How it is calculated
A TEM₀₀ beam has irradiance \(I(r) = I_0\,e^{-2r^2/w^2}\), where \(w = D_b/2\) is the 1/e² radius. Integrating over the open area gives closed-form results:
Here \(a\) is the aperture radius, \(s\) the full slit width, and \(x_0\) the distance from the beam center to the blade edge (positive while the blade has not yet reached the center). The inverse relations give the aperture needed for a target transmission \(T\):
Common clipping ratios
| Dt / Db | Circular aperture | Slit (width / Db) | Edge irradiance / peak |
|---|---|---|---|
| 0.5 | 39.35 % | 68.27 % | 60.7 % |
| 1.0 | 86.47 % | 95.45 % | 13.5 % |
| 1.25 | 95.61 % | 98.76 % | 4.39 % |
| 1.5 | 98.89 % | 99.73 % | 1.11 % |
| π/2 ≈ 1.571 | 99.28 % | 99.83 % | 0.72 % |
| 2.0 | 99.966 % | 99.994 % | 0.034 % |
| 2.5 | 99.9996 % | > 99.9999 % | 0.0004 % |
For a circular aperture: 99 % throughput needs \(D_t \ge 1.52\,D_b\), 99.9 % needs \(1.86\,D_b\), and 99.99 % needs \(2.15\,D_b\). A common design rule is to make every aperture at least \(\pi w \approx 1.57\,D_b\) across.
Clipping is more than lost power
A hard edge diffracts. Even 1 % clipping produces Fresnel ripples in the near field downstream and side lobes in the focal plane. Those can cause hot spots on optics, raise the measured M², and spread energy outside the processing spot. High-power and imaging systems therefore usually clear apertures at 2× the beam diameter or more. A deliberately truncated beam (scanner pupils) also changes the focused spot. See the focused spot tool for the APO factor. The power that is clipped is absorbed or scattered by the mount, which matters at kilowatt levels.
Knife-edge beam profiling
Translating a blade through the beam while recording transmitted power traces out \(T_\text{edge}(x_0)\), the error-function profile. For a Gaussian beam:
- The 15.9 % and 84.1 % points are separated by exactly \(w\), one 1/e² radius.
- The 10 % and 90 % points are separated by \(1.2816\,w\), so \(D_b = 2w \approx 1.561\,\Delta x_{10\text{–}90}\).
These clip-level conversions assume a Gaussian profile. For non-Gaussian beams, ISO 11146 specifies second-moment (D4σ) widths, which a knife edge can only approximate.
Worked example
An 8 mm beam passing a 12 mm iris (\(D_t/D_b = 1.5\)) transmits \(1 - e^{-4.5} = 98.89\) %. With 500 W in the beam, about 5.6 W lands on the iris. Getting to 99.9 % needs an aperture of \(1.858 \times 8 = 14.9\) mm.
Assumptions and limits
- Ideal TEM₀₀ profile, centered on the aperture. Pointing offsets reduce transmission.
- Transmission is evaluated at the aperture plane (power accounting only). It does not describe the diffracted field downstream.
- Multimode and top-hat beams have heavier or sharper edges. Use measured profiles for those.
References
- A. E. Siegman, Lasers, University Science Books (1986), ch. 17 (aperture transmission and diffraction of Gaussian beams).
- ISO 11146-1:2021, Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios.