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

被引:44
作者
Kostenbauder, A
Sun, Y
Siegman, AE
机构
[1] Edward L. Ginzton Laboratory, Stanford University, Stanford, CA
来源
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A-OPTICS IMAGE SCIENCE AND VISION | 1997年 / 14卷 / 08期
关键词
D O I
10.1364/JOSAA.14.001780
中图分类号
O43 [光学];
学科分类号
070207 ; 0803 ;
摘要
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 eigen modes, 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. (C) 1997 Optical Society of America.
引用
收藏
页码:1780 / 1790
页数:11
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