# 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
In order to calculate the error, the solution given by the Fresnel method and the rigorous one were calculated at each pair of values (_a,n_). The error was calculated at the target plane for every tested (_a,n_) pair using the L2-norm of the error under constant illumination. This error was divided with respect to the intensity obtained by the rigorous method integrated at the target plane. From the figure, two conclusions can be obtained: on the one hand the error in the range of the refractive index is \(\approx 0.01\). Figure 8: Intensity of the diffraction pattern as function of the distance to the center of the diffraction pattern for a dielectric cylinder with internal radius \(a=80\). (**a**) with refractive index \(n=1.85\); (**b**) with refractive index \(n=3.4\). The intensities were calculated assuming that the diffraction pattern was measured at an axial distance \(z=50\) mm from the fiber. Figure 7: Intensity of the diffraction pattern as a function of the distance to the center of the diffraction pattern for a dielectric cylinder with internal radius \(a=30\). (**a**) with refractive index \(n=1.85\); (**b**) with refractive index \(n=3.4\). The intensities were calculated assuming that the diffraction pattern was measured at an axial distance \(z=50\) mm from the fiber. values evaluated mainly increased with the radius of the cylinder; on the other hand, it can be seen that small differences existed between the two methods. To evaluate the sensitivity of the method to changes in the refractive index and radius, simulations were carried out starting with the rigorous solution, with values for the refractive index of \(n\) = 1.5 and \(a\) = 60 \(\upmu\)m. Then simulations were performed using the Fresnel method, slightly varying the radius and the refractive index. Figure 10 shows the intensity profile at a distance of 20 mm from the cylinder. The rigorous solution was calculated for a refractive index of \(n\) = 1.5 and \(a\) = 60 \(\upmu\)m, and the Fresnel method was performed for thickness values of 60, 62, 64, 66, 68, and 70 \(\upmu\)m. From the figure, it is clear that variations of 2 \(\upmu\)m in the radius created visible changes in the shape of the curves obtained using the Fresnel method. The distance from the maximum value to the minimum value of the curve also changed with an increase or decrease of 2 \(\upmu\)m in the radius of the cylinder. Figure 11 shows also the intensity profile at a distance of 20 mm from the cylinder, but in this case the modified parameter was the refractive index. The rigorous solution  
Figure 10: Intensity of the diffraction pattern as function of the distance to the center of the diffraction pattern for a dielectric cylinder with refractive index \(n\) = 1.50 and internal radii of 60, 62, 64, 66, 68, and 70 \(\upmu\)m.  
Figure 9: Relative error of the Fresnel method with respect to the rigorous one. was calculated for a refractive index of \(n=1.5\) and \(a=60\) um, and the Fresnel method was performed for values of the refractive index of 1.50, 1.51, 1.52, 1.53, 1.54, and 1.55. As in Figure 10, variations of 0.01 created visible changes in the shape of the curves and also in the difference between the maximum and minimum value of the curve. On the other hand, it is interesting to note that small changes in the radius provoked different, although subtle, variations in the theoretical curve than in the case of changing the refractive index. For instance, if one looks the behavior of the curves in the range of x: (\(-130\), \(-70\)) um, or symmetrically in the range of x: (\(70\), \(130\)) um, one can see that in the transformation of the curve at starting values (red curve) to that of the final values (light blue curve), those transforming through changes in the radius possessed a local maximum and a local minimum, whereas those transforming through changes in the refractive index did not.