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

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2.1 Sinusoidal Uniform Plane Waves 61

Maxwell’s equations (2.3) – (2.6) for uniform plane waves become

jk × E

= j ωµH

, (2.31)

jk × H

= −j ω²E

, (2.32)

jk · E

= 0, (2.33)

jk · H

= 0. (2.34)

It can be seen that in a uniform plane wave with an arbitrary direction of

propagation, E and H are both perpendicular to the direction of propagation

k, and E and H are perpendicular to each other. So a uniform plane wave

is a kind of transverse electric-magnetic wave, or simply a TEM wave.

multiplying (2.31) by −jk, substituting (2.32) in the right-hand side and

applying (B.30), (2.33), yields

= ω

µ²E

, (2.35)

= ω

µ²H

. (2.36)

These are Helmholtz’s equations for uniform plane waves, which can also be

obtained by applying (2.30) to general vector Helmholtz’s equations (2.1) and

(2.2).

The magnitude of the wave vector k is given by

k = ω

√

µ², (2.37)

which is just the angular wave number of the unbounded medium.

From equation (2.31) and (2.32) we have

ωµ

k × E

µ/²

× E

n × E

(2.38)

and

= −

ω²

k × H

= −

× H

= −η n × H

. (2.39)

The ratio of the complex amplitude of the electric ﬁeld to that of the magnetic

ﬁeld is equal to the wave impedance:

η =

ω²

ωµ

. (2.40)

Note that for a plane wave in a lossless simple medium, the wave impedance

is a real constant: the electric ﬁeld and the magnetic ﬁeld are in phase.

From (2.22), we have the phase velocity of a plane wave in an arbitrary

direction x:

d|x|

k cos θ

, (2.41)

62 2. Introduction to Waves

where θ denotes the angle between x and the wave vector k. Let θ = 0, we

then have the phase velocity in the direction of wave vector:

√

µ²

. (2.42)

In vacuum,

≈ 120π ≈ 377 Ω, v

√

= c, (2.43)

where η

and c are two universal physical constants.

The time parameters of a wave are the circular or angular frequency ω,

the time period T , and the frequency f, and the relations among them are

ω = 2πf =

2π

, T =

2π

. (2.44)

The space parameters of a wave are the angular wave number k, the

wavelength λ, and the wave number 1/λ, and the relations among them are

k =

2π

, λ =

2π

. (2.45)

The connection between time and space is the phase velocity:

= λf. (2.46)

Substituting (2.24) and (2.38) into the deﬁnition of the Poynting vector,

and applying the formula for a triple vector product (B.30) and equation

(2.33), we have the complex power ﬂow density in the plane wave:

S =

(x) × H

∗

(x) =

−jk·x

n × E

jk·x

(2.47)

This shows that in a simple medium the direction of power ﬂow is the same

as the direction of the wave vector. In a lossless medium, the Poynting vector

is real and the time-average power ﬂow density is independent of time and

the distance of propagation, which means that

∇ ·

S = 0.

From the complex Poynting theorem (and note that σ = 0, J = 0) we

have

²E

µH

. (2.48)

For a uniform plane wave in a lossless simple medium, the time-average elec-

tric energy density is equal to the time-average magnetic energy density.

They are both independent of time and the coordinates. The wave propa-

gates without damping, i.e., persistent wave.

The instantaneous power ﬂow density and the instantaneous energy den-

sities for a linear polarized uniform plane wave are alternate with frequency

2ω with respect to time and distance of propagation, refer to Problem 2.1.

2.1 Sinusoidal Uniform Plane Waves 63

2.1.3 Plane Waves in Lossy Media, Damped Waves

Except for a vacuum, all media have losses, including conductive loss or Joule

loss and dielectric loss or polarization loss. The complex permittivity of a

lossy medium was given in (1.96) as follows:

˙²(ω) = ²

(ω) − j

(ω) +

. (2.49)

When σ/ω ¿ ²

and ²

¿ ²

, the losses can be neglected and the medium is

identiﬁed as a lossless medium. The magnetization loss is not considered in

this section, but it can be added in if necessary.

In lossy media, the complex wave equations (1.156) and (1.157) become

∇

E + ω

(ω) − j

(ω) +

E = 0, (2.50)

∇

H + ω

(ω) − j

(ω) +

H = 0. (2.51)

These are also Helmholtz’s equations and the solutions for uniform plane

waves are similar to those for lossless media (2.18) and (2.19), but the phase

coeﬃcient becomes complex,

k = ω

˙² − j

= ω

− j

´i

. (2.52)

Let

k = β − jα , or ˙γ = j

k = α + jβ, (2.53)

where γ denotes the propagation coeﬃcient, α denotes the attenuation co-

eﬃcient in nepers per meter (Np/m) and β denotes the phase coeﬃcient in

radians per meter (rad/m). The relation between neper (Np) and decibel

(dB) is 1 Np = 8.686 dB.

From (2.53) and (2.52), we have

= (β

− α

) − j2αβ = ω

µ²

− jω

The real parts and the imaginary parts must be equal separately,

− α

= ω

µ²

, 2αβ = ω

and we have

β = ω

µ²

(

1 +

(²

+ σ/ω)

+ 1

1/2

, (2.54)

α = ω

µ²

(

1 +

(²

+ σ/ω)

− 1

1/2

. (2.55)

64 2. Introduction to Waves

This shows that the polarization loss and conductive loss not only introduce

attenuation but also change the phase coeﬃcient of the wave.

In lossy media, the wave impedance becomes complex as well,

˙η =

˙²

− j(²

+ σ/ω)

. (2.56)

The ﬁelds of a wave traveling in the +z direction become

(z) = E

− ˙γz

= E

−αz

−jβz

, (2.57)

(z) = H

− ˙γz

˙η

−αz

−jβz

. (2.58)

Thus the losses in a medium give rise to exponential decaying of the ﬁelds in

the direction of wave propagation, and the electric and magnetic ﬁelds are

no longer in phase. This kind of wave is called a damped wave.

The quantity δ = 1/α is the penetration depth or the skin depth over

which the amplitude of the ﬁeld decreases by a factor of 1/e ≈ 0.369,

δ =

√

µ²

(

1 +

(²

+ σ/ω)

− 1

1/2

. (2.59)

If the medium is conductive and the polarization loss is negligibly small,

σ/ω À ²

, ˙² ≈ ²

= ², (2.54) and (2.55) become

β = ω

√

µ²

(

1 +

ω²

+ 1

1/2

, (2.60)

α = ω

√

µ²

(

1 +

ω²

− 1

1/2

. (2.61)

The skin depth (2.59) becomes

δ =

√

µ²

(

1 +

ω²

− 1

1/2

. (2.62)

The wave impedance (2.56) becomes

˙η =

² − j(σ/ω)

. (2.63)

2.1 Sinusoidal Uniform Plane Waves 65

(1) Low-Loss Conductive Media

For the medium with low conductivity, and in a relatively high frequency

range, the conductive current is much less than the displacement current,

σ/ω² ¿ 1, so (2.60), (2.61), (2.62) and (2.63) reduce to

β ≈ ω

√

µ²

1 +

ω²

≈ ω

√

µ² α ≈

, (2.64)

δ ≈

, η ≈

. (2.65)

In such media the conductivity hardly aﬀects the phase coeﬃcient and wave

impedance, but it gives rise to an attenuation that is independent of frequency

and α ¿ β.

(2) Good Conductors

When the conductivity of a medium is rather high and the frequency is

relatively low, the conductive current is much larger than the displacement

current. Such a medium is identiﬁed as a good conductor and gives σ/ω² À 1.

In good conductors, equations (2.50) and (2.51) become diﬀusion equations,

the phase and attenuation coeﬃcients (2.60) and (2.61) and the skin depth

(2.62) reduce to

β ≈ α ≈

ωµσ

, δ ≈

ωµσ

. (2.66)

The attenuation coeﬃcient in good conductors is very large and equals the

phase coeﬃcient. At a distance of about λ/6, i.e., the phase shift of 1 rad, the

amplitude of the ﬁeld is attenuated to 1/e of the original value, or attenuated

to 1/e

2π

≈ 1/534 in one wavelength, see Fig. 2.4. For example, the skin depth

of copper is about 8.5 mm at 60 Hz and 1 µm at 3 GHz.

The wave impedance (2.63) reduces to

˙η =

jωµ

= (1 + j)

ωµ

2σ

ωµ

j(π/ 4)

. (2.67)

In good conductors, the phase diﬀerence between the electric and magnetic

ﬁelds is π/4.

In conductors, the electric energy density is no longer equal to the mag-

netic energy density; the ratio of w

to w

is given by

²E

µH

ω²

¿ 1,

which means that in good conductors, the magnetic ﬁeld is dominant in the

wave.

66 2. Introduction to Waves

0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

-.4

-.2

0.0

1.0

±2zz





ztEE  cose





4±cose



ztHH

Figure 2.4: Fields of a damped plane wave in a good conductor.

The tangential components of the electric and magnetic ﬁelds are contin-

uous across the boundary of a conductor, because there is no surface current.

The ratio of the tangential component of the electric ﬁeld to that of the mag-

netic ﬁeld just outside a conductor is still equal to the wave impedance of

the conductor (2.67):

jωµ

= (1 + j)

ωµ

2σ

= R

+ jX

, (2.68)

where Z

denotes the surface impedance of the good conductor, and

= X

ωµ

2σ

σδ

The complex Poynting vector of the wave in a good conductor is given by

S =

× H

∗

(1 + j)

ωµ

2σ

n = p + j q, (2.69)

where n is the unit vector of the wave vector k. The real part represents the

average power-ﬂow density entering into the conductor:

p = <

× H

∗

ωµ

2σ

n =

σδ

n. (2.70)

This is the power dissipation in a unit area on the surface of a good conductor.

It can be shown that it is equal to the Joule loss in a semi-inﬁnite conducting

cylinder of unit cross section. We leave the proof of this relation as an

exercise, see Problem 2.6.

2.2 Polarization of Plane Waves 67

(3) Perfect Conductors

In the electrostatic equilibrium state the electric ﬁeld inside a conductor

is always zero. In a time-varying state, the electric and magnetic ﬁelds in

a conductor are not zero but are damped waves. The exceptional case is

that in which the conductivity approaches inﬁnity, σ → ∞, such that the

skin depth approaches zero, δ → 0, i.e., the ﬁeld and current concentrate in a

inﬁnitesimal depth on the surface of the conductor and cannot penetrate into

the conductor. The above approximation is said to be a perfect conductor.

The surface impedance (2.68) of a perfect conductor approaches zero,

as does the tangential component of the electric ﬁeld at the surface. The

tangential component of the magnetic ﬁeld at the surface is equal to the

surface current density. This conclusion is just the boundary condition of the

short-circuit surface, given in Section 1.2.2:

n × E|

= 0, n × H|

= J

At the surface of a perfect conductor, the normal component of the Poynting

vector is zero, so there is no power loss in the perfect conductor.

Note that the perfect conductor is not identical to a superconductor, the

superconductor is a kind of speciﬁc material but the perfect conductor is only

an approximation of good conductors for simplifying the analysis in certain

conditions. However, the sup er conductor is the b est perfect conductor.

The concept of perfect conductors is successfully used in the analysis of

electromagnetic waves with conducting boundaries when the power loss on

the conductor surface is allowed to be negligible.

2.2 Polarization of Plane Waves

In the above section, the plane wave with a speciﬁc ﬁxed orientation of the

ﬁeld vector was presented. Since the wave equation is a linear equation, the

sum of solutions is also a solution to it. Many complex electromagnetic waves

may be considered as made up of a large number of simple plane waves with

diﬀerent magnitudes, frequencies, phases, orientations of the ﬁeld vector, and

directions of propagation.

In this section, we will discuss the combination of plane waves with the

same direction of propagation. The orientation of the ﬁeld vector of the

combined wave is not necessarily ﬁxed. The states of the ﬁeld vectors of these

waves are described by the polarization of the wave, which is designated as

the projection of the locus of the terminus of the instantaneous ﬁeld vector

on the wave front, i.e., the plane normal to the direction of propagation.

For the plane wave presented in the above section, the electric ﬁeld vec-

tor always lies in a given direction; it is said to be a linearly polarized or,

sometimes, plane polarized wave. The general form of the polarized wave is

68 2. Introduction to Waves

elliptic polarization; in special cases, it becomes linear polarization or circular

polarization.

For radio waves, including microwaves, it is common to describe the po-

larization by the orientations of the electric ﬁeld vector, but in optics, people

usually utilizes the magnetic ﬁeld vector to deﬁne the plane of polarization.

In this text, we use the former description.

2.2.1 Combination of Two Mutually Perpendicular

Linearly Polarized Waves

An arbitrary polarized electromagnetic wave can be explained by the sum of

two mutually p erpendicular linearly polarized waves with the same frequency

and the same propagation coeﬃcient but diﬀerent amplitudes and diﬀerent

phase angles.

(1) General Formulation

Suppose that the electric ﬁeld of a plane wave is the sum of vectors E

and

E =

jδ

y m

jδ

j(ωt−kz)

, (2.71)

where δ

and δ

are the phase angels of E

and E

, respectively, and the

phase diﬀerence is

∆ = δ

− δ

. (2.72)

The combined magnetic ﬁeld becomes

H =

−

jδ

−

y m

jδ

j(ωt−kz)

(2.73)

Let τ = ωt − kz and rewrite the two components of the electric ﬁeld in

instantaneous form:

= E

sin(τ + δ

), E

= E

y m

sin(τ + δ

). (2.74)

These are the parametric equations for an ellipse. It means that the combined

electric ﬁeld vector rotates and the terminus of it traces an elliptic path in a

plane normal to the direction of propagation.

The ﬁeld vector of a elliptically polarized wave rotates during the propa-

gation. The direction of the rotation depends upon the phase relation of the

two ﬁeld components. When an observer who transmits the wave, i.e., who

looks in the direction of propagation, the ﬁeld vector rotates in a clockwise

sense if δ

< δ

and ∆ is negative, and the ﬁeld vector rotates in a coun-

terclockwise sense if δ

> δ

and ∆ is positive. These are a clockwise (CW)

polarized wave and a counterclockwise (CCW) polarized wave, respectively.

Sometimes, they are called right-handed and left-handed waves instead. Most

2.2 Polarization of Plane Waves 69

Figure 2.5: An elliptically polarized plane wave.

literatures on electromagnetic waves or so called radio waves describe the di-

rection of the rotation in such a way.

In some literatures, especially in texts and papers on optics, the direction

of the rotation is determined by an observer who receives the wave, i.e., one

who looks in a direction opposite to the direction of propagation, and the

CW and CCW are exchanged.

From equations (2.74), and applying (2.72), yields

sin δ

−

y m

sin δ

= sin τ sin ∆,

cos δ

−

y m

cos δ

= −cos τ sin ∆.

Taking the sum of the square of the above two expressions, canceling τ

in it, gives

y m

−

2 cos ∆

y m

− sin

∆ = 0. (2.75)

This is the equation of an ellipse. The ellipse is internally tangential to a

rectangle with sides 2E

and 2E

y m

. The locus of the terminus of the electric

ﬁeld with respect to time and space is an elliptic helix, see Fig. 2.5(a). This

is the reason for the name elliptic polarization. Elliptical polarization is the

general form of polarized waves.

For the magnetic ﬁeld we have the similar expressions

sin(τ + δ

), H

= −

y m

sin(τ + δ

), (2.76)

and

y m

2 η

cos ∆

y m

− sin

∆ = 0. (2.77)

70 2. Introduction to Waves

Figure 2.6: Combination of two mutually perpendicular linearly polarized

waves.

The terminus of the magnetic ﬁeld vector traces an elliptical path perpen-

dicular to the ellipse traced by the terminus of the electric ﬁeld vector in the

same plane and rotates in the same sense, see Fig. 2.5(b).

In general, the trace is an inclined ellipse with respect to the x,y coordi-

nates. The semi-major axis and semi-minor axis and the orientation of the

ellipse are given as follows.

Suppose that the two principle axes of the ellipse are 2a and 2b. Place

new coordinates x

, such that x

is in the direction of 2a and y

is in the

direction of 2b. The angle between x

and x is denoted as ψ and is called

the orientation angle of the ellipse, see Fig. 2.6. The two components of the

electric ﬁeld in the new x

coordinates are E

and E

= E

cos ψ + E

sin ψ, E

= −E

sin ψ + E

cos ψ. (2.78)

In x

coordinates, the parametric equations of the ellipse are

= a sin(τ + θ), E

= ±b cos(τ + θ), (2.79)

where θ is an arbitrary phase angle of the ﬁelds and the signs + and −

correspond to CCW and CW, respectively.

(2) Relations Between E

, E

y m

, ∆ and a, b, ψ

Substituting (2.74) and (2.79) into (2.78) yields

a (sin τ cos θ + cos τ sin θ) = E

(sin τ cos δ

+ cos τ sin δ

) cos ψ

+ E

y m

(sin τ cos δ

+ cos τ sin δ

) sin ψ,

±b (cos τ cos θ + sin τ sin θ) = − E

(sin τ cos δ

+ cos τ sin δ

) sin ψ

+ E

y m

(sin τ cos δ

+ cos τ sin δ

) cos ψ.