What optical resonator configuration produces the most output power for a laser?

For a symmetric cavity (mode waist is largest for a confocal cavity, $R=L$. Mode volume then increases with length. But my guess is that you want the photons to spend more time in the gain medium so, unless you extend the gain medium, my guess is you don’t win for increasing length.

So that is the gain medium should entirely fill the cavity. You probably do better with more uniform mode weight so confocal is the best there again.

Id guess you get more power with a bigger gain medium but I’m sure something g limits that at some point. I’m not sure what.

I’ll post references for how to calculate cavity properties from mirror geometries and properties later.

Disclaimer: as you can see this is a lot of guesses. I’m not a laser engineer, but I do have experience with optical cavities.

*I think if you get the laser up to high enough power that you can saturate the atomic transitions even with a fraction of the circulating cavity light then second constraint will stop being relevant. This condition would be reached if the losses with the cavity round trip become small compared to the gain from the gain medium

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edit: With input from the OP I can give some more info now. The OP tells us that the geometry of the gas tube is fixed with radius $R_0$ and length $L_0$.

My hypothesis is that you will get the most power (as a function of cavity geometry) when the geometry of the cavity mode closely matches that of the gas tube. That is, 1) you want the light to be hitting as many He and Ne atoms as possible but 2) you don't want the light spending time where there aren't He and Ne atoms because then you won't be saturating the atomic transitions as strongly as possible.*

The formula for the waist of the cavity mode, $w_0$, and the waists on the mirror surfaces, $w_{text{mir}}$ in terms of mirror radius $R_1=R_2=R$ and $L$, are given by

begin{align} w_0^2 =& frac{lambda}{2pi} sqrt{2R-L}sqrt{L} w_{text{mir}} =& w_0 sqrt{frac{2R}{2R-L}} end{align}

We want a roughly cylindrical beam since the gas tube is cylindrical. This means, from the second equation, that we want $Rgg L$ so that $w_{text{mir}} approx w_0$. In this case the first equation can be approximated as

$$ w_0^2 approx frac{lambda}{2pi}sqrt{2RL} $$

From this you could estimate an optimarl $R$ such that $w_0 approx R_0$ to make sure you use the whole gas cylinder but don't waste power outside the cylinder.

The equations above are either given directly or can be derived from equations in Siegmann's Lasers textbook.