Preface ...................................................... xvii
Acknowledgments ............................................... xxi
List of Fundamental Constants ............................... xxiii
1 Ingredients .................................................. 1
1.1 Introduction ............................................ 1
1.2 Energy Levels and Bands in Solids ....................... 5
1.3 Spontaneous and Stimulated Transitions: The Creation
of Light ................................................ 7
1.4 Transverse Confinement of Carriers and Photons in
Diode Lasers: The Double Heterostructure ............... 10
1.5 Semiconductor Materials for Diode Lasers ............... 13
1.6 Epitaxial Growth Technology ............................ 20
1.7 Lateral Confinement of Current, Carriers, and Photons
for Practical Lasers ................................... 24
1.8 Practical Laser Examples ............................... 31
References .................................................. 39
Reading List ................................................ 40
2 A Phenomenological Approach to Diode Lasers ................. 45
2.1 Introduction ........................................... 45
2.2 Carrier Generation and Recombination in Active
Regions ................................................ 46
2.1 Spontaneous Photon Generation and LEDs ................. 49
2.4 Photon Generation and Loss in Laser Cavities ........... 52
2.5 Threshold or Steady-State Gain in Lasers ............... 55
2.6 Threshold Current and Power Out Versus Current ......... 60
2.6.1 Basic P-I Characteristics ....................... 60
2.6.2 Gain Models and Their Use in Designing Lasers ... 64
2.7 Relaxation Resonance and Frequency Response ............ 70
2.8 Characterizing Real Diode Lasers ....................... 74
2.8.1 Internal Parameters for In-Plane Lasers:
(αi), ηi, and g versus J ....................... 75
2.8.2 Internal Parameters for VCSELs: ηi and g versus
J, (αi), and αm ................................. 78
2.8.3 Efficiency and Heat Flow ........................ 79
2.8.4 Temperature Dependence of Drive Current ......... 80
2.8.5 Derivative Analysis ............................. 84
References .................................................. 86
Reading List ................................................ 87
Problems .................................................... 87
3 Mirrors and Resonators for Diode Lasers ..................... 91
3.1 Introduction ........................................... 91
3.2 Scattering Theory ...................................... 92
3.3 S and T Matrices for Some Common Elements .............. 95
3.3.1 The Dielectric Interface ........................ 96
3.3.2 Transmission Line with No Discontinuities ....... 98
3.3.3 Dielectric Segment and the Fabry-Perot
Etalon ......................................... 100
3.3.4 S-Parameter Computation Using Mason's Rule ..... 104
3.3.5 Fabry-Perot Laser .............................. 105
3.4 Three- and Four-Mirror Laser Cavities ................. 107
3.4.1 Three-Mirror Lasers ............................ 107
3.4.2 Four-Mirror Lasers ............................. 111
3.5 Gratings .............................................. 113
3.5.1 Introduction ................................... 113
3.5.2 Transmission Matrix Theory of Gratings ......... 115
3.5.3 Effective Mirror Model for Gratings ............ 121
3.6 Lasers Based on DBR Mirrors ........................... 123
3.6.1 Introduction ................................... 123
3.6.2 Threshold Gain and Power Out ................... 124
3.6.3 Mode Selection in DBR-Based Lasers ............. 127
3.6.4 VCSEL Design ................................... 128
3.6.5 In-Plane DBR Lasers and Tunability ............. 135
3.6.6 Mode Suppression Ratio in DBR Laser ............ 139
3.7 DFB Lasers ............................................ 141
3.7.1 Introduction ................................... 141
3.7.2 Calculation of the Threshold Gains and
Wavelengths .................................... 143
3.7.3 On Mode Suppression in DFB Lasers .............. 149
References ................................................. 151
Reading List ............................................... 151
Problems ................................................... 151
4 Gain and Current Relations ................................. 157
4.1 Introduction .......................................... 157
4.2 Radiative Transitions ................................. 158
4.2.1 Basic Definitions and Fundamental
Relationships .................................. 158
4.2.2 Fundamental Description of the Radiative
Transition Rate ................................ 162
4.2.3 Transition Matrix Element ...................... 165
4.2.4 Reduced Density of States ...................... 170
4.2.5 Correspondence with Einstein's Stimulated
Rate Constant .................................. 174
4.3 Optical Gain .......................................... 174
4.3.1 General Expression for Gain .................... 174
4.3.2 Lineshape Broadening ........................... 181
4.3.3 General Features of the Gain Spectrum .......... 185
4.3.4 Many-Body Effects .............................. 187
4.3.5 Polarization and Piezoelectricity .............. 190
4.4 Spontaneous Emission .................................. 192
4.4.1 Single-Mode Spontaneous Emission Rate .......... 192
4.4.2 Total Spontaneous Emission Rate ................ 193
4.4.3 Spontaneous Emission Factor .................... 198
4.4.4 Purcell Effect ................................. 198
4.5 Nonradiative Transitions .............................. 199
4.5.1 Defect and Impurity Recombination .............. 199
4.5.2 Surface and Interface Recombination ............ 202
4.5.3 Auger Recombination ............................ 211
4.6 Active Materials and Their Characteristics ............ 218
4.6.1 Strained Materials and Doped Materials ......... 218
4.6.2 Gain Spectra of Common Active Materials ........ 220
4.6.3 Gain versus Carrier Density .................... 223
4.6.4 Spontaneous Emission Spectra and Current
versus Carrier Density ......................... 227
4.6.5 Gain versus Current Density .................... 229
4.6.6 Experimental Gain Curves ....................... 233
4.6.7 Dependence on Well Width, Doping, and
Temperature .................................... 234
References ................................................. 238
Reading List ............................................... 240
Problems ................................................... 240
5 Dynamic Effects ............................................ 247
5.1 Introduction .......................................... 247
5.2 Review of Chapter 2 ................................... 248
5.2.1 The Rate Equations ............................. 249
5.2.2 Steady-State Solutions ......................... 250
Case (i): Well Below Threshold ................. 251
Case (ii): Above Threshold ..................... 252
Case (iii): Below and Above Threshold .......... 253
5.2.3 Steady-State Multimode Solutions ............... 255
5.3 Differential Analysis of the Rate Equations ........... 257
5.3.1 Small-Signal Frequency Response ................ 261
5.3.2 Small-Signal Transient Response ................ 266
5.3.3 Small-Signal FM Response or Frequency
Chirping ....................................... 270
5.4 Large-Signal Analysis ................................. 276
5.4.1 Large-Signal Modulation: Numerical Analysis
of the Multimode Rate Equations ................ 277
5.4.2 Mode Locking ................................... 279
5.4.3 Turn-On Delay .................................. 283
5.4.4 Large-Signal Frequency Chirping ................ 286
5.5 Relative Intensity Noise and Linewidth ................ 288
5.5.1 General Definition of RIN and the Spectral
Density Function ............................... 288
5.5.2 The Schawlow-Townes Linewidth .................. 292
5.5.3 The Langevin Approach .......................... 294
5.5.4 Langevin Noise Spectral Densities and RIN ...... 295
5.5.5 Frequency Noise ................................ 301
5.5.6 Linewidth ...................................... 303
5.6 Carrier Transport Effects ............................. 308
5.7 Feedback Effects and Injection Locking ................ 311
5.7.1 Optical Feedback Effects—Static
Characteristics ................................ 311
5.7.2 Injection Locking—Static Characteristics ....... 317
5.7.3 Injection and Feedback Dynamic
Characteristics and Stability .................. 320
5.7.4 Feedback Effects on Laser Linewidth ............ 321
References ................................................. 328
Reading List ............................................... 329
Problems ................................................... 329
6 Perturbation, Coupled-Mode Theory, Modal Excitations, and
Applications ............................................... 335
6.1 Introduction .......................................... 335
6.2 Guided-Mode Power and Effective Width ................. 336
6.3 Perturbation Theory ................................... 339
6.4 Coupled-Mode Theory: Two-Mode Coupling ................ 342
6.4.1 Contradirectional Coupling: Gratings ........... 342
6.4.2 DFB Lasers ..................................... 353
6.4.3 Codirectional Coupling: Directional Couplers ... 356
6.4.4 Codirectional Coupler Filters and Electro-
optic Switches ................................. 370
6.5 Modal Excitation ...................................... 376
6.6 Two Mode Interference and Multimode Interference ...... 378
6.7 Star Couplers ......................................... 381
6.8 Photonic Multiplexers, Demultiplexers and Routers ..... 382
6.8.1 Arrayed Waveguide Grating De/Multiplexers
and Routers .................................... 383
6.8.2 Echelle Grating based De/Multiplexers and
Routers ........................................ 389
6.9 Conclusions ........................................... 390
References ................................................. 390
Reading List ............................................... 391
Problems ................................................... 391
7 Dielectric Waveguides ...................................... 395
7.1 Introduction .......................................... 395
7.2 Plane Waves Incident on a Planar Dielectric
Boundary .............................................. 396
7.3 Dielectric Waveguide Analysis Techniques .............. 400
7.3.1 Standing Wave Technique ........................ 400
7.3.2 Transverse Resonance ........................... 403
7.3.3 WKB Method for Arbitrary Waveguide Profiles .... 410
7.3.4 2-D Effective Index Technique for Buried Rib
Waveguides ..................................... 418
7.3.5 Analysis of Curved Optical Waveguides using
Conformal Mapping .............................. 421
7.3.6 Numerical Mode Solving Methods for Arbitrary
Waveguide Profiles ............................. 424
7.4 Numerical Techniques for Analyzing PICs ............... 427
7.4.1 Introduction ................................... 427
7.4.2 Implicit Finite-Difference Beam-Propagation
Method ......................................... 429
7.4.3 Calculation of Propagation Constants in а
z-invariant Waveguide from a Beam Propagation
Solution ....................................... 432
7.4.4 Calculation of Eigenmode Profile from a Beam
Propagation Solution ........................... 434
7.5 Goos-Hanchen Effect and Total Internal Reflection
Components ............................................ 434
7.5.1 Total Internal Reflection Mirrors .............. 435
7.6 Losses in Dielectric Waveguides ....................... 437
7.6.1 Absorption Losses in Dielectric Waveguides ..... 437
7.6.2 Scattering Losses in Dielectric Waveguides ..... 438
7.6.3 Radiation Losses for Nominally Guided Modes .... 438
References ................................................. 445
Reading List ............................................... 446
Problems ................................................... 446
8 Photonic Integrated Circuits ............................... 451
8.1 Introduction .......................................... 451
8.2 Tunable, Widely Tunable, and Externally Modulated
Lasers ................................................ 452
8.2.1 Two- and Three-Section In-plane DBR Lasers ..... 452
8.2.2 Widely Tunable Diode Lasers .................... 458
8.2.3 Other Extended Tuning Range Diode Laser
Implementations ................................ 463
8.2.4 Externally Modulated Lasers .................... 474
8.2.5 Semiconductor Optical Amplifiers ............... 481
8.2.6 Transmitter Arrays ............................. 484
8.3 Advanced PICs ......................................... 484
8.3.1 Waveguide Photodetectors ....................... 485
8.3.2 Transceivers/Wavelength Converters and
Triplexers ..................................... 488
8.4 PICs for Coherent Optical Communications .............. 491
8.4.1 Coherent Optical Communications Primer ......... 492
8.4.2 Coherent Detection ............................. 495
8.4.3 Coherent Receiver Implementations .............. 495
8.4.4 Vector Transmitters ............................ 498
References ................................................. 499
Reading List ............................................... 503
Problems ................................................... 503
APPENDICES
1 Review of Elementary Solid-State Physics ................... 509
A1.1 A Quantum Mechanics Primer ........................... 509
A1.1.1 Introduction ................................. 509
A1.1.2 Potential Wells and Bound Electrons .......... 511
A1.2 Elements of Solid-State Physics ...................... 516
Al.2.1 Electrons in Crystals and Energy Bands ....... 516
A1.2.2 Effective Mass ............................... 520
A1.2.3 Density of States Using a Free-Electron
(Effective Mass) Theory ...................... 522
References ................................................. 527
Reading List ............................................... 527
2 Relationships between Fermi Energy and Carrier Density
and Leakage ................................................ 529
A2.1 General Relationships ................................ 529
A2.2 Approximations for Bulk Materials .................... 532
A2.3 Carrier Leakage Over Heterobarriers .................. 537
A2.4 Internal Quantum Efficiency .......................... 542
References ................................................. 544
Reading List ............................................... 544
3 Introduction to Optical Waveguiding in Simple Double-
Heterostructures ........................................... 545
A3.1 Introduction ......................................... 545
A3.2 Three-Layer Slab Dielectric Waveguide ................ 546
A3.2.1 Symmetric Slab Case .......................... 547
A3.2.2 General Asymmetric Slab Case ................. 548
A3.2.3 Transverse Confinement Factor, Гх ............ 550
A3.3 Effective Index Technique for Two-Dimensional
Waveguides ........................................... 551
A3.4 Far Fields ........................................... 555
References ................................................. 557
Reading List ............................................... 557
4 Density of Optical Modes, Blackbody Radiation, and
Spontaneous Emission Factor ................................ 559
A4.1 Optical Cavity Modes ................................. 559
A4.2 Blackbody Radiation .................................. 561
A4.3 Spontaneous Emission Factor, βsp ..................... 562
Reading List ............................................... 563
5 Modal Gain, Modal Loss, and Confinement Factors ............ 565
A5.1 Introduction ......................................... 565
A5.2 Classical Definition of Modal Gain ................... 566
A5.3 Modal Gain and Confinement Factors ................... 568
A5.4 Internal Modal Loss .................................. 570
A5.5 More Exact Analysis of the Active/Passive Section
Cavity ............................................... 571
A5.5.1 Axial Confinement Factor ..................... 572
A5.5.2 Threshold Condition and Differential
Efficiency ................................... 573
A5.6 Effects of Dispersion on Modal Gain .................. 576
6 Einstein's Approach to Gain and Spontaneous Emission ....... 579
A6.1 Introduction ......................................... 579
A6.2 Einstein A and В Coefficients ........................ 582
A6.3 Thermal Equilibrium .................................. 584
A6.4 Calculation of Gain .................................. 585
A6.5 Calculation of Spontaneous Emission Rate ............. 589
Reading List ............................................... 592
7 Periodic Structures and the Transmission Matrix ............ 593
A7.1 Introduction ......................................... 593
A7.2 Eigenvalues and Eigenvectors ......................... 593
A7.3 Application to Dielectric Stacks at the Bragg
Condition ............................................ 595
A7.4 Application to Dielectric Stacks Away from the
Bragg Condition ...................................... 597
A7.5 Correspondence with Approximate Techniques ........... 600
A7.5.1 Fourier Limit ................................ 601
A7.5.2 Coupled-Mode Limit ........................... 602
A7.6 Generalized Reflectivity at the Bragg Condition ...... 603
Reading List ............................................... 605
Problems ................................................... 605
8 Electronic States in Semiconductors ........................ 609
A8.1 Introduction ......................................... 609
A8.2 General Description of Electronic States ............. 609
A8.3 Bloch Functions and the Momentum Matrix Element ...... 611
A8.4 Band Structure in Quantum Wells ...................... 615
A8.4.1 Conduction Band .............................. 615
A8.4.2 Valence Band ................................. 616
A8.4.3 Strained Quantum Wells ....................... 623
References ................................................. 627
Reading List ............................................... 628
9 Fermi's Golden Rule ........................................ 629
A9.1 Introduction ......................................... 629
A9.2 Semiclassical Derivation of the Transition Rate ...... 630
A9.2.1 Case I: The Matrix Element-Density of
Final States Product is a Constant .......... 632
A9.2.2 Case II: The Matrix Element-Density of
Final States Product is a Delta Function .... 635
A9.2.3 Case III: The Matrix Element-Density of
Final States Product is a Lorentzian ........ 636
Reading List ............................................... 637
Problems ................................................... 638
10 Transition Matrix Element .................................. 639
A10.1 General Derivation ................................... 639
A10.2 Polarization-Dependent Effects ....................... 641
A10.3 Inclusion of Envelope Functions in Quantum Wells ..... 645
Reading List ............................................... 646
11 Strained Bandgaps .......................................... 647
A11.1 General Definitions of Stress and Strain ............. 647
A11.2 Relationship Between Strain and Bandgap .............. 650
A11.3 Relationship Between Strain and Band Structure ....... 655
References ................................................. 656
12 Threshold Energy for Auger Processes ....................... 657
A12.1 CCCH Process ......................................... 657
A12.2 CHHS and CHHL Processes .............................. 659
13 Langevin Noise ............................................. 661
A13.1 Properties of Langevin Noise Sources ................ 661
A13.1.1 Correlation Functions and Spectral
Densities ................................... 661
A13.1.2 Evaluation of Langevin Noise Correlation
Strengths ................................... 664
A13.2 Specific Langevin Noise Correlations ................ 665
A13.2.1 Photon Density and Carrier Density
Langevin Noise Correlations ................. 665
A13.2.2 Photon Density and Output Power Langevin
Noise Correlations .......................... 666
A13.2.3 Photon Density and Phase Langevin Noise
Correlations ................................ 667
A13.3 Evaluation of Noise Spectral Densities .............. 669
A13.3.1 Photon Noise Spectral Density ............... 669
A13.3.2 Output Power Noise Spectral Density ......... 670
A13.3.3 Carrier Noise Spectral Density .............. 671
References ................................................. 672
Problems ................................................... 672
14 Derivation Details for Perturbation Formulas ............... 675
Reading List ............................................... 676
15 Multimode Interference ..................................... 677
A15.1 Multimode Interference-Based Couplers ................ 677
A15.2 Guided-Mode Propagation Analysis ..................... 678
A15.2.1 General Interference ......................... 679
A15.2.2 Restricted Multimode Interference ............ 681
A15.3 MMI Physical Properties .............................. 682
A15.3.1 Fabrication .................................. 682
A15.3.2 Imaging Quality .............................. 682
A15.3.3 Inherent Loss and Optical Bandwidth .......... 682
A15.3.4 Polarization Dependence ...................... 683
A15.3.5 Reflection Properties ........................ 683
Reference .................................................. 683
16 The Electro-Optic Effect ................................... 685
References ................................................. 692
Reading List ............................................... 692
17 Solution of Finite Difference Problems ..................... 693
A17.1 Matrix Formalism .................................... 693
A17.2 One-Dimensional Dielectric Slab Example ............. 695
Reading List ............................................... 696
Index ......................................................... 697
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