Rana mentioned the possibility that the PR2 curvature makes the impact on the mode stability. Entry 7988
Here is the extended discussion.
Hypothesis:
The small but nonnegligible curvatures of the TT mirrors made the recycling cavity unstable or nearly unstable.
Conclusion:
If the RoC of the TT mirrors are 600 m (convex), the cavity would be barely stable.
If the RoC of the TT mirrors are less than 550m, the horizontal modes start to be unstable.
Assumption that all of the TT mirrors are concave should be confirmed.
Fact (I): Cavity stability
 The folded PRMI showed the mode stability issue. (L=6.78m from Jenne's entry 7973)
 The folded PRMPR2PR3flat mirror cavity also showed the similar mode issue. (L=4.34m)
 The unfolded PRMPR2 cavity demonstrated stable cavity modes. (L=1.91m)
Fact (II): Incident angle
 PRM 0deg
 PR2 1.5deg
 PR3 41deg
Fact (III): Mirror curvature
 RoC of PRM (PRMU02): +122.1m (measured, concave), or +115.6m (measured by the vendor)
 RoC of G&H mirrors: 600m ~ 700m (measured, I suppose the negative number means convex) (Jenne's entry 7851)
[Note that there is no measurement of the phase map for the PR2 mirror itself.]
 RoC of LaserOptik mirrors: 625m ~ 750m (measured, I suppose that the measurement shows the mirrors are convex.) (Jan's entry 7627 and 7638)
Let's assume that the TT mirrors are always convex and have a single number for the curvature radius, say RTT
Cavity mode calculation with Zach's arbcav
1) The unfolded PRMPR2 cavity:
The cavity becomes unstable when 0 > RTT > 122m (This is obvious from the gfactor calculation)
==> The measured RoC of the TT mirrors predicts the cavity is stable. (g=0.98, Transverse Mode Spacing 3.54MHz)
2) The folded PRMPR2PR3flat mirror cavity:
The cavity becomes unstable when RTT < 550 m
==> The measured RoC of the TT mirrors (RTT ~ 600m) predicts the cavity is barely stable (g=0.997, TMS ~600kHz).
 The instability occurs much faster than the unfolded case.
 The horizontal mode hits unstable condition faster than the vertical mode.
 The astigmatism mainly comes from PR3.
3) The folded PRMI:
The cavity becomes unstable when RTT < 550 m
==> The measured RoC of the TT mirrors (RTT ~ 600m) predicts the cavity is barely stable. (g=0.995, TMS ~500kHz)
 The instability occurs with almost same condition as the case 2)
The calculation result for the PRMI with RTT of 600 m. The code was also attached.
Q&A:
Q. What is the difference between unfolded and folded?
A. For the unfolded case, the PR2 reflect the beam only once in a roundtrip.
For the folded case, each TT mirror reflects the beam twice. Therefore the lens power by the mirror is doubled.
Q. Why the astigmatism mainly comes from PR3?
A. As the angle of incidence is much bigger than the others (41deg).
Q. Why the horizontal mode is more unstable than the vertical mode?
A. Offaxis reflection of a spherical mirror induces astigmatism. The effective curvature of the mirror in the horizontal direction
is R / Cos(theta) (i.e. longer), while it is R Cos(theta) (i.e. shorter). Indeed, the vertical and horizontal ROCs are factor of 2 different
for the 45deg incidence.
Q. Why the stability criteria for the case 2) and 3) similar?
A. Probably, once the effective curvature of the PRMPR2PR3 becomes negative when RTT < 550 m.
Q. You said the case 2 and 3 are barely stable. If the TMS is enough distant form the carrier, do we expect no problem?
A. Not really. As the cavity get close to the instability, the mode starts to be inflated and get highly astigmatic.
For the case 2), the waist radii are 5.0mm and 3.7mm for the horzontal and vertical, respectively.
For the case 3), they are 5.6mm and 4.1mm for the horzontal and vertical, respectively.
(Note: Nominally the waist radius is 3.1mm)
Q. What do you predict for the stability of the PRMPR2Flat_Mirror cavity?
A. It will be stable. The cavity is stable until RTT becomes smaller than 240 m.
Q. If the TT mirrors are concave, will the cavity stable?
A. Yes. Particularly if PR3 is concave.
Q. Rana mentioned the possibility that the mirrors are deformed by too tight mounting of the mirror in a ring.
Does it impact the stability of the cavity?
A. Possible. If the curvature is marginal and the mounting emphasizes the curvature, it may meet the unstable condition.
Q. How can we avoid this instability issue?
A.
1. Use flatter mirrors or at least concave mirrors.
2. Smaller incident angle to avoid emphasis of the RoC in the horizontal direction
3. Use weaker squishing force for mounting of the mirrors
4. Flip the PR3 mirror in the mounting ring by accepting the compromise that the AR surface is in the cavity.
