Abstract
Squeezed light is injected into the dark port of gravitational wave interferometers, in order to reduce the quantum noise. A fraction of the interferometer output light can reach the OPO due to sub-optimal isolation of the squeezing injection path. This backscattered light interacts with squeezed light generation process, introducing additional measurement noise. We present a theoretical description of the noise coupling mechanism and we prove the model with experimental results. We propose a control scheme to achieve a de-amplification of the backscattered light inside the OPO with a consequent reduction of the noise caused by it. The scheme was implemented at the GEO 600 detector and has proven to be crucial in maintaining a good level of quantum noise reduction of the interferometer for high parametric gain of the OPO. In particular, the mitigation of the backscattered light noise helped in reaching 6 dB of quantum noise reduction [Phys. Rev. Lett. 126, 041102 (2021) [CrossRef] ]. We show that the impact of backscattered-light-induced noise on the squeezing performance is phenomenologically equivalent to increased phase noise of the squeezing angle control. The results discussed in this paper provide a way for a more accurate estimation of the residual phase noise of the squeezed light field. Finally, the knowledge of the backscattered light noise coupling mechanism is a useful tool to inform the design of the squeezing injection path in terms of path stability and optical isolation.
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1. Introduction
In the framework of gravitational wave detectors [1–4], quantum-noise-limited interferometric measurements are very sensitive to the influence of scattered light. Stray light that reaches the photodetectors by another but the intended path will generally have randomly fluctuating phase and amplitude and will add a noisy contribution to the measured signal.
Typical scattering sources are, for example, small imperfections of optical surfaces, like point absorbers [5], and residual reflections from AR coatings. Scattered light coming from the environment, such as reflections from the vacuum envelope [6] or from baffles [7], has in general less impact. The propagation direction of this light does not coincide with the axis of the Gaussian beam carrying the signal, and just a small fraction of this stray light is scattered off of the optic surfaces back into the interferometer (IFO) mode [8]. In the special case of a squeezing-enhanced IFO [9], the squeezed light source itself can become an important scatterer.
The squeezed vacuum field is injected at the output port of the IFO with the help of a Faraday rotator ($\mathrm {FI_{inj}}$ in Fig. 1), but polarization imperfections can lead to some light from the IFO output being sent backwards along the squeezing-injection path. Stray light hitting the parametric amplifier is then re-injected into the IFO, creating unwanted phase fluctuations at the readout photodiode (hPD). This paper is focused on this kind of stray light, which will be called as backscattered light. To describe this process, special care must be taken to consider the nonlinear interaction of the stray-light field with the parametric amplifier. Of particular interest is the back-propagating light in the same polarization and spatial mode as the squeezed field because when scattered back toward the IFO, light in this mode will reach the main photodetector with high efficiency.
The squeezed light sources currently used in gravitational wave interferometers are called optical parametric oscillators (OPO). These are optical cavities with an embedded second-order nonlinear medium. The geometry of the OPO may vary for different detectors, as in the case of Virgo [10] and LIGO [11]. At GEO 600 [3], the OPO is a hemilithic cavity containing a plano-convex PPKTP crystal of about 10 mm length [12]. For the back-propagating stray-light field, the OPO is a highly over-coupled resonant cavity, which means that most of the field enters the cavity where it experiences the parametric gain before being reflected back toward the IFO. This can also be understood as seeding the OPO with the bright stray-light field instead of a vacuum field, thus creating a displaced squeezed state.
This paper is focused on the coupling of scattered light noise after the back-reflection off the OPO. Section 2 provides an analytical model of the parametric interaction. Experimental results confirming this model are presented in section 3. A new technique, developed at GEO 600, to actively control the backscattered light phase and reduce its contribution to the final noise is introduced in section 4. In section 5, finally, the impact of the new technique on the estimation of squeezed-light phase noise is discussed.
2. Description of backscatter coupling
The approach adopted in the following can be considered semi-classical since the extra-cavity fields that solve the equations of motion for the nonlinear interaction are described by complex numbers: the steady-state component of the linearization [13, §3.1.5]. Reference [14] describes a detailed model of the noise coupling using quantum operators but considering the assumption of constant phase between pump and seed field.
Furthermore, we consider a slowly-varying approximation: the equations of motion are solved for a steady state, but the seed is free to change its phase in the phasor diagram. However, the rotation rate is much slower than the cavity round-trip time, so at each measurement time, the phase relation between the seed and pump field can be considered fixed, and the cavity equations can be solved for the steady-state case.
Solving the steady-state equations of motion for a seeded OPO [15, §2.4.7] leads to the following expression for the reflected bright field in a reference frame rotating at the optical frequency:
The bright squeezed field is injected into the IFO and eventually reaches hPD, interfering with the output light from the IFO. Let the IFO output light amplitude be described by the real value $\mathrm {A_{IFO}}$ so that the other phases are referred to its reference frame. Assuming the backscattered light field to be small with respect to this field, the noise at the hPD is:
The phases can be split into two parts
where $\Phi (t)$ and $\Theta (t)$ are the slow varying contributions—at frequencies below the detection band—and $\delta \phi (t)$ and $\delta \theta (t)$ are the fast and small contributions. The squeezing angle is actively controlled to match the squeezed quadrature to the readout quadrature: $\Theta = 0$. Thus Eq. (2) can be expressed as:Equation (5) and (7) show that the noise due to backscattered light is made up of three contributions:
- • The first term describes a nonlinear coupling of the slow phase fluctuations $\Phi (t)$ of the stray-light field. This can be neglected as long as the absolute magnitude of these low-frequency fluctuations is not large enough to up-convert them into the detection bandwidth. At GEO 600, the calibrated detection bandwidth starts at $40\mathrm {Hz}$. A more detailed explanation of the up-convertion process is provided in section 3.1.
- • The second term describes a linear coupling of the fast small-amplitude phase fluctuations $\delta \phi (t)$ of the stray-light field.
- • The third term, finally, shows a linear coupling of residual squeezing-angle fluctuations $\delta \theta (t)$ in the presence of a stray-light field.
It is interesting to note that the first two noise coupling mechanisms actually get smaller when applying more squeezing: the parametric amplification process in the OPO deamplifies the backscattered light field (along with the vacuum fluctuations) in the correct quadrature to reduce its influence on the detected signal. The coupling of squeezing-angle fluctuations, however, increases with the pump power.
Equation (7) can be expressed as a power spectral density in units of relative intensity noise at the hPD:
The fact that the coupling of squeezing-angle fluctuations increases with increasing parametric gain leads to a situation where there is a maximum amount of squeezing that can be applied before the added backscatter noise outweighs the reduction of quantum noise. It is evident that this effect becomes a limiting factor if higher squeezing levels are desirable. Earlier theoretical and experimental investigations [16–18] of backscattered light noise have described the coupling of the stray light’s phase fluctuations, but have generally assumed the squeezing angle to be constant, thus neglecting the coupling mechanism described by the last term of Eq. (7).
3. Experimental results
The different coupling mechanisms described by Eq. (7) and their dependence on the parametric gain are investigated in this section.
3.1 Coupling of slow phase fluctuations
The effects of backscattering are observed by artificially introducing low-frequency path-length modulations of the squeezing injection path with the help of a mirror mounted on a longitudinal piezoelectric actuator. For sufficiently large amplitudes, larger than one wavelength, the nonlinear coupling—described by the first term in Eq. (5)—can up-convert the slow phase fluctuations into the detection band. This effect can be understood by considering the Jacobi-Anger expansion of the phase-modulated signal:
where $z$ is the modulation depth, $J_n(z)$ is the n-th Bessel function. The result is the characteristic ‘scattering shoulder’ in the spectrum of the detector output signal, as shown in Fig. 2. The measurement was done both with and without pumping the OPO. As predicted by Eq. (5), the amount of backscattering depends on the OPO’s nonlinear gain. With the squeezing angle tuned for reducing the shot-noise, the backscatter noise is also attenuated by the same factor. The plot shows also the simulated scattering shoulder, obtained usingAs already pointed out, applying higher squeezing factors only improves backscattering due to stray-light phase fluctuations. Thus, an upper limit on the impact of this form of backscattering can be set by comparing the detector’s output signal in the state where stray light reaches the OPO without any pump applied to a situation where the squeezer path is completely blocked with a high-quality beam damp. For GEO 600 no difference in the measured noise level was observed, showing that stray-light phase fluctuations are not a limiting source of noise at any frequency within the detection band. Experiments for the squeezing demonstration at LIGO Hanford had come to similar conclusions but could also show that for the more demanding requirements of future advanced detectors the backscatter immunity was not yet adequate at low frequencies [17]. This indicates that better optical isolation or more path-length stability will be needed at some point.
3.2 Coupling of fast phase fluctuations
Another form of backscatter-induced detection noise is the linear coupling of squeezing angle-fluctuations as described by the third term of Eq. (5). Experimental evidence for this effect had first been noticed at GEO 600 when an unexpected degradation of the squeezing level was observed for high nonlinear-gain settings of the OPO. Initially this was attributed to excess RMS phase noise of the squeezer which would lead to similar observations (see section 5). Backscattering was found to be the underlying mechanism when it became clear that the problem had started due to a lowered isolation of one of the Faraday isolators after a rerouting of the path trough the respective optics. Subsequent experiments could confirm this by intentionally lowering the backscatter isolation and observing the effect on the squeezing level.
The additional noise on the hPD caused by this form of backscattering is mostly white in GEO 600’s shot noise-limited frequency band (see Fig. 3). The coupling is additionally modulated with the sine of the randomly fluctuating mean phase of the stray-light field $\Phi$, this was observed experimentally as a random variation of the noise level on second timescales. This strong modulation of the fast phase-noise coupling spreads out features in the spectrum which therefore appears quite smooth when averaged over many periods of the modulation.
The predicted scaling of the backscatter noise with the OPO parametric gain was tested in the detection-band frequency range. To show this dependence experimentally, a harmonic signal with amplitude $\sim 30\,\textrm {mrad}$ and frequency in the detection band ($5.5\,\textrm {kHz}$) was sent through an actuation matrix to two different actuators. One was a piezo-mounted mirror in the in-air injection path (PZT in Fig. 1), the other was the frequency modulation input of the $80\,\textrm {MHz}$ RF generator used as the local oscillator for the phase locked loop (PLL) of the squeezer master laser with the IFO main laser. An actuation on the FM port of the RF generator affected only the squeezing angle $\delta \theta$. On the other hand, being the injection path traveled by both the squeezed vacuum and the backscattered light, the actuation on the piezo-mounted mirror alone would excite both $\delta \theta$ and $\delta \phi$ (twice as much). To excite only the backscattered phase $\delta \phi$, both the actuators were operated at the same time, with amplitudes and phases tuned in order to cancel the modulation on the squeezed vacuum phase. To achieve this, the line at $5.5\,\textrm {kHz}$ in the spectrum of the in-loop error signal for the squeezing phase control was minimized. The latter configuration was similar to what was done in [19], but with a different actuation on the squeezing phase: the FM input of the PLL’s LO was used instead of a piezo-mounted mirror in the the pump field path. The different excitations were performed for several values of the pump power and for both squeezing and anti-squeezing applied. Then the resulting band-limited root mean square (BLRMS) of the lines in the readout hPD signal was measured and normalized by the BLRMS of the line for the case where no pump light was injected into the OPO. The result is shown in Fig. 4. The predicted models—obtained using the second and third terms of Eq. (7)—are also plotted, after averaging out the slow backscattered-light phase fluctuations $\Phi$.
The excitation induced by the squeezing phase actuation $\delta \theta$ scales as $2|\sinh (r)|$, while the backscattered-light phase $\delta \phi$ excitation should scale as $\exp (-r)$. The dotted line in the figure takes into consideration a non-perfect cancellation of the squeezing angle coupling: $\exp (-r)+0.05\cdot 2\cdot \sinh (r)$.
4. Backscatter noise reduction by locking the backscattered light phase
To mitigate the effect of backscatter noise there are several general approaches. On the one hand, the amount of stray-light power reaching the squeezer needs to be limited as far as possible. For this reason, one or more additional Faraday isolators in the injection path (see Fig. 1) are used to suppress this back-propagating light with a typical isolation factor of $40\,\mathrm {dB}$ each [20]. Some small amount of the backscattered light can still reach the OPO, depending mainly on the imperfections of the injection Faraday rotator and on the number and quality of additional Faraday isolators. On the other hand, due to the linear coupling term in Eq. (7), residual fluctuations of the squeezing phase in the detection band must be kept as low as possible: the amount of residual phase noise will depend on the active control and on the intrinsic path stability of the overall optical setup. Also, the geometry of the OPO plays an important role: if it is built as a ring cavity the stray-light field is intrinsically decoupled from the squeezed vacuum field which propagates in the opposite direction. LIGO OPOs are in a bow-tie configuration for this exact reason [21]. Optical imperfections still lead to coupling of the stray light, but suppressed by over $40\textrm {dB}$ [22], which is roughly equivalent to one additional Faraday isolator. Finally, the active control of the backscattered light phase presented in this section can be employed in order to minimize the linear coupling of squeezing angle fluctuations.
Wade et al. [19] have suggested a technique for backscatter evasion by a combined modulation of the injection phase and pump phase to frequency shift the backscatter noise out of the detection band. The technique was demonstrated in a proof-of-principle experiment, but effective suppression without increasing the RMS phase noise of the squeezing injection appears challenging. A novel active control for the mitigation of backscattered-light-induced noise was developed and employed at GEO 600: the technique relies on the phase-locking of the backscattered light with the OPO pump field. The use of this technique makes it possible to achieve a higher level of squeezing, particularly towards a high parametric gain of the OPO. Figure 5 shows how the backscattered angle control was an important factor in reaching 6dB of squeezing level at GEO 600 [23]. At the optimal antisqueezing level (15.9dB), the squeezing level was improved by 0.2dB (from 5.8dB to 6.0dB) when the backscattering angle control loop was engaged.
By looking at Eq. (7), a way to mitigate the linear coupling of the residual squeezing angle fluctuations $\delta \theta$ would be to lock the slow backscattered light phase $\Phi$ to $0$ or $\pi$, this corresponds to a parametric deamplification of the backscattered light inside the OPO. The control system was called BSP (Back-Scattering Phase) loop. The error signal for this control system can be obtained by dithering the pump phase and demodulating the resulting line at the hPD. This signal, which tracks the backscattered-light phase, can be fed back to a piezo-mounted mirror to correct for the injection path length change, thus locking the backscattered light phase to that of the IFO carrier light. The squeezer pump is in turn phase-locked to the IFO carrier field through the coherent control loop [24]. Thus, a suitable hierarchy of control bandwidths ensures the phase-locking between backscattered light and pump.
The modulation is obtained via PLL. A harmonic signal at frequency above the detection band ($6868\,\mathrm {Hz}$) is fed to the FM port of the $80\,\mathrm {MHz}$ RF generator used as LO for the PLL between the squeezer master laser and the IFO main laser. A trade-off for the amplitude of this modulation must be found, since high values would directly lead to higher RMS phase noise of the squeezed light; on the other hand, the modulation must be strong enough for the line in $h$ to have a reasonable SNR. The optimal modulation amplitude was found to be around $6\,\textrm {mrad}$ at GEO 600.
After rearranging Eq. (2), approximating $\eta _\textrm{esc}=1$ and using Eq. (6), the contribution of backscattered light to the photocurrent from the hPD is
The squeezing angle $\theta (t)$ is actively kept at $0$ through the coherent control loop, so when a modulation is introduced it becomes $\theta (t)=\gamma \sin (\Omega t)$. For a small modulation depth $\gamma$:
When this signal is demodulated with an appropriate demodulation phase $\chi$, the error signal is obtained:
The control signal, obtained after digital filtering of the error signal, is fed to a piezo-mounted mirror (PZT in Fig. 1) in the squeezing injection path. The calibrated amplitude spectral density (ASD) of the control signal (Fig. 6) up to the unity gain frequency (UGF) of the control loop (UGF=$14\,\mathrm {Hz}$) gives an indication of the slow contribution to the backscattered light phase noise, which is due to the length fluctuations of the path between the OPO and the IFO dark port. The major contribution to the RMS of this noise is given by two peaks at $0.55\,\mathrm {Hz}$ and $0.78\,\mathrm {Hz}$, corresponding to the resonance frequencies of suspended optics in the squeezed light path such as the three beam directing optics (BDOs) and the signal recycling mirror (SRM) (see Fig. 1). The cumulative RMS of the calibrated control signal down to $10\,\mathrm {mHz}$ is $1.3\,\mathrm {rad}$ and it is dominated by the above mentioned suspended optics resonances. In the time domain, the control signal oscillates with an amplitude of $\sim 2\,\mathrm {rad}$ indicating that the intrinsic path stability between the OPO and the IFO dark port is good enough to avoid up-conversion of the slow phase fluctuations $\Phi$ into the detection-band frequency range. In fact, the scattering shoulder (described in section 3.1) has a maximum frequency given roughly by the product of the modulation frequency and the modulation amplitude. In the case of the natural oscillation of the squeezing injection path length, obtained from the calibrated BSP control signal, the scattering shoulder would extend up to $\lesssim 2\mathrm {Hz}$, below the detection bandwidth of GEO 600.
Above the UGF of the BSP loop, the error signal is dominated by the sensing noise. This corresponds to the shot noise at the hPD photodiode, being the error signal obtained from a demodulation at a frequency belonging to the shot-noise limited band of the detector. Figure 6 shows how the sensing noise can be reduced by increasing the modulation amplitude ($\gamma$ in Eq. (14)) of the squeezing angle. Due to the fact that higher modulation corresponds to higher RMS phase noise of the squeezed light, the loop is operating with a modulation amplitude of $6\mathrm {mrad}$, in order to not degrade the squeezing level. This means that the BSP loop is partially re-injecting the sensing noise in the squeezing injection path longitudinal control. This extra noise can be observed in the coherent control error signal, but it is not a limiting factor for the overall performance of the squeezing angle control.
5. Influence on phase noise estimation
Relative fluctuations of the angle between the squeezed quadrature and the measured quadrature degrade the shot noise reduction performance by coupling some of the noise from the anti-squeezed quadrature into the measurement quadrature. Even if the squeezing angle is actively controlled, there will remain some random fluctuations that degrade the squeezing level according to
The squeezing phase noise $\delta \theta$ couples to the RIN of the hPD signal also through the mechanism described in section 2. If this mechanism is neglected, the phase noise obtained by fitting squeezing level versus anti-squeezing level might be overestimated. Figure 5 shows this kind of plot, the different points were obtained by changing the power of the pump field entering the OPO. The two dotted lines represent the fitting of squeezing versus antisqueezing level obtained when the BSP loop was either engaged or not. The added RIN in terms of shot noise in the case of uncontrolled backscattered-light phase is given by
When the BSP loop is engaged, the factor $1/2$ in the above expression has to be replaced with $\sin ^2\phi _\mathrm {RMS}\sim \phi _\mathrm {RMS}^2$, where $\phi _\mathrm {RMS}$ is the RMS of the residual backscattered-light phase noise when $\Phi$ is controlled to be zero.
A fit of the data obtained with uncontrolled BSP, performed without considering the backscattered light noise contribution would give the following results:
Where $\eta _\mathrm {esc}=99{\% }$.Given $\mathrm {P_{bsc}}$, $S_{\delta \phi }$, and $S_{\delta \theta }$, by fitting the two curves an estimation of $\phi _\mathrm {RMS}$ can obtained, in addition to the injection efficiency $\eta _\mathrm {inj}$ and the residual squeezing phase noise $\theta _\mathrm {RMS}$. $\mathrm {P_{bsc}}$ can be obtained from a fit of the backscattering shoulder described by Eq. (10) when no squeeezing is injected ($r=0$).
As already pointed out in section 2, $S_{\delta \phi }$ and $S_{\delta \theta }$ are of similar magnitude at the frequency where the squeezing level is measured ($5.2\textrm {kHz}$), their value can be obtained from the calibrated error signal of the squeezing angle control [26]. The following values were obtained, for $\mathrm {P_{bsc}}=7\,\mathrm {fW}$ and $S_{\delta \phi }(f=5.2\textrm {kHz})=S_{\delta \theta }(f=5.2\textrm {kHz})=10^{-8}\mathrm {rad^2/Hz}$:
The estimation of the squeezing phase noise thus obtained is in agreement with the value expected from the incoherent sum of known sources: $17\,\mathrm {mrad}$ [27]. Not considering the effect of backsattered light would lead to an overestimation of the squeezing phase noise, as in Eq. (19).
The difference between the two RMS values of the residual phase fluctuations for the backscattered light and the squeezed light in Eqs. (21) and (22) is due to the different bandwidths of the two control loops. The UGF is $14\mathrm {Hz}$ for the BSP loop, and $4\mathrm {kHz}$ for the coherent control loop.
6. Conclusions
A theoretical description of the parametric amplification of the backscattered light was given. The model thus obtained provided a description of different coupling mechanisms of the noise introduced by backscattered light when the phase of the squeezed light is actively controlled. The dependence of these coupling mechanisms on the parametric gain was experimentally tested. A scheme to control the phase of the backscattered light by changing the length of the squeezed light injection path was proposed and demonstrated at GEO600. This method has proven to give a significant contribution in maintaining high levels of shot noise reduction, especially at high parametric gain of the OPO and when optical isolation was sub-optimal.
The RIN at the IFO readout photodiode caused by the backscatter coupling was described in Eq. (8). This knowledge is a useful tool to inform the design of the squeezing injection path in terms of path stability and optical isolation. Of particular interest is the fact that the coupling of squeezing angle fluctuations gets worse for higher parametric gain. This might set even more stringent requirements for the squeezing angle control to achieve the $10\mathrm {dB}$ squeezing level envisioned for future gravitational wave detectors, such as the Einstein Telescope [28] and Cosmic Explorer [29].
Funding
Max-Planck-Gesellschaft; Gottfried Wilhelm Leibniz Universität Hannover; Deutsche Forschungsgemeinschaft (EXC-2123, QuantumFrontiers - 390837967, SFB/Transregio 7); Bundesministerium für Bildung und Forschung; University of Glasgow; Science and Technology Facilities Council (ST/L000946/1).
Acknowledgments
The authors would like to thank Walter Graß for his years of expert support in the maintenance of critical infrastructure to the site to include the extensive $400\,\mathrm {m^3}$ vacuum system.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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