# Encoding the Intensity and Phase Gradient of Light Beams with Arbitrary Shapes
## 4 Discussion  
We have introduced and tested a method for beam shaping of arbitrary curves through the design of phase masks to display on a SLM. By using a SLM and few optical elements, we have been able to test the correct encoding of phase patterns, as presented in Figures 5 and 6. This test has been restricted to a visual comparison in shape for the intensity images, while, for phase gradient, it has been quantitative by testing through the interferograms obtained from pairs of vortex beams.  
Since the SLM is an active device, it offers a dynamic solution for beam shaping and additionally, given its large pixel array, it allows to encode more than one pattern over its active area. Moreover, we exploit the features of this device by designing beams that interfere in a common path interference setup. All these features may also be exploited from another active device, the digital micromirror device (DMD). Compared with a SLM, a DMD is usually chosen when faster modulation rates are required [25], at the expense of reducing the power efficiency. For optical trapping, this may be a drawback when the photon budget cannot be compromised. However, beam shaping at high speeds has been achieved when using a DMD in the off-axis regime, through diffusive media and through multimode fiber [25]. Additionally, with the development of optical nano printing [23], specifically through light controlled electric fields, the usage of optical imaging systems based on DMD as spatial light modulators offers a solution in nano-scale applications [25; 26; 27]. Considering that our approach for beam shaping may be applied using either a SLM or a DMD, a wider range of applications arise. In laser cutting [16], dynamic beam shaping increases the cutting speed and improves the cut edge quality. In optical trapping, manipulating the phase of light, both two-dimensional and three-dimensional structures can be programmed to line up, sort, or rearrange nano and micro-sized particles [15; 20; 28]. In biological cells,an indirect transportation of cells can be implemented through the design of an arbitrary curve or shape [29].  
Expanding the reach of beam shaping, we currently find emerging applications that can benefit from the presented method, such as optical nanolithography and optothermal manipulation. These techniques involve delicate processing of advanced substrates to remove very thin layers, using high quality beam shaping. In optical nanolithography, the manufacturing of microelectronics by the semiconductor industry and the remarkable properties of graphene [18,27] have led the interest in managing arbitrary 2D patterns with both high throughput and high resolution. Optical patterning of graphene nanostructures with thermal manipulation is studied to facilitate the development of terahertz photonic devices, useful in biosensing and photon detection [18]. Then, compared with conventional optical trapping, heat-mediated optical manipulation techniques allow versatile control of matter of different compositions, shapes, and sizes [30]. This is performed by designing and exploring optothermo-matter interactions for transporting mass, where it is of interest to implement arbitrary trajectories with thermal gradients [30]. Research in nanomanipulation techniques has been active in the last five years [15,18,19,20,28] and could be greatly benefited from methods for encoding arbitrary trajectories through beam shaping.