Theory of directional pulse propagation

We construct combined electric and magnetic field variables which independently represent energy flows in the forward and backward directions, respectively, and use these to reformulate Maxwell’s equations. These variables enable us to not only judge the effect and significance of backward-traveling field components, but also to discard them when appropriate. They thereby have the potential to simplify numerical simulations, leading to potential speed gains of up to 100% over standard finite difference time-domain (FDTD) or pseudospectral spatial-domain (PSSD) simulations. We present results for various illustrative situations, including an example application to second harmonic generation in periodically poled lithium niobate. These field variables are also used to derive both envelope equations useful for narrow-band pulse propagation, and a second order wave equation. Alternative definitions are also presented.

Figure 1

A $500\mathrm{nm}$ pulse in fused silica, represented with (a) a vacuum reference, (b) a fixed refractive index reference, (c) a perfectly matched dispersive reference. In all cases the initialization parameters are those that perfectly match the dispersive properties of the medium. Solid line: $G^{+}$ field. Dashed line: $G^{-}$ field.

Figure 2

The same pulse as in Fig. 1, after propagating $15\mu \mathrm{m}$; represented in (a) a vacuum reference, (b) a fixed refractive index reference (c) a perfectly matched dispersive reference. Solid line: $G^{+}$ field. Dashed line: $G^{-}$ field.

Figure 3

Simulation results showing $G^{\pm }$ for perfect reference $[{ϵ}_r={ϵ}_{\mathrm{silica}}\left(\omega \right)]$ and a matched dispersive frame.

Figure 4

A similar pulse as above, after propagating $10\mu \mathrm{m}$ through fused silica; represented in (a) a vacuum reference, (b) a fixed refractive index reference $(n=1.5)$, (c) a perfectly matched dispersive reference. Parameters have been adjusted to give a clearer final pulse shape. Solid line: $G^{+}$ field. Dashed line: $G^{-}$ field.

Figure 5

Second harmonic generation in $120\mu \mathrm{m}$ of $\mathrm{Li}{\mathrm{NO}}_3$, periodically poled at $6.05\mu \mathrm{m}$. Clockwise from top left: Initial pulse, final pulse, second harmonic power, final pulse spectrum. Solid line: $G^{+}$ field. Dot-dashed line: $G^{-}$ field.