Theoretical Problems in Cavity Nonlinear

The scientific and technological importance of lasers has generated great interest in the field of cavity nonlinear optics. This book provides a thorough description of this subject in terms of modern dynamical systems theory, with an emphasis on deriving analytical results and highlighting their physical significance. The book applies physical models for active and passive cavities to a variety of problems in laser theory, optical bistability and parametric oscillators. Subjects include scaling laws, Hopf bifurcations, passive Q-switching, and Turing instabilities. Several of the topics treated cannot be found in other books, including swept control parameter dynamics, laser stability, multimode rate equations, and antiphase dynamics. The book stresses the connections between theoretical work and actual experimental results, and will be of great interest to graduate students and researchers in theoretical physics, nonlinear optics, and laser physics.

Introduction

xi

1 Reduction of the Maxwell–Schrödinger equations

1

(14)

1.1 The Maxwell–Schrödinger equations

1

(3)

1.2 The Maxwell–Bloch equations

4

(1)

1.2.1 Decay processes

4

(1)

1.2.2 Incoherent optical pumping

5

(1)

1.3 Complex amplitude equations

5

(4)

1.4 The single-mode laser

9

(2)

1.5 Linear stability analysis

11

(4)

2 Parameter swept across a steady bifurcation I

15

(10)

2.1 Introduction

15

(1)

2.2 Sweeping across the laser first threshold

16

(9)

2.2.1 General formulation

16

(2)

2.2.2 Linear sweep

18

(2)

2.2.3 Exact solution of the linearized problem

20

(3)

2.2.4 Good cavity limit

23

(2)

3 Parameter swept across a steady bifurcation II

25

(12)

3.1 Influence of the initial condition

25

(4)

3.1.1 The regular problem

26

(1)

3.1.2 The critical case

27

(2)

3.2 Weak external field

29

(8)

3.2.1 Constant amplitude

29

(2)

3.2.2 Periodic amplitude

31

(3)

3.2.3 Stochastic amplitude

34

(3)

4 Optical bistability: Constant input

37

(17)

4.1 Introduction

37

(1)

4.2 Steady states

38

(2)

4.3 Reduction to one dimension

40

(2)

4.3.1 Good cavity limit

40

(1)

4.3.2 Bad cavity limit

41

(1)

4.4 The osculating parabola

42

(7)

4.4.1 Steady state solutions and nonlinear stability

43

(1)

4.4.2 Long pulses

44

(3)

4.4.3 Short pulses

47

(2)

4.5 Fully developed bistability

49

(5)

5 Optical bistability: Variable input

54

(12)

5.1 Swept parameter

54

(3)

5.2 Periodic modulation around a limit point

57

(3)

5.3 The onset of dispersive bistability

60

(6)

5.3.1 Steady states and linear stability

60

(1)

5.3.2 Periodic modulation: The optical transistor

61

(5)

6 Multimode optical bistability

66

(12)

6.1 The delay-differential equation

66

(5)

6.1.1 Reduction of the Maxwell–Bloch equations

66

(4)

6.1.2 Linear stability analysis

70

(1)

6.2 The discrete map

71

(7)

6.2.1 Constant control parameter

71

(4)

6.2.2 Swept parameter

75

(3)

7 Free-running multimode lasers

78

(13)

7.1 The semiclassical laser equations

79

(4)

7.2 The rate equations

83

(4)

7.2.1 The ring cavity rate equations

84

(1)

7.2.2 The Fabry–Pérot cavity rate equations

85

(2)

7.3 The Tang, Statz, and deMars equations

87

(4)

7.3.1 Steady state solutions

87

(1)

7.3.2 Stability of the single-mode solution

88

(3)

8 Antiphase dynamics

91

(14)

8.1 The reference model

92

(3)

8.2 Time-dependent solution

95

(2)

8.3 Power spectrum identities

97

(2)

8.4 How many frequencies?

99

(6)

8.4.1 Large mode number limit: One frequency

100

(1)

8.4.2 Large pump rate limit: One frequency

101

(1)

8.4.3 Linewidths inequalities

101

(1)

8.4.4 Close to threshold: Zero, one, or two frequencies

102

(1)

8.4.5 No degeneracy: N frequencies

103

(2)

9 Laser stability

105

(19)

9.1 Unidirectional single-mode ring laser

106

(7)

9.1.1 Formulation of the stability equation

106

(2)

9.1.2 Lorentzian broadening on resonance

108

(4)

9.1.3 Nonresonant steady state intensity

112

(1)

9.2 Unidirectional multimode ring laser

113

(6)

9.2.1 Formulation of the stability equation

113

(2)

9.2.2 The Risken–Nummedal instability

115

(2)

9.2.3 The inhomogeneous limit

117

(2)

9.3 Bidirectional single-mode ring laser

119

(5)

9.3.1 Formulation and steady states

119

(2)

9.3.2 Linear stability

121

(3)

10 Second harmonic generation

124

(16)

10.1 Introduction

124

(1)

10.2 Formulation

125

(4)

10.3 Steady state and stability

129

(2)

10.4 Analysis of the Hopf bifurcation

131

(6)

10.4.1 Straightforward integration

131

(2)

10.4.2 Multiple time scale expansion

133

(2)

10.4.3 The solvability condition

135

(2)

10.5 Extensions

137

(3)

11 Saturable absorbers

140

(20)

11.1 Introduction

140

(2)

11.2 Laser with a saturable absorber

142

(4)

11.3 Parametric oscillator with a saturable absorber

146

(9)

11.3.1 Formulation and steady states

146

(2)

11.3.2 Small amplitude periodic solutions

148

(2)

11.3.3 Passive Q-switching

150

(5)

11.4 Relaxation times in nonlinear optics

155

(5)

12 Transverse effects in optical bistability

160

(22)

12.1 The adiabatic limit

161

(1)

12.2 Nascent optical bistability

162

(3)

12.3 Negative detuning: The Ginzburg—Landau equation

165

(2)

12.4 Small detuning: The Swift—Hohenberg equation

167

(3)

12.5 Analysis of the Turing bifurcation

170

(10)

12.5.1 Linear stability analysis

170

(1)

12.5.2 Nonlinear stability analysis

171

(3)

12.5.3 Numerical results

174

(6)

12.6 The Glebsky—Lerman hamiltonian

180

(2)

Index

182

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Theoretical Problems in Cavity Nonlinear Optics

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