## 3 Analysis
### Diffraction-Based Compound System  
Let us again consider the compound system that is composed of two lenses (Figure 4). We intend to determine the output, means of the field \(g_{2}(x,z_{3})\) as a function of the object, and means of \(g(x)\) and the system parameters: \(d,f_{1}\), and \(f_{2}\). For simplicity, we decompose the calculation of the diffraction field in four successive steps (Figure 4). We go through the intermediate fields: \(g(x,z_{1})\), \(g_{1}(x,0)\), \(g_{1}(x,z_{2})\), \(g_{1}(x,d)\), \(g_{2}(x,0)\). To bypass heavy mathematical algebra by using the Fresnel transform (relation (1)), we suggest using the lens-based model of the diffraction system (Figure 3a). By using the model indicated in Figure 3a, Fresnel diffraction is performed in four steps. This leads to the setup of Figure 5. While the initial (original) lenses of Figure 4 are presented in dark gray color in Figure 5, the lenses of the lenses-based model are presented in light gray color. In step 1, two divergent lenses (in light gray), with focal length \(-z_{1}\), as well as a convergent spherical lens, with focal length \(z_{1}\), are involved in the diffraction process. These three lenses implement the Fresnel transform of \(g(x,0)\), at distance \(z_{1}\), means \(g(x,z_{1})=\text{FR}_{z1}[g(x,0)]\), according to the scheme of Figure 3a. Then, light goes through the first original lens (dark gray), with focal length \(f_{1}\). Step 2 puts three lenses into play, which  
Figure 4: System composed of two special lenses, separated by \(d\), and having different or equal focal lengths, i.e.,\(f_{1}\) and \(f_{2}\). For an object \(g(x,0)\), the output diffraction field \(g_{2}(x,z3)\) could be calculated in four steps. physically implement the Fresnel transform of \(g_{1}(x,0)\), at distance \(z_{2}\) and means \(g_{1}(x,z_{2})=\text{FR}_{22}\{g_{1}(x,0)\}\), according to the scheme of Figure 3a. Just behind the first initial lens (dark gray), we find the first divergent lens (light gray). Three side-by-side lenses are shown on the left-hand side of Figure 5. The three lenses cancel each other if:  
\[\frac{1}{f_{1}}=\frac{1}{z_{1}}+\frac{1}{z_{2}} \tag{12}\]  
For \(z_{2}\) satisfying Equation (12), the three lenses are without effect (see cross in Figure 5). Step 3 puts three lenses into play, which implement the Fresnel transform of \(g_{1}(x,z_{2})\), at distance \(d\)-\(z_{2}\) and means \(g_{1}(x,z_{2}+d-z_{2})=g_{1}(x,d)=\text{FR}_{d\to 2}\{g_{1}(x,z_{2})\}\), according to the scheme of Figure 3a. As depicted in Figure 5, we obtain two side-by-side lenses, with respective focal lengths -\(z_{2}\) and \(z_{2}\)-\(d\), which could be replaced by one lens with power: \(\text{F}_{\text{c}}=-1/z_{2}\)\(-1/(d-z_{2})\). The focal length is its inverse:  
\[f_{c}=\frac{1}{-1/z_{2}-1\ /(d-z_{2})}=z_{2}\frac{z_{2}-d}{d} \tag{13}\]  
Thus, light propagating until just behind this compound lens, with focal length \(f_{c}\), comes across four lenses, with respective focal lengths (from left to right in Figure 5): \(-z_{1}\), \(z_{1}\), \(z_{2}\), and \(f_{c}\), bearing in mind that three lenses on the left-hand side of Figure 5 are canceled out, and two other lenses are compounded together into one lens, with focal length \(f_{c}\). Among these four lenses, the first and fourth lens are divergent (focal lengths: \(-z_{1}\) and \(f_{c}\)), whereas the two lenses in the middle, namely the lenses with focal lengths \(z_{1}\) and \(z_{2}\), are convergent. Each of these two convergent lenses physically implement the Fourier transform (FT) of the wavefront in its front focal plane. The two successive FTs give us the zoomed version of the input wavefront of the first lens by the factor: \(z_{2}/z_{1}\). The first and fourth lens, among these four lenses, respectively introduce the two quadratic phase distributions \(\exp\left(j\pi\frac{x_{1}^{2}}{\lambda z_{1}}\right)\) and \(\exp\left(-j\pi\frac{x^{2}}{\lambda f_{c}}\right)\), which are the transmittances of two divergent lenses.