Diode Lasers and Photonic Integrated Circuits

Scott W. Corzine obtained his PhD from the University of California, Santa Barbara, Department of Electrical and Computer Engineering, for his work on vertical-cavity surface-emitting lasers (VCSELs). He worked for ten years at HP/Agilent Laboratories in Palo Alto, California, on VCSELs, externally modulated lasers, and quantum cascade lasers. He is currently with Infinera in Sunnyvale, California, working on photonic integrated circuits.

Milan L. Mashanovitch obtained his PhD in the field of photonic integrated circuits at the University of California, Santa Barbara (UCSB), in 2004. He has since been with UCSB as a scientist working on tunable photonic integrated circuits and as an adjunct professor, and with Freedom Photonics LLC, Santa Barbara, which he cofounded in 2005, working on photonic integrated circuits.

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

Problems 40

2 A Phenomenological Approach to Diode Lasers 45

2.1 Introduction 45

2.2 Carrier Generation and Recombination in Active Regions 46

2.3 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 a 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

A1.2.1 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, x 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 B 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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