A Numerical Method for One Dimensional Action Functionals of Photonic Band Gap Structures Feng Xie, Greg Reid, Sreeram Valluri October 4, 2003 Abstract Photonic band-gaps (PBG), photonic analogue of electronic semi-conductor band-gaps, have attracted much recent attention, because of numerous potential applications in commu- nications and computing. Ak¨ozbek and John, developed a variational model of such band- gaps, using action functionals, where solitary waves are expanded in terms of a finite or- thonormal basis. These expansions to finite order N converged to solitary waves. The non- linear polynomial equations for the coefficients in the expansions, have non-unique solutions. Our paper, makes a study of the multiplicity of the solutions for 1 dimensional photonic band-gap structures. It is found that the non-uniqueness grows dramatically with the order of the expansion N . We use homotopy, which continuously deforms the solutions of exactly solvable systems, into the solutions of the systems to be solved with new results in numeric algebraic geometry, such that all solutions are determined. We used Maple 7 to obtain the polynomial equations for the variational coefficients, extending Ak¨ozbek and John’s approach. A homotopy based package PHCpack was used to solve the systems for N ≤ 4 and a linearization-extrapolation method was developed to find real solutions for N ≥ 5. The results are compared with the exact soliton solutions and their convergence behavior is discussed. keywords: Photonic Band-gap, Variational parameters, Orthonormal basis, Solitons, Ho- motopy, Continuation, Newton’s Method, PHCpack, Maple. 1 Introduction Recently, there has been growing interest in the studies of the propagation of electromagnetic (EM) waves in disordered and periodic dielectric structures [1, 2], that have stimulated [3] theoretical and experimental work on the possible existence of photonic band gaps in 3D periodic dielectric structures [4, 5, 6]. Sajeev John [4] has proposed that optical localization for EM waves near a photonic band gap might be achieved by weak disordering of a periodic arrangement of spheres. Recent developments suggest that a clear experimental demonstration of optical localization is possible [4, 6, 8, 9]. Ak¨ ozbek and John [10] obtained solitary wave solutions for 2D square and triangular lattices, and 3D face centered cubic (fcc ) lattices. As in the 1D case, near the photonic band edges it is possible to establish a correspondence between the effective nonlinear electromagnetic wave equation and the equation for a particle moving in a classical potential well. This correspondence indicates the existence of higher-dimensional solitary waves. Unlike the one-dimensional case, where an exact analytical solution exists throughout the PBG, in higher dimensions, simple solutions are only known near the photonic band edges where the “effective mass approximation” for the photon 1