Solitons in Nonlocal Media

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1.4 Liquid Crystals interaction energy with an external electric field E due to the medium polarizability (ǫa is the dielectric anisotropy), Ki (i = 1, 2, 3) are the (three) Frank elastic constants, associated with splay, twist and bend, respectively (58). From the minimization of energy I find the Euler-Lagrange equations: ∂F − ∂qj 3 d ∂F = 0 (j = 1, 2) (1.8) dxi i=1 ∂ dqj dxi where x1 = x, x2 = y and x3 = z and q j are two generic angular coordinates which describes the director orientation in the laboratory frame. I assume hard boundary conditions, i.e. fixed director values at the edges (59). The director orientation close to the external walls, i.e. its boundary conditions, can be controlled by treating the surfaces which confine liquid crystals. There are two main possible alignments: homeotropic, with director normal to the surface edge, or planar, with molecules parallel to the interfaces (59). 1.4.3 Linear Optical Properties Macroscopically, nematic liquid crystals (NLC) behave as positive uniaxial crystals with optic axis parallel to the director. The dielectric tensor is given by (60) ǫij = ǫ⊥δij + ǫaninj (1.9) with ni the director component along the i-th direction, δij the Kronecker’s delta, ǫ⊥ and ǫ the dielectric constants perpendicular and parallel to ˆn, respectively, and ǫa = ǫ − ǫ⊥ the dielectric anisotropy. Both ǫ and ǫ⊥ depend on the order parameter S previously defined. In a homogeneous uniaxial medium, given a certain direction for the wavevector k, there are two independent plane eigenwaves: ordinary and extraordinary (61). Ordinary propagation resembles isotropic media with a refractive index √ ǫ⊥. The electric field of the extraordinary wave, conversely, is not parallel to the corresponding displacement field, which implies a Poynting vector S non parallel to k: in particular, S lies in the plane containing the optic axis and k, forming with the latter the walk-off angle 10
δ = arctan ǫa sin(2θ) ǫa+2ǫ⊥+ǫa cos(2θ) for the extraordinary plane wave is cos 2 θ ne = 1.4 Liquid Crystals , being θ the angle between k and ˆn. 1 The refractive index ǫ⊥ + sin2 −1 θ ǫ (1.10) Remarkably, while linear optical propagation in NLC is generally involved due to the lack of homogeneity, in most practical cases a description in terms of ordinary and extraordinary waves holds valid. I will deepen this point in the next chapters. Another important optical feature of NLC (which allows the experimental observation of optical propagation inside the NLC, as shown later) is their strong Rayleigh scattering 2 (58): in the visible range, light scattered by nematics is larger by a factor 10 6 than in isotropic fluids. In fact in NLC scattering is due to random variations in the dielectric tensor ǫ, caused by fluctuations in density, temperature, etc., or in orientation of ˆn (due to thermal agitation). The latter is the dominant effect in the nematic phase, being absent in isotropic fluids. Let me consider a plane wave with wavevector kin. The light scattered around the solid angle dΩ, centered around the direction of the output wavevector kout, can be evaluated through the scattering differential cross section dσ/dΩ (58) dσ dΩ = ǫak2 |nη(q)| 0 4π 2 2 î · âµ ˆf · ˆn + î · ˆn ˆf · âµ µ=1,2 (1.11) where k0 = 2π/λ is the wavenumber in vacuum, q is the scattering vector defined by kout = kin + q and I took a single value for all NLC elastic constants; î and ˆf are two unit vectors parallel to input and scattered fields, respectively; â1 and â2 are directions which diagonalize the NLC free energy for a fixed q (58), 〈〉 stands for thermal average and |nη(q)| 2 is the director component due to molecular fluctuations, with η any direction in the plane of â1 and â2. From (1.11) it is possible to deduce that scattering is strong for crossed polarizations, i.e., when incident and scattered field are orthogonal to each other, and is particularly strong for low q. Moreover, given that |nη(q)| 2 ∝ q −2 and being |q| ∝ k0, the scattered power in NLC shows a trend with the inverse square of incident wavelength (60). 1 Moreover, S lies between the optical axis and the wavevector k in positive uniaxials. 2 Rayleigh scattering implies no energy exchange between the electromagnetic field and the material, i.e. photons are elastically scattered. 11
Page 1: Solitons in Nonlocal Media Alessand
Page 4 and 5: To everybody who has supported me a
Page 6 and 7: Contents List of Figures vii 1 Intr
Page 8 and 9: CONTENTS 4.3.3 Soliton Trajectory .
Page 10 and 11: List of Figures 1.1 (a) In the isot
Page 12 and 13: LIST OF FIGURES 2.12 Numerical resu
Page 14 and 15: 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 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
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3.3.3 Highly Nonlocal Case 3.4 Soli
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3.4 Soliton Oscillations in a Finit
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(a) Equivalent potential Veq(〈ξ
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W NL 0 = ǫak0 2ne cos δ cos[2(θ0
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3.4 Soliton Oscillations in a Finit
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4 Vector Solitons in Nematic Liquid
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4.2 Cell Geometry and Basic Equatio
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4.3 Highly Nonlocal Limit about 2mW
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4.3 Highly Nonlocal Limit (a) Initi
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4.3 Highly Nonlocal Limit Figure 4.
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(a) Oscillation amplitude for beam
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4.4 Breathing as explained in secti
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4.4 Breathing experimental yz evolu
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5 Dissipative Self-Confined Optical
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5.2 Light Self-Confinement in Dye D
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(a) (b) 5.4 Role of Gain Saturation
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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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