Laser Physics

SIMON HOOKER and COLIN WEBB

Department of Physics, University of Oxford

OXFORD UNIVERSITY PRESS Contents

1 Introduction 1.1 The laser 1.2 Electromagnetic radiation in a closed cavity 1.2.1 The density of modes 1.3 Planck's law 1.3.1 The energy density of blackbody radiation Further reading Exercises

2 The interaction of radiation and matter 2.1 The Einstein treatment 2.1.1 Relations between the Einstein coefficients 2.2 Conditions for optical gain 2.2.1 Conditions for steady-state inversion 2.2.2 Necessary, but not sufficient condition 2.3 The semi-classical treatment ^ 2.3.1 Outline 2.3.2 Selection rules for electric dipole transitions 2.4 Atomic population kinetics^ 2.4.1 Rate equations 2.4.2 Semi-classical equations 2.4.3 Validity of the rate-equation approach Further reading Exercises

3 Broadening mechanisms and lineshapes 3.1 Homogeneous broadening mechanisms 3.1.1 Natural broadening 3.1.2 Pressure broadening 3.1.3 Phonon broadening 3.2 Inhomogeneous broadening mechanisms 3.2.1 Doppler broadening 3.2.2 Broadening in amorphous solids 3.3 The interaction of radiation and matter in the presence spectral broadening 3.3.1 Homogeneously broadened transitions 3.3.2 Inhomogeneously broadened atoms ^ 3.4 The formation of spectral lines: The Voigt profile^ viii contents

3.5 Other broadening effects 3.5.1 Self-absorption Further reading Exercises

4 Light amplification by the stimulated emission of radiation 4.1 The optical gain cross-section 4.1.1 Condition for optical gain 4.1.2 Frequency dependence of the gain cross-section 4.1.3 The gain coefficient 4.1.4 Gain narrowing 4.2 Narrowband radiation 4.2.1 Amplification of narrowband radiation 4.2.2 Form of rate equations 4.3 Gain cross-section for inhomogeneous broadening* 4.4 Orders of magnitude 4.5 Absorption 4.5.1 The absorption cross-section 4.5.2 Self-absorption 4.5.3 Radiation trapping Further reading Exercises

5 Gain saturation 60 5.1 Saturation in a steady-state amplifier 60 5.1.1 Homogeneous broadening 60 5.1.2 Inhomogeneous broadening1' 67 5.2 Saturation in a homogeneously broadened pulsed amplifier* 73 5.3 Design of laser amplifiers 77 Exercises 78

6 The laser oscillator 83 6.1 Introduction 83 6.2 Amplified spontaneous emission (ASE) lasers 83 6.3 Optical cavities 85 6.3.1 General considerations 85 6.3.2 Low-loss (or 'stable') optical cavities 89 6.3.3 High-loss (or 'unstable') optical cavities* 97 6.4 Beam quality* 103 6.4.1 The M2 beam-propagation factor 103 6.5 The approach to laser oscillation 106 6.5.1 The 'cold' cavity 106 6.5.2 The laser threshold condition 110 6.6 Laser oscillation above threshold 111 6.6.1 Condition for steady-state laser oscillation 112 6.6.2 Homogeneously broadened systems 113 contents ix

6.6.3 Inhomogeneously broadened systems^ 115 6.7 Output power 117 6.7.1 Low-gain lasers 117 6.7.2 High-gain lasers: the Rigrod analysis^ 120 6.7.3 Output power in other cases 123 Further reading 123 Exercises 123

7 Solid-state lasers 132 7.1 General considerations 132 7.1.1 Energy levels of ions doped in solid hosts ^ 132 7.1.2 Radiative transitions* 137 7.1.3 Non-radiative transitions* 138 7.1.4 Line broadening* 142 7.1.5 Three- and four-level systems 142 7.1.6 Host materials 146 7.1.7 Techniques for optical pumping 149 7.2 Nd3+:YAG and other trivalent rare-earth systems 157 7.2.1 Energy-level structure 157 7.2.2 Transition linewidth 157 7.2.3 Nd:YAG laser 158 7.2.4 Other crystalline hosts 163 7.2.5 Nd:glass laser 164 7.2.6 Erbium lasers 165 7.2.7 Praseodymium ions 169 7.3 Ruby and other trivalent iron-group systems 169 7.3.1 Energy-level structure* 169 7.3.2 The ruby laser 174 7.3.3 Alexandrite laser 177 7.3.4 CnLiSAF and CnLiCAF 180 7.3.5 Ti:sapphire 180 Further reading 184 Exercises 184

8 Dynamic cavity effects 188 8.1 Laser spiking and relaxation oscillations 188 8.1.1 Rate-equation analysis 190 8.1.2 Analysis of relaxation oscillations 190 8.1.3 Numerical analysis of laser spiking 192 8.2 Q-switching 193 8.2.1 Techniques for Q-switching 194 8.2.2 Rate-equation analysis of Q-switching 198 8.2.3 Comparison with numerical simulations 203 8.3 Modelocking 203 8.3.1 General ideas 204 8.3.2 Simple treatment of modelocking 206 8.3.3 Active modelocking techniques 208 8.3.4 Passive modelocking techniques 214 x contents

8.4 Other forms of pulsed output 221 Further reading 222 Exercises 222

9 Semiconductor lasers 226 9.1 Basic features of a typical semiconductor diode laser 226 9.2 Review of semiconductor physics 228 9.2.1 Band structure 228 9.2.2 Density of states and the Fermi energy (T = OK) 231 9.2.3 The Fermi-Dirac distribution {T ^ 0 K) 232 9.2.4 Doped semiconductors 233 9.3 Radiative transitions in semiconductors 235 9.4 Gain at a p-i-n junction 236 9.5 Gain in diode lasers 238 9.6 Carrier and photon confinement: the double heterostructure 241 9.7 Laser materials 243 9.8 Quantum-well lasers1" 244 9.9 Laser threshold 247 9.10 Diode laser beam properties 250 9.10.1 Beam shape 250 9.10.2 Transverse modes of edge-emitting lasers 250 9.10.3 Longitudinal modes of diode lasers 251 9.10.4 Single longitudinal mode diode lasers 253 9.10.5 Diode laser linewidth 254 9.10.6 Tunable diode laser cavities^ 255 9.11 Diode laser output power1" 257 9.12 VCSEL lasers1" 259 9.13 Strained-layer lasers 261 9.14 Quantum cascade lasers1^ 262 Further reading 264 Exercises 264

10 Fibre lasers 267 10.1 Optical fibres 267 10.1.1 The importance of optical-fibre technology 267 10.1.2 Optical-fibre properties: Ray optics 268 10.1.3 Optical-fibre properties: Wave optics 271 10.1.4 Dispersion in optical fibres 274 10.1.5 Fabrication of optical fibres 276 10.1.6 Fibre-optic components 277 10.2 Wavelength bands for fibre-optic telecommunications 280 10.3 Erbium-doped fibre amplifiers 282 10.3.1 Energy levels and pumping schemes 282 10.3.2 Gain spectra 282 10.3.3 EDFA design and layout 284 10.3.4 Fabrication of erbium-doped fibre amplifiers 285 10.4 Fibre Raman amplifiers 285 10.4.1 Introduction 285 contents xi

10.4.2 Raman scattering 285 10.4.3 Fibre Raman amplifiers 286 10.4.4 Long-haul optical transmission systems 287 10.5 High-power fibre lasers 289 10.5.1 The revolution in fibre-laser performance 289 10.5.2 Cladding-pumped fibre-laser design 290 10.5.3 Materials and mechanisms of cladding-pumped fibre-laser systems 291 10.5.4 High-power fibre lasers: Linewidth considerations 291 10.6 High-power pulsed fibre lasers 293 10.6.1 Large mode area (LMA) fibres 293 10.6.2 Q-switched fibre lasers 294 10.6.3 Oscillator-amplifier pulsed fibre lasers 294 10.7 Applications of high-power fibre lasers 295 Further reading 296 Exercises 296

11 Atomic gas lasers 298 11.1 Discharge physics interlude 298 11.1.1 Low-pressure and high-pressure discharges 298 11.1.2 Low-pressure glow discharge 299 11.1.3 Temperatures 300 11.1.4 The steady-state positive column 303 11.1.5 Ionization rates 306 11.1.6 Excitation rates 307 11.1.7 Second-kind or superelastic collisions 310 11.1.8 Excited-state populations in low-pressure discharges 311 11.2 The helium-neon laser 314 11.2.1 Introduction 314 11.2.2 Energy levels, transitions and excitation mechanisms 316 11.2.3 Laser construction and operating parameters 318 11.2.4 Output-power limitations of the He-Ne laser 319 11.2.5 Applications of He-Ne lasers 321 11.3 The argon-ion laser 321 11.3.1 Introduction 321 11.3.2 Energy levels, transitions and excitation mechanisms 322 11.3.3 Laser construction and operating parameters 325 11.3.4 Argon-ion laser: Power limitations 327 11.3.5 Krypton-ion lasers 328 11.3.6 Applications of ion lasers 329 Further reading 329 Exercises 329

12 Infra-red molecular gas lasers 332 12.1 Efficiency considerations 332 12.1.1 Energy levels of atoms and molecules 332 12.1.2 Quantum ratio 333 12.2 Partial population inversion between vibrational energy levels of molecules 335 12.3 Physics of the C02 laser 338 12.3.1 Levels and lifetimes 338 12.3.2 The effect of adding N2 341 12.3.3 Effect of adding He 342 12.4 CO2 laser parameters 343 12.5 Low-pressure c.w. CO2 lasers 344 12.6 High-pressure pulsed CO2 lasers 346 12.7 Other types of C02 laser 349 12.7.1 Gas-dynamic C02 lasers 349 12.7.2 Waveguide C02 lasers 351 12.8 Applications of CO2 lasers 351 Further reading 352 Exercises 352

13 Ultraviolet molecular gas lasers 355 13.1 The UV and VUV spectral regions 355 13.2 Energy levels of diatomic molecules 356 13.2.1 Separation of the overall wave function 356 13.2.2 Vibrational eigenfunctions 357 13.3 Electronic transitions in diatomic molecules: The Franck-Condon principle 358 13.3.1 Absorption transitions 358 13.3.2 The'Franck-Condon loop' 360 13.4 The VUV hydrogen laser 361 13.5 The UV nitrogen laser 364 13.6 Excimer molecules 364 13.7 Rare-gas excimer lasers 367 13.8 Rare-gas halide excimer lasers 370 13.8.1 Spectroscopy of the rare-gas halides 370 13.8.2 Rare-gas halide laser design 371 13.8.3 Pulse-length limitations of discharge-excited RGH lasers 373 13.8.4 Cavity design and beam properties of RHG lasers 373 13.8.5 Performance and applications of RGH excimer laser 375 Further reading 377 Exercises 378

14 Dye lasers 380 14.1 Introduction 380 14.2 Dye molecules 380 14.3 Energy levels and spectra of dye molecules in solution 382 14.3.1 Energy-level scheme 382 14.3.2 Singlet-singlet absorption 382 14.3.3 Singlet-singlet emission spectra 385 14.3.4 Triplet-triplet absorption 387 14.4 Rate-equation models of dye laser kinetics 387 contents xiii

14.5 Pulsed dye lasers 388 14.5.1 Flashlamp-pumped systems 388 14.5.2 Dye lasers pumped by pulsed lasers 389 14.6 Continuous-wave dye lasers 391 14.6.1 Population kinetics 391 14.6.2 Continuous waves dye laser design 393 14.7 Solid-state dye lasers 395 14.8 Applications of dye lasers 396 Further reading 398 Exercises 398

15 Non-linear frequency conversion 400 15.1 Introduction 400 15.2 Linear optics of crystals 400 15.2.1 Classes of anisotropic crystals 400 15.2.2 Vectors 402 15.2.3 Field directions for o- and e-rays in a uniaxial crystal 403 15.3 Basics of non-linear optics 405 15.3.1 Maxwell's equations for non-linear media 405 15.3.2 Second-harmonic generation in anisotropic crystals 406 15.3.3 The requirement for phase matching 408 15.4 Phase-matching techniques 409 15.4.1 Birefringent phase matching in uniaxial crystals 409 15.4.2 Critical and non-critical phase matching 412 15.4.3 Poynting vector walk-off in birefringent phase matching 414 15.4.4 Other factors affecting SHG conversion efficiency 414 15.4.5 Phase-matched SHG in biaxial crystals 415 15.4.6 Birefringent materials for SHG 416 15.4.7 Quasi-phase matching techniques 418 15.5 SHG: practical aspects 420 15.6 Three-wave mixing and third-harmonic generation (THG) 421 15.6.1 Three-wave mixing processes in general 421 15.6.2 Third-harmonic generation (THG) 423 15.7 Optical parametric oscillators (OPOs) 424 15.7.1 Parametric interactions 424 15.7.2 Optical parametric oscillators (OPOs) 425 15.7.3 Practical parametric devices 426 Further reading 428 Exercises 428

16 Precision frequency control of lasers^ 431 16.1 Frequency pulling 431 16.2 Single longitudinal mode operation 433 16.2.1 Short cavity 434 16.2.2 Intra-cavity etalons 435 16.2.3 Ring resonators 437 16.2.4 Other techniques 440 xiv contents

16.3 Output linewidfh 440 16.3.1 The Schawlow-Townes limit 441 16.3.2 Practical limitations 444 16.3.3 Intensity noise 446 16.4 Frequency locking 448 16.4.1 Locking to atomic or molecular transitions 450 16.4.2 Locking to an external cavity 452 16.5 Frequency combs 453 Further reading 456 Exercises 456

17 Ultrafast lasers 462 17.1 Propagation of ultrafast laser pulses in dispersive media 462 17.1.1 The time-bandwidth product 462 17.1.2 General considerations 463 17.1.3 Propagation through a dispersive system 466 17.1.4 Propagation of Gaussian pulses 469 17.1.5 Non-linear effects: self-phase modulation and the B-integral 472 17.2 Dispersion control 474 17.2.1 Geometric dispersion control 474 17.2.2 Chirped mirrors 478 17.2.3 Pulse shaping 480 17.3 Sources of ultrafast optical pulses 482 17.3.1 Modelocked lasers 482 17.3.2 Oscillators 483 17.3.3 Chirped-pulse amplification (CPA) 483 17.4 Measurement of ultrafast pulses 489 17.4.1 Autocorrelators 489 17.4.2 Methods for exact reconstruction of the pulse 492 Further reading 495 Exercises 495

18 Short-wavelength lasers 502 18.1 Definition of wavelength ranges 503 18.2 Difficulties in achieving optical gain at short wavelengths 503 18.2.1 Pump-power scaling 503 18.3 General properties of short-wavelength lasers 505 18.3.1 Travelling-wave pumping 505 18.3.2 Threshold and saturation behaviour in an ASE laser 506 18.3.3 Spectral width of the output 508 18.3.4 Coherence properties of ASE lasers 509 18.4 Laser-generated plasmas^ 510 18.4.1 Inverse bremsstrahlung heating 510 18.4.2 Generation of highly ionized plasmas from laser-solid interactions 511 18.4.3 Optical field ionization 514 18.5 Collisionally excited lasers 517 contents xv

18.5.1 Ne-likeionst 518 18.5.2 Ni-likeionst 520 18.5.3 Methods of pumping 520 18.5.4 Collisionally excited OFI lasers 528 18.6 Recombination lasers 530 18.6.1 H-like carbon 532 18.6.2 OFI recombination lasers 533 18.7 Other sources 535 18.7.1 High-harmonic generation 535 18.7.2 Free-electron lasers 537 Further reading 541 Exercises 541

Appendix A: The semi-classical theory of the interaction of radiation and matter 548 A.l The amplitude equations 548 A. 1.1 Derivation of the amplitude equations 548 A. 1.2 Solution of the amplitude equations 550 A.2 Calculation of the Einstein B coefficient 551 A.2.1 Polarized atoms and radiation 551 A.2.2 Unpolarized atoms and/or radiation 553 A.2.3 Treatment of degeneracy 554 A.3 Relations between the Einstein coefficients 555 A.4 Validity of rate equations 555

Appendix B: The spectral Einstein coefficients 557

Appendix C: Kleinman's conjecture 560

Bibliography 563

Index 579