## 2 MAS for Eye-Shaped Scatterers  
We assume an infinitely long cylinder with a cross section that resembles an eye (Figure 1). We investigate two separate cases: perfect electric conductor (PEC) or linear- homogeneous-isotropic dielectric. The geometry of the scatterer, depicted by a solid line, comprises two circular arcs with identical radii equal to \(\rho\), but with different centers. In particular, the Cartesian coordinates of the upper arc are given by (1):  
\[x_{u}=\rho\cos\varphi,\ y_{u}=\rho\sin\varphi-d, \tag{1}\]  
whereas those of the lower arc are given by (2):  
\[x_{l}=\rho\cos\varphi,\ y_{l}=\rho\sin\varphi+d, \tag{2}\]  
where \(\varphi\) is the azimuth angle and \(\pm d\) is the vertical displacement of each arc center, taken equal to the arc apothem (see Figure 2). Obviously, \(\varphi\) does not span the entire \([0,2\pi)\) interval, but it is limited by the arc width itself, which is given by (3):  
\[\varphi_{arc}=2\arccos\frac{d}{\rho}. \tag{3}\]  
The scatterer is illuminated by a TM plane wave impinging from an azimuth angle equal to \(\varphi_{inc}\). Therefore, the incident electric field \(E_{inc}\) is given by (4):  
\[E_{inc}(x,y)=E_{0}\exp[jk_{0}(x\cos\varphi_{inc}+y\sin\varphi_{inc})]\hat{ \mathbf{z}}, \tag{4}\]  
Figure 1: Geometry of the scatterer (solid line), including inner auxiliary sources (ASs) (diamonds) and outer auxiliary sources (ASs) (squares). The dots stand for collocation points (CPs) and circles for midpoints (MPs). Figure 2: Construction of the geometry. where \(E_{0}\) is the amplitude of the incident electric field, \(k_{0}\) is the free space wavenumber, and a hat denotes a unit vector along the corresponding direction. The incident magnetic field \(\mathbf{H}_{inc}\) is given by (5):  
\[\mathbf{H}_{inc}(x,y)=-\frac{E_{0}}{\zeta_{0}}(\sin\varphi_{inc}\hat{\mathbf{x}}-\cos \varphi_{inc}\hat{\mathbf{y}})\exp\{jk_{0}(x\cos\varphi_{inc}+y\sin\varphi_{inc})\}, \tag{5}\]  
where \(\zeta_{0}\) is the free space intrinsic impedance. To solve the scattering problem via MAS, two sets of ASs are generally defined, each one of multitude \(N\), as shown in Figure 1. In the PEC case, only the inner set is used; the outer set is necessary only in the dielectric configuration. In standard MAS formulation, both inner and outer auxiliary surfaces are conformal to the scatterer boundary. The electric field due to the \(n\)th inner AS, located at point \(\mathbf{r}_{n}\) and radiating in the outer space, is as follows:  
\[E_{sn}(\mathbf{r})=\hat{z}E_{n}H_{0}^{(2)}(k_{0}|\mathbf{r}-\mathbf{r}_{n}|), \tag{6}\]  
where \(E_{n}\) is the corresponding unknown weight, \((n=1,2,\dots,\,N)\), and \(H_{0}^{(2)}\) is the Hankel function of the zero order and second kind ([PERSON]'s function). The corresponding magnetic field of the \(n\)th auxiliary source is obviously proportional to the curl of (6), given explicitly in [20]. Similar expressions hold for the outer ASs, radiating in the inner space of the dielectric scatterer, except for \(k_{0}\) and \(\zeta_{0}\), which have to be replaced by \(k\) and \(\zeta\), respectively, corresponding to the scatterer's dielectric properties. The total scattered \(E\) field is expressed as the superposition of the fields in (6) and the \(H\) field accordingly. By applying the boundary conditions for both fields at the \(N\) collocation points (CPs) \((x_{m},y_{m})\)\((m=1,2,\dots,\,N)\) of the scattering boundary (blue solid dots in Figure 1), we cast a linear system of equations as in (7):  
\[[\mathbf{Z}][\mathbf{I}]=\{\mathbf{V}\}, \tag{7}\]  
where \(\{\mathbf{I}\}\) is the column vector of the unknown weights \(E_{n}\). In the PEC case, \([\mathbf{Z}]\) is a square matrix of size \(N\times N\) with elements determined by the interaction between ASs and CPs, and \(\{\mathbf{V}\}\) is the column vector of the incident \(E\) fields calculated at the CPs.