The Eigen Theory of Electromagnetic Waves in Complex Media

Since J. C. Maxwell presented the electromagnetic field equations in 1873, the existence of electromagnetic waves has been verified in various medium (Kong, 1986; Monk, 2003). But except for Helmholtz’s equation of electromagnetic waves in isotropic media, the laws of propagation of electromagnetic waves in anisotropic media are not clear to us yet. For example, how many electromagnetic waves are there in anisotropic media? How fast can these electromagnetic waves propagate? Where are propagation direction and polarization direction of the electromagnetic waves? What are the space patterns of these waves? Although many research works were made in trying to deduce the equations of electromagnetic waves in anisotropic media based on the Maxwell’s equation (Yakhno, 2005, 2006; Cohen, 2002; Haba, 2004), the explicit equations of electromagnetic waves in anisotropic media could not be obtained because the dielectric permittivity matrix and magnetic permeability matrix were all included in these equations, so that only local behaviour of electromagnetic waves, for example, in a certain plane or along a certain direction, can be studied. On the other hand, it is a natural fact that electric and magnetic fields interact with each other in classical electromagnetics. Therefore, even if most of material studies deal with the properties due to dielectric polarisation, magneitc materials are also capable of producing quite interesting electro-magnetic effects (Lindellm et al., 1994). From the bi-anisotropic point of view, magnetic materials can be treated as a subclass of magnetoelectric materials. The linear constitutive relations linking the electric and magnetic fields to the electric and magnetic displacements contain four dyadics, three of which have direct magnetic contents. The magnetoelectric coupling has both theoretical and practical significance in solid state physics and materials science. Though first predicted by Pierre Curie, magnetoeletric coupling was originally through to be forbidden because it violates time-reversal symmetry, until Laudau and Lifshitz (Laudau & Lifshitz, 1960) pointed out that time reversal is not a symmetry operation in some magnetic crystal. Based on this argument, Dzyaloshinskii (Dzyaloshinskii, 1960) predicted that magnetoelectric effect should occur in antiferromagnetic crystal Cr2O3, which was verified experimentally by Astrov (Astrov, 1960). Since then the magnetoelectric coupling has been observed in single-phase materials where simultaneous electric and magnetic ordering coexists, and in two-phase composites where the participating phase are pizoelectric and piezomagnetic (Bracke & Van Vliet,1981; Van Run et al., 1974) . Agyei and Birman (Agyei & Birman, 1990) carried out a detailed


Standard spaces of electromagnetic media
In anisotropic electromagnetic media, the dielectric permittivity and magnetic permeability are tensors instead of scalars. The constitutive relations are expressed as follows where the dielectric permittivity matrix ε and the magnetic permeability matrix µ are usually symmetric ones, and the elements of the matrixes have a close relationship with the selection of reference coordinate. Suppose that if the reference coordinates is selected along principal axis of electrically or magnetically anisotropic media, the elements at non-diagonal of these matrixes turn to be zero. Therefore, equations (1) and (2) are called the constitutive equations of electromagnetic media under the geometric presentation. Now we intend to get rid of effects of geometric coordinate on the constitutive equations, and establish a set of coordinate-independent constitutive equations of electromagnetic media under physical presentation. For this purpose, we solve the following problems of eigen-value of matrixes.
The above equations are just the modal constitutive equations in the form of scalar.

Eigen expression of Maxwell's equation
The classical Maxwell's equations in passive region can be written as Now we rewrite the equations in the form of matrix as follows or where [ ] ∆ is defined as the matrix of electric and magnetic operators. Substituting Eq. (1) into Eqs. (13) and (15) In another way, substituting Eqs. (5) and (6) into Eqs. (13) and (15) Rewriting the above in indicial notation, we get where, * i ∆ is the electromagnetic intensity operator, and i th row of

Electrically anisotropic media
In anisotropic dielectrics, the dielectric permittivity is a tensor, while the magnetic permeability is a scalar. So Eqs. (18) and ( Substituting Eqs. (4) -(6) into Eqs. (25) and (26), we have is defined as the eigen matrix of electromagnetic operators under the standard spaces. We can note from Appendix A that it is a diagonal matrix. Thus Eqs. (27) and (28) can be uncoupled in the form of scalar Eqs. (29) and (30) are the modal equations of electromagnetic waves in anisotropic dielectrics.

Magnetically anisotropic media
In anisotropic magnetics, the magnetic permeability is a tensor, while the dielectric permittivity is a scalar. So Eqs. (18) and (19) can be written as follows Substituting Eqs. (4) -(6) into Eqs. (31) and (32), we have where is defined as the eigen matrix of electromagnetic operators under the standard spaces. We can also note from Appendix A that it is a diagonal matrix. Thus Eqs. (33) and (34) The above can also be written as the standard form of waves www.intechopen.com

Behaviour of Electromagnetic Waves in Different Media and Structures
where, i is the electromagnetic operator. Eqs.(49) and (50) are just equations of electric field and magnetic field for bi-anisotropic media.

Applications 5.3.1 Bi-isotropic media
The constitutive equations of bi-isotropic media are the following 00 00 00 00 00 00 00 00 00 0 0 00 00 The eigen values and eigen vectors of those matrix are the following We can see from the above equations that there is only one eigen-space in isotropic medium, which is a triple-degenerate one, and the space structure is the following Then the eigen-qualities and eigen-operators of bi-isotropic medium are respectively shown as belows ( ) *T 23 So, the equation of electromagnetic wave in bi-isotropic medium becomes www.intechopen.com the velocity of electromagnetic wave is

Dzyaloshinskii's bi-anisotropic media
Dzyaloshinskii's constitutive equations of bi-anisotropic media are the following 00 00 00 00 00 00 00 00 00 0 0 00 00 The eigen values and eigen vectors of those matrix are the following We can see from the above equations that there are two eigen-spaces in Dzyaloshinskii's bianisotropic medium, in which one is a binary-degenerate one, the space structure is the following Then the eigen-qualities and eigen-operators of Dzyaloshinskii's bi-anisotropic medium are respectively shown as belows the velocities of electromagnetic wave are It is seen both from bi-isotropic media and Dzyaloshinskii's bi-anisotropic medium that the electromagnetic waves in bi-anisotropic medium will go faster duo to the bi-coupling between electric field and magnetic one.

The constitutive equation of lossy media
The constitutive equation of lossy media is the following It is equivalent to the following differential constitutive equation Eq.(73) can be written as

Eigen equations of electromagnetic waves in lossy media
Substituting Eqs. (10) and (79) into Eqs. (23) and (24), respectively, we have From them, we can get (82) and (83) are just equations of electric field and magnetic field for bi-anisotropic media.

Applications
In this section, we discuss the propagation laws of electromagnetic waves in an isotropic lossy medium. The material tensors in Eqs.(1) and (72) are represented by the following matrices 11 11 11 00 00 00 00 00 00 00 We can see from the above equations that there is only one eigen-space in an isotropic lossy medium, which is a triple-degenerate one, and the space structure is the following. Then the eigen-qualities and eigen-operators of an isotropic lossy media are respectively shown as follows ( ) where, c is the velocity of electromagnetic wave, τ is the lossy coefficient of electromagnetic wave 11 Then, the solutions of electromagnetic waves are the following It is an attenuated sub-waves.

The constitutive equation of dispersive media
The Eqs. (109) and (110) are just equations of electric field and magnetic field for general dispersive media.

Applications
In this section, we discuss the propagation laws of electromagnetic waves in an one-order dispersive medium. The material tensors in Eqs. (101) and (102) 11 11 1 11 00 00 00 The eigen values and eigen vectors of those matrix are the following We can see from the above equations that there is only one eigen-space in isotropic oneorder dispersive medium, which is a triple-degenerate one, and the space structure is the following Then, the solutions of electromagnetic waves are It is an attenuated sub-waves.

The constitutive equation of chiral media
The constitutive equations of chiral media are the following www.intechopen.com

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where χ is the matrix of chirality parameter, and a symmtric one.

Eigen equations of electromagnetic waves in chiral media
Substituting Eqs. (130) and (131) into Eqs. (23) and (24) (135) are just equations of electric field and magnetic field for chiral media.

Applications
In this section, we discuss the propagation laws of electromagnetic waves in an isotropic chiral medium. The material tensors in Eqs.(124) and (125) are represented by the following matrices 11 11 11 00 00 00 00 00 00 00 We can see from the above equations that there is only one eigen-space in isotropic medium, which is a triple-degenerate one, and the space structure is the following Then the eigen-qualities and eigen-operators of isotropic chiral medium are respectively shown as follows ( ) Then, the solution of electromagnetic waves is the following ( ) ( ) 12 Then, the solution of electromagnetic waves is the following It is seen that there only exists an electromagnetic sub-wave in opposite direction.

Conclusion
In this chapter, we construct the standard spaces under the physical presentation by solving the eigen-value problem of the matrixes of dielectric permittivity and magnetic permeability, in which we get the eigen dielectric permittivity and eigen magnetic permeability, and the corresponding eigen vectors. The former are coordinate-independent and the latter are coordinate-dependent. Because the eigen vectors show the principal directions of electromagnetic media, they can be used as the standard spaces. Based on the spaces, we get the modal equations of electromagnetic waves for anisotropic media, bianisotropic media, dispersive medium and chiral medium, respectively, by converting the classical Maxwell's vector equation to the eigen Maxwell's scalar equation, each of which shows the existence of an electromagnetic sub-wave, and its propagation velocity, propagation direction, polarization direction and space pattern are completely determined in the equations. Several novel results are obtained for anisotropic media. For example, there is only one kind of electromagnetic wave in isotropic crystal, which is identical with the classical result; there are two kinds of electromagnetic waves in uniaxial crystal; three kinds of electromagnetic waves in biaxial crystal and three kinds of distorted electromagnetic waves in monoclinic crystal. Also for bi-anisotropic media, there exist two electromagnetic waves in Dzyaloshinskii's bi-anisotropic media, and the electromagnetic waves in bi-anisotropic medium will go faster duo to the bi-coupling between electric field and magnetic one. For isotropic dispersive medium, the electromagnetic wave is an attenuated sub-waves. And for chiral medium, there exist different propagating states of electromagnetivc waves in different frequency band, for example, in low frequency band, the electromagnetic waves are composed of two attenuated sub-waves, in high frequency band, there only exists an electromagnetic sub-wave in opposite direction, and in the critical point, no electromagnetic can propagate. All of these new theoretical results need to be proved by experiments in the future.