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Abstract
A laser interferometer will be used in the spaceborne gravitational-wave detection missions to measure the inter-satellite optical pathlength variations. The phase readout system of the interferometer needs to be carefully designed and tested to accomplish a shot-noise-limited detection performance under the situation of pico-Watt level received lights. In this work, a scheme based on dual-tone acousto-optic diffraction is presented to verify the performance of the weak-light phase readout system. By optimizing the parameters of the photoreceiver and the local strong-light power, the signal-to-noise ratio of the beat-note signal is enhanced. Thanks to the scheme’s common-mode noise rejections for the laser frequency noise, and the optical-path noise, etc., the differential phase noise has achieved a performance of 2×10−4 rad/Hz1/2, which is dominated by the weak-light (∼13 pW) shot noise above the frequencies of 2 mHz.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser interferometers will be used to measure the displacement and the angular variation between two spacecraft separated by hundreds of thousands to millions of kilometers in the future spaceborne gravitational-wave detection missions such as LISA [1], TianQin [2], and Taiji [3]. This key technology has been successfully demonstrated by the Gravity Recovery and Climate Experiment Follow-On (GRACE-FO) mission [4].
One challenge of the large-baseline spaceborne laser interferometer is the detection of the extremely-weak light power received at the remote satellite, which is caused by the long arm-length, the beam divergence of the Gaussian beam, and the finite aperture of the telescope [5].
After the launch of the spacecraft, the laser links must be established first by proper acquisition strategy [6,7]. Then the beam steering method based on differential wavefront sensing (DWS) will be applied to further improve the pointing precisions of the laser beams [8,9]. A phase readout system with the ability to operate under the situation of pico-Watt level received light will relax the requirement to the acquisition and fasten the whole pointing process. Furthermore, verifying the performance of the weak-light phase readout system in the millihertz band will benefit different intersatellite laser interferometry missions in the future, such as missions with longer arm lengths, or limited power supply and volume like the GRACE-FO.
P. W. McNamara reported an experiment of weak-light phase locking for LISA in 2005 [10]. The weak-light power was 13 pW and the relative phase noise of the slave laser shown a shot-noise-limited performance (1.3×10−4 rad/Hz1/2) above 0.4 Hz. Excess noise at lower frequencies most probably came from thermal effects in the optical arrangement and phase-sensing electronics.
In 2009, C. Diekmann et al. implemented an analog OPLL with an offset frequency of about 20 MHz between two lasers, where the detected light powers were of the order of 31 pW and 200 µW [11]. The phase noise between the two lasers was a factor of two above the shot-noise-limit down to 60 mHz. The limiting noise source below 60 mHz was the analog phase measurement system.
Francis et al. demonstrated weak-light phase locking at a light power of 30 fW in 2014 [12]. In order to minimize the cycle slip rate with such weak light power, the loop bandwidth has been optimized to balance the laser phase fluctuation and the shot noise.
In this paper, a scheme of weak-light heterodyne interference detection based on dual-tone acousto-optic diffraction is presented to verify the performance of the weak-light phase readout system consisting of photoreceiver (PR) and digital phasemeter. This scheme shows an excellent performance in the common-mode noise rejections for both laser frequency noise and optical-path noise. However, due to the power shift caused by the cross-modulation effect of the diffracted beams, the weak-light power should be carefully calibrated. We improve the signal-to-noise ratio (SNR) of the heterodyne beat-note and reduce the phase noise by means of optimizing the parameters of PR and the power of the local strong-light. By applying this scheme with homemade PRs and a digital phasemeter, we have carried out the phase noise estimations with different weak-light powers and different heterodyne frequencies. The differential phase noise has achieved a performance of 2×10−4 rad/Hz1/2 at frequencies above 2 mHz with the weak-light and the strong-light of 13 pW and 500 µW, respectively.
2. Measurement principle
Due to the far propagation distance of the laser in an inter-satellite laser interferometer, the photon shot noise of the received light would set a limit for the smallest detectable single arm-length variation [13]. After the photoelectric conversion, the photon shot noise of the received light determines the maximum SNR of the heterodyne signal. Therefore, the phase noise of interference is limited by the received optical power as it is determined by the SNR [14].
In addition to shot noise, the SNR would be reduced by other noises including the laser amplitude noise and the electronic noise. Moreover, the optical-path noise and the laser frequency noise would influence the phase noise performance below 0.1 Hz. To accomplish a shot-noise-limited detection performance, the parameters of the PR and the laser power need to be optimized to reduce the impact of the laser amplitude noise and the electronic noise. Meanwhile, noise sources at lower frequencies should be verified separately.
Based on the principle of the dual-tone acousto-optic diffraction, we present the setup shown in Fig. 1 to test the noise sources mainly from the PRs and the phasemeter as this scheme has better common-mode rejection ratios for the laser frequency noise, optical-path noise, etc.
Fig. 1. Heterodyne detection using dual-tone acousto-optic diffraction. The AOM is driven by two combined and amplified RF signals to produce an interference light field. The interference light is split and focused on two PRs. The PRs’ output signals are sent to the phasemeter for phase extraction.
2.1 Experiment setup based on dual-tone acousto-optic diffraction
Interferometric phase extraction is a vital part of the laser interferometer metrology chain [15]. To verify its performance independently, we apply a heterodyne interference based on dual-tone acousto-optic diffraction.
According to [16], acousto-optic diffraction with multiple acoustic waves at different carrier frequencies generates multiple diffracted beams. In our experiment, the radio frequency (RF) signals with frequencies of f1 and f2 are superimposed to drive a fiber acousto-optic modulator (AOM). As a result, the sound waves with two different frequencies propagate in the acousto-optic crystal and interact with the light beam, herein generate two diffracted beams with different frequency shifts and diffraction angles.
Inside the fiber AOM, these two diffracted beams interfere with each other and are coupled into the same output fiber. The power of each light beam can be adjusted by changing the corresponding powers of the RF signals. The heterodyne frequency of the beat-note can be adjusted by changing the frequencies of f1 and f2. Thus, the dual-tone acousto-optic diffraction can be applied to simulate the inter-satellite laser interferometry, in which the heterodyne frequency is affected by the Doppler shift and the power of the received light is weak down to the level of nano-Watts or even pico-Watts.
The schematic diagram of the dual-tone acousto-optic diffraction is shown in Fig. 1. The signal generator 1 produces the RF signal with the frequency of f1 = 78 MHz and the power of 1.7 dBm. The signal generator 2 produces f2 = 98 MHz with much lower RF power, which is used to provide the weak-light. The above 2 RF signals are combined and amplified before sending to drive the AOM.
Laser light is produced by an ORION compact laser module and then passes through a fiber AOM with the center frequency of 80 MHz. The output light of the AOM is divided by a 50/50 cube beam splitter. One part of the output light is monitored by an optical power meter. Another part is equally split again and focused on two photodiodes with a sensing diameter of 300 µm. The output signals of the PRs are detected by the digital phasemeter based on the method of IQ demodulation [17].
In this scheme, the strong-light representing the local laser and the weak-light representing the receiving laser interfere inside the AOM, with almost equal optical pathlength. Due to the cross-modulation of multifrequency acousto-optic diffraction [16], the power of the weak-light cannot be measured correctly in advance. Alternatively, the weak-light power could be measured by using the average and peak voltage of a calibrated PR which will be discussed later.
2.2 Characterizations of the PR
A typical PR consists of a photodiode (PD) and a trans-impedance amplifier (TIA). The photodiode in the PR converts the receiving laser light to photocurrent and the TIA converts the photocurrent into voltage. For heterodyne interference detection, the key requirement of the PRs’ design is to maximize the SNR, while meeting the bandwidth requirements of the mission. This requires a PR circuit with high bandwidth and ultra-low noise [18].
2.2.1 Transfer function
The PRs’ circuit diagram is shown in Fig. 2. The key parameters affecting the circuit performance are the source capacitance Cs [the sum of the PR junction capacitance Cd, common-mode capacitance Ccm and differential mode capacitance Cdiff of operational amplifier (Op-Amp)], the Op-Amp gain-bandwidth product (GBP), the feedback resistance Rf, and the feedback capacitance Cf.
Fig. 2. Circuit diagram of PR. The interference light is absorbed by the InGaAs PD and converted to current. The TIA circuit converts current to voltage signal.
According to [19], The transfer function of the TIA is:
In the above formulas, Aol is the open-loop gain of the Op-Amp and ωA represents the dominant-pole frequency of the Op-Amp’s open-loop frequency response. These two parameters are used to describe the Op-Amp’s one-pole frequency response. ω0 is the eigenfrequency of the circuit, and Q is the quality factor. The bandwidth, the gain, and the stability of the PR can be adjusted by choosing different parameter values of the PR. Using the parameters in Table 1, the simulation results of the PRs’ transfer function is shown in Fig. 3. Under this set of parameters, the bandwidth, the gain, and the stability of the PR can meet the requirements of our experiment.
Table 1. Simulation Parameters of the Transfer Function
2.2.2 Noises of the PR
G. F. Barranco et al. introduced a system-level analysis of a PR designed for space-based interferometry [20]. To optimize the performance of the PR and finally improve its output SNR, the noise sources in the PR as well as their respective transfer functions are analyzed in detail in our study.
Under a certain optical power illumination, the main noises in the output signal of the PR include the shot noise of the photocurrent, the laser amplitude noise, and the electronic noise. The amplitude spectral densities of the shot noise and the laser amplitude noise can be written as:
The gains of N1(f) and N2(f) are the same as the signal gain of the PR:
where the signal gain GTIA(f) is defined as the real part of the transfer function of the PR.The main sources of the electronic noise are the feedback resistor’s Johnson noise and the noises of the operational amplifier. The power spectral density of the Johnson noise is
To ensure the stability of the circuit, a capacitor should be connected parallelly to the feedback resistor. Therefore, the bandwidth of the actual resistor thermal noise is limited. The gain of the resistance thermal noise is:
Manufacturers of operational amplifiers usually convert the noises inside the operational amplifiers into equivalent input current noise and equivalent input voltage noise, and the amplitudes of these two types of equivalent noise in the frequency domain are given in the datasheet. Taking ADA4817 manufactured by Analog Devices for example, above 100 kHz, the equivalent input current noise can be ignored and the power spectral density of the equivalent voltage noise is:
According to [19], in the trans-impedance amplifier circuit, the gain of the equivalent voltage noise can be simulated by using three key frequency points:
The gain of the equivalent voltage noise can be written as:
Through the above analysis, the theoretical noises of the PR are shown in Fig. 4.
Fig. 4. Noise analysis of the PRs’ output voltage signal. When the power of incoming light is set as 400 µW, the main noise source of the PRs’ output is shot noise.
2.3 Measuring the PRs’ gain
The signal gain is one of the key parameters of the PR and its frequency response indicates the PRs’ bandwidth. The DC and AC components of the PRs’ output can be used to calculate the gain of our homemade PR [21]. The output voltage of the measured PR in response to the receiving optical signal can be expressed as:
According to Eq. (10), GTIA(f) can be expressed as:
If P1=P2, the above formula could be simplified, as:
The interference efficiency γ of the beams inside the fiber-optic AOM can be conceived as 100% because of the small diameter of the fiber core. Hence, the gain of the PR can be calculated by the DC voltage and the peak voltage of the PRs’ output.
2.4 Calibrating the weak-light power
Generally, the laser powers of interference lights can be measured separately while turning off one of the lights. In dual-tone acousto-optic diffraction, this method is inapplicable because of the cross-modulation effect of the diffracted beams [16]. The existence of cross-modulation makes the powers of the two diffracted beams different from the powers when they exist separately.
However, the weak-light power in the dual-tone acousto-optic diffraction can be calculated by using the DC and AC output voltages of a calibrated PR.
If $P_1^{\prime} \gg P_2^{\prime}$, then Eq. (11) can be approximated as:
The weak optical power $P_2^{\prime}$ can be calculated as:
2.5 Maximizing the SNR
The photon shot noise of the received light determines the maximum SNR of heterodyne signal but the electronic noise and the laser amplitude noise also affect the SNR. According to [18,22], the junction capacitance Cd of PD needs to be minimized to reduce the effect of the electronic noise. However, for a certain photodiode with large capacitance, the SNR of PRs’ output signal can be maximized by choosing proper strong-light power and TIA circuit parameters.
The parameters of the PR used in the simulation are the same as mentioned above. The RIN of the laser is set as 1×10−8 Hz−1/2, and the heterodyne frequency is set as 20 MHz. Since the electronic noise and the laser amplitude noise have been determined, the relationship between the SNR and the interference lights can be shown in Fig. 5. The higher the weak-light is, the greater the SNR can be obtained. As the strong-light power increases in the range from 1 µW to 10 mW, the SNR gradually increases first and then decreases. The shape of the curves is determined by different growth trends of noises when the strong-light power increases. When the strong-light is relatively weak, the electronic noise dominates. As the strong-light power increases, the shot noise gradually exceeds the electronic noise and becomes the dominant noise. When the strong-light power continues to increase, the laser amplitude noise exceeds electronic noise, making the SNR gradually decreases.
Fig. 5. SNR influenced by laser powers. The greater the weak-light, the greater the SNR. As the strong-light power increases, the SNR gradually increases and then decreases.
Since the electronic noise is related to the resistance of the feedback resistor Rf, the SNR can be also influenced by Rf in addition to the strong-light power. Therefore, there are a set of parameters to make an optimal SNR.
Assuming the weak-light power is 13 pW, the analysis results in Fig. 6 show the expected SNR of the PRs’ output with different strong-light powers and different feedback resistors. According to the specification of ADA4817, the equivalent voltage noise is N4(f) = 4 nV/Hz1/2. However, the pre-test result shows that N4(f) equals 12 nV/Hz1/2, higher than the nominal value. The two contour plots in Figs. 6(a) and 6(b) are based on the nominal and experimental values of the equivalent voltage noise N4(f), respectively.
Fig. 6. SNR influenced by strong-light power and feedback resistance. (a) Results with the equivalent voltage noise N4(f) = 4 nV/Hz1/2 (from the specification of ADA4817). (b) Results with the equivalent voltage noise N4(f) = 12 nV/Hz1/2 (from experiment result of the PRs’ electronic noise). The upper right shaded parts represent that the output voltages of the PR reach the saturation output voltage of the Op-Amp.
The strong-light power and Rf determine the PRs’ DC output voltage, which is limited by the Op-Amp’s saturation voltage represented as the black dotted lines in Fig. 6. The values of strong-light power and Rf should be below these lines. Based on the result of Fig. 6(b), the maximum SNR of the PRs’ output could be obtained by setting the values of strong-light power and Rf as 500 µW and 10 kΩ, respectively.
3. Experimental results and discussions
3.1 PRs’ gain measurement
The PRs used in our experiment are designed to be the same with the feedback resistors 10 kΩ and the bandwidths larger than 30 MHz. However, due to the effect of the parasitic capacitances on the PCB board, the actual electronic components are slightly different from the parameters in Table 1.
The gains of the PRs are measured using the method mentioned above. The interference light field is generated by the dual-tone AOM with P1=P2. The PR1’s gain measurement result is shown in Fig. 7 as the blue line. Compared to parameter fitting in the red dotted line, the measurement result indicates that the PRs meet the design criteria.
Fig. 7. Gain measurement result of the PR1 in Fig. 1. The red dotted line represents the parameter fitting of the PRs’ gain taking the stray capacitances of the circuit into account. The blue line represents measuring results using the test method mentioned above.
3.2 Measuring the noises of the PR
Noises of the PRs in Fig. 1 are measured using diffracted beam generated by the fiber AOM. Because of the power fluctuation of AOM’s diffracted beam, the amplitude noise of the laser illuminating on the PR is higher than the theoretical estimation mentioned above. Hence the AOM induced laser power fluctuation is tested first in our study.
The power fluctuation of AOM’s diffracted beam is the summation of the laser RIN noise and the AOM induced laser power fluctuation [23]. The latter is mainly caused by the amplitude variation of the RF driving signal which induces changes in diffraction efficiency. To minimize the AOM induced laser power fluctuation, the usual way is to set the power of the driving signal to make AOM works at or near the maximum diffraction efficiency. The RIN noises of the laser beam and AOM diffracted beam are tested as showed in Fig. 8.
Fig. 8. RIN noises of the laser module and the combination of the laser module & AOM. The results show that the AOM induces extra power fluctuation mainly below 20 MHz. The extra noise reaches its maximum at 5 MHz, which is about double that of the laser module alone.
The results show that AOM induces extra power fluctuation mainly below 20 MHz. The maximum extra noise reaches about double that of the laser module alone around 5 MHz.
Figure 9 shows the noises of the PR when diffracted beam with power of 547 µW illuminates on the PRs’ photo surface. The red line represents the overall noise of the PRs’ output, the black line represents the electronic noise of the PR used, and the blue line is the shot noise of 547 µW laser light. As shown in the figure, the laser amplitude noise is slightly higher around 5 MHz because of the AOM induced laser power fluctuation. The electronic noise is higher than the theoretical value, most probably due to excess noise from the Op-Amp and stray capacitance from the PCB board.
Fig. 9. Noise measurement results of the PR. The electronic noise is tested without illumination and the total noise is tested under the illumination of 547 $\mathrm{\mu}$W laser power. The shot noise is calculated by laser power and the PRs' gain. The laser amplitude noise is calculated by the total noise, the electronic noise, and the shot noise.
3.3 Weak-light power measurement
During the process of the weak-light power measurement, the spectrum analyzer and the oscilloscope are needed to measure the amplitude of the AC and DC components of the output signal of PR. Self-calibration programs of the two instruments are operated before the measurement. A signal generator is used to calibrate the peak voltage signals of these two instruments at 20 MHz.
The resistance of the feedback resistor is 10 kΩ, the nominal tolerance of it is 1%. The responsivity of the PD (Re) is 0.65 A/W with a typical tolerance of 5%. The weak-light power’s error budget is shown in Table 2.
Table 2. Error Budget of the Weak-light Power Measurement
According to the estimation of Table 2, the combined relative standard uncertainty of $P_2^{\prime}$ is 13.66%. The uncertainties of AC and DC voltages contribute mainly to the total uncertainty.
3.4 Optimized SNR of the PR
Under the condition that the weak-light power is 15 pW, the strong-light power is changed and the SNR is recorded. The heterodyne interference frequency is set as 20 MHz. The parameters used for the theoretical prediction in Fig. 10 are modified to fit the results of the transfer function measurement and the noise measurement mentioned earlier. It can be seen from the results showed in Fig. 10 that the SNR variation trend is consistent with the theoretical prediction. The SNR rapidly decreases when the strong-light power is higher than 600 µW due to the saturation voltage of the Op-Amp.
Fig. 10. Influence of laser powers on SNR. The SNR variation trend is consistent with the theoretical prediction. The SNR rapidly decreases when the strong-light power is higher than 600 µW because of the saturation of the Op-Amp’s output.
Therefore, it can be concluded that under the condition of determining laser amplitude noise and electronic noise, an optimal strong-light power exists, so that the SNR of the signal can reach the maximum. In the phase measurement experiment below, we used a strong-light power of about 500 µW.
3.5 Weak-light heterodyne detection
By using the dual-tone AOM as the source of the heterodyne interference signal and adjusting the PRs’ resistance value and the strong-light power, the phase measurement amplitude spectrum curves are as shown in Fig. 11. The heterodyne interference frequency is set as 20 MHz, with light powers of 13 pW and 500 µW for the weak-light and strong-light respectively.
Fig. 11. Experiment result of the weak-light heterodyne detection. The two dash lines are the phase noises of the two PRs’ output signals compared with a digital reference signal. The solid blue line is the differential phase noise of the two PRs’ output signals. The black dotted line represents the theoretical phase noise due to 13 pW weak-light. The pink dash-dotted line represents the theoretical differential phase noise of two 13 pW weak-light. The differential signals of the two phases reach the limit of differential shot noise above 2 mHz.
According to [14,20], the noise spectral density of the phase can be obtained by the ratio of the noise to the signal. Therefore, the weak-light photocurrent shot noise induced phase noise density (ϕsn) and the electronic noise induced phase noise density (ϕen) can be respectively expressed as:
In Fig. 11, the black dashed line is the contribution of the electronic noise from the PR to the phase noise density (ϕsn). The black dotted line represents the contribution of the 13 pW weak-light photocurrent shot noise to the phase noise density (ϕen). The pink dash-dotted line represents the theoretical differential phase noise of two 13 pW weak-light powers. The theoretical differential shot noise is the square root of two times the shot noise of a single weak-light. It can be seen that the phases of the two PRs reach the limit of shot noise above 100 mHz, and the differential phase reaches the limit of differential shot noise above 2 mHz.
With the heterodyne interference frequency of 20 MHz and the strong-light power of 500 µW, we have tested the phase noises with different weak-light powers. The results of their differential phase noise are shown in Fig. 12. From top to bottom, the weak-light powers are 2.1 pW, 13 pW, 88 pW and 6.4 nW, respectively.
Fig. 12. Differential phase noises with different weak-light powers. The heterodyne frequency is 20 MHz and the strong-light power is 500 µW. The black dash-dotted line closest to each weak-light differential phase line represents the corresponding photocurrent shot noise contribution. The yellow dashed line represents the noise level of the phasemeter at 20 MHz.
The black dash-dotted line closest to each weak-light differential phase line represents the corresponding photocurrent shot noise contribution. The yellow dashed line represents the noise level of the phasemeter at 20 MHz. The thick dash-dotted line and the gray dashed line represent the interferometer requirements of TianQin {1 pm/Hz1/2 × [1+(6 mHz/f)4]1/2} and LISA {10 pm/Hz1/2 × [1+(2mHz/f)4]1/2}, respectively. To compare with the differential phase noises from the two PRs, the requirement curves are multiplied by the square root of two.
In Fig. 12, the experiment result with the weak-light power of 88 pW could roughly meet the requirement of LISA, except the noise level range from 1 mHz to 10 mHz. The weak-light power of 6.4 nW is the low limit to reach the 1 pm/Hz1/2 requirement of TianQin. The result of 6.4 nW will be discussed later. By increasing the weak-light power, the differential phase noise decreases gradually, but the inflection points of low-frequency noise increase. Since the shot noise is lower with higher power of weak-light, the other low-frequency noises are highlighted.
The differential phase measurement (PM) noise of the phasemeter is shown as the yellow dashed line (marked by ‘+’), from which it can be seen that the phasemeter noise is one of the main noises in the current system. The low-frequency noises of phasemeter include the amplitude noise of weak signal, the sampling time jitter noise, the environment thermal fluctuation, the phase shift of the reference clock, etc.
The differential phase noises with different heterodyne frequencies of 5 MHz, 10 MHz, and 20 MHz are tested with the weak-light power of 6.4 nW and the strong-light power of 500 µW. The PM noises with the above three frequencies are also tested. The experiment results are shown in Fig. 13.
Fig. 13. Results of differential phase noises (weak-light power of 6.4 nW) with different heterodyne frequencies and the PM noises corresponding to each frequency. Both the phasemeter and the weak-light phase measurements feature better performances at lower heterodyne frequencies. The phase readout system with the heterodyne frequency of 5 MHz achieves the photocurrent shot noise limit above 0.03 Hz.
The phase readout system with the heterodyne frequency of 5 MHz and the weak-light power of 6.4 nW achieves the photocurrent shot noise limit above 0.03 Hz. The current results cannot fully meet the requirement of TianQin, especially with the beat-note frequency higher than 5 MHz.
The results in Fig. 13 show that the phasemeter features a better performance with lower heterodyne frequencies. It indicates that the sampling jitter noise of the phasemeter should be decreased in the situation of weak-light interference.
In addition, the differential phase noise with the weak-light power of 6.4 nW is about twice times of the PM noise at the spectral frequencies below 0.01 Hz. The probable noise sources are the low-frequency noise of the PR, the fluctuations of light powers, and the motion of the optical mounts, etc. Further research will be carried out to decrease the low-frequency noise of the phase readout system.
4. Conclusion
In this paper, a scheme of weak-light optical heterodyne interference based on dual-tone acousto-optic diffraction is proposed to verify the performance of the phase readout system. The effects of electronic noise, laser amplitude noise and shot noise on SNR are analyzed theoretically. The measured SNR is improved by optimizing the circuit parameters and the strong-light power.
By applying this scheme with homemade PRs and a digital phasemeter, we have carried out the phase noise estimations with different weak-light powers and different heterodyne frequencies. Compared with the existing research results, the experimental setup of this scheme is simpler and has higher common-mode rejection ratios for laser frequency noise and optical-path noise. Using this scheme, the differential phase noise has achieved a performance of 2 × 10−4 rad/Hz1/2 at frequencies above 2 mHz, which is dominated by the 13 pW weak-light shot noise.
The phase readout system with the heterodyne frequency of 5 MHz and the weak-light power of 6.4 nW achieves the photocurrent shot noise limit above 0.03 Hz. The current results of the phase readout system cannot fully meet the requirement of TianQin, especially with the beat-note frequency higher than 5 MHz. The noise of the phasemeter, the low-frequency noise of the PR, and the fluctuations of light powers would be reduced and further improve the performance of the weak-light phase readout system used for intersatellite laser interferometry.
Funding
National Natural Science Foundation of China (91836104); Guangdong Major Project of Basic and Applied Basic Research (2019B030302001); National Key Research and Development Program of China (2020YFC2200200).
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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