# Validation of Fresnel-Kirchhoff Integral Method for the Study of Volume Dielectric Bodies
## 2 Theoretical Models
### Rigorous Solution
Outside the cylinder, the electric and magnetic fields are obtained by the sum of the scattered (denoted by the subscript \"\(s\)\"), and the incident (subscript \"\(i\)\") one:  
\[\overset{\rightarrow}{E}_{T}=\overset{\rightarrow}{E}_{i}+\overset{ \rightarrow}{E}_{s} \tag{18}\]  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]0  
We will also denote the fields inside the cylinder with the subscript (1). We will consider three expansions for the electric field: incident, scattered, and inside the cylinder. The proper Bessel functions \(Z_{v}(k\rho)\) will be chosen accordingly for each case. In particular, since the electric field must be finite at the origin, the Bessel functions of the first kind \(J_{v}(k\rho)\) will be chosen in the cases of the incident and the internal field. In the case of the scattered field, Hankel functions \(H_{v}(k\rho)\) will be chosen as the generating functions, since their asymptotic behavior is that of a decaying wave at large distances. The expansion of the electric field is:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]1  
where \(j=i\), \(1\), \(s\) for the incident, internal, and scattered fields, respectively. The corresponding magnetic fields are:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]2  
In this particular work, we will assume that light is polarized in the \(z\) axis direction. For this particular case, it is easy to see that the incident electric field depends only on the \(\stackrel{{\rightarrow}}{{M}}_{v}\) harmonics and the magnetic field on the \(\stackrel{{\rightarrow}}{{N}}_{v}\) harmonics. So, the electric field expansion for the incident light is:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]3  
and the expansion for the incident magnetic field is:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]4  
On the other hand, assuming a unit incident plane wave, the incident electric field can also be expressed in the form:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]5  
Now, making use of the expansion of the exponential function in terms of Bessel functions [17]:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]6  
So, comparing Equations (22) and (24) and making use of Equations (16) and (25), the expansion coefficients for \(E_{i}\) and \(H_{i}\) can be calculated as:  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]7  
For the case of the scattered and internal fields, the values of \(A_{v}^{(j)}\) and \(B_{v}^{(j)}\) are obtained imposing boundary conditions on the surface of the cylinder (\(\rho=a\)):  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]8  
\[\overset{\rightarrow}{M}=\overset{\rightarrow}{\
abla}\times\left(\overset{ \rightarrow}{e}\psi\right) \tag{11}\]9  
where \(\stackrel{{\rightarrow}}{{n}}\) is a unit vector perpendicular to the surface of the cylinder and directing outward, which in this case coincides with \(\stackrel{{\rightarrow}}{{e}}_{\rho}\). This is equivalent to saying that the \(z\) and \(\rho\) components of the electric and magnetic fields are continuous at the surface of the cylinder.