Tool · Beams and focusing

Knife-Edge Beam Width Calculator

How wide is a beam that a blade has been scanned across? Enter the blade travel between two power levels, such as 10 % and 90 %, or paste the whole scan, and the calculator returns the 1/e² radius, the diameter, the FWHM and the beam center, with the fitted error function drawn over the points. Background: knife-edge measurement, beam waist and beam diameter.

From a clip width

Power levels

Blade travel between them (µm)

The distance the blade moves while the power passing it goes from one level to the other, as a fraction of the unblocked power.

Fit a scan

Blade position (µm) and power, one pair per line

Paste two columns from a spreadsheet or power-meter log; any power unit works, and the scan may run in either direction. The example is computed for w = 50.2 µm with a little noise added.

Presets

Readouts

Scan and fitted error function

Dots are the pasted points, the blue line the fitted erfc and the dashed orange line the Gaussian intensity profile it implies, scaled to the plot. The 1/e² radius is where the profile falls to 13.5 % of its peak.

Clip width per unit w against clip level

For symmetric levels p and 1 − p the clip width is w·Φ⁻¹(1 − p), where Φ⁻¹ is the standard normal quantile. Wider clips use more of the tails and are more sensitive to background and noise.

Learn with it

Two short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The clip-width factors are compared with the standard normal quantiles, and the fit with a scan computed for known parameters.

Check	Expected	Computed	Tolerance

The expected values follow P(x) = (P₀/2) erfc(√2 (x − x₀)/w) for a Gaussian beam, evaluated by hand or from tabulated quantiles. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

A blade at position $x$ blocks one side of a Gaussian beam, and the power that passes is

P (x) = B + \frac{A}{2} e r f c ⁣ (\frac{\sqrt{2} (x - x_{0})}{w})

where $A$ is the unblocked power, $B$ a background or detector offset, $x_{0}$ the beam center and $w$ the 1/e² intensity radius. The intensity profile along the scan is a Gaussian with standard deviation $σ = w / 2$ , so the blade positions where a fraction $p$ and $1 - p$ of the power passes are $\pm σ Φ^{- 1} (1 - p)$ from the center, with $Φ^{- 1}$ the standard normal quantile. The clip width is therefore

Δ x_{p} = w Φ^{- 1} (1 - p), Δ x_{10 / 90} = 1.28155 w

and the 15.87 % to 84.13 % width equals $w$ exactly. A pasted scan is fitted for all four parameters by Levenberg-Marquardt least squares, which uses every point rather than two interpolated crossings and reports the RMS residual as a check on the Gaussian assumption. The model is one-dimensional: a knife edge integrates over the other axis, so an elliptical beam needs a scan along each axis, and a multimode or clipped beam gives a fitted $w$ that is no longer its second-moment radius.

Worked example

A 10–90 % blade travel of 64.08 µm gives $w = 64.08 / 1.28155 = 50.00 μ m$ , a 1/e² diameter of 100.0 µm and an intensity FWHM of $w \sqrt{2 \ln 2} = 58.87 μ m$ . The example scan, 21 points 10 µm apart computed for a 50.2 µm beam with a little noise added, fits to $w = 50.23 μ m$ with the center at 4.175 µm and an RMS residual of 0.2 % of the step; reading the 10 % and 90 % crossings from the same points by straight-line interpolation gives 50.93 µm, which is why the fit is the better estimate when the points are coarse.

Related: the knife-edge measurement entry describes the setup, the M² measurement procedure uses widths taken at several distances, and the Gaussian beam calculator propagates the radius found here.

References: J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Applied Optics 22, 3406–3410 (1983). ISO 11146-1:2021, Lasers and laser-related equipment: test methods for laser beam widths, divergence angles and beam propagation ratios. A. E. Siegman, Lasers, University Science Books (1986), chapter 17.