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 17 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, 2 ω L, 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.

FIG. 1.

(a) An opto-mechanical cavity with an optical parametric amplifier inside. The fixed input mirror and a movable end mirror form a cavity, in which there is a nonlinear crystal pumped by the green light. Only the infrared light is resonant inside the cavity. All cavity mirrors have high transmission to the green pump light. (b) A simplified optical feedback model, where the mechanical resonator scatters the main optical field into sidebands, OPA provides gains to the main optical field and sidebands, the beat-note of the main optical field and the sidebands creates a radiation pressure force acting back on the mechanical resonator to create the optical spring effect. The cavity has optical input and vacuum injection, and optical losses from mirror transmission and additional losses are taken into account.

FIG. 1.

(a) An opto-mechanical cavity with an optical parametric amplifier inside. The fixed input mirror and a movable end mirror form a cavity, in which there is a nonlinear crystal pumped by the green light. Only the infrared light is resonant inside the cavity. All cavity mirrors have high transmission to the green pump light. (b) A simplified optical feedback model, where the mechanical resonator scatters the main optical field into sidebands, OPA provides gains to the main optical field and sidebands, the beat-note of the main optical field and the sidebands creates a radiation pressure force acting back on the mechanical resonator to create the optical spring effect. The cavity has optical input and vacuum injection, and optical losses from mirror transmission and additional losses are taken into account.

Close modal

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.

The Hamiltonian of the system can be written as
(1)
where ωc is the resonant frequency of the cavity; ωL is the frequency of the laser; c ̂ and c ̂ are the annihilation and creation operators of the cavity field; p ̂ and q ̂ represent the momentum and position operators for the movable mirror; ε = 2 γ P / ( ω L ) is the cavity field driving strength with input laser power P and frequency ωL; γ = π c / ( 2 F L ) is the photon decay rate of the cavity with Finesse F and cavity length L; G is the non-linear gain of the OPA, which is proportional to the green pump power; and θ is the phase of the pump field. The first term in Eq. (1) corresponds to the intra-cavity power of the infrared light. The second term describes the coupling between the end mirror and the cavity infrared light through radiation pressure. The third term represents the energy of the end mirror. The fourth term represents the input infrared light coupling to the cavity. The last term represents the OPA amplification of the infrared optical field. The equations of motion of the system can be written as
(2)
where ξ ̂ is the Brownian noise operator with mean value of zero and c ̂ i n is the input vacuum noise operator with zero mean value. It satisfies the correlation relation
(3)
Solving Eq. (2) by setting the left side equal to zero, we obtain the steady-state values
(4)
(5)
where
(6)
is the effective cavity detuning and
(7)
is the phase of the cavity field.
After linearization, q ̂ q s + q ̂ , p ̂ p s + p ̂ , c ̂ c s + c ̂, and Fourier transformation of Eq. (2), we can solve the equations in frequency domain and derive the position fluctuations of the movable mirror as
(8)
The first term originates from the thermal noise while the second term arises from the vacuum noise. The d ( ω ) is introduced to simplify the expression of Eq. (8),
(9)
The susceptibility of the mechanical resonator is modified by the opto-mechanical interaction as
(10)
where χ 0 = 1 m ( ω 2 ω m 2 + i ω γ m ) is the intrinsic mechanical susceptibility (without optical spring) and the second term χ os 1 represents the opto-mechanical modification to the susceptibility with the OPA. To make modification clear, we rewrite this equation as
(11)
where the optical spring constant K os and optical damping Γ os are defined as
(12)
(13)
where γ is the cavity half linewidth without OPA, meff is the effective mass of the resonator, and ω is the observation frequency equal to the mechanical mode frequency in our case ω = ω m. It can be seen from Eq. (11) that the resonance frequency ω m and the damping rate γ m of the mechanical resonator are modified by the gain and phase of the OPA. The modified mechanical resonance frequency is
(14)
Hence, the optical spring frequency shift is
(15)
When the frequency shift is much smaller than the mechanical mode frequency, it can be approximated as
(16)
In the actual experiment described in the next paragraph, we monitored the OPA power gain by measuring the cavity transmission power ratio with/without OPA when the cavity and OPA are not detuned (i.e., Δ = 0 and θ = 0). For convenience, we define the OPA power gain as
(17)
By measuring the OPA power gain g OPA, we can determine the OPA nonlinear gain G according to Eqs. (17) and (5).

Equations (12) and (13) demonstrate that the optical spring and damping are proportional to the intra-cavity power (i.e., | c s | 2) 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., θ = 0 and G = 0, and hence ϕ = 0 in Eq. (7)]. The non-zero OPA gain and phase act like an effective cavity detuning.

We note that the intra-cavity power is also a function of OPA gain [Eq. (5)]. The figure of merit used in this paper is, therefore, the optical spring induced mechanical mode frequency shift [Eq. (16)] divided by the intra-cavity power at each cavity detuning. This ratio [Eq. (18)], therefore, represents the effect caused by only the OPA modification of the opto-mechanical interaction,
(18)
Here, I c = ω c C | c s | 2 / 2 L represents the cavity circulating power, C is the speed of light, θ = 0 for this experiment, and ϕ is obtained using Eq. (7).

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 pump beam. The s-polarized 1064 nm main beam 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 locking beam of the cavity (for cavity length control and detuning adjustment). Another p-polarized excitation beam 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.

FIG. 2.

Schematic of the experiment. SHG: second harmonic generation; AOM: acousto-optic modulator; BS: beam splitter; PBS: polarization beam splitter; DBS: dichroic beam splitter; TEC: temperature control; LIA: lock-in amplifier; membrane: mechanical resonator end mirror of the cavity. The excitation beam is amplitude modulated by AOM2 driven by a band limited noise source near the mechanical resonator frequency by AOM2. It drives the resonator to a higher amplitude and allows higher signal to noise ratio measurements.

FIG. 2.

Schematic of the experiment. SHG: second harmonic generation; AOM: acousto-optic modulator; BS: beam splitter; PBS: polarization beam splitter; DBS: dichroic beam splitter; TEC: temperature control; LIA: lock-in amplifier; membrane: mechanical resonator end mirror of the cavity. The excitation beam is amplitude modulated by AOM2 driven by a band limited noise source near the mechanical resonator frequency by AOM2. It drives the resonator to a higher amplitude and allows higher signal to noise ratio measurements.

Close modal

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 > 95 % 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 10 ppm) 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.

FIG. 3.

Sample of the mechanical resonator: There are 25 small windows in the silicon frame. The multi-layer AlGaAs/GaAs coating is bonded on one side of the silicon frame. The coating has high reflectivity at 1064 nm with a transmission of about 10 ppm. Each window is an independent free mechanical oscillator. In this experiment, we use the biggest one which is 1.16 × 1.16 mm2 with an effective mass of 5.7 μg.

FIG. 3.

Sample of the mechanical resonator: There are 25 small windows in the silicon frame. The multi-layer AlGaAs/GaAs coating is bonded on one side of the silicon frame. The coating has high reflectivity at 1064 nm with a transmission of about 10 ppm. Each window is an independent free mechanical oscillator. In this experiment, we use the biggest one which is 1.16 × 1.16 mm2 with an effective mass of 5.7 μg.

Close modal

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 γ / 2 π = ( 1.60 ± 0.02 ) MHz corresponding to a cavity finesse, F = 179 ± 2. 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 g OPA = 2.10 ± 0.15 for 10 mW input and g OPA = 1.52 ± 0.09 for 20 mW input at Δ = 0. From this, we infer a non-linear gain in Eq. (5), G / 2 π = 0.25 ± 0.02 MHz, and G / 2 π = 0.15 ± 0.02 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.

FIG. 4.

The power spectral density of mechanical mode measured from PDH error signal. The dots are measurement data while the solid lines are Lorentzian fitting to the measurement data. The right green dots/line: without detuning; left orange dots/line: red detuning.

FIG. 4.

The power spectral density of mechanical mode measured from PDH error signal. The dots are measurement data while the solid lines are Lorentzian fitting to the measurement data. The right green dots/line: without detuning; left orange dots/line: red detuning.

Close modal

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.

FIG. 5.

Optical spring as a function of the detuning of the cavity with and without OPA. The input power is 10 mW for both measurements. Dots and triangles represent the measured data, while solid and dashed curves represent the theoretical curves with and without OPA, respectively. The intra-cavity power in this figure is normalized to 1 W.

FIG. 5.

Optical spring as a function of the detuning of the cavity with and without OPA. The input power is 10 mW for both measurements. Dots and triangles represent the measured data, while solid and dashed curves represent the theoretical curves with and without OPA, respectively. The intra-cavity power in this figure is normalized to 1 W.

Close modal

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.

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.

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