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

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2.4 Oblique Reﬂection and Refraction of Plane Waves 101

Figure 2.20: Field maps of (a) n wave (TE) and (b) p wave (TM) for total

internal reﬂection.

102 2. Introduction to Waves

Figure 2.21: The phase shift of the ﬁeld in total internal reﬂection at the

dielectric boundary.

2.4.7 The Goos–H¨anchen Shift

Comparing the ﬁeld expressions in medium 1 for total internal reﬂection of a

n wave (TE) at the dielectric boundary, (2.231)–(2.233), with those of total

reﬂection at the perfect conductor surface, (2.165)–(2.167), we can see that

both of them are standing waves in the direction normal to the boundary

and traveling waves in the direction parallel to the boundary. In the case of

total reﬂection at the perfectly conductor surface, the null of the standing

wave of E is just on the boundary, but in the case of total internal reﬂection

at the dielectric boundary, the null of the standing wave of E extends into

the region of medium 2, see Fig. 2.21(a).

Kapany and Burke [48] propose that total reﬂection does not occur at the

material boundary but takes place slightly inside the lower-index medium as

shown in Fig. 2.21(b). This model is consistent with the result of experimen-

tal observation that a thin light beam is shifted laterally on total reﬂection.

This shift is known as the Goos–H¨anchen shift, after its ﬁrst observers [48, 69].

2.4.8 Reﬂection Coeﬃcients at Dielectric Boundary

As a conclusion, we will investigate the reﬂection coeﬃcients at dielectric

boundary with respect to the angle of incidence.

(1) Medium 2 is Optically Denser than Medium 1, n

> n

The magnitude and the phase angle of the reﬂection coeﬃcients with respect

to the angle of incidence for n

> n

are shown in Fig. 2.22(a).

The phase angle of the reﬂection coeﬃcient for the n wave (TE) is always

π. On the other hand, the phase angle of the reﬂection coeﬃcient for the p

wave (TM) is zero when the angle of incidence is less than the Brewster angle,

2.4 Oblique Reﬂection and Refraction of Plane Waves 103

0 10 20 30 40 50 60 70 80 90

-1.0

-.9

-.8

-.7

-.6

-.5

-.4

-.3

-.2

-.1

0.0

p(TM)

n(TE)

0 10 20 30 40 50 60 70 80 90

-.2

-.1

0.0

1.0

p(TM)

n(TE)

0 10 20 30 40 50 60 70 80 90

135

180

0 10 20 30 40 50 60 70 80 90

0.0

1.0

0 10 20 30 40 50 60 70 80 90

-180

-135

-90

-45

135

180

arg(

*

)

0 10 20 30 40 50 60 70 80 90

0.0

1.0

n(TE)

p(TM)

*

n(TE)

p(TM)

n(TE)

*

n(TE)

p(TM)

=1.5/1

p(TM)

n(TE)

p(TM)

arg(

*

)

(a) n

> n

(b) n

< n

p(TM)

5.11

15.1

Figure 2.22: Magnitude and phase angle of the reﬂection coeﬃcients at a

dielectric boundary for n

> n

(a) and n

< n

(b).

104 2. Introduction to Waves

< θ

, and π when the angle of incidence is larger than the Brewster angle,

> θ

. At the Brewster angle, the magnitude of the reﬂection coeﬃcient

for the p wave (TM) is zero and the phase angle changes from 0 to π.

The reﬂection coeﬃcients at θ

= 90

◦

for both modes are unity, because

no wave is transmitted into medium 2. The reﬂection coeﬃcients at θ

= 0

for the n wave (TE) and p wave (TM) are equal to each other and equal to

the reﬂection coeﬃcient of normal incidence.

(2) Medium 2 is Optically Rarer than Medium 1, n

< n

The magnitude and the phase angle of the reﬂection coeﬃcients with respect

to the angle of incidence for n

< n

are shown in Fig. 2.22(b). As θ

may

be larger than θ

for n

< n

, the curves shown on the left-hand side of

this ﬁgure are diﬀerent from those shown on the left. The magnitudes of the

reﬂection coeﬃcients for both modes are unity when the angle of incidence is

larger than the critical angle, and the phase angle change from 0 to −π; the

corresponding φ and ψ change from 0 to π/2.

In general, when a plane wave is obliquely incident upon a boundary, the

polarization states of the reﬂected and the refracted waves will be diﬀerent

from that of the incident wave, because the reﬂection co eﬃcient and the

refraction co eﬃcient are diﬀerent in magnitude or in phase for n wave (TE)

and p wave (TM).

2.4.9 Reﬂection and Transmission of Plane Waves at

the Boundary Between Lossless and Lossy Media

In a lossy medium, the permittivity and the angular wave number of the

plane wave become complex:

˙² = ²

− j

k = ω

− j

´i

= β − jα,

where the expressions of β and α are found in (2.54) and (2.55), respectively.

The phase velocity of the plane wave and the index of refraction for the

lossy medium are also complex and can be given by

˙v

β − jα

µ [²

− j (²

+ σ/ω)]

, (2.234)

˙n =

˙v

k =

(β − jα) = n

− jn

, (2.235)

where n

denotes the refractive index and n

denotes the absorption index of

the medium. Using (2.60) and (2.61), we have

= c

√

µ²

(

1 +

(²

+ σ/ω)

+ 1

1/2

, (2.236)

2.4 Oblique Reﬂection and Refraction of Plane Waves 105

= c

√

µ²

(

1 +

(²

+ σ/ω)

− 1

1/2

. (2.237)

The wave impedance of the lossy medium also becomes complex, see (2.56).

Substituting the above complex index and complex wave impedance into

the expressions for Snell’s law and Fersnel’s formulas, we have the relation

for the directions for wave vectors and the relation of the amplitudes of the

reﬂected, refracted, and incident waves when a plane wave is incident on the

surface of lossy media or on the boundary between lossy media [11, 96].

For the reﬂection and refraction of plane wave on the boundary between

two lossy media, please refer to [11].

(1) Boundary Between a Lossless and a Lossy Media

We are investigating the oblique incidence of a plane wave passing from a

lossless medium, through the boundary, into a lossy medium [5, 11]. The

index of medium 1 is real, n

, whereas the index of medium 2 is complex, ˙n

According to Snell’s law (2.158), we have the sine of the refraction angle,

sin θ

˙n

sin θ

− jn

sin θ

− jα

sin θ

. (2.238)

Hence sin θ

and θ

become complex. Then cos θ

becomes

cos θ

1 − sin

1 −

− jn

sin

1 −

− n

+ n

sin

− j

+ n

sin

. (2.239)

So cos θ

is also complex and can be written as

cos θ

= se

−jξ

= s(cos ξ − j sin ξ). (2.240)

The relation between (2.239) and (2.240) gives

cos 2ξ =1−

− n

+ n

sin

, s

sin 2ξ =

+ n

sin

. (2.241)

Applying the above results to (2.181) or (2.199), we have the expression

for the refracted electric ﬁeld:

(x, t) = E

exp{j[ωt −

(−x cos θ

+ z sin θ

)]}

= E

exp(−jk

sin θ

z) exp[j(β

− jα

)s(cos ξ − j sin ξ)z]e

jωt

= E

exp[s(α

cos ξ + β

sin ξ)x]

×exp{j[ωt + s(β

cos ξ − α

sin ξ)x − k

sin θ

z]}

= E

exp[j(ωt + q

x − q

z)], (2.242)

106 2. Introduction to Waves

Figure 2.23: Reﬂection and refraction of a plane wave at the boundary be-

tween a lossless and a lossy medium.

where

p = s(α

cos ξ + β

sin ξ), q

= s(β

cos ξ − α

sin ξ), q

= k

sin θ

The refracted wave becomes a nonuniform plane wave, refer to Fig. 2.23.

The amplitude of the wave attenuates along the −x direction, due to the

conductive and polarization loss, so the equiamplitude is the plane x = const,

but the equiphase is the plane with q

x + q

z = const. The direction of

the wave vector is normal to the equiphase, so the true angle of refraction,

denoted by θ

, becomes

sin θ

+ q

sin θ

sin

+ s

(β

cos ξ − α

sin ξ)

. (2.243)

Snell’s law becomes

sin θ

= n

21e

where

sin

+ s

cos ξ − n

sin ξ)

. (2.244)

The eﬀective refractive index n

is not equal to n

, the real part of the

complex refractive index of medium 2.

(2) Boundary Between a Lossless Medium and a Good Conductor

If medium 2 is a good conductor, σ

À ω²

, from (2.66), we have, β

≈ α

. According to (2.238) and (2.243), we have, sin θ

≈ 0, and sin θ

≈ 0.

This means that the wave transmitted into the good conductor is always

approximately along the direction normal to the surface of the conductor,

2.5 Transformation of Impedance for Electromagnetic Waves 107

regardless of the angle of incidence, and is damped rapidly. So, for electro-

magnetic devices enclosed by good conductors with any geometry, the ﬁelds

inside the conductor can always be considered as damped plane waves normal

to the surface of the conductor, and the surface impedance (2.68) and power

loss (2.70) in good conductors for plane waves can be used in most of such

problems.

2.5 Transformation of Impedance for

Electromagnetic Waves

We are interested in the ﬁelds in medium 1 for an arbitrary incidence and re-

ﬂection of uniform plan wave on the boundary of the two media, see Fig. 2.24.

Generally, the wave in medium 1 is a traveling-standing wave in the direc-

tion normal to the boundary. The composed tangential electric and magnetic

ﬁelds in medium 1 for the n wave (TE mode) can be found by substituting

Γ E

y m

for E

y m

in (2.179), (2.180), (2.182), and (2.183):

= E

y m

−jk

·x

+ E

y m

−jk

·x

= E

y m

+ Γ

−jk

, (2.245)

=−

y m

−jk

·x

−E

y m

−jk

·x

=−

y m

−Γ

−jk

(2.246)

Similarly, for the p wave (TM mode), from (2.194), (2.195), (2.197) and

(2.198):

= H

y m

−jk

·x

+ H

y m

−jk

·x

= H

y m

− Γ

−jk

, (2.247)

y m

−jk

·x

−H

y m

−jk

·x

y m

+Γ

−jk

(2.248)

where k

= k

, k

= k

. It can be seen that the composed

ﬁeld is a traveling wave in the direction z, tangential to the boundary, and a

traveling-standing wave in x, the normal direction.

Similar to what we did in Section 2.3.2 for normal incidence, it is conve-

nient to use the concept of impedance at an arbitrary cross section to describe

the relation between incident and reﬂected waves. Deﬁne

Z(x) =

⊥

(x)

⊥

(x)

, (2.249)

where E

⊥

and H

⊥

denote the ﬁeld components normal to the direction of

propagation of the traveling-standing wave. For the wave in the −x direction,

they are the ﬁeld components tangential to the boundary between the media,

108 2. Introduction to Waves

Figure 2.24: Impedance transformation for electromagnetic waves.

i.e., E

⊥

= E

, H

⊥

= −H

for the n wave (TE), and E

⊥

= E

, H

⊥

= H

for

the p wave (TM). Then, from (2.245)–(2.248) we have

Z(x) = Z

1 + Γ (x)

1 − Γ (x)

, Γ(x) =

Z(x) − Z

Z(x) + Z

, (2.250)

where Z

is the normal wave impedance, i.e., wave impedance or character-

istic impedance for the n wave (TE) or p wave (TM) for oblique incidence

in medium 1, (2.190),

= Z

cos θ

, or Z

= Z

= η

cos θ

and Γ (x) is the reﬂection coeﬃcient in the section x,

Γ (x) = Γ e

−j2k

and Γ = |Γ |e

−jφ

, (2.251)

where Γ = Γ (0) is the complex reﬂection coeﬃcient at the boundary, Γ = Γ

or Γ = Γ

, and k

= k

cos θ

= k

cos θ

The relation between the reﬂection coeﬃcients at two sections x

and x

Γ (x

) = Γ (x

−j2k

−x

)

= Γ (x

−j2k

, (2.252)

where l = x

− x

, refer to Fig. 2.24. The transformation relation for the

impedances with two sections x

and x

becomes

Z(x

) = Z

Z(x

) + jZ

tan k

+ jZ(x

) tan k

. (2.253)

This is just the impedance transformation formula in transmission-line the-

ory.

The composed tangential electric and magnetic ﬁelds for n wave (2.245),

(2.246) and those for p wave (2.247), (2.248) can be rewrite as the general

form of traveling-standing waves in the direction x,

⊥

= E

+ Γ e

−jk

jωt

= E

1 + |Γ |e

j(φ−2k

j(ωt+k

(2.254)

2.6 Dielectric Layers and Impedance Transducers 109

⊥

− Γ e

−jk

jωt

1 − |Γ |e

j(φ−2k

j(ωt+k

(2.255)

The amplitudes of the electric and magnetic ﬁelds are:

⊥

| = E

1 + |Γ |

+ 2|Γ |cos(φ − 2k

x), (2.256)

⊥

| =

1 + |Γ |

− 2|Γ |cos(φ − 2k

x). (2.257)

In general, the ﬁelds are traveling-standing waves. If Γ = 0, they become

traveling waves and if Γ = ±1, they become standing waves.

For θ

= 0, the relations (2.250)–(2.257) reduce the corresponding formu-

las (2.135)–(2.139) in Section 2.3.2 for plane waves normally incident at the

boundary.

The transformation relations for the impedance and reﬂection coeﬃcients

of plane waves are totally consistent with those in transmission-line theory,

see Section 3.2.1.

2.6 Dielectric Layers and

Impedance Transducers

The characteristics of the reﬂection and transmission of electromagnetic

waves at the surface of multi-layer media are important in many applica-

tions, such as the anti-reﬂecting and highly reﬂecting coatings in optics and

the concealment of targets in radar technology. From the view of networks,

the multi-layer coating is equivalent to the impedance transducer consists of

multi-section transmission lines or waveguides with diﬀerent characteristic

impedances.

2.6.1 Single Dielectric Layer,

The λ/4 Impedance Transducer

The reﬂection of a plane wave from a boundary of two media with diﬀerent

wave impedance η

and η

may be eliminated by coating the b oundary with

a intermediate layer with wave impedance η and thickness l. The incident

plane wave from the input medium, with the angle of incidence θ

produces

transmitted waves in the intermediate medium and the output medium with

the angles of refraction θ and θ

, respectively, and produces reﬂection waves

at the boundaries, as illustrated in Fig. 2.25.

The input impedance at the surface of the output medium Z(x

) is the

normal wave impedance of the output medium, given by (2.190) for n wave

(TE) or (2.205) for p wave (TM):

Z(x

) = Z

cos θ

, or Z(x

) = Z

= η

cos θ

110 2. Introduction to Waves

Figure 2.25: Single dielectric layer.

The input impedance and the reﬂection coeﬃcient at the surface of the

intermediate medium Z(x

) and Γ (x

) is given by the impedance transfor-

mation relation (2.253) and (2.250)

Z(x

) = Z

Z(x

) + jZ

tan k

+ jZ(x

) tan k

, Γ (x

) =

Z(x

) − Z

Z(x

) + Z

where Z

denotes the normal wave impedance of the input medium and Z

denotes the normal wave impedance of the intermediate medium.

= Z

cos θ

, or Z

= Z

= η

cos θ

= Z

cos θ

, or Z

= Z

= η cos θ,

If we make the thickness of the intermediate medium equal to an odd

number of quarter normal wavelengths, so that

l = (2n + 1)

, i.e., l = (2n + 1)

= (2n + 1)

cos θ

, (2.258)

then the impedance transformation relation becomes

Z(x

) =

Z(x

)

, i.e.,

Z(x

)

Z(x

)/Z

, or Z

Z(x

)Z(x

(2.259)

The normalized impedance at x

, Z(x

)/Z

, and at x

, Z(x

)/Z

, are re-

ciprocal with each other, We come to the conclusion that the characteristic

impedance of a quarter wavelength intermediate layer is equal to the geomet-

rical mean of its input impedance and output impedance.

If we make the normal wave impedance of the intermediate medium equal

to the geometrical mean of the normal wave impedance of the input and

output media

= Z

, (2.260)