Laser Beam Expander Calculator

Laser Beam Expander Calculator that finds magnifying power, output diameter and divergence, Galilean or Keplerian lens spacing, the spot at a target and the best power for a range.

Using the Laser Beam Expander Calculator

A laser beam expander is a small telescope that makes a collimated beam wider. The Laser Beam Expander Calculator works out how much wider, how much the divergence drops, how far apart to mount the two lenses, and what the beam looks like at your target.

Pick a mode in the Laser Beam Expander Calculator’s Solve For menu, then fill in the main inputs. The Beam, Distance & Focus accordion holds the input divergence, target distance, spread model, a focusing lens and the wavelength.

Four Ways to Solve

“MP from focal lengths” takes the two lens focal lengths and the design type, Galilean or Keplerian. “Output beam from MP” suits a store-bought expander rated 2×, 5× or 10×, where you only know its power.

“MP for an output size” finds the power that turns your beam into the diameter you need. “Best MP for far target” finds the power that makes the beam smallest at a set distance, which is the question most range users actually have.

Beam, Distance and Focus Inputs

Divergence can be entered as a half-angle, the Omni and Edmund Optics convention, or as the full angle most laser data sheets use. Units run from µrad and mrad to degrees and arcminutes.

The spread model is either a straight cone, as Omni uses, or a Gaussian beam with its waist at the expander output. The focusing lens field drives the focused spot card, and the wavelength feeds an M² check on your input beam.

Beam Expander Formula

A beam expander follows four short formulas. The Laser Beam Expander Calculator applies each one in order and lists them in the Step-by-Step Solution, worked with your own numbers.

Magnifying Power

Magnifying power MP is the objective focal length f_O divided by the input lens focal length f_I. For a Galilean design, use the size of the negative focal length. Magnification m is the inverse, 1 ÷ MP, and the tool shows both.

$$MP = \frac{f_O}{|f_I|}$$

Output Diameter and Divergence

The beam leaves MP times wider than it entered. Its divergence drops by the same factor, so the product of beam size and spread stays fixed. That product is the beam parameter product, and no passive optic can lower it.

$$D_O = MP \times D_I \qquad \theta_O = \frac{\theta_I}{MP}$$

Beam Diameter at a Distance

Past the expander, the beam grows by twice the output half-angle per unit of distance. Omni and Edmund Optics write the same thing as L·tan(2θ_O), which matches for the tiny angles lasers have.

$$D_L = D_O + 2L\tan\theta_O$$

Lens Spacing

For the output to leave collimated, the lenses sit one signed focal length sum apart. A Keplerian expander with two positive lenses has f_O + f_I between them. A Galilean expander has f_O − |f_I|, because its first lens is negative.

$$L_{lens} = f_O + f_I$$

That sign matters on the bench. The default 6× Galilean, with −25 mm and 150 mm lenses, needs 125 mm. Adding the magnitudes instead gives 175 mm, the Keplerian length, and the beam would leave badly out of collimation.

Galilean vs Keplerian Beam Expander

Both designs reach the same magnification with two lenses. They differ in the first lens and in whether the beam comes to a focus inside the tube. The Edmund Optics beam expander guide sets out the trade-offs.

Most commercial expanders are Galilean because they are shorter, cheaper and safe with high peak power. Pick Keplerian when you need a spatial filter. Its internal focus lets a pinhole strip noise from the beam, which a Galilean design cannot do.

A Keplerian expander is an astronomical telescope run backward, the same layout used in binoculars, which add prisms to turn the image upright. The Binoculars Range Calculator covers ranging through those optics with a mil reticle.

Worked Example: A 6× Galilean Expander

The default is the example used by Omni. A 2 mm Nd:YAG beam with 0.4 mrad half-angle divergence enters a Galilean expander with a −25 mm input lens and a 150 mm objective. The Laser Beam Expander Calculator returns MP = 6× and m = 0.1667.

The output beam is 12 mm wide with a half-angle of 66.67 µrad, and its area is 36 times larger. The lenses go 125 mm apart. At 5 m the expanded beam measures 12.67 mm, but the bare beam is only 6.00 mm there.

That is why the default run shows “Expander Not Paying Off Yet”. An expander starts wider, so it only wins once the bare beam has spread past it. For this beam the crossover is at 15 m, and the Laser Beam Expander Calculator lists it in the third card. At 100 m the expanded beam is 25.33 mm against 82.0 mm without it.

The hero line also checks the input beam. A 2 mm beam at 1064 nm spreading 0.4 mrad implies M² ≈ 1.18, a clean single-mode beam. A value under 1 triggers a warning, since no real beam can spread less than the diffraction limit.

Choosing the Best Magnification

More magnification is not always better at a fixed distance. A bigger MP shrinks the divergence, but it also starts the beam wider. The smallest spot comes when the two effects are equal, at this power.

$$MP_{best} = \sqrt{\frac{2L\tan\theta_I}{D_I}}$$

At that power the spot equals twice the expanded diameter, 2 × MP × D_I. Edmund’s own example, a 1 mm beam with 0.5 mrad half-angle aimed 100 m away, gives exactly 10×, the expander they chose. The table uses the default 2 mm, 0.4 mrad beam.

The gain grows with distance, which is why expanders matter most for long-range alignment, lidar and free-space links. The Spot Size With vs Without Expander table in the tool shows the same comparison at 1 m, 10 m, 100 m, 1 km and 10 km for your own beam.

Size matters at the receiving end too. A spot that fits inside the receiver lens loses no light at its edge. The Aperture Area Calculator gives the collecting area of a receiver lens or telescope for a beam of that size.

How to Calculate Spot Size of a Laser Beam

A wider beam focuses to a smaller spot. That is the second big reason to expand a beam before a focusing lens in laser cutting, marking and microscopy. The focused spot diameter is twice the focal length times the beam’s half-angle.

$$d = 2f\tan\theta_O \approx \frac{4M^2\lambda f}{\pi D}$$

The default sends the beam into a 70 mm lens. With the 6× expander it focuses to 9.33 µm, against 56.0 µm without it, a 6 times smaller spot and 36 times higher peak irradiance. Both forms of the formula agree here, since the input beam has M² ≈ 1.18.

The price is depth of focus. The expanded beam holds near its smallest size for only 108.9 µm along the axis, so the work surface has to sit precisely at the focus. The Laser Beam Divergence Calculator has a focused-spot mode for any beam, without an expander.

The card shows an ideal, diffraction-limited spot. Edmund Optics adds a spherical aberration term that grows with the beam diameter cubed, so a very wide beam through a short, simple lens focuses larger than this. Aspheric or corrected lenses close most of that gap.

The focal length often comes from a lens rated in diopters rather than millimeters. The Diopter Calculator converts that power into a focal length to enter in the Laser Beam Expander Calculator.

Fluence, Irradiance and Laser Safety

Expanding a beam spreads the same power over MP² times the area. The first card shows that factor, ×36 for the 6× default. Irradiance and fluence on the downstream optics drop by the same amount, which helps coatings survive high-power and pulsed lasers.

Fluence is pulse energy per unit area, usually in J/cm². A Gaussian beam’s peak fluence is 2E ÷ (πw²), with w the 1/e² radius. A 10 mJ pulse in a 2 mm beam peaks at 0.64 J/cm², and after a 6× expander it peaks at only 17.7 mJ/cm².

Laser safety eyewear is rated by optical density, OD = log₁₀(H₀ ÷ MPE). H₀ is the exposure the eye could receive and MPE is the maximum permissible exposure. An exposure 1,000 times the MPE needs OD 3. A wider beam lowers H₀ at the eye, but the full calculation should follow a laser safety standard.

Other Types of Beam Expanders

Lens-based expanders are called transmissive. Their magnification and collimation shift with wavelength, because glass bends each color by a different amount. The Index of Refraction Calculator shows that change through its Cauchy mode for common optical glasses.

Reflective expanders use curved mirrors instead of lenses. They have no chromatic error, so one unit works for several lasers, tunable lasers and ultrafast pulses. Variable or zoom expanders let you dial the magnification, which helps match beams from lasers that differ unit to unit.

Turned around, any expander becomes a beam reducer. Enter an output size smaller than the input in the “MP for an output size” mode, and the Laser Beam Expander Calculator reports MP below 1. A reducer is a common way to fit a beam onto a small detector or fiber, at the cost of faster divergence.

Beam Expander Questions

These answers cover the other questions people ask when they size an expander. For divergence measurement and Gaussian beam theory, see the Optics Calculators hub, home of the Laser Beam Expander Calculator.

What is the full form of LASER?

LASER stands for Light Amplification by Stimulated Emission of Radiation. The acronym is now written as an ordinary noun. Stimulated emission is what makes laser light coherent, single-colored and tightly collimated, which is what an expander then improves further.

What are the two types of laser beams?

Lasers run either continuous-wave, with steady output power, or pulsed, with energy packed into short bursts. Pulsed lasers reach very high peak power, which is why a Galilean expander, with no internal focus, is the usual pick for them. Fluence, not average power, sets the damage risk for pulses.

What is a four-level laser?

A four-level laser pumps atoms to a high energy level, from which they drop quickly to the upper laser level. The laser transition then ends on a level above the ground state that empties fast. This makes population inversion easy, and Nd:YAG, the default laser here, works this way.

What are type 2 and type 3 lasers?

In the U.S. FDA system, Class II, or IEC Class 2, covers products such as bar code scanners, a hazard only when viewed directly for long periods. Class IIIa, or 3R, includes laser pointers. Class IIIb, or 3B, is an immediate eye hazard from the direct beam.

What is the 1/e² beam diameter?

It is the width where intensity falls to 13.5% of the peak, and it contains 86.5% of a Gaussian beam’s power. Expander specs, divergence and spot size formulas all use it. The divergence calculator page also covers how to convert it to FWHM.

Is there a beam expander simulation?

The Expander Layout panel in the tool sketches your two lenses with the computed spacing, input and output diameters labeled, and the beam widening through a Galilean design or crossing at the internal focus of a Keplerian one. It is a schematic rather than a scale drawing, and it updates as you change the focal lengths, so it works as a quick visual check of your build.