I do not think a general solution exists for anisotropic media, though for some special cases it has been solved.

I guess Yuriy is right. http://cdn.intechopen.com/pdfs-wm/16302.pdf

http://www.slideshare.net/solohermelin/maxwell-equations-and-propagation-in-anisotropic-media-43285347

I believe you'll find the following reference useful:

John L. Tsalamengas, "Interaction of Electromagnetic Waves with General

Bianisotropic Slabs", IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. IO, OCTOBER 1992.

Also, reference [8] of the above paper specializes to anisotropic media.

Dear Dr. Yakhno,

if I understand your question correctly, you are looking for explicit formulae for the total electric and magnetic fields in terms of any external charges and currents, for a material with a possibly (bi)anisotropic electromagnetic response.

The constitutive relations of such a material may be given in the following form (cf. [1, Sec. 6.6] for a more general expansion):

(1)   Eind = χEEEext + χEBBext ,

(2)   Bind = χBEEext + χBBBext .

Here, the response functions χEE, χEB, etc. relate the induced electric and magnetic fields to the externally applied fields. They are in general tensorial integral kernels, hence for a general homogeneous material,

(3)   χEE = (χEE)ij(x – x', t – t') .

In certain limiting cases, the response functions may also be assumed to be constant (as you have specified). Alternatively, the constitutive relations may be given in terms of the permittivity tensor ε and the permeability tensor μ (and possibly some further magnetoelectric coefficients). These latter response functions can be expressed explicitly in terms of the above response functions (see [1, Eqs. (6.47)-(6.48)]).

Now, given the external charges and currents, one may first solve the (“vacuum”) Maxwell equations for the external electric and magnetic fields, such that one finds

(4)   Eext = Eext[ρext, jext] ,

(5)   Bext = Bext[ρext, jext] .

These functionals are explicitly known (see e.g. [1, Secs. 3.1-3.2]). From these, one can then calculate Eind and Bind by means of the constitutive equations (1)-(2), and finally obtain

(6)   Etot = Eext + Eind

(7)   Btot = Bext + Bind

for the total electric and magnetic fields. I hope this might be of help in answering your question. (Please let me know if I might provide any further details.)

[1] R. Starke, G.A.H.S., Functional Approach to electrodynamics of media, Phot. Nano. Fund. Appl. 14, 1-34 (2015); https://www.researchgate.net/publication/275208133_Functional_Approach_to_Electrodynamics_of_Media

Article Functional Approach to Electrodynamics of Media

Dear Dr.Schober,

Thanks alot for your comments and reference.

I looked your paper. It is very fundamental work. I will analyze a possibility to apply the result of your paper to my research.

Valery Yakhno

Checkout James Wait's work on the subject and Moran and Gianzero's paper attached.

 Dear Dr. Dean Homan,

Thank you for the Moran and Gianzero paper you sent me. It is interesting study. I did not know it. This work is devoted to the study subject to assumptions that \sigma is a tensor, \mu is a constant, i \omega\epsilon is ignored. My interest is related more to the case when \epsilon and \mu both of them are tensors, for example, symmetric positive definite matrices. Do you know where can be found the explicit formula of the fundamental solution of the Maxwell's system for this case?

With best regards,

Valery

Dear Dr. Yakhno,

thank you very much. I have just worked out the concrete formulae from the procedure described above. In Fourier space, they read as follows (using SI units):

E(k,ω) = (ε0 εr)-1 ( -ω2 + c2|k|2 )-1 ( -c2 ik ρ(k,ω) + iω j(k,ω) ) ,

B(k,ω) = μ0 μr ( -ω2 + c2|k|2 )-1 ( c2 ik x j(k,ω) ) ,

where εr and μr denote the relative permittivity and permeability tensors (which are assumed to be independent of k and ω), and in the second line x denotes the vector product. (Again, please let me know if any details of the derivation would be required.)

Dear Prof. Wagner,

Thanks for your comments and your work "The singular terms in the fundamental matrix of crystal optics "(Proc.R.Soc.A (2011)). It is nice that this paper has a good introduction with the wide description of the basic works about fundamental solutions. I understood that even for the case of bi-axial crystals the fundamental solution of the Maxwell's equation has a peculiar form which is not easy to be obtained. I will analyse your paper for the further understanding. Thanks a lot.

 Valery 

Dear Dr. Schober,

Thanks for your interest and comment for this topic. My understanding the fundamental solution of Maxwell's equations in explicit form in recent is the following. I think that the explicit formula of the Fourier image of fundamental solution (in the Fourier space) can be constructed. It is not clear for me how to calculate explicitly the inverse Fourier transform in the case of anisotropic media. It is possible for the case of isotropic media. For the case of biaxial crystals see Wagner's work "The singular terms in the fundamental matrix of crystal optics "(Proc.R.Soc.A (2011)).It is found an explicit formula of the singular terms in this study. Probably it is not easy. With best regards, Valery

Please allow me, Dr. Yakhno, to post here also the general solution in real space, which is obtained from the corresponding formulae in Fourier space by performing an inverse Fourier transformation. They are given in the attached file. With best regards, Giulio

Dear Dr. Schober,

I am not sure that we have a chance to use the fundamental solution of the wave equation. I think that we can consider the vector equations

$$

\bar{\bar{\epsilon}}\frac{\partial^2 \textbf{E}}{\partial

t^2}+{curl}_x({ \bar{\bar{\mu}}^{-1} curl}_x\textbf{E})=-\frac{\partial J}{\partial

t}

$$

$$

\bar{\bar{\mu}}\frac{\partial^2 \textbf{H}}{\partial

t^2}+{curl}_x({ \bar{\bar{\epsilon}}^{-1} curl}_x\textbf{H})= curl_x( \bar{\bar{\epsilon}}^{-1} J)

$$

It seems to me that these equations are not reducible to the wave equation if $\mu$ and $\epsilon$ are tensors (matrices) even for the case where $\mu$ and $\epsilon$ do not depend on $x$ and $t$ (wave vector and frequency).

Anyway I will be very glad if you will send me your reasoning in details.

Sincerely yours,

Valery

Dear Dr. Yakhno,

the last two contributions precisely reflect the different (microscopic vs. macroscopic) approaches to electrodynamics in media. In fact, in the traditional (macroscopic) approach, one introduces four field quantities, namely D, E, H and B. The corresponding Maxwell equations are then underdetermined and therefore have to be complemented by material relations. By contrast, in the modern microscopic approach used in ab initio materials physics, one instead formulates electrodynamics in media in terms of external, induced and total (total=external+induced) fields, all of which are fully determined by their respective Maxwell equations in terms of external, induced and total sources (i.e. charge and current densities). The definite solution for the total fields in terms of the external fields is then -- within the linear regime -- precisely the one given by Dr. Schober. It follows from expressing the total fields by linear response theory in terms of the external fields and the latter in terms of their own sources via the general solution formula for the microscopic Maxwell equations.

Correspondingly, in the microspic approach to electrodynamics in media all fields obey the ordinary inhomogeneous wave-equations of classical electrodynamics, but in terms of their respective sources. The "wave-equation in media" cannot obtained from these. Instead, it follows from a different logic which we have expounded in (it coincides with a form of the wave-equation in media which is already used in tracts on plasma physics):

http://arxiv.org/abs/1510.03404

Concerning your original question, you rightly remarked that the formula presented by Dr. Schober does not give the solution to that problem. Instead, it gives the answer to the corresponding question from the microscopic point of view. The decisive difference is that the procedure you have used in your last answer to obtain the wave-equations in media does not lead to new wave-equations in the microscopic setting. Instead, homogeneous equations like $\nabla\cdot(\mu H)=0$ yield identities which are always fulfilled (i.e. irrespective of $H$) on account of constraint conditions on the implied response functions. In particular, in this context $\mu, \epsilon$ etc. are always treated as response functions (i.e. they depend at least on the frequency and the wave-vector in the homogeneous case) and cannot be set constant in general. From their most general form one can then show that the homogeneous Maxwell equations are automatically fulfilled (see Sec. 6.5 in our article:  http://arxiv.org/abs/1401.6800).

With kind regards,

RS.

Dear Dr.Starke,

Thanks for your comments and the your interesting publications with Dr. Schober. I will try to understand how apply your approach to receive the explicit formula for fundamental solution of Maxwell's equations in anisotropic media with general structure of anisotropy. I would like to note that when I have questioned about explicit form for fundamental solution I supposed that somebody has obtained before this formula. But now I am not sure.

With best regards,

Valery

Dear Dr. Yakhno,

thank you, too, for your interest in our work and for your significant remarks on our previous answers. Please find attached a more detailed note on the general solution of the Maxwell equations in anisotropic media, which also contains some clarifications regarding the solution I had posted before.

With kind regards,

Giulio

Dear Dr. Schober,

Thank you for your comments and your work. I will try to use their.

With best regards

Valery

Dear Giulio,

I have been trying to follow your draft on the "Maxwell general solution" and it is very difficult to do it. For instance, you write the permittivity and permeability just with a sign and no saying what kind of tensor are they.

Notice that when you go to Maxwell equations (7-10) you mixed in a dangerous form the tensorial and vectorial calculus. For example, Gauss equation that you write a divergence operator on a product of a tensor by a vector is only defined for a certain kind of 2-tensor (antisymmetric). And the same happens with the rest of equations where the compativility conditions among the components is not trivial.

On the other hand, when you try to find the solution using the Green function, you apply it on a scalar wave function which is not contained in your Maxwell equations too.

Speaking in general, Maxwell equations is a system of partial differential equations with less equations that incognites due to contain the conservation of the sources a constraint identity. Therefore they are usually written in function of potentials where an extra equation is introduced. In the case of having extra equations due to constitutive tensorial equations, then it happens exactly the opposite, you can have more equations that variables if you do not relate the components properly and with a physical meaning. Unfortunatally this is a difficult task becuase you need to say what is the material that employ: ferroelectric, antiferroelectric, ferromagnetic, antiferromagnetic,metal, insulator, ferroic, multiferroic, etc...And their mixtures.

It is necessary to be very specific and clear on what are the features of the anisotropy used in the constitutive equations if you want to find an analytical solution.

Dear Prof. Baldomir,

Thank you very much for your comments. I agree with your opinion. Probably we need to have very specific anisotropy  for finding the fundamental solution of the Maxwell's equations in an explicit form.  

Sincerely yours,

Valery Yakhno

Dear Daniel,

thank you very much for your comments on our note. Please let us provide here some more details to clarify these issues and/or to resolve any possible misunderstandings:

-- In the first part of our note (“Macroscopic approach”) we have indeed calculated "naively" with some arbitrarily given tensors \epsilon and \mu. The purpose of this was to reproduce the wave equations derived by Dr. Yakhno. As we have made clear in the second part (“Microscopic approach”), however, this is actually not the approach that we adopt for finding the general solution of the Maxwell equations.

-- We agree with you that one has to be very careful to use only constitutive relations which are physically meaningful. In fact, this is precisely the aim of ab initio materials physics (see the References we cited), where one strives to calculate quantities like \epsilon and \mu from first principles. Thereby it turns out that one cannot simply choose \epsilon and \mu as arbitrary constant tensors.

-- By contrast, the analytical solution we have provided in the second part of our note (see Eqs. (37)-(38)) is indeed generally valid. This means, it is a valid solution of the microscopic Maxwell equations combined with any constitutive relations that can be physically realized. In particular, on a microscopic scale, constitutive relations always involve spatial and temporal convolutions instead of simple matrix multiplications.

-- Regarding the Green function, it is a standard fact in electrodynamics that the microscopic Maxwell equations imply scalar, inhomogeneous wave equations for the electric and magnetic fields in terms of the charge and current densities. These scalar wave equations can be solved by means of the scalar Green function and hence be used to construct the general solution of the Maxwell equations. (For a more detailed derivation, please see Sec. 3.2 in the attached article.)

I hope this might be a helpful answer.

With kind regards

Giulio

Article Functional Approach to Electrodynamics of Media

Dear Dr.Giulio Schober,

thanks for your comments. 

With best regards,

Valery

  Dear Giuliu,

  The main problem is that you write generic Maxwell equations impossible to solve with only the number of conditions that you introduce (for every tensorial constitutive equation).

  My advice is that you must distinguish the material medium (components of permittivity and permeability) that you have currents, charges and fields because things before defining a "tensorial" Green function.

  Every good wishes.

Dear Daniel,

this is precisely the difference between the macroscopic and the microscopic approach: while the "macroscopic Maxwell equations" are indeed underdetermined, in the microscopic approach all electric and magnetic fields (i.e., external or induced ones) are determined uniquely in terms of their respective charge and current sources. For any given constitutive relation (which can be physically realized), it is therefore possible to provide an analytical solution for the total (i.e., external plus induced) electric and magnetic fields in terms of the external charges and currents.

With kind regards

Giulio

   Dear Giuliu,

   I understand what are the macroscopic Maxwell equations using generic tensorial constitutive equations. But it seems that you assume that they always have good solutions (or even solutions) and that is not true. Calculete their Hessian when you substitute the electric induction field D and the magnetic B. Notice they are not vectors at all, but they 2-tensors and the number of independent Maxwell is different that in vacuum.

   Green functions are inverse operators which need to have very well defined the existence of domain too. Thus, besides the physical meanining, the mathematical contest is also fundamental if you want to find just a real solution.

Thank you again, Daniel, but I am afraid we are at cross-purposes. Our starting point are actually not the macroscopic, but the microscopic Maxwell equations (for the induced and the external fields) combined with some equally microscopic constitutive relations.

Dear Dr. Baldomir  and Dr. Schober,,

Thanks for your comments  and discussion about assumptions and restrictions which we need to use for the constructions of fundamental solutions for the Maxwell's equations in anisotropic media.

With kind regards,

Valery