Surface deformation recovery algorithm for reflector antennas based on geometric optics

Jianhui Huang; Huiliang Jin; Qian Ye; Guoxiang Meng

doi:10.1364/OE.25.024346

1. Introduction

Recently the 110-m radio telescope named QTT will be built in QiTai, XinJiang, China. The huge parabolic reflector of QTT is composed of thousands of panels, under which lots of fine actuators are installed to meet the strict requirement of surface adjustment. The adjustment is expected to exactly compensate deformations caused by gravity, wind and temperature. QTT works at the frequency from 100MHz to 105GHz, and its surface RMS should be less than 0.3mm (λ/10) for an acceptable efficiency. To reach the strict target both the measurement and the adjustment should be prominent. The error of actuators is less than 15 μm, thus the surface RMS depends primarily on the measurement.

Figure 1 describes the process and essence of the deformation of the antenna. There are no changes in the shape and surface of panels before and after deformation. We can find that the antenna deformation is essentially caused by changes in the space attitude of panels, which is controlled by the height of its four mounting points. The structure makes the antenna surface eventually becomes wrinkled. It will return to the ideal paraboloid after adjusting mounting points by the actuators. Therefore, the antenna deformation is in fact a discrete set of data, representing the displacement of the mounting points.

Fig. 1 Structure and deformation schematics of the reflector antenna QTT. Part “1” and “2” describe the antenna structure before and after deformation, respectively; part “3” is the brief diagram that indicates the essence of the deformation.

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There are many mature schemes for the measurement of large reflector antennas: radio holography [1–4], phase retrieval [5–8], photogrammetry [9], mechanooptical techniques [10], etc. In the above feasible schemes, radio holography and phase retrieval are probably the most convenient and time-saving measurement methods [11], which have made great achievements in applications. Theoretically, the final results that directly obtained from both phase retrieval and radio holography are just the phase on the aperture instead of the actual deformations. Hence, the accuracy of the two methods highly depends on the measuring frequency.

Radio holography method was first researched in 1960s. The core of the method is the relationship that the far-field complex amplitude equals exactly the Fourier transform of the aperture field [12]. With the measuring distance gets closer, errors brought by the relationship increases gradually. To make the method to be available in near-field, Baars [3] raised an effective improvement by including more items of Taylor expansions of the distance. In his study, a similar Fourier transform between the complex amplitudes of near-field and aperture field exists, after some processes to the measured near-field data. However, errors caused by the close distance are still not been avoided. From [13], when the measuring distance d is less than 10 times the antenna diameter, near-field Fourier transformation will also be unavailable. For phase retrieval, related algorithms were developed endlessly [14–16]. Their work greatly improved the practicability of the method. But, because of the similar core theories that established under the assumptions of paraxial approximation, the two methods are workable only in Fresnel region or Fraunhofer region, with the limitation that being available only for far distance (d>10D) measurement.

While, QTT needs to be measured at a small d. Due to the huge weight (>7000t), QTT’s distortion distribution varies greatly at different postures. There is expected a measurement scheme with the applicability to any elevation angles. Obviously it could be hardly achieved by traditional schemes because of the large d. This paper proposed a novel scheme to meet the expectation: a UAV carrying a source scans the antenna under test (AUT) by flying on a near plane with the distance 100m~300m, meanwhile the AUT remains stationary. Compared with conventional methods, the new scheme has many advantages: a larger range of elevation angle, a shorter measuring period, a stronger source and a smaller system error.

Figure 2 shows the schematic diagram of the UAV near-field measurement scheme, where the typical algorithms based on phase are not very suitable for the following reasons [17].

Fig. 2 Schematic diagram of UAV near-field measurement scheme.

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(1) UAV position errors will directly affect the final calculated phase (equivalent to the deformation), resulting in an inaccurate deformation recovery.
(2) To achieve a high accuracy, phase-based algorithms require a corresponding high test frequency, increasing electronic errors and system complexity.

2. Mathematics

2.1 Definitions

In this paper, propagation of the electromagnetic waves on panels was studied by GO method, based on the following facts: panels are made of aluminum, whose reflectivity to microwaves (<100G) nears 99%; panels are machined very flat, with its roughness less than 0.07 mm.

Thus, the radiation of the microwaves on the panels can be considered specular reflection of lights, with the intensity distribution (amplitude) depending on the shape of the antenna. Similar to [18], the amplitude data here is used to recover the surface shape, by calculating out the displacement of the mounting points.

For a brief analysis, a single-reflector antenna with 110m diameter and 33m focal length was studied. Dual-reflector antennas have similar results. Parameters and variables used for the analysis are listed in Table 1. All of the variables are matrices.

Table 1. List of Symbols

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Research in this paper is based on the following principles:

(1) Reciprocity theorem [19]. It will be the same result whether the AUT acts as a receiver or a transmitter.
(2) Smooth deformation. Surface deformation is mainly caused by the gravity. According to the mechanics theory, the skeletons of the antenna should be second-order continuous. Thus the displacements of panel’s mounting points (the set of δ) can be considered “smooth”.
In fact, δ is always discrete but “smooth”. In our discrete algorithm, once the set of δ are calculated, the deformation can be completely eliminated. To analyze the antenna model more conveniently, {δ} is considered a continuous derivative function δ(x, y). This does not bring in much error, because the solutions are also discrete, making δ_r almost equals to δ.
(3) Smooth amplitude. A₀ and A_δ are quite smooth, that means an appropriate interpolation will always be accurate.
(4) Tiny deformation. Thanks to the precision machining and assembling, deformation is tiny: panel’s space attitude changes, with the height of its mounting points being mm-level.

2.2 Projection equation

Lights travel linearly in a homogeneous medium, which can be described well by vectors. As shown in Fig. 3, suppose there is a smooth surface with its normal vector n. An incident ray p is reflected here and then the reflected light q is obtained.

Fig. 3 Schematics of the antenna model used for geometrical optics analysis. Part ‘1’ is used for the derivation of “projection equation”, and part ‘2’ and ‘3′ are for “energy equation”.

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As shown in Eq. (1), according to the Fermat principle, the reflection angle is equals to the incidence angle. Thus the expression of the reflected light q is solved as Eq. (2).

(\frac{\vec{p}}{| \vec{p} |} + \frac{\vec{q}}{| \vec{q} |}) = 2 (\frac{\vec{p} \cdot \vec{n}}{| \vec{p} | | \vec{n} |}) \frac{\vec{n}}{| \vec{n} |} .

\vec{q} = \frac{2}{{| \vec{n} |}^{2}} (\vec{p} \cdot \vec{n}) \vec{n} - \vec{p} .

The two-dimensional smooth function δ(x, y) represents the axial distorts of the panels. The antenna surface is expressed as Eq. (3), where f (x, y) is the ideal shape.

z (x, y) = f (x, y) + δ (x, y) = \frac{x^{2} + y^{2}}{4 F} + δ (x, y) .

The normal vector of the surface is calculated as follow. When the feed has no position errors, the incident light p is shown in Eq. (5).

\begin{array}{l} \vec{n} = (\begin{matrix} - \frac{\partial z}{\partial x} & - \frac{\partial z}{\partial y} & 1 \end{matrix}) \overset{w r i t t e n a s}{\to} (\begin{matrix} - z_{x} & - z_{y} & 1 \end{matrix}) . \\ \vec{p} = (\begin{matrix} - x & - y & F - z \end{matrix}) . \end{array}

Put Eq. (4) and Eq. (5) into Eq. (2), the reflected light q is solved below.

\vec{q} = \frac{2 (x z_{x} + y z_{y} + F - z)}{1 + z_{x}^{2} + z_{y}^{2}} \cdot (\begin{matrix} - z_{x} & - z_{y} & 1 \end{matrix}) - \vec{p} .

Introduce the parameter K to simplify Eq. (6), K represents the divergence characteristics of the surface to light. Thus the Eq. (6) is written as Eq. (7).

K = \frac{2 (x z_{x} + y z_{y} + F - z)}{1 + z_{x}^{2} + z_{y}^{2}} . \vec{q} = (\begin{matrix} x - K z_{x} & y - K z_{y} & z - F + K \end{matrix}) .

To facilitate subsequent derivations, K is simplified by ignoring its second-order items. Where, x, y, f and F are on the same magnitude 0~55; δ, δ_x and δ_y are on the same magnitude that less than 0.01. Remove the second-order smaller term of the denominators in Eq. (8) and finally we get the simplified K. P is the introduced function independent of δ, being 1.4~2.

\begin{matrix} K = \frac{2 (x z_{x} + y z_{y} + F - z)}{1 + z_{x}^{2} + z_{y}^{2}} = \frac{2 F + \frac{x^{2} + y^{2}}{2 F} + 2 (x δ_{x} + y δ_{x}) - 2 δ}{1 + \frac{x^{2} + y^{2}}{4 F^{2}} + \frac{x δ_{x} + y δ_{x}}{F} + \underset{m u c h s m a l l e r}{\underset{︸}{δ_{x}^{2} + δ_{x}^{2}}}} \\ \approx 2 F - [\frac{2 F}{F + f + \underset{m u c h s m a l l e r}{\underset{︸}{(x δ_{x} + y δ_{x})}}}] δ \approx 2 F - [\frac{2 F}{F + f}] δ = [2 F - P δ] . \end{matrix}

See Fig. 3 for the schematics. All the lights are projected onto the near field with the distance d. Record the ideal reflection point and the actual projection point as (x, y, h) and (x*, y*, h), respectively. Ignoring the diffraction, Eq. (9) holds. Then the unknown variable l can be solved in Z direction.

(\begin{matrix} x & y & z \end{matrix}) + l \cdot \vec{p} = (\begin{matrix} x^{*} & y^{*} & h \end{matrix}) .

l = (\frac{h - z}{z - F + K}) .

Substituting l and the simplified K into Eq. (9), the coordinate of the projection point on the near-field is calculated and further simplified.

{\begin{cases} x^{*} = x + l \cdot (x - K z_{x}) \approx x + [2 F \frac{f - h}{f + F}] (δ_{x} - \frac{P}{4 F^{2}} x δ) \\ y^{*} = y + l \cdot (y - K z_{y}) \approx y + [2 F \frac{f - h}{f + F}] (δ_{y} - \frac{P}{4 F^{2}} y δ) \end{cases} .

Equation (12) introduces coefficient functions G, P_new, U and V to simplify the equation. The maximum values of |G| and |U| (or |V|) are 46 and 0.33, respectively. All of them are matrices independent of δ. The final projection equation is expressed as follows. Where (x, y) is exactly the projection point of an ideal surface, and (x*, y*) is the point after the deformation.

{\begin{cases} G = [2 F \frac{f - h}{f + F}] = [2 F \frac{x^{2} + y^{2} - 4 F h}{x^{2} + y^{2} + 4 F^{2}}] \\ P_{n e w} = [\frac{P}{4 F^{2}}] = [\frac{2}{x^{2} + y^{2} + 4 F^{2}}] \\ U = - G P_{n e w} x = [\frac{- 4 F x (x^{2} + y^{2} - 4 F h)}{{(x^{2} + y^{2} + 4 F^{2})}^{2}}] \\ V = - G P_{n e w} y = [\frac{- 4 F y (x^{2} + y^{2} - 4 F h)}{{(x^{2} + y^{2} + 4 F^{2})}^{2}}] \end{cases} .

{\begin{cases} x^{*} = x + G \cdot δ_{x} + U \cdot δ \\ y^{*} = y + G \cdot δ_{y} + V \cdot δ \end{cases} .

2.3 Energy equation

The energy of microwaves in the propagation is the integral of its squared amplitude. From the energy transfer an equation can be established. Before and after the deformation, radiation of the feed (direction and scope) is unchanged.

As shown in Fig. 3, microwaves with the direction a and the angle φ spreads to the ideal panel and projects to the near-field plane in parallel after the reflection. Where, a and φ uniquely decide the total radiant energy W. For an ideal antenna, the average energy density obtained on its micro area dS is record as E. After the deformation, the same radiation with energy W spreads to the panels, and similarly, average energy density E* obtained on the deformed area dS*. According to the energy conservation law, Eq. (14) holds.

W = E \cdot d S = E^{*} \cdot d S^{*} .

Equation (15) holds only for a tiny deformation. In this case, dS* is almost the same to dS, and not affected by other micro areas. The average energy E and E* are actually the square of A₀ and A_δ, respectively. Thus we can derive Eq. (15) from Eq. (14).

\frac{E}{E^{*}} = {[\frac{A_{0} (x, y)}{A_{δ} (x^{*}, y^{*})}]}^{2} = \frac{d S^{*}}{d S^{}} .

To calculate the equation, exact values of dS and dS* are necessary. See Fig. 3, dS changes into dS* after the deformation. Where, u and v are respectively the coordinate offsets of the point A and C. Take B and B’ as the coordinate zero dots to calculate the area. Obviously, the value of dS is exactly dxdy/2, and the value of dS* is calculated as follows.

d S^{*} = \frac{1}{2} \cdot | \begin{matrix} u & d y^{*} & 1 \\ 0 & 0 & 1 \\ d x^{*} & v & 1 \end{matrix} | = \frac{d x^{*} d y^{*} - u v}{2} .

Look at Fig. 3, u and v means the shape changes of the area with length dx and width dy. We can calculate their values from Eq. (13). As shown in Eq. (17), in the case of smooth δ, δ_x, δ_y and δ_xy are close to 0. That means $u v$ in Eq. (16) is far less than dxdy, as well as dx*dy*.

{\begin{cases} u = [G \cdot δ_{x} + U \cdot δ] |_{A} - [G \cdot δ_{x} + U \cdot δ] |_{B} = [G \cdot δ_{x y} + U \cdot δ_{y}] \cdot d y ≪ d y \\ v = [G \cdot δ_{y} + U \cdot δ] |_{C} - [G \cdot δ_{y} + U \cdot δ] |_{B} = [G \cdot δ_{x y} + U \cdot δ_{x}] \cdot d x ≪ d x \end{cases}

Ignore $u v$ and put Eq. (16) into Eq. (15) and we get the core equation in our study.

{[\frac{A_{0}}{A_{δ}}]}^{2} = [\frac{d x^{*}}{d x^{}}] \cdot [\frac{d y^{*}}{d y^{}}] .

2.4 Deformation-amplitude equation

Putting the projection equation Eq. (13) into the energy equation Eq. (18) and calculating it by differential, the relationship between amplitude and deformation is obtained. Based on the small values (≈0) of δ, δ_x, δ_y, δ_xx and δ_yy, Eq. (19) has the simplified form below.

\begin{matrix} {[\frac{A_{0}}{A_{δ}}]}^{2} = [\frac{d (x + G δ_{x} + U δ)}{d x}] \cdot [\frac{d (y + G δ_{y} + V δ)}{d y}] \\ = [1 + G δ_{x x} + (U + G_{x}) δ_{x} + U_{x} δ] \cdot [1 + G δ_{y y} + (V + G_{y}) δ_{y} + V_{y} δ] \\ \approx 1 + G \nabla^{2} δ + [(U + G_{x}) δ_{x} + (V + G_{y}) δ_{y}] + (U_{x} + V_{y}) δ . \end{matrix}

Since the coordinate is uniform in the directions X and Y, matrix V is exactly the transpose of U. There are many similar relationships in Eq. (19). Introduce R and H to simplify it.

R = [U + G_{x}], R^{T} = [V + G_{y}], H = [U_{x} + V_{y}] .

The ranges of G, R and H are [0, 46], [-1.37, 1.37] and [-0.015, 0.042], respectively. Note that Eq. (19) is workable only within the circle of diameter D. The following step function S is used to revise it.

S (x, y) = {\begin{cases} 1, \sqrt{x^{2} + y^{2}} \leq D / 2 \\ 0, \sqrt{x^{2} + y^{2}} \geq D / 2 \end{cases} .

Introduce the amplitude parameter FA shown below for further analysis. We can see FA is actually the relative errors between the ideal and the measured amplitude.

F A = [{(\frac{A_{0}}{A_{δ}})}^{2} - 1] S = [{(1 + \underset{v e r y s m a l l}{\underset{︸}{(A_{0} / A_{δ} - 1)}})}^{2} - 1] S \approx [1 + 2 (\frac{A_{0}}{A_{δ}} - 1) - 1] S = 2 [\frac{A_{0}}{A_{δ}} - 1] S .

FA is obtained in the actual measurement. As shown in Eq. (23), the formula that connects the measured amplitude and the surface deformation finally gets established. It is called the Deformation-Amplitude equation (δ-FA).

F A = G \cdot \nabla^{2} δ + [R \cdot \frac{\partial δ}{\partial x} + R^{T} \cdot \frac{\partial δ}{\partial y}] + H \cdot δ .

From the range of the coefficient functions: G is much larger than R and H. Therefore, the terms of R and H in Eq. (23) can be ignored when there is a bumpy δ. That is

F A \approx G \cdot \nabla^{2} δ .

As shown in Fig. 4, based on the method of physic optics (PO), the simulation of a global smooth deformation shows the correctness of δ-FA. Where the expression of δ is as follow.

Fig. 4 An example to illustrate the correctness of the relationship δ-FA. δ is rotationally symmetric and ranges [-1.5, 2]mm. FA/G was obtained by PO method with the distance d = 0.

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δ (x, y) = \frac{\sin [(F / D) \sqrt{x^{2} + y^{2}}]}{400 \sqrt[4]{x^{2} + y^{2}}} (\frac{\sqrt{x^{2} + y^{2} + 4 F^{2}}}{2 F}) .

2.5 Numerical algorithm

According to Eq. (22), the ideal near field amplitude A₀ is needed for getting FA. Based on radiation pattern of the feed of the AUT, the ideal near-field amplitude A₀ can be precisely calculated by PO method. With the simulated A₀ and the measured A_δ, FA can be finally calculated from Eq. (22).

To recover the deformation, it seems necessary to seek out a solution to the equation δ-FA. However, this second order partial differential equation is hard to directly solve out because of its mathematical complexity. Considering that the principal component in the right side of Eq. (23) is the second order term G∇²δ, we can try to solve its simplified form Eq. (24), for the purpose of getting an approximate solution. In fact, we have found that the approximate solution is already very close to the real solution.

As a Poisson equation, Eq. (24) can be solved precisely. Based on Taylor expansions, a discrete Laplacian for the calculation of ∇²δ is introduced. As shown in Eq. (26), the suitable Laplacian L is a matrix of 3 × 3, with higher precision and less risk of occurring huge errors. In this way, ∇²δ can be calculated conveniently by convolution. Where, dx and dy are the step sizes in the calculation.

L = [\begin{matrix} 0 & 1 & 0 \\ 1 & - 4 & 1 \\ 0 & 1 & 0 \end{matrix}] \cdot d x d y, \nabla^{2} δ = δ * L .

By the convolution theorem, Eq. (26) can be calculated more quickly in frequency domain by means of FFT. Thus the unknown δ can be easily solved, as shown in Eq. (27). Where, the matrix L_N here is a matrix of N × N, which is produced from L by zero-padding: extends L in the lower right direction. This is to meet the requirements of the FFT calculation.

\tilde{δ} = F^{- 1} [\frac{F (F A / G)}{F (L_{N})}] .

2.6 Iterative approximation

Equation (27) gives only an approximate solution of the equation δ-FA. The paper found the exact solution can be gradually achieved by iteration from the approximate solution $\tilde{δ}$ . Note that in the right side of δ-FA, the second order term G∇²δ must be the main component. Rewrite δ-FA to get an iterative form as follow.

G \cdot \nabla^{2} δ = F A - [(R \cdot \frac{\partial δ}{\partial x} + R^{T} \cdot \frac{\partial δ}{\partial y}) + H \cdot δ] .

With a suitable initial $\tilde{δ}$ substituted into the right side, an updated solution is calculated out. Loop this operation until an acceptable result of δ is obtained.

In fact, iteration method for the solution of partial differential equation is quite different from the algebraic equation, with its convergence conditions and features are still remains to be studied. For the proposed relationship δ-FA in this paper, after many numerical simulations, it is considered to be convergent. However, δ converges slowly because of its large size. To accelerate the iteration, α is introduced to improve the iterative format. A suitable value of α should be 1~5, otherwise the iterative could be divergent. The final calculation format for the solution of Eq. (23) is expressed as follow.

δ = F^{- 1} [\frac{F {[F A - α (R δ_{x} + R^{T} δ_{y} + H δ)] / G}}{F (L_{N})}] .

Though the iteration format cannot converge to the real solution when α≠1, it can be used still to quickly achieve an acceptable approximate solution.

3. Methods

3.1 Measurement scheme

As shown in Fig. 2, the hardware applied in the scheme consists of the following parts.

(1) AUT. The AUT remains stationary at arbitrarily given elevation angle. The feed on the antenna works to receive signals from the UAV, collecting the near-field amplitude data A_δ.
(2) UAV. Carrying an electromagnetic wave emission source, the UAV flies parallel to the aperture at the distance d. Record its position when it scans the AUT along the S-shaped route.
(3) Source. With the working frequency matching the feed, the source radiates spherical waves. During the test the source needs to keep its radiation intensity constant.

With the above equipment, the deformation on the antenna surface can be measured quickly. In the scheme two kinds of data need to be collected.

(1) Position. When the UAV flies and hovers on the near-field plane, its spatial position needs to be measured and recorded. With the equipment of GPS-RTK technology, the UAV is able to obtain an accurate real-time location, with the errors less than 5cm. After processed by the onboard computer, the coordinates (X, Y, Z) of the UAV are finally obtained.
(2) Amplitude. Electromagnetic waves from the source are received by the precision feed, which integrates the received signal and then measures its average energy. From those energy data, A_δ can be calculated precisely, with its coordinates matching the UAV position (X, Y, Z).

With the obtained data, δ can be recovered from Eq. (29). It should be pointed out that the final result is related to the feed position. Hence after the subsequent surface adjustments by actuators, surface errors and feed position errors would be eliminated together.

During the measurement, the AUT is always in a quiescent state, the mechanical errors can be avoided. With the close distance, the source gets stronger, so that the actual measurement process will be more quickly. Compared with traditional schemes, the proposed scheme has the advantages of quickly measurements and arbitrary elevation angles.

3.2 Algorithm flow

The proposed scheme is conducted in near-field. Effective algorithm that connects the aperture field with the near-field is necessary. Due to the close distance, algorithms based on far field Fourier transformation are not suitable on account of the caused huge errors.

Based on the established relationship δ-FA, the paper developed a novel algorithm for the above scheme. The flow chart of this algorithm is shown in Fig. 5 and depicted below.

Fig. 5 Flow chart of the proposed algorithm.

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(1) Normalize the measured amplitude data into the matrix A_δ by interpolation.
(2) Obtain the corresponding standard near-field amplitude A₀.
(3) Calculate FA from Eq. (22).
(4) Get the approximate solution of the equation δ-FA from Eq. (27).
(5) Attempt to find a suitable inertia factor α.
(6) Improve the original approximate solution via iteration shown in Eq. (29) until an acceptable result δ_r is reached.

4. Simulations

4.1 Effectiveness of the algorithm

Numerical simulation was adopted in this paper to determine the effectiveness of the proposed algorithm. For generality δ was designed to be a random deformation.

Primary steps of the simulation are listed as follows.

(1) Design the randomly smooth deformation δ.
(2) Generate the near-field amplitude A_δ by the PO method.
(3) Recover δ from A_δ by the proposed algorithm.
(4) Compare the retrieved deformation δ_r with the given δ.

Based on the Maxwell equations, PO method [20] can precisely calculate the radiation intensity of an arbitrary point through numerical integral. It is widely considered an accurate theory in the field of antenna radiation. Besides, diffraction caused by the reflector edge is considered and calculated by PTD method.

In the comparison of δ_r with δ, RMS shown in Eq. (30) is used to evaluate their errors.

R M S = \sqrt{\frac{\sum_{m = 1}^{N} \sum_{n = 1}^{N} {[δ_{r} (m, n) - δ (m, n)]}^{2}}{N^{2}}} .

To describe the RMS value under δ with different magnitudes, relative RMS (RRMS) is introduced as shown in Eq. (31). Where, M_δ is the magnitude of δ, equals to the difference between its maximum and minimum value.

R R M S = \frac{R M S}{M_{δ}} .

As shown in Fig. 6, δ is randomly smooth, with a reasonable range [-1.4, 2] mm. Taking QTT as the model, A_δ is obtained by PO method. Calculation parameters are shown in Table 2.

Fig. 6 The designed randomly smooth deformation and the amplitude at d = 0. In the right figure, contours of the A_δ appear irregularly deflected, that is caused by the deformation.

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Table 2. Calculation Parameters in PO Method

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Put the simulated A_δ into the algorithm flow shown in Fig. 5. After series of process the retrieved deformation δ_r was finally obtained. See Fig. 7 for the whole simulation results. It can be seen that the proposed equation δ-FA is close to the real case; the developed algorithm especially the iteration approximation is efficient for the deformation recovery, with its RMS in 90% region about 0.3mm (9% of M_δ).

Fig. 7 The reconstructed deformation and the whole process.

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In Eq. (27) and Eq. (29), matrix G contains element 0 when d = 0, thus the region of the calculation cannot be 100%. That was the reason why 90% region was chosen. For a fast convergence, a suitable inertia factor that α = 2.2 was adopted. In practice, α needs many trials for its best value. Generally the best α appears when the iteration gets a critical convergence. The judgment of the convergence can be made by the RMS of neighboring δ_r.

With the accuracy of RRMS = 9%, it must be admitted that the proposed algorithm could hardly recover an absolutely accurate δ_r for the following several reasons.

(1) The basic reason is that the electromagnetic wave has the strong property of wave. The hypothesis completely based on GO method (pure geometry) will inevitably lead to errors. In fact only the Maxwell equation can accurately describe and calculate the propagation of the electromagnetic waves.
(2) An important reason is the diffraction. When d>0, there will be diffraction happens on the antenna aperture, called “pinhole diffraction”. The generation of A_δ took the diffraction into account, while the proposed algorithm did not.
(3) Caused by discrete calculation. There exist errors in the numerical calculation of δ-FA.

In actual measurements, test frequency f, deformation magnitude M_δ, flight height d and samples size N have many impacts on the accuracy of the algorithm. They were also analyzed by simulations based on the δ shown in Fig. 6.

4.2 M_δ-irrelevant and f-irrelevant

The proposed method is based on geometry. According to geometrical similarity principle, the accuracy of the algorithm depends only on the shape of δ, but has nothing to do with M_δ. Table 3 shows the performance of the algorithm under δ with different M_δ but the same shape. From the results, it can be concluded that the proposed method is M_δ-irrelevant indeed.

Table 3. Simulation Results for Different Deformation Magnitude and Frequency

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Frequency is not involved in the method and thus the method should be f-irralevant. This makes it possible to obtain a precise deformation recovery by a low frequency. Table 3 shows the results of the algorithm under different f, from which it can be seen that frequency has almost no impact on the accuracy of the algorithm.

Note that the test frequency has nothing to do with the operating frequency: we used 3G instead of 105G in the simulation. To some extent, a lower frequency makes better system stability. Therefore, we recommend in the actual use a frequency as low as possible.

4.3 The best measuring distance

Measurement distance d is important for the scheme. Obviously in the method, the farther the distance, the greater the effect of deformation on the near field amplitude, and thus algorithm becomes more accurate. However, due to the aperture diffraction and edge diffraction, A_δ will be blurred and lose some details when d gets far. From this point, a smaller d makes a better δ_r. Based on these two factors, there is an optimal d, which can be found by simulation. δ used in this simulation is shown in Fig. 6 and the parameters are:

M_{δ} = 3 . 4mm, f = 0. 3GHz, N = 256, α = 2.2, 100 % region .

When d>0, matrix G no longer contains element 0, thus the region of the algorithm can be set to 100%. With the increase of d, a large α can be used to get a better δ_r. By the simulations at different d, as shown in Fig. 8, a suitable measurement distance should be d<D, with the RRMS<8%. At this distance (d = D), FA and δ_r are shown in Fig. 9.

Fig. 8 Relationships of RRMS with measurement distance d and samples size N. In N-RRMS simulations, it did not converge when N = 128 and N = 200 for the calculation in 100% region, thus 98% region was chosen.

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Fig. 9 FA and the final reconstructed deformation at d = D.

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4.4 Size requirement of A_δ

Fewer sampling points can reduce the required measuring time, but a small N may makes the discrete calculation in this paper fail. It is necessary to determine the range of a suitable N.

M_{δ} = 3 . 4mm, f = 0. 3GHz, d = D, α = 3.9, 100 % region, 30 iterations

In the simulation, the above parameters were applied to get the size requirement of A_δ. As shown in Fig. 7, N-RRMS curves indicates that N should be [80, 128], with the advantage to get a high accuracy by as little as possible the size of A_δ.

4.5 Error analysis

For the proposed scheme, the scattering error of cloud and the tracking error of pointing angle are avoided after the application of UAV. Hence errors are mainly the feed noise and the UAV’s position error [21].

4.5.1 Feed noise

The noise produced by the feed is usually considered an additive white Gaussian noise (AWGN) [22]. Generally, noise gets lower when the distance gets closer. In the scheme with such a near distance, signal noise ratio (SNR) of the feed can easily be greater than 60dB. In the simulation, an AWGN with its SNR referred as ef was added into A_δ to evaluate the effects of the algorithm under feed noise.

4.5.2 UAV position error

UAVs in the air are always shaking because of the wind and its own movement. The position of the UAV is measured by the GPS-RTK technology, with its error less than 5cm. Positioning error can be divided into three directions: X, Y, Z. Since A_δ is basically unchanged on Z, there are only the errors on X and Y need to be considered. In the simulation, their coordinate errors are also considered to be AWGN, with its RMS referred as ep. By adding the noise, the coordinate system (x, y) transforms into a distorted one (x', y'). To evaluate the impact of position error, the input amplitude A_δ’ is generated on the distorted coordinate (x', y') by the interpolation from A_δ(x, y).

4.5.3 Simulation results

The parameters in the simulation are as follows.

M_{δ} = 3 . 4mm, f = 0. 3GHz, d = D, α = 3.9, 100 % region, 30 iterations

The effects of noise at d = D and d = 2D were investigated by the simulation based on the deformation shown in Fig. 6. As shown in Fig. 10, the relationship curves were obtained.

Fig. 10 The performances of the algorithm under feed noises and UAV position errors.

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From these curves, it can be seen that the algorithm is sensitive to noise, especially in the situation with small d. In practice, a large d would be more suitable if there is a strong noise, and vice versa. In fact, there would be ef>60dB, ep<5cm. Thus the RRMS is expected to be less than 9%, and the RMS <0.31mm. With a good performance to noises, it can be concluded that the proposed algorithm satisfies the accuracy requirement of the measurement for QTT.

5. Summary

QTT is a radio telescope for deep space exploration with a diameter of 110m and a weight of more than 7,000t. Its operating frequency reaches 105GHz, which makes the deformation on its aluminum surface needs less than 3mm. A suitable measurement scheme for the heavy QTT must be available for its arbitrary elevation angle and takes less time.

This paper preliminary proposed a scheme to meet the requirements of measurement for the QTT. By the application of UAV, the measurement could be conducted at any elevation angle, with the AUT remaining stationary and the UAV scanning it on a near-field plane. In this scheme, near-field amplitude is measured at a close distance.

To recover the deformation from the measured amplitude, the paper proposed a novel effective algorithm. Dealing the propagation of high frequency electromagnetic waves with GO, the Deformation-Amplitude equation that directly connects surface deformations with near-field amplitudes was established after precise mathematical derivations. In order to solve the equation, a fast and accurate algorithm based on FFT and iteration was developed and has been proven efficient and accurate.

In the simulations based on PO method, a randomly smooth deformation was considered. The results showed the proposed method could recover the deformation precisely from single near-field amplitude, with the RMS less than 10 percent of its deformation range. In addition, frequency, deformation magnitude, distance, data size, feed noise and UAV positioning error were taken into consideration too. Corresponding results showed frequency and deformation magnitude have almost none influences on the algorithm. For a higher accuracy the suitable measurement distance should be d<D. In error analysis, simulation results showed the good anti-noise ability of the algorithm.

In conclusion, requiring single near-field amplitude, the proposed method can accurately recover the deformation even in the case of a low frequency. It should be pointed out that the equation δ-FA proposed in this paper is applicable not only to a planar near-field, but also to other kinds such as the spherical near-field, as long as the corresponding coefficient matrices G, R and H are recalculated. The idea in this paper is also applicable to the optical field.

Funding

Ministry of Science and Technology of the People’s Republic of China (501100002855). National Basic Research Program of China (973 Program), 2015CB857100.

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Symbol	Unit	Meaning
D	(m)	Diameter of the antenna.
F	(m)	Focal length of the antenna.
d	(m)	Distance from the near-field to the aperture.
f	(GHz)	Frequency of the source.
δ	(m)	Surface deformation of the antenna.
$\tilde{δ}$	(m)	Approximate solution of the surface deformation.
δ_r	(m)	Retrieved surface deformation.
A_δ		Near-field amplitude after the deformation δ.
A₀		Near-field amplitude of an ideal antenna.
FA		Parameter defined by A₀ and A_δ.
G, R, H		Coefficient functions depending on D, F and d.
ef	(dB)	Intensity of the additive noise of the feed.
ep	(m)	Intensity of the position error of the UAV.
N		Size of the measured data A_δ.

Performances at different M_δ						Performances at different f
Conditions	f = 0.3GHz, d = 0, N = 256, α = 2.2, 90% region					Conditions	M_δ = 3.4mm, d = 0, N = 256, α = 2.2, 90% region
M_δ/μm	3.4	34	340	3400	34000	f /GHz	0.1	0.3	0.6	1.0	3.0
RMS/μm	0.373	3.193	31.73	318	3226	RMS/μm	320	318	319	318	319
RRMS/%	10.97	9.39	9.33	9.35	9.49	RRMS/%	9.41	9.35	9.38	9.35	9.38

Symbol	Unit	Meaning
D	(m)	Diameter of the antenna.
F	(m)	Focal length of the antenna.
d	(m)	Distance from the near-field to the aperture.
f	(GHz)	Frequency of the source.
δ	(m)	Surface deformation of the antenna.
$\tilde{δ}$	(m)	Approximate solution of the surface deformation.
δ_r	(m)	Retrieved surface deformation.
A_δ		Near-field amplitude after the deformation δ.
A₀		Near-field amplitude of an ideal antenna.
FA		Parameter defined by A₀ and A_δ.
G, R, H		Coefficient functions depending on D, F and d.
ef	(dB)	Intensity of the additive noise of the feed.
ep	(m)	Intensity of the position error of the UAV.
N		Size of the measured data A_δ.

Performances at different M_δ						Performances at different f
Conditions	f = 0.3GHz, d = 0, N = 256, α = 2.2, 90% region					Conditions	M_δ = 3.4mm, d = 0, N = 256, α = 2.2, 90% region
M_δ/μm	3.4	34	340	3400	34000	f /GHz	0.1	0.3	0.6	1.0	3.0
RMS/μm	0.373	3.193	31.73	318	3226	RMS/μm	320	318	319	318	319
RRMS/%	10.97	9.39	9.33	9.35	9.49	RRMS/%	9.41	9.35	9.38	9.35	9.38

Surface deformation recovery algorithm for reflector antennas based on geometric optics

Abstract

1. Introduction