New family of light beams and mirror shapes for future LIGO interferometers

Advanced LIGO’s present baseline design uses arm cavities with Gaussian light beams supported by spherical mirrors. Because Gaussian beams have large intensity gradients in regions of high intensity, they average somewhat poorly over fluctuating bumps and valleys on the mirror surfaces (thermal noise). Flat-topped light beams (mesa beams) are being considered as an alternative because they average over thermal noise more effectively. However, the proposed mesa beams are supported by nearly-flat mirrors, which experience a very serious tilt instability. In this paper we propose an alternative configuration in which mesa-shaped beams are supported by nearly-concentric spheres, which experience only a weak tilt instability. The tilt instability is analyzed for these mirrors in a companion paper by Savov and Vyatchanin. We also propose a one-parameter family of light beams and mirrors in which, as the parameter $\alpha$ varies continuously from 0 to $\pi$, the beams and supporting mirrors get deformed continuously from the nearly-flat-mirrored mesa configuration (FM) at $\alpha =0$, to the nearly-concentric-mirrored mesa configuration (CM) at $\alpha =\pi$. The FM and CM configurations at the endpoints are close to optically unstable, and as $\alpha$ moves away from 0 or $\pi$, the optical stability improves.

Figure 1

Optical axes of the families of minimal Gaussian beams used to construct: (a) an FM mesa beam [7], denoted in this paper $\alpha =0$; (c) our new CM mesa beam, denoted $\alpha =\pi$; (b) our new family of hyperboloidal beams, which deform, as $\alpha$ varies from 0 to $\pi$, from a FM beam (a) into a CM beam (c).

Figure 2

The correction $H_{\alpha }(r)$ to the mirror shape for hyperboloidal beams in a LIGO arm cavity ($L=4\mathrm{km}$) with $D=10\mathrm{cm}$ and with twist angles $\alpha$ between $\pi /2$ and $\pi$. For $\alpha =0$, the correction is the negative of that for $\alpha =\pi$; for $\alpha =0.1\pi$ it is the negative of that for $0.9\pi$; for any $\alpha$ between 0 and $\pi /2$, it is the negative of that for $\pi -\alpha$. For $\alpha =\pi$ (the Mexican-hat correction for our new CM mesa beam), $H_{\pi }(r)$ drops to about $-500\mathrm{nm}$ (half the wavelength of the light beam) at $r=16\mathrm{cm}$ (the mirror’s edge). These corrections are added onto the fiducial spheroidal shape $S_{\alpha }(r)$ [Eq. (13)].

Figure 3

The light beam’s un-normalized intensity $|U_{\alpha }{|}^2$ as a function of radius $r$ on the mirror, for hyperboloidal beams in a LIGO arm cavity ($L=4\mathrm{km}$) with $D=10\mathrm{cm}$ and various twist angles $\alpha$. For $\alpha =0$ and $\pi$, the intensity has the mesa shape; for $\alpha =0.5\pi$ it is a minimal Gaussian.

Figure 4

Geometric construction for computing the hyperboloidal field $U_{\alpha }(r,S_{\alpha },D)$ on the fiducial spheroid $S_{\alpha }$ (a segment of which is shown dotted).

Figure 5

The light beam’s un-normalized intensity $|U_{\alpha }{|}^2$ as a function of radius $r$ on the mirror, for hyperboloidal beams in a LIGO arm cavity ($L=4\mathrm{km}$) with fixed diffraction losses: 2.68 ppm in the clipping approximation, assuming mirror radii of 16 cm. The mesa beam ($\alpha =0)$ has $D=10\mathrm{cm}$ and is identical to that of Fig. 3. For $\alpha =0.1\pi$, to keep the diffraction losses at 2.68 ppm, $D$ has been increased to 10.5 cm and the physical beam diameter is, correspondingly, a bit larger than in Fig. 3. For $\alpha =0.2\pi$, $D$ has been increased to $D=13.0\mathrm{cm}$; for $\alpha =0.252\pi$, it has been increased to $D=\infty$, producing a beam shape that is Gaussian to the accuracy of our numerical computations, but is substantially larger than the minimal Gaussian of Fig. 3 and is approximately the same as the baseline design for advanced LIGO. (Our numerical computations suggest that for $D=\infty$ and all $\alpha \ne 0$ or $\pi$, the hyperboloidal beam is Gaussian, with width varying from minimal, $\sigma =b=\sqrt{\lambda L/2\pi }$ at $\alpha =\pi /2$ to $\sigma \to \infty$ as $\alpha \to 0$ or $\pi$, but we have not been able to prove this analytically.)