Fig. 1: From 1D to 2D space-time duality.

a Propagation of a monochromatic 1D spatial light beam, \(\psi \left({x;z}\right)\), over a distance \(\Delta z\) in free space results in a broadening of its spatial width due to diffraction, i.e., various spatial frequency components \({k}_{x}\) accumulating different propagation phases \(\exp \left(-i\left(\frac{{k}_{x}^{2}}{2{k}_{0}}\right)\Delta z\right)\). b Similarly, a 1D polychromatic short pulse, \(\psi \left(\tau {;z}\right)\), upon propagating a distance \(\Delta z\) in a dielectric with a 2-order dispersion of \({\beta }_{2}\), experiences a broadening of its pulse duration due to dispersion propagation, i.e., different temporal frequency components \(\Omega\) accumulating various propagation phases \(\exp \left(i\left(\frac{{\beta }_{2}{\Omega }^{2}}{2}\right)\Delta z\right)\). c Concept of 2D space-time duality. The propagation of a 2D ST Gaussian wavepacket \(\psi \left(x,\tau {;z}\right)\) exhibits a duality symmetry with a 2D spatial Gaussian beam \(\psi \left(x,{y;z}\right)\), when the material dispersion satisfies \({{\beta }_{2}=\beta }_{2}^{ * }\); otherwise, their behaviors diverge, entering the asymmetry regime. d Calculated similarity \(\gamma \left(z\right)\) between a 2D ST Gaussian wavepacket propagating through a medium over 20 times the Rayleigh range \({z}_{R}\) with varying material dispersion, compared to a 2D spatial Gaussian beam propagating the same distance in free space.