Silver ODT Calculations

I believe all of the waists are reported as the radius

Here is the beam waist for a -30, 200, 500 mm lens set up, assuming the first lens is 35 inches from the laser it is 69 micrometers, note the first divergence is 2450 micrometers (0.1 inches)

Here is the beam waist for -30, 250, 500 mm lens set up, assuming the first lens is 25 inches from the laser it is 55 micrometers, note the first divergence is 3050 micrometers (0.12 inches)

Calculating Trap Depth

The stark shift is

$u=\frac{1}{4}\alpha E^2$

We can back out the electric field from

$I=\frac{1}{2}c{ϵ}_0{E}^2$

So

$u=\frac{1}{2}\frac{\alpha 2I}{c{ϵ}_0}$

and

$I=\frac{2P}{\pi r^2}$

Thus

$u=\frac{\alpha }{c{ϵ}_0}\frac{P}{\pi r^2}$

The polarizability carries units of $4\pi {ϵ}_0{a}_0^3$ and for silver is 50. The power at the atoms will be roughly 80 Watts and r is the calculated beam waist. A better derivation is available in Optical Dipole Trap.

Here is the Mathematica calculation:

So the difference in trap depth is roughly 69 uK.

We decided to go with a 300mm lens to tighten the waist and increase the trap depth.