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I am confused on why would the propagation speed of any EM waves at ANY frequency is constant in the free space (vacuum) but they seem to disperse in any other materials as the propagation speed of EM waves in other materials depends on the frequency? Can someone give me a good proof of this phenomena with appropriate formulas?

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It's because most materials have (many) natural resonances. I assume you are alright with the phase velocity being different in different media, that is I assume you are alright with something of the form

$$ \frac{ \omega }{ k } = \frac{c}{n} = \frac{ c}{\sqrt{ \mu \epsilon }} \sim \frac{c}{\sqrt \epsilon} $$ where $k$ is the wavenumber of a wave, $\omega$ its frequency, $c$ the speed of light, $n$ is the index of refraction, $\mu$ the magnetic permeability of a substance and $\epsilon$ its dielectric constant. Since most materials have a magnetic permeability close to that of the vacuum, your question now becomes, why do most materials have a frequency dependent dielectric constant?

This is because most materials have some internal intrinsic frequencies. We can understand this better with some simple models. The dielectric constant measures the ratio of the electric field in a material versus what the electric field in vacuum would be, so what is important is the induced dipole field $ E' = E - 4\pi P$. If we assume the material is made up of a constant density $N$ of simple dipoles, consisting of charge $q$ separated by distance $x$, we have for out dielectric constant

$$ n^2 = \epsilon = \frac{ E + 4\pi P}{E} = 1 + 4\pi N q x $$

To proceed, we need a simple model for the response of the dipoles in our material. For that, let's just model them as simple driven damped harmonic oscillators

$$ \ddot x + \frac{ \omega_0 }{Q} \dot x + \omega_0^2 x = \frac{qE_0}{m} e^{i\omega t} $$ where $Q$ is the quality factor, $\omega_0$ is the resonant frequency, $q$ is the charge, $E_0$ is the amplitude of our incoming wave, $m$ is the mass of our charge and $\omega$ is the driving frequency.

For now, if we assume we are far from resonance, we can ignore the damping term and look for the amplitude of our oscillations at the driving frequency, so taking the ansatz $$ x = A e^{i \omega t} $$ we obtain $$ A = \frac{ q E_0 }{m } \frac{ 1 }{ \omega_0^2 - \omega^2 } $$

Giving us $$ n^2 = \epsilon = 1 + \frac{ 4 \pi N q^2 }{m } \frac{ 1 }{ \omega_0^2 - \omega^2 } $$

So, we can see the frequency dependence right here. As long as our material is made up of things like dipoles (which they are), that have some kind of natural frequency (which they do), we will get a frequency dependent dielectric constant, which will mean we have a frequency dependent phase velocity, which means dispersion.

Now, typically, molecules will have a whole bunch of resonant frequencies given by their slew of excitation frequencies, and each of these will contribute, but on the whole, the effect of all of these different frequencies will amount to something like the above with $\omega_0$ now to be interpreted as an "average" resonant frequency. For something like glass, this "average" frequency will be in the ultraviolet, meaning for optical phenomenon, $\omega < \omega_0$, so that $n > 1$ and in particular, the index of refraction for blue light should be higher than the index of refraction for red light. This is "normal" dispersion, where the index of refraction increases with increasing frequency. This is why the blue coming out of a prism is bent more than the red. If we had some kind of material with a particularly low average resonance, or we were looking at high frequency waves (like X-rays), we would have "anomalous" dispersion, where the index of refraction decreases with increasing frequency.

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  • $\begingroup$ +1 for mentioning intrinsic resonances and basically going through the entire first quarter of my undergrad emag class. OP: this model is known as the Drude Lorentz model of dielectrics if you want to look it up. $\endgroup$ – eqb Aug 4 '14 at 5:45

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