a single unit, rather than splitting it into pieces. Note that this approach still differs from solutions of Maxwell’s equations such as FDTD (finite difference time domain)[@Joseph-T-1997itap] or PSSD (pseudospectral spatial domain)[@Tyyrell-KN-2005jmo], because our single-field is based on a second-order wave equation, and uses a convenient choice of carrier function to define a field envelope.

Following these considerations, we now derive a [*single-field*]{} model for Raman generation, and, apart from that notable detail, follow an analogous path to that of Hickman, Paisner, and Bischel (HPB) [@Hickman-PB-1986pra]. In the model, we find that the coupling constants retain an oscillatory behaviour at the transition frequency, and that it is this that impresses the sideband modulation on the propagating field. Since the field is not only wideband, but contains significant sideband components, we need to propagate this (no longer slowly varying) field envelope using the GFEA. The necessity of allowing for these can be demonstrated by converting the single-field model into a multi-field counterpart – without the envelope-gradient corrections, we will not get a correct multi-field model.

Summary of the theory and the numerical implementation {#ss-intro-theorysummary}
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We model the wideband Raman generation process in the following way. We specify the field frequencies 