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Abstract
Multi-aperture optical telescopes have been extensively studied owing to their high resolution, low cost, and light weight. The next generation of optical telescopes is predicted to be equipped with dozens or even hundreds of segmented lenses; therefore, it is necessary to optimize the arrangement of the lens array. This paper proposes a new structure called the Fermat spiral array (FSA) to replace the conventional hexagonal or ring array for the sub-aperture arrangement of a multi-aperture imaging system. The point spread function (PSF) and modulation transfer function (MTF) of the imaging system are compared in detail at single and multiple incident wavelengths. The FSA can effectively weaken the sidelobe intensity of the PSF, which is 12.8 dB lower on average than conventional ones with a single incident wavelength in the simulation and 4.45 dB lower in the experiment. A new MTF evaluation function is proposed to describe the mean level of MTF at mid-frequencies. The FSA can improve the MTF of the imaging system and weaken the ringing effect in the images. The imaging simulation indicates that FSA has superior imaging quality compared to conventional arrays, with a higher peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). The imaging experiments also achieve a higher SSIM with the FSA, which agrees well with the simulation results. The proposed FSA multi-aperture will help improve the imaging performance of next-generation optical telescopes.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
According to the Rayleigh criterion, the resolution of a single-aperture optical telescope system is mainly limited by the aperture diameter [1]. Multi-aperture imaging systems use small apertures to achieve a resolution comparable to that of a larger single-aperture imaging system while avoiding the manufacturing challenges and high costs of large single-aperture optics [2–4]. These imaging systems can be divided into Michelson and Fizeau types based on their structures [5]. Fizeau-type systems can directly generate images covering the entire u-v plane. Therefore, they are suitable for detecting extended and dynamic targets and have attracted extensive research [6–9].
To achieve high image quality, optical path differences must be adjusted to a relatively small level [4,10]. Additionally, the structure of the aperture array affects the point spread function (PSF) and modulation transfer function (MTF); conventional aperture arrays are annular and ring structures [11,12]. Considering the quality of the MTF and practical operability, ring structures are widely employed, such as the Multiple Mirror Telescope (MMT) and the under-construction Giant Magellan Telescope (GMT) [13,14]. Golay, another structure, weakens the redundancy of the baselines [15,16]. The Golay structure is suitable for an optical sparse aperture, which aims to enhance the imaging resolution with a small light-collection area [17]. Therefore, Golay structures are helpful in land-based Michelson telescopes with long baselines and are not included in the array structure comparison in this study.
With the development of phase-controlling technology, a 50 nm root mean square (RMS) phasing for 18 lenses was reported [18], and a 30 nm RMS phasing result for 36 lenses was reported [19]. The next-generation giant optical telescopes are predicted to be equipped with dozens or even hundreds of segmented lenses. The PSF and MTF of a multi-aperture system are related to the distance between each pair of lenses; thus, conventional ring and hexagonal structures have repeated pairs that produce the same spatial frequency. A multicore fiber with a Fermat spiral has been reported and achieved a 10.9 dB reduction in the sidelobes in lensless imaging, which inspired us to introduce the Fermat spiral into the structure of multi-aperture imaging [20].
To the best of our knowledge, this study is the first to utilize a bio-inspired Fermat spiral array (FSA) arrangement to optimize the lens structure for multi-aperture imaging. Due to the aperiodic and asymmetric arrangement of Fermat spiral array, it has been proven to perform well in many studies, such as in reflectarray and optical phased arrays [21–23]. The PSF and MTF of the multi-aperture system have been detailly analyzed under the condition of single incident wavelength and multi wavelengths, respectively. The system conducted the structure of FSA possesses PSF with lower sidelobe intensity and MTF with a higher mean level in mid-frequency. Imaging simulation results show that FSA and conventional arrays have almost equal resolution, but FSA achieves a better image quality. The imaging experiments further demonstrated that FSA can improve the PSF and image qualities of the imaging system, which agreed well with the numerical simulation.
2. Theory of FSA arrangement, PSF, and MTF
2.1 FSA arrangement
FSA is derived from sunflowers, as demonstrated in detail in [24]. The positions of the elements in the FSA can be described by
The arrays’ area filling factor α is defined as the area of the multi-aperture to the area of the array’s circumcircle. With the development of optical telescopes for a large number of lenses, and considering the uniformity of the generated FSA, the array element number was set to 37, as shown in Fig. 1. The radius of the circumcircle and sub-apertures for the three structures are all the same, with the values of 10.28 mm and 1.14 mm, respectively, thus, the area filling factors of the three arrays are equal to 0.454. As shown in Fig. 1(c), the FSA has an aperiodic arrangement that can improve the coverage of the MTF because different pairs of lenses produce different spatial frequencies with low repeatability.
Fig. 1. Structure of multi-aperture with the circumcircle radius of 10.28 mm, the sub-aperture radius of 1.14 mm, 37-channel hexagonal array (a), 37-channel ring array (b), 37-channel FSA (c).
2.2 PSF in spatial domain and relative MTF in frequency domain
In the pupil plane, the multi-aperture ${P_{arr}}(x,y)$, composed of N identical elements, is described as follows
where ${P_{sub}}(x,y)$ is the modulus of the sub-apertures, $({x_n},{y_n})$ is the center coordinate of the n-th sub-aperture, ${\phi _n}(x,y)$ is the phase function of the n-th sub-aperture. Multi-aperture imaging can be considered an interference process in which different spatial frequencies are sampled by different pairs of sub-apertures. Assume all the sub-apertures are well in phase (i.e., all ${\phi _n} = 0$), an N-channel system has N(N-1)/2 combinations of vector distance, and the PSF of the imaging system can be expressed as:The PSF is the impulse response function of an optical imaging system; therefore, it is better to obtain a PSF with a narrower central peak and a lower intensity of sidelobes. In this study, the structures had the same circumcircle size; therefore, it was difficult to evaluate the PSF using the full width at half maximum (FWHM). Thus, we adopted the peak-to-integrated-sidelobe ratio (PISLR) as the evaluation function of the PSF, which is expressed as
MTF is also an effective evaluation criterion for imaging systems in the spatial frequency domain. The spatial frequency of the image is equal to the product of the spatial frequency of the ideal target image and the optical transfer function (OTF), where OTF is the normalized Fourier transform of the PSF. The MTF is equal to the modulus of the OTF and can be described as
According to Eqs. (3) and (5), the PSF and MTF of the multi-aperture system are highly related to the vector $(\Delta {x_j},\Delta {y_j})$, which is determined by the distance between each pair of sub-apertures. The repeated distance vector increases the sidelobe intensity of the PSF in the image plane. For the MTF in the spatial frequency domain, the repeated vector enhances some specific spatial frequencies but decreases the spatial frequency coverage in the entire u-v plane. The unrepeated combinations of the three array types were calculated. The 37-channel hexagonal array has 126 unrepeated combinations, and the 37-channel ring array has 345 unrepeated combinations; however, the 37-channel FSA has N(N-1)/2 = 666 (where N = 37) unrepeated combinations, that is, there are no repeat combinations for the FSA. It is predicted that the FSA can generate a PSF with a lower sidelobe and an MTF with better spatial frequency coverage.
3. Comparison of PSF and MTF and imaging qualities for different arrays
3.1 FSA and conventional arrays’ PSF, MTF
The actual imaging system usually has more than one incident wavelength, so we set several types of incident wavelength conditions, that is, one wavelength (600 nm), three wavelengths (457 nm, 532 nm, and 633 nm), and 11 wavelengths (380–780 nm with a band gap of 40 nm). The distance f from the pupil plane to the image plane was set to 400 mm to match the following experiments. Figure 2 shows the two-dimensional (2D) distributions of the PSF of the three arrays and the one-dimensional (1D) curves on a logarithmic scale under the condition of a single wavelength (600 nm). The 1D curves which are marked with dotted lines in the 2D distribution in Fig. 2 include the strongest secondary lobes.
Fig. 2. PSF of the three arrays, the first row shows the two-dimensional (2D) distributions of PSF, the second row shows the one-dimensional (1D) PSF in logscale along the direction indicated by the dotted line, 37-channel hexagonal array (a) (d), 37-channel ring array (b) (e), 37-channel FSA (c) (f).
As shown in Figs. 2(a) and (b), the PSF of the conventional arrays possesses an obvious sidelobe distributed symmetrically around the central lobe. Because the FSA has no repeat combinations of the distance vector, the sidelobes of the FSA’s PSF are distributed around the main lobe without any high-intensity ones and are calculated to be 12.8 dB lower than the conventional ones, on average. Furthermore, the PISLRs were calculated as -0.9888 for the hexagonal array, -2.0754 for the ring array, and 12.9329 for FSA, respectively. Therefore, FSA can decrease the sidelobe intensity and improve the energy encircled in the central lobe of the PSF for the imaging system, which is beneficial for the imaging system. The corresponding MTFs of the three arrays are shown in Fig. 3.
Fig. 3. MTF of the three arrays, the first row shows the 2D distributions of MTF, 37-channel hexagonal array (a), 37-channel ring array (b), 37-channel FSA (c); the second row shows the 1D MTF, along the x-direction (d), along the y-direction (e).
In Figs. 3(a) and (b), there are fluctuations in the MTF values of the conventional arrays, which can be observed in Figs. 3(d) and (e), and cause ringing effects in the images. The conventional array produced a fixed MTF pattern due to the repeated distance vectors, which weakened the ability of the imaging systems to transfer spatial frequencies and resulted in large fluctuations in the MTF values. Although the FSA’s MTF values were lower than the conventional ones at some specific enhanced spatial frequencies, they were distributed more uniformly in every direction, and they were higher at the spatial frequency where low MTF values were inevitable for conventional arrays. In addition, the $MT{F_{mid}}$ were calculated to be 3.1809 × 103 for the hexagonal array, 3.2798 × 103 for the ring array, and 3.6022 × 103 for FSA, respectively. The FSA structure is superior when considering the mean level of the midfrequency of the MTF.
For the multiwavelength condition, the PSF sidelobes of the imaging system were distributed in different positions corresponding to different incident wavelengths. As the wavelength increased, the sidelobes were distributed far from the central lobe. Because conventional arrays are symmetrical structures and have repeated distance vectors, the sidelobes of their PSF are distributed in a fixed direction. In contrast, the sidelobes of FSA’s PSF, distribute disorderly around the central lobe, have lower intensity than the conventional ones, calculated as 13.85 dB lower with three wavelengths and 13.95 dB lower with eleven wavelengths. The PISLRs are shown in Table 1, the FSA’s PISLR is much greater than the conventional ones, which means that the imaging system conducted with the FSA structure can obtain a better PSF with more energy encircled in the central lobe and a lower sidelobe intensity.
Table 1. The PISLR and $MT{F_{mid}}$
As depicted in Eq. (5), one distance vector samples more than one spatial frequency under the condition of multi-incident wavelengths. With the decrease in wavelength, the cutoff frequency, determined by the largest distance vector, increases. The distributions of MTFs became uniform because the system would sample more frequency points in the spatial plane, even the conventional array. The calculated $MT{F_{mid}}$ are shown in Table 1, the mean level of the FSA’s MTF still performs better than the other two arrays, but its advantage slightly decreases with the increase of incident wavelength number.
3.2 Imaging simulations
The USAF1951 resolution chart and pictures were imaged using the three array structures. The system resolution was determined by the order of distinguishable line pairs. The image quality was estimated using mean square error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM).
The simulated imaging results of the resolution chart for a single wavelength are shown in Fig. 4. The system resolution, PSNR, and SSIM are shown at the bottom of Fig. 4. The resolutions of the three arrays are close because they have the same circumcircle size, and their largest distance vectors are close. The same order of line pair, which is the 1st order in Group 5 (32 lp/mm), was distinguished with the three considered arrays, but FSA performed better in PSNR and SSIM. Notably, the ringing effect can be observed in the image of the conventional arrays, whereas it can barely be observed in the FSA image, as shown in Figs. 4(b2) and (c2). As depicted in Fig. 3, the MTF of the conventional arrays was distributed nonuniformly and had specific enhanced spatial frequency points, which caused the ringing effect. The MTF of FSA was distributed more uniformly, which effectively weakened the ringing effect.
Fig. 4. The imaging result of USAF1951 resolution chart, the first shows the entire scene, and the second row shows the zoomed-in part selected by the green box. The original images (a), images with 37-channel hexagonal array (b), images with 37-channel ring array (c), images with 37-channel FSA (d).
Peppers, Cameraman, and Goldhill were imaged using three arrays. The corresponding PSNR and SSIM values are marked at the bottom of each image in Fig. 5. The FSA still performed better than the conventional arrays in terms of the PSNR and SSIM. There is no steep value change in the adjacent spatial frequency points of the MTF of the FSA; therefore, the pictures imaged by the FSA are clearer in detail, as shown in Fig. 5(d).
Fig. 5. The imaging results of pictures, Peppers in the first row, Cameraman in the second row and Goldhill in the third row, respectively. The original images (a), images with 37-channel hexagonal array (b), images with 37-channel ring array (c), images with 37-channel FSA (d).
When the number of wavelengths was increased, the MTF of the conventional arrays also became uniform because a single distance vector produced more than one spatial frequency owing to the different imaging wavelengths. Therefore, the advantage of FSA is slightly lower than that of a single incident wavelength. The 37-channel array imaging results of the resolution chart and pictures with three and eleven wavelengths are listed in Tables 2 and 3, respectively. The “H” and “R” represent hexagonal and ring arrays, respectively. FSA showed an advantage in improving the imaging quality, as shown by the evaluation functions in Table 2 and Table 3. The image quality of FSA was higher than that of conventional methods, which agreed well with the results predicted by PSF and MTF in Sections 3.
Table 2. MSE, PSNR, SSIM, and resolution (lp/mm) of three arrays with three incident wavelengths.
Table 3. MSE, PSNR, SSIM, and resolution (lp/mm) of three arrays with eleven incident wavelengths.
4. Experiments
To verify the improvement in imaging quality by FSA, an experiment was conducted, as shown in Fig. 6. The light source was a halogen lamp with a central wavelength of 600 nm that was filtered to the desired wavelength before the target was illuminated. Lens 1 collimated the light passing through the target, then the collimated light carried the target information passed through the pupil mask and Lens 2 in sequence and was captured by a CCD (MC124CG-SY-UB, a 4112*3008 array of 3.45 µm pixel pitch), which was placed on the back focal plane of Lens 2. The focal lengths of lenses 1 and 2 were both 400 mm, which was equal to the condition of the numerical simulation. During the experiments, the system was covered with a black box after Lens 1 to weaken the influence of stray light and improve the signal-to-noise ratio (SNR) of the images.
Fig. 6. Schematic of experiment setup. 37-channel FSA and 37-channel hexagonal sample plates are displayed.
When capturing the PSF, the light source, the filter, and the target were replaced by a 632.8 nm laser and a 10 µm pinhole to obtain a point light source. The captured PSFs of the three arrays are shown in Fig. 7, where the first and second rows show the 2D and 3D results, respectively. The corresponding peak-to-sidelobe ratios (PSLR) and PISLR are marked at the bottom of Fig. 7. The sidelobe of the FSA is distributed around the central lobe with low intensity and is 4.45 dB lower than the conventional ones, on average. The PISLR of the FSA is higher than those of the other two arrays, indicating that more energy is encircled in the central lobe of the PSF. The FSA could effectively weaken the sidelobe intensity of the PSF and improve the imaging features of the system, which agrees well with the results of the numerical simulation; however, the values of PSLR and PISLR were not as good as those of the simulation because the CCD was not equipped with a cooling module and the inevitable system installation error.
Fig. 7. Captured PSF of three arrays, 37-channel hexagonal array (a) (d), 37-channel ring array (b) (e), 37-channel FSA (c) (f). The corresponding peak to sidelobe ratio (PSLR) and PISLR are marked below the figure.
A USAF 1951 resolution chart was used to conduct the imaging experiments. The images captured with a single aperture, which had the same radius as the circumcircle of the multi-aperture, were set as the reference images to calculate the image quality metrics. The captured images of the resolution chart are shown in Fig. 8, in which the PSNR and SSIM are marked under each image. The resolution of the three arrays is the same as that of the single aperture, which is the first order of group 5 (32 lp/mm). The images of the conventional arrays have a ringing effect, which can be observed in Figs. 8(b) and (c), especially in the zoomed-in central part. However, the ringing effect is weaker in Fig. 8(d), which has a higher SSIM and matches the human visual effects.
Fig. 8. Captured images of resolution chart, (a) single aperture, (b) 37-channel hexagonal array, (c) 37-channel ring array, (d) 37-channel FSA. The central part marked by the green box is zoomed-in and displayed in the upper left corner. The highest order of distinguishable fringe is marked by the red horizontal line and blue vertical line, and the corresponding gray values are plotted.
The resolution of the system operated with different arrays is similar, but it is degraded compared with the simulation results because of the adjustment error of the system and the photoelectric noise of the CCD. The PSNR and SSIM of the images captured with the FSA were higher than those of the conventional images, which was consistent with the simulation results.
5. Conclusion
We have demonstrated a multi-aperture imaging system with FSA lens sub-apertures. The proposed FSA performs well in improving the PSF and MTF of the multi-aperture imaging system when it has the same size of sub-apertures and circumcircles as conventional arrays. The PISLRs of the FSA are much higher than those of the conventional ones, indicating that the FSA can weaken the sidelobe of the PSF and improve the energy encircled in the central lobe. The FSA also achieves a higher mean MTF at mid-frequency under single incident wavelength and multi-wavelength conditions, effectively weakening the ringing effect in the images. Consequently, FSA can improve the quality of multi-aperture imaging, which is verified by the improvements in PSNR and SSIM. The FSA provides an option for next-generation giant optical telescopes.
Funding
National Natural Science Foundation of China (61701505, 62005207); Science and Technology on Electromechanical Dynamic Control Laboratory; State Key Laboratory of Pulsed Power Laser Technology (SKL 2019 KF 06); Natural Science Foundation of Shaanxi Province (2022JM-341, 2019JQ-648); Fundamental Research Funds for the Central Universities (ZYTS23129).
Acknowledgments
The authors thank the editor and anonymous reviewers for their valuable comments. The authors acknowledge the support from the Fundamental Research Funds for the Central Universities, and the Innovation Fund of Xidian Universities.
Disclosures
The authors declare that they have no competing financial interests or personal relationships that may have influenced the work reported in this study.
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.
References
1. H. M. Boffin, G. Hussain, J. Berger, and L. Schmidtobreick, Astronomy at High Angular Resolution: A Compendium of Techniques in the Visible and Near-Infrared (Springer, 2016).
2. S. Chung and D. W. Miller, “Design, implementation and operation of a sparse aperture imaging satellite testbed,” 16th Annual AIAA/USU Conference on Small Satellites (2002), pp. 1–16.
3. V. Zarifis, R. M. Bell Jr, L. R. Benson, P. J. Cuneo, A. L. Duncan, B. J. Herman, B. Holmes, R. D. Sigler, R. E. Stone, and D. M. Stubbs, “The multi aperture imaging array,” in Working on the Fringe: Optical and IR Interferometry from Ground and Space (1999), p. 278.
4. R. L. Kendrick, J. Aubrun, R. Bell, R. Benson, L. Benson, D. Brace, J. Breakwell, L. Burriesci, E. Byler, and J. Camp, “Wide-field Fizeau imaging telescope: experimental results,” Appl. Opt. 45(18), 4235–4240 (2006). [CrossRef]
5. A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University Press, 2006), Chap. 8.
6. A. Jiang, S. Wang, Z. Dong, J. Xue, J. Wang, and Y. Dai, “Wide-band white light sparse-aperture Fizeau imaging interferometer testbed for a distributed small-satellites constellation,” Appl. Opt. 57(11), 2736–2746 (2018). [CrossRef]
7. S. T. Thurman and J. R. Fienup, “Fizeau Fourier transform imaging spectroscopy: missing data reconstruction,” Opt. Express 16(9), 6631–6645 (2008). [CrossRef]
8. Q. Wu, J. Fan, B. Chen, L. Chen, and Y. Hou, “Optical imaging for Goaly3 Multi-mirror telescope system,” Opt. Laser Technol. 156, 108464 (2022). [CrossRef]
9. M. Metwally, F. Eltohamy, and T. M. Bazan, “Design of Very High-Resolution Satellite Telescopes Part III: Telescope Size Reduction,” IEEE Trans. Aerosp. Electron. Syst. 57(6), 4044–4050 (2021). [CrossRef]
10. Z. Zhou, J. Zhang, Q. Fu, and Y. Nie, “Phase-diversity wavefront sensing enhanced by a Fourier-based neural network,” Opt. Express 30(19), 34396–34410 (2022). [CrossRef]
11. N. Miller, B. Duncan, and M. P. Dierking, “Resolution enhanced sparse aperture imaging,” in Aerospace Conference (IEEE, 2006), p. 16.
12. R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002). [CrossRef]
13. C. B. Foltz, J. T. Williams, S. C. West, D. G. Fabricant, and H. M. Martin, “The rebirth of the MMT,” in Proceedings of the 16th IEEE Instrumentation and Measurement Technology Conference (Cat. No. 99CH36309) (1999), pp. 633–638.
14. M. Johns, P. McCarthy, K. Raybould, A. Bouchez, A. Farahani, J. Filgueira, G. Jacoby, S. Shectman, and M. Sheehan, “Giant magellan telescope: overview,” Ground-based and Airborne Telescopes IV 8444, 526–541 (2012). [CrossRef]
15. M. J. Golay, “Point arrays having compact, nonredundant autocorrelations,” J. Opt. Soc. Am. 61(2), 272–273 (1971). [CrossRef]
16. A. J. Stokes, B. D. Duncan, and M. P. Dierking, “Improving mid-frequency contrast in sparse aperture optical imaging systems based upon the Golay-9 array,” Opt. Express 18(5), 4417–4427 (2010). [CrossRef]
17. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007). [CrossRef]
18. D. S. Acton, J. S. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, and J. S. Smith, “Wavefront sensing and controls for the James Webb Space Telescope,” in Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave (2012), pp. 877–887.
19. F. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, and R. Mazzoleni, “On-sky testing of the active phasing experiment,” The Messenger 136, 25–31 (2009).
20. S. Sivankutty, V. Tsvirkun, O. Vanvincq, G. Bouwmans, E. R. Andresen, and H. Rigneault, “Nonlinear imaging through a Fermat’s golden spiral multicore fiber,” Opt. Lett. 43(15), 3638–3641 (2018). [CrossRef]
21. M. W. Niaz, Z. Ahmed, and M. B. Ihsan, “Reflectarray with logarithmic spiral lattice of elementary antennas on its aperture,” AEU Int. J. Electron. Commun. 70(8), 1050–1054 (2016). [CrossRef]
22. J. L. Pita, I. Aldaya, O. J. Santana, P. Dainese, and L. H. Gabrielli, “Side-lobe level reduction in bio-inspired optical phased-array antennas,” Opt. Express 25(24), 30105–30114 (2017). [CrossRef]
23. J. L. Pita, I. Aldaya, O. J. Santana, L. E. De Araujo, P. Dainese, and L. H. Gabrielli, “Side-lobe level reduction in two-dimensional optical phased array antennas,” in CLEO: Applications and Technology (2018), pp. W2A–W58A.
24. H. Vogel, “A better way to construct the sunflower head,” Math. Biosci. 44(3-4), 179–189 (1979). [CrossRef]
