Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Quantitative analysis and optimization design of the segmented planar integrated optical imaging system based on an inhomogeneous multistage sampling lens array

Weiping Gao, Ying Yuan, Xiaorui Wang, Lin Ma, Zhenshun Zhao, and Hang Yuan

Open Access Open Access

Abstract

The non-uniformity of the lens array in the segmented planar integrated optical imaging system has not been studied. We design an inhomogeneous multistage sampling lens array based on a hierarchical multistage sampling lens array of the segmented planar integrated optical imaging system by changing the radius of the lenslet for frequency-selective sampling to optimize the design of the UV spatial frequency distribution. The signal transfer model of the proposed imaging system was established considering the radius of the lenslet and key waveguide parameters. Simulation results show that the proposed system produces better imaging quality owing to the selective sampling of frequency. Compared with the homogeneous multistage sampling lens array (radius of 1.8 mm), the optimal radius parameters of the inhomogeneous multistage sampling lens array are given for the same longest lens array length. Detailed calculations of the parameters of the waveguide connected to the lens array were performed. Our results provide a theoretical basis for the optimal design and process of a new segmented planar integrated optical imaging system.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The choice of the diameter of the circular aperture is very important in optical remote sensing imaging systems. The quest for high resolution in astronomy leads to a large aperture because the resolution is proportional to the wavelength over the diameter of a circular aperture. Unfortunately, the diameter of the primary mirror for space telescopes is limited by the scaling laws of manufacturing costs [1]. The manufacturing and processing of large-diameter mirrors is very difficult, and the processing cost is proportional to the square of the diameter [2]. In addition, the large diameter makes it difficult to meet the requirements of existing mechanical support structures. Therefore, improving the imaging resolution without increasing the aperture of a single telescope is urgently needed by astronomers.

Interferometric techniques, using superimposed electromagnetic waves to extract source information, are one of the main technologies used to achieve high-resolution imaging [3,4]. These include the Very Long Baseline Interferometer (VLBI) [5], the Navy Precision Optical Interferometer (NPOI) [6], Very Large Telescope Interferometer (VLTI) [7], and the Georgia State University’s Center for High Angular Resolution Astronomy (CHARA) [8], which use meter-class telescopes to collect light and have interferometer baselines on the order of 10-100 meters. Segmented planar integrated optical imaging systems are advanced optical remote sensing imaging systems. Segmented Planar Imaging Detector for Electro-optical Reconnaissance (SPIDER) is an interference imager that uses multiple paired lenses to form an interference baseline combined with a photonic integrated circuit (PIC) to miniaturize the device structure but with high resolution [9–14]. The SPIDER samples the object visibility function in a UV plane and digitally reconstructs an image through an inverse Fourier transform relationship [9]. Therefore, optimizing the design of the UV spatial frequency distribution can effectively improve the imaging quality of the system. The optimal design of the UV spatial frequency distribution can be achieved by optimizing the arrangement and pairing method of the lens array, optimizing the design of the PIC, and improving the coupling efficiency of the lens array and PIC. Yu et al. devised the “checkerboard” imaging system design whose imaging effect is superior to that of the SPIDER imaging system [15]. We have proposed hierarchical multistage sampling lens arrays that improve the imaging quality of segmented planar integrated optical imaging systems by compensating for the middle and short radial lens arrays on both sides of the long radial lens array to increase the low-and medium-frequency information [16]. The current research is aimed at improving the lens array under the premise of uniform lenslet parameters of the segmented planar integrated optical imaging system. A study of the literature shows that reports on the UV spatial frequency distribution of the segmented planar integrated optical imaging system that consider the inhomogeneity of the lenslet on the lens array are absent.

In this study, the influence of the inhomogeneity of the lenslet on the imaging quality of the segmented planar integrated optical imaging system is studied. An inhomogeneous multistage sampling lens array was constructed by changing the lens radius for sampling different frequencies. The UV spatial frequency of the segmented planar integrated optical imaging system was optimized by selectively sampling the frequency. The signal transmission of the segmented planar integrated optical imaging system was modeled, and the pairing method of the middle radial lens array was improved. The imaging results of the segmented planar integrated optical imaging system with an inhomogeneous multistage sampling lens array were analyzed from the perspective of the UV spatial frequency distribution, point spread function (PSF), and imaging quality. Our results provide a theoretical basis for improving the new segmented planar integrated optical imaging system.

2. Imaging principle of the segmented planar integrated optical imaging system

2.1 Segmented planar integrated optical imaging system

A schematic of the segmented planar integrated optical imaging system is shown in Fig. 1. The segmented planar integrated optical imaging system consists of sensor array, processing module, and imaging reconstruction module. The sensor array consists of lens array, waveguide arrays, and array waveguide gratings (AWG). The processing module comprises balanced quadrature detectors and digital signal-processing parts. The waveguide array, AWG, and phase shifter are integrated on the PIC chip. The UV spatial frequency distributions obtained were reconstructed by the imaging reconstruction module.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the segmented planar integrated optical imaging system.

Download Full Size | PDF

The radiation of a long-range object is focused by different paired baselines. Then, the focused fields on different lens pupils are coupled and divided by the corresponding waveguide arrays. The obtained radiation fields with a broad spectrum were finely separated by the AWG. Two coherent optical signals interfere in a balanced quadrature detector. Both the ES and ER signals are divided into two signals. The two signals separated by ES are in phase, while the two signals separated by ER are in the quadrature-phase. Then, the in-phase current I and quadrature-current Q are obtained. The amplitude and phase of the interference fringes were extracted from I and Q. The UV spatial frequency distribution were obtained by a digital signal process. The light intensity distribution of the object can be obtained by the inverse Fourier transform of the UV spatial frequency distribution to achieve image reconstruction.

2.2 Design principle of an inhomogeneous multistage sampling lens array

In public reports, the design of the lens array of the segmented planar integrated optical imaging system was optimized to improve the arrangement and pairing method of the lens array under the condition of a uniform lenslet. Assuming that there are N lenslets on the lens array, the uniform pairing method pairs 1&N, 2&N-1, 3&N-2 … . As shown in Fig. 2(a), the UV spatial frequency distribution of the imaging system with a lens array with the same radius parameter is uniform. Figure 2(b) shows an inhomogeneous multistage sampling lens array. The UV spatial frequency distribution of the segmented planar integrated optical imaging system with an inhomogeneous multistage sampling lens array is nonuniform. Therefore, in comparison with the uniform sampling lens array, the inhomogeneous multistage sampling lens array can achieve selective sampling of different frequencies using the same pairing method.

 figure: Fig. 2.

Fig. 2. Sampling difference between uniform and inhomogeneous multistage sampling lens arrays.

Download Full Size | PDF

2.3 Improved segmented planar integrated optical imaging system

The description of the sampling difference between uniform and inhomogeneous multistage sampling lens arrays shows that the inhomogeneous multistage sampling lens array can achieve selective sampling of different frequencies to optimize the UV spatial frequency distribution. Therefore, we propose an inhomogeneous multistage sampling lens array to further improve the resolution of the segmented planar integrated imaging system without increasing the length of the lens array, instead of a hierarchical multistage sampling lens array.

The inhomogeneous multistage sampling lens array shown in Fig. 3 is an optimized design based on the hierarchical multistage sampling lens array Ι, which we proposed in [16] by changing the radius of the different sampling frequencies. Similarly, the inhomogeneous multistage sampling lens array comprises a long radial lens array and a middle radial lens array as the hierarchical multistage sampling lens array Ι. The middle radial lens array is arranged radially on both sides of the long radial lens array to compensate for the medium-and high-frequency information, which is different from the middle radial lens array of the hierarchical multistage sampling lens array Ι. The inhomogeneous multistage sampling lens array achieves better imaging quality using fewer lenses compared to the hierarchical multistage sampling lens array II [16].

 figure: Fig. 3.

Fig. 3. Inhomogeneous multistage sampling lens array of the segmented planar integrated optical imaging system.

Download Full Size | PDF

2.4 Baseline pairing method of inhomogeneous multistage sampling lens array

The inhomogeneous multistage sampling lens array shown in Fig. 3(c) is selected as an example. It is composed of 17 long radial lens arrays, each of which has a middle radial lens array on each side. The number of lenslets on the long and middle radial lens arrays were 30 and 22, respectively. The baseline pairing method of the long radial lens array is the same as that of the hierarchical multistage sampling lens array [16]. 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 paired lenses (9, 10), (5, 7), (3, 6), (22, 26) collect low-frequency information, and the radius of the lenslet is r1. The paired lenses (18, 23), (21, 27), (20, 28), and (19, 29) collect medium frequency information, and the radius of the lenslet is r2. The paired lenses (4, 17), (8, 25), (2, 24), and (1, 30) collect high-frequency information, and the radius of the lenslet is r3. The radius of the remaining unpaired lenses is r4. The pairing method of the middle radial lens array is (1, 22), (2, 21), (3, 18), (4, 20), (5, 19), (6, 17), (7, 16), and (8, 15). The lenses (8, 15), (7, 16), (6, 17), and (5, 19) collected medium frequency information, and the radius of the lenslet is r2. The paired lenses (4, 20), (3, 18), (2, 21), (1, 22) collect high-frequency information, and the radius of the lenslet is r3. Similarly, the radius of the remaining unpaired lenses is r4. The baseline length is not an integer multiple of the minimum baseline owing to the different radii of the lenslets on the inhomogeneous multistage sampling lens array.

Compared to the baseline pairing method of the middle radial lens array of the hierarchical multistage sampling lens array Ι, the sampling pairing method of the middle radial lens array of the inhomogeneous multistage sampling lens array mainly compensates for medium-and high-frequency information that cannot be collected by a long radial lens array.

3. Signal transfer modeling of the segmented planar integrated optical imaging system

The segmented planar integrated optical imaging system is mainly used for long-distance object imaging. The transmission process of the radiation signal of the segmented planar integrated optical imaging system is shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. Transmission process of the radiation signal of the segmented planar integrated optical imaging system.

Download Full Size | PDF

3.1 Coupling modeling of long-distance radiant light and lens array

U1(x1,y1) is the light field distribution of a long-distance object. In the simulation, the distance between the object and lens array planes is long, as in the case of astronomical observations. According to the Fraunhofer diffraction formula [12], regardless of atmospheric radiation attenuation, the input light of the lens array U(x,y) is

$${{\mathop{U}\limits^\rightharpoonup} (x,y)} = \frac{{{e^{ikz}} \cdot {e^{ik\frac{{{x^2} + {y^2}}}{{2z}}}}}}{{i\lambda z}}\int {\int_{ - \infty }^{ + \infty } {{{{\mathop{U}\limits^\rightharpoonup}}_1}(x,y)} } {e^{ - i\frac{{x{x_1} + y{y_1}}}{z}}}d{x_1}d{y_1}$$
where λ is the wavelength of the incident light, z is the distance between the object and lens array planes, and k is the wave number.

A phase delay occurs when light passes through the lens array. Ignoring the thickness of the lenslet, the transmittance of light through the jth lenslet is

$${t_{1j}}(x,y) = {e^{ - \frac{{ik}}{{2f}}({{x^2} + {y^2}} )}} \bullet circ\left[ {\frac{{\sqrt {{{(x - {x_j})}^2} + {{(y - {y_j})}^2}} }}{a}} \right]\;\;\;\;\textrm{ }j = 1,2, \cdots ,n\textrm{ }$$
where f is the focal length of the lenslet, (xj, yj) is the central coordinate of the jth lenslet, and a represents the radius of the lenslet.

After phase conversion of the lens array, the complex amplitude distribution of the light field U’(x, y) is

$${{\mathop{U}\limits^\rightharpoonup}^{\prime}} (x,y) = {\mathop{U}\limits^\rightharpoonup} (x,y) \cdot {t_{1j}}(x,y)\textrm{ }\quad j = 1,2, \cdots ,n\textrm{ }$$

3.2 Coupling modeling of the lens array and waveguide

The waveguide array is located on the focal plane of the lens array. According to the Fresnel diffraction formula [17], the light field, P (u, v), of the input waveguide array is

$$P(u,v) = \rho \cdot \frac{{{e^{ikf}}}}{{i\lambda f}} \cdot {e^{ik\frac{{{u^2} + {v^2}}}{{2f}}}} \cdot \int {\int_{ - \infty }^{ + \infty } {{{\mathop{U}\limits^\rightharpoonup}^{\prime}} (x,y){e^{ik\frac{{{x^2} + {y^2}}}{{2f}}}}} } {e^{ - ik\frac{{xu + yv}}{f}}}dxdy$$
where ρ is the instantaneous coupling efficiency of the focal electric field and the main mode of the waveguide. ρ is determined by the overlap integral between the distribution of the electric fields in the focal plane and the guided mode [18].
$$\rho = \frac{{{{\left|{\int_{{S_\infty }} {{E_{focus}}E_{waveguide}^ \ast dS} } \right|}^2}}}{{\int_{{S_\infty }} {{{|{{E_{focus}}} |}^2}dS\int_{{S_\infty }} {{{|{E_{waveguide}^{}} |}^2}dS} } }}$$
where Efocuse is the light amplitude of the long-distance radiation on the focal plane of the lenslet, and Ewaveguide is the electric field distribution of the TE wave in the waveguide. The integration domain extends to infinity in the transverse plane, and the symbol * denotes the complex conjugate.

The amplitude on the focal plane Efocus can be expressed as

$${E_{focus}}({x_i},{y_i}) = \frac{1}{{{{(\lambda f)}^2}}}\int {\int\limits_{ - \infty }^\infty {circ(\frac{{\sqrt {{x^2} + {y^2}} }}{a})} } {e^{ - i2\pi (\frac{{{x_i}}}{{\lambda f}}x + \frac{{{y_i}}}{{\lambda f}}y)}}dxdy$$
where f is the focus of the lenslet, and λ is the wavelength of the incident light.

A schematic diagram of the cross-section of the waveguide is shown in Fig. 5. The width and thickness of the waveguide were b and h, respectively. µ and ɛ represent the permeability and conductivity of the waveguide, respectively. Only the main-mode transmission was considered in the waveguide. The electric field distribution Ewaveguide of the TE10 wave in the waveguide is

$$\left\{ \begin{array}{l} {E_x} = {E_z} = 0\\ {E_y} ={-} \frac{{j\omega \mu }}{{k_c^2}}\frac{\pi }{b}{H_0}\sin \left( {\frac{\pi }{b}x} \right){e^{ - j{k_z}z}} \end{array} \right.$$
where $k_c^2 = {({{\raise0.7ex\hbox{$\pi $} \!\mathord{\left/ {\vphantom {\pi a}} \right.}\!\lower0.7ex\hbox{$a$}}} )^2}$, kz is the wave number along the z direction.

 figure: Fig. 5.

Fig. 5. Schematic diagram of the cross-section of the waveguide.

Download Full Size | PDF

3.3 Modeling of AWG transmission

The light propagating into the image plane was coupled into the waveguide array and then divided into multiple narrow-spectrum beams by the AWG. The optical signal divided by the AWG is then propagated into the phase shifter to adjust the phase of the incident light. Regardless of the inhomogeneity of the intensity between arrayed waveguides, the PSF of the central point of the input waveguide through the center of the AWG is as follows, if only the influence of the phase difference is considered [19].

$$Q(u^{\prime},v^{\prime}) \propto \frac{{{e^{ikR}}}}{{\sqrt {i\lambda R} }}F.T.{\left\{ {\frac{{{e^{ikR}}}}{{\sqrt {i\lambda R} }}F.T.\left\{ {P(u,v){e^{ - i\frac{k}{{2R}}({{u^2} + {v^2}} )}}} \right\}} \right\}_{\xi = \frac{{u^{\prime}}}{{\lambda R}},\eta = \frac{{v^{\prime}}}{{\lambda R}}}} \cdot \sum\limits_{L ={-} M}^{L = M} {\left\{ {K({Ld} ){e^{ - i2\pi Ld\frac{{u^{\prime} + v^{\prime}}}{{\lambda R}}}}} \right\}}$$
where R is the radius of the Roland circle of the AWG, M is the number of arrayed waveguides, and d is the spacing between the arrayed waveguides.

3.4 Modeling of the coherence intensity photoelectric detection and Image reconstruction

The light propagating from the AWG passes through the phase shifter to achieve the same phase. The corresponding coherent lights interfere in the balanced quadrature detector, as shown in Fig. 1. Assuming that the input light field of the balanced quadrature detector is ES and ER, the amplitude and phase relationship of the input and output can be expressed as follows [20]:

$$\left( \begin{array}{l} {E_A}\\ {E_B}\\ {E_C}\\ {E_D} \end{array} \right) = \left( \begin{array}{l} {T_{11}}\textrm{ }{T_{12}}\textrm{ }{T_{13}}\textrm{ }{T_{14}}\\ {T_{21}}\textrm{ }{T_{22}}\textrm{ }{T_{23}}\textrm{ }{T_{24}}\\ {T_{31}}\textrm{ }{T_{32}}\textrm{ }{T_{33}}\textrm{ }{T_{34}}\\ {T_{41}}\textrm{ }{T_{42}}\textrm{ }{T_{43}}\textrm{ }{T_{44}} \end{array} \right)\left( \begin{array}{l} {E_S}\\ 0\\ {E_R}\\ 0 \end{array} \right) = \frac{1}{2}\left( \begin{array}{l} 1\textrm{ }{e^{j\frac{{3\pi }}{4}}}\textrm{ }{e^{ - j\frac{\pi }{4}}}\textrm{ }1\\ {e^{j\frac{{3\pi }}{4}}}\textrm{ }1\textrm{ }1\textrm{ }{e^{ - j\frac{\pi }{4}}}\\ {e^{ - j\frac{\pi }{4}}}\textrm{ }1\textrm{ }1\textrm{ }{e^{j\frac{{3\pi }}{4}}}\\ 1\textrm{ }{e^{ - j\frac{\pi }{4}}}\textrm{ }{e^{j\frac{{3\pi }}{4}}}\textrm{ }1 \end{array} \right)\left( \begin{array}{l} {E_S}\\ 0\\ {E_R}\\ 0 \end{array} \right) = \frac{1}{2}\left( \begin{array}{l} {E_S} + {e^{ - j\frac{\pi }{4}}}{E_R}\\ {e^{j\frac{{3\pi }}{4}}}{E_S} + {E_R}\\ {e^{ - j\frac{\pi }{4}}}{E_S} + {E_R}\\ {E_S}\textrm{ + }{e^{j\frac{{3\pi }}{4}}}{E_R} \end{array} \right)$$

The power at the output ports is

$$\begin{array}{l} {P_1} = \frac{1}{4}{P_S} + \frac{1}{4}{P_R} + \frac{{\sqrt {{P_S}{P_R}} }}{2}\cos \left( {({{w_S} - {w_R}} )t + {\varphi_S}(t) - {\varphi_R}(t) + \frac{\pi }{4}} \right)\\ {P_2} = \frac{1}{4}{P_S} + \frac{1}{4}{P_R} - \frac{{\sqrt {{P_S}{P_R}} }}{2}\sin \left( {({{w_S} - {w_R}} )t + {\varphi_S}(t) - {\varphi_R}(t) + \frac{\pi }{4}} \right)\\ {P_3} = \frac{1}{4}{P_S} + \frac{1}{4}{P_R} + \frac{{\sqrt {{P_S}{P_R}} }}{2}\sin \left( {({{w_S} - {w_R}} )t + {\varphi_S}(t) - {\varphi_R}(t) + \frac{\pi }{4}} \right)\\ {P_4} = \frac{1}{4}{P_S} + \frac{1}{4}{P_R} - \frac{{\sqrt {{P_S}{P_R}} }}{2}\cos \left( {({{w_S} - {w_R}} )t + {\varphi_S}(t) - {\varphi_R}(t) + \frac{\pi }{4}} \right) \end{array}$$
where wS and wR are the frequencies of the input light, while ${\varphi _S}$ and ${\varphi _R}$ represent its initial phases.

After passing through the photodetector and differential amplifier, the in-phase current I and quadrature-current Q are

$$\begin{array}{l} I\textrm{ = }R\sqrt {{P_S}{P_R}} \sin ({\Delta \varphi } )\\ Q\textrm{ = }R\sqrt {{P_S}{P_R}} \cos ({\Delta \varphi } )\end{array}$$
where R is the responsivity of the detector. The amplitude and phase of the interferometer fringe can be obtained by
$$\begin{array}{l} \tan (\Delta \varphi ) = \frac{I}{Q}\\ \sqrt {{P_S}{P_R}} = \frac{I}{{R\sin (\Delta \varphi )}} \end{array}$$

By adjusting the control phase, the transmission phase information is finally converted into a level signal, and the information of the transmission signal is demodulated with respect to the level signal.

According to the Van Cittert-Zernike theory, for a source I(l, m), the mutual coherent intensity at the field points (xi, yi) and (xj, yj) is

$$V(u,v) = \int\!\!\!\int {I(l,m)\textrm{exp} [{2\pi i(ul + vm)} ]} dudv$$
$$u = \frac{{{x_i} - {x_j}}}{{\lambda z}},\;\; v = \frac{{{y_i} - {y_j}}}{{\lambda z}}$$

According to the above formula, an inverse Fourier transform is performed on the complex amplitude to obtain the intensity distribution of the target.

4. Simulation results and analysis

The imaging quality of the segmented planar integrated optical imaging system with hierarchical multistage sampling lens arrays and inhomogeneous multistage sampling lens arrays are simulated in the case of the same length of the longest lens array to verify the signal transfer model. The radii of the hierarchical multistage sampling lens arrays Ι and Π were 1.8 mm. The radii of the inhomogeneous multistage sampling lens array were r1=2.2 mm, r2=1.8 mm, r3=1.4 mm, and r4=1.8 mm. The simulation results were compared from three aspects: the UV spatial frequency distribution, PSF, and imaging results. The system parameters used for the simulations are listed in Table 1.

Tables Icon

Table 1. System parameters used for simulations.

View Table | View all tables in this article

The UV spatial frequency distribution of the segmented planar integrated optical imaging system with different lens arrays is depicted in Fig. 6. The simulation results show that the sampling information of the imaging system with the inhomogeneous multistage sampling lens array is significantly higher than that of the hierarchical multistage sampling lens array. The maximum sampling frequency radius of the imaging system with an inhomogeneous multistage sampling lens array is R = 1.122, which is larger than the sampling frequency radius R = 1.116 of the imaging system with a hierarchical multistage sampling lens array. The continuous sampling frequency radius of the segmented planar integrated optical imaging system with mutual intensity greater than 0.5, with inhomogeneous multistage sampling lens array, is r = 0.457, while those of the segmented planar integrated optical imaging system with hierarchical multistage sampling lens arrays Ι and П are r = 0.136 and r = 0.138 respectively. Owing to the improvement of the pairing method of the middle radial lens array and the selective sampling of frequency, compared to the hierarchical multistage sampling lens arrays, the inhomogeneous multistage sampling lens array increases the collection of medium-and high-frequency information without adding a short lens array.

 figure: Fig. 6.

Fig. 6. UV spatial frequency distribution of the segmented planar integrated optical imaging system with (a), (b) hierarchical multistage sampling lens arrays Ι and Π with a radius of 1.8 mm, (c) inhomogeneous multistage sampling lens array with radii r1 = 2.2 mm, r2 = 1.8 mm, r3 = 1.4 mm, r4 = 1.8 mm.

Download Full Size | PDF

Figure 7 shows the object PSF of the segmented planar integrated optical imaging system with different lens arrays. The full width at half maximum (FWHM) of the segmented planar integrated optical imaging system with hierarchical multistage sampling lens arrays Ι and Π are 0.2534 m and 0.251 m, respectively, while the FWHM of the imaging system with the inhomogeneous multistage sampling lens array is 0.2216 m. The width of the object PSF of the segmented planar integrated optical imaging system with the inhomogeneous multistage sampling lens array is clearly narrower than that of the hierarchical multistage sampling lens arrays.

 figure: Fig. 7.

Fig. 7. Object PSF of the segmented planar integrated optical imaging system with different lens arrays.

Download Full Size | PDF

The imaging quality of the segmented planar integrated optical imaging system with the above lens arrays were simulated, and the results are shown in Fig. 8. Figure 8(a) shows an example of a general source. Figures 8(b)–8(d) show the imaging results of the imaging system with hierarchical multistage sampling lens arrays and an inhomogeneous multistage sampling lens array assuming ideal waveguide coupling efficiency. It is obvious that the imaging results of the imaging system with the inhomogeneous multistage sampling lens array have a smaller blur area in the center of the image owing to the increased collection of high-frequency information. Moreover, as the lens radius for collecting low-frequency information on the inhomogeneous multistage sampling lens array increases, more low-frequency information is collected, and the contour of the image is clear. Therefore, an imaging system with the inhomogeneous multistage sampling lens array produces better imaging quality.

 figure: Fig. 8.

Fig. 8. (a) Light intensity distribution of object. (b) and (c) Imaging results of the segmented planar integrated optical imaging system with the hierarchical multistage sampling lens arrays. (d) Imaging results of the segmented planar integrated optical imaging system with the inhomogeneous multistage sampling lens array.

Download Full Size | PDF

5. Parameter optimization of the segmented integrated optical imaging system

The radius of the lenslet and the key parameters of the waveguide are observed to significantly affect the imaging quality of the segmented integrated optical imaging system from the radiation signal transmission model. To analyze the influence of these two factors on the imaging quality of the imaging system, simulations and comparisons were made from the perspective of UV spatial frequency distribution, PSF, and imaging quality.

5.1 Radius optimization of the inhomogeneous multistage sampling lens array

It can be seen that the sampling frequencies of lenslets with different radii are different from the above analysis. When the length of the longest lens array on the inhomogeneous multistage sampling lens array is the same, the inhomogeneous multistage sampling lens array can be divided into different types according to the difference between the radii of the sampling lenslet. The lens array with different parameters was simulated by the controlled variable method when the unsampled lens radius r4 remained unchanged. The imaging quality of the segmented integrated optical imaging system with an inhomogeneous multistage sampling lens array is better than that of the hierarchical multistage sampling lens array only when r2 and r4 are constant, r2 and r3 are variables, and r2+r3 is constant.

As shown in Fig. 3, the lenslets on the long radial lens array and middle radial lens array will overlap as the radius r1 of the low-frequency sampling lens continues to increase and the radius r3 of the high-frequency sampling lens continues to decrease. Considering the inhomogeneous multistage sampling lens array shown in Fig. 3(c) as an example, the 10th lenslet on the long radial lens array is tangential to the second lenslet on the middle radial lens array. The optimal lenslet radius parameters under this condition satisfy the following relationship:

$$\begin{array}{l} (\frac{{{r_3}}}{{\tan {\raise0.7ex\hbox{$\alpha $} \!\mathord{\left/ {\vphantom {\alpha 2}} \right.}\!\lower0.7ex\hbox{$2$}}}} + 7{r_3} + 12{r_1}) \bullet \tan {\raise0.7ex\hbox{$\alpha $} \!\mathord{\left/ {\vphantom {\alpha 8}} \right.}\!\lower0.7ex\hbox{$8$}} \ge {r_2}\\ {r_3} + (3{r_3} + {r_1}) \bullet \tan {\raise0.7ex\hbox{$\alpha $} \!\mathord{\left/ {\vphantom {\alpha 2}} \right.}\!\lower0.7ex\hbox{$2$}} \ge {r_1}\\ {r_1} + {r_3} = C \end{array}$$
where C is a constant, and α equals 2π/p. The optimal radius of r2 is 1.84 mm, which is calculated by the above formula when r1 and r3 are 2.2 mm and 1.4 mm, respectively. Considering the complexity of the actual processing, the optimal radius parameters of the inhomogeneous multistage sampling lens array are r1 = 2.2 mm, r2 = 1.8 mm, r3 = 1.4 mm, and r4 = 1.8 mm.

Figure 9 shows the UV spatial frequency distribution of the segmented integrated optical imaging system with inhomogeneous multistage sampling lens arrays with varying radius parameters. R represents the maximum frequency sampling radius, and r is the frequency sampling radius when the mutual intensity is greater than 0.3. Under the premise of the same longest lens array length, Compared with the UV spatial frequency distribution of the imaging system with hierarchical multistage sampling lens array showed in Fig. 9(b), when the radius of the medium frequency information sampling lenses and unpaired lenses on the inhomogeneous multistage sampling lens array remain unchanged, as the radius r1 of the low-frequency sampling lens increases and the radius r3 of the high-frequency sampling lens decreases, R and r gradually increase, as shown in Table 2. The reverse is also true. The maximum frequency sampling radius of the segmented integrated optical imaging system under the optimal radius parameters is 1.122. It is increased by 0.006 compared with the imaging system with the hierarchical multistage sampling lens array.

 figure: Fig. 9.

Fig. 9. UV spatial frequency distribution of the segmented planar integrated optical imaging system with different lens arrays. Inhomogeneous multistage sampling lens arrays with radii of (a) r1 = 1.6, r2 = 1.8, r3 = 2, r4 = 1.8 (mm), (c) r1 = 2, r2 = 1.8, r3 = 1.6, r4 = 1.8 (mm), (d) r1 = 2.2, r2 = 1.8, r3 = 1.4, r4 = 1.8 (mm). (b) Hierarchical multistage sampling lens array with radii of r1 = r2 = r3 = r4 = 1.8 (mm).

Download Full Size | PDF

Tables Icon

Table 2. Performance parameters of the segmented planar integrated optical imaging system with different lens arrays.

View Table | View all tables in this article

Figure 10 shows the object PSF of the segmented planar integrated optical imaging system with different lens arrays. The FWHM of the hierarchical multistage sampling lens array with a radius of 1.8 mm is 0.2558 m. As the radius r1 of the low-frequency sampling lens increases and the radius r3 of the high-frequency sampling lens decreases, the FWHM of the segmented planar integrated optical imaging system gradually decreases, and vice versa. The FWHM of the segmented integrated optical imaging system with an inhomogeneous multistage sampling lens array under the optimal radius parameters is 0.2216 m. It is reduced by 0.0342 m compared to the imaging system with the hierarchical multistage sampling lens array.

 figure: Fig. 10.

Fig. 10. Object PSF of the segmented planar integrated optical imaging system with different lens arrays.

Download Full Size | PDF

The imaging results of the segmented planar integrated optical imaging system with different lens arrays are shown in Fig. 11. It is evident that, in comparison with the imaging results of the imaging system with the hierarchical multistage sampling lens array in Fig. 11(c), the blur circle in the imaging center of the imaging system with an inhomogeneous multistage sampling lens array gradually decreases as the radius r1 of the low-frequency sampling lens increases and the radius r3 of the high-frequency sampling lens decreases. As the high-frequency sampling information increases, the center of the image becomes clearer gradually. As shown in Fig. 9, as the radius r1 of the low-frequency sampling lens increases and the radius r3 of the high-frequency sampling lens decreases, the maximum sampling frequency radius of the imaging system with an inhomogeneous multistage sampling lens array gradually increases.

 figure: Fig. 11.

Fig. 11. The imaging results of the segmented planar integrated optical imaging system with different lens arrays. (a) Light intensity distribution of the object. Inhomogeneous multistage sampling lens arrays with radii of (b) r1 = 1.6, r2 = 1.8, r3 = 2, r4 = 1.8 (mm), (d) r1 = 2, r2 = 1.8, r3 = 1.6, r4 = 1.8 (mm), (e) r1 = 2.2, r2 = 1.8, r3 = 1.4, r4 = 1.8 (mm). (c) Hierarchical multistage sampling lens array with radii of r1 = r2 = r3 = r4 = 1.8 (mm).

Download Full Size | PDF

5.2 Waveguide array parameter optimization

The waveguide array is an important element that spatially divides the light transmitted from the lens array. The signal transmission model in Chapter 3 shows that the light transmitted from the lens array is coupled to the waveguide array. Equations (5)–(7) show that the efficiency of radiant light transmission through the waveguide is related to the parameters of the waveguide. Therefore, the choice of waveguide parameters is very important to improve the imaging quality of the imaging system. The parameter of the corresponding waveguide array is optimized as follows when the radii of the inhomogeneous multistage sampling lens array are r1 = 2.2 mm, r2 = 1.8 mm, r3 = 1.4 mm, and r4 = 1.8 mm.

Waveguide arrays are located on the focal plane of the lens array, and each lenslet corresponds to N-arrayed waveguides. According to Rayleigh diffraction, the beam radius at the focal plane is

$${\rho _i} = \frac{{0.61f\lambda }}{{{r_i}}}\;\;\;\textrm{ }i = 1,2,3,4$$

Doped-SiO2 and SiO2 were selected as the materials of the waveguide core and cladding, respectively. To reduce the transmission loss of the waveguide, the refractive indices of the core and cladding of the waveguide were 1.46 and 1.44, respectively. The refractive index difference was 1.4%. According to the mode conditions of the slab waveguide and rectangular waveguide [21], the width and thickness of the waveguide can be calculated as follows:

$$m = \frac{{2h}}{{{\lambda _0}}}\sqrt {n_1^2 - n_2^2}\;\;\;\textrm{ }m = 0,1,2 \ldots $$
$${b_{\max }} = {2^{ - \frac{3}{2}}}{\Delta ^{ - \frac{1}{2}}}\frac{{{\lambda _0}}}{{{n_1}}}$$
where bmax is the maximum width of the waveguide to maintain a single mode, m is the number of modes, h is the waveguide thickness, and λ0 is the average wavelength.

According to Eq. (17), m < 2 for single-mode transmission. The thickness of the waveguide can be calculated to be less than 2.283 µm. The waveguide thickness h is selected as 2 µm, considering the actual processing complexity. According to Eq. (18), the maximum width of the waveguide during single-mode transmission is approximately 1 µm. However, the optical confinement factor was found to be low when using the film mode matching method (FEM). Light leakage can be reduced by increasing the waveguide width. As shown in Fig. 12, a width of 2 µm is selected as the waveguide width when the confinement factor reaches 0.89.

 figure: Fig. 12.

Fig. 12. (a) Light field with a waveguide width of 1 µm. (b) Light field with a waveguide width of 2 µm. (c) Relationship between the optical confinement factor and waveguide width.

Download Full Size | PDF

Figure 13 shows the power transmission process of the waveguide arrays with different spacings. The figure axes are X and Z. Z is the depth of the waveguide in the PIC. The monitor value in Fig. 13 is the intensity in the two waveguides. The waveguide in which the lens is coupled is represented by the blue curves, while its neighboring waveguide is represented by the green curves. In the simulation, Z was set to 2000 µm. The waveguide thickness h and width b were designated as 2 µm. The waveguide spacing is d. The coupling efficiency is large during the transmission process when the spacing of the waveguide arrays is 3 µm. When z is approximately 1000 µm, the light of the adjacent waveguides is completely coupled. As the spacing of the waveguide array increases, the coupling efficiency gradually decreases. The output power is almost unchanged, while the spacing between the waveguide arrays is 4 µm. At this separation, no cross coupling occurs. Therefore, considering the coupling between the waveguide arrays, the optimal waveguide spacing is 4 µm.

 figure: Fig. 13.

Fig. 13. The relation between array waveguide coupling and waveguide spacing.

Download Full Size | PDF

According to the Eq. (16), the diffraction spot radii of the low-frequency, medium-frequency, and high-frequency sampling lenses and unsampled lenses can be estimated as 2.29, 2.80, 3.59, and 2.80 µm, respectively. Based on the selected parameters of the waveguide array b = h = 2 µm and d = 4 µm, it can be observed that each diffraction spot of the lenslet on the lens array corresponds to a waveguide.

6. Conclusion

This study describes the imaging principle and provides a detailed theoretical analysis of the radiation signal transmission of the segmented planar integrated optical imaging system. An inhomogeneous multistage sampling lens array is proposed based on the hierarchical multistage sampling lens array by changing the radius for sampling of different frequencies to optimize the design of the UV spatial frequency distribution. A new baseline pairing method for a middle radial lens array is proposed to compensate for the unsampled information of the long radial lens array. When the lengths of the inhomogeneous multistage sampling lens array and hierarchical multistage sampling lens array are the same, the imaging quality of the segmented planar integrated optical imaging system with an inhomogeneous multistage sampling lens array is better than a hierarchical multistage sampling lens array by the selective sampling of the spatial frequency. Compared to the hierarchical multistage sampling lens array with a radius of 1.8 mm, the optimal parameters of the inhomogeneous multistage sampling lens array are r1 = 2.2 mm, r2 = 1.8 mm, r3 = 1.4 mm, and r4 = 1.8 mm. Meanwhile, the optimized parameters of the waveguide array connected to the lens array are calculated. The width and thickness of the waveguide are b = h = 2 µm. The spacing between adjacent waveguides is d = 4 µm. The results show that each diffraction spot after the lenslet corresponds to a waveguide under the selected parameter. The results provide a theoretical foundation for the development of integral optical imaging systems.

Funding

National Natural Science Foundation of China (61775174, 62005204, 62075176); Fundamental Research Funds for the Central Universities; Innovation Fund of Xidian University.

Disclosures

The authors declare no conflicts of interest.

References

1. S. J. Chung, D. W. Miller, and O. L. de Weck, “ARGOS testbed: study of multidisciplinary challenges of future space borne interferometric arrays,” Opt. Eng. 43(9), 2156–2167 (2004). [CrossRef]  

2. A. B. Meinel, “An overview of the technological possibilities of future telescopes,” ESO Conf. on Optical Telescopes of the Future, ESO, Geneva, 13–26 (1978).

3. J. D. Monnier, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66(5), 789–857 (2003). [CrossRef]  

4. R. S. Simon, K. J. Johnston, D. Mozurkewich, K. W. Weiler, D. J. Hutter, J. T. Armstrong, and T. S. Brackett, “Imaging optical interferometry,” International Astronomical Colloquium 19, 2483–2488 (2016). [CrossRef]  

5. J. P. Aufdenberg, A. Mérand, V. Coudé Du Foresto, O. Absil, E. Di Folco, P. Kervella, S. Ridgway, D. H. Berger, T. A. Ten Brummelaar, H. A. McAlister, J. Sturmann, L. Sturmann, and N. H. Turner, “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]  

6. J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. 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]  

7. R. G. Petrov, F. Malbet, G. Weigelt, P. Antonelli, U. Beckmann, Y. Bresson, A. Chelli, M. Dugué, G. Duvert, and S. Gennari, “AMBER, the near-infrared spectro-interferometric three-telescope VLTI instrument,” A&A 464(1), 1–12 (2007). [CrossRef]  

8. H. A. McAlister, T. A. ten Brummelaar, D. R. Gies, W. Huang, W. G. Bagnuolo Jr., M. A. Shure, J. Sturmann, L. Sturmann, N. H. Turner, S. F. Taylor, D. H. Berger, E. K. Baines, E. Grundstrom, and C. Ogden, “First results from the CHARA array. I. An interferometric and spectroscopic study of the fast rotator α Leonis (Regulus),” Astrophys. J. 628(1), 439–452 (2005). [CrossRef]  

9. A. L. Duncan, R. L. Kendrick, C. Ogden, D Wuchenich, S. T. Thurman, S. J. S. B. Yoo, T. H. Su, S. Pathak, and R Proietti, “SPIDER: Next generation chip scale imaging sensor,” Advanced Maui Optical and Space Surveillance Technologies Conference (2015).

10. 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 spacesurveillance sensor,” AMOS (2013).

11. R. P. Scott, T. Su, C. Ogden, S. T. Thurman, R. L. Kendrick, A. Duncan, R. Yu, and S. J. B. Yoo, “Demonstration of a photonic integrated circuit for multi-baseline interferometric imaging,” IEEE Photonics Conference (2014).

12. Q. Chu, Y. Shen, M. Yuan, and M. Gong, “Numerical simulation and optimal design of Segmented Planar Imaging Detector for Electro-Optical Reconnaissance,” Opt. Commun. 405, 288–296 (2017). [CrossRef]  

13. S. T. Thurman, R. L. Kendrick, A. Duncan, D. Wuchenich, and C. Ogden, “System design for a SPIDER Imager,” Frontiers in Optics, 2015.

14. T. H. Su, “Interferometric imaging using Si3N4 photonic integrated circuits for a SPIDER imager,” Opt. Express 26(10), 12801–12812 (2018). [CrossRef]  

15. Q. Yu, B. Ge, Y. Li, Y. Yue, F. Chen, and S. Sun, “System design for a “checkerboard” imager,” Appl. Opt. 57(35), 10218–10223 (2018). [CrossRef]  

16. W. P. Gao, X. R. Wang, L. Ma, Y. Yuan, and D. F. Guo, “Quantitative analysis of segmented planar imaging quality based on hierarchical multistage sampling lens array,” Opt. Express 27(6), 7955–7967 (2019). [CrossRef]  

17. J. H. Wang, W. Zhang, Y. W. Cui, and S. Y. Teng, “Fresnel diffraction by a square aperture with rough edge,” Optik 126(21), 3066–3071 (2015). [CrossRef]  

18. V. Coudé du Foresto, S. Ridgway, and J.-M. Mariotti, “Deriving object visibilities from interferograms obtained with a fiber stellar interferometer,” Astron. Astrophys., Suppl. Ser. 121(2), 379–392 (1997). [CrossRef]  

19. Y. S. Qin, X. M. Li, T. S. Lv, and F. Y. Guo, “Analysis of focal field and crosstalk simulation of array aperture for array waveguide grating multi/demultiplexer,” Chinese J. Lasers 29(9), 801–804 (2002).

20. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995). [CrossRef]  

21. Y. Xu, C. Liang, Y. Yang, and T. Zhao, “Silica-on-silicon optical waveguides,” Chinese Journal of Semiconductors 22(12), 1546–1550 (2001).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)
Fig. 1.
Fig. 1. Schematic diagram of the segmented planar integrated optical imaging system.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Sampling difference between uniform and inhomogeneous multistage sampling lens arrays.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Inhomogeneous multistage sampling lens array of the segmented planar integrated optical imaging system.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Transmission process of the radiation signal of the segmented planar integrated optical imaging system.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Schematic diagram of the cross-section of the waveguide.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. UV spatial frequency distribution of the segmented planar integrated optical imaging system with (a), (b) hierarchical multistage sampling lens arrays Ι and Π with a radius of 1.8 mm, (c) inhomogeneous multistage sampling lens array with radii r1 = 2.2 mm, r2 = 1.8 mm, r3 = 1.4 mm, r4 = 1.8 mm.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. Object PSF of the segmented planar integrated optical imaging system with different lens arrays.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. (a) Light intensity distribution of object. (b) and (c) Imaging results of the segmented planar integrated optical imaging system with the hierarchical multistage sampling lens arrays. (d) Imaging results of the segmented planar integrated optical imaging system with the inhomogeneous multistage sampling lens array.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. UV spatial frequency distribution of the segmented planar integrated optical imaging system with different lens arrays. Inhomogeneous multistage sampling lens arrays with radii of (a) r1 = 1.6, r2 = 1.8, r3 = 2, r4 = 1.8 (mm), (c) r1 = 2, r2 = 1.8, r3 = 1.6, r4 = 1.8 (mm), (d) r1 = 2.2, r2 = 1.8, r3 = 1.4, r4 = 1.8 (mm). (b) Hierarchical multistage sampling lens array with radii of r1 = r2 = r3 = r4 = 1.8 (mm).

View in Article | Download Full Size | PDF

Fig. 10.
Fig. 10. Object PSF of the segmented planar integrated optical imaging system with different lens arrays.

View in Article | Download Full Size | PDF

Fig. 11.
Fig. 11. The imaging results of the segmented planar integrated optical imaging system with different lens arrays. (a) Light intensity distribution of the object. Inhomogeneous multistage sampling lens arrays with radii of (b) r1 = 1.6, r2 = 1.8, r3 = 2, r4 = 1.8 (mm), (d) r1 = 2, r2 = 1.8, r3 = 1.6, r4 = 1.8 (mm), (e) r1 = 2.2, r2 = 1.8, r3 = 1.4, r4 = 1.8 (mm). (c) Hierarchical multistage sampling lens array with radii of r1 = r2 = r3 = r4 = 1.8 (mm).

View in Article | Download Full Size | PDF

Fig. 12.
Fig. 12. (a) Light field with a waveguide width of 1 µm. (b) Light field with a waveguide width of 2 µm. (c) Relationship between the optical confinement factor and waveguide width.

View in Article | Download Full Size | PDF

Fig. 13.
Fig. 13. The relation between array waveguide coupling and waveguide spacing.

View in Article | Download Full Size | PDF

Tables (2)

Tables Icon

Table 1. System parameters used for simulations.

View Table | View all tables in this article
Tables Icon

Table 2. Performance parameters of the segmented planar integrated optical imaging system with different lens arrays.

View Table | View all tables in this article

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

U ⇀ ( x , y ) = e i k z ⋅ e i k x 2 + y 2 2 z i λ z ∫ ∫ − ∞ + ∞ U ⇀ 1 ( x , y ) e − i x x 1 + y y 1 z d x 1 d y 1
t 1 j ( x , y ) = e − i k 2 f ( x 2 + y 2 ) ∙ c i r c [ ( x − x j ) 2 + ( y − y j ) 2 a ]   j = 1 , 2 , ⋯ , n  
U ⇀ ′ ( x , y ) = U ⇀ ( x , y ) ⋅ t 1 j ( x , y )   j = 1 , 2 , ⋯ , n  
P ( u , v ) = ρ ⋅ e i k f i λ f ⋅ e i k u 2 + v 2 2 f ⋅ ∫ ∫ − ∞ + ∞ U ⇀ ′ ( x , y ) e i k x 2 + y 2 2 f e − i k x u + y v f d x d y
ρ = | ∫ S ∞ E f o c u s E w a v e g u i d e ∗ d S | 2 ∫ S ∞ | E f o c u s | 2 d S ∫ S ∞ | E w a v e g u i d e | 2 d S
E f o c u s ( x i , y i ) = 1 ( λ f ) 2 ∫ ∫ − ∞ ∞ c i r c ( x 2 + y 2 a ) e − i 2 π ( x i λ f x + y i λ f y ) d x d y
{ E x = E z = 0 E y = − j ω μ k c 2 π b H 0 sin ⁡ ( π b x ) e − j k z z
Q ( u ′ , v ′ ) ∝ e i k R i λ R F . T . { e i k R i λ R F . T . { P ( u , v ) e − i k 2 R ( u 2 + v 2 ) } } ξ = u ′ λ R , η = v ′ λ R ⋅ ∑ L = − M L = M { K ( L d ) e − i 2 π L d u ′ + v ′ λ R }
( E A E B E C E D ) = ( T 11   T 12   T 13   T 14 T 21   T 22   T 23   T 24 T 31   T 32   T 33   T 34 T 41   T 42   T 43   T 44 ) ( E S 0 E R 0 ) = 1 2 ( 1   e j 3 π 4   e − j π 4   1 e j 3 π 4   1   1   e − j π 4 e − j π 4   1   1   e j 3 π 4 1   e − j π 4   e j 3 π 4   1 ) ( E S 0 E R 0 ) = 1 2 ( E S + e − j π 4 E R e j 3 π 4 E S + E R e − j π 4 E S + E R E S  +  e j 3 π 4 E R )
P 1 = 1 4 P S + 1 4 P R + P S P R 2 cos ⁡ ( ( w S − w R ) t + φ S ( t ) − φ R ( t ) + π 4 ) P 2 = 1 4 P S + 1 4 P R − P S P R 2 sin ⁡ ( ( w S − w R ) t + φ S ( t ) − φ R ( t ) + π 4 ) P 3 = 1 4 P S + 1 4 P R + P S P R 2 sin ⁡ ( ( w S − w R ) t + φ S ( t ) − φ R ( t ) + π 4 ) P 4 = 1 4 P S + 1 4 P R − P S P R 2 cos ⁡ ( ( w S − w R ) t + φ S ( t ) − φ R ( t ) + π 4 )
I  =  R P S P R sin ⁡ ( Δ φ ) Q  =  R P S P R cos ⁡ ( Δ φ )
tan ⁡ ( Δ φ ) = I Q P S P R = I R sin ⁡ ( Δ φ )
V ( u , v ) = ∫ ∫ I ( l , m ) exp [ 2 π i ( u l + v m ) ] d u d v
u = x i − x j λ z , v = y i − y j λ z
( r 3 tan ⁡ α / α 2 2 + 7 r 3 + 12 r 1 ) ∙ tan ⁡ α / α 8 8 ≥ r 2 r 3 + ( 3 r 3 + r 1 ) ∙ tan ⁡ α / α 2 2 ≥ r 1 r 1 + r 3 = C
ρ i = 0.61 f λ r i   i = 1 , 2 , 3 , 4
m = 2 h λ 0 n 1 2 − n 2 2   m = 0 , 1 , 2 …
b max = 2 − 3 2 Δ − 1 2 λ 0 n 1
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use