Short conclusion:
The roundtrip cavity length for the H1 OMC was adjusted to be 1.145m
instead of 1.132m such that the 19th HOMs of the lower sideband do not get resonant together with the carrier.
Background:
The purpose of the OMC is to transmit the carrier TEM00 mode while anything else is rejected.
As the optical cavity has infinite numbers of resonant modes, what we practically do is to select
the roundtrip accumulated gouy phase so that low order higher order modes for the carrier
as well as the sidebands (including the TEM00 modes).
The nominal round trip length of the OMC is 1.132m. The curvature of the mirror is 2.575m.
The nominal ratio between the TMS and FSR is 0.218791 and 0.219385 (TMS_V/TMS_H= 0.9973)
for the vertical and horizontal modes. This split comes from the nonzero angle (~4deg) of incidence on the curved mirrors.
In reality, the TMS/FSR ratio depends on the true curvature of the mirror. More importantly, astigmatism
of the mirror changes the difference of the ratios for the vertical and horizontal modes.
The mirror astigmatism can either reduce or increase the split. between the TMSs. For example,
the L1 OMC showed the TMS/FSR ratio of (0.218822, 0.219218) for the vertical and horizontal modes.
TMS_V/TMS_H is 0.9982 which is 0.18% from the unity. This suggests, roughly to say, that 0.27% of the
astigmatism coming from the AOI of 4deg was partially compensated by the mirror astigmatism. This was lucky.
Something unlucky happened to the case for the first choice of the H1OMC curved mirrors.
TMS_V/TMS_H is 0.990 which is indeed 1% away from the unity. This actually caused some problem:
As the modes spreads too wide, the 19th modes became unavoidable. (see the picture below)
Red  carrier, Blue  upper sideband (+45MHz), Green  lower sideband
After the replacing one of the PZT assembly with another one, 1TMS_V/TMS_H went down to 6%.
But still the 19th mode is on resonance. In order to shift the 19th mode from the resonance, the cavity length
had to be changed more than the range of the micrometer.
Simple simulation:
Attached Mathematica file calculates expected mode structure when the curved mirror position is
moved by DL (then the total roudtrip length changes 4*DL). This tells us that the 19th mode is
moved from the resonance by giving DL=0.003 or DL=0.0025.
It was impossible to make the cavity short enough as the gluing fixture interferes with the curved mirror.
In fact, it was also impossible to make the cavity long enough as it was. Therefore PEEK shims with
the thickness of 1.5mm was inserted.
Result:
The FSR and TMS were measured with the longer cavity. 50V was applied to PZT1.
FSR: 261.775MHz
TMS_V: 57.575MHz
TMS_H: 57.880MHz
=> Cavity round trip length of 1.1452m
=> TMS/FSR = {0.219941, 0.221106}
The 19th modes for the lower sidebands are successfully moved from the carrier resonance.
The first accidental resonance is the lower sideband at the 28th order modes.
