The Influence and Compensation of Microwave Holographic Measurement Errors on Antenna Measurement Accuracy (2024)

1. Introduction

The shape error of a reflector has a great influence on the electrical performance of an antenna, which will cause a loss of the antenna gain. In recent years, with the continuous improvement of the aperture and working frequency of the radio telescope antenna, the required antenna surface accuracy is getting higher and higher. Precisely gauging the surface precision of an antenna significantly contributes to the enhancement of its electrical performance.

Microwave antenna holography is a technology to accurately measure and evaluate the accuracy of the antenna surface. It has the characteristics of a simple principle, convenient operation, fast measurement speed, high measurement efficiency, and no special requirements for antenna attitude during the measurement process. It is widely used in large antenna surface measurements [1,2,3]: for example, GBT [4], SRT [5], Tianma [6], and so on. The technology mainly uses the Fourier transform relationship between the antenna aperture field and the radiation far field. The complex distribution of the far field is obtained by measuring the emission source signal, and then obtaining the aperture field distribution by the two-dimensional Fourier transform. The measured actual aperture distribution is compared with the ideal aperture distribution, and then the surface shape distribution of the antenna reflector is derived [7,8,9]. However, in the process of microwave holographic measurement, the existence of both feed offset error and pointing error will affect the reconstructed surface shape error distribution results and reduce the antenna measurement accuracy. Therefore, it is particularly important to study and compensate for the influence of these error sources in the process of microwave holographic measurement.

A large number of studies have been carried out in the academic community on the feed offset error in order to deeply explore its impact on antenna performance. Baars [10] used geometrical optics to derive expressions for the feed lateral offset error and feed axial offset error and investigated their effects on the antenna far-field pattern, respectively. Tarchi [11] studied the determination mothed of the feed position, using a combination of polynomial fitting and geometrical optics to derive the actual positional coordinates of the feed. Lian [12] proposed a new method to determine the amount of feed adjustment based on the far field without knowing the deformation. Zhang [13] proposed an innovative analytical method for assessing the directivity patterns of cluster-fed reflector antennas with random feed element positions and orientation displacements. Zhang [14] studied the comprehensive influence of different reflector subtended angles and feed displacements on the antenna. However, these studies have not thoroughly explored the underlying mechanisms that different feed offset errors have on the effect of the accuracy of the antenna surface shape in a microwave holography measurement.

The pointing error is divided into systematic error and random error [15]. The systematic error is the pointing deviation, which is mainly caused by the structural deformation of the antenna pedestal and the reflector. The random error is the scanning error mainly caused by inertial vibrations, wind load effects, and temperature effects during antenna movement. Many studies focus on the analysis of the influence of the pointing error caused by the systematic error on the electrical performance of antennas, as well as on the study of the systematic error compensation methods. Gawronski [16] derived an algorithm for pointing error estimation and azimuth axis tilt using tiltmeter data. Zhang [17] proposed a pointing error analysis model to estimate the pointing error caused by antenna structural deformation under wind loading based on the full working condition dynamic model (EWCDM). Xu [18] proposed a pointing error analysis method based on coordinate transformation and a dynamic model correction method based on modal weighting. Kong [19] proposed a pointing calibration error model for a radio telescope, considering the nonlinear error of the azimuth axis. However, there are relatively few studies on the scanning error. Most existing studies focus on reconstructing far-field patterns through interpolation methods [20] to mitigate the influence of the scanning error on the electrical performance of antennas. Nevertheless, the scanning error still influences the accuracy of antenna holographic measurements. Moreover, current studies do not provide detailed analyses of the scanning error in antenna holographic measurements. Specifically, there is still an insufficient number of studies on the mechanisms by which the scanning error influences antenna gain loss and surface shape measurement accuracy.

The purpose of this paper is to systematically study the influence of the feed offset error and the scanning error in microwave holographic measurement. The goals are to establish the mechanisms that influence gain loss and surface measurement accuracy and to identify effective compensation methods. Firstly, in Section 2, the relationship between the feed offset error and the scanning error on the far field of the antenna radiation is established by geometric optics and physical optics distribution. Secondly, in Section 3, the influence of the feed offset error and the scanning error on the far-field pattern and gain loss are studied, respectively. Then, in Section 4, through the two-dimensional Fourier transform relationship between the antenna far field and the aperture field, the influence mechanism of the feed offset error and the scanning error on the measurement accuracy of microwave holography is established. Finally, in Section 5, the compensation method and implementation steps of the influence of these two errors on the surface shape measurement accuracy are studied and verified by experimental cases.

2. Error Sources Analysis of Antenna Holographic Measurement

The aperture field distribution can be obtained by Fourier transform of the far-field data. In turn, we can also reverse the far-field data of the antenna by constructing the aperture field model. In this section, we introduce the theoretical foundation of holographic measurement and establish models of the influence of different error sources on the far-field pattern by using the aperture field.

2.1. Holographic Measurement Theory

As shown in Figure 1, it is a schematic diagram of the geometric relationship of the reflector antenna. According to the theoretical foundation of microwave holographic measurement, the far-field radiation pattern of the antenna can be expressed as [21]:

$T (θ, ϕ) = \iint_{S} A (r, ϕ^{'}) e^{j δ (r, ϕ^{'})} e^{j k r \sin θ \cos (ϕ - ϕ^{'})} r d r d ϕ^{'}$

(1)

where $A (r, ϕ^{'})$ is the aperture illumination function. $r$ , $ϕ^{'}$ is the polar coordinate $(r, ϕ^{'})$ on the aperture surface, $δ (r, ϕ^{'})$ is the phase error distribution function, $k = 2 π / λ$ , $λ$ is the wavelength. In Figure 1, S represents the aperture area, p is the far-field observation point, and $d S^{'}$ is the surface element.

Transform Equation (1) into a Cartesian coordinate system [22]:

$T (u, v) = e^{- j 2 k f} \iint_{S} A (x, y) e^{j δ (x, y)} e^{j k (u x + v y)} d x d y$

(2)

where $u = \sin θ \cos ϕ$ , $v = \sin θ \sin ϕ$ .

The far-field pattern is related to the distribution of the surface shape error by the equation [23]:

$ε (x, y) = \frac{λ}{4 π} \sqrt{1 + \frac{x^{2} + y^{2}}{4 f^{2}}} \times p h a s e \{e^{j 2 k f} F F T^{- 1} [T (u, v)]\}$

(3)

where f is the focal length and $F F T^{- 1} []$ is the two-dimensional inverse Fourier transform.

2.2. Error Sources

Figure 2 shows a 26 m radio telescope, of which the antenna type is the Cassegrain dual-reflector antenna and the reflector of which deforms under the gravitational load. The purpose of microwave holography measurement is to analyze the deviation between the actual shape and the ideal shape of the antenna surface according to the difference in the received signal, so that the surface deviation can be corrected and the received radio signal can be improved.

In this process, there are some factors that will lead to a difference between the received signal and the actual situation, thus affecting the accuracy of the antenna surface error reconstruction, among which the feed offset error and the pointing error are the two main interfering factors, as shown in Figure 3. Any small shift in the feed position may cause the phase center to deviate from the focus, making the received signal different from the actual situation, thus reducing the accuracy of the surface shape error reconstruction. The actual existence of surface shape errors on the antenna surface can also lead to misalignment of the focus with the phase center of the feed source. The purpose of antenna holography measurements is to measure such surface shape errors that actually exist on the antenna surface. Therefore, in this paper, we will only discuss the effect of feed source offset.

Similarly, if the pointing of the antenna is not precise enough, i.e., there is a pointing error, the antenna will also deviate when pointing to the actual radio source position, which also reduces the accuracy of the reconstructed antenna surface error. The systematic error is a constant pointing error that results in a linear distribution of the aperture phase, which is subtracted out by a two-dimensional fit to the aperture field phase. The scanning error is mainly manifested in the fact that the antenna pointing deviates from the predetermined scanning trajectory point during scanning, and the phase error of the aperture field caused by it is not regular and difficult to deal with. In the face of the increasing demand for the accuracy of large-aperture reflector antennas, even small scanning errors cannot be ignored. This paper focuses on the scanning error in the pointing error.

2.3. Feed Offset Error

In Equation (1), the feed offset error affects Δφ, so we need to construct the optical path difference model Δ caused by the feed offset through geometric optics and obtain Δφ through the equation Δφ = kΔ.

The optical path difference caused by the feed axial offset $δ_{a}$ is schematically shown in Figure 4a, and the relation can be expressed as [24]:

$Δ_{a} = δ_{a} (1 - \cos θ) = δ_{a} (1 - \frac{1 - \tan^{2} \frac{θ}{2}}{1 + \tan^{2} \frac{θ}{2}}) = 2 δ_{a} \frac{r^{2}}{4 f^{2} + r^{2}}$

(4)

where $δ_{a}$ is the axial focal offset of the feed.

As shown in Figure 4b, the optical path difference caused by the lateral offset of the feed is [25]:

$Δ_{l} = ρ - ρ^{'} \approx δ_{l} \sin θ \cos ϕ^{'} = δ_{l} \frac{4 f r}{4 f^{2} + r^{2}} \cos ϕ^{'}$

(5)

where $δ_{l}$ is the lateral focal offset of the feed.

2.4. Scanning Error

Scanning error occurs in the far-field measurement, resulting in each point on the u-v grid actually recording the far-field data on the neighboring position of the far-field coordinate corresponding to that point, as shown in Figure 5. The scanning error caused the data obtained from the measurement to form a non-regular grid distribution, and due to the fact that the data solving used the FFT algorithm to transform the regular grid data, the non-regular grid data was mistakenly processed as regular grid data, which in turn caused the measurement error.

The antenna samples the far-field in the presence of the scanning error, and the actual sampling position will deviate from the ideal sampling position. So, in the calculation of the far-field pattern by radiative integration, we use the amplitude and phase of the far-field pattern at the point $(u_{i} + Δ u, v_{i} + Δ v)$ instead of the amplitude and phase at the point $(u_{i}, v_{i})$ to simulate the effect of the scanning error on the microwave holography measurement process. So, scanning the error far-field pattern, we can use the radiation integral:

$T (u + Δ u, v + Δ v) = e^{- j 2 k f} \iint_{s} A (x, y) e^{j k δ (x, y)} e^{j k ((u + Δ u) x + (v + Δ v) y)} d x d y$

(6)

where $Δ u$ and $Δ v$ are the deviation values of the coordinates of the far-field sampling points.

The scanning deviations $Δ u$ and $Δ v$ can be expressed as:

$\begin{array}{l} Δ u = \sin (Δ E L) \cos (Δ A Z) \\ Δ v = \sin (Δ E L) \sin (Δ A Z) \end{array}$

(7)

where ΔEL and ΔAZ are the scanning elevation angle deviation and azimuth angle deviation during antenna scanning.

The term $e^{j k ((u + Δ u) x + (v + Δ v) y)}$ in Equation (6) is decomposed:

$T (u + Δ u, v + Δ v) = e^{- j 2 k f} \iint_{s} A (x, y) e^{j δ (x, y)} e^{j k (Δ u x + Δ v y)} e^{j k (u x + v y)} d x d y$

(8)

It can be seen from Equation (9) that the phase term $e^{j k (Δ u x + Δ v y)}$ directly reflects the phase change of the additional aperture field caused by the scanning error. Let $Δ φ (x, y) = k (Δ u x + Δ v y)$ , and the equation can be simplified as:

$T (u + Δ u, v + Δ v) = e^{- j 2 k f} \iint_{s} A (x, y) e^{j (δ (x, y) + Δ φ (x, y))} e^{j k (u x + v y)} d x d y$

(9)

where $(δ (x, y) + Δ φ (x, y))$ is the phase error distribution of the aperture field after adding the scanning error.

The phase error caused by the scanning error is linearly superimposed on the phase of the aperture field. The distribution of the aperture field can be expressed as:

$A (x, y) e^{j (δ (x, y) + Δ φ (x, y))} = e^{j 2 k f} F F T^{- 1} (T (u + Δ u, v + Δ v))$

(10)

After the phase of the aperture field is calculated by Equation (10) and brought into Equation (3), the surface measurement error caused by the scanning error can be reconstructed.

3. Effect of Error Sources on Antenna Gain

In the process of antenna holographic measurement, both feed offset error and scanning error will affect the far-field pattern of the antenna, resulting in a decrease in antenna gain and a reduction in antenna receiving efficiency. This section will use the different error models established in Section 2 to analyze the influence of different error sources on the far field of the antenna and establish the influence mechanism of different error sources on the antenna gain loss.

3.1. The Influence of Feed Offset Error on Antenna Gain

The amplitude distribution of the circular symmetric aperture field of the parabolic antenna can be defined as [22]:

$A (r, ϕ^{'}) = B + C {(1 - \frac{r^{2}}{a^{2}})}^{P}$

(11)

where B + C = 1, edge taper = 20 log B, 1 ≤ P ≤ 2.

The antenna is simulated and analyzed using Fast Fourier Transform (FFT) with the following input parameters: antenna diameter d = 12 m, wavelength λ = 5 cm, focal length f = 4.8 m, feed axial offset $δ_{a}$ = 0.5λ, feed lateral offset $δ_{l}$ = 0.5λ, edge taper = −12 dB.

Assuming that the antenna is an ideal parabolic surface, the Equations (4), (5), and (11) are substituted into Equation (2). The far-field pattern changes caused by the feed offset are shown in Figure 6. It can be seen from the figure that the axial offset of the feed causes the main lobe of the far-field pattern to decrease, the side lobe to rise, and the main lobe to fuse. The lateral offset of the feed causes the main lobe of the far-field pattern to shift and the side lobe to decrease, showing left–right asymmetry.

The antenna gain can be expressed as follows [26]:

$G = \frac{4 π}{λ^{2}} \frac{{|\int_{0}^{2 π} \int_{0}^{1} A (r, ϕ^{'}) e^{j δ (r, ϕ^{'})} r d r d ϕ^{'}|}^{2}}{\int_{0}^{2 π} \int_{0}^{1} A^{2} (r, ϕ^{'}) r d r d ϕ^{'}}$

3.2. The Influence of the Scanning Error on Antenna Gain

It can be seen from Equation (8) that Δu and Δv are related to the far-field coordinates (u, v), so the two must have one-to-one correspondence when using FFT, which is difficult to deal with. Therefore, this section uses the discrete Fourier transform (DFT) to solve the far-field pattern caused by the scanning error.

In the process of antenna microwave holographic measurement, the angle encoder measures and records the azimuth angle and elevation angle of the antenna in real time and compares them with the predetermined direction to obtain the scanning error ΔAZ and ΔEL of each scanning point. The offsets Δu and Δv of the coordinates (u, v) of each scanning point are calculated and brought into Equation (8) to solve the far-field pattern caused by the scanning error. Due to the randomness of the scanning error, it cannot be directly quantitatively analyzed. The scanning error of each scanning point in the whole scanning area can be statistically analyzed by the method of probability statistics, and the influence of the scanning error on the far-field pattern of the antenna can be studied. The mean represents the pointing error of the system, and the pointing is offset by a fixed value as a whole, which leads to the phase error of the linear distribution aperture surface, which can be deducted by two-dimensional linear fitting of the aperture surface phase. The standard deviation represents the scanning error, which describes the degree of deviation between the random scanning error and the mean.

In order to clearly demonstrate the effect of scanning error on the antenna far-field pattern, the antenna simulation parameters are changed so that the antenna diameter is 60 m and the rest of the parameters remain unchanged. The scanning errors ΔAZ and ΔEL of each sampling point are randomly generated by using the Gaussian distribution with the mean μ being zero and the standard deviation being 25 arcsec, and the far-field pattern is calculated, as shown in Figure 8. The absolute values of all the generated ΔAZ and ΔEL are averaged to get the average value of the elevation error and azimuth error for the whole scanning area, which is about 20 arcsec. The random scanning error causes the main lobe of the far-field pattern to become uneven, and the side lobes show left-right asymmetry. Due to the randomness of the scanning error, it may cause distortion in the far-field directional map, producing a large gain loss. With the increase in the antenna aperture surface, the effect of the scanning error will become more and more obvious.

The phase error of the antenna aperture field is solved by Equation (10) and then brought into Equation (12) to solve the relationship between the effect of the scanning error on the antenna gain loss by using the solved gain to make a ratio with the ideal gain.

In order to understand more deeply the effect of antenna diameter on the gain loss caused by scanning error, we conducted a series of simulation experiments, as shown in Figure 9. When the standard deviation of the scanning error is controlled at 30 arc seconds, the gain loss caused by the 30 m radio telescope antenna is about −0.014 dB, which is negligible. For a 110 m large-aperture, high-precision radio telescope antenna, the gain loss is about −0.36 dB, which is intolerable for the gain loss caused by a single error. This result indicates that the increase in antenna diameter can improve the signal reception ability of the antenna, and at the same time, it also makes the antenna more sensitive to the scanning error, which negatively affects the gain.

By fitting the feed offset error curve in Figure 9, the relationship equation between the scanning error and antenna gain loss is obtained as:

$η = s_{g l} σ^{2}$

(14)

where $s_{g l}$ is the gain loss coefficient with the values shown in the Table 2 and δ is the feed offset error.

From Equation (14), it can be seen that the antenna gain loss is proportional to the square of the standard deviation of the scanning error, by which the antenna gain loss caused by the scanning error can be quickly calculated.

4. The Influence of Error Sources on the Surface Shape Measurement Accuracy

In this section, the surface error distributions are reconstructed based on the far-field data, and then the relationships between the error sources and the antenna surface measurement accuracy are established.

4.1. The Influence of the Feed Offset on the Surface Shape Measurement Error

In the previous section, the far-field data caused by the feed offset error was obtained. The far-field data is subjected to the inverse Fourier transform, and the phase distribution is calculated. Substitute it into the Equation (3) to solve the surface measurement error distribution caused by the feed offset error, as shown in Figure 10.

Figure 10a shows the effect of feed axial offset $0.5 λ$ on the surface shape measurement accuracy after microwave holographic measurement, and the root mean square (RMS) error is about 4.97 mm. Figure 10b shows the effect of feed lateral offset $0.5 λ$ on the surface shape measurement accuracy after microwave holographic measurement, and the RMS error is about 6.98 mm. If there is no feed offset error, the surface shape measurement error distribution in Figure 10 will be an ideal planar distribution, which will not interfere with the actual surface error distribution of the antenna reconstructed by holographic measurement. However, when there is feed offset error, it is equivalent to adding the false error distribution shown in Figure 10 to the reconstructed actual surface error distribution, which interferes with the accuracy of holographic measurement results.

By calculating the surface measurement error RMS caused by the feed offset error, the relationships between the feed offset error and the surface measurement error RMS are established, as shown in Figure 11. Figure 11a shows the relationships between the RMS of the surface measurement error and the axial offset error of the feed when f/d takes different values. Figure 11b shows the relationships between the surface shape measurement error RMS and the lateral offset error of the feed when f/d takes different values. It can be seen from Figure 11a,b that the smaller the f/d value, the faster the RMS of the surface shape measurement error rises, and the larger the RMS of the surface shape measurement error caused by the feed offset error.

The relationships between the feed offset and the surface shape measurement error RMS are shown in Figure 11, and the relationship is:

$ε_{rms} = p \frac{δ}{λ}$

(15)

where $p_{a}$ is the RMS coefficient of measurement error of the feed axial offset surface shape and $p_{l}$ is the RMS coefficient of measurement error of the feed lateral offset surface shape, $δ$ is the feed offset value.

Table 3 shows the RMS coefficients $p_{a}$ and $p_{l}$ of feed offset and surface shape measurement error under different focal diameter ratio. According to the comparison of the data in the table, it can be concluded that the influence of feed offset on the surface shape measurement error decreases with the increase of antenna focal diameter ratio.

Excessive feed offset error leads to an increase in measurement error RMS and a decrease in measurement accuracy, which will seriously interfere with the deformation distribution of the reconstructed antenna surface structure.

4.2. The Influence of the Scanning Error on the Surface Shape Measurement Error

In the previous section, the far-field data caused by the scanning error were obtained, and the phase distribution was calculated by the inverse Fourier transform of the far-field data. The phase distribution is substituted into the Equation (3) to solve the surface measurement error distribution caused by the feed offset error. As shown in Figure 12, the RMS value is 0.108 mm. Although the error interference caused by the scanning error on the measurement results is not as regular as the feed offset error, its influence on the measurement accuracy of the antenna surface may be relatively small in many cases. However, in the face of the increasing demand for the accuracy of large aperture reflector antennas, even small scanning errors cannot be ignored.

In the following, the relationship between the random scanning error and the antenna gain loss is established by using the probability statistics method and adding the random scanning error to the variables u and v of Equation (8).

As shown in Figure 13, there are significant relationships between scanning error and surface shape measurement accuracy under different antenna diameter conditions. When the standard deviation of the scanning error is 15 arcsec, the RMS of the surface shape measurement error is approximately 0.25 mm for antennas with a diameter of 90 m and 0.37 mm for antennas with a diameter of 110 m. It is worth noting that the holographic measurement accuracy of the GBT 100 m antenna can reach 220 µm [4], indicating that the RMS of the surface shape measurement error due to scanning error already has a non-negligible impact on measurement accuracy. The larger-diameter antennas are more significantly affected when facing scanning errors, resulting in larger RMS errors. Compensation for scanning errors is more important in application scenarios that require high-precision surface shape measurements.

By fitting the relationship curve in Figure 13, the relationship equation between the scanning error and the face shape measurement error RMS is obtained as follows:

$ε_{r m s} = s_{r m s} σ$

(16)

where $s_{r m s}$ is the scanning error RMS coefficient, the value of which is shown in Table 4.

From Equation (16), it can be seen that the measurement error RMS is proportional to the standard deviation of the scanning error, and the measurement error RMS caused by the scanning error can be quickly calculated by this equation.

5. Errors Compensation

In the previous section, the influence of the feed offset error and the scanning error on surface shape measurement accuracy was studied. Both errors will affect the surface deformation of the reconstructed antenna in microwave holographic measurement, resulting in a decrease in measurement accuracy. It is very important to reduce or compensate for the influence of error on the measurement accuracy of antenna surface deformation.

5.1. Feed Offset Error Compensation

In this section, the nonlinear least squares fitting aperture field phase error is used to solve the axial offset of the feed, and the influence of the measurement accuracy caused by the feed offset error is compensated. Firstly, the data distribution of the surface shape measurement error with the feed offset error interference is obtained by holographic measurement or simulation. Then, the phase error is fitted in two dimensions by using the feed offset error model, and the offset of the feed is deduced in reverse. Finally, the measurement error caused by the feed offset is deducted.

The nonlinear least squares fitting is used to solve the actual offset error of the feed, and the main steps of the compensation for the influence of the surface measurement accuracy caused by the feed offset error are as follows:

The feed offset error model is first determined and the initial predicted feed offsets δ_a and δ_l are set:

$φ = \frac{4 π r (δ_{a} r + 2 δ_{l} f \cos ϕ^{'})}{λ (4 f^{2} + r^{2})}$

(17)

where δ_a is the feed axial offset error, δ_l is the feed lateral offset error, f is the antenna focal length, and r and $ϕ^{'}$ are the antenna aperture field coordinates.
Measure or simulate the radiation far-field data with the feed offset error and inverse the phase distribution of the antenna aperture surface, which is defined as $φ_r e a l$ ;
Establish the objective function, the sum of squares of the difference between the measured phase error and the model prediction:

$F (δ) = {\sum [φ_r e a l - φ]}^{2}$

(18)
The objective function is minimized by finding the feed offset $δ$ that can minimize the objective function $F (δ)$ to achieve the optimization goal;
The optimization process is repeated until the algorithm converges to a certain tolerance range or reaches the maximum number of iterations;
The δ_a, δ_l, and φ are calculated, and φ is subtracted from the phase error distribution of the aperture field obtained from the measured or simulated data.

5.2. Antenna Scanning Error Compensation

The existing compensation methods for the pointing error often focus on the pointing error of the system, such as the pointing error caused by the structural deformation of the main reflector, the structural deformation of the shafting, the gravity load, and other factors, ignoring the small random scanning error that may occur in the high-speed scanning process. These methods often fail to achieve the expected compensation effect when dealing with complex scanning scenarios, especially in application scenarios that require high-precision positioning. In view of this situation, this section assumes that after obtaining the far-field data in the case of scanning error, the surface measurement distribution of the inverse solution of the far-field data with the scanning error is compensated. We use the model correction method to compensate for the surface shape measurement error caused by the scanning error.

The main steps in using the model correction method to compensate for the surface shape measurement error caused by random error are as follows:

The measurement error distribution of the surface shape is solved by using the measurement data, which is defined as $ε_{r e a l}$ ;
During the antenna scanning process, the angle encoder is used to record the actual elevation pointing (EL) and azimuth pointing (AZ) of each scanning point in real time. Compared with the predetermined pointing, the scanning errors (ΔAZ and ΔEL) of each scanning point are obtained;
The offset errors Δu and Δv of the far-field coordinate points (u, v) are calculated by using Equation (9).
The offset errors Δu and Δv of each point in the far field are substituted into Equation (8) to simulate the random scanning error, and the surface shape measurement error caused by the random scanning error is solved, which is defined as $ε_{p}$ ;
The surface shape measurement error caused by the scanning error is deducted, $ε_{r e a l} - ε_{p}$ .

5.3. Compensation Examples

The feed offset error compensation method is verified by simulation experiments. The input parameters are as follows: the antenna aperture is d = 12 m, the wavelength is λ = 5 cm, the focal length is f = 4.8 m, and the edge illumination level is −12 dB.

The feed offset error in the aperture field phase is fitted using two methods: (1) polynomial fitting [11] and (2) the proposed compensation method. In the polynomial fitting method, the fitting coefficients are obtained by two-dimensional nonlinear fitting of the aperture field phase, and the aperture field phase error due to the feed offset error is solved by the coefficient values, and the offset position of the feed is calculated by the coefficients. In the proposed method, using Equation (18), the most suitable predicted phase φ is calculated. The inverse solution of δa and δl gives the feeder offset position directly without the need to solve complex geometrical relations.

Figure 14 displays the distribution of aperture field phase errors during the error compensation process for both methods. Table 5 lists the RMS errors of the antenna surface after compensation with the two methods. Under the influence of the feed offset error, the RMS error before compensation is 2.604 mm, and the RMS errors of method (1) and method (2) are reduced to 0.115 mm and 0.117 mm, respectively. Although the surface RMS error compensated with the polynomial method is smaller than that with the proposed method, it is evident from the compensation results in sub-figure (e) for the polynomial method and sub-figure (f) for the proposed method in Figure 14, that the surface error distribution in sub-figure (f) is more closely aligned with the original reflector surface error distribution (a), indicating a better compensation effect. This situation is because there is a big difference between the polynomial fitting aperture field phase and the actual result, resulting in too much deduction of the error phase after compensation, making the RMS error smaller.

In the following, the simulation experiment is carried out to verify the scanning error compensation of the 12 m antenna. The input parameters remain unchanged, and two circles of protrusions are set on the ideal parabolic surface. The amount and position distribution of the protrusions are shown in Table 6, and the RMS error is 0.27 mm.

Figure 15a shows the distribution of surface shape measurement error with scanning error, and the RMS error is 0.35 mm. Figure 15b shows the distribution of surface shape measurement error after compensation, and the RMS error is reduced to 0.27 mm. As shown in Table 7, the RMS error after compensation is equal to the RMS error of the actual setting, and the influence of the scanning error is basically eliminated.

6. Conclusions

In this paper, the influences of the feed offset error and scanning error on antenna microwave holographic measurement are simulated and analyzed, and compensation methods are proposed. The main conclusions and contributions include the following two parts:

First, the influence of feed offset error on far-field pattern is analyzed by geometric optics, and the influence mechanism of feed offset error on antenna gain and surface shape measurement error is established. Through simulation analysis, the relationship between feed offset error and antenna gain loss is quadratic. The greater the feed offset error, the more serious the antenna gain loss, and the smaller the antenna focal diameter ratio, the greater the influence of the feed offset error on the antenna gain loss. There is a linear relationship between the feed offset error and the RMS of the surface measurement error. The larger the feed offset error, the larger the RMS of the surface measurement error caused by the feed offset error, which leads to a decrease in measurement accuracy. The smaller the antenna focal diameter ratio, the greater the influence of the feed offset error on the RMS of the surface measurement error. Finally, the actual offset of the feed is solved by using the phase distribution of the aperture field with the feed offset error, and the actual feed offset obtained by the inverse solution is used to compensate the surface measurement results, which basically eliminates the influence of the feed offset error.

Secondly, the scanning error of the antenna during the scanning process is simulated by changing the coordinates of the far-field sampling points. Due to the randomness of the scanning error, it is difficult to directly perform quantitative analysis like the feed offset error. The influence of scanning error on antenna gain loss and surface shape measurement accuracy can only be studied by probability statistics. Through research, it was found that although the scanning error is very small in the actual measurement process, it will also affect the measurement results, reduce the antenna efficiency, and reduce the measurement accuracy. Finally, a compensation method to reduce the influence of scanning error is provided, and the feasibility of the method is verified by simulation cases. By comparing the results of surface shape measurement error before and after compensation, the influence of scanning error on surface shape measurement accuracy is effectively reduced.

Finally, in the actual scanning process of the antenna, there are many factors, such as feed offset error, misalignment of feed phase center and focus, pointing error, noise error, scanning error, and so on. This paper only analyzes the influence of feed offset error and scanning error, and the error influencing factors are not comprehensive enough. Future research should consider more comprehensive error sources, adopt more comprehensive error analysis methods, and consider more influencing factors, which will be the key to improving the accuracy of antenna performance evaluation and system optimization.

Author Contributions

Conceptualization, formal analysis, B.X. and Y.Z. (Yongqing Zhao); methodology, B.X., S.L. and Y.Z. (Yongqing Zhao); software, data curation, writing—original draft preparation, Y.Z. (Yongqing Zhao); investigation, Y.Z. (Yang Zhang) and W.W.; project administration, funding acquisition, B.X.; All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Natural Science Foundation of Xinjiang Uygur Autonomous Region (No. 2023D01C22), the National Natural Science Foundation of China (No. 12363011, 52275270, 52275269), the Tianchi Talents Program of Xinjiang, the National Key Basic Research Program of China (No. 2021YFC2203501) and the Xinjiang Postdoctoral Foundation.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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Figure 1. The schematic diagram of the geometric relationship of the reflector antenna.

Figure 2. The 26 m radio telescope.

Figure 3. Microwave holographic measurement error sources.

Figure 4. The geometry of the feed offset error: (a) the axial offset; (b) the lateral offset.

Figure 5. Antenna scanning error: (a) antenna far-field data processing flow; (b) u-v grid far-field data; black points are ideal far-field data; yellow points are far-field data with scanning error.

Figure 6. The far-field pattern caused by the feed offset. The blue curve is the ideal far-field pattern; the yellow curve is the far-field pattern caused by the axial offset of 0.5λ; and the orange curve is the far-field pattern caused by the lateral offset of 0.5λ.

Figure 7. The relationships between the gain loss of different antenna focal diameter ratios and the feed offset error: (a) the influence of the axial offset error of the feed with different focal diameter ratios on the gain loss of the antenna; (b) the influence of the lateral offset error of the feed with different focal diameter ratios on the gain loss of the antenna.

Figure 8. The far-field pattern caused by the scanning error: (a) u-plane intercept 2D far-field pattern; (b) v-plane intercept 2D far-field pattern.

Figure 9. Antenna gain loss due to scanning error.

Figure 10. The surface shape measurement error distribution caused by the feed offset: (a) the axial offset; (b) the lateral offset.

Figure 11. The relationships between the surface shape measurement error of different antenna focal diameter ratios and the feed offset error: (a) the influence of the axial offset error of the feed with different focal diameter ratios on the surface shape measurement error of the antenna; (b) the influence of the lateral offset error of the feed with different focal diameter ratios on the surface shape measurement error of the antenna.

Figure 12. The surface shape measurement error distribution caused by the scanning error.

Figure 13. Antenna surface shape measurement errors due to scanning error.

Figure 14. Compensation process for aperture field phase errors caused by feed offset errors: (a) aperture field phase errors caused by surface errors of the antenna; (b) aperture field phase errors induced after incorporating feed offset errors; (c) aperture field phase errors obtained from polynomial fitting; (d) aperture field phase errors obtained from fitting using the proposed method; (e) aperture field phase errors compensated using polynomial fitting; (f) aperture field phase errors compensated using the proposed method.

Figure 15. The scanning error compensation schematic diagram: (a) the distribution of surface shape measurement error under the combined action of the scanning error and the antenna surface error; (b) the distribution of surface shape measurement error after the scanning error compensation.

Table 1. Relationships between feed offset error gain loss coefficients $a_{g l}$ and focal diameter ratios.

f/d	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
lateral offset	−0.819	−0.248	−0.086	−0.034	−0.015	−0.007	−0.004	−0.002
axial offset	−6.816	−3.435	−1.794	−0.997	−0.589	−0.366	−0.238	−0.161

Table 2. Relationships between scanning error gain loss coefficients and focal diameter ratios.

Diameter (m)	30	50	70	90	110
$s_{g l}$	−1.37 × 10⁻⁵	−3.92 × 10⁻⁵	−8.61 × 10⁻⁵	−1.91 × 10⁻⁴	−3.92 × 10⁻⁴

Table 3. Feed offset error offset coefficients table.

f/d	$p_{a}$	$p_{l}$
0.3	0.016	0.017
0.4	0.010	0.014
0.5	0.007	0.011
0.6	0.005	0.010
0.7	0.003	0.008

Table 4. Relationships between the scanning error RMS coefficient and antenna diameter.

Diameter (m)	30	50	70	90	110
$s_{r m s}$	4.53 × 10⁻⁶	7.64 × 10⁻⁶	1.12 × 10⁻⁵	1.63 × 10⁻⁵	2.42 × 10⁻⁵

Table 5. The antenna surface shape measurement error RMS.

Terms	The Antenna Surface Error	Before Compensation	The Polynomial Compensation	The Proposed Compensation
RMS error/mm	0.120	2.604	0.115	0.117

Table 6. Simulation setting of convex position and convex amount.

Position (m)	Convexity (mm)
0 < r ≤ 1	0
1 < r ≤ 2	1
2 < r ≤ 3	0
3 < r ≤ 4	1
4 < r ≤ 5	0
5 < r ≤ 6	0

Table 7. RMS errors of the reflector surface shape before and after compensation.

Terms	Actual Setting	Before Compensation	After Compensation
RMS error/mm	0.27	0.35	0.27

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