  We extend the $t-z$ mapping formalism of time-dependent paraxial optics by
identifying configurations displaying a synthetic magnetic vector potential,
leading to a non-trivial band topology in propagating geometries. We consider
an inhomogeneous 1D array of coupled optical waveguides beyond the standard
monochromatic approximation, and show that the wave equation describing
paraxial propagation of optical pulses can be recast in the form of a
Schr\"{o}dinger equation, including a synthetic magnetic field whose strength
can be controlled via the transverse spatial gradient of the waveguide
properties across the array. We use an experimentally-motivated model of a
laser-written waveguide array to demonstrate that this synthetic magnetic field
can be engineered in realistic setups and can produce interesting observable
effects such as cyclotron motion, a controllable Hall drift of the wavepacket
displacement in space or time, and unidirectional propagation in chiral edge
states. These results significantly extend the variety of physics that can be
explored within propagating geometries and pave the way for exploiting this
platform for higher-dimensional topological physics and strongly correlated
fluids of light.
