# Encoding the Intensity and Phase Gradient of Light Beams with Arbitrary Shapes
## 3 Results
### Estimation of Parametric Equations  
A set of eight shapes, shown in Figure 4, was used for testing our method. These shapes were selected based on their common usage on beam shaping proposals [12; 14; 24], including some hand-drawn shapes too. This set includes common shapes with few harmonics; such as a ring (a) and an ellipse (b), known shapes with more harmonics; such as a droplet (c), a nephroid (d), a heart (e), an asteroid (f), and shapes drawn by hand with the contour of a ghost (g) and a fish (h).  
Every shape in Figure 4 is read an analyzed under the presented approach, extracting the harmonics to approximate the curve that describes the contour of the shape. According to our algorithm, we extract sufficient harmonics to reconstruct at least 95% of an original form. In the case of Figure 4c,g, this amount was reduced to 92%, since more than nine non-zero harmonics were estimated, increasing the complexity of phase extraction when applying Equation (2). In Table 1, the estimated parametric equations are organized.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Shape** & **Parametric Equation “x”** & **Parametric Equation “y”** \\ \hline (a) & Ring & \(\cos(t-3.08)\) & \(\cos(t+1.63)\) \\ (b) & Ellipse & \(\cos(t-3.06)+0.11\cos(3t+0.24)\) & \(\cos(t+1.65)+0.10\cos(3t-1.33)\) \\ (c) & Droplet & \(-0.04+\cos(t+3.11)+0.05\cos(2t-0.91)\) & \(0.24+\cos(t+1.59)+0.04\cos(2t+0.37)\) \\  & & \(+0.03\cos(3t-2.43)+0.01\cos(4t-2.83)\) & \(+0.05\cos(3t-1.25)+0.03\cos(4t-2.56)\) \\  & & \(+0.02\cos(5t+2.79)+0.02\cos(6t+0.71)\) \\ (d) & Nephroid & \(\cos(t-2.69)+0.11\cos(3t+1.39)+0.06\cos(5t+2.33)\) \\  & & \(+0.02\cos(7t-2.95)+0.02\cos(9t-2.12)+\) & \(\cos(t+2.03)+0.05\cos(3t-0.21)\) \\  & & \(0.02\cos(11t-1.26)+0.01\cos(13t-0.32)\) \\ (e) & Heart & \(\cos(t-3.04)+0.04\cos(2t-1.42)\) & \(-0.47+\cos(t+1.66)+0.23\cos(2t-2.96)+\) \\  & & \(+0.05\cos(3t-2.86)\) & \(0.04\cos(3t+1.82)+0.12\cos(4t+0.35)\) \\ (f) & Astroid & \(\cos(t-3.14)+0.31\cos(3t-3.13)+\) & \(\cos(t+1.57)+0.32\cos(3t-1.56)+0.07\cos(5t+1.59)\) \\  & & \(0.07\cos(5t-3.13)+0.05\cos(7t-3.12)+\) & \(+0.03\cos(7t-1.54)+0.02\cos(9t+1.61)\) \\  & & \(0.02\cos(9t-3.11)+0.01\cos(11t-3.1)\) & \(+0.01\cos(11t-1.53)\) \\  & & \(-0.03\cos+\cos(4t-2.88)+0.13\cos(2t+1.75)+\) & \(-0.25+\cos(t+1.83)+0.13\cos(2t-2.45)+\) \\ (g) & Ghost & \(0.05\cos(3t-3.04)+0.08\cos(4t+2.81)+\) & \(0.11\cos(3t-0.75)+0.03\cos(4t-1.96)+\) \\  & & \(0.05\cos(5t-1.53)+0.08\cos(6t+2.84)\) & \(0.07\cos(5t+2.8)+0.02\cos(6t+1.29)\) \\ (h) & Fish & \(-0.13+\cos(t-2.271)+0.06\cos(2t+0.9)+\) & \(+0.33\cos(3t-3.14)+0.2\cos(4t-2.78)+\) \\  & & \(0.03\cos(3t-2.25)+0.03\cos(4t+2.07)\) & \(0.02\cos(5t-2.02)+0.05\cos(6t+1.68)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Estimated parametric equations for shapes presented in Figure 4.  
Figure 4: Set of shapes used as target for engineering intensity and phase gradient of light beams by estimating parametric equations; (a) ring, (b) ellipse, (c) droplet, (d) nephroid, (e) heart, (f) astroid, (g) ghost, and (h) fish.  
Considering that all equations from Table 1 are expressed in terms of cosine functions, we can recognize how, for the simplest curve in Figure 4a, the parametric equations approximate a cosine waveform in the horizontal direction and a sine waveform in the vertical direction, as expected. For more complex waveforms, a greater number of harmonics are required. For example, since Figure 4c has a sharp edge, the parametric equation for 'y' requires more harmonics compared to a smooth shape like Figure 4b. Figure 4d-f have known parametric equations, but since we approximate their shape through a linear combination of cosine waveforms, several terms are required. The last expressions in Table 1, for Figure 4g,h, require several harmonics for reconstruction since they show no symmetry being drawn directly by hand.