# Validation of Fresnel-Kirchhoff Integral Method for the Study of Volume Dielectric Bodies
## 3 Results and Discussion  
### Validation of the Fresnel Method by Comparison with the Rigorous Solution  
The aim of this section is the validation of the more general Fresnel method for volume objects described in Section 2.1, which can be applied to a great number of situations, by the comparison with the rigorous solution for the dielectric cylinder obtained in Section 2.2. To make a proper comparison, the parameters in the formalism of Section 2.2 must be addressed adequately. We present in Figure 4 the geometric scheme we considered. In this case, we want to obtain the intensity pattern created by a cylinder when light impinges on it perpendicularly to its axis on a screen that is positioned at a distance of \(z_{p}\) from the cylinder. For the case of the rigorous solution, the intensity is obtained as:  
\[I=E_{T}^{2} \tag{31}\]  
where \(E_{T}\) is the modulus of the total electric field of Equation (20). The intensity for the Fresnel approach is obtained from Equation (4), and multiplying the amplitude by its complex conjugate. Figures 5 and 6 show the comparison of both theories for a circular cylinder. The intensity pattern was calculated at a screen that was positioned at an axial distance that was varied in the simulations. The wavelength of the incident light was chosen to be 633 nm, while the radius of the cylinder was considered as 30 \(\upmu\)m in the case of Figure 5 and 80 \(\upmu\)m in the case of Figure 6. It can be observed that the behavior of both curves, one obtained by using the Fresnel approximation and the other by using the rigorous solution, behaved in the same manner as a function of the axial distance. Both models gave basically the same results, but it was clear that the higher the distance to the screen, the better the agreement between both models. This is due to the fact that the Fresnel approximation works better for lower values of the Fresnel number (NF) [8], which is defined as:  
\[NF=\frac{a^{2}}{\lambda z} \tag{32}\]  
where \(a\) is dimension parameter of the object under study (the radius of the cylinder in this case), \(\lambda\) is the wavelength of light, and \(z\) is the distance to the screen. Increasing values of \(z\) give lower values of \(NF\), and therefore a better behavior for the Fresnel approximation. The increase of the refractive index also worsens the results of Fresnel method slightly, which can be observed when comparing parts (a) and (b) of Figures 5 and 6. Here, part (a) corresponds to a refractive index \(n=1.85\), whereas (b) corresponds to \(n=3.4\), which is considerably higher. Figure 4: Geometric scheme for the rigorous method. Despite the slight disagreement observed for both methods in the range of values considered, it must be said that the distances considered in the simulations of Figures 5 and 6 were rather conservative, since typical measuring distances from the object to the camera (CCD) were higher than 6 mm, which was the maximum distance considered in the simulations. Figures 7 and 8 show the diffraction pattern observed at a screen situated 50 mm from the cylinder for a radius of 30 \(\upmu\)m and 80 \(\upmu\)m, respectively. The agreement of both theories was clear in this case, thus validating the method proposed to the simulation of volume dielectric bodies. Figure 5: Intensity of the diffraction pattern as a function of the axial distance for a dielectric cylinder with internal radius \(a=30\)\(\upmu\)m: (**a**) with refractive index \(n=1.85\); (**b**) with refractive index \(n=3.4\). Figure 6: Intensity of the diffraction pattern as function of the axial distance z for a dielectric cylinder with internal radius \(a=80\)\(\upmu\)m: (**a**) with refractive index \(n=1.85\); (**b**) with refractive index \(n=3.4\). This is also clear from Figure 9, where the relative error of the Fresnel method with respect to the rigorous one is depicted for a range of values for the refractive index from 1.3 to 4 and values of the radius of the cylinder from 30 to 90.