Anomalous dynamic backaction in interferometers

We analyze the dynamic optomechanical backaction in signal-recycled Michelson and Michelson-Sagnac interferometers that are operated off the dark port. We show that in this case—and in contrast to the well-studied canonical form of dynamic backaction on the dark port—optical damping in a Michelson-Sagnac interferometer acquires a nonzero value on cavity resonance, and additional stability and instability regions on either side of the resonance, revealing additional regimes of cooling and heating of micromechanical oscillators. In a free-mass Michelson interferometer for a certain region of parameters we predict a stable single-carrier optical spring (positive spring and positive damping), which can be utilized for the reduction of quantum noise in future-generation gravitational-wave detectors.

Figure 1

Optomechanical setups. (a) The simplest setup for measuring the position of the test mass (mirror) via a reflected laser beam. (b) A Michelson interferometer with end mirrors performing antisymmetric motion. (c) Field amplitude at the output port of a Michelson interferometer as a function of arm-length difference. Points of zero amplitude correspond to dark fringes in the interference pattern. In a small vicinity near the dark fringe (marked with thick red) linear signal transduction is possible. (d) Resonant amplification of the optical field in a Fabry-Pérot cavity. (e) A signal-recycled Michelson interferometer. (f) A signal-recycled Michelson-Sagnac interferometer with an oscillating semitransparent membrane.

Figure 2

(a) Canonical $K$ and $\Gamma$ defined in Eq. (1) for $\xi =0$, normalized to their respective maximum values, for $\Omega <\gamma$. (b) Same for $\Omega >\gamma$. (c) and (d) respectively show the optical damping and spring in the signal-recycled MSI detuned from the dark port at $\delta l_{\mathrm{DP}}=3{\lambda }_0/4$ by $\xi \approx 0.01{\lambda }_0$, with $P_{\mathrm{in}}=200$ mW, ${\lambda }_0=1064$ nm, $\mathcal{L}=8.7$ cm, $R_{\mathrm{m}}^2=0.17$, $T_{\mathrm{SR}}^2=3\times {10}^{-4}$, and $\gamma \approx \Omega =2\pi \times 133$ kHz.

Figure 3

Anomalous $K$ and $\Gamma$ in a Michelson interferometer, defined by Eqs. (2a) and (2b), and normalized to their respective maximum values. (a) $K$ acquires three zeros starting from ${\gamma }_{\mathrm{m}}=\gamma /4$, or equivalently, ${\gamma }_{\mathrm{m}}={\gamma }_{\mathrm{SR}}/3$. (b) $\Gamma$ acquires three zeros starting from ${\gamma }_{\mathrm{m}}=\gamma /3$ or ${\gamma }_{\mathrm{m}}={\gamma }_{\mathrm{SR}}/2$. (c) ${\gamma }_{\mathrm{m}}=7\gamma /3$ or ${\gamma }_{\mathrm{m}}=3{\gamma }_{\mathrm{SR}}/4$ and (d) ${\gamma }_{\mathrm{m}}=\gamma /2$ or ${\gamma }_{\mathrm{m}}={\gamma }_{\mathrm{SR}}$ demonstrate the existence of stable optical spring ($K>0$ and $\Gamma >0$) in a certain range of detunings for larger offsets from the dark fringe.

Figure 4

Fields in a Michelson-Sagnac interferometer.

Figure 5

Normalized (nonsymmetrized) spectral densities of backaction noise for different membrane positions $\xi =\delta l-\delta l_{\mathrm{DP}}$ where $\delta l_{\mathrm{DP}}$ is defined by Eq. (A2). For better visualization we choose membrane power reflectivity $R_{\mathrm{m}}^2=0.3$, beam splitter asymmetry ${\delta }_{\mathrm{BS}}=-0.3$, and detuning $\Delta =0$. (a) $R_{\mathrm{SR}}^2=1$. (b) $R_{\mathrm{SR}}^2=0.7$.