Quantitative analysis of segmented planar imaging quality based on hierarchical multistage sampling lens array

Wei Ping Gao; Xiao Rui Wang; Lin Ma; Ying Yuan; Dan Feng Guo

doi:10.1364/OE.27.007955

1. Introduction

Recently, many advanced ultra-light, ultra-thin imaging systems have been developed [1,2]. As a typical representative, Segmented Planar Imaging Detector for Electro-optical Reconnaissance (SPIDER) offers significantly reduction in size, weight, and power consumption compared with the Very Long Baseline Interferometer (VLBI) [3], the Navy Precision Optical Interferometer (NPOI) [4], Very Large Telescope Interferometer (VLTI) [5] and Georgia State University’s Center for High Angular Resolution Astronomy (CHARA) [6] by using photonic integrated circuits and replacing the traditional lens with lenslet array to simultaneously form many interferometer baselines.

The concept of the segmented planar imaging based on the principle of the interference is proposed by Lockheed Martin of Segmented Planar Imaging Detector for Electro-optical Reconnaissance [7]. SPIDER is a small interferometric imager that samples the object being imaged in the frequency domain by substituting the traditional optical telescope with dense interference arrays. Light from an object is coupled into the waveguide through the paired lens array. Then the interference image is formed by the baseline interference. The intensity distribution of the object can be reconstructed by measuring the interference image [8]. Many researches have been done in recent years. Duncan et al. researched the system structure and design principle of SPIDER [7,9]. The working principle of PIC was demonstrated by Thurman et al [10]. Moreover, Qiuhui Chu et al verified that the imaging quality could be effectively improved by adjusting the Nyquist sampling density, optimizing the baseline pairing method and increasing the spectral channel of demultiplexer [11]. However, the reconstructed image quality of the segmented planar imaging system still has defects such as poor contrast, loss of medium and low frequency information, and low resolution. It is necessary to increase system sampling to improve image quality. This has become a key issue in developing and optimizing the performance of ultra-thin, high-resolution segmented planar imaging system.

According to the above mentioned problems of the segmented planar imaging system, a hierarchical multistage sampling lens array is proposed, which can increase the collection of medium and low frequency information and effectively improve the imaging quality. Meanwhile, a full-chain signal level model of the segmented planar imaging system is established considering the fill factor of hierarchical multistage sampling lens array and the wavelength spacing of the arrayed waveguide grating in PICs. Research results will provide theoretical supports for high performance segmented planar imaging system development.

2. Novel architecture of segmented planar imaging system

2.1 Structure of segmented planar imaging system

The structure of the segmented planar imaging system is shown in Fig. 1 [12]. It consists of lens array, PICs, digital signal processor (DSP) and image reconstruction modules. PICs contains optical wave-guide array, arrayed waveguide grating and balanced four-quadrature detection arrays (BFQDA). Because the original wheel-type lens array samples less medium and low frequency information and the imaging quality of the segmented planar imaging system is poor, so a hierarchical multistage sampling lens array is proposed to replace the original wheel-type lens array.

Fig. 1 The schematic diagram of the segmented planar imaging system.

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2.2 Hierarchical multistage sampling lens array

Aiming at solving the problem of low sampling of low and medium frequency information, hierarchical multistage sampling lens arrays of the segmented planar imaging system have been proposed which are depicted in Fig. 2. Figure 2(a) is the wheel-type lens array and Figs. 2(b) and 2(c) are the hierarchical multistage sampling lens array respectively. The hierarchical multistage sampling lens arrays are proposed based on the wheel-type lens array. The radial lens of the wheel-type lens array are used as the long radial lens, and the middle radial lens are compensated on both sides. As shown in Fig. 2(b), the innermost layer of the long radial lens and the innermost layer of the middle radial lens constitute the first layer of concentric circle. The innermost layer of the middle radial lens and the outermost layer of the long radial lens constitute the second layer of concentric circle. The long radial lens collect low, medium and high frequency information along its direction, and the middle radial lens collect low and medium frequency information along its direction. Thereby the hierarchical multistage sampling lens array Ι is formed. Similarly, according to the size of the actual lens array, the short radial lens can be continuously compensated on both sides of the middle radial lens, which can be seen in Fig. 2(c). The innermost layer of the long radial lens and the innermost layer of the middle radial constitute the first layer of concentric circle. The innermost layer of the middle radial lens and the innermost layer of the short radial lens constitute the second layer of concentric circle. And the innermost layer of the short radial lens and the outermost layer of the long radial lens form the third layer concentric circle. The short radial lens collect low frequency information along its direction. Therefore, the hierarchical multistage sampling lens array П is formed. N times of compensation form N + 1 concentric circles, and N-stage frequency information is collected. Hierarchical multistage sampling lens arrays improve the imaging quality of the segmented planar imaging system by increasing low and medium frequency information collection.

Fig. 2 The hierarchical multistage sampling lens arrays of the segmented planar imaging system. (a) The wheel-type lens array. (b) The hierarchical multistage sampling lens array Ι. (c) The hierarchical multistage sampling lens array П.

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2.3 Baseline pairing method of lens array

Taking the hierarchical multistage sampling lens array shown in Fig. 2(c) as an example, the lens array is composed of long, medium and short radial lens. The number of the lenslet on long, middle and short radial lens are 30, 22, 10 respectively. The pairing method of the long radial lens adopt the second generation pairing method. The pairing method is (1,30), (2,24), (3,6), (4,17), (5,7), (8,25), (9,10), (18,23), (19,29), (20,28), (21,27), (22,26). The baseline lengths are: 1, 2, 3, 4, 5, 6, 8, 10, 13, 17, 22, 29. The pairing method of the middle radial lens is (1,18), (2,4), (3,10), (5,17), (6,9), (7,8), (11,16), (12,22), (13,21), (14,20), (15,19) and mainly collect low and medium frequency information. The baseline lengths are: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 17. The pairing method of the short radial lens is (1,3), (2,5), (4,9), (6,10), (7,8) and mainly collect low frequency information. The baseline lengths are: 1, 2, 3, 4, 5. Compared with the wheel-type lens array and its pairing method, the hierarchical multistage sampling lens array and its pairing method increase the number of medium and low frequency information collection along its direction.

3. Full-chain imaging modeling of segmented planar imaging system

By analyzing the signal transmission process of the segmented planar imaging system, considering the lens array transmission, photonic integrated circuit transmission, the processing of the interference signal and image reconstruction process, the following full-chain imaging model is established.

The signal transmission process of the segmented planar imaging system is shown in Fig. 3. Light collected for the same field point with different lenslets. $b_{1}$ and $b_{n}$ pair form the longest baseline. $b_{i}$ and $b_{i + 1}$ pair form the shortest baseline. As an example of $b_{1}$ and $b_{n}$ paired baseline, light is coupled into the waveguide through lens and propagate into array waveguide grating $W_{1}, W_{n}$ . Then the light is spatially separated into different wavelength channels of $λ_{1}, λ_{2}$ and $λ_{3}$ , and then couple into phase shifter. For example, the output signals of $λ_{1}$ from phase shifter $P_{1}$ are $E_{1}$ and $E_{2}$ respectively. Two signals from phase shifter are then coupled into a balanced quadrature detector to process interference signals and the outputs are $I$ and $Q$ . The transfer process can be divided into following steps:

Fig. 3 Full-chain imaging model of the segmented planar imaging system.

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3.1 Modeling of lens array transmission

Suppose the distance between adjacent lenslets is D and the diameter of the lenslet is d. As shown in Fig. 4, the fill factor of the lens array is defined as

Fig. 4 The fill factor of the lens array.

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F F = \frac{d}{D}

For a single lenslet, considering the absence of aberrations, its pupil function can be expressed by:

P (x, y) = c i r c (\frac{\sqrt{x^{2} + y^{2}}}{\frac{d}{2}})

where

d

is the diameter of the lenslet. The amplitude on the focal plane can be expressed as:

\begin{array}{l} E (x_{i}, y_{i}) = \frac{1}{{(λ f)}^{2}} \int \int_{- \infty}^{\infty} c i r c (\frac{\sqrt{x^{2} + y^{2}}}{\frac{d}{2}}) e^{- i 2 π (\frac{x_{i}}{λ f} x + \frac{y_{i}}{λ f} y)} d x d y \\ = (\frac{π d^{2}}{4 λ f}) [\frac{2 J_{1} (π d r / λ f)}{π d r / λ f}] \end{array}

r = {(x^{2} + y^{2})}^{\frac{1}{2}}

where

f

is the focus of the lenslet,

λ

is the incident wavelength.

3.2 Modeling of array waveguide grating transmission

Array waveguide grating consists of input waveguides, output waveguides and array waveguides connecting the two free propagation slabs (FPS). Assume that the number of arrayed waveguides is $M$ and the adjacent waveguide spacing is $l$ . Light is coupled into waveguides through the lens array and then propagates into array waveguide grating. The normalized far-field distribution of the light input from the ith input channel waveguide through the array waveguide grating in the output slab waveguide is:

E = E_{0} \frac{E_{0} (0) + 2 \sum_{k = 1}^{M} E_{0} (k Δ θ) \cos (k φ)}{E_{0} (0) + 2 \sum_{k = 1}^{M} E_{0} (k Δ θ)}

Δ θ = \frac{m n_{g}}{n_{s} n_{c} l} Δ λ

where

E_{0}

is the far field distribution of the normalized center waveguide, the value of

k

ranges from 1 to

M

,

E_{0} (k Δ θ)

is the amplitude of light received by the kth array of waveguide,

φ

is the phase difference between adjacent arrayed waveguides,

m

is the diffraction order,

Δ λ

is the wavelength spacing of the array waveguide grating,

n_{c}

is the effective refractive index of the arrayed waveguide,

n_{s}

is the effective refractive index of a rectangular waveguide,

n_{g}

is the group refractive index.

3.3 Modeling of interference signal processing

The two signals $E_{1}$ and $E_{2}$ from phase shifter $P_{1}$ are then coupled into a balanced quadrature detector to process interference signals and the outputs are as follows:

\begin{array}{l} E_{A} = E_{1} + E_{2} \\ E_{B} = E_{1} - E_{2} \\ E_{C} = E_{1} + j E_{2} \\ E_{D} = E_{1} - j E_{2} \end{array}

Based on above equations, the output signals of $I$ and $Q$ can be expressed as:

\begin{array}{l} I = I_{1} - I_{2} = E_{A} \times E_{A}^{*} - E_{B} \times E_{B}^{*} = 4 E_{1} E_{2} \cos Δ φ \\ Q = I_{3} - I_{4} = E_{C} \times E_{C}^{*} - E_{D} \times E_{D}^{*} = 4 E_{3} E_{4} \sin Δ φ \end{array}

where

Δ φ

is the phase difference of input signals

E_{1}

and

E_{2}

.

3.4 Modeling of coherence intensity photoelectric detection

Light collected for the same field point with different lenslets is combined to create fringes. These digitized signals of $I$ and $Q$ are then Fourier-transformed to the frequency domain. The mutual coherent intensity $V (x_{1}, y_{1}; x_{n}, y_{n})$ of the paired lenslet $b_{1} (x_{1}, y_{1})$ and $b_{n} (x_{n}, y_{n})$ is:

V (x_{1}, y_{1}; x_{n}, y_{n}) = \iint I (ξ, η) ρ (ξ, η) \exp [- 2 π j (u ξ + v η)] d ξ d η

(u, v) = \frac{1}{λ z} (Δ x, Δ y)

where

(u, v)

is the spatial frequency coordinate,

ρ (ξ, η)

is the lenslet coupling efficiency projected onto the object plane,

z

is the object distance.

The frequency component of the spectrum of $I (ξ, η)$ at the spatial frequency $(u, v)$ can be expressed as:

μ (x_{1}, y_{1}; x_{n}, y_{n}) = \frac{\iint I (ξ, η) ρ (ξ, η) \exp [- 2 π j (u ξ + v η)] d ξ d η}{\iint I (ξ, η) d ξ d η}

3.5 Image reconstruction model

For a distributed source, Van Cittert-Zernike theorem states that the amplitude and phase of the interferometric fringes measure the Fourier transform of the brightness distribution. The brightness distribution of the source can be obtained by inverse Fourier transform the fringes.

4. Simulation results and analysis

In order to verify the correctness of the full-chain model, considering the spatial frequency distribution, the object point spread function (PSF) and the imaging results of the segmented planar imaging system, the imaging process of the novel integrated imaging system and the original integrated imaging system is simulated and analyzed. System parameters used for simulations are given in Table 1.

Table 1. System parameters used for simulations

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Based on the baseline pairing method and the Eqs. (9) and (10), the spatial frequency distributions of the wheel-type and hierarchical multistage sampling lens array are depicted in Fig. 5. The simulation results showed that the sampling points of the hierarchical multistage sampling lens array are significantly more than the wheel-type lens array. The radius of continuous sampling of the wheel-type lens array is $r = 0.164$ , while the hierarchical multistage sampling lens array Ι and П are $r = 0.24$ and $r = 0.33$ respectively. Due to the same longest baseline, the maximum sampling radius is $R = 0.785$ .

Fig. 5 The spatial frequency distributions of the segmented planar imaging system with (a) wheel-type and (b), (c) hierarchical multistage sampling lens array.

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The object PSF is then the inverse Fourier transform of spatial frequency distributions which can be seen in Fig. 6. The width of the object PSF of the segmented planar imaging system with wheel-type lens array and hierarchical multistage sampling lens array are 1.30m, 1.28m and 1.28m. The width of the object PSF of the segmented planar imaging system with hierarchical multistage sampling lens array are narrower than the wheel-type lens array. This phenomenon indicates that the hierarchical multistage sampling lens arrays reduce the width of the object PSF of the segmented planar imaging system, thereby improving the imaging performance of the system.

Fig. 6 The object PSF of the segmented planar imaging system with wheel-type and hierarchical multistage sampling lens array.

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A MATLAB model of the segmented planar imaging system based on above three lens arrays is constructed to simulate its imaging capabilities. Figure 7(a) shows an example of a general source. Figures 7(b)-7(d) are the imaging results of the imaging system with wheel-type and hierarchical multistage sampling lens arrays under the condition of the ideal waveguide coupling efficiency. As is clear can be seen that the imaging results of the system with hierarchical multistage sampling lens array have more detailed information than with wheel-type lens array. Because the system with hierarchical multistage sampling lens array П collects more low and medium frequency information in all directions, its image edge and texture is clearer than with other lens arrays which can be seen in the partial enlargement of the image. Moreover, the hierarchical multistage sampling lens array can improve the image contrast. Due to the same longest interferometer baseline, the system resolution is approximately given by

R = \frac{\bar{λ} z}{B_{\max}} = 1.29 m

where

B_{\max}

is the longest interferometer baseline,

\bar{λ}

is the mean wavelength and

z

is object distance.

Fig. 7 (a) Light intensity distribution and partial enlargement of source. (b) The imaging results and partial enlargement of the segmented planar imaging system with wheel-type lens array. (c) and (d) The imaging results and partial enlargement of the segmented planar imaging system with hierarchical multistage sampling lens array.

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5. Key parameters optimization for novel architecture

In the full-chain imaging model, many factors such as the fill factor of the lens array and the wavelength spacing of the arrayed waveguide grating will affect the imaging quality of the system. Now, for the above two parameters, the imaging quality of the imaging system is analyzed from three aspects: system spatial frequency distribution, object PSF and imaging results.

5.1 Fill factor of hierarchical multistage sampling lens array

In theory, the fill factor of the lens array is 1. Considering the error in the actual manufacturing process, the influence of different fill factor lens arrays on the imaging quality of the segmented planar imaging system is analyzed. The spatial frequency distributions of the segmented planar imaging system with hierarchical multistage sampling lens array with fill factors of 0.3, 0.5, 0.7, 0.9, 0.93, 0.95, 0.97 and 1 are shown in Fig. 8. As the fill factor of lens array increases, the frequency sampling point gradually increases.

Fig. 8 The spatial frequency distribution of the segmented planar imaging system with lens array with fill factors of 0.3, 0.5, 0.7, 0.9, 0.93, 0.95, 0.97 and 1.

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The object PSF of the segmented planar imaging system with hierarchical multistage sampling lens array with different fill factors can be seen in Fig. 9. The width of the object PSF of the segmented planar imaging system with lens array fill factor of 0.3, 0.5, 0.7, 0.9, 0.93, 0.95, 0.97 are 4.3m, 2.65m, 1.65m, 1.26m, 1.20m, 1.14m, 1.14m, 1.12m. As the fill factor of the lens array increases, the width of the object PSF gradually decrease and the imaging performance of the system is enhanced. The imaging results are shown in Fig. 10. With the increasing of the fill factor of lens array, imaging detail information increases and image quality is enhanced. Therefore, the imaging performance of the segmented planar imaging system can be improved by increasing the fill factor of the lens array. When the fill factor of the lens array exceeds 0.95, the image quality is substantially the same as the lens array with a fill factor of 1. As a consequence, the slight error(5%) of lens array fill factor has little effect on the image quality of the segmented planar imaging system and allow slight error(5%) in the preparation process.

Fig. 9 The object PSF of the segmented planar imaging system with hierarchical multistage sampling lens array with fill factors of (a) 0.3, 0.5, 0.7 and 0.9, (b) 0.93, 0.95, 0.97 and 1.

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Fig. 10 The imaging results of the segmented planar imaging system with hierarchical multistage sampling lens array with fill factors of 0.3, 0.5, 0.7, 0.9, 0.93, 0.95, 0.97 and 1.

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5.2 Wavelength spacing of arrayed waveguide grating

Arrayed waveguide grating is one of the most important devices in wavelength divided multiplexing technology. The imaging quality of the segmented planar imaging system is influenced by the wavelength spacing of arrayed waveguide grating. Adjusting the wavelength spacing of arrayed waveguide grating can optimize the imaging quality by improving the mutual coherent intensity. Based on the hierarchical multistage sampling lens array П, the imaging process of 5nm, 10nm, 20nm, 40nm and 64nm wavelength spacing of arrayed waveguide grating in segmented planar imaging system are simulated. The spatial frequency distributions are depicted in Fig. 11. The spatial frequency distribution consists of a continuously sampled solid circle and a discretely sampled concentric ring. By comparing the distribution of spatial frequency may reach a conclusion that with the increasing of the wavelength spacing of arrayed waveguide grating, the mutual coherent intensity and the number of effective frequency sampling points are increased. The radius of the solid circle and maximum ring in spatial frequency distribution of the segmented planar imaging system with the wavelength spacing of 5nm, 10nm, 20nm, 40nm and 64nm are $r = 0.33$ and $R = 0.785$ . However, with the increasing of wavelength spacing, the mutual coherent intensity and effective frequency sampling radius $r'$ with mutual coherent intensity exceeding 0.3 are decreased which can be seen in Table 2. In conclusion, as the wavelength spacing increases, the number of the effective sampling points for high frequency information decreases. Compared with the arrayed waveguide grating with wavelength spacing of 5nm, the effective sample radio referring to r' divided by R of arrayed waveguide grating with wavelength spacing of 10nm, 20nm, 40nm and 64nm are 1, 0.9, 0.66 and 0 respectively.

Fig. 11 The spatial frequency distribution of the segmented planar imaging system with arrayed waveguide grating with wavelength spacing of (a) 5nm, (b) 10nm, (c) 20nm, (d) 40nm and (e) 64nm.

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Table 2. Partial performance parameters of the segmented planar imaging system with arrayed waveguide gratings with different wavelength spacing

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The object PSF of the segmented planar imaging system with arrayed waveguide grating with different wavelength spacing can be seen in Fig. 12. The width of the PSF of the segmented planar imaging system with wavelength spacing of 5nm, 10nm, 20nm, 40nm and 64nm are 1.14m, 1.14m, 1.30m, 1.52m and 2.60m. As the wavelength spacing of the arrayed waveguide grating increases, the width of the object PSF gradually increases and the imaging performance of the system decreases. The imaging results of the system are shown in Fig. 13. The imaging performance of the segmented planar imaging system can be improved by reducing the wavelength spacing of the arrayed waveguide grating. Moreover, imaging details are gradually reduced because the effectively sample radio of imaging system with wavelength spacing of 20nm, 40nm and 64nm are gradually reduced compared with the arrayed waveguide grating with wavelength spacing of 5nm, and the high frequency sampling information is reduced. Due to the same effectively sample radio of 5nm and 10nm wavelength spacing, the imaging quality of 5nm and 10nm wavelength spacing is basically identical. Therefore, arrayed waveguide grating with wavelength spacing of 10nm is the optimal choice for the system.

Fig. 12 The object PSF of the segmented planar imaging system with arrayed waveguide grating with wavelength spacing of (a) 5nm and 10nm. (b) 5nm, 20nm, 40nm and 64nm.

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Fig. 13 The imaging results of the segmented planar imaging system with arrayed waveguide grating with wavelength spacing of (a) 5nm, (b) 10nm, (c) 20nm, (d) 40nm and (e) 64nm.

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6. Conclusion

In this paper, a hierarchical multistage sampling lens array is proposed and a full-chain signal level model of the segmented planar imaging system is established to improve the imaging quality. Based on the theory model of the imaging process of the segmented planar imaging system, the fill factor of the hierarchical multistage sampling lens array and the wavelength spacing of the arrayed waveguide grating in PICs of the system are quantitatively analyzed. The results show that the imaging quality of the hierarchical multistage sampling lens array is optimal compared with the wheel-type lens array because more medium and low frequency information have been obtained. With the increasing of the fill factor of lens array, imaging detail information increases and image quality is enhanced. Moreover, the imaging results show that 5% manufacturing errors are allowed in the manufacturing process of the lens array. The effectively sample radio of arrayed waveguide grating with wavelength spacing of 20nm, 40nm and 64nm are gradually reduced compared to the arrayed waveguide grating with wavelength spacing of 5nm. The results show that arrayed waveguide grating with wavelength spacing of 10 nm is the optimal choice for the system. The research results provide theoretical foundation for integral optical imaging system development.

Funding

National Natural Science Foundation of China (NSFC) (61377007, 61575152, 61775174); National Defense Basic Scientific Research program of China (JCKY2016208B001)

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Parameters	Symbols	Values
Range of wavelength	λ	380-700nm
Object distance	z	250km
Lenslet Diameter	d	3.6mm
Longest interferometric baseline	B_max	104.4mm
Number of the lenslet per column	N	30
Number of PIC	p	19
Channel number of arrayed waveguide grating Wavelength spacing of arrayed waveguide grating	SC Δλ	16 20nm
Optical path difference	Δ	7.2μm
Fill factor of the lens array	FF	1

Wavelength spacing	Mutual coherent intensity	Effective frequency sampling radius $r'$
5nm	0.9~1	0.785
10nm	0.67~0.96	0.785
20nm	0.11~0.86	0.71
40nm	0.01~0.52	0.52
64nm	0.06~0.22	0

Parameters	Symbols	Values
Range of wavelength	λ	380-700nm
Object distance	z	250km
Lenslet Diameter	d	3.6mm
Longest interferometric baseline	B_max	104.4mm
Number of the lenslet per column	N	30
Number of PIC	p	19
Channel number of arrayed waveguide grating Wavelength spacing of arrayed waveguide grating	SC Δλ	16 20nm
Optical path difference	Δ	7.2μm
Fill factor of the lens array	FF	1

Wavelength spacing	Mutual coherent intensity	Effective frequency sampling radius $r'$
5nm	0.9~1	0.785
10nm	0.67~0.96	0.785
20nm	0.11~0.86	0.71
40nm	0.01~0.52	0.52
64nm	0.06~0.22	0

Quantitative analysis of segmented planar imaging quality based on hierarchical multistage sampling lens array

Abstract

1. Introduction

2. Novel architecture of segmented planar imaging system

2.1 Structure of segmented planar imaging system

2.2 Hierarchical multistage sampling lens array

2.3 Baseline pairing method of lens array

3. Full-chain imaging modeling of segmented planar imaging system

3.1 Modeling of lens array transmission

3.2 Modeling of array waveguide grating transmission

3.3 Modeling of interference signal processing

3.4 Modeling of coherence intensity photoelectric detection

3.5 Image reconstruction model

4. Simulation results and analysis

5. Key parameters optimization for novel architecture

5.1 Fill factor of hierarchical multistage sampling lens array

5.2 Wavelength spacing of arrayed waveguide grating

6. Conclusion

Funding

References

Cited By

Figures (13)

Tables (2)

Equations (12)

Optics Express