Nonlinear optical beams in bounded nematic liquid crystal cells

Stable nonlinear beams, both solitary waves (nematicons) and optical vortices, can form in a nematic liquid crystal due to a balance between the nonlinear, nonlocal response of the nematic and the diffractive spreading of the light beam. The `huge' nonlinearity of a nematic liquid crystal makes it ideal for the experimental development of photonic devices as nonlinear effects occur over millimetre distances. In this work, a simple and fast method to analyse the trajectory of a nonlinear beam within a finite liquid crystal cell, based on a classical method not explored in this context, the method of images, is developed. With the orientation of the nematic molecules modelled using images, the evolution of the beam is obtained by using both asymptotics and modulation theory. The efficiency of this new method is shown by comparisons with a standard Fourier series solution for the nematic response and full numerical solutions of the governing equations. It is found that only a small number of images is required compared with the usual Fourier series technique in order to obtain excellent agreement with full numerical solutions. Finally, the contrasting effect of the cell boundaries on a nematicon and a vortex is explored. References A. Alberucci and A. Assanto, Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions, J. Opt. Soc. Amer. B, 24, 2314--2320 (2007), doi:10.1364/JOSAB.24.002314. A. Alberucci, G. Assanto, D. Buccoliero, A. S. Desyatnikov, T. R. Marchant, and N. F. Smyth, Modulation analysis of boundary-induced motion of optical solitary waves in a nematic liquid crystal, Phys. Rev. A, 79, 043816 (2009), doi:10.1103/PhysRevA.79.043816. A. Alberucci, M. Peccianti and G. Assanto, Nonlinear bouncing of nonlocal spatial solitons at boundaries, Opt. Lett., 32, 2795--2797 (2007), doi:10.1364/OL.32.002795. G. Assanto, A. Fratalocchi and M. Peccianti, Spatial solitons in nematic liquid crystals: from bulk to discrete, Opt. Express,, 15, 5248--5259 (2007), doi:10.1364/OE.15.005248. G. Assanto, M. Peccianti and C. Conti, Nematicons: optical spatial solitons in nematic liquid crystal, Opt. Photon. News, 14, 44--48 (2003), doi:10.1364/OPN.14.2.000044. C. Conti, M. Peccianti, G. Assanto, Route to nonlocality and observation of accessible solitons, Phys. Rev. Lett., 91, 073901 (2003), doi:10.1103/PhysRevLett.91.073901. R. Courant and D. Hilbert, Methods of Mathematical Physics Vol. 1, Interscience Publishers, New York (1965). W. L. Kath and N. F. Smyth, Soliton evolution and radiation loss for the nonlinear Schr{o}dinger equation, Phys. Rev. E, 51, 1484--1492 (1995), doi:10.1103/PhysRevE.51.1484. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena, Wiley, New York (1995). I. C. Khoo, Nonlinear optics of liquid crystalline materials, Phys. Rep., 471, 221--267 (2009), doi:10.1016/j.physrep.2009.01.001. Yu. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego, (2003). A. A. Minzoni, L. W. Sciberras, N. F. Smyth and A. L. Worthy, Propagation of optical spatial solitary waves in bias-free nematic-liquid-crystal cells, Phys. Rev. A, 84, 043823 (2011), doi:10.1103/PhysRevA.84.043823. A. A. Minzoni, N. F. Smyth and A. L. Worthy, Modulation solutions for nematicon propagation in nonlocal liquid crystals, J. Opt. Soc. Amer. B, 24, 1549--1556 (2007), doi:10.1364/JOSAB.24.001549. A. A. Minzoni, N. F. Smyth, A. L. Worthy and Yu. S. Kivshar, Stabilization of vortex solitons in nonlocal nonlinear media, Phys. Rev. A, 76, 063803 (2007), doi:10.1103/PhysRevA.76.063803. G. B. Whitham, Linear and Nonlinear Waves, J. Wiley and Sons, New York, (1974). A. I. Yakimenko, Yu. A. Zaliznyak and Yu. S. Kivshar Stable vortex solitons in nonlocal self-focusing nonlinear media, Phys. Rev. E, 71, 065603(R) (2005), doi:10.1103/PhysRevE.71.065603