β-separation line

A laser's frequency-noise spectral density contains everything about its coherence, but most applications still want one number \u2014 a linewidth. The catch: different parts of the spectrum contribute to the lineshape in qualitatively different ways, so integrating the PSD blindly gives nonsense.

Di Domenico, Schilt, and Thomann's geometric rule fixes that. Draw the \u03b2-separation line

on the frequency-noise plot ($S_{\Delta\nu}$ in Hz\u00b2/Hz versus offset frequency $f$). Spectral regions where the measured noise sits above the line are "slow, strong" modulations \u2014 they smear the carrier and build the linewidth. Regions below the line are "fast, weak" \u2014 they generate sidebands and wings but leave the line core intact. The approximate FWHM then follows from the area $A$ of the noise above the line alone:

The construction explains, in one picture, why linewidth depends on observation time: extend the measurement to lower Fourier frequencies and more $1/f$ noise crosses above the line, so the integrated $A$ \u2014 and the reported linewidth \u2014 grows. A "10 kHz laser at 1 ms" and the same device measuring 100 kHz over seconds are both telling the truth; they integrate different stretches of the same PSD.

Bench workflow: measure $S_{\Delta\nu}(f)$ with a discriminator or coherent receiver, overlay the line, integrate the region above it from $1/T_{obs}$ upward, apply the formula \u2014 and quote the observation time next to the result. The white-noise floor far below the line still matters separately: $\pi S_0$ is the intrinsic Lorentzian linewidth, the number tied to Schawlow–Townes physics.