Eigenmode expansions using biorthogonal functions: complex-valued Hermite–Gaussians

Abstract

A number of important optical systems, including gain-guided semiconductor lasers and unstable optical resonators, have governing equations that are linear but not Hermitian or self-adjoint. As a consequence, the propagation eigenmodes of these systems are not orthogonal in the usual 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 eigenmodes, conventional wisdom says that the expansion coefficients are given by the quadrature integrals between the input wave and the adjoint functions. Using a parabolic gain-guided system with complex Hermite–Gaussian eigenfunctions as a test case, we find that under a wide range of circumstances finite expansions using the quadrature integrals fail to converge properly, even for simple and realistic input functions. We then demonstrate that the coefficients for a finite expansion with minimum least-squares error in a biorthogonal system must be obtained from a more complex procedure based on inverting the eigenmode orthogonality matrix. Further tests on the complex Hermite–Gaussian system show that series expansions using these minimum-error coefficients converge and give much smaller errors under all circumstances.