Laser Beam Expander Calculator

Background.

A laser leaves its cavity as a narrow pencil, often only a millimetre or two across, and for many jobs that is too small. Focusing it to a tight spot, sending it a long way without spreading, or filling the aperture of a microscope objective all call for a wider beam. A beam expander does this with two lenses spaced so that a parallel beam goes in and a wider parallel beam comes out — a layout optical designers call afocal.

The enlargement is set entirely by the ratio of focal lengths. If the second lens has four times the focal length of the first, the beam comes out four times as wide. That ratio has a useful mirror image: the beam's divergence drops by the same factor, so a 4× expander turns a beam spreading at 1 milliradian into one spreading at 0.25. A wider beam is also gentler on the optics that follow, since the same power is spread over 16 times the area.

There are two classic layouts. A Keplerian expander uses two positive lenses separated by the sum of their focal lengths and brings the beam to a focus between them; a pinhole placed there can clean up the beam profile, but with a powerful pulsed laser that focus can ionise the air. A Galilean expander puts a negative lens first, has no internal focus and is shorter, the separation being the difference of the focal lengths.

Both follow the same magnification rule, so the page asks only for the input diameter and the two focal-length magnitudes. It is an ideal thin-lens model: it does not work out the spacing, lens sizes, aberrations or how a real Gaussian beam's waist moves through the system.

What is laser beam expander calculator?

A laser beam expander is an afocal optical system, usually two lenses, that takes a collimated input beam and returns a collimated output beam of larger diameter. Its expansion ratio, or magnification, is the focal length of the output lens divided by that of the input lens, M = |f₂| ÷ |f₁|, and the output diameter is the input diameter multiplied by M. Sending light through the same pair in reverse, longer-focus lens first, shrinks a beam by the same ratio.

How to use this calculator.

1. 01Enter the input beam diameter in millimetres, using the same definition as your laser's data sheet — for a Gaussian beam that is normally the 1/e² diameter.
2. 02Enter the focal lengths of the lens the beam meets first and of the output lens, in millimetres, as positive numbers. For a Galilean design with a negative first lens, leave off the minus sign.
3. 03Read the expansion ratio and the output diameter, then check that the output lens's clear aperture is comfortably larger than that diameter so the edges of the beam are not clipped.
4. 04Set the spacing yourself — f₁ + f₂ for two positive lenses, f₂ − |f₁| when the first lens is negative — and fine-tune it until the beam stays the same width near and far.

The formula.

M = |f₂| ÷ |f₁| and D_out = M × D_in. The two lenses are arranged to share a focal point. A ray entering parallel to the axis at height h is bent by the first lens towards that point; the second lens, one focal length f₂ beyond it, meets the ray at height h × f₂ ÷ f₁ and sends it out parallel again. Every ray height is scaled by the same ratio, so the whole beam is. Only magnitudes enter, because the sign of f₁ decides the layout — Keplerian with a positive first lens, Galilean with a negative one — and not the size of the output. The model is paraxial: thin, aberration-free lenses, rays close to the axis, and a perfectly collimated input. In practice, input divergence, lens aberrations and spacing errors appear as a slight residual focusing or spreading of the output beam.

A worked example.

Frequently asked questions.

References& sources.

1. [1]ISO 11145:2018, Optics and photonics — Lasers and laser-related equipment — Vocabulary (BIBLIOGRAPHIC/PAYWALL).
2. [2]RP Photonics Encyclopedia, Beam Expanders.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.