Deformation measurement by single spherical near-field intensity measurement for large reflector antenna

Let it first be assumed that the surface irregularities are described by function δ(r, φ) in the normal direction, and the main reflector can be expressed as a rotating paraboloid z(x, y) with deformation δ(r, φ), namely,

$$\begin{eqnarray}&&f=z+\delta (r,\varphi)=\displaystyle \frac{{r}^{2}}{4F}+\delta (r,\varphi).\end{eqnarray} \tag{ 1 }$$

Using the concept of the first and second fundamental form of a curve, let Z = Z(r cos φ, r sin φ, f(r, φ)) be a point of the main parabolic reflector, the first fundamental form of which at Z can be expressed as (Pressley 2010)

$$\begin{eqnarray}&&Ed{r}^{2}+2Fdrd\varphi +Gd{\varphi }^{2},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}\begin{array}{ll} & E=\langle {{\bf{Z}}}_{{\boldsymbol{r}}},{{\boldsymbol{Z}}}_{{\boldsymbol{r}}}\rangle =1+{f}_{r}^{2},\\ & F=\langle {{\boldsymbol{Z}}}_{{\boldsymbol{r}}},{{\boldsymbol{Z}}}_{{\boldsymbol{\varphi }}}\rangle ={f}_{r}{f}_{\varphi },\\ & G=\langle {{\boldsymbol{Z}}}_{{\boldsymbol{\varphi }}},{{\boldsymbol{Z}}}_{{\boldsymbol{\varphi }}}\rangle ={r}^{2}+{f}_{\varphi }^{2}.\end{array}\end{eqnarray} \tag{ 3 }$$

We suppose f_r = ∂ f/∂ r, f_φ = ∂ f/∂ φ. Note that coefficients E,F,G in Equation (3) are called the first fundamental quantities and Equation (2) allows one to compute lengths on a surface, and also angles and areas.

The second fundamental form (Pressley 2010) is

$$\begin{eqnarray}&&frac12(Ld{r}^{2}+2Mdrd\varphi +Nd{\varphi }^{2}),\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}\begin{array}{ll} & L=\langle {\boldsymbol{n}},{{\boldsymbol{Z}}}_{{\boldsymbol{r}}r}\rangle =\displaystyle \frac{r{f}_{rr}}{\sqrt{{r}^{2}{f}_{r}^{2}+{f}_{\varphi }^{2}+{r}^{2}}},\\ & M=\langle {\boldsymbol{n}},{{\boldsymbol{Z}}}_{{\boldsymbol{r}}\varphi }\rangle =\displaystyle \frac{r{f}_{r\varphi }-{f}_{\varphi }{}^{2}}{\sqrt{{r}^{2}{f}_{r}^{2}+{f}_{\varphi }^{2}+{r}^{2}}},\\ & N=\langle {\boldsymbol{n}},{{\boldsymbol{Z}}}_{{\boldsymbol{\varphi }}\varphi }\rangle =\displaystyle \frac{{r}^{2}{f}_{r}+r{f}_{\varphi \varphi }}{\sqrt{{r}^{2}{f}_{r}^{2}+{f}_{\varphi }^{2}+{r}^{2}}}.\end{array}\end{eqnarray} \tag{ 5 }$$

We suppose f_{r r} = ∂² f / ∂ r², f_{r φ} = ∂² f/(∂ r ∂ φ), f_{φ φ} = ∂² f / ∂ φ². Note that coefficients L,M,N in Equation (5) are called the second fundamental quantities and Equation (4) characterizes the degree of bend.

Additionally, the unit normal n at Z can be obtained.

$$\begin{eqnarray}\begin{array}{ll}{\boldsymbol{n}} & =\displaystyle \frac{\overrightarrow{{Z}_{r}}\times \overrightarrow{{Z}_{\varphi }}}{\left|\overrightarrow{{Z}_{r}}\times \overrightarrow{{Z}_{\varphi }}\right|}=\displaystyle \frac{1}{\sqrt{{r}^{2}{f}_{r}^{2}+{f}_{\varphi }^{2}+{r}^{2}}}\times \\ & ({f}_{\varphi }\sin \varphi -{f}_{r}r\cos \varphi,-{f}_{\varphi }\cos \varphi -{f}_{r}r\sin \varphi,r).\end{array}\end{eqnarray} \tag{ 6 }$$

Based on the well-known Weingarten maps (Pressley 2010), the curvature matrix of the main reflector at Z can be given as

$$\begin{eqnarray}\begin{array}{ll}W & =\left(\begin{array}{ll}{a}_{11} & {a}_{12}\\ {a}_{21} & {a}_{22}\end{array}\right)\\ & =\displaystyle \frac{1}{EG-{F}^{2}}\left(\begin{array}{ll}LG-MF & ME-LF\\ MG-NF & NE-MF\end{array}\right).\end{array}\end{eqnarray} \tag{ 7 }$$

Generally, the magnitude of deformation δ on the reflector is about 10⁻³ m. Here we suppose a global smooth deformation on the surface as shown in Equation (8) and Figure 3(a), with a magnitude of 1.2 × 10⁻³ m.

$$\begin{eqnarray}&&\delta =\displaystyle \frac{\sin \left(\displaystyle \frac{F}{D}\sqrt{{x}^{2}+{y}^{2}+\displaystyle \frac{F}{2}}\right)}{50F}\displaystyle \frac{1}{1+{e}^{[-0.6(0.4D-\sqrt{{x}^{2}+{y}^{2}})]}},\end{eqnarray} \tag{ 8 }$$

where the focal length F = 33 m, the diameter of antenna D = 110 m.

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We found that the first-order and second-order partial derivatives of δ are all about 10⁻⁴ m and the product of any two of them is about 10⁻⁷ m to 10⁻⁸ m, as shown in Figure 3(b) and Figure 3(c). Thus all the product-terms seem negligible for a₁₁ and a₂₂.

Substituting Equation (3) and Equation (5) into Equation (7), a₁₁ and a₂₂ can be solved below.

$$\begin{eqnarray}&&{a}_{11}=\displaystyle \frac{{f}_{rr}{r}^{3}+{f}_{\varphi }^{2}{f}_{rr}r-{f}_{\varphi }{f}_{r}{f}_{r\varphi }r+{f}_{\varphi }^{2}{f}_{r}}{{[{r}^{2}(1+{f}_{r}^{2})+{f}_{\varphi }^{2}]}^{\displaystyle \frac{3}{2}}}\\ &&\,=\displaystyle \frac{-\displaystyle \frac{{\delta }_{\theta }{\delta }_{r\varphi }}{2F}{r}^{2}-{\delta }_{r}{\delta }_{r\varphi }{\delta }_{\varphi }r+\displaystyle \frac{{\delta }_{\varphi }^{2}r}{F}+{\delta }_{rr}{\delta }_{\varphi }^{2}r+\displaystyle \frac{{r}^{3}}{2F}+{\delta }_{rr}{r}^{3}+{\delta }_{r}{\delta }_{\varphi }^{2}}{{(\displaystyle \frac{{r}^{4}}{4{F}^{2}}+\displaystyle \frac{{r}^{3}{\delta }_{r}}{F}+{r}^{2}{\delta }_{r}^{2}+{r}^{2}+{\delta }_{\varphi }^{2})}^{\displaystyle \frac{3}{2}}}\\ &&\,\approx \displaystyle \frac{\displaystyle \frac{{r}^{3}}{2F}+{\delta }_{rr}{r}^{3}}{{r}^{3}{(1+\displaystyle \frac{{r}^{2}}{4{F}^{2}})}^{\displaystyle \frac{3}{2}}}=\displaystyle \frac{\displaystyle \frac{1}{2F}+{\delta }_{rr}}{{(1+\displaystyle \frac{{r}^{2}}{4{F}^{2}})}^{\displaystyle \frac{3}{2}}}=-\displaystyle \frac{\displaystyle \frac{1}{2F}+{\delta }_{rr}}{{(1+\displaystyle \frac{z}{F})}^{\displaystyle \frac{3}{2}}},\\ &&{a}_{22}=\displaystyle \frac{-r({f}_{r}^{2}+1){f}_{\varphi \varphi }-{f}_{r}[{r}^{2}({f}_{r}^{2}+1)-r{f}_{\varphi }{f}_{\varphi r}+{f}_{\varphi }^{2}]}{{[{r}^{2}(1+{f}_{r}^{2})+{f}_{\varphi }^{2}]}^{\displaystyle \frac{3}{2}}}\\ &&\,\approx \displaystyle \frac{(\displaystyle \frac{z}{F}+1){\delta }_{\varphi \varphi }+\displaystyle \frac{{r}^{4}}{8{F}^{3}}+\displaystyle \frac{{r}^{2}}{2F}}{{r}^{2}{(1+\displaystyle \frac{z}{F})}^{\displaystyle \frac{3}{2}}}.\end{eqnarray} \tag{ 9 }$$

In the previous section the curvature matrix of the deformed antenna surface is obtained through the Weingarten maps. The detection method studied in this paper is that the plane wave incident on the antenna reflector, and the reflected electromagnetic wave radiates to the hemispherical detection field with the focus of the rotating paraboloid, namely, the feed position, as the spherical center.

Therefore, to study the electric field intensity of hemispherical detection field, we need to focus on the reflection wavefront of plane wave reflected by antenna reflector according to the principle of geometrical optics. This section mainly discusses the relation between the curvature matrix of incident wavefront, reflected wavefront and reflection surface.

The phase matching relation of incident and reflection wavefront on convex surface is given below based on GO theory (Lee 1974)

$$\begin{eqnarray}&&[a]\cos {\gamma }_{i}+{K}_{{\rm{i}}}^{T}{Q}^{{\rm{i}}}({P}_{0}){K}_{{\rm{i}}}=-[a]\cos {\gamma }_{{\rm{r}}}+{K}_{{\rm{r}}}^{T}{Q}^{{\rm{r}}}({P}_{0}){K}_{{\rm{r}}},\end{eqnarray} \tag{ 10 }$$

where [a] is the curvature matrix of the reflector surface, and γ_i is the incident angle while γ_r is the reflect angle. Qⁱ(P₀) is the curvature matrix of the incident wavefront, and Q^r(P₀) is the curvature matrix of the reflect wavefront, as shown in Figure 4.

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K_i and K_r are transformation matrices of orthogonal basis, as shown below. ${K}_{i}^{T}$ and ${K}_{r}^{T}$ are the transposes of K_i and K_r .

$$\begin{eqnarray}{K}_{{\rm{i}}}=\left(\begin{array}{cc}1 & 0\\ 0 & -\cos {\gamma }_{{\rm{i}}}\end{array}\right),\,{K}_{{\rm{r}}}=\left(\begin{array}{cc}1 & 0\\ 0 & \cos {\gamma }_{{\rm{r}}}\end{array}\right).\end{eqnarray} \tag{ 11 }$$

Supposed the incident wavefront is a plane, Qⁱ(P₀) = 0, and according to Fermat principle, we have γ_i = γ_r . Thus, the curvature matrix of the reflect wavefront Q^r(P₀) can be expressed as Equation (12) (The negative sign comes from the concave surface.).

$$\begin{eqnarray}{Q}^{r}=-2\cos ({\gamma }_{r}){({K}_{r}^{T})}^{-1}[a]{K}_{r}^{-1}=\left(\begin{array}{ll}{Q}_{11} & {Q}_{12}\\ {Q}_{21} & {Q}_{22}\end{array}\right),\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}\begin{array}{ll} & {Q}_{11}=-\displaystyle \frac{2}{\cos \gamma }{a}_{11},\,{Q}_{12}=2{a}_{12},\\ & {Q}_{21}=2{a}_{21},\,{Q}_{22}=-2\cos \gamma {a}_{22}.\end{array}\end{eqnarray} \tag{ 13 }$$

We have also

$$\begin{eqnarray}\begin{array}{ll}\cos (\gamma)= & \langle n,{[0,0,1]}^{T}\rangle =\displaystyle \frac{r}{\sqrt{{r}^{2}(1+{f}_{r}^{2})+{f}_{\varphi }^{2}}}\\ & \approx \displaystyle \frac{1}{{(\displaystyle \frac{z}{F}+1)}^{\displaystyle \frac{1}{2}}}.\end{array}\end{eqnarray} \tag{ 14 }$$

From the conclusion of differential geometry, principal curvatures of reflected wavefront $\displaystyle \frac{1}{{R}_{1}}$, $\displaystyle \frac{1}{{R}_{2}}$ are eigenvalues of curvature matrix (Lee et al. 1979)

$$\begin{eqnarray}\begin{array}{ll} & \displaystyle \frac{1}{{R}_{1}},\displaystyle \frac{1}{{R}_{2}}={\lambda }_{1},{\lambda }_{2}\\ & =\displaystyle \frac{1}{2}\left[({Q}_{11}+{Q}_{22})\pm \sqrt{{({Q}_{11}-{Q}_{22})}^{2}+4{Q}_{12}{Q}_{21}}\right].\end{array}\end{eqnarray} \tag{ 15 }$$

As shown in Figure 5, for the deformed reflector f (x, y), the magnitude of Q₁₁ and Q₂₂ calculated in MATLAB are in the order of 10⁻² m, while Q₁₂ and Q₂₁ are about 10⁻³ m. Additionally, the larger absolute values are outside the range of antenna aperture, and the absolute value near the aperture center is close to 0. Therefore, to facilitate subsequent derivations, Q₁₂ and Q₂₁ in Equation (15) are ignored.

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Thus, according to Equation (14), the two principal curvatures of the reflected wavefront 1/R₁,1/R₂ are equal to Q₁₁ and Q₂₂, respectively. Put Equation (9) and Equation (14) into Equation (13), Q₁₁ can be obtained as

$$\begin{eqnarray}&&{Q}_{11}=-\displaystyle \frac{2}{\cos \gamma }\displaystyle \frac{\displaystyle \frac{1}{2F}+{\delta }_{rr}}{{(1+\displaystyle \frac{z}{F})}^{\displaystyle \frac{3}{2}}}=-\displaystyle \frac{1+2F{\delta }_{rr}}{F+z}.\end{eqnarray} \tag{ 16 }$$

We have also

$$\begin{eqnarray}&&{Q}_{22}=-2\cos \gamma {a}_{22}=-\displaystyle \frac{2F{\delta }_{\varphi \varphi }+{r}^{2}}{{r}^{2}(F+z)}.\end{eqnarray} \tag{ 17 }$$

According to the intensity law of geometrical optics, namely, the total energy flow of light along a ray tube is constant (Deschamps 1972), as shown in Equation (18).

$$\begin{eqnarray}&&\displaystyle {\int }_{{F}_{1}}{\boldsymbol{W}}\cdot d\alpha =-\displaystyle {\int }_{{F}_{2}}{\boldsymbol{W}}\cdot d\alpha,\end{eqnarray} \tag{ 18 }$$

where F₁ and F₂ are the cross-sectional areas of the tube, respectively. W = 1 / 2 E × H is the energy flow density, namely, Poynting vector (Barbaraci 2020), whose direction is parallel to the surface F of the ray tube, as shown in Figure 6.

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Then the expression of the field propagating along the geometrical optics ray in a homogeneous medium is obtained as follows.

$$\begin{eqnarray}&&{E}_{2}={E}_{1}\sqrt{\displaystyle \frac{{\rho }_{1}{\rho }_{2}}{({\rho }_{1}+s)({\rho }_{2}+s)}}{e}^{-jks},\end{eqnarray} \tag{ 19 }$$

where E₁(V/m) is the electric field at s = 0. e^{−j k s} is spatial phase delay factor. ρ₁ and ρ₂ (m) are the two principal curvatures of the wavefront. Also k = ω/v = 2 π/λ is propagation constant.

Then, for the ideal antenna surface, as shown in Figure 7(a) above. Suppose a reflection point on the antenna reflector is M, and the incident plane wave is reflected by the reflector and propagates to point P of the spherical field with a distance of d, which can be obtained by d = F + z – R, from point M. The electric field at point P is given by

$$\begin{eqnarray}&&{E}_{P}={E}_{M}\sqrt{\displaystyle \frac{{R}_{1}{R}_{2}}{({R}_{1}+d)({R}_{2}+d)}}{e}^{-jkd}.\end{eqnarray} \tag{ 20 }$$

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Now put the expression of ideal reflector f = z = r²/4 F into Equation (16) and Equation (17), the two principal curvatures of the reflected wavefront can be obtained as

$$\begin{eqnarray}&&{{Q}_{11}|}_{\delta =0}={{Q}_{22}|}_{\delta =0}=-\displaystyle \frac{1}{F+z}.\end{eqnarray} \tag{ 21 }$$

Put Equation (21) into Equation (20), the amplitude of electric field at point P in ideal reflector can be expressed as below.

$$\begin{eqnarray}&&|{E}_{P}|=|{E}_{M}|\displaystyle \frac{F+z}{R}.\end{eqnarray} \tag{ 22 }$$

Considering the reflector with deformation as shown in Figure 7(b), we suppose a reflection point on the deformed surface is M^*, and then the wave reflected by the deformed reflector propagates to the point P^* with a distance of d^* from M^*.

For a large-scale rotating parabolic antenna, generally, the deformation of antenna surface is less than 10⁻³ order of magnitude, which is very small relative to its radius. Thus two assumptions can be made below:

• (1)  
  d^* = |P^* M^*| ≈|P M| = d = F + z – R.
• (2)  
  The point P^* can be approximately regarded as P.

Based on the assumptions above, the amplitude at point P^* on the spherical field can be given by

$$\begin{eqnarray}\begin{array}{ll}|{E}_{P}* | & =|{E}_{M}|\sqrt{\displaystyle \frac{{R}_{1}{R}_{2}}{({R}_{1}+d)({R}_{2}+d)}}\\ & =|{E}_{M}|\sqrt{\displaystyle \frac{1}{(1+{Q}_{11}d)(1+{Q}_{22}d)}}.\end{array}\end{eqnarray} \tag{ 23 }$$

Compare Equation (23) with Equation (22), we can get

$$\begin{eqnarray}&&{\left|\displaystyle \frac{{E}_{P}}{{E}_{P}^{* }}\right|}^{2}={\left(\displaystyle \frac{F+z}{R}\right)}^{2}(1+{Q}_{11}d)(1+{Q}_{22}d).\end{eqnarray} \tag{ 24 }$$

Then put Equation (16) and Equation (17) into Equation (24), and we also regard the product-terms of partial derivatives of δ much smaller.

$$\begin{eqnarray}&&{|\displaystyle \frac{{E}_{P}}{{E}_{P}* }|}^{2}={\left(\displaystyle \frac{F+z}{R}\right)}^{2}\left(1-\displaystyle \frac{1+2F{\delta }_{rr}}{F+z}d\right)\left(1-\displaystyle \frac{2F{\delta }_{\varphi \varphi }+{r}^{2}}{{r}^{2}\left(F+z\right)}d\right)\\ &&\,\,={\left(\displaystyle \frac{F+z}{R}\right)}^{2}\left[1-\displaystyle \frac{2{r}^{2}+2F\left({r}^{2}{\delta }_{rr}+{\delta }_{\varphi \varphi }\right)}{{r}^{2}\left(F+z\right)}d\right]\\ &&\,\,+{\left(\displaystyle \frac{F+z}{R}\right)}^{2}\left[\displaystyle \frac{{r}^{2}+2F\left({r}^{2}{\delta }_{rr}+{\delta }_{\varphi \varphi }\right)+4{{\rm{F}}}^{2}{\delta }_{rr}{\delta }_{\varphi \varphi }}{{r}^{2}{\left(F+z\right)}^{2}}{d}^{2}\right]\\ &&\,\,\approx {\left(\displaystyle \frac{F+z}{R}\right)}^{2}-\displaystyle \frac{2{r}^{2}+2F\left({r}^{2}{\delta }_{rr}+{\delta }_{\varphi \varphi }\right)}{{r}^{2}{R}^{2}}d\left(F+z\right)\\ &&\,\,+\displaystyle \frac{{r}^{2}+2F\left({r}^{2}{\delta }_{rr}+{\delta }_{\varphi \varphi }\right)}{{r}^{2}{R}^{2}}{d}^{2}.\end{eqnarray} \tag{ 25 }$$

A transformation from cylindrical coordinates to Cartesian coordinates, as shown in Equation (26) and Equation (27), is introduced.

$$\begin{eqnarray}&&{\delta }_{rr}={\delta }_{xx}\displaystyle \frac{{x}^{2}}{{x}^{2}+{y}^{2}}+2{\delta }_{xy}\displaystyle \frac{xy}{{x}^{2}+{y}^{2}}+{\delta }_{yy}\displaystyle \frac{{y}^{2}}{{x}^{2}+{y}^{2}}.\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{\delta }_{\varphi \varphi }={y}^{2}{\delta }_{xx}-2xy{\delta }_{xy}+{x}^{2}{\delta }_{yy}-x{\delta }_{x}-y{\delta }_{y}.\end{eqnarray} \tag{ 27 }$$

Then, put Equation (26) and Equation (27) into Equation (25).

$$\begin{eqnarray}\begin{array}{ll}{\left|\displaystyle \frac{{E}_{P}}{{E}_{P}* }\right|}^{2} & ={\left(\displaystyle \frac{d+R}{R}\right)}^{2}-\displaystyle \frac{2{r}^{2}+2F{r}^{2}({\delta }_{xx}+{\delta }_{yy})}{{r}^{2}{R}^{2}}d(d+R)\\ & +\displaystyle \frac{{r}^{2}+2F{r}^{2}({\delta }_{xx}+{\delta }_{yy})}{{r}^{2}{R}^{2}}{d}^{2}\\ & =\displaystyle \frac{{d}^{2}+2Rd+{R}^{2}}{{R}^{2}}-\displaystyle \frac{{d}^{2}}{{R}^{2}}-\displaystyle \frac{2d+2F\Delta \delta d}{R}\\ & =1-\displaystyle \frac{2Fd}{R}\Delta \delta,\end{array}\end{eqnarray} \tag{ 28 }$$

where d = F + z – R. We finally obtain the deformation-intensity equation of the spherical field around the feed as Equation (28).

According to Equation (28), the deformation-intensity equation can be written as

$$\begin{eqnarray}&&\Delta \delta =FAG,\end{eqnarray} \tag{ 29 }$$

where $FAG=FA/G,G=-2Fd/R,FA=|{E}_{P}{|}^{2}/|{E}_{P}^{* }{|}^{2}-1=EA-1$.

As shown in Equation (29), the deformation-intensity equation of spherical field is finally transformed into the form of a standard Poisson equation, where FA and G are discrete matrices relative to aperture coordinates (x, y). Thus we can only find the numerical solution of the Poisson equation above, which is a typical elliptic equation that can be solved precisely by a traditional method, finite difference method (FDM).

The main idea of FDM is to transform the Laplacian for the calculation of Δδ into a discrete scheme. Typically, the third-order Laplacian can be given by a five point difference scheme as shown in Equation (30) based on Taylor expansion (Sauer 2011).

$$\begin{eqnarray}L=\left(\begin{array}{ccc}0 & 1 & 0\\ 1 & -4 & 1\\ 0 & 1 & 0\end{array}\right).\end{eqnarray} \tag{ 30 }$$

Thus the matrix FAG in Equation (29) can be expressed as the convolution of L to the deformation δ.

$$\begin{eqnarray}&&FAG=\delta * L.\end{eqnarray} \tag{ 31 }$$

By means of FFT, δ in Equation (31) can be solved efficiently in frequency domain. However, the accuracy of results is inadequate for the demand of surface deformation recovery. In order to apply the FDM, discrete Laplacian L in Equation (30) has to be transformed into matrix T, as shown in Equation (32).

$$\begin{eqnarray}T=\displaystyle \frac{1}{{h}^{2}}{\left(\begin{array}{ccccc}{T}_{0} & -I & 0 & \cdots & 0\\ -I & {T}_{0} & \ddots & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0\\ \vdots & \ddots & \ddots & {T}_{0} & -I\\ 0 & \cdots & 0 & -I & {T}_{0}\end{array}\right)}_{{N}^{2}\times {N}^{2}},\end{eqnarray} \tag{ 32 }$$

where

$$\begin{eqnarray}{T}_{0}={\left(\begin{array}{ccccc}4 & -1 & 0 & \cdots & 0\\ -1 & 4 & \ddots & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0\\ \vdots & \ddots & \ddots & 4 & -1\\ 0 & \cdots & 0 & -1 & 4\end{array}\right)}_{N\times N},\end{eqnarray}$$

and h is step length. Thus we have

$$\begin{eqnarray}&&FA{G}_{{N}^{2}\times 1}={T}_{{N}^{2}\times {N}^{2}}\times {\delta }_{{N}^{2}\times 1}.\end{eqnarray} \tag{ 33 }$$

Finally, the deformation has been converted to the solution of a linear equation, which is then straightforward to employ many iterative methods such as Jacobian method, Gauss-Seidel method and successive over-relaxation (SOR) method (Sauer 2011). In our experiments we have employed Gauss-Seidel method that has demonstrated excellent performance. In fact, comparing with FFT and left division methods, we have proved it again.

We have determined the effectiveness of the numerical algorithm by simulation. The simulation conditions are shown in Table 2.

Primary steps are listed below.

(1) Design the global smooth deformation δ as Equation (8) and Figure 8(a).

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(2) Generate the amplitudes |E| and |E^*| of proposed spherical field with and without deformation on the main reflector by GO method, respectively. The simulated amplitude distribution of ideal antenna,|E|, is shown as Figure 8(b). The intensity ratio in Equation (28), |E|²/|E^*|², is given as Figure 8(c).

(3) Interpolate the simulated data EA(θ, φ) into Cartesian coordinate of aperture, namely, EA'(x, y). According to the antenna geometry as a rotating paraboloid, the relations between (θ, ϕ) and (x, y) are given as

$$\begin{eqnarray}\begin{array}{ll} & \theta =\arccos \left(\displaystyle \frac{4{F}^{2}-{x}^{2}-{y}^{2}}{4{F}^{2}+{x}^{2}+{y}^{2}}\right),\\ & \varphi =\left\{\begin{array}{l}\arctan (\displaystyle \frac{y}{x}),x\gt 0,y\gt 0\\ \arctan (\displaystyle \frac{y}{x})+\pi,x\lt 0\\ \arctan (\displaystyle \frac{y}{x})+2\pi,x\gt 0,y\lt 0\end{array}.\lt /0\gt \right.\end{array}\end{eqnarray} \tag{ 34 }$$

Then EA can be shown as Figure 8(d). Furthermore, we compared the simulated FA with the calculated value G ⋅ ∇² δ. By GO method, it was practically coincided with the theoretical values, as shown in Figure 9, which verified the effectiveness of the deformation-intensity equation we proposed above.

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(4) Recover δ from FA by the proposed algorithm and compare the results δ_r with the ideal deformation δ. In order to make a quantitative comparison, root-mean-square error (RMS) and relative root-mean-square error (RRMS) are introduced as below (Huang et al. 2017).

$$\begin{eqnarray}&&{\rm{RMS}}=\sqrt{\displaystyle \frac{\displaystyle \sum _{m}\displaystyle \sum _{n}{[{\delta }_{r}(m,n)-\delta (m,n)]}^{2}}{{N}^{2}}},\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{\rm{RRMS}}={\rm{RMS}}/{M}_{\delta }.\end{eqnarray} \tag{ 36 }$$

The inference proposed above was verified by a numerical calculation. As is shown in Figure 10, the solution of deformation by FDM has an accuracy of RRMS = 4.06%. As a comparison, the RRMS of solution by FFT is 7.47%.

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According to the Maxwell equations, the well-known physical optic (PO) method can be employed to calculate the electromagnetic field reflected by antenna based on the surface current integration, precisely (Umul 2020). Compared with GO method, which ignores the volatility of light, PO method is widely regarded as a more accurate theory when calculating the reflected field of antenna radiation. Furthermore, physical theory of diffraction (PTD) can be a supporting method to calculate the diffraction field caused by the reflector rim in simulation.

First, we keep the simulation condition unchanged as shown in Table 1 and suppose the same deformation as Equation (8). FA simulated by PO&PTD method is given as Figure 11, which has a relatively large difference with the ideal value according to the proposed equation, especially on the outer domain of the aperture.

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Then, the results of recovered deformation by PO&PTD are given as following Figure 12, the RRMS of which by FDM and FFT are 7.14% and 11.15%.

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Additionally, we have simulated the amplitudes of spherical field at different frequencies, and recovered the deformation by FDM, as shown in Table 3. Except for an invalid result at 0.1GHz, the RRMS at different frequencies are mainly consistent. Especially, when the radiation frequency of plane wave is 0.6 GHz or 1GHz, a more accurate result of deformation recovery is obtained.

Moreover, we have changed a new deformation with a lower amplitude, whose M_δ is 0.4 mm, to further check the effectiveness of the measurement method presented in this paper. As is shown in Figure 13, the red line is the deformation we set and the blue line is the solution solved by the FDM algorithm. Under this condition, the RMS and RRMS of the solution are respectively 16.95 μm and 4.20% while the RMS of surface with the new deformation is about 109.22 μm.

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(1) GO Limit

The basic errors in the derivation of the deformation-intensity equation is completely based on the theory of differential geometry and geometrical optics, and the wave character of electromagnetic wave is neglected in this process. However, the amplitude of spherical field by PO method is accurately calculated based on Maxwell equation, which leads to errors as shown in Figure 14, and the errors will be larger with a lower frequency.

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As a result, there exist differences between the equation proposed above and the simulation results by PO method as shown in Figure 11.

(2) Edge Diffraction

When the plane wave incident on the antenna reflector, diffraction occurs at the edge of the reflector, which will be calculated by PO&PTD method in simulation. While the equation proposed fails to take it into account. As shown in Figure 14, there exist subtle differences in simulation results of pure PO method (yellow line) and PO&PTD (red line).

(3) Lack of Boundary Conditions

The process of solving a partial differential equation (PDE) precisely needs boundary conditions no matter what method is used (Ponzellini Marinelli et al. 2021). In fact, the boundary conditions of deformation on the antenna surface are usually unknown. Then a hypothetical initial value has to be used during solving the equation and the algorithm will adjust it step by step, which will lead to an unknown complex function included in the final solution.