Second-Order Couplings in Ultrashort Pulses with Cylindrical Symmetry

With technological advancements in ultra-short laser pulses technology, we can generate pulses much shorter than we were able to previously. In the regime of the pico/femto-second (10-12 \ 10-15s), effects such as the Spatio-Temporal Couplings are not negligible, having impact in both the shape and spectral distribution of an ultra-short pulse. In this work, I tackle the effects of Second Order couplings, of the type (xi - ϒ xi) 2 (not studied until now), in a Gaussian pulse, for the following coordinate combinations: (x,t); (x,ω); (kx, ω); (kx,t), a total of 8 different coupling types.

These eight types of couplings were tested in a simulated pulse propagating from z=0 μm to z=30 μm. The visualization of said pulses was done through Intensity plots in the four different domains, and through Spectrograms, which allowed for an intuitive visualization of the pulses' spectral distribution.

An analysis using the numerical calculation of a quadratic Pearson Correlation coefficient was proposed and used in order to measure how strong the coupling between each coordinate was. This method was particularly useful for some domains, while for others it was deemed to not be effective (depending on the type of coupling). The script used for all of the above was written in Wolfram Mathematica.