The laser beam expander calculator above works out the geometry of a two-lens afocal telescope used to widen a laser beam. It gives the magnification, the separation the two lenses need, the output beam diameter, the divergence that results, and the Rayleigh range over which the expanded beam stays collimated. It handles both the Galilean and Keplerian arrangements, which behave identically in magnification and very differently in every practical respect.
Arb Digital publishes free engineering calculators that state their limits clearly. This one has an important limit that is not about optics. Laser radiation causes permanent eye injury, and the quantities on this page — beam diameter, divergence and irradiance — are the same quantities an exposure assessment uses. This page computes optics. It does not and cannot tell you whether a beam is safe.
Laser Safety Is Governed by a Standard, Not by This Page
Laser safety is governed by published standards: IEC 60825-1, Safety of laser products — Equipment classification and requirements, and in the United States ANSI Z136.1, American National Standard for Safe Use of Lasers. Those documents define the hazard classes, the exposure limits and the control measures, and they are the authority. Nothing on this page substitutes for them.
The class marked on the product label is what determines the controls that apply — enclosures, interlocks, eyewear of a specified optical density, controlled areas, training and administrative procedures. A laser's class is assigned by the manufacturer under the standard, and it is the starting point for every safety decision.
No beam should ever be viewed directly, and no beam should ever be aimed at another person. That holds regardless of power, class or apparent brightness. Specular reflections from a flat surface deserve the same caution as the direct beam, because a mirror-like reflection loses very little of the original irradiance.
Invisible wavelengths are more dangerous than visible ones, and the reason is physiological rather than optical. There is no blink reflex to radiation the eye cannot see. An infrared beam at 1,064 nm delivers its full energy to the retina with no aversion response at all, and injury can occur before the person knows anything happened. The same applies to ultraviolet. If you are working outside the visible band, treat every assumption about noticing a stray beam as false.
This page reports irradiance because it is a physical property of the beam that follows directly from power and area. Comparing that irradiance to a maximum permissible exposure is the job of a laser safety officer working from the governing standard, not the job of this page. No exposure limits are published here and no safety verdict is issued, because both depend on wavelength, exposure duration, pulse structure, viewing conditions and aperture in ways a general calculator cannot capture. University laser safety programmes such as those at Stanford Environmental Health and Safety and Cornell Environment, Health and Safety set out how that assessment is properly conducted.
What This Laser Beam Expander Calculator Does
A beam expander is an afocal telescope run backwards. Two lenses share a common focal point, so a collimated beam entering one side leaves collimated but wider. The magnification is the ratio of the focal lengths.
The value of doing this is divergence. A Gaussian beam's far-field divergence is inversely proportional to its waist radius, so widening the beam narrows the cone it spreads into by the same factor. Expanding ten times reduces divergence tenfold. This is why beam expanders appear in rangefinding, free-space optical communication and laser cutting, where a larger beam focused by a lens also produces a smaller focal spot.
The hero figure is the magnification. The grid gives the lens separation, the output beam diameter, the output divergence and the Rayleigh range, which is how far the expanded beam travels before it noticeably starts to spread.
How to Use It
- Enter both focal lengths as positive magnitudes. The tool applies the negative sign to the input lens in the Galilean case automatically.
- Use the 1/e² beam diameter. This is the convention laser datasheets use. If you have a full-width half-maximum figure, divide it by about 0.59 first.
- Set the wavelength. Divergence and Rayleigh range both depend on it, and an infrared beam diverges more than a visible one from the same aperture.
- Enter a realistic M². A value of 1 assumes an ideal single-mode beam. Real diode lasers are often well above that, and it multiplies the divergence directly.
- Check the output diameter against your lens aperture. A beam that overfills the output lens is clipped, causing diffraction rings and heating the mount.
The Formula: How the Expander Geometry Is Derived
Magnification is the ratio of focal lengths, M = f2 ÷ |f1|. For the afocal condition the lenses must share a focal point, so the separation is d = f1 + f2. In the Keplerian case both focal lengths are positive and that sum is their total. In the Galilean case the input lens is negative, so the separation becomes f2 − |f1| — a shorter assembly by twice the input focal length.
The output diameter is Dout = M Din, and divergence scales the opposite way: θout = θin ÷ M. Their product is conserved — the beam parameter product, and the reason no passive optic improves both at once.
For a Gaussian beam the half-angle divergence is θ = M²λ ÷ (πw0), where w0 is the waist radius, and the Rayleigh range is zR = πw0² ÷ (M²λ). The beam diameter at a distance z follows w(z) = w0√(1 + (z/zR)²). These are the standard published Gaussian beam relations; MIT OpenCourseWare's 6.013 Electromagnetics and Applications covers the wave optics they derive from.
Work the defaults through by hand. A Galilean expander with |f1| = 25 mm and f2 = 250 mm gives M = 10, and a separation of 250 − 25 = 225 mm. The Keplerian equivalent would need 275 mm. A 1 mm input beam becomes 10 mm. At 632.8 nm with a 0.5 mm input waist radius the input half-angle divergence is 632.8 × 10−9 ÷ (π × 5 × 10−4) = 4.03 × 10−4 rad, or 0.403 mrad. After tenfold expansion the waist is 5 mm and the divergence falls to 0.0403 mrad. The Rayleigh range rises from 1.24 m to π × (5 × 10−3)² ÷ 632.8 × 10−9 = 124 m, a hundredfold improvement, because it scales with the square of the waist.
Galilean or Keplerian: The Choice That Matters
Both produce the same magnification from the same focal length ratio, so the decision rests entirely on the internal focus. A Keplerian expander brings the beam to a real focus between the two lenses. A Galilean expander does not, because the diverging input lens never forms a real image.
That internal focus is a liability at high power. Concentrating a beam to a tight point inside the assembly can ionise the air and damage anything at that point. Galilean designs are therefore standard for pulsed and high-average-power work, and they are shorter into the bargain.
The internal focus is also an opportunity. A pinhole placed exactly at it turns the expander into a spatial filter, blocking the high-spatial-frequency components that show up as speckle and leaving a clean beam. That is the usual reason to accept the longer Keplerian assembly in low-power laboratory work.
What the Ideal Model Leaves Out
The magnification arithmetic is exact for thin lenses. Real lenses are not thin, and the collimating separation is measured between principal planes rather than glass surfaces, so a commercial expander's specified spacing differs slightly from the focal length sum.
Spherical aberration degrades the wavefront and raises the effective M², which is why real expanders use aspheric or multi-element designs rather than two singlets. Chromatic aberration matters if the source is not monochromatic.
Alignment is unforgiving. A small tilt between the two lenses steers the output beam by roughly the magnification times the input error, so a 10× expander amplifies pointing errors tenfold. Commercial expanders are supplied pre-aligned in a rigid housing for exactly this reason.
Where This Sits Next to the Other Laser Tools
Four laser pages divide the subject up. This one covers expander geometry: how two lenses turn a narrow beam into a wide one. The laser spot size calculator covers how a Gaussian beam grows with distance and what a focusing lens does to it. The laser brightness calculator covers radiance, which no passive optic can increase. The laser linewidth calculator covers spectral rather than spatial properties.
For the general optics, the thin lens equation calculator and the lens maker equation calculator handle imaging and lens design, and the telescope magnification calculator covers the same afocal geometry used for viewing rather than for beam shaping.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing 1/e² and full-width half-maximum diameters — they differ by a factor of about 1.7, and laser datasheets almost always use the 1/e² convention.
- Assuming M² is 1 — a multimode diode can be 10 or more, and the factor multiplies divergence directly, so the real far-field spot is much larger than the ideal calculation suggests.
- Using a Keplerian design at high power — the internal focus can ionise air and damage anything sitting at that point; Galilean avoids it entirely.
- Overfilling the output lens — a clipped beam produces diffraction rings and deposits energy in the mount, which at high power is a thermal and damage problem.
- Ignoring alignment sensitivity — an expander multiplies pointing errors by its magnification, so a tilt that was negligible at the input is not negligible at the output.
Related Free Tools From Arb Digital
Continue with the laser spot size calculator for propagation and focusing of the expanded beam, the laser brightness calculator for radiance, and the laser linewidth calculator for spectral width. The thin lens equation calculator and lens maker equation calculator cover the underlying lens behaviour, the telescope magnification calculator covers the same afocal pair used for viewing, and the wavelength calculator handles frequency and wavelength conversion. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
A Galilean expander uses a negative input lens and a positive output lens, has no internal focus, and is shorter by twice the input focal length. A Keplerian expander uses two positive lenses and brings the beam to a real focus between them. Both give the same magnification from the same focal length ratio.
Because the product of beam size and divergence angle is conserved for a given beam quality. A Gaussian beam's far-field divergence is inversely proportional to its waist radius, so widening the beam by a factor of ten narrows the divergence by the same factor. No passive optic can improve both at once.
Because the internal focus concentrates the whole beam into a very small volume between the lenses. At high peak power that can ionise the air, producing a visible spark and an audible crack, and it can damage dust or any component at that point. Galilean designs have no real internal focus and avoid the problem completely.
It is the distance from the beam waist at which the beam area has doubled, meaning the radius has grown by the square root of two. It is the practical measure of how far a beam stays collimated. Because it scales with the square of the waist radius, expanding a beam tenfold increases the Rayleigh range a hundredfold.
It is the beam quality factor, equal to 1 for an ideal single-mode Gaussian beam and larger for any real beam. It multiplies the divergence and divides the Rayleigh range, so a beam with M squared of 4 spreads four times faster than the ideal calculation predicts. Multimode diode lasers are frequently well above 1.
It changes the beam geometry, and safety is not something this page can determine. Spreading the same power over a larger area lowers irradiance at the output, but the beam also stays concentrated over a far longer distance, which changes the hazard at range. Classification and exposure assessment are governed by IEC 60825-1 and ANSI Z136.1 and belong to a laser safety officer.
Because there is no blink reflex to light the eye cannot see. A visible beam triggers an aversion response within a fraction of a second, which limits exposure. An infrared or ultraviolet beam produces no such response, so the eye can receive the full exposure with no warning at all and injury can occur before the person is aware of anything.
This tool is provided for educational and preliminary optical design use. It computes published Gaussian beam and thin-lens relations only and issues no safety assessment of any kind. Laser safety is governed by IEC 60825-1 and ANSI Z136.1; the class on the product label determines the required controls, and comparing any irradiance figure against a maximum permissible exposure is the responsibility of a qualified laser safety officer. Never view a beam directly and never aim one at any person.
