# Validation of Fresnel-Kirchhoff Integral Method for the Study of Volume Dielectric Bodies
## 3 Results and Discussion
### Experimental Validation  
Although the Fresnel method described was validated in Section 3.1, it is interesting to observe its ability to extract information from a determined volumetric dielectric object. In this section, in order to use the expressions of Section 2.1, a small cylindrical object will be studied, which in this case was chosen to be a human hair (which can be considered nearly cylindrical).  
Figure 12 shows the experimental setup used to obtain the diffraction pattern of the hair. The light coming from a He-Ne laser (633 nm) was collimated by using a system of lenses; the sample (hair) was placed between the laser and a CCD connected to a personal computer, which was used to process the data. Figure 13 shows the diffraction pattern obtained from the hair by using this setup.  
Figure 11: Intensity of the diffraction pattern as function of the distance to the center of the diffraction pattern for a dielectric cylinder with internal radius \(a=60\) μm and a refractive index of 1.50, 1.51, 1.52, 1.53, 1.54, 1.55.  
Figure 12: Experimental setup.  
In Figure 14, the normalized intensity obtained by dividing the intensity captured by the CCD by its maximum values is shown as a function of the distance to the center of the screen. In order to fit the theoretical function to the experimental data, we used the \"Isqcurvefit\" function of MATLAB, which implemented the \"trust region reflective\" algorithm [18]. The starting values for the algorithm were: radius \(a=50\)\(\upmu\)m and refractive index \(n=1.7\); the fitting of the theoretical curve was obtained by using the method in Section 2.1. The experimental data provided a refractive index of 1.55 and an internal radius of 25.4 \(\upmu\)m. The size of the hair was also measured using an optical microscope, which found a radius of 25 \(\pm\) 2 \(\upmu\)m, whereas the standard value of the refractive index of the human hair given in the literature is 1.55 [19]. The figure also demonstrates an experimental validation of the method proposed.  
Finally, in order to demonstrate the potential of the method, this was applied to obtain the dimensions of a crack artificially created in a piece of plastic. Figure 15 shows an image obtained from an optical microscope of the piece of plastic and the crack artificially created (a), and the diffraction pattern observed (b). Figure 16 shows the experimental data (extracted from the diffraction pattern) and a theoretical fit made by using the method described in Section 2.1. In this case, Equations (2) and (3) were changed to account for an elliptical crack with refractive index 1 (air), and the refractive index of the surroundings was set to 1.51 (refractive index of plastic). The varying parameter in this case was the width of the crack. As in the case of the hair, we used the \"Isqcurvefit\" function of MATLAB with  
Figure 14: Experimental data and theoretical fit of the intensity recorded on a CCD camera from the diffraction of a hair with a refractive index of 1.55 and a radius of 25.4 \(\upmu\)m.  
Figure 13: Diffraction pattern on the CCD of a hair.  
an initial guess of 100 \(\upmu\)m. The width of the crack obtained with the aid of the microscope was 175 \(\pm\) 5 \(\upmu\)m, whereas the fit gave a value of 172 \(\upmu\)m.