Nonlinear modal decomposition of structured light

Modal decomposition of structured light allows its unraveling into a basis of spatial modes, the superposition of which can be used to infer any physical property of the field. This is conventionally achieved by a linear projective system operated in measurement mode. Here, we show how modal decomposition of arbitrary structured light fields can be achieved with nonlinear optics, unraveling an unknown beam in the infrared by spatially overlapping it with a modal set encoded sequentially onto an <i>input</i> visible beam. Next, we show how the decomposition can be done in a single step by a nonlinear version of off-axis digital holography, again for full field reconstruction of the unknown beam. We verify both approaches with topical structured light examples, including orbital angular momentum and Hermite-Gaussian beams. Our nonlinear approach to modal analysis will benefit those applications where the structured beam is at an inconvenient wavelength, and extends our perspective on nonlinear optics from frequency conversion to modal detection.