## New approach to pulse propagation in nonlinear dispersive optical mediaPublished in JOSA B, Vol. 29 Issue 10, pp.2958-2963 (2012) |

**Spotlight summary:**Nonlinear optical propagation in waveguides is a rich and flourishing research field due to the simultaneous development of ever more powerful short-pulse lasers on the one hand, and novel waveguide types, such as microstructured fibers or silicon-based waveguides, on the other hand. Accurate modelling of complex nonlinear phenomena, e.g. supercontinuum generation or attosecond pulse propagation, becomes increasingly important, from both scientific and commercial perspectives, and so does the efficiency and versatility of numerical approaches and algorithms. Two recent papers by Xiao et al. present something rare: A method for nonlinear propagation modelling, which is substantially different from the conventional approaches in its derivation and implementation.

The time-honoured approach for nonlinear modelling is the 1+1D Nonlinear Schrödinger (NLS) equation, which, roughly speaking, is valid for pulses consisting of many optical cycles, in the absence of delayed-response (Raman) effects. The NLS is a first-order differential equation, which, given the temporal pulse profile at some point in the waveguide, specifies the derivative of this profile along the direction of propagation. Thus, it can be used to 'propagate' the pulse through the waveguide, which is usually done step by step in a numerical procedure, although some particular analytical solutions are found to exist. It is possible to generalize the equation to handle few- or single-cycle pulses, Raman response and the various effects of a complex waveguide structure. Indeed, many papers on this subject have appeared in recent years. Along with the generalization, an increase in the numerical complexity of the problem is typically seen. Furthermore, the derivations tends to be of a very formal nature, which makes it difficult to introduce the subject to students, and somewhat obscures the basic physics of the various phenomena.

By contrast, the approach taken by Xiao and co-workers is simple and highly intuitive. The basic idea is this: If the waveguide was purely linear, the temporal, or spatial, evolution of the field could be described by a linear response function, whose form (in the frequency domain) can be calculated trivially once the dispersion relation of the waveguide mode(s) is known. A Kerr-type nonlinear response will lead to a change in refractive index proportional to the local intensity of the pulse, thereby changing the velocity of the light and the time it takes to traverse the waveguide. The crucial insight of Xiao et al. is that the nonlinear propagation from one end of the waveguide to another, can be understood as a linear propagation with a modified velocity, which will be different for different parts of the pulse. Assuming a negligible change in the temporal intensity profile during the propagation, the output pulse can be calculated almost immediately.

For most of the nonlinear propagation problems that are really interesting, the assumption of an unchanging intensity profile is not satisfactory. To overcome this limitation, the authors slice up the waveguide into pieces small enough that the approximation will hold in an individual piece, and do the propagation calculation piece by piece, recalculating the intensity profile after each step. In this way, they regain a step-by-step numerical algorithm, similar to the ones currently used, but different in its details and justification. In vol. 37, of Optics Letters (p. 1271, 2012) the authors showed numerically that the new algorithm provide results equivalent to those of standard NLS solvers in the absence of linear dispersion effects, and for this case is significantly faster. In their latest paper, the equivalence of the two approaches is demonstrated also in the presence of dispersion, a crucially important fact for the applicability to realistic waveguide problems. Comparisons to the results of computationally heavy finite-difference time-domain simulations for single-cycle pulses also testify to the accuracy of the method.

It remains to be seen whether the formalism can be extended to include delayed nonlinear responses, and whether it will benchmark favorably against the established numerical schemes. Given the current interest in efficient modeling methods for complex nonlinear optics problems, we will probably not have to wait too long for these answers to appear.

--Jesper Lægsgaard

**Technical Division:**Light–Matter Interactions

**ToC Category:**Nonlinear Optics

OCIS Codes:
| (190.5530) Nonlinear optics : Pulse propagation and temporal solitons |

(320.7110) Ultrafast optics : Ultrafast nonlinear optics |

Posted on October 10, 2012

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