Laser Beam Expander Calculator (Galilean & Keplerian, magnification and divergence)

Understanding the laser beam expander calculator — concept and principle

This calculator designs a two-lens laser beam expander. From the focal lengths of the input and output lenses it gives the expansion ratio, output diameter, reduced divergence, lens spacing and design type, and compares the beam size 100 m away with and without the expander. It is used in laser alignment, material processing and free-space optics.

Making the beam 5 times wider makes it spread 5 times slower, so far away it ends up much smaller.

How it is calculated

Expansion ratio M = f₂ ÷ |f₁|

Output diameter = M × input diameter; output divergence = input divergence ÷ M

Lens separation = f₁ + f₂ (f₁ negative for a Galilean design)

Diameter after distance z = diameter + divergence × z

Focal lengths in mm, diameters in mm, divergence in mrad (full angle).

The principle

A beam expander is a telescope used backwards. The two lenses share a focal point, so a collimated beam enters and a wider collimated beam leaves. In a Galilean design the input lens is negative, so the beam never comes to a focus inside, which suits high-power lasers and keeps the device short. A Keplerian design uses two positive lenses with a real focus between them, where a pinhole can clean up the beam. Because a Gaussian beam's divergence is inversely proportional to its waist size, widening the beam M times cuts its divergence by the same factor.

Worked example

Default inputs: f₁ = −25 mm, f₂ = 125 mm, input diameter 1.0 mm, divergence 1.2 mrad.

M = 125 ÷ 25 = 5×.
Output diameter = 5 × 1.0 = 5 mm; output divergence = 1.2 ÷ 5 = 0.24 mrad.
Separation = −25 + 125 = 100 mm; f₁ is negative, so the design is Galilean (no internal focus).
After 100 m without the expander: 1 + 1.2 × 100 = 121 mm; with it: 5 + 0.24 × 100 = 29 mm.

Each point is the calculator re-run with only “Focal length of the input lens f₁ (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Focal length of the input lens f₁ (mm)	Expansion ratio (×)
-6.25	20×
-12.5	10×
-18.8	6.67×
-25	5×
-31.2	4×
-37.5	3.33×
-50	2.5×

How to read the result

An expander makes the beam bigger close to the laser but much smaller far away, because divergence drops. A larger beam can also be focused to a smaller spot by the next lens. Fine-tuning the lens spacing adjusts collimation.

Common mistakes

Using the sum of absolute focal lengths for a Galilean design; the spacing is f₂ − |f₁|.
Forgetting that the output aperture must be larger than the expanded beam.
Putting a pinhole in a Galilean expander, which has no internal focus.

Keep in mind: Expanded high-power beams can still be hazardous. Follow the laser safety rules for your laser class.

Frequently asked questions

What is the difference between Galilean and Keplerian expanders?

Galilean uses a negative input lens and is shorter with no internal focus; Keplerian uses two positive lenses and has a focus between them.

Why does an expander reduce divergence?

Divergence of a Gaussian beam is inversely proportional to the beam size.

How do I get 10× expansion?

Choose lenses with f₂ ÷ |f₁| = 10, for example −20 mm and 200 mm.

Can I use it in reverse?

Yes; reversed, it reduces beam size and increases divergence.

Sources: Melles Griot Optics Guide, Chapter 2 "Gaussian Beam Optics" (2005)

Checked 9 Nov 2026. Afocal two-lens telescope (lens spacing f₁ + f₂, magnification f₂/|f₁|): standard geometric optics. Divergence θ = λ/(πw₀), so θ falls as the waist grows: Melles Griot Optics Guide, Chapter 2 “Gaussian Beam Optics” (2005), as compiled in Michigan State University PHY 431 notes.

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