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.
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!