EIGENMODE EXPANSIONS USING BIORTHOGONAL FUNCTIONS - COMPLEX-VALUED HERMITE-GAUSSIANS

A number of important optical systems, including gain-guided semicondu ctor lasers and unstable optical resonators, have governing equations that are linear but not Hermitian or self-adjoint. As a consequence, t he propagation eigenmodes of these systems are not orthogonal in the u sual fashion but rather are biorthogonal to a set of adjoint functions . If one wishes to expand an arbitrary wave of such a system in terms of its eigen modes, conventional wisdom says that the expansion coeffi cients are given by the quadrature integrals between the input wave an d the adjoint functions. Using a parabolic gain-guided system with com plex Hermite-Gaussian eigenfunctions as a test case, we find that unde r a wide range of circumstances finite expansions using the quadrature integrals fail to converge properly, even for simple and realistic in put functions. We then demonstrate that the coefficients for a finite expansion with minimum least-squares error in a biorthogonal system mu st be obtained from a more complex procedure based on inverting the ei genmode orthogonality matrix. Further tests on the complex Hermite-Gau ssian system show that series expansions using these minimum-error coe fficients converge and give much smaller errors under all circumstance s. (C) 1997 Optical Society of America.