Minimising losses in an optical resonant cavity

Question

I am doing a question and I am asked to calculate the radius of curvature of two identical mirrors at either end of an optical cavity in order to minimise the losses in the cavity. Is the correct approach here to use gaussian optics and minimise the spot size of the light at one of the mirrors? I did this and got an answer that R=2L, where R is the radius of curvature and L is the length of the laser cavity. Does this sound feasible?

Answer 1

It's a bit of a strange question but one that I did encounter in my PhD work when I was trying to make as high finesse (low losses) cavity as possible. First off, you are asking about diffraction losses. That is, light lost because the cavity diffracts from the waist inside the cavity and and expands so that some of it falls off the edges of the mirrors. For most cavities diffraction losses are entirely negligible. These losses only matter if you have very very small cavity mirrors or are operating near concentric where the beam becomes large on the cavity mirrors.

Typically losses in a cavity are dominated by engineered transmission through one or more of the cavity mirrors, losses in some gain medium or other optics within the cavity, or scattering losses at the mirror surfaces. If you want as low losses as possible you won't put any optics in the cavity and you will design the coatings to be as low transmission as possible. It is possible to engineer losses as about 10 ppm without too much difficulty and as low as 2 ppm or perhaps 1 ppm with more work. In the visible/IR you require surface roughness at the 0.1 nm level to realize scattering losses at the 1 ppm level. This means that the lowest loss cavity can have losses around the 1 ppm level.

One way to estimate diffraction losses is to take a Gaussian beam with spot size $w$ on the cavity mirror with raidus $R_{\text{mir}}$ and determine how much of the power for the Gaussian beam is NOT contained within a circle of radius $R_{\text{mir}}$. Rough memory tells me that as long as $R_{\text{mir}} >= 3w$ the power losses from light "falling off the mirror" will be at the ppm level. However, this is pretty easy to arrange by just making the cavity mirrors larger.

Actually your question seems a little bit incomplete. The question only specifies $R$ and $L$, but the answer depends also on $R_{\text{mir}}$, the physical radius of the mirror. I guess we know that $R_{\text{mir}} <= R$. $R=2L$ corresponds to a confocal cavity configuration, a configuration well known for lots of different types of stability and general design "niceness". So your suggestion that $R=2L$ gives you the most favorable ratio for $w/R_{\text{mir}}$ doesn't sound too surprising to me, though I haven't explicitly confirmed it.