Eigenmodes of misaligned unstable optical resonators with circular mirrors.

It is shown numerically that the diffractive transverse (Fox-Li) eigenmodes supported by an unstable cavity with tilted end mirrors can be computed by expanding these modes in terms of the fully aligned (aberration-free) eigenmodes of the same cavity. Circular mirror resonators are considered in which the aligned cavity eigenmodes can be decomposed into different azimuthal components. The biorthogonality property of the aligned cavity eigenmodes is used to obtain the coefficients in the modal expansion of the misaligned modes. Results are given for two different resonators: a conventional hard-edge unstable cavity with a small tilt of the output coupler and one that uses a graded reflectivity output mirror with a small tilt of the primary mirror. It is shown that the series expansion of the misaligned modes in terms of the aligned modes converges, and the converged eigenvalues are virtually identical to those computed by using the Prony method. Symmetry considerations and other new insights into the effects of a mirror tilt on the modes of a resonator are also discussed.