## 3 Analysis
### A Scalar Theory-Based Compound System  
We have seen that the scalar system of theory enables encompassing the concept of the compound system just like geometrical optics. Figure 6 shows how the compound system works. For \(g(x,0)\), propagating and crossing two lenses spaced longitudinally (deviation of \(d\)). We start with the focal length (\(f\)), and then calculate the positions \(h\) and \(h^{\prime}\). To do so, we use the relations (8) and (9). To calculate the field \(g^{\prime}(x,z^{\prime})\), the Fresnel transform is applied. Then, the result is multiplied by the transmittance of the first lens. Then, the Fresnel transform is applied to cover the distance between the two lenses. The diffraction field obtained will be multiplied by the transmittance of the second lens. In the last step, the Fresnel transform is applied behind the second system. For avoiding all this complexity, the equivalent system is introduced. It replaces the two spherical lenses by a single lens, having the back focal length \(f^{\prime}\), which means having transmittance: \(\exp\left(-j\pi\frac{x^{2}}{\lambda f^{\prime}}\right)\).  
Figure 6: Diffraction-based compound system, corresponding to the architecture of Figure 4, to treat the aberration field \(a(x)\).  
All what we need is to calculate the field \(g(x,z)\) by using the Fresnel transform. This brings us to the system, the transmittance of which we know. We multiply this transmittance by \(g(x,z)\) to get \(g^{\prime}(x,0)\). Then, we apply the Fresnel transform on this product, which will result in \(g^{\prime}(x,z^{\prime})\). Note that the distance between H and H\({}^{\prime}\)(Figure 6) has no optical effect.  
The return path, i.e., from the diffraction field \(g^{\prime}(x,z^{\prime})\) to the object \(g(x)\), is feasible in the same way, which gives us an inverse transformation. As a result, iterative optimization methods, such as the Gerchberg-Saxton method [27; 28], become easily feasible. This enables going back and forth between the input and output of Figure 6, to satisfy certain constraints imposed by the application.  
In addition, our approach covers aberrations elegantly. Let us take the case where the first lens is affected by optical aberrations (Figure 6). Our approach remains the same. The wavefront aberration \(w(x)\) is included in the analysis. The diffraction field \(g(x,z)\) will be calculated in three steps: the Fresnel transform for \(z\)-\(h\), a multiplication by \(a\) (\(x\)), and the Fresnel transform for \(h\) (Figure 6). The field \(a(x)\) is expressed as a function of \(w(x)\):  
\[a(x)=\exp\left(j\frac{2\pi}{\lambda}w(x)\right) \tag{28}\]  
Our previous analysis limited attention to a system composed of two lenses. This analysis can be generalized straightforwardly.  
Aberration is treated as a diffractive element, which has a transmission function. This transmission function is considered a wavefront field \(a(x)=\ \exp\left(j\frac{2\pi}{\lambda}w(x)\right)\). We do not need any additional element. This field is treated as g(x) is in Equation (1).  
We also handle the effect of the finite extent of refractive and diffractive elements easily. An airy spot is observed when light goes across any size aperture, and diffraction occurs. Every lens has a finite aperture; therefore, diffraction occurs because of the borders. We consider the finite extent of the lens as a circular aperture, placed just behind an infinitely extended lens. This circular aperture acts as a diffractive amplitude element, which has a binary transmission function. We treat it as we treat \(a(x)\) in Figure 6.