CASE 2:
Determining the optimum focal length when lens input beam diameter is fixed.
If it’s impossible or undesirable to alter the system’s beam diameter, then knowing what focal length to use to produce a minimum spot size is beneficial.
To determine the focal length which will provide minimum spot size, we again take the equation for spot size, this time differentiating it with respect to focal length, and then setting it equal to zero to find the minimum value. This yields the following equation:

Optimum EFL 
Once the optimum focal length is chosen, choose the stock lens with the focal length closest to the optimum value, or for more critical applications, IIVI can readily fabricate an optic to the exact focal length and tolerances required.
As seen from the preceding discussion, there is a limit on the focus spot size which can be achieved when either focal length or input beam diameter is constrained. If the minimum spot size from the calculation is larger than required for the application at hand, then there is no choice but to change some optical system parameters.
NOTE
With higher power CO2 lasers, it is not generally advisable to use a lens with a diameter greater than 1.5 times the beam diameter (1/e2). Ratios greater than this increase the chance of inducing thermal distortions in the lens. This is caused by too great a thermal gradient across the optic as a result of the greater distance between the heated central beam region and the cooler edge of the lens. 