## 2 Mathematical Formalism  
In this section, we will show a mathematical formalism which supports the statement of a reduction in the edge illumination of a reflectarray and which involves fewer trap points between \(\mathcal{R}\) and \(\mathcal{M}\). Let us consider a complex matrix \(\left(E_{0x}^{\text{inc}}(x,y)e^{j\varphi(x,y)}\right)\) with \(M\times N\) entries. The \(mn\)th entry is the evaluation of the \(x\) component of the incident electric field, with a shifted phase given by \(\varphi(x,y)\) on the following partition of the spatial domain:  
\[x =x^{\prime}+m\Delta x-\frac{(M-1)\Delta x}{2};m=0,1,2,\ldots,M-1 \tag{3}\] \[y =y^{\prime}+n\Delta y-\frac{(N-1)\Delta y}{2};n=0,1,2,\ldots,N-1 \tag{4}\]  
where \(\Delta x\) and \(\Delta y\) are the steps of the partition of \(S_{\text{RA}}\) and \(M\times N\) is the number of elements of the partition. The variables \(x^{\prime}\) and \(y^{\prime}\) are bounded by \(0\leq x^{\prime}<\Delta x\) and \(0\leq y^{\prime}<\Delta y\). Now, let us consider a complex matrix \(\left(E_{\text{CP}}^{\text{rad}}\right)\) with \(M\times N\) entries. The \(mm\)th entry is the evaluation of the main component (co-polar component) of the radiated electric field on the angular sample, which is given by the following (see Equations (7) and (8) in [9]):  
\[\theta[p,q]=\arcsin(\sqrt{u^{2}[p]+v^{2}[q]}) \tag{5}\]\[\varphi[p,q]=\left\{ \begin{array}{ll}0&\text{if }\;v[q]=0\\ \frac{\pi}{2}&\text{if }u[p]=0,\;v[q]>0\\ \pi+\arctan\Bigl{(}|\frac{v[q]}{u[p]}|\Bigr{)}&\text{if }u[p]<0,\;v[q]>0\\ -\frac{\pi}{2}&\text{if }u[p]=0,\;v[q]<0\\ -\pi+\arctan\Bigl{(}|\frac{v[q]}{u[p]}|\Bigr{)}&\text{if }u[p]>0,\;v[q]<0\\ \arctan\Bigl{(}\frac{v[q]}{u[p]}\Bigr{)}&\text{if otherwise}\end{array} \right. \tag{6}\]  
where:  
\[u[p] =\frac{2\pi}{M\Delta xk_{0}}(p-\frac{M}{2});p=0,1,2,\ldots,M-1 \tag{7}\] \[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]  
The following theorem is satisfied:  
**Theorem 1**. : _If the complex matrix \(\left(E_{\text{cp}}^{\text{rad}}\right)\) contains samples of \(E_{\text{cp}}^{\text{rad}}\in\mathcal{R}\) as entries for the values of \(\theta,\phi\) given by Equations (5) and (6), respectively, and the set of matrices \(\left(E_{0x}^{\text{inc}}e^{i\phi}\right)\) contains samples of \(E_{0x}^{\text{inc}}(x,y)e^{i\varphi(x,y)}\) as entries for the values of \(x,y\), given by Equations (3) and (4), respectively, then the set of matrices \(\left(E_{0x}^{\text{inc}}e^{i\varphi}\right)\) and the set of matrices \(\left(E_{\text{cp}}^{\text{rad}}\right)\) are isomorphic._  
Proof of Theorem 1.: If the complex matrix \(\left(E_{\text{cp}}^{\text{rad}}\right)\) contains samples of \(E_{\text{cp}}^{\text{rad}}\in\mathcal{R}\) as entries, then  
\[E_{\text{cp}}^{\text{rad}}=jk_{0}\frac{e^{-jk_{0}r}}{2\pi r} \Bigl{[}\cos^{2}(\phi)+\sin^{2}(\phi)\cos(\theta)\Bigr{]} \tag{9}\] \[\iint_{S_{\text{BA}}}E_{0x}^{\text{inc}}(x,y)e^{i\varphi(x,y)}e^{ i\beta_{0}(\sin(\theta)\cos(\phi)x+\sin(\theta)\sin(\phi)y)}dxdy,\]  
Let us create a partition given by Equations (3) and (4) with \(0\leq x^{\prime}<\Delta x\) and \(0\leq y^{\prime}<\Delta y\). Therefore, Equation (9) leads to  
\[E_{\text{cp}}^{\text{rad}}=jk_{0}\frac{e^{-jk_{0}r}}{2\pi r} \Bigl{[}\cos^{2}(\phi)+\sin^{2}(\phi)\cos(\theta)\Bigr{]} \tag{10}\] \[\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}e^{i\beta_{0}(um\Delta x+yn\Delta y )}\int_{0}^{\Delta y}\int_{0}^{\Delta x}E_{0x}^{\text{ref}}(x^{\prime},y^{ \prime})e^{i\beta_{0}(ux^{\prime}+\mathbbm{y}^{\prime})}dx^{\prime}dy^{\prime},\]  
where \(u=\sin(\theta)\cos(\phi)\), \(v=\sin(\theta)\sin(\phi)\), and \(E_{0x}^{\text{ref}}(x^{\prime},y^{\prime})=E_{0x}^{\text{inc}}(x^{\prime},y^{ \prime})e^{i\varphi(x^{\prime},y^{\prime})}\). If the steps \(\Delta x\) and \(\Delta y\) are selected to be small enough, then \(E_{0x}^{\text{ref}}(x^{\prime},y^{\prime})\) can be approximated as the value of the sample in the center of each element of the partition.