Distributed feedback semiconductor lasers

Distributed feedback semiconductor lasers John Carroll, James Whiteaway & Dick Plumb The Institution of Electrical Engineers SPIE Optical Engineering Press

1 Preface Acknowledgments Principal abbreviations Principal notation xiii xv xvii xix 1 The semiconductor-diode laser 1 1.1 Background l 1.2 Early developments 1 1.2.1 The first semiconductor lasers 1 1.2.2 Fabry-Perot gain and phase requirements 2 1.2.3 Some characteristics of diode lasers 4 1.3 Improvements to reduce operating currents 6 1.3.1 Heterojunctions: carrier confinement 6 1.3.2 Heterojunctions: photon confinement 7 1.3.3 Structures for'horizontal'confinement 9 1.3.4 Degree of confinement the confinement factor 11 1.4 Variations on conventional Fabry-Perot laser design 12 1.4.1 High-low reflective facets 12 1.4.2 External cavities 13 1.4.3 Externalgrating 14 1.5 System requirements for single-frequency lasers 15 1.6 Introduction to lasers based on Bragg gratings 18 1.6.1 Introduction to Bragg gratings 18 1.6.2 Fabrication of gratings inside lasers 22 1.7 Some principal forms of grating laser 25 1.7.1 The distributed Bragg reflector laser 25 1.7.2 The distributed feedback (DFB) laser 26 1.7.3 More complex grating-based lasers 28 1.8 Summary 30 1.9 Bibliography 31 1.9.1 Semiconductor lasers 31 1.9.2 Optical communication systems 32 1.10 References 32

vi 2 Gain, loss and spontaneous emission 37 2.1 Introduction 37 2.2 Electronic processes in semiconductors 37 2.2.1 Energy states 37 2.2.2 Occupation probabilities 40 2.2.3 Radiative recombination and absorption 41 2.2.4 Transitions and transition rates 43 2.2.5 Auger recombination 44 2.3 Absorption, emission rates and spectra 46 2.3.1 Absorption, stimulated and spontaneous emission in a semiconductor 46 2.3.2 Stimulated-gain spectra in semiconductors 50 2.3.3 Homogenous and inhomogeneous broadening 53 2.3.4 Spontaneous-emission spectra from semiconductors 54 2.4 Semiconductor interactions with the lasing mode 55 2.4.1 Spontaneous-coupling factor 55 2.4.2 Petermann's '^factor' 58 2.4.3 Gain saturation in semiconductors 58 2.4.4 Spectral hole burning and carrier heating 59 2.4.5 Scattering losses 60 2.4.6 Free-carrier absorption 60 2.5 Henry's a factor (or linewidth enhancement factor) 61 2.6 Temperature-induced variations in semiconductor lasers 64 2.7 Properties of quantum-well-laser active regions 66 2.7.1 Introduction to quantum wells 66 2.7.2 Gain saturation and the need for multiple quantum wells 68 2.7.3 Strained quantum wells 69 2.7.4 Carrier transport 70 2.8 Summary 72 2.9 Bibliography 72 2.10 References 73 3 Principles of modelling guided waves 76 3.1 Introduction 76 3.1.1 Vertical and horizontal guiding 77 3.1.2 Index and gain guiding 77 3.1.3 Effective area and confinement factor 79 3.2 The slab guide 81 3.2.1 ТЕ and TM guided waves 81 3.2.2 Multilayer slab guides 82 3.3 Wave equations for the ТЕ and TM guided waves 83 3.4 Solving multislab guides 84 3.4.1 Effective refractive index 84 3.4.2 Reflection coefficient calculation 86 3.4.3 Gain-guiding example 89

vii 3.5 Scaling 90 3.6 Horizontal guiding: effective-index method 90 3.7 Orthogonality o$ fields 92 3.8 Far fields 92 3.9 Waveguiding with quantum-well materials 93 3.10 Summary and conclusions 95 3.11 References 96 4 Optical energy exchange in guides 97 4.1 The classic rate equations 97 4.1.1 Introduction 97 4.1.2 Rate of change of photon density 98 4.1.3 Rate of change of electron density 100 4.2 Some basic results from rate-equation analysis 103 4.2.1 Simplifying the rate equations 103 4.2.2 Steady-state results 104 4.2.3 Dynamic analysis 106 4.2.4 Problems of particle balance 110 4.3 Field equations and rate equations 111 4.3.1 Introduction 111 4.3.2 Wave propagation 111 4.3.3 Decoupling of frequency and propagation coefficient 114 4.4 Field equations with a grating 116 4.4.1 The periodic permittivity 116 4.4.2 Phase matching 117 4.4.3 Second-order gratings 120 4.4.4 Shape of grating 124 4.5 Summary 125 4.6 References 126 5 Basic principles of lasers with distributed feedback 128 5.1 Introduction 128 5.2 Coupled-mode equations for distributed feedback 129 5.2.1 Physical derivation of the coupling process 129 5.2.2 Complex gratings 131 5.3 Coupled-mode solutions and stopbands 132 5.3.1 Eigenmodes 132 5.3.2 The dispersion relationship and stopbands 133 5.4 Matrix solution of coupled-mode equations for uniform grating laser 135 5.4.1 The field input-output relationships 135 5.4.2 Reflections and the observed stopband 137 5.5 DFB lasers with phase shifts 139 5.5.1 Phase shifts 139 5.5.2 Insertion of phase shifts: the transfer-matrix method 142

viii 5.6 Longitudinal-mode spatial-hole burning 145 5.6.1 The phenomena 145 5.6.2 The stopband diagram 147 5.6.3 Influence of KL product on spatial-hole burning 148 5.6.4 Influence of phase shifts on spatial-hole burning 148 5.6.5 Spectrum and spatial-hole burning 150 5.7 Influence of series resistance 152 5.8 Simulating the static performance of DFB lasers 155 5.8.1 Light/current characteristics 155 5.8.2 Simulation of emission spectrum 159 5.9 Summary 161 5.10 References 162 6 More advanced distributed feedback laser design 165 6.1 Introduction 165 6.2 Linewidth 166 6.2.1 General 166 6.2.2 Calculation of linewidth under static conditions 168 6.2.3 Linewidth enhancement 171 6.2.4 Effective linewidth enhancement 172 6.2.5 Effective dynamic linewidth enhancement 176 6.2.6 Linewidth rebroadening 176 6.3 Influence of reflections from facets and external sources 177 6.3.1 Reflections and stability 177 6.3.2 Facet reflectivity and spectral measurements 179 6.3.3 Influence of facet reflectivity on SMSR for DFB lasers 180 6.4 Complex grating-coupling coefficients 183 6.4.1 General 183 6.4.2 Techniques for introducing complex grating-coupling coefficients 183 6.4.3 Influence of complex grating-coupling coefficient on static performance 184 6.4.4 Influence of complex grating-coupling coefficient on dynamic performance 187 6.4.5 Influence of facet reflectivity 189 6.5 High-power lasers with distributed feedback 189 6.5.1 General 189 6.5.2 Techniques for obtaining high front-to-back emission ratios 190 6.5.3 Laser-amplifier structures with distributed feedback 191 6.6 Dynamic modelling of DFB lasers 194 6.6.1 Uniform-grating DFB laser with reflective rear facet 194 6.6.2 Large signal performance of 2 x A /8 DFB lasers with strong and weak carrier-transport effects 197 6.7 Summary 202

ix 6.8 References 202 Numerical modelling for DFB lasers 209 7.1 Introduction 209 7.2 Ordinary differential equations 211 7.2.1 A first-order equation 211 7.2.2 Accuracy 213 7.3 First-order wave equations 214 7.3.1 Introduction 214 7.3.2 Step lengths in space and time central-difference method 215 7.3.3 Numerical stability 216 7.3.4 Gain and phase 218 7.4 Coupled reflections 219 7.4.1 Kappa coupling but no gain or phase changes 219 7.4.2 Matrix formulation 219 7.4.3 Phase jumps replacing scattering 221 7.4.4 Fourier checks 221 7.5 A uniform Bragg laser: finite difference in time and space 222 7.5.1 Full coupled-wave equations 222 7.5.2. MATLAB code 223 7.5.3 Analytic against numeric solutions 224 7.6 Spontaneous emission and random fields 226 7.6.1 Spontaneous noise and travelling fields 226 7.6.2 Null correlation for different times, positions and directions 228 7.6.3 Spontaneous magnitude 229 7.6.4 Tutorial programs 229 7.7 Physical effects of discretisation in the frequency domain 230 7.7.1 Discretisation process integrals to sums 230 7.7.2 Fast Fourier transform (FFT) 232 7.8 Finite-element strategies for a spectral filter 233 7.8.1 Lorentzian filter 233 7.8.2 Numerical implementation 235 7.9 Application of the filter theory to gain filtering 237 7.9.1 General 237 7.9.2 Filtering the gain in the travelling-wave equations 238 7.9.3 Numerical implementation 240 7.10 Basic DFB laser excited by spontaneous emission 241 7.10.1 Introduction and normalisation 241 7.10.2 Field equations 243 7.10.3 Charge-carrier rate equation 243 7.10.4 Numerical programs 246 7.11 Summary 248 7.12 References 249

x 8 Future devices, modelling and systems analysis 252 8.1 Introduction 252 8.2 Systems analysis 252 8.2.1 Introduction 252 8.2.2 Component modelling 253 8.2.3 System modelling 255 8.2.4 10 Gbit/s power amplification 256 8.2.5 Direct modulation: recapitulation 258 8.2.6 Simulation of integrated DFB laser and electroabsorption modulator 259 8.2.7 Cross-gain and four-wave-mixing wavelength conversion in an SOA 260 8.2.8 Simulation of cross-phase wavelength conversion in a Mach-Zehnder interferometer incorporating two SOAs 263 8.3 The push-pull laser 265 8.3.1 Introduction: push-pull electronics 265 8.3.2 Symmetrical push-pull DFB laser 266 8.3.3 Asymmetry and the push-pull DFB laser 269 8.3.4 Speed of response for a push-pull DFB laser 272 8.4 Tunable lasers with distributed feedback 274 8.4.1 Introduction 274 8.4.2 Simple multicontact tunable lasers 277 8.4.3 Wide-tuning-range lasers with nonuniform gratings 279 8.4.4 Other tunable-laser structures 282 8.4.5 Tunable-laser linewidth and modulation 284 8.4.6 Modelling tunable semiconductor lasers 284 8.4.7 Multiple DFB lasers with optical couplers for WDM 285 8.5 Surface-emitting lasers 286 8.5.1 Introduction to surface-emitting lasers 286 8.5.2 Operating parameters of VCSELs compared with edge emitters 287 8.5.3 Construction of VCSELs 291 8.5.4 Additional features of VCSELs 294 8.6 Summary 294 8.7 References 295 Appendix 1 Maxwell, plane waves and reflections ^ 304 Al.l The wave equation 304 Al.2 Linearly polarised plane waves (in a uniform 'infinite' material) 304 A1.3 Snell's law and total internal reflection 305 Al.4 Reflection amplitudes at surfaces: ТЕ fields 308 Al.5 ТЕ reflection amplitudes: three special cases 309 Al.6 Reflection amplitudes at surfaces: TM fields 309

xi Al.7 TM reflection amplitudes at surfaces: four special cases 310 Al.8 Reflection for waveguide modes at facets 311 Al.9 References 312 Appendix 2 Algorithms for the multilayer slab guide 313 A2.1 ТЕ slab modes 313 A2.2 TM slab modes 317 A2.3 Far fields 319 A2.4 Slab waveguide prog*>am 322 A2.5 References 323 Appendix 3 Group refractive index of laser waveguides 324 A3.1 References 328 Appendix 4 Small-signal analysis of single-mode laser 329 A4.1 Rate equations: steady-state and small-signal 329 A4.2 Carrier-transport effects 334 A4.3 Small-signal FM response of single-mode laser 336 A4.4 Small-signal FM response and carrier transport 337 A4.5 Photonic and electronic equations for large-signal analysis 339 A4.6 Reference 340 Appendix 5 Electromagnetic energy exchange 341 A5.1 Dielectric polarisation and energy exchange 341 A5.2 Electromagnetic-energy exchange and rate equations reconciled 344 A5.3 Electromagnetic-energy exchange and guided waves: field equations 348 A5.4 References 351 Appendix 6 Pauli equations 352 A6.1 Reference 356 Appendix 7 Kramers-Krönig relationships 357 A7.1 Causality 357 A7.2 Cauchy contours and stability 359 A7.3 A proper physical basis builds in causality 360 A7.4 Refractive index of transparent quaternary alloys 362 A7.5 References 364 Appendix 8 Relative-intensity noise (RIN) 366 A8.1 References 371 Appendix 9 Thermal, quantum and numerical noise 372 A9.1 Introduction 372 A9.2 Thermal and quantum noise 373

xii A9.3 Ideal amplification 374 A9.4 The attenuator 377 A9.5 Einstein treatment: mode counting 378 A9.6 Aperture theory 379 A9.7 Numerical modelling of spontaneous noise 380 A9.8 Higher-order noise statistics 384 A9.9 References 385 Appendix 10 Laser packaging 386 Al 0.1 Introduction 386 A10.2 Electrical interfaces and circuits 386 Al0.3 Thermal considerations 388 Al 0.4 Laser monitoring 388 A10.5 Package-related backreflections and fibre coupling 389 A10.6 References 391 Appendix 11 Tables of device parameters and simulated performance for DFB laser structures 392 Appendix 12 About MATLAB programs 396 Al 2.1 Instructions for access 396 Al2.2 Introduction to the programs 398 Index 405