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We extend the $t-z$ mapping formalism of time-dependent paraxial optics byidentifying configurations displaying a synthetic magnetic vector potential,leading to a non-trivial band topology in propagating geometries. We consideran inhomogeneous 1D array of coupled optical waveguides beyond the standardmonochromatic approximation, and show that the wave equation describingparaxial propagation of optical pulses can be recast in the form of aSchr\"{o}dinger equation, including a synthetic magnetic field whose strengthcan be controlled via the transverse spatial gradient of the waveguideproperties across the array. We use an experimentally-motivated model of alaser-written waveguide array to demonstrate that this synthetic magnetic fieldcan be engineered in realistic setups and can produce interesting observableeffects such as cyclotron motion, a controllable Hall drift of the wavepacketdisplacement in space or time, and unidirectional propagation in chiral edgestates. These results significantly extend the variety of physics that can beexplored within propagating geometries and pave the way for exploiting thisplatform for higher-dimensional topological physics and strongly correlatedfluids of light.

Artificial gauge fields in the t-z mapping for optical pulses: spatio-temporal wavepacket control and quantum Hall physics