Group refractive index ($n_g=n+\omega {\partial }_{\omega }n$)

The group refractive index is $n_g=c/v_g$ for a transparent isotropic propagation branch with forward group velocity $v_g=d\omega /dk>0$. Differentiating the local wavenumber $k=n(\omega )\omega /c$ givesThe last formula gives the propagation group delay along a path whose geometry is fixed as frequency varies; additional apparatus wave phases require their own frequency derivatives. For a narrow wave packet with little loss and negligible reshaping, it approximates the transit of its wave envelope. In the nondispersive case ${\partial }_{\omega }n=0$, it reduces to $\mathrm{OPL}/c$. An optical path length at one frequency contains no information about ${\partial }_{\omega }n$, so in a medium with optical dispersion it cannot by itself determine the group delay. ^[ref]

For an original local model near ${\omega }_{\ast }$, choose two equal geometric paths of length $L$, with $n_A(\omega )=3/2$ and $n_B(\omega )=3/2+(\omega -{\omega }_{\ast })/(10{\omega }_{\ast })$. At ${\omega }_{\ast }$ both optical path lengths equal $3L/2$, and matched launch and interface references give equal carrier wave phases. But $n_{g,A}=3/2$ and $n_{g,B}=8/5$, so their group delays differ by $L/(10c)$. This local stipulated model is a counterexample to identifying equal wave phase with equal pulse arrival; it asserts no material response outside the narrow band considered. Strong loss or pulse reshaping requires a fuller propagation model, and group delay need not time the earliest causal response.

For vacuum incidence at fixed angle ${\theta }_i$ into a transparent isotropic medium, Snell's law gives $n({\lambda }_0)=\sin {\theta }_i/\sin {\theta }_t({\lambda }_0)$. Thus $dn/d{\lambda }_0=-n\cot {\theta }_{t}d{\theta }_t/d{\lambda }_0$, andAngles must be differentiated in radians. This determines the local group velocity from a first derivative of measured refraction angles; pulse broadening needs further information about group velocity dispersion and absorption.

The group delay is the frequency derivative of an unwrapped transfer wave phase, using the $e^{-i\omega t}$ convention. It predicts the leading delay of a narrow wave envelope when the transfer wave amplitude and the delay vary little across its band; it is not generally a signal-front arrival time.

The second derivative of spectral phase with respect to angular frequency; it measures variation of group delay with angular frequency.

The cubic term in a component's spectral phase, conventionally written ${\varphi }_3{\Omega }^3/6$, where $\Omega$ is an angular frequency offset.