Like most engineers and scientists who were educated before PC’s were readily available, (in my case 1984 so only just before), I rely on a good “feel for how the numbers work”. In most cases I have pretty good results, not far wrong from what a rigorous numerical analysis produces. One area it’s very easy to come unstuck though is Gaussian Beam Propagation, or for the layman the size and properties of a laser beam as it propagates through space with or without optical elements such as lenses, mirrors, windows etc.
Predicting beam diameters, waist locations, and depth of field by some hand waving, for me at least, is not so easy. If there is one application prone to catching people out, it’s those using small sealed CO2 lasers, of which the best known are made by Synrad. These laser types have very small beam diameters ( say 3 – 4mm) ignoring the subtlelty of 1/e2 beam diameters, and combined with large beam divergences, say 4 – 5 mrad.
Don’t assume large beam divergences equal poor quality, these lasers can have M2 values of very close to 1, i.e perfection or “diffraction limited” in the jargon. A beam expander can be used to give a more traditional beam say 15mm diameter and perhaps just 1mrad of divergence.
I use an old spreadsheet routine, when I say old it runs on DOS, and it shows the waist location, waist size, Rayleigh range, wavefront curvature and all the things you need to know to design a laser beam delivery system. The little graphic in this blog post shows beam diameter vs distance from laser for an industrial CO2 laser and a 1.5X expander. I’ve found over the years excellent agreement between theory and practice, and I have a lot of successful beam expander designs under my belt. Talk to us at firstname.lastname@example.org if you have a CO2 laser beam delivery issue. There is some brief information on copper mirrors used in our reflective beam expanders our web site