Eigenmodes of misaligned, unstable optical resonators with circular mirrors

Abstract

The sensitivity of unstable optical resonators to intracavity phase aberrations is an important consideration in the design of these cavities. Mirror misalignment or tilt introduces a phase aberration that alters the near-field intensity and phase distributions and lowers the transverse-mode separation of the cavity. In this work it is shown, numerically, that the diffractive transverse (Fox-Li) eigenmodes, supported by an unstable cavity with tilted 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. Because the linear operator in the eigenvalue problem for the cavity modes is not Hermitian for open-sided resonators because of the boundary conditions, it cannot be rigorously guaranteed that the modes form a complete set. Furthermore, the modes are biorthogonal rather than orthogonal to one another. Thebiorthogonality 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-edged 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 modes and eigenvalues are virtually identical to those computed by the Prony or Krylov matrix methods.