Laser Beam Propagation in Nonlinear Optical Media

336 Pages·2013·10.593 MB·English

by  Shekhar Guha, Leonel P. Gonzalez

Laser Beam Propagation
in Nonlinear Optical
Media

Laser Beam Propagation
in Nonlinear Optical

Media
Shekhar Guha and Leonel P. Gonzalez
Air Force Research Laboratory
Wright-Patterson Air Force Base, OH, USA
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To
Sudeshna, Rahul and Rakesh – from Shekhar
and

Trina, Noah, Ian, Connor and Joshua – from Leonel

Contents
List of Figures xiii
List of Tables xxi
Preface xxv
Author Biographies xxvii
Acknowledgements xxix
1 Light Propagation in Anisotropic Crystals 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Vectors Associated with Light Propagation . . . . . . . . . . 2
1.2.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Non-plane waves . . . . . . . . . . . . . . . . . . . . . 4
1.3 Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 The principal coordinate axes . . . . . . . . . . . . . . 6
1.3.2 Three crystal classes . . . . . . . . . . . . . . . . . . . 7
1.3.3 The principal refractive indices . . . . . . . . . . . . . 7
1.4 Light Propagationin an Anisotropic Crystal . . . . . . . . . 8
1.4.1 Allowed directions of D and E in an anisotropic medium 9
1.4.2 Values of n for a given propagationdirection . . . . . 11
1.4.3 Directions of D and Eefor theeslow and fast waves . . 12
1.5 Characteristics of the Slow and the Fast Waves in a Biaxial
Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 n and n . . . . . . . . . . . . . . . . . . . . . . . . . 15
s f
1.5.2 ρ and ρ . . . . . . . . . . . . . . . . . . . . . . . . . 17
s f
1.5.3 The components of dˆ and dˆ . . . . . . . . . . . . . . 17
s f
1.5.4 The components of eˆ and eˆ . . . . . . . . . . . . . . 19
s f
1.6 Double Refraction and Optic Axes . . . . . . . . . . . . . . . 21
1.6.1 Expressions for components of dˆin terms of the angles
θ, φ and Ω . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6.2 Relating the angle δ to Ω, θ and φ . . . . . . . . . . . 27
1.6.3 Directions of E and S . . . . . . . . . . . . . . . . . . 29
1.6.4 The walk-off angles ρ and ρ . . . . . . . . . . . . . . 31
s f
1.6.5 An interim summary . . . . . . . . . . . . . . . . . . . 32
vii
viii Contents
1.7 Propagationalong the PrincipalAxes and along the Principal
Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 33
1.7.2 Propagationalong the principal axes X, Y and Z . . 34
1.7.3 Propagationalong the principal plane YZ . . . . . . . 35
1.7.4 k along YZ plane, Case 1: n < n < n . . . . . 35
X Y Z
1.7.5 k along YZ plane, Case 2: n > n > n . . . . . 37
X Y Z
1.7.6 Propagationalong the principal plane ZX . . . . . . . 38
1.7.7 k along ZX plane, Case 1a: n < n < n , θ <Ω 38
X Y Z
1.7.8 k along ZX plane, Case 1b: n < n < n , θ >Ω 40
X Y Z
1.7.9 k along ZX plane, Case 2a: n > n > n , θ <Ω 40
X Y Z
1.7.10 k along ZX plane, Case 2b: n > n > n , θ >Ω 41
X Y Z
1.7.11 Propagationalong the principal plane XY . . . . . . . 41

1.7.12 k along XY plane, Case 1: n < n < n . . . . . 42
X Y Z
1.7.13 k along XY plane, Case 2: n > n > n . . . . . 43
X Y Z
1.7.14 Summary of the cases of propagation along principal
planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.8 Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . . . . . 45
1.8.1 FielddirectionsoftheDandEvectorsforextraordinary
and ordinary waves . . . . . . . . . . . . . . . . . . . . 47
1.8.2 ρ=0 Case (extraordinary wave) . . . . . . . . . . . . 48
6
1.8.3 Another expression relating ρ and θ . . . . . . . . . . 50
1.8.4 ρ = 0 Case (ordinary wave) . . . . . . . . . . . . . . . 52
1.8.5 Two special cases: θ =0 and θ =90◦ . . . . . . . . . . 53
1.9 PropagationEquation in the Presence of Walk-off . . . . . . 54
1.9.1 Transformation between laboratory and crystal
coordinate systems . . . . . . . . . . . . . . . . . . . . 55
1.9.2 The propagationequation in the presence of walk-off . 55
2 Nonlinear Optical Processes 61
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2 Second Order Susceptibility . . . . . . . . . . . . . . . . . . . 62
2.3 Properties of χ(2) . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3.1 Properties of χ(2) away from resonance. . . . . . . . . 67
2.3.2 Kleinman’s symmetry . . . . . . . . . . . . . . . . . . 67
2.4 d Coefficients and the Contracted Notation . . . . . . . . . . 67
2.4.1 d Coefficients under Kleinman symmetry . . . . . . . 68
2.5 The Non-Zero d Coefficients of Biaxial Crystals . . . . . . . 69
2.6 The Non-Zero d Coefficients of Uniaxial Crystals . . . . . . . 70
2.7 Nonlinear Polarizations . . . . . . . . . . . . . . . . . . . . . 72
2.7.1 Nondegenerate sum frequency generation . . . . . . . 72
2.7.2 Difference frequency generation . . . . . . . . . . . . . 73
2.7.3 Second harmonic generation (SHG). . . . . . . . . . . 74
2.7.4 Optical rectification . . . . . . . . . . . . . . . . . . . 75
Contents ix
2.7.5 Convention used for numbering the three interacting
beams of light. . . . . . . . . . . . . . . . . . . . . . . 75
2.7.6 Summaryofpolarizationcomponentsfornon-degenerate
three wave mixing . . . . . . . . . . . . . . . . . . . . 76
2.7.7 Summary of polarization components for degenerate
three wave mixing (SHG and degenerate parametric
mixing) . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.8 Frequency Conversion and Phase Matching . . . . . . . . . . 78
2.8.1 Phase matching in birefringent crystals . . . . . . . . 80
2.8.2 Calculation of phase matching angles. . . . . . . . . . 83
2.9 Walk-Off Angles . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.9.1 Calculationofwalk-offanglesinthephasematchedcase
in KTP . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3 Effective d Coefficient for Three-Wave Mixing Processes 89
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1.1 Definition of d . . . . . . . . . . . . . . . . . . . . . 90
eff
3.1.2 Effective nonlinearity for nondegenerate three wave
mixing processes . . . . . . . . . . . . . . . . . . . . . 91
3.1.3 Effective nonlinearity for the degenerate three wave
mixing process . . . . . . . . . . . . . . . . . . . . . . 92
3.1.4 Type I degenerate three wave mixing process . . . . . 93
3.1.5 Type II degenerate three wave mixing process . . . . . 93
3.2 Expressions for d . . . . . . . . . . . . . . . . . . . . . . . 94
eff
3.2.1 d of biaxial crystals under Kleinman Symmetry
eff
Condition . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2.2 Reduction of d to expressions in the literature . . . 100
eff
3.3 d Values for Some Biaxial and Uniaxial Crystals of Different
eff
Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.1 d for KTP for propagationin a general direction . . 102
eff
3.3.1.1 d for KTP for a Type I (ssf) mixing process 103
eff
3.3.1.2 d for KTP for a Type II(sff) mixing process 104
eff
3.3.1.3 d for KTP for a Type II(fsf) mixing process 105
eff
3.3.2 d for KTP for propagationalong principal planes . . 106
eff
3.4 d for Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . 109
eff
3.5 d for Isotropic Crystals . . . . . . . . . . . . . . . . . . . . 114
eff
3.5.1 The direction of the nonlinear polarization . . . . . . 119
3.5.2 Propagationalong principal planes . . . . . . . . . . 121
3.5.3 Propagationthrough orientation patterned material . 124
4 Nonlinear Propagation Equations and Solutions 137
4.1 Nonlinear PropagationEquations . . . . . . . . . . . . . . . 137
4.1.1 Normalized form of the three wave mixing equations . 140
4.2 Solutions to the Three Wave Mixing Equations in the Absence
of Diffraction, Beam Walk-off and Absorption . . . . . . . . 141