I took the phase maps of the 40m X arm mirrors and calculated what is the loss of a gaussian beam due to a single bounce. I did it by simply calculating 1 - (overlap integral)^2 where the overlap is between an input Gaussian mode (calculated from the parameters of the cavity. Waist ~ 3.1mm) and the reflected beam (Gaussian imprinted with the phase map). The phase maps were prepared using PyKat surfacemap class to remove a flat surface, spherical surface, centering, etc. (Attachments 3, 4)

 

I calculated the loss map (Attachments 1,2: ~ 4X4 mm for ITM, 3X3mm for ETM) by shifting the beam around the phase map. It can be seen that there is a great variation in the loss: some areas are < 10 ppm some are > 80 ppm.

 

For the ITM (where the beam waist is) the average loss is ~ 23ppm and for the ETM its ~ 61ppm due to the enlarged beam. The ETM case is less physical because it takes a pure gaussian as an input where in reality the beam first interacts with the ITM.

 

I plan to do some first-order perturbation theory to include the cavity effects. I expect that the losses will be slightly lower due to HOMs not being completely lost, but who knows.