## 3 Intelligent Reconfigurable Surface Model
### IRS Incidence and Radiance Angles  
Prior to the derivation of the IRS incidence and radiance angles, it is important to mention that by means of the generalized reflection law (generalized Snell's law) a relativistic relationship can be provided between the direction of the reflected and incident rays with respect to the normal to the surface and the IRS phase discontinuity. Therefore, the generalized reflection law can be written as [9]  
\[\cos(\theta_{R})\sin(\phi_{R})=\frac{\lambda}{2\pi n_{R}}\frac{\delta\phi_{m}} {\delta r^{\prime}}, \tag{7}\]  
\[\sin(\theta_{R})-\sin(\theta_{I})=\frac{\lambda}{2\pi n_{R}}\frac{\delta\phi_{ m}}{\delta i^{\prime}}, \tag{8}\]  
where \(\theta_{R}\) is the angle between the reflected light beam and its projection on the plane orthogonal to the plane of incidence and the IRS patch, \(\phi_{R}\) is the angle between the vector normal to the IRS and the projection of the reflected light beam in the plane orthogonal to both the plane of incidence and the IRS, \(\lambda\) denotes the wavelength, \(n_{R}\) represents the refractive index of the incidence medium, \(r^{\prime}\) is the counterclockwise rotated version of the \(x\) axis about the \(y\) axis, \(\theta_{I}\) is the incidence angle on the IRS, which is considered as the angle between the incident light beam and the vector normal to the IRS, and \(i^{\prime}\) represents the counterclockwise rotated version of the \(z\) axis about the \(y\) axis.  
The generalized relativistic expressions of [PERSON]'s law can be developed in terms of the Cartesian coordinates of the scenario and use basic trigonometric identities. Therefore, we can obtain a closed-form representation of \(\theta_{R}\) and \(\phi_{R}\) as follows:  
\[\sin(\theta^{\prime})\sin(\phi^{\prime})-\sin(\theta_{R})\sin(\phi_{R})=\frac {\lambda}{2\pi n_{R}}\frac{\delta\phi_{m}}{\delta x}, \tag{9}\]  
\[\sin(\theta^{\prime})\sin(\phi^{\prime})-\sin(\theta_{R})\cos(\phi_{R})=\frac {\lambda}{2\pi n_{R}}\frac{\delta\phi_{m}}{\delta z}, \tag{10}\]  
or, finding the expressions in terms of \(\theta_{R}\) and \(\phi_{R}\),  
\[\theta_{R}=\sin^{-1}\Biggl{(}\left(\left(\left(\sin(\theta^{\prime})\sin( \phi^{\prime})-\frac{\lambda}{2\pi n_{R}}\frac{\delta\phi_{m}}{\delta x}\right) ^{2}+\left(\sin(\theta^{\prime})\cos(\phi^{\prime})-\frac{\lambda}{2\pi n_{R} }\frac{\delta\phi_{m}}{\delta z}\right)^{2}\right)^{\frac{1}{2}}\Biggr{)}, \tag{11}\]  
\[\phi_{R}=\tan^{-1}\Biggl{(}\frac{\sin(\theta^{\prime})\sin(\phi^{\prime})-\frac {\lambda}{2\pi n_{R}}\frac{\delta\phi_{m}}{\delta x}}{\sin(\theta^{\prime}) \cos(\phi^{\prime})-\frac{\lambda}{2\pi n_{R}}\frac{\delta\phi_{m}}{\delta z} }\Biggr{)}+\pi\mathbb{I}\biggl{(}\sin(\theta^{\prime})\cos(\phi^{\prime})- \frac{\lambda}{2\pi n_{R}}\frac{\delta\phi_{m}}{\delta z}\biggr{)}, \tag{12}\]  
where \(\theta^{\prime}\) denotes the angle between the normal to the IRS and the reflected ray, \(\phi^{\prime}\) represents the angle between the projection of the reflected ray onto the IRS and a ray parallel to the z-axis lying in that plane, and \(\mathbb{I}(s)\) is an indicator function with a binary behavior, depending on whether the condition \(s\) is satisfied (\(\mathbb{I}(s)=1\)) or not (\(\mathbb{I}(s)=0\)).