# A Polygonal Model to Design and Fabricate Ion-Exchanged Diffraction Gratings
## 3 Characterization of the Two Dimensional Ion-Exchange Process by the Polygonal Model  
The polygonal model presented in the previous section enables the characterization of the ion-exchange process, which takes place during the grating fabrication. From this characterization, specific grating elements can be designed and fabricated. The detailed procedure is described here, and it consists of three steps. First, a raw characterization of the one-dimensional ion-exchange process is made through the fabrication of some planar waveguides in suitable substrates. In our case, we fabricated these waveguides by using thermal ion-exchange in soda-lime glasses and a mixture 5% mole AgNO\({}_{3}\)/NaNO\({}_{3}\), at 340 \({}^{\circ}\)C, as the ion source. Next, the ion-exchange modification produced by this salt in this particular glass is analyzed by the inverse Wentzel-Kramers-Brillouin (IWKB) method. As is known, the IWKB method facilitates the determination of the in-depth refractive index profile resulting from the ion-exchange process. From this refractive index profile, the phase step and the penetration depth of the ions, for a given ion-exchange time, can be calculated [14]. In particular, we needed an ion-exchange time \(t=13\) min to get an in-depth penetration of about 7.5 \(\upmu\)m and a phase step around \(\pi\) for a wavelength \(\lambda_{0}=632.8\) nm. Second, the geometric characteristics of the gratings are chosen. In particular, their period is usually selected from diffraction needs. We chose a period \(\Lambda=30\)\(\upmu\)m and an opening width \(w=10\)\(\upmu\)m. The size of the opening was chosen to obtain a grating with the two sides of its phase profile similar in size, that is a quasi-antisymmetric phase profile. Note that most interesting phase profiles are of this kind. This opening width leads to \(v\approx 10\)\(\upmu\)m and \(s\approx 5\)\(\upmu\)m for a phase step \(\pi\), because the side diffusion of the ions is somewhat smaller than the in-depth diffusion. Next, the grating must be patterned on the substrate. We used for it a two-step photolithographic process. In the first step, a binary grating pattern with the aforementioned dimensions was plotted on a sheet of heavy paper and then reduced \(-1/13.5\) by using a lens Schneider XENON Sapphire 4.5/95. This led to a chromium master on a glass substrate that was then reduced by a Carl Zeiss S-planar lens working at magnification \(M=-1/5\) and \(f/\#=1.3\). The final result was an aluminum binary grating on the soda-lime glass chosen for the ion-exchange. The third step is the characterization of the two-dimensional ion-exchange process that takes place during the grating fabrication. For that, as we will see, it is enough to use three gratings with approximate phase steps \(\pi/2\), \(\pi\), and \(3\pi/2\). These phase steps cover the range of phases [0-27], which almost any interesting phase grating element will have. Then, from the characterization realized in the first step, we calculated the ion-exchange times (3.5 min, 14.5 min, and 32 min) needed to achieve, approximately, these phase steps. Next, after the fabrication of these three gratings, their diffraction efficiencies (\(\eta_{m}^{exp}\)) were measured and compared to those provided by the polygonal model (Equations (7) and (8)). This is done by calculating the RMS of the differences between measured and calculated efficiencies, that is:  
\[RMS(w,s,\delta,\Delta\varphi_{0})=\sqrt{\sum_{m=-(N-1)/2}^{(N-1)/2}(\eta_{m}^ {exp}-\eta_{m}^{Equation(7)})^{2}/N}, \tag{9}\]  
where \(N\) is the number of the diffraction orders measured. The minimization of this RMS permitted us to fit the polygonal model to the exact phase profile of each grating. Note that in addition to the intrinsic parameters of the model, we also considered the width of the opening as a free parameter. This was done to take into account the fabrication tolerances and/or to allow the approximate model to fit better to the real profile.