Fig. 3: Optimization of knotted fields.
Un-optimized (a–c) and optimized (d–f) trefoil knots sliced in two different cross-sections \({k}_{1}\) and \({k}_{2}\). Phase distribution at k1 (b, e) and \({k}_{2}\) (c, f) planes, where an example of the distance between two singularities \(\left|{{{\bf{r}}}}_{1}^{k}-{{{\bf{r}}}}_{2}^{k}\right|\) is illustrated in panel (e) and their respective amplitude distribution is shown in the inset. Red dots indicate the points where the singularity lines cross a particular \(k\)-plane at each plane. f This shows that, even though the singularities 3–5 are close to each other, their connection does not change the topology of the knot. In this plane, the algorithm focuses on optimizing the distance between singularities 7 and 3−6. The coefficients for the un-optimized [optimized] optical trefoil knot for each LG mode \((p,l)\) are (0,0) 1.71 [1.29]; (1,0) −5.66 [−3.95]; (2,0) 6.38 [7.49]; (3,0) −2.3 [−3.28]; (0,3) −4.36 [−3.98].
