2005 European Quantum Electronics Conference

True uni-directional pulse propagation using Fleck fieldvariables

P. Kinsler, S.B. Radnor, G.H.C. New.Blackett Laboratory, Imperial College London, Prince Consort Road,

London SW7 2BW, UK.DrPaul.Kinsler@physics.orghttp:/hvww.qols.ph.ic.ac.uAl

Abstract: Pulse propagation theories continue to ignore the backward-propagating field component. Byusing directional field variables, we derive wave equations allowing us to accurately judge the effect ofthe backward components.

We use a pair of combined electro-magnetic field variables G' = \/ E ±i \/iH to describe forward and backwardpropagating optical signals separately[1]. Maxwell's equations can now be re-expressed without approximation toform wave equations describing the field propagation in fully vectorised and plane-polarized versions. In addition, wealso present second order and envelope-based wave equations for G±, in both stationary and moving frames. UsingG±, it is as easy to include magneto-optic as electro-optic effects (i.e. dispersion and nonlinearity). For example, theplane polarized first order wave equations describing the z-propagation of G± in the spectral (o) domain are -

DzG± = }iFtw ioG'o 2 [G- + GJ 2-/ G+ ] (1)

where ax contains the dispersion and nonlinearity affecting the electric field; Oix is its magnetic counterpart.

Using these directional G± variables has three main advantages:* Since most pulse propagation studies are interested only in the forward propagating part of the field, the forwardpropagating field variable G+ is the appropriate physical choice to use in such a description.* Most approaches assume the backward propagating parts of the field are negligible. Since our G- describes thisbackward part, we have a clear and physically appropriate basis on which to test such approximations.* Simulating only the forward pulse G+ yields a 2x speed advantage. Adding a z-propagated PSSD algorithm[2] wealso get a fast and flexible treatment of dispersive and nonlinear effects outperforming standard FDTD approaches.

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Fig. 1. Generation of a backward G- wave when a forward propagating wave G+ encounters a medium with a differentrefractive index (n = 1.5). The dotted line is the initial pulse. the heavy line is the transmitted G+, and the light line is the"reflected" G- wave. See the text for further remarks.

As a demonstration, figure 1 shows an intial forward propagating wave G+ encountering a medium with a differentrefractive index. Note that in the z propagated picture, the interpretation of G- is rather subtle, and we will expandon this in our presentation. The G- contribution in the region of the main G+ pulse plays an important role, since itensures the amplitudes of the transmitted electric and magnetic fields are correct.

We will also present a range of numerical simulations covering various dispersive and nonlinear cases in the moretypical forward-propagating regime - second and third harmonic generation, soliton propagation, parametric ampli-fication, birefringent media, instantaneous and dispersive nonlinearities. On-going calculation of the backward wavecomponents allows us to monitor their role and justify (or not) the common forward-only approximations.

[1] J. A. Fleck, Phys. Rev. Bl, 84 (1970).[2] J.C.A. Tyrrell, P. Kinsler, G.H.C. New, J. Mod. Opt., in press.

0-7803-8973-5/05/$20.00 ©2005 IEEE 88'