Solitons in Nonlocal Media

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2.2 Set-Up Figure 2.1: Sketch of the NLC sample: (a) side view, (b) top view, (c) front view. Voltage at 1kHz is applied along x via two transparent electrodes deposited onto the glass slides that define the yz plane. ITO stands for Indium Tin Oxide. 2.2 Set-Up Fig. 2.2(a) is a sketch of the set-up employed in the experimental work. A He-Ne laser emitting at λ = 633nm is the light source. An optical system, composed by a half-wave plate and a polarizer, is used to control beam power and polarization. Specifically, the polarization at the input interface is always linear: rotating the half-wave plate I can vary angle β, as defined in fig. 2.2(b). The beam passes through an objective lens, so that the input waist is of the order of a few microns. Light impinges normally to the input interface, with its wavevector parallel to ˆz and equally far from the two glasses normal to ˆx. The field inside the sample is analyzed by the light scattered from NLC (see section 1.4.3) and collected by a microscope and a CCD 1 camera. 2.3 Effect of the Input Interface In this section I will study, both experimentally and theoretically, the manner in which the input interface affects beam coupling in the sample, in particular the field polarization which reaches the bulk NLC. 1 Charge Coupled Device. 16
2.3 Effect of the Input Interface Figure 2.2: (a) Experimental set-up. (b) Input field E and its polarization in the plane xy at z = 0. i.e. at the interface between air and NLC. Sign convention is such that β shown in the figure is positive. In order to describe the director orientation, I introduce the two angles ξ and γ, as shown in fig. 2.3. As the bulk NLC is preceded by a transition layer of thickness d following the input interface in z = 0, I model this transition layer as an anisotropic struc- ture stratified along z, with optical properties con- stant in each layer, i.e. with a dielectric tensor which does not depend on transverse coordinates xy 1 . To model finite beam behavior I take the director value in x = a/2, 2 given that the experimental beam width is much smaller than the cell thickness a. Under such Figure 2.3: Director ˆn and the two angles ξ and γ used to describe its orientation in space. hypotheses I can apply Berreman’s method (78) to describe the propagation of electro- magnetic plane waves with k = kˆz, i.e. impinging normally on the sample. I note that the wavevector k cannot change direction in the transition layer under these hypotheses. Hence I cast Maxwell’s equations in the form: dΦ = iω∆(z)Φ (2.1) dz being ω the optical angular frequency and 1 This implies neglecting optical reorientation; such approximation will be justified later. 2 Having neglected the optical reorientation, the profile does not change along y owing to the symmetry. 17
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 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 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
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(a) Equivalent potential Veq(〈ξ
Page 88 and 89:
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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