Solitons in Nonlocal Media

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4 Vector Solitons in Nematic Liquid Crystals 4.1 Vector Solitons: an Introduction The simplest vector solitons (VS) are shape-preserving self-localized solutions of cou- pled Schröedinger nonlinear evolution equations (6). Among them, Manakov spatial solitons (88) can be derived by the inverse scattering technique and were first ob- served in AlGaAs with orthogonally-polarized collinear beams interacting incoherently (89). Two-wavelength vector solitons in Kerr media were predicted by De La Fuente and co-workers (90), whereas VS consisting of bright and dark solitons were reported by Shalaby and Barthelemy (91). Quadratic solitons belong to the class of VS, be- cause they encompass the parametric interaction of waves at different wavelengths (48; 92; 93). The resulting self-guided beams, in general, have energy flows along directions depending on relative powers and birefringence (92). In photorefractives, VS were demonstrated in various forms, ranging from incoherent VS (94) to VS with bright and dark solitary components (95), soliton dipoles (96) and multimode solitons (94; 97). The term molecule soliton was introduced to embrace the rich and complex VS phenomenology (35). In this chapter I study the propagation of vector solitons in NLC encompassing two beam of two different wavelengths, i.e. a bi-color soliton. I will show how numerical results are in good agreement with experimental observations. The results discussed here were partially published in Ref. (98). 72
4.2 Cell Geometry and Basic Equations 4.2 Cell Geometry and Basic Equations Let me consider two beams at two different wavelengths, λ1 and λ2 respectively, which impinge with arbitrary angles on a cell in an anisotropic configuration, with wavevectors k1 and k2 belonging to the plane yz (see fig. 4.1). I define θj = θ0j + Ψ the angle between the director ˆn and the wavevector kj (j = 1, 2) [see fig. 4.1(c)], with θ0j the value when the optical reorientation is negligible, i.e. in the linear regime, and Ψ the perturbation induced by light. I get θ0j = θ0 − θ rif j (if both wavevectors are parallel to z θ0i = θ0) 1 , where θ rif j is the angle between axis z and kj, and θ0 the angle between ˆn and ˆz [fig. 4.1(d)], imposed by the rubbing on the glass slides along yz (fig. 4.1). I assume, without loss of generality, that λ2 > λ1, i.e. δ2 < δ1. Furthermore, I define the axis s as the direction in the midst of the individual Poynting vectors 2 . The equations describing soliton propagation are K∇ 2 ǫ0ǫa1 ⊥θ + ∂Aj 2ik0jnej cos δj ∂sj 4 |A1| 2 sin[2(θ1 − δ1)] + ǫ0ǫa2 4 |A2| 2 sin[2(θ2 − δ2)] = 0 (4.1) + Dtj ∂ 2 Aj ∂t 2 j ∂ + Dxj 2Aj ∂x2 j + k 2 0jδǫttjAj = 0 (j = 1, 2) (4.2) being θ the angle formed by ˆn and z. 3 Eq. (4.1) is expressed in the reference system xts as defined above: the second derivative of the director angle distribution along s was neglected 4 . Moreover, in eq. (4.1) terms coming from the interference between the two beams were neglected due to their fast variation in time, with a frequency 5 much larger than the typical cut-off in NLC(58; 60). Eqs. (4.2) are written respectively in the frameworks xjtjsj, obtained by rotating xyz by an angle βj around ˆx [fig. 4.1(d)]; additionally, in eq. (4.2) the relation between nonlinear perturbations on the dielectric tensor ǫ and angles θj is δǫttj = 2 sin (θj − δj) − sin2 (θ0j − δj) (j = 1, 2). For small perturbations, i.e. for low beam ǫaj 1 Angles are considered positive (negative) when y > 0(y < 0), for absolute values below π. 2 With reference to fig. 4.1(d) the angle β between s and z is given by (β1 + β2) /2. 3 Angles θ, θ1 and θ2 differ each other for a constant so their derivatives are equal. 4 Clearly, this approximation is good as long as the angle between s1 and s2 is small. 5 Obviously this frequency is proportional to the difference c λ1 vacuum. 73 c − , being c the light-speed in λ2
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Solitons in Nonlocal Media Alessand
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To everybody who has supported me a
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Contents List of Figures vii 1 Intr
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CONTENTS 4.3.3 Soliton Trajectory .
Page 10 and 11:
List of Figures 1.1 (a) In the isot
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LIST OF FIGURES 2.12 Numerical resu
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LIST OF FIGURES 3.7 As in fig. 3.6,
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LIST OF FIGURES 3.14 (a) 3D sketch
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LIST OF FIGURES 4.7 Left column: ac
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LIST OF FIGURES C.1 Plot of V υ m(
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1.2 Optical Solitons cialized liter
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1.3 Nonlocality width 1 . Nonlocali
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1.3 Nonlocality ∆n(x) = I(x ′
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(a) Isotropic phase (b) Nematic pha
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1.4 Liquid Crystals interaction ene
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(a) E = 0 (b) E = 0 1.4 Liquid Crys
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1.5 Spatial Solitons in Nematic Liq
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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
Page 58 and 59: 3.1 Definition 2 √ ln2w x/t; more
Page 60 and 61: 3.2 Role of the Boundary Conditions
Page 78 and 79: 3.3 Soliton Trajectory 3.3.1 Genera
Page 80 and 81: x = 〈x〉, 3.3 Soliton Trajectory
Page 82 and 83: 3.3.3 Highly Nonlocal Case 3.4 Soli
Page 84 and 85: 3.4 Soliton Oscillations in a Finit
Page 86 and 87: (a) Equivalent potential Veq(〈ξ
Page 88 and 89: W NL 0 = ǫak0 2ne cos δ cos[2(θ0
Page 90 and 91: 3.4 Soliton Oscillations in a Finit
Page 94 and 95: 4.2 Cell Geometry and Basic Equatio
Page 96 and 97: 4.3 Highly Nonlocal Limit about 2mW
Page 98 and 99: 4.3 Highly Nonlocal Limit (a) Initi
Page 100 and 101: 4.3 Highly Nonlocal Limit Figure 4.
Page 102 and 103: (a) Oscillation amplitude for beam
Page 104 and 105: 4.4 Breathing as explained in secti
Page 106 and 107: 4.4 Breathing experimental yz evolu
Page 108 and 109: 5 Dissipative Self-Confined Optical
Page 110 and 111: 5.2 Light Self-Confinement in Dye D
Page 112 and 113: (a) (b) 5.4 Role of Gain Saturation
Page 114 and 115: γ [m −1 ] w t [µm] 80 40 60 40
Page 116 and 117: 5.4 Role of Gain Saturation having
Page 118 and 119: 5.6 Co-Propagating Pump form γ =
Page 120 and 121: (a) λ = 633nm (b) λ = 532nm 5.6 C
Page 122 and 123: Appendix A Optical Properties of NL
Page 124 and 125: A.2 Derivation of the Electromagnet
Page 126 and 127: A.2 Derivation of the Electromagnet
Page 128 and 129: Appendix B Numerical Algorithm B.1
Page 130 and 131: B.1 Simulations of Nonlinear Optica
Page 132 and 133: Appendix C Analysis of the Index Pe
Page 134 and 135: Eq. (C.7) can be expressed as V υ
Page 136 and 137: C.4 Computation of V υ m C.4 Compu
Page 138 and 139: Appendix D List of Publications D.1
Page 140 and 141: 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, “
Page 150 and 151:
REFERENCES [93] G. Assanto and G. S
Page 152:
REFERENCES [111] E. Rosencher and B
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Solitons in Nonlocal Media

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