How to Measure M² Beam Quality: Camera and Knife-Edge Methods

Scope

This article describes the procedure for measuring the M² beam quality parameter of a laser beam according to ISO 11146 (Test methods for laser beam widths, divergence angles, and beam propagation ratios). The measurement has two parts: measuring the beam's width at a series of positions through a focus, and fitting those widths to the propagation law. The widths can be measured with a beam-profiling camera, using the second-moment (D4σ) method, or with the knife-edge method; both are given here in procedural detail, together with the hyperbolic fit, a worked example, automated M² systems, and the dominant sources of error. Scanning-slit profilers follow the knife-edge principle and are mentioned for comparison. Measuring divergence alone, without the full caustic, is covered in How to Measure Laser Beam Divergence.

Definition

The beam quality parameter $M^2$ (also called the beam propagation ratio) compares the divergence of a real beam to that of an ideal Gaussian beam of the same waist size. For a Gaussian beam of waist radius $w_0$ propagating in the $z$ direction, the beam radius evolves as:

$$w(z)=w_0\sqrt{1+{\left(\frac{z-z_0}{z_R}\right)}^2},z_R=\frac{\pi w_0^2}{\lambda },$$

where $z_0$ is the position of the waist and $z_R$ is the Rayleigh range. The far-field divergence half-angle is $\theta =\lambda \mathrm{/}(\pi w_0)$.

For a real (non-Gaussian) beam, the same hyperbolic functional form applies, but the divergence is scaled by $M^2\ge 1$:

$$\begin{array}{rl}{\displaystyle w(z)}& {\displaystyle =w_0\sqrt{1+M^4{\left(\frac{z-z_0}{z_R^G}\right)}^2},}\\ {\displaystyle z_R^G}& {\displaystyle =\frac{\pi w_0^2}{\lambda },}\end{array}$$

equivalently expressed in terms of beam waist and divergence:

$$M^2=\frac{\pi w_0\theta }{\lambda }\mathrm{.}$$

$M^2=1$ corresponds to an ideal diffraction-limited Gaussian beam (the Gaussian Beam Calculator computes its waist, Rayleigh range and divergence). Higher values indicate the beam diverges faster than the corresponding ideal Gaussian, requiring tighter optics or larger apertures to deliver the same beam quality at distance.

Typical $M^2$ values:

For asymmetric beams (such as edge-emitter outputs), $M^2$ is reported separately for the two principal axes, $M_x^2$ and $M_y^2$.

Beam width definition

ISO 11146 specifies the second-moment beam width (the D4σ width) as the standard width for $M^2$ extraction:

$$w^2=4{\sigma }^2=\frac{4\iint (x-x_c{)}^{2}I(x,y)dxdy}{\iint I(x,y)dxdy},$$

where $I(x,y)$ is the intensity distribution and $x_c$ is the centroid. Here $w=2\sigma$ is the second-moment radius; the D4σ diameter is $2w=4\sigma$. For a Gaussian beam, $w$ equals the conventional $1\mathrm{/}{e}^2$ intensity radius exactly (and the D4σ diameter equals the $1\mathrm{/}{e}^2$ diameter). For non-Gaussian beams, the D4σ width remains well-defined and produces an $M^2$ value consistent with the divergence-based definition.

Alternative width definitions include full-width-half-maximum (FWHM), $1\mathrm{/}e$, and $1\mathrm{/}{e}^2$. These produce different numerical values for non-Gaussian beams and must not be used for $M^2$ extraction without explicit conversion. ISO 11146 is unambiguous: use D4σ.

Choosing how to measure the width

The camera is the method of choice when the beam fits on the sensor and the wavelength suits it, because it measures the second moment the standard specifies and shows the whole profile. The knife edge covers the cases the camera cannot: wavelengths outside the sensor's range, very small or very large beams, and powers that are awkward to attenuate.

Camera-based measurement (second-moment method)

Setting up the camera

1. Attenuate before the camera. Use reflective attenuation (uncoated wedge pickoffs) or neutral-density filters of good optical quality, placed where they do not distort the beam; a filter placed in a converging beam near a focus can add aberration, and multiple reflections between filters can add ghost images. Set the attenuation so the peak pixel sits well within the sensor's linear range, below saturation but using most of its dynamic range.
2. Match the beam to the sensor. The beam's $1\mathrm{/}{e}^2$ diameter should cover many pixels (a common rule of thumb is at least ten across the diameter, more for structured beams) and fit within the sensor with room for the wings: the integration region below extends to about three times the beam width.
3. Take a dark frame. With the beam blocked, record the background and subtract it from every image. Ambient light, sensor offset and fixed-pattern noise all appear in it.
4. Remove the protective window if the maker allows it, or use a camera with a wedged window; interference between the window's surfaces and the sensor produces fringes in a coherent beam.

Why the baseline matters

The second moment weights each pixel by the square of its distance from the centroid, so a small uniform residual far from the beam contributes heavily. For a Gaussian beam of diameter $D$ integrated over a square region three times $D$ on a side, a residual background equal to a fraction of the peak intensity inflates the D4σ width as follows (computed from the second-moment integral):

A background that is invisible in the image therefore makes the width, and the M² computed from it, far too large, and a larger integration region makes it worse. For this reason ISO/TR 11146-3 describes baseline correction and restricts the integration to a region a few times the beam width, found iteratively: compute the width, set the region from it, recompute, and repeat until the width stops changing. Profiler software implements this; its settings, not the default display width, decide whether the reported value is the ISO second-moment width.

Measuring the caustic

The camera is moved along the beam through a focus, as in the procedure below, and the D4σ width is recorded in both principal axes at each position. For a beam whose axes are rotated relative to the sensor, the software's principal-axis widths should be used rather than the widths along the sensor's rows and columns. A camera behind a lens on a motorized rail, stepping through the focus and fitting the result, is what commercial M² measurement systems automate; the physics, the ISO point distribution and the error sources are the same as for a manual measurement.

Knife-edge method principle

A knife-edge is translated transversely across the beam at fixed propagation distance $z$. As the edge moves from fully blocking to fully unblocking the beam, the transmitted power $P(x)$ traces a sigmoid curve. The intensity profile is the derivative of this curve:

$$I(x)=-\frac{dP}{dx}\mathrm{.}$$

For a Gaussian beam, the transmitted power as a function of edge position has the form of an erf function:

$$P(x)=\frac{P_{\text{max}}}{2}[1-\text{erf}\left(\frac{\sqrt{2}(x-x_c)}{w}\right)],$$

where $w$ is the $1\mathrm{/}{e}^2$ intensity radius. Fitting the measured $P(x)$ to this form extracts $w$ directly.

For non-Gaussian beams, the second-moment width can be extracted from the knife-edge data either by numerical differentiation followed by second-moment integration, or by fitting to a generalized lineshape. For near-Gaussian beams the Gaussian erf fit gives a width within a few percent of the true D4σ; for flat-top, multimode or pedestal-bearing beams the two can differ by 10 to 20% or more, and the second-moment route should be used.

The knife-edge method has two important practical advantages over camera-based methods:

1. No pixel saturation for high-power beams: the integrating detector sees the whole unmasked beam, so only its total-power linear range matters (see Detector saturation below).
2. No wavelength-limited sensor: knife-edge measurements work at wavelengths where CCDs are not available (deep UV, mid-IR, THz).

The corresponding limitation is that the knife-edge method measures one-dimensional integrated profiles. Two orthogonal scans (knife edges along $x$ and $y$) give the widths along the two axes, and the full two-dimensional distribution only for a separable beam; for highly non-symmetric beams, beams with rotated principal axes, or beams with complex spatial structure, camera methods are preferred. The knife-edge measurement entry gives the relation between the 10–90% edge distance and the beam radius used for quick single-plane checks.

Equipment

The focusing lens transforms the input beam to produce a measurable waist. For a high-quality lens (f-number $>4$, low aberrations), the lens preserves $M^2$ and the measured waist parameters apply to the input beam after correction by the lens focal length.

For input beams that are already focused (such as fiber outputs), the lens may be unnecessary; the input beam waist is measured directly in the existing focus region.

Procedure

ISO 11146-1 specifies the measurement procedure in detail. The essential elements:

1. Configure the focused beam

Place the focusing lens in the beam path. Identify the approximate waist position $z_0$ by observing the beam diameter through the focus region. The stage should reach well beyond $\pm 2z_R$ on either side of the waist (about $\pm 3z_R$), where $z_R=w_0\mathrm{/}\theta =z_R^G\mathrm{/}{M}^2$ is the Rayleigh range of the real beam, $M^2$ times shorter than the ideal-Gaussian value $z_R^G$ above, so that half of the positions can lie beyond $2z_R$.

For a collimated input beam and lens focal length $f$, the focused waist has radius approximately

$$w_0\approx \frac{M^2\lambda f}{\pi w_{\text{in}}},$$

where $w_{\text{in}}$ is the input beam radius at the lens. Choose $f$ so that $w_0$ is in the range $20-200\mu$m for convenient knife-edge measurement.

2. Measure the width at multiple z positions

ISO 11146-1 requires at least 10 measurement positions distributed as follows:

• Half within $\pm z_R$ of the waist (the near-field region)
• Half beyond $\pm 2z_R$ from the waist (the far-field region)

Even distribution is not required; the asymmetric distribution (more points near the waist where small width variations are visible, more points in the far field where divergence is established) gives the best fit.

At each $z$ position, record a camera frame and compute the D4σ widths as described above, or perform a knife-edge scan:

1. Translate the knife from fully blocking to fully unblocking the beam
2. Record transmitted power $P(x)$ at $\ge 30$ knife positions
3. Subtract any background (with knife fully blocking)

Record both the absolute $z$ position and the resulting transmitted power curve.

3. Repeat for orthogonal axis

Rotate the knife edge by 90° (or use a second knife edge oriented perpendicular to the first). Repeat the scan series for the $y$ axis. The orthogonal axis must be measured at the same $z$ positions to enable proper $M_x^2$ and $M_y^2$ extraction for asymmetric beams.

For circularly symmetric beams, only one orientation is necessary; for any beam from a non-symmetric source (edge-emitter, fiber-coupled with poor mode matching, etc.), both axes are required.

4. Extract beam width at each z

Camera measurements give the D4σ width directly. For each knife-edge scan, fit the transmitted power to the erf function:

$$P(x)=\frac{P_{\text{max}}}{2}[1-\text{erf}\left(\frac{\sqrt{2}(x-x_c)}{w(z)}\right)]+P_{\text{bg}}\mathrm{.}$$

The fit has four free parameters: $w(z)$, $x_c$, $P_{\text{max}}$, and $P_{\text{bg}}$. The fitted $w(z)$ is the $1\mathrm{/}{e}^2$ intensity radius at this $z$ position.

For non-Gaussian beams, the erf fit may not capture the lineshape well; in this case, numerically differentiate $P(x)$ to obtain $I(x)$ and compute the second moment directly:

$$w(z{)}^2=4\cdot \frac{\sum _i(x_i-x_c{)}^{2}I(x_i)}{\sum _{i}I(x_i)}\mathrm{.}$$

5. Fit the hyperbolic propagation

Plot $w(z{)}^2$ versus $z$ for each axis. The data follows:

$$w(z{)}^2=w_0^2+{\left(\frac{M^2\lambda }{\pi w_0}\right)}^2(z-z_0{)}^2\mathrm{.}$$

Equivalently, in terms of the asymptotic divergence half-angle $\theta =M^2\lambda \mathrm{/}(\pi w_0)$:

$$w(z{)}^2=w_0^2+{\theta }^2(z-z_0{)}^2\mathrm{.}$$

Perform a least-squares fit on $w(z{)}^2$ versus $z$, with three free parameters: $w_0$, $z_0$, and $\theta$ (or equivalently $M^2$).

6. Compute M²

From the fit:

$$M^2=\frac{\pi w_0\theta }{\lambda }\mathrm{.}$$

Report:

• $M^2$ for both axes ($M_x^2$ and $M_y^2$)
• Waist radius $w_0$ for both axes
• Waist position $z_0$
• Rayleigh range $z_R$
• 95% confidence interval on $M^2$ from the fit
• Number of data points $N$ and range $z_{\text{min}}$ to $z_{\text{max}}$ in units of $z_R$

Worked example

A 1064 nm fiber-coupled solid-state laser is focused through an aspheric lens with $f=50$ mm. The focused beam is scanned over $\pm 4$ mm around the apparent focus position, with 10 knife-edge scans per axis at:

$z=-4.0,-3.0,-2.0,-1.0,-0.4,0.0,0.4,1.0,2.0,4.0$ mm

(relative to nominal focus). At each $z$, the erf fit to the knife-edge scan gives:

Fitting $w^2$ versus $z$ to the hyperbolic form gives:

For the x-axis, $M_x^2=\pi \times 56.7\mu \text{m}\times 0.069\mathrm{/}1064\text{ nm}=11.6$, and the fitted waist lies at $z_{0,x}$ = 0.043 mm.

The fitted $z_0$ differs slightly from nominal because the operator's reference position was not exactly at the true waist; the fit recovers the true position. The real-beam Rayleigh range is $w_0\mathrm{/}\theta$ = 0.82 mm for x and 0.84 mm for y, so three of the ten positions lie within $\pm z_R$ and five beyond $\pm 2z_R$; a set that follows ISO 11146 would move the two positions at $\pm 1.0$ mm inside $\pm 0.8$ mm.

The resulting $M^2\approx 11$ is consistent with a multi-mode fiber-coupled solid-state laser. The slight asymmetry ($M_x^2>M_y^2$) reflects the residual asymmetry of the fiber-coupled output mode.

Sources of measurement error

Insufficient points outside the Rayleigh range. The hyperbolic fit's divergence parameter is constrained primarily by far-field data points ($\mathrm{\mid }z-z_0\mathrm{\mid }>z_R$). Measurements that cluster near the waist produce poorly-constrained divergence and large $M^2$ uncertainty. ISO 11146 specifies the asymmetric distribution (half near, half far) for this reason.

Lens aberrations. Spherical aberration in the focusing lens redistributes intensity in the focal region and can artificially inflate $M^2$. Use of a low-aberration aspheric or apochromatic lens, and avoidance of operation at large fractions of the lens NA, mitigate this contribution. Standard guideline: lens $f$-number $>4$.

Beam clipping. If the beam diameter exceeds the lens aperture at any point in the optical path, the clipped beam has artificially elevated divergence and $M^2$. The clear aperture must accommodate $\ge 3w$ for negligible clipping.

Background light. Stray light reaching the detector during the knife-edge scan adds a baseline that shifts the apparent beam centroid and broadens the apparent profile. Subtract background measured with knife fully blocking; for very weak beams, use lock-in detection.

Knife-edge diffraction. As the waist approaches a few wavelengths, diffraction at the edge and the edge's finite sharpness distort the measured profile and the erf model loses accuracy. With a sharp edge and a detector that collects all of the transmitted light, waists of a few micrometers can still be measured.

Camera baseline and saturation. An unsubtracted background inflates the D4σ width, as tabulated above; a saturated peak flattens the profile and does the same. Check the dark frame and the peak pixel value at every position, since the beam's peak intensity changes by orders of magnitude through the focus and the attenuation may need to change with it.

Detector saturation. During the unblocked portion of a knife-edge scan, the detector receives the full beam power. Operation in the linear range of the detector must be verified, particularly for pulsed lasers, where peak power may saturate even at low average power.

Beam position drift during the measurement. A 10-position $M^2$ measurement takes minutes; if the beam drifts laterally during this time, the measured profile centroids shift and the apparent width may be inflated. Active beam stabilization or short measurement times mitigate this.

Mistaken use of FWHM or $1\mathrm{/}{e}^2$ width. Reporting $M^2$ using a non-D4σ width is a common literature error. The numerical $M^2$ value from a Gaussian-fit $1\mathrm{/}{e}^2$ width and from a true D4σ measurement differ for non-Gaussian beams. For diffraction-limited or near-Gaussian beams, the difference is small (a few percent); for high-$M^2$ beams with complex structure, the difference can be 20% or more.

Polarization-dependent reflectivity at high incidence angles. For beams measured at non-normal incidence on lenses or beam splitters, polarization-dependent transmission may distort the measured profile. Use S- or P-polarized input where possible, or characterize the polarization dependence.

Validation

The fit residuals on $w^2$ versus $z$ should be randomly distributed about zero with no systematic trend. Visible curvature in the residuals indicates either lens aberration, beam clipping, or fit failure.

For a diffraction-limited reference source (single-mode HeNe or single-mode fiber output), the measured $M^2$ should fall in the range 1.0–1.1. Values significantly higher indicate measurement error, not source quality.

Repeated measurement on the same source under nominally identical conditions should agree within $\pm 5\mathrm{\%}$. Drift larger than this indicates source instability, beam drift, or measurement chain issues.

For asymmetric beams, the geometric mean $\sqrt{M_x^2{M}_y^2}$ is often quoted as a single quality metric; report both axes to enable proper system design.

References

For the controlling international standard, see ISO 11146-1:2021, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles, and beam propagation ratios, and ISO/TR 11146-3 for the intrinsic and geometrical beam classification, propagation and details of test methods, including baseline correction for camera measurements. For the original theoretical treatment of beam quality and the M² formalism, see Siegman (1990, 1993) on new developments in laser resonators and beam quality. For practical comparison of knife-edge, slit, and CCD methods, see Roundy (1999) and the SPIE proceedings on laser beam profiling.