1. Introduction

In 2016, the Laser Interferometer Gravitational-Wave Observatory (LIGO) announced the first detection of gravitational wave, predicted by Einstein’s theory of general relativity [1–2], which was then honoured by the Nobel Prize in 2017. The direct observation of GWs provides a pathway to study the behaviors of massive celestial bodies, such as coalescences and mergers of binary black holes and neutron stars, supernovae explosions, and gamma-ray bursts, which is of great importance for the fields of fundamental physics and cosmology. Analogous to the electro-magnetic spectrum, GWs also cover a very wide frequency range. The ground-based GW detectors, such as LIGO, are sensitive to the GWs in the range from tens of Hz to several kHz, limited by the baseline length, the Earth’s seismic, and gravity-gradient noises. Space-borne GW detectors are able to open the low-frequency window. Laser Interferometer Space Antenna (LISA) is an European Space Agency (ESA) lead mission in collaboration with National Aeronautics and Space Administration (NASA) [3], aiming to detect the GWs in the range of 0.1 mHz to 0.1 Hz. Similar space-based projects include DECIGO [4], TianQin [5], and TaiJi [6–8].

The space-borne GW detector is generally composed of a nearly equilateral triangle with an average baseline of ${10^8}$-${10^9}$ m. Each spacecraft houses two drag-free proof masses controlled by the capacitive sensors and micro-Newton thrusters, which act as the end points of the long arm. The acceleration noise of the proof mass itself should be well minimized via a disturbance reduction system, since it is actually a part of the sensitivity floor of the whole system. The GWs will modulate the relative length between the proof masses (generally pm level), and the length information can be detected by exchanging the coherent laser beams between the spacecraft. To maintain the signal-to-noise ratio, the optical transponder scheme is recommended, in which one laser works as the master laser while all the other lasers are offset phase locked to this master laser. The offset frequency is related to the Doppler shifts of the laser fields, normally 1 MHz per m/s for a 1064 nm laser source. Equivalently, the whole constellation shares one laser source. To detect the GWs, the strain sensitivity in the interested band should be better than ${10^{-21} \rm /Hz^{1/2}}$ , which is much better than the state of the art of ultra-stable lasers and ultra-stable oscillators. In the case of LIGO, the laser phase noise can be cancelled in real time with the equal-arm configuration of Michelson interferometer, leading to an excellent noise floor. However, for LISA, TaiJi, and TianQin, it is not an easy work to fly an equal-arm interferometer in space. One technique to reduce the laser phase noise will be arm locking [9], where the laser frequency is synchronized to the arm length of the constellation. Since 2003, various schemes of arm locking have been proposed both theoretically [10–15] and experimentally [16–20], to reduce the potential risk of time delay interferometry (TDI). At the very beginning, scientists show that single arm locking is useful to reduce the laser phase noise in spite of the time-of-flight delay because of the long arm [9]. However, the first null locates in the science band (0.1 mHz - 1 Hz), leading to a significant noise amplification. Subsequently, dual arm locking has been designed exploiting two arms, to efficiently remove the first null out of the science band [12]. In addition, the start-up transients and the noise suppression at low frequencies can be well improved. To further improve the performance, modified dual arm locking has been finally presented, which is a smooth combination of common and dual arm-locking sensors [13]. The modification of dual arm locking allows the high gain of dual arm locking, and preserves the frequency pulling characteristics and low-frequency noise coupling of common arm locking. In this case, the clock noise (0.1 mHz to 20 mHz) and the spacecraft motion (20 mHz to 1 Hz) together contribute to the final noise floor. This level of performance can well satisfy the capability of time delay interferometry. However, complex design of the arm-locking controller is required, and the noise suppression performance will be highly affected by the arm length mismatch. Even, arm locking could be invalid when the arm length mismatch equals to zero. We try to modify the configuration of single arm locking, and hope to obtain a robust method with high gain, no nulls in the science band, low frequency pulling, and ease to use.

Optical frequency combs, as a powerful tool, have seen a vast number of applications, such as optical frequency measurement [21], optical communication [22], absolute distance measurement [23,24], frequency division [25], and precision spectroscopy [26], etc. These combs feature a series of precisely spaced, narrow emission lines in frequency space, and are with pulsed nature in the time domain. The two parameters, the repetition frequency and the carrier-envelope-offset frequency, can be tightly synchronized to an external clock. Consequently, all the frequency markers will share the same frequency precision as the clock [27]. In the converse manner, a microwave frequency can be also generated via frequency comb from optical frequency [28]. Consequently, this microwave frequency actually tracks the optical frequency via a down-conversion factor with high coherence. In this way, time delay interferometry with optical frequency combs has been demonstrated recently, which can simultaneously reduce the laser phase noise and the clock noise by a single step of TDI [29,30]. Yet, the deployment of frequency comb in arm locking has not been reported.

Here, we describe an updated version of single arm locking, where the noise amplification at the nulls in the science band can be well circumvented by careful control of the noise of the signals. Optical frequency comb is used to precisely divide the laser phase noise by a certain factor into the microwave domain, and the obtained signals, which are immune with the comb frequency noises, are electrically mixed to further generate the arm-locking sensor. We demonstrate the principle of this new kind of single arm locking, and examine the performance of the noise suppression. The results show that, the requirement of TDI can be well satisfied.

2. Laser source of frequency combs

In this section, we briefly describe the current status of laser frequency combs. For more information, several comprehensive reviews [31–33] can be found. Frequency combs are able to provide a number of mutually coherent, evenly-spaced narrow lines in a wide frequency band, and thus are capable of coherently linking the optical and microwave frequencies, as shown in Fig. 1. One comb line can be expressed as ${N \times f_{rep} + f_{ceo}}$, where ${N}$ is an integer, ${f_{rep}}$ is the repetition frequency, and ${f_{ceo}}$ is the carrier-envelope-offset frequency. ${T}$ is equal to ${1/f_{rep}}$, and ${f_{ceo}}$ is ${\varphi _{ceo} f_{rep} / 2 \pi }$. To date, there are several kinds of laser frequency combs, including combs of mode-locked lasers, electro-optic combs, and microresonator-based combs, etc., and these combs can support a range of applications and fundamental science.

 

Fig. 1. Concept of frequency comb, ${f_{rep}}$: repetition frequency, ${f_{ceo}}$: carrier-envelope-offset frequency, ${\varphi _{ceo}}$: carrier phase slip rate because of the difference between the group and phase velocities .

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Combs of passive mode-locked lasers. Two kinds of combs of passive mode-locked lasers have been developed for long periods, which are solid-state combs and fiber-based combs. Solid-state combs, such as that based on Ti:Sapphire, can offer a compact cavities and with direct diode pumping. However, the precise free-space alignment is required due to the solid state, which is challenging and expensive in the possible space-borne applications. Fiber-based combs, such as that based on Er-doped fiber, are the most widely-used comb sources all over the world. The fiber beam work can prevent the misalignment in the free space, and can be assembled using common telecommunications fiber handling tools. Recently, the reliable operations out of the lab and in space have been reported [34,35], showing a powerful capability in the practical applications. The combs of passive mode-locked laser are now commercially available.

Electro-optic combs. When one optical carrier is phase modulated by a sinusoidal signal, a series of sidebands can be stimulated around the center frequency, forming a comb-like structure. This is the so-called electro-optic (EO) comb. Unlike the passive mode locking, the repetition frequency of the EO combs is the driving frequency of the EO modulators, which can be tuned in a large range from several MHz to tens of GHz due to the large bandwidth of the modulator. One challenge is that the direct output of the EO modulators is not ultrashort pulse, since the phases of the frequency components are not precisely aligned. Therefore, the dispersion compensation is required to generate the femtosecond-class pulses. Today, one company is able to provide the product of EO comb generators [36], and the all-fiber structure could support the operation on a sounding rocket and the space missions.

Microresonator-based combs. Now, the microresonator-based combs are standing at the frontier in the field, which feature compact device footprints, low power consumption, and high repetition rates. In general, after a series of nonlinear processes in the micro cavity, the soliton state (i.e., the low noise comb state) can be reached resulting from the double balance between the gain and loss, as well as dispersion and Kerr nonlinearity. Due to the micro size of the resonators, the repetition frequency of the microcombs can be many tens of GHz, resulting in an ultrafast measurement speed. In recent years, microresonator-based combs have found a number of applications, and the chip-scale comb system based on the ${\rm Si_3\ N_4}$ platform has been reported [37]. Microcombs can be the qualified candidates for the future space mission because of the low power consumption and the micro footprints.

3. Updated single arm locking with precise control of the laser phase noise

At first, we briefly introduce the principle of the updated single arm locking. As shown in Fig. 2, the laser output, whose phase noise is initially ${\phi ({t})}$, is split into two parts at BS1. One part is the local oscillator. The other part is delayed with the time of flight ${\tau }$ (i.e., the incoming beam), and noted as ${\phi (t - \tau )}$. In our scheme and particularly, an optical frequency divider is involved into the incoming beam, which actually serves as an optical frequency synthesizer. In this case, the laser phase noise can be scaled down by a certain factor, and therefore the laser phase noise of the output of the optical frequency divider can be expressed as ${\alpha \cdot \phi (t - \tau )}$, with the downcoversion factor ${\alpha < 1}$. The two beams are combined at BS2, and detected by a photodetector linked to a phase meter. The output signal ${\phi _{pm}(t)}$ of the phase meter is injected into the arm-locking controller, which finally feedback adjusts the laser frequency to reduce the phase noise.

 

Fig. 2. Schematic of the updated single arm locking, BS: beam splitter.

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The output signal ${\phi _{pm}}$ of the phase meter can be written as:

In the frequency domain, Eq. (1) can be transformed into:

When the feedback loop is closed, the phase noise at point A can be expressed as:

 

Fig. 3. Nyquist diagram and Bode diagram of the single arm-locking sensor with different ${\alpha }$ values.

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We find that, the Nyquist diagram is circular, with (1,0) as the center. When decreasing the value of ${\alpha }$, the radius of the Nyquist diagram decreases. It is clear that, the real part of the sensor can not be equal to zero, when ${\alpha }$ is less than 1. For all the ${\alpha }$ values, the systems are unconditionally stable. Let us focus the Bode diagram. It is well known that, when ${\alpha }$ equals to 1, the magnitude rolls up from zero to 2, and then rolls off. The phase features the differentiator (advanced phase) and the integrator (lag phase), sequentially, and experiences a jump from ${-\pi /2}$ to ${\pi /2}$ at each nulls. In sharp contrast, with ${\alpha < 1}$, the magnitude can not reach zero as the lower bound, which changes in the range from ${1 - \alpha }$ to ${1 + \alpha }$. This implies that, the noise amplification at the nulls can not reach infinite, and the peaks could be inhibited. In the case of the phase frequency characteristics, the phase becomes more stable when ${\alpha }$ is less than 1. When ${\alpha }$ is set to 0.1, the phase change range is only from ${-\pi /20}$ to ${\pi /20}$. We consider that, this fact could relax the phase compensation following in the design of the arm-locking controller.

4. Updated single arm locking with an optical frequency comb

4.1 Precise control of the laser phase noise

In Sec. 4, we use the optical frequency divider to scale down the laser phase noise for convenience of the explanation of the updated single arm locking. However, limited by the state of the art of the photodetector bandwidth, ${\alpha }$ can not be sufficiently small. For example, the 300 GHz bandwidth corresponds to the ${\alpha }$ value of about 0.9989, when the laser wavelength is 1064 nm. This means, if the frequencies of the local oscillator and the incoming light are far different from each other, we can not observe the beat note at the photodetector. It is obvious that, smaller ${\alpha }$ value can result in a better suppression of the peak (Please note that, ${\alpha }$ can not be zero, since if that there is not returned signal from the distant spacecraft. We will discuss the bound of the ${\alpha }$ value hereafter). Therefore, one challenge is emerging, which is how to coherently make ${\alpha }$ small enough. One solution could be the technique of optical frequency comb.

Optical frequency combs are able to work as a bidirectional bridge, with the optical frequency and the microwave frequency standing at the two sides. We can use frequency comb to coherently transfer the optical noise into the microwave regime, with an appropriate downconversion factor [38,39]. Consequently, the mixing of the microwave signals could become easier, which actually corresponds to the heterodyne beat of the optical signals. By scaling up, the electrical mixing signal can serve as the input of the arm-locking controller to suppress the phase noise of the laser source, after the possible transfer to the phase term by an integrator.

For convenience of explanation, we do not involve the clock noise, the spacecraft motion, and the shot noise into the derivation, and the distant spacecraft just works as a perfect mirror in this subsection. The detailed method is shown in Fig. 4. The local oscillator ${f(t)}$ beats with the two nearby comb lines (e.g., the ${N}$th and ${N+1}$th line), and the beat notes can be expressed as

 

Fig. 4. Schematic of the laser phase noise control using optical frequency comb, DDS: direct digital synthesizers.

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The mode number ${N}$ and ${N+1}$ can be precisely measured with uncertainty better than 0.5, by a wavelength meter. ${{f_{b1}}}$ and ${{f_{b1}^{* \prime }}}$ are then divided by ${K_1}$ and ${K_2}$ based on two direct digital synthesizers (DDS), where

The outputs of the two DDS are then mixed, and we can get

We find that, the signal ${\Delta _1}$ in the electrical domain is immunity to the comb frequency noise, and can be coherently traceable to the local oscillator ${f(t)}$ with a downconversion factor of ${(K_2-K_1)/K_1 K_2}$.

In the similar way, we can get another signal corresponding to the incoming beam from the distant spacecraft, as:

Then, an additional DDS is added into the signal link, which is of key importance in the configuration of the updated single arm locking. ${\Delta _2}$ is divided by ${K_3}$, ${K_3=1/ \alpha }$, and we achieve

Based on Eq. (11), we find that, the laser phase noise can be coherently and arbitrarily scaled with a certain factor by the virtue of the optical-to-electrical network. This is rather difficult in the optical domain actually. ${\Delta _1}$ and ${\Delta _3}$ are mixed, and then multiplied by ${K_4}$. ${K_4=K_1K_2/(K_2-K_1)}$. Please note that, ${K_4}$ could be very large, and the commercial devices could not meet the requirement. However, this DDS can be customized using FPGA. The final signal can be expressed as:

In the phase term (i.e., both sides integrated with respect to time), Eq. (12) can be rewritten as:

The parameters can be easily determined. We assume that the laser wavelength is 1064 nm in vacuum, i.e., about 281.759 THz. Recently, the dissipative Kerr soliton (DKS) combs have found a wealth of applications. If the repetition frequency and the carrier-envelope-offset frequency are 5 GHz and 2 GHz, respectively. ${f_{b1}}$ can be measured to be 2.8 GHz, and ${{f_{b1}^{*}}}$ is therefore 2.2 GHz. ${N}$ can be determined as 56351. If we set ${K_1}$ as 200, ${K_2}$ can be calculated as 200.0035. ${\Delta _1}$ is then roughly 27.9 MHz, while ${\Delta _2}$ is at the same level. In this case, ${K_4}$ is about ${1.1 \times 10^{7}}$, which can be realized via a FPGA-based signal processor.

4.2 Updated single arm locking with an optical frequency comb

In this subsection, we introduce the clock noise, the spacecraft motion, the shot noise, and the distant spacecraft into the system. The schematic is shown in Fig. 5. Along the line in Ref. [13], the clock noise, the noise due to the spacecraft motion, and the shot noise can be estimated as:

 

Fig. 5. Schematic of the single arm locking with optical frequency comb, S/C: spacecraft.

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The phase noise at the phase meter output on spacecraft 3 is given by:

When the phase locking loop is closed on spacecraft 3, the phase noise at point C is

Please note that, the gain ${G_3}$ is much larger than 1. Therefore, the coefficients of ${1/(1+G_3)}$ and ${G_3/(1+G_3)}$ are approximately equal to 0 and 1, respectively. The phase noise at the phase meter output on the spacecraft 1 is given by

 

Fig. 6. Noise floor of updated single arm-locking at the laser output with different ${\alpha }$ values. The total noise comprises clock noise, spacecraft motion, and shot noise.

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Also plotted in Fig. 5 is the expected TDI capability (green curve), equal to the frequency noise requirement before TDI, which is given by

We find that, when ${\alpha }$ equals to 1, the arm-locking sensor works as the traditional single arm locking. There are a series of the peaks at the nulls, which actually corresponds to the infinite noise amplification. In addition, the noise floor is relatively steep during the whole science band (${0.45 \, {\rm Hz}/Hz^{1/2}}$ @ 0.1mHz, ${6.42 \, {\rm Hz}/Hz^{1/2}}$ @ 10mHz, ${1.8 \, \times 10^{-3} {\rm Hz}/Hz^{1/2}}$ @ 1Hz ). With decreasing the ${\alpha }$ value, the peaks are becoming smooth, whose magnitudes are proportional to the coefficient of ${1/(1-\alpha )}$. For example, the ${\alpha }$ value of 0.5 means 2 times of amplification of the noises at the nulls. When ${\alpha }$ is reduced to 0.95, the peaks can not reach the curve of the TDI capability already. As ${\alpha }$ is as small as 0.1, the noise floor becomes relatively flat (${3.6 \times 10^{-3} \, {\rm Hz}/Hz^{1/2}@0.1mHz}$, ${3.5 \times 10^{-4} \, {\rm Hz}/Hz^{1/2}@10mHz}$, ${4 \times 10^{-4} \, {\rm Hz}/Hz^{1/2}@1Hz}$), and all below ${10^{-2}\rm \, Hz/Hz^{1/2}}$ across the science band. This is because the clock noise and the noise due to the spacecraft motion from the distant spacecraft has been scaled down by ${\alpha }$, as shown in Eq. (20). It is clear that a smaller ${\alpha }$ is preferable. We can evaluate the upper bound of ${\alpha }$ via the noise magnitude at the nulls (i.e., the peaks), whose band corresponds to the noise caused by the spacecraft motion. If we assume that the noise floor should be below ${1 \rm \, Hz/Hz^{1/2}}$, this condition is given by

Therefore, the maximum value of ${\alpha }$ can be estimated as:

Consider the lower bound of ${\alpha }$, it is obvious that ${\alpha }$ can not be infinitely small, since there is not the delayed length information in that case. Based on Fig. 5, the design of ${\alpha }$ should maintain that the frequency ${\alpha \cdot \left ( {\frac {1}{{{K_1}}} - \frac {1}{{{K_2}}}} \right ) \cdot g\left ( t \right )}$ can be distinguished by the following electrical network, while it is reasonable that the frequency resolution can achieve 1 mHz for most of the electrical devices. ${g(t)}$ is the optical frequency of the incoming beam. Therefore, we reach

The minimum value of ${\alpha }$ can be thus estimated as:

Along this guideline, the ${\alpha }$ value, greater than 0.1 and less than 0.8, is suggested.

5. Performance of updated single arm locking with an optical frequency comb

5.1 Design of the updated single arm locking controller

The arm locking controller is required to provide sufficient gain to suppress the laser phase noise to be below the TDI capability curve within the science band. The design of controller can be divided into two parts: low-frequency part (< 1 mHz) and high-frequency part (> 1 Hz). For the low-frequency part, to control the Doppler frequency pulling at low-frequency, the controller is required to be AC coupled with high-pass filtering at very low frequencies. The high-pass filter is based on a classical lead-lag design. Since an arm locking controller has been designed successfully in Ref. [13], we adopt this controller but with some modification in the frequency poles and zeros. The frequency response of this controller can be written as [13]

 

Fig. 7. Bode plot of the open-loop gain ${G_L}$ with ${\alpha }$ = 0.80.

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5.2 Performance assuming free-running laser noise and pre-stabilized laser noise

In this section, we use three types of laser frequency noises, which are free-running (FR) laser noise, Mach-Zehnder (MZ) stabilization, and Fabry-Perot (FP) cavity stabilization [13]. The laser frequency noises can be expressed as:

Figure 8 shows the noise budget of updated single arm-locking with free-running laser noise. Benefitting from the introduce of the coefficient ${\alpha }$, the peaks in the science band is not infinite. The peaks get slighter with smaller ${\alpha }$. We can find that the laser frequency noise is the limiting noise source (the system is gain limited) with the other system noise sources well below the laser frequency noise. It also shows that even without any form of laser pre-stabilization, arm locking will meet the TDI capability across the entire science band while ${\alpha }$ < 1. The noise level is above the floor shown in Fig. 5, due to the limited gain in practice. Compared with the modified dual arm-locking sensor, this updated single arm-locking sensor is able to work continuously for the full year, and the noise suppression performance is not related to the arm length mismatch, showing a high robustness.

 

Fig. 8. The noise budget of updated single arm-locking with different ${\alpha }$ values. The performance was calculated with free-running laser noise as an initial condition.

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In Fig. 9, the total noise after arm locking is plotted in the cases of no pre-stabilization, Fabry-Perot cavity stabilization, and Mach-Zehnder stabilization. It is clear that either pre-stabilization type in combination with arm locking will deliver performance several orders of magnitude better than the TDI capability.

 

Fig. 9. Noise of updated single arm-locking with different laser frequency noises for different ${\alpha }$ values.

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5.3 Frequency pulling estimation

On the spacecraft, the beat frequencies are measured through the known offset in the phase locking loop at the distant spacecraft and the Doppler frequency caused by the relative spacecraft motion. The phase meter demodulates the beat signal with the difference frequency of the two lasers. In practice, the preset frequency at the phase meter should be updated in real time, since the Doppler frequency is changing all the time. However, the Doppler frequency can not be determined accurately, resulting in a frequency pulling effect to the master laser. Further, all the six lasers will be pulled in the constellation. The solution is to set the arm-locking controller into the AC-coupling mode, which means the gain at low frequencies is small. The frequency pulling in the dual arm locking is significant, because the frequency response to differential Doppler error is inversely proportional to the mismatch of the arm length. Fortunately, the single or common arm locking features a better frequency pulling characteristics. As demonstrated in Ref. [13], at the lock acquisition, the maximum pulling is 460 and 90 MHz, for the free-running and pre-stabilized lasers, respectively. In the case of steady state, the maximum pulling is less than 8 MHz for over 800 days, when the mode is common arm locking. We consider that, the maximum pulling of the updated single arm locking presented here is at the same level as the common arm locking. In general, the mode-hopping free tuning range for NPRO lasers is about 10 GHz, much larger than 460 MHz. Therefore, there is not potential risk of the mode hopping in the operation.

6. Time domain simulation

In this section, we carry out the time domain simulation with Simulink to verify the performance of the updated version of single arm locking. Based on the schematic shown in Fig. 5, the block diagram of Simulink is depicted in Fig. 10. It is worth emphasizing that the role of product in the left of Fig. 10(a) is to control the start time of feedback and reduce the turn-on transient. More details about this part can be found in Ref. [9].

 

Fig. 10. (a): Block diagram of the updated single arm locking with optical frequency comb; (b): block diagram of the optical frequency comb subsystem.

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To approach the real situations in space, we measure several noises in the lab for 10000 s with 1 s sampling interval. Figure 11(a) shows the noises of the free-running NPRO laser (home made) and the FP-stabilized laser (home made). The noise of the free-running NPRO laser is measured by beating two identical lasers, and that of the FP-stabilized laser is measured by beating with the frequency comb referenced to the hydrogen maser (T4Science iMaser 3000). The clock (Rakon HSO14, 5 MHz) noise measured by beating with the hydrogen maser, the noise due to the spacecraft motion, and the shot noise are shown in Fig. 11(b). Considering the noises due to the frequency comb (Menlosystem FC1500), the noises of the repetition frequency and the carrier-envelope-offset frequency are shown in Fig. 11(c) in the stabilized and free running cases, respectively. In the simulation, the repetition frequency is about 250 MHz, and the carrier-envelope-offset frequency is about 20 MHz. Therefore, the mode number N can be calculated to be 1127036. If we set ${K_1}$ equal to 1000, ${K_2}$ turns out to be 1000.00088728. Finally, ${\alpha }$ is 1 and 0.5 for convenience of comparison.

 

Fig. 11. (a): Noises of the free running NPRO laser and the cavity-stabilized laser. (b): Noises of the ultra-stable oscillator, spacecraft motion, and the shot noise. (c): Noises due to the frequency comb.

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In addition, to shorten the computation time for the simulation we only adopt a simple controller design. This can be achieved by placing zeros and poles alternately with a frequency spacing ratio of 10, i.e., poles at 0, 0.01, 0.1, 1, 10 Hz and zeros at 0.005, 0.05, 0.5, 5, 50 Hz. Similar design can be seen in Refs. [9] and [11]. The simulation results are shown in Fig. 12. Please note that, the frequency comb is free running here, since the results are exactly the same for a stabilized comb or a free-running comb in the simulation. However, in practice, the comb noise can indeed make contributions to the final results, and the stability at ${10^{-12}}$ level is sufficient to transfer the laser noise [35]. Figure 12(a) indicates the results with the free-running NPRO laser while ${\alpha }$ equals to 1 (i.e., the traditional single-arm locking). We find that, a series of the nulls exist in the science band. If ${\alpha }$ is 0.5, the nulls can be significantly suppressed as shown in Fig. 12(b). The laser phase noise has been reduced by about 5 orders of magnitude at 1 mHz, better than the TDI capability. Nonetheless, the performance can not reach the floor determined by the clock noise and the noise due to the spacecraft motion. Figures 12(c) and 12(d) show the results with the cavity-stabilized laser when ${\alpha }$ equal to 1 and 0.5, respectively. Similarly, the nulls have been removed when ${\alpha }$ is set to 0.5. We find that, the laser phase noise can be reduced to be below the noise floor, which means that the clock noise and the noise due to the spacecraft motion dominate the ultimate performance. These results show a good agreement with the theory.

 

Fig. 12. Time domain simulation results in Simulink. (a) and (b): Results with the free-running NPRO laser when ${\alpha }$ equal to 1 and 0.5, respectively; (c) and (d): Results with the cavity-stabilized laser when ${\alpha }$ equal to 1 and 0.5, respectively.

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7. Discussion

In Sec. 3, only the returned signal from the far spacecraft is tuned by a certain factor, to better understand this presented method. We would like to describe a unified approach in this section, to demonstrate the updated single arm locking comprehensively. Actually, the local oscillator can be also precisely controlled based on the same architecture. The scheme is shown in Fig. 13, and we find an optical frequency multiplier has been involved in the local beam, with ${\alpha _1}$ > 1 as the factor. The scaling factor at the incoming beam is ${\alpha _2}$, and ${\alpha _2}$ < 1.

 

Fig. 13. Schematic of the updated single arm locking. Both the local oscillator and the incoming beam are tuned by a certain factor.

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In this case, Eq. (3) can be updated to

The arm-locking sensor is therefore ${{\alpha _1} - {\alpha _2} \cdot {e^{ - i\omega \tau }}}$, i.e., ${{\alpha _1} \cdot (1 - {\alpha _2} / {\alpha _1}{e^{ - i\omega \tau }})}$. ${{\alpha _2}/{\alpha _1}}$ <1. It is obvious that, this system is unconditionally stable. Considering the magnitude frequency response, a factor of ${\alpha _1}$ is additionally involved, compared with the sensor in Sec. 3. This can bring an additional gain in advance before the controller, which can improve the noise floor. The practical scheme shall be modified slightly. In contrast to the electrical network shown in Fig. 4, a DDS with ${K_5}$ factor should be placed in the local branch, as depicted in Fig. 14. ${K_5 = 1 / \alpha _1}$.

 

Fig. 14. Modified local branch with an additional DDS with ${K_5}$ factor.

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We examine the noise floor determined by the clock noise, the spacecraft motion, and the shot noise. (${\alpha _1}$, ${\alpha _2}$) are set to (1, 1), (10, 1), (10, 0.5), and (100, 0.5), respectively. The results are shown in Fig. 15. It is clear that, the peaks have been effectively reduced when the local oscillator or the incoming beam are tuned with factors ${\alpha _1}$ and ${\alpha _2}$. With ${\alpha _1}$ > 1, it could be valuable that the noise floor can be improved, which means that the clock noise, the noise due to the spacecraft motion, and the shot noise can be reduced in the arm locking. Nevertheless, these noise can be only kept the same in the traditional single arm locking, and amplified at the nulls. In the case of ${\alpha _1}$ = 100 and ${\alpha _2}$ = 0.5, the total noise can already be below ${10^{-4} \, {\rm Hz}/Hz^{1/2}}$ in the band from 0.1 mHz to 1 Hz, which is far better than the TDI capability. The performance of the noise reduction is similar with that in Sec. 5.2, other than the total noise with the free-running laser can acquire an additional reduction in advance due to the factor of ${\alpha _1}$.

 

Fig. 15. Noise floor with different configurations of (${\alpha _2}$, ${\alpha _2}$).

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In discussion, the deployment of laser frequency combs can indeed bring advantages for arm locking. In spite of this, the system could become more complicated, and the equipment size and power consumption should be carefully considered in the practical situations. We suggest that the microcombs can be utilized in the future space-borne missions due to their compact footprints and low power consumption.

8. Conclusion

In this work, we develop an updated version of single arm locking, which enables the suppression of the noise amplification at the nulls in the science band. To remove the peaks in the science band, we precisely control the laser phase noise by using laser frequency comb. The laser phase noise can be coherently scaled down into the microwave regime, which is immune to the comb frequency noises. We detailed analyze the principle of how the updated single arm locking can suppress the noise amplification at the nulls, and provide a guideline of the design of the ${\alpha }$ value. The updated single arm-locking sensor is unconditional stable when ${\alpha }$ < 1. The frequency response of the sensor shows the magnitude and the phase become more stable with decreasing the ${\alpha }$ value, which may bring advantages in the following design of the locking controller.

The analytical results show that, with the consideration of the clock noise, the spacecraft motion, and the shot noise, the peaks in the noise floor have been well suppressed, related to the factor of ${1/(1-\alpha )}$. We find that, smaller ${\alpha }$ is able to bring better noise floor. When the laser phase noise is from the free-running laser, the suppression performance can well satisfy the requirement of TDI capability, but can not achieve the noise floor owing to the limited gain. In spite of this, as long as the laser source is pre stabilized to a Mach-Zehnder interferometer or a Fabry-Perot cavity, the performance can immediately approach the noise floor, and significantly reduce the risk of TDI. Since the updated single arm locking only uses one arm of the constellation, the potential failure of the dual arm locking does not exist. This implies that, our method can serve for full year. We also give a unified description of the updated single arm locking, where both the local and the incoming beams are controlled based on optical frequency comb. It could be useful that, the noise floor can be improved as well. To verify the performance of the presented scheme, we carry out the simulation in Simulink, where the noises including the laser phase noise, clock noise, noise due to the spacecraft motion and the comb noise are measured in the lab. The simulation results show a good agreement with the theory, and indicate that our method is reasonable and feasible.

Arm locking will be a valuable solution of the laser phase noise for the space-borne GW detector in future. No additional hardware is required for the realization of arm locking, but the possible risk of TDI can be greatly reduced. Recently, the space-borne frequency combs have been reported, showing the powerful capability in the space mission [40]. Meanwhile, various comb sources are developing at a rapid pace [41,42], which can be the candidates for the future GW detectors. Our work provides a back-up scheme for the arm locking, and can be used in the operation in orbit.