How it is calculated
Two lenses separated by the sum of their focal lengths form an afocal telescope: collimated light in, collimated light out. With \(f_1\) the input lens and \(f_2\) the output lens (signed, positive for converging),
A Galilean expander pairs a negative input lens with a positive output lens. \(M\) is positive (upright beam) and the tube is shorter than \(f_2\). A Keplerian expander uses two positive lenses. \(M\) is negative (the beam is inverted) and there is a real focus between the lenses.
The product of diameter and divergence is conserved, so expansion does not change beam quality. It only redistributes it. If you leave the input divergence blank, the calculator uses the diffraction-limited value \(\Theta = 4M^2\lambda/(\pi D_\text{in})\). If you enter a measured divergence, it reports the implied M² so you can check the datasheet for consistency.
Effect on the focused spot
A focusing lens maps the far field to its focal plane, so the spot diameter is the focal length times the full divergence of the beam entering the lens:
A 3× expander therefore gives a spot one-third the size, nine times the peak fluence, and one-ninth the depth of focus (\(z_R \propto d_F^2\)). Before buying a shorter focal length lens, check whether the expanded beam still fits through the scanner and focusing optics. The focused spot tool includes aperture truncation.
Galilean or Keplerian?
- Galilean has no internal focus, so it is preferred for pulsed lasers with high peak power: there is no risk of air breakdown or of damaging an optic placed near the focus. It is also shorter and has fewer components.
- Keplerian has an accessible internal focus. A pinhole placed there acts as a spatial filter that strips high-spatial-frequency noise from the beam. The spot is about \(f_1\Theta_\text{in}\), and a pinhole of roughly 1.5 times the 1/e² focus diameter passes about 99 % of a TEM₀₀ beam. At high peak power the internal focus can ionize air, so evacuate the filter or use a Galilean design instead.
Worked example
A 3 mm laser beam with a 1.2 mrad full-angle divergence (implied M² ≈ 2.66 at 1064 nm) passes through a Galilean expander with \(f_1 = -50\) mm and \(f_2 = 150\) mm:
- \(|M| = 3\) and \(L = 100\) mm. The output is 9 mm in diameter with 0.40 mrad divergence.
- With a 100 mm focusing lens the spot shrinks from \(100\,\text{mm}\times 1.2\,\text{mrad} = 120\) µm to 40 µm. The Rayleigh range at focus becomes 0.44 mm.
- A Keplerian version (\(f_1 = +50\) mm) needs \(L = 200\) mm and has a 60 µm internal focus. A 90 µm pinhole would make a suitable spatial filter.
Assumptions and limits
- Thin, aberration-free lenses. Real expanders have a lens spacing that differs slightly from \(f_1 + f_2\) because of principal-plane positions, and they are usually adjustable to set collimation. Use the vendor's mechanical data.
- Collimated input. If the input beam's waist is far from the expander, the output waist moves too. The divergence ratio still holds, but the location of the minimum spot does not. See Gaussian beam propagation.
- No clipping. Both lenses and every downstream aperture must be at least about 1.5× the local beam diameter. See beam clipping.
- Focused-spot estimate assumes an untruncated beam with no lens aberrations, and the focal plane close to the beam waist.
References
- A. E. Siegman, Lasers, University Science Books (1986), ch. 17–20.
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019), ch. 3.
- W. J. Smith, Modern Optical Engineering, 4th ed., McGraw-Hill (2008) — afocal systems.