Solitons in Nonlocal Media

More documents

Recommendations

Info

Φ = ⎛ ⎜ ⎝ Ex Hy Ey −Hx ⎞ ⎡ ⎟ ⎢ ⎟ ⎢ ⎠ ,∆ = ⎢ ⎣ 2.3 Effect of the Input Interface 0 µ0 0 0 0 ǫxx − ǫxzǫzy ⎤ 0 ⎥ ǫzz ⎥ 0 0 0 µ0⎦ ǫxx − ǫ2 zx ǫzz ǫxx − ǫxzǫzy ǫzz 0 ǫyy − ǫ2 zy ǫzz 0 (2.2) with ǫij (i, j = x, y, z) the elements of the dielectric tensor, which depend on z due to the director rotation along the transition layer. To numerically compute eq. (2.1) I can divide the region 0 < z < d into N sections, each of thickness h. In order to get a good approximation, h must be chosen so that dielectric tensor variations in each section are negligible. Hence, the vector Φ at z = d can be found by solving Φ(d) = Π N−1 ν=0 Ph(νh) Φ(0) = TdΦ(0) (2.3) where I set Ph(z) = e iωh∆(z) . The matrix Ph(z) is the transfer function (for the field vector Φ) which models the section limited by z and z + h. Therefore Td is the transfer function for the whole transition layer. The next step is to link the dielectric tensor [related to ∆ by means of (2.2)] to the director profile (into the transition layer). To this purpose, I can use eq. (1.9) with ny = cos ξ cos γ, nx = sinξ and nz = cos ξ sinγ. Finally, I assume a certain director profile in 0 < z < d, being z = 0 the input interface. Specifically, I take a linear trend for ξ and γ. A direct computation from eq. (2.3) confirms that specific profiles do not affect polarization, which mainly depends on how fast the director angle varies along z, i.e. from 0 to d, if the total variation is constant. Now I have to establish the director position in z = 0 and z = d. It is easily seen that, given the boundary conditions at the input interface, γ(x, y, z = 0) = 0 and ξ(x, y, z = 0) = π/4, independently from the applied voltage V. At z ≥ d the bulk NLC has γ(x, y, z = d) = γbulk = π/4. Conversely, ξ(x, z = d) = ξbulk(x) changes with V and is found by solving the reorientational equation derived from eq. (1.8) together with the associated electrostatic equation for z > d. Therefore, I have to solve the ODE (Ordinary Differential Equation) 1 system (79; 80) 1 The unknown quantities V and ξbulk exclusively depend on x. 18
ǫ sin 2 ξbulk + ǫ⊥ cos 2 d ξbulk 2V 2.3 Effect of the Input Interface dx ′2 + ∆ǫLF sin(2ξbulk) dξbulk dx ′ dx 2 dξbulk K3 sin 2 ξbulk + K1 cos 2 d ξbulk 2ξbulk dx ′2 + K3 − K1 sin(2ξbulk) 2 + ∆ǫLF 2 dV sin(2ξbulk) dx ′ dV = 0 (2.4) ′ dx ′ 2 = 0 (2.5) being V the electrostatic potential. I introduced a new reference system x ′ y ′ z ′ defined by ˆ y ′ = ˆn(V = 0), ˆ x ′ = ˆx and ˆ z ′ = ˆ x ′ × ˆ y ′ . In eqs. (2.4)-(2.5) V is the root mean square (RMS) of the voltage, K1 and K3 are the Frank’s elastic constants for splay and bend, respectively, and ∆ǫLF is the dielectric anisotropy at low frequencies. The boundary conditions are V (x ′ = 0) = 0, V (x ′ = a) = V and ξbulk(x ′ = 0) = ξbulk(x ′ = a) = 2π/180, the last one stemming from the pre-tilt (see section 2.1). The system composed by eqs. (2.4) and (2.5) has to be solved numerically: results for various V are shown in fig. 2.4. As stated above, I need to know ξ in x = a/2 in bulk NLC. From fig. 2.4(a) this corresponds to the maximum ξ for a fixed V, i.e. ξmax(V). Taking a linear input polarization at the first slice (see section 2.2), I can split the elec- tric field vector into ordinary (o) and extraordinary (e) components; the latter matches the director orientation in z = 0. From eq. (2.3) and taking d = 20µm, I find that the input e component transfers nearly all its power to the e-wave in the bulk NLC for every V; similarly, o-wave components in z = 0 remain o-waves in bulk. This is represented by the two straight horizontal lines in fig. 2.5(b) at β = 45 ◦ (e input) and β = 135 ◦ (o input): as the bias increases, for every z > d the director elevation increases as well [see fig. 2.4(c)], and the NLC principal axes rotate. e- and o-wave input components, however, remain decoupled as they evolve through the transition layer. Otherwise stated, if the rotation of the NLC dielectric tensor with z is slow enough that the index change is adiabatic, the e-wave displacement field De rotates with z, remaining parallel to the director projection onto xy (58). It is clear how this result maintains its validity also for larger d; hence, given that transition layers d are about 100µm, the numerical results retain their validity also for the cell in fig. 2.1. To validate these predictions, I varied V and experimentally studied the input polarization maximizing energy coupling on the e (o) component. For low V, it is easy 19
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 36 and 37: 2.2 Set-Up Figure 2.1: Sketch of th
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
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
show all

Solitons in Nonlocal Media

Create successful ePaper yourself

Delete template?

Save as template?