Lecture 16: Waves in Layered Media

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Lecture 16

Waves in Layered Media

16.1 Waves in Layered Media

Because of the homomorphism between the transmission line problem and the plane-wave

reflection by interfaces, we will exploit the simplicity of the transmission line theory to arrive

a formulas for plane wave reflection by layered media. This treatment is not found in any

other textbooks.

16.1.1 Generalized Reflection Coefficient for Layered Media

Because of the homomorphism between transmission line problems and plane waves in layered

medium problems, one can capitalize on using the multi-section transmission line formulas

for generalized reflection coefficient, which is

˜Γ12 = Γ12 + ˜Γ23e−2jβ2l2

1 + Γ12 ˜Γ23e−2jβ2l2

(16.1.1)

This reflection coefficient includes multiple reflections from the right of the 12 junction. It

can be used to study electromagnetic waves in layered media shown in Figures 16.1 and 16.2.

Using the result from the multi-junction transmission line, we can write down the gener-

alized reflection coefficient for a layered medium with an incident wave at the 12 interface,

including multiple reflections from the right. It is given by

˜ R12 = R12 + ˜ R23e−2jβ2z l2

1 + R12 ˜ R23e−2jβ2z 2l2

(16.1.2)

where l2 is now the thickness of the region 2. In the above, we assume that the wave is incident

from medium 1 which is semi-infinite, the generalized reflection coefficient above is defined at

the media 1 and 2 interface.1 It is assumed that there are multiple reflection coming from the

23 interface, so that the 23 reflection coefficient is the generalized reflection coefficient ˜ R23.

1We have borrowed Figure 16.1 from Kong’s book, where the first region is Region 0. But in our lecture,

the first region is Region 1.

151

152 Electromagnetic Field Theory

Figure 16.1: Figure for layered media borrowed from Kong’s book. Please note that in our

notes, the first region is Region 1. We shall also, replace x with z and vice versa (courtesy of

J.A. Kong, Electromagnetic Wave Theory).

Figure 16.2 shows the case of a normally incident wave into a layered media. For this

case, the wave impedance becomes the intrinsic impedance.

Waves in Layered Media 153

Figure 16.2: The equivalence of a layered medium problem to a transmission line prob-

lem. This equivalence is possible even for oblique incidence. For normal incidence, the wave

impedance becomes intrinsic impedances (courtesy of J.A. Kong, Electromagnetic Wave The-

ory).

We shall discuss finding guided waves in a layered medium next using the generalized

reflection coefficient. For a general guided wave along the longitudinal direction parallel to

the interfaces (x direction in our notation), the wave will propagate in the manner of

e−jβxx

For instance, the surface plasmon mode that we found previously can be thought of as a

wave propagating in the x direction. This wave has very interesting phase and group velocity.

Hence, it is prudent to understand phase and group velocity better before doing this.

154 Electromagnetic Field Theory

16.2 Phase Velocity and Group Velocity

Now that we know how a medium can be frequency dispersive in the Drude-Lorentz-Sommerfeld

(DLS) model, we are ready to distinguish the difference between the phase velocity and the

group velocity

16.2.1 Phase Velocity

The phase velocity is the velocity of the phase of a wave. It is only defined for a mono-

chromatic signal (also called time-harmonic, CW (constant wave), or sinusoidal signal) at one

given frequency. A sinusoidal wave signal, e.g., the voltage signal on a transmission line, can

take the form

V (z, t) = V0 cos(ωt − kz + α) (16.2.1)

This sinusoidal signal moves with a velocity

vph = ω

k (16.2.2)

where, for example, k = ω√με, inside a simple coax. Hence,

vph = 1/√με (16.2.3)

But a dielectric medium can be frequency dispersive, or ε(ω) is not a constant but a function

of ω as has been shown with the Drude-Lorentz-Sommerfeld model. Therefore, signals with

different ω’s will travel with different phase velocity.

More bizarre still, what if the coax is filled with a plasma medium where

ε = ε0

(

1 − ωp2

ω2

)

(16.2.4)

Then, ε < ε0 always meaning that the phase velocity given by (16.2.3) can be larger than

the velocity of light in vacuum (assuming μ = μ0). Also, ε = 0 when ω = ωp, implying that

k = 0; then in accordance to (16.2.2), vph = ∞. These ludicrous observations can be justified

or understood only if we can show that information can only be sent by using a wave packet.2

The same goes for energy which can only be sent by wave packets, but not by CW signal;

only in this manner can a finite amount of energy be sent. These wave packets can only travel

at the group velocity as shall be shown, which is always less than the velocity of light.

2In information theory, according to Shannon, the basic unit of information is a bit, which can only be

sent by a digital signal, or a wave packet.

Waves in Layered Media 155

16.2.2 Group Velocity

Figure 16.3: A Gaussian wave packet can be thought of as a linear superposition of monochro-

matic waves of slightly different frequencies. If one Fourier transforms the above signal, it

will be a narrow-band signal centered about certain ω0 (courtesy of Wikimedia [101]).

Now, consider a narrow band wave packet as shown in Figure 16.3. It cannot be mono-

chromatic, but can be written as a linear superposition of many frequencies. One way to

express this is to write this wave packet as an integral in terms of Fourier transform, or a

summation over many frequencies, namely

V (z, t) =

∞ ˆ

−∞

dωV (z, ω)ejωt (16.2.5)

Assume that V (z, t) is the solution to the dispersive transmission line equations with ε(ω),

then it can be shown that V (z, ω) is the solution to the one-dimensional Helmholtz equation3

dz2 V (z, ω) + k2(ω)V (z, ω) = 0 (16.2.6)

3In this notes, we will use k and β interchangeably for wavenumber. The transmission line community

tends to use β while the optics community uses k.

156 Electromagnetic Field Theory

When the dispersive transmission line is filled with dispersive material, then k2 = ω2μ0ε(ω).

Thus, upon solving the above equation, one obtains that V (z, ω) = V0(ω)e−jkz , and

V (z, t) =

∞ ˆ

−∞

dωV0(ω)ej(ωt−kz) (16.2.7)

In the general case, k is a complicated function of ω as shown in Figure 16.4.

Figure 16.4: A typical frequency dependent k(ω) albeit the frequency dependence can be

more complicated than shown.

Since this is a wave packet, we assume that V0(ω) is narrow band centered about a

frequency ω0, the carrier frequency as shown in Figure 16.5. Therefore, when the integral

in (16.2.7) is performed, it needs only be summed over a narrow range of frequencies in the

vicinity of ω0.

Waves in Layered Media 157

Figure 16.5: The frequency spectrum of V0(ω).

Thus, we can approximate the integrand in the vicinity of ω = ω0, and let

k(ω) ∼ = k(ω0) + (ω − ω0) dk(ω0)

dω + 1

2 (ω − ω0)2 d2k(ω0)

dω2 + · · · (16.2.8)

To ensure the real-valuedness of (16.2.5), one ensures that −ω part of the integrand is exactly

the complex conjugate of the +ω part. Another way is to sum over only the +ω part of the

integral and take twice the real part of the integral. So, for simplicity, we write (16.2.5) as

V (z, t) = 2<e

∞ ˆ

dωV0(ω)ej(ωt−kz) (16.2.9)

Since we need to integrate over ω ≈ ω0, we can substitute (16.2.8) into (16.2.9) and rewrite

it as

V (z, t) ∼ = 2<e







ej[ω0t−k(ω0)z]

∞ ˆ

dωV0(ω)ej(ω−ω0)te−j(ω−ω0) dk

dω z

︸︷︷︸

F (t− dk

dω z)







(16.2.10)

where more specifically,

(

t − dk

dω z

)

∞ ˆ

dωV0(ω)ej(ω−ω0)te−j(ω−ω0) dk

dω z (16.2.11)

158 Electromagnetic Field Theory

It can be seen that the above integral now involves the integral summation over a small range

of ω in the vicinity of ω0. By a change of variable by letting Ω = ω − ω0, it becomes

(

t − dk

dω z

)

ˆ +∆

−∆

dΩV0(Ω + ω0)ejΩ(t− dk

dω z) (16.2.12)

The above itself is a Fourier transform integral that involves only the low frequencies of the

Fourier spectrum. Hence, F is a slowly varying function. Moreover, this function F moves

with a velocity

vg = dω

dk (16.2.13)

Here, F (t− z

vg ) in fact is the velocity of the envelope in Figure 16.3. In (16.2.10), the envelope

function F (t − z

vg ) is multiplied by the rapidly varying function

ej[ω0t−k(ω0)z] (16.2.14)

before one takes the real part of the entire function. Hence, this rapidly varying part represents

the rapidly varying carrier frequency shown in Figure 16.3. More importantly, this carrier,

the rapidly varying part of the signal, moves with the velocity

vph = ω0

k(ω0) (16.2.15)

which is the phase velocity.

16.3 Wave Guidance in a Layered Media

Now that we have understood phase and group velocity, we are at ease with studying the

We have seen that in the case of a surface plasmonic resonance, the wave is guided by an

interface because the Fresnel reflection coefficient becomes infinite. This physically means that

a reflected wave exists even if an incident wave is absent or vanishingly small. This condition

can be used to find a guided mode in a layered medium, namely, to find the condition under

which the generalized reflection coefficient (16.1.2) will become infinite.

16.3.1 Transverse Resonance Condition

Therefore, to have a guided mode exist in a layered medium, the denominator of (16.1.2) is

zero, or that

1 + R12 ˜ R23e−2jβ2z l2 = 0 (16.3.1)

where t is the thickness of the dielectric slab. Since R12 = −R21, the above can be written as

1 = R21 ˜ R23e−2jβ2z l2 (16.3.2)

Waves in Layered Media 159

The above has the physical meaning that the wave, after going through two reflections at

the two interfaces, 21, and 23 interfaces, which are R21 and R23, plus a phase delay given

by e−2jβ2z l2 , becomes itself again. This is also known as the transverse resonance condition.

When specialized to the case of a dielectric slab with two interfaces and three regions, the

above becomes

1 = R21R23e−2jβ2z l2 (16.3.3)

The above can be generalized to finding the guided mode in a general layered medium. It

can also be specialized to finding the guided mode of a dielectric slab.

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