# Green Scalar Function Method for Analyzing Dielectric Media
## 3 Numerical Results  
First, the analytical solutions obtained by the Fourier series development of the angular harmonics are compared with the scalar formalism of [PERSON]'s function. In order to achieve this, consider the following coordinate scheme presented in Figure 1. When a plane wave, polarized perpendicularly, impinges on the cylinder in the \(xz\)-plane with a tilt angle \(\zeta\), the intensity pattern of the scattered light is measured on a screen. On the other hand, the screen where the intensity pattern is collected can be moved along the axial axis \(y\), and the intensity can be gathered at different distances from the cylinder. In the following simulations, we can see the axial evolution of the normalized intensity of the scattered light with \(\lambda=633\) nm in Figure 1(a), moving toward the screen with respect to the cylinder, and the pattern of the scattered light on the screen at a distance of \(y=10\) mm, as shown in Figure 1(b). We can see that the proposed numerical method with the Green's function approximates and behaves in an analogous way to the analytical solution proposed by Equation (17). Moreover, in Figure 1(a), we observe that for values close to \(y=0\), the behavior of the axial intensity curve presents some abrupt changes in the intensity, which is in agreement with the scalar theory of diffraction being in the Fresnel regime close to the source. Modifying the dimensions of the dielectric medium, in this case reducing its dimensions by half (\(a=20\)\(\upmu\)m), we can observe how the pattern of the scattered beam on the screen widens with respect to the intensity pattern for a cylinder twice the size; see Figure 2(b) with respect to Figure 1(b). This result shows that the light scattered by the dielectric medium actually has its origin in the diffraction that the cylinder itself performs when incident on light [29]. On the other hand, the axial behavior of the intensity of the scattered beam, as seen in Figure 2(a), shows less variation as it approaches the source and decays faster than a larger scattering medium in comparison with Figure 1(a). These two comments are consistent with the scalar theory of light diffraction. On the other hand, the solution obtained by the scalar Green's method fits the analytical solution better (17). Figure 1: Coordinate system for a scattering cylinder. Figure 2: Cylinder with circular base \(a=40\)\(\upmu\)m and with refractive index of \(n=1.5\). Once it has been demonstrated that the numerical method fits the analytical prediction, we decided to try different geometries and to analyze the scattered field in the plane. Thus, the following results represent the light scattering of different dielectric surfaces. For this purpose, a 2D scalar Green's function given by Equation (9) is used. In Figure 3(a), we consider a cylinder with an elliptical base with a semi-major axis \(a=30\)\(\upmu\)m and semi-minor axis \(b=10\)\(\upmu\)m with refractive index \(n=1.5\). A planar wave front with a wavelength of \(\lambda=633\) nm propagating in air impinges normally on the dielectric surface, \(\zeta=\pi/2\). We can observe the pattern of the scattered field \(E^{(3)}\) once the electromagnetic wave interacts with the dielectric cylinder. It should be noted that the incident beam is on the \(y\)-axis according to the coordinate system of Figure 4. This presents two clear scattered wave fronts, which refers to the reflection of the incident wave \(E^{(1)}\). On the other hand, in Figure 3(b), the intensity of the refracted field inside the dielectric cylinder is represented. Trying other geometries, we opted for a prism with a rectangular base \(20\times 30\)\(\upmu\)m\({}^{2}\). We have in the following Figure 4(a) the scattered field by this new dielectric medium for a refractive index \(n=1.5\) and the internal field for different refractive indices, \(n=1.5\) in Figure 4(b) and \(n=1.05\) in Figure 4(c).