# Ultra-short pulse propagation

The study of the propagation of light pulses in dielectric media starts from Maxwell's equations. However, given the particularities of the system, several approximations are made with varying orders of success. The most common approximations are: reduce to 1+1 dimensional systems (like in optical fibers), unidirectional propagation (no counter-propagation given the short interaction time), and slowly-varying envelope approximation (SVEA). They lead to the paradigmatic equation of the field: the nonlinear Schrödinger Equation (NLSE).

The NLSE has an exact solution that keeps its form during propagation: the optical soliton. The solitons are localized waves sustained by the opposite effects of nonlinearity and dispersion in optical fibers. Such pulses have been understood through theoretical studies and numerical simulations, which have successfully explained some features of nonlinear pulse dynamics, including the emission of dispersive waves, soliton fission, and supercontinuum generation.

We are interested in ultra-short pulses with ~10-100 fs in the visible spectrum. In this case, several higher-order nonlinear and dispersive effects come into play, and the approximations used to derive the NLSE are not valid anymore. In this case, other nonlinear effects take further importance, especially the soliton fission, which immediately brakes down the pulses by emission of dispersive waves (DW), also known as resonant radiation (RR), Cherenkov radiation, or non-solitonic radiation (NSR). We usually consider a more complete equation, the unidirectional pulse propagation equation (UPPE).

On the other hand, photonic crystal fibers (PCFs) or microstructured fibers, have a solid core surrounded by a regular array of air holes in the propagation direction that changes the dispersion relation of the fiber core. These holes can be fabricated with certain degrees of freedom to engineer this relation beyond material dispersion. The PCFs made of silica have been very successful at shedding supercontinuum light, which has found a wide range of applications.

Another feature of PCFs is that the ability to support a wide range of frequencies opens the possibility of creating very steep pulses close to the single-cycle regime. These pulses are helpful in the search for optical analogues of Hawking radiation because the effective temperature and the emission rate depend crucially on the rate of change of the refractive index due to the Kerr effect, which ultimately depends on the steepness of the pulses.