# Validation of Fresnel-Kirchhoff Integral Method for the Study of Volume Dielectric Bodies
## 2 Theoretical Models  
### Fresnel Integral Method  
The basic ideas of the method proposed are explained in this section. We are interested in obtaining some particular parameters, such as refractive index, size, etc. of a particular object. The parameters to be evaluated must have the ability to change the phase of the incident light. In Figure 1, a basic scheme for the situation proposed is shown. We will assume an object with a refractive index \(n\) and a thickness \(d\), allowing both parameters to vary inside the object. If plane B is described with coordinates (\(x^{\prime}\),\(y^{\prime}\)), and the amplitude of the incident light is \(U_{A}\), then the amplitude of light at plane B can be calculated as \(U_{B}=U_{A}\exp(iknd)=U(x^{\prime},y^{\prime})\), where \(k\) is the wavenumber, related to the wavelength of light, \(\lambda\), as \(k=2\pi/\lambda\). Once the amplitude \(U(x^{\prime}\),\(y^{\prime})\) is evaluated at plane B, the field \(U(P)\) at a point \(P\) (\(x\),\(y\)) (Figure 2) can be obtained by applying the Fresnel approximation to the Fresnel-Kirchhoff integral [8]:  
\[U(P)=\frac{e^{ikz}}{i\lambda z}\iint_{-\infty}^{\infty}U\big{(}x^{\prime},y^{ \prime}\big{)}\exp\bigg{\{}i\frac{k}{2z}\Big{[}\big{(}x-x^{\prime}\big{)}^{2 }+\big{(}y-y^{\prime}\big{)}^{2}\Big{]}\bigg{\}}dx^{\prime}dy^{\prime} \tag{1}\]  
The intensity of the field at point \(P\)(\(x\),\(y\)) is computed as \(I(x\),\(y)=1\)\(U(x\),\(y)\)\(|\)\({}^{2}\). By fitting the theoretical curve of the intensity to experimental data, one can extract information from the object under study. We will demonstrate this in Section 3. It is noticeable that in this method, the features of a volumetric object are described by a phase function, which is introduced into Equation (1). Therefore, the three-dimensional information of the object is incorporated in a two-dimensional integral. This, in our opinion, is an advantage of this method over other numerical methods existing in the literature [13], since we have reduced a three-dimensional problem to a two-dimensional one, with the correspondent saving of  
Figure 1: The volume object. computational time. On the other hand, since the method is based on the evaluation of the integral (1), which is closely related to the Fresnel transform [14], it is well suited for solving inverse problems such as in [14]. It has nonetheless some limitations; for instance, since the method makes use of the Fresnel-Kirchhoff integral, the method could be inaccurate for large values of the Fresnel number, defined in Section 3. For the particular case of a dielectric circular cylinder, we will assume that the phases accumulated by light from a plane just in front of the cylinder, A, and another just after it, B, are [12]:  
\[\varphi_{a}=2 ka,\qquad\qquad|x^{\prime}|>a \tag{2}\]  
\[\varphi_{C}=2k(n_{C}-1)\sqrt{a^{2}-x^{\prime 2}},\ |x^{\prime}|<a \tag{3}\]  
where \(n_{c}\) is the refractive index of the cylinder and \(a\) is the radius of a circle representing the cross-section of the cylinder. \(\varphi_{a}\) takes into account the phase accumulated in air, and \(\varphi_{c}\) is the phase accumulated by the cylinder. Now, assuming that a unit amplitude wave is incident onto the cylinder, \(U(x^{\prime})\) can be evaluated as \(U(x^{\prime})=\exp(\mathrm{i}k\varphi)\), where \(\varphi\) takes into account at each point \(x^{\prime}\) the different contributions of \(\varphi_{a}\) and \(\varphi_{C}\).