Hi, In deriving a complex wave vector for waves in media, an equation D = ε E is typically used. Later though, it is common to relate D and E through a complex ε which is generally written as εcomplex = ε + 4 pi i σ / ω. It seems a contradiction to insert one relationship into the Maxwell equations and then explain how really it's another, where a complex epsilon allows for a phase shift between D and E. I'm also confused whether the σ here is necessarily complex; in the constituitive relation used to derive it, we had j = σ E and from the Dressel Electrodynamics of Solids book and the Wooten online optical solid state book he identifies σ as real. See, for example, http://web.mit.edu/course/6/6.732/www/6.732-pt2.pdf pages 2-3. Thanks!
The definition of D and H, in contrast to E and B, is convention. What is physical is the current density j. Part of it may be expressed as either the divergence of polarization P or rotation of the magnetization M. In different situations different conventions are useful. Nowadays in optics one mostly uses a convention where H=B, see e.g. http://siba.unipv.it/fisica/articoli/P/PhysicsUspekhi2006_49_1029.pdf As to your question whether conductivity sigma is real or complex, it is clearly complex in general. That reflects the fact that the current may be out of phase with the electric field.
I did some reading on this. As long as you epsiloncomplex is a scalar, one often splits it into it's real part epsilon and the imaginary term depending on real sigma. However, generally these quantities are tensors. Then it is usual to take the epsilon as the hermitian part of epsiloncomplex and the sigma dependent part as the antihermitian part. Both epsilon and sigma are then hermitian but eventually complex. There may be many different definitions which does not matter as long as you use it consistently.
This didn't quite answer my question. In the Wooten text, he inputs a constant of proportionality between j and E as sigma and then defines a complex sigma and that is then the constant of proportionality between j and E, reflecting that they are out of phase. I think the answer is also alluded to by him, the in the space/time representation sigma (and epsilon) for that matter is really a linear operator (he shows this after 2.67). In fourier space, we have the more simple relationship j = sigma E where sigma is some number which is obtained by plugging in for omega. I think the answer is here, but I haven't been able to see through it yet.
Yes, due to temporal and spatial dispersion, the product form j=sigma E makes only sense in Fourier space. I took this for granted as it is in Fourier space where complex quantities appear most naturally.