)

0

and then insert the above result into (6.130a)  t d  2 c˜e,0 (t) = − |gk,μ | ei(ωeg −ωk )(t−t ) c˜e,0 (t  )dt  . k,μ dt 0  Replacing the sum k with an integral, one has

(6.132)

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6 Fundamentals of Quantum Optics



|deg |2 d c˜e,0 (t) = − 2 dt 6π ε0 c3

∞ 0

 dωk ωk3

t



ei(ωeg −ωk )(t−t ) c˜e,0 (t  )dt  ,

(6.133)

0

where the following result (see Problem 6.1)  μ

|deg · ek,μ |2 dΩ =

8π |deg |2 , 3

(6.134)

has been applied. It is common to use the assumption that the spontaneous emission has no memory of the history, i.e., dtd c˜e,0 (t) depends only on the value of c˜e,0 at the time t. Such an assumption is referred to as the Markov approximation. Under this approximation, we replace c˜e,0 (t  ) in (6.133) with c˜e,0 (t) and take it out of the integral, |deg |2 d c˜e,0 (t) = − 2 dt 6π ε0 c3



∞ 0

 dωk ωk3

t

 ei(ωeg −ωk )τ dτ c˜e,0 (t).

(6.135)

0

−1 Since the time scale of interest is much longer than the transition time ωeg , the upper  limit of the integral dτ may be extended to ∞ and one obtains



∞

e

i(ωeg −ωk )τ

 dτ = πδ(ωk − ωeg ) − iP

0

1 ωk − ωeg

 .

(6.136)

Here P denotes the Cauchy principal part, which only gives rise to a frequency shift, i.e., the Lamb’s shift. This shift can be incorporated into ωeg through the renormalization. As a result, (6.135) is reduced to a simple form γ d c˜e,0 (t) = − c˜e,0 (t), dt 2

(6.137)

where γ has been defined in (6.126). It is seen that the probability amplitude decays at a rate that is half of the spontaneous emission rate γ.

6.11 Master Equation In reality, a quantum system denoted by S cannot be completely isolated from the environment that is generally described by a bosonic reservoir (also called the bath) H R / =

k,μ

† ωk bk,μ bk,μ .

(6.138)

The symbol R here is used to denote the reservoir. The Hamiltonian of the quantum system is defined as HS and we use ρ S R to represent the density operator for the quantum system combined with the reservoir. The quantum system-reservoir Hamil-

6.11 Master Equation

327

tonian may be separated into two parts: the free Hamiltonian of the quantum system and the reservoir H0 = HS + H R and their interaction V . In the interaction picture, the system’s Hamiltonian is given by VI (t)/ = ei H0 t/ (V /)e−i H0 t/ . We formally integrate the Liouville equation d ρ S R (t) = −i[VI (t)/, ρ S R (t)], dt 

and obtain

t

ρ S R (t) = ρ S R (0) − i

[VI (t  )/, ρ S R (t  )]dt  .

(6.139)

(6.140)

0

Inserting the above expression back into the Liouville equation, one obtains d ρ S R (t) = −i[VI (t)/, ρ S R (0)] − dt



t

[VI (t)/, [VI (t  )/, ρ S R (t  )]dt  ]. (6.141)

0

In general, the number of degrees of freedom of the reservoir is so big that the quantum system-reservoir interaction hardly affects the reservoir. Thus, the combined density operator ρ S R (t) may be approximated by ρ S R (t) = ρ S (t) ⊗ ρ R , where ρ S (t) = Tr R (ρ S R ) is the density operator for the quantum system while ρ R (t) = Tr S (ρ S R ) corresponds to the density operator for the reservoir. Additionally, we apply the assumption that the quantum system-reservoir interaction makes the quantum system lose the memory of its history, i.e., the Markov approximation. As a result, the Liouville equation is reduced to d ρ S (t) = −iTr R [VI (t)/, ρ S (0) ⊗ ρ R ] dt  t − Tr R [VI (t)/, [VI (t  )/, ρ S (t) ⊗ ρ R ]dt  ].

(6.142)

0

The above equation is our basis of discussing the dissipative process of a quantum system. The density operator ρ S may denote an emitter, an optical cavity, or an emitter-cavity interacting system.

6.11.1 Two-Level Emitter We first study the dissipation for a two-level (excited |e and ground |g) quantum emitter. The free Hamiltonian of the emitter-reservoir system is expressed as † σ− + H0 / = ωeg σ−

k,μ

† ωk bk,μ bk,μ ,

(6.143)

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6 Fundamentals of Quantum Optics

and the interaction operator is V / =

k,μ

† † ∗ (gk,μ σ− bk,μ + gk,μ bk,μ σ− ).

(6.144)

Using the Hadamard’s lemma, e A Be−A = B + [A, B] +

1 1 [A, [A, B]] + [A, [A, [A, B]]] + . . . , 2! 3!

(6.145)

one finds the following expressions †

†

eiωeg σ− σ− t σ− e−iωeg σ− σ− t = σ− e−iωeg t , e e e

† iωeg σ− σ− t

† † −iωeg σ− σ− t σ− e

† iωk bk,μ bk,μ t † iωk bk,μ bk,μ t

bk,μ e

=

† −iωk bk,μ bk,μ t

† † bk,μ e−iωk bk,μ bk,μ t

(6.146a)

† iωeg t σ− e ,

= bk,μ e =

−iωk t

(6.146b) ,

† bk,μ eiωk t .

(6.146c) (6.146d)

Thus, the Hamiltonian in the interaction picture is derived as VI (t)/ =



k,μ

 † † ∗ gk,μ σ− bk,μ e−i(ωk −ωeg )t + gk,μ bk,μ σ− ei(ωk −ωeg )t .

(6.147)

It is natural to assume that the reservoir follows the thermal equilibrium, ρ R = † bk ,μ  = n¯ k,μ δk,k δμ,μ and ρthermal . One can easily derive the expectation values bk,μ bk,μ  = bk,μ bk ,μ  = 0. From (6.142), we have 

  † † |gk,μ |2 n¯ k,μ [σ− σ− ρ(t) − σ− ρ(t)σ− ]ei(ωk −ωeg )(t−t ) k,μ 0  † † −i(ωk −ωeg )(t−t  ) + H.c., (6.148) +(1 + n¯ k,μ )[σ− σ− ρ(t) − σ− ρ(t)σ− ]e

d ρ(t) = − dt

t

dt 



where the subscript S in ρ S (t) has been omitted for simplicity and ρ(t  ) has been replaced with its value at the time t in the Markov approximation. Using (6.137) and k,μ

γ |gk,μ |2 n¯ k,μ πδ(ωk − ωeg ) = n¯ th , 2

(6.149)

Equation (6.148) is reduced to   d 1 † † ρ(t) = (n¯ th + 1)γ σ− ρσ− − {σ− σ− , ρ} dt 2   1 † † +n¯ th γ σ− ρσ− − {σ− σ− , ρ} , 2

(6.150)

6.11 Master Equation

329

where n¯ th is the thermal average number of the bosons with ωk = ωeg , γ is the spontaneous emission rate of |e, and the anticommutator takes the form {A, B} = AB + B A.

(6.151)

Generally, the environment temperature is set at zero, n¯ th = 0, and the dissipation of the density operator is further simplified as d ρ(t) = γD[σ− ]ρ, dt

(6.152)

with the definition of the Lindblad superoperator 1 D[o]ρ = oρo† − {o† o, ρ}. 2

(6.153)

Using (6.152), the equations of motion of the density matrix elements for a two-level emitter are derived as 1 d ρee (t) = − ρee (t), dt T1

d 1 ρeg (t) = − ρeg (t). dt T2

(6.154)

Here the so-called relaxation and dephasing rates are given by 1 = γ, T1

1 γ = , T2 2

(6.155)

respectively. Indeed, T1 denotes the lifetime of the emitter’s excited state |e [see Fig. 6.5a] while T2 measures the coherence time of the |g − |e transition. In many cases, the spontaneous emission of the excited state is not the only decay source to the emitters. Other dissipative mechanisms, such as inter-emitter collisions, may strongly affect the phase of the emitter’s polarization, leading to 1 ≥ 2T1 1 . Therefore, the Liouville equation takes a more general form as T2 1 d 1 ρ(t) = −i[H/, ρ] + D[σ− ]ρ + D[σz ]ρ, dt T1 2Tϕ

(6.156)

where we have included the coherent evolution governed by the Hamiltonian H of the quantum system. The parameter 1/Tϕ denotes the pure dephasing rate and the total dephasing rate reads 1 1 1 = + . (6.157) T2 2T1 Tϕ Involving the interaction between the emitter and a classical light field, the dissipative equations of motion of the density matrix elements are

330

6 Fundamentals of Quantum Optics

Fig. 6.5 Dissipative dynamics of a light-driven emitter. a Relaxation of a free emitter initially prepared in |e. The decay follows e−t/T1 . b Damped Rabi oscillatory behavior of the resonantlydriven emitter with Ωr T1 = 30 and Ωi T1 = 20. The emitter is initially prepared in the ground state |g. c Ramsey  fringes as a function of the free-evolution time T . The π-pulse duration τ is given by τ = π/ Ωr2 + Ωi2 . The decay of the envelope follows e−T /T2 . d Steady-state polarizability α of the emitter versus the detuning Δ in the weak-driving limit. For all curves, T2 /T1 = 0.4

d 1 Ω Ω∗ ρee = − ρee − i ρge + i ρeg , dt T 2 2  1  d 1 Ω ρeg = − + iΔ ρeg + i (2ρee − 1), dt T2 2

(6.158a) (6.158b)

and the corresponding equations of motion of three Bloch-vector components take the form 1 d u = − u + Δv + Ωi w, dt T2 1 d v = − v − Δu − Ωr w, dt T2 1 d w = − (w + 1) + Ωr v − Ωi u. dt T1

(6.159a) (6.159b) (6.159c)

6.11 Master Equation

331

The set of equations (6.159) are known as the Bloch equations that are fundamental to study the NMR (nuclear magnetic resonance) spectroscopy. They can be numerically solved by using the matrix algebra (see Appendix F). Due to the dissipation, the Rabi oscillation of the emitter between |g and |e exhibits a damping behavior [see Fig. 6.5b]. A similar behavior is also found in the Ramsey oscillation, whose envelope decays at a constant rate of 1/T2 [see Fig. 6.5c]. Thus, the Ramsey method is often employed to measure the dephasing time T2 . As the time t goes to ∞, the dissipative system approaches a steady state (denoted by “ss”) with u (ss) = (ΔΩr − Ωi /T2 )/Θ, (ss)

v = (ΔΩi + Ωr /T2 )/Θ, (ss) = −(1/T22 + Δ2 )/Θ. w

(6.160a) (6.160b) (6.160c)

Here we have defined the coefficient Θ=

T1 1 + Δ2 + |Ω|2 . 2 T2 T2

(6.161)

The population difference w(ss) is always negative, i.e., the population in |e is smaller than that of |g, because of the relaxation of the |e state. The steady-state den(ss) (ss) + 1)/2 and ρ(ss) − iv (ss) )/2 are obtained sity matrix elements ρ(ss) ee = (w eg = (u from the Bloch vector, ρ(ss) ee =

T1 |Ω|2 Ω , ρ(ss) . eg = (Δ − i/T2 ) T2 2Θ 2Θ

(6.162)

Figure 6.2b depicts an example of the damped Rabi oscillation in the Bloch-sphere representation, where, as one can see, the movement trajectory of the Bloch vector approaches an steady-state point that is located inside the sphere. We are interested in the steady-state solutions in the limit of |Ω| ∼ 0 since the absorption spectrum is often performed in this weak-driving regime. In this case, the parameter Θ is approximated by Θ ≈ 1/T22 + Δ2 , which is independent of the light (ss) field. The steady-state ρ(ss) ee and ρeg are then re-expressed as ρ(ss) ee ≈

I T1 T2 1 deg · e , ρ(ss) E 0 eikL ·r0 , eg ≈ − 2 I 2 2 Δ + i/T2 1 + Δ T2 s

(6.163)

where I = cε0 E 0∗ E 0 /2 is the light intensity and the saturation intensity corresponding to the |g − |e transition is defined as Is =

2 cε0 . (deg · e)(dge · e∗ )

(6.164)

332

6 Fundamentals of Quantum Optics

It is seen that the probability ρ(ss) ee of the emitter in |e depends linearly on the drivingfield intensity while the coherence ρ(ss) eg is proportional to the light amplitude. Note that, the above equations are only valid in the limit of I  Is . The expectation value of the dipole operator gives the emitter’s dipole moment, p = d = p(+) + p(−) with (+) in the form p(+) = αe E20 eikL ·r0 , p(+) = dge ρeg . Using ρ(ss) eg we further express p where the electric polarizability of the emitter reads α=−

(deg · e)(dge · e∗ ) α0 with α0 = . T2 (Δ + i/T2 ) T2−1

(6.165)

The real part Re(α), which reaches its maximum at |Δ| = T2−1 , corresponds to the emitter dispersion of the light field [see Fig. 6.5d]. In contrast, the imaginary part Im(α) measures the emitter absorption, which is maximized at the resonance Δ = 0 and strongly decays when |Δ| exceeds T2−1 .

6.11.2 Dynamical Polarizability of an Emitter In above, we have discussed the steady-state polarizability α of the emitter in the limit of the near-resonant light-emitter interaction. However, in a more common situation the light frequency ω L may be far-off-resonant to the emitter’s transition frequency ωeg = ωe − ωg . Thus, the RWA is no longer valid and the density matrix of the emitter cannot reach a steady state. Now we investigate the emitter’s polarizability in E∗ an arbitrary light field E = e E20 e−iωL t + e∗ 20 e−iωL t . Here the position of the emitter has been set at the origin of coordinates, r0 = 0. In the semiclassical framework, the Hamiltonian of a light-driven emitter is expressed as H = ωe σee + ωg σgg − d · E,

(6.166)

with the light-induced dipole vector operator d. Using the master equation (6.156), one obtains the equation of motion of the off-diagonal density matrix element  deg · e d ρeg = −i(ωe − ωg − i/T2 )ρeg − i E 0 e−iωL t dt 2  deg · e∗ ∗ iωL t (ρee − ρgg ). E0 e + 2

(6.167)

The above equation involves the counter-rotating term of the light-emitter interaction. We are interested in the weak light-emitter coupling, for which the emitter rarely stays in the upper state |e, and one can make the approximation ρee ≈ 0 and ρgg ≈ 1. Substituting ρeg = ρ˜eg e−i(ωe −ωg −i/T2 )t into (6.167), we obtain

6.11 Master Equation

333

deg · e d ρ˜eg = i E 0 ei(ωe −ωg −ωL −i/T2 )t dt 2 deg · e∗ ∗ i(ωe −ωg +ωL −i/T2 )t +i . E0 e 2

(6.168a)

Integrating the above equation from −∞ to t, one arrives at ρeg (t) =

deg · e/ E 0 −iωL t e ωe − ωg − ω L − i/T2 2 deg · e∗ / E 0∗ iωL t e . + ωe − ωg + ω L − i/T2 2

(6.169a)

In deriving the above equation, we have used ρeg (−∞) = 0. The emitter’s polarization p = p(+) + p(−) is given by the expectation value of the electric dipole operator d = deg ρge + dge ρeg . For the linear polarization of medium, the positive-frequency amplitude vector p(+) can be written as αe E20 , where the emitter’s dynamical polarizability α is derived as α=

|deg · e|2 



1 1 + ωe − ωg − ω L − i/T2 ωe − ωg + ω L + i/T2

 .

(6.170)

For the near-resonant interaction ω L ∼ (ωe − ωg ), the first term dominates and (6.170) returns to (6.165).

6.11.3 EIT in Λ-Type Emitters In Sect. 4.5, we have introduced the EIT of the coupled optical cavities. Such a coherent nonlinear behavior is caused by the classical interference between the optical waves in two cavities. Actually, the EIT was firstly introduced to the interaction between lights and multilevel (e.g., three-level Λ-, - and V -type) emitters, where the quantum interference between different transition pathways of an emitter induces a transparency window to the probe beam. As an example, we consider two laser beams coupling to a Λ-type emitter as shown in Fig. 4.15a. The emitter is composed of three levels |u = 1, 2, 3, where |1 is the ground state and the energy of |3 is highest. The |1 − |3 and |2 − |3 transitions are electric dipole allowed while the |1 − |2 transition is electric dipole forbidden. The light-emitter interaction is described by the following Hamiltonian H/ = Δ1 σ11 + Δ2 σ22 +

 1  1 ∗ † † Ω1 σ13 + Ω1 σ13 Ω2∗ σ23 + Ω2 σ23 + . (6.171) 2 2

The operators σuv with u, v = 1, 2, 3 are defined as σuv = |u v|. The Rabi frequency Ω1 (Ω2 ) measures the coupling strength between the probe (coupling) beam

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Fig. 6.6 EIT of the Λ-type emitters. a Re(α/α0 ) versus (Δ, δ). b Re(α/α0 ) as a function of the Raman detuning δ with Δ = 0. c Im(α/α0 ) versus (Δ, δ). d Im(α/α0 ) as a function of δ with Δ = 0. Here γ31 = γ32 = γ, γ21 = 10−3 γ and Ω2 = 2γ with an arbitrary units of frequency γ

and the |1 − |3 (|2 − |3) transition of the emitter with the corresponding detuning Δ1 (Δ2 ). The dissipative dynamics of this open quantum system is governed by the master equation d ρ = −i[H/, ρ] + γ31 D[σ13 ]ρ + γ32 D[σ23 ]ρ + γ21 D[σ12 ]ρ, dt

(6.172)

with the decay rates γ31 and γ32 of the corresponding emitter’s transitions. Generally, the decay rate γ21 is much smaller than γ31 and γ32 . Due to the applied weak-probe and strong-coupling beams, the emitters mainly stay in the ground state |1. Thus, one may make the reasonable approximations ρ11 ≈ 1 and ρ22 ≈ ρ33 ≈ ρ32 ≈ 0. The master equation (6.172) then gives us ⎛ ⎞ iΩ2∗ γ21     & 0 ' − iδ d ρ21 ρ21 ⎜ 2 ⎟ 2 iΩ1 . (6.173) = − ⎝ iΩ + ⎠ ρ γ31 + γ32 2 − dt ρ31 31 − i(Δ + δ) 2 2 2 where we have defined the Raman detuning δ = Δ1 − Δ2 and replaced Δ2 with Δ for simplicity. Solving the above linear differential equations (see Appendix F), one obtains the steady-state solution

6.11 Master Equation

ρ(ss) 31

335

γ21 − iδ iΩ1 2 =− .     2 γ21 γ31 + γ32 |Ω2 |2 − iδ − i(Δ + δ) + 2 2 4

(6.174)

The positive-frequency polarizability of the emitter’s |1 − |3 transition is then recognized as α(δ, Δ) = −

2|d31 · e1 |2 (ss) ρ31 Ω1

= α0 

iγ31

γ

21

− iδ



2 ,   γ + γ |Ω2 |2 γ21 31 32 − iδ − i(Δ + δ) + 2 2 4

(6.175)

with the constant α0 = |d31 · e1 |2 /γ31 and the polarization e1 of the probe beam. Figure 6.6a–d illustrate the dependence of α on δ and Δ. The absorption of the probe field is proportional to Im(α). One can see that a narrow absorption dip exhibits near the Raman resonance δ = 0 when the coupling beam is tuned near resonance. Comparing Fig. 6.6b, d, a strong change in absorption is accompanied by a sharp dispersion over a narrow spectral range. Such a rapid change in refractive index has been utilized to slow the group velocity of the light down to 17 m/s [10]. Besides, the EIT can be also applied to store the optical information in an atomic medium [11].

6.11.4 Cavity Next, we consider the master equation for the light field in a lossy cavity. Due to the incomplete reflection of the cavity’s mirrors, the photons inside the cavity can escape into the environment outside the cavity. This decay may be viewed as the consequence of the single-mode cavity interacting with the reservoir full of bosonic oscillators, † ∗ (gk,μ a † bk,μ + gk,μ abk,μ ). (6.176) V / = k,μ

Here a and a † are the annihilation and creation photon operators associated with the cavity mode. The parameter gk measures the coupling strength between one photon and one boson that is characterized by (k, μ). Following the same derivation process in Sect. 6.11.1, the dissipative dynamics of the density matrix operator of the cavity mode is given by     1 1 d ρ(t) = κ(n¯ th + 1) aρa † − {a † a, ρ} + κn¯ th a † ρa − {aa † , ρ} , (6.177) dt 2 2

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6 Fundamentals of Quantum Optics

where κ = κin + κout is the summation of the incident-mirror κin and output-mirror κout loss rates. The value of n¯ th corresponds to the average number of the thermal bosons with ωk = ω L (ω L is the cavity-mode frequency). Indeed, the master equation (6.177) is equivalent to the Fokker–Planck equation (see Problem 6.2). Under the assumption of the zero-temperature thermal reservoir, the master equation of the photon density operator takes the form d ρ(t) = −i[H/, ρ] + κD[a]ρ. dt

(6.178)

In many cases, the cavity is driven by an external classical light field, whose frequency and input power are ω and P, respectively, from the incident mirror. The corresponding Hamiltonian is written as √  κin P † −iωt−iφ H/ = ω L a a + (a e + aeiωt+iφ ), 2 ω †

(6.179)

with the phase difference φ between the driving and cavity fields. The master equation (6.177) leads to the equations of motion of the amplitude A = Tr(aρ) and photon number N = Tr(a † aρ) √  κin P −iωt−iφ d , A = −(κ/2 + iω L )A − i e dt 2 ω √  κin P d N = −κN − i (A∗ e−iωt−iφ − Aeiωt+iφ ). dt 2 ω

(6.180a) (6.180b)

˜ −iωt−iφ into the above equations, one finds the steady-state soluSubstituting A = Ae tions of the intracavity field ˜ (ss)

A

N (ss)

√  κin P 1 , = −i 2 ω κ/2 + i(ω L − ω) κ2 /4 κin N0 2 = . κ κ /4 + (ω L − ω)2

(6.181a) (6.181b)

The factor N0 = P/κω denotes the number of photons entering the cavity within the characteristic time scale κ−1 . One can prove that N (ss) = | A˜ (ss) |2 , which is because the cavity is driven by a classical light field. The steady-state value of the intracavity photon number N (ss) is maximized at the resonant driving, ω = ω L .

6.11 Master Equation

337

6.11.5 Interaction-Induced Spectral Shift and Broadening We are ready now to discuss the effect of the weak coupling between a two-level emitter and a single-mode cavity on the cavity mode. The master equation for the combined system is written as 1 1 d ρ(t) = −i[H/, ρ] + D[σ− ]ρ + D[σz ]ρ + κD[a]ρ, dt T1 2Tϕ

(6.182)

where the system’s Hamiltonian is expressed as H = ω L a † a + ωe σee + ωg σgg  

 † − deg σ− + dge σ− · E0 ea + e∗ a † .

(6.183)

Again, the position of the emitter is set at the origin of coordinates, r0 = 0. The density operator ρ includes both emitter and cavity mode. From (6.182) one may derive the equations of motion of the observable expectations a(t) = Tr[aρ(t)] and σ− (t) = Tr[σ− ρ(t)] as d a = −(κ/2 + iω L )a dt E0 E0 †  + i (dge · e∗ )σ− , +i (deg · e∗ )σ−   d σ−  = −(1/T2 + iωe − iωg )σ−  dt E0 E0 −i (deg · e)σz a − i (deg · e∗ )σz a † .  

(6.184a)

(6.184b)

Since the emitter is weakly driven by the cavity mode, the emitter stays mainly in ˜ −iωL t and the lower state |g, i.e., σz ≈ −1. We make the substitutions a = ae −(1/T2 +iωe −iωg )t , where the amplitudes a ˜ and σ˜ −  vary much slowly σ−  = σ˜ − e in comparison with the terms e−iωL t and e−i(ωe −ωg )t . The above equations are then re-expressed as κ E0 d † (−1/T2 +iωe −iωg +iω L )t a ˜ = − a ˜ + i (deg · e∗ )σ˜ − e dt 2  E0 +i (dge · e∗ )σ˜ − e(−1/T2 −iωe +iωg +iωL )t ,  d E0 σ˜ −  = i (deg · e∗ )a˜ † e(1/T2 +iωe −iωg +iωL )t dt  E0 +i (deg · e)ae ˜ (1/T2 +iωe −iωg −iωL )t . 

(6.185a)

(6.185b)

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6 Fundamentals of Quantum Optics

Integrating (6.185b) from −∞ to t yields σ− (t) =

(deg · e)a(t) E0 (deg · e∗ )a † (t) E0 + ,  ωe − ωg + ω L − i/T2  ωe − ωg − ω L − i/T2

(6.186)

where we have used σ− (−∞) = Tr[σ− ρ(−∞)] = 0. For simplicity, we assume e is a real vector. Inserting (6.186) into (6.184a) leads to   ωL ωL d κ α a + i α∗ a † . a = − + iω L − i dt 2 2ε0 Veff 2ε0 Veff

(6.187)

The polarizability α has been defined in (6.170). Equation (6.187) can be further written as a second-order differential equation,   ωL d d2 a a + κ + Im(α) dt 2 ε0 Veff dt   ωL  κ Im(α) − ω L Re(α) a = 0. (6.188) + |κ/2 − iω L |2 + ε0 Veff 2 Using the fact ω L κ, the above equation is approximated as d2 a + ω L dt 2



κ Im(α) + ωL ε0 Veff



  d Re(α) a = 0, a + ω 2L 1 − dt ε0 Veff

(6.189)

Indeed, the observable expectation a corresponds to the complex amplitude of a classical light field. Comparing to (1.149), we find the quality factor of the perturbed cavity 1 1 Im(α) = + , (6.190) Q Q0 ε0 Veff where Q 0 = ω L /κ corresponds to the quality factor of a naked cavity. The dissipative dynamics of the emitter (e.g., the spontaneous emission) introduces an extra decay channel to the intracavity energy, reducing the system’s Q factor. One may also obtain the frequency shift of the cavity mode ω L caused by the emitter-cavity interaction δω L = ωL

 1−

Re(α) Re(α) −1≈− . ε0 Veff 2ε0 Veff

(6.191)

Therefore, in the weak-coupling limit (i.e., σz ≈ −1) the effect of the emitter-cavity interaction on the cavity mode can be summarized as follows: The real part of the emitter’s polarizability Re(α) shifts the mode’s resonant frequency while the imaginary part Im(α) broadens the mode’s spectral linewidth.

6.11 Master Equation

339

6.11.6 Diagonalization Method Let us write the Markovian master equation in a more general form d γ J D[J ]ρ, ρ = −i[H/, ρ] + J dt

(6.192)

where the operator J denotes a possible channel of the irreversible loss of energy and coherence of a quantum system with the corresponding rate γ J . Defining the non-Hermitian operator Λ = −i H/ −

γJ J † J, J 2

(6.193)

the master equation (6.192) is rewritten as d ρ = Λρ + ρΛ† + γ J J ρJ † . J dt

(6.194)

We then choose an orthonormal set of vectors {|v ; v = 1, 2, 3, ..., N } to span the Hilbert space. Thus, (6.194) is converted to N 2 linear differential equations  d ρv1 v2 = Λv1 v3 δv4 v2 + (Λ† )v4 v2 δv1 v3 v3 ,v4 dt  + γ J Jv1 v3 (J † )v4 v2 ρv3 v4 , J

(6.195)

with, for example, ρv1 v2 = v1 | ρ |v2 . The N × N density matrix ρ may be mapped into a N 2 -dimensional vector R, i.e., Rα = ρv1 v2 , via the one-to-one correspondence between {α = 1, 2, ..., N 2 } and the pairs {(v1 , v2 ); v1,2 = 1, 2, ..., N }. By further defining a N 2 × N 2 matrix M with the elements Mαβ = M(v1 v2 ),(v3 v4 ) = −Λv1 v3 δv4 v2 − (Λ† )v4 v2 δv1 v3 −

J

γ J Jv1 v3 (J † )v4 v2 ,

(6.196)

the set of differential equations (6.195) is reduced to a matrix differential equation d R = −MR, dt

(6.197)

The above equation can be solved by using the diagonalization method (see Appendix F). It is worth noting that no matter what the initial condition is, (6.197) arrives at the same steady state R(ss) eventually. For a system composed of Ne two-level quantum emitters, the dimension N is equal to 2 Ne . An arbitrary

state of the system  can be written as a superposition state of 2 Ne emitter states { u 1 , u 2 , ..., u i , ..., u Ne ; u i = e, g}. In contrast, the dimension

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6 Fundamentals of Quantum Optics

of a system consisting of photons is not fixed since the number of Fock states can go

to  the infinity. Thus, the Fock-state basis must be truncated at a maximal state

n p in the practical calculation and the dimension is given by N = (n p + 1). When n p is large enough, the probability of the system in the Fock states {|n ; n > n p } is negligible and the Hilbert space is approximately spanned by {|n ; n = 0, 1, ..., n p }. Although the diagonalization approach gives the exact (or quasi-exact) solution of ρ(t), the solvable system size is limited by the finite amount of computer memory and computational time.

6.11.7 Two-Time Correlation Functions Once the density matrix ρ(t) is obtained, the single-time expectation of an arbitrary physical observable O can be calculated straightforwardly, O(t) = Tr[Oρ(t)], which is directly linked to the experimental measurement. Additionally, in many practical cases one needs to calculate the two-time correlation function A(t)B(t + τ ) between two operators A and B so as to derive the spectrum, from which the frequency shift and spectral linewidth (coherence time) of the system can be read out directly. Now, we consider how to evaluate A(t)B(t + τ ) from ρ(t). Below, ρ S and ρ R are used to denote the density matrix operators of a quantum system and the reservoir, respectively. The expectation of B at the time τ reads B(τ ) = Tr S [Bρ S (τ )] (  ) = Tr S BTr R U † (τ )ρ S (0) ⊗ ρ R (0)U (τ ) ,

(6.198)

where the trace Tr S (Tr R ) operates on the quantum system (reservoir) and U (t) is the unitary time-evolution operator for the whole system (including both quantum system and reservoir). Similarly, A(t)B(t + τ ) is written as A(t)B(t + τ ) = Tr S [B(t + τ )ρ S (t)A] (  ) = Tr S B(t)Tr R U † (τ )(ρ S (t)A) ⊗ ρ R (0)U (τ ) .

(6.199)

As t goes to infinity, ρ S (t) approaches the steady state ρ(ss) due to the inevitable S dissipation and we have  †  [ρ(ss) S A](τ ) = lim Tr R U (τ )(ρ S (t)A) ⊗ ρ R (0)U (τ ) . t→∞

(6.200)

Since the system loses its memory on its initial condition when t → ∞, A(t)B(t + τ ) becomes independent of t. Consequently, the two-time correlation function is reduced to   (6.201) lim A(t)B(t + τ ) = Tr S B[ρ(ss) S A](τ ) . t→∞

6.11 Master Equation

341

Fig. 6.7 Resonance fluorescence of a two-level emitter driven by an optical plane wave. a Firstorder correlation. b Second-order correlation

The only difference between the right hand sides of (6.198) and (6.201) is to replace ρ S (τ ) with [ρ(ss) S A](τ ). According to the quantum regression theorem, the evolution of [ρ(ss) S A](τ ) also follows (6.197). As a result, the two-time correlation A(t)B(t + τ ) can be evaluated by using ρ(t). We should point out that the quantum regression theorem does not hold true for a non-Markovian process. As an example, let us consider the correlation functions of the resonance fluorescence from a laser-driven two-level atom (transition frequency ωeg ) located at r0 . The driving laser is assumed to be perfectly coherent and the Rabi frequency is

342

6 Fundamentals of Quantum Optics

Ω. The only decay source in the system is the spontaneous emission (rate γ) of the atom. In the far-zone limit (i.e., the distance between the atom and the position of the photodetector r is much larger than the radiation wavelength |r − r0 | λ), the positive-frequency field operator E(+) (r, t) of the fluorescent light is proportional to the lowering operator of the atom, (+)

E

  |r − r0 | 1 σ− t − . (r, t) ∝ |r − r0 | c

(6.202)

Figure 6.7a shows the normalized first-order correlation function g (1) (τ ) =

E(−) (r, t)E(+) (r, t + τ ) , E(−) (r, t)E(+) (r, t)

(6.203)

derived from the quantum regression theorem. It is seen that the function g (1) (τ ) does not decay to zero in the weak-driving limit of Ω ∼ 0. That is, the elastic Rayleigh scattering of the driving light from the atom plays the main role in the light-atom interaction. As Ω is increased to a large enough value, the inelastic scattering takes the main effect and g (1) (τ γ −1 ) approaches zero. Interestingly, when Ω γ, g (1) (τ ) presents an oscillatory behavior, which actually is related to the Autler–Townes effect. The corresponding fluorescence spectrum S(ω) = F [g (1) (τ )] exhibits three peaks, where the Rayleigh peak is located at ω = ωeg and the blue- and red-side peaks are separated from ωeg by the Rabi frequency Ω [12]. Such a fluorescence spectrum is known as the Mollow triplet. One may further study the normalized second-order correlation function g (2) (τ ) =

E(−) (r, t)E(−) (r, t + τ )E(+) (r, t + τ )E(+) (r, t) . E(−) (r, t)E(+) (r, t)E(−) (r, t + τ )E(+) (r, t + τ )

(6.204)

As displayed in Fig. 6.7b, we have g (2) (τ ) > g (2) (0) = 0, i.e., the photon antibunching, for an arbitrary driving strength Ω. The antibunching behavior generally refers to the sub-Poissonian photon statistics, where the variance of the photon number is less than its mean value. A direct signature of the antibunching effect is the singlephoton emission, that is, an emitter can only emit one photon at a time. This is understandable that after emitting a photon, the single atom returns to the ground state and has to wait for some time to be re-excited into the upper level. Thus, two successive photon emission events never overlap temporally. As τ goes to the infinity, the function g (2) (τ ) approaches unity.

Problems

343

Problems 6.1 A plane wave with an arbitrary real wavevector k = k x ex + k y e y + k z ez has two independent polarization directions with the complex unit vectors ek,1 = (1) k,x ex + (1) (2) (2) (2) e +  e and e =  e +  e +  e . We map the Cartesian compo(1) k,2 k,y y k,z z k,x x k,y y k,z z nents of k into the spherical coordinates ⎛ ⎞ ⎛ ⎞ kx sin θ cos ϕ ⎝k y ⎠ = |k| ⎝ sin θ sin ϕ ⎠ . cos θ kz

(6.205)

Two transverse polarization vectors are then written as ⎛

⎞ √ ⎛ eiα sin ϕ + cos θ cos ϕ ⎞ (1) k,x 2 ⎝ iα ⎜ (1) ⎟ −e cos ϕ + cos θ sin ϕ⎠ , ⎝k,y ⎠ = 2 (1) − sin θ k,z ⎛ (2) ⎞ √ ⎛ eiα sin ϕ − cos θ cos ϕ ⎞ k,x 2 ⎝ iα ⎜ (2) ⎟ −e cos ϕ − cos θ sin ϕ⎠ ⎝k,y ⎠ = 2 (2) sin θ k,z

(6.206a)

(6.206b)

with an arbitrary phase α. Verify the orthogonality among k, ek,1 and ek,2 , k · ek,1 = k · ek,2 = e∗k,1 · ek,2 = 0, e∗k,1 · ek,1 = e∗k,2 · ek,2 = 1.

(6.207a) (6.207b)

Also verify the following relation  ∗  ∗ ki k j (2) (1) (2) , (1) k,i k, j + k,i k, j = δi, j − |k|2

(6.208)

with i, j = x, y, z and further (6.134). 6.2 The density matrix ρ(t) of the light field in the P-representation is expressed as  ρ(t) =

P(α, α∗ , t) |α α| dαdα∗ ,

(6.209)

Substituting (6.209) into (6.177), one may obtain the following Fokker–Planck equation associated with the function P(α, α∗ , t) ∂ κ P(α, α∗ , t) = ∂t 2



 ∂ ∂ ∗ ∂2 P(α, α∗ , t) + κn¯ th α+ α P(α, α∗ , t), ∗ ∂α ∂α ∂α∂α∗ (6.210)

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where the two terms on the right side of the equal sign correspond to the drift and diffusion, respectively, of the probability density function P(α, α∗ , t). Derive (6.210) by using the following identities  a |α α| = †

 ∂ ∗ + α |α α| , ∂α

 |α α| a =

 ∂ + α |α α| . ∂α∗

(6.211)

References 1. I.I. Rabi, Space quantization in a gyrating magnetic field. Phys. Rev. 51, 652–654 (1937). https://doi.org/10.1103/PhysRev.51.652 2. S.H. Autler, C.H. Townes, Stark effect in rapidly varying fields. Phys. Rev. 100, 703–722 (1955). https://doi.org/10.1103/PhysRev.100.703 3. W.M. Itano, J.C. Bergquist, J.J. Bollinger, J.M. Gilligan, D.J. Heinzen, F.L. Moore, M.G. Raizen, D.J. Wineland, Quantum projection noise: population fluctuations in two-level systems. Phys. Rev. A 47, 3554–3570 (1993). https://doi.org/10.1103/PhysRevA.47.3554 4. N.F. Ramsey, A molecular beam resonance method with separated oscillating fields. Phys. Rev. 78, 695–699 (1950). https://doi.org/10.1103/PhysRev.78.695 5. P.A.M. Dirac, The quantum theory of the emission and absorption of radiation. Proc. R. Soc. Lond. A 114, 243–265 (1927). https://doi.org/10.1098/rspa.1927.0039 6. L. Mandel, Sub-Poissonian photon statistics in resonance fluorescence. Opt. Lett. 4, 205–207 (1979). https://doi.org/10.1364/OL.4.000205 7. E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89–109 (1963). https://doi.org/10.1109/ PROC.1963.1664 8. T. Holstein, H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098–1113 (1940). https://doi.org/10.1103/PhysRev.58.1098 9. R. Chikkaraddy, B. de Nijs, F. Benz, S.J. Barrow, O.A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, J.J. Baumberg, Single-molecule strong coupling at room temperature in plasmonic nanocavities. Nature 535, 127–130 (2016). https://doi.org/10.1038/nature17974 10. L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Light speed reduction to 17 metres per second in an ultracold atomic gas. Nature 397, 594–598 (1999). https://doi.org/10.1038/17561 11. C. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Observation of coherent optical information storage in an atomic medium using halted light pulses. Nature 409, 490–493 (2001). https://doi.org/ 10.1038/35054017 12. B.R. Mollow, Power spectrum of light scattered by two-level systems. Phys. Rev. 188, 1969– 1975 (1969). https://doi.org/10.1103/PhysRev.188.1969

Chapter 7

Molecular Cavity QED

Abstract Cavity quantum electrodynamics (cavity QED) is the study of the strong coupling between two-level emitters and high-Q optical cavities, where the quantum nature of the light plays a crucial role. Thus far, the relevant cavity-QED experiments have been performed by using various particles, including neutral atoms, trapped ions, quantum dots, and nitrogen-vacancy centers. Here, we focus on the cavity QED with the molecules placed inside or close to a microcavity. The molecules have complex energy structures and short energy-level lifetimes, posing the experimental challenges. Nevertheless, the strong molecule–cavity coupling regime is still accessible by carefully designing the cavity structure and controlling the ambient environment. The strong emitter–cavity coupling does not only shorten the measurement time in quantum metrology but also enhances the sensitivity of a sensor. The fundamental limit of the measurement uncertainty is set by the Heisenberg uncertainty principle.

7.1 Introduction In the cavity-QED paradigm, the two-state emitters interact with a single-mode light field that is well confined in a high-Q optical cavity. The system’s size should be strongly reduced so that the quantum nature of the light becomes dominant. An excellent cavity-QED platform is the key to explore the fundamental aspects of the light–matter interaction, such as the modified spontaneous emission of an emitter, collective super- and sub-radiance of multiple emitters, dissipative dynamics of open quantum systems, quantum information processing, and quantum metrology. The cavity–emitter coupling strength is essential to be large enough that the physical system exhibits the quantum behavior well before the decay sources take effect. The cavity QED based on the atoms and ions has been profoundly investigated in both theory and experiment because of the simple structure of the internal energy levels of atoms [1]. In contrast, the intrinsic mechanical and electronic degrees of freedom of molecules significantly complicate the molecular energy spectrum. Additionally, the strong interaction between the molecules and the environment (reservoir) dramatically broadens the molecular transition lines. These two factors mostly © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2_7

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impede the molecular cavity-QED platform reaching the strong-coupling regime. To address these issues, various methods have been exploited, such as the plasmonic local-field enhancement and the cavity QED at cryogenic temperature. So far, the strong coupling has been achieved between a large variety of quantum emitters, including atoms [2, 3], semiconductor quantum dots [4, 5], nitrogen-vacancy centers in diamond nanocrystals [6, 18] and superconducting nanocrystals [7], and different kinds of microwave and optical cavities, for instances, whispering-gallery-mode microcavities [8], photonic nanocavities [9], and superconducting resonators [10]. It is worth noting that the interaction between a superconducting qubit and a superconducting resonator (i.e., the so-called circuit QED) can go beyond the strong-coupling regime, reaching the ultrastrong-coupling regime [11] or even the deep-strong-coupling regime [12]. Here, we restrict ourselves to the cavity-QED systems composed of single molecules interacting with microcavities. In the first section, we aim to convey the general concept of the cavity QED. The emitters studied here are the molecules, whose structure and relevant physical properties are introduced in the second section. Next, we bring the recent experimental developments in labs. In the last section, we consider the Heisenberg-limited detection, which is one of the most active fields in quantum metrology.

7.2 Cavity QED In Sect. 6.8, we have discussed the simplest light–matter interaction model, where a two-level (upper |e and lower |g) quantum emitter with the transition frequency ωeg is coherently coupled to a single-mode quantized cavity field at the frequency ω L . The eigenstates of the cavity field are the Fock states {|n ; n ∈ Z}. The cavity– emitter coupling strength g is inversely proportional to the square root of the effective −1/2 quantization volume Veff of the cavity mode, g ∝ Veff . In the resonant case, ωeg = ω L , two states |e, n and |g, n + 1 are degenerate when g = 0. The cavity–emitter interaction g = 0 lifts this degeneracy, leading to an energy-level splitting. As a result, the eigenstates of the quantum system are the dressed states, i.e., the emitter’s electronic states are dressed by the clouds of photons. Tuning the detuning Δ = ω L − ωeg around the resonant interaction results in an avoided crossing between the energy levels associated with |e, n and |g, n + 1. In √ particular, the anticrossing gap between two dressed states |± = (|e, 0 + |g, 1)/ 2 is equal to twice of the coupling strength g. The inevitable decay sources, e.g., the spontaneous emission of the emitters (rate γ) and the cavity loss (rate κ), randomly interrupt the coherent cavity–emitter interaction, which erases the quantum properties of the physical system after a timescale t > min(γ −1 , κ−1 ). The size of the quantum system, measured by the number of emitters and photons participating in the interface, must be large enough that the system can exhibit the quantum behavior before the decay sources take effect. For example, the quantum system is initialized with a two-level emitter in its excited |e

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state and n resonant photons inside the cavity. The emitter can devote its √ internal energy ωeg either to the cavity field via the stimulated emission at a rate 2g n + 1 or to the reservoir via the √ spontaneous emission at a rate γ. The occurrence of the former is favored by 2g n + 1 > γ, resulting in a characteristic photon number Np =

γ2 . 4g 2

(7.1)

For N p  1, the size of the quantum system reaches the macroscopic regime, challenging the feasibility of a macroscopic Fock state of the photons. In contrast, for N p  1 a resonant intracavity field with its energy even less than one photon energy ω L can efficiently induce the stimulated emission of the |e-state emitter. Usually, N p is called the saturation photon number. In addition, for a quantum system composed of an ensemble of |e-state emitters interacting with an optical cavity, two processes affect the intracavity-photon number: the cavity decay √ at a rate κ and the photon generation via the stimulated emission at a rate 2g Ne . The former causes photon loss while the latter raises the number of coherent photons. Due to the spontaneous emission, only a portion of emitters contributes to the stimulated emission. Thus, the |e-state population of the emitters must be larger than a critical value Ne =

κγ , 4g 2

(7.2)

so as to maintain the gain-loss balance of the intracavity photons [13]. For Ne  1, a large number of emitters, i.e., the macroscopic medium, are required to compensate for the cavity loss and the emitter’s spontaneous emission. In contrast, when Ne  1 the stimulated emission based on even one quantum emitter can well exceed the intracavity-photon loss. In the weak-coupling regime (g  γ, κ), exploring the quantum properties of the cavity–emitter interaction requires a large-size system because of N p  1 and Ne  1. In comparison, in the strong-coupling regime (g  γ, κ) with N p  1 and Ne  1, the required system size is significantly reduced down to a scale of one emitter interacting with one photon. One of the intrinsic signatures of a stronglycoupled cavity–emitter system is the vacuum Rabi splitting [14]. As illustrated in Fig. 7.1a, a single two-level emitter is situated inside an optical cavity. The emitter’s transition is resonantly coupled to the cavity mode. A weak probe light passes through the cavity. The transmission spectrum is measured via scanning the frequency of the probe beam. The strong cavity–emitter interaction induces the spectral splitting [see Fig. 7.1b], compared to the transmission spectrum of an empty cavity. The inter-peak distance is twice the cavity–emitter coupling strength. Two spectral peaks are further pushed away from each other as the number of emitters is increased. As illustrated by (6.64) and (6.106), enhancing the coupling strength g between one emitter and one photon may be potentially implemented from two aspects: one way is to choose a large transition dipole moment |deg | of the emitter but with a low

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Fig. 7.1 Schematic diagram of the cavity QED. a One quantum emitter resonantly couples to the cavity mode with a strength g. The decay rate of the emitter’s transition is γ. The loss rates of photons through two cavity mirrors are κin and κout , respectively. A weak probe beam passes the cavity and its transmission is recorded by a photodetector (PD). b Transmission spectrum with κ = κin + κout = 2γ, g = 3γ and the number of emitters inside the cavity Ne = 1, 2, 3. In comparison, the transmission spectrum of the empty cavity (without the emitter) is also presented

spontaneous emission rate γ; and the other way is to suppress the cavity-mode volume Veff while still maintaining a high Q factor. Unfortunately, the former approach is usually infeasible because γ also scales as |deg |2 [see (6.126)]. That is to say, a large dipole moment |deg | does not only enhance g but also increases γ. The latter method can be accomplished through the exquisite design on the fundamental properties of the optical cavity. The single-emitter cooperativity C=

2g 2 1 , = 2Ne κγ

(7.3)

is commonly employed to measure the cavity–emitter interaction. Increasing the cavity Q factor raises C linearly while C is inversely proportional to the mode volume

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Veff . The strong (weak) interaction between one emitter and one photon leads to a value of C much greater (lower) than unity. Now, we come to the question: how to suppress the effective cavity-mode volume Veff ? For a common Fabry–Pérot interferometer, the light power of its lowest order transverse mode (i.e., TEM00 ) follows a Gaussian distribution around the cavity’s central axis. The light power within the beam waist w0 takes over 86% of the total value. The emitter approximately interacts with a homogeneous light field when it is located within the waist w0 of the Gaussian beam. Thus, it is naturally to choose the cavity quantization volume as Veff = πw02 l,

(7.4)

with the cavity length l. However, it is not such straightforward to derive Veff for an irregular cavity structure, for example, photonic microcavities, whose cavity modes have a strongly inhomogeneous distribution. When the emitter size is much smaller than the cube of the mode wavelength λ3L (more precisely Veff ), the emitter may be viewed as a point-like dipole located at r0 , which couples to the local electric field of a cavity mode E(r0 ). The effective mode volume Veff may be evaluated by averaging the whole energy of the cavity mode based on this local light intensity,  Veff =

 |E(r)|2 dr |E(r0 )|2 .

(7.5)

Since the cavity-mode energy is fixed, i.e., the energy of one photon ω L , placing the emitter at the maximum of |E(r)|, i.e., |E(r0 )| = max(|E(r)|), leads to the minimal mode volume and further the maximal cavity–emitter coupling strength.

7.3 Purcell Effects The spontaneous emission of an emitter depends strongly on the environment it resides in. As illustrated by (6.123), the value γ is computed based on the density of electromagnetic modes in free space ρ(ω). The emitter’s environment may be tailored by, for instance, using a high-Q cavity. The resulting density of electromagnetic modes is in a Lorentzian profile ρcav (ω) =

κ2 /4 2 . πκ (ω − ω L )2 + κ2 /4

(7.6)

Thus, the spontaneous emission rate of an emitter located inside a cavity is given by γcav = γ FP

κ2 /4 , (ωeg − ω L )2 + κ2 /4

(7.7)

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Table 7.1 Selected Purcell factors FP for the Fabry–Pérot, microtoroidal, microspherical, micropillar, and photonic crystal cavities and the nanoparticle-on-mirror structure interacting with different quantum emitters λeg (nm) Veff /(λeg /n)3 2g/(2π) Q FP Ref (GHz) Fabry–Pérot cavity & atom, ion 852 3.2 × 106 6.4 × 10−3 852.4 2.9 × 104 68 × 10−3 Micropillar cavity & quantum dot 937 15 33.9 Photonic crystal cavity & quantum dot 1182 1 41.2 942.5 151.7 36.8 Microsphere & atom, nitrogen-vacancy center 852 4.9 × 104 0.048 634.2 7.4 × 104 0.11 Microdisk & quantum dot 744 6 96.7 Nanoparticle-on-mirror structure & molecule 665 3.5 × 10−7 9.2 × 104 702 6.2 × 10−6 1.2 × 104 −6 730 1.6 × 10 2.9 × 104 Bowtie structure & quantum dot 677.5 1.3 2.9 × 104

2.0 × 108 4.3 × 107

4.7 111

[14] [15]

7.4 × 103

36

[4]

6.0 × 103 1.3 × 104

455 6.6

[5] [16]

5 × 107 1 × 107

82 9.7

[17] [18]

8.3 × 103

106

[19]

15.9 7.0 7.5

3.5 × 106 8.5 × 104 3.7 × 105

[20] [21] [22]

7.3

0.44

[23]

where the parameter1 FP =

3Q 3 (λ /Veff ), 4π 2 eg

(7.8)

is the so-called Purcell factor [24]. In a more general expression of FP , λ3eg should be replaced by (λeg /n) with the refractive index n of the circumstance medium. When the spontaneous emission dominates the decay of the quantum emitter, one finds the relation between FP and the cooperativity FP = 2C.

(7.9)

In Table 7.1, we list several selected Purcell factors FP for different types of optical cavities that are commonly applied in experimental QED as well as the nanoparticleon-mirror (NPoM) structure. It is seen that the NPoM structure possesses the features of extremely small mode volume, strong electromagnetic-field-emitter coupling, and very high Purcell factor compared to other cavity-QED schemes. Nonetheless, as we 1 In

some literature, a factor 1/3 is introduced to account for a randomly oriented dipole moment.

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351

have pointed out in Sect. 2.3.4.3, treating the NPoM structure as an optical cavity is problematic due to its very low Q factor.

7.4 Molecular Emitters In Sect. 6, we have introduced the mathematical framework of the light–matter interaction developed in quantum optics. We have not specified the species of quantum emitters, which may be atoms, ions, and molecules. Here we mainly focus on the molecular emitters. In comparison to the atoms and ions, the energy-level structure of molecules is more complex due to their intrinsic mechanical (e.g., vibration and rotation) and electronic degrees of freedom. In addition, the substantial degree of decoherence strongly broadens the absorption and emission spectra of molecules. In this section, we aim to give the readers a general picture of the molecular structure.

7.4.1 Dipole Moment The physical and chemical properties of a zero-net-charge molecule are primarily determined by its electric dipole moment, which is normally represented by the quantum mechanical operator d=

 j

q j r j with

 j

q j = 0.

(7.10)

The sum in the above definition extends over all nuclei and electrons and the position vectors r j are referred to an arbitrary origin. We apply an extra light field E to interact with the molecule. The corresponding interaction operator is given by the scalar product of the electric dipole and the electric field V = −d · E.

(7.11)

For a given pair of molecular states Ψi (initial) and Ψ f (final), this light–molecule interaction leads to a transition from Ψi to Ψ f with an amplitude   V f i = Ψ f  V |Ψi  =



Ψ f∗ V Ψi dτ .

(7.12)

 Here dτ denotes an integral over all coordinates. Generally, the size of a molecule is much shorter than the light wavelength, for which the magnitude of E is approximately constant over the molecule. Then, we obtain

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Fig. 7.2 Structures of simple molecules. The carbon dioxide and benzene molecules do not own permanent electric dipole moment. The nonzero dipole moments d0 for water and ammonia molecules are 1.85 D and 1.47 D. The directions of d0 are marked by the arrows

V f i = −d f i · E with d f i

 = Ψ f  d |Ψi  = 



Ψ f∗ dΨi dτ .

(7.13)

We first consider the case of Ψi = Ψ f , i.e., the molecular state does not change after the dipole transition. Indeed, dii corresponds to the expectation value of the electric dipole moment for the molecule in a specific Ψi state. For the molecule in different electronic states, |dii | may vary by a few tens of percent. In contrast, for a given electronic state, the variation of |dii | with respect to different vibrational states is only about 1 percent and the change of the rotational states hardly affects the value of |dii |. Another important source of the variation of dii lies in the isotope effect. Thus, the permanent dipole moment of a molecule d0 may be defined based on the molecular ground electronic and vibrational state Ψ0 for a specific isotopic specie d0 = Ψ0 | d |Ψ0  .

(7.14)

Although the net charge of a molecule is zero, the positive and negative charges in most molecules do not completely overlap, resulting in d0 = 0 (see Fig. 7.2). The molecules with nonzero permanent dipole moments are called polar molecules. For Ψ f = Ψi , d f i denotes the transition dipole moment. The emission (Ψ f is higher than Ψi in energy) and absorption (Ψ f is lower than Ψi in energy) transitions have the same probability that is proportional to |d f i |2 . The zero value d f i = 0 indicates that the Ψi − Ψ f transition is forbidden under the electric dipole interaction. Nonetheless, it does not mean that the transition between Ψi and Ψ f never occurs because the Ψi − Ψ f transition might be allowed under the higher order electric and magnetic multipole (e.g., quadrupole and octupole) moments.

7.4.2 Jablonski Diagram The molecular spectroscopy can be generally illustrated by the Jablonski energy diagram (see Fig. 7.3). The Born–Oppenheimer (adiabatic) approximation, where the nuclei are assumed to be frozen due to the huge difference between the masses

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353

Fig. 7.3 Jablonski energy diagram. The electronic states are arranged vertically by energy and grouped horizontally by the spin multiplicity. The radiative (nonradiative) processes are marked by the straight (wavy) arrows

of nuclei and electrons in a molecule, has been widely applied in the analysis of the molecular energy structure in quantum chemistry. Under this approximation, the molecular energy is expressed as a sum of three terms, E tot = E ele + E vib + E rot .

(7.15)

The dominant part E ele consists of the kinetic energy of electrons and inter-electron, inter-nucleus, and electron–nucleus Coulomb interactions. The electrons move with various orbital angular momenta characterized by different adiabatic potentials. The molecular transition between two electronic states typically occurs in the visible and ultraviolet regime. Then the kinetic energy of the nucleus motion is considered, which leads to a set of quantized vibrational states (denoted by the quantum number v) superimposed on the orbital movement of electrons. The origin of the vibrational structure may be interpreted based on the following classical description: Due to the relatively heavy masses of nuclei, the movement of nuclei is much slower than that of electrons. When

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Fig. 7.4 Vibronic transition in classical (a) and quantum mechanical (b) pictures. For simplicity, the Born–Oppenheimer potentials are assumed to be harmonic

an electronic transition happens, the electronic configuration changes instantly while the nuclei are still in their initial stationary positions. The new distribution of the electrons imposes a force on the nuclei, resulting in an oscillatory movement of the nuclei around their original positions [see Fig. 7.4(a)]. In the quantum mechanical picture, the oscillation of the nuclei is quantized [see Fig. 7.4b]. The vibrational energy E vib is much weaker than E ele . Typically, the energy separation between two adjacent vibrational states is in the infrared regime (wavelength of 1 µm ∼ 0.1 mm). Finally, the energy of the molecular translation and rotation E rot ( E vib ) is added into E tot . As a result, each vibrational energy level is further subdivided into a ladder of rotational states with an adjacent energy-level spacing in the microwave regime (wavelength of 1 mm ∼ 1 dm). As shown in Fig. 7.3, the electronic states of a molecule can be separated into two groups: singlet (symbol: S) states with all paired electron spins and triplet (symbol: T) states with a set of unpaired electron spins. Various coherent and incoherent processes of the molecular transitions occur among different states. Some are accompanied by absorbing and emitting the radiation while the others transfer the heat and thermal energy, affecting the external motion of molecules. 1. Absorption. The molecules in the low-lying levels (e.g., S0 ) acquire the energy from the light field and then transit up to the excited levels (e.g., S1 ). The typical time scale of such processes is of the order of 10−15 s. 2. Fluorescence. The excited molecules may spontaneously return back to the lower levels through releasing the energy in the form of photons. This radiation process occurs within a time scale 10−9 ∼ 10−6 s. 3. Phosphorescence. This process denotes the spontaneous emission transition between singlet and triplet states (e.g., S0 and T1 ). Due to the spin-forbidden nature of the relevant transition, such a molecular luminescence is a long-lived light emission process (10−3 ∼ 103 s).

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4. Intersystem crossing. When the vibrational levels of one singlet and one triplet state (e.g., S1 and T1 ) overlap, the spin–orbit coupling enables the molecules non-radiatively transiting between two electronic states with different multiplicity (10−10 ∼ 10−6 s). 5. Internal conversion. The radiationless transitions happen between two electronic states with the same multiplicity, i.e., singlet-to-singlet or triplet-to-triplet states. It is also known as a radiationless de-excitation process (10−14 ∼ 10−10 s). 6. Vibrational relaxation. For a given electronic state, the energy deposited in the electrons may be given away as the kinetic energy of molecules via the transitions from the higher vibrational states to the lower ones. Those rapid transitions (10−12 ∼ 10−10 s) are caused by the thermal collisions among solvent molecules, leading to a slight increase in the temperature of the sample. 7. Quenching. It refers to any processes, such as the excited-state reactions, molecular rearrangements, energy transfer, complex formation, and collision, which weaken the fluorescence intensity of the sample. The typical time scale of the quenching processes is about 10−7 ∼ 10−5 s.

7.4.3 Franck–Condon Principle The molecule can transit between different electronic states in a radiative or nonradiative way. Generally, the electronic-state transitions are accompanied by vibrational transitions. The relevant processes are referred to as the vibronic transitions. For a given initial state Sn,v , the unrestrictive selection rule on the vibrational transitions allows all values of v of a final state Sn ,v , and the vibronic transitions Sn,v → Sn ,v form a progression associated with a pair of n and n . In a progression, the transition intensities of different Sn,v → Sn ,v lines are not equal. They present a distribution along the progression. Now we meet a question, how to analyze which vibronic transition is strongest in intensity and mostly happens. Answering this question relies on the so-called Franck–Condon principle [25–27]. We consider the vibronic transitions between two Born–Oppenheimer potentials as shown in Fig. 7.4b, where the equilibrium density distributions of nuclei on different vibrational energy levels are also presented. A molecule is initially prepared in the lowest vibrational state of the lower electronic state. When an electronic transition occurs, the nuclei cannot make a response in time. We draw a vertical line starting from the location of the maximum of the nuclei density distribution in the lower electronic state. This vertical line cuts through different vibrational states of the upper electronic state. This vertical transition has the most probability to be terminated at a vibrational state with a maximum local density of the nuclei. More precisely speaking, the final vibrational state should have a wavefunction that maximally overlaps the wavefunction of the initial vibrational state. For other vibrational states, the corresponding transition intensities are relatively weak.

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Fig. 7.5 Absorption and emission spectra of molecules. a Schematic diagram of the absorption– emission cycle. The molecules are excited from S0,v=0 to S1,v and then transit back to S0,v=0 via the radiative and nonradiative relaxation, leading to the quasi-continuous absorption and emission spectra. b Absorption (dash) and emission (solid) spectra of the fluorescent dye PM546 in mixture with a volume percentage of 10% glycerol. The inset displays the corresponding molecular structure. The transition dipole moment is along the long axis of the molecule

7.4.4 Absorption and Emission Spectroscopy Figure 7.5a illustrates the general schematics of the molecular absorption–emission cycle. In the absorption process, the molecules in the ground state S0,v=0 are excited to a higher energy level (for example, S1,v ) by absorbing the external light quanta, Absorption: S0,v=0 + ω → S1,v .

(7.16)

According to the Franck–Condon principle, in a molecular entity the electronic transition cannot occur without changing the nuclear vibrational dynamics. The maximum of the absorption spectrum may not be positioned at the resonant transition between two electronic states with the same vibrational quantum number, S0,v=0 → S1,v =0 . The instantaneous change of the electronic state of a molecule inevitably affects the initially stationary nuclei. The nuclei must reorganize their positions so as to match the new electronic configuration. As the motion of nuclei approaches a new equilibrium, the excess kinetic energy of nuclei is relaxed via the >1012 collisions per second as well as the electrostatic interactions with the surrounding solvent molecules. Thus, each vibronic-transition line is strongly broadened, resulting in a quasi-continuous absorption spectrum with a typical width of tens of nm (or tens of THz) as shown in Fig. 7.5b. This is very unlike the atomic spectroscopy, where the typical spectral width is less than tens of fm (or less than tens of MHz). The excited molecules can transit back to the lower electronic states through the spontaneous emission of the fluorescence photons, Emission: S1,v → S0,v + ω .

(7.17)

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However, as displayed in Fig. 7.5a, b, the band-maximum location of the emission spectrum is higher (lower) than that of the absorption spectrum in units of wavelength (frequency), i.e., the red shift. This wavelength/frequency difference is caused by the loss of the vibrational excitation energy and is usually called the Stokes shift. According to the Kasha’s rule [28], the molecules in an excited electronic state can rapidly relax to its lowest vibrational level within 10−14 ∼ 10−12 s. Consequently, the photon emission is always expected to occur from the lowest lying electronically excited state S1,v =0 . Besides the spontaneous emission, the nonradiative mechanisms, e.g., intersystem crossing and internal conversion, may also induce the molecules in S1,v =0 decaying back to S0,v . The quantum yield Φ, defined as Φ=

number of photons emitted , number of photons absorbed

(7.18)

is normally used to measure the efficiency of S1,v =0 being deactivated by the fluorescence rather than by the nonradiative processes. In practice, Φ is usually computed by γf Φ= , (7.19) γ f + γnr where γ f is the radiative (fluorescence) rate and γnr denotes the nonradiative rate constant. As one can see, the value of Φ ranges between 0 and 1. The lifetime of the excited state S1,v =0 is given by τ = (γ f + γnr )−1 , shorter than its fluorescence lifetime τ f = γ −1 f . It is worth noting that although τ strongly depends on the environment of the molecules, such as concentration (refractive index) and temperature of solvent, τ f reflects the natural (intrinsic) feature of the molecules. According to the Frack–Condon picture, the absorption and emission spectra are expected to be the mirror images of each other [see Fig. 7.2b]. However, in some real molecular systems, this mirror image rule may be broken down because of the different geometric arrangements of nuclei in the ground and excited states or in the excited-state reactions. Additionally, some molecular absorption/emission spectra may exhibit several narrow bands (corresponding to different vibrational progressions) and they are usually referred to as the structured spectra. Otherwise, the spectra are unstructured.

7.4.5 Beer–Lambert Law The transition probability of an molecule from the initial electronic state |1 to the final electronic state |2 is determined by the square of the magnitude of the dipole moment |d21 |2 , whose quantum mechanical derivation is complex and is not our focus. Alternatively, the transition probability can be obtained experimentally from the measurement of the light absorption. For a light beam at a frequency

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7 Molecular Cavity QED

ν = ω/2π passing through a homogeneous solution sample along the x-direction, the transmitted intensity of the light Io (ν) is smaller than its incident intensity Ii (ν) due to the inevitable absorption of the sample. The dimensionless absorbance of the sample is defined as Ii (ν) . (7.20) A(ν) = log10 Io (ν) If A(ν) is close to zero at some frequency ν (wavelength λ = c/ν), it means that the sample hardly absorbs the light at this particular frequency (wavelength). The attenuation of the light is related to the optical properties of the sample via the Beer–Lambert law A(ν) = ε(ν)Cl, (7.21) with the molar absorptivity/extinction coefficient ε(ν), the concentration of solution C, and the length of the sample l. In addition, we assume the light intensity at x is I (ν, x) and d I (ν, x) represents the intensity attenuation after the light traveling for a small distance d x. One may obtain the following differential equation: d I (ν, x) = −α(ν)C I (ν, x), dx

(7.22)

where the minus symbol denotes the light attenuation and the absorption coefficient α(ν) is given by (7.23) α(ν) = σ(ν)N A , with the cross-sectional area σ(ν) of the absorbers in solution and the Avogadro’s number N A = 6.022 × 1020 . Setting I (ν, x = 0) = Ii (ν) and I (ν, x = l) = Io (ν), ε(ν) is linked to α(ν) via α(ν) α(ν) ≈ . (7.24) ε(ν) = ln 10 2.303 In practice, the dimensions of ε(ν), C, l and σ(ν) are commonly chosen to be L · mol−1 · cm−1 , mol · L−1 , cm and cm2 , respectively, in the chemical analysis. Here L is the symbol of liter and mol stands for mole. To distinguish ε(ν) in units ˜ to represent the absorptivity coefficient in units of of m3 · mol−1 · m−1 , we use ε(ν) L · mol−1 · cm−1 . As we have mentioned, the transition between two electronic states is always accompanied by a large number of vibrational and rotational transitions, giving rise to a wide spectral broadening. Integrating ε(ν) ˜ over the whole spectrum leads to the so-called dimensionless oscillator strength  4m e cε0 ln 10 ∞ ε(ν)dν, ˜ N A e2 n 10 0  1.44 × 10−19 ∞ = ε(ν)dν, ˜ n 0

f1→2 =

(7.25)

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with the refractive index n of the sample. The oscillator strength ranges from 0 to 1 and a strong transition has an oscillator strength close to 1. In some cases, the molar absorptivity ε˜ is measured as a function of the wavenumber ν˜ = ν/c = λ−1 . The oscillator strength f1→2 is then re-expressed as f1→2

4.32 × 10−9 = n



∞

ε( ˜ ν)d ˜ ν, ˜

(7.26)

0

with ν˜ in units of cm−1 . In quantum mechanics, f1→2 is related to the dipole moment d21 via 8π 2 m e ν21 f1→2 = |d21 |2 , (7.27) 3 he2 with the frequency ν21 of the |1 − |2 transition. Hence, the transition dipole moment |d21 | is given by (in units of Debye)  |d21 | = 0.2525 × 109 ·

f1→2 , ν21

(7.28)

which may be derived from the measurement of ε(ν). ˜ For the visible wavelength λ = 500 nm, the maximal oscillator strength f1→2 = 1 corresponds to a transition dipole moment |d21 | = 10.3 D = 3.4 × 10−29 C · m.

7.4.6 Einstein Coefficients We now consider the relation between the absorption and emission spectra and the Einstein coefficients. As we will see below, the Einstein coefficient for the spontaneous emission A2→1 between two relevant molecular states |1 (lower) and |2 (upper) may be derived from the molar absorptivity coefficient ε(ν) and the fluorescence spectrum F(ν). We assume that all molecules in solution are in the |1 state. A light beam at the frequency ν travels in the sample along the x-direction. According to the Beer–Lambert law, the light intensity at x is given by I (ν, x) = I (ν, 0)10−ε(ν)C x , = I (ν, 0)e−(ln 10)ε(ν)C x .

(7.29)

The intensity attenuation of the light beam traveling with a short distance Δx at x, i.e., ΔI (ν, x) = I (ν, x + Δx) − I (ν, x), is expressed as ΔI (ν, x) = −(ln 10)ε(ν)I (ν, x)CΔx.

(7.30)

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The concentration of the sample is equal to C = n d /N A with the number density of molecules n d . We assume the cross-sectional area of the light beam is s. The number of molecules within the volume V = sΔx is then calculated to be N = n d V . Due to the absorption, the light energy is reduced by −ΔI (ν, x)V within the time duration Δt = nΔx/c. Here n is the sample’s refractive index. The reduced light energy is converted into the internal energy of a portion of molecules, exciting them into the upper state |2. The corresponding excited-state population reaches ΔN (ν, x) = −ΔI (ν, x)V /(hν) with the single-photon energy hν. Consequently, the excitation rate of the molecules is given by (ln 10)c ε(ν) 1 ΔN (ν) = I (ν, x). N Δt hn N A ν

(7.31)

Integrating over the whole absorption spectrum, one obtains  0

∞

1 ΔN (ν) dν ≈ B1→2 I (ν21 , x), N Δt

(7.32)

where we have replaced I (ν, x) by its value at ν21 because of the fact that the light with ν around ν21 mainly contributes the absorption. The Einstein coefficient B1→2 for the photon absorption (in units of m3 · J−1 · s−2 ) is then derived as B1→2 =

(ln 10)c hn N A



∞ 0

ε(ν) dν. ν

(7.33)

Using ε(ν) ˜ in units of L · mol−1 · cm−1 , B1→2 may be rewritten as B1→2 =

ln 10 c 10 hn N A



∞ 0

ε(ν) ˜ dν. ν

(7.34)

On the other hand, we recall the relation between the Einstein coefficient B(0) 1→2 derived based on the Planck distribution and the absorption oscillator strength f1→2 . B(0) 1→2 =

e2 f1→2 . 4ε0 m e hν21

(7.35)

Using B1→2 = f B2 (n)B(0) 1→2 with the local-field factor f B (n), we arrive at the transition dipole moment

 1/2  ∞ 1 ε(ν) ln 10 · 3hc 1/2 dν |d21 | = 4πε0 · . 8π 3 N A ν n f B2 (n) 0

(7.36)

The f B (n) comes from the local-field correction. The above equation may be written in a more practical form

7.4 Molecular Emitters

361

|d21 | = 9.58 × 10

−2



1 · 2 n f B (n)



∞

0

ε(ν) ˜ dν ν

1/2 ,

(7.37)

where  ∞ ε(ν)|d21 | is in units of Debye. It∞is worth noting that (7.36) implies f1→2 ∝ dν while f1→2 depends on 0 ε(ν)dν in (7.26). This is because we have 0 ν assumed that the absorption band is sharp in the derivation of (7.26). However, such an assumption is only valid for the atomic transitions, rather than the molecular transitions. −1 Further, the Einstein coefficient for the photon emission A(0) 2→1 (in units of s ) derived based on the Planck distribution is expressed as A(0) 2→1 (ν) =

2πν 2 e2 f2→1 , ε0 m e (c/n)3

(7.38)

with the emission oscillator strength f2→1 =

8π 2 m e ν21 |d12 |2 , 3 he2

(7.39)

and the dipole transition matrix element d12 = 1| d |2. We use the function F(ν) to denote the fluorescence spectrum of the molecules and define a characteristic frequency for the fluorescence emission

ν

−3

∞ em =

ν −3 F(ν)dν ∞ . 0 F(ν)dν

0

(7.40)

Then, A(0) 2→1 is simplified as A(0) 2→1 =

64π 4 |d12 |2 1 . 3 4πε0 3h(c/n) ν −3 em

(7.41)

The Einstein coefficient A2→1 for the sample is related to A(0) 2→1 via A2→1 = f A2 (n)A(0) 2→1 .

(7.42)

with the local-field factor f A (n). Consequently, the transition dipole moment |d12 | is derived as

 1/2 1 3h 1/2 −3 · ν˜ em · A2→1 , |d12 | = 4πε0 · 64π 4 n 3 f A2 (n)

(7.43)

with the wavenumber ν˜ = ν/c. Choosing the dimension of ν˜ in cm−1 leads to the following expression:

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7 Molecular Cavity QED

Table 7.2 Transition dipole moment of organic dyes PM546 and rhodamine 123 measured in experiment [30] Molecules Absorption peak Emission peak Oscillator Dipole moment (nm) (nm) strength (D) PM546 rhodamine 123

493.5 507

505 532

 |d12 | = 1.786 × 103 ·

0.453 –

7.1 8.1

1 · ν˜ −3 em · A2→1 3 n f A2 (n)

1/2 .

(7.44)

Due to |d21 | = |d12 | and f (n) ≡ f A (n) = f B (n), A2→1 is related to the absorption and emission spectra via the Strickler–Berg equation [29] ∞

A2→1 = 2.881 × 10−9 · n 2 ·  ∞0 0

F(ν)d ˜ ν˜

ν˜ −3 F(ν)d ˜ ν˜

 · 0

∞

ε( ˜ ν) ˜ d ν. ˜ ν˜

(7.45)

We see that the radiative rate constant A2→1 is proportional to the square of the refractive index n. The above equation provides a way to derive the Einstein A coefficient from the measurement of the absorption and emission spectra of the molecules, which further enables one to evaluate the dipole moment |d12 | from (7.44). The transition dipole moments of a few selected molecules are listed in Table 7.2.

7.5 Molecular Cavity QED in Experiment So far, we have briefly introduced the cavity-QED framework by using the language of quantum optics and also the quantum mechanical treatment of the molecular emitters. In this section, we introduce several relevant experiments, whose main goals are to build up the molecular cavity-QED platforms that reach the strong-coupling regime.

7.5.1 Single Molecules Interacting with Plasmonic Oscillations In [20], the NPoM structure is employed, where a gold mirror (film) is placed underneath a gold nanosphere (radius r ) with a gap thickness l as shown in Fig. 7.6a. The real nanosphere and its virtual image form a dimer-like nanostructure and the electromagnetic field within the gap region is strongly enhanced [31]. Varying r and l changes the LSPR wavelength λ p and the field enhancement factor. The methylene-

7.5 Molecular Cavity QED in Experiment

363

Fig. 7.6 Strong coupling between molecules and plasmon modes. a NPoM structure, where a gold nanoparticle (NP) is placed above a gold film. b Scattering spectra of the emitters with different orientations. c Energy-level structure as a function of the detuning Δ = ω p − ω0 . d Resonant Autler–Townes splitting versus the number of emitters Ne . Figures b–d are reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature, [20], © 2016

blue molecules with the transition dipole moment |d| = 3.8 D at λ0 = 665 nm are situated in the gap region, playing the role of quantum emitters. For r = 20 nm and l = 0.9 nm, the plasmonic oscillation of the surface electrons occurs at λ p ≈ λ0 and the local-field enhancement can reach as high as 103 . A p-polarized light field illuminates the sample at an angle of incidence θinc = 55◦ and the scattered light within a cone of half-angle θcol = 53◦ is collected, depending on the numerical aperture of the objective. Necessary measures should be taken to align the transition dipole moment d of the molecules in a certain direction and also prevent the molecular aggregation. Figure 7.6b plots the experimentally measured dark-field scattering spectrum for the quantum emitters in different orientations. When d is perpendicular to the polarization of the local electric field, the molecules hardly interact with the plasmon mode. Consequently, only one peak, which actually corresponds to the LSPR in the absence of molecules, is presented at λ0 in the scattering spectrum. The quality factor

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7 Molecular Cavity QED

of the LSPR is derived to be ∼13 with a spectral width γ p = 2π × 33 THz at the room temperature. In contrast, the single scattering peak is split into two, whose central wavelengths are located at λ− = 756 nm and λ+ = 595 nm, respectively, when the dipole moment d is in the vertical direction to the gold mirror plane, indicating the strong molecule-local-field coupling. The frequency distance between λ± and λ0 approximates (ω+ − ω0 ) ≈ (ω0 − ω− ) ≈ 2π × 53 THz with ω± = 2πc/λ± and ω0 = 2πc/λ0 . Such an interaction-induced spectral splitting may be interpreted via the Autler–Townes doublet, ω± =

1 1 (ω p + ω0 ) ± N e Ω 2 + Δ2 , 2 2

(7.46)

with the plasmonic resonance frequency ω p = 2πc/λ p , the light-emitter Rabi frequency Ω and the detuning Δ = ω p − ω0 . Here we have generalized the Autler– Townes splitting to the interaction between the local electromagnetic field and Ne identical emitters with at most one excitation. Figure 7.6c displays the spectrum of the energy levels ω± versus the light-emitter detuning Δ. It is seen that an anticrossing occurs between ω± branches around ω0 . At the resonance, one obtains √ Ne Ω = ω+ − ω+ = 2π × 106 THz, larger than γ p , with Ne = 10. Such a fast Rabi flopping cannot be measured directly. The dependence of the resonant splitting (ω+ − ω+ ) on the mean number of emitters Ne is verified in Fig. 7.6d. Reducing Ne narrows the splitting. The singlemolecule emission can be testified by performing the Hanbury Brown–Twiss photon correlation, which has been done in the further work [22]. Besides enhancing the local electromagnetic field, the NPoM structure strongly affects the density distribution of electromagnetic modes around the molecular dipole, speeding up the decay of the molecular electronic dipole transition by the Purcell factor FP . Such an acceleration of the spontaneous decay has also been confirmed in [22] and the Purcell factor is estimated to exceed 103 . Nonetheless, the limitation of the relatively long instrumental response time (0.1 ∼ 1 ns) hinders the direct measurement of FP . Indeed, the singlemolecule fluorescence enhancement has been demonstrated in various plasmonic structures, such as bowtie nanoantenna [32] and the film-coupled nanocube [33]. It should be noted that all experimental observations listed above may be explained within the semiclassical framework of the light–emitter interaction, where the NPoM structure is not treated as an optical cavity. As pointed out in [34], the concept of a closed (or quasi-closed) cavity is problematic to the plasmonic structures because of the very low quality factor. Actually, the interaction-induced spectral splitting can be also observed when the molecules interact with single-metal nanoparticles, like nanorod [35] and nanoprism [36]. Obviously, single-metal nanoparticles cannot form an optical cavity due to their negative relative permittivity.

7.5 Molecular Cavity QED in Experiment

365

7.5.2 Strong Coupling Between Molecules and Microcavities The strong coupling between an optical microcavity and a molecular electric dipole transition has been performed in, for example, [37, 38]. As illustrated in Fig. 7.7a, the microcavity is composed of a pair of Ge/ZnS distributed Bragg reflectors (DBRs) with a separation of 26 µm. The cavity-mode linewidth is about 2π × 0.5 THz (2.16 meV). A thin film of poly(methyl methacrylate) (PMMA) is sandwiched between two dielectric mirrors. An infrared laser beam passes through the cavity and its transmission spectrum is measured. As shown in Fig. 7.7b, the PMMA layer has an absorption dip located at the wavelength 5.8 µm (1731 cm−1 , 0.214 eV) with a spectral linewidth of 2π × 0.9 THz (3.85 meV). This spectral dip corresponds to a vibrational transition of the organic molecules. The resonant wavelength of the microcavity is tuned by changing the parallelism between two Bragg mirrors. When the microcavity is nearly resonant to the vibrational transition of the molecules, two distinct dips can be identified in the transmission spectrum, corresponding to the energy-level splitting. Figure 7.7c depicts the dependence of the positions of two spectral dips on the cavity wavelength, i.e., the energy spectrum. Two bands are presented and the minimal interband spacing (∼12 meV, 2.9 THz) occurs at the resonant cavity–molecule interaction, forming the vibrational polaritons. This microcavity-molecule structure may be extended to a microcavity interacting with a hybridized ensemble of molecules. For example, the organic layer consists of two kinds of molecules, PMMA and dimethylformamide (DMF). The transmission spectrum of the hybridized layer is shown in Fig. 7.7d. The DMF has a dipole transition with a wavelength of 6.0 µm (1677 cm−1 , 0.208 eV) that is close to the PMMA’s dipole transition. The wavelength of the cavity mode may be tuned by changing the incident angle of the probing beam relative to the normal direction of the cavity mirrors. Three spectral peaks can be observed in the angle-resolved transmission spectrum [see Fig. 7.7e]. Mapping the transmission spectrum onto the energy spectrum, one finds three energy bands with two anticrossings as shown in Fig. 7.7f. Actually, these two avoided crossings occur at 0.208 and 0.214 eV, respectively, manifesting the strong coupling between the microcavity and both kinds of molecules.

7.5.3 Molecular Microlasers The microcavity-molecule structure can be also used to perform the lasing action. Among various microlaser structures, the vertical cavity surface emitting lasers (VCSELs) are particularly useful for the telecommunications and the integrated photonic networks. As demonstrated in [39], an optical microcavity consists of two distributed Bragg reflectors, which are separated by a distance of half wavelength λ/2 [see Fig. 7.8a]. The intensity reflection coefficient R of either planar mirror exceeds 99.9% within the visible spectral range of 525 ∼ 645 nm. The cavity’s quality factor

366

7 Molecular Cavity QED

Fig. 7.7 Strong coupling between molecules and a microcavity. a Scheme of the microcavity– molecule interaction. The transmission spectrum of the infrared (IR) probe beam is recorded. b Transmission spectrum of a thin film of PMMA. c Anticrossing induced by the strong coupling between the microcavity and PMMA molecules. d Transmission spectrum of the hybridization of PMMA and DMF. e Angle-resolved transmission spectrum of the microcavity-hybridized-layer structure. The corresponding energy spectrum is summarized in f. Figures a, d, e and f are reprinted with permission from [38]. Copyright (2016) American Chemical Society. Figures b and c are reprinted with permission from [37]. Copyright (2017) American Chemical Society

is estimated to be at least Q = 6 × 103 at the resonant wavelength λ = 592 nm. Two thin films of PMMA are coated on the top of two Bragg mirrors and serve as the spacer layers. Either PMMA layer has a thickness of λ/4. A monolayer of the amphiphilic fluorescent dye (Lissamine rhodamine B sulfonyl didodecyl amine, LRSD) is inserted between two PMMA layers and acts as the laser gain medium. Figure 7.8b displays the absorption and emission spectra, which peak at 575 and 588 nm, respectively, of the LRSD. Such a separation between absorption and emission spectra makes the dye molecules a standard four-level lasing system (see Problem 7.1). The relaxation time of the LRSD dye emission in monolayer is measured to be 3.2 ns. The VCSEL device is pumped by a light pulse at 532 nm with a pulse duration of 8 ns. The pump pulse is focused to a spot with a diameter of 250 µm. To enhance the coupling efficiency, the incident angle of the pump beam relative to the normal of the cavity mirrors is adjusted to 45◦ . Figure 7.8c shows the dependence of the light output from the cavity on the pump intensity. It is seen that the output light contains the components having the polarizations same (||) and orthogonal (⊥) to that of the pump pulse. The ||-component clearly exhibits a threshold behavior with a threshold pump intensity E th = 4.4 µJ/cm2 . In contrast, the slope of the ⊥-component retains the same as the pump intensity is increased. This phenomenon has also been observed in other VCSELs [40, 41]. The spectrum of the ||-component above the threshold

7.5 Molecular Cavity QED in Experiment

367

Fig. 7.8 Single-molecule monolayer laser. a Structure of VCSEL. The optical microcavity is formed by two DBRs. An organic thin-film gain layer is sandwiched between two PMMA layers. b Absorption and emission spectra of the LRSD monolayer. c Output intensities of the ||- and ⊥-components as a function of the pump intensity. d Spectrum of the ||-component below (0.5E th ) and above (5E th ) the threshold. The sharp peak’s linewidth is < 0.05 nm. Reprinted with permission from [39]. Copyright (2017) American Chemical Society

presents a sharp spectral peak at the resonant wavelength [see Fig. 7.8d]. Also, the corresponding spectral lineshape manifests an inhomogeneous broadening.

7.5.4 Single Molecule Interacting with a Microcavity In order to achieve the strong coupling between one molecule and an optical cavity, the linewidth of the molecular spectrum must be suppressed. According to the Franck–Condon principle, the ground vibrational level of an electronic excited state |e, v = 0 may couple to both vibrational levels |g, v = 0 and |g, v = 0 in the ground electronic state. The zero-phonon |g, v = 0 − |e, v = 0 line can be narrowed by 105 -fold to the Fourier limit in the cryogenic environment. In contrast, the radiationless relaxation from |g, v = 0 to |g, v = 0 strongly broadens the emis-

368

7 Molecular Cavity QED

Fig. 7.9 Single molecule interacting with a microcavity. a Scheme of transmission measurement. b Transmission spectra in the absence of molecules (Abs), with the existence of one molecule (Exi), and the reference floor when the cavity is far-detuned (∼20 GHz) from the |g, v = 0 − |e, v = 0 transition (Ref). c Fluorescence spectrum of the molecule when the cavity is far detuned from the molecular transition. d The second-order intensity auto-correlation function g (2) (τ ). e Positions of transmission peaks as a function of the cavity-molecule detuning. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Physics, [42], © 2019

sion spectrum of the |g, v = 0 − |e, v = 0 line. A potential way of narrowing the emission spectrum is to modify the branching ratio out of |e, v = 0 in a selective manner that enhances the zero-phonon |g, v = 0 − |e, v = 0 transition line. Despite the strong coupling between molecules and plasmonic nanostructures, the large bandwidth of the plasmon resonances does not allow for selective addressing of the narrow transitions [42]. In comparison, the optical microcavities are the more promising candidates. In [42], an open, tunable, and laterally scannable Fabry–Pérot microcavity was designed particularly for this use. As illustrated in Fig. 7.9a, a curved micromirror with a radius of curvature of 10 µm and nominal reflectivity of 99.996% is fabricated at the end of an optical fiber. This curved micromirror faces against a planar mirror (i.e., DBR), forming an optical cavity with a length of 4.7 µm. A thin anthracene (AC) crystal with a thickness of about 600 nm is placed into the cavity. The dibenzoterrylene (DBT) molecules are embedded in AC. The DBT density is low enough that at most only one organic molecule is situated inside the cavity. The environment is cooled down to the liquid helium temperature T ∼ 4K. Consequently, the homogeneous linewidths of the molecular emitters become such narrow that they no longer overlap. The |g, v = 0 − |e, v = 0 transition, whose wavelength λ lies in the range of 783 ∼ 785 nm, reaches its Fourier limit of 40 MHz. At the low temperature, the cavity mode presents a Voigt profile with a linewidth of κ = 2π × 3.3 GHz, corresponding to a quality factor Q = 1.2 × 105 , and has a volume of 4.4 × λ3 . A light beam at 783 nm produced by a Ti:sapphire laser passes through the Fabry– Pérot cavity for the spectral measurement. As demonstrated in Fig. 7.9b, the cavity transmission drops by 99% when the cavity is resonantly coupled to a single

7.5 Molecular Cavity QED in Experiment

369

molecule. The FWHM of the transmission dip reaches about 604 MHz. This value much exceeds the Fourier limit (40 MHz), resulting from the cavity–molecule interaction. To remove the effect of the cavity on the linewidth of the molecular transition, the cavity is detuned from λ and the red-shifted fluorescence from the molecule is recorded as a function of the excitation frequency. The resulting spectrum reveals a FWHM of γ = 44 MHz [see Fig. 7.9c], which coincides with the value for a bulk DBT:AC system. The intensity auto-correlation g (2) (τ ) confirms that the fluorescent photons originate from the single-molecule emission as shown in Fig. 7.9d. Figure 7.9e displays the dependence of the positions of the transmission peaks on the molecule-cavity detuning. An anticrossing occurs at the resonant molecule–cavity interaction with a gap of 2g = 1.6 GHz. The cooperativity factor of this singlemolecule QED system reaches C = 2g 2 /κγ = 8.8, right at the onset of strong coupling.

7.6 Heisenberg-Limited Quantum Metrology Quantum metrology, also known as quantum sensing, makes use of the features of quantum mechanics, such as discrete energy levels of quantum emitters, superposition principle, and entanglement, to measure a physical quantity or enhance the sensitivity beyond the classical limit [43]. In a quantum sensing process, the detector interacts with the object and generates a response signal that is measured by the readout device. The size of objects may vary from microscopic (e.g., single-quantum emitters and photons) to universe (e.g., the gravitational waves) scale. The detectors can be either classical (e.g., micro- and optical-frequency waves) or quantum (e.g., internal energy levels of emitters and nonclassical properties of light) ones. The response signal produced by the detector must be converted into a physical quantity that is measurable to the readout system. For example, the population of quantum emitters in a certain state is indirectly measured via mapping it onto a photon flux that can be read out by a photodetector. Typically, a quantum sensing process includes three steps [see Fig. 7.10]: Preparation.The object  and detector are initially separated and prepared in the well-defined states ψobj (0) and |ψdet (0), respectively. These two states then evolve freely for a time duration t0 ,     ψobj (t0 ) = e−i Hobj t0 / ψobj (0) , |ψdet (t0 ) = e

−i Hdet t0 /

|ψdet (0) ,

(7.47a) (7.47b)

where Hobj and Hdet are the Hamiltonians of the object and detector, respectively. Detection. The detector interacts with the object for a time duration T . Due to the inevitable dissipative sources, the detector–object interaction V should be strong enough that the detection can be finished well before the dissipation takes effect. Readout. The detector produces a response signal that is measurable to a readout

370

7 Molecular Cavity QED

Fig. 7.10 Quantum sensing protocol

apparatus. The fundamental noises, such as quantum projection noise and shot noise, limit the detection uncertainty. The nature of quantum mechanics impedes to capture the full picture of a physical observable through single measurement. It is essential to repeat the whole detection process for multiple times and then evaluate the observable’s expectation value. For different specific physical systems, the quantum sensing process needs to be adjusted accordingly.

7.6.1 Heisenberg Uncertainty and Squeezed States We consider two observables associated with the operators A and B, respectively. These two operators satisfy the commutation relation [A, B] = iC.

(7.48)

The standard deviations (uncertainties) of measuring A and B are given by ΔA =



A2  − A2 , ΔB =



B 2  − B2 .

(7.49)

The Heisenberg uncertainty principle tells us ΔAΔB ≥

| C| . 2

(7.50)

7.6 Heisenberg-Limited Quantum Metrology

371

The above inequality limits to what extent of precision a pair of physical quantities can be measured simultaneously. When the quantum system is in a state that leads to  | C|  min (ΔA)2 , (ΔB)2 < , (7.51) 2 this state is called the squeezed state. Further, if the following condition ΔAΔB =

| C| , 2

(7.52)

is also satisfied, this state is called an ideal squeezed state.

7.6.1.1

Squeezed Light

As an example, let us consider a quantized single-mode light field E(t) = 2eE0 (X 1 cos ωt + X 2 sin ωt),

(7.53)

with the quadrature operators X 1,2 and their commutation relation X1 =

1 † i i (a + a), X 2 = (a † − a), [X 1 , X 2 ] = . 2 2 2

(7.54)

When the light is in the coherent state |α (i.e., a |α = α |α), it is easy to obtain ΔX 1 = ΔX 2 =

1 . 2

(7.55)

Although ΔX 1 ΔX 2 = 1/4, the coherent state is not squeezed in either ΔX 1 or ΔX 2 direction. In contrast, one has √ ΔX 1 = ΔX 2 =

2n + 1 , 2

(7.56)

for the light field in the Fock state |n. The fluctuations in ΔX 1 and ΔX 2 are enhanced as the photon number n grows. Thus, the Fock state is not a squeezed state either. The situation becomes different for the light state generated from the degenerate parametric process. The relevant Hamiltonian is expressed as H/ = ig ∗ a 2 − ig(a † )2 ,

(7.57)

with the coupling constant g. The system is assumed to be initialized in the coherent state, |ψ(0) = |α. The light state at the time t is given by |ψ(t) = e−i H t/ |α. We define the so-called squeeze operator

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7 Molecular Cavity QED

 S(ξ) = exp

ξ∗ 2 ξ † 2 a − (a ) , 2 2

(7.58)

with ξ = 2gt. It is easy to prove that S † (ξ) = S −1 (ξ) = S(−ξ).

(7.59)

The state |ψ(t) is then rewritten as |α, ξ ≡ |ψ(t) = S(ξ) |α ,

(7.60)

and the expectation value of an operator O reads

O = α, ξ| O |α, ξ = α| S † (ξ)O S(ξ) |α .

(7.61)

Using (6.145), one may find the following unitary transformations S † (ξ)aS(ξ) = a cosh r − a † eiθ sinh r, S † (ξ)a † S(ξ) = a † cosh r − ae−iθ sinh r,

(7.62a) (7.62b)

with ξ = r eiθ . We further introduce the rotated quadrature operators Y1,2  1  † iθ/2 a e + ae−iθ/2 , 2  i  † iθ/2 a e − ae−iθ/2 . Y2 = 2 Y1 =

(7.63a) (7.63b)

Using (7.62), one obtains the standard deviations  e−r ,

Y12  − Y1 2 = 2  r e ΔY2 = Y22  − Y2 2 = . 2 ΔY1 =

(7.64a) (7.64b)

It is seen that ΔY1 ΔY2 = 1/4. For r > 1 (r < 1), ΔY1 (ΔY2 ) is suppressed at the expense of increasing ΔY2 (ΔY1 ). Hence, |α, ξ is called the squeezed coherent state, which is also an ideal squeezed state.

7.6.1.2

Spin Squeezing

As discussed in Sect. 6.3, a two-level quantum emitter interacting with a radiation is equivalent to a spin-1/2 particle in a magnetic field. Let us consider a quantum system composed of N spin-1/2 particles. We make the assumptions that none dissipative sources influence the quantum system and all particles are in the same state. Thus,

7.6 Heisenberg-Limited Quantum Metrology

373

the system’s state is given by the following product state |θ, ϕ =

 j

⊗R j (θ, ϕ) |↓ j ,

(7.65)

with the rotation operator

θ ( j) R j (θ, ϕ) = exp −i n · σ , n = sin ϕ ex − cos ϕ e y , 2

(7.66)

acting on the j-th particle. The operator σ = σx ex + σ y e y + σz ez is the Pauli matrix vector and has the commutators [σx , σ y ] = 2iσz , [σ y , σz ] = 2iσx and [σz , σx ] = 2iσ y . Equation (7.65) can be rewritten in the following form: |θ, ϕ =



θ θ ⊗ sin |↑ j + cos eiϕ |↓ j . j 2 2

(7.67)

We map the |θ, ϕ state into the angular momentum representation. One may define the total spin operator S = Sx ex + S y e y + Sz ez with three components Sx =

1  ( j) 1  ( j) 1  ( j) σx , S y = σ y , Sz = σ . j j j z 2 2 2

(7.68)

The commutation relations between Sx,y,z are [Sx , S y ] = i Sz , [S y , Sz ] = i Sx , [Sz , Sx ] = i S y .

(7.69)

Since all particles are in the same state at any time, the Hilbert space may  be spanned  by the common eigenstates of S2 and Sz , i.e., the so-called Dicke states { N2 , m ; m = − N2 , − N 2−1 , · · · , N2 } with  

  N N N N   + 1  ,m , S  ,m = 2 2 2 2     N N Sz  , m = m  , m . 2 2 2

(7.70a) (7.70b)

It is easy to prove the following expressions:    N  ,−N = ⊗ |↓ j , 2 j 2  ⊗R j (θ, ϕ). exp (−iθn · S) = j

and (7.67) is re-expressed as

(7.71a) (7.71b)

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7 Molecular Cavity QED

  N N |θ, ϕ = exp (−iθn · S)  , − 2 2    N /2 N = e−imϕ dm,−N /2  , m , m=−N /2 2

(7.72)

with the Wigner small d-matrix  dm,−N /2 =





θ N /2−m θ N /2+m N! cos sin . (N /2 − m)!(N /2 + m)! 2 2

(7.73)

The |θ, ϕ state is usually referred to as the coherent atomic state (CSS) [44]. Using (7.67) and (7.68), one may derive the following expectation values: N (N − 1) 2 N N sin θ cos ϕ, Sx2  = + sin θ cos2 ϕ, 2 4 4 N (N − 1) 2 N N sin θ sin ϕ, S y2  = + sin θ sin2 ϕ,

S y  = 2 4 4 N (N − 1) N N

Sz2  = + cos2 θ.

Sz  = − cos θ, 2 4 4

Sx  =

(7.74a) (7.74b) (7.74c)

and also test the Heisenberg uncertainty ΔSα ΔSβ ≥

| Sγ | , 2

(7.75)

with α, β, γ ∈ (x, y, z) and α = β = γ. Choosing the appropriate (θ, ϕ) can squeeze the CSS. For instance, in |θ = π/4, ϕ = 0 we have (ΔSx )2 < Sz /2. Thus, the squeezing criterion (ΔSα )2 < Sγ /2 with α, γ ∈ (x, y, z) and α = γ is coordinate dependent. Nevertheless, the squeezing of CSS should not happen since it is the product state of independent particles. Kitagawa and Ueda gave an alternative squeezing criterion in [45]. The spin system has a prior direction, i.e., the mean-spin direction, e0 =

S . | S|

(7.76)

The total spin operator in the direction perpendicular to e0 is expressed as Se⊥ = e⊥ · S,

(7.77)

with the unity vector e⊥ · e0 = 0. The squeezing criterion is then given by     min ΔSe⊥ < ΔSe⊥ CSS =

√

N , 2

(7.78)

7.6 Heisenberg-Limited Quantum Metrology

375

 where ΔSe⊥ = Se2⊥  − Se⊥ 2 . The standard deviation ΔSe⊥ in the CSS reaches √   ΔSe⊥ CSS = N /2 (see Problem 7.2). In [46, 47], another squeezing criterion was proposed to measure the squeezing in Ramsey spectroscopy. The spin system is initialized (t = 0) in the Dicke state |N /2, −N /2, in which we have Sz  = −N /2, Sx  = S y  = 0, Sz2  = N 2 /4, and

Sx2  = S y2  = N /4. Then, the spin system is rapidly rotated around the y-axis by π/2 in the clockwise direction, i.e., the first π/2-pulse. The new system is further rotated around the new z-axis by a degree φ in the clockwise direction, i.e., the free-evolution period. For the laser frequency locking, φ is set at φ = 2nπ + π/2 with n ∈ Z. Next, the spin system is rapidly rotated around the new y-axis by π/2 in the clockwise direction, i.e., the second π/2-pulse. Finally, the expectation value (f)

Sy (φ) is measured, where the superscript “ f ” denotes the final coordinate system. The noise sources in the system introduces an error Δφ, which is given by (f)

(ΔS y )t=0 ΔSz (φ) 1 = =√ , (Δφ)CSS =   ∂ S ( f ) (φ)  (| Sz |)t=0 N y       ∂φ

(7.79)

to φ. It is seen that (Δφ)CSS is limited by the quantum projection noise, and the squeezing criterion is naturally defined as Δφ < (Δφ)CSS .

(7.80)

7.6.2 Heisenberg-Limited Phase Measurement The high-precision phase measurement is of particular importance in the gravitational-wave detection, high-resolution spectroscopy, atom interferometer, and atomic frequency standard. Figure 7.11a illustrates the Mach–Zehnder interferometer configuration, which is commonly employed to detect the light phase fluctuation. The incident light beam is divided into two sub-beams by a beam splitter and recombined later by another beam splitter. The phase difference between two optical paths is measured via the balanced detection. Another common method to measure the phase is the Ramsey interferometry that has been a standard technique in atomic clocks. The quantum Rosetta stone [48] tells us that these two approaches represent the equivalent physical process. A general phase estimation protocol consists of three steps: (i) The system is well prepared in an initial state described by the density matrix operator ρ(0); (ii) Then, an extra phase φ is introduced, ρ(φ) = U (φ)ρ(0)U −1 (φ) with a unitary operator U (φ); and (iii) The system is subjected to a generalized measurement, i.e., a positive operator valued measure (POVM), whose elements are the operators E x with the measurement outcome x. The outcome x may vary continuously or discretely.

376

7 Molecular Cavity QED

Fig. 7.11 Heisenberg-limited phase detection. a Scheme of the optical Mach–Zehnder interferometer. The dashed line denotes the vacuum field. b Shot-noise and Heisenberg limit versus the number of emitters/photons

The probability distribution of the measurement is given by p(x|φ) = Tr[E x ρ(φ)]. According to [49], the standard deviation Δφ of the measurement of φ has the following inequality 1 Δφ ≥ √ , (7.81) ν F(φ) with the Fisher information  F(φ) =

 ∂ p(x|φ) 2 1 . dx p(x|φ) ∂φ

(7.82)

Here ν is the number of times the measurement procedure is repeated and ν = 1 corresponds to the single-shot experiment. Equation (7.81) is referred to as the quantum Cramér–Rao bound. We consider that a POVM is described by a set of nonnegative self-adjoint discrete operators  {E i ; i = 1, 2, · · · } [50]. The sum of these elements gives the identity operator, i E i = I . All projection operators, E i E j = δi, j , are commonly chosen to form a POVM. However, it should be noted that the elements of a POVM are not necessarily orthogonal. Hence, the number of elements in the POVM may be larger than the dimension of the Hilbert space. As we have shown in Sect. 1.5.5, the detection uncertainty for √ a coherent-state ensemble of photons is limited by the shot noise, i.e., Δφ = 1/ N , which is also

7.6 Heisenberg-Limited Quantum Metrology

377

called the standard quantum limit. The same scaling is obtained for the projection measurement performed on a CSS ensemble of quantum emitters [see (7.79)]. Nonetheless, the shot-noise limit is not fundamental since the squeezed and entangled (as we will see below) states enable the sub-shot-noise phase measurement. Actually, the fundamental limit is set by the Heisenberg uncertainty principle. We first consider the Heisenberg limit in atomic spectroscopy. The fully entangled GHZ (Greenberger–Horne–Zeilinger) state [51] for an ensemble of N two-level quantum emitters is written as  N 1  N |GHZ = √ ⊗ |g j + ⊗ |e j . j=1 j=1 2

(7.83)

We apply the unitary rotation operator U (φ) =

φ ( j) , ⊗ exp i σz j=1 2

N

(7.84)

on |GHZ and obtain  N N 1  |Ψ  = U (φ) |GHZ = √ e−i N φ/2 ⊗ |g j + ei N φ/2 ⊗ |e j . j=1 j=1 2 (7.85) The Hilbert space may be reduced to a subspace spanned by |g, g, · · · , g and |e, e, · · · , e. The elements of a POVM can be chosen as E + = |+ +| and E − = |− −| with  N 1  N |± = √ ⊗ |g j ± ⊗ |e j . j=1 j=1 2

(7.86)

The sum of E ± is equal to the identity operator. The measurement of E ± in |Ψ  leads to Nφ Nφ , p(−|φ) = E −  = sin2 . (7.87) p(+|φ) = E +  = cos2 2 2 The Fisher information is then given by F(φ) =

  1 ∂ p(+|φ) 2 ∂ p(−|φ) 2 1 + = N 2. p(+|φ) ∂φ p(−|φ) ∂φ

(7.88)

The minimal standard deviation of φ in the single-shot measurement reads min(Δφ) =

1 . N

(7.89)

The above result is called the Heisenberg limit and can be also obtained via the error-propagation formula

378

7 Molecular Cavity QED

Δp(±|φ) Δφ =  .  ∂ p(±|φ)    ∂φ with the standard deviation Δp(±|φ) =

(7.90)



E ±2  − E ± 2 . In comparison with the √ shot-noise limit, the phase-measurement sensitivity is enhanced by a factor of 1/ N [see Fig. 7.11b]. The Heisenberg limit in the atomic spectroscopy has been verified in the proof-of-principle experimental work [52], where three ions are entangled to enhance the spectroscopic sensitivity. In the Mach–Zehnder interferometer, the incident light in one path a is in the Fock state |N  while the vacuum state |0 in the path b [see Fig. 7.11a]. We assume that the first beam splitter is a “magic” one, which leads to the output state  1  |N00N = √ |N , 0 + ei N φ |0, N  . 2

(7.91)

The state |N1 , N2  = |N1 a ⊗ |N2 b denotes the photon numbers in two arms are N1 and N2 , respectively. The N00N state is also known as the maximally pathentangled number state. One may choose the elements of a POVM as E + = |+ +| and E − = |− −| with 1 |± = √ (|N , 0 ± |0, N ) . 2

(7.92)

and the Heisenberg limit (7.89) is obtained. Alternatively, one may define the observable with the associated Hermitian operator A = |N , 0 0, N | + |0, N  N , 0|. The expectation value of this operator is A = cos N φ. The standard deviation of the phase is given by 1 ΔA = . (7.93) Δφ =   N  ∂ A    ∂φ

AQ1

The enhanced sensitivity in phase resolution based on the N00N state has been already demonstrated in [53].

Problems 7.1 The dye laser is a four-level system as shown in Fig. 7.12. The pump light excites the dye molecules from the ground-vibrational level |1 = S0,v=0 to the excited vibrational level |4 = S1,v>0 . The excited molecules then relaxes rapidly to |3 = S1,v=0 in a radiationless manner. The optical cavity is resonantly coupled to the molecular transition between |3 and a certain |2 = S0,v=0 . The transition frequency is ω. The

7.6 Heisenberg-Limited Quantum Metrology

379

Fig. 7.12 Scheme of energy levels in a four-level laser

molecules release their internal energy to the cavity field and transit to |2. The |2 level is quickly depopulated through the radiationless relaxation to |1. Repeating the above processes maintains the laser action. The populations in different molecular levels are Ni=1,2,3,4 . It is easy to obtain the following rate equations: d N4 = W p (N1 − N4 ) − γ4 N4 , dt d rad N3 = γ43 N4 − (γ3 + γ32 )N3 , dt d rad N2 = γ42 N4 + (γ32 + γ32 )N3 − γ21 N2 , dt

(7.94a) (7.94b) (7.94c)

where W p is the pumping rate and γi j with i, j = 1, 2, ..., 4 and i < j denotes the  rad decay rate from |i to | j, and γi = i−1 j=1 γi j . Here γ32 is the rate of the molecules radiating the photons the cavity via the stimulated emission. The conservation into 4 Ni = N with the total number of molecules N . Similarly, of molecules gives i=1 the rate equation of the number of photons inside the optical cavity is derived as d rad N p = −κN p + γ32 N3 , dt

(7.95)

rad N3 is proportional to the with the photon loss rate κ. The stimulated emission γ32 rad photon number N p and the population inversion ΔN = N3 − N2 , i.e., γ32 N3 = ω B N ΔN with the Einstein B coefficient, the cross-sectional area of the laser beam p Sv S, and the light velocity v. In the limit of γ43  γ3 , γ42 ∼ 0, and W p  γ4 , derive the following steady-state solutions:

380

7 Molecular Cavity QED

ΔN (ss) =

κ , N p(ss) ≈ Ns ω B Sv



N −1 , Nth

(7.96)

where the saturation photon number Ns and the threshold Nth are Ns ≈

γ43 γ21 + γ32 κ γ21 + γ32 , Nth ≈ . ω B γ21 − γ32 ω B γ21 + γ43 2 Sv Sv

(7.97)

It is seen that the laser action requires γ21 > γ32 and N > Nth . 7.2 When the spin system is in the CSS (7.67), the mean-spin direction is given by e0 =

S = sin θ cos ϕ ex + sin θ sin ϕ e y − cos θ ez . | S|

(7.98)

Two independent unit vectors that are perpendicular to e0 can be chosen as e1 = cos θ cos ϕ ex + cos θ sin ϕ e y + sin θ ez ,

(7.99a)

e2 = − sin ϕ ex + cos ϕ e y .

(7.99b)

An arbitrary unity vector perpendicular to e0 is written as e⊥ = cos α e1 + sin α e2 .

(7.100)

√   Prove the standard deviation ΔSe⊥ in the CSS is ΔSe⊥ CSS = N /2. 7.3 A system is composed of an ensemble of N independent two-level quantum emitters. The system is initialized in the following product state |Ψ i =

N j=1

|+ j ,

(7.101)

√ with |+ = (|g + |e) / 2. Then, the unitary rotation operator (7.84) is applied on |Ψ i and one obtains |Φf = U (φ) |Ψ i . (7.102) Derive the Fisher information F(φ) for the system in |Φf and prove the minimal standard deviation Δφ √ of the single-shot phase measurement is limited by the shot noise min(Δφ) = 1/ N .

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Appendix A

Vector Calculus

The dot (scalar) product of two vectors A = A x ex + A y e y + A z ez and B = Bx ex + B y e y + Bz ez in the Cartesian coordinates is expressed as A · B = |A||B| cos θ = A x Bx + A y B y + A z Bz ,

(A.1)

with the angle θ between two vectors. The dot product of two orthogonal vectors is zero. The cross product of A and B is given by A × B = (A y Bz − A z B y )ex − (A x Bz − A z Bx )e y + (A x B y − A y Bx )ez .

(A.2)

The vector V = A × B is perpendicular to the plane containing both A and B. Indeed, the vectors A, B, and V form a right-hand configuration. The module of V reads |V| = |A||B| sin θ. The cross product of two parallel vectors is zero. The following two triple product identities are often used:    Ax A y Az    A · (B × C) =  Bx B y Bz  = B · (C × A) = C · (A × B), C x C y C z  A × (B × C) = B × (A · C) − C × (A · B).

(A.3) (A.4)

The differential operator ∇ in the Cartesian coordinates takes the form ∇ = ex

∂ ∂ ∂ + ey + ez . ∂x ∂y ∂z

(A.5)

There exist four typical field operations associated with ∇: • Gradient of a scalar field f (r) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2

385

386

Appendix A: Vector Calculus

∇ f (r) =

∂ f (r) ∂ f (r) ∂ f (r) ex + ey + ez ; ∂x ∂y ∂z

(A.6)

• Divergence of a vector field F(r) = Fx (r)ex + Fy (r)e y + Fz (r)ez ∇ · F(r) =

∂ Fx (r) ∂ Fy (r) ∂ Fz (r) + + ; ∂x ∂y ∂z

(A.7)

• Curl of a vector field 

   ∂ Fz (r) ∂ Fy (r) ∂ Fz (r) ∂ Fx (r) ∇ × F(r) = − ex − − ey ∂y ∂z ∂x ∂z   ∂ Fy (r) ∂ Fx (r) − ez ; + ∂x ∂y

(A.8)

• Laplacian of a vector field  f (r) = ∇ 2 f (r) = (∇ · ∇) f (r) =

∂ 2 f (r) ∂ 2 f (r) ∂ 2 f (r) + + . ∂x 2 ∂ y2 ∂z 2

(A.9)

It can be verified that (1) the divergence of the curl of a vector field is zero ∇ · (∇ × F) = 0,

(A.10)

and (2) the curl of the gradient of a scalar field is zero ∇ × (∇ f ) = 0.

(A.11)

In addition, one may find the following useful identities: • Gradient ∇( f 1 f 2 ) = f 1 ∇ f 2 + f 2 ∇ f 1 , ∇(F1 · F2 ) = (F1 · ∇)F2 + (F2 · ∇)F1 +F1 × (∇ × F2 ) + F2 × (∇ × F1 );

(A.12) (A.13)

• Divergence ∇ · ( f F) = f (∇ · F) + (∇ f ) · F, ∇ · (F1 × F2 ) = (∇ × F1 ) · F2 − F1 · (∇ × F2 ); • Curl

(A.14) (A.15)

Appendix A: Vector Calculus

∇ × ( f F) = f (∇ × F) + (∇ f ) × F, ∇ × (F1 × F2 ) = F1 (∇ · F2 ) − F2 (∇ · F1 ) +(F2 · ∇)F1 − (F1 · ∇)F2 ;

387

(A.16) (A.17)

• Second derivatives ∇ × (∇ × F) = ∇(∇ · F) − ∇ 2 F, ∇ 2 ( f 1 f 2 ) = f 1 ∇ 2 f 2 + 2(∇ f 1 ) · (∇ f 2 ) + (∇ 2 f 1 ) f 2 , ∇ 2 ( f F) = F∇ 2 f + 2(∇ f · ∇)F + f (∇ 2 F).

(A.18) (A.19) (A.20)

Appendix B

Cylindrical and Spherical Coordinate Systems

In cylindrical coordinates, a point is specified by r = ρ eρ + h e z ,

(B.1)

[see Fig. B.1], where ρ ∈ [0, ∞) is the radial distance from the z-axis, the azimuthal angle ϕ ∈ [0, 2π) is measured from the x-axis in the x − y plane, and h ∈ (−∞, ∞) gives the height of the cylinder along the z-axis. The unit vectors eρ , eϕ and ez are normal, tangent and parallel to the cylinder’s surface, respectively. The variables (ρ, ϕ, h) are related to the variables (x, y, z) in the Cartesian coordinate system via ρ=



y x 2 + y 2 , ϕ = (π+) arctan , h = z, x

(B.2)

and the inverse transformation x = ρ cos ϕ, y = ρ sin ϕ, z = h.

(B.3)

The extra phase π in the expression of ϕ is taken when x < 0. Further, the relationships between (eρ , eϕ , ez ) and (eρ , eϕ , ez ) are given by eρ = cos ϕ ex + sin ϕ e y , eϕ = − sin ϕ ex + cos ϕ e y ,

(B.4a) (B.4b)

ez = ez ,

(B.4c)

and the inverse transformation ex = cos ϕ eρ − sin ϕ eϕ , e y = sin ϕ eρ + cos ϕ eϕ ,

(B.5a) (B.5b)

ez = ez ,

(B.5c)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2

389

390

Appendix B: Cylindrical and Spherical Coordinate Systems

Fig. B.1 Cylindrical coordinate systems

In the cylindrical coordinate system, the gradient and Laplacian of a scalar function f (r) are expressed as 1∂f ∂f ∂f eρ + eϕ + ez , ∂ρ ρ ∂ϕ ∂z   ∂f 1 ∂2 f 1 ∂ ∂2 f ρ + 2 + , ∇2 f = ρ ∂ρ ∂ρ ρ ∂ϕ2 ∂z 2 ∇f =

(B.6) (B.7)

and the divergence and curl of a vector function F(r) = Fρ eρ + Fϕ eϕ + Fz ez are written as  1 ∂ ∂ 1 ∂  ρFρ + Fϕ + Fz , ρ ∂ρ ρ ∂ϕ ∂z     1 ∂ ∂ ∂ ∂ Fz − Fϕ eρ + Fρ − Fz eϕ ∇ ×F = ρ ∂ϕ ∂z ∂z ∂ρ   1 ∂ ∂ (ρFϕ ) − Fρ ez . + ρ ∂ρ ∂ϕ ∇ ·F =

(B.8)

(B.9)

In the spherical coordinate system, a point r in the three-dimensional space is specified by the radial distance 0 ≤ r < ∞, polar angle 0 ≤ θ ≤ π, and azimuthal angle 0 ≤ ϕ < 2π, i.e., (B.10) r = r er ,

Appendix B: Cylindrical and Spherical Coordinate Systems

391

Fig. B.2 Spherical coordinate systems

as shown in Fig. B.2. Here, er , eθ and eϕ are three orthogonal unit vectors in the directions of increasing r , θ, and ϕ, respectively. The spherical coordinates (r, θ, ϕ) can be obtained from the Cartesian coordinates (x, y, z) via the following relations: r=



 x2

+

y2

+

z2,

θ = (π+) arctan

x 2 + y2 y , ϕ = (π+) arctan . (B.11) z x

The extra phase π in the expression of θ (ϕ) is taken when z < 0 (x < 0). The Cartesian coordinates (x, y, z) may be expressed in terms of (r, θ, ϕ) as x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ.

(B.12)

The linear transformation from three orthogonal unit vectors (ex , e y , ez ) in the Cartesian coordinate system to the spherical unit vectors (er , eθ , eϕ ) is derived as er = sin θ cos ϕ ex + sin θ sin ϕ e y + cos θ ez , eθ = cos θ cos ϕ ex + cos θ sin ϕ e y − sin θ ez ,

(B.13a) (B.13b)

eϕ = − sin ϕ ex + cos ϕ e y .

(B.13c)

The reverse transformation is given by ex = sin θ cos ϕ er + cos θ cos ϕ eθ − sin ϕ eϕ , e y = sin θ sin ϕ er + cos θ sin ϕ eθ + cos ϕ eϕ ,

(B.14a) (B.14b)

ez = cos θ er − sin θ eθ .

(B.14c)

392

Appendix B: Cylindrical and Spherical Coordinate Systems

In spherical coordinates, the gradient of an arbitrary scalar field f (r, θ, ϕ) is written as 1∂f 1 ∂f ∂f er + eθ + eϕ . (B.15) ∇f = ∂r r ∂θ r sin θ ∂ϕ The Laplacian of f (r, θ, ϕ) has the form 1 ∂ ∇ f = 2 r ∂r 2

    ∂2 f 1 ∂ ∂f 1 2∂ f r + 2 sin θ + 2 2 . ∂r r sin θ ∂θ ∂θ r sin θ ∂ϕ2

(B.16)

The divergence of an arbitrary vector field F = Fr er + Fθ eθ + Fϕ eϕ reads ∇ ·F=

∂ 1 ∂ 2 1 ∂ 1 (r Fr ) + Fϕ , (Fθ sin θ) + r 2 ∂r r sin θ ∂θ r sin θ ∂ϕ

(B.17)

and the curl of F takes the form ∇ ×F =

 ∂ ∂ 1 (Fϕ sin θ) − Fθ er r sin θ ∂θ ∂ϕ  1 ∂ 1 ∂ Fr − (r Fϕ ) eθ + r sin θ ∂ϕ ∂r  1 ∂ ∂ (r Fθ ) − Fr eϕ . + r ∂r ∂θ

(B.18)

Appendix C

Bessel Functions

Many problems in the electromagnetic field analysis are connected to the following second-order differential equation: x2

d2 y dy +x + (x 2 − ν 2 )y = 0, dx2 dx

(C.1)

which is known as the Bessel’s equation of order ν. Here, we only consider ν = n or ν = n + 1/2 with n ∈ Z. The solutions of the above equation can be found via the Fröbenius method. Bessel functions of the first kind Jν (x). The solutions to (C.1) are expressed by the power series Jν (x) =

∞ k=0

 x ν+2k (−1)k . k!Γ (ν + k + 1) 2

(C.2)

Here, Γ (z) is the gamma function.1 Below, we list several properties of Jν (x) that are commonly used: (1) In the limit of |x| ∼ 0, Jν (x) is approximated as Jν (x) ∼

1 For

 x ν 1 . Γ (ν + 1) 2

(C.3)

the argument z = n or z = 1/2 ± n with n ∈ Z, Γ (z) reads Γ (n) = (n − 1)!, Γ (1/2 + n) =

where Γ (1/2) =

(2n)! (−4)n n! Γ (1/2), Γ (1/2 − n) = Γ (1/2), n 4 n! (2n)!

√ π.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2

393

394

Appendix C: Bessel Functions

(2) The Bessel functions Jν (x) satisfy the following recurrence relations and firstorder derivatives: Jν−1 (x) − Jν+1 (x) = 2Jν (x), d ν [x Jν (x)] = x ν Jν−1 (x), dx

Jν−1 (x) + Jν+1 (x) =

2ν Jν (x), x

d −ν [x Jν (x)] = −x −ν Jν+1 (x), dx

and the second-order derivative d2 1 Jν (x) = (Jν−2 (x) − 2Jν (x) + Jν+2 (x)). 2 dx 4

(C.5)

In the special case, we have J0 (x) = −J1 (x).

(C.6)

(3) For ν ∈ Z, (C.1) can be obtained from the Laplace’s equation (i.e., ∇ 2 f (r) = 0) in cylindrical coordinates through the separation of variables method. Thus, the Bessel functions Jν (x) are also called the cylindrical harmonics and we have J−ν (x) = (−1)ν Jν (x),

(C.7)

i.e., Jν (x) and J−ν (x) are not linearly independent. In addition, at x = 0 one obtains Jν (0) = δν,0 , (see Fig. C.1). (4) The Jacobi–Anger expansion ei x cos θ =

∞ ν=−∞

i ν Jν (x)eiνθ , ei x sin θ =

∞ ν=−∞

Jν (x)eiνθ ,

(C.8)

is usually utilized in the phase modulation spectroscopy. (5) For ν = ±1/2, we have

J−1/2 (x) =

2 cos x, πx

J1/2 (x) =

2 sin x. πx

(C.9)

(6) In the limit of |x| ∼ ∞, the asymptotic form for Jν (x) is given by

Jν (x) ∼

  νπ π ∞ 2 a2k (ν) cos x − − (−1)k 2k k=0 πx 2 4 x  π ∞ a (ν) νπ 2k+1 − , − sin x − (−1)k 2k+1 k=0 2 4 x

for | arg(x)| < π, where

(C.10)

Appendix C: Bessel Functions

395

Fig. C.1 Bessel functions Jν (x) and Nν (x)

ak (x) =

(4ν 2 − 12 )(4ν 2 − 32 ) · · · (4ν 2 − (2k − 1)2 ) with k ≥ 1. (C.11) k!8k

(7) The following integral is useful: 

a 0

x Jν2 (bx/a)d x =

a2 2 J (b). 2 ν+1

(C.12)

Bessel functions of the second kind Nν (x). The solutions Nν (x) are also known as Weber or Neumann functions. For a non-integer ν, Nν (x) is related to Jν (x) via Nν (x) =

Jν (x) cos(νπ) − J−ν (x) . sin(νπ)

(C.13)

In the case of ν ∈ Z, Nν (x) is defined by Nν (x) = lim

→0

Jν+ (x) cos[(ν + )π] − J−(ν+ ) (x) . sin[(ν + )π]

(C.14)

The functions Nν (x) defined in such a way are linearly independent on Jν (x) for all values of ν. The following properties of Nν (x) are useful: (1) For an integer order ν ∈ Z, Nν (x) and N−ν (x) are not linearly independent, N−ν (x) = (−1)ν Nν (x).

(C.15)

As x approaches 0, Nν (x) goes to the infinity (see Fig. C.1). (2) The Bessel functions Nν (x) satisfy the recurrence relations and first-order derivatives

396

Appendix C: Bessel Functions

Nν−1 (x) − Nν+1 (x) = 2Nν (x), d ν [x Nν (x)] = x ν Nν−1 (x), dx

Nν−1 (x) + Nν+1 (x) =

2ν Nν (x), x

d −ν [x Nν (x)] = −x −ν Nν+1 (x), dx

and the second-order derivatives d2 1 Nν (x) = (Nν−2 (x) − 2Nν (x) + Nν+2 (x)). 2 dx 4

(C.17)

(3) For the argument |x| → ∞, one has the following asymptotic behavior

  νπ π ∞ 2 a2k (ν) sin x − − (−1)k 2k Nν (x) ∼ k=0 πx 2 4 x 

∞ π νπ k a2k+1 (ν) , − + cos x − (−1) k=0 2 4 x 2k+1

(C.18)

for | arg(x)| < π. The coefficients ak (ν) are given by (C.11). Hankel functions of the first and second kind Hν(1,2) (x). The Hankel functions are defined as Hν(1) (x) = Jν (x) + i Nν (x), Hν(2) (x) = Jν (x) − i Nν (x).

(C.19)

For ν ∈ Z, the above definitions need to be modified, like (C.14). The Hankel functions have the following properties: (1) For ν = n or ν = n + 1/2 with n ∈ Z, we have (1) (2) (x) = eiνπ Hν(1) (x), H−ν (x) = e−iνπ Hν(2) (x). H−ν

(C.20)

(2) For a large argument |x| → ∞, the Hankel functions have the asymptotic behaviors −π < arg x < 2π

  νπ π  ∞ k ak (ν) 2 exp i x − − i , Hν(1) (x) ∼ k=0 πx 2 4 xk for −2π < arg x < π

  νπ π  ∞ 2 ak (ν) (2) exp −i x − − Hν (x) ∼ (−i)k k . k=0 πx 2 4 x for

(C.21a)

(C.21b)

Equation (C.21a) denotes the outgoing wave while (C.21b) corresponds to the incoming wave. Modified Bessel functions of the first and second kind Iν (x) and K ν (x). The modified Bessel functions are defined as

Appendix C: Bessel Functions

397

Iν (x) = i −ν Jν (i x), K ν =

π I−ν (x) − Iν (x) , 2 sin(νπ)

(C.22)

where the limit definition similar to (C.14) should be used when ν ∈ Z. The modified Bessel functions are the solutions of the modified Bessel’s equation x2

d2 y dy − (x 2 + ν 2 )y = 0. +x 2 dx dx

(C.23)

Below, we list some common properties of the modified Bessel functions (1) Iν (x) and K ν (x) satisfy the following recurrence relations and first-order derivatives: Iν−1 (x) + Iν+1 (x) = 2Iν (x), d (x ν Iν (x)) = x ν Iν−1 (x), dx −K ν−1 (x) − K ν+1 (x) = 2K ν (x), d (x ν K ν (x)) = −x ν K ν−1 (x), dx

Iν−1 (x) − Iν+1 (x) = 2ν I (x), x ν d −ν −ν (x Iν (x)) = x Iν+1 (x), dx −K ν−1 (x) + K ν+1 (x) = 2ν K ν (x), x d −ν −ν (x I (x)) = −x I (x), ν ν+1 dx

and the second-order derivatives d2 1 Iν (x) = (Iν−2 (x) + 2Iν (x) + Iν+2 (x)) , dx2 4 d2 1 K ν (x) = (K ν−2 (x) + 2K ν (x) + K ν+2 (x)) . 2 dx 4 When ν = 0, we have I0 (x) = I1 (x) and K 0 (x) = −K 1 (x). (2) For the order ν = 1/2 and complex argument x ∈ C, we have

I1/2 (x) =

2 sinh x, K 1/2 (x) = πx

π −x e . 2x

(C.24)

(3) For ν ∈ Z, we have Iν (x) = I−ν (x) and K ν (x) = K −ν (x). (4) For the complex order ν ∈ C and complex argument x ∈ C, the modified Bessel functions have the following asymptotic expansions:  4ν 2 − 1 (4ν 2 − 1)(4ν 2 − 9) ex 1− Iν (x) ∼ √ + 8x 2!(8x)2 2πx 2 2 (4ν − 1)(4ν − 9)(4ν 2 − 25) − + ··· for | arg z| < 3!(8x)3

 4ν 2 − 1 (4ν 2 − 1)(4ν 2 − 9) π −x 1+ e + K ν (x) ∼ 2x 8x 2!(8x)2 (4ν 2 − 1)(4ν 2 − 9)(4ν 2 − 25) + + · · · for | arg z| < 3!(8x)3

π , 2

3π , 2

398

Appendix C: Bessel Functions

in the limit |x| → ∞. (5) In the limit of |x| ∼ 0, Iν (x) is approximated by Iν (x) =

 x ν 1 , Γ (ν + 1) 2

(C.26)

while K ν (x) approaches the infinite. (6) The following integral is commonly used 

∞ x0

x K ν2 (ax)d x =

 x02  K ν−1 (ax0 )K ν+1 (ax0 ) − K ν2 (ax0 ) . 2

(C.27)

Spherical Bessel jν (x) and n ν (x) and Hankel h (1,2) (x) functions. When solving ν the Helmholtz equation (i.e., ∇ 2 f (r) = −k 2 f (r) with the eigenvalue k) in spherical coordinates by the separation of variables, the radial equation takes the form x2

d2 y dy + (x 2 − ν(ν + 1))y = 0, + 2x dx2 dx

(C.28)

with ν ∈ Z. The spherical Bessel functions jν (x) and n ν (x)

jν (x) =

π Jν+1/2 (x), n ν (x) = 2x

π Nν+1/2 (x), 2x

(C.29)

are two linearly independent solutions to (C.28). Below, we list the expressions of the first few spherical Bessel functions j0 (x) = j1 (x) = j2 (x) = j3 (x) = n 0 (x) = n 1 (x) = n 2 (x) = n 3 (x) =

sin x , x sin x cos x , − 2 x  x 3 cos x 3 sin x − −1 , 2 x x x2     15 15 6 sin x cos x − , − −1 3 2 x x x x x cos x , − x cos x sin x , − 2 −  x x 3 sin x 3 cos x − − 2 +1 , x x x2     15 15 6 cos x sin x − . − 3 + −1 x x x x2 x

Appendix C: Bessel Functions

399

The spherical Bessel functions ( f ν (x) = jν (x), n ν (x)) have the properties d dx d dx d (2ν + 1) dx 2ν + 1 x

f ν (x) = f ν−1 (x) −

ν+1 f ν (x), x

(C.31a)

ν f ν (x) − f ν+1 (x), x

(C.31b)

f ν (x) = ν f ν−1 (x) − (ν + 1) f ν+1 (x),

(C.31c)

f ν (x) = f ν−1 (x) + f ν+1 (x),

(C.31d)

f ν (x) =

with ddx f 0 (x) = − f 1 (x) and ddx f −1 (x) = f −2 (x). In the scattering theory, the following expansion is frequently used eikr cos θ =

∞ l=0

i l (2l + 1) jl (kr )Pl (cos θ),

(C.32)

with the Legendre polynomials Pl (cos θ). The Riccati–Bessel functions are defined as

πx πx Jν+1/2 (x), χν (x) = −xn ν (x) = − Nν+1/2 (x). ψν (x) = x jν (x) = 2 2 (C.33) One may also define the spherical Hankel functions as (2) h (1) ν (x) = jν (x) + in ν (x), h ν (x) = jν (x) − in ν (x),

whose derivatives and recurrence relations are same to (C.31).

(C.34)

Appendix D

Diagonalization of the Two-Dimensional Matrix

We derive the eigenvectors of a 2 × 2 matrix, whose the eigenvalue equation is expressed as      a a A Beiϕ = λ , (D.1) Be−iϕ D b b with the real numbers A, B, and D. By doing the transformation b˜ = beiϕ , the above equation is rewritten as      a a A B =λ ˜ . (D.2) B D b˜ b Both a and b˜ are real numbers. The eigenvalue λ is obtained by solving the characteristic equation   A − λ B    (D.3)  B D − λ = 0, which leads to the characteristic polynomial λ2 − (A + D)λ + AD − B 2 = 0.

(D.4)

The eigenvalues are given by λ± =

 1 [(A + D) ± (A − D)2 + 4B 2 ]. 2

(D.5)

After obtaining the eigenvalues, we need to derive the corresponding eigenvectors. Substituting λ+ back into (D.3), one has   1 (A − D) − (A − D)2 + 4B 2 a + B b˜ = 0. 2 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2

(D.6) 401

402

Appendix D: Diagonalization of the Two-Dimensional Matrix

In addition, the normalization condition requires a 2 + b˜ 2 = 1.

(D.7)

Thus, for the eigenvalue λ+ the corresponding eigenvector is ⎞ 2B −  2 1/2 ⎟  ⎜   2 + 4B 2 2 ⎟ ⎜ (A − D) − (A − D) + 4B a ⎟ ⎜  =⎜ ⎟. (A − D) − (A − D)2 + 4B 2 b + ⎜ −iϕ ⎟ e ⎠ ⎝  2 1/2  (A − D) − (A − D)2 + 4B 2 + 4B 2 ⎛

(D.8)

Similarly, one can derive the eigenvector ⎞ 2B −     2 1/2 ⎟ ⎜   ⎟ ⎜ (A − D) + (A − D)2 + 4B 2 + 4B 2 a ⎟ ⎜  =⎜ ⎟. (A − D) + (A − D)2 + 4B 2 b − ⎜ −iϕ ⎟ e ⎠ ⎝     2 1/2 (A − D) + (A − D)2 + 4B 2 + 4B 2 ⎛

corresponding to the eigenvalue λ− .

(D.9)

Appendix E

Eigenmodes of Optical Fiber

Figure E.1 shows the general structure of an optical fiber, which consists of a circular light-conducting core with a relative permittivity i and a cladding with a relative permittivity o . The fiber core is along the z-direction and has a diameter 10 ∼ 100 µm. The existence of the optical bound states requires that the refractive index of √ √ the core i is higher than that of the cladding o . In what follows, we consider the eigenmodes of the optical fiber.

E.1 Wave Theory of Step-Index Fibers It is natural to choose the cylindrical coordinates for the analysis. The electric and magnetic fields are written as E(r, t) = E ρ (r, t) eρ + E ϕ (r, t) eϕ + E z (r, t) ez , H(r, t) = Hρ (r, t) eρ + Hϕ (r, t) eϕ + Hz (r, t) ez .

(E.1a) (E.1b)

The Faraday’s law of induction ∇ × E(r, t) = −μ0 ∂H(r, t)/∂t leads to ∂ 1 ∂ ∂ Hρ (r, t) = − E z (r, t) + E ϕ (r, t), ∂t ρ ∂ϕ ∂z ∂ ∂ ∂ E z (r, t) − E ρ (r, t), μ0 Hϕ (r, t) = ∂t ∂ρ ∂z ∂ ∂ 1 ∂ E ρ (r, t), μ0 Hz (r, t) = − E ϕ (r, t) + ∂t ∂ρ ρ ∂ϕ μ0

(E.2a) (E.2b) (E.2c)

and the Ampère’s circuital law ∇ × H(r, t) = ∂D(r, t)/∂t gives

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2

403

404

Appendix E: Eigenmodes of Optical Fiber

Fig. E.1 Schematic diagram of a step-index fiber

∂ 1 ∂ ∂ E ρ (r, t) = Hz (r, t) − Hϕ (r, t), ∂t ρ ∂ϕ ∂z ∂ ∂ ∂ (r)ε0 E ϕ (r, t) = − Hz (r, t) + Hρ (r, t), ∂t ∂ρ ∂z ∂ ∂ 1 ∂ Hϕ (r, t) − Hρ (r, t). (r)ε0 E z (r, t) = ∂t ∂ρ ρ ∂ϕ (r)ε0

(E.3a) (E.3b) (E.3c)

Writing the longitudinal components as E z (r, t) = E z (r)e−iωt and Hz (r, t) = Hz (r)e−iωt with the light frequency ω, one obtains 

 ∂2 1 ∂2 1 ∂ ∂2 2 + + + + ω (r)ε μ 0 0 (r) = 0, ∂ρ2 ρ ∂ρ ρ2 ∂ϕ2 ∂z 2

(E.4)

with (r) = E z (r) or Hz (r) from the standard wave equations ∇ 2 E(r, t) = (r)ε0 μ0 ∂ 2 E(r, t)/∂t 2 and ∇ 2 H(r, t) = (r)ε0 μ0 ∂ 2 H(r, t)/∂t 2 . Equation (E.4) can be solved via the approach of separation of variables, i.e., (r) = (ρ)eiηνϕ eiβz ,

(E.5)

with η = ±1, ν ∈ Z and the propagation constant β of the light wave along the z-direction. The radial function (ρ) follows   1 ∂ ν2 ∂2 2 (ρ) = 0, (ρ) + q (ρ) + − ∂ρ2 ρ ∂ρ ρ2

(E.6)

√ with q 2 = (ρ)k02 − β 2 , the vacuum wavenumber k0 = 2π/λ = ω ε0 μ0 , and (ρ < R) = i and (ρ > R) = o . We assume that the optical fiber is lossless, i.e., the amplitude of the light does not decay when it travels along the fiber. Thus, both (ρ) and β take the real values. The value of q 2 can be positive or negative. We use “i” and “o” to denote the core and cladding regions, respectively. Within the core, the light field must be finite at ρ = 0. Consequently, qi2 is positive and (ρ) ∼ Jν (qi ρ). The z-components of the light hereby take the forms

Appendix E: Eigenmodes of Optical Fiber

E z(i) (r, t) = A Jν (qi ρ)eiξ , Hz(i) (r, t) = B Jν (qi ρ)eiξ ,

405

(E.7)

with the shorthand ξ = ηνϕ + βz − ωt. In contrast, in the cladding region (ρ) decays to zero as ρ is increased. Hence, qo2 should be negative and the longitudinal components of the light outside the fiber core are expressed as E z(o) (r, t) = C K ν

q ρ q ρ o o eiξ , Hz(o) (r, t) = D K ν eiξ . i i

(E.8)

The parameters (A, B, C, D) can be determined by the boundary conditions at the interface ρ = R. Since qo2 is negative, the frequency cannot be higher than the cutoff frequency β . (E.9) ωc = √ o ε0 μ0 Actually, the components (E ρ , E ϕ , E z ) and (Hρ , Hϕ , Hz ) are not independent with each other. The transverse (E ρ , E ϕ , Hρ , Hϕ ) components are related to the longitudinal (E z , Hz ) ones via (E.2) and (E.3). As a result, we obtain the following expressions of the inside-core components: E ρ(i) (r, t)

=

E ϕ(i) (r, t) = Hρ(i) (r, t) = Hϕ(i) (r, t) =

i qi2 i qi2 i qi2 i qi2

 ωμ0  βqi A Jν (qi ρ) + iην B Jν (qi ρ) eiξ , ρ  β  iην A Jν (qi ρ) − ωμ0 qi B Jν (qi ρ) eiξ , ρ  ω i ε0 A Jν (qi ρ) eiξ , βqi B Jν (qi ρ) − iην ρ  β iην B Jν (qi ρ) + ω i ε0 qi A Jν (qi ρ) eiξ , ρ

(E.10a) (E.10b) (E.10c) (E.10d)

and the inside-cladding components E ρ(o) (r, t)

=

E ϕ(o) (r, t) = Hρ(o) (r, t) = Hϕ(o) (r, t) =

i qo2 i qo2 i qo2 i qo2



  q ρ qo ωμ0 o  qo ρ βC K ν + iην D Kν eiξ , i i ρ i  q ρ q  q ρ β o o o iην C K ν − ωμ0 D K ν eiξ , ρ i i i  q ρ  q ρ qo ω o ε0 o o β D K ν − iην C Kν eiξ , i i ρ i  q ρ q  q ρ β o o o iην D K ν + ω o ε0 C K ν eiξ , ρ i i i

(E.11a) (E.11b) (E.11c) (E.11d)

where, for instance, Jν (qi ρ) = (∂ Jν (x)/∂x)x=qi ρ . Using the appropriate boundary conditions, one may solve the effective propagation constant β and the parameters (A, B, C, D) for a certain wavelength λ.

406

Appendix E: Eigenmodes of Optical Fiber

TE modes. Setting E z = 0, A = C = 0 and ν = 0, we obtain the TE modes with Hz = 0. Due to ν = 0, Hz is independent of ϕ, i.e., ∂ Hz /∂ϕ = 0. Thus, we have E ρ = Hϕ = 0. The boundary conditions require E ϕ(i) = E ϕ(o) and Hz(i) = Hz(o) at the interface ρ = R. The propagation constant β is determined by the following dispersion equation: 1 K 0 (qo R/i) 1 J0 (qi R) + = 0. (E.12) qi R J0 (qi R) qo R/i K 0 (qo R/i) Using the recurrence relation of Bessel functions, we arrive at 1 J1 (qi R) 1 K 1 (qo R/i) + = 0. qi R J0 (qi R) qo R/i K 0 (qo R/i)

(E.13)

The effective propagation constant β may have multiple values, which are sorted in the descending order. The m-th β corresponds to the m-th TE mode of the optical fiber that is denoted by TE0,m with m ∈ N. TM modes. Similarly, the TM modes with E z = 0 are given by setting Hz = 0 and B = D = 0. Due to ν = 0, E z is independent of ϕ, i.e., ∂ E z /∂ϕ = 0, and we have Hρ = E ϕ = 0. The boundary conditions require E z(i) = E z(o) and Hϕ(i) = Hϕ(o) at the interface ρ = R lead to the relevant dispersion equation i J0 (qi R) o K 0 (qo R/i) + = 0. qi R J0 (qi R) qo R/i K 0 (qo R/i)

(E.14)

This equation can be re-expressed as i J1 (qi R) o K 1 (qo R/i) + = 0, qi R J0 (qi R) qo R/i K 0 (qo R/i)

(E.15)

by using the recurrence relation of Bessel functions. From the above equation, one may derive the effective propagation constant β, and the symbol TM0,m is used to denote the m-th TM mode. Hybrid modes. If both longitudinal components E z and Hz are nonzero, the fiber modes are referred to as the hybrid modes, i.e., HEν,m or EHμmν,m , depending upon which of E z or Hz dominates. The dispersion equation is derived as β2 ν2 2 k0



1 1 + 2 (qi R) (qo R/i)2

2

1 K ν (qo R/i) 1 Jν (qi R) + = qi R Jν (qi R) qo R/i K ν (qo R/i) (E.16)  o K ν (qo R/i) i Jν (qi R) + , × qi R Jν (qi R) qo R/i K ν (qo R/i) 

with ν ≥ 1. The V -number of an optical fiber is defined as V = k0 R · NA,

(E.17)

Appendix E: Eigenmodes of Optical Fiber

407

Fig. E.2 b − V diagram of an optical fiber with λ = 1550 nm, i = 1.44 and o = 1

√ where NA = i − o is the so-called numerical aperture. Since V is proportional to ω, it is also called the normalized frequency parameter. In addition, one may define the normalized propagation constant b=

β 2 /k02 − o , i − o

(E.18)

which varies between 0 and 1. Figure E.2 illustrates the dependence of b on V . It is seen that when V < Vc ≈ 2.405 only one mode, i.e., HE1,1 , can be guided inside the fiber. The HE1,1 mode is also called the fundamental mode. Actually, Vc corresponds to the cutoff frequency ωc . Figure E.3 exhibits several examples of the intensity distribution of eigenmodes as well as the corresponding polarization.

408 Fig. E.3 Intensity distribution of HE1,1 , TE0,1 , and TM0,1 modes in the transverse plane. The core radius is R = 5 µm. The arrows denote the corresponding mode polarization

Appendix E: Eigenmodes of Optical Fiber

Appendix E: Eigenmodes of Optical Fiber

409

E.2 Linearly Polarized Modes In above, we have discussed the rigorous bound states of the light inside the fiber. In many practical cases, the relative difference |( i − o )/ i | is of the order of 1%. The light inside such an optical fiber is not tightly confined and is almost linearly polarized, i.e., the linearly polarized (LP) modes. By applying the weakly guiding approximation i ≈ o to (E.13) and (E.14), the dispersion equation for the TE and TM modes is given by 1 J1 (qi R) 1 K 1 (qo R/i) + = 0, qi R J0 (qi R) qo R/i K 0 (qo R/i)

(E.19)

Similarly, the dispersion equations for the EH and HE modes are derived as EH modes: HE modes:

1 qi R 1 qi R

Jν+1 (qi R) 1 + Jν (qi R) qo R/i Jν−1 (qi R) 1 − Jν (qi R) qo R/i

K ν+1 (qo R/i) = 0, K ν (qo R/i) K ν−1 (qo R/i) = 0, K ν (qo R/i)

(E.20a) (E.20b)

with ν ≥ 1. Due to the weakly guiding approximation, the LP modes labeled by LPν−1,m with m ∈ N are degenerate. Below, we list the comparison of the LP modes with the conventional modes ν=1 ν=2 ν≥3

LP0,m LP1,m LPν−1,m

HE1,m TE0,m , TM0,m , HE2,m . EHν−2,m , HEν,m

The intensity pattern of the LPν−1,m mode exhibits 2(ν − 1) intensity maxima along the azimuthal direction and (m − 1) zero crossings (or m intensity maxima) in the radial direction. The intensity patterns of several selected LP modes are listed in Fig. (E.4). The fundamental mode LP0,1 corresponds to HE1,1 , which may be approximated by two perpendicularly polarized modes ⎧ J (q ρ) 0 i ⎪ eiβz−iωt for ρ ≤ R ⎨ J (q R) 0 i E(r, t) ∼ ex,y K (q ρ/i) ⎪ ⎩ 0 o eiβz−iωt for ρ > R K 0 (qo R/i)

.

(E.21)

In many cases, for the sake of simplicity the outside-core field is further replaced with e−γ(ρ−R) , which decays exponentially. The constant γ is fitted to the exact field at the core boundary, K 1 (qo R/i) . (E.22) γ = (qo /i) K 0 (qo R/i)

410

Appendix E: Eigenmodes of Optical Fiber

Fig. E.4 Intensity distribution of several selected LP modes in the transverse plane. The core radius is R = 5 µm

Appendix F

Solving the First-Order Linear Matrix Differential Equation

When studying open quantum systems, one often needs to solve a set of first-order linear differential equations, which can be written in a matrix form d R(t) = −MR(t) + B. dt All the variables are listed in the column matrix ⎛ ⎞ R1 (t) ⎜ R2 (t)⎟ ⎟ R(t) = ⎜ ⎝ ... ⎠. Rn (t)

(F.1)

(F.2)

The relations among different variables are given by the square n × n matrix ⎞ M1,1 (t) . . . M1,n ⎜ . ⎟ .. M(t) = ⎝ ... . .. ⎠ . Mn,1 (t) . . . Mn,n ⎛

(F.3)

In many cases, the matrix M is time independent, M(t) = M. The steady-state solutions of R(t → ∞) are related to the constant column matrix ⎛

⎞ B1 ⎜ B2 ⎟ ⎟ B=⎜ ⎝. . . ⎠ . Bn

(F.4)

We formally derive the exact solution of this matrix differential equation. First, we diagonalize the time-independent M via an invertible matrix D, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2

411

412

Appendix F: Solving the First-Order Linear Matrix Differential Equation

EM = D−1 MD, where all the eigenvalues {λk , k = 1, 2, ..., n} ments of EM , ⎛ λ1 . . . ⎜ .. . . EM = ⎝ . .

(F.5)

of M are given by the diagonal ele⎞ 0 .. ⎟ . .⎠

(F.6)

0 . . . λn

Using EM and D, (F.1) is rewritten as  d  EM t −1 e D R(t) = eEM t D−1 B. dt

(F.7)

After formally integrating the above equation, one obtains the solution −EM t R(t) = De−EM t D−1 R(0) + DE−1 )D−1 B, M (I − e

with the inverse matrix E−1 M

⎞ 0 .. ⎟ . . ⎠ −1 0 . . . λn

⎛ −1 λ1 . . . ⎜ .. . . =⎝ . .

(F.8)

(F.9)

We consider two cases: 1. Re(λk ) > 0 and B = 0. The real parts of eigenvalues {λk , k = 1, 2, ..., n} are all positive and the constant B is nonzero. We obtain the steady- state (“ss”) solutions as t → ∞, −1 R(ss) = R(t → ∞) = DE−1 M D B,

(F.10)

which is independent on the initial condition R(0). 2. λk0 = 0, Re(λk=k0 ) > 0 and B = 0. Only one of the eigenvalues is zero, λk0 = 0, while the real parts of others are positive. The constant B is zero. The master equation of the density matrix operator ρ falls in this case. The steady-state matrix elements of R(ss) is given by [R(ss) ]k1 = [D]k1 ,k0

k2

[D−1 ]k0 ,k2 [R(0)]k2 .

(F.11)

Index

A Absorbance, 358 Absorbing boundary condition, 105 Absorption coefficient, 358 Absorptivity, 358 Activity, 249 Affinity, 242 Allan deviation, 48 Annihilation operator, 313 Antibunching, 342 Antiunitary operator, 196 Aptamers, 242 Association reaction, 242 Atomic clocks, 48 Autler–Townes doublet, 305, 342, 364 Avogadro’s number, 249, 358

Bulk sensing, 235

C Canonical ensemble, 161 Cardano’s method, 228 Cauchy principal, 326 Cauchy–Schwarz inequality, 30 Cavity QED, 299, 345 Chaotic light, 31 Chemical potential, 249 Circuit QED, 346 Clausius–Mossotti relation, 16, 85 Coherent atomic state, 374 Coherent state, 318 Conservation of momentum, 175 Cooperativity, 348 Coulomb gauge, 6 Coupled spring-mass system, 107 Cramér–Rao bound, 376 Creation operator, 313 Critical coupling, 144, 167 Curvature loss, 136 Cylindrical harmonics, 394

B Baker–Hausdorff formula, 318 Balanced heterodyne method, 44 Beer–Lambert law, 358 Bessel’s equation, 393 Bistability, 220 Blackbody radiation, 49, 299 source, 318 D Bloch Dark count rate, 46 sphere, 307 Dead time, 46 vector, 306 Debye, 13 Bohr radius, 13 Degree of protonation, 246 Boltzmann Dephasing constant, 316 rate, 329 factor, 316 time, 331 Depolarization factor, 15 Born–Oppenheimer approximation, 352 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. Vollmer and D. Yu, Optical Whispering Gallery Modes for Biosensing, Biological and Medical Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-60235-2

413

414 Diabolic point, 271, 273 Dick effect, 49 Dicke states, 373 Dipole approximation, 301 Dirac comb, 214 delta function, 6 Displacement operator, 318 Dissociation process, 243 Dressed states, 320 Drude–Lorentz model, 67 Drude–Sommerfeld model, 63 Dynamical polarizability, 333

E Einstein’s A coefficient, 325 Electromagnetically Induced Transparency (EIT) , 210, 333, 335 Electro-optic effect, 176 Elementary charge, 13 Energy conservation, 175 Enthalpy, 250 Entropy, 250 Equilibrium association constant, 244 Equilibrium dissociation constant, 244 Exceptional point, 199, 271, 273 Extinction coefficient, 358

F Fermi’s golden rule, 324, 325 Fermi velocity, 63 Fiber-optic-telecommunication wavelength, 136 Finesse, 34, 130 Finite-difference time-domain, 102 Finite element method, 102 Fisher information, 376 Fluorescence, 354 Fock state(s), 316 Fokker–Planck equation, 336 Four-wave mixing, 192 Franck–Condon principle, 355 Fröhlich condition, 85 Free spectral range, 33, 130 Frequency chain, 213 Frequency tripling, 191 Fresnel theorem, 77 Full permutation symmetry, 180 Full width at half maximum, 24 Fundamental mode, 124, 407

Index G Gauge invariance, 5 Gaussian process, 27 Geiger-mode operation, 46 Gibbs free energy, 249 Greenberger–Horne–Zeilinger state, 377

H Hadamard’s lemma, 328 Hanbury Brown–Twiss experiment, 30 Hansch–Couillaud method, 54 Heat diffusion equation, 162 Heisenberg limit, 377 picture, 317 uncertainty principle, 370, 377 Hermite–Gaussian functions, 133 Heterodyne, 43 Holstein–Primakoff transformation, 323 Homodyne, 45 Hotspot, 285 Hybridization, 251 Hybrid modes, 406 Hybrid plasmonic waveguide, 80 Hyperparametric oscillations, 194

I Index ellipsoid, 176 Inversion symmetry, 180

J Jablonski energy diagram, 352 Jacobi–Anger expansion, 394 Jacobi–Anger identity, 39 Johnson noise, 42

K Kasha’s rule, 357 Kerr effect, 189 Kirchhoff’s voltage law, 36 Kretschmann configuration, 75

L Label-free sensing, 254 Lamé constants, 221 Lamb–Dicke parameter, 49 regime, 49 Lamb’s shift, 326

Index Landau–Zener method, 51 Langmuir isotherm, 245 Langmuir model, 243 Lasing droplets, 172 Levi–Civita symbol, 196 Lindblad superoperator, 329 Linear dynamic range, 40 Liouville–von Neumann equation, 306 Lock-in amplifier, 38 Longitudinal mode, 33 Lorentz local-field correction, 85 Lorentz local-field factor, 16 Lorentz–Lorenz equation, 16, 85 Lorentz oscillator model, 16, 65 Lorenz gauge, 5 Lowering operator, 304

M Mach–Zehnder interferometer, 375, 378 Magic wavelength, 21, 49 Mandel parameter, 318, 319 Markov approximation, 326, 327 Material loss, 136 Maxwell–Boltzmann distribution, 26 Mean free path, 63 Melting, 251 Michelson–Morley interferometer, 47 Microbubble, 262 Microfluidic channels, 264 Mie theory, 91 Mixed state, 305 Modal equation, 122 Mode splitting, 267 Mollow triplet, 342 Mott insulating state, 51

N Nanoantennas, 110 Noble metals, 62 Non-Markovian process, 341 Nuclear Magnetic Resonance (NMR), 331 Numerical aperture, 407 Nyquist formula, 42

O Occupancy, 244 Ohmic loss, 68 Onsager local-field correction, 16 Optical bistability, 189 Optical indicatrix, 176

415 Optical parametric oscillator, 186 Optical theorem, 99 Optomechanical (spring) oscillation, 271 Optomechanics, 217 Orbital angular momentum, 124 Oscillator strength, 20, 358 Otto configuration, 75 Over-coupling, 167

P Parametric down-conversion, 186 Parity transformation, 194 Partition coefficient, 249 Pauli matrices, 304 Periodic poling, 184 Permanent dipole moment, 352 Phase matching, 175, 181, 186 Phosphorescence, 354 Photonic molecules, 277 Photon recoil energy, 49 Physisorption, 242 Planck’s constant, 13 Plasma frequency, 63 Plasmon hybridization model, 108 Plasmonic focusing, 78 lasers, 80, 81 waveguides, 78 Pockels effect, 176 Point-dipole limit, 13 Poisson bracket, 313 Poissonian statistics, 318 Polarizability, 15 Polar molecules, 352 Pound–Drever–Hall method, 39 Power spectral density, 22 Poynting’s theorem, 9 Poynting vector, 8, 66 Principal diameter, 156 Purcell factor, 350 Pure dephasing rate, 329 Pure state, 305

Q Quantization volume, 311, 320 Quantum harmonic oscillator, 313 Langevin approach, 221 metrology, 369 projection noise, 309 regression theorem, 341

416 sensing, 369 yield, 357 Quantum Rosetta stone, 375 Quasi-static approximation, 83

R Rabi flopping, 304 frequency, 304 Radiation loss, 136 Raising operator, 304 Raman detuning, 334 resonance, 335 Raman lasing, 269 Ramsey fringes, 310, 330 method, 331 oscillation, 331 separated oscillatory field method, 309 Recovery time, 46 Refractive index units, 235 Relaxation rate, 329 Responsivity, 46 Riccati–Bessel functions, 94 Rotating-Wave Approximation (RWA), 303

S Saturation photon number, 347 Scalar potential, 5 Schawlow–Townes linewidth, 24 Schottky formula, 42 Second-harmonic generation, 183 Second quantization, 310 Sellmeier formula, 137 Shot noise limit, 280 Single-shot experiment, 376 Skin depth, 71 Slowly varying envelope approximation, 37, 53, 179 Spaser, 82 Square-law photodetector, 42 Squeeze coherent state, 372 operator, 371 state, 371 Standard chemical potential, 249 Standard quantum limit, 280, 377 Stimulated Raman scattering, 269 Stokes photon, 269 Stokes shift, 357

Index Strickler–Berg equation, 362 Sub-Poissonian statistics, 318 Supercontinuum generation, 215 Super-Poissonian statistics, 318 Surface absorption, 138 Surface plasmon optics, 78 Surface plasmon polaritons, 67 Surface scattering, 137 Surface sensing, 235

T Thermal conductivity, 162 diffusivity, 162 equilibrium, 328 Third-harmonic generation, 191 Thomson cross-section, 66 Three-wave mixing, 180 Time dilation, 50 Time reversal transformation, 196 Total internal reflection, 119

U Under-coupling, 167 Universal gas constant, 249

V Vacuum Rabi splitting, 321, 324, 347 state, 316 Van’t Hoff equation, 251 Vector potential, 4 Vector spherical harmonics, 120 Vibronic transitions, 355 Voigt integral, 26 Voigt’s contracted notation, 177

W Wave–particle duality, 2 Weighted residual method, 107 Weisskopf–Wigner theory, 325 Wigner 3 j symbols, 183 Wiener–Khintchine theorem, 22 Wigner small d-matrix, 374 Wigner’s theorem, 196

Z Zero-point energy, 316