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

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2.2 Polarization of Plane Waves 71

The coeﬃcients of sin τ in the two sides of each equation must be equal, and

so must be the coeﬃcients of cos τ . We have

a cos θ = E

cos δ

cos ψ + E

y m

cos δ

sin ψ, (2.80)

a sin θ = E

sin δ

cos ψ + E

y m

sin δ

sin ψ, (2.81)

±b cos θ = −E

sin δ

sin ψ + E

y m

sin δ

cos ψ, (2.82)

±b sin θ = E

cos δ

sin ψ − E

y m

cos δ

cos ψ. (2.83)

The sum of the squares of the four equations gives

+ b

= E

+ E

y m

. (2.84)

The sum of the product of (2.80) and (2.82) and the product of (2.81) and

(2.83) gives

±ab = E

y m

sin ∆. (2.85)

The quotient of (2.82) to (2.80) and the quotient of (2.83) to (2.81) are both

equal to ±b/a and must be equal to each other. Thus

tan 2ψ =

y m

− E

y m

cos ∆. (2.86)

It is convenient to introduce the ratios of E

y m

to E

and b to a. Let

tan φ =

y m

, then tan 2φ=

y m

−E

y m

, sin 2φ =

y m

; (2.87)

tan χ =

, then sin 2χ = ±

2ab

+ b

, (2.88)

where χ denotes the elliptic angle and φ denotes the orientation angle of a

linear polarized wave composed by E

and E

y m

Dividing twice (2.85) by (2.84) gives

sin 2χ = ±

2ab

+ b

y m

+ E

y m

sin ∆. (2.89)

Then we have the relations between χ and ψ and φ and ∆:

sin 2χ = sin 2φ sin ∆, and tan 2ψ = tan 2φ cos ∆. (2.90)

(3) Special Cases

(1) ∆ = 0. E

and E

are in phase. From (2.90) we note that χ = 0, the

minor-axis of the ellipse is zero, and the wave is linearly polarized with the

orientation angle ψ:

ψ = φ = arctan

y m

72 2. Introduction to Waves

The ﬁeld expressions of an arbitrary oriented linearly polarized wave are given

E = (

y m

j(ωt−kz+θ)

, (2.91)

H = (

y m

) e

j(ωt−kz+θ)

−

y m

j(ωt−kz+θ)

. (2.92)

(2)∆ = ±π/2. From (2.90) we note that ψ = 0 and χ = φ. The wave becomes

an ortho-elliptically polarized wave, the major and minor axes coincide with

x, y axes.

(3) ∆ = ±π/2 and E

= E

y m

. Thus χ = φ = π/4, a = b, and the major

and minor axes are equal. The wave b ecomes circularly polarized. ∆ = −π/2

represents the clockwise (CW) or right-handed circularly polarized wave and

∆ = π/2 represents the counterclockwise (CCW) or left-handed wave.

The ﬁelds of a clockwise circularly polarized wave are given by

= E

x +

−j

j(ωt−kz)

= E

(

x − j

y)e

j(ωt−kz)

, (2.93)

= H

y −

−j

j(ωt−kz)

= H

(

y + j

x)e

j(ωt−kz)

y −

−j

j(ωt−kz)

(

y + j

x)e

j(ωt−kz)

. (2.94)

The ﬁelds of a counterclockwise circularly polarized wave are given by

CCW

= E

CCW

x +

j(ωt−kz)

= E

CCW

(

x + j

y)e

j(ωt−kz)

, (2.95)

CCW

= H

CCW

y −

j(ωt−kz)

= H

CCW

(

y − j

x)e

j(ωt−kz)

CCW

y −

j(ωt−kz)

CCW

(

y − j

x)e

j(ωt−kz)

. (2.96)

It can be shown that, for a circularly polarized wave, both the average

Poynting vector and the instantaneous Poynting vector are independent of

distance of propagation, refer to Problem 2.1.

2.2.2 Combination of Two Opposite Circularly

Polarized Waves

An arbitrary polarized electromagnetic wave may also be explained by the

sum of two circular polarized waves rotating in opposite directions with the

same frequency and the same propagation coeﬃcient but diﬀerent amplitudes

and diﬀerent phase angles.

2.2 Polarization of Plane Waves 73

Figure 2.7: Combination of two opposite circularly polarized waves.

(1) General Formulation

The composed electric and magnetic ﬁelds for CW and CCW circularly po-

larized waves are written as

E = E

+ E

CCW

(

x−j

y) e

jα

+ E

CCW

(

x + j

y) e

jα

j(ωt−kz)

(2.97)

H = H

+ H

CCW

(

y+j

x) e

jα

CCW

(

y − j

x) e

jα

j(ωt−kz)

(2.98)

where α

and α

denote the phase angles of the CW and CCW waves, re-

spectively. The phase diﬀerence is

∆α = α

− α

. (2.99)

Expressions (2.97) and (2.98) represent the ﬁeld of an arbitrary elliptically

polarized wave, see Fig. 2.7. The principle semi-axes a and b are given by

a = E

+ E

CCW

, b =

− E

CCW

. (2.100)

The orientation angle ψ and the elliptic angle χ are given by

ψ =

+ α

, χ = arctan

= arctan

− E

CCW

+ E

CCW

. (2.101)

(2) Special Cases

(1) E

= E

CCW

, then b = 0 and χ = 0 and the wave is linearly polarized.

(2) E

CCW

= 0, then a = b, E = E

, and a clockwise circularly polarized

74 2. Introduction to Waves

wave results.

(3) E

= 0, then a = b, E = E

CCW

, and a counterclockwise circularly

polarized wave results.

(4) α

= −α

, then ψ = 0, to give an ortho-elliptically polarized wave.

In conclusion, we have three sets of parameters for describing the state of

polarization of a wave:

(1) The amplitudes of two mutually perpendicular linearly polarized waves

, E

y m

, and their phase diﬀerence ∆;

(2) The amplitudes of two opposite circularly polarized waves E

, E

CCW

and their phases α

, α

;

(3) Two principle semi-axes a, b, and the orientation angle ψ.

The ﬁrst and the second sets of parameters include phases, which are diﬃ-

cult to measure experimentally. On the contrary, the third set of parameters

is easy to measure. A rotary polarization analyzer and a detector can detect

the maximum and the minimum of the wave intensity, a

and b

, and the

orientation angle of the maximum intensity, ψ.

2.2.3 Stokes Parameters and the Poincar´e Sphere

Deﬁne four parameters S

, S

, and S

as follows

= E

− E

y m

= 4E

CCW

cos ∆α, (2.102)

= 2E

y m

cos ∆ = 4E

CCW

sin ∆α, (2.103)

= 2E

y m

sin ∆ = 2

−

CCW

, (2.104)

= E

+ E

y m

= 2

CCW

+ S

. (2.105)

These parameters are known as the Stokes parameters, and only three

of them are independent. The relations between the Stokes parameters and

parameters a, b, ψ, and χ are given by

= S

cos 2χ cos 2ψ, (2.106)

= S

cos 2χ sin 2ψ, (2.107)

= S

sin 2χ, (2.108)

= a

+ b

, (2.109)

where χ is the elliptic angle given in (2.101).

It can be seen from the above relations that S

, S

and S

are the Carte-

sian coordinates of a point on the spherical surface of radius S

. This sphere

is known as the Poincar´e sphere. The radius of the sphere represents the

intensity of the wave, i.e., the square of the ﬁeld strength, and each point

on the sphere surface corresponds to a state of polarization. The polar axis

of the sphere is in the direction of S

, and the equatorial plane is the S

-S

plane, see Fig. 2.8(a).

2.2 Polarization of Plane Waves 75

Figure 2.8: (a) Poincar´e sphere and (b) its expanded view.

The points on the equator of the Poincar´e sphere, where S

= 0, i.e., χ = 0

and b = 0, represent linearly polarized waves. The north and south poles,

= S

= 0, i.e., 2χ = ±π/2, b = a, represent circularly polarized waves.

The north pole, where S

is positive, 2χ = +π/2, ∆ = +π/2, represents a

CCW or left-handed circularly polarized wave, and the south pole, where S

negative, 2χ = −π/2, ∆ = −π/2, represents a CW or right-handed circularly

polarized wave. Points on the north semi-sphere, except at the equator and

north pole, represent a CCW elliptically polarized wave and points on the

south semi-sphere, except at the equator and south pole, represent a CW

elliptically polarized wave. The latitude represents the ratio of the major to

minor axes and the longitude represents the orientation, see Fig. 2.8(b).

As we already know, the parameters a, b, and ψ are easy to measure, so

the Stokes parameters and the coordinates on the Poincar´e sphere can be

determined experimentally by means of (2.106)–(2.109) [11, 58].

76 2. Introduction to Waves

2.2.4 The Degree of Polarization

A steady-state sinusoidal traveling wave with a certain frequency is called a

monochromatic wave. Monochromatic waves are necessarily polarized waves.

An arbitrary time-dependent ﬁeld in a linear system can be treated

by a superposition of sinusoidal ﬁelds with diﬀerent frequencies. This is

done by means of the Fourier transform. Thus an arbitrary wave can be

composed of mono chromatic waves with diﬀerent frequencies, so-called non-

monochromatic waves. The frequency distribution of a non-monochromatic

wave is known as the spectrum of the wave. For periodic waves, the spec-

trum is discrete and for the aperiodic waves, the spectrum is continuous. It is

usual to speak of a wave that includes a range of frequencies of the spectrum

which is very small compared with the center frequency of the spectrum as

a quasi-monochromatic wave.

The monochromatic wave and the combination of monochromatic waves

with ﬁnite spectrum are polarized waves.

The non-monochromatic waves are not necessarily polarized waves.

(1) Non-polarized wave. The amplitude, frequency, phase, and the orienta-

tion of the ﬁeld vectors of a non-p olarized wave are random. Natural light

including solar light is an example of a non-polarized wave.

(2) Quasi-polarized wave. The ﬁeld vector of a quasi-polarized wave is po-

larized, but its amplitude, frequency, and phase are random. Natural light

passing through a polarizer becomes quasi-polarized.

(3) Partially polarized wave. The parameter to evaluate the partially polar-

ized wave is the degree of polarization, which denotes the ratio of the wave

intensities of the polarized and non-polarized components, i.e., the square of

the ratio of the ﬁelds.

2.3 Normal Reﬂection and Transmission of

Plane Waves

An incident electromagnetic wave passing through an boundary surface of

diﬀerent media usually gives rise to both a reﬂected wave and a transmitted

wave. The reason is that the boundary conditions cannot be satisﬁed by the

ﬁelds of a single traveling wave. The composed ﬁelds of the incident, reﬂected

and transmitted waves have to satisfy the boundary conditions.

Firstly, we begin with the simplest example, a uniform plane wave nor-

mally incident to a metal surface, i.e., approximately a perfect-conductor

surface or so called short-circuit surface from a nonconducting medium.

2.3 Normal Reﬂection and Transmission of Plane Waves 77

Figure 2.9: Normal incidence and reﬂection of uniform plane wave at a

perfect-conductor surface.

2.3.1 Normal Incidence and Reﬂection at a

Perfect-Conductor Surface, Standing Waves

Suppose there is a uniform plane wave incident from medium 1 through a

plane boundary into medium 2. When medium 2 is a perfect conductor, there

is no transmitted wave in it, because both the electric ﬁeld and the magnetic

ﬁeld must vanish in a perfect conductor.

The ﬁelds of the linearly polarized incident plane wave propagating along

−x and the reﬂected wave along +x, refer to Fig. 2.9, are given by

y m

jkx

jωt

, (2.110)

y m

−jkx

jωt

, (2.111)

jkx

jωt

= −

y m

jkx

jωt

, (2.112)

−jkx

jωt

y m

−jkx

jωt

, (2.113)

where k =

√

µ² is the phase coeﬃcient and η =

µ/² is the wave impedance

of plane waves in the nonconducting medium 1.

The composed ﬁelds can be obtained by adding the incident and the

reﬂected ﬁelds:

= E

+ E

= (E

y m

jkx

+ E

y m

−jkx

jωt

, (2.114)

= H

+ H

−

y m

jkx

y m

−jkx

jωt

. (2.115)

The tangential component of the composed electric ﬁeld must be zero on

the plane x = 0 to satisfy the boundary condition of the perfect-conductor

surface.

x=0

= 0, E

y m

+ E

y m

= 0.

78 2. Introduction to Waves

The reﬂection coeﬃcients of electric ﬁeld and magnetic ﬁeld are given by

y m

= −1, (2.116)

= −

y m

= +1. (2.117)

This means that total reﬂection occurs on the surface of a perfect-conductor

or short-circuit surface.

The composed ﬁelds (2.114) and (2.115) become

= E

y m

jkx

+ e

−jkx

jωt

= 2jE

y m

sin kxe

jωt

= −

y m

jkx

+ e

−jkx

jωt

= −2

y m

cos kxe

jωt

Let

= 2jE

y m

the composed ﬁelds can be rewritten as

= E

sin kxe

jωt

, (2.118)

= j

cos kxe

jωt

. (2.119)

The corresponding instantaneous values of the ﬁelds are

= =E

= =(E

sin kxe

jωt

) = E

sin kx sin ωt, (2.120)

= =H

= =

cos kxe

jωt

cos kx cos ωt. (2.121)

Electric and magnetic ﬁelds in the form of (2.118) and (2.119) or (2.120)

and (2.121) are standing waves, resulting from the combination of incident

and reﬂected waves with the same amplitudes. The standing wave does not

travel in any direction, it is just an oscillation with a sinusoidal amplitude

distribution shown in Fig. 2.10.

Deﬁne the ratio of the complex amplitude of the electric ﬁeld and that of

the magnetic ﬁeld at an arbitrary cross-section x as the input impedance, or

simply, impedance Z(x). From (2.118) and (2.119), we have

Z(x) =

(x)

= jη tan kx = jX(x). (2.122)

The impedance of a standing wave is reactance, see Fig. 2.11.

We can see from Fig. 2.10 and Fig. 2.11 that when kx = nπ, n = 0, 1, 2, ···

or x = n(λ/2), the amplitude of the electric ﬁeld is zero and the amplitude

2.3 Normal Reﬂection and Transmission of Plane Waves 79

Figure 2.10: Electric and magnetic ﬁelds of a standing wave.

of magnetic ﬁeld reaches its maximum, consequently, the input impedance

is zero, i.e., the eﬀective short-circuit. When kx = (n + 1/2)π or x = (n +

1/2)(λ/2), the amplitude of the electric ﬁeld reaches its maximum and the

amplitude of magnetic ﬁeld is zero, and the input impedance reaches inﬁnity,

i.e., the eﬀective open-circuit. The distance between the short-circuit and

the open-circuit is λ/4, while the distance between two neighboring short-

circuit or two neighboring open-circuit is λ/2. The impedance between the

short-circuit and the open-circuit is an inductance or a capacitance.

The complex Poynting vector of the standing wave is imaginary,

S =

E × H

∗

2η

sin kx cos kx

. (2.123)

It means that there is only reactive power ﬂow oscillation and no active power

transmission in the standing wave.

For a clockwise (CW) circularly polarized incident wave along −x, the

incident electric ﬁeld and the reﬂected electric ﬁeld (Γ

= −1) are

= (

y + j

z)E

jkx

jωt

= −(

y + j

z)E

−jkx

jωt

The reﬂected wave is a counterclockwise (CCW) circular polarized wave along

+x.

80 2. Introduction to Waves

Figure 2.11: The input reactance of a standing wave.

The composed ﬁeld is

E = E

+ E

= (

y + j

z)E

jkx

− e

−jkx

jωt

= (

y + j

z)2jE

sin kx e

jωt

= (

y + j

z)E

sin kx e

jωt

. (2.124)

The composed electric ﬁeld is a rotation vector with its amplitude sinusoidally

distributed along x, i.e., a circularly polarized standing wave. If the incident

wave is elliptically polarized, the composed ﬁeld will be an elliptically polar-

ized standing wave.

2.3.2 Normal Incidence, Reﬂection and Transmission at

Nonconducting Dielectric boundary,

Traveling-Standing Waves

Consider a uniform plane wave normally incident from medium 1 through a

plane boundary into medium 2 and the two media are nonconductive with

diﬀerent constitutional parameters. There are incident wave, reﬂected wave

in medium 1 and refracted wave in medium 2, see Fig. 2.12.

The ﬁelds of incident and reﬂected waves in medium 1 are given by

y m

jωt

, (2.125)

y m

−jk

jωt

, (2.126)