A method for suppressing laser phase noise using improved common mode lock arm technique

Abstract

The application provides a method for suppressing laser phase noise by using improved common mode lock arm technology, comprising: establishing three data streams of a satellite-borne GW detector; combining the three data streams to eliminate noise from optical platform motion; improving the principle of the lock arm, designing a lock arm controller, and using the output of one closed loop phase meter on the main spacecraft minus the output of another closed loop phase meter to further eliminate laser phase noise. The application has the beneficial effect that the output of one closed loop phase meter on the main spacecraft minus the output of another closed loop phase meter can further eliminate laser phase noise in real time. On the one hand, even under the arm lock closed loop, the phase meter output of each arm also contains the GW signal of the other arm, which means that the GW signal can be well preserved while eliminating the remaining noise. On the other hand, since the data at the phase meter output point is studied, the data can be processed in real time, and this method is easier to implement.

Description

Technical Field

[0001] This invention relates to the field of common-mode locking arms, and more particularly to a method for suppressing laser phase noise using improved common-mode locking arm technology. Background Technology

[0002] In 2016, the Laser Interferometer Gravitational-Wave Observatory (LIGO) reported the first direct evidence of the existence of gravitational waves (GWs) predicted by Einstein's theory of general relativity. [1,2] Direct observations of GWs will open a window for testing gravitational theories under strong fields and for studying the behavior of massive celestial bodies, which is of great significance to the fields of fundamental physics and cosmology.

[0003] Typically, spaceborne GW detectors are designed with an average baseline of 10. 8 -10 9 Michelson interferometer [6-8] The coherent laser beams between spacecraft contain relative length information caused by GW. Gravitational wave signals are extremely weak, and the corresponding relative length change within the scientific frequency band is approximately... To improve detection accuracy, it is necessary to suppress various types of noise entering laser interferometry. Among the many noise sources, laser phase noise introduced by laser frequency fluctuations is dominant, and this noise is typically tens of orders of magnitude higher than the expected GW signal.

[10] For ground-based GW detectors, constructing an equal-arm configuration for the Michelson interferometer is relatively easy, allowing laser phase noise to be reduced to the required level. However, for space-based GW detectors, maintaining an equal-arm configuration during flight in space is difficult. If we assume the two arms of LISA differ in length by a few percent, the noise fluctuations from the ultra-stable laser would be approximately five orders of magnitude greater than the sensitivity of the LISA project.

[11] .

[0004] In arm-locking technology, by transferring the stability of the arm length to frequency stability, single-arm locking helps reduce laser phase noise.

[12] Subsequently, dual-arm locking and improved dual-arm locking were proposed. [14,15] Previous literature has already conducted various theoretical analyses and experiments. [16-28] The zero-point peak is shifted out of the LISA scientific band to avoid significant amplification of noise at zero points within the scientific band. However, the locking arm will fail when the arm length mismatch is zero. Typically, due to frequency pull and bandwidth limitations, the locking arm can reduce laser phase noise by 3-5 orders of magnitude (~20kHz). [15,18,27]Typical locking arm techniques cannot suppress noise other than laser phase noise. Optical platform motion noise and clock noise can be suppressed by using additional techniques on the spacecraft. For example, optical platform motion noise can be eliminated by combining the test quality data stream and reference data stream in the remote and main spacecraft.

[22] By using sideband modulation

[31] or optical frequency comb [29,30] It can effectively reduce clock noise.

[0005] Traditional arm-locking techniques typically analyze the noise floor of the closed-loop phase at the laser output point, but neglect the noise floor of the phase meter output. In fact, even under arm-locked closed-loop conditions, the phase meter output of each arm contains the GW signal of the other arm, meaning that further processing of the phase meter output can be performed simultaneously. Summary of the Invention

[0006] To address the aforementioned problems, this invention provides a method for suppressing laser phase noise using an improved common-mode locking arm technique. The spaceborne GW detector comprises three spacecraft, six optical stations, six test mass blocks, and six lasers. The locking arm technique reduces laser phase noise in real time by locking the laser frequency to the arm length of the spacecraft. The method mainly includes:

[0007] Establish a data stream for the spaceborne GW detector, which includes scientific interferometry, test quality measurements, and reference measurements;

[0008] To eliminate noise from the motion of the optical platform, the above data streams are combined;

[0009] Improve the principle of the locking arm and design a locking arm controller to ensure the needs of different frequency bands;

[0010] The laser phase noise is further eliminated by subtracting the output of the other closed-loop phase meter from the output of one closed-loop phase meter on the main spacecraft.

[0011] Furthermore, the spaceborne GW detector has three types of data streams: scientific interferometry... i (t) and s i' (t), Test quality measurement ε i (t) and ε i' (t) and reference measurement τ i (t) and τ i' (t):

[0012]

[0013]

[0014] τ i (t)=pi' (t)-p i (t)+μ i' (t) (1c)

[0015]

[0016]

[0017] τ i' (t)=p i (t)-p i' (t)+μ i (t), (3c)

[0018] Where i and i' represent two optical stages in the i-th spacecraft, namely the i-th and i'-th optical stages, and h i (t), h i' (t) represents the GW signal received by the i-th and i'-th optical stages at time t, p i (t), p i' (t) represents the laser phase noise of the i-th and i'-th optical stages at time t, p i-1 (t) represents the laser phase noise of the (i-1)th optical stage at time t, p (i+1)' (t) represents the laser phase noise of the (i+1)'th optical stage at time t. Let represent the shot noise at the photodetector of the i-th optical stage at time t. Let represent the vector random process related to the mechanical vibration of the optical stage relative to the local inertial reference frame at time t for the i-th and i'-th optical stages. Let represent the vector random process related to the mechanical vibration of the optical stage relative to the local inertial reference frame at time t for the (i-1)th optical stage. Let represent the vector random process related to the mechanical vibration of the optical stage relative to the local inertial reference frame at time t for the (i+1)'-th optical stage. μ represents the vector random process relating the test quality of the i-th and i'-th optical stages at time t to the mechanical vibration of the local inertial reference frame. i (t), μ i' (t) represents the fiber noise of the i-th and i'-th optical stages at time t, D (i+1)' Let D represent the delay operator of the (i+1)'-th optical stage. i-1 This represents the delay operator for the (i-1)th optical stage.

[0019] Furthermore, the linear combination of data streams is:

[0020]

[0021] By introducing two observable values ​​η i and η i' ,get:

[0022]

[0023]

[0024] in, These represent the unit vectors for the positive and negative directions of laser beam propagation between the two spacecraft on the optical platform i-1, respectively.

[0025] Furthermore, two lasers in the same spacecraft were used by using z i (t) Combination locking, i.e.:

[0026]

[0027] Where i and i' represent two optical stages in the i-th spacecraft, namely the i-th and i'-th optical stages, p i' (t), p i (t) represents the laser phase noise of the i-th and i'-th optical stages at time t, τ i (t) and τ i' (t) represents the reference measurement of the i-th and i'-th optical stages at time t, μ i (t), μ i' (t) represents the fiber noise of the i-th and i'-th optical stages at time t.

[0028] Furthermore, assuming all control loops are open, the output of the phase meter on the third spacecraft in the improved locking arm is given by the following formula.

[0029]

[0030] Where s3 represents scientific interferometry, and h3(ω) represents the GW signal of the third optical stage. p represents the shot noise at the photodetector of the third optical stage. O1 (ω), p O3 (ω) represents the closed-loop phase at point O1 on the phase meter of the first spacecraft and point O3 on the phase meter of the third spacecraft, respectively. This represents the unit vector along the direction of laser beam propagation between the first and third spacecraft. Let L1 and L2 represent the vector random processes related to the mechanical vibration of the optical stage of the 1' and 3' optical stages relative to the local inertial reference frame, respectively, and let L2 represent the distance between the 1st and 3rd spacecraft.

[0031] For ease of calculation, Θ is used. i (ω) and Θ i' (ω) respectively represent and Let represent the vector random processes related to the mechanical vibration of the optical stage frame of the i-th, (i-1)-th, and (i+1)-th optical stages relative to the local inertial reference frame, respectively. This represents the unit vector along the direction of laser beam propagation from the previous spacecraft to the current spacecraft. Let represent the unit vector along the direction of laser beam propagation from the current spacecraft to the next spacecraft (counterclockwise is the positive direction). If the laser on the third spacecraft is phase-locked with the incident light, then the closed-loop phase at point O3 of the phase meter on the third spacecraft is:

[0032]

[0033] Where G3 represents the closed-loop gain of the controller in the third spacecraft, h3 represents the GW signal received by the third spacecraft, and the phase p output by the phase meter on the first spacecraft is... A13 (ω) is given by the following formula:

[0034]

[0035] Among them, s 1' (ω) represents the scientific interferometry of the 1'th optical stage, h 1' This indicates the GW signal received by the 1'th optical stage. Let represent the shot noise at the photodetector of the 1'th optical stage; substituting equation (16) into equation (17), we get:

[0036]

[0037] Similarly, the phase meter output A on the first spacecraft in the improved locking arm is improved. 12 The phase at that point is:

[0038]

[0039] Where G2 represents the closed-loop gain of the second spacecraft, L3 represents the distance between the second spacecraft and the first spacecraft, and h 2' p1 represents the GW signal received by the 2' optical stage, and p2 represents the laser phase noise of the 2nd spacecraft at time t.

[0040] Furthermore, the design of the locking arm controller is divided into three stages:

[0041] (i) In the low-frequency range, the controller should have appropriate filtering to limit the Doppler frequency shift, while the phase of this range should also maintain an appropriate phase margin;

[0042] (ii) In the scientific band, the controller should maintain a high constant gain as much as possible;

[0043] (iii) In the high-frequency range, the gain should increase with frequency f. -k To reduce attenuation, k must be less than 1 in order to limit excessive signal band noise and provide phase margin.

[0044] Furthermore, in stage (i), Doppler frequency shift is limited by using a high-pass filter below the scientific band.

[0045] Furthermore, in stage (iii), the frequency response is given by the following equation:

[0046]

[0047] Among them, z i and z j To indicate zero, l i and l j Indicates an extreme point.

[0048] The beneficial effects of the technical solution provided by this invention are as follows: This invention uses the output of one closed-loop phase meter on the main spacecraft to subtract the output of another closed-loop phase meter to further eliminate laser phase noise. On the one hand, even under arm-locked closed-loop conditions, the phase meter output of each arm still contains the GW signal of the other arm, which means that the GW signal can be well preserved while eliminating other noise. On the other hand, since the data from the phase meter output points is used, the data can be processed in real time, and this method is easier to implement. Attached Figure Description

[0049] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0050] Figure 1 This is a schematic configuration diagram of the spaceborne GW detector in an embodiment of the present invention.

[0051] Figure 2 This is a schematic diagram illustrating the phase-locking configuration in an embodiment of the present invention.

[0052] Figure 3 This is a graph showing the GW sensitivity of the Michelson interferometer for different arm length mismatches in an embodiment of the present invention.

[0053] Figure 4 This is a schematic diagram of an improved locking arm in an embodiment of the present invention.

[0054] Figure 5 This is a Bode plot of the locking arm controller in the low-frequency and high-frequency components in an embodiment of the present invention.

[0055] Figure 6 This is an open-loop Bode diagram of the updated common-mode locking arm and the updated and improved dual-arm locking in this embodiment of the invention.

[0056] Figure 7 This is a graph showing the GW sensitivity of the Michelson interferometer in an embodiment of the present invention.

[0057] Figure 8 This is a block diagram of the updated common-mode locking arm in Simulink in an embodiment of the present invention.

[0058] Figure 9 This is a relative frequency fluctuation diagram of the time-domain simulation results in Simulink used in this embodiment of the invention. Detailed Implementation

[0059] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0060] In existing methods, the common-mode locking arm exhibits a peak at the zero point, resulting in high zero-point noise. This necessitates using dual locking to shift the zero point outside the scientific band. However, the improved common-mode locking arm of this invention does not exhibit a zero-point peak in the scientific band. Existing locking arm technologies do not utilize the phase meter position (i.e.,...) Figure 1 The output data (at point O in the diagram) is indirect and cannot be processed in real time. However, embodiments of this invention provide a method for suppressing laser phase noise using an improved common-mode locking arm technique. This method uses the output of one closed-loop phase meter on the main spacecraft to subtract the output of another closed-loop phase meter to further eliminate laser phase noise in real time. This is called the improved locking arm technique, which uses arm locking to suppress laser phase noise. The improved method obtains the data at point A of the phase meter output, which can be processed in real time, and the method is easy to implement.

[0061] 1. Theoretical Analysis and Simulation

[0062] First, this invention demonstrates that the various noise and signal terms under closed-loop control are the same as those under open-loop control, except that laser noise is suppressed. Furthermore, the improved common-mode locking arm of this invention does not require the construction of a complex locking arm sensor to move the zero point outside the scientific band. Then, a locking arm controller is designed to ensure high gain in the scientific band and meet other requirements. Results show that laser phase noise can be suppressed to below the secondary noise level for most of the scientific band. Finally, time-series simulations were performed in Simulink to verify the feasibility of this method.

[0063] Typically, a spaceborne GW detector consists of three spacecraft, six optical stations, six test mass blocks, and six lasers.

[0064] like Figure 1 As shown, the three spacecraft are labeled as spacecraft SC1, SC2, and SC3, respectively. The two optical stages within each spacecraft are labeled i and i', where i = 1, 2, and 3. The spacecraft and optical stages correspond to each other. The separation distance between the spacecraft is denoted by L. i L i' This indicates that, relative to optical platforms i and i', because the distance the laser travels outward and the distance it travels back are different, the distances between spacecraft are constantly changing. Therefore, L1, L2, and L3 represent the distances between the 3rd and 2nd spacecraft, the 1st and 3rd spacecraft, and the 2nd and 1st spacecraft, respectively. 1' L 2' L 3' These represent the distances between the second and third spacecraft, the third and first spacecraft, and the first and second spacecraft, respectively. It is the unit vector of the laser beam propagation direction between the two spacecraft represented by optical platform i. It is the unit vector along the direction of laser beam propagation between the third and second spacecraft. It is the unit vector along the direction of laser beam propagation between the first and third spacecraft. It is the unit vector along the laser beam propagation direction between the second and first spacecraft. The onboard GW detector has three types of data streams: scientific interferometry data from the i-th and i'-th optical stages. i (t) and s i' (t), Test quality measurement ε i (t) and ε i' (t) and reference measurement τ i (t) and τ i' (t). These measurements from optical platform i are:

[0065]

[0066]

[0067] τ i (t)=p i' (t)-p i (t)+μ i' (t) (1c)

[0068] Where h is the GW signal, p is the laser phase noise, and N is the shot noise at the photodetector. Item and The terms are vector random processes related to the mechanical vibrations of the optical stage and the test mass relative to the local inertial reference frame, respectively; μ is the fiber noise; and D is the delay operator. The delay operator D for any data stream x(t) is... i Defined by the following formula:

[0069] D i x(t)=x(tL i (2)

[0070] Where the unit length is the speed of light c, in this embodiment, the speed of light is used as the benchmark, and the unit speed of light is considered as unit 1, that is, c = 1. A delay of one second equals the distance c, which is 1. A delay of t seconds equals the distance ct. As shown in Equation 2, for the distance L between spacecraft... i The delay time is L. i No further transformation by dividing L by the speed of light c is needed. The expressions in this method do not include clock noise. Similarly, for optical platform i', we have:

[0071]

[0072]

[0073] τ i' (t)=p i (t)-p i' (t)+μ i (t),(3c)

[0074] Note the differences ε between each optical platform i (t)-τ i (t) contains displacement information of the optical platform relative to the test mass. Therefore, it is necessary to combine these data streams in an appropriate manner to eliminate noise from the motion of the optical platform.

[11] Therefore, the following linear combination is defined:

[0075]

[0076] The subscript indicates a cycle: 1-2-3-1-2-3-1···, that is, when i=1, i+1 is 2, when i=2, i+1 is 3, when i=3, i+1 is 1, and so on.

[0077] By introducing two observable values ​​η from the i-th spacecraft i and η i' ,get:

[0078]

[0079]

[0080] Each spacecraft contains two lasers. i and p i' Typically, other lasers are phase-locked with the main laser.

[32] This embodiment uses Figure 2 In the phase-locked configuration, laser p 1' It was selected as the primary laser. Two lasers in the same spacecraft were selected using z... i Combination locking, i.e.:

[0081]

[0082] Assuming fiber noise is reciprocal, therefore p i' equals p i Phase-locking conditions between spacecraft are determined using scientific interferometry. i (t) can be used as the error signal to obtain:

[0083]

[0084] Therefore, by combining equations 5-8, we can obtain the observable values ​​η1 and η in the main spacecraft (i.e., the first spacecraft). 1' Data flow:

[0085]

[0086]

[0087] From (9a)-(9b), we can obtain:

[0088]

[0089] Equation (10) shows that if the interferometer arms in the lock arm are equal, i.e., the distances between the spacecraft are L1 = L2 = L3, the laser phase noise can be precisely eliminated. In fact, the lock arm lengths will differ by a few percent. The greater the difference in length between the two arms, the greater the amplitude of the laser phase noise. To determine whether the laser phase magnitude meets the requirements, obtaining the sensitivity limit of the instrument is crucial. Typically, the sensitivity of the detector in the lock arm can be quantified by the ratio of signal strength to instrument noise, which is called the signal-to-noise ratio (SNR) and is generally expressed in the frequency domain. [33 – 35] The combined power spectral density (PSD) of the residual noise in equation (10) can be expressed in terms of relative frequency fluctuations, i.e.:

[0090]

[0091] μ=2πfL,ΔL=L3-L2, Where f represents the laser frequency, L represents the arm length, and S represents the laser frequency. a S x δ v The amplitude spectral densities (ASDs) of mass, shot noise, and laser phase noise were measured separately. The combined transfer function R(f) can be obtained when the direction of the GW signal is perpendicular to the plane of the onboard GW detector.

[0092]

[0093] Therefore, the desired expression for the sensitivity function h(f) of the spaceborne GW detector can be obtained:

[0094]

[0095] If Fabry-Perot cavity pre-stabilized laser phase noise is used as input [15,36] ,have:

[0096]

[0097] For the LISA task, the arm length of the locking arm is L = 2.5 × 10⁻⁶. 6 The maximum arm length difference can reach 35,000 km (approximately 1%). The initial targets for testing mass noise and shot noise are respectively... In this embodiment, the laser frequency is selected as v0 = 3 × 10⁻⁶. 14 Hz. Substituting the above parameters into equation (13), we can obtain the GW sensitivity limit, such as Figure 3 As shown, the solid line assumes an arm length mismatch ΔL / L = 0, which can also be considered the sensitivity limit of the instrument. The dashed line assumes an arm length mismatch ΔL / L = 1%. Clearly, the current laser phase noise cannot meet the requirements of GW detection, with a difference of approximately five orders of magnitude. Therefore, this invention designs an alternative method for eliminating laser noise—an improved arm-locking technique. To date, proposed methods include TDI (Thin-Dip Resonance).

[11] and locking arm technology

[12] The basic principle of TDI is to synthesize virtual equal-arm-length interference through data post-processing.

[11] Arm-locking technology is a real-time technique that locks the laser frequency to the arm length of the constellation.

[12] This invention employs locking arm technology. Furthermore, from... Figure 3 As can be seen, in the case of the transfer function of this simple Michelson combination, the zero value always exists.

[0098] 2. Improved Locking Arm Technology Principle

[0099] First, the principle of the improved locking arm is as follows: Figure 4As shown, the locking arms include common-mode locking arms, differential-mode locking arms, and dual-arm locking arms, etc. Different locking arms correspond to different sensors. Subtraction is a subtractor, and the laser p... 1' As the master laser, therefore p 1' It serves as the phase reference for all other lasers. Figure 4 In this diagram, G1, G2, and G3 represent the closed-loop gains of the first controller (located in the first spacecraft SC1), the second controller (located in the second spacecraft SC2), and the third controller (located in the third spacecraft SC3), respectively. Clock noise is not considered in this method. For simplicity, ω is used to represent f. Assuming all control loops are open, then A on the third spacecraft... 31 The phase meter output p at point (i.e., s3) A31 It is given by the following formula:

[0100]

[0101] For ease of calculation, Θ is used. i (ω) and Θ i' (ω) respectively represent and The subscripts here represent a cycle: 1-2-3-1-2-3-1..., that is, when i = 1, i+1 is 2, when i = 2, i+1 is 3, when i = 3, i+1 is 1, and so on. If the laser on spacecraft 3 is phase-locked with the incident light, then the closed-loop phase at point O3 is:

[0102]

[0103] The phase output by the phase meter on spacecraft 1 is given by the following formula:

[0104]

[0105] Substituting equation (16) into equation (17), we get:

[0106]

[0107] Similarly, point A 12 The phase at that point is:

[0108]

[0109] In the high-gain transponder limit, G j / (1+G j → 1, 1 / (1+G j→0,j=2,3,G2 represents the closed-loop gain of the second spacecraft, and G3 represents the closed-loop gain of the third spacecraft. Therefore, equations (18) and (19) can be rewritten in matrix form:

[0110]

[0111] Table 1: Transfer function matrix of different locking arm sensors

[0112]

[0113] in For the transfer function, if a vector L is used X =[αβ] represents the form of different types of lock arm sensors, and at point O1 we have: p1(ω)-G1L X ·p A1 (ω)=p O1 (ω), G1 represents the closed-loop gain of the first spacecraft, p O1 (ω) represents the closed-loop phase at point O1 of the phase meter on the first spacecraft. The transfer functions of different sensors are shown in Table 1.

[15] Then the signal at point O1 can be obtained:

[0114]

[0115] in It is a lock arm sensor.

[0116] In existing technologies, the focus is on the signal output at point O1, rather than point A1. Clearly, the phase meter output at point A1 is also of significant research value. Based on equation (20), the closed-loop signal output at point A1 can be given by the following equation:

[0117]

[0118] Where N p The laser phase noise detected at the main spacecraft's optical receiver, N s It is shot noise, N Θ It is the equivalent optical stage motion noise, N h It's a GW signal:

[0119]

[0120] By subtracting the outputs of the two closed-loop phase meters at point A1, the output at point B is obtained.

[0121]

[0122] in Based on the transfer function in Table 1, the results for different sensors can be obtained. If single-arm locking and differential-mode locking are selected, the GW signal is suppressed by the closed-loop gain G1. Therefore, these two sensors are not considered here. When the other three arm locking sensors are selected, the output of equation (24) is similar. Taking a common common-mode locking arm as an example:

[0123]

[0124] In the formula, h1 is h1(ω).

[0125] Similarly, test quality measurements and reference measurements can be used to eliminate optical stage motion noise. Therefore, combining equations 4 and 25 yields the new data stream after eliminating optical stage noise:

[0126]

[0127] Note that there is an approximation 1+2G1T 13 ≈1+2G1T 12 ≈1+G1T 12 +G1T 13 Equation (26) can be approximated as:

[0128]

[0129] Except for the laser noise term, the other terms in equation (27) are the same as in equation (10), and the laser noise is factored by 1 / (1+G1Y). X Suppression. The zero point of this factor will result in a peak in laser phase noise suppression. However, the transfer function of this Michelson combination also reaches an extreme value at the same frequency; therefore, in this scheme, it is not necessary to use a double-locked or improved double-locked sensor to move the zero point outside the scientific band. According to Figure 3 The comparison shows that the control system G1Y X The open-loop gain should be designed to be approximately 10 in the frequency band. 5 In addition, the controller design must meet the requirements of the controller, namely, the requirements of low frequency, high frequency and scientific band.

[0130] GW performance after 3-lock arm treatment

[0131] 3.1 Design of the Lock Arm Controller

[0132] The design of the locking arm controller can be divided into three stages:

[0133] (i) In the low-frequency range, the controller should have appropriate filtering to limit the Doppler shift. This can be achieved by using a high-pass filter below the scientific band. Simultaneously, the phase in this range should also maintain an appropriate phase margin.

[0134] (ii) In the scientific band, the controller should maintain a high constant gain O(10) as much as possible. 5 However, it should be noted that the gain at 1 Hz is limited by the lock-arm bandwidth.

[0135] (iii) In the high-frequency range, the gain should increase with f -k To reduce attenuation, k must be less than 1 in order to limit excessive signal band noise and provide phase margin.

[0136] Generally, the unity-gain frequency is set to approximately 20kHz, which is also the lock-arm bandwidth. If an f0 is chosen... -2 / 3 With roll-off, the gain at 1 Hz is approximately 1000, and the phase margin is approximately 30°. This can be achieved by placing zeros and poles with a frequency spacing ratio of 5. Then, two pairs of zeros and poles with a frequency spacing ratio of 9 are placed to increase the gain by 0.01 Hz. The frequency response of this part is given by the following equation:

[0137]

[0138] Among them, zero point z i and pole l i They are 2π×5Hz and 2π×1Hz, 2π×50Hz and 2π×10Hz, etc., with zero point z. j and pole l j They are 2π×0.01Hz and 2π×0.9Hz, respectively.

[0139] Figure 5 The Bode plot for this section is shown on the left. For better demonstration, a constant gain for the low-frequency range has been added. As shown, all requirements are met. For the low-frequency range, reference [reference missing] is used.

[15] The design was modified, but some changes were made to the values ​​of the zeros and poles. The frequency response of this part is given by the following equation:

[0140]

[0141] Among them, zero point z 12 = 2π × 18.3 × 10 -6 Extreme point l 10 =2π×3×10 -7 Hz, pole l 11 =2π×2×10 - 4 Hz, pole l 12 = 2π × 9.3 × 10 -5 Hz, and gain g 11 =15.8g 12 =5. Figure 5 The Bode plot for this part is shown in (right). The first part of equation (29) is the limiting Doppler frequency pull, and the last part is the convolution gain. The second part provides a phase margin of less than 180° at a lower unity-gain frequency. Then, the locking arm sensor Y can be obtained. X The open-loop gain of the control system for (ω) is:

[0142]

[0143] in It is the locking arm sensor Y X The approximation of (ω) aims to eliminate the sensor's gain drop. For common-mode sensors and dual-arm sensors, the goal is to obtain... and therefore, Figure 6 The open-loop Bode plot of the common-mode locking arm is shown on the left. It can be seen that the higher and lower unity-gain frequencies are 3.6 μHz and 20 kHz, respectively, with a phase margin of approximately 30°, and the phase at the lower unity-gain frequency is approximately 150°, ensuring the stability of the control system. To better compare different locking arm sensors, controllers were designed for both the common-mode locking arm sensor and the improved dual-arm sensor. The open-loop Bode plots of these two sensors are shown below. Figure 6 As shown on the right.

[0144] 3.2 GW strain performance after arm locking

[0145] exist Figure 7 In this paper, the performance of the detector with the updated common-mode locking arm is plotted for the FP cavity. In the frequency range of 0.1 mHz to 0.01 Hz, the laser phase noise is suppressed by 5 orders of magnitude and meets the detector sensitivity requirements. In the range of 0.01 Hz to 1 Hz, the laser phase noise is suppressed by 3-4 orders of magnitude, but does not meet the detector sensitivity requirements, which are limited by the 20 kHz bandwidth. Furthermore, it is noted that the results for the updated common-mode locking arm are almost identical to those for the updated modified dual-arm locking, concluding that it is unnecessary to use a dual-lock sensor to shift the null point outside the scientific band. Figure 7 The graph was drawn with a maximum arm length mismatch of 1%, which means that performance will be better for smaller length mismatches.

[0146] 4. Performance of the method in time-domain simulation

[0147] Temporal simulations were performed using Simulink to verify the lower bound of the laser phase noise for the improved common-mode locking arm scheme. Figure 4 The schematic diagram shown is equivalent to the Simulink block diagram. Figure 8As shown in the figure. In this simulation, it is assumed that clock noise and optical platform motion noise are removed, shot noise has no effect on the final result, and only the effect of laser phase noise is considered. The goal is to verify the system's ability to suppress laser phase noise; therefore, these additional noise sources are not included in this simulation.

[0148] First, to achieve faster computation time, the laser phase noise is modeled as having an amplitude of 30 Hz / Hz. 1 / 2 The white noise. The controller design in SC1 is the same as the controller design in Section 3.1. The gain of the controllers in SC2 and SC3 is sufficient to allow the spacecraft to act as a reflector. In this embodiment, the scientific band is (10 -4 (Hz-1Hz), with the sampling time and total simulation time set to 0.1s and 5×10⁻¹⁰ respectively. 4 After the simulation, the laser phase noise output at points O and B can be obtained.

[0149] Convert the result to 3×10 14 The relative frequency fluctuation pattern of Hz laser frequency. Figure 9 The closed-loop relative frequency fluctuations of the time-domain data at points O (line 2) and B (line 3) are compared with the input relative frequency fluctuations (line 1). Furthermore, the corresponding theoretical results obtained through equations (21) and (26) are shown (dashed lines). The relative frequency fluctuations at 10... -4 The frequency decreases by approximately 7-8 orders of magnitude from Hz to 0.01Hz, and by approximately 4-7 orders of magnitude from 0.01Hz to 1Hz. Furthermore, a sinusoidal signal with an amplitude of 0.05μHz and a frequency of 1mHz is added to the simulation to model the GW signal. The corresponding output is also... Figure 9 As shown, a 1 mHz GW signal is sufficiently clear. Furthermore, the output at point B is obtained in real time, and the close consistency between theoretical and simulated noise performance validates the theoretical modeling scheme. This also demonstrates that laser phase noise suppression can be achieved in real time, and that the scheme is easy to implement.

[0150] Aside from laser phase noise, the output after locking the main spacecraft arm is identical to that without arm locking. Then, within the frequency range of 0.1 mHz to 0.01 Hz, laser phase noise can be suppressed to the desired level through rigorous controller design. Simulink simulation results agree well with theoretical results, demonstrating the feasibility of the method. Performance in the 0.01 Hz to 1 Hz frequency range is limited by a 20 kHz bandwidth; once the bandwidth is widened, noise suppression performance improves. Furthermore, zero-point peaks always exist in the presence of the GW transfer function of the Michelson interferometer, meaning that complex arm-locking sensors are unnecessary.

[0151] The beneficial effects of this invention are as follows: Existing arm locking technologies mainly include single-arm, dual-arm, differential-mode arm locking, common-mode arm locking, and improved dual-arm locking technologies. Typically, the noise floor of the closed-loop phase at the laser output point is analyzed, while the noise floor of the phase meter output is ignored. The noise floor of the closed-loop phase cannot be displayed in real time, let alone processed in real time. This invention proposes a new method that uses the output of one closed-loop phase meter on the main spacecraft to subtract the output of the other closed-loop phase meter to further eliminate laser phase noise in real time. On the one hand, even under arm-locked closed-loop conditions, the phase meter output of each arm still contains the GW signal of the other arm, meaning that the GW signal can be well preserved while eliminating other noise, which has been verified in simulations. On the other hand, since the data from the phase meter output point is used, the data can be processed in real time, and this method is easier to implement.

[0152] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

[0153] Finally, the relevant references cited in this invention are uniformly explained as follows:

[0154] [1] B. Abbott et al., (LIGO Scientific and Virgo Collaborations), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett., 116, 061102 (2016).

[0155] [2] B. Abbott et al., (Virgo and LIGO Scientific Collaborations), GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary BlackHole Coalescence, Phys. Rev. Lett., 116, 241103 (2016).

[0156] [3]J.Aasi et al.,(LIGO Scientific Collaboration),Advanced LIGO,Class.Quant.Grav.,32,074001(2015).

[0157] [4]F.Acernese et al.,(VIRGO Collaboration),Advanced Virgo:a second-generation interferometric gravitational wave de-tector,Class.Quant.Grav.,32,024001(2015).

[0158] [5]K.Somiya(KAGRA Collaboration),Detector configuration of Japanese cryogenic gravitational-wave de-tector,Class.Quant.Grav.,29,124007(2012).

[0159] [6]P.Amaro-Seoane et al.,Laser Interferometer Space Antenna,arXiv:1702.00786(2017).

[0160] [7]J.Luo et al.,TianQin:a space-borne gravitational wave detector,Class.Quant.Grav.,33,035010(2016).

[0161] [8]X.Gong et al.,Descope of the ALIA mission,J.Phys.Conf.Ser.,610,012011(2015).

[0162] [9]S.Kawamura et al.,The Japanese space gravitational wave antennaDECIGO,Class.Quant.Grav.,23,S125(2006).

[0163]

[10] M.Tinto and J.Armstrong,Cancellation of laser noise in anunequal-arm interferometer detector of gravitational radiation,Phys.Rev.D,59,102003(1999).

[0164]

[11] M.Tinto and S.V.Dhurandhar,Time-delay interferometry.LivingRev.Relativity,24,1(2021).

[0165]

[12] B.S.Sheard,M.B.Gray,D.E.McClelland,D.A.Shaddock,Laser frequencystabilization by locking to a LISA arm,Phys.Lett.A,320,9(2003).

[0166]

[13] M.Tinto,F.B.Estabrook,J.W.Armstrong,Time delay interferometrywith moving spacecraft arrays,Phys.Rev.D,69,082001(2004).

[0167]

[14] A.Sutton,D.A.Shaddock,Laser frequency stabilization by dual armlocking for LISA,Phys.Rev.D,78,082001(2008).

[0168]

[15] K.McKenzie,R.E.Spero,and D.A.Shaddock,Performance of arm lockingin LISA,Phys.Rev.D,80,102003(2009).

[0169]

[16] J.Sylvestre,D.A.Shaddock,Simulations of laser locking to a LISAarm,Phys.Rev.D,70,102002(2004).

[0170]

[17] A.F.Garcia Marin,G.Heinzel,R.Schilling,A.Rudiger,V.Wand,F.Steier,F.G.Cervantes,A.Weidner,O.Jennrich,F.Meca ,and K.Danzman,Phase locking to aLISA arm:first results on a hardware model,Class.Quantum Grav.,22,S235(2005).

[0171]

[18] J.Thorpe and G.Mueller,Experimental verification of arm-lockingfor LISA using electronic phase delay,Phys.Lett.A,342,199(2005).

[0172]

[19] B.S.Sheard,M.B.Gray,D.A.Shaddock,et al.,Laser frequency noisesuppression by arm-locking in LISA:progress towards a bench-topdemonstration,Class.Quant.Grav.,22,S221(2005).

[0173]

[20] B.S.Sheard,M.B.Gray,and D.E.McClelland,High-bandwidth laserfrequency stabilization to a fiber-optic delay line,Appl.Opt.,45,8491(2006).

[0174]

[21] Yu Y,Wand V,Mitryk S,et al.,Arm locking with Doppler estimationerrors,Journal of Physics:Conference Series.,228,012044(2010).

[0175]

[22] J.Thorpe,P.Maghami,and J.Livas,Time domain simulations of armlocking in LISA,Phys.Rev.D,83,122002(2011).

[0176]

[23] Y.Yu,S.Mitryk,and G.Mueller,Experimental validation of dual / modified dual arm locking for LISA,Class.Quantum Grav.,28,094009(2011).

[0177]

[24] Y.Yu,S.Mitryk,and G.Mueller,Arm locking for space-based laserinterferometry gravitational wave observatories,Phys.Rev.D,90,062005(2014).

[0178]

[25] J.I.Thorpe,K.McKenzie,Arm locking with the GRACE follow-on laserranging interferometer,Phys.Rev.D,93,042003(2016).

[0179]

[26] H.Wu,J.Ke,P.P.Wang,et al.,Arm locking using laser frequency comb,Optics Express,30,8027(2022).

[0180]

[27] J.T.Valliyakalayil,J.H.Sutton,R.E.Spero,et al.,Enhanced frequencynoise suppression for LISA by combining cavity and arm locking controlsystems,Phys.Rev.D,105,062005(2022).

[0181]

[28] S.Ghosh,J.Sanjuan,G.Mueller,Arm locking performance with the newLISA design,Class.Quantum Grav.,39,115009(2022).

[0182]

[29] M.Tinto andN.Yu,Time-delay interferometry with optical frequencycomb,Phys.Rev.D,92,042002(2015).

[0183]

[30] H.Z.Wu,P.P.Wang,P.Hao,et al.,Time delay interferometry usinglaser frequency comb as the direct signal source,Optics and Lasers inEngineering,151,106938(2022).

[0184]

[31] P.P.Wang,Y.J.Tan , et al.,Refined clock-jitter reduction in theSagnac-type time-delay interferometry combinations,Phys.Rev.D,104,082002(2021).

[0185]

[32] M.Tinto,D.A.Shaddock,J.Sylvestre , et al.,Implementation oftime-delay interferometry for LISA,Phys.Rev.D,67, 122003(2003).

[0187]

[33] J.Y.Vinet,Some basic principles ofa LISA,C.R.Phys.,14,366(2013).

[0188]

[34] X.Y.Lu,Y.J.Tan,C.G.Shao,Sensitivity functions for space-bornegravitational wave detectors.Phys.Rev.D,100,044042(2019).

[0189]

[35] P.P.Wang,Y.J.Tan ,et al.,Sensitivity functions ofspace-bornegravitational wave detectors for arbitrary time-delay interferometrycombinations regarding nontensorial polarizations,Phys.Rev.D,104,023002(2021).

[0190]

[36] J.Thorpe,Lisa long-arm interferometry,Class.Quantum Grav.,27,11(2009).

Claims

1. A method for suppressing laser phase noise using an improved common-mode lock-arm technique, wherein the spaceborne GW detector comprises three spacecraft, each spacecraft containing two optical stages, two test mass blocks, and two lasers, characterized in that: include: Establish a data stream for the spaceborne GW detector, which includes scientific interferometry, test quality measurements, and reference measurements; To eliminate noise from the motion of the optical platform, the above data streams are combined; The principle of the locking arm was improved, and a locking arm controller was designed to ensure that the controller meets the different requirements of low frequency, scientific band and high frequency. Specifically: Assuming all control loops are open, the output of the phase meter on the third spacecraft in the improved locking arm is given by the following formula. Where s3 represents scientific interferometry, and h3(ω) represents the GW signal of the third optical stage. p represents the shot noise at the photodetector of the third optical stage. O1 (ω), p O3 (ω) represents the closed-loop phase at point O1 on the phase meter of the first spacecraft and point O3 on the phase meter of the third spacecraft, respectively. This represents the unit vector along the direction of laser beam propagation from the first spacecraft to the third spacecraft. Let L1 and L2 represent the vector random processes related to the mechanical vibration of the optical stage of the 1' and 3' optical stages relative to the local inertial reference frame, respectively, and let L2 represent the distance from the 1st spacecraft to the 3rd spacecraft. For ease of calculation, Θ is used. i (ω) and Θ i' (ω) respectively represent and Let represent the vector random processes related to the mechanical vibration of the optical stage frame of the i-th, (i-1)-th, and (i+1)-th optical stages relative to the local inertial reference frame, respectively. This represents the unit vector along the direction of laser beam propagation from the previous spacecraft to the current spacecraft. Let represent the unit vector along the direction of laser beam propagation from the current spacecraft to the next spacecraft, with counterclockwise as the positive direction. If the laser on the third spacecraft is phase-locked with the incident light, then the closed-loop phase at point O3 of the phase meter on the third spacecraft is: Where G3 represents the closed-loop gain of the controller in the third spacecraft, h3(ω) represents the GW signal received by the third spacecraft, and the phase p output by the phase meter on the first spacecraft is... A13 (ω) is given by the following formula: Among them, s 1' (ω) represents the scientific interferometry of the 1'th optical stage, h 1' This indicates the GW signal received by the 1'th optical stage. Let represent the shot noise at the photodetector of the 1'th optical stage; substituting equation (16) into equation (17), we get: Similarly, the phase meter output A on the first spacecraft in the improved locking arm is improved. 12 The phase at that point is: Where G2 represents the closed-loop gain of the second spacecraft, L3 represents the distance between the second spacecraft and the first spacecraft, and h 2' p1 represents the GW signal received by the 2' optical stage, and p2 represents the laser phase noise of the 2nd spacecraft at time t. Laser phase noise can be further eliminated by subtracting the output of another closed-loop phase meter from the output of one closed-loop phase meter on the main spacecraft.

2. The method for suppressing laser phase noise using an improved common-mode locking technique as described in claim 1, characterized in that: The spaceborne GW detector has three types of data streams: scientific interferometry... i (t) and s i' (t), Test quality measurement ε i (t) and ε i' (t) and reference measurement τ i (t) and τ i' (t): τ i (t)=p i' (t)-p i (t)+μ i' (t) (1c) τ i' (t)=p i (t)-p i' (t)+μ i (t), (3c) Where i and i' represent two optical stages in the i-th spacecraft, namely the i-th and i'-th optical stages, and h i (t), h i' (t) represents the GW signal received by the i-th and i'-th optical stages at time t, p i (t), p i' (t) represents the laser phase noise of the i-th and i'-th optical stages at time t, p i-1 (t) represents the laser phase noise of the (i-1)th optical stage at time t, p (i+1)' (t) represents the laser phase noise of the (i+1)'th optical stage at time t. Let represent the shot noise at the photodetector of the i-th optical stage at time t. Let represent the vector random process related to the mechanical vibration of the optical stage relative to the local inertial reference frame at time t for the i-th and i'-th optical stages. Let represent the vector random process related to the mechanical vibration of the optical stage relative to the local inertial reference frame at time t for the (i-1)th optical stage. Let represent the vector random process related to the mechanical vibration of the optical stage relative to the local inertial reference frame at time t for the (i+1)'-th optical stage. μ represents the vector random process relating the test quality of the i-th and i'-th optical stages at time t to the mechanical vibration of the local inertial reference frame. i (t), μ i' (t) represents the fiber noise of the i-th and i'-th optical stages at time t, D (i+1)' Let D represent the delay operator of the (i+1)'-th optical stage. i-1 This represents the delay operator for the (i-1)th optical stage.

3. The method for suppressing laser phase noise using an improved common-mode locking technique as described in claim 2, characterized in that: The linear combination of data streams is: By introducing two observable values ​​η i and η i' ,get: in, These represent the unit vectors for the positive and negative directions of laser beam propagation between the two spacecraft on the optical platform i-1, respectively.

4. The method for suppressing laser phase noise using an improved common-mode locking technique as described in claim 1, characterized in that: Two lasers in the same spacecraft use z i (t) Combination locking, i.e.: Where i and i' represent two optical stages in the i-th spacecraft, namely the i-th and i'-th optical stages, p i' (t), p i (t) represents the laser phase noise of the i-th and i'-th optical stages at time t, τ i (t) and τ i' (t) represents the reference measurement of the i-th and i'-th optical stages at time t, μ i (t), μ i' (t) represents the fiber noise of the i-th and i'-th optical stages at time t.

5. A method for suppressing laser phase noise using an improved common-mode locking technique as described in claim 1, characterized in that: The design of the locking arm controller is divided into three stages: (i) In the low-frequency range, the controller should have appropriate filtering to limit the Doppler frequency shift, while the phase of this range should also maintain an appropriate phase margin; (ii) In the scientific band, the controller should maintain a high constant gain as much as possible; (iii) In the high-frequency range, the gain should increase with frequency f. -k To reduce attenuation, k must be less than 1 in order to limit excessive signal band noise and provide phase margin.

6. A method for suppressing laser phase noise using an improved common-mode locking technique as described in claim 5, characterized in that: In stage (i), Doppler frequency shift is limited by using a high-pass filter below the scientific band.

7. A method for suppressing laser phase noise using an improved common-mode locking technique as described in claim 5, characterized in that: In stage (iii), the frequency response is given by the following equation: Among them, z i and z j To indicate zero, l i and l j Indicates an extreme point.

Images