Macroscopic Maxwell's equations and speed of light in media

[Reignbeaux]

So I followed the derivation of the Macroscopic Maxwell's equations by averaging the fields / equations and doing a taylor series to separate the induced charges and currents from the free ones. But why does light now "suddenly" travel slower in dielectric media? I mean, sure, it comes out from the macroscopic equations, but what is happening here?

If you think about it from a microscopic point of view, the response of the medium, so it's polarisation (or radiation resulting from the polarisation) have to kind of cancel out the original wave traveling at c and create a "new" one going at c/n. Right?

How does this now drop out of the macroscopic equations, without having to worry about what is actually happening? Is it the averaging? Is there a better way to think about this?

 

[Charles Link]

It's been a while since I've seen this derivation in an Optics class in college, but the electric field gives the molecules of the material get an induced sinusoidal (in time) polarization that causes them to be radiating dipoles, and this radiated electric field gets superimposed on the incident electric field. Using diffraction theory, (Huygens principle, etc.), the resulting wave front can be computed, and it is found that the additional field from the dipoles causes an overall decrease in the speed of the wave front. I don't recall precisely how this comes about, except that the new speed of light through the material is then given by $v=c/n$ where $n=\sqrt{\epsilon}$. $\\$ Editing: Macroscopically, it is simpler than that. Mawell's $\nabla \times E=-\frac{1}{c} \dot{B}$ along with $\nabla \times B=\frac{4 \pi J_{total}}{c}+\frac{1}{c} \dot{E}$ with $J_{total}=J_{free}+J_p +J_m$ , where $J_p=\dot{P}$, will give this result macroscopically in the derivation of the wave equation.

 

[blue_leaf77]

To add what Charles wrote above, the induced dipole moment and thus the polarization field is emitted with a certain phase lag with respect to the incident field. Microscopically, this delay in the photon emission is connected to the lifetime of various levels in the medium which were excited due to the passage of the incident light. In the atom-to-atom space, the emitted photons still travel with the speed of c, but macroscopically due to the delay in the polarization field, the net propagation through the medium looks slower.

 

[Charles Link]

To add what Charles wrote above, the induced dipole moment and thus the polarization field is emitted with a certain phase lag with respect to the incident field. Microscopically, this delay in the photon emission is connected to the lifetime of various levels in the medium which were excited due to the passage of the incident light. In the atom-to-atom space, the emitted photons still travel with the speed of c, but macroscopically due to the delay in the polarization field, the net propagation through the medium looks slower.

Thank you @blue_leaf77. I think I have this part correct, that $P=\chi E$ and basically $\dot{P}=i \omega P$. The radiated $E$ from $P$ in a lossless material, (where $\chi$ is completely real), will come from $J_p=\dot{P}$, and will automatically lag in phase from the incident $E$. If, in fact, $\chi$ has an imaginary component, where $\chi(\omega)=\chi'(\omega)+i \chi''(\omega)$, then it will also be a lossy material. $\\$ I think I have this correct, but it has been a while since I studied these derivations in detail.

 

[blue_leaf77]

The radiated P P in a lossless material, (where χ \chi is completely real), will automatically lag in phase from the incident E E . If, in fact, χ \chi has an imaginary component, it will also be a lossy material.

Yes that's true. In fact, after checking out my old notes, for an incident field of the form $E(t) = E_0 \cos \omega t$, the induced polarization looks like
$$
P(t) \propto n E_0 \left( (\omega_0^2 - \omega^2)\cos \omega t + 2\gamma\omega \sin\omega t \right)
$$
where $n$ the density of the medium, $\omega_0$ the peak absorption frequency, and $\gamma$ the width of the absorption peak. In frequency domain, this results from $\chi$ being complex as you said (its imaginary part contains $\gamma$).

 

[Charles Link]

Yes that's true. In fact, after checking out my old notes, for an incident field of the form $E(t) = E_0 \cos \omega t$, the induced polarization looks like
$$
P(t) \propto n E_0 \left( (\omega_0^2 - \omega^2)\cos \omega t + 2\gamma\omega \sin\omega t \right)
$$
where $n$ the density of the medium, $\omega_0$ the peak absorption frequency, and $\gamma$ the width of the absorption peak. In frequency domain, this results from $\chi$ being complex as you said (its imaginary part contains $\gamma$).

Right at resonance, where $\omega=\omega_o$, I believe the amplitude of the $P$ will be limited by $\gamma$ , but that is a very fine detail. Also, additional item, I think the $\omega_o^2-\omega^2$ belongs in the denominator. (It's a polarization response where a resonance is present). See also http://nptel.ac.in/courses/113104005/59 and page (2), equation (4.80) and equation (4.81).

 

[blue_leaf77]

Yes, indeed the more complete expression involves $(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2$ in the overall denominator but I omit this in my previous post because I only want to focus on the time-dependence of the polarization.