What M² means
\(M^2\) is pronounced “M-squared.” It compares the real beam parameter product—waist radius \(\omega_0\) times far-field divergence half-angle \(\theta\)—with the diffraction-limited value:
\[ M^2 =\frac{\omega_0\theta}{\lambda/\pi} =\frac{\pi\omega_0\theta}{\lambda}\ge1 \]An ideal TEM₀₀ beam has \(M^2=1\). If \(M^2=1.2\), the beam parameter product is 20% above the diffraction limit—the “squared” is already part of the parameter's name, so do not square 1.2 again.
What it changes
\[ \theta=\frac{M^2\lambda}{\pi\omega_0}, \qquad z_R=\frac{\pi\omega_0^2}{M^2\lambda} \]At a fixed waist, the beam diverges \(M^2\) times faster than an ideal Gaussian. For a fixed collimated beam diameter and lens, its best focused radius is approximately \(M^2\) times larger. Gaussian-like propagation can therefore be modeled with \(\lambda\to M^2\lambda\).
Non-circular beams generally have separate values \(M_x^2\) and \(M_y^2\) along their principal axes.
How to obtain M²
First check the laser manufacturer's datasheet, but use the value only if it applies to the same operating power, alignment, and configuration. If it is not specified—or the operating point has changed—measure the beam caustic yourself.
How to measure M²
- Focus the beam with a known lens.
- Measure its width at multiple axial positions through the waist and into both diverging regions.
- Use a camera profiler, scanning slit, or knife edge and fit the resulting caustic.
The ISO convention uses the second-moment (D4σ) diameter. Define \(\omega_\sigma=D_{4\sigma}/2\) and fit
\[ \omega_\sigma^2(z)=\omega_{\sigma,0}^2 +\left(\frac{M^2\lambda}{\pi\omega_{\sigma,0}}\right)^2(z-z_0)^2 \]The fit returns the waist position \(z_0\), waist radius \(\omega_{\sigma,0}\), and \(M^2\). The 1/e² and D4σ diameters agree for an ideal Gaussian; do not mix them for a non-Gaussian beam.