Laser Beam Diameter which Minimizes Volume?

[LucidDragon]

Homework Statement

Consider a 10 meter long gas column. We interrogate the gas molecules with a HeNe laser (lambda=633nm) at the minimum possible gas volume. If we focus the beam tightly, it will eventually diverge and the sampled gas volume will increase. Consider a minimum beam diameter D. Derive an expression for the beam diameter D that minimizes the sampled volume.

Homework Equations

Gaussian Beam: Theta=ω/z=λ/(π*w0)
Rayleigh Range: z0=[π*(w0)^2 ] / λ
Beam Waist = 2*w0
2*w0=(4*λ*f) / (π * D)

The Attempt at a Solution

Assuming that gas column refers to a cylindrical column, then the Volume would be: V=L*π*r^2=10*π*r^2. However I'm not sure where to go from here. I tried rewriting volume as: V=10*π*w0^2, then taking the derivative and setting it to zero to minimize volume, but that leads nowhere and is likely wrong. Any help or even hints on where to start would be greatly appreciated. Thank you!

 

[mfb]

I would expect the ideal beam to converge up to the middle, then diverge again. The width at the center is a free parameter - a smaller width leads to a smaller volume close to the center but a larger volume at the edges as divergence increases. Express the volume as function of this parameter (probably via an integral) and minimize.