Thermal phase noise in Fabry-Pérot resonators and fiber Bragg gratings

Abstract

We report a rigorous derivation of the power spectral density (PSD) spectrum of the phase noise in the transmitted and reflected signals of a Fabry-P\'erot (FP) interferometer. In transmission off and on resonance, the PSD at low frequencies is found to be simply equal to the PSD of the single-pass phase noise (the phase noise acquired by a signal that has traveled once through the FP interferometer) times ${({n}_{g}/n)}^{2}$, where ${n}_{g}$ is the group index and $n$ the phase index of the light inside the FP interferometer. Equivalently, the phase noise (in radians) is equal to the single-pass phase-noise times ${n}_{g}/n$. This result states that the phase noise is proportional to the number of passes light makes through the FP interferometer, as expected physically. This simple expression provides a quick means of calculating the phase noise in a linear Fabry-P\'erot interferometer of any length in the range of frequencies of interest to the most applications. At high frequencies, we show that the PSD spectrum can also be expressed in a simple closed-form expression on resonance, but that off resonance it must be calculated numerically. Numerical simulations show that the PSD oscillates between 3 dB below the low-frequency PSD and 3 dB below the single-pass PSD. Equivalent expressions are derived for the phase noise in the signal reflected by a Fabry-P\'erot interferometer, which is found to have the same magnitude at low frequencies, but to be typically significantly lower at high frequencies.