Open access peer-reviewed chapter

The Eigen Theory of Electromagnetic Waves in Complex Media

By Shaohua Guo

Submitted: October 8th 2010Reviewed: April 11th 2011Published: July 5th 2011

DOI: 10.5772/16313

Downloaded: 2158

1. Introduction

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 (Landau & Lifshitz, 1960) pointed out that time reversal is not a symmetry operation in some magnetic crystal. Based on this argument, Dzyaloshinskii (Dzyaloshinskii, 1959) 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 analysis of the linear magnetoelectric effect, which showed that the effect should occur not only in some magnetic but also in some electric crystals. Pradhan (Pradhan, 1993) showed that an electric charge placed in a magnetoelectric medium becomes a source of induced magnetic field with non-zero divergence of volume integral. Magnetoelectric effect in two-phase composites has been analyzed by Harshe et al. ( Harshe et al., 1993), Nan (Nan, 1994) and Benveniste (Benveniste, 1995). Broadband transducers based on magnetoelectric effect have also been developed (Bracke & Van Vliet, 1981). Although the development mentioned above, no great progress in the theories of electromagnetic waves in bi-anisotropic media because of the difficulties in deal with the bi-coupling in electric field and magnetic one of the Maxell’s equation and the bi-anisotropic constitutive equation by classical electromagnetic theory.

Recently there is a growing interest modeling and analysis of Maxwell’s equations (Lee & Madsen, 1990; Monk, 1992; Jin et al., 1999). However, most work is restricted to simple medium such as air in the free space. On the other hand, we notice that lossy and dispersive media are ubiquitous, for example human tissue, water, soil, snow, ice, plasma, optical fibers and radar-absorbing materials. Hence the study of how electromagnetic wave interacts with dispersive media becomes very important. Some concrete applications include geophysical probing and subsurface studied of the moon and other planets (Bui et al., 1991), High power and ultra-wide-band radar systems, in which it is necessary to model ultra-wide-band electromagnetic pulse propagation through plasmas (Dvorak & Dudley, 1995), ground penetrating radar detection of buried objects in soil media (liu & Fan, 1999). The Debye medium plays an important role in electromagnetic wave interactions with biological and water-based substances (Gandhi & Furse, 1997). Until 1990, some paper on modeling of wave propagation in dispersive media started making their appearance in computational electromagnetics community. However, the published papers on modeling of dispersive media are exclusively restricted to the finite-difference time-domain methods and the finite element methods (Li & Chen, 2006; Lu et al., 2004). To our best knowledge, there exist only few works in the literature, which studied the theoretical model for the Maxwell’s equation in the complex anisotropic dispersive media, and no explicit equations of electromagnetic waves in anisotropic dispersive media can be obtained due to the limitations of classical electromagnetic theory.

Chiral materials have been recently an interesting subject. In a chiral medium, an electric or magnetic excitation will produce simultaneously both electric and magnetic polarizations. On the other hand, the chiral medium is an object that cannot be brought into congruence with its mirror image by translation and rotation. Chirality is common in a variety of naturally occurring and man-made objects. From an operation point of view, chirality is introduced into the classical Maxwell equations by the Drude-Born-Fedorov relative constitutive relations in which the electric and magnetic fields are coupled via a new materials parameter (Lakhtakia, 1994; Lindell et al., 1994), the chirality parameter. These constitutive relations are chosen because they are symmetric under time reversality and duality transformations. In a homogeneous isotropic chiral medium the electromagnetic fields are composed of left-circularly polarized (LCP) and right- circularly polarized (RCP) components (Jaggard et al., 1979; Athanasiadis & Giotopoulos, 2003), which have different wave numbers and independent directions of propagation. Whenever an electromagnetic wave (LCP, RCP or a linear combination of them) is incident upon a chiral scatterer, then the scattered field is composed of both LCP and RCP components and therefore both LCP and RCP far-field patterns are derived. Hence, in the vector problem we need to specify two directions of propagation and two polarizations. In recent years, chiral materials have been increasingly studied and there is a growing literature covering both their applications and the theoretical investigation of their properties. It will be noticed that the works dealing with wave phenomena in chiral materials have been mainly concerned with the study of time-harmonic waves which lead to frequency domain studies (Lakhtakia et al., 1989; Athanasiadis et al., 2003).

In this chapter, the idea of standard spaces is used to deal with the Maxwell’s electromagnetic equation (Guo, 2009, 2009, 2010, 2010, 2010). By this method, the classical Maxwell’s equation under the geometric presentation can be transformed into the eigen Maxwell’s equation under the physical presentation. The former is in the form of vector and the latter is in the form of scalar. Through inducing the modal constitutive equations of complex media, such as anisotropic media, bi-anisotropic media, lossy media, dissipative media, and chiral media, a set of modal equations of electromagnetic waves for all of those media are obtained, each of which shows the existence of electromagnetic sub-waves, meanwhile its propagation velocity, propagation direction, polarization direction and space pattern can be completely determined by the modal equations.This chapter will make introductions of the eigen theory to reader in details. Several novel theoretical results were discussed in the different parts of this chapter.

2. 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

D=εE,   B=μHE1

Rewriting Eq.(1) in form of scalar, we have

Di=εijEj,  Bi=μijHjE2

where the dielectric permittivity matrix εand the magnetic permeability matrix μare 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.

(ελI)ϕ=0,   (μγI)φ=0E3

where λi(i=1,2,3)and γi(i=1,2,3)are respectively eigen dielectric permittivity and eigen magnetic permeability, which are constants of coordinate-independent. ϕi(i=1,2,3)and φi(i=1,2,3)are respectively eigen electric vector and eigen magnetic vector, which show the electrically principal direction and magnetically principal direction of anisotropic media, and are all coordinate-dependent. We call these vectors as standard spaces. Thus, the matrix of dielectric permittivity and magnetic permeability can be spectrally decomposed as follows

ε=ΦΛΦΤ,   μ=ΨΠΨΤE4

where Λ=diag[λ1,λ2,λ3]and Π=diag[γ1,γ2,γ3]are the matrix of eigen dielectric permittivity and eigen magnetic permeability, respectively. Φ={ϕ1,ϕ2,ϕ3}and Ψ={φ1,φ2,φ3}are respectively the modal matrix of electric media and magnetic media, which are both orthogonal and positive definite matrixes, and satisfyΦTΦ=I,ΨTΨ=I.

Projecting the electromagnetic physical qualities of the geometric presentation, such as the electric field intensity vectorE, magnetic field intensity vectorH, magnetic flux density vectorBand electric displacement vectorDinto the standard spaces of the physical presentation, we get

D*=ΦΤD,E*=ΦΤEE5
B*=ΨΤB,H*=ΨΤHE6

Rewriting Eqs.(5) and (6) in the form of scalar, we have

Di*=ϕiΤDi=1,2,3,Ei*=ϕiΤEi=1,2,3E7
Bi*=φiΤBi=1,2,3,Hi*=φiΤHi=1,2,3E8

These are the electromagnetic physical qualities under the physical presentation.

Substituting Eq. (4) into Eq. (1) respectively, and using Eqs.(5) and (6) yield

Di*=λiEi*i=1,2,3E9
Bi*=γiHi*i=1,2,3E10

The above equations are just the modal constitutive equations in the form of scalar.

3. Eigen expression of Maxwell’s equation

The classical Maxwell’s equations in passive region can be written as

×H=tD,×E=tBE11

Now we rewrite the equations in the form of matrix as follows

[0zyz0xyx0]{H1H2H3}=t{D1D2D3}E12

or

[Δ]{H}=t{D}E13
[0zyz0xyx0]{E1E2E3}=t{B1B2B3}E14

or

[Δ]{E}=t{B}E15

where [Δ]is defined as the matrix of electric and magnetic operators.

Substituting Eq. (1) into Eqs. (13) and (15) respectively, we have

[Δ]{H}=t[ε]{E}E16
[Δ]{E}=t[μ]{H}E17

Substituting Eq. (16) into (17) or Eq. (17) into (16), yield

[]{H}=t2[μ][ε]{H}E18
[]{E}=t2[μ][ε]{E}E19

where []=[Δ][Δ]is defined as the matrix of electromagnetic operators as follows

[]=[(z2+y2)xy2xz2yx2(x2+z2)yz2zx2zy2(x2+y2)]E20

In another way, substituting Eqs. (5) and (6) into Eqs. (13) and (15), respectively, we have

[Δ][Φ]{E*}=t[Ψ]{B*}E21
[Δ][Ψ]{H*}=t[Φ]{D*}E22

Rewriting the above in indicial notation, we get

{Δi*}Ei*=t{φi}Bi*i=1,2,3E23
{Δi*}Hi*=t{ϕi}Di*i=1,2,3E24

where, Δi*is the electromagnetic intensity operator, and ith row of[Δ*]=[Δ][Φ].

4. Electromagnetic waves in anisotropic media

4.1. Electrically anisotropic media

In anisotropic dielectrics, the dielectric permittivity is a tensor, while the magnetic permeability is a scalar. So Eqs. (18) and (19) can be written as follows

[]{H}=t2μ0[ε]{H}E25
[]{E}=t2μ0[ε]{E}E26

Substituting Eqs. (4) - (6) into Eqs. (25) and (26), we have

[*]{H*}=t2μ0[Λ]{H*}E27
[*]{E*}=t2μ0[Λ]{E*}E28

where[*]=[Φ]T[][Φ]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

i*Hi*+μ0λit2Hi*=0i=1,2,3E29
i*Ei*+μ0λit2Ei*=0i=1,2,3E30

Eqs.(29) and (30) are the modal equations of electromagnetic waves in anisotropic dielectrics.

4.2. 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

[]{H}=t2ε0[μ]{H}E31
[]{E}=t2ε0[μ]{E}E32

Substituting Eqs. (4) - (6) into Eqs. (31) and (32), we have

[*]{H*}=t2ε0[Π]{H*}E33
[*]{E*}=t2ε0[Π]{E*}E34

where [*]=[Ψ]T[][Ψ]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) can be uncoupled in the form of scalar

i*Hi*+ε0γit2Hi*=0i=1,2,3E35
i*Ei*+ε0γit2Ei*=0i=1,2,3E36

Eqs.(35) and (36) are the modal equations of electromagnetic waves in anisotropic magnetics.

5. Electromagnetic waves in bi-anisotropic media

5.1. Bi-anisotropic constitutive equations

The constitutive equations of bi-anisotropic media are the following (Lindellm & Sihvola, 1994; Laudau & Lifshitz, 1960)

D=εE+ξHE37
B=ξE+μHE38

whereξis the matrix of magneto-electric parameter, and a symmetric one.

Substituting Eqs. (5) and (6) into Eqs. (37) and (38), respectively, and multiplying them with the transpose of modal matrix in the left, we have

ΦTD=ΦTεΦE*+ΦTξΨH*E39
ΨΤB=ΨΤξΦE*+ΨΤμΨH*E40

LetG=ΦTξΨ=ΨTξΦ, that is a coupled magneto-electric matrix, and using Eq. (4), we have

D*=ΛE*+GH*E41
B*=GE*+ΠH*E42

Rewriting the above in indicial notation, we get

Di*=λiEi*+gijHj*i=1,2,3j=1,2,3E43
Bi*=γiHi*+gijEj*i=1,2,3j=1,2,3E44

Eqs. (43) and (44) are just the modal constitutive equations for bi-anisotropic media.

5.2. Eigen equations of electromagnetic waves in bi-anisotropic media

Substituting Eqs. (43) and (44) into Eqs. (23) and (24), respectively, we have

{Δi*}Ei*=t{φi}(γiHi*+gijEj*)E45
{Δi*}Hi*=t{ϕi}(λiEi*+gijHj*)E46

From them, we can get

({Δi*}t{ϕi}gijδij)T({Δi*}+t{φi}gijδij)Ei*=t2{φi}{ϕi}TλiγiEi*E47
({Δi*}+t{φi}gijδij)T({Δi*}t{ϕi}gijδij)Hi*=t2{ϕi}{φi}TγiλiHi*E48

The above can also be written as the standard form of waves

i*Ei*+t{Δi*}T({φ}{ϕ})giiEi*+t2{ϕ}T{φ}(λiγigii2)Ei*=0i=1,2,3E49
i*Hi*+t{Δi*}T({φ}{ϕ})giiHi*+t2{ϕ}T{φ}(λiγigii2)Hi*=0i=1,2,3E50

where, i*={Δi*}T{Δi*}is the electromagnetic operator. Eqs.(49) and (50) are just equations of electric field and magnetic field for bi-anisotropic media.

5.3. Applications

5.3.1. Bi-isotropic media

The constitutive equations of bi-isotropic media are the following

D=[ε000ε000ε]E+[ξ000ξ000ξ]HE51
B=[ξ000ξ000ξ]E+[μ000μ000μ]HE52

The eigen values and eigen vectors of those matrix are the following

Λ=diag[ε,ε,ε],Π=diag[μ,μ,μ]E53
Φ=Ψ=[100010001]E54

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

W=W1(3)[ϕ1,ϕ2,ϕ3]E55
ϕ1*=φ1*=33[1,1,1]TE56

Then the eigen-qualities and eigen-operators of bi-isotropic medium are respectively shown as belows

E1*=ϕ1*TE=33(E1+E2+E3)E57
1*=(x2+y2+z2),g11=ξE58

So, the equation of electromagnetic wave in bi-isotropic medium becomes

(x2+y2+z2)E1*=(μεξ2)t2E1*E59

the velocity of electromagnetic wave is

c(1)=1μεξ2E60

5.3.2. Dzyaloshinskii’s bi-anisotropic media

Dzyaloshinskii’s constitutive equations of bi-anisotropic media are the following

D=[ε000ε000εz]E+[ξ000ξ000ξz]HE61
B=[ξ000ξ000ξz]E+[μ000μ000μz]HE62

The eigen values and eigen vectors of those matrix are the following

Λ=diag[ε,ε,εz],Π=diag[μ,μ,μz]E63
Φ=Ψ=[100010001]E64

We can see from the above equations that there are two eigen-spaces in Dzyaloshinskii’s bi-anisotropic medium, in which one is a binary-degenerate one, the space structure is the following

W=W1(2)[ϕ1,ϕ2]W21[ϕ3]E65

Then the eigen-qualities and eigen-operators of Dzyaloshinskii’s bi-anisotropic medium are respectively shown as belows

E2*=ϕ2TE=E3,|E1*|=(Eϕ2TE2*)T(Eϕ2TE2*)=E12+E22E66
1*=(x2+y2+2z22xy2),2*=(x2+y2),g11=ξ,g22=ξzE67

So, the equations of electromagnetic wave in Dzyaloshinskii’s bi-anisotropic medium become

(x2+y2+2z22xy2)E12+E22=(μεξ2)t2E12+E22E68
(x2+y2)E3=(μzεzξz2)t2E3E69

the velocities of electromagnetic wave are

c(1)=1μεξ2E70
c(2)=1μzεzξz2E71

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.

6. Electromagnetic waves in lossy media

6.1. The constitutive equation of lossy media

The constitutive equation of lossy media is the following

D=εE+tσdEdτdτE72

It is equivalent to the following differential constitutive equation

D˙=εE˙+σEE73

Let

De=εE,D˙d=σEE74

Eq.(73) can be written as

D˙=D˙e+D˙dE75

or

t{D}=([ε]t+[σ]){E}E76

Using Eq.(5), the above becomes

t{D*}=([Φ]T[ε][Φ]t+[Φ]T[σ][Φ]){E*}E77

According to Appendix B and Eq.(77), we have

t{D*}=([Λ]t+[Γ]){E*}E78

Rewriting the above in indicial notation, we get

tDi*=(λit+ηi)Ei*E79

Eq.(79) is just the modal constitutive equations for lossy media.

6.2. Eigen equations of electromagnetic waves in lossy media

Substituting Eqs. (10) and (79) into Eqs. (23) and (24), respectively, we have

{Δi*}Ei*=t{φi}γiHi*i=1,2,3E80
{Δi*}Hi*={ϕi}(λit+ηi)Ei*i=1,2,3E81

From them, we can get

i*Ei*+ttξiγiλiEi*+tξiγiηiEi*=0i=1,2,3E82
i*Hi*+ttξiγiλiHi*+tξiγiηiHi*=0i=1,2,3E83

whereξi={ϕi*}T{φi*}. Eqs.(82) and (83) are just equations of electric field and magnetic field for bi-anisotropic media.

6.3. 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

ε=[ε11000ε11000ε11],μ=[μ11000μ11000μ11],σ=[σ11000σ11000σ11]E84

The eigen values and eigen vectors of those matrix are the following

Λ=diag[ε11,ε11,ε11],Π=diag[μ11,μ11,μ11],Γ=diag[σ11,σ11,σ11]E85
Φ=Ψ=Θ=[100010001]E86

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.

Wmag=W1(3)[ϕ1,ϕ2,ϕ3],Wele=W1(3)[φ1,φ2,φ3]E87

where, ϕ1*=33{1,1,1}T, φ1*=33{1,1,1}T,ξ1=1.

Then the eigen-qualities and eigen-operators of an isotropic lossy media are respectively shown as follows

E1*=33(E1+E2+E3)E88
H1*=33(H1+H2+H3)E89
1*=13[(x2+y2+z2)]E90

So, the equation of electromagnetic wave in lossy media becomes

(x2+y2+z2)E1*=1c2t2E1*+1τ2tE1*E91

Rewriting it in the component form, we have

(x2+y2+z2)E1=1c2t2E1+1τ2tE1E92
(x2+y2+z2)E2=1c2t2E2+1τ2tE2E93
(x2+y2+z2)E3=1c2t2E3+1τ2tE3E94

where, cis the velocity of electromagnetic wave, τis the lossy coefficient of electromagnetic wave

c=1μ11ε11,  τ=1μ11σ11E95

Now, we discuss the the propagation laws of a plane electromagnetic wave in x-axis. In this time, Eq. (92) becomes

x2E1=1c2t2E1+1τ2tE1E96

Let the solution of Eq. (96) is as follows

E1=Aexp[i(kxωt)]E97

Substituting the above into Eq. (96), we have

k2=ω2c2+iωτ2E98

From Eq.(96), we can get

k¯=k1+ik2E99

where

k1=ωc[1+(1+c4ω2τ4)122]12,k2=ωc[1+(1+c4ω2τ4)122]12E100

Then, the solutions of electromagnetic waves are the following

E1=Aek2xei(k1xωt)=A¯ei(k1xωt)E101

It is an attenuated sub-waves.

7. Electromagnetic waves in dispersive media

7.1. The constitutive equation of dispersive media

The general constitutive equations of dispersive media are the following

D=εE+ε1E˙+ε2E¨+E102
B=μH+μ1H˙+μ2H¨+E103

whereεi,(i=1,2,)andμi,(i=1,2,)are the higher order dielectric permittivity matrix and the magnetic permeability matrix respectively, and all symmtric ones.

Substituting Eqs. (5) and (6) into Eqs. (101) and (102), respectively, and multiplying them with the transpose of modal matrix in the left, we have

D*=ΦTεΦE*+ΦTε1ΦE˙+ΦTε2ΦE¨+E104
B*=ΨTμΨH+ΨTμ1ΨH˙+ΨTμ2ΨH¨+E105

It can be proved that there exist same standard spaces for various order electric and magnetic fields in the condition close to the thermodynamic equilibrium. Then, we have

Di*=λiEi*+λi(1)E˙i*+λi(2)E¨i*+E106
Bi*=γiHi*+γi(1)H˙i*+γi(2)H¨i*+E107

Eqs. (105) and (106) are just the modal constitutive equations for the general dispersive media.

7.2. Eigen equations of electromagnetic waves in dispersive media

Substituting Eqs. (105) and (106) into Eqs. (23) and (24), respectively, we have

{Δi*}Ei*=t{φi}(γiHi*+γi(1)H˙i*+γi(2)H¨i*+)E108
{Δi*}Hi*=t{ϕi}(λiEi*+λi(1)E˙i*+λi(2)E¨i*+)E109

From them, we can get

1ξii*Ei*+γiλittEi*+(γiλi(1)+γi(1)λi)tttEi*+(γiλi(2)+γi(1)λi(1)+γi(2)λi)ttttEi*+=0E110
1ξii*Hi*+γiλittHi*+(γiλi(1)+γi(1)λi)tttHi*+(γiλi(2)+γi(1)λi(1)+γi(2)λi)ttttHi*+=0E111

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

7.3. 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) are represented by the following matrices

ε=[ε11000ε11000ε11],μ=[μ11000μ11000μ11],ε1=[ε11000ε11000ε11],μ1=[μ11000μ11000μ11]E112

The eigen values and eigen vectors of those matrix are the following

Λ=diag[ε11,ε11,ε11],Λ1=diag[ε11,ε11,ε11]E113
Π=diag[μ11,μ11,μ11],Π1=diag[μ11,μ11,μ11]E114
Φ=Ψ=[100010001]E115

We can see from the above equations that there is only one eigen-space in isotropic one-order dispersive medium, which is a triple-degenerate one, and the space structure is the following

Wmag=W1(3)[ϕ1,ϕ2,ϕ3],Wele=W1(3)[φ1,φ2,φ3]E116

where, ϕ1*=33{1,1,1}T, φ1*=33{1,1,1}T,ξ1=1. Thus the eigen-qualities and eigen-operators of isotropic one-order dispersive medium are known as same as Eqs. (88) – (90). The equations of electromagnetic wave in one-order dispersive medium become

(x2+y2+z2)E1*=1c2ttE1*+(μ11ε11(1)+μ11(1)ε11)tttE1*E117
(x2+y2+z2)H1*=1c2ttH1*+(μ11ε11(1)+μ11(1)ε11)tttH1*E118

in which cis the velocity of electromagnetic wave

c=1μ11ε11E119

Now, we discuss the propagation laws of a plane electromagnetic wave in x-axis. In this time, Eq.(116) becomes

x2E1=1c2ttE1+(μ11ε11(1)+μ11(1)ε11)tttE1E120

Let

E1=Aexp[i(kxωt)]E121

Substituting the above into Eq.(119), we have

k2=ω2c2iω3(μ11ε11(1)+μ11(1)ε11)E122

From the above, we can get

k¯=k1+ik2E123

where

k1=ωc[1+(1+c4(μ11ε11(1)+μ11(1)ε11)2ω2)122]12,k2=ωc[1+(1+c4(μ11ε11(1)+μ11(1)ε11)2ω2)122]12E124

Then, the solutions of electromagnetic waves are

E1=Aek2xei(k1xωt)=A¯ei(k1xωt)E125

It is an attenuated sub-waves.

8. Electromagnetic waves in chiral media

8.1. The constitutive equation of chiral media

The constitutive equations of chiral media are the following

D=εE-χtHE126
B=χtE+μHE127

whereχis the matrix of chirality parameter, and a symmtric one.

Substituting Eqs. (5) and (6) into Eqs. (124) and (125), respectively, and multiplying them with the transpose of modal matrix in the left, we have

D*=ΦTεΦE*-ΦTχΨtH*E128
B*=ΨTχΦtE*+ΨTμΨH*E129
LetΓ=ΨTχΦ, that is a coupled chiral matrix, and using Eq. (4), we have
D*=ΛE*-ΓTtH*E130
B*=ΓTtE*+ΠH*E131

For most chiral,Γ=diag[ς1,ς2,ς3]. Then we have

Di*=λiEi*ςitHi*E132
Bi*=ςitEi*+γiHi*E133

Eqs.(130) and (131) are just the modal constitutive equations for anisotropic chiral media.

8.2. Eigen equations of electromagnetic waves in chiral media

Substituting Eqs. (130) and (131) into Eqs. (23) and (24), respectively, we have

{Δi*}Ei*=t{φi}(ςitEi*+γiHi*)i=1,2,3E134
{Δi*}Hi*=t{ϕi}(λiEi*ςitHi*)i=1,2,3E135

From them, we can get

i*Ei*+ξiςi2ttttEi*+(2ςii*+ξiλiγi)ttEi*=0i=1,2,3E136
i*Hi*+ξiςi2ttttHi*+(2ςii*+ξiλiγi)ttHi*=0i=1,2,3E137

whereξi={ϕi}T{φi}=1,i={Δi*}T{φi}. Eqs.(134) and (135) are just equations of electric field and magnetic field for chiral media.

8.3. 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

ε=[ε11000ε11000ε11],μ=[μ11000μ11000μ11],χ=[χ11000χ11000χ11]E138

The eigen values and eigen vectors of those matrix are the following

Λ=diag[ε11,ε11,ε11],Π=diag[μ11,μ11,μ11],Γ=diag[χ11,χ11,χ11]E139
Φ=Ψ=[100010001]E140

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

Wmag=W1(3)[ϕ1,ϕ2,ϕ3],Wele=W1(3)[φ1,φ2,φ3]E141

where, ϕ1*=33{1,1,1}T, φ1*=33{1,1,1}T,ξ1=1.

Then the eigen-qualities and eigen-operators of isotropic chiral medium are respectively shown as follows

E1*=33(E1+E2+E3),H1*=33(H1+H2+H3)E142
1*=13[(x2+y2+z2)],1*=33(x+y+z)E143

So, the equations of electromagnetic wave in isotropic chiral medium become

(x2+y2+z2)E1*=χ112ttttE1*+[23χ11(x+y+z)+1c2]ttE1*E144
(x2+y2+z2)H1*=χ112ttttH1*+[23χ11(x+y+z)+1c2]ttH1*E145

where, cis the velocity of electromagnetic wave

c=1μ11ε11E146

Now, we discuss the propagation laws of a plane electromagnetic wave in x-axis. In this time, Eq.(142) becomes

x2E1=χ112ttttE1+[23χ11x+1c2]ttE1E147
Let
E1=Aexp[i(kxωt)]E148

Substituting the above into Eq.(145), we have

k2=(1c2+i23χ11k)ω2χ112ω4E149
or
k2i23χ11ω2k+(χ112ω2μ11ε11)ω2=0E150
  1. when

    ω2μ11ε11χ1120E151

    By Eq.(148), we have

    k1=k1+ik1=iχ11ω2[3+(x+iy)]E152
    k2=k2+ik2=iχ11ω2[3(x+iy)]E153

    From them, we can get

    k1=χ11ω2y,k1=(3+x)χ11ω2E154
    k2=χ11ω2y,k2=(3x)χ11ω2E155

    where, x=223+[9+(μ11ε11χ112ω2)2ω4]12,

    y=223+[9+(μ11ε11χ112ω2)2ω4]12E156
    .

    Then, the solution of electromagnetic waves is the following

    E1=A1ek1xei(k1xωt)+A2ek2xei(k2xωt)E157

    It is composed of two attenuated sub-waves.

  2. when

    ω2μ11ε11χ1120E158

    By Eq.(148), we have

    k1=k1+ik1=iχ11ω2[3+3+1ω2(ω2μ11ε11χ112)]E159
    k2=k2+ik2=iχ11ω2[33+1ω2(ω2μ11ε11χ112)]E160

    where

    k1=0,k1=χ11ω2[3+3+1ω2(ω2μ11ε11χ112)]E161
    k2=χ11ω2[3+1ω2(ω2μ11ε11χ112)3],k2=0E162

    Then, the solution of electromagnetic waves is the following

    E1=A1ek1xeiωt+A2ei(k2xωt)E163

    It is seen that there only exists an electromagnetic sub-wave in opposite direction.

  3. when

    ω2μ11ε11χ112=0E164

    By Eq.(148), we have

    k=k+ik=i23χ11ω2E165

    where

    k=0,k=23χ11ω2E166

    Then, the solution of electromagnetic waves is the following

    E=AekxeiωtE167

    No electromagnetic sub-waves exist now.

9. 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, bi-anisotropic 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.

10. Appendix A: Proof of the eigenmode of electromagnetic operator matrix

The Maxwell’s equation of anisotropic dielectrics is the following

[]{H}=t2μ0[ε]{H}E168

Using the representation transform relationship Eq. (6), we have

[][Φ]{H*}=t2μ0[ε][Φ]{H*}E169

Substituting the spectral decomposition matrix of dielectric permittivity Eq. (4) into above, we have

[][Φ]{H*}=t2μ0[Φ][Λ]{H*}E170

Comparing the both sides of above equation, we can get

[][Φ]=t2μ0[Φ][Λ]E171

Multiplying the both sides of above with the transpose of modal matrix in the left, we have

[Φ]T[][Φ]=t2μ0[Λ]E172

It is seen that the right side above is a diagonal matrix, which shows that the electromagnetic operators matrix can also be spectrally decomposed in standard spaces, then we get

[*]=t2μ0[Λ]E173

Rewriting above in the form of scalar, we have

i*=t2μ0λiE174

11. Appendix B: Spectrally decomposition of lossy matrix

The Helmholtz’s free energy of electromagnetic system with lossy property is the following

ψ=ψ(D,B,Dd)=12(DDd)ε1(DDd)+12Bμ1BE175

Differentiating the above with lossy variable, and using Eq.(1), we have

R=ψDd=(DDd)ε1=EE176

According to the Onsager’s principle, for the process of closing to equilibrium, the rate of lossy variable is proportion to the driving force, that is

ψDid+βijdDjddt=0E177

where, βij=βjiis the general friction coefficient. Rewritting the above in matrix form, we have

{R}=[B]{D˙d}E178

Projecting the lossy electric displacement vectorDinto the standard spaces of the physical presentation, we get

{Dd}=a{φ}E179

Using Eq.(B2), we have

{R}=ωa˙{φ}E180

Substituting Eqs. (B5) and (B6) into Eq. (B4), the condition of non-zero solution to αis the following

([B]ω[I]){φ}=0E181

It is seen that the general friction coefficient matrix can also be spectrally decomposed in standard spaces, so we have

[B]=[Φ][Ω][Φ]TE182

Comparing Eq. (B4) with Eq. (74), it is known that we can also spectrally decompose the lossy matrix in standard spaces

[σ]=[Φ]T[Γ][Φ]E183

where,

Γ=Ω1=diag[η1,η2,η3]E184
.

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Shaohua Guo (July 5th 2011). The Eigen Theory of Electromagnetic Waves in Complex Media, Behaviour of Electromagnetic Waves in Different Media and Structures, Ali Akdagli, IntechOpen, DOI: 10.5772/16313. Available from:

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