## 2 Layer-Based Shifted Fresnel Diffraction  
The Fresnel diffraction of the 2D optical wave field can be expressed by:  
\[U(u,v)=\frac{\exp(jkz)}{j\lambda z}\exp\!\left[j\frac{k}{2z}\!\left(u^{2}+v^{2 }\right)\right]\!\iint\limits_{\infty}U(x,y)\exp\!\left[j\frac{k}{2z}\!\left( x^{2}+y^{2}\right)\right]\exp\!\left[-j2\pi\!\left(f_{x}x+f_{y}y\right)\right]dxdy, \tag{1}\]where \((x,y)\) and \((u,v)\) denote the coordinates in the source plane and observation plane, respectively. \(z\) is the wave propagation distance, \(k=2/\lambda\) is the wave number, and \(f_{x}\) and \(f_{y}\) are calculated as:  
\[f_{x}=\frac{u}{\lambda z},\ f_{y}=\frac{v}{\lambda z}. \tag{2}\]  
By imposing a 2D Fourier transform, Equation (1) can be represented as:  
\[U(u,v)=\frac{\exp(jkz)}{j\lambda z}\exp\left[j\frac{k}{2z}\Big{(}u^{2}+v^{2} \Big{)}\right]F\Bigg{\{}U(x,y)\exp\left[j\frac{k}{2z}\Big{(}x^{2}+y^{2}\Big{)} \right]\Bigg{\}}, \tag{3}\]  
where \(F\) denotes the Fourier transform. According to the sampling rule of the discrete Fourier transform, the following relation exists:  
\[\Delta f_{x}=\frac{1}{N_{x}\Delta x},\ \Delta f_{y}=\frac{1}{N_{y}\Delta y}, \tag{4}\]  
where \(N_{x}\) and \(N_{y}\) are the sampling numbers in \(x\) and \(y\) coordinates, \(\Delta x\) and \(\Delta y\) are the sampling intervals in the \(x\) and \(y\) directions, and \(\Delta f_{x}\) and \(\Delta f_{y}\) are the frequency sampling intervals, respectively. Hence, the sampling distances in the observation plane can be deduced as:  
\[\Delta u=\frac{\lambda z}{N_{x}\Delta x},\ \Delta v=\frac{\lambda z}{N_{y} \Delta y}. \tag{5}\]  
where \(N_{x}\Delta x\) and \(N_{y}\Delta y\) denote the size of the sampling window in the source plane. From Equation (5), we can see that the sampling interval in the observation plane is proportional to the wave propagation distance, and inversely proportional to the size of the sampling window. The sampling ranges in the observation plane can be given by:  
\[L_{u}=N_{x}\Delta u=\frac{\lambda z}{\Delta x},L_{v}=N_{y}\Delta v=\frac{ \lambda z}{\Delta y}, \tag{6}\]  
where \(L_{u}\) and \(L_{v}\) are the sampling ranges along the \(u\) and \(v\) axes in the observation plane. We can see that the sampling parameters in the observation plane cannot be adjusted freely in the Fresnel diffraction calculation, which are determined by the sampling parameters of the source plane and the wave propagation distance. Shifted Fresnel diffraction can overcome the limitations of traditional Fourier-based Fresnel diffraction, by allowing for arbitrary sampling intervals in the source plane and the observation plane. Figure 1 is the geometric setup of the shifted Fresnel diffraction. (\(x_{0}\),\(y_{0}\)) is the center of the sampling grid in the source plane (\(x\),\(y\)), with a resolution of \(P\times Q\), and (\(u_{0}\),\(v_{0}\)) is the center of the sampling grid in the observation plane (\(u\),\(v\)), with a resolution of \(M\times N\). \(\Delta x\) and \(\Delta y\) are the sampling intervals in the source plane, and \(\Delta u\) and \(\Delta v\) are the sampling intervals in the observation plane. The coordinates of these two grids are determined by:  
\[ \begin{split} x_{p}&=x_{0}+p\Delta x,\ y_{q}=y_{0} +q\Delta y,\\ u_{m}&=u_{0}+m\Delta u,\ v_{n}=v_{0}+n\Delta v, \end{split}  \tag{7}\]  
where \(p\), \(q\), \(m\), \(n\) are the integer indices:  
\[ \begin{split}-\frac{P}{2}&\leq p\leq\frac{P}{2}-1,- \frac{Q}{2}\leq q\leq\frac{Q}{2}-1,\\ -\frac{M}{2}&\leq m\leq\frac{M}{2}-1,\ -\frac{N}{2} \leq n\leq\frac{N}{2}-1.\end{split}