Solitons in Nonlocal Media

More documents

Recommendations

Info

1.3 Nonlocality ∆n(x) = I(x ′ dG ) G|x=x ′ + d(x − x ′ ) (x − x x=x ′ ′ ) + 1 d 2 2G d(x − x ′ ) 2 (x − x x=x ′ ′ ) 2 + . . . dx ′ = = G0P + G2 I(x ′ ) x − x ′ ∞ 2 x ′ ′ dx + . . . = P G2m − x 2m being Gm ≡ 1 d m! mG d(x−x ′ ) m m=0 I(x ′ ) (1.5) x=x ′ , 〈f(x)〉 I(x) = I(x)f(x)dx/ I(x)dx and I(x)dx = P, with P the beam power. Furthermore, I assumed that xI(x)dx = 0, i.e. the point x = 0 is the beam center. In deriving eq. (1.5) I used the relationship G2m+1 = 0, i.e. I supposed that beam peak and the maximum of the Green function overlap. I underline the linear relationship between ∆n and the power P, as expected from the initial hypothesis. Eq. (1.5) is the power series (in space) of the nonlinear index perturbation ∆n. In the highly nonlocal case, i.e. when the beam width is negligible with respect to the extension of the response function, taking an even parity for the intensity I get ∆n ∼ = G0 + G2 x ′2 P + G2Px2 (53), i.e. the index perturbation is parabolic I(x ′ ) in space, with concavity proportional to power through a material dependent param- eter G2; moreover, G2 < 0 (G2 > 0) in focusing (defocusing) media 1 . Substituting into (1.3), I retrieve the Schröedinger equation for a parabolic potential, i.e. the well known quantum harmonic oscillator (49). In essence, I have transformed the nonlinear problem into a well known linear problem, largely studied in quantum mechanics and in optics; given the simple mathematics needed to describe this family of solitons (as compared, for example, to inverse scattering technique) they were named accessible solitons (31). The eigenfunctions corresponding to solitons are Hermite-Gauss functions 2 . Since the eigenfunction width depends on G2, which is power-dependent, solitons with a certain width exist only for a value of the power. These solitons are stable, i.e. small pertur- bations do not destroy their nature in propagation. Accessible solitons have firstly observed in liquid crystals (30; 47; 51) and, later, in lead-glasses with a thermal nonlinearity (39; 57). 1 If I consider a Dirac function I = Pδ(x) for the intensity profile I get ∆n = PG(x) which, expanded around x = 0, gives the same result because δ(x)x 2 dx = 0. 2 In cartesian coordinates. Other coordinate systems yield different eigenfunctions; for example, a cylindrical system gives Laguerre-Gauss functions [see (56)]. 6
1.3.2 Weak Nonlocality 1.4 Liquid Crystals In eq. (1.4), expanding the intensity profile I centered in x = 0 and with even parity in a power series around x ′ = x, I get ∆n = ∞ m=0 I2m(x) t 2m G (1.6) where I2m(x) = 1 ∂ (2m)! 2mI(x) ∂x2m and t2m G = t2mG(t)dt. In the weakly nonlocal case, i.e. when the intensity is wider than the medium response function G, in eq. (1.6) terms corresponding to m > 2 can be neglected, providing ∆n = I + I2 t2m G = I + 1 2∂2I/∂x 2 t2m : the self-induced waveguide is smoother, stabilizing the soliton in G (2+1)D (32). 1.4 Liquid Crystals In this thesis, in order to investigate the role of nonlocality in nonlinear optical propagation, I examine liquid crystals. In this section I will remind the physical properties which explain nonlocal nonlinear optical propagation in this kind of media. 1.4.1 Liquid Crystal Phases Three states of matter are the most diffused in nature: solid, liquid and gas. Some organic compounds named liquid crystals show intermediate phases between liquid and solid, featured by specific properties 1 . Liquid crystals are characterized by disorder in at least one direction and some degree of anisotropy; for a particle or a specific pattern in a certain position, the probability to find a similar one depends on direction 2 (58). Given the definition above, liquid crystal phases group in three main families, according to the degree of long range positional order exhibited by the molecules: • nematic: the gravity centers of the molecules are totally disordered, but their orientation is correlated; 1 A rigorous definition refers to mesomorphic phases. 2 This means that the density-density correlation function is anisotropic with respect to some macro- scopic axes. 7
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 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 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 76 and 77:
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?