Analysis of diffraction gratings displayed in spatial light modulators at the Nyquist limit: Application to the triplicator grating

In this work we analyze diffraction gratings displayed on a pixelated spatial light modulator (SLM) at its spatial resolution limit (Nyquist limit) i.e., with largest diffraction angle, where the binary phase profile is the only alternative. Their implementation is influenced by the effect of a $\text{sinc}$ envelope and the multiple replicas of the diffraction pattern that arise from the SLM pixelation, where the fill factor plays an essential role. We use the Fourier transform theory to analyze the binary phase grating in a pixelated device in terms of the pixel size, fill factor and phase difference between the two levels in the grating. This convolutional approach probes very useful to gain physical insight of the different contributions to the diffraction orders, and analytical expressions for the complex amplitude coefficients and intensities are obtained. Experimental verification is provided by encoding binary phase triplicator gratings with different periods on a high-resolution liquid-crystal on silicon SLM. The effect of the device pixelation and pixel crosstalk on the diffraction efficiency and on the conditions to obtain a Nyquist triplicator phase grating are studied. These results can be interesting for applications requiring programmable and large beam steering angles.