Unstable resonator modes for lasers with circular mirrors and high Fresnel numbers

The design of unstable resonators for large lasers with high Fresnel numbers and circular mirrors requires an ability to calculate their mode structures. Four methods for obtaining mode structure by solving the complex integral equation are analyzed. Included are a numerical method, two hybrid methods, and a virtual-source method. The hybrid methods are basically analytical methods with special numerical integration of analytical solutions (over the feedback mirror) to obtain improved solutions in the output annulus. The hybrid methods are designed for use with high-Fresnel-number resonators. However, their applicability extends into the low-Fresnel-number regime, where a comparison shows one of the hybrid methods agreeing exceptionally well with the numerical method. For analysis at high Fresnel numbers, the hybrid and virtual-source methods are compared with each other. The two hybrid methods are expected to differ from each other in the central core region when the Fresnel number is low, but they are expected to agree with each other when the Fresnel number is high. For the hybrid comparison at a high Fresnel number, the next to lowest loss modes show a similar structure. However, lack of agreement for the lowest loss mode shows that approximations in the development of the second hybrid method cause the selection of the wrong geometrical mode.