Lecture 15: Interesting Physical Phenomena in Electromagnetic Field

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Lecture 15

Interesting Physical Phenomena

Though simple that it looks, embedded in the TM Fresnel reflection coefficient are a few more

interesting physical phenomena. These are the phenomena of Brewster’s angle [96, 97] and

the phenomena of surface plasmon resonance, or polariton [98, 99].

15.1 Interesting Physical Phenomena–Contd.

We will continue with understanding some interesting phenomena associated with the single-

interface problem. Albeit rather simple, embedded in the equations lie deep interesting phe-

nomena that we shall see.

15.1.1 Brewster Angle

Brewster angle was discovered in 1815 [96, 97]. Furthermore, most materials at optical fre-

quencies have ε2 6 = ε1, but μ2 ≈ μ1. In other words, it is hard to obtain magnetic materials

at optical frequencies. Therefore, the TM polarization for light behaves differently from TE

polarization. Hence, we shall focus on the reflection and transmission of the TM polarization

of light, and we reproduce the TM reflection coefficient here:

RT M =

( β1z

ε1

− β2z

ε2

) / ( β1z

ε1

+ β2z

ε2

)

(15.1.1)

The transmission coefficient is easily gotten by the formula T T M = 1 + RT M . Observe that

for RT M , it is possible that RT M = 0 if

ε2β1z = ε1β2z (15.1.2)

Squaring the above, making the note that βiz = √β2

i − β2

x, one gets

ε22(β12 − βx2) = ε12(β22 − βx2) (15.1.3)

143

144 Electromagnetic Field Theory

Solving the above, assuming μ1 = μ2 = μ, gives

βx = ω√μ

√ ε1ε2

ε1 + ε2

= β1 sin θ1 = β2 sin θ2 (15.1.4)

The latter two equations come from phase matching at the interface. Therefore,

sin θ1 =

√ ε2

ε1 + ε2

, sin θ2 =

√ ε1

ε1 + ε2

(15.1.5)

or that

sin2 θ1 + sin2 θ2 = 1, (15.1.6)

Then, assuming that θ1 and θ2 are less than π/2, and using the identity that cos2 θ1 +sin2 θ1 =

1, then it can be shown that

sin θ2 = cos θ1 (15.1.7)

or that

θ1 + θ2 = π/2 (15.1.8)

This is used to explain why at Brewster angle, no light is reflected back to Region 1. Figure

15.1 shows that the induced polarization dipoles in Region 2 always have their axes aligned in

the direction of reflected wave. A dipole does not radiate along its axis, which can be verified

heuristically by field sketch and looking at the Poynting vector. Therefore, these induced

dipoles in Region 2 do not radiate in the direction of the reflected wave. Notice that when

the contrast is very weak meaning that ε1 ∼ = ε2, then θ1 ∼ = θ2 ∼ = π/4, and (15.1.8) is satisfied.

Interesting Physical Phenomena 145

Figure 15.1: A figure showing a plane wave being reflected and transmitted at the Brewster’s

angle. In Region t, the polarization current or dipoles are all pointing in the kr direction, and

hence, there is no radiation in that direction (courtesy of J.A. Kong, EM Wave Theory [31]).

Because of the Brewster angle effect for TM polarization when ε2 6 = ε1, |RT M | has to

go through a null when θi = θb. Therefore, |RT M | ≤ |RT E | as shown in Figure 15.2. Then

when a randomly polarized light is incident on a surface, the polarization where the electric

field is parallel to the surface (TE polarization) is reflected more than the polarization where

the magnetic field is parallel to the surface (TM polarization). This phenomenon is used to

design sun glasses to reduce road glare for drivers. For light reflected off a road surface, they

are predominantly horizontally polarized with respect to the surface of the road. When sun

glasses are made with vertical polarizers, they will filter out and mitigate the reflected rays

from the road surface to reduce road glare. This phenomenon can also be used to improve

the quality of photography by using a polarizer filter as shown in Figure 15.3.

146 Electromagnetic Field Theory

Figure 15.2: Because |RT M | has to through a null when θi = θb, therefore, |RT M | ≤ |RT E |

for all θi as shown above.

Figure 15.3: Because the TM and TE lights will be reflected differently, polarizer filter can

produce remarkable effects on the quality of the photograph [97].

15.1.2 Surface Plasmon Polariton

Surface plasmon polariton occurs for the same mathematical reason for the Brewster angle

effect but the physical mechanism is quite different. Many papers and textbooks will introduce

this phenomenon from a different angle. But here, we will introduce it from the Fresnel

reflection coefficient for the TM waves.

The reflection coefficient RT M can become infinite if ε2 < 0, which is possible in a plasma

medium. In this case, the criterion for the denominator to be zero is

−ε2β1z = ε1β2z (15.1.9)

When the above is satisfied, RT M becomes infinite. This implies that a reflected wave exists

when there is no incident wave. Or Href = HincRT M , and when RT M = ∞, Hinc can be

Interesting Physical Phenomena 147

zero, and Href can assume any value.1 Hence, there is a plasmonic resonance or guided mode

existing at the interface without the presence of an incident wave. It is a self-sustaining wave

propagating in the x direction, and hence, is a guided mode propagating in the x direction.

Solving (15.1.9) after squaring it, as in the Brewster angle case, yields

βx = ω√μ

√ ε1ε2

ε1 + ε2

(15.1.10)

This is the same equation for the Brewster angle except now that ε2 is negative. Even if

ε2 < 0, but ε1 + ε2 < 0 is still possible so that the expression under the square root sign

(15.1.10) is positive. Thus, βx can be pure real. The corresponding β1z and β2z in (15.1.9)

can be pure imaginary, and (15.1.9) can still be satisfied.

This corresponds to a guided wave propagating in the x direction. When this happens,

β1z =

√

β12 − βx2 = ω√μ

[

ε1

(

1 − ε2

ε1 + ε2

)]1/2

(15.1.11)

Since ε2 < 0, ε2/(ε1+ε2) > 1, then β1z becomes pure imaginary. Moreover, β2z = √β22 − βx2

and β22 < 0 making β2z becomes even a larger imaginary number. This corresponds to a

trapped wave (or a bound state) at the interface. The wave decays exponentially in both

directions away from the interface and they are evanescent waves. This mode is shown in

Figure 15.4, and is the only case in electromagnetics where a single interface can guide a

surface wave, while such phenomenon abounds for elastic waves.

When one operates close to the resonance of the mode so that the denominator in (15.1.10)

is almost zero, then βx can be very large. The wavelength becomes very short in this case,

and since βiz = √β2

i − β2

x, then β1z and β2z become even larger imaginary numbers. Hence,

the mode becomes tightly confined or bound to the surface, making the confinement of the

mode very tight. This evanescent wave is much more rapidly decaying than that offered by

the total internal reflection. It portends use in tightly packed optical components, and has

caused some excitement in the optics community.

1This is often encountered in a resonance system like an LC tank circuit. Current flows in the tank circuit

despite the absence of an exciting voltage.

148 Electromagnetic Field Theory

Figure 15.4: Figure showing a surface plasmonic mode propagating at an air-plasma interface.

As in all resonant systems, a resonant mode entails the exchange of energies. In the case

of surface plasmonic resonance, the energy is exchanged between the kinetic energy of the

electrons and the energy store in the electic field (courtesy of Wikipedia [100]).

15.2 Homomorphism of Uniform Plane Waves and Trans-

mission Lines Equations

It turns out that the plane waves through layered medium can be mapped into the multi-

section transmission line problem due to mathematical homomorphism between the two prob-

lems. Hence, we can kill two birds with one stone: apply all the transmission line techniques

and equations that we have learnt to solve for the solutions of waves through layered medium

problems.2

For uniform plane waves, since they are proportional to exp(−jβ · r), we know that with

∇ → −jβ, Maxwell’s equations becomes

β × E = ωμH (15.2.1)

β × H = −ωεE (15.2.2)

for a general isotropic homogeneous medium. We will specialize these equations for different

polarizations.

15.2.1 TE or TEz Waves

For this, one assumes a TE wave traveling in z direction with electric field polarized in the y

direction, or E = ˆyEy , H = ˆxHx + ˆzHz , then we have from (15.2.1)

βz Ey = −ωμHx (15.2.3)

βxEy = ωμHz (15.2.4)

2This treatment is not found elsewhere, and is peculiar to these lecture notes.

Interesting Physical Phenomena 149

From (15.2.2), we have

βz Hx − βxHz = −ωεEy (15.2.5)

Then, expressing Hz in terms of Ey from (15.2.4), we can show from (15.2.5) that

βz Hx = −ωεEy + βxHx = −ωεEy + β2

ωμ Ey

= −ωε(1 − β2

x/β2)Ey = −ωε cos2 θEy (15.2.6)

where βx = β sin θ has been used.

Eqns. (15.2.3) and (15.2.6) can be written to look like the telegrapher’s equation by letting

−jβz → d/dz to get

dz Ey = jωμHx (15.2.7)

dz Hx = jωε cos2 θEy (15.2.8)

If we let Ey → V , Hx → −I, μ → L, ε cos2 θ → C, the above is exactly analogous to the

telegrapher’s equation. The equivalent characteristic impedance of these equations above is

then

Z0 =

√ L

C =

√ μ

cos θ =

√ μ

βz

= ωμ

βz

(15.2.9)

The above is the wave impedance for a propagating plane wave with propagation direction

or the β inclined with an angle θ respect to the z axis. When θ = 0, the wave impedance

becomes the intrinsic impedance of space.

A two region, single-interface reflection problem can then be mathematically mapped to

a single-junction two-transmission-line problem discussed in Section 13.1.1. The equivalent

characteristic impedances of these two regions are then

Z01 = ωμ1

β1z

, Z02 = ωμ2

β2z

(15.2.10)

We can use the above to find Γ12 as given by

Γ12 = Z02 − Z01

Z02 + Z01

= (μ2/β2z ) − (μ1/β1z )

(μ2/β2z ) + (μ1/β1z ) (15.2.11)

The above is the same as the Fresnel reflection coefficient found earlier for TE waves or RT E

after some simple re-arrangement.

Assuming that we have a single junction transmission line, one can define a transmission

coefficient given by

T12 = 1 + Γ12 = 2Z02

Z02 + Z01

= 2(μ2/β2z )

(μ2/β2z ) + (μ1/β1z ) (15.2.12)

The above is similar to the continuity of the voltage across the junction, which is the same

as the continuity of the tangential electric field across the interface. It is also the same as the

Fresnel transmission coefficient T T E .

150 Electromagnetic Field Theory

15.2.2 TM or TMz Waves

For the TM polarization, by invoking duality principle, the corresponding equations are, from

(15.2.7) and (15.2.8),

dz Hy = −jωεEx (15.2.13)

dz Ex = −jωμ cos2 θHy (15.2.14)

Just for consistency of units, since electric field is in V m−1, and magnetic field is in A m−1

we may chose the following map to convert the above into the telegrapher’s equations, viz;

Ey → V, Hy → I, μ cos2 θ → L, ε → C (15.2.15)

Then, the equivalent characteristic impedance is now

Z0 =

√ L

C =

√ μ

ε cos θ =

√ μ

βz

β = βz

ωε (15.2.16)

The above is also termed the wave impedance of a TM propagating wave making an inclined

angle θ with respect to the z axis. Notice again that this wave impedance becomes the

intrinsic impedance of space when θ = 0.

Now, using the reflection coefficient for a single-junction transmission line, and the appro-

priate characteristic impedances for the two lines as given in (15.2.16), we arrive at

Γ12 = (β2z /ε2) − (β1z /ε1)

(β2z /ε2) + (β1z /ε1) (15.2.17)

Notice that (15.2.17) has a sign difference from the definition of RT M derived earlier in the

last lecture. The reason is that RT M is for the reflection coefficient of magnetic field while

Γ12 above is for the reflection coefficient of the voltage or the electric field. This difference

is also seen in the definition for transmission coefficients.3 A voltage transmission coefficient

can be defined to be

T12 = 1 + Γ12 = 2(β2z /ε2)

(β2z /ε2) + (β1z /ε1) (15.2.18)

But this will be the transmission coefficient for the voltage, which is not the same as T T M

which is the transmission coefficient for the magnetic field or the current. Different textbooks

may define different transmission coefficients for this polarization.

3This is often the source of confusion for these reflection and transmission coefficients.

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