Some requirements for the PMC:

==Cavity pole==

 For intensity filtering. The modulation frequencies for the refcavs is ~ 15-25 MHz, we want the intensity fluctuation at this frequency to be shot noise limited.  We have to determine what should be the frequency pole. Intensity noise around 1MHz - 30MHz will be ~ 1/f^2, see the paper by Harb etal, eq1 and fig9, get the paper from psl:1156. Under the assumption that RIN remains constant, at 20MHz the laser will already by shot noise limited (@ 1mW input).  laser intensity noise / shot noise ~ 0.16. (laser intensity noise here means intensity noise from spontaneous emission/ pump-source intensity noise/ dipole fluctuation noise/ noise from intra cavity losses, any thing except shot noise)

  Thie pole can change with the cavity length and Finesse, [ Finesse = FSR/(2*cavity Pole)] , so our choices for mirror reflectivity, cavity length will affect this number as well. So for a fixed set of mirrors (fixed finesse), longer perimeter means lower cavity pole, but the cavity will be more susceptible to acoustic coupling.

==First longitudinal body mode==

  It should be at high frequency ( for high UGF servo). The shorter the length, the higher the frequency. See PSL:1134.

== g-factor==

 For a stable cavity, g factor has to be between 0 and 1.  Another reason: We should choose g-factor such that HOMs do not coincide with other cavity axial modes (FSR apart). For a ring cavity with 2 curve mirror R1,and R2, g = (1- p/R1) x (1 - p/R2) where p is the round trip length. (For 3-mirror cavity, g = (1 - p/(R))^2 . See HOM calculation.

==Stiffness==

 we want a solid, bulk shape PMC, not thin long one. This will make the PMC less susceptible to acoustic noise.

==Higher order mode suppression== 

Other transverse modes will be suppressed by a factor of (1-r)^2 / (1 +r^2 -2rcos(2*pi* df_mn/ FSR)  where df_mn is the gouy phase shift of m+n mode, r =r1*r2*r3.. (reflectivity of each mirror in the cavity) see evan's note. Transverse modes of the output of the NPRO can be found by scanning the PMC and measure the transmitted beam. Other modes beside TEM00, will be reflected back from the refcav and incident on the RFPD. This will cause the mode mismatch and increase shot noise level. Usually, higher r (higher Finesse), will suppress more HOMs.

==Build up power==:

= Pin x Finesse/ pi. CVI mirrorsfor high damage threshold power have maximum power for cw around 10MW/cm². So I use this number as an upper limit for the power threshold. Assuming the power input is ~ 30 mW, average spotsize is 350 um. This gives ~ 8W/cm². So Finesse can be up to ~ 3e6.  (10 MW/cm² > (Finesse/pi) x 8 W/cm²) .

Some assumptions:

• Losses(scatter/absorption) on each mirror is ~ 100 ppm. It seems that a super polished mirrors in vacuum has ~ 10 ppm loss. This comes from a Finesse measurement of the previous 8" refcavs, see psl:1046. The calculation shows that loss in one cavity is 25 ppm (for 2 mirrors), and 160ppm for another cavity. Since the PMC mirrors will be in air, and probably not as good as refcav mirrors, dust in air might accumulate over time and causes extra loss on the mirrors, 100 ppm loss assumption might be ok for this calculation.
• PZT range is about 15um @1000V, as shown in the catalog, see PSL 1052 for the details, (we can drive it with ~0-300 V, so ~ 4um displacement),see PSL:1052

Let's see some of the designs that are available. Then we can decide which one we should modify to suit our requirement.

1. Design1 iLIGO PMC: Isosceles triangular PMC, fused silica, perimeter = 0.42m, flat-curve (1m ROC)-flat mirrors. Round Trip = 0.42m See T-080195,here (it says the pole is 7 MHz).
2. Design2 (Dmass' PMC): stainless steel PMC, perimeter =0.4m , same mirrors as those of design1, so its finesse is the same.
3. Design3, AdvLIGO PMC style (4 mirrors, bow-tie): stainless steel (see PSL:)