ZnSe lenses for CO2 laser: plano-convex vs convexo-concave

Some time ago I've bought CO2 laser engraver from China. It had lots of issues, including bad lens, which supplier had to replace. I've chosen 102mm replacement lens that time. A month ago I noticed that price of 20mm ZnSe lenses on ebay dropped to 15$, so I ordered more suitable lens for my engraver with 51mm focal length. When it arrived I was pleased with quality of edges and coating, but was surprised to see that it is an convexo-concave lens instead of usual plano-convex type.

On the left - first bad lens, on the right - replacement sent by supplier and in the middle - this new one.

Is there any practical difference between plano-convex and convexo-concave lenses for CO2 laser focusing? Also, which way one should mount it? Let's try to figure it out...

Plano-convex lens, 51mm focal distance, 20mm beam diameter. Spot is almost the same as airy disk (i.e. close to be diffraction limited):

Convexo-concave. Spot diameter is ~1.4x smaller. Without diffraction - energy density would have been almost twice higher, but now we might only get +10..+20% extra energy density at the center of the spot.

If we mount lens the other way - spot gets really big due to spherical aberration. So the right way is convex side closer to the laser.

So, convexo-concave lens starts having benefits over plano-convex at >20mm beam diameter (51mm FL) and >10mm (25.4mm FL). One can only get such big beam diameter with beam expanders. When working with usual CO2 tubes without beam expanders (beam diameter 3-5mm) - there is literally no difference between the lenses, it is diffraction limited for both.

Spot size also gets lower by ~30% for off-axis illumination (for example, when XY mirrors are slightly misaligned), but again, system remains diffraction limited until we are using very thick beams.

Resume: When working with thick beams (>20mm beam diameter for 51mm FL and >10mm for 25.4mm FL) convexo-concave lens could deliver twice higher energy density in the spot over plano-convex one. Usual CO2 cutters/engravers without beam expanders will not see any benefit as both lenses will be difraction limited and will have the same spot sizes.

Focusing lens should be installed convex side closer to the laser. But for thin beams (3-5mm) "wrong" way is still good enough to be diffraction limited, only focal length will change by ~1.15ΠΌΠΌ.

PS. This is relevant to CO2 lasers only! Solid state/fiber lasers have ~10 times smaller airy diameter and all the math will be totally different.
November 15, 2014