Kerr-Enhanced Optical Spring

We propose and experimentally demonstrate the generation of enhanced optical springs using the optical Kerr effect. A nonlinear optical crystal is inserted into a Fabry-Perot cavity with a movable mirror, and a chain of second-order nonlinear optical effects in the phase-mismatched condition induces the Kerr effect. The optical spring constant is enhanced by a factor of $1.6\pm 0.1$ over linear theory. To our knowledge, this is the first realization of optomechanical coupling enhancement using a nonlinear optical effect, which has been theoretically investigated to overcome the performance limitations of linear optomechanical systems. The tunable nonlinearity of demonstrated system has a wide range of potential applications, from observing gravitational waves emitted by binary neutron star postmerger remnants to cooling macroscopic oscillators to their quantum ground state.

Figure 1

(a) Schematic of an optomechanical system whose coupling is enhanced by the optical Kerr effect. An optical crystal with third-order nonlinearity ${\chi }^{(3)}$ (Kerr medium) is inserted into a laser-driven optical cavity with a movable-end mirror. (b) Theoretical intracavity power curve. Linear and Kerr systems correspond to the cases of Kerr gains at $\zeta =0$ and $\zeta =-1$, respectively. The vertical axis is normalized by the intracavity power at resonance $P_{\max }=2\mathcal{F}{P}_0/\pi$. (c) Schematic of the mechanical and enhanced optical springs, whose complex spring constants are $K_{\mathrm{m}}$ and $K_{\mathrm{opt}}$, respectively. The positive optical spring resulting from the radiation pressure restoring force is enhanced by signal amplification due to the Kerr effect with $\zeta <0$.

Figure 2

(a) Experimental setup. Light emitted from a 1064 nm Nd:YAG laser passed through a Faraday isolator (FI) and an electro-optic modulator (EOM) and was attenuated by a half-wave plate (HWP), polarizing beam splitter (PBS), and beam damper (BD). A periodically poled ${\mathrm{KTiOPO}}_4$ (PPKTP) crystal was inserted into the cavity to induce NOEs. (b) Image of a small mirror suspended by a double spiral spring. A magnet was attached to the back of the mirror, and the cavity length was varied through a coil magnet actuator. (c),(d) Transmitted light power at resonance versus crystal temperature $T_{\mathrm{C}}$. The vertical axes are normalized by the transmitted light power at resonance in the first phase-mismatched condition ($39.6°\mathrm{C}$). The red dotted lines indicate the setting temperature for the optical spring constant measurement. The temperatures producing the phase-mismatched conditions remain periodically present in the high-temperature region, as shown in (d). (e) Measured cavity spectra. The horizontal axis was normalized by the half width at half maximum of the counterpropagating light power.

Figure 3

(a),(b) Estimation of the optical spring constant $k_{\mathrm{opt}}=K_{\mathrm{opt}}(0)$. The input power $P_0$ was set to 600, 450, 300, and 150 mW, and the cavity detuning $\xi$ was finely varied. The crystal temperature $T_{\mathrm{C}}$ was $39.6°\mathrm{C}$ for (a) and $45.4°\mathrm{C}$ for (b). The circles with error bars represent the estimated optical spring constants obtained from the transfer function measurements, and the solid lines in each color show the fitting results with the Kerr gain $\zeta$ and the maximum value of the optical spring constant for the linear theory $k_{\mathrm{opt}-0}=K_{\mathrm{opt}}(0){|}_{\zeta =0,\xi =1/\sqrt{3}}$ as parameters.

Figure 4

Net signal amplification ratio obtained by normalizing the estimated maximum value of the optical spring constant using the relative input light power. The filled circles with error bars represent the estimated signal amplification ratio, and the solid lines in each color show the fitting with the Kerr theory. The vertical axis is normalized by $k_{\mathrm{opt}-0}$ estimated for $T_{\mathrm{C}}=39.6°\mathrm{C}$ and $P_0=600\mathrm{mW}$, and the gray-filled area corresponds to its estimated error.