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That's darn close to what I said. With typical beams diameters of hand held lasers a beam diameter of 10mm is close to what you'll have using a 10x expander. And a reduction of divergence about 0.12 mrd. That will put more light into a given area when compared to a 3 or 2x expander.
Using commonly available lenses with focal lengths of -6mm and 60 for a 10x expander and
-6mm and 18 for a 3x expander one can see that the Rayleigh length (RL) is greater with the 10x expander.
10x expander
0.12 mrd
RL 83337.648 mm or 273.41748 feet
10mm exit beam dia
3x expander
0.40mrd
RL 7500.348 mm or 24.607441 feet
3mm Exit beam dia.
So what this means is this. The RL is the length over which the beam will expand the square root of 2 or 1.4 times it's initial diameter. After that distance the beam expands in a linear fashion. The 10 x expander will out perform a 3x expander once past the rayleigh length of the 3x expander
I see what you're saying here. Thanks for laying it out in mathematical terms..