where F(ω) is the complex spectral distribution, ε
0 is the permittivity of free space, z
i>0 is the initial pulse position of the backward-propagating pulse, and the frequency dependence of each component is explicitly shown. Subscripts denoting layer have been omitted for convenience. A large confinement factor normally implies a very small group velocity v
g (e.g., vg=2×10
5 m/s for confinement factor n=300 and a substrate of SiO
2 of refractive index 1.4446 at free space wavelength 1.5 μm), which can be negligible compared to the speed of free electrons from standard electron microscopes and DC electron guns. Hence, the counter-propagating pulses practically approximate a stationary, standing wave grating.
When the GP pulse duration is large, a simplified form for Equation (12) can be: