# Validation of Fresnel-Kirchhoff Integral Method for the Study of Volume Dielectric Bodies
## 2 Theoretical Models
### Rigorous Solution  
The aim of this section is to solve Maxwell equations for the particular problem of the scattering of an infinite circular dielectric cylinder. These solutions are obtained without any approximation and will serve to test the validity of the Fresnel approximation made in Section 2.1. It is important to say that solutions of this kind exist in the literature [15; 16], but in this work we have chosen another route to obtain the analytical expressions for the scattering coefficients, and we believe that this derivation is interesting on its own for the scientific community. Needless to say, although the scattering coefficients are different in this work from those obtained in other derivations, the final scattering and internal electric fields are the same. In this derivation, the starting point is the scalar Helmholtz wave equation:  
\[\
abla^{2}\psi+k^{2}\psi=0 \tag{10}\]  
Once scalar solutions of previous equation are obtained, vector solutions of Maxwell equations can be found in terms of the scalar solutions by building the so-called vector harmonics:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]  
\[\overset{\rightarrow}{N}=\frac{\overset{\rightarrow}{\
abla}\times\overset {\rightarrow}{M}}{k} \tag{12}\]  
Here, \(\overset{\rightarrow}{e}\) is an arbitrary vector. In our derivation, we chose this arbitrary vector to be the unitary vector \(\overset{\rightarrow}{e}_{\rho}\), according to Figure 3, whereas in other works [15; 16] the unit vector was chosen to be \(\overset{\rightarrow}{e}_{z}\). Although \(\overset{\rightarrow}{e}_{z}\) is a natural choice for axial symmetric objects, we believe that the choice of \(\overset{\rightarrow}{e}_{\rho}\) is a more general option, allowing for the simulation of dielectric bodies with other shapes. In this way, we give another expansion of the electric and magnetic fields in terms of the new vector harmonics calculated in this work (Equations (16) and (17)). In Figure 3, the axis of the cylinder under study was chosen to be the \(z\) axis. Figure 3: Reference frame and unit vectors for cylindrical coordinates. From the particular configuration that we are treating here, it is clear that a proper choice of coordinates are the cylindrical coordinates (\(\rho\), \(\varphi\), \(z\)). In these coordinates, the Helmholtz equation takes the form:  
\[\frac{1}{\rho}\frac{\partial}{\partial\rho}\bigg{(}\rho\frac{\partial\psi}{ \partial\rho}\bigg{)}+\frac{1}{\rho^{2}}\frac{\partial^{2}\psi}{\partial\varphi ^{2}}+\frac{\partial^{2}\psi}{\partial z^{2}}+k^{2}\psi=0 \tag{13}\]  
Separable solutions of this equation can be found in the form:  
\[\psi_{v}(\rho,\varphi,z)=Z_{v}(k\rho)e^{iv\varphi}e^{i\hbar z} \tag{14}\]  
where \(v=0\), \(1\), \(2\),... and \(h\) is dictated by the form of the incident wave. In this work, we will assume that the electric field is incident parallel to the axis of the cylinder (in the \(z\) direction), so that we can consider that \(h=0\). On the other hand, \(Z_{v}\) (\(k\rho\)) satisfy the following Bessel equation [17]:  
\[\rho^{2}\frac{d^{2}}{d\rho^{2}}Z_{v}(k\rho)+\rho\frac{d}{d\rho}Z_{v}(k\rho)+( k^{2}\rho^{2}-v^{2})Z_{v}(k\rho)=0 \tag{15}\]  
For each solution of type (14), we can apply Equations (11) and (12) to calculate the corresponding vector harmonics, giving:  
\[\overset{\rightarrow}{M}_{v}=\frac{-iv}{\rho}Z_{v}(k\rho)e^{iv\varphi} \overset{\rightarrow}{e}_{z} \tag{16}\]  
\[\overset{\rightarrow}{N}_{v}=\frac{e^{iv\varphi}}{k\rho}\bigg{[}\frac{v^{2}}{ \rho}Z_{v}(k\rho)\overset{\rightarrow}{e}_{\rho}+iv\bigg{(}-\frac{Z_{v}(k \rho)}{r}+kZ_{v}^{\prime}(k\rho)\bigg{)}\overset{\rightarrow}{e}_{\varphi} \bigg{]} \tag{17}\]  
where \(Z_{v}^{\prime}(k\rho)=\frac{dZ_{v}(k\rho)}{d(k\rho)}\); that is, the prime denotes the absolute derivative of z with respect its argument. Now the electric and magnetic fields can be expanded in terms of the cylindrical harmonics.