Fig. 2: Comparison between phase gradient maximization and field minimization to obtain phase singularities.

Both methods can be used to obtain phase singularities, but they produce different field behavior in terms of its real (blue) and imaginary (red) zero-isosurfaces. Yellow dots label the positions at which the field and phase gradients are optimized. Inset surface plots are the logarithmically scaled field intensities at z = 0 μm over the same XY domain. The z = 0 μm plane is indicated with the gray plane in each isosurface plot. a When the phase gradient in a specified direction is maximized, the two zero-isosurfaces align approximately tangentially and in the direction normal to that specified gradient. This produces a flat low field intensity structure along these aligned zero-isosurfaces. b Minimizing the field amplitude at a point to produce a singularity merely enforces a crossing of the zero-isosurfaces without any alignment, producing a 1D line singularity. c Simultaneously optimizing two nearby points with directed phase gradients can extend the range of the singularity sheet. d Minimizing the field amplitude at two points simultaneously does not guarantee alignment of the zero-isosurfaces and can instead produce multiple crossing lines, each producing a 1D line singularity.