This elog is meant to summarize my numerical simulations for looking into the effects of curvature on the RC mirrors. I've tried to go through my reasoning (which may or may not be correct) and once this gets a bit more refined, I will put all of this into a technical note.

Motivation: 

• Both the G&H (PR2, SR2) and Laseroptik (PR3 SR3) are convex on the HR side with RoCs of approximately -600m and -700m (though as stated in the linked elog, I'm not actually sure if there are measurements of this number) EDIT AUG15: There are measurements for the Laseroptik mirrors here
  GV April 8 2017: This elog by Jenne suggests that the installed PR2 has an RoC of approximately -700m. Koji has uploaded the phase map data for the RC TT mirrors to /users/public_html/40m_phasemap/40m_TT and /users/public_html/40m_phasemap/40m_TT2. The G&H mirror data seems to be in the former folder, and it looks like there are two mirrors, one with RoC of ~ -700m and the other with RoC of ~ -500m. Does this mean PR2 has RoC -700m and SR2 has RoC -500m?
• As a result, both the PRC and SRC were close to instability
• By flipping the folding mirrors, the instability has been mitigated, but at the expense of the non-ideal situation where the AR coated side and the substrate are now inside the recycling cavity
• We would like to order some new folding mirrors. In order to avoid receiving convex mirrors from the vendor, we want to specify a concave curvature for the HR side
• The aim of this investigation is to look at how concave we should make these mirrors, because although the cavity stability improves with concavity of the HR side, possible disadvantages of having too convex mirrors are:
  • ​Mode-mismatch between the recycling cavities and the arms
  • Astigmatism

The study:

• I've built a Finesse model for the 40m, which has been used for all the numerical studies quoted here
• In constructing this Finesse model, I've used the following sources to specify various paramaters:
  • ​RoCs, R, T and physical dimensions of 4 test-masses, PRM, SRM and BS: Core optics wiki page
  • Losses - arm losses from Yutaro's measurements in elog11857 and elog11818 (distributed equally between ITM and ETM). For other optics, a generic value of 25ppm was used
  • "Ideal" lengths for our current modulation frequency were used for the various cavities (37.795m for the arms, 6.753m for PRC, 5.399 for SRC)
  • The folding mirrors (PR2, PR3, SR2, SR3) are initialized as flat in the model
• I performed some low-level checks (e.g. arm linewidth, PRC FSR etc) to check that the model was sensible
• I then proceeded to investigate the effects of curvature on the folding mirrors. Specifically, I investigated the following:
  • What is the mode mismatch between the recycling cavity mode and the arm as a function of the RoC of the folding mirror?
  • What is the effect of the RoC of the folding mirrors on the round-trip gouy phase accumulated (and hence the transverse mode spacing) in the recycling cavities?
• For now, the parameter space explored is from 300m concave to 1000m concave. An RoC of 1km for a 2" optic corresponds to a sag of ~0.3 microns. I will explore the 1km-10km concave space and update the results shortly

Results:

• Attachments #1 and #2 show the mode mismatch between the recycling cavity and the arm for various curvatures. The colorbars have been normalized to span the same range in all the plots
• For both the PRC and the SRC, if we have folding mirrors with an RoC of 1000m concave, we will have a mode mismatch of 2-3%. The number gets worse the more convex the mirror
• Attachments #3 and #4 show the one-way accumulated Gouy phase. Here, I have varied the curvature of the folding mirrors along a specific axis at a time (i.e. I've assumed that the folding mirrors are identical). I've also added the transverse mode spacing as a second y-axis. I have yet to check how these numbers compare with the linewidth of the 00-mode for the various fields, but for 1km concave folding mirrors, the TMS is in the region of 2MHz 

To do:

• I will extend the range of RoCs explored to 10km concave and post results - but I will have to check with EricG to make sure that it is feasible for us to specify curvatures in this range
• I was trying to use the RT gouy phase as calcluated by my Finesse simulations to plug into some analytical expressions to try and generate plots like this for various RoCs of the folding mirrors, but if the TMS calculations suffice, I will abandon these efforts
• What are the other specifications we need to worry about before placing an order? Some thoughts from Rana's earlier elog:
  • The coatings need to be dichroic to allow extraction of the green beam (but only PR3/SR3 is currently dichroic?)
  • Wedge angle on the AR side?
• Are there any other obvious sanity checks I should carry out?