Revisiting the relation between laser spectrum and phase noise spectral density with new outcomes

We revisit an expansion series introduced a long time ago by D. Middleton to link the phase noise power spectral density (PN-PSD) of an oscillator to its power spectrum, which is poorly known and used in the laser community. In contrast, a different approach proposed by D. Elliott is widely used to compute the spectrum of a laser from its PN-PSD. We show some benefits brought by the use of Middleton's series both on the conceptual and quantitative points of view and we introduce a simple guideline to implement this approach with an arbitrary PN-PSD. We illustrate these considerations with several experimental examples.