Jianke Yang Dept of Mathematics and Statistics, University of Vermont

Dipole and Vector Solitons in 2D Photonic Lattices Jianke Yang Dept of Mathematics and Statistics, University of Vermont Igor Makasyuk, Anna Bezryadina, Zhigang Chen Dept of Phys. & Astronomy, San Francisco State University

Discrete solitons in waveguide arrays D. N. Christodoulides et al., , Optics Letters 13, 794 (1988). H. S. Eisenberg et al., , Physical Review Letters, 81, 3383 (1998).

Optically-induced lattices in photorefractive crystals SBN v From multiple o-beam interference Linear waveguides Efremidis et al., PRE 2002 Fleischer, et al., PRL, Nature 2003 Nashev, et al., OL 2003 O-beam E-beam Spatial modulation of a partially coherent o-beam Chen, et al. PRL2004 Amplitude mask

So far, fundamental and vortex solitons in a 2D lattice have been reported: Fleischer, et al., PRL, Nature 2003 Martin, et al., PRL 2004 Malomed and Kevrekidis, PRE 2001 Yang and Musslimani, OL 2003 Neshev, et al., PRL 2004 Fleischer, et al., PRL 2004 Yang, New J. Phys. 2004

In this talk, we report both theoretically and experimentally dipole and vector solitons in a 2D photonic lattice

Dipole solitons in a 2D lattice Theoretical model: Here U: electric field; z: propagation distance; E0 : applied DC field; D: lattice spacing; I0: lattice intensity; r33: electro-optic coefficient; k0= 2p/l0; k1= k0 ne;

Out-of phase dipole-solitons Moderate intensity High intensity Low intensity Lattice

In-phase dipole solitons High intensity Moderate intensity Low intensity Lattice always unstable

Note: the above dipole solitons arise due to a balance of discrete diffraction nonlinearity, and lobe interactions They can not exist without the lattice.

Simulations of a pair of Gaussian beams Output Input Out-of phase In-phase Low NL High NL High NL No lattice

Quadrupole solitons Out-of-phase In-phase Always unstable Can be stable

Dipole solitons: experimental results Output Input Out of Phase In Phase Low NL High NL High NL No lattice

Anisotropic effect: out-of-phase case Output Input Low NL Low NL High NL No lattice with lattice with lattice These dipole solitons are robust against anisotropic effects

Anisotropic effect: in-phase case Output Input Low NL Intermediate NL High NL These dipole solitons are sensitive to anisotropic effects

Vector solitons in a 2D lattice If we make the two beams of the dipole incoherent, and launch into the same lattice site, then we can study vector lattice solitons

2D vector lattice solitons: experiment Input Output Expt. results Num. results Low NL High NL High NL Coupled Decoupled Mutually Incoherent

2D vector lattice solitons: theory Vector solitons can be derived from scalar ones by a polarization rotation: (x, y) : scalar lattice soliton; : polarization Scalar 2D lattice solitons have been studied before: Yang and Musslimani, Opt. Lett. 2003 Efremidis, et al. PRL 2004

Dipole-like vector solitons in a 2D lattice If we make the two beams incoherent, and launch into different lattice sites, then we can study dipole-like vector lattice solitons Comb. input Low NL High NL 1st comp. 2nd comp. Expt. results Num. results

Conclusions • 1. We have demonstrated the formation of dipole, • quadrupole, vector, and dipole-like vector solitons in a • 2D photonic lattice for the first time. • 2. These solitons arise due to a balance of discrete • diffraction, nonlinearity, and lobe interactions. • 3. These solitons are stable in certain parameter regimes.

A scalar lattice soliton They are stable in a large parameter space

Dipole-like vector solitons in a 2D lattice If we make the two beams incoherent, and launch into different lattice sites, then we can study dipole-like vector lattice solitons Comb. input Low NL High NL 1st comp. 2nd comp. Expt. results Num. results

Jianke Yang Dept of Mathematics and Statistics, University of Vermont