Beam expander

A beam expander is an afocal telescope used backwards: two lenses whose focal points coincide, so that parallel light in gives parallel light out, with the beam diameter multiplied by the ratio of focal lengths, $M = f_2/f_1$. Because the beam parameter product is conserved, the divergence falls by the same factor, and the Rayleigh range grows as $M^2$. Used in reverse, the same optic reduces a beam.

Keplerian and Galilean designs

A Keplerian expander uses two positive lenses separated by $f_1 + f_2$. The beam comes to a focus between them and the output is inverted. That internal focus is useful, because a pinhole placed there turns the expander into a spatial filter, and harmful at high power, because air can break down at the focus. A Galilean expander uses a negative lens followed by a positive one, separated by $f_2 - |f_1|$; there is no real focus and the assembly is shorter. For a 10× expander with 20 mm and 200 mm focal lengths, the Keplerian version is 220 mm long and the Galilean 180 mm. High-power laser expanders are usually Galilean, and many adjust the lens spacing to set the output collimation, or to compensate the input beam's own divergence.

Why expand

Expanding a helium-neon beam with a 0.4 mm waist radius by a factor of 10 lowers the half-angle divergence from 0.50 mrad to 0.050 mrad and raises the Rayleigh range from 0.79 m to 79 m, which matters for alignment over long paths and for free-space links. The other use is focusing: a lens of focal length $f$ focuses a collimated Gaussian beam of radius $w$ to a spot of radius about $\lambda f/(\pi w)$, so enlarging the beam before a focusing lens shrinks the spot in proportion, which is why laser machining heads and scanning systems expand the beam before the objective. The limit is the aperture of the following optics, and the growth of aberrations with beam size.

Setting up

The lens spacing is adjusted until the output beam neither converges nor diverges, checked by measuring its size near the expander and several metres away or with a shear-plate interferometer. The input beam should be centred on both lenses, since an off-axis beam picks up coma and astigmatism. Wavefront distortion from the expander is specified as a peak-to-valley or rms wavefront error.

References: W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008), Ch. 9; A. E. Siegman, Lasers (University Science Books, 1986), Ch. 17.