Knife-edge measurement

The knife-edge technique measures the transverse profile of a focused or collimated laser beam by translating a sharp opaque blade across the beam path and recording the power passing the blade as a function of blade position.

Geometry.

The transmitted power $P(x)$ as the blade moves across the beam at position $x$ is the complementary error function for a Gaussian beam:

$$P(x) \;=\; \frac{P_0}{2} \, \text{erfc}\!\left( \frac{\sqrt{2} \, (x - x_0)}{w} \right),$$

where $P_0$ is the total beam power, $x_0$ is the beam center, and $w$ is the $1/e^2$ beam radius. The function transitions from $P_0$ (blade fully out) to 0 (blade fully blocking) over a characteristic width set by $w$.

Width extraction. The 10–90 % transition width $\Delta x_{10/90}$ is most commonly used:

$$\Delta x_{10/90} \;=\; x_{90\%} - x_{10\%} \;\approx\; 1.28 \, w,$$

where $x_{P\%}$ is the blade position at which the transmitted power is $P\%$ of the unblocked value. The 1.28 factor is for an ideal Gaussian; real beams may deviate by a few percent.

Other commonly used measures:

Measure	Relation to $w$	Notes
16–84 % width	$1.00 \, w$	Equals $2 \sigma$ (1-sigma full width)
10–90 % width	$1.28 \, w$	Most common; standard convention
5–95 % width	$1.65 \, w$	More tolerant to background but more sensitive to dark counts

Practical setup. A typical knife-edge measurement uses a razor blade or polished metal edge mounted on a motorized linear translation stage with sub-micron resolution. A large-area photodetector (much larger than the beam) is placed immediately behind the blade, removing any sensitivity to beam steering or detector position. The blade is stepped across the beam in $\sim$ 100 to 1000 steps, recording transmitted power at each position.

Advantages. Robust to beam imperfections (insensitive to non-Gaussian shape in the orthogonal direction). Single-axis at a time, but two orthogonal scans (X and Y blade orientations) suffice for separable beams. Requires no specialized hardware beyond a translation stage and a photodetector. Works for beams from UV to far-IR (limit set by detector and blade material).

Limitations. Single-axis at a time — does not capture full 2D profile. Assumes the beam is separable (Gaussian or near-Gaussian) in both axes. For non-Gaussian or strongly astigmatic beams, the orthogonal-axis profile is implicitly averaged and useful information is lost. Multi-mode beams produce non-erfc transitions and require numerical fitting rather than simple width extraction.

Comparison to other techniques.

Technique	1D vs 2D	Resolution	Sensitive to non-Gaussian?
Knife-edge	1D, repeat for orthogonal	sub-μm	No (averages orthogonal axis)
Slit scan	1D, repeat for orthogonal	sub-μm	No (better tolerance than knife-edge for asymmetric beams)
Pinhole scan	2D (raster)	depends on pinhole	Yes — captures full 2D profile
Camera/CCD	2D direct	pixel-limited	Yes
$M^2$ measurement	Multi-Z 1D	combines knife-edge or camera	Yes — quantifies non-Gaussian content

ISO 11146 specifies the second-moment width ($D4\sigma$) as the formal beam diameter for all beam-quality characterization. The knife-edge $1.28 w_{10/90}$ equals $D4\sigma$ for an ideal Gaussian but deviates by 5–20 % for real beams. For research and qualification, $M^2$ measurement (which combines knife-edge or camera widths at multiple Z positions) is the standard; for quick bench measurement, knife-edge alone is sufficient and is the workhorse of laser laboratories.