# A Polygonal Model to Design and Fabricate Ion-Exchanged Diffraction Gratings
## 1 Introduction  
Diffractive Optical Elements (DOEs) are widely used in different fields such as holography [1], spectroscopy [2], astronomy [3], and so on. Among DOEs, diffraction gratings stand out for their versatility, being a common part of devices for wavelength division multiplexing, optical pulse compression, or laser tuning. Gratings modulate the amplitude or the phase of the light incident on them. As a result, light is diffracted into the so-called diffraction orders. In applications where the diffraction efficiency is important, phase modulation is preferred to amplitude modulation. The most investigated phase gratings are those with sinusoidal profiles. The gratings with this kind of profile have significant advantages. First, they can be mathematically analyzed by closed-form expressions. Secondly, these profiles can give rise to many interesting diffraction patterns including the cancellation of the zero order, splitting in several orders, and so on. Lastly, sinusoidal gratings can be fabricated by interference of two plane waves on photosensitive substrates. However, gratings fabricated in this way are delicate, and specific phase profiles are difficult to obtain. Here, we use the ion-exchange technique to fabricate robust and precise sinusoidal-like phase gratings by taking advantage of its simplicity, low cost, and easy control of the fabrication parameters [4]. Moreover, the flat surface of the elements fabricated by this technique permits straightforward cleaning and antireflection coating, as well as integration in stacked optical systems [5]. As is known, ion-exchange changes the refractive index of suitable substrates, typically in a region of some micrometers under its surface. This change can be made selectively in only some regions of the substrate, provided that a mask for the ions is used. In particular, by designing appropriate masks, as well as the ion-exchange process itself, different kinds of phase gratings have been fabricated [6, 7]. In these works, the authors successfully designed and fabricated grating beam splitters by using a numerical approach for both the ion-exchange process and the diffraction calculations. However, the design of such elements using numerical methods is not easy when trying to make complex elements. In the present work, we propose asimple polygonal model for the phase profile achieved following thermal ion-exchange processes. This model enables the design and fabrication of grating elements in a simple way. After characterization of the ion-exchange process by using such a model, different grating elements can be designed and fabricated easily. As examples, we design and fabricate a diffraction grating with suppressed zero order and a three splitter. Cancellation of the zero order of diffraction is desirable in many applications of diffractive elements, such as digital holography [8], spatial light modulation [9], and grating fabrication [10], among others. On the other hand, light splitting is useful, in general, for power division tasks [11], and in particular, it has been used, combined with other phase functions, in the fabrication of vortex gratings [12].  
In Section 2, we explain the characteristics of the model and obtain expressions for the transmission function of gratings with such a phase profile. Next, we describe the characterization procedure of the two-dimensional ion-exchange process by using this model. This characterization provides the characteristic parameters of the model as a function of the ion-exchange time. Finally, in Section 4, we check the potential of the model by designing and fabricating a grating with suppressed zero order and a three splitter.