Effects of mirror aberrations on Laguerre-Gaussian beams in interferometric gravitational-wave detectors

A fundamental limit to the sensitivity of optical interferometers is imposed by Brownian thermal fluctuations of the mirrors’ surfaces. This thermal noise can be reduced by using larger beams which “average out” the random fluctuations of the surfaces. It has been proposed previously that wider, higher-order Laguerre-Gaussian modes can be used to exploit this effect. In this paper, we show that susceptibility to spatial imperfections of the mirrors’ surfaces limits the effectiveness of this approach in interferometers used for gravitational-wave detection. Possible methods of reducing this susceptibility are also discussed.

Figure 1

Power spectral densities of uncoated mirror surface roughness. The dashed lines are the measured spectra of three Advanced LIGO arm cavity mirrors. A model approximating these spectra [black trace, see (2)] was created to generate the random maps used in our work.

Figure 2

Surface figure (in nm) of a typical generated phase map with piston, tilt, power, and astigmatism terms subtracted.

Figure 3

Normalized intensity distribution of the ${\mathrm{LG}}_{3,3}$ mode at the mirror position (${\omega }_0=0.021\mathrm{m}$, $z=1997.25\mathrm{m}$).

Figure 4

Fabry-Perot cavity with a perturbation $\delta h(x,y)$ on the end mirror.

Figure 5

Frequency shift of LG modes introduced as a result of realistic mirror perturbations.

Figure 6

Intensity distributions of the total field reflected from an arm cavity whose end mirror was perturbed by the reference map. These distributions were calculated independently via two different techniques. Top—analytic method described in Sec. 3b. Bottom—fast-Fourier-transform(FFT)-based numerical simulation (see Sec. 4b).

Figure 7

Contrast defect with the ETM of one cavity perturbed by rescaling the reference phase map: solid red is from the analytical calculation, blue marker is from the FFT calculation.

Figure 8

Ratio, numerical/analytical, of single-cavity contrast defects calculated with the end mirror perturbed by a 0.3 nm rms figure error.

Figure 9

Interferometer contrast defect as a function of test mass surface roughness (all four mirrors are perturbed with random phase map at the same level of rms). Solid markers report mean values of numerical results with the corresponding shaded regions illustrating 1 standard deviation (see 4b), which is roughly 4 times higher than the trace in Fig. 7.

Figure 10

Analytic calculation with different conditions for reducing the contrast defect: solid curve is the original contrast defect; dashed line has corrective rings added to the phase map; dotted curve is with detuned injecting laser frequency.

Figure 11

Frequency shift in LG modes introduced by adding rings to the reference phase map.