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 $S_{\varphi }(f)$ (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 $S_{\nu }(f)=f^2{S}_{\varphi }(f)$.

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 $\mathrm{\Delta }\nu =\pi S_0$. 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 ($S_{\nu }(f)=8\ln (2)f\mathrm{/}{\pi }^2$) 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 ($\delta \nu \mathrm{/}\nu =\delta L\mathrm{/}L$); 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.

References: G. Di Domenico, S. Schilt, and P. Thomann, "Simple approach to the relation between laser frequency noise and laser line shape," Applied Optics 49, 4801 (2010). C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE Journal of Quantum Electronics 18, 259 (1982). L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). K. Kikuchi, "Fundamentals of coherent optical fiber communications," Journal of Lightwave Technology 34, 157 (2016).