Solitons in Nonlocal Media

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Eq. (C.7) can be expressed as V υ √ π m(υ) = 2 C.3 Computation of V ξ m πm ωte ( 2 ) 2 ω2 t e πmυ υ erfc + ωt mπ 2 ωt + e −πmυ erfc − υ + ωt mπ 2 ωt (C.8) being erfc(x) ≡ 2 ∞ √ π x e−y2dy. Eq. (C.8) is the searched result. Fig. C.1 shows the found profile of V υ m(υ). Figure C.1: Plot of V υ m(υ) versus υ for m = 1,10,20,30,40,50. Smaller values for m correspond to higher peaks. C.3 Computation of V ξ m From eq. (3.23) Eq. (C.9) can be written as V ξ 1 m = 0 fξ(ζ)sin (πmζ)dζ (C.9) V ξ m = 1 ∞ e 2i −∞ iπmζ − e −iπmζ fξ(ζ)rect1(ζ − 0.5)dζ (C.10) 114
C.3 Computation of V ξ m where rectb(x) is 1 for x ∈ (−b/2 b/2), 0 elsewhere. Using the Fourier transform properties I obtain V ξ m = −ℑ F [rect1(ζ − 0.5)fξ(ζ)] ν = m 2 (C.11) where operators F and ℑ stand for Fourier transformation and imaginary part, re- spectively. Convolution theorem provides F [rect1(ξ − 0.5)fξ(ξ)] = F [rect1(ξ − 0.5)] ∗ F [fξ(ξ)], being ∗ the convolution operator. Defining fξ(ξ) = g(ξ − 〈ξ〉) and F[g](ν) = ˜g(ν), knowing that F [rect1(ξ − 0.5)](ν) = e −iπν Ca(πν) and F [fξ(ξ)] (ν) = e −2πi〈ξ〉ν ˜g(ν), eq. (C.11) becomes V ξ ∞ m = −ℑ e −∞ −iπν m −2πi〈ξ〉( Ca(πν)e 2 −ν) m ˜g − ν dν 2 (C.12) If Ca(πν) is very narrow with respect to ˜g (ν), i.e., the beam is much smaller than the cell width a, if 〈ξ〉 is not too close to the cell edges, eq. (C.12) yields V ξ m ∼ = − ℑ e −iπm〈ξ〉 m ˜g 2 ∞ −∞ m = sin (πm 〈ξ〉)˜g 2 Ca(πν)dν (C.13) ξ2 − For a Gaussian beam along ξ, i.e. fξ(ξ) = e ω2 x , I can easily derive ˜g(ν) = √ πωxe−π2ω2 xν2 . Therefore, eq. (C.13) becomes Eq. (C.14) is the result I looked for. V ξ m(〈ξ〉) ∼ = √ πωx sin(πm 〈ξ〉)e −π2 ω 2 x( m 2 ) 2 115 (C.14)
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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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shape Ee ∝ exp − (x−a/2)2 +t
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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
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
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 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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