Phase retrieval problem

Microwave holography is the most widely used technique to reconstruct the surface error of antennas. It can be divided into two classes of techniques: phase coherent holography (PCH) and phase retrieval holography (PRH). As PCH needs an accurate phase measurement for the surface reconstruction, which is difficult to realize in high frequency, it is more suitable for low-frequency range. Unlike PCH, PRH is aimed at reconstructing the phase information in aperture plane only from the far-field amplitude, which can be easily measured in high-frequency ranges. Moreover, PRH does not need any additional equipment such as a reference antenna and/or a phase-sensitive receiver. Due to those advantages, PRH is widely applied in the surface reconstruction for large reflector antennas working in high-frequency ranges [Reference Y.6].

According to physical optics, the electrical field distribution $F\lpar u\comma\; v\rpar$ in the small angle far-field region is the 2-D Fourier transform of the aperture distribution $f\lpar x\comma\; y\rpar$.

(1)$$\vert F\lpar u\comma\; v\rpar \vert e^{i\phi\lpar u\comma v\rpar } = \int\int \vert f\lpar x\comma\; y\rpar \vert e^{i\varphi\lpar x\comma y\rpar }e^{-i\lpar ux + vy\rpar }d x\, d y$$

where $\phi \lpar u\comma\; v\rpar$ is the phase distribution of the far-field, and $\varphi \lpar x\comma\; y\rpar$ is the phase distribution of the aperture plane. Phase retrieval problem is a class of problem to reconstruct the aperture phase distribution $\varphi \lpar x\comma\; y\rpar$ from the only measured far-field amplitude $\vert F\lpar u\comma\; v\rpar \vert$. We note that phase retrieval is also commonly applied in some other synthesis problems, such as astronomy, electron microscopy, X-ray crystallography, lensless imaging, etc.

CMAES-HIO algorithm

To solve the phase retrieval problem, Gerchberg and Saxton first proposed the Gerchberg–Saxton (GS) algorithm, which iterates back and forth between the spatial domain and the Fourier domain [Reference Gerchberg7]. It requires the accurate amplitude measurements of the spatial domain and the Fourier domain, which is difficult to realize in antenna problems. In 1980s, Fienup proposed a class of the modified GS algorithms to improve the convergent characteristics of the traditional GS algorithm. Among these algorithms, HIO algorithm obviously accelerates the convergent rate of the GS algorithm by adding a negative feedback in the iterative process according to the support constraints [Reference Fienup8, Reference Fienup9]. However, as it is still a greedy algorithm, which does not solve the issue of the underdetermination of the phase retrieval problem, the algorithm's results are still sensitive to the initial guesses and can easily stagnate at a local minimum.

To cope with the local stagnation problem, a hybrid global phase retrieval algorithm named CMAES-HIO, which is a combination of the hybrid-input-output (HIO) algorithm and covariance matrix adaptation evolution strategy (CMA-ES), was proposed in Xu et al. [Reference Xu, Ye and Hoorfar5]. The schematic diagram of the CMAES-HIO algorithm is shown in Fig. 1. In step 1, we first utilize a phase parameterization process to decrease the optimized variables and then apply the CMA-ES algorithm, which is an evolutionary algorithm for difficult non-linear non-convex black-box optimization problems in continuous domain [Reference BouDaher and Hoorfar10], to obtain a near-global phase guess parameterized by the Zernike polynomials. Then, in step 2, we set the optimized phase guess as the initial phase of the one-plane HIO algorithm and utilize the HIO algorithm to iterate back and forth between the aperture and the far-field until it satisfies the convergence criterion.

Fig. 1. Diagram of different steps in CMAES-HIO algorithm.

Zernike parameterization

The Zernike polynomials are a sequence of orthogonal polynomials on the unit circle. They are widely used in problems such as astronomy, optics, and optometry to describe functions on a circular domain [Reference Wang and Silva11]. In CMAES-HIO algorithm, we utilize the Zernike polynomials to parameterize the aperture initial phase guess in order to decrease the number of the optimized variables. It will effectively accelerate the convergent rate of the CMA-ES algorithm. The detailed Zernike parametrization is defined as follows:

(2)$$\varphi_0\lpar r\comma\; \theta\rpar = \sum{a_{n\comma m}}{Z_{n\comma m}}\lpar r\comma\; \theta\rpar $$

(3)$$\eqalign{{Z_{n\comma m}}\lpar r\comma\; \theta\rpar & = \left\{\matrix{ N_n^mR_n^{\vert m\vert }\lpar r\rpar \cos\lpar m\theta\rpar \comma\; m\ge0\comma\; \cr -N_n^mR_n^{\vert m\vert }\lpar r\rpar \sin\lpar m\theta\rpar \comma\; m\le0\cr }\right.}$$

(4)$$\eqalign{N_n^m& = \sqrt{{2\lpar n + 1\rpar \over 1 + \delta_{m0}}}\comma\; \quad \delta_{ij} = \left\{\matrix{ 1\comma\; \quad i = j\comma\; & \cr 0\comma\; \quad i\neq{\,j}. }\right.}$$

(5)$$R_n^{\pm{m}}\lpar r\rpar = \sum_{s = 0}^{{\lpar n-m\rpar }/{2}}{{\lpar -1\rpar ^s\lpar n-s\rpar !\over s!\lsqb {\displaystyle{n + m} \over {2}}-s\rsqb !\lsqb {\displaystyle {n-m} \over {2}}-s\rsqb !}}r^{n-2s}$$

where ${Z_{n\comma m}}\lpar r\comma\; \theta \rpar$ is the Zernike polynomials, $a_{n\comma m}$ is the Zernike coefficients $m$ and $n$ that are non-negative integers with $n-\vert m\vert = {\rm even}$, $n\ge {m}$, $\theta$ is the azimuthal angle, and $r$ is the radial distance $0\le {r}\le {1}$. In the parameterization, the maximum $n$ is defined as the Zernike order $n^\prime$, thus the number of the Zernike coefficients is equal to ${\lpar n^\prime + 1\rpar \lpar n^\prime + 2\rpar }/{2}$.

Monte-Carlo method

Monte-Carlo method is a broad class of computational algorithms that utilizes repeated random sampling to obtain numerical results. To obtain the optimal Zernike order used in the Zernike parameterization, we investigate the effect of the Zernike order on the algorithm performance and utilize the Monte-Carlo method to simulate the random surface distortions in order to find the optimal Zernike order under an unknown distortion.

We introduce a hybrid evaluation parameter combining algorithm accuracy and the required CPU time to quantify the algorithm performance, which is defined as below.

(6)$$P_i = \omega_1{a_i-\min\lpar a_i\rpar \over \max\lpar a_i\rpar -\min\lpar a_i\rpar } + \omega_2 {\displaystyle{{\displaystyle{{1} \over {t_i}}} -\min \left( {\displaystyle {{1} \over {t_i}}}\right )} \over {\max \left( {\displaystyle{{1} \over {t_i}}}\right ) -\min \left( {\displaystyle {{1} \over {t_i}}}\right)}}$$

where $a_i$, defined in (7), is the global success rate of the CMAES-HIO under $N_0$ trials for Zernike order $i$, $t_i$ is the time cost, and $\omega _1$ and $\omega _2$ are weight factors with $\omega _1 + \omega _2 = 1$.

(7)$$a_i = {N_i\lpar RMSE\lpar phase\rpar \lt \sigma\rpar \over N_0}$$

where $\sigma$ is a given threshold value and $RMSE\lpar phase\rpar$ is the Root Mean Square Error of the retrieved aperture phase.

In this paper, we consider the simulation of Shanghai Sheshan TM-65 m Cassegrain telescope at the frequency of 3 GHz in S band. The surface distortions are simulated as random continuous distortions. The beam amplitude distribution is given as a Gaussian type with a $-16$ dB edge taper. For each simulated distortion, we utilize the CMAES-HIO algorithm to retrieve the distortion under different given Zernike orders. For each Zernike order, we apply several trials of random aperture phase initialization to obtain the time cost and the algorithm's global success rate, which is defined in (7). Then, we calculate the hybrid evaluation parameter $P$ defined in (6) for the given Zernike order. By repeating the simulation of the CMAES-HIO algorithm under different random distortions, we can reveal the hybrid algorithm performance of the CMAES-HIO algorithm under different Zernike orders for an unknown distortion. Finally, we obtain the optimal Zernike order which we should utilize when we aim to reconstruct an unknown surface distortion.

Simulation results

In this work, we simulated 100 types of random continuous distortions. The search range of the Zernike order to be optimized in this paper is set from $1$ to $10$. For each Zernike order, the trials of random aperture phase initialization is set to 100. The success threshold value is set as 0.1rad, and the weight factors $\omega _1$, $\omega _2$ are both set as 0.5 to consider the effect of the algorithm accuracy and time cost simultaneously.

Plots of success rate and time cost versus Zernike orders for the 1st, 50th, 100th random distortions and the average curves are shown in Fig. 2, and plots of the parameter $P$ versus Zernike orders for the 1st,50th, 100th random distortions and the average curves are shown in Fig. 3(e). The simulated aperture amplitude is shown in Fig. 3(a). The distributions of the 1st, 50th, and 100th distortions are shown in Figs 3(b)–(d). The results of Fig. 2 illustrate that a Zernike order in the range of 5 or 6 can obtain the highest global success rate, and a larger Zernike order will increase the time cost. The results of Fig. 3 illustrate that the value of 5 and 6 also correspond to the maximum of the hybrid evaluation parameter $P$ for random continuous distortions. The presented numerical results clearly demonstrate that in order to obtain the optimal comprehensive algorithm performance of CMAES-HIO technique in the surface reconstruction of large reflector antennas for an unknown surface distortion, the Zernike order should be set to a value of 5 or 6.

Fig. 2. Plots of success rate and time cost versus Zernike orders.

Fig. 3. (a) Simulated aperture amplitude; (b, c, d) distributions of the 1st, 50th, and 100th simulated surface distortions; (e) plots of the parameter $P$ versus Zernike orders.