# Dispersion vs. time-dependence

1. Dec 15, 2011

### Niles

Hi

I have two books about electrodynamics that solve Maxwell's Equations. The first one uses the assumptions

1) Linear regime (i.e. not strong fields)
2) Isotropic medium (so disregard tensor nature of ε)
3) Transparent medium (i.e. a real ε)
4) No dispersion of ε

In the second book, they use (1)-(3) as well, but (4) is now stated as

4) ε is time-invariant

Now, my questions is: How can time-invariance of ε be the same as ε not having dispersion? Because if ε is constant in time, then Fourier-transforming it will give me a delta-function. So ε *will* depend on ω. What is wrong with my reasoning so far?

Thanks for any help.

Best,
Niles.

2. Dec 15, 2011

### Born2bwire

You do not need to assume these assumptions for Maxwell's equations (though it does greatly simplify the solving of the equations) so I don't think you need to assume that the two are the same. I would say that real \epsilon and dispersionless \epsilon are equivalent. The real and imaginary parts of the permittivity are related by a Hilbert transform called the Kramers-Kronig relation. If you have loss (imaginary part) this requires that the real part be frequency dependent (dispersion). So a dispersionless permittivity has to be lossless.

But I guess you can say they are equivalent because if you had dispersion and you had the situation where you sent a wave of 1MHz and then say 25 MHz then you would see that the \epsilon must change according to the frequency at hand (of course since we are doing time limited pulses there will be a bandwidth of frequencies in fact). So if you took the Fourier Transform you would get two pulses that would be associated with the two frequencies that you used. So if the permittivity does not change in time, then it can't change in response to a changing frequency in the incident waves and thus if you would find that for all frequencies that the epsilon would be constant. I guess you have to think of the Fourier Transform being done with respect to the behavior of the epsilon in time with respect to the frequency of the waves.