Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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8.2 Velocities of Waves in Dispersive Media 491

For a plane wave propagating along z in a dispersive medium,

P =

<(E × H

∗

) =

∗

E × E

∗

, (8.57)

where E = E

−αz

−jβz

, E ×E

∗

= |E|

= |E

−2αz

∗

√

∗

and

√

∗

= n

∗

= n

+ jn

. Then we have

P =

−2αz

. (8.58)

The average density of stored energy is given by

w =

|E|

|H|

Nmω

|x|

Nmω

|x|

, (8.59)

where the ﬁrst and the second terms represent the average density of stored

energy of electromagnetic ﬁelds in vacuum;

Nmω

|x|

is the volume density

of the potential energy of the molecular resonator, where −mω

x denotes the

restoring force; and

Nmω

|x|

denotes the volume density

of the kinetic energy of the molecular resonator.

The ratio of E to H is equal to the wave impedance. Applying (8.13)

and (8.14), we have

|H|

|E|

|²

||E|

|E|

1 +

NZe

− ω

(ω

− ω

)

+ ω

− j

NZe

ωγ

(ω

− ω

)

+ ω

|E|

− ω

NZe

+ ω

(ω

− ω

)

+ ω

. (8.60)

Substituting (8.2) and (8.60) into (8.59), we obtain

w =







NZe

+ ω

(ω

− ω

)

+ ω

− ω

NZe

+ ω

(ω

− ω

)

+ ω







−2αz

(8.61)

Substituting (8.58) and (8.61) into (8.56), we obtain the velocity of energy

ﬂow

NZe

+ ω

(ω

− ω

)

+ ω

− ω

NZe

+ ω

(ω

− ω

)

+ ω

c. (8.62)

492 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Figure 8.6: The propagation of a signal in disp ersive medium.

In the region of normal dispersion, |ω

− ω

| À ωγ and |ω

− ω

| À

NZe

/²

m, it reduces to

≈ n

1 +

NZe

2²

+ ω

(ω

− ω

)

= n

. (8.63)

At the high-frequency range, n

< 1, v

< v

, the velocity of energy ﬂow

and the group velocity approaches c at ω → ∞. At the low-frequency range,

> 1, v

> v

and when ω ¿ ω

, v

≈ v

In the region of anomalous dispersion, the velocity of energy ﬂow is much

diﬀerent from the group velocity and remains less than c. The ratio of c to

is plotted as the dash-dotted line in Fig. 8.5.

8.2.4 Signal Velocity

In the region of normal dispersion, group velocity represents the velocity of

the signal propagation. But in the region of anomalous dispersion, the group

velocity v

will lead to a velocity greater than the velocity of light in vacuum,

8.3 Anisotropic Media and Their Constitutional Relations 493

c, which is relativistically impossible. The above-deﬁned group velocity loses

its meaning as a signal velocity in the region of anomalous dispersion.

The propagation of a signal in a dispersive medium has been carefully

investigated by L. Brillouin [17, 96]. This investigation is of a much more

delicate nature and we would rather give here a statement of the conclusions.

According to Brillouin, a signal is a disturbance in the form of a train of

oscillations starting at a certain instant as shown in Fig. 8.6(a). In the course

of propagation in a dispersive medium, the signal is deformed; see Fig. 8.6(b).

It was found that after p enetrating to a certain depth in the medium, the

main body of the signal is preceded by a forerunner which travels with the

velocity c. The ﬁrst forerunner arrives with small period and zero amplitude,

and then grows slowly both in period and in amplitude. The amplitude then

decreases while the period approaches the natural period of the electrons.

Then the second forerunner arrives with the velocity c(ω

+ a

) < c,

where a = NZe

/m. The period of the second forerunner is at ﬁrst very large

and then decreases, while the amplitude rises and then falls in a manner

similar to that of the ﬁrst forerunner. These two forerunners can partly

overlap and their amplitudes are very small but increase rapidly as their

periods approach that of the signal. With a sudden rise of amplitude the

principle part of the disturbance arrives, traveling with a velocity v

, which

Brillouin deﬁnes as the signal velocity. The time variation of the signal

propagating a certain distance in a dispersive medium is shown in Fig. 8.6(c).

An explicit and simple expression for v

cannot be given, but physically its

meaning is quite clear. For a detector with normal sensitivity, a measurement

should, in fact, indicate a velocity of propagation approximately equal to v

However, as the sensitivity of the detector is increased, the measured velocity

increases, until in the limit of inﬁnite sensitivity we should record the arrival

of the front of the ﬁrst forerunner, which travels with the velocity c. The

ratio of c to v

given by Brillouin is plotted as the dotted line in Fig. 8.5.

8.3 Anisotropic Media and Their

Constitutional Relations

In the previous chapters, we have studied the ﬁelds and waves in isotropic

media, in which the orientation of polarization or magnetization is in the

same direction as that of the ﬁeld vector, and the responses to ﬁelds with dif-

ferent orientations are the same. Hence for isotropic media, the permittivity

and permeability are scalars which may be complex and may be frequency

dependent or nonlinear.

In a number of technically important materials, responses of polarization

and magnetization to ﬁelds with diﬀerent orientations may diﬀer, so that the

orientation of p olarization or magnetization can be in the diﬀerent direction

to that of the ﬁeld vector. Such media are known as anisotropic media. In

494 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

these cases the permittivity and/or permeability must be tensors or matrices

[38, 53, 84].

8.3.1 Constitutional Equations for Anisotropic Media

Rewrite the constitutional equations for general anisotropic media given in

Section 1.1.2:

D = ² · E, B = µ · H, (8.64)

where ² is the tensor permittivity and µ is the tensor permeability. For

steady-state sinusoidal time-varying ﬁelds, the constitutional tensors are com-

plex tensors, i.e., the elements of the matrices are complex and depend on

frequency. Although a medium can be both electrically and magnetically

anisotropic, but in fact, most anisotropic media are either electric anisotropic

or magnetic anisotropic.

For electric anisotropic or so called ²-anisotropic medium, the permittivity

is a tensor and the permeability is a scalar, so

D = ² · E, B = µH, (8.65)

where

² =









. (8.66)

For magnetic anisotropic or so called µ-anisotropic medium, the perme-

ability is a tensor and the permittivity is a scalar, so

D = ²E, B = µ · H, (8.67)

where

µ =









. (8.68)

The alternative expressions of the constitutional relations for electric

anisotropic media are

E = κ · D , H = νB, (8.69)

and for magnetic anisotropic media are

E = κD, H = ν · B, (8.70)

where κ = ²

−1

is the impermittivity tensor and ν = µ

−1

is the imperme-

ability tensor.

8.3 Anisotropic Media and Their Constitutional Relations 495

8.3.2 Symmetrical Properties of the Constitutional

Tensors

We are now going to explain the properties of the constitutional tensors for

passive and lossless media.

In source-free and nonconducting media, J = 0, J

= 0, and σ = 0, the

complex Poynting theorem (1.179) becomes

∇·

S = ∇·

E × H

∗

= ∇·

S + j q

= j ω

E · D

∗

−

B · H

∗

. (8.71)

The divergence of the average Poynting vector is

∇ ·

S = ∇ ·

<(E × H

∗

)

<[ j ω (E · D

∗

− B · H

∗

)]. (8.72)

For an arbitrary complex quantity, we have

<z =

(z + z

∗

Then the above equation becomes

∇ ·

S =

{[j ω(E · D

∗

− B · H

∗

)] + [j ω(E · D

∗

− B · H

∗

)]

∗

}

j ω

[E · D

∗

− B · H

∗

− E

∗

· D + B

∗

· H]. (8.73)

Substituting (8.64) into it, we obtain

∇ ·

S =

j ω

[E · ²

∗

· E

∗

− µ · H · H

∗

− E

∗

· ² · E + µ

∗

· H

∗

· H]. (8.74)

Applying the rule of tensor algebra (E.43),

A · b

∗

· A

∗

= A

∗

· b

†

· A,

where b

∗

is the conjugate tensor of b and b

†

= b

∗T

is the conjugate trans-

posed tensor or associate tenor of b, then (8.74) becomes

∇ ·

S =

j ω

∗

· (²

†

− ²) · E + H

∗

· (µ

†

− µ) · H]. (8.75)

In source-free and nonconducting media, ∇ ·

S = 0. This must be true

for an arbitrary choice of E and H, and therefore

†

= ², µ

†

= µ, or ²

= ²

∗

, µ

= µ

∗

. (8.76)

We conclude that the transpose of the constitutional tensor for lossless

anisotropic media is the complex conjugate of the tensor itself, hence both

496 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

tensors

² and

µ are Hermitian tensors. The relations among the complex

elements of a Hermitian constitutional tensor are given by

= ²

∗

, µ

= µ

∗

, (8.77)

and

= ²

∗

, µ

= µ

∗

, (8.78)

The diagonal elements have to be real and the non-diagonal elements have

to be conjugate symmetry.

(1) Reciprocal Media

If ²

and µ

are real, (8.78) becomes

= ²

, µ

= µ

. (8.79)

The constitutional tensors ² and µ are real and are symmetrical tensors,

= ², µ

= µ, (8.80)

and the media are known as reciprocal media.

Every symmetrical tensor of rank two can be transformed, by rotation

of the coordinate system, to a diagonal tensor. A diagonal tensor is one in

which all of the oﬀ-diagonal elements are zero, i.e., ²

= 0, µ

= 0 when

i 6= j. Hence by proper orientation of the coordinate system with respect to a

given reciprocal medium, the tensor permittivity and the tensor permeability

of the reciprocal medium can be expressed in the following form:

² =





0 0

0 ²

0 0 ²





, µ =





0 0

0 µ

0 0 µ





, (8.81)

This special coordinate system, which is chosen to have the constitutional

tensor in the diagonal form, is called the principle coordinate system and the

three coordinate axes of the system are said to be the principle axes.

The reciprocity theorem given in Section 1.8 is suitable for the ﬁelds in

reciprocal media. The isotropic media are reciprocal media with ²

= ²

or µ

= µ

(2) Nonreciprocal Media

If ²

and µ

are imaginary, (8.78) becomes

= −²

, µ

= −µ

. (8.82)

The constitutional tensors ² and µ are not symmetrical tensors which cannot

be transformed, by rotation of the coordinate system, to a diagonal tensor.

8.4 Characteristics of Waves in Anisotropic Media 497

Hence the general forms of the constitutional tensors are

² =





j²

−j²

j²

−j²





, µ =





jµ

−jµ

jµ

−jµ





, (8.83)

and the media are known as nonreciprocal anisotropic media or gyrotropic

media. The reciprocity theorem fails for nonreciprocal media or gyrotropic

media with asymmetrical constitutional tensors. Most nonreciprocal media

must have magnetic properties because of the gyrotropic nature of the mag-

netic force e(v × B) and magnetic torque m ×B.

For some nonreciprocal media, by rotation of the coordinate system, it is

possible to make the constitutional tensor in the following simplest form

² =





j²

−j²

0 0 ²





, µ =





jµ

−jµ

0 0 µ





. (8.84)

This special coordinate system can also be called the principle co ordinate

system, and the axis z is known as gyrotropic axis.

8.4 Characteristics of Waves in Anisotropic

Media

In anisotropic media, the eﬀective indices and the propagation coeﬃcients

for waves in diﬀerent directions may be diﬀerent. In addition, the electric

induction D and the electric ﬁeld E in electric anisotropic media or the

magnetic induction B and the magnetic ﬁeld H in magnetic anisotropic

media may be in the diﬀerent directions. Consequently, the wave vector and

Poynting’s vector may be in diﬀerent directions. In this section, the general

equations and rules of wave propagation in anisotropic media will be given

and the behavior of the waves in diﬀerent kinds of anisotropic media will be

discussed in detail in the next sections.

8.4.1 Maxwell Equations and Wave Equations

in Anisotropic Media

The Maxwell equations for electric anisotropic and magnetic anisotropic me-

dia are

Electric anisotropic media, ², µ Magnetic anisotropic media, ², µ

∇ × E = −jωµH, ∇×E = −jωµ · H,

∇ × H = −jω² · E + J , ∇ × H = −jω²E + J ,

∇ · (² · E) = %, ∇ · (²E) = %,

∇ · (µH) = 0, ∇ · (µ · H) = 0.

498 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

As an example, we handle the case of the ﬁelds and waves in electric

anisotropic media. For source-free and nonconductive media, the Maxwell

equations become

∇ × E = −jωµH, (8.85)

∇ × H = −jω² · E, (8.86)

∇ · (² · E) = 0, (8.87)

∇ · (µH) = 0. (8.88)

Taking the curl of (8.85) and substituting ∇ × H from (8.86), we obtain

∇ × ∇ × E = −jωµ∇ × H = ω

µ² · E. (8.89)

The left-hand side may be expanded by using the vector identity for ∇×∇×A

(B.45), as we did to obtain the wave equation for isotropic media, yields

∇

E − ∇(∇ · E) + ω

µ² · E = 0. (8.90)

This is the governing equation for electromagnetic waves in ²-anisotropic

media.

For plane waves with space factor e

−jk·x

, we have ∇ = −jk, and he

Maxwell equations become

−jk × E = −jωµH, −jk × H = jω² · E, (8.91)

and the wave equation for E becomes

E − k(k · E) − ω

µ² · E = 0. (8.92)

Similarly, for magnetic anisotropic media, the wave equation for H are

∇

H − ∇(∇ · H) + ω

²µ · H = 0, (8.93)

H − k(k · H) − ω

²µ · H = 0. (8.94)

8.4.2 Wave Vector and Poynting Vector in Anisotropic

Media

In Section 2.1, we showed that, for plane waves with space factor e

−jk·x

, the

nabla operator, ∇, becomes −jk and

∇ · A = −jk ·A, ∇ × A = −jk × A,

Then the Maxwell equations become

k × E = ωB, (8.95)

k × H = −ωD, (8.96)

8.4 Characteristics of Waves in Anisotropic Media 499

k · D = 0, (8.97)

k · B = 0, (8.98)

It is clear from the above equations that the wave vector k is perpendicular

to the plane determined by D and B, which is called the DB plane.

Another important direction of wave propagation is the direction of power

ﬂow, which is deﬁned by the Poynting vector,

S =

E × H

∗

So the direction of power ﬂow is perpendicular to the plane determined by

E and H. In optics the unit vector in the direction of S is called the ray

vector. Finally we have

k ⊥ D, k ⊥ B, S ⊥ E, S ⊥ H.

The plane that contains D and B and is perpendicular to k is the constant-

phase surface or phase front and the plane that contains E and H and is

perpendicular to S is the constant-power or constant-strength surface.

In situations for which D k E and B k H, the ray vector and the wave

vector are in the same direction. This is the same as the behavior of waves

in isotropic media. In anisotropic media, it is possible that D 6kE or B 6kH

and, as a consequence, S is not parallel to k.

In electric anisotropic media, it is possible that D 6k E, and S is not

parallel to k. The angle between S and k is the same as that between D

and E. In this situation, B k H, so the four vectors D , E, k, and S are

coplanar and normal to both B and H. Similarly, in magnetic anisotropic

media, the four vectors B, H, k, and S are coplanar and normal to both D

and E. The above-mentioned relations are shown in Fig. 8.7.

In conclusion, the Poynting vector S is not necessarily in the same direc-

tion as the wave vector k inside an anisotropic media. The wave with k kS is

called the ordinary wave or, in abbreviation, o wave and the wave with k 6kS

is called the extraordinary wave or e wave.

8.4.3 Eigenwaves in Anisotropic Media

In isotropic media, the indices of refraction and the propagation coeﬃcients of

plane waves in diﬀerent directions are the same, so, for waves with any state

of polarization and any direction of propagation, the state of polarization of

the wave remains unchanged during the propagation. However, in anisotropic

media, because the constitutional parameters in diﬀerent directions may be

diﬀerent, the eﬀective indices and the propagation coeﬃcients in diﬀerent

directions may be diﬀerent, so, in general, the state of polarization of the

wave may change during the propagation.

In anisotropic media, only the waves with speciﬁc direction of propaga-

tion and with speciﬁc polarization state can keep the state of polarization

500 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Figure 8.7: The relations among the directions of wave vector, energy ﬂow

and ﬁeld vectors.

unchanged during the propagation. These speciﬁc waves are called eigen-

waves or characteristic waves, and in some literature, natural modes. In a

speciﬁc direction of propagation, there are two eigenwaves with diﬀerent state

of polarization. In reciprocal media, there are two linearly polarized eigen-

waves, and in nonrecipro cal media or so called gyrotropic media, there are

generally two elliptically or circularly polarized eigenwaves.

8.4.4 kDB Coordinate System

For further discussions on the solutions for the ﬁelds and waves in anisotropic

media, it is convenient to establish a new orthogonal coordinate system called

the kDB system, which consists of the DB plane and the wave vector k [53].

(1) The Construction of the kDB System and the Relations Between

the kDB System and xyz System

The kDB system (η, ξ, ζ) has unit vectors

η,

ξ,

ζ. Let the unit vector

be in the direction of k in such a way that

k =

ζk, (8.99)

and the plane perpendicular to

ζ is the DB plane. Let the unit vector

lie in the x-y plane and be normal to the projection of k on the x-y plane.

Hence

η is determined by the intersection of the x-y plane and the DB plane

and perpendicular to k-

z plane. Then the unit vector

ξ lies in both the DB

plane and the k-

z plane and perpendicular to the ζ-η plane. The relation

between coordinates η, ξ, ζ and x, y, z is given in Fig. 8.8.

Suppose that the angle between

ζ and

z is γ and the angle between the

projection of

ζ on the x-y plane and

x is φ; see Fig. 8.8. One easily ﬁnds

that

η =

x sin φ −

y cos φ, (8.100)

ξ =

x cos γ cos φ +

y cos γ sin φ −

z sin γ, (8.101)

ζ =

x sin γ cos φ +

y sin γ sin φ +

z cos γ. (8.102)