Solitons in Nonlocal Media

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4.3 Highly Nonlocal Limit Figure 4.3: Plot of vector soliton trajectory. Angle between vector soliton direction and z is given by β + ρ, where β is the angle between axis s and z (see note 2) and ρ = arctan k3 from eq. (4.23). Single solitons oscillate sinusoidally around this direction, keeping a phase shift equal to π. s1 and s2 represent single beam energy direction when other beam is lacking. In this plot beams are launched at the same point, i.e. their positions are identical in z = 0. d〈t2〉 ds = tan ∆β, where ∆β = s=0 δ2−δ1 1 (I suppose that for each beam the derivative in 2 s = 0 is unchanged with respect to the single beam case). After some simple algebra I find k1 = k4 = 0, k3 = tan∆β 1 + and k2 = tan∆β in eq. (4.23) the beams trajectories are ⎧ 1 ⎪⎨ 〈t1〉 = − α ⎪⎩ 〈t2〉 = − 1 α 2 + m2 2γ2Ψ (1) 2 (1) 2 tan ∆β4γ2Ψ 2 αm2 4γ2Ψ (1) 2 α 2 m2 tan ∆β 4γ2Ψ (1) 2 αm2 sin(αs) + tan ∆β sin(αs) + tan∆β 1 + (1) 4γ2Ψ 2 . Replacing these αm2 (1) 4γ2Ψ 2 α2 s m2 1 + (1) 4γ2Ψ 2 α2 s m2 (4.24) Therefore, the vector soliton propagates along a direction at an angle ∆ with z, given by: ∆ = β + ρ = β + arctan tan ∆β 1 In my case ∆β > 0 because I have λ1 > λ2. 1 − 80 m1γ2Ψ (1) 2 2m1γ2Ψ (1) 2 + m2γ1Ψ (2) 2 (4.25)
4.3 Highly Nonlocal Limit If beam 1 is not present, I have Ψ (1) 2 = 0, i.e. ∆ = δ2; if beam 2 is absent, I have Ψ (2) 2 = δ1, i.e. ∆ = δ1; in both cases the oscillation amplitudes go to zero and the period to infinity, as expected. The results obtained for θ0 = π/6, λ1 = 1064nm and λ2 = 633nm from eqs. (4.24) and (4.25) are shown in fig. 4.4: every vector soliton property is established by the balance between individual beam powers. Let me discuss the results for the oscillation amplitudes. For Ired > 89Wmm −2 the absolute value of the amplitudes increases monotonically with IIR; physically, at low IIR the red attraction is dominant and the infrared beam collapses into red without oscillations, whereas for higher IIR infrared force begins to contrast the red attraction and the oscillation amplitude increases. This process continues until the infrared beam becomes stronger than the red: at this point, the amplitude decreases as IIR is grows. This behavior is visible in the range displayed in figs. [4.4(a)]-[4.4(b)] for Ired = 7 and 48Wmm −2 . Moreover, the amplitudes are in phase opposition having the sign inverted: in particular, the red (infrared) oscillation amplitude is positive (negative) being its energy direction above (below) ∆. To conclude the discussion about oscillation amplitudes, it is important to note that their size is a few microns, making their observation very hard due to the blur caused by scattered photons. The oscillation period is graphed in fig. 4.4(c): for a given Ired the period diminishes as IIR increases, and vice versa, owing to the larger attractive force between the beams. Fig. 4.4(d) plots the vector soliton propagation angle ∆: when the infrared is negligible I have ∆ = δ1, i.e. propagation along the red walk-off (∆ → δ2), when the red is negligible with respect to the infrared, the propagation is along the infrared walk-off (∆ → δ1). Comparing, for example, blue and green curves it is easy to see that the transition between these two limits is as sharp as the parameter Ired is low, being the red strength weaker. The red oscillations and the infrared beam act asymmetrically on the interaction, being the refractive index well (induced at a fixed power) larger for red wavelengths than for infrared due to the larger coupling between the optical field and the NLC at higher frequencies. 81
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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
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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
Page 44 and 45:
2.5 Theory of Nonlinear Optical Pro
Page 46 and 47:
2.5 Theory of Nonlinear Optical Pro
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shape Ee ∝ exp − (x−a/2)2 +t
Page 50 and 51: 2.5 Theory of Nonlinear Optical Pro
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 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,
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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Solitons in Nonlocal Media

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