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Abstract
To improve the effectiveness of spatial spectrum sampling for the photonic-integrated interferometric imaging, an array forming scheme is proposed with evenly distributed interferometric baselines, which is referred to as the even sampling photonic-integrated interferometric array (ESPIA). The subaperture array of ESPIA is configured as equi-spaced concentric rings. The subaperture beams are coupled and transmitted to the photonic integrated circuit through fiber optic channels and paired into baselines by the interferometric beam combination. The characteristics of ESPIA are analyzed with the discrete modulation transfer function (D-MTF) and multi-resolution mutual information (MR-MI). The simulation results show that it can realize the even sampling coverage of spatial spectrum effectively. With the same scale of synthetic aperture and subaperture array, it can also improve the capabilities of information acquisition for the interferometric array.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development in the fields such as space exploration and remote sensing, the requirements for optical imaging with long range and high resolution are growing rapidly. However, the conventional imaging technologies are increasingly approaching the capability limits. Consequently, it is necessary to develop new technological mechanisms for breaking the bottlenecks [1]. Optical aperture synthesis interferometric imaging has distinctive advantages in the resolving power and system SWaP etc., which has begun to replace the conventional optical imaging methods in some applications of astronomical observation and remote sensing. With the technological advances, optical aperture synthesis has the potentials to meet the requirements of super-large aperture, super-long range, and super-high resolution [2,3].
Because of the complexities and difficulties especially caused by the implementation scheme based on bulk optical components, the development and application of optical aperture synthesis is constrained severely for a long time [4]. Recently, integrated photonics provides a new approach to improving the capabilities of optical aperture synthesis. By adopting photonic integrated circuits (PICs) for the interferometric beam combination, the capabilities in real-time, sensitivity and extensibility can be improved significantly. Also, the stability and reliability of interferometric arrays can be improved effectively [5–7]. The photonic-integrated interferometric aperture synthesis is adopted by the astronomical interferometric array first. The black holes and exoplanets have been observed successfully by the instrument GRAVITY/VLTI [8]. Furthermore, the SPIDER project explores the scheme of large-scale photonic-integrated interferometric imaging, and proposes an ultra-thin and ultra-light synthetic aperture imager with an array of PICs [9]. The possible application to the Europa exploration task is also investigated, where the size and weight of space-borne imager can be reduced more than an order of magnitude by using SPIDER [10].
Based on the van Cittert-Zernike theorem, SPIDER proposes to construct a large-scale Michelson-type interferometric array. The subapertures of SPIDER are densely distributed on a hollow disk with a spoke-like configuration, where the longest baseline is equivalent to the spoke length [11,12]. Each spoke of subapertures corresponds to a multi-channel multi-spectral PIC, which consists of an array waveguide grating (AWG) and hundreds of phase shifting interferometers (PSIs). Based on the pairwise ABCD scheme, PSIs can realize the multi-channel multi-spectral beam combination together with AWG [13–16]. In spite of the advantages of the array forming scheme of SPIDER, the synthetic aperture is smaller than a half of the physical aperture, which in turn limits the imaging resolution. Therefore, at the cost of doubling the physical aperture size, a direct mapping relation is established between the subapertures and the PICs, where the subaperture array is not exploited efficiently. In addition, the sampling density of the u-v plane decreases linearly when the spatial frequency increases, and the sampling distribution becomes rather non-uniform, which has significant influence to the quality of image reconstruction. For example, the sparse sampling of high spatial frequencies will decrease the capability in resolving details [17,18].
To improve the subaperture array of SPIDER, various array forming schemes have been proposed, such as SPIDER [15], HMSLA [17], RLA [19], and HLA [20]. These schemes prefer to enhance the capability of information acquisition for low spatial frequencies, but the amelioration to the problems about the sparsity of high spatial frequencies and the unevenness of sampling distribution is limited. Therefore, the Gibbs ringing and other artificial effects arise in the image reconstruction [21,22]. To further improve the capability of information acquisition and the quality of image reconstruction, it is necessary to solve the sparsity and unevenness problems of sampling distribution in the photonic-integrated interferometric arrays.
In this paper, we propose an array forming scheme with the evenly distributed interferometric baselines for synthetic aperture imaging, which is referred to as the Even Sampling Photonic-integrated Interferometric Array (ESPIA).
The subsequent sections of this paper are organized as follows. Section 2 presents the mechanism of ESPIA array forming scheme. Section 3 describes the simulation experiments for ESPIA characterization. Section 4 summarizes and discusses the analysis results. Section 5 concludes the paper.
2. Array forming scheme
To realize the even sampling of spatial spectrum, the subaperture array of ESPIA is configured as the equi-spaced concentric rings, as shown in Fig. 1(a). Within each ring the subapertures are evenly distributed, and the intervals between neighboring rings are approximately equal to each other. In the following, all of the results are acquired under the ideal conditions, i.e., all the subapertures have ideal lenses with the same size and pitch.
Let $m = 1,2,3\ldots $ represents the order of concentric rings (from center to outside), then the number of subapertures in ring m is ${N_m} = 8m - 4$, and the total number of subapertures in the interferometric array is Narray=Σk-1…m Nk = (2m)2. Let $k = 1\ldots {N_m}$ represents the index of a subaperture in ring m, then the subaperure can be expressed as $S(m,k)$. Let $\Delta {D_m} = {D_m} - {D_{m - 1}}$ represents the intervals between neighboring rings, where ${D_m}$ represents the diameter of ring m, with ${D_0} = 0$. As shown in Fig. 1(b), when the order m increases, $\Delta {D_m}$ will approach to a fixed value $\Delta {D_\infty } \approx 2.546d$, where d represents the subaperture pitch within all rings, and the intervals are roughly equal to each other except for $\Delta {D_1}$.
The array forming scheme of ESPIA adopts two pairing modes for subapertures, that is, the basic pairing mode (BPM) and extended pairing mode (EPM). In the BPM, the subapertures are distributed with the central symmetry, which means that each subaperture is only paired with the opposite one in the same ring to constitute an interferometric baseline. It is expressed as $S(m,k) \leftrightarrow S(m,k^{\prime})$, with $k^{\prime} = ({k + {{{N_m}} / 2}} )\bmod ({{N_m}} )$, and the number of u-v plane samplings is equal to that of interferometric baselines. As shown in Fig. 2, with the BPM, the distributions of u-v plane samplings are the same as those of subaperture array (ignoring the coordinate units), which means that the even sampling of u-v plane is realized. However, although the sampling evenness can be improved effectively by the BPM, the problem of sparse sampling is still to be solved.
The EPM is also realized within the same rings, where each subaperture is paired with two others of the specific subaperture pitch to constitute interferometric baselines. This mode can be expressed as S(m,k)↔ S(m,k′′), with $k^{\prime\prime} = ({k \pm {\delta_m}} )\bmod ({{N_m}} )$ and ${\delta _m}$ represents the pairing pitch. In order to realize the dense and even sampling, the subaperture pitches of each order are simulated and analyzed thoroughly. The pairing pitch of each order is selected so that EPM samplings are distributed in the middle of BPM samplings as even as possible. We summarize the empirical formula of pairing pitches as
Since ${\pm} {\delta _m}$ samplings are coincident to each other, the number of u-v plane samplings is only 1/2 of that of interferometric baselines. As shown in Fig. 3(a)-(c), with the EPM, the u-v plane samplings are roughly distributed in the middle of concentric circles formed by the BPM samplings, which means that the dense and even sampling of u-v plane is realized. As shown in Fig. 3(d), the analysis results show that when $m > 3$, compared with the middle circle diameters ${D_m} - \Delta {D_m}/2$ between neighboring circles in the BPM sampling patterns, the relative errors are less than 2% for the circle diameters $D_m^ + $ in the EPM sampling patterns. As shown in Fig. 4, by synthesizing two pairing modes, the equivalent pupil and sampling density can be improved significantly.
Based on the array forming scheme of ESPIA, we propose a system architecture of synthetic aperture imager similar to SPIDER. As shown in Fig. 5, the beam intensity of each subapeture is equally divided into three parts by the fiber couplers, one for the BPM and the other two for the EPM. Compared with the system architecture of SPIDER, differences lie mainly in the configurations of subaperture array and the coupling between subapertures and PICs.
3. Simulation and analysis
To validate the effectiveness of ESPIA, it is analyzed by simulation with regards to the characteristics of optical transfer and capability of information acquisition. Using the measures of discrete modulation transfer function (D-MTF) and multi-resolution mutual information (MR-MI), ESPIA is compared to other array forming schemes including SPIDER [15] and HMSLA [17] with the benchmark of ideal monolithic aperture, where HMSLA is the optimal scheme of information acquisition for low spatial frequencies in the literatures. The simulation parameters are designed mainly for consistent comparisons as shown in Table 1.
Table 1. Main simulation parameters of array forming schemes
SPIDER scheme is shown in Fig. 6(b). There are 37 subaperture spokes with the equal angular interval distributed on a disk of radius $R = d/2\sin (\alpha /2)$, where $d$ is subaperture diameter and $\alpha = 2\pi /37$. Within each spoke, there are 30 subapertures densely packed with the equal space. HMSLA scheme is shown in Fig. 6(c), there are three kinds of subaperture spokes, with the spoke number of 19, 38, 76 and the subaperture number of 30, 22, 10 per spoke respectively. ESPIA scheme is shown in Fig. 6(d), there are 12 orders of the concentric rings, with the subaperture number of 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92 respectively.
The characteristics of optical transfer is evaluated with the D-MTF, where MTF is acquired by the autocorrelation of equivalent pupil [23]. The simulation experiments are carried out under the condition of the same equivalent pupil diameter ${D_{pupil}} = 30d$ ($d$ is subaperture diameter). As shown in Fig. 7, from the normalized D-MTF of four schemes, it is clear that the characteristics of modulation transfer of ESPIA is similar to that of IDEAL, and the modulation contrast and sampling evenness over the full range are better than those of SPIDER and HMSLA. For the characteristics of HMSLA, it is better than that of SPIDER in low spatial frequencies and worse than that of SPIDER in high spatial frequencies. The overall characteristics of SPIDER are not as good as the other schemes. Furthermore, we use the Area Under the Curve (AUC) to evaluate the performance of four schemes quantitatively. With the ideal MTF as a benchmark, the normalized AUCs of MTF curves by the piecewise linear fitting are 0.606 (SPIDER), 0.582 (HMSLA), and 0.996 (ESPIA) respectively.
The capability of information acquisition is evaluated with the MR-MI, where MR-MI is acquired from the multi-resolution histograms of object images and reconstructed images [24]. The simulation experiments are carried out with USAF 1951 images to evaluate the performance of the mentioned schemes comprehensively. As shown in Fig. 8, the images of synthetic aperture schemes are all reconstructed by using the Non-Uniform Fourier Transform (NUFT) [25]. The image of multi-resolution decomposition is based on the Gaussian pyramids [26,27], where ${G_{n + 1}}$ is constructed by the decimation of the Gaussian kernel convolution with ${G_n}$. In the simulation experiments, the object image is decomposed successively into ${G_0}$, ${G_1}$, ${G_2}$, ${G_3}$, ${G_4}$ and ${G_0}$ is the original object image.
As shown in Fig. 9, the MR-MI of each synthetic aperture scheme is normalized by that of ideal monolithic aperture respectively. From the normalized MR-MI of three schemes, it is conspicuous that the MI of ESPIA is larger than or equal to that of SPIDER and HMSLA for different resolutions of USAF1951 object image. Concerning the overall performance, the AUCs of MR-MI curves by the piecewise linear fitting are 0.599 (SPIDER), 0.616 (HMSLA), and 0.705 (ESPIA) respectively. It is demonstrated that ESPIA has more powerful capability of information acquisition over the full range, and SPIDER and HMSLA are similar with regards to it. For different kinds of objects, there exist certain differences in terms of MR-MI because of the diversity of information contents. However, ESPIA can still realize equilibrium optimization over all aspects owing to its even sampling.
4. Result and discussion
Although the above simulation experiments are based on simplified models, it is obvious that ESPIA has distinctive advantages in terms of the characteristics of modulation transfer and capability of information acquisition over the full range. For the comprehensive analysis of ESPIA, several performance indexes are further compared under the same simulation parameters. As shown in Table 2, the simulation results are summarized for five schemes, including two additional schemes RLA and HLA.
Table 2. The performance index comparisons of different schemes
From Table 2, it is apparently seen that the performances of ESPIA are the best except the sensitivity, which means that ESPIA realizes the equilibrium optimization of overall capabilities at the cost of lower sensitivity. Compared to other schemes, with the densely and evenly distributed interferometric baselines, ESPIA can increase the aperture utilization rate and u-v sampling density significantly, which accounts for the improvement to the modulation transfer characteristics and the overall information acquisition capability. However, because of the partial redundancy of interferometric baselines, the sensitivity is decreased and the larger interferometer array is needed. By comparing the results simulation and analysis, the main characteristics of five schemes can be summarized in Table 3.
Table 3. Summary of main characteristics of five schemes
5. Conclusion
In this paper, an array forming scheme referred to as ESPIA is proposed with the evenly distributed interferometric baselines, based on which the system architecture of synthetic aperture imager is constructed. With the subaperture array configuration of equi-spaced concentric rings, ESPIA realizes dense and even sampling of u-v plane by combining two modes of subaperture pairing. The simulation results show that ESPIA can improve the effectiveness of spatial spectrum sampling for synthetic aperture imaging. With the ideal MTF as a benchmark, the normalized AUC of ESPIA reaches 0.996. Compared to other schemes in the literatures, ESPIA has the distinctive advantages in the modulation transfer characteristics and the overall information acquisition capability.
Funding
National Key Scientific Instrument and Equipment Development Projects of China (YJKYYQ20200057); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2020221); National Natural Science Foundation of China (62005279, 62105327); Jilin Scientific and Technological Development Program (20200201294JC).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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