Ultrafast OpticsFranz X. KaertnerSpring Term 2005
Ultrafast Optics Franz X. Kaertner Spring Term 2005
Contents11Introduction11.1Course Mission11.2Pulse Characteristics31.3Applications.71.4Review of Laser Essentials1.512History.1.615 Laser Materials212Maxwell-Bloch Equations212.1Maxwell'sEquations222.2Linear Pulse Propagation in Isotropic Media232.2.1Plane-Wave Solutions (TEM-Waves)242.2.2Complex Notations2.2.3Poynting Vectors,EnergyDensityandIntensity for25Plane Wave Fields252.2.4Dielectric Susceptibility272.3Bloch Equations.272.3.1The Two-Level Model2.3.230The Atom-Field Interaction In Dipole Approximation322.3.3Rabi-Oscillations2.3.435The Density Operator372.3.5Energy- and Phase-Relaxation2.3.6The Two-Level Atom with a Coherent Classical Exter-nal Field39412.4Dielectric Susceptibility442.5RateEquations2.645Pulse Propagation with Dispersion and Gain3
Contents 1 Introduction 1 1.1 Course Mission . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Pulse Characteristics . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Review of Laser Essentials 7 1.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Laser Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Maxwell-Bloch Equations 21 2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Linear Pulse Propagation in Isotropic Media . . . . . . . . . . 22 2.2.1 Plane-Wave Solutions (TEM-Waves) . . . . . . . . . . 23 2.2.2 Complex Notations . . . . . . . . . . . . . . . . . . . . 24 2.2.3 Poynting Vectors, Energy Density and Intensity for Plane Wave Fields . . . . . . . . . . . . . . . . . . . . 25 2.2.4 Dielectric Susceptibility . . . . . . . . . . . . . . . . . 25 2.3 Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 The Two-Level Model . . . . . . . . . . . . . . . . . . 27 2.3.2 The Atom-Field Interaction In Dipole Approximation . 30 2.3.3 Rabi-Oscillations . . . . . . . . . . . . . . . . . . . . . 32 2.3.4 The Density Operator . . . . . . . . . . . . . . . . . . 35 2.3.5 Energy- and Phase-Relaxation . . . . . . . . . . . . . . 37 2.3.6 The Two-Level Atom with a Coherent Classical External Field . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4 Dielectric Susceptibility . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.6 Pulse Propagation with Dispersion and Gain . . . . . . . . . . 45 3 .
4CONTENTS482.6.1Dispersion522.6.2Loss and Gain2.755Kramers-KroenigRelations572.8Pulse Shapes and Time-Bandwidth Products633Nonlinear Pulse Propagation633.1The Optical Kerr-effect643.2 Self-Phase Modulation (SPM)3.367 The Nonlinear Schrodinger Equation673.3.1TheSolitonsof theNSE683.3.2TheFundamental Soliton703.3.3Higher Order Solitons733.3.4Inverse Scattering Theory773.4Universality of the NSE3.577Soliton Perturbation Theory3.684Soliton Instabilities by Periodic Perturbations893.7Pulse Compression893.7.1General Pulse Compression Scheme913.7.2Spectral Broadening with Guided Modes923.7.3Dispersion Compensation Techniques973.7.4Dispersion Compensating Mirrors.1123.7.5Hollow Fiber Compression Technique3.8113Appendix: Sech-Algebra1143.9Summary127Laser Dynamics (single-mode)41274.1 Rate Equations1324.2 Built-up of Laser Oscillation and Continuous Wave Operation1334.3Stability and Relaxation Oscillations1364.4Q-Switching.1374.4.1ActiveQ-Switching4.4.2 140Single-Frequency Q-Switched Pulses.1424.4.3Theory of Active Q-Switching4.4.4PassiveQ-Switching:1464.5Example: Single Mode CW-Q-Switched Microchip Lasers1554.5.1Set-up of the Passively Q-Switched Microchip Laser:：1554.5.2157DynamicsofaQ-SwitchedMicrochipLaser4.6163Q-Switched Mode Locking
4 CONTENTS 2.6.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6.2 Loss and Gain . . . . . . . . . . . . . . . . . . . . . . . 52 2.7 Kramers-Kroenig Relations . . . . . . . . . . . . . . . . . . . . 55 2.8 Pulse Shapes and Time-Bandwidth Products . . . . . . . . . . 57 3 Nonlinear Pulse Propagation 63 3.1 The Optical Kerr-effect . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Self-Phase Modulation (SPM) . . . . . . . . . . . . . . . . . . 64 3.3 The Nonlinear Schrödinger Equation . . . . . . . . . . . . . . 67 3.3.1 The Solitons of the NSE . . . . . . . . . . . . . . . . . 67 3.3.2 The Fundamental Soliton . . . . . . . . . . . . . . . . 68 3.3.3 Higher Order Solitons . . . . . . . . . . . . . . . . . . 70 3.3.4 Inverse Scattering Theory . . . . . . . . . . . . . . . . 73 3.4 Universality of the NSE . . . . . . . . . . . . . . . . . . . . . 77 3.5 Soliton Perturbation Theory . . . . . . . . . . . . . . . . . . . 77 3.6 Soliton Instabilities by Periodic Perturbations . . . . . . . . . 84 3.7 Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . 89 3.7.1 General Pulse Compression Scheme . . . . . . . . . . . 89 3.7.2 Spectral Broadening with Guided Modes . . . . . . . . 91 3.7.3 Dispersion Compensation Techniques . . . . . . . . . . 92 3.7.4 Dispersion Compensating Mirrors . . . . . . . . . . . . 97 3.7.5 Hollow Fiber Compression Technique . . . . . . . . . . 112 3.8 Appendix: Sech-Algebra . . . . . . . . . . . . . . . . . . . . . 113 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4 Laser Dynamics (single-mode) 127 4.1 Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2 Built-up of Laser Oscillation and Continuous Wave Operation 132 4.3 Stability and Relaxation Oscillations . . . . . . . . . . . . . . 133 4.4 Q-Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.1 Active Q-Switching . . . . . . . . . . . . . . . . . . . . 137 4.4.2 Single-Frequency Q-Switched Pulses . . . . . . . . . . . 140 4.4.3 Theory of Active Q-Switching . . . . . . . . . . . . . . 142 4.4.4 Passive Q-Switching . . . . . . . . . . . . . . . . . . . 146 4.5 Example: Single Mode CW-Q-Switched Microchip Lasers . . . 155 4.5.1 Set-up of the Passively Q-Switched Microchip Laser . . 155 4.5.2 Dynamics of a Q-Switched Microchip Laser . . . . . . . 157 4.6 Q-Switched Mode Locking . . . . . . . . . . . . . . . . . . . . 163
5CONTENTS4.7167Summary1735Active Mode Locking.1745.1The Master Equation of Mode Locking：1775.2Active Mode Locking by Loss Modulation5.3:182Active Mode-Locking by Phase Modulation5.4：183Active Mode Locking with Additional SPM5.5.186ActiveModeLockingwith SolitonFornation5.5.1188StabilityCondition5.5.2196Numerical simulations:2015.5.3Experimental Verification:2035.6Summary5.7Active Modelocking with Detuning2075.7.1Dynamics of the Detuned Actively Mode-locked Laser . 2125.7.2215Nonnormal Systems and Transient Gain:2175.7.3The Nonormal Behavior of the Detuned Laser2256 Passive Modelocking.2276.1Slow Saturable Absorber Mode Locking2326.2Fast Saturable Absorber Mode Locking6.2.1Without GDD and SPM:233: 2376.2.2With GDD and SPM: 2416.3 Soliton Mode Locking .-: 2466.4 Dispersion Managed Soliton Formation2577 Kerr-Lens and Additive Pulse Mode Locking：2577.1 Kerr-Lens Mode Locking (KLM) 7.1.1 Review of Paraxial Optics and Laser Resonator Design 2582617.1.2Two-Mirror Resonators2707.1.3Four-Mirror Resonators.2757.1.4The Kerr Lensing Effects:2807.2AdditivePulseModeLocking2898Semiconductor Saturable Absorbers:2918.1CarrierDynamicsand SaturationProperties:2958.2High Fluence Effects8.3：299Break-up into Multiple Pulses8.4Summary:306
CONTENTS 5 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5 Active Mode Locking 173 5.1 The Master Equation of Mode Locking . . . . . . . . . . . . . 174 5.2 Active Mode Locking by Loss Modulation . . . . . . . . . . . 177 5.3 Active Mode-Locking by Phase Modulation . . . . . . . . . . . 182 5.4 Active Mode Locking with Additional SPM . . . . . . . . . . 183 5.5 Active Mode Locking with Soliton Formation . . . . . . . . . 186 5.5.1 Stability Condition . . . . . . . . . . . . . . . . . . . . 188 5.5.2 Numerical simulations . . . . . . . . . . . . . . . . . . 196 5.5.3 Experimental Verification . . . . . . . . . . . . . . . . 201 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.7 Active Modelocking with Detuning . . . . . . . . . . . . . . . 207 5.7.1 Dynamics of the Detuned Actively Mode-locked Laser . 212 5.7.2 Nonnormal Systems and Transient Gain . . . . . . . . 215 5.7.3 The Nonormal Behavior of the Detuned Laser . . . . . 217 6 Passive Modelocking 225 6.1 Slow Saturable Absorber Mode Locking . . . . . . . . . . . . . 227 6.2 Fast Saturable Absorber Mode Locking . . . . . . . . . . . . . 232 6.2.1 Without GDD and SPM . . . . . . . . . . . . . . . . . 233 6.2.2 With GDD and SPM . . . . . . . . . . . . . . . . . . . 237 6.3 Soliton Mode Locking . . . . . . . . . . . . . . . . . . . . . . . 241 6.4 Dispersion Managed Soliton Formation . . . . . . . . . . . . . 246 7 Kerr-Lens and Additive Pulse Mode Locking 257 7.1 Kerr-Lens Mode Locking (KLM) . . . . . . . . . . . . . . . . . 257 7.1.1 Review of Paraxial Optics and Laser Resonator Design 258 7.1.2 Two-Mirror Resonators . . . . . . . . . . . . . . . . . . 261 7.1.3 Four-Mirror Resonators . . . . . . . . . . . . . . . . . . 270 7.1.4 The Kerr Lensing Effects . . . . . . . . . . . . . . . . . 275 7.2 Additive Pulse Mode Locking . . . . . . . . . . . . . . . . . . 280 8 Semiconductor Saturable Absorbers 289 8.1 Carrier Dynamics and Saturation Properties . . . . . . . . . . 291 8.2 High Fluence Effects . . . . . . . . . . . . . . . . . . . . . . . 295 8.3 Break-up into Multiple Pulses . . . . . . . . . . . . . . . . . . 299 8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
6CONTENTS3099 Noise and Frequency Control9.1The Mode Comb：3109.2 Noise in Mode-locked Lasers: 314.3179.2.1The Optical Spectrum3209.2.2The Microwave Spectrum. 3219.2.3 Example: Yb-fiber laser:.3249.3Group-and Phase Velocity of Solitons3269.4 Femtosecond Laser Frequency Combs33310PulseCharacterization33310.1IntensityAutocorrelation10.2InterferometricAutocorrelation (IAC)33610.2.1 Interferometric Autocorrelation of an Unchirped Sech-Pulse..34110.2.2 Interferometric Autocorrelation of a Chirped Gaussian342Pulse.34210.2.3 Second OrderDispersion34310.2.4 Third Order Dispersion34510.2.5Self-PhaseModulation10.3 Frequency Resolved Optical Gating (FROG)34734910.3.1 Polarization Gate FROG10.3.2 FROG Inversion Algorithm：351.：35410.3.3SecondHarmonicFROG10.3.4 FROG Geometries35510.4 Spectral Interferometry and SPIDER35710.4.1 Spectral Interferometry.35710.4.2SPIDER.35910.4.3 Characterization of Sub-Two-Cycle Ti:sapphire Laser365Pulses.36710.4.4Pros and Consof SPIDER37111 Ultrafast Measurement Techniques:37111.1 Pump Probe Measurements11.1.1 Non-ColinearPump-ProbeMeasurement:.371.37211.1.2 Colinear Pump-Probe Measurement:37411.1.3 Heterodyne Pump Probe11.2Electro-Optic Sampling:37637811.3 THz Spectroscopy and Imaging
6 CONTENTS 9 Noise and Frequency Control 309 9.1 The Mode Comb . . . . . . . . . . . . . . . . . . . . . . . . . 310 9.2 Noise in Mode-locked Lasers . . . . . . . . . . . . . . . . . . . 314 9.2.1 The Optical Spectrum . . . . . . . . . . . . . . . . . . 317 9.2.2 The Microwave Spectrum . . . . . . . . . . . . . . . . 320 9.2.3 Example: Yb-fiber laser: . . . . . . . . . . . . . . . . . 321 9.3 Group- and Phase Velocity of Solitons . . . . . . . . . . . . . 324 9.4 Femtosecond Laser Frequency Combs . . . . . . . . . . . . . . 326 10 Pulse Characterization 333 10.1 Intensity Autocorrelation . . . . . . . . . . . . . . . . . . . . . 333 10.2 Interferometric Autocorrelation (IAC) . . . . . . . . . . . . . . 336 10.2.1 Interferometric Autocorrelation of an Unchirped SechPulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 10.2.2 Interferometric Autocorrelation of a Chirped Gaussian Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 10.2.3 Second Order Dispersion . . . . . . . . . . . . . . . . . 342 10.2.4 Third Order Dispersion . . . . . . . . . . . . . . . . . . 343 10.2.5 Self-Phase Modulation . . . . . . . . . . . . . . . . . . 345 10.3 Frequency Resolved Optical Gating (FROG) . . . . . . . . . . 347 10.3.1 Polarization Gate FROG . . . . . . . . . . . . . . . . . 349 10.3.2 FROG Inversion Algorithm . . . . . . . . . . . . . . . 351 10.3.3 Second Harmonic FROG . . . . . . . . . . . . . . . . . 354 10.3.4 FROG Geometries . . . . . . . . . . . . . . . . . . . . 355 10.4 Spectral Interferometry and SPIDER . . . . . . . . . . . . . . 357 10.4.1 Spectral Interferometry . . . . . . . . . . . . . . . . . . 357 10.4.2 SPIDER . . . . . . . . . . . . . . . . . . . . . . . . . . 359 10.4.3 Characterization of Sub-Two-Cycle Ti:sapphire Laser Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 10.4.4 Pros and Cons of SPIDER . . . . . . . . . . . . . . . . 367 11 Ultrafast Measurement Techniques 371 11.1 Pump Probe Measurements . . . . . . . . . . . . . . . . . . . 371 11.1.1 Non-Colinear Pump-Probe Measurement: . . . . . . . . 371 11.1.2 Colinear Pump-Probe Measurement: . . . . . . . . . . 372 11.1.3 Heterodyne Pump Probe . . . . . . . . . . . . . . . . . 374 11.2 Electro-Optic Sampling: . . . . . . . . . . . . . . . . . . . . . 376 11.3 THz Spectroscopy and Imaging . . . . . . . . . . . . . . . . . 378
iCONTENTS11.4 Four-Wave Mixing:38012PulseAmplification385
CONTENTS i 11.4 Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . 380 12 Pulse Amplification 385
Chapter 1Introduction1.1 Course Mission·Generation of ultrashort pulses:Nano-,Pico-,Femto-,AttosecondPulses. Propagation of ultrashort pulses●Linearandnonlineareffects. Applications in high precision measurements, nonlinear optics, opticalsignal processing, optical communications, x-ray generation,....1.2Pulse CharacteristicsMost often, there is not an isolated pulse, but rather a pulse train.nT.(n-1)Ta(n+1)T,Figure 1.l: Periodic pulse train1
Chapter 1 Introduction 1.1 Course Mission • Generation of ultrashort pulses: Nano-, Pico-, Femto-, Attosecond Pulses • Propagation of ultrashort pulses • Linear and nonlinear effects. • Applications in high precision measurements, nonlinear optics, optical signal processing, optical communications, x-ray generation,. 1.2 Pulse Characteristics Most often, there is not an isolated pulse, but rather a pulse train. Figure 1.1: Periodic pulse train 1
2CHAPTER1.INTRODUCTIONTr: pulse repetition timeW : pulse energyPave = W/Tr : average powerTrwHM is the Full Width at Half Maximum of the intensity envelope of thepulse in the time domain.ThepeakpowerisgivenbyWTRPp =(1.1)PaveTFWHMTFWHMand the peak electric field is given byPp2ZF0(1.2)Ep =oAeffAef is the beam cross-section and Zro= 3772 is the free space impedance.Time scales:1ns~30cm(high-speed electronics,GHz)1 ps~300μm1 fs~300nm1as = 10-18s ~ 0.3nm = 3A (typ-lattice constant in metal)The shortest pulses generated to date are about 4 - 5fs at 800nm (/c =2.7fs),less than two optical cycles and 250 as at 25 nm.For few-cycle pulsesthe electric field becomes important, not only the intensity!5fsFigure 1.2: Electric field waveform of a 5 fs pulse at a center wavelength of800 nm. The electric field depends on the carrier-envelope phase
2 CHAPTER 1. INTRODUCTION TR: pulse repetition time W : pulse energy Pave = W/TR : average power τFWHM is the Full Width at Half Maximum of the intensity envelope of the pulse in the time domain. The peak power is given by Pp = W τFWHM = Pave TR τFWHM , (1.1) and the peak electric field is given by Ep = r 2ZF0 Pp Aeff . (1.2) Aeff is the beam cross-section and ZF0 = 377 Ω is the free space impedance. Time scales: 1 ns ∼ 30 cm (high-speed electronics, GHz) 1 ps ∼ 300 µm 1 fs ∼ 300 nm 1 as = 10−18 s ∼ 0.3 nm = 3 ˚A (typ-lattice constant in metal) The shortest pulses generated to date are about 4 − 5 fs at 800 nm (λ/c = 2.7 fs), less than two optical cycles and 250 as at 25 nm. For few-cycle pulses, the electric field becomes important, not only the intensity! Figure 1.2: Electric field waveform of a 5 fs pulse at a center wavelength of 800 nm. The electric field depends on the carrier-envelope phase
31.3.APPLICATIONSaverage power:Pave ~1W, up to 100 W in progress.kW possible, not yet pulsedrepetition rates:Trl = fr= mHz - 100 GHzpulse energy:W = lpJ- 1kJpulse width:5 fs - 50 ps,modelockedTFWHM=30 ps -100 ns,Q- switchedpeak power:1kJ~1PW,Pp1psobtained with Nd:glass (LLNL - USA, [1][2][3])For a typical lab pulse, the peak power is10nJPp~1MW10 fspeak field of typical lab pulse:V106 × 1012 V10V~10101/2×377Ep=元× (1.5)mmnm1.3Applications High time resolution: Ultrafast Spectroscopy, tracing of ultrafast phys-ical processes in condensed matter (see Fig. 1.3), chemical reactions,physical and biological processes, influence chemical reactions with fem-tosecond pulses: Femto-Chemistry (Noble Prize, 2000 to A. Zewail),high speed electric circuit testing and sampling of electrical signals, seeFig. 1.4
1.3. APPLICATIONS 3 average power: Pave ∼ 1W, up to 100 W in progress. kW possible, not yet pulsed repetition rates: T −1 R = fR = mHz − 100 GHz pulse energy: W = 1pJ − 1kJ pulse width: τFWHM = 5 fs − 50 ps, modelocked 30 ps − 100 ns, Q − switched peak power: Pp = 1 kJ 1 ps ∼ 1 PW, obtained with Nd:glass (LLNL - USA, [1][2][3]). For a typical lab pulse, the peak power is Pp = 10 nJ 10 fs ∼ 1 MW peak field of typical lab pulse: Ep = s 2 × 377 × 106 × 1012 π × (1.5)2 V m ≈ 1010 V m = 10 V nm 1.3 Applications • High time resolution: Ultrafast Spectroscopy, tracing of ultrafast physical processes in condensed matter (see Fig. 1.3), chemical reactions, physical and biological processes, influence chemical reactions with femtosecond pulses: Femto-Chemistry (Noble Prize, 2000 to A. Zewail), high speed electric circuit testing and sampling of electrical signals, see Fig. 1.4
4CHAPTER1.INTRODUCTIONPump-probemeasurementPumppuTetdavioeShortpulselaserBeamsplittere(4t)SlowdetectoProbepAtTime delay1fs0.15μmpumpand probe pulseSpiegels(4t)Computer-controlledtranalation stageatComputersoreenCharge carrler recombination:nsThermallzatlonelectronswithlattice:psThemalization electron gas:10-100fsFigure 1.3: Pump-probe setup to extract time constants relevant for thecarrierdynamicsinsemiconductorsHighSpeedA/D-Conversion(100 GHz)VoltegeVoltage16TimeAPPAVModulatoT,TimeAV: 10 bitAVAtVoPulse jitterAt:2元=>At~1 fsNoTo=100GHzloFigure 1.4: High speed A/D conversion with a high repetition rate pico- orfentosecondlaser
4 CHAPTER 1. INTRODUCTION Figure 1.3: Pump-probe setup to extract time constants relevant for the carrier dynamics in semiconductors. Figure 1.4: High speed A/D conversion with a high repetition rate pico- or femtosecond laser