An arm length stabilization system for KAGRA and future gravitational-wave detectors

Our lock acquisition scheme is similar to that of Advanced LIGO [16]. It is divided into three steps as follows: (1) lock both arm cavities with the green lasers and keep the arm cavity lengths at off-resonant points for the main laser by the ALS system, (2) lock the vertex interferometer (i.e. PRCL, SRCL and MICH) and keep them locked using a combination of the interferometric signals that are less sensitive to the carrier field of the main laser e.g. third harmonic demodulation signals [26] or beat note signals using sideband fields non-resonant in the main interferometer [25, 27], and finally (3) bring the arm cavity lengths to their resonances and switch the error signals of the arm length degrees of freedom to that obtained by using the carrier field of the main laser. In the step (3), the main laser frequency needs to be controlled with respect to the resonant frequency of the arm cavities with a precision better than the linewidths of the arm cavities. Otherwise, the control of the three degrees of freedom of the vertex interferometer would be disturbed.

The ALS of KAGRA is different from that of Advanced LIGO in their concepts, designs, and control strategy. Table 1 summarizes the main differences between the ALS systems of KAGRA and Advanced LIGO. The features of the KAGRA ALS can be noted in the following. The first feature is that two green beams are injected from the central area, as opposed to the end stations. Thanks to this feature, the necessary length of the optical fibers is not on the same order of the arm length but approximately 60 m, which makes it scalable to those with longer arm length such as the third generation detectors [10, 11]; phase noise and optical loss associated with the fiber do not increase as the arm length gets long. The second is that the configuration is simple in two aspects: less number of the sensors and less number of SHG setups. One possible drawback coming from this feature is that phase noise the lasers pick up as they propagate through the optical fibers directly becomes sensing noise of the ALS system. However, this issue is not critical because fiber phase noise is not too large since the fiber length is short; the noise design and characterization revealed that the ALS system could satisfy the noise requirements, as described in the following sections. Additionally, the fiber noise cancellation technique [28] can be implemented independently if needed, though it would indeed add complexity to the system. We can expect the suppression of a factor of more than 10³ below 100 Hz by setting the cancellation bandwidth to around $10~\mathrm{kHz}$. The third feature is that the ALS CARM and ALS DARM signal are produced by summing the two signals from two arms in electronics or real-time controllers, respectively. This increases the flexibility of the ALS control loops.

The working principle of the KAGRA ALS can be summarized as follows. The two auxiliary lasers are phase-locked to the PSL with certain frequency offsets, which are on the order of tens of megahertz for ease of beat note detection. The frequency of the green laser coming from each SHG output is locked to the corresponding arm cavity by using the combination of a double-path AOM and a VCO as a frequency actuator. At this point, each control signal applied to the corresponding VCO is equivalent to the difference between the PSL frequency and the resonant frequency of the corresponding arm cavity, on the assumption that the auxiliary laser is tightly locked to the PSL via the phase-locked loop (PLL). Therefore, the sum of the two control signals works as an error signal for CARM, while the difference of the two works as an error signal for DARM. Let us call such error signals ALS CARM and ALS DARM. By feeding them back to the PSL frequency and arm lengths, respectively, the main laser frequency is fixed to certain points with respect to the resonances of the arms.

A detailed optical and control diagram of the KAGRA ALS is shown in figure 2. For the PLLs of the auxiliary lasers, beat notes between the PSL and the auxiliary laser are detected by photodetectors (PLL PD X and Y). For this purpose, we chose silicon photodiodes, which have fast response but relatively low responsivity for 1064 nm laser, because we can easily have a lot of optical power on the photodiodes. The signals of the beat notes are demodulated at phase-frequency discriminators by mixing them with local oscillator (LO) signals from radio frequency (RF) signal generators. One can change the frequency offset between the PSL and the auxiliary laser by changing the frequency of the LO. Each demodulated signal is processed by an analogue filter circuit and then fed back to the frequency of the auxiliary laser via the piezoelectric transducer of the laser, accompanied with the laser temperature control for slow drift compensation. The green laser from each auxiliary laser is frequency-shifted by a double path AOM. Phase modulation for the Pound–Drever–Hall (PDH) technique [29] is successively applied to it by an electro-optic modulator (EOM) in the laser room enclosure. Then the green lasers are injected to the X and Y arms from the back side of PR2 and SR2, respectively. The reflection from each arm cavity is picked off using a Faraday isolator and detected by a photodetector (PDH PD X or Y) to provide the PDH error signal. Each PDH error signal is filtered by another analogue circuit and fed back to a low-noise VCO that generates RF signals driving the AOM. The control signals, the voltages applied to the VCOs, are used to obtain the ALS CARM and ALS DARM error signals. Because of the different control bandwidths, the CARM loop partially involves analogue control filters, while the DARM loop is entirely realized by a digital control. The control signals are added with each other and then low-passed in analogue filters to obtain the ALS CARM error signal; the low-pass filtering makes the frequency response of the ALS CARM matched with the one for the main laser CARM signal (REFL CARM). The ALS CARM and the REFL CARM are sent to the same CARM servo board so that the input signals for the servo can be gradually switched. The servo has two output ports named slow and fast outputs, which are fed back to the IMC length and the error point of the IMC control loop, respectively, to form a dual loop control [30]. Meanwhile, for the DARM control, the two control signals driving the VCOs are sampled and digitally processed by real-time controllers, and then fed back to the differential motion of ETMs. Once the ALS CARM and ALS DARM loops are fully engaged, the resonant frequency of each arm cavity can be tuned to a desired value with respect to the PSL frequency by changing the frequency of the LO for the corresponding PLL.

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Here the noise design of the KAGRA ALS system is described. The primary noise requirement is that the root mean square (RMS) frequency fluctuation of the main laser with respect to the resonant frequency of one arm cavity is smaller than the linewidth of the arm cavity. This ensures smooth handing over from the ALS system to the main laser signals with a help of the signal derived from the optical power in transmission as done in Advanced LIGO [16]. In KAGRA, the designed value of the full width of an arm cavity is 33 Hz [25].

A more ambitious target can be given by the linewidth of the CARM cavity, which virtually serves as a single optical cavity corresponding to the CARM, with the full interferometer locked. If the RMS frequency fluctuation is smaller than the CARM linewidth, the CARM control can be directly handed over from the ALS CARM to the REFL CARM. This will greatly simplify the arm locking process and contribute to increasing the duty cycle of the detector. The full interferometer CARM linewidth is narrowed to 1.7 Hz for KAGRA [25] due to the double cavity pole of the power recycling.

We numerically simulated and designed servo loops of the KAGRA ALS system in the frequency domain. The results of the simulated noise contribution to frequency fluctuations of the main laser with respect to the CARM resonant frequency are shown in figure 3. Since the PDH signal of a cavity senses low-passed fluctuations of the input laser frequency with respect to the cavity resonant frequency with the corner frequency of the low-pass equal to the cavity pole frequency [31], such low-passed fluctuations need to be considered for the evaluation of noise performance of the ALS system. Thus for the CARM control, the noise spectra have been computed by $[\Delta f_\mathrm{main}(\,f)-\Delta f_\mathrm{CARM}(\,f)]L(\,f;\kappa_\mathrm{CARM})$, where $\Delta f_\mathrm{main}$ and $\Delta f_\mathrm{CARM}$ are the frequency of the main laser entering the main interferometer and the CARM resonant frequency, respectively, and $L(\,f;\kappa_\mathrm{CARM})$ is a low-pass filter with a corner frequency of $\kappa_\mathrm{CARM}$, the cavity pole frequency of CARM [32]. $L(\,f;\kappa_\mathrm{c})$ is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L(\,f;\kappa_\mathrm{c})= \frac{1}{1+{\rm i}f/\kappa_\mathrm{c}} . \nonumber \end{align} \tag{ 1 }$$

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Here, $\Delta f_\mathrm{main}$ is the laser frequency fluctuation at the PRM, while $\Delta f_\mathrm{CARM}$ is the averaged fluctuation of the resonant frequencies of the X and Y arms. We adopt the spectra of $[\Delta f_\mathrm{main}(\,f)-\Delta f_\mathrm{CARM}(\,f)]L(\,f;\kappa_\mathrm{CARM})$ because frequency fluctuations measured by the common mode of the arm cavities includes the effect from the CARM cavity pole and its RMS has to be compared with the CARM linewidth. Figure 3 shows that the RMS of frequency fluctuations controlled by the ALS system is designed to be as low as 0.65 Hz, which is smaller than the CARM linewidth. The dominant noise source above approximately 1 Hz is expected to be frequency noise of the VCOs driving the AOMs. The dominant noise source below 1 Hz is expected to be phase noise of the green lasers introduced by the optical fibers. Fiber phase noise had been estimated based on a measurement of phase noise of an optical fiber of length of 5 m placed on an optical table. We scaled the measured level of phase noise in such a way that the power spectral density of phase noise of a fiber is proportional to the length of the fiber. We also confirmed that the primary noise requirement, which is the target for the single arm, can be achieved; the RMS of $[\Delta f_\mathrm{main}(\,f)-\Delta f_\mathrm{ARM}(\,f)]L(\,f;\kappa_\mathrm{ARM})$ is 2.0 Hz, which is smaller than the linewidth of an the arm cavity. Here, $\Delta f_\mathrm{ARM}$ is the resonant frequency of an arm cavity and $\kappa_\mathrm{ARM}$ is the cavity pole frequency of the arm cavity. The low-pass filter with the single arm pole $\Delta f_\mathrm{ARM}$ is used because this is the evaluation of the single arm performance of the ALS system.

Figure 4 shows the set-up for the length sensing and control of the X arm cavity during the experimental period. This experimental set-up was almost identical to the design shown in figure 2. The differences were the following. Only the X arm was involved, and therefore the optics and electronics related to the Y arm were not involved. The PRM was intentionally misaligned. Unrelated laser beams were dumped so that they did not reach the signal recycling mirrors.

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The optical power of the main laser incident on the IMC was set to about $270~\mathrm{mW}$. The laser power incident on the X arm was approximately $10~\mathrm{mW}$, after the partial transmission of the PRM (10%) and the BS (50%). The main laser was phase-modulated at $45.0~\mathrm{MHz}$ in the laser room. The power of the green laser after the optical fiber was approximately $10~\mathrm{mW}$. The LO signal at about $40.0~\mathrm{MHz}$ for the PLL was provided by a stable signal generator, E8663D from Keysight Technologies. The frequency of the oscillator for the PDH method of the X arm green laser was set to $33~\mathrm{MHz}$. The center frequency of the VCO was approximately $80~\mathrm{MHz}$. The PDH signal of the main laser and the X arm, which is labeled as 'CARM Error' in figure 4, was obtained by the demodulation of the signal of the photodetector at the REFL port at $45.0~\mathrm{MHz}$.

Table 2 summarizes the measured parameters of the X arm cavity. The cavity length, mode matching ratio of the main laser to the arm, and transverse mode spacing were measured by scanning the main laser frequency through free spectral ranges of the arm cavity utilizing the ALS system. The finesse for the main laser was measured by a ring-down method [33]. The roundtrip loss of the arm cavity was obtained by the measurement of the reflectivity of the cavity combined with the information from the cavity scan measurement [33]. The finesse for the green laser was obtained by the measurement of the open loop transfer function of the PDH lock loop.

Table 2. Optical parameters of the X arm cavity. The values measured in the X arm experiment are shown along with the design values.

Parameter name	Designed	Measured
Cavity lengtha	3000 m [25]	$2999.990(2)$ m
Finesse for 1064 nmb	1530 [25]	$1410(30)$
Roundtrip loss for 1064 nma,c	<$100~\mathrm{ppm}$ [25]	$86(3)~\mathrm{ppm}$
Mode matching ratio for 1064 nma	—	$0.91(1)$
Transverse mode spacinga	$34.80~\mathrm{kHz}$ [25]	$34.79(5)~\mathrm{kHz}$
Finesse for 532 nmd	49.2	$41.0(3)$

^aMeasured by cavity scan. ^bMeasured by ring-down. ^cMeasured by the reflectivity of the cavity. ^dMeasured by the transfer function of the green lock loop. Note. Refer to the main body for the description of each measurement method.

We achieved and demonstrated lock acquisition of the arm, where the error signal was handed over from the ALS CARM signal to the REFL CARM signal. Figure 5 shows how the power of the main and green laser at the TRX port (transmission of the X arm; depicted in figure 4) evolved in time, along with the steps taken throughout the entire process. At point I, the resonance of the arm only for the green laser was achieved by locking the green laser frequency to the arm using the VCO only as an actuator. At point II, the feedback of the ALS CARM signal to the IMC length was turned on. From this point, the difference between the main laser frequency and the arm resonant frequency was controlled via the ALS system so that no longer the main laser stochastically resonated in the arm. At point III, the ALS CARM control with the full frequency bandwidth was engaged using both slow and fast outputs of the CARM servo. The difference between the main laser frequency and the arm resonant frequency was gradually tuned by sweeping the LO frequency of the PLL. At point IV, the position of the resonance of the arm cavity was spotted in terms of the LO frequency. In the shaded area labeled as V, the main laser was kept at a resonant point via the control by the ALS system. We stayed for approximately $20~\mathrm{s}$ in this stage for the purpose of demonstrating the stability of the ALS system. At point VI, the input of the CARM servo for 'CARM Error' was enabled to start the hand-over of the control paths. At point VII, after $1~\mathrm{s}$ from point VI, the input of the CARM servo from the ALS CARM signal was disabled to finish the hand-over. Consequently, lock acquisition of the main laser to the X arm was achieved. These lock acquisition processes did not fail unless the control of the green laser frequency to the arm lost its lock. Since the duration of lock of the green laser frequency to the arm was typically a few hours, it can be said that the processes were reliable. We tested these processes several times on different days, and succeeded in achieving lock acquisition of the main laser to the arm every time in the same way. This result clearly shows that the performance of the ALS system was demonstrated and the lock acquisition scheme of an arm cavity with it was established.

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Here we present a characterization of the main noise sources of the ALS system. To evaluate residual frequency fluctuations of the PSL with respect to the arm cavity, we locked the main laser frequency to the X arm with 'CARM Error' used as an error signal while the frequency of the green laser injected to the X arm was also locked to the X arm without feeding any signals back to the main laser or test masses. In this configuration, sensing noise of the ALS system can be inferred by measuring the ALS CARM signal under the assumption that residual frequency fluctuations of the main laser with respect to the X arm is small enough to be ignored for the noise estimation, which should be validated.

The results of the ALS sensing noise are shown in figure 6. The black trace shows the measured amplitude spectral density (ASD) of the ALS sensing noise calibrated to $[\Delta f_\mathrm{main}(\,f)-\Delta f_\mathrm{X}(\,f)]L(\,f;\kappa_\mathrm{X})$, where $\Delta f_\mathrm{X}$ is the resonant frequency of an arm cavity and $\kappa_\mathrm{X}$ is the cavity pole frequency of the arm cavity. The grey dotted trace shows the cumulative RMS of the black trace integrated from the high frequency side. The RMS was measured to be 8.2 Hz, which is smaller than the primary requirement, i.e. the linewidth of the X arm cavity. This indicates that the ALS system was sensitive enough to hold the main laser within the resonance width of one arm cavity, as is consistent with the time series shown in the previous section. Note that here we used the measured parameters of the cavity described in section 4.2.

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In order to investigate the origin of measured sensing noise of the ALS system, we made a number of supplemental measurements to estimate the contributions of the various noise sources. Figure 6 shows the results. The blue dashed line shows total noise, which is quadrature sum of the contributions of all the considered noise sources. In the frequency band approximately between 10 Hz and 1000 Hz, phase noise caused by the optical fiber for the green laser limited total noise. It shows a good agreement with measured sensing noise. In contrast, there is discrepancy between total and measured noise at low frequencies below 10 Hz. This indicates there were noise sources that are not taken into account. The magenta trace includes the residual fluctuations of the main laser frequency and was lower than the black trace, which validated the assumption of this noise evaluation scheme.

Let us discuss here the implications of the noise analysis shown in figure 6. Comparing figures 3 and 6, it is clear that we had underestimated the fiber phase noise. This probably came from the difference of the environment between the KAGRA site and the fiber testing setup. The optical fibers in KAGRA were laid on cable racks and the wall of the laser room enclosure, and they suffer from vibration and air flow. Since it is known that phase noise caused by an optical fiber can be largely suppressed by a fiber noise cancellation technique [28], we can suppress this noise by implementing such a technique if further improvement of the noise performance is required.

The possible noise sources that might account for the discrepancy between total and measured noise at low frequencies below 10 Hz are the couplings from the motion of the suspended optics. We observed significant coherence between measured sensing noise of the ALS system and the longitudinal motion of PR2 sensed by its local sensors [34] at the peak at 0.45 Hz (Figure 7). PR2 reflects the main laser but transmits the green laser. Therefore, its longitudinal motion must have caused Doppler shift only in the frequency of the main laser. The amount of the PR2 motion sensed by the local sensors was 0.85 $\mu$m $(\sqrt{\rm Hz}){}^{-1}$ at 0.45 Hz. Since the amount of the Doppler shift can be expressed as $2v/\lambda$, where $v$ is velocity of the mirror and $\lambda$ is wavelength of the main laser, the Doppler shift caused by PR2 is estimated to be 4.5 Hz $(\sqrt{\rm Hz}){}^{-1}$, which is comparable to the peak height in the sensing noise at 0.45 Hz. On this Doppler noise issue, one strategy for improvement is an online noise subtraction utilizing the local sensors. Similarly, the motion of the steering mirror behind PR2 that is installed on a suspended breadboard might have introduced Doppler noise because this mirror is relevant only to the green laser. Although there was no direct way to prove it because we had no sensors for this mirror unfortunately, it is not likely that the Doppler shift caused by the steering mirror limited the noise performance; it can be inferred that the contribution of such a Doppler shift was smaller the sensing noise level of the ALS system, on the assumption that the amount of the motion of the steering mirror is on the same order of the seismic motion. In any case, we need further investigation on the discrepancy.

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Next, we discuss the demonstrated performance of the ALS system in the context of the full interferometer of KAGRA. As mentioned in section 3.3, the measured RMS of the ALS sensing noise was well within the primary requirement (33 Hz). Therefore, assuming that we can achieve the same noise performance of the ALS system also for the Y arm, we will be able to lock the full interferometer using the ALS system with a help of the arm transmission signals [16]. Keeping the same assumption, we expect that the RMS of $[\Delta f_\mathrm{main}(\,f)-\Delta f_\mathrm{CARM}(\,f)]L(\,f;\kappa_\mathrm{CARM})$ will be 2.4 Hz (Figure 8). Figure 8 indicates that we need to improve the noise level of the ALS system in the frequency band lower than approximately 1 Hz to satisfy the more ambitious target, which is that the RMS of $[\Delta f_\mathrm{main}(\,f)-\Delta f_\mathrm{CARM}(\,f)]L(\,f;\kappa_\mathrm{CARM})$ is smaller than the full width of the CARM cavity. Although the current noise performance does not satisfy the more ambitious target, a previous work [35] suggested that a self-amplification process of the PDH signal might enable us to hand over the control paths directly from the ALS CARM to the REFL CARM signal, even if frequency fluctuations are larger than the linewidth. Achieving direct lock of CARM with the ALS system for the first time is a future work.

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