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DELaser Spot Size Calculator
Calculate the focused 1/e² laser spot diameter, beam waist, Rayleigh range and depth of focus from wavelength, focal length, input beam diameter and M² or BPP. It covers free-space and fiber-coupled laser systems, offers three depth-of-focus criteria, and works backwards from a required spot size or depth of focus.
How to calculate a focused laser spot size
A lens focuses a collimated laser beam to a waist. Four quantities set its 1/e² diameter: the wavelength, the focal length, the collimated beam diameter arriving at the lens, and the beam quality. For a beam that is Gaussian apart from a beam propagation ratio M²:
Keep the units consistent. With λ in µm and f and D in mm, d₀ comes out in µm. The most common mistake is mixing a radius into a formula written for diameters: swapping w₀ for d₀ puts every result out by a factor of two. This calculator works in diameters throughout, and the beam-path diagram labels which diameter is which.
Everything else collapses onto one number. Substituting the f-number N = f/D gives d₀ = (4λM²/π)·N. Once λ and M² are fixed, the spot depends on the ratio of focal length to beam diameter and on nothing else. A 100 mm lens with a 5 mm beam and a 200 mm lens with a 10 mm beam produce the same spot.
Spot size, beam waist, Rayleigh range and depth of focus
The beam does not stay at d₀. Away from the waist it grows as
Substituting d₀ gives the identity zR = d₀·f/D. The Rayleigh range is the spot diameter multiplied by the f-number, which is a useful thing to carry in your head. Because the spot is linear in the f-number and the Rayleigh range is quadratic in it, the spot scales as f/D while the depth of focus scales as (f/D)². Halving the spot costs three quarters of the depth. No ordinary refractive optic escapes that trade-off, and it is the reason extended-depth-of-focus beam shaping exists.
Depth of focus is a tolerance, not a physical edge. It only means something once you state how much diameter growth you accept. The general form is DOF = 2·zR·√((1+X)²−1) for a fractional diameter growth X, and the calculator lets you switch between the three criteria that come up in practice:
| Criterion | Diameter growth | Depth of focus | Peak intensity at the edge |
|---|---|---|---|
| Standard, the textbook value | +41.4% (√2) | 2·zR | 50% |
| Precision | +10% | 0.917·zR | 83% |
| High precision | +5% | 0.640·zR | 91% |
Most published spot-size calculators quote only the standard figure. That number is generous compared with what a marking, welding or drilling process actually tolerates, which is why the switch sits directly on the result: pick +10% or +5% and the plot, the bracket and the number all follow. At the standard criterion the half-depth is exactly the Rayleigh range, so the two are the same statement in different words.
Read the last column before you commit to a tolerance. A diameter criterion and an intensity criterion are different things, and for ablation or welding it is often the intensity that sets the window.
Entering beam quality as M² or BPP
M² and the beam parameter product carry the same information in different units. BPP is the product of the waist radius and the far-field half-angle divergence:
The relation runs through λ, so the same BPP means a different M² at a different wavelength. That is why the unit switch next to the beam-quality field converts the value instead of relabelling it. It also sets a floor: no beam can have a BPP below λ/π, and the calculator refuses values below it rather than returning an M² under 1.
Diode and fiber lasers are usually specified by BPP, solid-state and ultrafast sources by M². Both are defined by the ISO 11146 second-moment method, so a supplier’s M² should already be the D4σ value this formula expects.
Fiber-coupled laser spot size
In a fiber-coupled system the beam quality is not a free parameter. The fiber sets it, from the core diameter and the fiber NA:
There is a useful sanity check here. A fiber-coupled chain is the core imaged onto the work plane, so the spot should equal the core diameter times the magnification:
That is not an independent formula. Substituting the expressions above reduces the Gaussian result to exactly this. It is also the number fiber-laser engineers already carry in their heads, so the calculator shows both. They agree to within the small-angle approximation (tan θ ≈ θ), a few tenths of a percent at NA 0.12 and growing with NA.
Three caveats. First, check which NA definition your datasheet uses: 1/e², 5% and full-aperture values differ enough to move both M² and the collimated diameter noticeably. Second, this is the standard step-index multimode estimate, which treats the core as uniformly filled. For a single-mode fiber it is a rough guide only, and the calculator warns when the numbers you enter imply M² ≈ 1.
Third, read the result for what it is. For a fully filled multimode fiber the number is close to the imaged core diameter, and that profile is nearer to a flat top with rolled edges than to a Gaussian. It is not a Gaussian 1/e² diameter and the enclosed-power fractions in the table below do not carry over. Use it as the width of the illuminated area, then measure if the process depends on the edge steepness.
Worked examples
Fiber laser marking head, 1064 nm
- Fiber core Ø
- 50 µm
- Fiber NA
- 0.12
- Collimator f
- 25 mm
- Focusing f
- 160 mm
- Derived M²
- 8.88 (BPP 3.01 mm·mrad)
- Collimated Ø
- 6.04 mm
- Focused spot Ø
- 318 µm
- Standard DOF
- 16.9 mm
The imaging cross-check gives 50 µm × 160/25 = 320 µm, 0.5% from the Gaussian result. Note how generous the depth of focus is. That comes from the high M², not from good optics. Switch the criterion to +10% and it drops to 7.7 mm.
Free-space UV micromachining, 355 nm
- Wavelength
- 355 nm
- Beam Ø at lens
- 12 mm
- M²
- 1.3
- Focusing f
- 100 mm
- Focused spot Ø
- 4.90 µm
- Rayleigh range
- 40.8 µm
- Standard DOF
- 81.6 µm
Compare that with the example above: a 65 times smaller spot comes with a 200 times shorter depth of focus. This is the (f/D)² scaling in practice, and it is why UV micromachining needs real focus control rather than a nominally flat part.
Working backwards: which focal length gives a 50 µm spot?
- Wavelength
- 1064 nm
- Beam Ø at lens
- 8 mm
- M²
- 1.1
- Target spot Ø
- 50 µm
- Required focal length
- 268 mm
- Resulting standard DOF
- 3.36 mm
This inverts in closed form, with no iteration. Every reverse problem reduces to a target spot diameter, and a target depth of focus converts to one via zR = DOF/2 and d₀ = √(4λM²zR/π).
1/e², FWHM and D4σ: which diameter is quoted?
Spot sizes are quoted at several different definitions, and mixing them is the second most common source of factor-level errors after radius versus diameter. For an ideal Gaussian:
| Definition | Relative to 1/e² | Enclosed power |
|---|---|---|
| 1/e² diameter (13.5% of peak) | 1.000·d₀ | 86.5% |
| D4σ, the second-moment width of ISO 11146 | 1.000·d₀ | 86.5% |
| FWHM (50% of peak) | 0.589·d₀ | 50% |
| Diameter enclosing 99% of the power | 1.517·d₀ | 99% |
D4σ and 1/e² coincide only for a clean Gaussian. For a real beam with wings the second-moment width is the larger of the two, and the more useful one. The calculator shows the FWHM equivalent under the spot result, so a datasheet quoted either way can be compared without arithmetic.
This decides which diameter you should type in. M² is itself defined through second moments, so a measured M² from a beam-profiler report belongs with D4σ diameters, in and out. Entering a 1/e² diameter is the near-Gaussian approximation: fair for a single-mode source, weaker as M² rises. The two inputs are the same number often enough that the distinction rarely changes a lens choice, and it is worth knowing which one you made.
Assumptions and limitations
Read these before quoting a number from any Gaussian spot-size calculator, including this one.
- Ideal thin lens, no aberrations. Spherical aberration and coma in a real objective enlarge the spot. Treat the result as the best case the geometry permits.
- No truncation. The formula assumes the whole beam passes. Keep every clear aperture in the path at 1.5 to 2 times the beam diameter, because clipping a Gaussian both widens the focus and adds diffraction rings.
- Paraxial. Above roughly NA 0.5 the closed-form expressions lose accuracy, and near the wavelength scale vector-field effects matter. The calculator flags both cases rather than pretending otherwise.
- No thermal focus shift. At high average power, focus drift in the optics is often larger than the depth of focus computed here.
- Circular, astigmatism-free beam, wavelength-independent M². For an elliptical or two-axis beam, run each axis separately and expect two waists at two different positions.
- One diameter convention throughout. Whatever you enter, the output is the same convention: put in D4σ and you get D4σ out, put in 1/e² and you get 1/e² out. Since ISO 11146 defines M² through second moments, a measured M² pairs strictly with D4σ. For a fiber-coupled chain the spot approximates the imaged core, which is not a Gaussian diameter at all.
- A diameter criterion is not a process criterion. Your window may be set by fluence or intensity instead. See the intensity column in the depth-of-focus table above.
How we validated this calculator
The tool carries a built-in test suite of 19 assertions, runnable in the browser by appending
?selftest=1 to the page URL. It is not a claim of correctness by assertion. Each check
compares against something independent:
- Textbook values: d₀ for λ = 1064 nm, f = 100 mm, D = 10 mm, M² = 1 against the hand calculation.
- Internal identities: zR = d₀·(f/D), d(zR) = √2·d₀, standard DOF = 2·zR, DOF@10% = 0.917·zR, DOF@5% = 0.640·zR, BPP = λM²/π, and the far-field divergence recovering D/f.
- Scaling laws: spot ∝ M² and zR ∝ M² at fixed λ, f and D.
- Round trips: all four reverse solves fed back through the forward calculation.
- An independent physical route: the fiber result checked against plain geometric imaging, Dcore·ffocus/fcoll, which agrees to within the small-angle approximation.
- Unit equivalence: a beam quality entered as BPP producing the identical beam to the equivalent M².
Every formula and unit convention is documented rather than implied. The criterion factor K(X) is implemented in general form, which is what lets the interface offer all three depth criteria from the same code path.
Sources
- ISO 11146, Lasers and laser-related equipment: test methods for laser beam widths, divergence angles and beam propagation ratios. The normative definition of M² and D4σ.
- A. E. Siegman, Lasers, University Science Books, 1986. The standard treatment of Gaussian beam propagation and the M² formalism.
- S. A. Self, “Focusing of spherical Gaussian beams”, Applied Optics 22(5), 658 to 661 (1983). The reference for Gaussian beam imaging through a lens. doi:10.1364/AO.22.000658
- R. Paschotta, RP Photonics Encyclopedia. Accessible cross-checks on Gaussian beams and beam quality.
- Newport, Gaussian Beam Optics. A vendor tutorial that works the same relations in the notation most integrators meet first.
Frequently asked questions
What is the formula for laser spot size?
For a focused Gaussian beam, d₀ = 4λfM²/(πD), where d₀ is the 1/e² spot diameter, λ the wavelength, f the focal length, D the collimated 1/e² beam diameter at the lens and M² the beam propagation ratio. In radius form the same relation reads w₀ = 2λfM²/(πD).
Does a higher M² give a longer or a shorter depth of focus?
It depends on what you hold fixed, and this trips people up. At fixed wavelength, focal length and input beam diameter, which is the usual situation when you swap a laser on an existing head, a higher M² gives both a larger spot and a longer depth of focus, both in proportion to M² (DOF = 8λf²M²/πD²). Only if you hold the spot size constant, by shortening the focal length or expanding the beam, does a higher M² shorten the Rayleigh range.
Should I use the 1/e² diameter, D4σ or the FWHM?
Not the FWHM, for anything that goes into a propagation formula. It turns up in process descriptions and it is 0.589 times the 1/e² diameter for a Gaussian, but no propagation formula is written for it.
Between the other two: for a near-Gaussian beam the 1/e² and D4σ diameters coincide, so either convention gives the same answer. For a real non-Gaussian beam they do not. ISO 11146 defines beam width and M² through second moments, so a measured M² belongs with D4σ diameters. Enter a 1/e² diameter instead and the result is a near-Gaussian approximation. Either way, say which one you mean whenever you quote a spot size.
How do I convert BPP to M²?
M² = π·BPP/λ. With BPP in mm·mrad and λ in µm the numbers work out directly: 3.0 mm·mrad at 1064 nm is M² ≈ 8.9. The conversion depends on wavelength, so a BPP figure alone does not fix M².
What spot size do I get from a fiber-coupled laser?
To a good approximation, the core imaged by the magnification of the optics: spot ≈ core diameter × ffocus/fcoll. A 50 µm core with a 25 mm collimator and a 160 mm focusing lens gives about 320 µm. The full Gaussian treatment gives 318 µm for the same system.
Which depth-of-focus criterion should I use?
Start from the process, not from the textbook. The standard 2·zR figure allows the beam to grow by 41%, which most marking, welding and drilling windows do not tolerate. If your process holds to 10% diameter growth, the usable depth is 0.917·zR, roughly half the standard number. At 5% it is 0.640·zR. Switch the criterion on the result and compare, then check the intensity column in the table above, since a fluence-limited process has a different window again.
Why is my measured spot larger than the calculated value?
Usually one of five things, in this order of likelihood. The beam is clipped somewhere in the path, so check every clear aperture against 1.5 to 2 times the beam diameter. The real M² is worse than the datasheet value at your operating point. Lens aberrations, which this model excludes. Thermal focus shift at high average power. Or a measurement taken at a different diameter definition than the calculation.
Can I share a configured calculation?
Yes. Every input is encoded in the page URL, so copying the address bar reproduces the exact calculation for someone else.