The laser spot size calculator above applies the Gaussian beam propagation relation to work out how wide a laser beam is at a given distance from its waist, together with the Rayleigh range, the far-field divergence half-angle and the radius of curvature of the wavefront. In its second mode it does the more common practical job: given a collimated beam entering a lens of known focal length, it returns the radius of the spot formed at the focus and the depth of focus around it.
Arb Digital publishes free engineering calculators, and this one comes with a warning that is not decoration. Spot size is the denominator of every irradiance figure, and irradiance is exactly the quantity a laser exposure assessment is built from. Laser safety is governed by IEC 60825-1 internationally and by ANSI Z136.1 in the United States; the class printed on the equipment label determines the controls that apply. Comparing any number this page produces against a maximum permissible exposure is a laser safety officer's job, not this page's. No beam should ever be viewed directly and no beam should ever be aimed at a person.
What This Laser Spot Size Calculator Does
A Gaussian beam does not travel as a cylinder and it does not spread as a cone from a point. It has a narrowest cross-section, the waist, and it expands hyperbolically either side of that. Close to the waist the growth is negligible; far from it, the beam edge approaches a straight line whose slope is the far-field divergence. The tool computes where you are on that curve.
In propagation mode you supply the waist radius, the wavelength and the beam quality factor, and it returns the radius at your chosen distance. In focus mode you supply the beam radius at the lens and the focal length, and it returns the focused waist and the depth of focus. Both modes report the on-axis peak irradiance in that plane if you give it a power, and both use the 1/e² radius convention throughout.
How to Use It
- Choose a mode — propagate a waist forward, or focus a collimated beam through a lens.
- Enter the wavelength in nanometres and the beam quality M². Use one for a fundamental-mode source and the datasheet figure otherwise.
- Enter the beam size and say whether your figure is a radius or a diameter. This is the waist in propagation mode and the beam at the lens in focus mode.
- Enter the distance in metres, or the focal length in millimetres, depending on the mode you selected.
- Optionally add a power in milliwatts to get the on-axis peak irradiance in that plane, and read the Rayleigh range and divergence from the grid.
The Formula: How Gaussian Beam Radius Is Calculated
For a beam of wavelength λ with waist radius w₀ and beam quality M², the Rayleigh range is
zR = π w₀² / (M² λ)
and the radius at a distance z from the waist follows the hyperbolic law
w(z) = w₀ √(1 + (z / zR)²)
The far-field divergence half-angle is θ = M² λ / (π w₀), which is exactly w₀/zR, and the radius of curvature of the wavefront is R(z) = z(1 + (zR/z)²). The wavefront is flat at the waist, most strongly curved at exactly one Rayleigh range, and approaches a spherical wave centred on the waist far away. The beam-width and divergence definitions used here follow ISO 11146, "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios"; the measurement side of that work is the province of national metrology institutes such as the NIST Applied Physics Division, which maintains laser power and energy calibration services.
In focus mode, for a collimated beam of radius win entering a lens of focal length f, the focused waist is
wf = M² λ f / (π win)
which holds when the incoming beam is large compared with the focused spot, as it almost always is. The depth of focus is the Rayleigh range of that new waist, computed the same way.
Peak on-axis irradiance for a Gaussian profile is I₀ = 2P / (π w²), not P divided by the area. The factor of two is there because a Gaussian is peaked: the on-axis value is twice the value you would get by spreading the same power uniformly over πw². Forgetting it understates the centre of the beam by a factor of two.
Work the defaults through by hand. A 632.8 nm beam, waist radius 0.5 mm, M² of one. The Rayleigh range is π × (5 × 10⁻⁴)² / 6.328 × 10⁻⁷ = 1.241 m. At ten metres, z/zR is 8.057, so w = 0.5 × √(1 + 64.91) = 0.5 × 8.119 = 4.059 mm, a diameter of 8.119 mm. The divergence half-angle is 6.328 × 10⁻⁷ / (π × 5 × 10⁻⁴) = 4.029 × 10⁻⁴ rad, or 0.403 mrad. The wavefront radius is 10 × (1 + 0.01540) = 10.15 m. With 5 mW of power, the peak irradiance at that plane is 2 × 0.005 / (π × (4.059 × 10⁻³)²) = 193.2 W/m², which is 0.0193 W/cm².
Laser Safety Is Governed by a Standard, Not by This Page
Laser radiation damages eyes and skin, and retinal injury from a visible or near-infrared beam can be instantaneous and permanent. The international framework is IEC 60825-1, "Safety of laser products — Part 1: Equipment classification and requirements", and in the United States use is governed by ANSI Z136.1, "American National Standard for Safe Use of Lasers". Both are cited here by number and title rather than reproduced, because the numerical limits inside them are only meaningful within their full methodology.
The class marked on the equipment is the practical entry point. Classification is performed by the manufacturer under the standard, and the class is what dictates the controls: enclosures, interlocks, key switches, beam stops, controlled areas, wavelength-specific eyewear and documented training. A Class 4 laser can injure from a diffuse reflection off a matte surface and can ignite materials; a Class 3B beam is hazardous on direct intrabeam viewing.
Two points cause real injuries and deserve emphasis. Invisible wavelengths are more dangerous than visible ones, not less: below roughly 400 nm and above roughly 700 nm there is no visible spot, no blink reflex and no aversion response, and near-infrared light between about 700 and 1400 nm still reaches the retina. And no beam should ever be viewed directly or aimed at a person, a vehicle or an aircraft, whatever the label says.
If you need to know whether an irradiance is within a maximum permissible exposure, that determination belongs to a qualified laser safety officer working from the governing standard with the full exposure conditions in hand. The Berkeley Office of Environment, Health & Safety describes how class-based controls and the laser safety officer role fit together, and any organisation running Class 3B or Class 4 equipment is expected to have that structure in place.
Why a Smaller Waist Means a Faster-Spreading Beam
The most counter-intuitive result on this page is that you cannot have both a small beam and a slow-spreading one. Divergence goes as λ/(πw₀), so halving the waist doubles the divergence. Squeeze a beam down and it fights back by opening out faster.
This is why long-range beam delivery uses a beam expander rather than a tighter source. Expanding a beam by a factor of ten reduces its divergence by a factor of ten, so the spot at a kilometre shrinks by nearly ten times even though the beam leaving the aperture is ten times wider. The product of waist and divergence is fixed at M²λ/π, and that is the entire trade.
The same arithmetic explains why a shorter wavelength focuses to a smaller spot for the same optics, which is the reason ultraviolet lithography exists, and why a long-wavelength carbon dioxide laser cannot match a one-micron fibre laser for fine cutting at equal beam quality.
Depth of Focus Is Usually the Number You Actually Need
For anything that processes material — cutting, welding, marking, drilling — the focused spot radius is only half the story. The other half is how far the workpiece can move along the beam axis before the spot has grown enough to matter, and that is set by the Rayleigh range of the focused waist.
At one Rayleigh range from focus, the radius has grown by a factor of √2 and the peak irradiance has therefore halved. That is a useful practical definition of the working window. The awkward part is that the Rayleigh range scales as the square of the focused spot radius, so halving the spot to double the irradiance cuts the depth of focus to a quarter. Tight focusing buys intensity and pays for it in tolerance on part height, fixturing and thermal drift.
A short focal length gives a small spot and a shallow working window; a long focal length gives a larger spot and a much more forgiving one. Which is right depends on the tolerances of the machine rather than on the laser.
Where This Sits Next to the Other Optics Tools
This page answers "how big is the beam here". The laser brightness calculator answers a different question — how concentrated the source is in radiance terms, which is the quantity no optic can improve. The laser beam expander calculator handles the telescope that trades diameter against divergence, and the laser linewidth calculator covers the spectral side of beam quality rather than the spatial one.
For the underlying optics, the lens maker equation calculator gives the focal length of a lens from its surfaces, the Snell's law calculator handles refraction at an interface, and the optical density calculator covers attenuation through a filter. The wavelength calculator converts between wavelength and frequency, and the angular resolution calculator covers the diffraction limit for imaging apertures, which is the same physics in a different guise.
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
- Entering a beam diameter as a radius — the factor of two propagates into a factor of four in area and irradiance, and into a factor of four in Rayleigh range.
- Using P divided by area for peak irradiance — a Gaussian peak is twice that, because the profile is peaked rather than flat-topped.
- Assuming M² is one — it rarely is outside a single-mode source, and it multiplies both the divergence and the focused spot directly.
- Treating a diode as circular — edge-emitting diodes have very different beam quality in the fast and slow axes, and each axis has to be computed separately.
- Reading an irradiance figure as a safety verdict — it is not one. Comparison against an exposure limit requires the full conditions and belongs to a laser safety officer under IEC 60825-1 or ANSI Z136.1.
Related Free Tools From Arb Digital
Pair this with the laser brightness calculator for radiance and beam parameter product, the laser beam expander calculator for magnification and the divergence trade, and the laser linewidth calculator for coherence length. The lens maker equation calculator and Snell's law calculator cover the optics, the wavelength calculator and photon energy calculator cover the light itself, and the angular resolution calculator covers the diffraction limit. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the radius at which the irradiance has fallen to one over e squared of its on-axis peak, which is about 13.5 per cent. That circle contains roughly 86.5 per cent of the total power. It is the convention used in the Gaussian propagation relations and in ISO 11146, and it is not the same as full width at half maximum, which is smaller by a factor of about 1.18.
Because a Gaussian beam is a solution of the wave equation, not a bundle of rays. It has a finite narrowest cross-section called the waist, and it expands hyperbolically either side of it. Extrapolating the far-field cone backwards gives you a virtual point that the beam never actually reaches, which is why the hyperbolic relation is needed near the waist.
No. The focused radius is proportional to the wavelength, the beam quality factor and the focal length, and inversely proportional to the beam radius at the lens. Diffraction sets a floor at roughly the wavelength itself, and in practice lens aberrations, aperture limits and beam quality stop you well short of it.
It is the natural length scale of a beam. At one Rayleigh range from the waist the radius has grown by the square root of two and the peak irradiance has halved. Inside it the beam is effectively collimated; well outside it the beam grows almost linearly. For material processing, twice the Rayleigh range is the usual working definition of depth of focus.
No, and it must not be used as one. It reports beam geometry and, optionally, irradiance. Laser safety is governed by IEC 60825-1 and ANSI Z136.1, the class marked on the equipment determines the required controls, and comparing an irradiance figure against a maximum permissible exposure is the work of a qualified laser safety officer with the full exposure conditions in hand.
Because nothing warns you. Below about 400 nm and above about 700 nm there is no visible spot, so there is no blink reflex and no aversion response, yet near-infrared light between roughly 700 and 1400 nm still passes through the eye and reaches the retina. A damaging dose can be absorbed with no sensation at all.
Only one axis at a time. An edge-emitting diode is strongly astigmatic, with very different waist sizes and beam quality factors in the fast and slow axes, so it has to be treated as two independent beams. Run the tool twice with each axis's own waist and M squared, and remember that the two waists are not in the same plane.
It multiplies it directly. The focused radius is proportional to M squared, so a beam with M squared of ten focuses to a spot ten times larger than a diffraction-limited beam through the same lens, and the peak irradiance is a hundred times lower. Beam quality, not power, usually decides whether a source can do a fine-feature job.
This tool is provided for educational and preliminary engineering use only. It computes Gaussian beam geometry and does not assess laser safety, does not publish maximum permissible exposure values and does not issue any safety verdict. Laser safety is governed by IEC 60825-1 and ANSI Z136.1, and the class marked on the equipment determines the required controls. Never view a laser beam directly and never aim one at a person. Any comparison of these figures against an exposure limit must be made by a qualified laser safety officer working to the governing standard.
