- #1

- 136

- 6

Dealing about wave propagation in a medium and dispersion, wavenumber [itex]k[/itex] can be considered as a function of [itex]\omega[/itex] (as done in Optics) or vice-versa (as maybe done more often in Quantum Mechanics). In the first case,

[itex]k (\omega) \simeq k(\omega_0) + (\omega - \omega_0) \displaystyle \left. \frac{dk(\omega)}{d \omega} \right|_{\omega = \omega_0} + \frac{1}{2} (\omega - \omega_0)^2 \left. \frac{d^2 k(\omega)}{d\omega^2} \right|_{\omega = \omega_0}[/itex]

where first derivative is the inverse of group velocity [itex]1/v_g[/itex] and the second derivative is (maybe?) inverse of group velocity dispersion [itex]1 / \alpha[/itex].

In the second case,

[itex]\omega (k) \simeq \omega(k_0) + (k - k_0) \displaystyle \left. \frac{d\omega(k)}{d k} \right|_{k = k_0} + \frac{1}{2} (k - k_0)^2 \left. \frac{d^2 \omega(k)}{dk^2} \right|_{k = k_0}[/itex]

where the first derivative is the group velocity itself [itex]v_g[/itex] and the second derivative is the group velocity dispersion itself [itex]\alpha[/itex].

But what about the limit-cases, when for some reason [itex]v_g = 0[/itex] or [itex]\alpha = 0[/itex]? The latter case is maybe more common: it would imply that, in a medium, for a signal whose group velocity doesn't vary with frequency, the second order term in the [itex]k(\omega)[/itex] expansion has an infinite coefficient.