Sun, Z. X., Wang, J. Q., Yu, L. F., et al. 2024. Surface shape detection methods for large radio telescopes. Astronomical Techniques and Instruments, 1(5): 275−283. https://doi.org/10.61977/ati2024035.
Citation:
Sun, Z. X., Wang, J. Q., Yu, L. F., et al. 2024. Surface shape detection methods for large radio telescopes. Astronomical Techniques and Instruments, 1(5): 275−283. https://doi.org/10.61977/ati2024035.
Sun, Z. X., Wang, J. Q., Yu, L. F., et al. 2024. Surface shape detection methods for large radio telescopes. Astronomical Techniques and Instruments, 1(5): 275−283. https://doi.org/10.61977/ati2024035.
Citation:
Sun, Z. X., Wang, J. Q., Yu, L. F., et al. 2024. Surface shape detection methods for large radio telescopes. Astronomical Techniques and Instruments, 1(5): 275−283. https://doi.org/10.61977/ati2024035.
The surface accuracy of a radio telescope is directly related to its operational efficiency and detection sensitivity. This is crucial under high-frequency observation conditions, where surface shape errors need to be controlled to within 1/16 of the working wavelength. In addition, the primary reflector of large radio telescopes is subject to dynamic deformation, caused by factors such as gravity and thermal effects. This paper presents a method for detecting the surface shape of radio telescopes using radio interferometry techniques combined with active reflector adjustment technology. This enables accurate assessment and correction of surface errors, ensuring the electrical performance of the radio telescope. This study investigates the practical applications of high-precision measurement techniques, such as microwave holography, out-of-focus holography, and wavefront distortion methods at the Tianma 65 m radio telescope (TMRT). Furthermore, the study presents the construction method of gravity models at different elevation angles and demonstrates the efficacy of the active reflector model. The results of the measurements indicate that the application of these methods to the TMRT has led to a notable enhancement of the accuracy of the primary reflector and a substantial improvement in efficiency in the Q-band. Through a process of iterative measurements and adjustments, the surface shape error is ultimately reduced to 0.28 mm root mean square (RMS).
The primary reflector of a radio telescope plays an essential role in focusing the radio waves emitted by distant objects in space. The precision of the shape of this surface determines the extent to which the waves can be effectively converged to a single point. A significant discrepancy in the primary reflector will result in imperfect convergence of the radio waves, thereby reducing the signal strength of the receiving system and consequently affecting the detection sensitivity and the final recorded observation. In the field of radio astronomy and communication, ensuring that large reflector antennas can perform at an optimal level, at all operating frequency ranges, represents a significant challenge. The primary reflector of an antenna is usually precisely assembled from a number of panels, each requiring a surface profile that matches the ideal state, and is particularly important for radio telescopes used for high-frequency observations. In reality, the surface accuracy of the panels is limited by two contributions: Mechanical errors during the panel manufacturing process, and displacement of the primary reflector of the antenna due to ambient temperature variations, gravitational loads and other external factors[1]. Therefore, to maintain the efficient performance of the radio telescope, the surface shape of the primary reflector of the antenna must be inspected periodically and adjusted in a timely manner based on the inspection results. In addition, an irregular surface shape may introduce additional system noise, such as side lobe amplification or main lobe distortion, which increase the background noise and internal system interference, affecting the detection capability for weak signal sources.
The most important indices of high-precision radio telescopes are the surface accuracy and the pointing accuracy. The former is directly related to the working frequency and the efficiency of the antenna; the higher the surface accuracy, the higher the working frequency can be, and consequently the higher the efficiency. The general requirement is that the surface error should be less than 1/20 of the working wavelength[2], but with increasing observation frequency and aperture size, this difficulty is increasing correspondingly. For example, working at 100 GHz on the 65 m radio telescope, this surface error must be 0.15 mm. Because of temperature and other environmental factors, this is still a challenge.
At present, the main methods for measuring the surface shape of a large antenna are microwave holography, wavefront perturbation[3], photogrammetry, total station measurement, and laser scanning. Each of these methods has its own merits[4]. Of these, the microwave holography and wavefront perturbation methods use radio signals to measure the surface shape, which have at least the following advantages over the industrial measurement methods mentioned above: These techniques can directly use the low-noise receiving link system of the antenna, to measure the primary reflector, sub-reflector, and pointing. In addition, the measurements can be made at the full elevation angles of the antenna. Currently, some well-known international large-scale radio telescopes use microwave holography as a means of high-precision measurement to determine the accuracy of their primary reflector. Phase interferometry was utilized at the Delingha 13.7 m telescope, the Urumqi 26 m telescope, and the Shanghai 65 m telescope in China[5]. Using satellite holography measurements and active reflector adjustments, the Shanghai 65 m telescope achieved a primary reflector RMS accuracy of 0.3 mm at 53°. In addition, at high and low elevations, we studied out-of-focus holography (OOF)[6]. The obvious advantage of the OOF technique is the ability to use sky radio sources as signals using a common astronomy receiver in observation mode, applicable over a wide range of elevations. Actual measurements have shown that the Q-band primary reflector model can offer effective improvement over the efficiency of the antenna by more than 3 times at high and low elevations. The wavefront perturbation method has recently been verified in the Q-band, using the TMRT. During the measurement process, the antenna tracks a strong radio source. However, the antenna does not need to scan the antenna beam. Instead, it loads several sets of known orthogonal deformations through the active reflector. The gain maximum tracks the current optimal deformation, and experimental results show that the effect is significant, consistent with deformation due to gravity, measured by defocus holography[7].
New large reflector antennas typically use a main reflector control system to quickly and easily adjust the shape of the primary reflector, to maintain the efficiency of the antenna's high-frequency observations. The TMRT is a fully solid panel Cassegrain antenna with an aperture of 65 m. To solve the defocusing problem caused by gravity deformation at high and low elevation angles, the hexapod mechanism attached to the antenna drives the sub-reflector. This mechanism maintains the electrical performance of the antenna with respect to elevation, by adjusting the attitude of the sub-reflector in real time[8]. In addition, to compensate for the loss in efficiency caused by the gravitational deformation at high and low elevation angles, the main reflector of the TMRT is equipped with a control system to adjust the antenna reflector [9].
2.
REFLECTOR ANTENNA SURFACE MEASUREMENT METHODS
2.1
Microwave Interferometric Holography
There are two types of microwave holographic measurements, namely phase retrieval and phase correlation (also known as the interferometer method). The former does not require a reference signal; instead, the phase characteristics are derived from the antenna's radiation model, based on measured far-field radiation characteristics. A signal source with a very high signal-to-noise ratio (SNR) is required during the measurement. The phase correlation method requires a reference antenna to track the changes in the signal source, and the phase information is obtained by cross-correlation processing.
Microwave holography is an important method for parabolic antenna shape detection. The theoretical basis behind the holographic measurement is that there is a two-dimensional Fourier transform relationship between the aperture field and the far field of a parabolic antenna. For an ideal paraboloid, the optical distance from the focal point to the aperture plane is equal. However, in reality, the primary reflector of the antenna will not be an ideal paraboloid, so the phases in the aperture plane must not be equal. Detecting this phase difference, when the wavelength of the signal is known, theoretically determines the small difference between the antenna surface and the ideal paraboloid.
For the relationship between the far-field radiation of the parabolic antenna and the tiny deformation of the surface, a mathematical relationship [10] can be obtained as
ε(x,y)=λ4π√1+x2+y24F2Phase{ej2kFF−1[T(u,v)]}.
(1)
Where, F−1[···] is the two-dimensional inverse Fourier transform, Phase refers to taking the phase value, (x, y) is the coordinate value on the paraboloid, F is the equivalent focal length of the parabolic antenna, and λ is the wavelength of the measured signal. During the actual observation process of the antenna, as the accuracy of the reflector decreases, the efficiency of the antenna will decrease significantly, which will directly affect the aperture efficiency of the radio telescope, particularly at high frequencies. The effect of a parabolic antenna with surface precision ε (mm) on the efficiency is given by [11]
η=exp[−(4πελ)2],
(2)
where, ε is the RMS of the deviation between the actual reflector of the antenna and the ideal parabola, and λ is the wavelength of the radio signal received by the antenna. Therefore, for an error of only 1/16 of the wavelength, sensitivity is reduced to half the maximum. To limit the loss to 10%, the error must not exceed 1/40 of the wavelength [12]. In addition, the structure of the main lobe and side lobe of the antenna pattern is also affected by the antenna surface error. Therefore, accurately measuring the surface accuracy of the antenna reflector and compensating the surface shape error are of great significance to the reflector antenna. For the TMRT, the antenna surface shape needs to be adjusted in real time to compensate for efficiency loss during high-frequency observations. According to the physical parameters of the TMRT, assuming that the corresponding panel is convex, the simulation calculation is as follows: The signal wavelength is 0.1315 m, the antenna diameter is 65 m, the diameter of antenna sub-reflector is 6.5 m, the Cassegrain antenna equivalent focal length is 142.625 m, and the feed half cone angle θm = 13°[13]. On an ideal paraboloid, there are three circle of panel bulges, with the sizes and positions as given in Table 1.
Table
1.
Simulated bulge position and size for the antenna panel
The illumination distribution, far-field radiation amplitude pattern and phase pattern on the aperture surface are shown in Fig. 1. The phase pattern is very different, due to surface distortion. The amplitude pattern mainly shows a broadening of the main lobe and side lobes. After obtaining the actual radiation pattern, the surface shape can be restored using inverse Fourier transform. Several convolution kernel functions used in holography are the spherical harmonic function (sphere), exponential function (exp), rectangular window function (pill-box), sinc function, and the product of the exponential and sinc functions (exp*sinc). We use the same resolution and different convolution kernels for the simulation, and perform surface shape restoration and slicing on the far-field data of the three circle bulges. At the same resolution, using the exponential function and the spherical harmonic function, the recovered surface shape has steep convexities and is flat within the convex region, which is close to the simulation input. The convexity of the rectangular window, while steepest, introduces significant oscillations within the convex region.
Figure
1.
(A) Amplitude distribution on the aperture surface. (B) Three-ring protrusion on the simulated paraboloid surface. (C) Amplitude pattern of ideal and convex surface shapes. (D) Phase pattern of ideal and convex surface shapes. Here, k is the sampling factor, D is the antenna diameter, and θ is the half beam width.
The maximum observation frequency of the TMRT is 43 GHz. Therefore, the surface accuracy error is required to be less than 350 µm (1/20 of the minimum observation wavelength of 7 mm). The adjustable control system assembled on the primary reflector can be used to control the antenna surface shape and compensate for any efficiency loss caused by the deformation of the antenna reflector. The deformation of the primary reflector includes systematic effects and non-repeatable effects. Currently, non-repeatable effects are difficult to correct, but can be avoided or mitigated if high-frequency observations are performed in good weather conditions, such as nights with low winds. Manufacturing errors cannot be corrected, and determine the limit of primary reflector accuracy. Phase coherence holography technology has the advantages of high spatial resolution and high precision, and is used to correct the assembly error of the primary reflector of the Tianma 65 m antenna. Through the use of satellite holography measurements and active reflector adjustments on the Tianma 65 m antenna, the accuracy of the primary reflector is optimal at an elevation of 53°. Then we use OOF at high and low elevation. The model is measured and extracted at each elevation. Actual measurements show that the Q-band primary reflector model can effectively increase the efficiency of the antenna by more than three times at both high and low elevation. The obvious advantage of OOF technology is that it uses cosmic radio sources as signals. It has a common astronomical receiver and observation mode, and can be applied at a wide range of elevation angles.
The OOF algorithm also uses the far-field amplitude information of the parabolic antenna for phase recovery. At the beginning of the algorithm, the antenna aperture field distribution under focusing is Tap1. The amplitude distribution is represented by a Gaussian function, and the phase distribution is described by a Zernike polynomial. The defocused aperture field distribution Tap2 is obtained using the phase factor. Fourier transform is performed on the aperture field distribution under focus and defocus to obtain the corresponding focus and defocus in the far-field. It then compares the transformed far-field amplitude with the observed value, and takes its RMS as the objective function. If the convergence conditions are met, the algorithm stops, otherwise the Levenberg-Marquard (LM) algorithm is used to optimize the Zernike polynomial coefficients. Results are formed into a new aperture field distribution and brought into the next cycle. The OOF algorithm flow chart is shown in Fig. 2.
Although the OOF technique also uses Fourier transform and the corresponding iterative process, its core is the LM algorithm. This is a nonlinear least squares algorithm, and is also a local search algorithm. In addition, since the LM algorithm is used to optimize the corresponding parameters, the number of parameters of the Zernike polynomial determines the resolution of the aperture field phase[14]. Therefore, the resolution of the OOF algorithm is low and can only obtain the phase distribution on a large scale.
2.3
Wavefront Perturbation Method
Wavefront perturbations are measured and corrected for gravitational deformations of main reflector surfaces[15]. Compared with other existing methods, this has the following characteristics:
(1) Receiving link and observation frequency are the same for radio frequencies.
(2) The reference antenna is not used, unlike with the traditional interference holography method.
(3) Based on the signal amplitude, the detection method is simple, and multiple calibration methods are used to mitigate signal error caused by factors such as atmospheric fluctuations.
(4) For a perturbed wavefront method, a telescope must have an active mechanism that locally changes wavefront position, such as an active reflector, a deformable sub-reflector, or a beam waveguide plane reflector[16].
(5) There is no need for beam scanning during the observation process, as long as the target is tracked, simplifying observations.
for the coordinate (x, y), where fn(x, y) is a set of orthogonal functions and an is the corresponding coefficients of the expansion. Then, the post-perturbation surface error (RMS) , {\delta }{{'}}\left(x,y\right) , is derived as
by superimposing a wavefront perturbation function, {f}_{i}\left(x,y\right) , of the same form as the above orthogonal basis function with a perturbation amplitude of εi. Then, the surface error, where λ and G0 are the working wavelength and ideal gain, respectively, is related to the antenna gain by the Ruze formula,
The perturbed antenna gain is maximized when εi = −ai, by substituting (3) and (4) into (5). Therefore, the error expansion coefficients can be determined by measuring the variation of the antenna gain as a function of the perturbation.
Telescope deformations, including thermal and gravitational, are typically dominated by large lower-order surface errors. By selecting lower-order orthogonal basis functions, such as lower-order Zernike polynomials, this method can efficiently measure and correct telescope dynamic deformation errors. In addition, surface error compensation can be performed quickly and conveniently during telescope operation. Wavefront perturbation on the TMRT is achieved by the main active reflector. A Q-band (43 GHz) heterodyne receiver chain is used to measure the antenna gain variation with respect to the perturbations. The Q-band dual beam power data are used as observation data. Regular displacement changes are made by the actuator. The wavefront perturbation mode uses 18 low-order orthogonal Zernike functions to describe the change pattern, given by
Z_n^m\left(\rho,\varphi\right)=\left\{\begin{array}{c}R_n^m\left(\rho\right)\mathrm{cos}\left(m\varphi\right),\text{f}\text{o}\text{r}\; m\geqslant0 \\ R_n^m\left(\rho\right)\mathrm{sin}\left(m\varphi\right),\text{f}\text{o}\text{r}\; m < 0\end{array}\right.
where (ρ, φ) represent polar coordinates, and m and n are the azimuthal and radial orders of the Zernike polynomial, respectively. To ensure that the main beam is accurately pointed at the target source during the measurement, local pointing errors are corrected prior to observation. An offset pointing away from the target source is observed at the start and end of each surface measurement to provide accurate background power data required for calibration. After loading each perturbation function, we wait for all the actuators to be positioned, record the power data output by the receiver, and switch to the following perturbation function. Each data acquisition requires approximately 25 minutes of total measurement time.
3.
MEASUREMENT OF THE REFLECTOR ERROR FOR THE TMRT
We set up an interferometer with a 1.8 m antenna and the TMRT. Asia Satellite 4's 12.25 GHz Ku-band beacon is selected as the measurement signal. The elevation angle of the satellite at Tianma 65 m is approximately 53°. The antenna provides 1 m aperture resolution by scanning 65 columns of elevation around the satellite. The scan interval is set close to the width of the beam. If this value is set inappropriately, it will lead to aliasing of the samples, resulting in a distortion of the left and right edges of the final shape recovered from the antenna reflector. The azimuth is first shifted by a certain amount before the antenna scans each column in elevation, then the antenna scans the elevation from low to high. The number of columns scanned is increased to the left and right successively. The antenna is pointed at the satellite for calibration processing after each scan reaches the highest elevation position. Simultaneously, to reduce the phase variation in the received link, we keep the 1.8 m reference antenna located as close as possible to the TMRT. The separation is approximately 30 m to minimize atmospheric jitter. In addition, to minimize the difference in phase variation caused by temperature effects on the receiving link, the reference antenna and the TMRT follow the same path when laying the local oscillator and intermediate frequency (IF) cables. We also inject the calibration signal close to the satellite signal into the measurement and reference antennas simultaneously, and then perform phase processing on the double differential phase of the connected interferometer calibration signal and satellite signal.
The main reflector system of the TMRT consists of 1104 actuators, installed under the antenna panel, and a control system. There are 4 bolts on each actuator, with each bolt connecting to a corner of 4 adjacent panels. As the 1104 actuators rise and fall, the 1008panels also correspondingly.
The measurement process obtains the amplitude and phase of the far field of the antenna. The two-dimensional inverse Fourier transform of the far-field radiation is used to calculate the amplitude and phase at the antenna aperture surface. During data processing, the far field data needs to be "gridded" with the number of grid points set to an integer power of 2 to apply the fast two-dimensional inverse Fourier transform. Finally, holographic measurements of the TMRT are used to obtain a 512 × 512 antenna surface error data matrix. Determining the positional adjustment amount for the 4 corners of each panel (raising or lowering the panel) is the ultimate goal of the holographic measurement. The resulting 512 × 512 data matrix must first be mapped to each panel to find the adjustment values of each. Every panel is mapped to several adjustable points, but in reality each panel only has 4 adjustable corners. This is because each panel is much smaller than the primary reflector of the antenna, and it is also made of rigid material. The antenna panel can be regarded as a plane, and the plane equations of several adjustment points of each panel can be obtained using the least-squares method. Then the adjustment amount at the four corners of the panel can be calculated. However, one actuator simultaneously controls 2–4 panels. Averaging multiple adjustments for each actuator to complete the active reflector model is obviously not optimal. Based on this, we can add conditional constraints that force the adjustment amounts of adjacent panels to match.
Through an iterative process of repeated measurements and adjustments, we gradually bring the surface shape closer to the ideal state. The process of improving the surface shape is shown in Fig. 3, omitting shielding and support legs. The RMS of the surface shape is gradually increased from 0.58 mm to 0.28 mm. Verification measurements must be made after each adjustment of the main surface due to the influence of environmental factors such as weather and wind on the measurements. After many measurements and corrections, the final surface errors are obtained, with the results of multiple independent measurements in close agreement. To ensure accuracy, three subsidence marks are placed during the measurement.
Figure
3.
Measurement of shape error after successive adjustments of the active reflector. The color bar has units of millimeters. The numbers 0.58 mm, 0.41 mm, 0.29 mm and 0.33 mm are the surface errors (RMS).
Table 2 shows the surface shape error statistics of the three measurements. Within the range of 65 m in diameter, the accuracy of the surface shape error reaches 0.3 mm. As the statistical radius is reduced, the surface shape accuracy increases. In principle, the surface shape accuracy reaches 0.25 mm within 58 m. The decrease in measurement accuracy of the outer circle panel is mainly caused by several aspects. First of all, the illumination function of the 65 m antenna decreases by 10 dB at the edge. Second, the outer edge of the backframe is more strongly affected by environmental factors. Each measurement takes over 2 hours, and it is difficult to ensure complete consistency over multiple measurements. Consequently, the consistency of the outer edge of the surface shape will be relatively poor after multiple measurements.
Verification of the performance of the adjusted main surface at the optimum elevation angle of 53° was performed on a clear day for Ka-band measurements. To avoid loss of efficiency due to gravitational deformation, the elevation range we measured is limited to approximately 53°. We also use alternate measurements to avoid atmospheric differences and pointing errors in the Ka-band caused by weather factors.
At the same azimuth and elevation position, the main reflector model is set to two states, on and off, and the antenna efficiency is measured in both states. To avoid the influence of the main reflector model on the pointing error before and after loading, a pointing correction is performed for each elevation position, measuring the efficiency of the main reflector model before and after loading. Results are shown in Fig. 4. At the optimum elevation angle of 53°, the efficiency before activating the main reflector is about 37%, and the efficiency after activating the active reflector is about 56%, with an improvement ratio of 1.5. In addition, when the active reflector is not used, the antenna efficiency measurement results near the optimal elevation angle should be consistent. Fluctuations in recorded data are caused by measurement errors. With decreasing elevation, the efficiency improvement gradually decreases, mainly due to the gravitational deformation of the main reflector at high and low elevations. The efficiency starts to decrease due to gravitational deformation when the elevation is below 45° and above 60°.
Figure
4.
Comparing efficiency with and without the active reflector model.
During the measurement process, measurement errors also make a contribution. The pointing of the antenna is subject to real-time correction. Therefore, the errors are mainly due to power variations and pointing errors caused by the Ka-band atmosphere. Fig. 5 shows the linear power diagram obtained by scanning the antenna azimuth before and after activation of the active surface, near an elevation of 53°. After activation, the scanning signal power increases to about 1.5 times the original value. During the test process, the main reflector model is alternately turned on and off for short periods of time, and the power data are obtained by azimuth scanning, which can minimize the influence caused by pointing errors, atmospheric variations, and other factors.
Figure
5.
The linear power diagram obtained by scanning the antenna azimuth before and after the active reflector is activated.
We have constructed an active reflector model for the TMRT using the OOF method and the wavefront perturbation method at other elevation angles. Fig. 6 shows a comparison of the scans of the same radio source at an elevation angle of 35°, with the active reflector model switched off, with the OOF model loaded, and with the wavefront perturbation model loaded. The performance of the wavefront perturbation and OOF methods is equivalent or better. Fig. 7 shows the normalized gain change tested at the full elevation angle after loading the wavefront perturbation gravity model described above, showing significant improvement in efficiency at high and low elevation angles, compared with efficiency without the gravity model. Notably, the improvement is approximately 1.3 times at a low elevation of 30° and 1.6 times at a high elevation of 70°.
Figure
6.
Antenna scanning comparison of the same radio source, 3C84. The relative height of the Gaussian waveform shows the SNR.
In the main methods of reflector shape detection technology for large radio telescopes, surface shape accuracy plays a crucial role in performance, particularly for high-frequency observations. Microwave holography and wavefront perturbation have become the preferred measurement methods, providing full elevation angle measurement, and allowing integration with the existing receiver link system. The effectiveness of these methods in improving reflector shape accuracy and antenna efficiency is demonstrated through reflector shape measurement analysis of the TMRT. The telescope achieved a surface shape error of 0.28 mm RMS at an elevation angle of 53°, increasing the antenna efficiency in the Ka-band from 37% to 56% through several rounds of measurements and adjustments. Furthermore, when applied at other elevation angles, the defocus holography and wavefront perturbation methods confirm the ability of these models to improve efficiency over a wide range of elevations. Specifically, the antenna efficiency is improved by approximately 1.3 times and 1.6 times at low and high elevation angles, respectively. These results not only confirm the practicability of the methods described above, but also provide an important reference for the efficient operation and maintenance of large radio telescopes.
ACKNOWLEDGEMENTS
This work was supported by the National Key R&D Program of China (2018YFA0404702, 2019YFA0708904, 2021YFC2203501), Shanghai Key Laboratory of Space Navigation and Positioning Techniques, the National Natural Science Foundation of China (12273097, 11903068).
AUTHOR CONTRIBUTIONS
Zhengxiong Sun conceived the ideas, designed and implemented the study, and wrote the paper. Jingqing Wang performed the statistical analysis, and revised the paper. Linfeng Yu and Yongchen Jiang developed the hardware and did the measurement of holography. Zheng Lou designed the wavefront perturbation measurement and developed the software. Qian Ye designed the actuators. Yongbin Jiang and Rongbin Zhao designed the control system for measurement. Qinghui Liu and Guangli Wang reviewed the paper and gave constructive suggestions. All authors read and approved the final manuscript.
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