Optical spatiotemporal Fourier synthesis: tutorial

Fourier synthesis is one of the foundations of physical optics. Spatial Fourier optics is a basis for understanding optical imaging, microscopy, and holography. In conventional Fourier optics, the complex spatial field distribution in the Fourier plane constitutes the spatial spectrum of the field to be realized in physical space. Analogously, in temporal Fourier optics, the complex temporal spectrum can be manipulated for ultrafast pulse-shaping. We present here a tutorial on the emerging field of spatiotemporal Fourier optics whereby the spatial and temporal spectra are manipulated jointly to produce spatiotemporally structured optical fields that display unique propagation characteristics. In this tutorial, we focus on a subset of the overall class of nonseparable spatiotemporally structured fields, namely cylindrically symmetric fields in which each radial spatial frequency is associated with a single wavelength. This subset of fields comprises propagation-invariant wave packets that travel rigidly in linear media at a tunable group velocity and includes space-time wave packets and other closely related structured fields. We describe a spatiotemporal Fourier synthesis system capable of preparing arbitrary optical fields belonging to this subclass.