I sent this to Tara in an email, but I thought I'd include it here for posterity:
So if you compare the low frequency and high frequency equations in the Cole paper, they're different by a factor of:
sqrt(pi)*r_G/r_T,
where r_G is the radius of the beam spot (r_G = w/sqrt(2)), and r_T is the thermal diffusion length (r_T = sqrt(kappa/(2*pi*C*f)).
Plus, if you look at the definition of low and high frequency:
w^{^2}*C*pi*f/kappa,
that is equal to (r_G/r_T)^{^2}. After giving the low and high frequency thermo-optic equations, the cole paper cites Matt Evan's paper and a Braginski paper from 2000. In the conclusion to the Braginski paper, they mention that when the frequency is high, or the spot size is low, defined as r_G<r_T or r_G/r_T < 1, the adiabatic assumption that they use breaks down. Then, in Equation 9 of the Braginski paper, they indicate that the breakdown results in an error on the order of r_T/r_G.
Going back to the Cole paper, it appears as though for the high frequencies, they've just adjusted the low frequency equation by the adiabatic breakdown error. What I still don't understand is where the extra factor of sqrt(pi) came from, and why it's the inverse of the adiabatic breakdown error. Some of it might be typo. I'll check with Garrett to see what he has to say about it. |