How the focused spot size is calculated
For a beam with 1/e² diameter \(D_b\) at a lens of focal length \(f'\), the focused 1/e² spot diameter is
The apodization factor APO depends only on the truncation ratio \(T = D_b/D_t\), where \(D_t\) is the smallest clear aperture the beam passes through. For an untruncated Gaussian (\(T \lesssim 0.5\)), \(\mathrm{APO} = 4/\pi \approx 1.273\) and the formula reduces to the textbook result \(d_F = 4\lambda f' M^2/(\pi D_b)\), or \(w_0 = M^2\lambda f'/(\pi w_L)\) in radius form.
Truncation and the APO factor
A larger input beam gives a smaller spot, but only until the aperture starts to clip it. Clipping diffracts light out of the central lobe. The focal profile then moves away from a Gaussian toward an Airy pattern, and the spot stops shrinking in proportion to \(1/D_b\). This calculator does not use a lookup table. It evaluates the Fraunhofer diffraction integral of a truncated Gaussian,
and finds the radius at which \(|U|^2\) falls to \(e^{-2}\) of its peak.
| T = Db/Dt | APO | Power transmitted | Comment |
|---|---|---|---|
| ≤ 0.5 | 1.27–1.29 | > 99.9 % | Effectively untruncated Gaussian |
| 0.7 | 1.45 | 98.3 % | Common compromise for scanners |
| 0.9 | 1.69 | 91.5 % | Noticeable side lobes |
| 1.0 | 1.83 | 86.5 % | Aperture at the 1/e² diameter |
| → ∞ | 1.645·T | → 0 | Uniform illumination (Airy disk); \(d_F \to 1.645\,\lambda f'/D_t\) |
In practice, T ≈ 0.6–0.8 is a reasonable design point for galvo scanners and focusing heads. Past that, enlarging the beam gives little reduction in spot size and costs throughput and pointing sensitivity.
Depth of focus
The Rayleigh range \(z_R = \pi w_0^2/(M^2\lambda)\) is the distance over which the beam radius grows by \(\sqrt{2}\) (the area doubles and the peak fluence halves). Most processes tolerate much less than that. For a maximum allowed spot growth \(\varepsilon\) (e.g. 5 %), the usable depth of focus is
Because \(z_R \propto w_0^2\), halving the spot size cuts the depth of focus by four. Tight-focus ablation and scribing processes therefore need height sensing or tracking optics.
Worked example
A fiber laser at 1064 nm with M² = 1.1 and an 8 mm beam, focused by a 100 mm lens with a 12 mm clear aperture: T = 0.667, so APO = 1.412 and
That gives \(z_R \approx 0.29\) mm, and the ±5 % depth of focus is only about 0.18 mm in total. Without the aperture the spot would be 18.6 µm. The 12 mm aperture costs about 11 % in spot size and 1.1 % in power.
Assumptions and limits
- Paraxial, aberration-free lens. Real lenses add spherical aberration, which matters at low f-numbers (roughly f/# < 5 for singlets). Use a ray-trace or the lens vendor's spot data when θ is large.
- Collimated input with the waist near the lens. The focus then lies close to \(f'\). A diverging or converging input shifts the focus. See the Gaussian beam propagation tool.
- M² is applied as a multiplier. This is the embedded-Gaussian model. It is exact for untruncated beams and a good approximation with truncation.
- Truncated beams are no longer Gaussian. When T > ~0.8, \(z_R\) and DOF from the 1/e² radius are approximate, and side lobes carry a few percent of the energy.
References
- A. E. Siegman, Lasers, University Science Books (1986), ch. 17–18 (Gaussian beams, aperture diffraction).
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019), ch. 3.
- ISO 11146-1:2021, Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios.