= Group refractive index
{title2=$n_g=n+\omega\,\partial_\omega n$}
= Group index
{synonym}
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> $k=n(\omega)\omega/c$ gives
$$
n_g=n+\omega\frac{\partial n}{\partial\omega}=n-\lambda_0\frac{\partial n}{\partial\lambda_0},\qquad
\tau_g=\frac{\partial\phi_{\mathrm{prop}}}{\partial\omega}=\frac1c\int_\gamma n_g(\mathbf r,\omega)\,ds.
$$
The 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 $\operatorname{OPL}/c$. 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>. <RP Photonics Encyclopedia by Rüdiger Paschotta/Group Index>{ref}
For an original local model near $\omega_*$, choose two equal geometric paths of length $L$, with $n_A(\omega)=3/2$ and $n_B(\omega)=3/2+(\omega-\omega_*)/(10\omega_*)$. At $\omega_*$ 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.
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