Thermal-Noise Limit in the Frequency Stabilization of Lasers with Rigid Cavities

We evaluate thermal noise (Brownian motion) in a rigid reference cavity used for frequency stabilization of lasers, based on the mechanical loss of cavity materials and the numerical analysis of the mirror-spacer mechanics with the direct application of the fluctuation dissipation theorem. This noise sets a fundamental limit for the frequency stability achieved with a rigid frequency-reference cavity of order $1\mathrm{H}\mathrm{z}/\sqrt{\mathrm{H}\mathrm{z}}$ ($0.01\mathrm{H}\mathrm{z}/\sqrt{\mathrm{H}\mathrm{z}}$) at 10 mHz (100 Hz) at room temperature. This level coincides with the world-highest level stabilization results.

Figure 1

Assumed cavity shapes. Common parameters are axial hole diameter: 10 mm, mirror diameter: 25.4 mm, mirror thickness: 5 mm, and coating thickness: $2\mu \mathrm{m}$. The thick arrows show positions of Gaussian forces in the calculation.

Figure 2

Calculation result (case No. 1) and the experimental results (converted from Refs. 20, 22 into frequency noises at 563 nm). In this case, the contribution from the mirror substrate is dominant, because it is made of ULE. The experimental results agree well with our calculation. The discrete calculation performed at frequency intervals of $\sim 10\mathrm{H}\mathrm{z}$ was interpolated to give the expected smooth $f^{-1/2}$ dependence in this plot.