Ground-based gravitational wave detectors use laser interferometry to realize ultra-precise displacement measurements between free-floating test masses to detect signals in the frequency band of 10 Hz to 10 kHz. Light scattered out of and back into the main beam path creates noise in these detectors that non-linearly up-converts low-frequency, out-of-band signals into the measurement band. This scattered light induced noise becomes more relevant as the sensitivity at lower frequencies is improved in current and future detectors. To suppress this noise source, we study how the strong spatial coherence of the laser light can be reduced to a few centimeters by introducing high-speed pseudo-random noise phase modulations into the Michelson interferometer topology. We simulate the interferometer signals in the presence of scattered light in the time domain. Our simulations show that tuning of the coherence can reduce the coupling of scattered light by orders of magnitude, and that the phase modulations are, in principle, compatible with resonant cavities that are employed in the complex interferometer topologies of modern detectors to achieve ultra-low quantum noise levels. We outline the currently expected limitations of this approach and discuss the applicability to detectors.

Laser interferometric gravitational wave detectors like Advanced LIGO, Advanced Virgo, and KAGRA^{1} provide further insight in astrophysical and cosmological phenomena.^{2–5} Upgrades and next-generation observatories, like the Einstein Telescope^{6} and Cosmic Explorer,^{7} will further increase sensitivity, making more events observable with higher accuracy. Especially, improvements at low frequencies, down to a few Hz, are desired to, among other benefits, observe higher-mass events and increase the time-window between detection and merger. The latter, in particular, is crucial for multi-messenger astronomy.^{8} Achieving these improvements is driving research for the reduction of several types of noises, like the here central scattered light induced noise.

All current as well as most designs of next-generation detectors share a common topology. The core of which is a combination of stable continuous-wave lasers and enhanced Michelson interferometers that use optical resonators to increase sensitivity and circulating power. Detrimentally, the high coherence length of these lasers allows light scattered from the main beam to interfere. Back-scattering from a moving surface may lead to broad bandwidth noise. Even in today's detectors, this is a limiting factor at low frequencies^{9,10} and might contribute to as-yet unidentified noise. Currently, scattered light is mitigated in multiple ways that include baffles and the reduction of scattering sources or their seismically and acoustically driven motion.^{11,12} As even single scattered light photons are troublesome,^{13,14} reaching the design sensitivities of future detectors is a significant challenge.

We investigate a scheme using pseudo-random noise (PRN) sequences for phase modulation on the laser to intentionally reduce the coherence where it is not explicitly needed. Similar approaches were studied and implemented in interferometers to change the dynamics of the scattered light,^{15–17} where it itself is phase modulated. Imprinting a modulation on the laser was considered before the modern detectors were built but deemed unfeasible.^{18} However, technical advances in today's electro-optical components, drivers, and measurement equipment combat the then identified challenges. By assuming state-of-the-art technology, successfully used in digitally enhanced interferometry,^{19,20} we show that the coupling of scattered light can be suppressed, significantly reducing, or even removing, its impact on the sensitivity. We present and discuss numerical simulations of scattered light in a PRN modulated Michelson interferometer. Furthermore, we study the effect tunable coherence has on the operation of optical resonators such as Fabry-Perot cavities used in enhanced Michelson interferometers.

In a Michelson interferometer operated with stable, continuous-wave laser light, the power at the output ports depends only on the relative phase between the optical paths and is thereby independent of the macroscopic length difference between the arms. This independence allows light, scattered out of the main beam, with a large range of delays relative to the arms to interfere at the beam splitter. As the larger delay involves a reflection somewhere outside the interferometer, this light picks up an additional, possibly non-static, phase. The extremely long coherence lengths of the lasers favor this scattered light coupling coherently back into the readout. This then creates a non-linear phase noise component, which spreads to higher frequencies than the source due to fringe-wrapping and upconversion. Here, we consider that the frequencies of the disturbances are in the lower range of, or below, the measurement bandwidth of the detector.

We model a Michelson interferometer in which a fraction of the light is scattered out of the main beam in one of the arms, as shown in Fig. 1. We use a beam splitter with low reflectivity *r _{sc}* to model this. The light is then back-reflected from a moving surface and reenters the interferometer so that the field coming from this arm contains the sum of the main and the scattered light.

*P*

_{0}is the input power,

*ω*is the angular frequency of the laser (we assume $ \lambda = 2 \pi c / \omega = 1064 \u2009 nm$ in our simulations),

*τ*is the microscopic difference in propagation time between the arms, and

*τ*is the additional delay of the scattered light, which can be time-dependent. We implement a DC-readout scheme in the simulation, calculating a time-series of measured phase difference as shown in Fig. 2(a), from which we calculate the amplitude spectral density (ASD). For simplicity, we omit other noise sources, such as shot noise. Two main features of scattered light can be observed, the frequency upconversion and scatter-shoulder in the ASD, which are extended for increased sinusoidal dynamics, and the constant envelope due to fringe-wrapping in the time-series.

_{sc}*π*, the second equality holds with $ c \xb1 1 ( t ) = \xb1 1$. This then leads to the output power from Eq. (1) to depend on the auto-correlation function $ R ( \tau )$ of the sequence $ c \xb1 1 ( t )$, which, for, e.g., a maximum length sequence (

*m*-sequence), is an ideal two-level auto-correlation function, if the measurement by the photodiode is averaged over full sequences,

In Fig. 2(b), we show the scattered light effects now with and without PRN modulation present. We simulate a strain signal of $ 2 \xd7 10 \u2212 19 \u2009 m$ in amplitude with 100 Hz, which is masked by the sum of two scatter sources with some *μ*m scale movement range. The PRN modulation sequence has two adjustable parameters: the sequence length $ l seq$, for which we use 63 chips, and the modulation frequency $ f mod$, which here is $ 1 \u2009 GHz$. While the former determines the suppression factor, the latter determines the interval of the remaining interferometric coherence length. The chips of the modulation sequence have a duration of $ t chip = 1 / f mod$, which corresponds to an optical path of $ c \xb7 t chip$. As an m-sequence is periodic, this gives a time delay corresponding to certain optical length for which auto-correlation, and thereby coherence is regained as the sequence repeats. We call this the recoherence length $ d coh = c \xb7 l seq / f mod$.

As we study and simulate phase modulation frequencies in the GHz-range, we can bring the coherence length down to centimeters, making, e.g., ghost-beams originating at the back-surface of mirrors incoherent. In our example shown in Fig. 2(b), we reach a suppression of $ 17.76 \u2009 dB$ (further results available in the supplementary material). As the reachable suppression in decibel is given by $ 10 \u2009 \u2009 log 10 ( 1 / l seq )$ and, therefore, only depends on the sequence length, using longer sequences would be optimal, but further limitations need to be considered.

*τ*is the round trip delay of the cavity and $ r 1 , 2$ and $ t 1 , 2$ are the amplitude reflectivity and transmittance of the first (input) and second mirror, respectively. In the same manner, we can obtain equations for the transmitted and circulating fields.

As expected for reduced coherence lengths, the random modulation suppresses stable constructive and destructive interference around the cavity mirrors for an arbitrary macroscopic cavity length, inhibiting power buildup in the cavity. In all cavity field equations, such as Eq. (4), the PRN term $ c \xb1 1$ causes the sum to converge randomly around zero. It is, therefore, not trivial to understand the interaction of the PRN modulation and a two-mirror-resonator for arbitrary lengths. The area we study more closely is the one, where the length of the cavity is close or equal to the recoherence-length of the PRN code or an integer multiple of it.

This shows, in principle, that we can operate cavities in the same way as without modulation. However, operation close to, but not exactly at this ideal macroscopic length is much more practical and is investigated in more detail in the following simulations.

In complex interferometer topologies, such as in gravitational wave detectors, optical cavities fulfill several different tasks, which means different configurations of cavities are used. We, therefore, use our simulation to demonstrate that we are still able to meet the requirements needed for the detector to work in the same way as without modulation.

We first verified our simulations by comparing the reflected, circulating, and transmitted fields in an overcoupled cavity with ideal length (finesse of $ \u2248 57$) with PRN modulation (63 chips, $ 1 \u2009 GHz$) to a cavity response without modulation for microscopic detuning. To study the field in the simulation, we used an additional PRN demodulation of the fields to recover the phase response we expect to achieve with optical demodulation in a Michelson. The resulting field amplitudes and phases correspond exactly to the well-known case without PRN modulation,^{21} indicating that microscopic detuning on the order of one wavelength behaves as expected (simulation results shown in the supplementary material). Hence, we are confident that both operations on resonance, anti-resonance, or at other detunings are, in principle, possible with tunable coherence, as required for arm-cavities, power, and signal recycling. We note that overcoupled, undercoupled, and impedance-matched cavities respond analogously to the operation with PRN modulation.

The most critical aspect in our simulation is how far off resonance a cavity can be tuned before the PRN modulation interferes negatively, assuming that the macroscopic cavity length is matched to an integer multiple of the recoherence-length. For high modulation frequencies—and thus short coherence lengths—it is also relevant, whether the chip-length is an integer multiple of the laser wavelength, as this determines the achievable overlap of microscopic tuning and macroscopic coherence.

To investigate these effects, we simulate power buildup in the cavity and the Pound–Drever–Hall (PDH) error signals, which are suitable metrics for the operation of cavities. (A simulation depicting the time-dependent behavior of circulating and reflected power in a cavity with a length unmatched to the recoherence-length is available in the supplementary material.)

Power buildup should not decrease for a slight mismatch of modulation sequence length and cavity length in the presence of PRN modulation. We simulate the power buildup in an impedance-matched cavity for varying cavity length and three modulation frequencies. For a modulation frequency of about $ 10 \u2009 GHz$, one chip has a length of about $ 2.99 \u2009 cm$, resulting in a minimum cavity length matched to the recoherence-length of a 63 chip sequence of about $ 1.88 \u2009 m$. The results, depicted in Fig. 3, show a clear repeating pattern, with the maximum power buildup repeating at integer multiples of the matched cavity length. For various ratios, we observe a fraction of the power build up. For instance, at a cavity length of 1.5 sequence lengths, only half of the buildup is realized, since only every second round trip (equaling a shift of exactly three sequences) the PRN modulation allows for coherent interference. The peak around a length matched cavity has a FWHM of only around $ 1.89 \u2009 mm$ for a modulation frequency of about $ 10 \u2009 GHz$, equaling about 1776 wavelengths. Furthermore, we observe a higher dependency of the power buildup on the length-matching for a higher cavity finesse (not shown). Physically, this is due to the higher number of cavity field round-trips, which allows even a slight mismatch to add up to a wrong modulation phase. Thus, depending on the modulation frequency and cavity finesse, this FWHM can correspond to not only far less than a centimeter but also down to *μ*m of offset. On a microscopic scale, we observe that the cavity can still be tuned through several fringes, the number depending on cavity finesse, and modulation frequency, before the buildup starts to drop significantly. We note that here the modulation frequency was chosen such that one chip-length always equals exactly an integer multiple of a wavelength for zero code mismatch, the above-mentioned ideal situation.

Highly precise readout of the cavity detuning is critical, especially for the arm-cavities, and should ideally be possible even if one of the mirrors moved through several fringes before lock is acquired. To verify this, we investigate how the Pound–Drever–Hall locking scheme responds to the PRN modulation by adding phase modulation onto our modulated electric field to simulate the error signal. For this, a modulation depth of $ \delta = 0.01$ and a modulation frequency of $ f PDH = 50 \u2009 MHz$ were assumed. We then compute the field reflected from an impedance-matched cavity ( $ r 1 , 2 2 = 0.9$) by scanning the tuning and, thus, extracting the PDH error signal. To investigate the influence of code mismatch, we repeat this for various cavity lengths at a fixed modulation sequence length of 63 chips at about $ 1 \u2009 GHz$. Here, we note that the local oscillator phase is not newly optimized for unmatched cavities. As the shape of the PDH signal varies for different phases, this opens another degree of freedom. The result is shown in Fig. 4 where we observe that again a length matched cavity works without limitations. The error signal shows some unexpected behavior. In addition to a shift of the zero crossing away from the resonance condition with increasing mismatch, we observe a slight increase in the slope, which we can currently not exclude as an artifact from our simulation. This shows that more simulations and especially experiments are necessary to fully understand the impact of the PRN modulation on the interferometer operation.

We also performed simulations of a Fabry–Pérot–Michelson interferometer with km-scale arm length, medium-finesse cavities, and a PRN modulation rate of $ 1 \u2009 GHz$ where the cavity length is matched to the recoherence-length, and both arms are delay-matched. We injected a gravitational wave signal and verified that the output in the presence of a PRN modulation is indistinguishable from the case without modulation. Microscopic length changes induced by the injected GW-signal do not lead to any artifacts in the signal, thus allowing us to conclude that our scheme is, in principle, compatible with this interferometer topology.

Our simulations are currently limited to assumptions such as perfect modulation depth and infinite modulation bandwidth. While there are approaches to solve this, they require even higher sampling rates and precision, which require more excessive computing than used so far. Numerical effects have to be taken into account, especially since the simulations require precise microscopic and macroscopic delay values for many round-trips in the cavities. We expect that these constraints do not change the overall picture of the presented results. However, they remain to be confirmed in a laboratory experiment.

We conclude that simulations show how pseudo-random noise phase modulations of the laser can significantly reduce the scattered light induced noise in the readout of gravitational wave detectors. While the suppression factor only depends on the modulation sequence length, the complex topology of real interferometers restricts the choice of suitable sequences or has to be adapted to be compatible. A periodic *m*-sequence seems suitable here. With this, we can simulate not only the use of tunable coherence in a Fabry–Pérot–Michelson interferometer successfully but also key features of optical resonators commonly used in the detectors. We are optimistic that also further advanced topologies, like the dual-recycled Fabry–Pérot–Michelson interferometer, can realistically be equipped with tunable coherence. As main challenges, we currently identify small length differences between cavities used in the layout and in auxiliary functions, such as the Schnupp Asymmetry^{22} and mode-cleaners, as these strongly limit the usable sequence length. In order to not negatively impact the setup, all asymmetries and cavity lengths should be chosen as multiple integers of a chosen minimal recoherence, or code-repetition, length. This could be, e.g., about 1 m optical path difference with a modulation frequency of $ 20 \u2009 GHz$, using a code-length of 63 chips a code-length of 63 chips, which still allows for a suppression factor of more than 17 dB. While this could also allow more scatter sources, located exactly at multiple of this length, to potentially again couple coherently into the interferometer, the likelyhood of this happening shrinks with increasing modulation frequency as the critical remaining coherence-length shrinks down. Furthermore, to avoid small mismatches accumulating to macroscopic differences over many round-trips in high-finesse cavities, it might be necessary to stabilize the modulation frequency relative to the laser wavelength. For this, we identify the use of a frequency comb as a possible solution as this would allow for the stabilization of the GHz modulation frequency relative to the much higher THz laser frequency.

The compatibility of our approach with other established noise reduction techniques, such as squeezed light, needs to be investigated to ensure the scheme is competitive in its ability to combat quantum noise effects. Nevertheless, our simulation results substantiate our effort to study tunable coherence for scattered light suppression in gravitational wave detectors with our progressing laboratory experiment. In our estimate, future detectors like the Einstein Telescope low-frequency detector (ET-LF)^{6} could benefit strongly from this suppression of scattered light and face easier implementation, as the input power (3 W in the case of ET-LF) lies in an achievable range with available high-speed phase modulators. While not analyzed here, the scheme we propose can also be adapted to other interferometer topologies studied for future detectors, like speed-meter configurations,^{23} and might be especially well suited for Sagnac topologies in which arm-lengths are intrinsically matched.^{24} Finally, the use of very long m-sequences over the kilometer long arms could, in principle, reduce scattered light coupling so strongly that there is no further need for other scattered light mitigation in the detectors. This extreme approach is obviously at odds with some of the standard techniques, like the short power- and signal-recycling cavities, but is worth to be mentioned to stimulate further investigations and discussions. As the *m*-sequence modulated continuous-wave laser acts as a hybrid between continuous-wave and a white-light laser source, it can be used to realize new interferometer schemes.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional simulation results.

We would like to thank Harald Lück and David Shoemaker for fruitful discussion. This research is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy—EXC 2121 “Quantum Universe”—390833306.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Daniel Voigt:** Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Supervision (supporting); Validation (lead); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead). **André Lohde:** Conceptualization (supporting); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (lead); Writing – original draft (supporting); Writing – review & editing (supporting). **Oliver Gerberding:** Conceptualization (lead); Formal analysis (supporting); Funding acquisition (lead); Investigation (supporting); Methodology (supporting); Project administration (lead); Resources (lead); Supervision (lead); Validation (supporting); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.