Optical pulses with spatial dispersion – solitons & relativity

Christian, JM; McDonald, GS; Hodgkinson, TF; Chamorro-Posada, P

Abstract

We propose a new model for describing the evolution of scalar optical pulses in Kerr-type waveguides. The (normalized) wave envelope u satisfies a governing equation that is of the nonlinear Helmholtz type, where the space and time coordinates are denoted by ζ and τ, respectively, α is related to the group velocity, and s = ±1 flags the anomalous/normal temporal dispersion regime. The first term includes spatial dispersion. There are two main contributions to this term in semiconductor waveguides: the propagation contribution (inherent to any electromagnetic mode propagating off-axis) and a recently-proposed material contribution that arises from field-exciton coupling [1].

We will report new solutions and pulse properties, arising from when the universal slowly-varying envelope approximation and the consequent Galilean boost to a local time frame are abandoned. Together, these two simplifications lead to a theory of optical pulses based on the nonlinear Schrödinger equation, with all its advantages and disadvantages. Here, we develop a Helmholtz formalism and uncover a broad range of new physical predictions.
The model is a temporal analogue of the spatial nonlinear Helmholtz equation [2]. Hence, one may deploy mathematical and computational techniques that are similar to those used over recent years to analyse broad scalar nonlinear beams.

We have derived exact analytical bright and dark solitons of
the model equation. The geometry of these new pulse solutions, which complement their spatial counterparts,
has been explored in detail. They exhibit generic features (for instance, one encounters both forward and backward-propagating solution families), and map directly onto a Lorentz-type transformation. More specifically, we have discovered that the velocity combination rule for Helmholtz soliton pulses is formally identical to that encountered in relativistic particle mechanics. Further analytical work has led to the derivation of new invariance laws and conserved quantities. Importantly, the predictions of conventional pulse theory can be recovered in an appropriate simultaneous multiple limit.

Recent computations, in conjunction with linear analysis and nonlinear stability criteria, have predicted that the soliton pulses of the model equation tend to be robust against perturbations to their temporal shape. This key result provides compelling evidence for the stability of Helmholtz pulses in general.

<FIGURE 1>

Figure 1. Perturbed (a) bright and (b) dark pulses evolve into stationary solitons of the model equation.

References
[1] Biancalana F and Creatore C, Opt. Exp. 16, 14882–93 (2008).
[2] Christian J M, McDonald G S and Chamorro-Posada P, J. Opt. Soc. Am. B 26, 2323–30 (2009).

Citation

Christian, J., McDonald, G., Hodgkinson, T., & Chamorro-Posada, P. (2010, August). Optical pulses with spatial dispersion – solitons & relativity. Poster presented at National Photonics Conference, Photon 10, Southampton, UK

• Presentation Conference Type: Poster
• Conference Name: National Photonics Conference, Photon 10
• Conference Location: Southampton, UK
• Start Date: Aug 23, 2010
• End Date: Aug 26, 2010
• Publication Date: Aug 23, 2010
• Deposit Date: Oct 13, 2011
• Publicly Available Date: Apr 5, 2016
• Publisher URL: http://photon10.iopconfs.org/
• Additional Information: Event Type : Conference