# New Possibilities for Constructing Heuristic Solutions to Problems of Electromagnetic Diffraction
## 3 Formulas Used in the MFC
### Modifying Functions for a Perfectly Conducting Plane Angular Sector  
An analytical solution to the problem of perfectly conducting quarter-plane diffraction appeared only a few years ago [17]. The geometry of the problem is shown in Figure 3.  
The polarization transmission coefficients \(D\) are determined by the expressions in [17], as follows:  
\[\left( \begin{array}{c}E_{\theta}^{\infty}\\ E_{\varphi}^{\infty}\end{array} \right)=\frac{\exp(ikR)}{kR}\left( \begin{array}[] {cc}D_{\theta\theta}&D_{\theta\varphi}\\ D_{\varphi\theta}&D_{\varphi\varphi}\end{array} \right)\left( \begin{array}[] {c}E_{\theta}^{inc}\\ E_{\varphi}^{inc}\end{array} \right). \tag{21}\]  
In [3,20,21], a heuristic solution to the problem of diffraction by a perfectly conducting plane angular sector is presented. In this case, the principle of \"imaginary edge\" was used. The imaginary edge is the direction in space in relation to which the directions to the source and to the observation point are disposed on the diffraction cone. A diffraction cone is a set of directions that make up the same angles with the edge [3,9]. In Figure 4, the dotted line shows an imaginary edge making an angle \(\gamma/2\) with the \(-z\) direction, while the real edge makes an angle \(\beta/2\) with this direction. The currents of physical optics are designated as \(j_{\varphi}\) (a) and \(j_{\theta}\) (b), depending on the types of polarization corresponding to Figure 3a,b. Components of currents directed perpendicularly and parallel to the imaginary edge are designated \(cr\) and \(cp\), respectively. Figure 5 illustrates the application of an imaginary edge. Figure 5a shows a scheme for solving 2D problem for diffraction of a plane wave at an infinite edge. Waves are scattered only in the directions of the diffraction cones. On the diffraction cones, the GTD and the MEW solutions coincide. Figure 4: Decomposition of the PO current for the “imaginary” edge. \(cr\): current component perpendicular to the imaginary edge, \(cp\): current component parallel to the imaginary edge. Figure 3: Geometry of the problem of diffraction on a plane angular sector. (**a**) first type of polarisation, (**b**) second type of polarisation. The wave vector of a wave incident on an edge can be divided into two components: one going along the edge, the other is perpendicular to it. This decomposition is shown in Figure 5c. Figure 5b shows an infinite edge, source position and viewpoint that are outside the diffraction cone, relative to the source. Diffraction cones are also shown, corresponding to situations where the source and observation point are swapped. If the positions of the source and observation points are interchanged, then decomposition becomes different, which leads to a violation of the principle of reciprocity. This violation has to be fought in the classical version of the MEW [9]. In Figure 5d a lilac dotted line shows an imaginary edge. This is the direction on the surface of the scatterer, determined so that the source and receiver are disposed on the diffraction cone in relation to the imaginary edge. Thus, for a solution using an imaginary edge, the reciprocity principle is fulfilled. The concept of an imaginary edge unites the approaches of the GTD and the MEW, since the GTD solution on an imaginary edge is used in order to obtain the MEW solution in directions outside the diffraction cones. A more complete set of formulas describing the heuristic solution of the diffraction problem on a plane angular sector, as well as the calculation results, are given in [3, 20, 21] and in [35, 36, 37]. Calculated data from [17] were used as a verification solution. In this case, we used the linear integral in the PO approximation (13) and the diffraction coefficients (16) and (17), which replaced the diffraction coefficients (19) and (20), contained in the PO expression for the vector potential (10) [3].