## 2 Analysis of the Macroscopic FP Principle
### Forward Model  
The laser point light source emits a wave filed that is monochromatic with wavelength \(\lambda\) and is spatially coherent across the surface of the object. The illumination light field interacts with the object, and part of the light field reaches the camera sensor through the imaging system. The light field emitted from an object is a two-dimensional complex field \(O(\xi,\eta)\).  
The light filed from the object \(O(\xi,\eta)\) propagates a distance \(z_{2}\) toward the imaging system to satisfy the far field Fraunhofer approximation. The field at the pupil plane \(U(u,v)\) is related to the light field at the object \(O(\xi,\eta)\) through a Fourier transform.  
\[U(u,v)=F\{O(\xi,\eta)\} \tag{3}\]  
Because of the finite diameter of the photographic lens, only a portion of the light filed \(U(u,v)\) can be imaged onto the camera sensor. Let the pupil function be given by \(P(u,v)\). For an ideal circle pupil, \(P(u,v)\) is given by  
\[P(u,v)=\left\{\begin{array}{l}1,u^{2}+v^{2}\leq r^{2}\\ 0,otherwise\end{array}\right. \tag{4}\]  
where \(r\) is the semi-diameter of the entrance pupil of the photographic lens.  
The light field at the pupil plane \(U(u,v)\) is low-passed by pupil function (or coherent transfer function) to form an image of the object at the camera sensor.  
\[\begin{array}{ll}U^{\prime}(u,v)&=U(u,v)\bullet P(u,v)\\ &=F\{O(\xi,\eta)\}\bullet P(u,v)\end{array} \tag{5}\]  
Because the camera sensor only detects optical intensity, the LR image measured by the camera is  
\[I(x^{\prime},y^{\prime})=\left|F^{-1}\{F\{O(\xi,\eta)\}\bullet P(u,v)\}\right| ^{2} \tag{6}\]  
where \(F\) is the Fourier transform, and \(F^{-1}\) is the inverse Fourier transform.  
The equivalent coherent transfer function (CTF) after aperture synthesis most appropriately reflects the characteristics of FP imaging, and thus the equivalent coherent transfer function corresponding to FP synthetic aperture imaging is derived herein.  
\[H(u,v)=\left\{\begin{array}{l}1\,\ if\ circ(\sqrt{u^{2}+v^{2}}/r)\otimes\sum \limits_{n=1}^{N_{shift}}\delta(u-u_{n},v-v_{n})>0\\ 0\,\ if\ circ(\sqrt{u^{2}+v^{2}}/r)\otimes\sum\limits_{n=1}^{N_{shift}}\delta(u- u_{n},v-v_{n})\leq 0\end{array}\right. \tag{7}\]  
where the relationship of relative position between the individual sub-aperture centers can be expressed as Equation (8):  
\[\left\{\begin{array}{l}u_{n}=u_{n-1}+d(1-overlap)\\ v_{n}=v_{n-1}+d(1-overlap)\end{array}\right. \tag{8}\]  
where \(overlap\) the overlap denotes the spectrum overlap ratio, \(N_{shift}\) represents the total number of camera scans, and \(d\) is the diameter of the pupil of the photographic lens.  
The synthetic aperture range corresponding to a single aperture after a two-dimensional scanning translation stage is presented in the left side of Figure 2, whereas the right side represents the derived equivalent synthetic aperture coherence transfer function expressed by Equation (7). As observed, the two results were highly similar.  
The iterative algorithm of FP imaging shows that the quality of the reconstructed image is also related to the amount of overlap between adjacent images. The effect of overlap ratio on the reconstruction of the ptychography iterative engine (PIE) has been described clearly [10; 11]; meanwhile, we directly ported the results from real space PIE to spectrum plane of FP and studied the effect of the root mean square error (RMSE) and resolution of high-resolution image. Therefore, in this study, we scanned the spectrum plane of the object using a camera to acquire low-resolution images with a varying amount of overlap between adjacent images, which determined the distance the camera + image lens movements and the image reconstruction quality and is the important parameter we have to consider.