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

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8.1 Classical Theory of Dispersion and Dissipation in Material Media 481

.5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5

0.0

1.0

1.2

1.4

1.6

n" 



n' 



n' 



n" 



Figure 8.2: Frequency responses of n

(ω) and n

(ω).

8.1.4 Normal and Anomalous Dispersion

The general features of the real and imaginary parts of complex permittivity

with respect to frequency around two successive resonant frequencies are

shown in Fig. 8.3.

The damping constants γ

are generally small compared with the natural

frequencies ω

. This means that |ω

− ω

| À ωγ

and, as a consequence,

˙²(ω) is approximately real and ²

is approximately zero for most frequencies

not too close to ω

. The factor

− ω

−1

is positive for ω < ω

and

negative for ω > ω

. Thus, at low frequencies, below the lowest natural

resonant frequency, ω < min(ω

), all the terms in the sum in (8.7) and (8.10)

are positive so that χ

must be positive and ²

must be larger than ²

. If

ω is greater than min(ω

) but less than the other ω

, the term in the sum

containing min(ω

) is negative but the other terms are still positive. As the

frequency is increased so that successive values of ω

are passed, more and

more terms in the sum in (8.7) and (8.10) become negative, until ﬁnally the

whole sum becomes negative when ω > max(ω

), then χ

becomes negative

and ²

becomes less than ²

As the frequency approaches any of the ω

, refer to Fig. 8.3, both ²

and ²

increase rapidly. The real part ²

reaches a maximum at a value

of ω slightly less than ω

, then decreases rather sharply to ²

at ω = ω

Thereafter it reaches a minimum at a value of ω slightly larger than ω

and,

ﬁnally, increases slowly to ²

. The imaginary part ²

reaches its maximum

sharply at ω = ω

. These frequencies correspond to the absorption lines of

the medium. At low pressure, gases are almost transparent except in narrow

regions close to the absorption lines.

In the regions where the slope of the ²

curve is positive, which means that

(ω) increases with ω, the dispersion is said to be normal dispersion. There

482 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Figure 8.3: Frequency responses of ²

(ω) and ²

(ω) around two successive

resonant frequencies.

exist small ranges of frequency near the resonant frequencies at which the

slope of the ²

curve is negative, and the dispersion in this situation is said to

be anomalous dispersion. Normal dispersion occurs everywhere except in the

neighborhood of a resonant frequency. Only if there is anomalous dispersion

does the imaginary part of ˙² become appreciable. A positive imaginary part

to ˙² represents dissipation of energy from the electromagnetic wave to the

medium. The frequency intervals in which ²

(ω) > 0 are called regions

of resonant absorption and the medium is known as a passive medium. If

(ω) < 0, energy is supplied to the wave by the medium and ampliﬁcation

occurs as in a maser or laser. The medium with ²

(ω) < 0 is known as an

active medium.

At the low-frequency end, the lowest resonant frequency for insulators

is diﬀerent from zero. The electric susceptibility and permittivity given by

(8.6) and (8.9), respectively, tend to their static values in the limit as ω → 0

and no singularity arises. However, for conductors, the behavior of the free

electrons must be considered.

8.1.5 Complex Index for Metals

When particles are packed together at densities of the order of those of liquids

or solids, the inﬂuence of the polarization on the local ﬁeld can no longer be

ignored. The ideal gas model is then no longer valid. As a consequence of

the disturbance to electron behavior by surrounding particles in liquids and

solids, the absorption regions are broader than those predicated by the ideal

gas model.

For conducting materials, we need to include consideration of the free

electrons or conduction electrons. We make two reasonable assumptions: (1)

that free electrons with no restoring force acting on them have zero resonant

8.1 Classical Theory of Dispersion and Dissipation in Material Media 483

frequency i.e., −mω

x = 0 so that ω

= 0, and (2) that the eﬀect of the

ions in a conducting “sea” of electrons removes the need for the local ﬁeld

correction. Under these assumptions, the expression of the permittivity for

the ideal gas model (8.9) becomes valid again but a singularity arises at

ω = 0.

Assume that a fraction f

of the electrons per molecule are free electrons,

and the damping factor which allows for collisions of free electrons with the

lattice ions is denoted by γ

. It is convenient to express (8.9) in the following

form:

˙²(ω) =





i(i6=0)

−ω

+jωγ





− j

mω

+jω

= ²

(0)

(ω)−j

mω

+jω

(8.29)

where the contributions of all dipoles are collected together under the square

brackets and are denoted by ²

(0)

(ω),

(0)

(ω) = ²

i(i6=0)

− ω

+ jωγ

and the contribution of the free electrons is given separately in the last term.

In terms of band theory, the terms in the summation correspond to ex-

citation from the valence band to the conduction band whereas the term

including f

and γ

covers excitation from occupied to unoccupied conduc-

tion band states.

8.1.6 Behavior at Low Frequencies,

Electric Conductivity

Assuming that the conducting medium obeys Ohm’s law, rewriting the ex-

pression of ˙²(ω) (1.96),

˙²(ω) = ²

(ω) − j

(ω) +

= [²

(ω) − j²

(ω)] − j

then comparing it with (8.29), we obtain

(ω) − j²

(ω) = ²

(0)

(ω), σ =

m(γ

+ jω)

. (8.30)

This is essentially the result obtained from Drude’s model (in 1900) for the

electrical conductivity, with f

N being the number of free electrons per unit

volume in the medium.

The damping constant γ

can be determined empirically from experi-

mental data on the low-frequency conductivity of the conductor, which gives

ω ¿ γ

and

σ =

mγ

, or

mσ

. (8.31)

484 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

For copper, N = 8 × 10

atoms/m

and at normal temperatures the low-

frequency conductivity is σ ≈ 5.8×10

S/m. This gives γ

≈ 3×10

−1

If we assume that f

≈ 1 and thus γ

≈ 3 × 10

−1

, this shows that up

to frequencies well beyond the microwave region, i.e., up to 10

Hz(s

−1

conductivity of metals is essentially real, i.e., the current density is in phase

with electric ﬁeld and independent of frequency.

At higher frequencies in and beyond the infrared region, the conductivity

is complex and depends on the frequency in a way described qualitatively by

(8.30). For a proper understanding of electrical conductivity for a solid in a

high frequency, a quantum approach is necessary, since the Pauli principle

plays an important role.

At low frequencies, a medium containing an appreciable number of free

electronics can be regarded as a conductor, otherwise, even for the medium

containing a large number of free electronics, for example metal, must be

regarded as an insulator, and the dispersive properties of the medium can be

attributed to a complex permittivity or a refractive index and an extinction

coeﬃcient. It should be noted that the meaning of the term “low frequency”

here is diﬀerent to those in electronic engineering. It covers the entire fre-

quency band from d-c up to sub-millimeter waves. The demarcation between

“low frequency” and “high frequency” is roughly between the sub-millimeter

wave and far-infrared wave, i.e., in THz range. It is just the frequency of

demarcation between electronics and optics.

8.1.7 Behavior at High Frequencies,

Plasma Frequency

At frequencies far above the highest resonant frequency, i.e., ω À max{ω

ω À γ

, the expression for ²(ω) (8.9) reduces to

²(ω) = ²

−

= ²

−

NZe

where NZ denotes the total number of electrons per unit volume. This

expression can also be written in the following form:

²(ω)

1 −

, (8.32)

where

NZe

, (8.33)

where ρ

= NZe denotes the volume charge density of electrons and ω

known as the plasma frequency of the medium. Since each of e, m, and ²

is a

universal constant, the plasma frequency ω

is determined only by the total

number of electrons per unit volume NZ or the volume charge density of

8.2 Velocities of Waves in Dispersive Media 485

electrons ρ

. The physical meaning of ω

is the natural oscillating frequency

of electrons in a neutral plasma when the ions remain at rest. The discussion

of the permittivity of plasma and the waves in plasma will be given later in

Sections 8.9 and 8.10.

The permittivity of metal is given by (8.29). At high frequencies, ω À γ

this takes the following approximate form

˙²(ω)

(0)

(ω)

−

∗

(0)

(ω)

−

, where ω

∗

(8.34)

is the plasma frequency of the conduction electrons, given an eﬀective mass

∗

to include partially the eﬀects of binding.

It is clear that the behavior of the interaction of an electromagnetic wave

with any material medium including metal approaches the behavior of the

interaction with a plasma when the frequency of the wave is suﬃciently high,

whereas the relative permittivity approaches unity when ω À ω

For ω ¿ ω

, the light penetrates only a very short distance into the metal

and is almost entirely reﬂected, because the extinction coeﬃcient is large.

But when the frequency is increased to the range ω > ω

, the metal becomes

transparent and the reﬂectivity of the metal surface changes drastically. This

occurs typically in the ultraviolet region and is known as the ultraviolet

transparency of the metal. This is just the frequency of demarcation between

optics and high-energy physics or fundamental-particle physics.

8.2 Velocities of Waves in Dispersive Media

In the previous chapters, the propagation of electromagnetic waves in un-

bounded and bounded systems with non-dispersive ﬁlling media is discussed.

Generally, in a guided wave system, all of the waves other than the TEM

mode are dispersive modes. This kind of dispersion is known as waveguide

dispersion. The characteristics of waveguide dispersion are determined by

the propagating mode of interest and the geometry of the guided-wave sys-

tem. In a multi-mode waveguide the phase velocities of diﬀerent modes are

diﬀerent. This leads to inter-mode dispersion.

In this section, the dispersion of electromagnetic waves caused by the

medium itself will be discussed. It is known as material dispersion.

For plane waves, in non-dispersive media, the phase velocity is equal to the

group velocity, but in dispersive media, the phase velocity is no longer equal

to the group velocity. In the region of week dispersion, the signal velocity

and energy velocity are both approximately equal to the group velocity, but

in the region of strong disp ersion, the group velocity, energy velocity and

signal velocity are no longer equal to each other [17, 43, 96].

486 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

8.2.1 Phase Velocity

In dispersive media, the propagation coeﬃcient of a plane wave is complex,

k = ω

µ˙² =

˙n =

− jn

) = β − jα, (8.35)

where

β = ωn

/c, α = ωn

/c (8.36)

denote the phase coeﬃcient and the attenuation coeﬃcient, respectively. The

phase velocity of a plane wave in a disp ersive medium becomes

(ω)

. (8.37)

Generally, in dispersive media, the phase velocity depends on the frequency.

For the medium where ²

¿ ²

and if n

of the medium is only slightly

larger than 1, substituting (8.27) into the above expression of v

, we obtain

≈

1 +

NZe

2²

− ω

+ ω

(8.38)

It is easily seen that when ω < ω

the phase velocity is less than the speed

of light in vacuum, c, whereas when ω > ω

, the phase velocity is larger than

c. In the region of normal dispersion, |ω

− ω

| À ωγ, it reduces to

≈

1 +

NZe

2²

− ω

. (8.39)

At the high frequency end, the phase velocity approaches c at ω → ∞.

The phase velocity that enters into the wave solution has a steady-state

time dependence of e

jωt

, i.e., a pure monochromatic wave with a deﬁnite

frequency and wave number or a wave train of inﬁnite duration which exists

only if the source is turned on at t = −∞ and kept on for all future time

as well. In practice, a steady-state sinusoidal wave will be observed after a

suitable period of time has elapsed and the transient of switching on of the

source has died out. Once steady-state conditions prevail, the phase velocity

can be introduced to describe the velocity at which a constant phase point

appears to move along the medium or the system. However, there is no

information being transmitted along the system once steady-state conditions

have been established. The term signal is used to denote a time function

that can convey information to the observer. Thus any step change of the

wave or a wave train with ﬁnite duration is a signal, but once steady-state

conditions are achieved, there is no more signal because the observer does

not receive any more information. Thus the phase velocity is not associated

with any physical entity such as a signal, wavefront, or energy ﬂow. Hence

the fact that in dispersive media, in some frequency bands, the phase velocity

is larger than c does not violate Einstein’s theory of special relativity.

8.2 Velocities of Waves in Dispersive Media 487

8.2.2 Group Velocity

In reality, idealized monochromatic waves do not arise. Even in the most

sharply tuned radio transmitter or most monochromatic light source, waves

with ﬁnite frequency ranges are generated and transmitted. Furthermore,

any signal must consist of wave trains of ﬁnite extent or waves with ﬁnite

frequency spectra. Since the basic equations are linear, any time-dependent

process can be treated by a superposition of sinusoidal waves with diﬀerent

frequencies and wave numbers.

In a dispersive medium or disp ersive guided wave system, the phase veloc-

ity is not the same for each frequency component of the wave. Consequently

diﬀerent components of the wave travel with diﬀerent speeds and tend to

change phase with respect to one another during the propagation and give

rise to phase distortion of the waveform. The velocity of the signal, i.e., the

velocity of the envelope of the wave train, is diﬀerent from the phase velocity

of the monochromatic wave.

The appropriate tool for the analysis of the propagation of ﬁnite wave

trains or wave packets is the Fourier integrals. Suppose that the phase co-

eﬃcient k of the wave is a general smoothly varying function of the angular

frequency ω,

k = k(ω). (8.40)

Then an arbitrary component of the ﬁelds in a wave along z can be expressed

as the following Fourier integral:

u(z, t) =

√

2π

∞

−∞

A(ω)e

j(ωt−kz)

dω, (8.41)

where A(ω) is the amplitude of the monochromatic wave with frequency

ω, which describes the properties of the linear superposition of the waves

with diﬀerent frequencies. It is given by the inverse Fourier transform of the

function u(z, t), evaluated at z = 0:

A(ω) =

√

2π

∞

−∞

u(0, t)e

−jωt

dt. (8.42)

The above two equations deﬁne a Fourier transform pair, see Fig. 8.4(a).

The circular frequency ω can also be expressed as a function of k:

ω = ω(k), (8.43)

and the Fourier integral of u(z, t) becomes

u(z, t) =

√

2π

∞

−∞

A(k)e

j(ωt−kz)

dk, (8.44)

see Fig. 8.4(b), where A(k) is the amplitude of the monochromatic wave with

wave number k, which describes the properties of the linear superposition

488 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Figure 8.4: A ﬁnite wave train and its Fourier spectrum in frequency (a) and

in wave number (b).

of the waves with diﬀerent wave numbers. It is given by the inverse Fourier

transform of the function u(z, t), evaluated at t = 0:

A(k) =

√

2π

∞

−∞

u(z, 0)e

jkz

dz. (8.45)

Now we turn to the motion of the wave packet u(z, t). The function of

frequency ω(k) can be expanded around the center value k

ω(k) = ω

dω

(k − k

) + ···, (8.46)

where ω

= ω(k

). If the distribution of amplitude A(k) is fairly sharply

peaked around k

, ∆k ¿ k

, only two terms in the above series have to be

considered, and (8.44) becomes

u(z, t) =

(

√

2π

∞

−∞

A(k) exp

−j(k−k

)

z−

dω

)

j(ω

t−k

. (8.47)

Hence the wave packet u(z, t) is explained in the form of a mo dulated

monochromatic wave traveling in the positive z direction and the integral

in the braces represents its envelope U (z, t):

u(z, t) = U(z, t)e

j(ω

t−k

, (8.48)

U(z, t) =

√

2π

∞

−∞

A(k) exp

−j(k − k

)

z −

dω

dk. (8.49)

8.2 Velocities of Waves in Dispersive Media 489

The phase factor of the modulated wave is e

j(ω

t−k

and the phase velo city

. (8.50)

The condition for a constant envelope shape is

z −

dω

t = const,

and the wave packet travels along z with a velocity

dω

dk/dω|

, (8.51)

which is deﬁned as the group velocity. When the wave number is complex,

k = β − jα, the group velocity becomes

dω

dβ

dβ/dω|

, (8.52)

By virtue of (8.36),

β(ω) =

ωn

(ω)

we get

(ω)

, v

(ω) + ω[dn

(ω)/dω]

. (8.53)

The phase velocity and the group velocity in the neighborhood of a reso-

nant peak of a dispersive medium can be obtained by applying the frequency

responses of n

(ω) given in Fig. 8.2 in the ab ove expressions. They are plot-

ted, as ratios of c, as the dashed line and the solid line, respectively, in

Figure 8.5.

In the medium where ²

¿ ²

and if n

of the medium is only slightly

larger than 1, by using (8.27) in (8.53), we obtain

≈

1 +

NZe

2²

(ω

+ ω

)[(ω

− ω

)

− ω

]

[(ω

− ω

)

+ ω

]

(8.54)

In the region of normal dispersion, |ω

− ω

| À ωγ, this reduces to

≈

1 +

NZe

2²

+ ω

(ω

− ω

)

. (8.55)

Comparing with (8.39), we see that in this region v

< v

and v

< c. The

group velocity also approaches c at ω → ∞ .

490 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Figure 8.5: The plots of v

, v

, and v

as ratios of c in the neighborhood

of a resonant peak of a dispersive medium.

In the region of normal dispersion, ω < ω

and ω > ω

, where (dn

/dω) >

0 and n

> 1, the group velocity is smaller than the phase velocity and also

smaller than c, i.e., v

< v

and v

< c. In this case, the group velocity

represents the velocity of the signal propagation. In the region of anomalous

dispersion, ω

< ω < ω

, where (dn

/dω) < 0 and |dn

/dω| can b ecome large,

then v

> v

and the group velocity diﬀers greatly from the phase velocity.

When n

+dn

/dω become less than 1 or even negative then the group velocity

becomes larger than c or even negative. This result does not mean that the

ideas of special relativity are violated, rather that the group velocity deﬁned

here no longer represents the velocity of a signal which propagates in the

medium with strong anomalous dispersion, because a large value of |dn

/dω|

is equivalent to a rapid variation of ω as a function k and consequently the

two term approximation made in (8.47) is no longer valid. Refer to [17, 96].

8.2.3 Velocity of Energy Flow

The deﬁnition of the energy ﬂow velocity is

, (8.56)

where P denotes the density of average power ﬂow and w denotes the average

density of energy stored in the electromagnetic ﬁelds. They can be obtained

by means of Poynting’s theorem given in Section 1.4.