Coatingless, tunable finesse interferometer for gravitational wave detection

Future generation gravitational wave interferometric detectors will require a strong reduction of thermal noise, and will be limited by quantum fluctuations of laser light, namely, shot noise and radiation pressure noise. One of the factors which currently constrain the thermal noise performances of this kind of instruments is the dissipation introduced by the optical coating of the mirrors. In a recent article Braginsky and Vyatchanin proposed to reduce the contribution of thermoelastic noise in the coating building an optical cavity with corner reflectors instead of spherical mirrors. In this work we study a solution which, avoiding completely any kind of coating, allows for injection and extraction of light. We discuss the proposed optical scheme, evaluating the expected thermal noise performances.

Figure 1

A possibility for coupling a coatingless cavity with the laser source and the readout. The schematic idea is shown in the upper diagram. With a naive implementation spurious modes are generated in the general case. In the lower figure we show a more careful implementation which avoids losses.

Figure 2

The basic optical setup for a coatingless resonant cavity. Each triangle is the section of a dielectric prism, and the path of the beam is indicated. Several solutions can be derived from it, as discussed in the text. In the left bottom of the picture a graphical representation of the optical properties of the central beam splitter. Each thick arrow is equivalent to a free propagation, while the thin ones represent a transmission or a reflection through the interface. Filled circles stand for the different representative values of the input ($i_1,i_2,i_3,i_4$) and output ($o_1,o_2,o_3,o_4$) fields.

Figure 3

The four schemes considered for a coatingless semitransparent mirror equivalent. Two reflectors are added to the beam splitter and used to reduce the number of input and output ports to two. Note that in some cases one or both of the added reflectors could be attached to the beam splitter, in order to obtain a monolithic object. A dashed line indicate a particular internal resonant path, see the text for a discussion.

Figure 4

The effective reflectivity $r$ minimized over the phase shifts connected to the geometry in the four schemes analyzed. The reflection coefficient $R$ on the surface $\mathcal{S}$ is on the horizontal axis. The reflectivity $\rho$ is fixed to the value of ${\mathrm{SiO}}_2$.

Figure 5

On the left plot the effective minimum reflectivities as a function of the input beam’s incidence angle on the beam splitter surface are compared among the different schemes. On the right plot we show the finesse of the corresponding single ended coatingless resonant cavity. Here $n=1.42$ (silica) and the polarization is parallel to the incidence plane.

Figure 6

The same of the Fig. 5 for a beam polarized perpendicularly to the incidence plane.

Figure 7

The function $G(R,\mathcal{F})$ defined in Eq. (22). For a given value of the finesse $R$ can be tuned in such a way to minimize ${\mathcal{F}}^{-1}\partial \mathcal{F}/\partial T$.

Figure 8

The scheme of a Virgo-like, coatingless interferometer. The equivalent of the long Fabry-Perot cavities $F{P}_{1,2}$ are obtained here using a reflector $R_i$ and a type I CSM $P_i$. The $CSM$ depicted here is not completely monolithic, see the text for a discussion.

Figure 9

Geometrical construction for the curvature of reflectors. Dashed contours are sections of paraboloid (rotation simmetry around the $y$ axis). From this geometrical construction we can see that the reflector is equivalent to a mirror with a given radius of curvature.

Figure 10

Contributions to the thermal noise amplitude for fused Silica 300 K. On horizontal axis the frequency, on the vertical one the equivalent strain amplitude of the noise, in unit $H{z}^{-1/2}$.

Figure 11

Contributions to the thermal noise amplitude for Sapphire at $T=300\mathrm{K}$. On horizontal axis the frequency, on the vertical one the equivalent strain amplitude of the noise, in unit $H{z}^{-1/2}$.

Figure 12

Contributions to the thermal noise amplitude for Sapphire at $T=10\mathrm{K}$. On horizontal axis the frequency, on the vertical one the equivalent strain amplitude of the noise, in unit $H{z}^{-1/2}$.