# A Polygonal Model to Design and Fabricate Ion-Exchanged Diffraction Gratings
## 2 Polygonal Model for the Phase Profile of Ion-Exchanged Gratings
between the theoretical and experimental efficiencies of the gratings fabricated and will show the utility of this simple model. For more complex periodic structures, the phase profile would be given by the following general polygonal function:  
\[\Delta\varphi(x)=\{a_{j}+b_{j}x,\quad x_{j-1}\leq x\leq x_{j},\quad j=1,...,M, \quad x_{0}=-\frac{\Lambda}{2},x_{M}=\frac{\Lambda}{2}\}, \tag{3}\]  
where \(\Lambda\) is the period of the grating, \(M\) the number of segments, and \(a_{j}\), \(b_{j}\) would be determined by the specific shape of the profile. The substitution of this expression in the equation for the transmission of the order \(m\) of a grating, in the thin-grating approximation,  
\[t_{m}=\frac{1}{\Lambda}\int_{-\Lambda/2}^{\Lambda/2}\exp(i\Delta\varphi(x)) \exp(-im\frac{2\pi}{\Lambda}x)dx, \tag{4}\]  
leads to a general expression for the diffraction coefficients of such a grating:  
\[t_{m}=\sum_{j}\frac{\exp(ia_{j})}{i\left(\Lambda b_{j}-2\pi m\right)}\{\exp[i( b_{j}-\frac{2\pi m}{\Lambda})x_{j}]-\exp[i(b_{j}-\frac{2\pi m}{\Lambda})x_{j-1}]\}. \tag{5}\]  
We must stress that in a standard selective ion-exchange process, the parameters \(a_{j}\) and \(b_{j}\) will be functions of ion-exchange time \(t\). As shown later, a linear dependence between these parameters and \(\sqrt{t}\) is found in a first approximation, that is,  
\[a_{j}(t)=a_{j1}+a_{j2}\sqrt{t}\quad\text{and}\quad b_{j}(t)=b_{j1}+b_{j2}\sqrt {t}, \tag{6}\]  
where \(a_{j1}\), \(a_{j2}\), \(b_{j1}\), and \(b_{j2}\) would be obtained after characterization of the ion-exchange process. In general, different cases are possible: \(a_{j1}=a_{j2}=0\); \(a_{j1}=0\), \(a_{j2}\
eq 0\); \(a_{j1}\
eq 0\), \(a_{j2}=0\); and the same cases for \(b_{j1}\) and \(b_{j2}\). This allows us to analyze the temporal evolution of diffraction coefficients, \(t_{m}(t)\), so that an energy transfer is observed between diffraction orders \(m\), giving rise to different optical transformations. As for the profile given by Equation (2), it corresponds to the particular case \(M=6\) with \(b_{2}=0\), where the period \(\Lambda\) equals \(w+v+2s\). In this case, by introducing Equation (2) into Equation (4) and integrating, we obtain, after a long but straightforward calculation, the following expression for the diffraction coefficient of order \(m\):  
\[t_{m}= \frac{s}{\Lambda}\frac{\sin(\Delta\varphi_{0}-\pi ms/\Lambda)}{ \Delta\varphi_{0}-\pi ms/\Lambda}+\frac{s}{\Lambda}\exp(i\frac{\pi m}{\Lambda} (v-w))\frac{\sin(\Delta\varphi_{0}+\pi ms/\Lambda-m\pi)}{\Delta\varphi_{0}+ \pi ms/\Lambda}\] \[-\exp(i\frac{\pi m}{\Lambda}(v-w))\frac{\sin(\Delta\varphi_{0}+ \pi ms/\Lambda-m\pi)}{m\pi}+\frac{\sin(\Delta\varphi_{0}-\pi ms/\Lambda)}{m\pi}\] \[+\frac{w}{\Lambda}\exp(i\Delta\varphi_{0})\exp(-i\frac{\pi m}{ \Lambda}(s+w))\bigg{(}\exp(i(\frac{\pi mw}{2\Lambda}+\frac{\delta}{2}))\frac{ \sin((\pi mw/\Lambda-\delta)/2)}{\pi mw/\Lambda-\delta}\bigg{)}\] \[+\frac{w}{\Lambda}\exp(i\Delta\varphi_{0})\exp(-i\frac{\pi m}{ \Lambda}(s+w))\bigg{(}\exp(-i(\frac{\pi mw}{2\Lambda}-\frac{\delta}{2}))\frac {\sin((\pi mw/\Lambda+\delta)/2)}{\pi mw/\Lambda+\delta}\bigg{)}\] Figure 2:0  
The first two lines are the transmission for \(\delta=0\) and the last ones the correction to take into account the triangle defined in the interval \([s/2,w+s/2]\). Note that the diffraction efficiency of the order \(m\) can be easily calculated as:  
Figure 2:1