Peter Vingaard Larsen
Ph.D. in Applied Mathematics
Valdemarsgade 83,
1665 Copenhagen V
Telephone (Direct): (+45) 6171 0927
E-mail:  peter.vingaard(a)


An important entity in optical communication is the so called light bullet, as these serve as carriers of information. Light bullets are localized in both time and space and are thus fine examples of spatio-temporal solitons! However, the solitons supported in the so-called quadratic nonlinear materials are depending on the sign of a material parameter, dispersion. To obtain the light bullet shape, the dispersion has to be anomalous - for normal dispersion, another localized structure appears: the X-wave!

This entity is in fact biconically shaped, but longitudinal cuts reveals an X-shape; hence the name. The X-wave is harder to handle than the light bullet, because the tails that make up the X is much lower in intensity than the central peak. X waves (and light bullets) have previously been described using nonlocality in 1 dimension, but we extend the method with another dimension. With the introduction of nonlocality in this setting, one is able to reduce two coupled partial differential equations to only one!

The 2 dimensional nonlocal description, however, reveals a few difficulties with singularities of the so-called response function, which are discussed, but the main features of the system is captured. As a novel tool, the nonlocal description is used to investigate cases where the dispersion is different at the light's fundamental frequency compared to its second harmonic wave (which is generated by the nonlinearity of these crystals at the double frequency). We predict the new effect of X-shapes in cases of anomalous dispersion!

The numerics involved was a Fourier split-step method in combination with Fast Fourier Transform and a Runge-Kutta solver for the nonlinear step to solved 2 coupled continuous nonlinear partial differential equations. This was implemented in ForTran and MatLab (for plotting).