Solitons in Nonlocal Media

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3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation Figure 3.4: Perturbation profiles for (a,c) ω = 0.01 and (b,d) ω = 0.1 for (a,b) 〈ξ〉 = 0.5 and (c,d) 〈ξ〉 = 0.75. The dashed (solid) lines correspond to profiles along υ(ξ − 〈ξ〉). The profiles are chosen such to contain the perturbation peak. Squares (triangles) are the corresponding values computed with a full numerical approach, which completely agree with the theoretical predictions from eq. (3.20). 46
3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation with K and µ < Kπ2 a 2 given constants. Eq. (3.27) governs reorientation in liquid crystals in an isotropic configuration (30) and is largely used as a ruling equation for refractive index in nonlinear nonlocal media (34; 83). I consider the same geometry investigated in section 3.2.2. Using the same normalizations I get κ∇ 2 ξυ ∆ρ + µ∆ρ + |A(ξ, υ)|2 = 0 (3.28) In the next section I compute the corresponding Green function, necessary to eval- uate the nonlinear perturbation through eq. (3.11). 3.2.3.1 Green Function for the Screened Poisson Equation To find the Green function G I have to solve κ∇ 2 G + µG = δ(ξ − ζ, υ − η) (3.29) with boundary conditions as in section 3.2.2.1. Developing G in a Fourier series re- spect to ξ as previously done for the Poisson equation, I can write ∞ m=1 bm(υ, ζ, η)sin(πmξ). Substitution of the latter into eq. (3.31) leads to ∞ m=1 κ ∂2bm ∂υ2 + µbm − κbm (πm) 2 sin(πmξ) = δ(ξ − ζ, υ − η) (3.30) Every coefficient bm is determined by ∂2bm ∂υ2 − (πm) 2 − µ κ bm = 2 κ sin(πmζ)δ(υ −η), which is equal to eq. (3.17) with the transformation (πm) 2 → (πm) 2 − µ 1. κ From eq. (3.19) it is easy to compute 3.2.3. G(ξ, |υ − η|, ζ) = − ∞ m=1 1 (πm) 2 − µ κ 1 To have a real square root ∀m, the relationship µ < Kπ 2 sin(πmζ) e − (πm) 2 − µ κ |υ−η| sin(πmξ) (3.31) 47 a 2 must hold true, as anticipated in section
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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,
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
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 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
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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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