Solitons in Nonlocal Media

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5 Dissipative Self-Confined Optical Beams in Doped Nematic Liquid Crystals 5.1 Gain and Solitary Waves Propagation Since losses are a detrimental factor in the propagation and use of spatial solitons, the investigation of solitary waves in the presence of dissipative terms, e.g. gain or absorption, was carried out mostly within the context of the complex Ginzburg Lan- dau equation, (99; 100) with attention to the potential applications in lasers (101). In previous chapters I discussed nonlinear optical propagation in undoped nematic liquid crystals; in this chapter I study nonlinear wave propagation when a luminescent dye is added to the NLC, inducing an optical gain. To model dye effects in NLC I will use various models, beginning with the simplest case: a gain dependent neither from signal intensity (gain saturation) nor space coordinates (for example, pump spatial profile). Interestingly, a negative gain can be used to describe the scattering losses, that play an important role in the nematic phase due to the orientational order [see chapter 1 and references (58; 60)]. I will later consider a gain dependent on intensity, i.e. gain saturation, and a gain dependent on pump profile, i.e. varying in space. In every case the optical pump propagates along the x axis [see 5.1(a)]; eventually, I will consider a pump co-propagating with the signal beam; then, I will study the interaction of two solitons at two different wavelengths, i.e. a vectorial soliton (see chapter 4), with power 88
5.2 Light Self-Confinement in Dye Doped Nematic Liquid Crystals: Model exchange between them induced by the dye. I will assume a suitable concentration of dopants in order to obtain the desired gain. Gain and luminescence have been reported in several dye-doped liquid crystals, includ- ing nematics (102; 103; 104) cholesterics (105; 106; 107; 108) and blue phases (109) in planar and cylindrical geometries. Results, although referred to a particular class of highly nonlocal nonlinear media such as NLC, are qualitatively valid in all highly nonlocal media where spatial solitons can be generated [see (31) and chapter 3]. 5.2 Light Self-Confinement in Dye Doped Nematic Liquid Crystals: Model I refer to a nematic liquid crystalline cell as sketched in fig. 5.1. Two glass slides sandwich a liquid crystal layer [fig. 5.1(a)] of thickness d = 100 µm. I take that, in the absence of an external excitation, the molecular director ˆn (i.e., the optic axis) is homogeneously distributed in the sample and lies on the yz plane at an angle θ0 with the longitudinal z-axis [fig. 5.1(b)]. For the sake of simplicity, I limit my investigation to the case of zero voltage applied across the cell, in order to avoid beam-shift in the xs plane due to birefringent walk-off (47; 81). The latter would require the wave propagation to be treated vectorially. Figure 5.1(c) shows the reference system xts after rotating xyz by the walk-off angle δ around the x-axis. In this way, the versor ˆs corresponds to the direction of the Poynting vector. The propagation of extraordinary polarized light in the paraxial approximation along s is governed by a nonlocal nonlinear Schröedinger equation (NNLSE) (see chapter 1) 2ik ∂A ∂s + ∇2 ⊥ A + k2 0∆n(|A|)A − 2ik0neγA = 0 (5.1) where A is the slowly-varying envelope of the electric field, ne the (linear) extraordinary refractive index, k0 the vacuum wavenumber, D x/t the anisotropic diffraction coefficients, γ ∈ R the amplitude gain (loss) coefficient and δǫtt the nonlinear optical perturbation induced on the dielectric tensor ǫ via reorientation. The last term takes into account the power exchange between the light beam and the external environment. Clearly, no soliton propagation is admitted by eq. (5.1) due to the lack of a loss (gain) 2 2 sin (θ − δ) − sin (θ0 − δ) term balancing the gain (loss). Hereafter, I take δǫtt = ǫa , with ǫa the optical anisotropy, δ the walk-off angle and θ the angle between the 89
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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 .
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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
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Φ = ⎛ ⎜ ⎝ Ex Hy Ey −Hx ⎞
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(a) Reorientation angle versus x
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2.4 Soliton Observation to the Free
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2.5 Theory of Nonlinear Optical Pro
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Page 48 and 49:
shape Ee ∝ exp − (x−a/2)2 +t
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Page 54 and 55:
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 92 and 93: 4 Vector Solitons in Nematic Liquid
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 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,
Page 142 and 143: REFERENCES [10] Y. Nakamura and I.
Page 144 and 145: REFERENCES [33] N. I. Nikolov, D. N
Page 146 and 147: REFERENCES [52] ——, “Spatial
Page 148 and 149: 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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