November 9, 2012 11:1 WSPC/S0218-8635 145-JNOPM 1250031 Journal of Nonlinear Optical Physics & Materials Vol. 21, No. 3 (2012) 1250031 (13 pages) c  World Scientific Publishing Company DOI: 10.1142/S0218863512500312 TIME DOMAIN ANALYSIS OF HELMHOLTZ SOLITON PROPAGATION USING THE TLM METHOD P. CHAMORRO-POSADA Departamento de Teor´ ıa de la Se˜ nal y Comunicaciones e Ingenier´ ıa Telem´ atica, Universidad de Valladolid, ETSI Telecomunicaci´ on, Paseo Bel´ en 15, E-47011 Valladolid, Spain pedcha@tel.uva.es G. S. McDONALD Joule Physics Laboratory, Materials and Physics Research, School of Computing, Science and Engineering, University of Salford, Salford M5 4WT, United Kingdom Received 15 August 2012 The transmission line matrix method is used to study Helmholtz solitons as solutions of the two-dimensional time-domain Maxwell equations in nonlinear media. This approach permits to address, in particular, the propagation and intrinsic stability properties of subwavelength soliton solutions of the scalar nonlinear wave equation and the behavior of optical solitons at arbitrary interfaces. Various numerical issues related to the analysis of soliton beams using the time-domain method are also discussed. Keywords : Helmholtz solitons; TLM method; subwavelength solitons. 1. Introduction The two-dimensional (2D) nonlinear scalar wave (SW) equation is conventionally used for the study of soliton light beams. This equation describes exactly the non- linear evolution of pure 2D transverse electric (TE) electromagnetic fields, but it is also a very good theoretical framework for most common planar experimental setups. 1 For continuous-wave (CW) beams under paraxial propagation conditions, the nonlinear SW equation reduces to the nonlinear Schr¨ odinger (NLS) equation that is a standard mold for the analysis of spatial optical solitons. 2 The NLS equa- tion can be integrated analytically, 2 for a certain class of material responses, or numerically, using highly efficient algorithms. 3 If paraxial propagation is not assumed, the scalar nonlinear Helmholtz (NLH) equation is obtained from the SW equation in the CW case without the resource of any approximation. The NLH framework allows to study optical solitons under 1250031-1