In short
- M² measurement needs a caustic, not a snapshot. Measure the beam diameter at 10 or more planes through a focus and fit a hyperbola. One beam image cannot give M².
- Place the planes correctly. About half within one Rayleigh length of the waist, about half beyond two Rayleigh lengths (ISO 11146-1).
- Use second-moment widths. Subtract the background and integrate over an area about three times the beam width, or the result will drift.
- Fit d² = a + bz + cz². Then M² = (π / 8λ)·√(4ac − b²). The calculator below does the fit and checks your plane placement.
M² measurement (often written “M2 measurement”) answers one practical question: how close is your laser to an ideal Gaussian beam, and so how small a spot and how long a focus can you get from it? The number appears on most laser datasheets, yet it is easy to measure wrongly in-house because of a few avoidable mistakes. This guide walks through the ISO 11146 method with a camera beam profiler, step by step, with a worked example and a free fitting tool.
What M² tells you
M², the beam propagation ratio, compares a real beam with a perfect TEM₀₀ Gaussian beam of the same wavelength. For any beam, the product of waist diameter d₀ and full far-field divergence angle Θ is fixed by propagation:
d₀ · Θ = M² · 4λ / π
An ideal Gaussian has M² = 1. A good single-mode diode-pumped laser is typically close to 1. Multimode, high-power and diode lasers can reach 10, 50 or more. The related beam parameter product (BPP) is the waist radius times the half-angle divergence, BPP = M²·λ/π, usually given in mm·mrad.
Why it matters in practice: for a given focusing lens, the focused spot grows in proportion to M², and the depth of focus (Rayleigh length) at a given spot size shrinks in proportion to M². A beam with M² = 2 needs a lens twice as fast to reach the same spot, and then has half the depth of focus. That is why material processing, medical laser and laser-ranging specifications set M² limits.
Why one beam profile cannot give M²
A beam profiler image at one plane gives the beam diameter, centroid and shape at that plane. It does not show how the beam will spread. Two beams with the same diameter on the sensor can have very different M², because one may be at its waist and the other far from it.
ISO 11146-1 therefore defines M² from the caustic: the beam diameter measured at many planes along the propagation axis, through a waist. If the laser’s own waist is inside the cavity or far away, a lens with low aberration creates an accessible artificial waist. A thin, aberration-free lens changes d₀ and Θ but not their product, so M² is preserved.
M² (M2) measurement step by step, per ISO 11146
- Create an accessible waistUse a long-focal-length, low-aberration lens (often 200–500 mm) so the Rayleigh length is several millimetres and the waist is a few times larger than the pixel pitch.
- Attenuate before the sensorUse wedges, a beam sampler or reflective ND filters. Keep the peak signal below saturation at the waist, where irradiance is highest.
- Find the waist and estimate the Rayleigh lengthScan coarsely, locate the smallest diameter, and find where the diameter grows by √2. That distance is the Rayleigh length zR.
- Measure at 10 or more planesAbout half within ±zR of the waist, about half beyond 2·zR. Re-adjust exposure at each plane.
- Use second-moment diametersSubtract the background, integrate over an area about 3× the beam width, iterate until the width converges. For asymmetric beams, measure both principal axes.
- Fit and reportFit d²(z) = a + bz + cz², report d₀, z₀, Θ, zR and M² for each axis, with the wavelength and lens used.
The fit equations
ISO 11146-1 fits the square of the second-moment diameter as a quadratic in z, by least squares. From the coefficients a, b and c:
- Waist position: z₀ = −b / (2c)
- Waist diameter: d₀ = √(4ac − b²) / (2√c)
- Full divergence angle: Θ = √c (radians, when z and d are in the same units)
- Rayleigh length: zR = d₀ / Θ
- Beam propagation ratio: M² = (π / 8λ) · √(4ac − b²)
When the measurements are taken after a focusing lens, d₀, z₀ and Θ describe the artificial waist. M² is the property of the laser; the original waist and divergence can be recovered with the lens transformation if needed.
Worked example: a 1064 nm laser after a 300 mm lens
A 1064 nm laser is focused by a 300 mm lens. A rough scan puts the waist near z = 150 mm (measured from a reference on the rail) with a diameter of about 0.2 mm and a Rayleigh length of about 22 mm. Ten planes are measured with second-moment widths:
| Plane z (mm) | Diameter d (mm) | Distance from waist |
|---|---|---|
| 60 | 0.827 | > 2 zR |
| 80 | 0.643 | > 2 zR |
| 100 | 0.488 | > 2 zR |
| 130 | 0.263 | within zR |
| 140 | 0.222 | within zR |
| 150 | 0.198 | within zR |
| 160 | 0.220 | within zR |
| 170 | 0.262 | within zR |
| 205 | 0.529 | > 2 zR |
| 230 | 0.728 | > 2 zR |
The least-squares fit gives a = 1.8026 mm², b = −0.02346 mm and c = 7.81 × 10⁻⁵. From these:
- z₀ = 150.3 mm, d₀ = 0.198 mm
- Θ = 8.84 mrad (full angle), zR = 22.4 mm
- M² = 1.29, BPP = M²·λ/π = 0.44 mm·mrad
Five planes lie within one Rayleigh length and five beyond two, so the placement meets ISO 11146-1. Paste the same numbers into the calculator below to reproduce the result.
At the waist, a 0.198 mm beam on a sensor with 5.5 µm pixels covers 36 pixels, comfortably above the usual minimum of about 10. At z = 60 mm the beam is 0.83 mm wide, so the sensor must be at least about 2.5 mm wide (three times the beam) to keep the integration area on the chip. Both checks belong in the plan before the first frame is taken.
M² caustic fit calculator
Paste one plane per line: position z and diameter d, both in mm (second-moment diameters, one axis). The tool fits d² = a + bz + cz² and checks plane placement against ISO 11146-1.
Common M² measurement mistakes and how to avoid them
| Mistake | What happens | Fix |
|---|---|---|
| All planes close to the waist | The divergence term is poorly defined; M² can come out too high, too low or even below 1 | Put about half of the planes beyond 2 zR |
| Background not subtracted | Noise far from the beam inflates the second moment; widths grow with the integration area | Subtract a dark frame or baseline, and integrate over about 3× the beam width |
| Too few pixels at the waist | The waist diameter is overestimated, so M² is biased | Aim for at least 10 pixels across d₀; use a longer lens if needed |
| Beam clipped at far planes | Wings are cut off, far-field widths are underestimated | Keep the beam under about one third of the sensor width |
| Saturated pixels at the waist | The peak is flattened, the width is overestimated | Re-adjust exposure or attenuation at every plane |
| Aberrating focusing lens or filters | The lens adds its own aberration to the beam; measured M² is higher than the laser’s | Use a long focal length, a well-corrected lens and wedged, high-quality attenuators |
| Only one axis measured | Astigmatic or elliptical beams are misreported | Fit x and y (principal axes) separately and report both |
| Clip-level widths used instead of second moments | Matches second moments only for a clean Gaussian; differs for multimode beams | Use second-moment widths for the M² value, clip widths for quick alignment |
How much does plane placement matter? We simulated the same 1064 nm beam (true M² = 1.30) 5,000 times with 2% random noise on each width. With ten planes crowded within ±10 mm of the waist, 90% of the results fell between 1.07 and 1.49, and about 2% came out below 1. With the ISO placement of the worked example, 90% fell between 1.24 and 1.35.
Pulsed lasers add two points: trigger the profiler to the laser so every frame captures a pulse, and use a global-shutter camera so the whole frame is exposed at the same moment. Average several frames per plane to reduce the effect of pulse-to-pulse fluctuation.
Choosing a beam profiler for M² work
The profiler is the core of a caustic measurement, and the wavelength chooses the family. IZAK camera-based laser beam profilers cover four bands: UV–VIS–NIR from 300 to 1100 nm, SWIR up to 2.05 µm, MWIR from 1.5 to 5.7 µm and LWIR from 7.7 to 12 µm.
For M², check three things beyond the wavelength:
- Pixel pitch against the waist. At least about 10 pixels across d₀. In the infrared, pixels are larger, so plan a longer focal length to make a larger waist.
- Sensor size against the far planes. The beam is about 2.2 times wider than the waist at two Rayleigh lengths and about 3.2 times wider at three, and it must stay on the chip with margin.
- Dynamic range and exposure control. Irradiance changes by roughly (d/d₀)² between the waist and the far planes, often a factor of 10 or more, so exposure and attenuation must follow.
The Gaussian Beam Calculator helps plan the lens and the planes before you start, and the laser beam profiler selection guide covers sensor fit and attenuation in more detail. For the basics of profiling, see The Essentials of Laser Beam Profiling, and for 2–5 µm sources, MWIR beam profiling.
Frequently asked questions
QHow many measurement planes does ISO 11146 require for M²?
QCan I measure M² without a focusing lens?
QWhy did my measurement give M² below 1?
References
- ISO 11146-1:2021, Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams.
- ISO 11146-2:2021, Part 2: General astigmatic beams.
- ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods.
- A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends in Optics and Photonics, vol. 17, 1998.
Tzachi Sabati
CEO, IZAK Scientific
Physicist specializing in photonics and quantum technologies, with deep expertise in quantum sensors and advanced optical systems. Leads the Advanced Quantum Lab course at the Technion, bridging academic excellence with industry innovation. At IZAK Scientific, provides cutting-edge photonics-based solutions, developing customized inspection and sensing systems for R&D and production. Passionate about advancing quantum sensing applications and integrating novel technologies to meet industry needs.
