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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Foundations of Nonlinear Optical Microscopy Concise yet comprehensive resource presenting the foundations of nonlinear optical microscopy Foundations of Nonlinear Optical Microscopy brings together all relevant principles of nonlinear optical (NLO) microscopy, presenting NLO microscopy within a consistent framework to allow for the origin of the signals and the interrelation between different NLO techniques to be understood. The text provides rigorous yet practical derivations, which amount to expressions that can be directly related to measured values of resolution, sensitivity, and imaging contrast. The book also addresses typical questions students ask, and answers them with clear explanations and examples. Readers of this book will develop a solid physical understanding of NLO microscopy, appreciate the advantages and limitations of each technique, and recognize the exciting possibilities that lie ahead. Foundations of Nonlinear Optical Microscopy covers sample topics such as: * Light propagation, focusing of light, pulses of light, classical description of light-matter interactions, and quantum mechanical description of light-matter interactions * Molecular transitions, selection rules, signal radiation, and detection of light * Multi-photon fluorescence and pump-probe microscopy * Harmonic generation, sum-frequency generation, and coherent Raman scattering Senior undergraduate and graduate students in chemistry, physics, and biomedical engineering, along with students of electrical engineering and instructors in both of these fields, can use the information within Foundations of Nonlinear Optical Microscopy and the included learning resources to gain a concise yet comprehensive overview of the subject.
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
Acknowledgments
1 Light: Electromagnetic Radiation
1.1 Introduction
1.2 Electromagnetic Fields
1.3 Transverse Waves
1.4 Polarization States
1.5 Reflection and Transmission at Interfaces
1.6 Transformation of the Field by a Lens
1.7 Intensity and Energy
Bibliography
Note
2 Focused Light
2.1 Introduction
2.2 Interference of Multiple Waves
2.3 The Scalar Focal Field
2.4 The Focused Vector Field
2.5 Aberrations
Bibliography
3 Ultrafast Pulses
3.1 Introduction
3.2 Optical Pulses in the Frequency Domain
3.3 Optical Pulses in the Time Domain
3.4 Measurement of Pulse Duration
3.5 Pulse Properties and Nonlinear Optical Signals
3.6 Noise
Bibliography
4 Classical Model of Nonlinear Optics
4.1 Introduction
4.2 Linear Optical Response
4.3 Nonlinear Optical Response
4.4 Properties of Nonlinear Susceptibilities
Bibliography
5 Semiclassical Model of Nonlinear Optics
5.1 Introduction
5.2 Quantum Mechanical Concepts
5.3 Density Matrix Formalism
5.4 Perturbation Expansion for the Density Matrix
5.5 Linear and Nonlinear Susceptibilities
5.6 Quantization of the Electromagnetic Field
Bibliography
6 Nonlinear Optical Signals: Molecular Transitions, Signal Radiation, Propagation, and Detection
6.1 Introduction
6.2 Molecular Transitions
6.3 Spatial Properties of Radiated Signal
6.4 Signal Radiation and Detection
Bibliography
Notes
7 Multiphoton Absorption and Fluorescence Microscopy
7.1 Introduction
7.2 Physics of Nonlinear Absorption
7.3 Optical Sectioning and Imaging Resolution
7.4 Direct Detection of Multiphoton Absorption
7.5 Fluorescence Detection of Multiphoton Absorption
Bibliography
8 Pump–Probe Microscopy
8.1 Introduction
8.2 Pump–Probe Signals in Microscopy
8.3 Electronic Pump–Probe Spectroscopy
8.4 Detection of Pump–Probe Signals
8.5 Imaging Based on the Photothermal Effect
Bibliography
9 Harmonic Generation Microscopy
9.1 Introduction
9.2 Molecular Response in Harmonic Imaging
9.3 SHG Imaging of Fibrillar Structures
9.4 THG Imaging of Discontinuities
Bibliography
Note
10 Sum‐Frequency Generation Microscopy
10.1 Introduction
10.2 Molecular Vibrations
10.3 Molecular Vibrations in Sum‐Frequency Generation
10.4 Collinear Sum‐Frequency Generation Microscopy
10.5 Third‐Order Sum‐Frequency Generation Microscopy
Bibliography
11 Coherent Raman Scattering Microscopy
11.1 Introduction
11.2 The Raman Effect
11.3 Coherent Raman Scattering
11.4 Raman Spectroscopy of Lipids
11.5 Imaging Properties of CARS
11.6 Imaging Properties of SRS
11.7 Spectral Resolution
Bibliography
Appendix A: Fourier Transform Relationships
Appendix B: Wavenumbers Unit
Appendix C: Power Spectral Density of White Noise
Appendix D: Fermi's Golden Rule
Appendix E: Population Dynamics with Relaxation
Appendix F: Stimulated Raman Scattering with Quantized Fields
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 Generalized Jones vectors of common polarization states.
Table 1.2 Phase shift properties of the propagator when multiplied by a f...
Chapter 2
Table 2.1 First nine Zernike polynomials. The second column gives the ANSI i...
Chapter 3
Table 3.1 Corrections to expressed in terms of the wavelength‐dependent i...
Table 3.2 GVD in fs
2
/mm and TOD in fs
3
/mm for several optical glasses.
Table 3.3 Measured group delay dispersion (GDD) for several objective lense...
Table 3.4 Properties of intensity autocorrelation function of Gaussian and ...
Table 3.5 Forms of voltage noise relevant to optical measurements in a micr...
Chapter 4
Table 4.1 Nonresonant third‐order nonlinear susceptibilities of various mat...
Chapter 6
Table 6.1 Absorption cross section and fluorescence lifetime of selected mol...
Table 6.2 Gouy phase mismatch for several coherent nonlinear optical imagin...
Chapter 7
Table 7.1 Two‐photon absorption cross sections () of selected molecular fl...
Table 7.2 Full width at half maximum (FWHM) of the intensity excitation vol...
Table 7.3 Emission properties of various fluorophores used in multiphoton e...
Chapter 8
Table 8.1 Thermal properties of materials at , including density , specif...
Chapter 11
Table 11.1 Measured first differential Raman cross sections of vibrational ...
Table 11.2 Selected Raman spectral signatures of lipids.
Appendix A
Table A.1 Fourier transform relationships.
List of Illustrations
Chapter 1
Figure 1.1 The electric field as a vector field. The vector is a position ...
Figure 1.2 Plane wave in vacuum. The electric and magnetic field vectors are...
Figure 1.3 Dipole radiation. (a) Reference frame for the radiating dipole, w...
Figure 1.4 Sketch of a spherical wave. The wave originates from a source pla...
Figure 1.5 Linear (a) and circular (b) polarization states of a plane wave t...
Figure 1.6 Reflection and transmission of an incident plane wave with an ele...
Figure 1.7 Transmission of a plane wave at an interface for the condition ....
Figure 1.8 A plane wave incident on a plano‐convex lens with radius of curva...
Figure 1.9 The incident planar wavefront of the field is transformed by a ...
Figure 1.10 Rotation of the field vector at the reference surface. (a) Incom...
Figure 1.11 Electric field distribution on the spherical surface for an ‐po...
Chapter 2
Figure 2.1 Radiation produced by an infinite sheet of dipoles at is observ...
Figure 2.2 Wave propagation according to the Huygens–Fresnel principle. (a) ...
Figure 2.3 Diffraction of waves launched from the spherical reference surfac...
Figure 2.4 Focal field in the scalar approximation, evaluated using equation...
Figure 2.5 Lateral dependence of the scalar focal field and intensity. (a) N...
Figure 2.6 Axial dependence of the scalar focal field and intensity. (a) Nor...
Figure 2.7 Paraxial Gaussian beam near the focal region. (a) Beam radius (...
Figure 2.8 Amplitude and phase of the focused field in the focal plane of a ...
Figure 2.9 Magnitude of the ‐ and ‐polarized components of the focal field...
Figure 2.10 Amplitude (a) and phase (b) of the component in the plane ne...
Figure 2.11 Amplitudes of various focused beams in the focal plane of a NA =...
Figure 2.12 (a) Schematic of the optical path of the Schwarzschild–Cassegrai...
Figure 2.13 Comparison of focal field of a refractive objective and a reflec...
Figure 2.14 Amplitude distribution in the focal plane of the field focused b...
Figure 2.15 A spherical wavefront is aberrated with a combination of tilt ...
Figure 2.16 Wavefront aberrations at the exit pupil of (a) , (b) , and (c)...
Chapter 3
Figure 3.1 Spectrum of a laser pulse: (a) Spectral intensity profile of an...
Figure 3.2 Optical properties of common silica glasses: (a) Index of refract...
Figure 3.3 Power spectrum of Gaussian and hyperbolic secant laser pulses. (a...
Figure 3.4 Sketch of the amplitude (black) and phase (gray) of a given spect...
Figure 3.5 (a) Frequency domain representation of a laser field modeled as t...
Figure 3.6 Temporal profile of a Gaussian pulse (black line) and a secant hy...
Figure 3.7 Frequency and temporal properties of a linearly chirped Gaussian ...
Figure 3.8 Broadening factor of a Gaussian pulse, defined as , as a functio...
Figure 3.9 General approaches for measuring the temporal pulse width. (a) Ma...
Figure 3.10 Linear autocorrelation. (a) The amplitude profile of two Gaussia...
Figure 3.11 Second‐harmonic generation intensity autocorrelation measurement...
Figure 3.12 Interferometric autocorrelation of a Gaussian pulse of spectra...
Figure 3.13 Laser noise. (a) Sketch of relative intensity noise (), includi...
Chapter 4
Figure 4.1 Harmonic oscillator. A particle with charge and mass is bound...
Figure 4.2 Spectral dependence of the displacement near a resonance of fre...
Figure 4.3 The field is incident upon a thin slab of material with thickne...
Figure 4.4 (a) Asymmetric anharmonic potential (solid black) obtained when t...
Figure 4.5 Nonlinear polarization density as a function of the incoming el...
Figure 4.6 A structure with a symmetry axis (a) and a structure with a s...
Chapter 5
Figure 5.1 Spatial distribution of a wavefunction representing a particle bo...
Figure 5.2 (a) Schematic representation of the time variables in the integra...
Figure 5.3 Elements of double‐sided Feynman diagrams for the calculation of
Figure 5.4 Double‐sided Feynman diagrams for a first‐order light–matter inte...
Figure 5.5 Spectral dependence of the linear susceptibility. (a) Real (black...
Figure 5.6 Three‐level system used to model the SFG signal, with . Incident...
Figure 5.7 Double‐sided Feynman diagrams that contribute to the SFG signal w...
Figure 5.8 Four‐level system used to model the THG signal. The signal freque...
Figure 5.9 Double‐sided Feynman diagrams that contribute to the THG signal i...
Figure 5.10 Double‐sided diagrams for the process of linear absorption for a...
Chapter 6
Figure 6.1 Light‐induced transitions between states. The transition rate of ...
Figure 6.2 Inelastic light scattering. (a) The scattered field () has a low...
Figure 6.3 Vibrational Raman (black) and Fourier transform infrared (FTIR, r...
Figure 6.4 Variation in dipole moment and polarizability for vibrational mod...
Figure 6.5 Two radiating dipoles driven by an incoming plane wave . The rad...
Figure 6.6 Phase matching between two dipoles that are driven at and radia...
Figure 6.7 Index of refraction for two types of tissue: serosa from colorect...
Figure 6.8 Relative intensity of the nonlinear signal as a function of , wh...
Figure 6.9 Gouy phase mismatch along the optical axis for the SHG process fo...
Figure 6.10 Gouy phase mismatch along the optical axis for the CARS process ...
Figure 6.11 A molecular dipole at the location in the focal volume radiate...
Figure 6.12 Radiation profiles of the coherent nonlinear polarization. The C...
Figure 6.13 SHG radiation profiles for (a) a thin slab in the plane and (b...
Figure 6.14 Radiation profiles in fluorescence microscopy. (a) Radiation pat...
Figure 6.15 Comparison of the light/matter energy flow in SRS and homodyne d...
Figure 6.16 Dependence of heterodyne NLO signals on the intensity of the loc...
Chapter 7
Figure 7.1 Multiphoton absorption. Jablonski diagrams of the 2PA (a) and 3PA...
Figure 7.2 Evolution of the combined radiation‐matter density operator durin...
Figure 7.3 Intensity loss as a function of propagation distance due to absor...
Figure 7.4 Electronic transitions in molecules for 1PA and 2PA. (a) Schemati...
Figure 7.5 Intensity of the focal excitation volume as a function of the dim...
Figure 7.6 Total lateral excitation intensity as a function of axial positio...
Figure 7.7 Image formation in multiphoton fluorescence microscopy. (a) Objec...
Figure 7.8 Detection of single color two‐photon absorption using a sinusoida...
Figure 7.9 Contributions to the NTA loss signal. (a) Jablonski diagrams corr...
Figure 7.10 Dual color 2PA imaging of astaxanthin in shrimp internal organs....
Figure 7.11 Sketch of two‐photon excited fluorescence in the context of the ...
Figure 7.12 Depth resolved imaging with multi‐photon excited fluorescence. (...
Figure 7.13 Calculated relative fluorescence signals under 1PE, 2PE, and 3PE...
Figure 7.14 Signal‐to‐background ratio () as a function of penetration dept...
Chapter 8
Figure 8.1 Changes in the probe beam transmission due to pump‐induced refrac...
Figure 8.2 Schematic of absorptive pump–probe microscopy. (a) Layout of the ...
Figure 8.3 Principles of excited state absorption, in which a pump pulse tra...
Figure 8.4 Principle of the stimulated emission process. A pump pulse transf...
Figure 8.5 Imaging of chromoproteins with stimulated emission contrast, usin...
Figure 8.6 Principle of the ground state depletion process. (a) A pump pulse...
Figure 8.7 Sketch of the time‐resolved contributions to the electronically r...
Figure 8.8 Transient absorption microscopy (TAM) of melanins in tissue. (a) ...
Figure 8.9 Sketch of various pump–probe processes on the ground state and ex...
Figure 8.10 Principle of modulation transfer, shown here for a loss signal o...
Figure 8.11 Distribution of temperature along the radial coordinate in an aq...
Figure 8.12 Temperature change of a point removed from a heated point sour...
Figure 8.13 Effect of the position of a small thermal lens in the focal volu...
Figure 8.14 Photothermal (PT) imaging in the visible range. (a) PT image of ...
Figure 8.15 High‐resolution infrared (IR) imaging with photothermal microsco...
Chapter 9
Figure 9.1 Harmonic generation. (a) Jablonski diagram for SHG. (b) Jablonski...
Figure 9.2 Selected spectral features in the near ultraviolet absorption spe...
Figure 9.3 Schematic illustration of the effect of molecular orientation on ...
Figure 9.4 Orientation of the monomer and the supra‐molecular structure. (a)...
Figure 9.5 Simulation of the SHG signal intensity for a water immersion ob...
Figure 9.6 Structure of collagen. Tropocollagen consists of three polypeptid...
Figure 9.7 Several relevant orientations of fibrillar collagen in microscopy...
Figure 9.8 SHG imaging and polarity of fibrillar structures. (a) Signals fro...
Figure 9.9 SHG polarimetry. (a) Alignment of collagen fibril in the laborato...
Figure 9.10 SHG imaging of cholesterol monohydrate microcrystals in atherosc...
Figure 9.11 Model for estimating the number of SHG photons from a single col...
Figure 9.12 THG radiation profiles. Normalized radiation profiles from a sph...
Figure 9.13 Sample geometries that give rise to nonvanishing THG signals. (I...
Figure 9.14 Multiharmonic imaging of the spinal cord from a mouse. (a) SHG i...
Chapter 10
Figure 10.1 Molecular bond vibration as an harmonic oscillator. (a) Two mass...
Figure 10.2 (a) Methylene mode where the position of the carbon atom is fixe...
Figure 10.3 Stretching vibration of the hydroxyl group, assuming that the ox...
Figure 10.4 Overtones and combination bands. (a) Spectral representation of ...
Figure 10.5 Fermi resonance. (a) A vibrational mode mixes with the first o...
Figure 10.6 (a) Jablonski diagram of the SFG process, where and are inpu...
Figure 10.7 Two Feynman diagrams (a) and (b) of pathways that contribute to ...
Figure 10.8 Carbon–hydrogen groups in the molecular frame . (a) Methanetriy...
Figure 10.9 Double‐sided Feynman diagrams of the TSFG process with a resonan...
Figure 10.10 Spectral response of the nonlinear susceptibility. (a) Real and...
Figure 10.11 Several implementations of the SFG microscope. (a) Wide‐field i...
Figure 10.12 Excitation field for SFG, using an reflective objective with in...
Figure 10.13 Simulated SFG radiation profiles for a slab of thickness (a),...
Figure 10.14 Spectrally resolved mapping. (a) collection of a three‐dimensio...
Figure 10.15 Spectral mapping with SFG microscopy. (a) SFG image of a cellul...
Figure 10.16 Tensor elements of molecular collagen. (a) Orientation of a met...
Figure 10.17 (a) SFG image of collagen in rat tail tendon tissue, when is ...
Figure 10.18 SFG anisotropy of collagen I in rat tail tendon as a function...
Figure 10.19 Excitation field for TSFG, using an reflective objective with i...
Figure 10.20 Simulated TSFG radiation profiles for spherical object of diame...
Figure 10.21 (a) TSFG spectral dependence of mineral oil, water, and heavy w...
Chapter 11
Figure 11.1 Raman‐shifted scattering contributions. The Raman effect gives r...
Figure 11.2 Quantum pathways in the Raman process. (a) Jablonski diagram and...
Figure 11.3 Frequency contributions in (dual‐color) coherent Raman scatterin...
Figure 11.4 Jablonski diagrams for the (a) dual‐color CARS and (b) SRS proce...
Figure 11.5 Eight double‐sided diagrams that represent the quantum pathways ...
Figure 11.6 Four of the eight double‐sided diagrams that represent the quant...
Figure 11.7 SRS spectrum of two simulated Raman lines () with , , , and
Figure 11.8 CARS spectra for the same Raman lines as in Figure 11.7. (a) Rea...
Figure 11.9 Relative resonant CARS signal as a function of , the fraction o...
Figure 11.10 Resonant enhancement effects in CARS and SRL. Simulations assum...
Figure 11.11 Raman spectra of selected fatty acids, a wax ester, cholesterol...
Figure 11.12 Phospholipids and lipid bilayers. (a) Fatty acid in the molecul...
Figure 11.13 Simulated CARS intensity for a spherical particle of isotropic ...
Figure 11.14 (a) Spectral phase effects in CARS. CARS lineshape (solid black...
Figure 11.15 Shadowing effects in forward‐detected CARS. (a) Cross‐sectional...
Figure 11.16 Difference between the forward and backward detecte...
Figure 11.17 Polarization dependence of the CARS signal for the symmetric
Figure 11.18 CARS signal strength estimated from a cylindrical sample volume...
Figure 11.19 Predicted SRL signal in photons/ for excitation and sample par...
Figure 11.20 Effective spectral excitation profile for different pulse con...
Figure 11.21 Spectral resolution in CRS. The first column shows (black), w...
Figure 11.22 Relative resonant and nonresonant contributions to the CARS...
Figure 11.23 Principle of increasing the spectral resolution of the effectiv...
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Acknowledgments
Begin Reading
Appendix A: Fourier Transform Relationships
Appendix B: Wavenumbers Unit
Appendix C: Power Spectral Density of White Noise
Appendix D: Fermi's Golden Rule
Appendix E: Population Dynamics with Relaxation
Appendix F: Stimulated Raman Scattering with Quantized Fields
Index
End User License Agreement
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Foundations of Nonlinear Optical Microscopy
Eric Olaf Potma
University of California,
Irvine,
United States
Copyright © 2024 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
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Preface
Nonlinear optical microscopy combines nonlinear optics (NLO) with high numerical aperture objective lenses, making it possible to examine the nonlinear optical response of materials at a spatial resolution of a few micrometers or better. One of the earliest implementations of this idea dates back to 1975 when second‐harmonic generation signals were collected from crystalline grains with micrometer‐scale resolution. Other early examples include a study from 1982, which used coherent anti‐Stokes Raman scattering signals to generate microscopic images of plant cells. Yet, despite setting impressive precedents, most of these original studies remained isolated achievements that did not immediately galvanize new technological developments or applications. That changed in 1990, almost thirty years after the invention of the laser light source, when a group of researchers from Cornell University published their now seminal paper on two‐photon excited fluorescence microscopy. By clearly showing the utility of NLO excitation for rapid depth‐resolved imaging of biological samples, the 1990 work set the stage for further developments, thereby inaugurating the budding field of nonlinear optical microscopy.
The rapid developments that followed the genesis of the two‐photon excited fluorescence microscope have resulted in reliable commercial nonlinear optical imaging solutions that can now be found in public and private research facilities across the globe. The impact of nonlinear optical microscopy is vast and particularly palpable in biological imaging, where the NLO microscope has grown into a standard research tool. Current imaging platforms are relatively easy to use so that advanced knowledge of the mechanisms that give rise to the observed nonlinear optical signals is optional. However, the underlying physics on which NLO imaging techniques are based is fascinating. A better understanding of the processes that give rise to the recorded signals is helpful for interpreting the information contained in the image while also driving technical improvements and stimulating new developments. For a student new to NLO microscopy, learning about the origins of the signals can be daunting. The collection of theoretical models that describes the signals in NLO microscopy borrows elements from electromagnetic theory, wave optics, and nonlinear optical spectroscopy. Although such elements can be found in the literature, a single textbook that presents the foundations of NLO microscopy within a cohesive framework has been absent thus far. This lack of a single resource for the physics of NLO microscopy formed the motivation for this book.
As a graduate student embarking on a mission to develop new nonlinear optical imaging tools in the late 1990s, the number of resources on the topic was scant and scattered. To get a better handle on the origin of nonlinear optical signals, I spent copious hours studying textbooks on nonlinear optics and spectroscopy. In particular, the excellent book on nonlinear optics by Robert Boyd provided several pillars on which to build a framework for NLO microscopy. Similarly, my copy of Shaul Mukamel's book on nonlinear optical spectroscopy is tattered due to my excessive thumbing through its pages. Both books left their indelible mark on my writings. I also took a course on nonlinear optics by Jasper Knoester, which offered additional tools and context. His course notes inspired several passages in this book. I further absorbed some of the advanced knowledge about NLO spectroscopy within the group of Douwe Wiersma at the University of Groningen, the Netherlands. Parts of this knowledge have percolated into this volume. During my graduate career, the works by Tony Wilson and Colin Sheppard provided key insights into focusing of light and image formation in the optical microscope. Later on, I learned a lot about propagation of light through microscope systems from the works by Peter Török, and elements of his models can be found in this publication. Phase matching is another important concept in NLO microscopy, yet a corresponding theory was lacking in the late 1990s. As a postdoc, I was fortunate to be part of an inquisitive research team headed by X. Sunney Xie at Harvard University, through which I came to appreciate the intriguing and somewhat fortuitous phase‐matching conditions in the focus of a microscope objective lens. Among many other contributions, Ji‐Xin Cheng deserves credit for his insights into phase matching, and his influence can be seen throughout this work. The many discussions with Ara Apkarian at the University of California, Irvine, have further refined my understanding of light, matter and their mutual interaction, which helped me tie up a few of the remaining loose ends for this book.
The topics in this book are not exhaustive, and this volume does not cover additional aspects of NLO microscopy, such as optical design, super‐resolution, light scattering in tissues, practical biological applications, and image analysis. Instead, I have tried to focus on the fundamentals of light propagation, signal generation, and detection – topics that take a little effort to comprehend and leave many students in this field with questions, the same questions that I wrestled with as a student. In collecting several of the pieces mentioned above and weaving them together into a cohesive tapestry, I hope to have delivered a resource that was missing on my bookshelf when I started my PhD career.
Eric Olaf Potma
Irvine18 April 2023
Acknowledgments
I had plenty of scattered notes on nonlinear optical microscopy. These could serve as the basis for a book on the topic, at least that is what I thought when I submitted the book proposal to my publisher. It quickly became clear that the notes were rather useless, and that random class notes are no substitute for a structured textbook. It was also evident that my initial estimate of one year of writing time was pure folly, as it took me more than twice as long before I had compiled a decent draft of the book. And as the manuscript progressed, I started to doubt the wisdom of the whole project. I needed a reality check on the approach and usefulness of the book, as well as scrutiny of the notation and validity of the many mathematical expressions. I have been fortunate to be part of a supportive community of students, colleagues, and friends, and this project has benefitted tremendously from the help they offered. My coworkers and colleagues at the University of California, Irvine, have provided valuable feedback on the selection of topics and have made many important suggestions for improving the book's contents. In particular, I would like to thank Dave Knez, Yryx Luna Palacios and Salile Khandani for their careful reading and comments on several book chapters. Abid Anjum Sifat was instrumental in checking some of the derivations as well as calculating radiation profiles that ended up as figures in the book. Yong Li collected SRS data for another figure. Janaka Ranasinghesagara wrote the code for calculating far‐field radiation and provided critical feedback on Chapters 1 and 2. I am also grateful to Daryl Preece for going through the first two chapters of the book with surgical precision. Similarly, I thank Dima Fishman for his careful evaluation of Chapter 3 and for sharing his thoughts on Chapter 4. I also enjoyed several discussions with Shaul Mukamel that have improved my treatment of nonlinear optical signals with quantized fields in Chapter 5.
Numerous members of the optical microscopy community have helped with fact‐checking and proofreading of chapter drafts. A big thank you goes out to Faezeh Tork Ladani, who scrutinized parts of Chapter 1. I greatly appreciate a careful read through of Chapter 3 by Jeff Squier (Colorado School of Mines), and the extensive notes on Chapter 4 by Randy Bartels (Morgridge Institute). I am thankful to Chris Xu (Cornell University) for making several suggestions and comments on Chapter 7. Discussions with Jesse Wilson (Colorado State University, Fort Collins) informed some of the material discussed in Chapter 8. I truly appreciate the critical notes by Yeran Bai (University of California, Santa Barbara), Delong Zhang (Zhejiang University), Herve Rigneault (Institute Fresnel) and Guillaume Baffou (Institute Fresnel) on the topic of photothermal imaging in Chapter 8. I am also indebted to Israel Rocha‐Mendoza (CICESE) and Pablo Loza‐Alvarez (ICFO), who spent time to sift through Chapter 9 and offered many suggestions for improvements. Garth Simpson (Purdue University) combed through Chapter 10 and caught several oversights. I am extremely thankful for his help. Dan Fu (University of Washington) and Chi Zhang (Purdue University) pointed out various refinements for Chapter 11, and Charles Camp (NIST) filtered out several mistakes in the same chapter, for which I am grateful. The conversations with Wei Min (Columbia University) about the connection between 2PA and SRS were stimulating, and some of these ideas have contributed to elements of Chapter 11 and Appendix F. Finally, I thank Andrea Miller for providing art direction for the book's front cover.
1Light: Electromagnetic Radiation
1.1 Introduction
In order to understand the imaging properties of the nonlinear optical microscope, we first have to have a basic understanding of light itself. In particular, a description of light in terms of propagating waves is needed to model the formation of the tightly focused volume. Fortunately, such a description is well established, and in this chapter, we review two useful forms of propagating light, namely the plane wave and the spherical wave. We also summarize helpful notations for the polarization state of light, and briefly discuss relevant expressions for reflected and transmitted light. The final aim of this chapter is to study the way in which a thin lens modifies an incident plane wave.
1.2 Electromagnetic Fields
The study of electromagnetic radiation is fascinating, and many aspects of electromagnetic radiation are worthy topics of discussion. In this book, however, we focus only on the bare essentials. Our goal is to find good descriptions of propagating light, which we can then use to model the tightly focused volume in the microscope. To arrive at such descriptions, we first have to glance at Maxwell's equations and the wave equation that follows from them.
1.2.1 Vector Fields
Light is electromagnetic radiation. In a classical description, light radiates through space as propagating electromagnetic waves. A wave is defined through its electric and magnetic fields, which oscillate in time in a synchronized manner. In vacuum, the electric and magnetic fields are indicated as and , respectively, which are position dependent vector fields that vary as a function of time. In Cartesian coordinates, defined by the axes , and , the electric field takes on the following form
where are unit vectors that point in the directions, respectively. The electric field is expressed in SI units of . At a given point in space, the electric field is a vector with projections of magnitude along the respective Cartesian coordinates, see Figure 1.1. The projections are also referred to as the orthogonal polarization components of the field. The corresponding expression for the magnetic field is similar, with replaced by , which has units of V · s/m2.
Figure 1.1 The electric field as a vector field. The vector is a position vector indicating the location at which the field is considered. The field vector at location has projections of magnitude , and along the () coordinates, respectively.
Electromagnetic waves in vacuum propagate at the speed of light, defined as . The quantity is called the vacuum permeability, which in classical terms relates to the magnetic inductance of a vacuum. Similarly, , called vacuum permittivity, is a measure of the capacitance of a vacuum. Together, and pose a limit to how fast an electromagnetic disturbance can travel through a vacuum. The established value for the vacuum permeability is . Using , the value for the vacuum permittivity is .
1.2.2 Wave Equation in Vacuum
Electromagnetic waves, in the form of the electric and magnetic fields, are not arbitrarily defined. Instead, the fields are described by a set of equations known as Maxwell's equations. In vacuum, the equations in differential form are written as
where the curl operator indicates the circulation density of the field and the divergence operator denotes the flux density of the field. Here, we have written the electric field and magnetic field in shorthand form as and , respectively. Maxwell's equations show that the electric and magnetic fields are interdependent. For instance, equation (1.2), known as Maxwell–Faraday's law, states that a time‐varying magnetic field induces an electric field. Similarly, a time‐varying electric field gives rise to a magnetic field, as described by equation (1.3). The remaining two expressions, equations (1.4) and (1.5), indicate that in vacuum the flux density of the electric and magnetic fields is zero.
Maxwell's equations can be rewritten to bring out the wave character of the electromagnetic field. For this purpose, we take the curl of equation (1.2) and use the vector identity . We then use the fact that is zero, as per equation (1.4), and use equation (1.3) to write the curl of in terms of the time derivative of . These operations result in the following equation
This expression shows that the second‐order derivative of the field in space is proportional to the field's second‐order derivative in time, a characteristic of a wave equation. Equation (1.6), therefore, is known as Maxwell's wave equation in vacuum. A similar form can be derived for the magnetic field.
We are interested in time‐harmonic solutions of the form , in which the spatial part of the solution is expressed as , a complex quantity, whereas the temporal part is described by .1 More generally, we can write a monochromatic, time‐harmonic field mode that oscillates at angular frequency as
where the quantity is introduced as a matter of convenience, in order to avoid the explicit use of the prefactor. The electric field is a real quantity, but it is expressed here as a sum of complex functions. For mathematical purposes, it is often more convenient to work with the complex function than the full expression of the field given in (1.7). The actual (real) electric field can then be obtained by taking the real part of the complex function .
Example 1.1 Show that expression (1.7) represents a real quantity.
Solution The complex conjugate of is , which means that expression (1.7) can be written as
The field in equation (1.7) thus equals , which is a real quantity.
By substituting the complex time‐harmonic field into equation (1.6), the wave equation can be rewritten as
where is called the angular wave number. Equation (1.8) is known as the vector Helmholtz equation, which expresses the spatial properties of the field. If a solution for can be found that complies with equation (1.8), it is also a valid solution of Maxwell's equations. Section 1.3 discusses several useful solutions of the Helmholtz equation.
1.2.3 Fields and Matter
We can measure the presence of electromagnetic waves because its electric and magnetic fields exert a force on electric charges. In general, the Lorentz force experienced by a charge moving at a velocity in the presence of an electromagnetic field is given as
Because the electromagnetic field interacts with charges, it can bring about change to matter. Of particular relevance to the topic of this book is the force experienced by the electrons bound to atoms that make up materials, such as optical glasses or biological samples inspected in microscopy experiments. Due to the action of the field, the electrons will move under the influence of the time‐periodic electromagnetic force, thereby inducing a time‐varying polarization in the material. Vice versa, the presence of charges can also alter the properties of the electromagnetic field. For instance, the induced polarization in the material forms the basis for the exchange of energy between fields and matter, as is the case in the process of optical absorption. In addition, the induced motion of charges in matter is also responsible for the observed propagation effects of electromagnetic waves as they encounter materials, such as the redirection of the wave's propagation direction at interfaces or the focusing of waves by lenses.
To understand these effects, we first need to consider the behavior of fields in the presence of charges and currents in a certain volume, as well as how the fields might, in turn, alter the material properties within that volume. Maxwell's equations (1.2)–(1.5) are only valid for electromagnetic fields in vacuum. To include the effects of current density and charge density on the fields, as well as the response of the material to the presence of the fields, the equations can be expanded as
The two new quantities are the electric displacement field and the magnetizing field , which are defined through the following so‐called “constitutive relations”
The electric displacement field describes the combined effect of the electric field and the polarization density in the material (in units of ) caused by . The field in equation (1.15) now includes both the magnetizing field as well as the material's magnetization density (in units of ) in the presence of the magnetizing field. In vacuum , i.e. the field is directly proportional to the magnetizing field. This is no longer the case in matter, and to indicate this difference, the field is often referred to as magnetic induction or magnetic flux density. In this book, we refer to as the magnetic field, i.e. the field that results from the magnetizing field and/or the magnetization of matter.
For nondispersive and isotropic materials, the linear response of the material to applied electric and magnetic fields is given as
Equation (1.16) shows that the polarization of the material grows as the electric field grows. The strength of the polarization is further determined by , a dimensionless quantity called the electric susceptibility of the material. Materials that are more responsive to the electric field have higher values, producing a higher polarization density. Equation (1.17) shows a similar relation for the magnetization of the material, which is directly proportional to the magnetizing field and , the magnetic susceptibility of the material. Using the definitions for and , equations (2.84) and (1.15) can be recast as
where the relative permittivity and the relative permeability are defined through
If the material is spectrally dispersive, i.e. the material properties change with , then the quantities above are functions of as well. The resulting quantities and are important material parameters, as they describe how the electric and magnetic fields permeate the medium relative to how the fields distribute in a vacuum. Note that and are macroscopic quantities in that they describe the field properties within a volume of matter that is much larger than that occupied by a single charge or atom, i.e. they represent quantities averaged over numerous atoms that make up a volume.
For a homogeneous material, the joint effect of the relative permittivity and permeability is captured by the refractive index, defined as
The refractive index is an important material‐dependent quantity in the context of electromagnetic field propagation. The tilde indicates that the refractive index is a complex quantity, and this quantity can generally be written in terms of its real and imaginary parts
where the index of refraction is related to the speed of light propagation in matter and the extinction coefficient is related to the dissipation of light energy. Recalling that , we see that there is a clear connection between and the speed of light. Whereas quantifies the speed by which the electromagnetic field permeates the vacuum, the refractive index provides a measure of how the field permeates a material relative to its permeation in vacuum. The refractive index, and in particular its real part , can be understood as a correction to the speed of light when the electromagnetic field propagates in a given medium.
1.2.4 Prominence of Electric Field Interactions
It is helpful to establish the relative magnitude by which and are able to drive the movement of charges in matter. For this purpose, we consider the forces exerted by the electromagnetic field on a single charge by the and fields. From equation (1.9), we can identify the electric force as and the magnetic force as . We next assume that the charge is driven by a time‐harmonic electromagnetic field in the form of a propagating plane wave. Section 1.3.1 discusses the properties of plane waves. Here, we are interested in the magnitudes and produced by a plane wave as it acts on the charge . It is not difficult to show that and that for plane waves the magnitude of the magnetic force can be written as , where . The ratio between the magnitude of the electric and magnetic forces experienced by is then [1]
We next assume that the charge is an electron that is bound to an atom. The velocity of such an electron can be estimated as , where is Planck's constant, is the electron mass, and is the Bohr radius. The ratio in equation (1.24) can now be written as , where is the fine structure constant. Since , where is the electron charge, we see that the electric force exerted on the electron is more than a hundred times larger than the magnetic force. Therefore, we can quite generally state that optical effects induced by the electric field component of propagating electromagnetic fields are dominant over effects mediated by the magnetic field component. Note that this conclusion pertains to propagating fields only, and that the situation for confined optical fields in the near‐zone can be quite different from that stated in equation (1.24).
For the phenomena described in this book, we can safely ignore the light–matter interactions mediated through the magnetic field component and primarily focus on the properties and interactions of the electric field. In terms of the distribution of the electromagnetic field in nonmagnetic materials, we can also assume that and that . Under these conditions, the refractive index can be simplified to . The complex relative permittivity can now be written as
which allows us to relate the real and imaginary parts of to the optical quantities and
In the small dissipation limit, and thus . Using equation (1.26), we then find that , which provides a useful relation between the index of refraction and the real part of the susceptibility. This relation states that, in the limit of a linear material response, is responsible for the retardation of electromagnetic waves in matter relative to their propagation in vacuum.
1.2.5 Wave Equation in Matter
Maxwell's wave equation (1.6) describes the properties of the electric field in vacuum. This equation is no longer valid when the field permeates matter, and corrections are needed. Starting from equations (1.10)–(1.13), we can derive a new wave equation by taking the curl of equation (1.10), using relations (1.14) and (1.15), and substituting equation (1.11). The result is the inhomogeneous wave equation for the electric field
The right‐hand side of equation (1.28) describes how changes in current densities affect the electric field, in the form of the electrical current density (), the polarization current density (), and the circulation current density due to the material magnetization ().
With reference to Section 1.2.4, we only consider materials in which the optical response is dominated by electrons bound to atoms and molecules. Such materials lack free or very loosely bound electrons and can be classified as dielectric materials. This includes, for instance, virtually all biological materials, but it excludes metals and strongly conducting semiconducting materials. For dielectric materials, the absence of free charges ensures that and the lack of electrical current densities, i.e. . In addition, the considered materials are assumed nonmagnetic so that . Under these conditions, the wave equation can be rewritten as
For the materials considered the condition holds. This implies that we can use the same vector identity that led to equation (1.6). After expressing the polarization density as in equation (1.16), we obtain
This result can be simplified by moving the second‐order time derivative on the right‐hand side to the left‐hand side of the equation
and by using . This produces the following form for the wave equation in a homogeneous medium
Note that the only difference between equation (1.6), which holds for fields in vacuum, and equation (1.30) is the effect of the refractive index in the latter. Assuming a time‐harmonic form of the electric field, equation (1.30) reduces to the vector Helmholtz equation given by (1.8), with the important difference that the angular wave number is now defined as
where is the angular wavenumber in vacuum. Because is complex, is now a complex number as well. In the absence of light absorption by the material, , and the wave number is given as . In the limit that the material responds linearly to the incident light, the propagation of light in matter is thus remarkably similar to propagation in vacuum, with the difference taken up by the new definition of . In vacuum, and the wave number is identical to its former form. In matter, the speed of light is altered as , which, in the absence of light absorption, reduces the angular wave number to . We are now ready to explore solutions to the Helmholtz equation for the electric field in isotropic, homogeneous materials.
1.3 Transverse Waves
Depending on the boundary conditions, finding solutions to the Helmholtz equation can be challenging. However, it is helpful to consider several particular solutions that are physically intuitive and that can be used as a starting point for analyzing the propagation of electromagnetic radiation in more complex situations. In this section, we study fields in an isotropic, homogeneous medium in the absence of any material boundaries. In this case, the solutions of the Helmholz equation are propagating waves in which the electric and magnetic field components oscillate in a direction perpendicular to the wave's propagation direction. Such waves are also called transverse waves. The most prominent example of a transverse wave is the plane wave, discussed in Section 1.3.1. Another example of a transverse wave is the spherical wave, discussed in Section 1.3.2. Both wave forms are relevant to the discussion of wave propagation in optical microscopes.
1.3.1 Plane Waves
Using the field notation defined in equation (1.7), the plane wave solution to the vector Helmholz equation can be written as
where
Here, is a real vector and is the wave vector of the propagating wave. The scalar product determines the spatial propagation phase of the wave. The wave vector is expressed in terms of its Cartesian components as
The magnitude of the wave vector is
which equals the angular wave number defined in equation (1.31). The direction of the wave vector is , which defines the direction of wave propagation. The electric field of a plane wave is governed by the (real) field vector , whose direction and magnitude are defined by the radiation source. Although the field vector itself is independent of location and time , it is modulated by the function , which is a harmonic function of space and time.
Example 1.2 The plane wave expression in equation (1.32) is written in terms of a complex exponential function. Reformulate the plane wave expression in terms of a real function.
Solution Using and the fact that is real, we can write
Note that the factor 2 is a consequence of our definition of the complex function in equation (1.7).
The harmonic nature of the plane wave implies that the wave repeats itself after propagating over a certain distance during the time period of an oscillation. Therefore, for the spatial part of the oscillation, there must be a distance along the propagation direction for which equals . We thus require that
which implies that . This leads to the following expression of the angular wave number
where is called the wavelength of the electromagnetic wave. This expression underscores that the angular wave number can be understood as a spatial frequency. The quantity represents the number of wavelengths per unit distance, or spatial frequency, also commonly referred to as the wave number. The angular wave number is radians times the wave number. Comparing equations (1.31) with (1.36), and assuming a nonabsorbing medium, gives
where is the oscillation frequency of the electromagnetic field. In vacuum, where and the speed of light is , the wavelength is . In a homogeneous dielectric medium, the wavelength of the propagating wave is
from which we see that the effective wavelength of the field is shortened in media relative to . Note that the angular frequency of the field, and thus its color, is independent of the propagation medium.
Let us next examine a couple of key properties of the plane wave.
Transverse wave
. In a medium free of unbound charges, i.e. a dielectric medium, we have . Substituting the plane wave solution in this expression yields , and thus which means that the projection of onto is zero. In other words, the wave vector of a plane wave is perpendicular to the direction of the electric field vector. Since the wave advances in the direction of the wave vector, the electric field oscillations are perpendicular to the direction of propagation. Waves that have this property are called
transverse
waves. Consequently, plane waves are transverse waves. Using Maxwell's equation (
1.11
), and applying a similar procedure, we find that . This can be rewritten as which means that the field is perpendicular to both and . Hence, both the electric and magnetic fields are orthogonal to the wave vector. Furthermore, the expression above shows that the and fields are oscillating in phase, which is also evident in
Figure 1.2
.
Planar wavefront
. A plane perpendicular to is formed by all the points for which is a constant. The electric field vector is constant throughout such a plane. The idealized plane wave, where the field is constant over the infinite extent of the plane, derives its name from this property. In
Example 1.2
, the plane wave is seen to propagate according to the function , an oscillatory function that evolves with a phase , a function of both space and time. The
wavefront
of the wave is defined by the surface formed by all points () that give rise to the same value , i.e. a surface of constant phase. For a given point in time, the factor is a constant, which means that for a plane wave the wavefront is formed by the points , which is a plane perpendicular to . We thus recognize that plane waves have planar wavefronts.
Figure 1.2 Plane wave in vacuum. The electric and magnetic field vectors are oriented perpendicular to the propagation direction . The gray area indicates a plane of constant phase . The same phase is found for a parallel plane that is located a distance away.
Phase velocity
. In the direction of propagation, we can write the spatial coordinate as the scalar , and the spatial propagation phase as . What is the speed of propagation of the wavefront, i.e. a surface of constant phase , along ? Using the triplet product rule for partial derivatives, we may write the rate of change of location of the wavefront as [
1
]
Since in the direction of propagation , we find that and that , and thus . The speed of propagation of the wavefront, called the phase velocity, is then found as
Recall that in the absence of light absorption, , and thus from equation (1.38), we find . The phase velocity tells us the speed of propagation of a monochromatic wave of angular frequency in a material with an index of refraction . This analysis also provides a simple and intuitive definition of the index of refraction as , i.e. the ratio between the speed of light propagation in vacuum over the speed of phase propagation in matter.
The plane wave solution of equation (1.32) is a harmonic wave written in vectorial form. In certain cases, it is sufficient to express the plane wave solution in scalar form. Assuming that the electric field can be described by a single polarization component in the transverse plane, the scalar plane wave is written as , where the complex function is defined as
Here, we have written the complex field in italics to distinguish it from the real field (nonitalic). In the case of a single plane wave, we may define a coordinate system in which the direction of propagation coincides with one of the Cartesian axes. If we choose the propagation to be along , and the electric field to be oriented in the direction, then equation (1.39) further simplifies to . The latter form of the wave is a solution of the one‐dimensional, homogeneous wave equation. Although the one‐dimensional wave equation can be used for solving many problems in optics, it is less useful for describing the propagation and interactions of light in the tightly focusing conditions relevant to microscopy. Therefore, in general, we shall retain the three‐dimensional character of wave propagation and only refer to one‐dimensional problems where appropriate.
Example 1.3 A plane wave of is passing a point in space. What is the duration of a full oscillation of the electric field vector measured at this point?
Solution If the position remains constant, the change in phase is determined solely by the time evolution of the wave. Since a full phase cycle corresponds to , we have . Using , we find , which gives
Example 1.4 A plane wave of is traveling through a material with an index of refraction . What is the time it takes for a plane of constant phase, i.e. the wavefront, to travel a distance in this material?
Solution The wavefront travels with a phase velocity . It takes to cover a distance , so that
This is, of course, the same answer we found above, because it takes one full oscillation period for to repeat itself, during which the wavefront travels a distance in the material.
1.3.2 Spherical Waves
Spherical waves form a second class of waves that is useful for describing the propagation of wavefronts. A spherical wave exhibits a curved wavefront that emanates from a source located at . From a mathematical point of view, a spherical wave is an idealized wave form that exhibits spherical symmetry, i.e. it shows no dependence on the angles and that define the spherical surface . Instead, the spatial part of the spherical wave only depends on the radius . The assumption of spherical symmetry is at odds with the notion of propagating (transverse) field vectors, as a transverse vector is not spherically symmetric: a spherical surface cannot be covered uniformly with tangential vectors that retain a single polarization direction. Ignoring these problems for now and proceeding regardless, the three‐dimensional wave equation reduces to a one‐dimensional differential equation that only depends on . The scalar solution to this equation is given by harmonic spherical waves of the form
where relates to the source strength and has units of . The wavefront is defined by the condition , which results in spherically symmetric wavefronts. Contrary to the plane wave, the amplitude of the spherical wave is not invariant as a function of propagation distance. Instead, the amplitude decreases as , which implies that the energy per surface area scales as . Since the area of the wavefront expands as , the radial amplitude dependence complies with the notion that the total energy carried by the wave is conserved.
