mi‘.
`
`'." iii
`
`L‘?-~¢H‘i:.i
`
`[i'I~_Jji§‘h a_ti'1-J‘
`
`Clmstopher C. Davis
`
`ASML 1126
`
`1
`
`ASML 1126
`
`

`

`This comprehensive textbook provides a detailed introducti
`on to the basic physics and
`engineering aspects of lasers,
`as well as to the design and operational principles of a
`wide range of optical systems and elec
`tro-optic devices. Throughout, full details of
`important derivations and results are
`given, as are many practical examples of the
`FIDBHCC
`design, construction, and perfo
`characteristics of diflerent types of lasers and
`electro-optic devices.
`
`parametric processes, phase conjugation an
`with chapters on optical detection, coheren
`ce theory. and the applications of lasers.
`Covering a broad range of topics in no
`odern optical physics and engineering, this
`book will be invaluable to those taking
`undergraduate courses in laser physics,
`optoelectronics, photonics, and optical e
`ngineering. It will also act as a useful
`reference for graduate students and rese
`archers in these fields.
`
`2
`
`

`

`
`
`Lasers and Electro-Optics
`Fundamentals and Engineering
`
`C H R I S T O P H E R C. D A V IS
`
`CAMBRIDGE
`L;-;;@;:_:~-° UNIVERSITY PRESS
`
`
`
`3
`
`

`

`PUBLISHED BY THE aaass SYNDICATE or THE UNIVERSITY or CAMBRIDGE
`The Pitt Building, Trumpington Street, Cambridge, United Kingdom
`CAMBRIDGE UNIVERSITY PRESS
`
`The Edinburgh Building, Cambridge CB2 ZRU. UK
`40 West 20th Street. New York, NY 10011-4211, USA
`477 Williamstown Road, Port Melbourne, VIC 3207, Australia
`Ruiz de Alarcon 13, 28014 Madrid, Spain
`Dock House, The Waterfront, Cape Town 8001, South Africa
`
`http:IIwww.cambridge.org
`
`O Cambridge University Press I996
`
`This book is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without
`the written permission of Cambridge University Press.
`
`First published 1996
`Reprinted (with corrections) 2000, 2002
`
`Printed in the United Kingdom at the University Press, Cambridge
`
`ijapefizce 9.5il2pt Monotype Times System TEX [UPI-I]
`
`A catalogue recordfiJr this book is availablefrom the British Library
`
`Library of Congress Cataloguing in Publication data
`Davis, Christopher C., 1944-
`Lasers and electro-optics : fundamentals and engineering!
`Christopher C. Davis.
`p.
`cm.
`Includes bibliographical references.
`ISBN 0-521-30831-3 (hardback.) — ISBN 0-521-48403-0 (pblc)
`1. Lasers.
`2. Electrooptics.
`I. Title.
`TAl675.D38
`1995
`621.36-dc2O 94-43230 CIP
`
`ISBN 0 521 30831 3 hardback
`ISBN 0 521 48403 0 paperback
`
`4
`
`

`

`
`
`
`
`Contents
`
`1.1
`1.2
`1.3
`1.3.1
`1.3.2
`1.4
`1.4.1
`1.5
`
`1.6
`1.7
`1.8
`1.9
`1.10
`
`2.1
`2.2
`2.2.1
`2.3
`2.3.1
`2.4
`
`2.4.1
`2.5
`
`2.6
`2.6.1
`2.6.2
`2.7
`2.8
`2.9
`2.10
`2.1 1
`
`Preface
`
`Spontaneous and Stimulated 'Il'ansitions
`Introduction
`
`Why ‘Quantum’ Electronics?
`Amplification at Optical Frequencies
`Spontaneous Emission
`Stimulated Emission
`
`The Relation Between Energy Density and intensity
`Stimulated Absorption
`Intensity of a Beam of Electromagnetic Radiation in Terms of Photon
`Flux
`
`Black-Body Radiation
`Relation Between the Einstein A and B Coefiicients
`The Efl'ect of Level Degeneracy
`
`Ratio of Spontaneous and Stimulated Transitions
`Problems
`
`Optical liequency Amplifiers
`Introduction
`
`Homogeneous Line Broadening
`Natural Broadening
`Inhomogeneous Broadening
`Doppler Broadening
`Optical Frequency Amplification with a Homogeneously Broadened
`Transition
`
`The Stimulated Emission Rate in a I-lomogeneously Broadened System
`Optical Frequency Amplification with Inhomogeneous Broadening
`Included
`
`Optical Frequency Oscillation — Saturation
`Homogeneous Systems
`Inhomogeneous Systems
`Power Output from a Laser Amplifier
`The Electron Oscillator Model of a Radiative Transition
`
`What Are the Physical Significanoes of f and 1"?
`The Classical Oscillator Explanation for Stimulated Emission
`Problems
`
`References
`
`5
`
`...o-.ua-:--\.u---:-
`
`ll
`16
`
`19
`20
`
`22
`
`1312313
`
`27
`
`30
`33
`
`34
`35
`35
`38
`
`45
`49
`52
`54
`
`55
`
`5
`
`

`

`x
`
`Contents
`
`3 Introduction to ‘No Practical Laser Systems
`3.1 Introduction
`
`3.1.1 The Ruby Laser
`3.2 The HeIiurn—Neon Laser
`
`References
`
`4 Passive Optical Resonators
`4.1 Introduction
`
`4.2 Preliminary Consideration of Optical Resonators
`4.3 Calculation of the Energy Stored in an Optical Resonator
`4.4 Quality Factor of a Resonator in Terms of the Transmission of its End
`Reflectors
`
`4.5 Fabry—Perot Etalons and Interferometers
`4.6 Internal Field Strength
`4.7 Fabry—Perot Interferometers as Optical Spectrum Analyzers
`4.7.1 Example
`4.8 Problems
`
`References
`
`5 Optical Resonators Containing Amplifying Media
`5.1 Introduction
`
`5.2 Fabry—Perot Resonator Containing an Amplifying Medium
`5.2.1 Threshold Population Inversion — Numerical Example
`5.3 The Oscillation Frequency
`5.4 Multimode Laser Oscillation
`
`5.5 Mode-Beating
`5.6 The Power Output of a Laser
`5.? Optimum Coupling
`5.8 Problems
`
`References
`
`6 Laser Radiation
`
`6.1 Introduction
`6.2 Dilftaction
`6.3 Two Parallel Narrow Slits
`6.4 Single Slit
`6.5 Two-Dimensional Apertures
`6.5.1 Circular Aperture
`6.6 Laser Modes
`6.7 Beam Divergence
`6.8 Linewidth of Laser Radiation
`6.9 Coherence Properties
`6.10 Interference
`6.11 Problems
`
`References
`
`7 Control of Laser Oscillators
`
`7.1 Introduction
`
`57
`
`Cha\UnLn'--Ila!‘-I--I
`3%6'a‘oo
`
`er.
`
`72
`73
`79
`BI
`84
`86
`
`87
`
`38
`
`88
`88
`91
`92
`93
`99
`101
`105
`106
`
`107
`
`108
`
`108
`108
`110
`M0
`111
`111
`113
`111
`118
`119
`121
`124
`
`124
`
`126
`
`126
`
`
`
`6
`
`

`

`
`
`Contents
`
`xi
`
`
`
`7.2 Multimode Operation
`7.3 Single Longitudinal Mode Operation
`7.4 Mode-Locking
`7.5 Methods of Mode-Locking
`7.5.1 Active Mode-Locking
`7.6 Pulse Compression
`
`References
`
`126
`127
`131
`134
`134
`138
`
`139
`
`141
`8 Optically Pumped Solid-State Lasers
`141
`8.1 Introduction
`141
`8.2 Optical Pumping in Three- and Four-Level Lasers
`141
`8.2.1 Eflective Lifetime of the Levels Involved
`142
`8.2.2 Threshold Inversion in Three- and Four-Level Lasers
`143
`8.2.3 Quantum Etiiciency
`143
`8.2.4 Pumping Power
`144
`8.2.5 Threshold Lamp Power
`144
`8.3 Pulsed Versus CW Operation
`145
`8.3.1 Threshold for Pulsed Operation of a Ruby Laser
`145
`8.3.2 Threshold for CW Operation of a Ruby Laser
`8.4 Threshold Population Inversion and Stimulated Emission Cross-Section 146
`8.5 Paramagnetic Ion Solid-State Lasers
`147
`8.6 The Nd :YAG Laser
`147
`8.6.1 Effective Spontaneous Emission Coeflicient
`152
`8.6.2 Example — Threshold Pump Energy of a Pulsed Nd :YAG Laser
`153
`8.7 CW Operation 01' the Nd:YAG Laser
`154
`8.8 The Nd“ Glass Laser
`154
`8.9 Geometrical Arrangements for Optical Pumping
`159
`8.9.1 Axisymmetric Optical Pumping of a Cylindrical Rod
`159
`8.10 High Power Pulsed Solid-State Lasers
`166
`8.11 Diode-Pumped Solid-State Lasers
`167
`8.12 Relaxation Oscillations (Spiking)
`168
`8.13 Rate Equations for Relaxation Oscillation
`170
`8.14 Undarnped Relaxation Oscillations
`174
`8.15 Giant Pulse (Q-Switched) Lasers
`175
`8.16 Theoretical Description of the Q-Switching Process
`179
`8.16.1 Example Calculation of Q-Switched Pulse Characteristics
`182
`8.17 Problems
`183
`
`References
`
`9 Gas Lasers
`
`9.1 Introduction
`9.2 Optical Pumping
`9.3 Electron Impact Excitation
`9.4 The Argon Ion Laser
`9.5 Pumping Saturation in Gas Laser Systems
`9.6 Pulsed Ion Lasers
`9.7 CW Ion Lasers
`9.8 ‘Metal’ Vapor Ion Lasers
`
`183
`
`185
`
`185
`185
`187
`188
`190
`191
`192
`196
`
`_
`,
`:r
`
`7
`lm_
`
`7
`
`

`

`XII
`Contents
`"
`
`
`9.9 Gas Discharges for Exciting Gas Lasers
`9.10 Rate Equations for Gas Discharge Lasers
`9.11 Problems
`
`References
`
`10 Molecular Gas Lasers I
`
`10.1 Introduction
`10.2 The Energy Levels of Molecules
`10.3 Vibrations of a Polyatomic Molecule
`10.4 Rotational Energy States
`10.5 Rotational Populations
`10.6 The Overall Energy State of :1 Molecule
`10.7 The Carbon Dioxide Laser
`10.8 The Carbon Monoxide Laser
`10.9 Other Gas Discharge Molecular Lasers
`
`References
`
`11 Molecular Gas Lasers II
`
`11.1 Introduction
`11.2 Gas Transport Lasers
`11.3 Gas Dynamic Lasers
`11.4 High Pressure Pulsed Gas Lasers
`11.5 Ultraviolet Molecular Gas Lasers
`11.6 Photodissociation Lasers
`11.7 Chemical Lasers
`11.8 Far-Infrared Lasers
`11.9 Problems
`References
`
`'
`
`12 Tunable Lasers
`12.1 Introduction
`12.2 Organic Dye Lasers
`12.2.1 Energy Level Structure
`12.2.2 Pulsed Laser Excitation
`12.2.3 CW Dye Laser Operation
`12.3 Calculation of Threshold Pump Power in Dye Lasers
`12.3.1 Pulsed Operation
`12.3.2 CW Operation
`12.4 Inorganic Liquid Lasers
`12.5 Free Electron Lasers
`12.6 Problems
`References
`
`13 Semiconductor Lasers
`13.1 Introduction
`13.2 Semiconductor Physics Background
`13.3 Carrier Concentrations
`13.4 Intrinsic and Extrinsic Semiconductors
`13.5 The p-n Junction
`
`199
`201
`204
`
`205
`
`201
`
`207
`207
`212
`214
`214
`216
`217
`222
`224
`
`224
`
`225
`
`225
`225
`228
`232
`238
`241
`241
`244
`244
`246
`
`248
`248
`248
`248
`251
`252
`253
`256
`259
`260
`260
`266
`266
`
`267
`267
`267
`271
`274
`275
`
`8
`
`

`

`
`
` Contents xiii
`
`13.6 Recombination and Luminescence
`
`13.6.1 The Spectrum of Recombination Radiation
`13.6.2 External Quantum Efliciency
`13.7 I-leterojunctions
`13.7.1 Ternary and Quaternary Lattice-Matched Materials
`13.7.2 Energy Barriers and Rectification
`13.7.3 The Double Heterostructure
`13.8 Semiconductor Lasers
`13.9 The Gain Coefficient of a Semiconductor Laser
`13.9.1 Estimation of Semiconductor Laser Gain
`
`13.10 Threshold Current and Power—VoItage Characteristics
`13.11 Longitudinal and Transverse Modes
`13.12 Semiconductor Laser Structures
`
`l3.12.l Distributed Feedback (DFB) and Distributed Bragg Reflection (DBR)
`Lasers
`
`13.13 Surface Emitting Lasers
`13.14 Laser Diode Arrays and Broad Area Lasers
`13.15 Quantum Well Lasers
`13.16 Problems
`
`References
`
`14 Analysis of Optical Systems I
`14.1 Introduction
`
`14.2 The Propagation of Rays and Waves through Isotropic Media
`14.3 Simple Reflection and Refraction Analysis
`14.4 Paraxial Ray Analysis
`14.4.1 Matrix Formulation
`
`14.4.2 Ray Tracing
`14.4.3 Imaging and Magnification
`14.5 The Use of Impedances in Optics
`14.5.1 Reflectance for Waves Incident on an Interface at Oblique Angles
`14.5.2 Brewster's Angle
`14.5.3 Transformation of Impedance through Multilayer Optical Systems
`14.5.4 Polarization Changes
`14.6 Problems
`
`References
`
`15 Analysis of Optical Systems I]
`15.1 Introduction
`
`15.2 Periodic Optical Systems
`15.3 The Identical Thin Lens Waveguide
`15.4 The Propagation of Rays in Mirror Resonators
`15.5 The Propagation of Rays in Isotropic Media
`15.6 The Propagation of Spherical Waves
`15.? Problems
`
`References
`
`9
`
`280
`
`281
`283
`285
`285
`286
`236
`290
`292
`293
`
`295
`296
`29'!
`
`299
`
`304
`306
`307
`310
`
`311
`
`312
`312
`
`312
`313
`316
`316
`
`324
`325
`327
`331
`332
`332
`334
`335
`
`336
`
`337
`337
`
`337
`339
`340
`342
`346
`347
`
`347
`
`9
`
`

`

`
`
`xiv Contents
`
`16 Optics of Gaussian Beams
`16.1 Introduction
`
`16.2 Beam-Like Solutions of the Wave Equation
`16.3 Higher Order Modes
`
`16.3.1 Beam Modes with Cartesian Symmetry
`16.3.2 Cylindrically Symmetric Higher Order Beams
`16.4 The Transformation of a Gaussian Beam by a Lens
`16.5 Translbrmation of Gaussian Beams by General Optical Systems
`16.6 Gaussian Beams in Lens Waveguides
`
`16.7 The Propagation of a Gaussian Beam in a Medium with a Quadratic
`Refractive Index Profile
`
`16.8 The Propagation of Gaussian Beams in Media with Spatial Gain or
`Absorption Variations
`16.9 Propagation in a Medium with a Parabolic Gain Profile
`16.10 Gaussian Beams in Plane and Spherical Mirror Resonators
`16.11 Symmetrical Resonators
`16.12 An Example of Resonator Design
`16.13 Difiraction Losses
`16.14 Unstable Resonators
`16.15 Problems
`
`References
`
`1'7 Optical Fibers and Waveguides
`17.1 Introduction
`
`17.2 Ray Theory-of Cylindrical Optical Fibers
`17.2.1 Meridional Rays in a Step-Index Fiber
`17.2.2 Step-Index Fibers
`17.2.3 Graded-Index Fibers
`
`17.2.4 Bound. Refracting, and Tunnelling Rays
`17.3 Ray Theory of a Dielectric Slab Guide
`17.4 The Gocs—Hiincben Shift
`
`17.5 Wave Theory of the Dielectric Slab Guide
`
`17.6 P-Waves in the Slab Guide
`17.7 Dispersion Curves and Field Distributions in a Slab Waveguide
`17.3 S-Waves in the Slab Guide
`
`17.9 Practical Slab Guide Geometries
`17.10 Cylindrical Dielectric Waveguides
`
`17.l0.1 Fields in the Core
`17.10.2 Fields in the Cladding
`17.1D.3 Boundary Conditions
`
`17.11 Modes and Field Patterns
`17.12 The Weakly-Guiding Approximation
`17.13 Mode Patterns
`17.14 Cutoff Frequencies
`17.14.] Example
`17.15 Multimode Fibers
`17.16 Fabrication of Optical Fibers
`17.17 Dispersion in Optical Fibers
`
`348
`348
`
`343
`354
`
`354
`355
`357
`371
`371
`
`372
`
`372
`373
`375
`377
`379
`331
`382
`384
`
`386
`
`387
`387
`
`387
`387
`390
`392
`
`393
`395
`397
`
`399
`
`400
`404
`405
`
`407
`408
`
`413
`414
`414
`
`415
`416
`417
`419
`421
`423
`423
`425
`
`I
`_
`3‘
`"
`
`:
`:
`
`1
`
`5
`:
`'
`
`;
`-"I
`'
`
`.
`.
`1
`5
`
`
`
`10
`
`10
`
`

`

`_.
`
`Contents
`
`17.17.1 Material Dispersion
`17.172 Waveguide Dispersion
`17.18 Solitons
`17.19 Erbium-Doped Fiber Amplifiers
`17.20 Coupling Optical Sources and Detectors to Fibers
`17.20.] Fiber Connectors
`17.21 Problems
`
`References
`
`I8 Optics of Anisotropic Media
`18.1 Introduction
`18.2 The Dielectric Tensor
`18.3 Stored Electromagnetic Energy in Anisotropic Media
`18.4 Propagation of Monochromatic Plane Waves in Anisotropic Media
`18.5 The Two Possible Directions of D for a Given Wave Vector are
`Orthogonal
`18.6 Angular Relationships between D, E, H, In, and the Poynting Vector S
`18.7 The lndicatrix
`18.8 Uniaxial Crystals
`18.9 Index Surfaces
`18.10 Other Surfaces Related to the Uniaxial Indicatrix
`18.11 Huygenian Constructions
`18.12 Retardation
`18.13 Biaxial Crystals
`18.14 Intensity Transmission Through Polarizer/Waveplate/Polarizer Combin-
`ations
`18.14.] Examples
`18.15 The Jones Calculus
`l8.15.1 The Jones Vector
`18.152 The Jones Matrix
`18.16 Problems
`
`References
`
`3:.-
`
`427
`428
`430
`430
`433
`434
`435
`
`437
`
`438
`438
`438
`4-40
`441
`
`443
`444
`446
`448
`450
`452
`453
`457
`461
`
`464
`465
`465
`466
`467
`470
`
`471
`
`.
`
`19 The Electro-Optic and Acousto-Optic Effects and Modulation of
`Light Beams
`472
`19.1 Introduction to the Electro-Optic Effect
`472
`19.2 The Linear Electro-Optic Eflect
`472
`19.3 The Quadratic Electro-Optic Efiect
`479
`19.4 Longitudinal Electro-Optic Modulation
`480
`19.5 Transverse Electro-optic Modulation
`482
`19.6 Electro-Optic Amplitude Modulation
`486
`19.7 Electro-Optic Phase Modulation
`488
`19.8 High Frequency Waveguide Electro-Optic Modulators
`489
`19.8.1 Straight Electrode Modulator
`490
`19.9 Other High Frequency Electro-Optic Devices
`493
`19.10 Electro-Optic Beam Deflectors
`495
`19.11 Acousto-Optic Modulators
`495
`19.12 Applications of Acousto-Optic Modulators
`502
`19.l2.1 Difiraction Efficiency of TeO;
`502
`
`11
`
`11
`
`

`

`-
`
`Contents
`
`19.12.2 Acousto-Optic Modulators
`19.123 Acousto-Optic Beam Deflectors and Scanners
`19.12.4 RF Spectrum Analysis
`19.13 Construction and Materials for Acousto—Optic Modulators
`19.14 Problems
`
`References
`
`20 Introduction to Nonlinear Processes
`
`20.1 Introduction
`20.2 Anharmonic Potentials and Nonlinear Polarization
`
`20.3 Nonlinear Susceptibilitics and Mixing Coeflicients
`20.4 Second Harmonic Generation
`
`20.4.1 Symmetries and K.leinman's Conjecture
`20.5 The Linear Electro-Optic Effect
`20.6 Parametric and Other Nonlinear Processes
`
`20.7 Macroscopic and Microscopic Susceptibilities
`20.8 Problems
`
`References
`
`21 Wave Propagation in Nonlinear Media
`21.1 Introduction
`
`21.2 Electromagnetic Waves and Nonlinear Polarization
`21.3 Second Harmonic Generation
`
`21.4 The Effective Nonlinear Coefficient dd;
`21.5 Phase Matching
`21.5.] Second Harmonic Generation
`
`21.5.2 Example
`21.5.3 Phase Matching in Sum-Frequency Generation
`21.6 Beam Walk-Ofl‘ and 90‘ Phase Matching
`21.7 Second Harmonic Generation with Gaussian Beams
`
`21.7.1 lntracavity SHG
`21.7.2 External SHG
`
`21.7.3 The Efiects of Depletion on Second Harmonic Generation
`21.8 Up-Conversion and Difference-Frequency Generation
`21.9 Optical Parametric Amplification
`21.9.1 Example
`21.10 Parametric Oscillators
`
`21.10.! Example
`21.11 Parametric Oscillator Tuning
`21.12 Phase Conjugation
`21.12.] Phase Conjugation in CS;
`21.13 Optical Bistability
`21.14 Practical Details of the Use of Crystals for Nonlinear Applications
`21.15 Problems
`
`References
`
`22 Detection of Optical Radiation
`22.1 Introduction
`22.2 Noise
`
`502
`503
`504
`504
`507
`
`507
`
`508
`
`508
`508
`
`512
`514
`
`516
`516
`517
`
`518
`522
`
`522
`
`524
`524
`
`524
`528
`
`530
`532
`533
`
`533
`535
`535
`536
`
`537
`538
`
`538
`541
`542
`544
`545
`
`547
`548
`550
`553
`554
`557
`558
`
`559
`
`561
`561
`561
`
`
`
`12
`
`12
`
`

`

`Contents
`
`22.2.1 Shot Noise
`22.2.2 Johnson Noise
`22.2.3 Generation—Recornbination Noise and 1/3' Noise
`22.3 Detector Performance Parameters
`22.3.1 Noise Equivalent Power
`22.3.2 Detectivity
`22.3.3 Frequency Response and Time Constant
`22.4 Practical Characteristics of Optical Detectors
`22.4.1 Photoemissive Detectors
`22.4.2 Photoconductive Detectors
`22.4.3 Photovoltaic Detectors (Photodiodes)
`22.4.4 p—i-n Photodiodes
`22.4.5 Avalanche Photodiodes
`22.5 Thermal Detectors
`2.6 Detection Limits for Optical Detector Systems
`22.6.1 Noise in Photomultipliers
`22.6.2 Photon Counting
`22.6.3 Signal-to-Noise Ratio in Direct Detection
`22.6.4 Direct Detection with p—i-n Photodiodes
`22.6.5 Direct Detection with APDS
`22.7 Coherent Detection
`22.8 Bit-Error Rate
`References
`
`23 Coherence Theory
`23.1 Introduction
`23.2 Square-Law Detectors
`23.3 The Analytic Signal
`23.3.1 Hilbert Transforms
`23.4 Correlation Functions
`23.5 Temporal and Spatial Coherence
`23.6 Spatial Coherence
`23.? Spatial Coherence with an Extended Source
`23.8 Propagation Laws of Partial Coherence
`23.9 Propagation from a Finite Plane Surface
`23.10 van Cittert-Zernike Theorem
`23.11 Spatial Coherence of a Quasi-Monochromatic, Uniform, Spatially
`Incoherent Circular Source
`23.12 Intensity Correlation Interferometry
`23.13 Intensity Fluctuations
`23.14 Photon Statistics
`23.14.] Constant Intensity Source
`2114.2 Random Intensities
`23.15 The Hanbury-Brown—T‘wiss Interferometer
`23.16 Hanbury-Brown—'I‘wiss Experiment with Photon Count Correlations
`References
`
`24 Laser Applications
`24.! Optical Communication Systems
`
`13
`
`xvii
`
`561
`564
`56?
`568
`568
`569
`569
`570
`570
`576
`582
`586
`587
`589
`591
`592
`593
`594
`595
`597
`598
`603
`605
`
`607
`607
`607
`608
`610
`61]
`614
`618
`620
`622
`625
`630
`
`632
`634
`635
`638
`639
`640
`643
`645
`646
`
`647
`647
`
`13
`
`

`

`
`
` xfiii Contents
`
`24.1.1 Introduction
`
`24.1.2 Absorption in Optical Fibers
`24.1.3 Optical Communication Networks
`24.1.4 Optical Fiber Network Architectures
`24.1.5 Coding Schemes in Optical Networks
`24.1.6 Line-of-Sight Optical Links
`24.2 Holography
`24.2.1 Wavefront Reconstruction
`
`24.2.2 The Hologram as a Diffraction Grating
`24.2.3 Volume Holograms
`24.3 Laser Isotope Separation
`24.4 Laser Plasma Generation and Fusion
`
`24.5 Medical Applications of Lasers
`24.5.1 Laser Angioplasty
`
`References
`
`Appendix 1 Optical Terminology
`
`Appendix 2 The 6-Function
`
`Appendix 3 Black-Body Radiation Formulas
`
`Appendix 4 RLC Circuit
`A4.l Analysis of a Driven RLC Circuit
`
`647
`
`649
`650
`651
`653
`654
`656
`656
`
`660
`661
`664
`669
`
`671
`673
`
`673
`
`676
`
`679
`
`681
`
`683
`683
`
`Appendix 5 Storage and Transport of Energy by Electromagnetic Fields 686
`
`Appendix 6 The Reflection and Refraction of a Plane Electromagnetic
`Wave at the Boundary Between Two Isotropic Media of Different
`Refractive Index
`
`Appendix 7 The Vector Differential Equation for Light Rays
`
`689
`
`692
`
`Appendix 8 Symmetry Properties of Crystals and the 32 Crystal Classes 695
`696
`AB.1
`Class 6rnrn
`696
`A82
`Class 42m
`697
`Class 222
`A83
`
`Appendix 9 Tensors
`
`Appendix 10 Bessel Function Relations
`
`Appendix 11 Green's Functions
`
`Appendix 12 Recommended Values of Some Physical Constants
`
`Index
`
`698
`
`701
`
`702
`
`705
`
`706
`
`14
`
`14
`
`

`

`
`
`1 S
`
`pontaneous and Stimulated Transitions
`
`1.1
`
`Introduction
`
`A laser is an oscillator that operates at very high frequencies. These optical fre-
`quencies range to values several orders of magnitude higher than can be achieved
`by the ‘conventional’ approaches of solid-state electronics or electron tube technol-
`ogy. In common with electronic circuit oscillators, a laser is constructed from an
`amplifier with an appropriate amount of positive feedback. The acronym LASER,
`which stands for light amplification by stimulated emission of radiation, is in reality
`therefore a slight misnomerl".
`In this chapter we shall consider the fundamental processes whereby amplifi-
`cation at optical frequencies can be obtained. These processes involve the funda-
`mental atomic nature of matter. At the atomic level matter is not a continuum, it
`
`is composed of discrete particles - atoms, molecules or ions. These particles have
`energies that can have only certain discrete values. This discreteness or quantiza-
`tion, of energy is intimately connected with the duality that exists in nature. Light
`sometimes behaves as if it were a wave and in other circumstances it behaves as
`
`if it were composed of particles. These particles, called photons, carry the discrete
`packets of energy associated with the wave. For light of frequency v the energy
`of each photon is hv, where in is Planck’s constant — 6.6 x 10*“ J 5. The energy
`hv is the quantum of energy associated with the frequency v. At the microscopic
`level the amplification of light within a laser involves the emission of these quanta.
`Thus, the term quantum electronics is often used to describe the branch of science
`that has grown from the development of the maser in 1954 and the laser in 1960.
`The widespread practical use of lasers and optical devices in applications such
`as communications, and increasingly in areas such as signal processing and image
`analysis has lead to the use of the term pliotonics. Whereas, electronics uses
`electrons in various devices to perform analog and digital functions, photonics
`aims to replace the electrons with photons. Because photons have zero mass, do
`not interact with each other to any significant extent, and travel at the speed of
`light photonic devices promise small size and high speed.
`
`1.2 Why ‘Quantum’ Electronics?
`
`In ‘conventional’ electronics, where by the word ‘conventional’ for the present
`purposes we mean frequencies where solid-state devices such as transistors or
`diodes will operate, say below 10“ Hz, an oscillator is conveniently constructed
`by applying an appropriate amount of positive feedback to an amplifier. Such an
`
`1The more truthful acronym LOSER. was long ago deemed inappropriate.
`
`15
`
`15
`
`

`

`2
`
`Spontaneous and Stimulated Transitions
`
`Amplifier
`fifiifldu
`
`Fig. l.l. Circuit diagram of
`a simple amplifier with
`feedback.
`
`III
`
`
`Vc=Ao(Vt+ |3Vo)
`
`arrangement is shown schematically in Fig. (1.1). The input and output voltages
`of the amplifier are V; and V9 respectively. The overall gain of the system is A,
`where A = V'o/ V]. Now,
`
`so
`
`and
`
`Vo = .400’! + I3 Vol
`
`V —_-
`0
`
`Aalfi
`I-fl/Io
`
`A =
`
`A0
`1—flAa
`
`.
`
`1.1
`
`)
`
`(
`
`If 13.49 = +1 then the gain of the circuit would apparently become infinite,
`and the circuit would generate a finite output without any input.
`In practice
`electrical ‘noise’, which is a random oscillatory voltage generated to a greater
`or lesser extent in all electrical components in any amplifier system, provides
`a finite input. Because BA.) is generally a function of frequency the condition
`flAo = +1 is generally satisfied only at one frequency. The circuit oscillates at this
`frequency by amplifying noise at this same frequency that appears at its input.
`However, the output does not grow infinitely large, because as the signal grows,
`/lo falls — this process is called saturation. This phenomenon is fundamental
`to all oscillator systems. A laser (or maser) is an optical (microwave) frequency
`oscillator constructed from an optical (microwave) frequency amplifier with positive
`feedback, shown schematically in Fig. (1.2). Light waves which become amplified
`on traversing the amplifier are returned through the amplifier by the reflectors
`and grow in intensity, but this intensity growth does not continue indefinitely
`because the amplifier saturates. The arrangement of mirrors (and sometimes other
`components) that provides the feedback is generally referred to as the laser cavity
`or resonator.
`
`We shall deal with the full characteristics of the device consisting of amplifying
`medium and resonator later, for the moment we must concern ourselves with the
`problem of how to construct an amplifier at optical frequencies. The frequencies
`involved are very high, for example lasers have been built which operate from
`
`
`
`16
`
`16
`
`

`

`
`
`3
`1.3 Amplification at Optical Frequencies
`j
`
`Fig, 1.2. Schematic
`diagram of a basic laser
`structure incorpflffil-I118 3'1
`amplifying medium and
`two feedback mirrors. M-
`
` L
`M
`M’
`
`
`
`Fig 1.3. Simple schematic
`energy level diagram for a
`particle.
`
`Energy
`
`V01 is the frequency of the
`emitted photon
`
`E; ——:%
`_
`Excited
`l
`states
`ll /\f\/\/¥> .llVg
`e ——«i‘.———
`
`
`5!:
`
`Ea j____j The level with the lowest energy — E0
`is the ground stale
`
`very short wavelengths, for example 109.8 nm, using para-hydrogen gas as the
`amplifying medium, to 2650 pm using methyl bromide as the amplifying medium.
`This is a frequency range from 2.73 x 10” Hz down to 1.13 x 10" Hz. The
`operating frequencies of masers overlap this frequency range at the low frequency
`end, the fundamental difference between the two devices is essentially only one of
`scale. If the length of the resonant cavity which provides feedback is L, then for
`L > 3., where A is the wavelength at which oscillation occurs, we have a laser: for
`L. ~ I. we have a maser.
`
`1.3 Amplification at Optical Frequencies
`
`How do we build an amplifier at such high frequencies? We use the energy delivered
`as the particles that consitute the amplifying medium make jumps between their
`different energy levels. The medium may be gaseous, liquid, a crystalline or glassy
`solid, an insulating material or a semiconductor. The electrons that are bound
`within the particles of the amplifying medium, whether these are atoms, molecules
`or ions, can occupy only certain discrete energy levels. Consider such a system of
`energy levels, shown schematically in Fig. (1.3). Electrons can make jumps between
`these levels in three ways.
`
`17
`
`17
`
`

`

`Fig. 1.4. Schematic
`representation of
`spontaneous emission
`between two levels of
`energy E. and E1.
`
`4
`
`Population
`density
`
`N‘ :_Qj 5‘
`
`Spontaneous and Stimulated Transitions
`
` hVy
`
`III |
`
`III
`
`Population
`density
`
`NJ
`
`E
`-'
`
`E1
`
`1.3.]
`Spontaneous Enrission
`An electron spontaneously falls from a higher energy level to a lower one as shown
`in Fig. (1.4), the emitted photon has frequency
`
`V” =
`
`E; — E]
`(1.2)
`.
`h
`This photon is emitted in a random direction with arbitrary polarization (except
`in the presence of magnetic fields. but this need not concern us here). The photon
`carries away momentum it/1 = hv/c and the emitting particle (atom, molecule or
`ion) recoils in the opposite direction. The probability of such a spontaneous jump
`is given quantitatively by the Einstein A coeflicient defined as A” = ‘probability’
`per second of a spontaneous jump from level i to level j.
`For example. if there are N; particles per unit volume in level i then N;Au per
`second make jumps to level j. The total rate at which jumps are made between
`the two levels is
`div
`7'’ = -—N..4.,.
`There is a negative sign because the population of level i is decreasing.
`Generally an electron can make jumps to more than one lower level, unless it is
`in the first (lowest) excited level. The total probability that the electron will make
`a spontaneous jump to any lower level is A. s“ where
`
`(1.3)
`
`A. = Zn”.
`J
`
`(1.4)
`
`The summation runs over all levels j lower in energy than level i and the total rate
`at which the population of level i changes by spontaneous emission is
`
`JN1
`T‘ = -Nrfh.
`which has the solution
`
`N; = constant x e“"‘.
`
`If at time t= 0,N; = N? then
`
`N; = NPe“".
`
`(1.5)
`
`(1.6)
`
`i falls exponentially with time as electrons leave by
`so the population of level
`spontaneous emission. The time in which the population falls to 1/2 of its initial
`value is called the natural lifetime of level i, 1.1, where 1!] = I/A1. The magnitude
`of this lifetime is determined by the actual probabilities of jumps from level
`i
`
`
`
`18
`
`18
`
`

`

`
`
`5
`1.3 Amplification at Optical Frequencies
`=
`— _ r##__._.
`
`Jumps which are likely to occur are called allowed
`by spontaneous emission.
`transitions, those which are unlikely are said to be forbidden. Allowed transitions in
`the visible region typically have A” coeflicients in the range 105-10‘ 5*‘. Forbidden
`transitions in this region have Ag coefficients below 10‘ s“. These probabilities
`decrease as the wavelength of the jump increases. Consequently, levels that can
`decay by allowed transitions in the visible have lifetimes generally shorter than 1
`as, similar forbidden transitions have lifetimes in excess of 10-100 as. Although no
`jump turns out to be absolutely forbidden, some jumps are so unlikely that levels
`whose electrons can only fall to lower levels by such jumps are very long lived.
`Levels with lifetimes in excess of 1 hour have been observed under laboratory
`conditions. Levels which can only decay slowly. and usually only by forbidden
`transitions, are said to be metastable.
`When a particle changes its energy spontaneously the emitted radiation is not,
`as might perhaps be expected, all at the same frequency. Real energy levels are not
`infinitely sharp. they are smeared out or broadened. A particle in a given energy
`level can actually have any energy within a finite range. The frequency spectrum
`of the spontaneously emitted radiation is described by the lineshape function, g(v).
`This function is usually normalized so that
`
`-W
`
`(1.7)
`/an g(v)dv = 1.
`g(v)dv represents the probability that a photon will be emitted spontaneously in the
`frequency range v + dv. The lineshape function g(v) is a true probability function
`for the spectrum of emitted radiation and is usually sharply peaked near some
`frequency ‘llo, as shown in Fig. (1.5). For this reason the function is frequently
`written g(vg,v) to highlight this. Since negative frequencies do not exist in reality
`the question might properly be asked: ‘Why does the integral have a lower limit of
`minus infinity?‘ This is done because g(v) can be viewed as the Fourier transform
`of a real function of time, so negative frequencies have to be permitted mathemat-
`ically. In practice g(v) is only of significant value for some generally small range
`of positive frequencies so
`
`j: g(v)dv : 1.
`
`(1.3)
`
`The amount of radiation emitted spontaneously by a collection of particles can
`be described quantitatively by their radiant intensity I.(v). The units of radiant
`intensity are watts per steradiansl The total power (watts) emitted in a given
`frequency interval dv is
`
`(1.9)
`W(v)dv = LI,(v)dvdQ.,
`where the integral
`is taken over a closed surface S surrounding the emitting
`particles.
`The total power emitted is
`
`W0 = no W(v)dv.
`—an
`
`(1.10)
`
`1'The steradian is the unit of solid angle, (1. The surface ofa sphere encompasses a solid angle of 4:: steradians.
`
`19
`
`_
`
`19
`
`

`

`
`
`6 Spontaneous and Stimulated Transitions
`
`Fig. l.5. A lineshape
`function g(vo,v).
`
`80')
`
`
`
`W(v) is closely related to the lineshape function
`
`WM = W030’).
`
`(1-11)
`
`For a collection of N; identical particles the total spontaneously emitted power per
`frequency interval is
`
`W(v) = N;Aghvg(v).
`
`(1.12)
`
`Clearly this power decreases with time if the number of excited particles decreases.
`For a plane electromagnetic wave we can introduce the concept of intensity,
`which has units of W m‘2. The intensity is the average amount of energy per
`second transported across unit area in the direction of travel of the wave. The
`spectral distribution of intensity, I(v), is related to the total intensity, In, by
`
`I(v) = 1030').
`
`(1.13)
`
`It is worth pointing out that in reality perfect plane waves do not exist, such waves
`would have a unique propagation direction and infinite radiant intensity. However,
`to a very good degree of approximation we can treat the light from a small source
`as a plane wave if we are far enough away from the source. The light coming from
`a star represents a very good example of this.
`
`1.3.2 Stimulated Emission
`
`Besides being able to make transitions from a higher level to a lower one by
`spontaneous emission, electrons can also be stimulated to make these jumps by
`the action of an external radiation field, as shown in Fig. (1.6).
`Let the energy density of the externally applied radiation field at frequency v
`be p[v) (energy per unit volume per unit frequency interval; i.e., J m‘3 Hz”). If v
`is the same frequency as a transition between two levels labelled 2 and 1, the rate
`at which stimulated emissions occur is N28510:} s" Hz" m‘3 where B§,(v) is a
`function specific to the electron jump between the two levels and N2 is the number
`of particles per unit volume in the upper level of the transition. The frequency
`
`
`
`20
`
`20
`
`

`

`
`
`1.4 The Relation Between Energy Density and Intensity
`
`
`
`
`E1—E|=hV11
`N; -——Q—— E;
`
`Fig. 1.6. Schematic
`representation of
`stimulated emission
`bcmm two levels of
`energy E; and Ei-
`
`II I
`
`/\/\/\/\->
`
`""'*
`
`/\J\/\.f\-) Inn:
`I A/vv-> M:
`I
`
`~1:é‘.—— E1
`
`dependence of B§,(v) is the same as the lineshape function
`
`3510’) = 32i3(Vo.V)-
`
`(1-14)
`
`B21 is called the Einstein coefficient for stimulated emission.