Solitons in Nonlocal Media

More documents

Recommendations

Info

(a) E = 0 (b) E = 0 1.4 Liquid Crystals Figure 1.4: (a) In absence of external electric fields the director lies on the plane yz, forming an angle θ0 with ˆz. (b) When an electric field is applied parallel to ˆy, a dipole is induced in the molecules, which rotates towards the electric field in order to minimize their energy. The equilibrium angle θ is reached when the total torque acting on molecules becomes null. I note how rotations take place in a plane defined by the excitation geometry. 1.4.4 Reorientational Nonlinearity In section 1.4.2 I showed that the interaction energy between the electric field and the NLC is FE = − ǫa 2 (ˆn · E)2 . This term, inserted into eq. (1.8), gives a torque ΓE acting on the molecules and equal to ΓE = 2D × E, being D the electric field displacement 1 . Physically, when an external electric field is applied to the NLC, every molecule (excluding those normal to the field) becomes an induced dipole parallel to the long axis. The torque between E and the induced dipoles tends to rotate the molecules until they are parallel to E (see figure 1.4). The equilibrium position for ˆn corresponds to ΓE perfectly counterbalanced by the interaction forces between molecules, stemming from Fd (58). When E and ˆn are perpendicular to each other, no reorientation takes place below a threshold in the field: this value is called Freedericskz threshold (58). Let me now discuss the case of E(t) varying in time at frequencies larger than the cut-off for a reorientational NLC response. In this case, assuming monochromatic fields [i.e. E = Esin(ωt)], the torque induced by E is the temporal average of its instantaneous value, ΓE = 2 〈D × E〉 t = D × E, with 〈〉 t indicating the temporal average. Although E varies its sign periodically in time with a sinusoidal behavior, the torque rotates the molecules always in the same direction because when E changes sign so do the induced dipoles. Typically, the NLC response time is about 10ms, well above the optical range. 1 Equations (1.8) stem from the balance between the torques which act on the molecules (59). 12
1.5 Spatial Solitons in Nematic Liquid Crystals I can now address the reorientational nonlinearity of NLC. For a finite size beam, since the director reorientation in a given point is larger for larger beam intensities, there is a nonlinear refractive index change ∆n which depends on excitation. From eq. (1.10), for positive uniaxial NLC, the nonlinearity is self-focusing and the index larger for stronger intensities. Moreover, the perturbation in director distribution is more extended than the beam width due to the nonlocal interactions between the molecules, as modeled in (1.8) by the terms derived from Fd. 1.5 Spatial Solitons in Nematic Liquid Crystals The nonlinear optical properties of liquid crystals have been extensively studied owing to some unique features. First of all, they possess a nonlinearity which is about eight orders of magnitude larger than in isotropic liquids such as CS2 (18; 62; 63), allowing the formation of spatial solitons at very low powers (≈ 1 − 10mW for waist of a few microns) with continuous wave lasers. At variance with media exhibiting an electronic response, they are highly nonlinear in a wide wavelength range (15). Moreover, in NLC the optical beam creates a waveguide able to guide other low power signals, even at a different wavelength (64). Their response time, however, is 10 − 100ms with respect to a few fs in electronic media (60). First direct observation of self-focusing in NLC was carried out by Braun (65); after- wards self-localization was observed in capillaries (66) with dye doped liquid crystals. Dyes have two effects: they enhance the reorientational nonlinearity by the Janossy effect (67) and induce a temperature increase due to absorption 1 . The same group investigated higher order solitons in capillaries (68) and spatial solitons in the presence of a thermal nonlinearity (69; 70) 2 , both in cylindrical and planar cells. Another group focused its attention to planar waveguides (71; 72), demonstrating solitary wave prop- agation in (1+1)D geometries with undoped NLC. In 2000, Peccianti et al. demonstrated optical spatial solitons in bulk undoped NLC (46) in a (2+1)D geometry with planar alignment, proving their stability via nonlocality. They overcame the Freedericksz threshold by applying a low frequency electric 1 Undoped NLC are transparent in the visible and near infrared wavelength ranges. 2 In this case the soliton is due to the ordinary component, being the thermal nonlinearity focusing for this component. 13
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 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
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?