Solitons in Nonlocal Media
Solitons in Nonlocal Media
Solitons in Nonlocal Media
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CONTENTS<br />
2.5.2.2 Soliton Profile . . . . . . . . . . . . . . . . . . . . . . . 30<br />
3 <strong>Nonlocal</strong>ity and Soliton Propagation 36<br />
3.1 Def<strong>in</strong>ition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36<br />
3.2 Role of the Boundary Conditions on the Nonl<strong>in</strong>ear Index Perturbation . 39<br />
3.2.1 Poisson 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39<br />
3.2.2 Poisson 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41<br />
3.2.2.1 Green Function <strong>in</strong> a F<strong>in</strong>ite Rectangular Geometry . . . 41<br />
3.2.2.2 Perturbation Profile . . . . . . . . . . . . . . . . . . . . 43<br />
3.2.3 Screened Poisson Equation . . . . . . . . . . . . . . . . . . . . . 44<br />
3.2.3.1 Green Function for the Screened Poisson Equation . . . 47<br />
3.2.3.2 Perturbation Profile . . . . . . . . . . . . . . . . . . . . 48<br />
3.2.4 Reorientational Equation for the NLC <strong>in</strong> Anisotropic Configuration 48<br />
3.2.4.1 Perturbative Approach for the Director Profile Compu-<br />
tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49<br />
3.2.4.2 Solution <strong>in</strong> a F<strong>in</strong>ite Rectangular Geometry . . . . . . . 51<br />
3.2.5 Highly <strong>Nonlocal</strong> Limit for the 2D Case . . . . . . . . . . . . . . . 54<br />
3.3 Soliton Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58<br />
3.3.1 General Expression for the Equivalent Force . . . . . . . . . . . . 58<br />
3.3.2 Power series for the Equivalent Force . . . . . . . . . . . . . . . . 61<br />
3.3.3 Highly <strong>Nonlocal</strong> Case . . . . . . . . . . . . . . . . . . . . . . . . 62<br />
3.4 Soliton Oscillations <strong>in</strong> a F<strong>in</strong>ite-Size Geometry . . . . . . . . . . . . . . . 62<br />
3.4.1 Poisson 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62<br />
3.4.2 Poisson and Screened Poisson 2D . . . . . . . . . . . . . . . . . . 63<br />
3.4.3 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65<br />
3.4.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65<br />
3.4.3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 69<br />
4 Vector <strong>Solitons</strong> <strong>in</strong> Nematic Liquid Crystals 72<br />
4.1 Vector <strong>Solitons</strong>: an Introduction . . . . . . . . . . . . . . . . . . . . . . 72<br />
4.2 Cell Geometry and Basic Equations . . . . . . . . . . . . . . . . . . . . 73<br />
4.3 Highly <strong>Nonlocal</strong> Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
4.3.1 Reorientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
4.3.2 Optical Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 76<br />
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