The cavity modes , where q is the complex beam parameter and m,n is the mode index, are the eigenmodes of the cavity propagator. That is:
where is the mirror reflection matrix. At the 40m, ITM is flat, so . ETM is curved, so , where R is the ETM's radius of curvature.
is the Gouy phase.
is the free-space field propagator. When acting on a state it propagates the field a distance L.
The phase maps perturb the reflection matrices slightly so:
Where h_12 are the height profiles of the ITM and ETM respectively. The new propagator is
, where is the unperturbed propagator and
To find the perturbed ground state mode we use first-order perturbation theory. The new ground state is then
Where N is the normalization factor. The (0,1) and (1,0) modes are omitted because they can be zeroed by tilting the mirrors. Gouy phase of TEM00 mode is taken to be 0.
Some simplification can be made here:
The last step is possible since the beam parameter q matches the cavity.
The loss of the TEM00 mode is then: