**5. Scattering inside the dielectric medium**

For each such element, the far-zone field is computed and then superposed over all elements, as follows:

$$E\_{\rm far}^{\rm scat} = \eta \sqrt{\frac{k}{8\pi\rho}} e^{-j3\pi\zeta\_{\rm f}} e^{-jk\rho} \sum\_{i=1}^{K} f\_x^{(i)} e^{jk\left(\mathbf{x}\_m^{(i)} cos\rho + \mathbf{y}\_m^{(i)} sin\rho\right)} \Delta l^{(i)},\tag{25}$$

where *K* is the number of elements adjacent to the boundary, *x*ð Þ*<sup>i</sup> <sup>m</sup>* , *y*ð Þ*<sup>i</sup> m* � � the i-th segment midpoint, and *Δl* ð Þ*<sup>i</sup>* the i-th segment length.

Since FEM is formulated in terms of the scattered electric field, computation of the derivatives of the scattered field requires additional effort. This can simply be achieved by using the weighted sum of derivatives of shape functions in terms of nodal scattered fields.

#### **5.1 Scattering inside the dielectric medium, TM case**

For the dielectric object, the TM case, the interior part of the object should be included. Since the magnetic current density becomes nonzero, the integral containing the magnetic current density should be evaluated. The magnetic current density has *x*- and *y*-components, which is defined as M ¼ *a*^*xMx* þ *a*^*yMy*. Hence, it shows that ^*ρ* � M ¼ cos *φMy* � sin *φMx* � �*a*^*z*. The components of the magnetic current density can be determined in terms of the scattered electric field as follows:

$$\mathbf{M} = \mathbf{E} \times \hat{n} = -n\_{\mathbf{y}}E\_{\mathbf{z}}\hat{a}\_{\mathbf{x}} + n\_{\mathbf{x}}E\_{\mathbf{z}}\hat{a}\_{\mathbf{y}},\tag{26}$$

Finally, the scattered electric field can be obtained as follows:

$$\begin{split} E\_{\rm far}^{\rm exact} &= \sqrt{\frac{k}{8\pi\rho}} e^{-j\lambda\tau\_{4}} e^{-jk\rho} \sum\_{i=1}^{K} \left( n\_{\rm x} \cos\rho + n\_{\rm y} \sin\rho \right) E\_{\rm x}^{(i)} e^{jk\left(\mathbf{x}\_{m}^{(i)} \cos\rho + \mathbf{y}\_{m}^{(i)} \sin\rho \right)} \Delta l^{(i)} \\ &+ \eta \sqrt{\frac{k}{8\pi\rho}} e^{-j\lambda\tau\_{4}} e^{-jk\rho} \sum\_{i=1}^{K} I\_{\rm x}^{(i)} e^{jk\left(\mathbf{x}\_{m}^{(i)} \cos\rho + \mathbf{y}\_{m}^{(i)} \sin\rho \right)} \Delta l^{(i)} \end{split} \tag{27}$$

Here, *E*ð Þ*<sup>i</sup> <sup>z</sup>* can be determined as the average of nodal field values connected to the boundary.

#### **5.2 Scattering inside the dielectric medium, TE case**

For dielectric object, TE case: The magnetic current density is nonzero and has only *z*-component, M ¼ *a*^*zMz* because ^*ρ* � ð Þ¼� ^*ρ* � M *Mza*^*z*. The scattered magnetic field is determined as:

$$\begin{split} H\_{\text{far}}^{\text{scat}} &= \sqrt{\frac{k}{8\pi\rho}} e^{-j3\pi\not{q}} e^{-jk\rho} \sum\_{i=1}^{K} \left( J\_{\text{x}}^{(i)} \sin\rho - J\_{\text{y}}^{(i)} \cos\rho \right) e^{jk\left(\mathbf{x}\_{n}^{(i)} \cos\rho + \mathbf{y}\_{n}^{(i)} \sin\rho \right)} \Delta l^{(i)} \\ &+ \eta \sqrt{\frac{k}{8\pi\rho}} e^{-j3\pi\not{q}} e^{-jk\rho} \sum\_{i=1}^{K} M\_{\text{x}}^{(i)} e^{jk\left(\mathbf{x}\_{n}^{(i)} \cos\rho + \mathbf{y}\_{n}^{(i)} \sin\rho \right)} \Delta l^{(i)} \end{split} \tag{28}$$
