Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations

Unidirectional pulse propagation equations [UPPE, Phys. Rev. E 70, 036604 (2004)] have provided a theoretical underpinning for computer-aided investigations into dynamics of high-power ultrashort laser pulses and have been successfully utilized for almost a decade. Unfortunately, they are restricted to applications in bulk media or, with additional approximations, to simple waveguide geometries in which only a few guided modes can approximate the propagating waveform. The purpose of this work is to generalize the directional pulse propagation equations to structures characterized by strong refractive index differences and material interfaces. We also outline a numerical solution framework that draws on the combination of the bulk-media UPPE method with single-frequency beam-propagation techniques.

Figure 1

Energy loss experienced by an initially collimated, pulsed Gaussian beam with the beam waist smaller than the capillary inner diameter. The gUPPE approach (solid black line) includes proper modeling of loss due to light leaking into the cladding. The ALMEx method (dashed red line) incorrectly predicts loss vs propagation distance. Left and right panels present results for different values of $n_{cl}$.

Figure 2

An example of the spatial profile of the optical field (taken at the central time slice $\tau =0$ of the pulse) after propagation in the nonlinear regime. The dip in the center is caused by defocusing due to electrons freed by the leading edge of the pulse. The light leaking into the cladding is evident beyond $r=200$ $\mu$m, as is the damping in the PML. The grid used to sample the radial dimension was chosen equidistant, with 800 points over 0.4 mm total radius. The propagation distance of this snapshot is $z=$ 1.5 cm.

Figure 3

Spatiotemporal reshaping of a midinfrared pulse in a hollow capillary waveguide. The trailing edge of the pulse in the on-axis region is depleted due to generated plasma. The diagonal texture visible for $r>200$ $\mu$m is the outgoing radiation eventually absorbed in the PML. The propagation distance of this snapshot is $z=0.5$ cm.

Figure 4

Extreme spectral broadening (supercontinuum). The propagation regime in the hollow waveguide at high pressure is akin to femtosecond filamentation with the concomittant supercontinuum generation reflecting the extreme nonlinear evolution.