Open Access
Add to BibSonomy
Abstract
The traditional microlens array imager lacks content information and has low imaging quality when restoring the image of the target, which limits the development of photon integrated detection and imaging technology. Therefore, this paper proposes a new structure of the microlens array optical frequency detection imager, wherein the main microlens array is embedded into the auxiliary microlens array to achieve high-quality restoration of the target. The simulation results show that the proposed structure increased the peak signal-to-noise ratio (PSNR) of the restored image by 21.05% and reduced the mean square error (MSE) by 29.39%. The microlens array with the main and auxiliary nested structure can achieve high-quality imaging of the target and has great application potential in the field of target detection.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A photonic integrated circuit (PIC)-based interferometric imaging system has been developed by the Lockheed Martin Center for Advanced Technology and UC Davis. The split plane imaging detector (SPIDER) [1–4] for photoelectric reconnaissance is an important device in this system. SPIDER is an interferometric imaging system, which consists of one-dimensional interferometric arrays (interference arms) in multiple directions. Each interferometric arm has the same design and is composed of microlenses. The waveguide couples the light into the PIC [1]. For complex visibility of each interferometric baseline, light from the extended scene of array, which is collected by the microlenses and sent through the waveguides, is measured by detectors.
Interferometry is an important principle used by SPIDER interferometer for target detection. The principle is to use the superposition of the electromagnetic waves to obtain the content information of the target [5,6]. The SPIDER combines brings together multiple microlenses to form a synthetic aperture to image the target, producing a high-resolution image. We refer to the space vector distance between the lens pairs as the interference baseline. SPIDER utilizes the Van Cittert-Zernike theorem [6] for measurements, each of which corresponds to the Fourier component of the target. The baseline measurement corresponds to the angular spatial frequency υ=B/λ, where B is the interference baseline (spatial distance vector between microlenses) and λ is the wavelength of the light. The cutoff spatial frequency is determined by the maximum baseline length and the minimum wavelength. Each baseline pair creates interference fringes for each spectral channel. Measuring the target with baselines in all the directions of the PIC maps out the 2D Fourier plane, effectively providing a 2D Fourier transform of the object. Examples of such interferometer arrays include the CHARA array [7,8], the Very Large Telescope Interferometer [9] and the Navy Precision Optical Interferometer [10]. These systems use far-field spatial coherence measurements to form intensity images of astronomical sources [11]. In addition, Tiehui Su and Guangyao Liu [15] et al. designed and fabricated a Si3O4 type photonic integrated circuit (PIC) and completed the reconstruction of the target image through experiments. Antosh Kumar, Lokendra Singh et al [16]. successfully proposed a novel and compact design of a one-bit magnitude comparator using plasmonic MZI and verified using FDTD . Santosh Kumar, Chanderkanta et al [17]. successfully designed an optical parity checker and generator circuit using the electro-optic effect of lithium niobate MZI in the beam propagation method along with mathematical description. The study is verified using beam propagation method (BPM). Dalai G. Sankar Rao, Sandip Swarnakar and Santosh Kumar et al [18]. reported all-optical NNX logic gates using two-dimensional PhCWs (photonic crystal waveguides). The flexible devices presented here satisfy the functionality of NAND (NOT-AND), NOR (NOT-OR) and XNOR (exclusive NOR) logic gates using only one structure with proper changes in the phase of an applied light signal. Santosh Kumar, Lokendra Singh and Nan-Kuang Chen et al. [19] proposed a novel design of all optical universal gates using optical Kerr-effect and optical bistability of a plasmonics-based Mach-Zehnder interferometer (MZI). The nonlinear Kerr-material provides ultrafast switching which can be used to develop switching components for WDM applications. Santosh Kumar, Ashish Bisht and Gurdeep Singh et al [20]. demonstrates the structure and working principle of an optical 2-bit multiplier using lithium niobate (LiNbO3) based Mach-Zehnder interferometer (MZI). They carriy out by simulating the proposed device with Beam propagation method (BPM). In 2020, an update of CS-CPCIT (CS-CPCIT+) was proposed. The arrangement of photonic integrated circuits greatly simplified their structure [21].
The new structure microlens array detector based on the principle of interference detection can replace the large optical telescope system. The traditional telescope system uses the physical principles of reflection and refraction to detect long-distance targets. It requires a huge rigid structure to carry a large optical lens or mirror, which introduces defects and problems such as large volume, large mass and high energy consumption. For example, the Hubble telescope is 13.3 m long, weighs 27,000 pounds and uses a primary mirror with a diameter of 2.4 m [12]. We can achieve the same resolution with a smaller size, lower weight and lower power making it an attractive option.
However, in the traditional SPIDER interferometer, the microlens array structure still has shortcomings. In the photonic integrated interference imaging system, the microlens array is the acquisition or receiving unit of the optical information of the target and its function is to receive the spatial frequency information sent by the target [13]. The disadvantage of the traditional SPIDER interferometer is that the spatial frequency information emitted by the target cannot be fully collected, which leads to the problem of lack of content information or low definition in the restored image when the target is detected and imaged. Aiming at the above problems, we have improved a microlens array using a new structure.
In this study, a novel microlens array structure is proposed to perform experimental simulation of image restoration on target images. In the experimental simulation, the restored image of the new structure microlens array and the traditional structure microlens array are evaluated by the image quality evaluation function. The micro-lens array is the core component of the detection imaging system. The simulation results show that the new type of micro-lens array of the main and auxiliary nested type can widely collect the optical information emitted by the target and achieve high-quality image restoration.
2. Imaging and working principle of microlens array
In the photon integrated interferometric imaging system, a large number of microlenses are formed into a two-dimensional plane receiving array to realize the collection of the spatial frequency information emitted by the target. Since each microlens in the array is in a different position, spatial frequency information of different sizes can be collected. The restored image of the target can be obtained by superimposing the frequency information collected by each microlens. We can regard the process of light waves propagating from the detected target to the front surface of the microlens array as a Fraunhofer diffraction process, thus the light field distribution on the front surface of the microlens array can be expressed as [14]:
Since the position of the observed image is located at the focal plane of the microlens array, the focal length of each microlens on the array is equal and small, therefore, the light field propagation from the back surface of the microlens array to the focal plane can be regarded as Fresnel diffraction process. Combined with Eq. (3), the light field distribution on the focal plane is:
From Eqs. (4) and (5), we can obtain the light field distribution of the output image of each micro-element pair on the target. The output image on the focal plane can be obtained by superimposing the light field distributions of all the object-plane micro-elements to the output image. Therefore, the point spread function connects the relationship between the light field of the object surface to that of the image surface in a quantitative form. Thus, the light field distribution (point spread function) of the output image of each micro-element on the target is expressed as:
Figure 1 shows the schematic diagram of a photonic integrated system based on the main and auxiliary nested microlens arrays. A pair of microlenses in a microlens array forms a set of coherent baselines that couple optical signals into single-mode fibers. The size of the optical waveguide used in the photonic integrated interference imaging system is less than 10 µm, so we use a single-mode fiber to transmit the optical signal data. The role of the arrayed waveguide grating (AWG) is to separate quasi-monochromatic light waves into signals of different wavelengths and transmit them in their respective channels. In the simulation, the center wavelength is 1310 nm, while in the experiment, the light wave emitted by the target is quasi-monochromatic light in the center wavelength range, so we use the arrayed waveguide grating to align the monochromatic light wave decomposition. The wavelength represents the type of light wave. The purpose is that two light waves of the same type of wavelength can form coherent light. The role of the KG-IDPM series LiNbO3 electro-optical phase modulator is to phase-modulate two light waves, its model is SN.1299958, the applicable wavelength range is 750-2000nm. Its working principle is that under the action of an external electric field, the phase modulation of the light wave signal is carried out by using the effect that the effective refractive index of the electro-optic material is proportional to the applied voltage and it changes according to a certain law. The working principle of the electro-optic phase modulator is expressed as ${\textrm{E}_{\textrm{out}}}(\textrm{t} )= {\textrm{E}_{\textrm{in}}}(\textrm{t} )\cdot {\textrm{e}^{\textrm{j}{\mathrm{\varphi }_{\textrm{PM}}}(\textrm{t} )}} = {\textrm{E}_{\textrm{in}}}(\textrm{t} )\cdot {\textrm{e}^{\textrm{j}\frac{{\textrm{u}(\textrm{t} )}}{{{\textrm{V}_\mathrm{\pi }}}}\mathrm{\pi }}}$, where ${\mathrm{\varphi }_{\textrm{PM}}} = \frac{{2\mathrm{\pi }}}{\mathrm{\lambda }}\cdot \triangle {\textrm{n}_{\textrm{eff}}} \cdot {\textrm{l}_{\textrm{el}}} \propto \textrm{u}(\textrm{t} )$ is the phase modulated by the phase modulator, ${\propto} $ means proportional to the symbol and Vπ means the half-wave voltage. The purpose is to form a set of coherent lights that interfere within a 90-degree optical mixer. The model of the 90-degree optical mixer is COH24-X, which has two inputs and four outputs. Assuming that the two lightwave signals modulated by the phase modulator are ${\textrm{E}_1} = {\textrm{E}_{01}}{\textrm{e}^{\textrm{i}({{\mathrm{\omega }_1} + {\mathrm{\varphi }_1}} )}}$ and ${\textrm{E}_2} = {\textrm{E}_{02}}{\textrm{e}^{\textrm{i}({{\mathrm{\omega }_2} + {\mathrm{\varphi }_2}} )}}$ respectively, then the output signals are respectively ${\textrm{E}_{\textrm{out}1}} = \frac{1}{2}({{\textrm{E}_1} + {\textrm{E}_2}} )$, ${\textrm{E}_{\textrm{out}2}} = \frac{1}{2}({{\textrm{E}_1} - {\textrm{E}_2}} )$, ${\textrm{E}_{\textrm{out}3}} = \frac{1}{2}({{\textrm{E}_1} + \textrm{i}{\textrm{E}_2}} )$, ${\textrm{E}_{\textrm{out}4}} = \frac{1}{2}({{\textrm{E}_1} - \textrm{i}{\textrm{E}_2}} )$. The signal output by the optical mixer is transmitted to the balanced detector for intensity information and phase information measurement. The photoelectric detection of PIN/InGaAs balanced detector produced by THORLABS company is suitable for 800-1700nm wavelength range, its model is PDB450C. The photodetected data is then transmitted into the information processing system to reconstruct the image.
Fig. 1. Schematic diagram of photonic integrated system based on main and auxiliary nested microlens array.
Assuming that the coordinates of the center of the lens D1 in the microlens array are (a1, b1) and the coordinates of the center of the lens D2 are (a2, b2), when the far-field conditions are met, the complex coherence [14] between the lenses D1 and D2 is
3. Structural design of microlens array
Figure 2 is a structural diagram of the main and auxiliary nested microlens arrays. We embed the main micro-lens array lens sleeve into the center of the auxiliary micro-lens array flat plate to form a new-structured micro-lens array imager. Regarding the size parameters of the main micro-lens and the auxiliary micro-lens, we choose the sizes of 52.33 mm and 31.40 mm respectively, their focal length parameters f are both 1 mm. In the simulation part, we use the above main microlens size parameters and auxiliary microlens size parameters. For the optimization of pupil diameter, we set the radius parameter R in the simulation, which is the distance from the inner circle of the auxiliary microlens array to its center point, we use it as an intermediate parameter to control the number of sampling points of the aperture function. The aperture function of the microlens is determined by the number of sampling points N. Then, we can conclude that the aperture function of the main microlens is determined by the sampling point N1 and the aperture function of the auxiliary microlens is determined by the sampling point N2. The optimization problem of the pupil diameter is transformed into the problem of establishing a microlens array through the radius parameter R to obtain high-quality imaging. We continuously change the parameter R to optimize the pupil diameter to achieve the best imaging effect.
The sleeve is embedded in the central position because: Fig. 3(b) is the spectrogram of the original image, the spatial frequency is mainly at the low frequency and the DC component, so that the spatial frequency information sent by the object can be fully received.
3.1 Structure of the main microlens array
Figure 4 shows the arrangement structure of the main microlens array. The main microlens array consists of an inner structure and an outer structure. The internal structure is an N×N square matrix structure composed of lenses and the external structure is a pyramid structure in which the number of lenses gradually decreases from the inside to the outside. Assuming that the aperture function of the main microlens is P1(x,y), we can express the light field at each point (xi,yi) on the image plane by formula (4) as:
According to Eqs. (8) and (9), we can obtain the point spread function from the target to the image plane and explore the collection of spatial frequency information by the photonic integrated system from the perspective of the frequency domain. The point spread function of the main microlens array is expressed as:
The lenses in the main microlens array can arbitrarily form a pair of coherent baselines, which makes up for the lack of collected information due to information redundancy. The two microlenses of the internal structure of the microlens array can form an interference baseline, the lenses of the external structure can also arbitrarily form an interference baseline, the microlenses of the internal and external structures can still form a set of coherent baselines. This free matching method can form baselines of different lengths and different vector directions.
3.2 Structure of the auxiliary microlens array
Figure 5 is a structural diagram of the auxiliary microlens array. The secondary microlens array is a two-dimensional spatial frequency receiving surface composed of one-dimensional interference arms in multiple directions. The pupil diameter of the microlens on the one-dimensional interference arm is smaller than that of the main microlens. The size parameters of the main microlens and the auxiliary microlens are selected as 52.33 mm and 31.40 mm respectively, the focal length parameter f is both 1 mm.
The reason we adopt the 2D spatial frequency receiving plane is that the 2D spatial frequency receiving plane maps the 2D spatial frequency Fourier plane, effectively providing the 2D Fourier transform of the target. In its one-dimensional interference arm, the long interference baseline BMAX collects information with higher spatial frequency in order to restore the details or contour information of the target; while the short interference baseline BMIN collects information with lower spatial frequency for the purpose of content information is restored. On the other hand, we can use the short baseline BMIN in the 2D plane to assist the main microlens array to collect target content information; at the same time, use its long baseline to collect target outline or edge information.
To sum up, the high-frequency components of an image can be understood as mainly measuring the edges and contours of the image; the low-frequency components are mainly a comprehensive measure of the intensity of the entire image. Figure 3(a) shows the image of the target, we perform Fourier transform on it, the spectrogram in Fig. 3(b) shows that the low-frequency information components occupy the main part, so we embed the main microlens array into the auxiliary microlens array, the main micro-lens array is used as the main spatial frequency information receiving array, which has the main function of imaging the target content. Using the 2D plane mentioned above, the detailed information of the target is collected through its long baseline; the acquisition of the target content information is performed by the auxiliary main microlens array through its short baseline.
4. Simulation restoration and result analysis
In the simulation software, we established the main and auxiliary nested microlens arrays and the traditional microlens arrays by writing aperture function codes. For the main and auxiliary nested microlens arrays, the number of sampling points N1 used by the main microlens is 100, their size parameters are all 52.33 mm and the focal length parameters are all 1 mm; the number of sampling points N2 of the auxiliary microlenses is 60, their size parameters are all 31.40 mm and their focal length parameters are also 1 mm. The lens-related parameters in the traditional structure microlens array are the same as those of the auxiliary lens in the main and auxiliary nested microlens array. To verify the imaging effect of the main and auxiliary nested microlens arrays, we selected the case with an object distance of d = 75 m and the same focal length of the microlens arrays and imaging a target with a size of 1 meter. Figure 6(a) and Fig. 6(b) are the simulation result diagrams of the image restoration of the target by the main-auxiliary nested structure microlens array and the traditional structure microlens array respectively. From the figure, it can be observed that the image restoration of the main and auxiliary nested microlens array can clearly reflect the content information of the target.
Fig. 6. Restoration image simulation result diagram. (a) restored image of the main and auxiliary nested microlens array; (b) restored image of the traditional structure of the microlens array.
The spatial frequency information by the microlens array is under-acquisition. Therefore, when the target is imaged and restored, some content of the restored image will be missing due to missing information. To evaluate the quality of restored images, we introduce peak signal-to-noise ratio PSNR and mean square error MSE. The expressions of PSNR [13] and MSE values [13], respectively.
Table 1. Parameters of the main and auxiliary nested microlens arrays
Table 2 shows the MSE value and the PSNR value evaluated by the image quality function after the image restoration of the target by the two microlens arrays was performed. The MSE value of the image restored by the main and auxiliary nested microlens array is reduced by 29.39% compared with that of the image restored by the traditional microlens array; the PSNR value of the image restored by the main and auxiliary nested microlens array has increased by 21.05% compared with the mean square error value of the image restored by the traditional microlens array. We can conclude that the main and auxiliary nested microlens array can be used as the spatial frequency receiver of the photon interference imaging system to achieve high-quality image restoration of the detection target.
Table 2. MSE and PSNR of the restored images of the main and auxiliary nested and traditional microlens arrays
To explore the influence of the main and auxiliary nested microlens arrays on the image quality of the restored image, the next section specifically discusses the influence of the main and auxiliary microlens arrays on the restored image quality, the influence of the size of the main microlens array on the restored image quality.
4.1 Influence of the main and auxiliary microlens arrays on the restored image quality
Figure 7(a) and Fig. 7(b) show the main microlens array and the main and auxiliary nested microlens arrays. To explore the influence of the main and auxiliary microlens arrays on the quality of the restored image, we selected the case with an object distance of d = 300 m and the focal length of the microlens array was the same, the image restoration was carried out on a target with a size of 1 m.
Fig. 7. (a) Main microlens array with an internal structure size of 7×7; (b) Main and auxiliary nested microlens array.
Figure 8(a) and Fig. 8(b) are simulation result diagrams of the two kinds of microlens arrays for the restoration of the target. The simulation results show that the restored image effect of the main and auxiliary nested microlens array reflects the content information of the target more clearly. Table 3 shows the MSE value and the PSNR value of the restored images of the two arrays. The mean square error values (MSE) of the restored image of the main and auxiliary nested microlens arrays are reduced by 28.98% compared to that of the restored image of the 7×7 main microlens array with the internal structure scale structure; The PSNR value of the image restored by the conventional microlens array increased by 22.73% compared to the mean square error value of the image restored by the traditional microlens array. The simulation results show that the image quality of the target restored by the main and auxiliary nested microlens arrays is better than the image quality of the target restored by the main microlens array alone lens array.
Fig. 8. (a) Restoration results of the main microlens array with an internal structure scale of 7×7; (b) Restoration results of the main and auxiliary nested microlens arrays.
Table 3. Results of the mean square error value MSE and the peak signal-to-noise ratio value PSNR of the restored images of the two arrays
Table 4 shows the respective contribution of the main and auxiliary microlens arrays to the decrease in the MSE value and the increase of the PSNR value. The simulation data results show that, in the main and auxiliary nested microlens arrays at this scale, the main microlens array contributed 71.02% to the MSE reduction of the restored image, which is 2.5 times the contribution of the auxiliary microlens array; Furthemore, it contributed 81.48% to the increase of the PSNR of the restored image, which was 4.4 times the contribution of the secondary microlens array. The simulation results show that the contribution of the main microlens array to the image of the target is greater than that of the auxiliary microlens array. Therefore, we used the traditional microlens array as the auxiliary microlens array.
Table 4. Contribution data table of main and auxiliary microlens arrays to mean square error MSE and peak signal-to-noise ratio PSNR
4.2 Influence of the main microlens array scale on the quality of restored images
Figure 9 shows that when we choose the object distance as d = 300 m and the focal length of the microlens is the same, image restoration is performed on the target with the size of 1 m using three main microlenses of different scales. Table 5 shows the MSE and PSNR values of the restored images after the main microlens arrays of different scales restore the image of the target. The larger the PSNR is, the better is the imaging quality; the smaller the mean square error is, the better is the imaging quality. The simulation data results in the table show that the internal structure scale of the main microlens array becomes larger, the mean square error value becomes smaller, the peak signal-to-noise ratio value increases and the restored image quality improves gradually. The simulation results show that the larger the size of the main microlens array is, the more DC component information and low-frequency information is emitted by the acquisition target, the better is the imaging quality.
Fig. 9. Image restoration results of three scales of primary microlens arrays. (a) restored image of the main microlens array with an internal structure scale of 7×7; (b) restored image of the main microlens array with an internal structure scale of 9×9; (c) main microlens array with an internal structure scale of 11×11 restoration image of microlens array.
Table 5. PSNR and MSE of restored images with different main microlens array scales
In summary, the conclusion of simulation 4.1 shows that the main microlens array plays a major role in the contribution of the target imaging; moreover, the use of the main and auxiliary microlens array nested structures can improve the imaging quality. The conclusion of simulation 4.2 shows that, the greater the increase of the size of the main microlens array is, the more the DC information and low frequency information is sent by the acquisition target. In practical applications, we can use the main microlens and the auxiliary microlens arrays of appropriate sizes for nesting, which can improve the imaging quality of the target.
5. Conclusion
In the photon integrated detection imaging, the collection efficiency of the spatial frequency information emitted by the target determines the quality of the restored image. The traditional microlens array integrated on the photonic integrated circuit (PIC) interference imaging system can be used as a receiver for collecting spatial frequency information. However, there is still the problem of low efficiency of spatial frequency information collection due to the structure. In this paper, a microlens array based on the main and auxiliary nested structures is proposed. The simulation results show that the image restoration quality of the main-auxiliary nested microlens array is better than that of the traditional microlens array, The PSNR increased by 21.05%; the MSE value decreased by 29.39%. In practical applications, we can use the main micro-lens array and the auxiliary micro-lens array of appropriate size for nesting and use the main micro-lens array to collect the DC component and low-frequency information emitted by the target, the high-frequency component information of the target can be collected to improve the imaging quality of the target. In conclusion, the main-auxiliary nested structure microlens array can achieve high quality image restoration of the target. The focus of this work is the theoretical simulation verification. In the next stage, we are ready to realize the high-quality reconstruction of the target image by the photonic integrated interferometric imaging system based on the main and auxiliary nested microlens arrays through experiments. We have completed the selection of devices and the optimization of lenses. We have tried our best to contact merchants to purchase equipment and instruments for manufacturing chips. We complete the experiments on the theory as soon as possible and elaborate and explain in the next paper.
Funding
Natural Science Foundation of Shaanxi Province (2020JM-206); State Key Laboratory of Laser Interaction with Matter (SKLLIM2103); 111 Project (B17035).
Acknowledgments
The authors thank the optical sensing and measurement team of Xidian University for their help.
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 maybe obtained from the authors upon reasonable request.
References
1. K. Badham, R. L. Kendrick, D. Wuchenich, C. Ogden, G. Chriqui, A. Duncan, S. T. Thurman, T. Su, W. Lai, J. Chun, S. Li, G. Liu, and S. J. B. Yoo, “Photonic integrated circuit-based imaging system for SPIDER,” in CLEO Pacific Rim (Optical Society of America, 2017), paper #3–3C-2.
2. R. P. Scott, T. Su, C. Ogden, S. T. Thurman, R. L. Kendrick, A. Duncan, R. Yu, and S. Yoo, “Demonstration of a photonic integrated circuit for multi-baseline interferometric imaging,” in 2014 IEEE Photonics Conference (IPC) (IEEE, 2014), 1–2.
3. R. L. Kendrick, A. Duncan, C. Ogden, J. Wilm, D. M. Stubbs, S. T. Thurman, T. Su, R. P. Scott, and S. J. B. Yoo, “Flat panel space based space surveillance sensor,” in Advanced Maui Optical and Space Surveillance Technologies (AMOS) Conference (2013).
4. Richard L. Kendrick, Alan Duncan, Chad Ogden, Joe Wilm, and Samuel T. Thurman, Segmented planar imaging detector for electro-optic reconnaissance, (Google Patents, 2014).
5. P. Hariharan, Optical Interferometry (Academic2003).
6. J. W. Goodman, Statistical Optics (John Wiley & Sons, 2015).
7. J. Aufdenberg, A. Mérand, V. C. Du Foresto, O. Absil, E. Di Folco, P. Kervella, S. Ridgway, D. Berger, T. Ten Brummelaar, and H. McAlister, “First results from the CHARA Array. VII. Long-baseline interferometric measurements of Vega consistent with a pole-on, rapidly rotating star,” Astrophys. J. 645(1), 664–675 (2006). [CrossRef]
8. T. A. Brummelaar, H. A. McAlister, S. T. Ridgway, W. G. Bagnuolo Jr., N. H. Turner, L. Sturmann, J. Sturmann, D. H. Berger, C. E. Ogden, R. Cadman, W. I. Hartkopf, C. H. Hopper, and M. A. Shure, “First Results from the CHARA Array. II. A Description of the Instrument,” Astrophys. J. 628(1), 453–465 (2005). [CrossRef]
9. R. Petrov, F. Malbet, G. Weigelt, et al., “AMBER, the near-infrared spectro-interferometric three-telescope VLTI instrument,” Astron. Astrophys. 464(1), 1–12 (2007). [CrossRef]
10. J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. Bowers, N. Elias II, C. Hummel, K. Johnston, D. Buscher, J. H. Clark III, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The navy prototype optical interferometer,” Astrophys. J. 496(1), 550–571 (1998). [CrossRef]
11. T. Pearson and A. Readhead, “Image formation by self-calibration in radio astronomy,” Annu. Rev. Astron. Astrophys. 22(1), 97–130 (1984). [CrossRef]
12. National Astronautics and Space Administration, “About the Hubble space telescope,” https://www.nasa.gov/mission_pages/hubble/Story/index.html.
13. Hong Liu Ma, Kang Huang, Yong Liu, Hong Chang, Shou Qian Chen, and Wang Zhang, “Research on photonic integrated interferometric imaging mechanism and space spectrum coverage,” Opt. Lett. J. 41(4), 0411003 (2014).
14. Nai Guang Lv, Fourier Optics (Machinery Industry Press, Beijing, 2006), p. 44.
15. Tie Hui Su, Guang Yao Liu, Katherine E Badham, Samuel T. Thurman, Richard L. Kendrick, Alan Duncan, Danielle Wuchenich, Chad Ogden, Guy Chriqui, Shao Qi Feng, Jaeyi Chun, Wei Cheng Lai, and S. J. B. Yoo, “Interferometric imaging using Si3N4 photonic integrated circuits for a SPIDER imager,” Opt. Express 26(10), 12801–12812 (2018). [CrossRef]
16. Santosh Kumar, Lokendra Singh, and Sandip Swarnakar, “Design of One-Bit Magnitude Comparator Using Nonlinear Plasmonic Waveguide,” Plasmonics 12(2), 369–375 (2017). [CrossRef]
17. Chanderkanta Santosh Kumar and Angela Amphawan, “Design of parity generator and checker circuit using electro-optic effect of Mach–Zehnder interferometers,” Opt. Commun. 364, 195–224 (2016). [CrossRef]
18. Dalai G. Sankar Rao, Sandip Swarnakar, and Santosh Kumar, “Performance analysis of all-optical NAND, NOR, and XNOR logic gates using photonic crystal waveguide for optical computing applications,” Opt. Eng. 59(5), 057101 (2020). [CrossRef]
19. Santosh Kumar, Lokendra Singh, and Nan-Kuang Chen, “Design of All-Optical Universal Gates Using Plasmonics Mach-Zehnder Interferometer for WDM Applications,” Plasmonics 13(4), 1277–1286 (2018). [CrossRef]
20. Santosh Kumar, Ashish Bisht, Gurdeep Singh, and Angela Amphawan, “Implementation of 2-bit multiplier based on electro-optic effect in Mach–Zehnder interferometers,” Opt. Quantum Electron. 47(12), 3667–3688 (2015). [CrossRef]
21. G. Liu, D. Wen, Z. Song, and T. Jiang, “System design of an optical interferometer based on compressive sensing: an update,” Opt. Express 28(13), 19349–19361 (2020). [CrossRef]
