# Electro-Optical Imaging Technology Based on Microlens Array and Fiber Interferometer
## 3 Theory and Algorithm  
Figure 2 shows the principle and processes of the electro-optical imaging technology; they consist of four parts: interference between microlens pairs, calculation of complex degree of coherence according to fringes, spectral coverage, and image reconstruction. Figure 1: (a) Rectangular microlens array using a symmetric pairing method; (b) microlens array using a complex rectangular sub-aperture arrangement method; (c) structure of the fiber interferometer. Figure 2: Schematic of the cutting-edge electro-optical imaging technology. Using the [PERSON] theorem in partial coherence theory [8], this study analyzes the mutual intensity of a light field illuminated by an incoherent source and reaches an important conclusion: when the linearity of the source and observation area are much smaller than the distance between them, the complex coherence coefficient over the observation area is proportional to the normalized Fourier transform of the source intensity. According to the principle shown in Figure 2, the complex coherence coefficient between Q\({}_{1}\) and Q\({}_{2}\) in aperture planes can be expressed as [14]:  
\[\mu(Q_{1},Q_{2})=\frac{\iint_{-\infty}^{+\infty}o(\alpha,\beta)exp\big{[}-j \frac{2\pi}{\lambda z}((x_{2}-x_{1})\alpha+(y_{2}-y_{1})\beta)\big{]}d\alpha d\beta} {\iint_{-\infty}^{+\infty}o(\alpha,\beta)d\alpha d\beta} \tag{1}\]  
\[\left\{ \begin{array}{l}u=\frac{\Delta x}{\Delta z}=\frac{(x_{2}-x_{1})}{ \Delta z}\\ v=\frac{\Delta y}{\Delta z}=\frac{(y_{2}-y_{1})}{\Delta z}\end{array} \right. \tag{2}\]  
According to Equation (2), the distance between the sub-apertures Q\({}_{1}\) and Q\({}_{2}\) in a pair is called the interferometer baseline, which is closely related to the spatial frequency \((u,v)\) of the detection scene. Therefore, we can optimally arrange the microlens array and use a better baseline-pairing method to realize spatial frequency coverage. Equations (1) and (2) provide a theoretical basis for interference imaging, which is different from the aberration imaging theory and the diffraction imaging principle. The first step in reconstructing the image is to solve the complex coherence coefficient of a pair of lenses. According to the partial coherence theory, the complex coherence coefficient of two apertures on the observation plane can be determined by measuring the interference fringes. The second step is to change the interference baseline and repeat the first step to achieve spectral coverage. Finally, the solution result is substituted into the Equations (1) and (2), and the inverse intensity of the target scene is obtained by using the inverse Fourier transform. The detailed process and the simplified version of the program are shown in Algorithm 1. ```
1: Initial(obj,aperture,...);
2:fori,j = 1: (size(aperture)) //Step2: Change the interference baseline
3:fors,t = 1: max max(size(obj)) //Step1: Solve one complex coherence coefficient
4:\(\Delta\)phase = 2\(\pi\)/\(\lambda\)*d(s,t);
5:\(\Pi\) = \(\Sigma(1+10^{\circ}\)cos(\(\Delta\)phase)); \(\Pi\) = \(\Sigma(1+10^{\circ}\)cos(\(\Delta\)phase + \(\pi\)/2)); //Interference fringe
6:\(\Pi\) = \(\Sigma(1+10^{\circ}\)cos(\(\Delta\)phase + \(\pi\))); \(\Pi\) = \(\Sigma(1+10^{\circ}\)cos(\(\Delta\)phase + \(3\pi\)/2));
7:end
8: absIM(i,j) = \(\mathrm{atan}((\Pi\) - \(\Pi\))/(\(\Pi\) - \(\Pi\))); //Amplitude and phase
9: angIM(i,j) = \((\Pi\) - \(\Pi\))/2/\(\mathrm{sin}(\)angIM);
10:end
11: IM = absIM*exp(angIM);
12: im = fftshift(ift2(IM)); //Step3: Image reconstruction
```  
**Algorithm 1** Imaging process and reconstruction algorithm. Solving fringes and image reconstruction  
If the interference fringe coordinate system is omitted, the fiber interference fringe intensity of any interference baseline can be expressed as [15]:  
\[I=I_{0}(1+|\mu|\cos(\varphi)) \tag{3}\]  
In current interferometric measurements, the most commonly used phase calculation method is the four-step phase shift algorithm.