Lecture 17

Dielectric Waveguides

Before we embark on the study of dielectric waveguides, we will revisit the transverse reso-

nance again. The transverse resonance condition allows one to derive the guidance conditions

for a dielectric waveguide easily without having to match the boundary conditions at the

interface again: The boundary conditions are already used when deriving the Fresnel reflec-

tion coefficients, and hence they are embedded in these reflection coefficients. Much of the

materials in this lecture can be found in [31, 38, 75].

17.1 Generalized Transverse Resonance Condition

The guidance conditions, the transverse resonance condition given previously, can also be

derived for the more general case. The generalized transverse resonance condition is a powerful

condition that can be used to derive the guidance condition of a mode in a layered medium.

To derive this condition, we first have to realize that a guided mode in a waveguide is due

to the coherent or constructive interference of the waves. This implies that if a plane wave

starts at position 1 (see Figure 17.1)1 and is multiply reflected as shown, it will regain its

original phase in the x direction at position 5. Since this mode progresses in the z direction,

all these waves (also known as partial waves) are in phase in the z direction by the phase

matching condition. Otherwise, the boundary conditions cannot be satisfied. That is, waves

at 1 and 5 will gain the same phase in the z direction. But, for them to add coherently or

interfere coherently in the x direction, the transverse phase at 5 must be the same as 1.

Assuming that the wave starts with amplitude 1 at position 1, it will gain a transverse

phase of e−jβ0xt when it reaches position 2. Upon reflection at x = x2, at position 3, the wave

becomes ˜ R+e−jβ0xt where ˜ R+ is the generalized reflection coefficient at the right interface

of Region 0. Finally, at position 5, it becomes ˜ R− ˜ R+e−2jβ0xt where ˜ R− is the generalized

reflection coefficient at the left interface of Region 0. For constructive interference to occur

1The waveguide convention is to assume the direction of propagation to be z. Since we are analyzing a

guided mode in a layered medium, z axis is as shown in this figure.

161

162 Electromagnetic Field Theory

Figure 17.1: The transverse resonance condition for a layered medium. The phase of the wave

at position 5 should be equal to the transverse phase at position 1.

or for the mode to exist, we require that

˜ R− ˜ R+e−2jβ0xt = 1 (17.1.1)

The above is the generalized transverse resonance condition for the guidance condition for a

plane wave mode traveling in a layered medium.

In (17.1.1), a metallic wall has a reflection coefficient of 1 for a TM wave, hence if ˜ R+ is

1, Equation (17.1.1) becomes

1 − ˜ R−e2−jβ0xt = 0. (17.1.2)

On the other hand, in (17.1.1), a metallic wall has a reflection coefficient of −1, for TE wave,

and Equation (17.1.1) becomes

1 + ˜ R−e2−jβ0xt = 0. (17.1.3)

17.2 Dielectric Waveguide

The most important dielectric waveguide of the modern world is the optical fiber, whose

invention was credited to Charles Kao [91]. He was awarded the Nobel prize in 2009 [102].

However, the analysis of the optical fiber requires analysis in cylindrical coordinates and the

use of special functions such as Bessel functions. In order to capture the essence of dielectric

waveguides, one can study the slab dielectric waveguide, which shares many salient features

with the optical fiber. This waveguide is also used as thin-film optical waveguides (see Figure

17.2). We start with analyzing the TE modes in this waveguide.

Dielectric Waveguides 163

Figure 17.2: An optical thin-film waveguide is made by coating a thin dielectric film or sheet

on a metallic surface. The wave is guided by total internal reflection at the top interface, and

by metallic reflection at the bottom interface.

17.2.1 TE Case

Figure 17.3: Schematic for the analysis of a guided mode in the dielectric waveguide. Total

internal reflection occurs at the top and bottom interfaces. If the waves add coherently, the

wave is guided along the dielectric slab.

We shall look at the application of the transverse resonance condition to a TE wave guided

in a dielectric waveguide. Again, we assume the direction of propagation of the guided mode

to be in the z direciton in accordance to convention. Specializing the above equation to the

dielectric waveguide shown in Figure 17.3, we have the guidance condition as

1 = R10R12e−2jβ1xd (17.2.1)

where d is the thickness of the dielectric slab. Guidance of a mode is due to total internal

reflection, and hence, we expect Region 1 to be optically more dense (in terms of optical

refractive indices)2 than region 0 and 2.

To simplify the analysis further, we assume Region 2 to be the same as Region 0. The

new guidance condition is then

1 = R2

10e−2jβ1xd (17.2.2)

2Optically more dense means higher optical refractive index, or higher dielectric constant.

164 Electromagnetic Field Theory

Also, we assume that ε1 > ε0 so that total internal reflection occurs at both interfaces as

the wave bounces around so that β0x = −jα0x. Therefore, for TE polarization, the single-

interface reflection coefficient is

R10 = μ0β1x − μ1β0x

μ0β1x + μ1β0x

= μ0β1x + jμ1α0x

μ0β1x − jμ1α0x

= ejθT E (17.2.3)

where θT E is the Goos-Hanschen shift for total internal reflection. It is given by

θT E = 2 tan−1

( μ1α0x

μ0β1x

)

(17.2.4)

The guidance condition for constructive interference according to (17.2.1) is such that

2θT E − 2β1xd = 2nπ (17.2.5)

From the above, dividing it by four, and taking its tangent, we get

tan

( θT E

2

)

= tan

( nπ

2 + β1xd

2

)

(17.2.6)

or

μ1α0x

μ0β1x

= tan

( nπ

2 + β1xd

2

)

(17.2.7)

The above gives rise to

μ1α0x = μ0β1x tan

( β1xd

2

)

, n even (17.2.8)

−μ1α0x = μ0β1x cot

( β1xd

2

)

, n odd (17.2.9)

It can be shown that when n is even, the mode profile is even, whereas when n is odd, the

mode profile is odd. The above can also be rewritten as

μ0

μ1

β1xd

2 tan

( β1xd

2

)

= α0xd

2 , even modes (17.2.10)

− μ0

μ1

β1xd

2 cot

( β1xd

2

)

= α0xd

2 , odd modes (17.2.11)

Using the fact that −α2

0x = β2

0 − β2

z , and that β2

1x = β2

1 − β2

z , eliminating βz from these two

equations, one can show that

α0x = [ω2(μ11 − μ00) − β2

1x] 1

2 (17.2.12)

Dielectric Waveguides 165

and (17.2.10) and (17.2.11) become

μ0

μ1

β1xd

2 tan

( β1xd

2

)

= α0xd

2

=

√

ω2(μ11 − μ00) d2

4 −

( β1xd

2

)2

, even modes (17.2.13)

− μ0

μ1

β1xd

2 cot

( β1xd

2

)

= α0xd

2

=

√

ω2(μ11 − μ00) d2

4 −

( β1xd

2

)2

, odd modes (17.2.14)

We can solve the above graphically by plotting

y1 = μ0

μ1

β1xd

2 tan

( β1xd

2

)

even modes (17.2.15)

y2 = − μ0

μ1

β1xd

2 cot

(

β1x

d

2

)

odd modes (17.2.16)

y3 =

[

ω2(μ11 − μ00) d2

4 −

( β1xd

2

)2] 1

2

= α0xd

2 (17.2.17)

Figure 17.4: A way to solve (17.2.13) and (17.2.13) is via a graphical method. In this method,

both the right-hand side and the left-hand side of the equations are plotted on the same plot.

The solutions are the points of intersection of these plots.

166 Electromagnetic Field Theory

In the above, y3 is the equation of a circle; the radius of the circle is given by

ω(μ11 − μ00) 1

2

d

2 . (17.2.18)

The solutions to (17.2.13) and (17.2.14) are given by the intersections of y3 with y1 and

y2. We note from (17.2.1) that the radius of the circle can be increased in three ways: (i)

by increasing the frequency, (ii) by increasing the contrast μ11

μ00 , and (iii) by increasing the

thickness d of the slab.3 The mode profiles of the first two modes are shown in Figure 17.5.

Figure 17.5: Mode profiles of the TE0 and TE1 modes of a dielectric slab waveguide (courtesy

of J.A. Kong [31]).

When β0x = −jα0x, the reflection coefficient for total internal reflection is

RT E

10 = μ0β1x + jμ1α0x

μ0β1x − jμ1α0x

= exp

[

+2j tan−1

( μ1α0x

μ0β1x

)]

(17.2.19)

and ∣

∣RT E

10

∣

∣ = 1. Hence, the wave is guided by total internal reflections.

Cut-off occurs when the total internal reflection ceases to occur, i.e. when the frequency

decreases such that α0x = 0.

From Figure 17.4, we see that α0x = 0 when

ω(μ11 − μ00) 1

2

d

2 = mπ

2 , m = 0, 1, 2, 3, . . . (17.2.20)

3These features are also shared by the optical fiber.

Dielectric Waveguides 167

or

ωmc = mπ

d(μ11 − μ00) 1

2

, m = 0, 1, 2, 3, . . . (17.2.21)

The mode that corresponds to the m-th cut-off frequency above is labeled the TEm mode.

Thus TE0 mode is the mode that has no cut-off or propagates at all frequencies. This is

shown in Figure 17.6 where the TE mode profiles are similar since they are dual to each

other. The boundary conditions at the dielectric interface is that the field and its normal

derivative have to be continuous. The TE0 or TM0 mode can satisfy this boundary condition

at all frequencies, but not the TE1 or TM1 mode. At the cut-off frequency, the field outside

the slab has to become flat implying the α0x = 0 implying no guidance.

Figure 17.6: The TE modes are dual to the TM modes and have similar mode profiles.

At cut-off, α0x = 0, and from the dispersion relation that α2

0x = β2

z − β2

0 ,

βz = ω√μ00,

for all the modes. Hence, both the group and the phase velocities are that of the outer region.

This is because when α0x = 0, the wave is not evanescent outside, and most of the energy of

the mode is carried by the exterior field.

When ω → ∞, the radius of the circle in the plot of y3 becomes increasingly larger. As

seen from Figure 17.4, the solution for β1x → nπ

d for all the modes. From the dispersion

relation for Region 1,

βz =

√

ω2μ11 − β2

1x ≈ ω√μ11, ω → ∞ (17.2.22)

168 Electromagnetic Field Theory

Hence the group and phase velocities approach that of the dielectric slab. This is because

when ω → ∞, α0x → ∞, implying that the fields are trapped or confined in the slab and

propagating within it. Because of this, the dispersion diagram of the different modes appear

as shown in Figure 17.7. In this figure,4 kc1, kc2, and kc3 are the cut-off wave number or

frequency of the first three modes. Close to cut-off, the field is traveling mostly outside

the waveguide, and kz ≈ ω√μ0ε0, and both the phase and group velocities approach that

of the outer medium as shown in the figure. When the frequency increases, the mode is

tightly confined in the dielectric slab, and kz ≈ ω√μ1ε1. Both the phase and group velocities

approach that of Region 1 as shown.

Figure 17.7: Here, we have kz versus k1 plot for dielectric slab waveguide. Near its cut-off,

the energy of the mode is in the outer region, and hence, its group velocity is close to that of

the outer region. At high frequencies, the mode is tightly bound to the slab, and its group

velocity approaches that of the dielectric slab (courtesy of J.A. Kong [31]).

17.2.2 TM Case

For the TM case, a similar guidance condition analogous to (17.2.1) can be derived but with

the understanding that the reflection coefficients in (17.2.1) are now TM reflection coefficients.

4Please note again that in this course, we will use β and k interchangeably for wavenumbers.

Dielectric Waveguides 169

Similar derivations show that the above guidance condition, for 2 = 0, μ2 = μ0, reduces to

0

1

β1x

d

2 tan β1x

d

2 =

√

ω2(μ11 − μ00) d2

4 −

(

β1x

d

2

)2

, even modes (17.2.23)

− 0

1

β1x

d

2 cot β1x

d

2 =

√

ω2(μ11 − μ00) d2

4 −

(

β1x

d

2

)2

, odd modes (17.2.24)

Note that for equation (17.2.1), when we have two parallel metallic plates, RT M = 1, and

RT E = −1, and the guidance condition becomes

1 = e−2jβ1xd ⇒ β1x = mπ

d , m = 0, 1, 2, . . . , (17.2.25)

17.2.3 A Note on Cut-Off of Dielectric Waveguides

The concept of cut-off in dielectric waveguides is quite different from that of hollow waveguides

that we shall learn next. A mode is guided in a dielectric waveguide if the wave is trapped

inside, in this case, the dielectric slab. The trapping is due to the total internal reflections at

the top and the bottom interface of the waveguide. WShen total internal reflection ceases to

occur at any of the two interfaces, the wave is not guided or trapped inside the dielectric slab

anymore. This happens when αix = 0 where i can indicate the top-most or the bottom-most

region. In other words, the wave ceases to be evanescent in Region i.

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