The Physics and Technology of Laser Resonators

Edited by

D R Hall and P E Jackson

Department of Physics,

Heriot-Watt University, Edinburgh

Institute of Physics Publishing Bristol and Philadelphia

CONTENTS

List of Contributors xii

Preface xiii

PART ONE FUNDAMENTALS OF RESONATOR PHYSICS

1 Laser Resonators and Gaussian Beams l А С WALKER 1.1 Introduction 1 1.2 The Fabry-Perot Resonator 1 1.3 Geometric Ray Analysis of Optical Resonators 5

1.3.1 Ray Transfer Matrices 5 1.3.2 Stability Conditions for an Open Resonator 7 1.3.3 Examples of Stable Open Resonators 8

1.4 Diffraction Approach to Stable Resonators 9 1.4.1 Gaussian Spherical Waves 9 1.4.2 Lowest-Order Mode Diameters in Stable

Resonators 10 1.4.3 Fresnel Number and Diffraction Losses 12 1.4.4 Higher Order Transverse Modes and their

Suppression 13 1.5 Properties of Gaussian Beams 14

1.5.1 The Paraxial Wave Equation and the. Complex Beam Parameter 14

1.5.2 Propagation of Gaussian Beams 16 1.5.3 Power Distribution in Gaussian Beams 18 1.5.4 Gaussian Beam Focussing 18

1.6 References 20

2 Unstable Resonators 21 p E DYER

2.1 Introduction 21 2.2 Advantages 21 2.3 Basic Concepts 22 2.4 Geometrical Mode Analysis 23 2.5 Practical Resonators 26 2.6 Near and Far-Field Pattern 27 2.7 Experimental Studies 29 2.8 Wave Analysis 30 2.9 Design Procedure 33

vi

2.10 2.11

2.12 2.13

Mode Control in Short Gain-Life Lasers Special Techniques 2.11.1 Injection Locking 2.11.2 Novel Geometries 2.11.3 Variable Reflectivity Mirror Resonators Conclusions References

34 36 36 37 37 38 39

3 Theory of Waveguide Laser Resonators 40 С A HILL

3.1 Introduction 40 3.2 Propagation in Hollow Dielectric Waveguides 41

3.2.1 Waveguide Mode Expressions 42 3.3 Waveguide Resonator Analysis 44

3.3.1 The Concept of Resonator Modes 45 3.3.2 Waveguide Modes 47 3.3.3 Mode Coupling, Coupling Losses and Mode

Losses 48 3.3.4 Single-Mode, Few Mode, and Multimode Theory 49

3.4 First Order Theory and its Limits SO 3.4.1 Coupling Loss Theory of Single Mode

Waveguide Resonators 50 3.4.2 Dual Case I Waveguide Lasers 53 3.4.3 Rigrod Analysis for Waveguide Lasers 55

3.5 Real Waveguide Resonators: Experiment and Theory 56 3.5.1 Distant Mirrors 56 3.5.2 Tilted Mirrors and Folded Lasers 57 3.5.3 Tunability and Line Selection 58 3.5.4 Resonator Mode Degeneracies: Hopping and

Hooting 59 3.6 Summary 60 3.7 References 60

4 Ring Laser Resonators 62 T A KING

62 62 62 64 65 65 67 67 68 68 69

4.1 4.2

4.3

Introduction Features of Ring Cavities 4.2.1 Cavity Optics 4.2.2 Image Rotation in Nonplanar Rings 4.2.3 Spatial Hole Burning 4.2.4 Unidirectional Oscillation 4.2.5 Unstable Ring Resonators 4.2.6 Passive Ring Resonators Nonplanar Rings 4.3.1 Polarization Rotation 4.3.2 Nonplanar Ring and Frequency Modes

vu

4.4 Applications 72 4.4.1 Ring Laser Gyroscope 72 4.4.2 Nonplanar Solid State Ring Oscillator 75 4.4.3 Other Applications 76

4.5 References 78

5 Multifold Laser Resonators 80 H J BAKER 5.1 Introduction 80 5.2 Plane Mirror Folded Cavities 82 5.3 Resonator Support Structure 83 5.4 Mode-Mirror Interactions 85 5.5 Mode-Medium Interactions 86 5.6 Curved Mirror Folding of Cavities 88 5.7 Off-Axis Curved Mirror Folding 89 5.8 Summary 93 5.9 References 93

6 Resonators with Variable Reflectivity Mirrors 94 V MAGNI, S DE SILVESTRI and A CYBO-OTTONE 6.1 Introduction 94 6.2 Unstable Resonators with Gaussian Mirrors 95 6.3 Unstable Resonators with SuperGaussian Mirrors 98 6.4 Practical Variable Reflectivity Mirrors and

Couplers 100 6.5 References 104

7 Numerical Resonator Calculations 106 P E JACKSON 7.1 Introduction 106 7.2 Scalar-Diffraction Iteration Algorithms 106

7.2.1 Fox and Li Calculations 106 7.2.2 High-Order Modes 110 7.2.3 Modes in the Presence of Saturable Gain 112 7.2.4 Other Algorithms 114

7.3 Matrix Techniques 115 7.4 The Open-Ended Waveguide Approach 116 7.5 References 116

8 Frequency Stabilisation of Lasers 117 К M ABRAMSKI and D R HALL 8.1 Introduction 117

VUl

8.2 Laser Frequency Fluctuations 118 8.2.1 Elementary Considerations 118 8.2.2 Intrinsic Fluctuations 119 8.2.3 Environmental Fluctuations 120

8.3 Characterisation of Frequency Stability 120 8.3.1 Measurement Techniques 120 8.3.2 Definition of Frequency Stability 121

8.4 Principles of Laser Frequency Stabilisation 123 8.4.1 Passive Stabilisation 123 8.4.2 Active Frequency Stabilisation 124

8.5 Representative Frequency Stabilisation Schemes 128 8.5.1 He-Ne Lasers 128 8.5.2 C 0 2 Lasers 129 8.5.3 Semiconductor Lasers 129 8.5.4 Solid State Lasers 130

8.6 Conclusions 130 8.7 References 131

9 Propagation of Multimode Laser Beams - The M2 Factor 132

M W SASNETT

9.1 Introduction 132 9.2 Higher-Order Beams 132 9.3 A Numerical Evaluation of Beam Quality 135 9.4 Experimental Measurements 135 9.5 Higher-Order Beams in Laser Applications 140 9.6 Summary and Conclusion 141 9.7 References 142

10 Laser Beam Analysis by Greometrical Optics 143

W A E GOETHALS

10.1 Introduction 143 10.2 Presentation of an Optical System in the

Configuration Phase Space 143 10.2.1 Paraxial Single Ray Tracing 144 10.2.2 Ray Ensembles 144 10.2.3 Liouville's Theorem 145

10.3 Paraxial Optics of Gaussian and Higher Order Laser Beams 146 10.3.1 Paraxial Beam Tracing 147 10.3.2 Twiss Parameters 147 10.3.3 Examples 148

10.4 Beam Quality 149 10.5 Exact Beam Tracing 150

10.5.1 Laser Beams from Stable Resonators 150 10.5.2 Laser Beams from Unstable Resonators 150

10.6 Discussions 153 10.7 References 153

PART TWO LASER RESONATOR TECHNOLOGY

11.1 11.2

11.3 11.4 11.5 11.6

Introduction Thermal Effects and the Rod Geometry 11.2.1 Thermal-Birefringence 11.2.2 Thermal Focussing The Basic Resonator Compensated Resonators Summary References

11 YAG Laser Resonator Technology 154 G BURROWS

154 155 156 158 158 161 162 162

12 Resonators for Dye Lasers and Optical Parametric Oscillators 163 M H DUNN

12.1 Introduction 163 12.2 Dye Laser Cavity with Transform-Limited Linewidth 163 12.3 Visible Optical Parametric Oscillators 167 12.4 References 175

13 Resonators for High Power CO2 Lasers 176 D J DYSON

13.1 Introduction 176 13.2 The Requirements of the Cavity Design 176 13.3 Thermal Distortion 178 13.4 Thermal Lensing 181 13.5 Thermal Damage 184 13.6 Alignment Tolerance 186 13.7 Refraction Effects 187 13.8 Designing a Laser Cavity 188 13.9 Conclusion 189 13.10 References 189

14 Diode Laser Resonators 190 M J ADAMS and I D HENNING

14.1 Introduction 190 14.2 Diode Laser Fundamentals 190

15.1 15.2 15.3 15.4 15.5 15.6

Introduction The Monolithic Cavity The Hybrid Cavity Mode-Locked Cavities Ring Lasers References

14.3 Diode Laser Structures 191 14.4 Basic Diode Laser Resonators 192 14.5 More Complicated Resonators 193 14.6 Resonators for Optical Integration 195 14.7 Conclusion 196 14.8 References 197

15 Diode Pumped Solid State Laser Resonators 198 A I FERGUSON

198 199 200 201 206 208

16 Fibre Laser Resonators 209 P URQUHART

16.1 Introduction 209 16.2 Single-Mode Doped Fibre 209 16.3 Fibre Fabry-Perot Laser 210 16.4 Connectors and Splices 212 16.5 Direction Couplers 213 16.6 Other Intra-Cavity Components 215 16.7 Alternative Resonator Designs 215 16.8 Conclusion 218 16.9 References 218

17 CO2 Waveguide Laser Resonators 220 / E ROSS 17.1 Introduction 220 17.2 Resonators For Waveguide C 0 2 Lasers 221

17.2.1 Design of the Waveguide 221 17.2.2 Waveguide Resonator Design 222

17.3 The Technology of C 0 2 Waveguide Lasers 224 17.3.1 Excitation Techniques 224 17.3.2 Waveguide Laser Construction 225 17.3.3 Influence of the Gain Medium on Waveguide

Mode Control 227 17.4 Folded C 0 2 Waveguide Lasers 227 17.5 Resonators for Slab Waveguide Lasers 230 17.6 References 231

18 Ezcimer Laser Resonators 232

232 232 232 235 236 239 239 240 242 243 244 233 245 245

M J 18.1 18.2

18.3 18.4

18.5 18.6

18.7 18.8

SHAW Introduction Plane Resonators 18.2.1 Extraction Efficiency 18.2.2 Cavity Build-up Time Narrow-Line Operation Transverse Mode Selection 18.4.1 Unstable Resonators 18.4.2 Injection Seeding Short Pulses Practical Considerations 18.6.1 Windows 18.6.2 Mirrors Conclusions References

INDEX 247