Phase noise and frequency noise

A single "linewidth" number compresses, and often distorts, what a laser's phase is actually doing. The full description is a noise spectral density: either the phase-noise PSD $S_\phi(f)$ (rad²/Hz, or the RF convention $\mathcal{L}(f)$ in dBc/Hz) or the frequency-noise PSD $S_\nu(f)$ (Hz²/Hz). The two carry identical information, related by $S_\nu(f) = f^2 S_\phi(f)$.

The frequency-noise PSD of a real semiconductor laser has a canonical shape: a 1/f region at low offset frequencies (technical noise: current source, temperature, mechanics, plus intrinsic flicker) descending to a white floor $S_0$ at high offsets (the quantum/spontaneous-emission limit). Each region maps to lineshape differently:

• The white floor alone would produce a Lorentzian line of width $\Delta\nu = \pi S_0$. This is the intrinsic (instantaneous) linewidth, the number tied to Schawlow–Townes physics and the one modern datasheets increasingly quote.
• The 1/f region adds a Gaussian-like envelope whose width grows with observation time: the same laser measures broader over 1 s than over 100 µs. Any linewidth claim without an observation time or method attached is incomplete.

The β-separation line ($S_\nu(f) = 8\ln(2)\,f/\pi^2$) is the standard recipe for connecting a measured PSD to an effective linewidth: noise above the line contributes to the line core; noise below contributes wings only.

Why systems engineers care: coherent communication scales the phase-noise tolerance with symbol rate and QAM order (DSP carrier recovery absorbs slow noise; fast noise near the symbol rate is fatal); interferometric fiber sensing converts laser frequency noise directly into apparent strain via path imbalance ($\delta\nu/\nu = \delta L/L$); LIDAR and optical clocks live and die on the PSD's low-offset behavior.

Measurement: frequency discriminators (imbalanced interferometer, cavity slope) convert $\delta\nu \to$ voltage for direct PSD capture; cross-correlation between two discriminators digs below instrument noise; DSH provides the legacy single-number check.