Solitons in Nonlocal Media

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Contents List of Figures vii 1 Introduction 1 1.1 Solitons in Nonlinear Physics . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Optical Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Strong Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Weak Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.1 Liquid Crystal Phases . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.2 Continuum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.3 Linear Optical Properties . . . . . . . . . . . . . . . . . . . . . . 10 1.4.4 Reorientational Nonlinearity . . . . . . . . . . . . . . . . . . . . 12 1.5 Spatial Solitons in Nematic Liquid Crystals . . . . . . . . . . . . . . . . 13 2 Scalar Solitons in Nematic Liquid Crystals 15 2.1 Cell Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Effect of the Input Interface . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Soliton Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Theory of Nonlinear Optical Propagation in NLC . . . . . . . . . . . . . 24 2.5.1 Ruling Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.1.1 The Highly Nonlocal Case . . . . . . . . . . . . . . . . 26 2.5.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.2.1 Nonlinear Propagation . . . . . . . . . . . . . . . . . . 28 iii
CONTENTS 2.5.2.2 Soliton Profile . . . . . . . . . . . . . . . . . . . . . . . 30 3 Nonlocality and Soliton Propagation 36 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation . 39 3.2.1 Poisson 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Poisson 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.2.1 Green Function in a Finite Rectangular Geometry . . . 41 3.2.2.2 Perturbation Profile . . . . . . . . . . . . . . . . . . . . 43 3.2.3 Screened Poisson Equation . . . . . . . . . . . . . . . . . . . . . 44 3.2.3.1 Green Function for the Screened Poisson Equation . . . 47 3.2.3.2 Perturbation Profile . . . . . . . . . . . . . . . . . . . . 48 3.2.4 Reorientational Equation for the NLC in Anisotropic Configuration 48 3.2.4.1 Perturbative Approach for the Director Profile Compu- tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.4.2 Solution in a Finite Rectangular Geometry . . . . . . . 51 3.2.5 Highly Nonlocal Limit for the 2D Case . . . . . . . . . . . . . . . 54 3.3 Soliton Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 General Expression for the Equivalent Force . . . . . . . . . . . . 58 3.3.2 Power series for the Equivalent Force . . . . . . . . . . . . . . . . 61 3.3.3 Highly Nonlocal Case . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Soliton Oscillations in a Finite-Size Geometry . . . . . . . . . . . . . . . 62 3.4.1 Poisson 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.2 Poisson and Screened Poisson 2D . . . . . . . . . . . . . . . . . . 63 3.4.3 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 69 4 Vector Solitons in Nematic Liquid Crystals 72 4.1 Vector Solitons: an Introduction . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Cell Geometry and Basic Equations . . . . . . . . . . . . . . . . . . . . 73 4.3 Highly Nonlocal Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.1 Reorientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.2 Optical Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 76 iv
Page 1: Solitons in Nonlocal Media Alessand
Page 4 and 5: To everybody who has supported me a
Page 8 and 9: CONTENTS 4.3.3 Soliton Trajectory .
Page 10 and 11: List of Figures 1.1 (a) In the isot
Page 12 and 13: LIST OF FIGURES 2.12 Numerical resu
Page 14 and 15: LIST OF FIGURES 3.7 As in fig. 3.6,
Page 16 and 17: LIST OF FIGURES 3.14 (a) 3D sketch
Page 18 and 19: LIST OF FIGURES 4.7 Left column: ac
Page 20 and 21: LIST OF FIGURES C.1 Plot of V υ m(
Page 22 and 23: 1.2 Optical Solitons cialized liter
Page 24 and 25: 1.3 Nonlocality width 1 . Nonlocali
Page 26 and 27: 1.3 Nonlocality ∆n(x) = I(x ′
Page 28 and 29: (a) Isotropic phase (b) Nematic pha
Page 30 and 31: 1.4 Liquid Crystals interaction ene
Page 32 and 33: (a) E = 0 (b) E = 0 1.4 Liquid Crys
Page 34 and 35: 1.5 Spatial Solitons in Nematic Liq
Page 36 and 37: 2.2 Set-Up Figure 2.1: Sketch of th
Page 38 and 39: Φ = ⎛ ⎜ ⎝ Ex Hy Ey −Hx ⎞
Page 40 and 41: (a) Reorientation angle versus x
Page 42 and 43: 2.4 Soliton Observation to the Free
Page 44 and 45: 2.5 Theory of Nonlinear Optical Pro
Page 48 and 49: shape Ee ∝ exp − (x−a/2)2 +t
Page 56 and 57:
3 Nonlocality and Soliton Propagati
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3.1 Definition 2 √ ln2w x/t; more
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3.2 Role of the Boundary Conditions
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3.3 Soliton Trajectory 3.3.1 Genera
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x = 〈x〉, 3.3 Soliton Trajectory
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3.3.3 Highly Nonlocal Case 3.4 Soli
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3.4 Soliton Oscillations in a Finit
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(a) Equivalent potential Veq(〈ξ
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W NL 0 = ǫak0 2ne cos δ cos[2(θ0
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3.4 Soliton Oscillations in a Finit
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4 Vector Solitons in Nematic Liquid
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4.2 Cell Geometry and Basic Equatio
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4.3 Highly Nonlocal Limit about 2mW
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4.3 Highly Nonlocal Limit (a) Initi
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4.3 Highly Nonlocal Limit Figure 4.
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(a) Oscillation amplitude for beam
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4.4 Breathing as explained in secti
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4.4 Breathing experimental yz evolu
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5 Dissipative Self-Confined Optical
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5.2 Light Self-Confinement in Dye D
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(a) (b) 5.4 Role of Gain Saturation
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γ [m −1 ] w t [µm] 80 40 60 40
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5.4 Role of Gain Saturation having
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5.6 Co-Propagating Pump form γ =
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(a) λ = 633nm (b) λ = 532nm 5.6 C
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Appendix A Optical Properties of NL
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A.2 Derivation of the Electromagnet
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A.2 Derivation of the Electromagnet
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Appendix B Numerical Algorithm B.1
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B.1 Simulations of Nonlinear Optica
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Appendix C Analysis of the Index Pe
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Eq. (C.7) can be expressed as V υ
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C.4 Computation of V υ m C.4 Compu
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Appendix D List of Publications D.1
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D.2 Conference Papers M. Peccianti,
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REFERENCES [10] Y. Nakamura and I.
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REFERENCES [33] N. I. Nikolov, D. N
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REFERENCES [52] ——, “Spatial
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REFERENCES [72] M. A. Karpierz, “
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REFERENCES [93] G. Assanto and G. S
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REFERENCES [111] E. Rosencher and B
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Solitons in Nonlocal Media

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