# Two-Dimensional Digital Beam Steering Based on Liquid Crystal Phase Gratings
## 3 Results and Discussion  
Figure 11 shows the diffraction pattern in the far field for different integer topologies. The same topology is applied to each LC cell, resulting in a diagonal deviation in the plane perpendicular to the propagation direction (XY). It is clear from Figure 11 that the diffraction efficiency is best for smaller deviation angles and decreases as the diagonal deviation angle increases towards the maximum angle of 1.65 degrees. This is in accordance with diffraction theory, which predicts a 40% diffraction efficiency for a perfect 1D binary diffraction grating [23]. The almost perfect  
Figure 11: Representative photographs taken of the diffraction pattern for topologies applied in each cell. The distance between the LC BS and the screen is \(\mathrm{z}=44\) cm: (**a**) 0; (**b**) +2; (**c**) +4; (**d**) +6; (**e**) +12; (**f**) +18; (**g**) +24; (**h**) +36. Corresponding to diagonal deviation angles of: (**a**) 0\({}^{\circ}\); (**b**) 0.065\({}^{\circ}\); (**c**) 0.18\({}^{\circ}\); (**d**) 0.27\({}^{\circ}\); (**e**) 0.73\({}^{\circ}\); (**f**) 0.83\({}^{\circ}\); (**g**) 1.1\({}^{\circ}\); (**h**) 1.65\({}^{\circ}\). Figure 10: Normalized transmission intensity measured (blue) from the calibration, versus the fitted intensity (orange) as a function of the PWM duty cycle applied. calibration of the system can be appreciated in the images of Figure 11 by the absence of the central 0 order peak, in all but the first OFF images. The diffraction efficiency has been mapped (Figure 12a) for all combinations of integer topologies (i.e., where the 72 electrodes span an integer number of \(2\pi\) retardations) ranging from \(+36\) to \(-36\) in each cell. A photo of every configuration was taken, and the ratio between the intensity power of the deviated spot and the full frame was calculated afterwards. This procedure does ignore any power diffracted outside the imaged area but is insensitive towards intensity fluctuations in the laser light source. A conservative estimate of the background intensity reading, as well as occasional supersaturated camera pixels, may have led to a slight underestimate of the diffraction efficiency. In Figure 12a, one appreciates similar diffraction efficiency for both horizontal and vertical axes, indicating that the vertical rubbing direction along the electrode of one device, and perpendicular to the next device, is only marginally affecting the grating efficiency. The beam steeper response has been simulated using Fast Fourier Transform of a simple step-like phase variation, with 25% transmission reduction in the interpixel space (Figure 12b) caused by the DLW. The simulated diffraction efficiency, using this simple model, has been mapped for all combinations of integer topologies ranging from \(+36\) to \(-36\). A 2D Fast Fourier Transform (2D FFT) is used to calculate the evolution of the electric field in the far field [35]:  
\[E_{far\ field}=X(\lambda,z)*2D\ FFT(A_{P}(\varnothing_{Total})), \tag{5}\]  
where \(A_{P}(\varnothing)\) represents the aperture in the grating plane as a function of the phase profile. \(X(\lambda,z)\) is a quadrature function with elements: the wave length of the incident laser beam (\(\lambda\)), and the distance (\(z\)) from the system to the screen. Including crosstalk, fringe effects or higher order effects had no positive impact on the similarity between model and measurements. Similarly, the simple model coincides with observation of the absence of XY asymmetry, which could be assigned to the rubbing being either parallel or perpendicular to the pixels. The PWM working range spans approximately a quarter of the full 12 bits PWM range, i.e., a 10 bits working resolution. Thus, an almost infinitely small angular step size may be used as a consequence of the full independence of each electrode (Figure 13). Figure 12: (**a**) Experimental \(72\times 72\) power diffraction efficiency map (%) as a function of the diffraction angle.