Laser Beam Spot Size Calculator for the 1/e² focus of a Gaussian beam, its depth of focus, the beam size at any distance and the lens, input beam or M² that your target spot needs.
Enter the beam size at the lens, the focal length, wavelength and M².
Advanced
Beam Profile Around Focus
Beam radius near the focus, drawn to scale against itself. The shaded band is the depth of focus (±zR), where the spot stays within √2 of its smallest size.
Spot Size by Focal Length
| Focal Length | Spot (1/e²) | DOF | f-Number |
|---|---|---|---|
| 25.00 mm | 2.069 µm | 10.35 µm | f/2.50 |
| 50.00 mm | 4.138 µm | 41.38 µm | f/5.00 |
| 75.00 mm | 6.207 µm | 93.11 µm | f/7.50 |
| 100.0 mm | 8.276 µm | 165.5 µm | f/10.0 |
| 150.0 mm | 12.41 µm | 372.4 µm | f/15.0 |
| 200.0 mm | 16.55 µm | 662.1 µm | f/20.0 |
| 300.0 mm | 24.83 µm | 1.490 mm | f/30.0 |
Your beam through common lens focal lengths. A shorter lens makes a smaller spot, but depth of focus shrinks with the square of the spot size.
Step-by-Step Solution
- λ = 650.0 nm, D = 10.00 mm, f = 100.0 mm, M² = 1.000
- S = 4·M²·λ·f ÷ (π·D)
- S = 4 × 1.000 × 6.500×10⁻⁷ m × 0.1000 m ÷ (π × 0.01000 m)
- Waist radius w₀ = 4.138 µm; Rayleigh range zR = π·w₀² ÷ (M²·λ) = 82.76 µm; depth of focus = 2·zR = 165.5 µm
- Spot diameter S = 2·w₀ = 8.276 µm
Using the Laser Beam Spot Size Calculator
The Laser Beam Spot Size Calculator solves six related problems from the Solve For list. All of them treat the beam at the lens as collimated with its waist there, the setup used by the standard spot size formula.
| Mode | You enter | Main result |
|---|---|---|
| Focused spot size | λ, beam diameter D, focal length f, M² | 1/e² spot diameter |
| Beam diameter at a distance | The same four, plus distance from the lens | Beam diameter there |
| Focal length for a spot | λ, D, target spot, M² | Lens focal length |
| Beam diameter for a spot | λ, f, target spot, M² | Beam size needed at the lens |
| M² from a measured spot | λ, D, f, measured spot | M² and beam parameter product |
| Spot from a fiber | λ, fiber core and NA or MFD, collimator f, lens f | Spot, collimated beam size, M² |
Four result cards follow the main answer. They give the FWHM diameter and spot area, then the depth of focus with a tighter tolerance band that you set under Advanced. The other two show the working f-number with NA and focal shift, and the peak intensity once you enter laser power under Advanced.
The Advanced panel also holds the Focus Model. The standard formula suits most collimated beams, while Exact Gaussian finds the true waist when the beam is small for the lens. A beam profile chart, a table of common focal lengths and a step list sit below the cards.
Spot Size of a Laser Beam Formula
A collimated Gaussian beam focused by a lens forms a waist close to the focal point. Its 1/e² diameter follows from four numbers, and this is the spot size formula most laser makers quote.
$$S = \frac{4 M^2 \lambda f}{\pi D}$$
Here λ is the wavelength, f the focal length, D the 1/e² beam diameter at the lens, and M² the beam quality factor. The beam waist w₀ is the radius, half of S. The same result written in radius form is w₀ = M²λf ÷ (πw), with w = D/2.
Take a red 650 nm beam, 10 mm wide, through a 100 mm lens with M² = 1. The spot is 4 × 650 nm × 100 mm ÷ (π × 10 mm) = 8.276 µm, with a waist radius of 4.138 µm. The working f-number is f/10.
M² scales the spot directly. A beam with M² = 1.3 focuses 30% larger through the same lens. A 1064 nm beam 8 mm wide with M² = 1.3 gives 22.01 µm through a 100 mm lens. A CO₂ beam at 10.6 µm, 20 mm wide with M² = 1.2 and a 127 mm lens, focuses to 102.8 µm.
Three levers shrink the spot. Use a shorter focal length, a wider beam at the lens, or a shorter wavelength. Widening the 650 nm beam three times with a beam expander cuts the spot from 8.276 µm to 2.759 µm on the same lens.
Exact Gaussian Focus
The standard formula assumes the beam’s own Rayleigh range at the lens is much longer than f. When the beam is small or the lens is long, the real waist forms before the focal point and comes out smaller than the formula predicts.
$$w_0 = \frac{D/2}{\sqrt{1 + \left(z_{R,in}/f\right)^2}}, \quad z_{R,in} = \frac{\pi (D/2)^2}{M^2 \lambda}$$
A 1 mm beam at 632.8 nm through a 1 m lens shows the gap. The standard formula gives 806 µm, but the exact model gives 627 µm, and the waist sits 606 mm from the lens instead of 1 m. Pick Exact Gaussian under Advanced, and the Laser Beam Spot Size Calculator warns you when the two models differ by more than 1%.
For the 10 mm red beam on a 100 mm lens the two models agree within 0.01%. The true waist sits only about 68 nm before f, the focal shift shown on the third card.
Depth of Focus and Rayleigh Range
The Rayleigh range zR is the distance from the waist where the beam grows by √2 and its area doubles. Depth of focus is twice that distance, measured across the waist.
$$z_R = \frac{\pi w_0^2}{M^2 \lambda}, \qquad DOF = 2 z_R$$
The 8.276 µm red spot has zR = 82.75 µm, so its depth of focus is 165.5 µm. Precision cutting and imaging often hold the spot within +5% instead. That band is 2zR√(1.05² − 1), or 0.640 zR, and it gives only 52.99 µm for the same spot.
$$DOF_t = 2 z_R \sqrt{(1 + t)^2 – 1}$$
The Spot Size Tolerance field under Advanced sets t, from just above 0% up to the 41.4% that marks the standard depth of focus. A 10% tolerance gives 0.917 zR, or 75.84 µm for the red spot, which suits work that can accept a slightly larger spot.
Depth of focus grows with the square of the spot size. The table below follows the same 650 nm, 10 mm, M² = 1 beam through common lenses. The Laser Beam Spot Size Calculator draws the same table for your own beam and highlights your lens.
| Focal length | Spot (1/e²) | Depth of focus | f-number |
|---|---|---|---|
| 25 mm | 2.069 µm | 10.35 µm | f/2.5 |
| 50 mm | 4.138 µm | 41.38 µm | f/5 |
| 75 mm | 6.207 µm | 93.11 µm | f/7.5 |
| 100 mm | 8.276 µm | 165.5 µm | f/10 |
| 150 mm | 12.41 µm | 372.4 µm | f/15 |
| 200 mm | 16.55 µm | 662.1 µm | f/20 |
| 300 mm | 24.83 µm | 1.490 mm | f/30 |
Halving the spot cuts the usable depth by four. That trade decides most lens choices, since a fine spot is only useful if the part stays inside its depth of focus.
Beam Diameter at Any Distance
Away from the waist the beam grows along a hyperbola. The distance mode measures z from the waist and returns the 1/e² diameter at your plane. That lets the Laser Beam Spot Size Calculator work as a beam diameter calculator along the whole axis.
$$D(z) = S\sqrt{1 + \left(z/z_R\right)^2}$$
Enter the distance from the lens, not from the focus. For the red example, a plane 100.1 mm from the lens sits 100 µm past the waist, and the beam there is 12.98 µm wide, 1.568 times the spot.
Far from the waist the beam spreads at a fixed half-angle, θ = M²λ/(πw₀). For the 8.276 µm spot that is 0.05 rad, which equals D/(2f) and matches the numerical aperture on the third card. The Laser Beam Divergence Calculator takes over for long-range spot sizes, such as a 1 mm He–Ne beam that grows to about 80.6 mm over 100 m.
1/e² vs FWHM Beam Diameter
The 1/e² diameter is where intensity falls to 13.5% of the peak. It is the width used in the spot size formula and in most datasheets, as defined in the RP Photonics beam radius entry.
FWHM is the width at half the peak, which many camera profilers report by default. For a Gaussian beam it is 0.589 of the 1/e² diameter, so the 8.276 µm spot reads 4.872 µm FWHM. The 1/e² circle holds 86.5% of the power, and the FWHM circle holds exactly half.
| Width | Relative to 1/e² | Power inside | Red example |
|---|---|---|---|
| 1/e² diameter | 1.000 | 86.5% | 8.276 µm |
| FWHM diameter | 0.589 | 50.0% | 4.872 µm |
| 2 × 1/e² aperture | 2.000 | 99.97% | 16.55 µm |
That last row is why the first card suggests a lens clear aperture of at least twice the beam diameter. Clipping less than 0.04% of the power keeps the focus clean of diffraction rings.
Non-Gaussian beams need the second-moment D4σ width from ISO 11146, which equals the 1/e² width for a true Gaussian. Mixing a FWHM reading with this calculator understates the spot by about 41%.
How to Measure Focal Spot Size
Laser beams are measured with a knife edge, a scanning slit or a camera profiler. A knife edge or slit moved through the beam suits Gaussian spots and simple equipment. A camera gives the full profile and the ISO width in one shot, but it usually needs an attenuator that does not distort the beam.
To test a focal spot, scan the profiler through the focus and keep the smallest reading. A spot of only a few microns can fall below the camera’s pixel size, so magnify it with a relay lens or switch to a slit scanner.
M² and Beam Parameter Product
M² compares a beam with a perfect Gaussian of the same waist. It sets the minimum beam size a lens can reach, since no beam can focus smaller than the M² = 1 limit for its λ and f. The beam parameter product, the waist radius times the divergence half-angle, carries the same information in mm·mrad.
Diode and fiber laser datasheets often quote BPP rather than M². Switch the beam quality unit to BPP and enter it directly; the calculator converts with M² = π·BPP/λ. A 0.5 mm·mrad beam at 1064 nm, for example, has M² = 1.476.
BPP has a floor of λ/π, the value for a perfect Gaussian beam. That is 0.207 mm·mrad at 650 nm and 0.341 mm·mrad at 1070 nm, and any entry below the floor is rejected as physically impossible.
$$M^2 = \frac{\pi D S}{4 \lambda f}, \qquad BPP = \frac{M^2 \lambda}{\pi}$$
The M² mode of the Laser Beam Spot Size Calculator reverses the spot formula. A 650 nm beam, 10 mm wide, that measures 12 µm behind a 100 mm lens has M² = 1.450 and a BPP of 0.300 mm·mrad. A perfect beam through the same optics would reach 8.276 µm.
One spot reading assumes it was taken right at the waist. The ISO 11146 method fits the beam radius at several planes, about half within one Rayleigh length of the focus and half beyond two. Values below 1 are not physical, so the calculator flags them as a likely 1/e² or focus error.
Focus Lens and Beam Size for a Target Spot
Two modes of the Laser Beam Spot Size Calculator plan optics backward from the spot you need. For a 12 µm target with the same 650 nm, 10 mm beam, the focal length mode returns a 145.0 mm lens.
The beam diameter mode keeps the lens and changes the beam. A 12 µm spot through a 100 mm lens needs a 6.897 mm beam at the lens. That number sizes an expander, or a fiber collimator as the fiber section below shows. The Diopter Calculator converts a chosen focal length to lens power.
Collimated Beam Size From a Fiber
Many lasers deliver light through a fiber, then a collimator and a focusing lens. The fiber mode of the Laser Beam Spot Size Calculator works out the collimated beam size first, so you do not need to measure it.
A single-mode fiber emits a near-Gaussian beam whose waist is its mode field diameter (MFD). The fiber collimator entry gives the collimated radius as f·λ/(πw), which as a diameter reads as follows.
$$D = \frac{4 \lambda f_{col}}{\pi \, MFD}$$
A 650 nm fiber with a 4 µm MFD and a 30 mm collimator gives a 6.207 mm beam with M² = 1. A 100 mm focusing lens then images the mode down to 13.33 µm, the MFD scaled by f ÷ f_col.
A multimode fiber is treated with geometry. The beam leaves at the fiber’s NA, so the collimated diameter is 2·f_col·tan(asin NA), and the core radius times that angle sets the BPP.
$$M^2 = \frac{\pi (d_{core}/2) \tan(\arcsin NA)}{\lambda}$$
A 1070 nm fiber laser with a 50 µm core that fills an NA of 0.12 has M² = 8.87 and a BPP of 3.02 mm·mrad. With a 100 mm collimator the beam is 24.17 mm wide, and a 150 mm lens focuses it to 75 µm with a 931 µm depth of focus.
That spot equals the core diameter times f ÷ f_col, an imaging rule that holds whatever the NA. The beam size and M² depend on how the light was launched, and real fiber lasers often fill less than the fiber’s rated NA. Enter the beam NA from the laser datasheet when it is given, not the fiber’s maximum. If the core and NA are too small to carry more than one mode, the calculator sets M² to 1 and asks for single-mode input instead.
Spot Size, Intensity and Area
The area of a beam at its 1/e² radius is πw₀², which is 53.79 µm² for the red spot. The peak intensity at the center of a Gaussian spot is twice the average over that circle, a factor the Gaussian beam article notes is often forgotten.
$$I_{peak} = \frac{2P}{\pi w_0^2}$$
A 5 mW red beam focused to 8.276 µm reaches a peak of 18.59 kW/cm². The 10 mm beam at the lens carried about 1.46 million times less intensity, which is the concentration gain on the fourth card.
Increasing the focal spot size spreads the same power over a larger area, so intensity drops with the square of the diameter. A small spot suits cutting, drilling and fine marking. A larger spot gives a longer depth of focus and a gentler heat input for welding and surface treatment. The Laser Brightness Calculator shows how power and M² set the radiance behind these numbers.
Laser Spot Size Questions
These answers cover the questions searched most often next to a laser spot size result. Each one uses figures you can reproduce in the Laser Beam Spot Size Calculator with the same inputs.
What is the spot size of a laser?
Spot size is the 1/e² diameter of the beam at its focus, the smallest point along its path. It ranges from a few microns for a tightly focused visible beam to over 100 µm for a CO₂ beam on a long lens.
An unfocused laser has no single spot size. Its beam diameter changes with distance, so a beam that leaves the laser about 1 mm wide can be several centimeters wide 100 m away.
What is the diameter of a laser beam?
Beam diameter is usually the 1/e² width, measured where intensity falls to 13.5% of the peak. It depends on where you measure, since the beam narrows toward a focus and widens past it, so a datasheet value applies at the output aperture only.
Always check whether a datasheet quotes 1/e², FWHM or D4σ. A FWHM figure is only 0.589 of the 1/e² width, so it must be converted before it goes into the spot formula.
How do you calculate the divergence and spot size of a laser beam?
For a focused beam, the spot comes from S = 4M²λf/(πD), and the divergence half-angle past the focus is θ = M²λ/(πw₀). For the red example both point to the same cone, 0.05 rad, set by D/(2f).
For an unfocused beam, start from its waist and divergence. A 1 mm He–Ne beam with M² = 1 has θ = 0.403 mrad, so its spot reaches about 80.6 mm at 100 m.
How do you calculate OD for a laser?
Optical density for safety eyewear is OD = log₁₀(H₀ ÷ MPE), where H₀ is the worst-case exposure and MPE is the maximum permissible exposure in the same units. The OSHA Technical Manual works it through the 7 mm limiting aperture of the eye.
Its example puts a 5 W argon beam through that aperture for 12.99 W/cm² and arrives at OD 5.9. A focused spot raises H₀ sharply, which is why the eyewear rating must cover the beam where people can actually be exposed.
What is a Class 3 or Class 4 laser?
Laser class sets the hazard level from power and exposure. For continuous beams, OSHA lists Class IIIa at 1 to 5 mW, Class IIIb at 5 to 500 mW and Class IV above 500 mW. The FDA class guide matches these to IEC 3R, 3B and 4.
Class IIIb is an immediate eye hazard in the direct beam. Class IV adds hazards from reflected beams, skin exposure and fire, and focusing either one to a small spot multiplies its intensity.
What are the main types of lasers?
Lasers are grouped by gain medium into gas, solid-state, fiber, semiconductor diode and dye lasers. Each family has its own typical beam quality, which shows up here through M² and sets how small a spot it can reach.
The basic parameters to know before focusing any of them are wavelength, power, beam diameter, divergence and M². These are the same numbers the Laser Beam Spot Size Calculator asks for, with power used only for intensity. Many run from a steady DC drive, as covered in the Laser Linewidth and Bandwidth Calculator guide.
The diffraction limit behind every spot here also limits telescopes and microscopes, which the Angular Resolution Calculator covers through the Airy disk.