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-Pérot (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-Pérot 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-Pérot interferometer, which is found to have the same magnitude at low frequencies, but to be typically significantly lower at high frequencies.

Thermal Noise-Limited Fiber-Optic Sensing at Infrasonic Frequencies

“…However, their claim of reaching the thermal noise limit at infrasonic frequencies (below 20 Hz) has been shown to be inconsistent with theoretical predictions [3], [11], [15]. In fact, a thorough analysis recently reported by Skolianos et al shows that the predicted thermal noise is nearly an order of magnitude below the measured noise in [15], indicating the reported strain sensitivity may still be limited by other noises, such as the laser phase noise. Nevertheless, this debate prompts the question of whether the current technology is ready to attain thermal noise-limited fiber-optic sensing in the infrasonic range [3], [4].…”

“…In a similar effort, Gagliardi et al demonstrated the highest strain sensitivity below 10 Hz with a fiber Fabry-Perot (FP) cavity and a frequency combstabilized diode laser [2]. However, their claim of reaching the thermal noise limit at infrasonic frequencies (below 20 Hz) has been shown to be inconsistent with theoretical predictions [3], [11], [15]. In fact, a thorough analysis recently reported by Skolianos et al shows that the predicted thermal noise is nearly an order of magnitude below the measured noise in [15], indicating the reported strain sensitivity may still be limited by other noises, such as the laser phase noise.…”

“…of n g /n when the laser is on resonance with the cavity and the noise frequency is much smaller than the cavity free spectral range [15]. Here n g refers to the group index of the FP cavity, which is related to the cavity finesse F through the relation n g ≈ (2/π)n F when F 1.…”

Observation of fundamental thermal noise in optical fibers down to infrasonic frequencies

“…For the range of infrasonic frequencies (10 mHz to 30 Hz), Gagliardi et al 4 reported strain measurements with the highest sensitivity obtained by using a fiber Fabry-Perot (FP) cavity. However, the sources of the noise floor are still in doubt 18,24,25 . Recently, in the experiment for ultra-stable fiber-stabilized laser, Dong et al 8 observed that the frequency noise PSD range from 1 Hz to 100Hz exhibits a 1 f characteristic seems consistent with Duan's theory.…”

“…Over the past three decades, such thermal noise has been investigated theoretically [9][10][11][12][13][14][15][16][17][18] and experimentally 1,7,[19][20][21][22][23] . Glenn 9 first calculated the fluctuations of thermodynamic phase in optical medium and showed that they can be larger than shot noise over a wide frequency range.…”