We use a nonlinear optical process to write optical vortices from the infrared (IR) to the near-IR, and the same nonlinear process to track the initial IR optical vortices using a novel form of phase reconstruction by nonlinear digital holography. This also allows us to monitor the creation and annihilation of phase singularities, their individual topological charges and to extract the modal components of an optical field. In particular, we show that singularities can be tracked with a 3 \(\times \) 3-pixel precision, limited not by the technique itself, but by the stability of the experimental setup. By carefully calibrating the magnification of both the encoding and recording devices, we are able to prescribe and implement well-defined trajectories for the singularities. This control over singularity motion allows one to illustrate algebraic operations through physical overlap, for example, adding or canceling charges (+1 and –1). Finally, we exploit the ability to track singularity positions as a quantitative probe of mode composition: When combining a vortex and a Gaussian mode, the displacement of the singularity provides a direct measure of the relative modal contribution. Our findings enhance the toolkit for monitoring the dynamics of phase singularities, particularly in wavelengths that are challenging to detect.
