# A Polygonal Model to Design and Fabricate Ion-Exchanged Diffraction Gratings
## 2 Polygonal Model for the Phase Profile of Ion-Exchanged Gratings  
The phase profile of a grating is conditioned by its fabrication process. In particular, in the fabrication of gratings by thermal ion-exchange through a mask, the phase profile generated is determined not only by the shape of the mask, but also by the ion-exchange process itself. As a first approach, the phase profile of a grating fabricated by this technique would be a square profile. However, when a thermal ion-exchange takes place through a mask opening, ions diffuse beneath the masked part generating a transition region (s), which extends beyond the opening (\(w\)) (see Figure 1). Then, the phase profile of an ion-exchanged grating, \(\Delta\varphi(x)\), would be more like a trapezoid. Moreover, the openings are usually similar in size to the transition region. This makes the ions spread deeper in the middle of the openings than close to their edges. As a result, the phase profile should be rounded in the region corresponding to the opening, just as predicted by the diffusion equation that governs the ion-exchange process [7] and confirmed experimentally [13]. In a first approximation, we can assume, for this region, a triangular phase profile and a linear profile for the transition region or side diffusion. Therefore, all the phase profile can be modeled by using straight lines giving place to the so-called polygonal model (see Figure 2). In order to gain more insight into this shape of the phase profile, we plot, in Figure 3, typical profiles of in-depth refractive index changes of a substrate, \(\Delta n(z)\), for three side positions: at the middle of the opening, at the edge of the mask, and at the middle of the transition region. As is known, this change of the refractive index is proportional to the concentration C(\(x\),\(z\)) of in-diffused ions, which can be calculated by solving a two-dimensional non-linear diffusion equation [7]. From \(\Delta n(x,z)\), the phase profile is calculated by [14]:  
Figure 1: Scheme of the fabrication of an ion-exchanged grating. \(s\) and \(\Lambda\) indicate the transition region (or side diffusion) and the period of the grating, respectively. \[\Delta\varphi(x)=\frac{2\pi}{\lambda}\int_{0}^{\infty}\Delta n(x,z)dz, \tag{1}\]  
where \(\lambda\) is the wavelength. From this equation, it can be easily checked that the shape of the polygonal phase profile is compatible with the curves shown in Figure 3. On the other hand, the third dimension (y-axis) has little effect on the phase profile, and hence on the grating behavior, because the diffusion length of ions is very short compared to the length of this grating dimension. In other words, the ion-diffusion is not limited in the y-direction. According to Figure 2, the phase profile can be expressed as follows:  
\[\Delta\varphi(x)=\left\{ \begin{array}{ccc}-2\Delta\varphi_{0}\left(1+v/s+x/s \right)&-(v+s)\leq x\leq-(v+s/2)\\ -\Delta\varphi_{0}&-(v+s/2)\leq x\leq-s/2\\ 2\Delta\varphi_{0}\ x/s&-s/2\leq x\leq s/2\\ \Delta\varphi_{0}+\delta(-s/w+2x/w)&s/2\leq x\leq(w+s)/2\\ \Delta\varphi_{0}+\delta(2+s/w-2x/w)&(w+s)/2\leq x\leq w+s/2\\ 2\Delta\varphi_{0}\left(1+w/s-x/s\right)&w+s/2\leq x\leq w+s\\ \end{array} \right. \tag{2}\]  
where all the variables are defined in that figure. This model allows the characterization of the two-dimensional ion-exchange process, which takes place in the fabrication of the grating through the determination of the parameters of the model (\(s\), \(\delta\), and \(\Delta\varphi_{0}\)). In Section 3, we will explain the procedure that we used. It is based on the comparison  
Figure 3: Typical in-depth refractive index changes after an ion-exchange process through an opening for three side positions: middle of the opening (green curve), the edge of the mask (blue line), and the middle of the transition region (red line). Figure 2: Modeling of the phase profile of an ion-exchanged grating by the polygonal model.