Fig. 4
Fractional frequency instability measurements. a Measured fractional frequency instabilities (FI) of the beat frequency between the Fabry-Pérot (FP) cavity reference laser and the free-running laser (squares in black), and the beat frequency between the reference laser and the laser stabilized to the WGM resonator without evacuation (squares in olive), respectively. The red squares and bars are the mean \((\mu ={n}^{-1}\sum _{i=1}^{n}{{\rm{FI}}}_{I})\) and standard deviation \((\sigma =\sqrt{{n}^{-1}\sum _{i=1}^{n}{({{\rm{FI}}}_{I}-\mu )}^{2}})\) of the frequency instability of the laser stabilized to the resonator calculated from 10 sets of frequency counting data at 1 s averaging time, respectively, measured with a frequency counter where n is the number data set. The red circles and bars are μ and σ of the frequency instability of the laser stabilized to the resonator derived from the 74 frequency noise power spectral density (FNPSD) measurement traces. The magenta area represents the expected Allan deviation bound imposed by random walk frequency noise. b The noise statistics of 74 FNPSD traces. The red curve is the mean value and the cyan area is the connected standard deviations. The blue line is the thermorefractive noise limit. Both mean frequency noise and standard deviation are reduced along the offset frequency and the frequency noise reaches the thermorefractive noise limit at 6 Hz. However, the thermo-mechanical noise, induced by the residual thermal expansion, is still concentrated near the carrier causing the standard deviation of the Allan deviation measurements in a
