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Abstract
The Segmented Planar Imaging Detector for Electro-Optical Reconnaissance (SPIDER) is a small volume, lightweight, low energy consumption, and high-resolution system expected to replace traditional large aperture telescopes for long-distance detection. In this paper, an improved SPIDER system is proposed, which uses a multimode interference (MMI) coupler instead of an orthogonal detector, and successfully doubles the space spectrum coverage. We present a three-point configuring method to configure lenslets, calculate spatial spectrum values from the output currents obtained by MMI. By comparing the performance of the MMI-SPIDER and SPIDER systems through simulations, we demonstrate that the former has more complete spatial spectrum coverage, resulting in better image restoration quality.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Long-distance detection is a challenging task that requires high-resolution imaging systems. In recent years, the development of space optical systems has focused on achieving large fields of view, high image quality, and compact size. However, Traditional optical telescopes have limited spatial resolution due to the Rayleigh and diffraction effects. One way to overcome these effects and improve resolution is to increase the aperture size, this also results in increased volume, weight, energy consumption, and system complexity, which limit its range of applications. For example, the James Webb Space Telescope (JWST) has an aperture diameter of 6.5 m, a weight of 6.2 Mg, and a power consumption of 2 kw [1].
Synthetic aperture technology provides a solution for large imaging systems by using multiple sub-apertures arranged to interfere and form an equivalent system with a large effective aperture. The resolution of the resulting imaging is proportional to the longest baseline formed between sub-apertures, overcoming the limitation of a single aperture size. One example of this technology is the Segmented Planar Imaging Detector for Electro-Optical Reconnaissance (SPIDER), proposed by Lockheed Martin in 2013 [2,3]. Figure 1 illustrates the Payload Design Concept and Function diagram of SPIDER, which is a novel imaging technology based on photonic integrated circuits (PICs) and synthetic aperture technology. The top layer consists of a radial array of lenslets covered with a shield plate that eliminates stray light. Below each axial lenslet array is a PIC card, all of which are radially arranged. A PIC card comprises couplers such as optical delay lines, phase modulators, arrayed waveguide gratings, and optical couplers, followed by optical-to-electrical converters and photo detectors. The signal is output as an electrical signal and processed. The arrangement of the lenslet array and the PIC arrays primarily serves to collect as much spatial spectrum information from the target as possible [4]. Compared to traditional telescopes, SPIDER reduces volume, mass, and power consumption by a factor of 10-100 [5–8].
Fig. 1. SPIDER payload design concept and the function diagram. (a) Layer-by-layer concept of SPIDER [4]. (b) Head-to-tail baseline pairing method. (c) Schematic functional diagram.
In the SPIDER system, the use of lenslets to construct more reasonable baselines is crucial for improving the quality of the reconstructed image. As shown in Fig. 1(b), the system employs a Head-to-Tail pairing method. For a SPIDER array with 2N lenslets, only N baselines can be formed, resulting in a conversion efficiency of just 50%. However, increasing baseline coverage requires greater complexity as each baseline necessitates a set of transmission, interference, and measurement devices. To overcome these limitations, Gang Liu et al. proposed a compressive sensing-based CPCIT structure (CS-CPCIT) consisting of 2N lenslets [4], increasing the number of baselines from N to 2N-1 compared to SPIDER. Building upon this improvement, G. Liu et al. introduced CS-CPCIT + and SA-CPCIT to further increase the number of baselines to N × N and N × (2N-1) respectively [9,10]. However, none of these techniques can achieve imaging within a single shot as measurements must be performed sequentially. K. Cao et al. proposed a pseudo two-layer parity combination, while W. Zhang introduced an innovative lenslet array arrangement designed to optimize the system’s structure and significantly enhance the quality of reconstructed images. Nevertheless, these approaches also doubled the number of required interference devices, thereby increasing the processing complexity of the PIC [11,12]. M. Yao et al. introduced a lenslet array structure that combines coherent detection with traditional imaging, whereas Q. Yu et al. and W. Gao et al. dedicated their efforts to optimizing the placement structure of the lenslet array and proposed a range of solutions aimed at improving imaging quality. However, these efforts did not prioritize the effective optimization of the PIC's structure itself [13–15].
In this paper, we propose an improved SPIDER structure that replaces the system’s orthogonal detectors with 8 × 8 Multimode Interference (MMI) couplers. The effective baseline sampling is doubled while maintaining immediate imaging capabilities. The paper is structured as follows. Section 2 introduces the concept of the improved imaging system. Section 3 outlines two fundamental theories of the system, including the basic imaging principle: Van Cittert–Zernike theorem, and the N × N Multimode Interference (MMI) coupler. Section 4 presents the signal processing procedure of the MMI-SPIDER system in detail, and proposes a Three-Point Configuring Method to configure lenslets. Section 5 conducts numerical simulations to evaluate the system’s performance. Finally, Section 6 provides our conclusions.
2. Concept of MMI-SPIDER
The SPIDER imaging system utilizes lenslet arrays instead of traditional large aperture optical telescopes to sample the target in the Fourier domain. As shown in Fig. 1(c), a pair of lenslets capture and couple the optical signal into PIC, after detecting the output of the orthogonal detector, the spatial spectrum information corresponding to the baseline formed by this pair of lenslets can be obtained.
Figure 2 shows the conceptual diagram of the Multimode Interference couplers-based SPIDER system (MMI-SPIDER), comparing with the SPIDER system shown in Fig. 1, the lenslet array arrangement is similar in both systems, but MMI-SPIDER uses MMI couplers instead of orthogonal detectors to enable direct interference of more optical signals.
Fig. 2. (a) MMI-SPIDER concept. (b) Diagram of the signal transmission procedure of one group of lenslets.
In our design, each group of three lenslets (marked as 1,2,3) forms three different baselines: B12, B23, and B13, as shown in Fig. 2(b). The light collected by these lenslets is coupled into the optical waveguide at the focal plane and split into multiple wavelengths by an Arrayed waveguide grating (AWG). The optical signals are then input into the three ports of the MMI. The MMI output ports detect the currents and transmit it to the Digital Signal Processing unit.
3. Theoretical foundation
3.1 Van Cittert-Zernike theorem
In the process of passive light source interference imaging, the light source is generally an extended independent radiation light source with weak coherence. However, after the transmission of this extended light source for a distance, the coherence will increase with the transmission distance. The light source will change from being completely incoherent to partially coherent, and the coherence area will increase. This kind of light source cross-intensity propagation relationship can be described by Van Cittert-Zernike theorem [16,17]. As illustrated in Fig. 3, for an incoherent optical source with an intensity distribution $I({\xi ,\eta } )$, the mutual intensity $J({u,v} )$ between two points with coordinates ${Q_1}({{x_1},{y_1}} )$ and ${Q_2}({{x_2},{y_2}} )$ on the exit pupil plane at a distance z can be expressed as:
Here, $k = 2\pi /\lambda $ is a normalization constant, and the coordinates $I({\xi ,\eta } )$ lie within the optical source region. $\lambda $ represents the average incident light wavelength, $\mathrm{\Delta }x = {x_1} - {x_2}$ and $\mathrm{\Delta }y = {y_1} - {y_2}$ are the baseline vectors of two points on the exit pupil plane.
Equation (1) reveals the Fourier transform relationship between $J({u,v} )$ and $I({\xi ,\eta } )$. By acquiring the mutual intensities, the target image can be computed through inverse Fourier transform.
Since the Fourier transform of the real function is conjugate with respect to the origin:
the spectrum of $F({ - u, - v} )$ can be obtained by knowing the spectrum of $F({u,v} )$, so baseline $({\Delta x,\Delta y} )$ and baseline $({ - \Delta x, - \Delta y\textrm{}} )$ are actually equivalent.3.2 Theoretical model of N × N MMI
Bryngdahl discovered the self-imaging effect (SIE) in uniform dielectric waveguides in 1973, which enables a multi-mode waveguide to periodically image the input signal along the transmission direction [18,19]. Based on this effect, a flat waveguide-based Multimode Interference (MMI) coupler can be designed. This type of coupler has several advantages, such as small size, low loss, uniform optical splitting, large working bandwidth, and simple fabrication. Since the introduction of the MMI coupler, it has attracted considerable attention and has been applied in various components of silicon-based photonic integrated circuits [20–22].
The MMI coupler consists of an input access waveguide, a multimode interference section, and an output access waveguide. The optical signal incident from one or more input waveguides can form single or multiple image points on the output end face after passing through the multimode interference region, and these image points can be coupled to the output waveguide. The power and phase of the image point depend on the distribution of the input optical field and the size of the multimode interference region. By adjusting these parameters, the desired output optical field can be obtained. Typically, light from one input channel is distributed to one or more output channels.
For an N × N MMI coupler, when the device length satisfies the minimum length that can achieve multiple self-imaging, N self-images are formed at the output end, each with an intensity of $1/\sqrt N $ [23–25]. Although the coupler evenly distributes the incident optical signal's power to all output ports, the output signals possess different phases depending on the port.
The self-imaging properties of generalized N × N multimode interference couplers are derived in detail in [23–25]. The positions, amplitudes, and phases of the self-images are directly linked to the coupler's lengths and widths. Analytically solving the eigenmode superposition equation for any arbitrary length establishes a general formalism applicable to practical N × N couplers used in integrated optics, enabling the extraction of simple phase relations.
Considering a coupler with N inputs and N outputs, with ports evenly distributed as shown in Fig. 4, the width of the multimode interference region ${W_{MMI}}$ is crucial. We focus on the shortest length ($L = 3{L_\pi }/N$) that can produce N imaging points, where ${L_\pi }$ represents the beat length of the two lowest-order modes. The phase change of N image points when the optical signal is incident from the r channel and exits from the s channel can be expressed using the following equations [24,25]:
Fig. 4. Diagram of N × N MMI coupler. Inputs are numbered bottom-up with index r and outputs are numbered top-down with s.
So the input-output transmission equation of N × N MMI can be expressed as:
When multiple optical signals are incident simultaneously, their image points at the same position of the output end will interfere with each other. This interference can be modeled as a linear superposition of two image points. In this study, we detect the output port currents after multiple input optical signals are interfered and superimposed, and calculate the optical mutual intensities of the input ports based on this.
4. Analysis of MMI-SPIDER system
4.1 Signal processing procedure
In this study, we use 8 × 8 MMI as the interference device. Based on Eq. (2), for the N = 8 MMI, the phase relationship ${\varphi _{sr}}$ between its output and input waveguides is summarized in Table 1.
Table 1. Phase relationship between output and input
Specifically, in the proposed MMI-SPIDER setup, three lenslets are configured as one group to collect optical signals as the three inputs to the 8 × 8 MMI, while the remaining five input ports of the MMI are left unused. Figure 2(b) is a schematic of this process.
As shown in Fig. 5, a quasi-monochromatic incoherent extended light source is observed in plane ${\Sigma _1}$. The light field originating from ${\Sigma _1}$ is transmitted to ${\Sigma _2}$. Three lenslets in plane ${\Sigma _2}$ as receiving apertures, denoted as ${Q_1}({{x_1},{y_1}} )$, ${Q_2}({{x_2},{y_2}} )$ and ${Q_3}({{x_3},{y_3}} )$ respectively. The initial phases of ${Q_1}$, $\textrm{}{Q_2}$, and ${Q_3}$ are denoted as ${\emptyset _1}$, ${\emptyset _2}$ and ${\emptyset _3}$ respectively. Then the oscillations of the light can be mathematically represented as follows [26]:
As indicated in Table 1, distinct phase modulations are obtained with different combinations of input ports. Using combination of ports 1,3, and 7 as an example, the distributions of complex amplitude of output ports can be expressed as:
The current of each output port measured in an infinite long period of time is:
Therefore, the primary objective is to extract $J({{Q_1},{Q_2}} )$, $J({{Q_1},{Q_3}} )$ and $J({{Q_2},{Q_3}} )$ from Eq. (8). To accomplish this, we define:
Uniquely deterministic solutions for $J({{Q_1},{Q_2}} )$, $J({{Q_1},{Q_3}} )$, and $J({{Q_2},{Q_3}} )$ can be obtained by solving Eq. (10).
We provide an illustrative example that demonstrates the process of positioning and restoring a single point light source. In Fig. 5, considering a scenario where a point light source exists solely in the plane ${\Sigma _1}$. The parameters of the point light source are defined as follows: the amplitude is set to ${a_0} = 1$, the wavelength is $\lambda = 700\textrm{nm}$. The position coordinates of the point light source on the $\xi $ and $\eta $ axes are $({0.1\textrm{m},0} )$, and the distance between the two planes is $\textrm{z} = 500\textrm{m}$. Furthermore, we position the receiving apertures, denoted as ${Q_1}$, ${Q_2}$, and ${Q_3}$, on the x and y axes as (0, 0), (D, 0), and (3D, 0), respectively, with $\textrm{D} = 1\textrm{mm}$. Subsequently, the distributions of complex amplitude for ${Q_1}$, ${Q_2}$, and ${Q_3}$ can be expressed as:
Following the coupling of light into the multimode interference (MMI) device through the three receiving apertures, ${Q_1}$, ${Q_2}$, and ${Q_3}$, the simulation of output currents detected by eight output ports of MMI is depicted in Fig. 6.
Based on the output currents and utilizing Eq. (10), we can calculate the mutual intensities $J({{Q_1},{Q_2}} )$, $J({{Q_1},{Q_3}} )$, and $J({{Q_2},{Q_3}} )$. The corresponding spatial spectrum point coordinates in the spatial spectrum domain are determined as ($\textrm{D}/\lambda z$, 0), ($2\textrm{D}/\lambda z$, 0), and ($3\textrm{D}/\lambda z$, 0). Employing Eq. (2), the spectral information for six points, excluding the origin of coordinates (a real number which solely represents the brightness), can be derived. The specific values of these points and the result of performing an inverse Fourier transform is presented in Fig. 7.
Figure 7 demonstrates that with only three receiving apertures and one 8 × 8 Multimode Interference (MMI) coupler, three spectral spectrum points (6 points after conjugate symmetry) can be successfully recovered, basically realizing the positioning of the point light source.
4.2 Lenslet configuring method
Due to the conjugate symmetry of the spectrum as shown in Eq. (2), central symmetry should be avoided when designing the lenslet configuration. The arrangement of the lenslet array is depicted in Fig. 8, consisting of 39 interference arms with the same number of lenslets on each arm. Assuming that the lenslets are placed in close proximity, the distance between two adjacent lenslets is equal to the lenslet diameter d. The angle between two neighboring lenslets can be calculated as $\mathrm{\alpha } = 2\pi /p$, where $p = 39$ denotes the number of interference arms. Additionally, the radius of the circle enclosed by the inner layer of lenslets array can be expressed as follows:
As shown in Fig. 8, the 39 interferometer arms are numbered clockwise, there are a total of q lenslets on each arm, numbered in an outward to inward direction. The $n$th lenslet on the $m$th interference arm is denoted as $lens({m,n} )$, its abscissa is denoted as $x({m,n} )$ and its ordinate as $y({m,n} )$.
The position coordinates of the $n$th lenslet in the $m$th interference arm ($x({m,n} )$, $y({m,n} )$) can be express as follows:
We propose a lenslets combination method named the Three-Point Configuring Method with odd-even alternating, each lenslet is configured only once. The $n$th lenslet on the $m$th interference arm is denoted as $lens({m,n} )$. As illustrated in Fig. 9(a), when n is odd, $lens({m,n} )$ is combined with $lens({m + 1,n} )$ and $lens({m + 2,n} )$, resulting in the spatial spectrum point $({u,v} )$ distribution:
When n is even, as shown in Fig. 9(b), $lens({m,n} )$ is combined with $lens({m + 12,n} )$ and $lens({m + 25,n} )$, also resulting in the acquisition of three spatial spectral points, but with a different spatial spectrum point $({u,v} )$ distribution given by:
In this setup, the optical signals collected by the lenslets are transmitted through the optical waveguide as the three inputs of an 8 × 8 MMI coupler. The mutual intensities of the optical signals can be parsed utilizing Eq. (10). Due to the asymmetry of the lenslets configuration, six points of spatial spectrum values can be obtained.
5. Numerical simulation
This section presents a comparison of the imaging quality between MMI-SPIDER and SPIDER systems that have the same number and layout of lenslets. Table 2 summarizes the system parameters used for the simulations. To examine the effects of the improved system structure and the novel lenslet configuration method on the imaging system, this study employs single wavelength light as a simplifying assumption and sets the number of spectral bins to 1. The lenslet array consists of 39 arms with 60 lenslets each in a 2D arrangement, as Fig. 10(a) illustrates. The SPIDER system uses the Head-to-Tail pairing method and creates 30 baselines per arm with lengths from 1 mm to 59 mm. The MMI-SPIDER system applies the Three-Point Configuring Method shown in Fig. 9. For even values of n, as Fig. 9(b) sketches, the three configured lenslets form three baselines, in the shape of an equilateral triangle. When $n = 2$, 39 lenslets generate 39 longest baselines of 112.6 mm, when $n = 60$, the innermost lenslets produce 39 shortest baselines of 10.4 mm. For odd values of n, as Fig. 9(a) shows, the three lenslets create shorter baselines to compensate for the low-frequency parts when n is even. Figure 10(b) and Fig. 10(c) display the spatial spectrum coverage of the SPIDER and MMI-SPIDER systems, respectively. In our MMI-SPIDER design, $39 \times 60 = 2340$ lenslets yield a total of 2340 baselines, which is twice as many as the SPIDER system. The Three-Point Configuring method that we propose extends the length of the longest baseline and enhances low-frequency coverage.
Fig. 10. Schematic diagram of lenslet arrangement and uv-spatial spectrum coverage: (a) lenslet array :39 interference arms with 130 lenslets on each arm. (b) uv-spatial spectrum coverage of the MMI-SPIDER system. (c) uv-spatial spectrum coverage of the SPIDER system.
Table 2. Designed system simulation parameters
We use three discrete models of 256 × 256 pixels as test images: the Resolution Board, Katrina-2005, Ranger 7-Moon and Crab Pulsar [27], shown in Fig. 11(a), Fig. 12(a), Fig. 13(a) and Fig. 14(a), respectively. The images reconstructed by the MMI-SPIDER and SPIDER systems are shown in Fig. 11(b), Fig. 12(b), Fig. 13(b), Fig. 14(b) and Fig. 11(c), Fig. 12(c), Fig. 13(c), Fig. 14(c), respectively. The coordinates represent the pixels of the 256 × 256 image. The quality of the images is evaluated based on both the peak signal-to-noise ratio (PSNR) and the structural similarity index (SSIM) [28].
Fig. 11. Simulation results. (a) Resolution board. (b) Dirty image restored by MMI-SPIDER. (c) Dirty image restored by SPIDER.
Fig. 12. Simulation results. (a) Test image: Katrina-2005. (b) Dirty image restored by MMI-SPIDER. (c) Dirty image restored by SPIDER.
Fig. 13. Simulation results. (a) Test image: Ranger 7-Moon. (b) Dirty image restored by MMI-SPIDER. (c) Dirty image restored by SPIDER.
Fig. 14. Simulation results. (a) Test image: Crab Pulsar. (b) Dirty image restored by MMI-SPIDER. (c) Dirty image restored by SPIDER.
The MMI-SPIDER system covers the spatial spectrum more comprehensively than the SPIDER system, which leads to better image quality in the MMI-SPIDER imaging system. This is confirmed by the evaluation results in Table 3, where the PSNR and SSIM values of the reconstructed images are significantly improved in the MMI-SPIDER system compared to the SPIDER system.
Table 3. Evaluation results (unit: dB)
6. Conclusion
In this paper, we proposed an improved SPIDER system utilizes 8 × 8 MMI couplers. In the SPIDER system, two fixed lenslets are paired to measure the mutual intensity of the two received light beams, which corresponds to one spatial spectral point according to the Van Cittert-Zernike theorem. In the MMI-SPIDER system, we use three inputs in the 8 × 8 MMI and recover the mutual intensities of the three input light beams from the eight output currents. By configuring three lenslets as a group, spectral information of three points can be obtained, and the efficiency of spatial spectrum sampling is doubled compared to the original system. We propose the Three-Point Configuring method with odd-even alternating, which increases the length of the longest baseline and enhances the low-frequency part compared to the head-to-tail pairing method of the SPIDER system. The improved system provides twice the number of spatial spectrum coverage points, and we compare and verify the quality of the dirty map recovered by the MMI-SPIDER and SPIDER system through numerical simulation. The simulation results show that the PSNR and SSIM of the dirty map are improved in the MMI-SPIDER system compared to the SPIDER system.
Acknowledgments
We sincerely thank the editors and anonymous reviewers for their contributions to this paper.
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.
Supplemental document
See Supplement 1 for supporting content.
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