Phase and Group Velocity
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## Definition
**Phase velocity:** _(means only when mentioning one wave with a specific frequency)_ is the velocity at which points of *constant* phase move through the medium (space).
Mathematically write: $v_{p}=\frac {\omega}{k}$
**Group velocity:** _(means when there are multiple waves travel together forming ++a group of wave++ or ++a wave packet)++_ is the speed at which the energy of a wave packet propagates, telling us how fast is the wave envelope of that wave packet.
## Derivation
**Phase velocity**
**#1 point of view**
We first start from the form of a wave:
$w_{1}=A_{0}Cos(k_{}.x-\omega_{}.t)=A_{0}Cos\phi$
where $A_{0}$ is the amplitude of a wave and $\phi$ represents the phase.
Look at the wave propagating in space ($x$-direction as indicated by the formula), if we want to see how fast the phase of wave propagates in space, we choose two consecutive values of the wave of the same phase $\phi$ at different time and space, i.e. $\phi_{1}=\phi_{2}=\phi_{c}$.
Explicitly write: $\phi_{1}=\phi_{2}=\phi_{c}$
$\Leftrightarrow k_{}.x_{1}-\omega_{}.t_{1}=k_{}.x_{2}-\omega_{}.t_{2}
\Rightarrow k.(x_{2}-x_{1})=\omega_{}.(t_{2}-t_{1})\Rightarrow \frac{(x_{2}-x_{1})}{(t_{2}-t_{1})}=\frac {\omega}{k}=v_{p} \rightarrow$ phase velocity.
**#2 point of view**
A monochromatic wave of angular frequency $\omega$ propagates through a medium of refractive index n.
$E(z,t)=E_0 e^{i(k.z-\omega.t)}+c.c$
where $k=n\omega/c$.
Clearly, the phase is $\phi=k.z-\omega.t$
points of constant phase move a distance $\Delta z$ in a time $\Delta t$, + definitionition of the phase velocity we have,
$k\Delta z=\omega\Delta t$.
Thus $v_p=\frac{\Delta z}{\Delta t}=\frac{\omega}{k}=\frac{c}{n}$
**Group velocity**
a pulse = a group of waves = a wave packet = waves add up in phase at the peak.
If this pule propagates through a medium without distortion, the constituent waves must add in phase for $z$.
We first write the phase $\phi=\frac{n\omega z}{c}-{\omega t}$
No distortion = no change in phase = $\frac{d\phi}{d\omega}=0$ or
$\frac{dn}{d\omega}\frac{\omega z}{c}+\frac{nz}{c}=t \Leftrightarrow z=(\frac{c}{n+\omega \frac{dn}{d\omega}})t=v_g t$,
where $v_g=\frac{c}{n+\omega \frac{dn}{d\omega}}=\frac{d\omega}{dk}$ is the group velocity.
The last equality is obtained from the relation $k=n\omega/ c$
$dk=dn \frac{\omega}{c}+\frac{n}{c}d\omega \rightarrow \frac{dk}{d\omega}=\frac{\omega}{c}\frac{dn}{d\omega}+\frac{n}{c} \rightarrow \frac{d\omega}{dk}=\frac{c}{n+\omega\frac{dn}{d\omega}}=v_g$.
Alternatively, we can express
$v_g=\frac{c}{n+\omega\frac{dn}{d\omega}}=\frac{c}{n_g}$
where $n_g=n+\omega\frac{dn}{d\omega}$ represents the group index.