We experimentally demonstrate that the optical spring effect can be modified using an optical parametric amplifier in an opto-mechanical cavity. The theoretical analysis shows that both the gain and phase of the optical parametric amplifier can modify the frequency of a mechanical resonator in an opto-mechanical cavity. This modification could be used to tune the frequency of peak sensitivity of gravitational wave detectors. The experimental results show a factor of 1.2 ± 0.8 increase in mechanical resonator frequency shift induced by optical spring by tuning the optical parametric amplifier gain.
The sensitivities of current gravitational wave detectors (GWDs), such as Advanced LIGO/Virgo,1,2 are limited by quantum shot noise at high frequencies and are limited or close to being limited by quantum radiation pressure noise at low frequencies.3,4 Manipulating the input quantum noise using squeezed vacuum injection5,6 is one of the techniques to improve the GWD's quantum noise limited sensitivity. Phase squeezed vacuum has been routinely used in Advanced LIGO/Virgo and has improved the sensitivity by about 3 dB in the shot noise limited band.3,4 By detecting a combination of two quadratures of the output signal, one may optimize the signal-to-noise ratio and improve the GWD sensitivity.7 On the other hand, changing the dynamics of the GWD test mass from a free-mass to a harmonic oscillator in the detection band can also improve the quantum noise limited sensitivity. Braginsky et al.8,9 realized that low noise rigidity created by radiation pressure in a detuned optical cavity can enhance quantum noise limited sensitivity of position sensing of an otherwise free mass. Buonanno and Chen10,11 analyzed in detail the sensitivity of a laser interferometer gravitational wave detector with a detuned signal recycling cavity (SRC), where the radiation pressure force acting on the suspended test masses will change their dynamics and, hence, their response to gravitational waves. The detuned SRC also creates ponder-motive squeezing induced by cross correlation between the phase and amplitude quadrature of the optical field. At the opto-mechanical resonant frequency of the test mass, the sensitivity of the GWD can be improved significantly.6,10 The frequency of peak sensitivity can be tuned by manipulating the parameters of the SRC to tune the opto-mechanical resonance frequency. A tunable peak sensitivity frequency would allow deterministic signals like in-spiral events to be tracked as their frequency evolves, improving detection signal to noise ratio. The idea of dynamic tuning was first proposed by Meers et al.,12 however, by tuning the optical resonance frequency of the SRC. Simakov13 did time-domain analysis of a dynamically tuned recycling interferometer for detection of chirp gravitational wave signals and showed signal-to-noise improvement of for a shot noise limited GEO 600-like detector. Here, we experimentally demonstrated another way of dynamically tuning the interferometer, by tuning the optomechanical resonance frequency using an optical parametric amplifier (OPA) inside the recycling cavity.
There are a few ways to tune the frequency of the opto-mechanical resonance to the detection band. Increasing the arm-cavity intra-cavity power is one of them. However, higher laser power results in adverse effects, such as thermal deformation14–17 and parametric instabilities18,19 in the detector, which need to be controlled. The opto-mechanical resonance frequency can also be tuned by tuning the phase of the signal recycling cavity or recycling mirror (SRM) position.12,13 However, it is difficult to move the SRM position by the order of light wavelength while maintaining the detector observation due to the multi-degree control associated with the detuned SRC. A variable reflectivity SRM20–22 was proposed to replace the conventional SRM to allow change in both detuning and bandwidth of the GWD. A scheme with an optical parametric amplifier (OPA) inside the SRC was proposed23 by Somiya et al. to enhance the optical spring effect and to tune the opto-mechanical resonance frequency to the GW detection band. Korobko et al.24 analyzed the scheme in detail to include the internal squeezing effect and show that the detector sensitivity spectrum can be modified by engineering the optical spring. In this paper, we experimentally studied an opto-mechanical cavity containing an OPA and demonstrated the OPA-enhanced optical spring effect. Different from the original proposal,23 the OPA in our experiment is contained within the cavity where there is a strong carrier field, simplifying the experiment. Similar schemes have been proposed and analysed25–27 for other applications, such as ground-state cooling of mechanical resonators.
We consider a simplified model as shown in Fig. 1(a), representing our experimental setup to demonstrate the OPA enhanced optical spring effect. A nonlinear crystal is placed inside an optical cavity, with a movable end mirror. The 1064 nm (infrared) laser beam with angular frequency ωL is injected into the cavity that is locked to the laser frequency with a tunable frequency offset so that the movable end mirror experiences an optical spring effect. The non-linear crystal is pumped by a 532 nm (green) laser light of frequency, , which is injected through the fixed input mirror, but is not resonant inside the cavity as mirrors have high transmission at 532 nm. Figure 1(b) represents a simplified optical feedback model of the system. The movable mirror (mechanical resonator) scatters part of the main infrared laser into sidebands. The OPA provides gain to the optical fields of the main laser and sidebands. The radiation pressure force of the beat-note between the main infrared laser light and sidebands acts back on the movable mirror to create an optical spring effect and modify the mechanical resonator dynamics. The OPA amplifies both the main infrared laser light and sidebands, hence modifying the optical spring effect.
Huang and Agarwal28 analyzed the opto-mechanical cavity with OPA inside the cavity in the context of cavity cooling of the mechanical resonator. They proved that the OPA can enhance the effective cooling. Later on, it was realized that the OPA can be tuned to suppress the Stokes scattering process and relax the resolved sideband requirement for ground-state cooling.29,30 We adopt the formalism of Huang and Agarwal28 to analyze the effect of the OPA enhanced optical spring effect.
Equations (12) and (13) demonstrate that the optical spring and damping are proportional to the intra-cavity power (i.e., ) and are a function of OPA gain and phase. When the OPA gain is zero (i.e., G = 0), the optical spring and optical damping are the same as that in a simple optomechanical cavity.31 Without cavity detuning, but with non-zero OPA gain, the optical spring occurs when the OPA phase is non-zero, but the optical spring disappears when the OPA phase is zero [i.e., , and hence in Eq. (7)]. The non-zero OPA gain and phase act like an effective cavity detuning.
A schematic of the experimental setup is shown in Fig. 2. The laser source is a monolithic nonplanar Nd:YAG laser with 2 W single mode output power at 1064 nm. Half of the laser power goes into the second harmonic generator to generate the 532 nm light. The s-polarized 532 nm laser beam of about 300 mW is used as the OPA . The s-polarized 1064 nm of ∼10–20 mW is injected into the OPA and interacts with the mechanical oscillator. Some of the p-polarized 1064 nm laser passes through an acousto-optic modulator (AOM1) to be frequency-shifted and used as the of the cavity (for cavity length control and detuning adjustment). Another p-polarized is used to drive the mechanical resonator with radiation pressure force. We use the traditional Pound–Drever–Hall (PDH) locking technique32 to lock the cavity to the laser frequency with a control bandwidth less than the resonant frequency of the mechanical resonator. The PDH error signal is sent to a spectrum analyzer for monitoring the displacement signal of the mechanical oscillator as well as a data acquisition system to record data for post-processing and to analyze the optical spring effect.
The cavity has a V-shape design and sits in a vacuum tank to reduce the air-damping of the mechanical oscillator and to isolate acoustic noises. There is a PPKTP crystal with a size of 1 × 2 × 10 mm3 close to the input mirror. Both end faces of the crystal are anti-reflective (AR) coated to reduce intra-cavity loss. The flat fixed input mirror has AR coatings at both 1064 and 532 nm on the outside surface, 98% reflectivity for 1064 nm, and AR coating for 532 nm on the inside surface. The concave fixed folding mirror has a reflectivity 99.98% at 1064 nm and a transmission at 532 nm with a radius of curvature of 100 mm. The movable mirror at the end is a multi-layer AlGaAs/GaAs coating material with high reflectivity (transmission ) at 1064 nm. The thickness of the material is about 6 μm. The total cavity length is 262 mm and has a beam waist of 50 μm on the surface of the input mirror and another beam waist of 180 μm on the surface of the mechanical resonator.
Figure 3 shows one of the samples of the mechanical resonator. It consists of a 420 μm thick silicon frame with multi-layer AlGaAs/GaAs coating bonded to one side. The frame is through-etched to form 25 small AlGaAs/GaAs coating membrane windows of different sizes within an area of 7.3 × 7.3 mm2. The biggest window has a size of 1160 × 1160 μm2 and the smallest 175 × 175 μm2. Each window can be regarded as a mechanical oscillator. Here, we use the biggest window whose drum-mode resonant frequency is 35.5 kHz with an effective mass of 5.7 μg. The coating for 1064 nm has a power transmission of ∼10 ppm.
The PPKTP is temperature-controlled to about 33 C to maintain the phase matching condition for the OPA. The OPA gain depends on the power of the green pump light and the optical cavity loss. The OPA can work on either amplification or de-amplification determined by the phase of the 532 nm pump beam. When working on amplification, high signal light results in pump depletion and thermal effects and reduces OPA gain in our experiments. A PZT-mounted steering mirror in the green beam path is used to control the pump beam phase. The phase locking was achieved by dither locking. The PZT was driven with a small signal at 19.6 kHz. The cavity transmission photodetector signal is demodulated with a lock-in amplifier phase locked to the PZT drive signal. The lock-in amplifier output is the phase error signal for the green pump light. This error signal drives the PZT to maintain the green pump phase. The green pump light phase is tuned by adjusting the DC offset of the error signal. The s-polarized main infrared beam inside the cavity can be detuned from the cavity resonance by tuning the frequency of AOM1 in the p-polarized cavity locking light beam.
We first measured the cavity linewidth without OPA to be MHz corresponding to a cavity finesse, . With main input beam power of 10 and 20 mW, respectively, we measured the cavity transmission power (proportional to intra-cavity power). Comparing the cavity transmission power with and without the 300 mW green OPA pump light injection, we determined the OPA power gain for 10 mW input and for 20 mW input at Δ = 0. From this, we infer a non-linear gain in Eq. (5), MHz, and MHz, respectively. The OPA gain with 20 mW input is smaller than that with 10 mW input due to OPA pump depletion, thermal effects, and possible other unknown effects. It needs further investigation to have a conclusive answer. To avoid confusion, we show here only the optical spring results with 10 mW input.
We measured the mechanical resonator's frequency by monitoring the PDH error signal. The mechanical resonator was driven to a high amplitude by an independent amplitude-modulated excitation beam at the resonator back surface through radiation pressure to allow higher signal to noise ratio measurements. AOM2 in Fig. 2 driven by a band limited noise source at frequencies near the resonator frequency provides amplitude modulation on the excitation beam. The cavity detuning for the main laser beam was created by frequency shifting the locking beam from the main laser beam through AOM1 in Fig. 2. The optical spring effect was measured as a change in resonator frequency. The cavity is blue-detuned when the main laser beam frequency is higher than the cavity resonant frequency, while the opposite is called red-detuned. Figure 4 shows typical mechanical mode power spectral density with and without red detuning. The solid lines are Lorentzians fit to the measured data.
Figure 5 shows the resonant frequency shift of the mechanical resonator caused by the optical spring effect as a function of the cavity detuning without and with OPA and with main laser input power of 10 mW. Here, we show the frequency shift divided by the intra-cavity power [Eq. (18)], to show the OPA effect. The yellow dots in Fig. 5 show the measured frequency shift without OPA, while the yellow dashed line represents the theoretical prediction. The maximum normalized frequency shift is about 52 Hz per Watt. The green dots show measured frequency shift with OPA and the main beam power of 10 mW. The maximum frequency shift is close to 71 Hz per Watt. The green solid lines represent the theoretical prediction. The results indicate that the OPA enhances the optical spring effect.
The phase of the green pump light is kept on resonance to maintain the maximum OPA gain even when the infrared beam is detuned from resonance. However, when the detuning is close to or larger than the cavity linewidth, this phase locking loop is hard to maintain because the feedback loop gain and phase change with the cavity detuning. This is the reason we have limited data points in Fig. 5 in measurements with the OPA. It would be possible to extend the measurement to the far detuned regime by adding a variable gain amplifier in the control loop, or by increasing the input power while detuning to maintain a constant intra-cavity power.
In summary, we demonstrated the OPA enhanced optical spring effect, which could potentially be used in the signal recycling cavity of gravitational wave detectors to tune the opto-mechanical frequency and, thereby, the frequency of peak sensitivity for targeting known GW events. We have shown a factor of 1.2 ± 0.8 increase in optical spring effect by tuning the OPA gain. In theory, tuning can be achieved by tuning either the OPA gain or OPA phase or both of them. In our experiment, the OPA gain is small, because a relatively strong carrier is required to create observable optical spring effect, which induced the OPA pump depletion and thermal effects, and limited the OPA gain. This will not happen in the laser interferometer gravitational wave detectors where there is almost no carrier light in the signal recycling cavity. The limited OPA gain prevented us from observing the optical spring effect when the cavity is on resonance, but with non-zero OPA gain and phase, as well as the effect by tuning OPA phase. Another point to note is that the optical spring demonstrated in the experiment was stable. However, the optical spring in the gravitational wave detectors will be unstable, which will require an appropriate control scheme to stabilize.11 The results showing here cannot be directly compared with a result of a signal-recycled interferometer because of the different configurations. Subject to further experimental investigation with signal recycled interferometer, OPA enhanced optical spring as a dynamic tuning approach could be applied to the third generation gravitational wave detectors, such as the Einstein Telescope.33
This work was supported by the Australian Research Council Discovery Project via No. DP160102447 and the Center of Excellence for Gravitational Wave Discovery Project via No. CE170100004. We would like to acknowledge that Professor Shiuh Chao at the National Tsing Hua University of Taiwan provided the silicon windows and Dr. Garrett Cole at Thorlabs, Inc. provided AlGaAs/GaAs coatings.
Conflict of Interest
The authors have no conflicts to disclose.
Jue Zhang: Formal analysis (lead); Investigation (lead); Writing – original draft (equal). Chunnong Zhao: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Hengxin Sun: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Hui Guo: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Carl Blair: Investigation (equal); Writing – original draft (equal). Vladimir Bossilkov: Data curation (supporting); Formal analysis (supporting); Writing – original draft (equal). Michael Page: Data curation (supporting); Formal analysis (supporting). Xu Chen: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Supervision (supporting). Jiangrui Gao: Funding acquisition (supporting); Investigation (supporting); Methodology (supporting); Resources (supporting); Supervision (supporting). Li Ju: Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.