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## Refraction of power-law spatial solitons -the Helmholtz-Snell law

### Citations

13 |
Theory of light-beam propagation at nonlinear interfaces.
- Aceves, Moloney, et al.
- 1989
(Show Context)
Citation Context ...guration is a key “building block” structure that facilitates more sophisticated device designs and architectures for a diverse range of applications. The seminal papers of Aceves, Moloney and Newell =-=[1,2]-=- considered perhaps the simplest scenario, where a spatial soliton was incident on the boundary between two different Kerr-type materials. Their intuitive approach reduced the full complexity of the e... |

11 | Helmholtz solitons at nonlinear interfaces,” - Sánchez-Curto, Chamorro-Posada, et al. - 2007 |

9 |
Non-paraxial beam propagation methods,”
- Chamorro-Posada, McDonald, et al.
- 2001
(Show Context)
Citation Context ...tween these two regimes is determined by δ = 0, in which case θt = θi and the incident beam passes undeviated across the interface. Thus, δ = 0 defines the transparency condition, in which linear and nonlinear mismatches in refractive index exactly cancel each other. One can also note that the “no interface” scenario, where Δ = 0 and α = 1, satisfies the transparency condition, as one would expect intuitively. IV. SIMULATIONS Extensive simulations have been used to test the analytical predictions of the Helmholtz-Snell law (6) for q = 1, 2 and 3 against direct numerical integration of Eq. (4) [11]. The Kerr case of q = 2 is “sandwiched” between two regimes where the focusing properties of the two media are weaker and stronger than the Kerr type (q = 1 and 3, respectively). Various sets of simulations have considered linear interfaces (defined by α = 1), nonlinear interfaces (defined by Δ = 0), and mixed interfaces (with arbitrary values of α and Δ). Computations have considered arbitrary incidence angles and regimes where Δ < 0, neither of which are accessible to paraxial theory [1–4]. A full account of new phenomena in various parameter regimes (see Fig. 2) will be presented. REFERENC... |

8 | Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides,” - Aceves - 1990 |

8 | Helmholtz solitons in power-law optical materials,”
- Christian, McDonald, et al.
- 2007
(Show Context)
Citation Context ...at describe the mismatch between the linear and nonlinear characteristics of the two media: 2 02 01 andn n α α α 2 1 ⎛ ⎞ Δ ≡ 1− ≡⎜ ⎟ ⎝ ⎠ (5a,b) From these relations, three distinct scenarios emerge: (i) linear interfaces (defined by α = 1 so that there is no mismatch in the non-linear coefficients), (ii) non-linear interfaces (defined by Δ = 0, so the two media have the same linear index), (iii) mixed interfaces (with arbitrary choices of α and Δ). Model (4) in medium 1 is just the conventional power-law Helmholtz equation, for which the exact analytical bright soliton solutions are now known [10]. The forwardpropagating solutions have a sech2/q profile, ( ) 20 2 2 , sech 1 2 1 4exp exp , 2 21 2 q Vu a V i V i V ξ ζξ ζ η κ κβ ζ ζξ κ κκ ⎛ ⎞− = ⎜ ⎟⎜ ⎟+⎝ ⎠ ⎡ ⎤+ ⎛ ⎞ ⎛ ⎞× + −⎢ ⎥⎜ ⎟ ⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎣ ⎦ (6) where η0 is the peak amplitude and V is the conventional transverse velocity. Here, V is related to the propagation angle θ in the laboratory (x,z) frame (measured with respect to the longitudinal, i.e., z, axis) through the relation, tan 2 Vθ κ= . (7) Propagation angles are thus bounded by 90− ° ≤ θ ≤ 90+ ° , and they may be arbitrarily large. The additional parameters in solution (6) are β ≡... |

7 | Nonlinear interfaces: intrinsically nonparaxial regimes and effects,” - Sánchez-Curto, Chamorro-Posada, et al. - 2009 |

5 | Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media,” - Varatharajah, Newell, et al. - 1989 |

5 |
Nonlinear wave propagation in planar structures,”
- Mihalache, Bertolotti, et al.
- 1989
(Show Context)
Citation Context ...finite-beam effects and (linear and nonlinear) medium discontinuities. In this presentation, we report the first steps toward extending our Kerr analyses to regimes involving wider classes of materials. In our systematic approach, we first consider media whose nonlinear refractive index nNL has a generic power law-type dependence on the electric field amplitude E, ( )NL 02 qn E E n α = , (1) where α is a (small) positive coefficient, n0 is the linear index (n0 >> α), and the exponent q > 0. The single power-law model (1) is perhaps the simplest non-Kerr nonlinearity one might care to consider [8,9]. Materials that fall within this category include some semiconductors (e.g., InSb and GaAs / Fig. 1. Arbitrary-angle spatial soliton refraction. The refraction process here is external (θt > θi; see text in Sec. III). GaAlAs), doped filter glasses (e.g., CsSxSex-2) and liquid crystals. Non-Kerr regimes (i.e., where q deviates from the value of 2) have been found to give rise to a diverse range of new qualitative phenomena. II. POWER-LAW INTERFACES MODEL We consider the TE-polarized continuous-wave scalar electric field ( ) ( ) ( ) ( ) ( ) *, , , exp , exp ,E x z t E x z i t E x z i tω ω= − + ... |

3 |
Spatial solitons of the power-law nonlinearity,”
- Snyder, Mitchell
- 1993
(Show Context)
Citation Context ...finite-beam effects and (linear and nonlinear) medium discontinuities. In this presentation, we report the first steps toward extending our Kerr analyses to regimes involving wider classes of materials. In our systematic approach, we first consider media whose nonlinear refractive index nNL has a generic power law-type dependence on the electric field amplitude E, ( )NL 02 qn E E n α = , (1) where α is a (small) positive coefficient, n0 is the linear index (n0 >> α), and the exponent q > 0. The single power-law model (1) is perhaps the simplest non-Kerr nonlinearity one might care to consider [8,9]. Materials that fall within this category include some semiconductors (e.g., InSb and GaAs / Fig. 1. Arbitrary-angle spatial soliton refraction. The refraction process here is external (θt > θi; see text in Sec. III). GaAlAs), doped filter glasses (e.g., CsSxSex-2) and liquid crystals. Non-Kerr regimes (i.e., where q deviates from the value of 2) have been found to give rise to a diverse range of new qualitative phenomena. II. POWER-LAW INTERFACES MODEL We consider the TE-polarized continuous-wave scalar electric field ( ) ( ) ( ) ( ) ( ) *, , , exp , exp ,E x z t E x z i t E x z i tω ω= − + ... |

2 | Helmholtz dark solitons at nonlinear interfaces,” - Sánchez-Curto, Chamorro-Posada, et al. - 2010 |

1 | et al., “Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides - Aceves - 1990 |