This afternoon I measured the distances required to calculate the mode matching, and then I came up with a solution using code that I wrote. In doing so, I realized that there is a problem with the way we have been calculating the overlap in the past.
Last summer, Jenna used the overlap function to show that an overlap of > 99% could be achieved with a non-astigmatic beam injected into a cavity with an elliptical eigenmode (like our gyro cavity, whose mode has a waist twice as big in one direction as in the other). This is fine, but there are two things to note:
Jenna calculated such a nice overlap because she assumed that our input beam was not astigmatic. The coupling is far more sensitive to curvature match at the input (i.e. making the beam flat at the flat mirror) than it is to the area of the beam cross section, and a non-astigmatic beam allows you to do that arbitrarily well.
If I take the solution I calculated for the average of the X and Y components of our input beams and apply it to both components individually, the overlap I get with the cavity eigenmode is < 1%, despite the fact that it is >99% if we assume that the average is a real, physical, circular beam.
I think we may need to do the cylindrical mode matching after all.