Finite mirror effects in advanced interferometric gravitational wave detectors

Thermal noise is expected to be the dominant source of noise in the most sensitive frequency band of second-generation, ground-based gravitational-wave detectors. Reshaping the beam to a flatter, wider profile which probes more of the mirror surface reduces this noise. The “Mesa” beam shape has been proposed for this purpose and was subsequently generalized to a family of hyperboloidal beams with two parameters: twist angle $\alpha$ and beam width $D$. Varying $\alpha$ allows a continuous transition from the nearly flat ($\alpha =0$) to the nearly concentric ($\alpha =\pi$) Mesa beam configurations. We analytically prove that in the limit $D\to \infty$ hyperboloidal beams become Gaussians. The ideal beam choice for reducing thermal noise is the widest possible beam that satisfies the Advanced LIGO (Laser Interferometer Gravitational-wave Observatory) diffraction loss design constraint of 1 part per million (ppm) per bounce for a mirror radius of 17 cm. In the past the diffraction loss has often been calculated using the clipping approximation that, in general, underestimates the diffraction loss. We develop a code using pseudospectral methods to compute the diffraction loss directly from the propagator. We find that the diffraction loss is not a strictly monotonic function of beam width, but has local minima that occur due to finite mirror effects and leads to natural choices of $D$. For an $\alpha =\pi$ Mesa beam a local minimum occurs at $D=10.67\mathrm{cm}$ and leads to a diffraction loss of 1.4 ppm. We then compute the thermal noise for the entire hyperboloidal family. We find that if one requires a diffraction loss of strictly 1 ppm, the $\alpha =0.91\pi$ hyperboloidal beam is optimal, leading to the coating thermal noise (the dominant source of noise for fused-silica mirrors) being lower by about 10% than for a Mesa beam while other types of thermal noise decrease as well. We then develop an iterative process that reconstructs the mirror to specifically account for finite mirror effects. This allows us to increase the $D$ parameter to 11.35 cm for a nearly concentric Mesa beam and lower the coating noise by about 30% compared to the original Mesa configuration.

Figure 1

(a) The beam intensity profile $|U_{\alpha }{|}^2$ and (b) corrections $H_{\alpha }$ are shown at fixed $D=10\mathrm{cm}$ for different twist parameters $\alpha$.

Figure 2

(a) The diffraction loss for a Mesa configuration ($\alpha =\pi$) is shown as a function of $D$ on a logarithmic plot for several different mirror radii: $R=16\mathrm{cm}$, $R=17\mathrm{cm}$, and $R=18\mathrm{cm}$. The diffraction loss computed numerically using Eq. (25) (solid, dashed, and dotted-dashed lines) exhibits local minima due to finite mirror effects. It can be seen that the minima get narrower as $R$ increases and that they go below the values estimated using the clipping approximation (dotted lines). (b) The fractional difference $|U{|}_{\mathrm{finite}}^2/|U{|}_{\mathrm{theory}}^2-1$ between the theoretical infinite mirror beam intensity profile and the actual profile given by the first eigenvector of the propagator in Eq. (22) is plotted for $D=10.67\mathrm{cm}$ and $R=16$, 17, 18, and 20 cm. The deviation decreases with $R$ as expected.

Figure 3

The diffraction losses of the fundamental mode and the first excited mode are plotted as a function of $D$ for the $\alpha =\pi$ Mesa configuration. The two curves cross due to finite mirror effects causing an anomalous diffraction loss for the first excited mode.

Figure 4

The noise ratios ${\mathrm{\text{Noise}}}_{\alpha }/{\mathrm{\text{Noise}}}_{\mathrm{Mesa}}-1$ for three types of noise are shown as a function of $\alpha$ for fixed $D=10\mathrm{cm}$. The minimal Gaussian $\alpha =\pi /2$ has the highest noise and Mesa ($\alpha =0$, $\pi$) has the lowest noise.

Figure 5

Diffraction loss as a function of $D$ is displayed for $\alpha =\pi$, $\alpha =0.95\pi$, $\alpha =0.90\pi$. It can be seen that as $\alpha$ decreases the minimum diffraction loss is lower and occurs for larger $D$.

Figure 6

The minimum diffraction loss in ppm (represented by crosses) and the corresponding $D$ (dots) are shown as a function of $\alpha$. The solid line represents the best exponential fit of the form $a+\exp (b+c{sin}^2\alpha )$ with $a=0.094$, $b=-4.34$, and $c=2.20$.

Figure 7

(a) The largest values of $D$ giving a strict 1 ppm diffraction loss are shown as a function of $\alpha$. The discontinuity is caused by finite mirror effects as explained in the text. (b) The corresponding values of the noise, normalized so that the noises for the 1 ppm Mesa ($\alpha =\pi$, $D=9.62\mathrm{cm}$) are all equal to 1. The discontinuity in allowed maximum $D$ leads to a sharp drop in noise at $\alpha =0.91\pi$.

Figure 8

The phase of the fundamental eigenbeam as a function of radius is shown for iteration zero and 250. It can be seen that the iteration scheme drives it closer to zero as expected.

Figure 9

The diffraction loss in ppm calculated using the clipping approximation is compared to that using the propagator eigenvalues for the iterated and original mirrors as a function of $D$. As before, the configuration studied is $\alpha =\pi$ Mesa with $R=17\mathrm{cm}$. The iteration process lowers the diffraction loss by a factor of 30 to 100.

Figure 10

The diffraction loss is shown as function of iteration number for a $\alpha =\pi$ Mesa configuration with $D=11.35\mathrm{cm}$. The original beam (not shown) has diffraction loss at 46.5 ppm, and it can be seen that after a few iterations the diffraction loss begins to converge to an exponential with lower bound $\sim 1\mathrm{ppm}$. The best-fit exponential is given by $0.96+1.616\exp (-0.013i)\mathrm{ppm}$, where $i$ is the iteration number.

Figure 11

The intensity profile $|U_{\pi }{|}^2$ for the mirror with $R=\infty$, $R=17\mathrm{cm}$ uniterated and $R=17\mathrm{cm}$ at iteration 250 are compared for the $\alpha =\pi$ Mesa configuration with $D=11.35\mathrm{cm}$. The finite mirror effects induce oscillations in the intensity profile that do not disappear when the mirror is corrected. An inset shows the central 8 cm “plateau” of the beam in detail.

Figure 12

The correction to the mirror $H_{\pi }$ at iteration 0 and at iteration 250 are compared for the $\alpha =\pi$ Mesa configuration with $D=11.35\mathrm{cm}$, $R=17\mathrm{cm}$. The iteration scheme introduces some bumps on the mirror of the size $\sim 2\mathrm{nm}$. The inset shows the central 8 cm of the mirror in more detail.