# A Polygonal Model to Design and Fabricate Ion-Exchanged Diffraction Gratings
## 4 Design and Fabrication of a Grating with Suppressed Zero Order and a Three Splitter
### Design and Fabrication of a Three Splitter  
The procedure that we used to fabricate a three splitter was the same as described in the previous paragraphs, but now the condition \(\eta_{0}=\eta_{1}\) or \(\eta_{0}=\eta_{-1}\) must be fulfilled. Therefore, from Figure 5, we must calculate the intersection between the blue curve (efficiency of the zero order) and the red curve (efficiency of the first order). This intersection is at \(t\approx 8.1\) min; however, we finally rounded this time to \(t=8\) min because of the experimental uncertainties of the fabrication process. Indeed, for such a short time, the transient regimes are relevant, and we observed that they gave rise to uncertainties of about 3% in the ion-exchange time. After fabrication, we measured the diffraction efficiencies of the first three orders. They were \(\eta_{0}=25.5\%\) and \(\eta_{\pm 1}=31.0\%\). It is clear from these values that our goal was not achieved. Three reasons can be given to explain such a discrepancy. First, is the inaccuracy of the polygonal model; second is the error in the opening width resulting from the photolithographic process; and third is the high sensitivity, in this region, of the efficiencies of the first two orders with the duration of the ion-exchange process, which along with the aforementioned uncertainties made the prediction very difficult. Both the second and third reasons can be minimized by using a thorough lithography and lower temperatures for the ion-exchange process, respectively. Despite the differences obtained between the efficiencies, note that the model gave a result close to the desired one, as well as clues to find a better solution. Indeed, according to Figure 5, the time was overestimated because the efficiencies obtained were on the right side of the intersection, at \(t\approx 9\) min. Therefore, we considered a new ion-exchange time \(t=7.5\) min, expecting to achieve a better result. The diffraction efficiencies of this last grating are shown in Table 2. The values were not yet optimal, but there was no point in doing more trails due to the reasons given before. Anyway, this result can be considered good: the difference between the efficiencies of the first three orders was small (3%), and their sum (87.3%) was near the theoretical maximum (92.5%). This last fact was a consequence of the similarity between the phase profile resulting from this fabrication process and the profile necessary to have an optimal three splitter [16]. It is true that the difference between the experimental and predicted efficiencies was a little high (about 6% for the zero order). However, this  
\begin{table}
\begin{tabular}{c c c} \hline \(m\) & \(|t_{m}^{2}|_{exp}\) & \(|t_{m}^{2}|_{Equation}(7)\) \\ \hline
0 & 0.5\% & 0.35\% \\ \(\pm 1\) & 32.55\% & 31.4\% \\ \(\pm 2\) & 12.2 \% & 13.6\% \\ \(\pm 3\) & 3.25 \% & 3.0\% \\ \(\pm 4\) & 1.3\% & 1.3\% \\ \(\pm 5\) & 0.2\% & 0.3\% \\ \(\pm 6\) & 0.05\% & 0.05\% \\ \hline \end{tabular}
\end{table}
Table 1: Experimental and predicted diffraction efficiencies for \(\lambda_{0}=632.8\) nm of the grating designed to suppress its zero order.  
polygonal model was not conceived of to predict exactly the shape of the phase profile and hence the efficiencies of the gratings. Our goal was to provide a simple model to design optical elements made of ion-exchanged gratings.  
Finally, we show in Figure 8 the image of a red pencil (left) through the grating with suppressed zero order (middle) and the three splitter (right). Of course, this result agreed with the data given in the previous tables. With regard to the grating that suppressed its zero order, only the blue component of the pencil remained at this order because the red component was completely removed. Moreover, the \(\pm 1\) orders were the most relevant ones. As for the three splitter, the three first orders were approximately equal in brightness, and the rest of the orders almost vanished.