How this instrument works
A beam expander is an afocal pair of lenses set apart by the sum of their focal lengths, so the system carries no net focusing power: a collimated beam enters travelling parallel and leaves travelling parallel, just at a different diameter. The scale factor is the magnification M, and it depends on nothing but the two focal lengths — the longer one divided by the shorter one, M = f2 ⁄ f1. Put a short lens first and a long one second and every ray bundle that crosses the system comes out wider by exactly that ratio, whatever diameter it started at.
That ratio isn't a rule of thumb — it falls straight out of the Lagrange invariant that governs every paraxial optical system: at any plane, beam radius multiplied by local ray angle stays constant as light crosses lens after lens. Because that product can't change, widening the beam by a factor of M forces the divergence half-angle to shrink by the same factor. One formula therefore fixes two things at once, diameter and spread, which is why f2/f1 turns up as often in laser range-finding as it does in projector optics.
Two families share this formula and differ only in layout. A Keplerian expander uses two converging lenses and forms a real focus midway between them, convenient for parking a pinhole there to spatially filter the beam — but that focus packs the laser's full power into a point of air, and a strong pulsed source can ionize it there. Galilean expanders swap the input lens for a diverging one, keep no internal focus at all, and are what laser-cutting and marking heads use almost exclusively past modest power. Range-finding engineers reach for the same ratio from the other direction: expanding a 10 mm diode beam tenfold narrows its divergence tenfold too, often the difference between a usable return at 200 metres and none at all.
- Enter Input (objective) lens focal length — the shorter lens the beam meets first, typically given in millimetres.
- Enter Output (eyepiece) lens focal length — the longer lens the widened beam exits through.
- Read Beam expansion magnification: the output beam diameter is this many times the input diameter.
- Keep f2 larger than f1 for an expander; reverse the order and the same formula gives a beam reducer instead.
Worked example — a 10x expander from 10 mm and 100 mm lenses
Take a diode-laser bench that needs its raw 2 mm beam both widened and straightened before it reaches a distant target. Enter 10 mm — 0.01 m — into Input (objective) lens focal length and 100 mm — 0.1 m — into Output (eyepiece) lens focal length. The instrument divides: M = f2 ⁄ f1 = 0.1 ⁄ 0.01 = 10. Beam expansion magnification reads 10, meaning the beam leaves ten times as wide as it entered — a 2 mm input grows to a 20 mm output.
That factor of ten does double duty. Because beam diameter and divergence trade off exactly through this system, the same 10x also divides the beam's divergence angle by ten — a raw diode divergence of 2 milliradians leaves the expander at 0.2 milliradians. Over a 200 m range that difference is the gap between a spot that has grown to 40 cm and one that has barely spread past 4 cm, precisely why range-finding and lidar heads carry an expander sized by this same ratio.
Questions
What's the difference between a Galilean and a Keplerian beam expander?
A Keplerian design uses two converging lenses and forms a real focus between them, useful for spatial filtering with a pinhole but risky at high power because that focus concentrates the beam into a point of air or glass. A Galilean design replaces the input lens with a diverging one, has no internal focus at all, and is the standard choice for laser cutting, marking, and other high-power continuous or pulsed systems. Both obey the same M = f2 ⁄ f1 ratio; only the sign of f1 and the physical layout differ.
Why does expanding the beam also reduce its divergence?
Because beam radius times ray angle is conserved through any lossless paraxial system — the Lagrange invariant. Multiply the diameter by M and the divergence half-angle must divide by the same M, since their product can't change. A 10x expander that turns a 2 mm beam into 20 mm also turns 2 milliradians of divergence into 0.2 milliradians, which is the entire reason expanders exist: keeping a laser beam usably tight over distance.
Can the magnification be less than 1?
Yes — set f1 larger than f2 and M = f2 ⁄ f1 comes out below 1, turning the same two-lens system into a beam reducer rather than an expander. That's how a telescope run in reverse behaves: light entering the eyepiece side and leaving through the objective shrinks in diameter while its divergence grows, the mirror image of the expanding case.
Does swapping the two lenses change the answer?
Yes, and it's the most common mistake with this formula. Feeding the long lens in as f1 and the short one as f2 inverts the ratio — a 100 mm and 10 mm pair meant to give 10x instead reports M = 0.1, a tenfold reduction. Input (objective) lens focal length must be the lens the raw beam meets first; matching field to physical position, not just plugging in two numbers, is what keeps the result correct.
How do I size an expander for a focusing lens or aperture?
Match the expanded beam diameter to the aperture you're filling, since a diffraction-limited focused spot scales with wavelength times focal length divided by beam diameter — a wider beam at the focusing lens gives a smaller, tighter spot. Divide the focusing lens's clear aperture by your raw beam diameter to get the magnification you need, then pick f1 and f2 whose ratio matches it; a 5 mm raw beam filling a 50 mm aperture calls for M = 10, the same ratio this instrument's worked example uses.
What separation do the two lenses need?
For a Keplerian pair, the lenses sit apart by f1 + f2, so the output lens's front focal point coincides with the input lens's back focal point and no net power remains in the system. A Galilean pair, whose input lens is diverging, separates instead by f2 minus the size of f1, always shorter than the Keplerian equivalent for the same magnification — one reason Galilean expanders are also the more compact of the two.