Alexei, Dan Brown

The design spec for the OMC has an astigmatic eigenmode. This is generally ignored when analyzing mode scans since the finesse of the OMC is not large enough to resolve the second order peaks of the x/y eigenmodes and they are assumed to be close enough as to basically sit on top of each other.

Recent independent FINESSE models by Dan and I (mine is at the IFOSIM git) have found that the mismatch (as estimated by the ratio of the second order peak to the zeroth order [A2/A0]) is always underestimated by about ~ 18.6% (i.e. a factor of 0.814) for any mismatch we put in.

My current working model is that the small separation between the x and y peaks causes the second order peak to appear smaller and wider than it normally would if the OMC eigenmode was not astigmatic. Therefore in an ideal environment (no misalignments, perfect input gaussian beam) the mismatch computed from an OMC mode scan will ALWAYS UNDERESTIMATE the actual mismatch between the beam and the cavity eigenmode by a constant factor.

Figure 1 demonstrates the effect of what I am talking about. Lx(fwhm,2*delta_fx) and Ly(fwhm,2*delta_fy) are lorentzians (corresponding to a second order resonance) with a fwhm and offset defined by the design OMC parameters.
fwhm = 643943 Hz
delta_fx = 5.813185e+07 Hz (mode spacing of xaxis modes)
delta_fy = 5.797750e+07 Hz (mode spacing of yaxis modes)

The height of the Lx+Ly peak is 18.69% (a factor of 0.814) smaller than the height of 2*Lx (what it would be without astigmatism).

In order to counteract this effect multiplying the measured peak height for the second order peak by 1.23 is sufficient.
This factor can be computed from evaluating the ratio of 2*Lx at its center to  Lx + Ly at the midpoint of their centers by the following equation

(x^2 + 1)

where x = |2*delta_fx - 2*delta_fy| / fwhm.

The complete derivation of this will be posted up on the DCC at some point in the near future.

Please note that this only meant to be a first order approximation to what is going on. This breaks down for tiny mismatches (10^-4 and smaller) as the peak location stops being at the midpoint between Lx and Ly. If the cavity astigmatism or finesse is high enough to resolve the individual axes then this model is also invalid. But in this particular case of the OMC it should hold, and so my recommendation is for future mismatch calculations (the ones that are computed by A2/A0) from OMC scans to be scaled by a factor of 1.23 to avoid underestimating the actual mismatch.