I am trying to understand the physical meaning of the dispersion relation. Is it how inhomogeneous a media is ? Or how much the electromagnetic fields spread in the media? Or ?
Electromagnetism – Understanding the Dispersion Relation in Optics
dispersionelectromagnetismoptics
Related Solutions
Dispersion can be locally nought and effectively nought over quite wide bands in engineered devices.
In an optical fiber, for example, the phase delays of the modal fields depends on two things:
The dispersion of the material making up the fiber and
The frequency dependence of the modal eigenvalue equation, which depends on the refractive index profile and is there even if the fiber could be made out of dispersionless materials.
The second order term in the phase as function of wavelength Taylor series determines, to first order, the total dispersion. In optical fibers, the two mechanisms above contribute exactly mutually cancelling dispersions from effects 1 and 2 in certain wavelength bands. For example, a step index profile fiber typically has quite a wide, low dispersion band at 1300nm wavelength. This band can be shifted by engineering the refractive index profile (the so called depressed cladding index profile) into the 1550nm communications band.
Grating devices can hold a system dispersionless over stunningly wide wavelength bands. For instance, carefully designed "chirped " fiber Bragg gratings are used to exactly oppose the dispersion of optical systems over hundreds of nanometers bandwidth, and thus are useful in setting up femtosecond long pulses.
Dispersion relations connect the energy to wavelength (or momentum) of a particle/wave.
For example:
$$\hbar \omega=\hbar c k=\hbar c \frac{2\pi}{\lambda}$$
Would be the dispersion relation of light, and it shows that energy and momentum are linearly proportional. Waves with zero momentum have zero energy.
Compare this to:
$$\hbar \omega=\hbar \omega_0 + a k^2$$
Now, zero momentum does not imply zero energy, and there is a non-linear relation between energy and momentum.
(Answering OP's edited question) key property that can be obtained from the dispersion relation is the speed of propagation, or group velocity. This is given by the slope $d\omega/dk$, which for acoustic phonons gives the speed of sound, and for optical phonons is typically quite small.
Best Answer
A dispersion relation tells you how the frequency $\omega$ of a wave depends on its wavelength $\lambda$--however, it's mathematically better to use the inverse wavelength, or wavenumber $k = 2\pi/\lambda $ when writing equations because the phase velocity is
$v_{\rm phase}\ \ = \omega / k$
and the group velocity is
$v_{\rm group}\ \ = d\omega/dk$.
These apply to all types of waves. Regarding electromagnetic waves in vacuum:
$ \omega(k) = ck $
so that
$v_{\rm phase} \ \ = v_{\rm group} \ \ = c$.
The waves are dispersionless. In a medium, even a homogeneous medium, such as glass, the index of refraction increases with frequency (in the visible, of course) so that light is dispersed by color.