Thermal lensing formula:
from (T090018 by A. Abramovici (which references another doc).
In the above equation:
w 1/e^2 beam radius
k thermal conductivity (not the wave vector) = 1.3 W / m/ K
alpha absorption coefficient (~10 ppm/cm for our glass)
NP power in the glass (alpha*NP = absorbed power)
dn/dT index of refraction change per deg (12 ppm/K)
d mirror thickness (25 mm for all of our SOS)
I'm attaching a plot showing the focal length as a function of recycling cavity power for both our current MOS and future SOS designs.
I've assumed a 10 ppm/cm absorption here. It may actually be less for our current ITMs which are made of Heraeus low absorption glass - our new ITMs are Corning 7980-A (measured to have an absorption of 13 ppm/cm ala the iLIGO COC FDD). I expect that our thermal lens focal length will always be longer than 1 km and so I guess this isn't an issue.