How it is calculated
A beam with 1/e² waist radius \(w_0\) and beam quality \(M^2\) keeps the hyperbolic profile of an ideal Gaussian, but spreads \(M^2\) times faster. Every quantity follows from the Rayleigh range:
\(\theta\) is the far-field half-angle divergence of the 1/e² radius. The full angle \(\Theta = 2\theta\) is what most laser datasheets quote. The beam parameter product \(\mathrm{BPP} = w_0\theta = M^2\lambda/\pi\) is invariant through ideal optics. A lens can trade waist size for divergence but cannot reduce their product, which makes BPP the natural figure of merit for fiber and multimode lasers.
At \(z = \pm z_R\) the radius has grown by \(\sqrt{2}\), the area has doubled, and the peak intensity has halved. The wavefront curvature is strongest there (\(R = 2z_R\)) and is flat both at the waist and in the far field. Over the whole focus the Gouy phase advances by π relative to a plane wave.
Depth of focus for a spot-size tolerance
The full Rayleigh range (\(\sqrt{2}\) growth) is too loose for most processes. If the spot radius may grow by at most a fraction \(\varepsilon\), the usable depth is
A 5 % tolerance gives only 0.64 \(z_R\) in total. Because \(z_R \propto w_0^2\), halving the spot cuts the depth of focus by a factor of four.
Worked example
A 1064 nm beam with M² = 1.1 focused to \(w_0\) = 25 µm:
- \(z_R = \pi (25\,\mu\text{m})^2 / (1.1 \times 1.064\,\mu\text{m}) = 1.678\) mm.
- At z = 5 mm the radius is \(25\sqrt{1 + (5/1.678)^2} = 78.6\) µm. The peak intensity is 10.1 % of its value at the waist, and \(R = 5.56\) mm.
- The full-angle divergence is \(2 \times 1.1 \times 1.064 / (\pi \times 25) = 29.8\) mrad (1.708°). BPP = 0.373 mm·mrad.
- The ±5 % depth of focus is 1.07 mm in total.
Assumptions and limits
- Paraxial propagation. The model assumes \(\theta \ll 1\). Above roughly 0.1–0.2 rad (NA ≳ 0.15), the 1/e² radius from these formulas becomes increasingly approximate.
- M² is an embedded-Gaussian model. For a multimode beam, \(w(z)\) and \(z_R\) describe the second-moment (D4σ) radius, as defined in ISO 11146. \(R(z)\) and the Gouy phase strictly apply to the embedded TEM₀₀ component.
- Uniform medium. In a medium of index \(n\), use \(\lambda/n\) for the wavelength. Thermal lensing and aberrations are not included.
- Symmetric, stigmatic beam. For astigmatic or elliptical beams (diode lasers, some fiber lasers), apply the calculation separately to each axis with its own waist position and M².
References
- A. E. Siegman, Lasers, University Science Books (1986), ch. 17.
- H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
- ISO 11146-1:2021, Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios.