Formula and Method for the Laser Beam Expander Calculator
A laser beam expander is a two-lens afocal (telescope-like) optical system that increases the diameter of a collimated input laser beam by a magnification factor M, while reducing the beam's angular divergence by that same factor. It is used to lower power density on downstream optics, shrink a laser's far-field spot at long range, or match a beam to the clear aperture of a telescope, scanner, or lens system. This calculator uses the standard thin-lens beam-expander relations M = D_out / D_in = f2 / f1, f2 = M × f1, and θ_out = θ_in / M, with lens spacing depending on whether the design is Galilean or Keplerian.
How the calculation works
Enter the input and output beam diameters — their ratio sets the required magnification, M = D_out / D_in. Enter the focal length of the first lens (f1); the calculator finds the focal length the second lens must have, f2 = M × f1, so both lenses share a common focal point and the system stays afocal (collimated in, collimated out). The lens separation follows from that shared focal point: L = f2 − f1 for a Galilean design (diverging lens 1, converging lens 2), or L = f1 + f2 for a Keplerian design (two converging lenses). Finally, because expanding a beam's diameter by M shrinks its divergence by the same factor — étendue is conserved by an ideal afocal system — the output full-angle divergence is θ_out = θ_in / M.
Galilean vs. Keplerian design
In a Galilean beam expander, lens 1 is diverging (negative focal length) and lens 2 is converging (positive focal length); the beam never comes to a real focus inside the housing, which keeps the assembly compact (L = f2 − f1) and avoids the intense focal spot that can ionize air or damage optics at high pulse energies. In a Keplerian beam expander, both lenses are converging and the beam passes through a real internal focus (L = f1 + f2); that design is longer overall, but the internal focus is a point where a pinhole spatial filter can remove diffraction rings and clean up the beam's intensity profile — a common step in high-quality laser systems.
Practical limits and common mistakes
This calculator uses paraxial (thin-lens) ray optics, which assumes small angles and lenses that are thin compared with their focal lengths — real multi-element expander lenses need slightly different spacing to correct for thickness and aberrations. Real laser beams are Gaussian, not uniform-diameter rays, so "diameter" should be measured the same way (commonly the 1/e² diameter) on both the input and output sides. Also remember that θ_out = θ_in / M is the theoretical, diffraction-limited reduction in divergence — a misaligned or aberrated expander can add divergence back, so treat the calculated value as a best-case target to verify with a beam profiler.