Jennie W, Sheila D

I have been updating my mode-matching calculations for the output chain of the interferometer to include the measurements we took where we heated up and cooled down SR3 (alog #84432), doing a single bounce measurement of the interferometer in each state.

Making a grid of possible values for the q parameter just before OM2 (8.8cm before, as this is the refernce I used when I was looking at single bounce measurements with hot OM2)we get the blue dots shown in the first image. If we assume that SR3 when heated has a curvature of 36.0087m, as calculated in alog #86184, then we can work out an ABCD matrix from just before SR3 to just before OM2 for both cases (SR3 cold and SR3 hot).

The beam parameter is a function of the distance to the waist  z - z₀ and the Rayleigh range z_R.

q_in = z - z₀ + jz_R

q_in = [ q_in

            1]

F_SR3 M q_in = q_out          when F is the transfer matrix of the SR3 in its cold state, and M is the transfer matrix after SR3 to the point at which we get the beam parameter q_out (8.8 cm before OM2).

q_in = M ^-1F_SR3 ^-1 q_out 

F_SR3,hot M q_in = q_out,hot when F_SR3,hot is the transfer matrix of the SR3 in its hot state and q_out,hot is the beam parameter 8.8cm before OM2 for this case.

If we substitute in the equation for q_in we can get an expression for q_out,hot:

q_out,hot  = F_SR3,hot M M ^-1 F_SR3 ^-1 q_out

In the first image the red dots are the possible q_out,hot values obtained from my grid of q_out values given in blue. The other colored dots are possible q values that are consistent with single bounce measurements while changing the heating of the SR3 (see alog #85988 for details) and OM2 (see alog #84255 for details).

The second plot shows arrows which start at all the grid points/measurements for the cold state, and end at the grid points/measurements when SR3 is heated.