Nonlinear Optics: Theory, Numerical Modeling,

Nonlinear Opticsprobes in great depth quadratic and cubic nonlinearities, photorefractive nonlinear optics, the nonlinear optical properties of nematic liquid crystals, and photonic bandgap structures. This reference places core physical principles and theoretical concepts in dialogue with contemporary applications and research and presents the essential physics behind the products and promise of this very fertile field. It presents new developments in the dispersion management in nonlinear optical fibers and the combined effects of quadratic and cubic nonlinearities on harmonic generation. Coverage includes a review of linear wave propagation and the origins of optical nonlinearity.

Partha P. Banerjee is Professor and Chair of Electrical and Computer Engineering at the University of Dayton, Ohio.

Preface

iii

1 Optical Propagation in Linear Media

1

(36)

1.1 Maxwell's Equations

1

(3)

1.2 Linear Wave Propagation in Homogeneous, Linear Isotropic Media

4

(8)

1.3 Wave Propagation in Anisotropic Media

12

(13)

1.4 Diffraction

25

(6)

1.5 Dispersion

31

(3)

1.6 Problems

34

(1)

References

35

(2)

2 The Origin and Modeling of Optical Nonlinearity

37

(22)

2.1 A Simple Physical Model for Optical Nonlinearity

38

(4)

2.2 Physical Effects of Nonlinear Polarization

42

(1)

2.3 Mathematical Modeling of Optical Nonlinearities

43

(11)

2.4 An Alternative Approach

54

(2)

2.5 Summary

56

(1)

2.6 Problems

57

(1)

References

57

(2)

3 Second Harmonic Generation

59

(20)

3.1 The Physics of Second Harmonic Generation

59

(3)

3.2 SHG in Crystals-Mathematical Formulation

62

(8)

3.3 Phase Matching in Anisotropic Crystal

70

(1)

3.4 Nonlinear Transverse Effects in Second Harmonic Generation

71

(2)

3.5 SHG in a Medium with Second-Order and Third-Order Nonlinear Susceptibilities

73

(3)

3.6 Summary

76

(1)

3.7 Problems

77

(1)

References

77

(2)

4 Self Refraction of Optical Beams

79

(22)

4.1 The Nonlinear Refractive Index Coefficient n2

80

(1)

4.2 Self Refraction of Gaussian Beams

81

(9)

4.3 Focused Gaussian Beam in a Kerr-Slice: The z-Scan Method

90

(6)

4.4 Self-Bending of Optical Beams

96

(1)

4.5 Self-Phase Modulation Due to X(2): Effective x(3)

97

(1)

4.6 Summary

98

(1)

4.7 Problems

99

(1)

References

100

(1)

5 Optical Bistability

101

(22)

5.1 The Nonlinear Ring Cavity

101

(8)

5.2 Transverse Optical Bistability

109

(1)

5.3 Linear-Nonlinear Interface

110

(9)

5.4 Conclusion

119

(1)

5.5 Problems

120

(1)

References

120

(3)

6 Optical Phase Conjugation

123

(22)

6.1 Comparison with Holography

124

(1)

6.2 The k-Space Formalism

125

(7)

6.3 Semiclassical Analysis: CW Case

132

(5)

6.4 Phase Conjugation of Pulses and Transient Phase Conjugate Response

137

(5)

6.5 Discussion

142

(1)

6.6 Problems

142

(2)

References

144

(1)

7 Stimulated Brillouin and Raman Scattering

145

(10)

7.1 Stimulated Brillouin Scattering

145

(6)

7.2 Stimulated Raman Scattering

151

(2)

7.3 Problems

153

(1)

References

154

(1)

8 Solitons in Optical Fibers

155

(20)

8.1 Background on Linear Optical Fibers

156

(2)

8.2 Fiber Dispersion and Nonlinearity

158

(3)

8.3 Fiber-Optic Solitons and the NLS Equation

161

(3)

8.4 Dispersion Managed Soliton Communication Systems

164

(2)

8.5 Dark Solitons

166

(5)

8.6 Optical Shocks and Self Steepening of Pulses

171

(2)

8.7 Problems

173

(1)

References

173

(2)

9 Photorefractive Nonlinear Optics

175

(34)

9.1 Introduction

175

(2)

9.2 The Kukhtarev Equations and Their Simplification

177

(4)

9.3 Beam Fanning and Distortion in Photorefractive Materials

181

(13)

9.4 Two-Wave Mixing in Photorefractive Materials

194

(2)

9.5 Four-Wave Mixing and Phase Conjugation in Photorefractive Materials

196

(4)

9.6 Self Phase Conjugation and Edge Enhancement

200

(2)

9.7 Organic Photorefractive Materials

202

(4)

9.8 Problems

206

(1)

References

207

(2)

10 Nonlinear Optical Properties of Nematic Liquid Crystals

209

(34)

10.1 The Liquid Crystalline State of Matter

210

(1)

10.2 Classification of Liquid Crystals

210

(1)

10.3 Liquid Crystal Alignment

211

(2)

10.4 Principles of the Continuum Theory

213

(2)

10.5 Director Distribution of Homogeneously Aligned Nematic Liquid Crystal Under an External Electrical Field

215

(9)

10.6 Nonlinear Optical Properties from Optically Induced Molecular Reorientation

224

(4)

10.7 Optically Induced Reorientational Nonlinearity with External Voltage

228

(7)

10.8 Analysis of Beam Propagation in Liquid Crystals and z-Scan

235

(5)

10.9 Problems

240

(1)

References

241

(2)

11 Self Organization in Nonlinear Optical Systems

243

(28)

11.1 Introduction

243

(2)

11.2 Nonlinear Two-Dimensional Systems with Feedback

245

(4)

11.3 Self Organization in Photorefractive Materials

249

(6)

11.4 Theory of Self Organization in Photorefractive Materials

255

(2)

11.5 Instability Criterion and the Dispersion Relation

257

(2)

11.6 Nonlinear Eigenmodes in the Steady State

259

(5)

11.7 Model of Hexagonal Formation Based on Transverse Electrical Instability

264

(1)

11.8 Potential Applications

265

(2)

11.9 Problems

267

(1)

References

268

(3)

12 Nonlinear Optics of Photonic Bandgap Structures

271

(18)

12.1 Introduction

271

(1)

12.2 The Linear Dispersion Relation

272

12.3 The Effect of Cubic Nonlinearity

216

(66)

12.4 Second Harmonic Generation in Photonic Bandgap Structures

282

(5)

12.5 Problems

287

(1)

References

288

(1)

Appendix A The Split Step Beam Propagation Method

289

(6)

A.1 Examples

290

(4)

References

294

(1)

Appendix B Wavelet Transforms and Application to Solution of Partial Differential Equations

295

(12)

B.1 Introduction to Wavelets

295

(1)

B.2 Wavelet Properties and Scaling Functions

296

(2)

B.3 Digital Filters and Multi-Resolution Analysis

298

(4)

B.4 Application of Wavelet Transform to the NLS Equation

302

(4)

References

306

(1)

Index

307

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Nonlinear Optics: Theory, Numerical Modeling, and Applications

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