Stable double-resonance optical spring in laser gravitational-wave detectors

We analyze the optical spring characteristics of a double pumped Fabry-Perot cavity. A double-resonance optical spring occurs when the optical spring frequency and the detuning frequency of the cavity coincide. We formulate a simple criterion for the stability of an optical spring and apply it to the double-resonance regime. Double-resonance configurations are very promising for future gravitational-wave detectors as they allow us to surpass the Standard Quantum Limit. We show that stable double resonance can be demonstrated in middle scale prototype interferometers such as the Glasgow 10 m-Prototype, Gingin High Optical Power Test Facility or the AEI 10 m Prototype Interferometer before being implemented in future gravitational-wave detectors.

Figure 1

Scheme of an Advanced LIGO interferometer pumped by two lasers. The main laser is detuned to give a positive optical rigidity and negative damping (it is tuned on the right slope of resonance curve), while the auxiliary laser is detuned to give negative optical rigidity and positive damping.

Figure 2

Roots correspond to a stable (slightly damped) oscillation if $Y_r^{\prime }$ and $Y_i$ have the same signs near roots of $Y_r$, i.e. plots of $Y_r$ and $Y_i$ correspond to each other as shown on plot.

Figure 3

For the case of 3 pumps the roots are stable if in accordance with criterion (16) the plots of $Y_r$ and $Y_i$ correspond to each other as shown.

Figure 4

“Ideal” double resonance: curve $Y_r$ touches $z$ axis in point $z_1$: a) left root ${\overline{z}}_1$ of $Y_i$ coincides with $z_1$; b) curve $Y_i$ touches $z$ axis in the same point $z_1$.

Figure 5

Detailed plot of two close roots of $Y_r$ (14c). The roots of $Y_i$ (14d) have to be shifted to the right relative to the roots of $Y_r$ in order to fulfill the stability criterion (16).

Figure 6

Trace 1: susceptibility $|\Psi (f)|$ as function of frequency $f=\Omega /2\pi$ for the parameters given in Eq. (19). Traces 2,3: same as trace 1 but with $ϵ\to ϵ/2$ and with $ϵ\to ϵ/4$. Trace 4: susceptibility of free mass ($4/m(2\pi f{)}^2$).

Figure 7

Scheme of the optical rigidity experiment within the Glasgow Prototype Interferometer. Only the end mirror can move, while the mirrors PRM and IM can be considered as unmovable because their masses are 30 times larger than that of the end mirror.

Figure 8

Detailed plot of the two close roots of $Y_r$ (14c): $z_{1,2}=z_0\pm ϵ$. The roots of $Y_i$ (14d) are separated by the same distance $2ϵ$, but the trace of $Y_i$ itself is shifted by the distance ${ϵ}_1$.