Dispersion Relation in Different Media

  • #1
deuteron
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TL;DR Summary
does the dispersion relation hold for electromagnetic waves in all media, or just in non-dispersive media?
In my lectures, we have derived the dispersion relation
$$ |\vec k|^2 = \frac {n^2 \omega^2}{c^2}$$
by substituting in a plane wave solution for the electromagnetic wave, into the wave equation derived from the Maxwell equations
$$\Delta\vec E= \mu_0\epsilon_0 \frac {\partial^2 \vec E}{\partial t^2}$$

But we also derived the same wave equation for classical waves
$$\Delta f = \frac 1 {v^2} \frac {\partial^2 f}{\partial t^2}$$
from observing a wave propagating with velocity ##v##, and then saying that the wave would be constant in a coordinate system that was moving with the wave with the speed ##v##. From that requirement, we got that the most general form of the wave solution is
$$f(x,t)= A_1 h(x-vt) + A_2 g(x+vt)$$
and by twice partial differentiating, we got the above wave equation.

But this derivation from classical physics requires the assumption of non-dispersion, since we say that the wave does not change its shape for a coordinate system moving with the speed of the wave.
However, we get the same wave equation from the Maxwell equations, even though electromagnetic waves are dispersed in some media
How is that possible that 1. we derive an equation by making an assumption on non-dispersion, 2. we derive the same equation from Maxwell equations, 3. but the solution of the wave equations describing EM waves are allowed to be dispersed?

And if the wave equation holds for dispersed waves too, does that mean that the dispersion relation we derive by substituting in a plane wave solution, hold for both dispersed and non-dispersed plane waves, thus essentially for all waves?
Or is the above dispersion relation only valid for non-dispersive media? (which would make me more confused, since we can substitute in a solution for the wave equation that is dispersed, into the wave equation; since by requirement all waves are supposed to fulfill the wave equation, and there are dispersed waves)
 
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  • #2
If I understand you correctly, you are confusing "dispersion relation" with "dispersion", which may not be surprising.

A "dispersion relation" is the relationship between momentum and energy, while "dispersion" usually refers to a frequency/wavelength -dependent material property (for example, n(λ)).

Does that help?
 
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  • #3
Andy Resnick said:
If I understand you correctly, you are confusing "dispersion relation" with "dispersion", which may not be surprising.

A "dispersion relation" is the relationship between momentum and energy, while "dispersion" usually refers to a frequency/wavelength -dependent material property (for example, n(λ)).

Does that help?

we called the relationship between ##\vec k## and ##\omega## the dispersion relation in my lectures, and derived the above relationship from the wave equation, which we derived by assuming that the wave packet doesn't get spread, fall apart, in time; so by assuming a non-dispersive medium

I was curious if the relation we derived that way holds also for dispersive media, since we get the same wave equation from the Maxwell equations too
 
  • #4
deuteron said:
we called the relationship between ##\vec k## and ##\omega## the dispersion relation in my lectures, and derived the above relationship from the wave equation, which we derived by assuming that the wave packet doesn't get spread, fall apart, in time; so by assuming a non-dispersive medium

I was curious if the relation we derived that way holds also for dispersive media, since we get the same wave equation from the Maxwell equations too
It seems you are asking the same question again?

Rather than give the same answer again, let me try this way: a dispersive medium is one in which the material response to an applied field is non-instantaneous and nonlocal:

D(t,x) = ∫dτ∫dξ ε(t,τ;ξ,x)E(τ,ξ)

From this, one can impose causality and derive the Kramers-Kronig relations (or equivalently Hilbert transforms) relating the real and imaginary components of (say) the permittivity.

Another response (from "Formal structure of electromagnetics" by Post):
"The traditional real algebraic relation between the fields is not adequate to represent dispersion even if one makes the coefficients ε and μ functions of the frequency or wave number, because the phase shift between cause and effect is not accounted for by a real algebraic relation. It was noted quite early, as in circuit theory, that the formalism of complex field variables enables one to remove this inadequacy."

So, in general, the answer is no. For example, the Pierce dispersion relation looks very different than E = cp or E = p2/2m:

https://www.egr.msu.edu/~pz/tutorial-TWT.pdf (eqn 5)

https://ece-research.unm.edu/FY12MURI/pdf_Files/Schamiloglu_EAPPC_BEAMS_2012.pdf (an example of 'dispersion engineering')
 
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