CONTENTS Preface xii 1 Introduction and Review 1.1 Introduction to Ultrashort Laser Pulses,1 1.2 Brief Review of Electromagnetics,4 1.2.1 Maxwell's Equations,4 1.2.2 The Wave Equation and Plane Waves,6 1.2.3 Poynting's Vector and Power Flow,8 1.3 Review of Laser Essentials,10 1.3.1 Steady-State Laser Operation,10 1.3.2 Gain and Gain Saturation in Four-Level Atoms,15 1.3.3 Gaussian Beams and Transverse Laser Modes,17 1.4 Introduction to Ultrashort Pulse Generation Through Mode-Locking.22 1.5 Fourier Series and Fourier Transforms,25 1.5.1 Analytical Aspects,25 1.5.2 Computational Aspects,28 Problems.30 2 Principles of Mode-Locking 32 2.1 Processes Involved in Mode-Locking,32 2.2 Active Mode-Locking,33 2.2.1 Time-Domain Treatment,34 2.2.2 Frequency-Domain Treatment,40 2.2.3 Variations of Active Mode-Locking.43 2.3 Passive Mode-Locking Using Saturable Absorbers,44 2.3.1 Saturation Model,47 2.3.2 Slow Saturable Absorber Mode-Locking,50 2.3.3 Fast Saturable Absorber Mode-Locking,54 viiCONTENTS Preface xiii 1 Introduction and Review 1 1.1 Introduction to Ultrashort Laser Pulses, 1 1.2 Brief Review of Electromagnetics, 4 1.2.1 Maxwell’s Equations, 4 1.2.2 The Wave Equation and Plane Waves, 6 1.2.3 Poynting’s Vector and Power Flow, 8 1.3 Review of Laser Essentials, 10 1.3.1 Steady-State Laser Operation, 10 1.3.2 Gain and Gain Saturation in Four-Level Atoms, 15 1.3.3 Gaussian Beams and Transverse Laser Modes, 17 1.4 Introduction to Ultrashort Pulse Generation Through Mode-Locking, 22 1.5 Fourier Series and Fourier Transforms, 25 1.5.1 Analytical Aspects, 25 1.5.2 Computational Aspects, 28 Problems, 30 2 Principles of Mode-Locking 32 2.1 Processes Involved in Mode-Locking, 32 2.2 Active Mode-Locking, 33 2.2.1 Time-Domain Treatment, 34 2.2.2 Frequency-Domain Treatment, 40 2.2.3 Variations of Active Mode-Locking, 43 2.3 Passive Mode-Locking Using Saturable Absorbers, 44 2.3.1 Saturation Model, 47 2.3.2 Slow Saturable Absorber Mode-Locking, 50 2.3.3 Fast Saturable Absorber Mode-Locking, 54 vii