Optical Interferometers with Reduced Sensitivity to Thermal Noise

A fundamental limit to the sensitivity of optical interferometry is thermal noise that drives fluctuations in the positions of the surfaces of the interferometer’s mirrors, and thereby in the phase of the intracavity field. Schemes for reducing this thermally driven phase noise are presented that rely upon the coherent character of the underlying displacements and strains. Although the position of the physical surface fluctuates, the optical phase upon reflection can have reduced sensitivity to this motion. While practical implementation of such schemes for coherent compensation face certain challenges, we hope to stimulate further work on this important thermal noise problem.

Figure 1

(a) The undistorted shape of a substrate of radius $a=1.5\mathrm{mm}$ and thickness $l=1\mathrm{mm}$ is depicted by the shaded region and wire frame [28]. Axial displacements $u_z(x,y,z)$ and strains ${ϵ}_{zz}(x,y,z)$ are shown across planes at $z=0$, $l/2$, $l$ (top to bottom) for the eigenmode ${\mathit{\xi }}_0(\mathit{r})$ with eigenfrequency ${\omega }_0/2\pi =2.22\mathrm{MHz}$. Plotted are contours for $u_z(x,y,z)$ on which ${ϵ}_{zz}(x,y,z)$ is color coded at a time of maximum axial extension, where $u_z(x,y,z)=0$ absent excitation. The phase for propagation from $z=z_R$ to the substrate and back is modified by surface motion $u_z$ at $z=0$. (b) Coating stack for a high reflectivity mirror with embedded FP cavity for high strain sensitivity.

Figure 2

Phase shift $\delta \Phi (k)$ and transmission coefficient $T(k)$ for two different coatings illustrated in Fig. 1b and specified in the text. (a),(b) The case $\zeta =-1600/\mathrm{m}$ for increasing phase $j{\eta }_{\mathrm{FP}}$ of the internal resonant structure. (a) Total phase $\delta \Phi (k)$ with “magic” wave vectors $k_{\pm }$ and (b) associated transmission coefficient $T(k)$ for $j=1$, 4, 16 with ${\eta }_{\mathrm{FP}}=\pi$ from widest to narrowest curves. (c),(d) $\delta \Phi (k)$ and $T(k)$ for the case $\zeta =-5000/\mathrm{m}$ for two coatings with slightly different internal resonances ${\eta }_{\mathrm{FP}}^{(A,B)}$ with distinct values $k_{-}^B$, $k_{+}^A$ for nulling noise from mirrors $B$, $A$. In all cases, $\delta \Phi (k)/|\delta \theta (k_0)|$ is plotted for $q_0>0$, with $\delta \theta (k)$ given by the dashed lines in (a),(c).

Figure 3

(a) Correlation of thermally driven axial displacements $C(z_1,z_2)$ and (b) displacement-strain correlation $Q(z_1;z_2,\Delta z_2)/\sqrt{\Delta z_2/w_0}$ versus $z_2/w_0$ for $z_1/w_0=0$, 1, 3, 10, with $\Delta z_2/w_0={10}^{-3}$ in (b). The dashed trace in (a) is $N(z_2,z_2)$. (c) Illustration of the mirror geometry discussed in the text. (d) Spectral density of phase fluctuations $\mathit{F}(z_2)$ for the reflected field ${\mathit{E}}_{\mathit{r}}$ in (c) for an ${\mathrm{SiO}}_2$ substrate. From top to bottom, $\alpha =1.5$, 0.3, 1.0, 0.7 with the blue trace for $\alpha ={\alpha }_{\min }$ overlaying the curve for $\alpha =0.7$. The dashed curve is from a simple model that incorporates incoherent contributions from transverse strains to $\delta \beta$. All curves are for $\sigma =0.2$.