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In a vacuum, electromagnetic waves of all frequencies travel with the same phase speed, so they propagate with a fixed shape once determined. In a dispersive medium, waves of different frequencies travel with different phase speeds and this causes the wave packet to change shape when propagating. Certainly, the dispersion phenomenon is due to the medium, but what is its property responsible for dispersion?

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Electromagnetic field induces polarisation and magnétisation in the media, which are not an instantaneous response. This results in k-vector being frequency-dependent, hence the group velocity, $$ v_g=\frac{d\omega}{dk}=\left(\frac{dk}{d\omega}\right)^{-1} $$ is different from the phase velocity $$ v_{ph}=\frac{\omega}{k}, $$ which is what we call dispersion.

Update
Dispersion and causality section of the Wikipedia article on permittivity gives a rather good review of the relevant .EM equations

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  • $\begingroup$ Do you mean the inability of the medium to respond to the waves instantaneously causes the wavenumber to be frequency dependent? $\endgroup$
    – Kksen
    Commented Aug 6, 2021 at 6:52
  • $\begingroup$ @KamKahSen yes, indeed - the media responds with different speed to different frequencies $\endgroup$
    – Roger V.
    Commented Aug 6, 2021 at 8:18
  • $\begingroup$ How can we show this fact (If the response to waves is not instantaneous, then the properties must be frequency-dependent) mathematically? $\endgroup$
    – Kksen
    Commented Aug 6, 2021 at 9:10
  • $\begingroup$ @KamKahSen you can check section Dispersion and causality in the Wikipedia article on permittivity : en.m.wikipedia.org/wiki/Permittivity $\endgroup$
    – Roger V.
    Commented Aug 6, 2021 at 10:05
  • $\begingroup$ That's very useful,thanks for the info. $\endgroup$
    – Kksen
    Commented Aug 6, 2021 at 10:38
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. . . . . the dispersion phenomenon is due to the medium, but what is its property responsible for dispersion?

The incoming electromagnetic wave interacts with the medium and the result of that interaction depends of the frequency of the electromagnetic wave and the properties of the medium.

The situation is very complex as one has to consider the effect on a single charge of not only the effect of the electric field of the incoming electromagnetic wave but the electric field produced by all the electric fields produced by all the other charge in the medium.
To reduce this complexity it is often assumed that the effect on the charge under consideration of the electric fields produced by the electric fields of all the other charges is negligible as compared with the electric field produced by the incoming electromagnetic wave.

The medium is a very complex system with very many natural frequencies of oscillation so let me pick one component of the system, the electrons orbiting the nucleus of an atom which can be polarised and assume that the electron shell fastened elastically to the atom, ie, liken it to being like a mass on the end of a spring with a natural frequency of oscillation of $\omega_0$.

That electrons will be undergoing forced oscillation due to the electric field of the incoming electromagnetic wave at the frequency of the wave $\omega$. In general this will mean that the electrons will be oscillating out of phase with the incoming electromagnetic wave and so producing an oscillating electric field which is out of phase. Here is an example of the origin of the apparent slowing down of electromagnetic waves within a medium.

For light and glass one of natural frequencies (resonant frequency) of glass is in the ultra-violet region of the electromagnetic spectrum so the change of phase is smaller for red light than for blue light light because the frequency of blue light is greater than that of red light and hence closer to $\omega_0$.
This in turn means that the phase velocity of a given frequency of red light is greater than that of blue light which is also true of the group velocities over a small range of frequencies and thus the refractive index of glass for red light is less than that for blue light, ie red light appears to travel faster through glass than blue light.

Another interesting fact is that in certain circumstances the phase velocity can actually exceed the speed of light in free space but that would be for a single frequency and the group velocity would still be less than the speed of light.

Much more here from Feynman, The Origin of the Refractive Index.

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Dispersion is phenomenon in which the temporal form of the signal pulse is altered. It is called dispersion because the pulse usually expands over time ("disperses").

The pulse has a spectral content The different spectral components of the pulse travel at different speeds and therefore reach the output at different times (expansion)

Cause of Dispersion

Spectral components reach the output at different times ... Pulse time segments also occur at different times

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