Solitons in Nonlocal Media

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3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation is given by (3.19). So g1 = sin[2(θ0 − δ)] ∞ m=1 1 πm Vm(υ)sin(πmξ) where for Vm(υ) the equations derived above keep their validity. If the excitation field |A| 2 is not the product of two functions, both of them dependent only from one transverse variable, I have to compute g2(ξ, υ) through the general expressions (3.21) and (3.22), with forcing term |A| 2 θ1 [see eq. (3.43)]. In the opposite case, i.e. when |A(ξ, υ)| 2 = fξ(ξ)fυ(υ), I have that |A(ξ, υ)| 2 θ1(ξ, υ) = ∞ Al(ξ)Bl(υ) (3.45) l=0 being Al(ξ) = 1 ξ πlfξ(ξ)V l sin(πlξ) and Bl(υ) = fυ(υ)V υ l having defined g2(ξ, υ) = sin[4(θ0 − δ)] F m ∞ l (υ) = −∞ G m l = 1 0 ∞ m=0 1 πm sin(πmξ) (υ). It is easy to demonstrate ∞ l=0 G m l m Fl (υ) (3.46) Bl(η)e −πm|(υ−η)| dη (3.47) sin(πmζ)Al(ζ)dζ (3.48) Equation (3.48) is formally identical to eq. (3.23) and can be computed by means of eq. (C.13), as demonstrated in C.3. Equation (3.47) has no analytical expression, even in the Gaussian case, so I must evaluate them numerically. Computed g1 and g2 for Gaussian profiles with 〈ξ〉 = 0.5 and 〈ξ〉 = 0.75 are shown in fig. 3.6 and fig. 3.7, respectively: the two functions possess the same shape with very good approximation, i.e. g2(ξ, υ) ∼ = cg1(ξ, υ). This means that the reorientation angle in NLC and the solu- tions of Poisson equation behave nearly the same for ω
3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation Figure 3.6: Plots of g1 (a,d) and g2 (b,e) for 〈ξ〉 = 0.5. The corresponding profiles are plotted in (c) and (f) versus ξ (g1 solid line, g2 squares) and υ (g1 dashed line, g2 triangles), normalized to one (the cross sections are in υ = 0 and ξ = 0.5, respectively). Results for g1 and g2 perfectly overlap. Excitation waists are ω = 0.03 (a,b,c) and ω = 0.1 (d,e,f), respectively. Figure 3.7: As in fig. 3.6, but for 〈ξ〉 = 0.75. 53
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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(
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
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
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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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