Ultrafast Lasers: A Comprehensive Introduction to Fundamental Principles with Practical Applications

Preface
Acknowledgements
Contents
1 Plane Wave Propagation in Dispersive Media
	1.1 Maxwell's Equations in SI Units
	1.2 Material Equations
	1.3 Wave Equation with Refractive Index
		1.3.1 Derivation of the Wave Equation
		1.3.2 Solution of the Wave Equation: Plane Wave
		1.3.3 Summary of the Notation in Vacuum and in a Dispersive Medium
		1.3.4 TEM Wave and Impedance
		1.3.5 Polarization
		1.3.6 Energy Density, Poynting Vector, and Intensity
	1.4 Dispersion
		1.4.1 Dispersion for Electromagnetic Waves
		1.4.2 Sellmeier Equation in the Visible and Near-Infrared
		1.4.3 Refractive Index from the VUV to the X-Ray Region
2 Linear Pulse Propagation
	2.1 Motivation
	2.2 Wave Equation in the Spectral Domain: Helmholtz Equation
		2.2.1 Fourier Transform
		2.2.2 Derivation of the Helmholtz Equation
	2.3 Linear Versus Nonlinear Wave Propagation
		2.3.1 Superposition Principle
		2.3.2 Linear System Theory
		2.3.3 Nonlinear Systems
	2.4 Ultrafast Pulses
		2.4.1 Pulse Wave Packet and Pulse Envelope
		2.4.2 Time–Bandwidth Product. Analogy with Heisenberg's Uncertainty Relation
		2.4.3 Spectral Phase Yielding Shortest Pulse
	2.5 Linear Pulse Propagation in a Dispersive Material
		2.5.1 Linear Pulse Propagation Versus Linear Dispersion
		2.5.2 Slowly-Varying-Envelope Approximation
		2.5.3 First and Second Order Dispersion
		2.5.4 Phase Velocity and Group Velocity
		2.5.5 Dispersive Pulse Broadening
		2.5.6 Dispersion as a Function of Frequency and Wavelength
		2.5.7 Optical Communication
		2.5.8 Can a Pulse Propagate Faster Than the Speed of Light in Vacuum?
		2.5.9 Definition of the Group Index
		2.5.10 Higher Order Dispersion
		2.5.11 Slowly-Evolving-Wave Approximation
3 Dispersion Compensation
	3.1 Introduction and Motivation
	3.2 Prism Compressor
		3.2.1 Second Order Dispersion of the Four-Prism Compressor
		3.2.2 Third Order Dispersion of the Four-Prism Compressor
		3.2.3 Continuous Adjustment of Dispersion
	3.3 Grating Compressor, Stretcher, and Pulse Shaper
		3.3.1 Diffraction Grating Compressor
		3.3.2 Turning a Grating Compressor into a Stretcher
		3.3.3 Grating-Based Pulse Shaper
	3.4 Gires–Tournois Interferometer (GTI)
	3.5 Summary of Dispersion Compensation with Angular Dispersion and GTI
	3.6 Mirrors with Controlled Phase Properties
		3.6.1 Bragg Mirror
		3.6.2 Dielectric GTI-Type Mirrors
		3.6.3 Chirped Mirrors
		3.6.4 Design of Chirped Mirrors
	3.7 Dazzlers
	3.8 Dispersion Measurements
4 Nonlinear Pulse Propagation
	4.1 Self-Phase Modulation (SPM)
		4.1.1 Kerr Effect and SPM
		4.1.2 Pulse Compressors Such as Fiber-Grating, Fiber-Prism, and Fiber-Chirped-Mirror Compressors
		4.1.3 Nonlinear Optical Pulse Cleaner
		4.1.4 Average Power Scaling of Pulse Compressors
	4.2 Self-Focusing and Filamentation Compressor
		4.2.1 Self-Focusing via a Kerr Lens
		4.2.2 Filament Formation
		4.2.3 Filament Pulse Compression
	4.3 Solitons
		4.3.1 Discovery of the Soliton
		4.3.2 Solution of the NSE: The Fundamental Soliton
		4.3.3 Solution of the NSE: Higher-Order Solitons
		4.3.4 Optical Communication with Solitons
		4.3.5 Periodic Perturbations of Solitons
	4.4 Self-Steepening
		4.4.1 Higher-Order Nonlinear Effects
		4.4.2 Optical Shock Front
		4.4.3 Effect of GDD on Optical Shock
	4.5 Nonlinear Propagation in a Saturable Absorber or Saturable Amplifier
		4.5.1 Saturable Amplifier
		4.5.2 Saturable Absorber
		4.5.3 Nonlinear Pulse Propagation in a Saturable Absorber or Amplifier
5 Laser Rate Equations, Steady-State Solutions, Relaxation Oscillations, and Transfer Functions
	5.1 What Do We Need to Know About Lasers?
		5.1.1 Diode-Pumped Solid-State Laser
		5.1.2 Rate Equations for an Ideal Four-Level Laser
		5.1.3 Steady-State Solutions (Four-Level Laser)
		5.1.4 Gain Saturation (Four-Level Laser)
		5.1.5 Three-Level Laser
	5.2 Relaxation Oscillations in a Four-Level Laser
		5.2.1 Linearized Rate Equations
		5.2.2 Ansatz for Solution After Perturbation
		5.2.3 Over-Critically Damped Lasers
		5.2.4 Under-Critically Damped Lasers
		5.2.5 Examples of Relaxation Oscillations Using Different Laser Materials
		5.2.6 Measurement of the Small-Signal Gain
	5.3 Transfer Function Analysis
		5.3.1 Rate Equations for Power and Gain
		5.3.2 Relaxation Oscillations
		5.3.3 Transfer Function
		5.3.4 Transfer Function Measurement
7 Saturable Absorbers for Solid-State Lasers
	7.1 Introduction
	7.2 Slow and Fast Saturable Absorbers
		7.2.1 Saturable Absorber Parameters and Rate Equation
		7.2.2 Justification for the Simplified Saturable Absorber Rate Equation
		7.2.3 Slow Saturable Absorber
		7.2.4 Fast Saturable Absorber
		7.2.5 Summary of Relevant Equations
	7.3 Nonlinear Reflectivity Model Functions
		7.3.1 Approximations
		7.3.2 Time-Dependent Reflectivity
		7.3.3 Pulse-Averaged Reflectivity
		7.3.4 Correction for Gaussian Beam Profile
		7.3.5 Inverse Saturable Absorption (ISA)
		7.3.6 Summary of Relevant Model Functions for Nonlinear Reflectivity
	7.4 Semiconductor Saturable Absorbers
		7.4.1 Semiconductor Saturable Absorber Materials
		7.4.2 Introduction to Semiconductor Relaxation Dynamics
		7.4.3 Fast Saturable Absorbers with Carrier Trapping Engineering
		7.4.4 Fast Saturable Absorbers with Quantum-Confined Stark Effect
		7.4.5 Saturable Absorber Optimization with Quantum Confinement
		7.4.6 Derivation of the Density of States
	7.5 Semiconductor Saturable Absorber Mirror (SESAM)
		7.5.1 SESAM Design: A Historical Perspective
		7.5.2 Resonant Versus Antiresonant SESAM
		7.5.3 Antiresonant High-Finesse SESAM
		7.5.4 Reflectivity, Phase, Dispersion, and Penetration Depth of a DBR
		7.5.5 Antiresonant Low-Finesse SESAM
		7.5.6 Dispersive SESAM
		7.5.7 Ultrabroadband SESAMs
		7.5.8 SESAM Optimization with Standing Wave Field Enhancement
	7.6 SESAM Damage
		7.6.1 SESAM Damage Measurements
		7.6.2 SESAM Damage Theory
		7.6.3 SESAM Design for High Average Power Thin-Disk Lasers
	7.7 SESAM Characterization
		7.7.1 One-Beam Measurement of Nonlinear Reflectivity
		7.7.2 Pump–Probe Measurement of Recovery Time
		7.7.3 Pump–Probe Measurement of Nonlinear Reflectivity
	7.8 Novel Saturable Absorber Materials
8 Q-Switching
	8.1 Active Q-Switching
		8.1.1 Fundamental Principle of Active Q-Switching
		8.1.2 Acousto-Optic Q-Switched Diode-Pumped Solid-State Laser
		8.1.3 Pulsed Single-Frequency Laser
		8.1.4 Actively Q-Switched Microchip Laser
	8.2 Theory for Active Q-Switching
		8.2.1 Rate Equations
		8.2.2 Inversion Build-Up Phase
		8.2.3 Pulse Build-Up Phase: Leading Edge of the Pulse
		8.2.4 Dynamics During the Pulse Duration
		8.2.5 Pulse Depletion Phase: Trailing Edge of the Pulse
	8.3 Passive Q-Switching
		8.3.1 Fundamental Principle of Passive Q-Switching
		8.3.2 Passively Q-Switched Microchip Laser
		8.3.3 Passively Q-Switched Monolithic Ring Laser
	8.4 Theory for Passive Q-Switching
		8.4.1 Rate Equations
		8.4.2 Model for SESAM Q-Switched Microchip Laser
		8.4.3 Pulse Energy
		8.4.4 Pulse Duration and Pulse Shape
		8.4.5 Pulse Repetition Rate
		8.4.6 Remarks on Three-Level Lasers
	8.5 Passively Q-Switched Microchip Lasers Using SESAMs
		8.5.1 SESAM Design
		8.5.2 Laser Setup
		8.5.3 SESAM Q-Switching Results
		8.5.4 Design Guidelines
		8.5.5 Summary of the Q-Switched Microchip Laser
9 Passive Modelocking
	9.1 Introduction and Basic Principle
		9.1.1 Basic Principle
		9.1.2 Starting Passive Modelocking
		9.1.3 Historical Development
	9.2 Coupled Cavity Modelocking
	9.3 Passive Modelocking with a Slow Saturable Absorber and Dynamic Gain Saturation
		9.3.1 Modelocking Conditions
		9.3.2 Example: Colliding Pulse Modelocking (CPM)
		9.3.3 Pulse Formation through Saturable Absorption and Gain
		9.3.4 Master Equation and Solution
	9.4 Passive Modelocking with a Fast Saturable Absorber
		9.4.1 Definition of an Ideally Fast Saturable Absorber
		9.4.2 Example: Kerr Lens Modelocking (KLM)
		9.4.3 Master Equation
		9.4.4 Solution without SPM and GDD
		9.4.5 Comparison with Active Modelocking without SPM and GDD
		9.4.6 Solution with SPM and GDD for Soliton Formation
		9.4.7 Problem for Self-Starting Modelocking
	9.5 Passive Modelocking with Slow Saturable Absorber and Without Dynamic Gain Saturation
		9.5.1 Soliton Modelocking
		9.5.2 Example: Soliton Modelocked Ti:sapphire Laser
		9.5.3 Soliton Modelocking with GDD > 0-.4 and n2 < 0-.4
		9.5.4 What Happens Without Soliton Formation
	9.6 Summary of Analytical Solutions for all Modelocking Techniques
	9.7 Q-Switching Instabilities of Passively Modelocked Solid-State Lasers
		9.7.1 Q-Switching Instabilities: A More Serious Issue for Solid-State Lasers
		9.7.2 Q-Switching Instabilities Without ISA: Derivation and Discussion of (9.126)
		9.7.3 Q-Switching Instabilities with Soliton Modelocking but Without ISA: Derivation and Discussion of (9.127)
		9.7.4 Q-Switching Instabilities with ISA: Discussion of (9.129)
		9.7.5 Special Cavity Designs to Prevent Q-Switching Instabilities
		9.7.6 Negative n2-.4 to Prevent Q-Switching Instabilities
	9.8 Passively Modelocked Diode-Pumped Semiconductor Lasers
		9.8.1 Optically Pumped VECSELs
		9.8.2 Optically-Pumped MIXSELs and SESAM-Modelocked VECSELs
	9.9 Dual-Comb Modelocking
	9.10 Performance Frontiers in Ultrafast Lasers
		9.10.1 Pulse Generation in the Few-Cycle Regime
		9.10.2 High Average Power
		9.10.3 Gigahertz Pulse Repetition Rates
10 Pulse Duration Measurements
	10.1 Electronic Measurements of the Pulse Duration
		10.1.1 Cables and Connectors
		10.1.2 Fast Photodiode
		10.1.3 Estimating the Time Resolution and Measurement Bandwidth
		10.1.4 Sampling Oscilloscope
		10.1.5 Microwave Spectrum and Signal Analyzers
		10.1.6 Equivalent-Time Sampling
	10.2 Optical Autocorrelation
		10.2.1 Pulse Spectrum
		10.2.2 Intensity Autocorrelation
		10.2.3 Interferometric Autocorrelation (IAC)
		10.2.4 High-Dynamic-Range Autocorrelation
		10.2.5 Temporal Smearing in Noncollinear Autocorrelation
	10.3 Frequency-Resolved Optical Gating (FROG)
		10.3.1 Basic Principle
		10.3.2 Second Harmonic Generation FROG (SHG-FROG)
	10.4 Spectral Phase Interferometry for Direct Electric Field Reconstruction (SPIDER)
		10.4.1 Basic Principle
		10.4.2 Experimental Realization
11 Intensity Noise and Timing Jitter of Modelocked Lasers
	11.1 Introduction
		11.1.1 Definition of Intensity Noise and Timing Jitter
		11.1.2 Basic Mathematical Principles for Noise
	11.2 Measurement Techniques for Intensity Noise and Timing Jitter
		11.2.1 General Remarks on Intensity Noise
		11.2.2 General Remarks on Timing Jitter
		11.2.3 Measurement Based on Microwave Spectrum Analyzers
		11.2.4 Measurement Based on Electronic Reference Signals
		11.2.5 Measurement Based on Optical Cross-Correlations
		11.2.6 Measurement with an Indirect Phase Comparison Method
	11.3 Noise Measurements with Power Spectral Densities
		11.3.1 Ideal Laser Without Amplitude Fluctuations and Timing Jitter
		11.3.2 Pulse Train with Intensity Noise
		11.3.3 Pulse Train with Timing Jitter
		11.3.4 Summary of Intensity Noise and Timing Jitter
	11.4 Noise Characteristics of Modelocked Lasers
		11.4.1 Some Basic Remarks on Noise, Stabilization, and Coupling Mechanisms
		11.4.2 Ultrafast Dye Lasers
		11.4.3 Flashlamp-Pumped Solid-State Lasers
		11.4.4 Diode-Pumped SESAM-Modelocked Solid-State Lasers
		11.4.5 Argon-Ion Versus Diode-Pumped Ti:Sapphire Lasers
	11.5 Some Words of Caution About Noise Characterization
		11.5.1 General Remarks About Nonstationary Processes and Finite Measurement Durations
		11.5.2 Noise Measurements in the Frequency Domain
		11.5.3 Noise Measurements in the Time Domain
	11.6 Signal-to-Noise Ratio (SNR) Optimization Techniques
		11.6.1 Basic Principle of Pump–Probe Measurements
		11.6.2 High Frequency Chopping with Lock-in Detection
		11.6.3 Minimal Detectable Signal for Shot-Noise-Limited Detection
		11.6.4 Short Measurement Durations for Lower SNR
12 Optical Frequency Comb from Modelocked Lasers
	12.1 Introduction
	12.2 Carrier Envelope Offset (CEO) Phase and Frequency
	12.3 Measurement of the CEO Frequency
		12.3.1 Basic Principle with Heterodyne (Interference) Signals
		12.3.2 Method 1 : f-to-2f Interferometer
		12.3.3 Method 2: Frequency-Doubled Transfer Oscillator with Lower Bandwidth Requirement
		12.3.4 Method 3: Frequency-Tripled Transfer Oscillator
		12.3.5 Method 4: SHG and THG of Two Auxiliary Oscillators and Self-Referenced 2f-to-3f Interferometer
		12.3.6 Method 5: Frequency Interval Bisection
	12.4 Phase and Frequency Noise
		12.4.1 Spectrally Resolved CEO Frequency Noise
		12.4.2 Basic Mathematical Principle of Phase and Frequency Noise
		12.4.3 Integrated Phase and Frequency Noise
		12.4.4 Phase Noise from Cavity Mirror Vibrations
		12.4.5 Intensity Noise, Timing Jitter, and CEO Frequency Noise
		12.4.6 Phase-Time and Fractional (or Relative) Frequency Noise
		12.4.7 Polynomial Model:  White, 1/f (Flicker), and 1/f2 Noise
		12.4.8 Allan Variance
	12.5 Connection Between the Optical Laser Spectrum and the Phase Noise
		12.5.1 Optical Laser Spectrum
		12.5.2 Variance of the Phase Noise Change and Optical Autocorrelation
		12.5.3 Optical Spectral Linewidth from Frequency and Phase Noise
		12.5.4 Optical Linewidth for (Low-Pass Filtered) White Frequency Noise
		12.5.5 β-Separation Line
		12.5.6 Optical Linewidth from Flicker Noise
		12.5.7 Quantum Noise Limit
		12.5.8 Optical Interferometers, Coherence, and Phase Noise
	12.6 Stabilized Optical Frequency Combs
		12.6.1 Basic Principle of Active Stabilization
		12.6.2 CEO Frequency Stabilization with Laser Feedback Control
		12.6.3 Full Optical Frequency Comb Stabilization
		12.6.4 External CEO Frequency Stabilization
	12.7 Laser Technology for Optical Frequency Combs
A Fourier Transform
B Dispersion for Quantum Mechanical Particles
C Delta-Function
D Delta-Comb
E Soliton Algebra
F Linearized Operators for the Master Equation
References
Index