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

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2.3 Normal Reﬂection and Transmission of Plane Waves 81

Figure 2.12: Normal incidence, reﬂection and transmission of uniform plane

wave at a nonconducting media boundary.

= −

y m

jωt

, (2.127)

y m

−jk

jωt

. (2.128)

The composed ﬁelds in medium 1 are given by the combination of the incident

and the reﬂected ﬁles,

y 1

y(E

+ E

) =

y m

+ E

y m

−jk

jωt

, (2.129)

z(H

+ H

) =

−

y m

−jk

jωt

. (2.130)

The ﬁelds in medium 2 are transmitted ﬁelds only,

y 2

= E

y m

jωt

, (2.131)

= H

= −

y m

jωt

. (2.132)

In the above equations, k

= ω

√

and k

= ω

√

denote the phase

coeﬃcient of plane wave in media 1 and 2, and η

/²

and η

/²

denote the wave impedances of media 1 and 2, respectively.

On the boundary of the two media, there is no surface current, which

means that neither of the two media is a perfect conductor. Both the tangen-

tial component of the composed electric ﬁeld and the tangential component

of the composed magnetic ﬁelds must be continuous at the boundary between

medium 1 and medium 2. The boundary equations are given by

n × E

x=0

= n × E

x=0

, i.e., E

y m

+ E

y m

= E

y m

, (2.133)

n × H

x=0

= n × H

x=0

, i.e., −

y m

= −

y m

. (2.134)

82 2. Introduction to Waves

Deﬁne a reﬂection coeﬃcient of electric ﬁeld that E

y m

= Γ E

y m

, from

(2.133) and (2.134), we have

Γ = |Γ |e

jφ

y m

− η

+ η

. (2.135)

The reﬂection coeﬃcient Γ is real when medium 1 and medium 2 are b oth

lossless media, while it become imaginary when medium 2 is lossy medium

and medium 1 is still lossless.

The composed ﬁelds in medium 1 become

y 1

y m

+Γ e

−jk

jωt

y m

1+|Γ |e

−j(φ−2k

jωt

, (2.136)

=−

y m

−Γ e

−jk

jωt

=−

y m

1−|Γ |e

−j(φ−2k

jωt

. (2.137)

The amplitudes of the ﬁelds are

y 1

| = E

y m

1 + |Γ |

+ 2|Γ |cos(φ − 2k

x) , (2.138)

| =

y m

1 + |Γ |

− 2|Γ |cos(φ − 2k

x) . (2.139)

The ﬁeld amplitude distribution along x is shown in Fig. 2.13.

We can see from equations (2.138), (2.139) and Fig. 2.13 that, the ﬁeld

maximum and ﬁeld minimum appear alternatively. The distance between the

maximum and the minimum is λ/4, while the distance between two neigh-

boring maximum or two neighboring minimum is λ/2. The ﬁeld minimum

is no longer zero as it is for the standing wave. This kind of wave is called

traveling-standing wave.

The ﬁeld maximum and minimum are

max

= E

y m

(1 + |Γ |), E

min

= E

y m

(1 − |Γ |).

Equations (2.138), (2.139) can be rewritten as follows,

y 1

= E

y m

+ Γ e

−jk

jωt

= 2Γ E

y m

cos k

x e

jωt

+ (1 − Γ )E

y m

j(ωt+k

, (2.140)

= −

y m

− Γ e

−jk

jωt

= −j

2Γ E

y m

sin k

x e

jωt

− (1 − Γ )

y m

j(ωt+k

. (2.141)

It shows that a traveling-standing wave is the combination of a traveling wave

and a standing wave.

2.3 Normal Reﬂection and Transmission of Plane Waves 83

0.00 .25 .50 .75 1.00 1.25 1.50 1.75 2.00

0.0

1.0

1.2

1.4

1.6

1.8

2.0

*

= 0.6

Figure 2.13: Field amplitude distribution of a traveling-standing wave.

Deﬁne the ratio of E

max

to E

min

as the standing-wave ratio, or SWR for

short, denoted by

ρ = SWR =

max

min

1 + |Γ |

1 − |Γ |

or |Γ | =

ρ − 1

ρ + 1

. (2.142)

The state of |Γ | = 0 and ρ = 1 corresponds to non-reﬂection or matching i.e.,

a traveling wave, and the state of |Γ | = 1 and ρ → ∞ corresponds to total

reﬂection, i.e., standing wave.

Usually the standing wave and the traveling-standing wave are together

denoted as the standing wave with speciﬁc standing-wave ratio.

The input impedance for a traveling-standing wave is given by the ratio

of (2.136) to (2.137),

Z(x) =

y 1

= η

+ Γ e

−jk

− Γ e

−jk

. (2.143)

Using (2.135), yields

Z(x) = η

+ jη

tan k

+ jη

tan k

. (2.144)

We can see from (2.135), (2.136), (2.137) and (2.144) that, for a plane

wave incident from a rare medium into a dense medium with permittivity

much larger then that of the rare medium, ²

À ²

, η

¿ η

, it leads to

Γ ≈ −1, E

y 1

≈ 0, and Z(0) ≈ 0, i.e., the tangential component of electric

ﬁeld is vanished. the boundary corresponds to a short-circuit boundary. On

the contrary, when ²

¿ ²

, η

À η

, it leads to Γ ≈ +1, H

≈ 0 and

Z(0) ≈ ∞, the boundary corresponds to an open-circuit boundary. Generally,

it corresponds to an impedance boundary.

84 2. Introduction to Waves

Figure 2.14: Incident, reﬂected, and refracted wave vectors.

2.4 Oblique Reﬂection and Refraction of

Plane Waves

A uniform plans wave incident obliquely into the boundary between two

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

Generally, the transmitted wave does not propagate in the same direction as

that of the incident wave, and is called the refracted wave. The laws governing

the reﬂection and refraction of plane waves are Snell’s law for the directions

of propagation and the Fresenel’s law for the amplitudes and phases of the

waves.

2.4.1 Snell’s Law

Consider an incident uniform plane wave obliquely passing through a plane

boundary between two media. According to the formulation of a plane wave

propagating in an arbitrary direction, (2.24), the electric ﬁeld vectors of the

incident, reﬂected, and transmitted waves are

(x, t) = E

j(ω

t−k

·x)

, (2.145)

(x, t) = E

j(ω

t−k

·x)

, (2.146)

(x, t) = E

j(ω

t−k

·x)

, (2.147)

where the subscripts i, r, and t denote the quantities of the incident, reﬂected,

and transmitted waves, respectively, refer to Fig. 2.14.

Suppose the boundary between medium 1 and medium 2 is placed on the

plane x = 0, and the unit normal directed from 2 to 1 is n. The boundary

condition for the tangential components of the electric ﬁelds is given by

n ×

j(ω

t−k

·x)

+ E

j(ω

t−k

·x)

x=0

= n × E

j(ω

t−k

·x)

x=0

2.4 Oblique Reﬂection and Refraction of Plane Waves 85

This boundary condition must be satisﬁed at all points on the boundary

plane at all time. It implies that the time and spatial variations of all ﬁelds

must be the same at x = 0:

= ω

= ω and k

· x|

x=0

= k

· x|

x=0

= k

· x|

x=0

i.e.,

y + k

z = k

y + k

z = k

y + k

z, at x = 0. (2.148)

Place the coordinates such that the incident wave vector is on the x,z

plane. Then we have

= 0.

The equation (2.148) must be satisﬁed at all y and z, which yields

= 0, k

= 0.

The incident, reﬂected, and refracted wave vectors are coplanar vectors, laid

on the plane of incidence deﬁned by the boundary normal n and the incident

wave vector k

, i.e., the x,z plane. Then the equation (2.148) becomes

= k

. (2.149)

The three wave vectors can then be written in component forms:

= −

sin θ

cos θ

, (2.150)

sin θ

cos θ

, (2.151)

= −

sin θ

cos θ

. (2.152)

where θ

, θ

, and θ

denote the angles of incidence, reﬂection, and refraction,

respectively, see Fig. 2.14.

The incident and reﬂected waves propagate in medium 1 and the refracted

wave propagates in medium 2. The phase coeﬃcients of the plane waves are

= k

= ω

√

, k

= k

= ω

√

. (2.153)

Substituting (2.150)–(2.153) into (2.149) yields

= θ

, (2.154)

i.e., the angle of reﬂection is equal to the angle of incidence, known as the

law of reﬂection, and

sin θ

√

, (2.155)

86 2. Introduction to Waves

i.e., the sine of the angle of refraction and the sine of the angle of incidence

is proportional to the phase velocity of plane wave in the media, known as

the law of refraction.

Deﬁne the index of refraction or simply the index of a medium as the

ratio of the phase velocity of plane wave in vacuum to the phase velocity in

the medium:

n =

√

. (2.156)

The indices of medium 1 and medium 2 are

√

and n

√

respectively. The relative index of medium 2 to medium 1 is

. (2.157)

Then (2.155) becomes

sin θ

= n

. (2.158)

In conclusion, for a uniform plane wave incident obliquely from medium 1

into medium 2, the incident, reﬂected, and refracted wave vectors are copla-

nar, the angle of reﬂection is equal to the angle of incidence, and the ratio

of the sine of the angle of incidence to the sine of the angle of refraction is

equal to the ratio of the phase velocities of uniform plane waves in the two

media, or the inverse ratio of the indices of the two media. This is known as

Snell’s law.

2.4.2 Oblique Incidence and Reﬂection at a

Perfect-Conductor Surface

For the reﬂection and refraction of waves, the relations among the directions

of propagation, on the one hand, follow from the wave nature of the phenom-

ena but do not depend on the detailed nature of the ﬁelds and their boundary

conditions. On the other hand, the relations among the intensities and phases

of the waves depend entirely on the speciﬁc nature of electromagnetic ﬁelds

and their boundary conditions.

For a plane wave obliquely incident upon a plane boundary, the boundary

conditions for waves with diﬀerent polarization states are diﬀerent. In section

2.2.1, we have shown that a plane wave with an arbitrary polarization state

can be decomposed into two mutually perpendicular line polarized waves. It

is convenient to separate a wave into two modes, the mode with its electric

ﬁeld normal to the plane of incidence, called n wave, and the one with its

electric ﬁeld parallel to the plane of incidence, called p wave. For the n wave,

the electric ﬁeld vector has only the component parallel to the boundary

2.4 Oblique Reﬂection and Refraction of Plane Waves 87

Figure 2.15: Oblique incidence and reﬂection at a p erfect-conductor surface,

(a) n wave and (b) p wave.

plane, which is the transverse component with respect to the wave vectors k

and k

, and is denoted as the TE mode. For the p wave, the magnetic ﬁeld

vector has only the transverse component, and is denoted as the TM mode

[38].

In this section, we suppose that the medium 2 is metal, and can be seen

as a perfect conductor, there is no refracted wave in it. The ﬁelds of the

incident wave and the reﬂected wave must satisfy the short-circuit bound-

ary conditions on the metal surface. The n wave and p wave incident and

reﬂected at the perfect-conductor surface are shown in Fig. 2.15(a) and (b),

respectively.

(1) The n Wave or TE Mode

The electric ﬁeld of the incident n wave has only a y component, normal to

the x-z plane, the plane of incidence. To satisfy the boundary condition, the

electric ﬁelds of the reﬂected waves must also have only y components, refer

to Fig. 2.15(a). They can be given by (2.24) as follows,

(x) =

y m

−jk

·x

, (2.159)

(x) =

y m

−jk

·x

. (2.160)

The magnetic ﬁelds can be derived by (2.38):

(x) =

−jk

·x

×E

(x)

−

sin θ

y m

−

cos θ

y m

−jk

·x

, (2.161)

(x) = (

−jk

·x

×E

(x)

−

sin θ

y m

cos θ

y m

−jk

·x

, (2.162)

88 2. Introduction to Waves

where η =

µ/² denotes the wave impedances of the nonconducting medium,

in which the incident wave propagates.

The boundary equation on the plane surface of a perfect conductor at

x = 0 is given by

n ×

+ E

x=0

= 0, i.e., E

y m

+ E

y m

= 0. (2.163)

We then have the coeﬃcient of reﬂection for the electric ﬁeld of the n wave:

y m

= −1. (2.164)

Total reﬂection occurs on the surface of a perfect conductor for oblique inci-

dence and reﬂection of a n wave.

Applying the above relations (2.164) in the electric ﬁeld expressions

(2.159) and (2.160) we have the composed electric ﬁeld:

= E

+ E

= E

y m

− e

−jk

j(ωt−k

= 2jE

y m

sin k

j(ωt−k

Let

= 2jE

y m

yields

= E

sin k

j(ωt−k

, (2.165)

The composed magnetic ﬁeld components are derived from (2.161),

(2.162), and (2.164) as follows:

= −

sin θ

sin k

j(ωt−k

, (2.166)

= j

cos θ

cos k

j(ωt−k

. (2.167)

We can see from (2.165)–(2.167) that the composed ﬁeld of the incident

and the reﬂected waves outside the perfect conductor forms a traveling wave

in the z direction and a standing wave in the x direction. The equiphase

is a plane perpendicular to z, i.e., the x-y plane, but the amplitude in this

plane is not uniform, so it is a nonuniform plane wave in the z direction.

The electric ﬁeld of the wave has only a transverse (y) component and is

normal to the direction of propagation z, but the magnetic ﬁeld has both a

transverse (x) and a longitudinal (z) component. This type of traveling-wave

mode (n wave) is called the transverse electric mo de and is denoted by TE

mode. The electric and magnetic ﬁeld lines of the n wave (TE mode) are

shown in Fig. 2.16.

The phase velocity and the wavelength in longitudinal (z) direction are

k sin θ

sin θ

, λ

2π

k sin θ

sin θ

, (2.168)

2.4 Oblique Reﬂection and Refraction of Plane Waves 89

Figure 2.16: Field maps of the n wave (TE mode) for the total reﬂection

from the surface of a perfect conductor.

and the wavelength in transverse (x) direction is

2π

k cos θ

cos θ

, (2.169)

where λ

denotes the wavelength in free space.

We can see that, λ

> λ

and v

> c. The phase velocity is larger than

the velocity of light in free space, so it is a fast wave. For a fast wave, the

distribution of ﬁelds in the transverse direction must be a standing wave.

The planes at x = nλ

/2, n = 1, 2, 3, ···, are equivalent short-circuit

planes, and the planes at x = (n + 1/2)λ

/2 are equivalent open-circuit

planes. If we put a perfect-conductor plane at x = nλ

/2, it has no inﬂuence

on the wave propagation, because the tangential electric ﬁeld component is

zero at those planes. It forms the TE

modes in a parallel-plate transmission

line.

We can also put two parallel p erfect-conductor planes perpendicular to

y, and they have no inﬂuence on the wave propagation, because the electric

ﬁeld is normal to those planes. It forms the TE

modes in a rectangular

waveguide.

(2) The p Wave or TM Mode

The magnetic ﬁelds of the incident and reﬂected p waves have only a y

component, refer to Fig. 2.15(b), and can be given by

(x) =

y m

−jk

·x

, (2.170)

90 2. Introduction to Waves

(x) =

y m

−jk

·x

, (2.171)

The electric ﬁelds can be given by (2.39):

(x) =

−jk

·x

=−η

×H

(x)

xη sin θ

y m

zη cos θ

y m

−jk

·x

, (2.172)

(x) = (

) e

−jk

·x

=−η

×H

(x)

−

xη sin θ

y m

zη cos θ

y m

−jk

·x

, (2.173)

The boundary equation on the plane surface of a perfect conductor at

x = 0 becomes

n ×

+ E

x=0

= 0, i.e., E

− E

= 0. (2.174)

We then have the coeﬃcient of reﬂection for electric ﬁeld of the p wave:

= +1. (2.175)

Total reﬂection occurs on the surface of a perfect conductor for a p wave

also, but the reﬂection coeﬃcient is +1 for p wave instead of −1 for n wave.

Then we have the composed magnetic ﬁeld:

= H

y m

−jk

j(ωt−k

= 2H

y m

cos k

j(ωt−k

Let

= 2H

y m

yields

= H

cos k

j(ωt−k

, (2.176)

= η

sin θ

cos k

j(ωt−k

, (2.177)

= jη

cos θ

sin k

j(ωt−k

. (2.178)

The composed ﬁeld of the incident and the reﬂected waves outside the

perfect conductor is also a nonuniform plane wave in the z direction. The

magnetic ﬁeld of the wave has only a transverse (y) component and is normal

to the direction of propagation z. The electric ﬁeld has transverse (x) and

longitudinal (z) components. This type of traveling-wave mode (p wave) is

called the transverse magnetic mode, denoted by TM mode. The electric and

magnetic ﬁeld lines of the p wave (TM mode) are shown in Fig. 2.17.

The planes at x = nλ

/2, n integer, are equivalent short-circuit planes,

and the planes at x = (n +1/2)λ

/2 are equivalent open-circuit planes. If we

put a perfect-conductor plane at x = nλ

/2, it has no inﬂuence on the wave

propagation, because the tangential electric ﬁeld component is zero at those

planes. It forms the TM

modes in a parallel-plate transmission line.