Taiji Program in Space of the Chinese Academy of Sciences mainly uses the space laser interferometry to measure the middle and low frequency gravitational waves(0.1 mHz-1 Hz). In addition to covering ESA′s LISA/eLISA planned detection frequency bands, this band includes sources such as coalescence of supermassive and intermediate mass black holes, EMRI system, Hanoi white dwarfs orbiting, and other cosmic gravitational radiation processes. The program focuses on the frequency range of 0.01 Hz-1 Hz, which has a higher detection sensitivity than LISA/eLISA. Unlike the scientific goal of LISA/eLISA, the Taiji program has a significant detection advantage in focusing on intermediate mass, dual black hole orbiting systems of masses ranging from several hundred to one hundred thousand solar masses. The main scientific goal of the Taiji program is to determine the mass, spin and distribution of black holes and the polarization of gravitational waves through the accurate measurement of gravitational waves; to explore how medium-sized seed black holes are formed; to determine whether dark matter forms a seed black hole, and how seed black holes grow into massive black holes and supermassive black holes? In addition, traces of the formation, evolution and death of the first generation of stars are investigated, direct limits on the strength of the original gravitational waves are given，direct observation data for revealing the nature of gravitation are provided.
中科院空间太极计划(Taiji Program in Space)主要采用空间激光干涉法测量中、低频段引力波(0.1 mHz~1 Hz)。此频段覆盖了欧空局的LISA/eLISA计划探测频段，其波源包括超大质量和中等质量黑洞的并合、极大质量比绕转系统、河内白矮星绕转、以及其它宇宙引力波辐射过程。太极计划探测重点在0.01~1 Hz频段，具有比LISA/eLISA更高的探测灵敏度。另外，有别于LISA/eLISA的科学目标，太极计划将重点瞄准质量在几百至十万太阳质量范围内的中等质量双黑洞绕转并合系统，对此太极计划有显著的探测优势。太极计划的主要科学目标是通过精确测量引力波，测定黑洞的质量、自旋以及分布和引力波的极化；探索中等质量种子黑洞是如何形成的，暗物质能否形成种子黑洞，种子黑洞是如何成长为大质量黑洞和超大质量黑洞的；寻找第一代恒星形成、演化、死亡的遗迹，对原初引力波强度给出直接限制，为揭示引力本质提供直接的观测数据。
Laser Interferometer Space Antenna(LISA) is a deep space satellite exploration mission co-operated by Europe and the United States, which is mainly used to detect and study the gravitational waves at frequency band of 10-4-10-1 Hz. The basic mission concepts are described in many documents and all LISA-like mission designs use optical telescopes to send and receive beams, although the task proposals vary from country to country. The detection system adopts an equilateral triangular formation of satellites with three identical satellites at the apexes of the triangle, respectively. Each satellite contains two identical interferometer platforms, with two suspended test loads serving as the endm points of the Michelson interferometer(as shown in Fig. 1). The length of the equilateral triangle is on the order of one million kilometers and the detection of gravitational waves requires that the distance between two drag-free test masses at both ends of the measuring side should be within 10 picometers(10-4 to 10-1 Hz, arm length 5 million km). Due to the divergence angle of the Gaussian beam, the telescope's field of view and the effect of diffraction, the laser at the transmitter 2 W reaches the receiver by the order of about 100 pW. If the arm length changes, the laser emission capacity and the telescope aperture must be adjusted accordingly to ensure that the telescope in the distant spacecraft receives the same amount of energy.
图 1 空间引力波探测天文台，三个航天器构成等边三角形，通过端点航天器间的激光链路来测量位于端点的测试质量间距离的变化
Figure 1. A space-based gravitational wave observatory, consists of an equilateral triangle of three spacecraft with laser links between endpoint spacecraft, to measure the change in the distance between the test masses at the endpoint
LISA是一个欧美合作的深空卫星探测任务，主要是为了探测和研究频段在10-4~10-1 Hz的引力波。基本的任务概念在很多的文献都有描述，尽管不同国家的任务建议多少有差别，但所有LISA-like的任务设计都使用光学望远镜来收发光束。探测系统采用等边三角形的卫星编队构型，三颗相同的卫星分别位于三角形的3个顶点。每颗卫星含2套相同的干涉仪平台、2个悬浮的测试质量作为Michelson干涉仪的端点(如图 1所示)。等边三角形的边长在百万公里量级，引力波的探测要求测量边长两端2个自由悬浮测试质量间距离的变化在10皮米量级(10-4~10-1 Hz，臂长500万公里)。由于高斯光束的发散角、望远镜视场以及衍射作用的影响，发射端2 W的激光到达接收端大约100 pW的量级。因此，如果臂长发生变化，发射激光能力及望远镜口径也需相应调整以保证远处航天器的望远镜接收的能量在同样的量级。
The three spacecraft of the LISA mission move in the Keplerian orbit around the sun respectively, the angle between the plane formed by the three spacecraft and the ecliptic is 60° (Fig. 2), the formation of triangles will make a complete revolution within a year while the spacecraft is spinning around the Sun. LISA′s orbital design is the simplest and the most mature design, and the configuration of the three spacecraft ensures that the mission will be in stable operation for five years, and even up to 8 years.
图 2 LISA轨道及航天器编队示意图，航天器组成的平面与黄道面夹角约为60°, 航天器星座质心落后地球约20°，航天器之间的距离为5×106 km
Figure 2. Schematic diagram of LISA orbit and spacecraft formation. The angle between the spacecraft plane and the ecliptic plane is about 60°. The constellation trails earth by about 20°, and the distance between spacecraft is 5×106 km
In the theory of relativity, the influence of gravitational waves passes upon space-time can be regarded as the perturbation in the flat space-time background. The influence of gravitational waves on the triangular configuration of the LISA-like spacecraft is shown in Fig. 3, where “+” and “×” are two independent polarization directions of the gravity wave. The propagation of photons or other matter particles in time and space can be described by the metric line elements. When the gravitational waves pass through the space-time, the space-time metric tensor changes, resulting in changes in the propagation state of particles such as photons. At the same time, changes of the metric tensor will cause changes in space-time curvature, and then change the level of tidal forces between two points in space-time.
图 3 引力波经过测试质量组成的三角形编队。三角形编队中3个测试质量代表3个LISA航天器，红色臂长的时变量即需要测量的引力波引起的距离变化量
Figure 3. Gravity waves pass through the triangle formation formed by the test masses. The three test masses in formation represent three LISA spacecraft, the time variable of the red arm length represents the distance variation caused by the gravitational wave that needs to be measured
Based on the above principle, the space gravitational wave detector uses the free-floating test mass as a sensor, and adopts heterodyne laser interferometry to obtain small changes caused by the space-time structure when the gravitational wave passes through. For convenience, we only consider one polarization direction “+” when the gravitational wave propagates. Without considering the effects of gravitational waves(h+=0), the time for light to travel back and forth between two test masses is[4-5]
In equation (1), L is the distance |x1-x2| between two test masses. When the gravitational wave passes through the space-time, the round-trip time of the light between the two test masses will change. The relationship between the amount of time change and the amplitude of the gravitational wave is
The amount of change in distance caused by the amount of time change is:
Assuming an orbiting system consists of two black holes with a mass of one hundred thousand solar masses, the gravitational wave intensity h is on the order of 10-13 when gravitational radiation released by the orbiting system is in the frequency band of 1 mHz. If the orbiting system is 400 million light-years from the earth, the intensity h is about 10-21 when gravitational waves propagate to the earth. If the two test masses are 5×106 km apart, the distance between the test masses caused by the gravitational waves is in the order of 10 pm.
假设有一绕转系统是由两个十万太阳质量的黑洞构成，绕转系统释放的引力辐射在1 mHz频段时，引力波的强度h为10-13量级。如果该绕转系统距离地球为4亿光年，当引力波传播到地球，强度h约变为10-21。若两测试质量相距5×106 km，则由引力波引起的测试质量间距离变化量约为10 pm量级。
There are two main functions of the telescope in the laser interferometry ranging system: one is to expand the small-diameter beam used in the interference optical platform to a collimated beam close to the diffraction limit; the other is to enable the interferometric laser beam to be efficiently transmitted between the two spacecraft and to receive the incident beam from the far-end spacecrafts while emitting the beam. Fig. 4 shows the concept maps of LISA interferometry system and that of binary satellite interferometry, from which the effect of the telescope can be found. The laser light is emitted from spacecraft 1, expanded by a telescope, and propagated to a spacecraft 2 of 5×106 km. Only a small portion of the light energy can be received by the far-end telescope. Fig. 5 shows one of the optical payload systems on the spacecraft[6-13].
激光干涉测距系统中的望远镜的主要功能有两个：一是将干涉光学平台用的小直径光束扩束成接近衍射极限的准直光束；二是使干涉测量用激光束在两个航天器间有效的传输，发射光束的同时接收来自远端航天器的入射光束。图 4表示的是LISA干涉测量系统概念图以及双星干涉测量概念图，图中可以看出望远镜的作用。激光从航天器1发出，经望远镜扩束后，传播到5×106 km的航天器2，只有很少一部分光能被远端望远镜接收。图 5为航天器上其中的一个光学有效载荷系统[6-13]。
Assuming that the laser power emitted by the telescope of spacecraft 1 is P0, the laser power received by the spacecraft 2 telescope will be P, when the laser power is propagated to the remote spacecraft 2. The laser waist is designed at telescope exit pupil, and the waist size should be the same as the size of the exit pupil of the telescope, then we obtain
Where D is the telescope aperture, L is the distance between two spacecrafts, λ is the laser wavelength. For LISA missions,
. LISA is currently planning to use a laser with a power of 2 W and a remote spacecraft 2 telescope with a laser power of the order of 100 pW. This order of light can not be returned directly to spacecraft 1 for measurement. Therefore, the working mode of space gravitational wave detection is different from that of the ground gravitational wave detector, weak-light phase-locking amplification technology must be adopted. A hundred picowatt-order laser received by remote spacecraft 2 interferes with the local laser, the phase difference information is derived from the interference signal, the phase of the local laser is locked to the received phase of the laser, and then the phase locked local laser is retransmitted back to the spacecraft 1. The laser intensity received by the spacecraft 1 telescope is also on the order of hundred picowatt and its phase information is maintained.
。LISA目前计划使用的激光器功率为2 W，此时远端航天器2望远镜接收到的激光功率约为100 pW量级。由于这个量级的光无法直接返回航天器1进行测量。所以，空间引力波探测的工作方式不同于地面引力波探测器，必须采用弱光锁相放大技术，使远端航天器2接收到的百皮瓦量级激光与本地激光进行干涉，从干涉信号解出相位差信息，将本地激光器的相位锁定在接收到的激光相位，然后将锁定相位后的本地激光再发射回航天器1。航天器1望远镜接收到的激光光强同样在百皮瓦量级，并且其相位信息得以保持。
It can be seen from Fig. 4 that the interferometric measurement of the space gravitational wave detection task is jointly implemented by three parts:one is the interference measurement of the change in the distance between the spacecraft 1 and the test mass 1; the other is the interferometric measurement of the change of the distance between the spacecraft 1 and the spacecraft 2; and the third is the interferometric measurement of the change in the distance between spacecraft 2 and test mass 2. Spacecraft 1 and spacecraft 2 are equipped with independent lasers, denoted as E1cos(ω1t+φ1) and E2cos(ω2t+φ2) respectively. After being received by the spacecraft 2 telescope, the spacecraft 1 emits a laser beam to interfere with the local laser to generate an interference signal Ecos(Δωt+Δφ), where Δφ=(φ1-φ2)+
. The phase difference information between the received light and the local laser is obtained from the interference signal and fed back to the light source controller of the spacecraft 2. The phase of the laser light of the spacecraft 2 is locked with the phase of the received light and the local laser of the spacecraft 2 is:
Laser after phase-locked on the spacecraft 2 is emitted back to the spacecraft 1, it interferes with the local laser on the spacecraft 1 after being reflected by the test mass of the spacecraft 1 to generate an interference signal cos[Δωt+
(2L+2LR+2LL)]. It can be seen that the phase of the interference signal contains information about the distance between the test masses.
Table 1 shows the key technical indicators of the principle prototype of the telescope in Taiji Program in Space(refer to the eLISA task). The principle prototype of the telescope is used as the design input according to this[6-13]. The same design considerations also apply to other LISA-like tasks, and this paper discusses only the technical requirements that affect the design of the optical system and other important factors.
表 1 望远镜关键技术指标
Table 1. Key technology requirements of telescope
Characteristics Requirements Aperture 20 cm Optical efficiency ≥0.853 Field of view acquisition mode 400μrad full angle Science mode(out of plane) ±7 μrad (in plane) ±4.2 μrad in-plane Optical path length stability Magnification 40 Far-field wavefront quality λ/20
The choice of the working band of the telescope is determined by the laser used by the space gravitational wave detector. Taiji program plans to run in orbit for 5 years, the laser should have good frequency stability, phase stability and power stability, and should meet the needs of high-power and space working environment. Through comprehensive consideration, we decided to adopt the relatively mature 1 064 nm Nd:YAG solid state laser.
望远镜工作波段的选择是由空间引力波探测器使用的激光器决定的。太极计划拟在轨运行5年，激光器需要有很好的频率稳定性、相位稳定性和功率稳定性，并且尽量满足大功率以及空间工作环境的需求。综合考虑，计划采用技术相对成熟的1 064 nm Nd:YAG固体激光器。
(1) Far-field system wavefront
Due to the presence of respiration angles between spacecrafts, the boresight between the two spacecraft telescopes will jitter, and this jitter will be directly coupled into a TTL(Tilt-To-Length coupling), which is the main source of noise other than shot noise. To minimize the TTL, center the telescope′s outgoing wavefront at the spacecraft′s center of mass, with far field spherical wave and as smooth as possible. Such TTL noise will not be introduced when the spacecraft is jittered along the telescope′s visual axis due to the uniform radii of the spherical wavefront in all directions. However, if there are aberrations in the telescope system, such as the first three aberrations(Piston, Tilt x, Tilt y), the far-field wavefront pointing at millions of kilometers will jitter and then TTL noise will be introduced(the contribution of telescopes to the TTL interferometry system, this topic will be elaborated in another paper). Although this error term associated with spacecraft pointing can partly eliminated by the alignment between spacecraft, the most effective way to minimize the TTL noise generated by the telescope is to design the wavefront near the diffraction limit.
由于航天器之间呼吸角的存在，两航天器望远镜之间的视轴会发生抖动，这个抖动将直接耦合成为TTL(tilt-to-length coupling)噪声，它是除散粒噪声之外最主要的噪声源。为了尽量减小TTL，使望远镜出射波前的中心位于航天器的质心，远场为球面且尽量光滑。这样当航天器沿望远镜视轴方向抖动时，由于球面波前所有方向半径一致，故不会引入此类TTL噪声。而如果望远镜系统存在像差，如像差前三项(Piston, Tilt x, Tilt y)，则在百万公里处的远场波前指向就会产生抖动，继而引入TTL噪声(望远镜对干涉测量系统TTL的贡献，该工作正在开展，拟另文详述)。尽管这个与航天器指向有关的误差项可以通过航天器之间的对准消除一部分，但最有效的方法是通过设计接近衍射极限的波前来尽量减小望远镜产生的TTL噪声。
Another reason why telescopes require high wavefront quality is to maximize the transmission efficiency of beam energy between the two spacecrafts. The Strehl ratio is usually adopted as a criterion for this feature. In the interfering arm of the space gravitational wave detection system, there are two telescope systems, which are respectively responsible for transmitting and receiving, and the beam energy transmission efficiency is proportional to the square of the Strehl ratio. If the system wavefront error of the single-link measurement is σ=λ/20, the corresponding Strehl ratio is:1-(2πσ)2=0.9. Therefore, if two sets of telescope systems for transmitting and receiving are considered, the beam energy transfer efficiency is proportional to 0.92=0.8, which can be used as a criterion for the diffraction limit of the optical system. Considering that there must be aberrations in the optical components on the interferometric optical platform, while leaving a certain margin for the assembly of the interferometer optical system, the permissible wavefront error of the telescope system is λ/30.
(2) Optical transmission efficiency
Telescope transmission efficiency is determined by the shot noise of the telescope system. The shot noise of the system is also one of the most important noise sources in the laser interferometry system. This noise is mainly due to the fluctuation of photon number, and the fluctuation of photon number follows Poisson statistics. If the laser power is P, the number of photons per unit time can be expressed as N=
, and the number of fluctuations in the number of photons follows Poisson statistics, which can be expressed as:
According to the uncertainty principle of quantum mechanics, the phase fluctuation of laser Δφ and photon number fluctuation ΔN have the following relations:
There is a conversion relationship
between distance variation δ1 and phase variation, thus we get the expression of shot noise as follows:
If the spacecraft 1 telescope emits laser power P0, the exiting light is TEM00 mode Gaussian beam. In order to reduce the divergence of the beam, it is necessary to increase the beam waist of the outgoing beam ω, for the case of LISA, ω=0.446D, D is taken as the aperture of the telescope system. Then at a distance of 5×106 km from spacecraft 1 telescope, the laser power that telescope of spacecraft 2 can receive is calculated by equation (4).
Taking into account the losses caused by optical components when the laser propagates in the interferometer, the actual received light intensity from the spacecraft 2 telescope will be lower. Let the total optical efficiency of the entire laser interferometry system be ε, and the optical efficiency given by LISA is ε≈0.3. If the laser light intensity emitted by the spacecraft 1 telescope is 2 W, the wavelength is 1 064 nm and the telescope aperture is 400 mm, the shot noise of the laser interferometry system is δ1=
考虑到激光在干涉仪中传播时由光学元件产生的损耗，航天器2望远镜实际接收到的光强会更低，设整个激光干涉测量系统的总光学效率为ε，参考LISA给出的光学效率ε≈0.3。如果航天器1望远镜发射的激光光强为2 W，波长为1 064 nm，望远镜口径为400 mm，则激光干涉测量系统的散粒噪声为：
Shot noise is the intrinsic noise of an interferometric system. From the previous analysis it can be concluded that this noise can only be reduced by increasing the telescope aperture or increasing the laser power. In determining the telescope aperture and emitting laser power, if the shot noise is to be reduced, only the light intensity received by the spacecraft 2 can be increased, that is, the energy transfer efficiency of the optical system can only be increased as much as possible. For four-mirror system used in the prototype telescopes, a technical index of > 0.853 requires that the reflectance of each mirror be better than 0.96, but stray light specifications may require a higher reflectance of the mirror.
散粒噪声是干涉测量系统的内禀噪声，从前面的分析可以得出，该噪声只能通过增大望远镜口径或增大激光器功率来减小。在望远镜口径以及发射激光器功率确定的情况下，要降低散粒噪声，只能增大航天器2接收到的光强，也就是说只能尽量提高光学系统的能量传输效率。对于望远镜原理样机采用的四反系统， > 0.853的技术指标要求每个反射镜的反射率优于0.96，但杂散光的技术指标会要求反射镜的反射率更高。
(3) Field of view
Telescope′s field of view is divided into science and acquisition field of view.
The scientific field of view of telescopes is the range of field of view required by the telescope in scientific measurement. The selection of the arm length and orbit of the interferometric system determines the instantaneous field of view of the telescope. As each spacecraft turns around the sun in its own orbit, the launched telescope should consider the location of the receiving telescope in advance due to millions of kilometers of arm length and limited speed of light to ensure initial inter-satellite alignment. The scientific field of view of the telescope is determined from the orbital stability index of the advance pointing angle.
Interferometric measurements of changes in the distance between two spacecrafts require that the inter-satellite laser link be established by the telescope first. Since the capture between the spacecraft is open-loop, the capture field of view of the telescope is determined based on the selected arm length and orbit and the acquisition strategy angle.
(4) Optical length stability
The optical path stability of the telescope is determined by the allocation of noise budget of the single-link measurement system, which varies from one spacecraft to another, to the telescope subsystem. The total noise budget of the single link measurement system is
0.1 mHz≤f≤1 Hz，f0=2.8 mHz.
Noise of the optical path stability distributed by the telescope system is 1
, 0.1 mHz≤f≤1 Hz.
，0.1 mHz≤f≤1 Hz。
According to the technical requirements of the telescope principle prototype(see Tab. 1), and the functions and effects of the telescopes analyzed in the previous section, and considering the harsh stray light requirements, the initial program of the telescope optical system adopts the structure of the off-axis and four-mirror(as shown in Fig. 6)[6-13]. The main mirror M1 is an off-axis paraboloidal mirror, the secondary mirror M2 is an off-axis hyperbolic mirror, the rear third mirror M3 and the fourth mirror M4 are spherical. An image plane is designed between M2 and M3. Setting an liminayinh stray light stop at this position effectively suppresses stray light. M3, M4 components can be adjusted when the telescopeis in orbit to compensate the vibration due to satellite launch, orbit transfer and system wavefront quality worsening due to changes in the structure parameters of the telescope caused by stress release and temperature gradient in micro gravity after entering orbit.
根据望远镜原理样机的技术要求(见表 1)，结合前面对望远镜的功能及作用分析，考虑苛刻的杂散光要求，望远镜光学系统初步方案采用离轴四反的结构形式(如图 6所示)[6-13]。主镜M1是离轴抛物面，次镜M2为离轴双曲面，后组三镜M3和四镜M4均为球面。光学系统在M2和M3之间设计有一次像面，在此位置设置消杂光光阑可以有效抑制杂散光。望远镜入轨后可以调节M3、M4组件，以补偿由于卫星发射、变轨带来的振动，以及入轨后微重力环境下的应力释放、温度梯度造成的望远镜结构参数变化导致的系统波前质量下降。
From the analysis of 2.1, we can see that the telescope has a very good wavefront quality whether it is to minimize the TTL noise or to improve the energy transmission efficiency of interferometric link beam. Fig. 7 shows the wavefront error at the exit pupil of the system, with an RMS value of 0.013λ. Fig. 8 and Fig. 9 show the intensity distribution and phase distribution at the exit pupil.
From Fig. 4 and above, it can be concluded that the LISA mission needs to measure the variation of 10 pm over a distance of 5×106 km. For space gravitational wave detection, LISA program consists of three spacecrafts with two telescopes on each spacecraft, thus a total of six sets of identical telescope systems. Many factors affect the interferometric measurement of the order of the picometer. Here, we mainly analyze the influence of the change of structural parameters of the telescope on the far-field phase. In this case, it is assumed that the telescope acts as a beam expander transmitting system(in practice, receiving and emitting are performed simultaneously).
从图 4以及前面所述可知，LISA任务需要测量5×106 km距离上10 pm的变化量。为了实现空间引力波探测，LISA计划由3个航天器组成，每个航天器上有两套望远镜，一共6套相同的望远镜系统。许多因素会影响皮米量级的干涉测量，这里主要分析望远镜结构参数变化对远场相位的影响，此时，假设望远镜作为扩束发射系统(实际工作时，接收和发射同时发生)。
(1) Calculation method
In the space gravitational wave detection mission, the laser is emitted from the spacecraft 1 telescope and received by the far-end spacecraft 2 telescope. The wavefront distribution can be obtained based on Kirchhoff′s diffraction formula:
Where E(x, y) is the wavefront distribution of the outgoing laser at the exit pupil of the spacecraft 1 telescope, E(x1, y1) is the wavefront distribution of the laser propagating to the far spacecraft 2 telescope, point (x, y) is any point on the S surface, (x1, y1) is any point on the S′ plane, r is the distance from point (x, y) to point (x1, y1), and k(θ) is the tilt factor. Fig. 10 is the laser wavefront propagation diagram.
式中，E(x, y)是发射激光在航天器1望远镜出瞳的波前分布，E(x1, y1)是激光传播到远端航天器2望远镜处的波前分布，点(x, y)是S面上任意一点，点(x1, y1)是S′面上任意一点，r是点(x, y)到点C(x1, y1)的距离，k(θ)是倾斜因子。激光波前传播示意图见图 10。
As shown in the figure, the transmitting telescope in spacecraft and the receiving telescope spacecraft are respectively in the S-and S′-planes, the X-and Y-axis are respectively parallel to the X1-and Y1-axes, the Z-axis represents the propagation direction of the laser light, and OC is the line of center of transmitting and receiving telescopes, the distance between OC is u. In equation (10),
According to the above equation, based on the aperture of the telescope, the wavefront deformation caused by the first three terms of the Zernike polynomials is mainly considered. At the distance of 5×106 km, the intensity distribution and the wavefront in the far field are as shown in Fig. 11.
图 11 发射望远镜波前λ/60，5×106 km处强度(a)和波前(b)分布图
Figure 11. Diagrams of intensity and wavefront distributions at 5×106 km with telescope wave front of λ/60
由上述计算方法，根据望远镜口径，主要考虑泽尼克多项式前三项引起的波前变形，5×106 km处，远场的强度分布和波前如图 11所示。
(2) Far-field wavefront sensitivity analysis
Tab. 2 shows the variation of the structural parameters of the telescope given by the reference error. Tab. 3 shows the PV value of the far-field wavefront at 5×106 km, and Fig. 12-Fig. 19 show the far-field wavefront distribution and variation(As a result of length reason, only the data of most influential secondary mirror is listed here).
表 2 望远镜结构参数变化量
Table 2. Variations of telescope parameters
Type of variations Variations M1 M2 M3 M4 position X decenter/μm 0.5 0.5 0.5 0.5 Y decenter/μm 0.5 0.5 0.5 0.5 Z decenter/μm 0.5 0.5 0.5 0.5 X tilt/(″) 0.4 0.4 0.4 0.4 Y tilt/(″) 0.2 0.4 0.4 0.4 Z tilt/(″) 0.4 0.4 - - surface Radius/μm 1 1 1 1 Conic 0.00001 0.001 - -
表 3 5×106 km处远场波前变化量PV值
Table 3. Far field wavefront variations(PV) at 5×106 km
Type of variations Variations Variation PV(λ) M1 M2 M3 M4 position X decenter/μm 8.26×10-6 1.12×10-6 1.33×10-6 1.47×10-6 Y decenter/μm 4.34×10-7 5.35×10-8 1.05×10-7 2.93×10-6 Z decenter/μm 4.04×10-7 1.47×10-6 1.31×10-7 2.70×10-8 X tilt/(″) 3.33×10-6 1.31×10-6 1.10×10-6 6.16×10-7 Y tilt/(″) 2.42×10-7 5.91×10-8 7.94×10-7 6.61×10-8 Z tilt/(″) 3.46×10-9 1.30×10-6 - - surface Radius/mm 8.86×10-7 1.87×10-6 1.26×10-7 1.99×10-8 Conic 4.07×10-7 1.06×10-6 - -
图 12 次镜M2在X方向偏心0.5 μm时5×106 km处波前分布和变化量
Figure 12. Wavefront distribution and variation at 5×106 km with X decenter of M2 of 0.5 μm
图 13 次镜M2在Y方向偏心0.5 μm时5×106 km处波前分布和变化量
Figure 13. Wavefront distribution and variation at 5×106 km with Y decenter of M2 of 0.5 μm
图 14 次镜M2在Z方向偏心0.5 μm时5×106 km处波前分布和变化量
Figure 14. Wavefront distribution and variation at 5×106 km with Z decenter of M2 of 0.5 μm
图 15 次镜M2绕X轴旋转0.4″时5×106 km处波前分布和变化量
Figure 15. Wavefront distribution and variation at 5×106 km with M2 rotating around X by 0.4″
图 16 次镜M2绕Y轴旋转0.4″时5×106 km处波前分布和变化量
Figure 16. Wavefront distribution and variation at 5×106 km with M2 rotating around Y by 0.4″
图 17 次镜M2绕Z轴旋转0.4″时5×106 km处波前分布和变化量
Figure 17. Wavefront distribution and variation at 5×106 km with M2 rotating around Z by 0.4″
图 18 次镜M2半径变化1 μm时5×106 km处波前分布和变化量
Figure 18. Wavefront distribution and variation at 5×106 km with radius change of M2 of 1 μm
图 19 次镜M2二次曲面系数变化0.001时5×106 km处波前分布和变化量
Figure 19. Wavefront distribution and variation at 5×106 km with conic change of M2 of 0.001
Due to the change in the structural parameters of the telescope system is small, it can be considered that the variation of the structural parameters of the telescope is linear with the variation of the far-field wavefront. Considering that the expected LISA-like orbital temperature stability is on the order of 10-5 K, the analysis results of Fig. 12-Fig. 19 and Tab. 3 indicate that the variation of the far-field wavefront under the structural parameters of the telescope system can meet the requirements of the space gravity wave detection.
(3) Integrated opto-mechanical-thermal analysis
The principle prototype of the telescope is shown in Fig. 20(a). The reflective mirror is made of grade 0 crystallite and adopts invar which matches the coefficient of linear expansion. Fig. 20(b) is the structural deformation nephogram due to in-orbit temperature and gravity release. Fig. 21 shows the fitting nephogram of the each reflective mirror obtained by the surface fitting. The position change of each mirror is shown in Tab. 4. Returning the surface shape and position of each mirror to the optical system for iteration, the wavefront of each field of view of the telescope is obtained, as shown in Fig. 22.
表 4 各反射镜的平动和转动
Table 4. Translaton and rotation of mirrors
M1 M2 M3 M4 PV/nm 45.721 1.663 0.121 0.711 RMS/nm 8.927 0.503 0.032 0.183 ΔX/μm 0.001 -0.369 -0.005 -0.024 ΔY/μm -0.403 0.977 0.497 -1.271 ΔZ/μm 0.855 1.407 0.540 -0.988 Δθx/(″) -0.363 -0.009 2.570 2.063 Δθy/(″) 0.104 -0.403 0.011 0.027
The analysis shows that under the orbital environment, the wavefront of the telescope system changes from λ/60(RMS) to λ/50(RMS), which is better than λ/30(RMS) required for interferometric measurement, so the system wavefront changes caused by the telescope entering the orbital environment from the ground environment meets the requirements of space laser interferometry gravitational wave detection.