Mirror coating-thermal-noise mitigation using multi-spatial-mode cavity readout

We present an approach to mitigating coating thermal noise in optical cavities by using multiple transverse electromagnetic (TEM) spatial modes to read out and stabilize laser frequency. With optimal weightings we synthesize a wider sampling of mirror surfaces, improving the averaging of Brownian thermal fluctuations. We show a thermal noise improvement factor of 1.57, comparable to a mesa beam of a nominal 12 m prototype cavity, and a factor 1.61 improvement over the ${\text{TEM}}_{00}$ using three modes in a 0.1 m cavity for a practical laboratory experiment, equivalent to cooling mirrors to 120 K from room temperature.

Figure 1

The proposed multimode readout scheme. Light from a laser is frequency-shifted to match the higher-order mode resonances of a reference cavity. Each frequency component is phase-modulated with a unique RF frequency and injected. Readout is made on a single reflection photodetector, and a Pound-Drever-Hall error signal is extracted for each mode in the cavity. The resulting signals are combined with a weighting that optimizes canceling of thermal noise sampled by each unique mode profile. The synthesized error signal spreads the effective sampled area of the mirrors. The combined error signal is fed back to the laser, via a proportional-integral-derivative (PID) controller, tying laser frequency to that of the optical resonator. Comparison of frequency noise performance is made with a beat note detection with a phasemeter against a reference laser of identical or better performance.

Figure 2

Thermal noise (upper plots) and thermal noise fractional improvement (lower plots) for combinations of Hermite-Gauss (left) and Laguerre-Gaussian (right) modes as compared to the fundamental mode. Cavity parameters: 0.1 m length with a 320 $\mu \mathrm{m}$ beam waist. Thermal noise of single modes of highest order available are also plotted for comparison (${\text{HG}}_{n0}$ and ${\text{LG}}_{p0}$). Improvement is unbounded in the absence of a clipping criterion and unlimited injection and readout resources.

Figure 3

Example Hermite-Gaussian mode weightings for the case of short aspect ratio cavities in the non-clipping-limited case. Optimally combined modes flatten the central region of the beam creating steeper cutoff at the edges. Here the total number of modes is 25, i.e., limited to a maximum mode order number of 4 in both directions. The best weighting places most of the emphasis on the highest-order modes available, but lower order modes still make contributions to effective sampling of the mirror. Table 1 gives the values for this example mode combination.

Figure 4

Ratio of improvement for an optimal combination of the first 25 Hermite-Gaussian modes over the clipping limited ${\text{HG}}_{00}$ fundamental mode as the beam waist is swept. The dark blue line gives the mean value, and the shaded region gives the standard deviation. Maximum improvement to thermal noise peaks at the mesa beam limit of 1.58. Here we use a 12 m cavity and compare against a mesa type beam ($D=4\sqrt{L_{\text{cav}}\lambda /2\pi }$), each clipping at the same 11.2 mm clipping aperture. The sweep is performed by stepping the end mirrors' radius of curvature and recomputing the thermal noise and optimal combination at each point. Combined losses are bound to less than 1 ppm.

Figure 5

Ratio of thermal noise improvement for a three-mode configuration over the fundamental mode. The faction of total weighting in the two higher modes (${\text{HG}}_{02}$ and ${\text{HG}}_{20}$) is swept with the remainder in the fundamental mode (${\text{HG}}_{00}$). Thermal noise improvement peaks at 1.61 with 49% of the power in each HOM.