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

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7.9 Distributed Feedback (DFB) Structures 471

Figure 7.41: A pair of dielectric layers in a uniform medium of index n.

and the reﬂection coeﬃcient is given by

Γ =

− Z

+ Z

(1 + 4∆n/n) − 1

(1 + 4∆n/n) + 1

≈

2∆n

Comparing this approximation for Γ with (7.287), while l = p and |κ|p ¿ 1

for one layer pair,

Γ (p)|

= tanh |κ|p ≈ |κ|p,

we have

|κ|p ≈

2∆n

. (7.290)

Substituting this relation into (7.289) gives

≈

1 +

4∆n

(l/p)

1 +

4∆n

(n/4∆n)

2|κ|l

. (7.291)

Utilizing the fact that the term in brackets approaches the numerical constant

e as ∆n/n is made very small, i.e.,

lim

∆n/n→0

1 +

4∆n

(n/4∆n)

= e,

we deduce that

= e

2|κ|l

. (7.292)

It is the same result as obtained with the DFB approach (7.288), where Z

corresponds to Z

and Z

corresponds to Z

. We may conclude that both

approaches give the same result in the limit of small ∆n/n, i.e., the weak

coupling limit. The coupled-mode approach is notably simpler and gives

the frequency dependence of the reﬂection coeﬃcient as shown in Fig. 7.37.

The structure is no longer a reﬂector when used outside the stop band, or

472 7. Periodic Structures and the Coupling of Modes

Figure 7.42: (a) Problem 7.1. Disk-loaded coaxial line, (b) Problem 7.2.

Disk-loaded conducting cylinder.

when |δ| > |κ|, because the reﬂections from individual layers are no longer

interferences.

The multi-layer reﬂector as a DFB structure is successfully used in the

vertical cavity surface emitting semiconductor lasers (VCSEL), refer to [116].

Problems

7.1 Find the eigenvalue equation of the azimuthal uniform TM modes in the

disk-loaded coaxial line shown in Fig. 7.42a, (1) as a uniform system,

(2) as a periodic system.

7.2 Find the eigenvalue equation of the azimuthal uniform TM modes in the

disk-loaded conducting cylinder shown in Fig. 7.42b, (1) as a uniform

system, (2) as a periodic system.

7.3 Find the eigenvalue equation of the azimuthal uniform TE modes in

the disk-loaded circular waveguide with a center coupling hole, using

uniform system approach.

7.4 Show that the azimuthal nonuniform TE or TM mode by itself cannot

satisfy the boundary conditions of the disk-loaded waveguide with a

center coupling hole, using uniform system approach.

7.5

Show that the azimuthal nonuniform TE or TM mode by itself cannot

satisfy the boundary conditions of the disk-loaded coaxial line and the

disk-loaded conducting cylinder, using uniform system approach.

Problems 473

Figure 7.43: Problem 7.6. Parallel-plate transmission line made of a inclined

conducting plate and a common conducting plate.

Figure 7.44: Problem 7.8. Helix enclosed by a rigid conducting tube.

7.6 Find the ﬁeld components, eigenvalue equation, and the propagation

characteristics of the parallel-plate transmission line shown in Fig. 7.43,

in which the upper plate is an inclined conducting plate and the lower

plate is made of common conducting material. The angle between the

direction of conduction and the direction of the transverse axis x is ψ.

7.7 Find the eigenvalue equation of a sheath helix of radius a enclosed by a

perfect conducting tube of radius b (b > a).

7.8 Find the eigenvalue equation of a sheath helix of radius a enclosed by a

longitudinal conducting tube of radius b (b > a) shown in Fig. 7.44(b),

which is the physical model of a helix enclosed by a longitudinal rigid

conducting tube, shown in Fig. 7.44(a).

7.9 Assume a periodic structure with the spatial period equal to 5 cm; the

frequency at which β

p = π/2 is 3 GHz. Find the phase constants, the

phase velocities, and the longitudinal wavelengths of the fundamental

(n = 0) and the n = −1 space harmonics.

474 7. Periodic Structures and the Coupling of Modes

7.10 Find the ratio of β

−1

to β

in a periodic structure, when β

= 0,

= π/4, β

= π/2, β

= 3π/4 and β

= π.

7.11 Prove the two theorems given in Section 7.4.5. Refer to [107].

7.12 Find the natural frequencies of coupled resonators. The natural fre-

quencies of the two uncoupled resonators are ω

and ω

, and the

coupling coeﬃcients are κ

and κ

Hint: The mode-coupling equations of coupled resonators are given by,

= −jω

a + κ

= −jω

b + κ

7.13 Find the general solutions a(t) and b(t) of coupled resonators given in

the last problem. Suppose that the initial value of a and b at t = 0 are

a(0) and b(0), respectively.

7.14 Prove the power conservation for the coupled waves of co-directional

mode coupling and contra-directional mode coupling.

Chapter 8

Electromagnetic Waves

in Dispersive Media and

Anisotropic Media

Up to this point, we have concentrated our attention on issues concerning

the ﬁelds and waves in simple media, i.e., linear, non-dispersive, and isotropic

media, For a simple medium, the constitutional parameters are assumed to be

constant scalars and frequency independent. In fact, no medium can always

be a simple medium except vacuum. Most media can only be approximated

as simple media under certain conditions.

Recently, the wave-propagation in dispersive, and anisotropic media be-

came more imp ortant in microwave, THz (tera-hertz) and light-wave tech-

nologies. The eﬀects of dispersion, and anisotropy of the media must be taken

into account in the design of many devices and a number of new devices with

unique characteristics were developed by applying these eﬀects.

The propagation of electromagnetic waves in material media is a micro-

scopic process of the interaction between ﬁelds and matter. In principle, it

must be investigated theoretically by means of quantum mechanics and sta-

tistical mechanics rather than macroscopic electrodynamics. In classical elec-

trodynamics, macroscopic constitutional parameters are deﬁned as we have

discussed in Section 1.1.2 and these parameters can be found experimentally.

In addition, classical theories based on macroscopic models for some special

matters, for example, ideal gas, conductors, electron beams, plasmas, and

ferrites, have been developed successfully.

In this chapter, the constitutional relations of dispersive media and

anisotropic media are given and the analysis of wave propagation phenomena

in such media are introduced. The problems of ﬁelds and waves in nonlin-

ear media [38, 116] are not included in this book, they belong to the course

“Nonlinear Optics” [6, 16, 89].

475

476 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

8.1 Classical Theory of Dispersion and

Dissipation in Material Media

The microscopic interaction between electromagnetic waves and particles re-

sults in dispersion and dissipation of waves in material media. All media in

reality show a certain amount of dispersion. However, within a limited fre-

quency range, on the speciﬁc kinds of media involved, this dispersion turns

out to be suﬃciently small so the permittivity ², the permeability µ of the

medium and the velocity of propagation for a TEM wave can be regarded

as constant and independent of the frequency of the waves. Even when ²

and µ do depend on the frequency, the treatment for time-harmonic ﬁelds

given in the previous chapters for a non-dispersive medium remains valid for

each frequency component. However, for electromagnetic wave trains that

are a superposition of sinusoidal waves in a range of frequencies, dispersive

eﬀects can no longer be ignored. In the radio wave and microwave bands, the

dispersion and dissipation in most media are rather weak but in the THz,

infrared, visible light, and ultraviolet bands, the dispersion and dissipation

are usually strong and strongly depend upon the frequency.

In Sections 1.1.2 and 1.1.4, we point out that, in dispersive media, the

response of polarization and magnetization are not instantaneous, and for

sinusoidal time-dependent ﬁelds, the permittivity and permeability become

complex and depend upon the frequency. In order to examine these conse-

quences we need a simple model of dispersion [17, 96].

8.1.1 Ideal Gas Model for Dispersion and Dissipation

It is assumed that the material media are composed of small particles which

can be polarized under the inﬂuence of an electric ﬁeld. These particles can

be molecules or atoms. Consider a medium acted on by the time-harmonic

electric ﬁeld E(x, t) = E

jωt

of an incident wave. Assume that the wave

has a wavelength much larger than atomic dimensions, which is true even for

ultraviolet radiation, so that the ﬁeld acting on the electron cloud in a particle

of the medium is independent of its position relative to the nuclear or positive

ion core of the particle. For simplicity, we will neglect the diﬀerence between

the applied electric ﬁeld and the local ﬁeld, i.e., the ﬁeld that arises from

other polarized particles acting on the electron cloud will be neglected. The

model is therefore appropriate only for substances of relatively low density

or gasses of low pressure and is known as the ideal gas model.

In the classical model of dispersion, the center of mass of the electron

cloud in a particle is displaced by an amount x as a result of the action of

the electric ﬁeld. Any displacement of the electron cloud from its central

ion core produces a restoring force −mω

x, where ω

denotes the natural

angular frequency or so called binding frequency of the oscillating electron

and m is its mass. Also a damping force, denoted by −mγ(dx/dt), where

8.1 Classical Theory of Dispersion and Dissipation in Material Media 477

γ denotes the damping factor, is included. This damping force arises from

radiation and interaction with other charges. The interaction of the restoring

force and damping force with the inertia of the moving charge cloud produces

a resonance as in a mechanical spring–mass system or a harmonic oscillator.

The equation of motion of an electron with charge e can be written as

+ γ

+ ω

= eE

jωt

. (8.1)

Magnetic ﬁeld eﬀects are neglected in the equation because the velocity of

the electron is much lower than the speed of light.

In the steady state, the displacement is also time harmonic with the same

frequency as that of the ﬁeld and can be expressed in phasor form x = x

jωt

We obtain the solution of the above equation:

x =

− ω

+ jωγ

E. (8.2)

The dipole moment caused by the displacement of the electron is

p = ex =

− ω

+ jωγ

E. (8.3)

Suppose that the medium is consists of molecules of the same type, there

are N molecules per unit volume, with Z electrons per molecule, and that

there are f

electrons per molecule with natural circular frequency ω

and

damping factor γ

. Then the dipole moment of each molecule becomes

− ω

+ jωγ

E, (8.4)

where f

measures the strength of the ith resonance, and

= Z. The

polarization vector, i.e., the dipole moment per unit volume is

P = Np

− ω

+ jωγ

E. (8.5)

From the deﬁnition of electric susceptibility given in (1.29), i.e., P =

χE, we obtain the complex susceptibility

˙χ(ω) =

− ω

+ jωγ

= χ

(ω) − jχ

(ω). (8.6)

(ω) =

− ω

+ ω

, (8.7)

(ω) =

ωγ

− ω

+ ω

. (8.8)

478 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

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

-.8

-.6

-.4

-.2

0.0

1.0

1.2

max

Figure 8.1: Frequency responses of χ

(ω) and χ

(ω).

The permittivity of the medium is deﬁned in (1.33) and also becomes

complex:

˙²(ω) = ²

[1 + ˙χ(ω)] = ²

(ω) − j²

(ω) = ²

− ω

+ jωγ

, (8.9)

(ω) = ²

[1 + χ

(ω)] = ²

− ω

+ ω

, (8.10)

(ω) = ²

(ω) =

ωγ

− ω

+ ω

. (8.11)

The real part, χ

(ω) or ²

(ω), describes the dispersion and the imaginary

part, χ

(ω) or ²

(ω), describes the dissipation of the medium. All of them

are functions of frequency. The parameters f

, ω

and γ

are well deﬁned in

quantum mechanics and when they have appropriate quantum deﬁnitions, the

above expressions represent fairly accurate descriptions of the polarization of

the medium. The normalized frequency responses of χ

(ω) and χ

(ω) for the

ith terms with ω

= ω

in (8.7) and (8.8) are shown in Fig. 8.1, which is a

typical responses of a damped resonant system.

In practice, there are diﬀerent molecules and electrons with diﬀerent ω

and γ

in the medium. Then a number of discrete resonant peaks appear

on the response curve, which is similar to the resp onse of a resonator with

a number of modes. Similarly, the displacement of one ion from another

produces ionic resonance, but it is much less strong than that for the electron

resonance because of the much larger masses.

8.1 Classical Theory of Dispersion and Dissipation in Material Media 479

If the medium consists of the same type of electrons with resonant fre-

quency ω

and damping factor γ, the expressions for the permittivity reduce

˙²(ω) = ²

NZe

− ω

+ jωγ

. (8.12)

(ω) = ²

NZe

− ω

+ ω

, (8.13)

(ω) =

NZe

ωγ

− ω

+ ω

. (8.14)

8.1.2 Kramers–Kronig Relations

The analytic properties of the complex permittivity and the assumption of

the causal connection between the polarization and the electric ﬁeld provide

interesting and important relationships between the real part and the imagi-

nary part of the complex permittivity, so the frequency behavior of one part

is not independent of that of the other. These relationships are known as

Kramers–Kronig relations as they were ﬁrst derived by H. A. Kramers and

R. de L. Kronig independently [43].

(ω) =

∞

(ω

)

− ω

dω

, χ

(ω) =−

2ω

∞

(ω

)

− ω

dω

, (8.15)

Experimental values of χ

(ω) or ²

(ω) from absorption measurement allow

the calculation of χ

(ω) or ²

(ω) by using the Kramers–Kronig relations.

8.1.3 Complex Index of Refraction

The deﬁnition of the refraction index is given in Section 2.4:

n =

√

. (8.16)

For nonmagnetic media, µ

= 1, and for dispersive medium, the dielectric

constant ²

and the index of refraction n becomes complex,

˙²

= ²

− j²

, ˙n =

˙²

− j²

= n

− jn

, (8.17)

where

+ ²

, n

+ ²

− ²

, (8.18)

denotes the index of refraction and n

denotes the extinction coeﬃcient

for dispersive media. Both of them are functions of frequency.

480 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Substituting (8.9) into (8.17), we obtain

˙n =

1 +

− ω

+ jωγ

. (8.19)

Generally ²

is much less than ²

, then n

and n

reduce to

≈

, n

≈

. (8.20)

Substituting (8.10) and (8.11) into (8.20), we obtain the index and the

extinction coeﬃcient for the ideal gas model:

1 +

(ω

− ω

)

− ω

+ ω

, (8.21)

2²

ωγ

− ω

+ ω

1 +

(ω

− ω

)

− ω

+ ω

. (8.22)

For an ideal gas or most optical materials, n

is only slightly larger than 1.

Then the above expressions can be simpliﬁed as the following approximate

formulas:

≈ 1 +

2²

− ω

+ ω

, (8.23)

≈

2²

ωγ

− ω

+ ω

. (8.24)

If the medium consists of the same type of molecules with resonant frequency

and damping factor γ, the above expressions reduce to

1 +

NZe

− ω

+ω

, (8.25)

NZe

2²

ωγ

− ω

+ ω

NZe

− ω

+ ω

, (8.26)

and if n

of the medium is only slightly larger than 1, they reduce to

≈ 1 +

NZe

2²

− ω

+ ω

, (8.27)

≈

NZe

2²

ωγ

− ω

+ ω

. (8.28)

The frequency responses of n

(ω) and n

(ω) are shown in Fig. 8.2.