aLIGO LHO Logbook

H1 ISC

nicolas.smith@LIGO.ORG - posted 17:58, Wednesday 10 September 2014 (13869)

Simple analytic model of PRC recycling gain in the presence of differential ITM lenses

Lisa asked me to think about the problem of how the radius of curvature error of the ITMs affects the recycling gain.

I’ve made a very simple modal analysis of a power recycling cavity with mismatched ITMs. To calculate the scattering from the 00 mode to the bull’s eye (mode mismatch) mode, I used Modal Model Update 4.

The scattering matrix element from the 00 mode to the bull’s eye is:

$langle B | D | 00 angle = i frac{kw^2Delta R}{2R^2}$

where $k$ is the wavenumber, $w$ is the beam width radius, $\Delta R$ is the ITM radius error, and $R$ is the ITM radius.

So computing the resonant mode in the cavity, including the scattering to second order, I calculate the amplitude recycling gain (assuming only a differential error in the ITM lenses) to be:

$frac{t_p}{1-r_pr_i(1-frac{k^2w^4Delta R^2}{32 R^4})}$

where $t_p$ and $r_p$ are the PRM amplitude transmission and reflection coefficients, and $r_i$ is the ITM amplitude reflection coefficient. The expression reduces to the standard FP cavity formula for $\Delta R=0$ .

The full mathematica notebook includes losses, and a common ITM radius error, but those effects are small.

Plugging in numbers from Lisa, I get the recycling gain with no lens to be 61 and with the ITM lenses it’s 57.

Non-image files attached to this report

prcmismatch.nb