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

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6.2 Symmetrical Planar Dielectric Waveguides 331

Region 2, x ≤ −h:

= Ce

τx

−jβz

, (6.81)

y 2

= jωµ

τCe

τx

−jβz

, (6.82)

= −jβτ Ce

τx

−jβz

, (6.83)

= −τ

τx

−jβz

; (6.84)

Region 3, x ≥ h:

= De

−τx

−jβz

, (6.85)

y 3

= −jωµ

τDe

−τx

−jβz

, (6.86)

= jβτ De

−τx

−jβz

, (6.87)

= −τ

−τx

−jβz

. (6.88)

The relations for T , τ, and β expressed by (6.62) still valid.

As with the TM modes, applying the ﬁeld-matching conditions on the

boundaries x = h and x = −h,

y 2

(−h, z)=E

y 1

(−h, z) → µ

τCe

−τh

=µ

T (A sin T h+B cos T h), (6.89)

y 3

(h, z)=E

y 1

(h, z) → µ

τDe

−τh

=µ

T (A sin T h−B cos T h). (6.90)

(−h, z)=H

(−h, z) → −τ

−τh

(A cos T h−B sin T h), (6.91)

(h, z)=H

(h, z) → −τ

−τh

(A cos T h+B sin T h). (6.92)

The eigenvalue equation for TE modes is obtained,

T h tan

T h −

mπ

= µ

τh, (6.93)

where m is 0 or even for even modes and odd for odd modes.

The expressions for transverse wave numbers T and τ and the longitudinal

wave number β for TE modes are determined by (6.93), (6.73), and (6.62).

The relations for the coeﬃcients can also be obtained from the boundary

conditions. For even modes:

A = 0, B =

−τh

sin T h

C, D =−C. (6.94)

For odd modes:

A=−

−τh

cos T h

C, B = 0, D =C. (6.95)

The surface impedance of the slab for TE modes becomes

(TE)

= −

(h)

(−h)

= −j

ωµ

. (6.96)

This is a capacitive reactance. We can see that an arbitrary cylindrical system

enclosed by a capacitive surface can support TE surface waves.

332 6. Dielectric Waveguides and Resonators

Figure 6.3: Graphical solution of the eigenvalue equations for the symmetrical

planar dielectric waveguide.

6.2.3 Cutoﬀ Condition, Guided Modes and Radiation

Modes

Either the eigenvalue equation for TM modes (6.72) or that for TE modes

(6.93) individually combined with (6.73) can be solved to give the transverse

angular wave number in the core, T , and that outside the core, τ, in terms

of the angular frequency ω. These equations can be solved graphically as

shown in Fig. 6.3. The curves representing (6.72) or (6.93) for given modes

are plotted as solid lines and the curve representing (6.73) is plotted as a

dashed line, which is exactly a quadrant of a circle. The intersections of

the two sets of curves are the solutions of the two equations, which give T h

and τ h for given ω for guided modes. We can see from the ﬁgure that the

higher the frequency, the larger the circle, and we have more intersection

points. This means that more modes are guided modes when the frequency

becomes higher. The modes that have no intersection with the circle are

radiation modes; in other words, for those modes, the transverse angular wave

numbers outside the core are imaginary, τ = jk

, therefore the transverse

dependent parts of the ﬁelds become traveling waves e

±jk

. Such modes

are radiation modes. For radiation modes, accompanied with the radiation

in the transverse direction, which leads to power losses, the waves in the

longitudinal direction will be attenuated. The lower limit in the frequency

of a certain mode as a guided mode is known as the cutoﬀ frequency and is

denoted by ω

. Note that the physical meaning of “cutoﬀ” for a dielectric

waveguide is entirely diﬀerent from that for a metallic waveguide. When

the frequency is lower than the cutoﬀ frequency of a certain mode, decaying

ﬁelds along the longitudinal direction in a metallic waveguide cause no power

6.2 Symmetrical Planar Dielectric Waveguides 333

loss. Such modes are known as the cutoﬀ modes or evanescent modes. But

in dielectric waveguides, when the frequency is lower than the cutoﬀ value,

the modes become radiation modes and the slab becomes a radiator. For

a dielectric waveguide operating at a certain frequency, a ﬁnite number of

guided modes and an inﬁnite number of radiation modes exist in it.

The cutoﬀ condition for a dielectric waveguide is τ = 0. From (6.72) and

(6.93), we get the cutoﬀ condition for even TM and TE modes:

τh = 0, tan T h = 0, T h =

mπ

, m = 0, 2, 4, 6, ···.

Similarly, we get the cutoﬀ condition for odd TM and TE modes:

τh = 0, cot T h = 0, T h =

mπ

, m = 1, 3, 5, ···.

Using (6.73), we obtain the expression for the cutoﬀ frequency

h =

mπ

= ω

√

− µ

, ω

mπ

√

− µ

. (6.97)

The cutoﬀ conditions for TE modes and TM modes of the same order

are the same but the corresponding eigenvalue equations, i.e., the dispersion

characteristics, are diﬀerent from each other. Thus they are not degenerate

modes.

6.2.4 Dispersion Characteristics of Guided Modes

The longitudinal phase coeﬃcient of a guided mode, β, is determined by

(6.62) when τ and T for a given ω are found. Then the dispersion curves,

i.e., ω–β or k–β diagrams, are plotted as shown in Fig. 6.4(a). The dispersion

curves for all modes are limited in an interval set by the lower bound β =

√

and the upper bound β = ω

√

. At the cutoﬀ frequency, ω →

, the longitudinal phase coeﬃcient β approaches its lower bound which

corresponds to k

. At the same time v

→ 1/

√

, i.e., the phase velocity

of the guided wave approaches that of the plane wave in the medium of

the cladding. This corresponds to the critical angle of incidence of a plane

wave on the boundary. As the frequency increases, the longitudinal phase

coeﬃcient β approaches its upper bound that corresponds to k

. At the same

time v

→ 1/

√

, i.e., the phase velocity of the guided wave becomes that

of the plane wave in the medium in the core. This is the situation of 90

◦

incidence of the plane wave on the boundary, i.e., the incident wave vector is

parallel to the boundary. As the frequency increases, more and more guided

modes propagate in the slab.

In most optical waveguides, µ

is very close to µ

. In this case, the

dispersion curves are limited in a very narrow interval and are therefore

diﬃcult to read. To obtain a more convenient scaled diagram that shows the

334 6. Dielectric Waveguides and Resonators

Figure 6.4: Dispersion curves of the symmetrical planar dielectric waveguide.

ω–β relationship, we deﬁne a normalized frequency V by

V = h

− k

= ωh

√

− µ

, (6.98)

and a normalized guided index b by

b =

− k

(τh)

. (6.99)

The index b is zero at cutoﬀ and approaches unity at values far away from

cutoﬀ.

Then the eigenvalue equations for TM modes (6.72) and TE modes (6.93)

become

tan

√

1 − b −

mπ

1 − b

, for TM modes, (6.100)

tan

√

1 − b −

mπ

1 − b

, for TE modes. (6.101)

The normalized dispersion curves of b versus V are shown in Fig. 6.4(b).

6.2.5 Radiation Modes

It may be recalled from Section 5.1 that in metallic waveguides, there can be

a ﬁnite number of guided modes and an inﬁnite number of cutoﬀ modes or

evanescent modes. The characteristic impedance of a cutoﬀ mode is reactive,

so the property of any discontinuity in a metallic waveguide in which higher

evanescent modes are excited is reactive and gives rise to reﬂection of waves.

6.2 Symmetrical Planar Dielectric Waveguides 335

Figure 6.5: Transverse dependence of ﬁelds in a symmetrical planar dielectric

waveguide.

In a dielectric waveguide, all modes with a cutoﬀ frequency higher than

the operating frequency become radiation modes. In any dielectric waveguide,

for any operating frequency, there must exist a ﬁnite number of guided modes

and an inﬁnite number of radiation modes. When a dielectric waveguide is

excited by a source, or a imperfection or discontinuity is located in the guide,

all guided modes and radiation modes are excited to satisfy the boundary

conditions of the waveguide and the source or discontinuity. Guided modes

propagate along the guide a long way, whereas radiation modes radiate in the

transverse direction and are attenuated in the longitudinal direction. This

phenomenon gives rise to loss of energy.

For example, a bend in a metallic waveguide gives rise to reﬂection but in

a dielectric waveguide it gives rise to bending loss if the radius of curvature

is suﬃciently small. If the waveguide is uniform and inﬁnitely long and the

source is located at plus or minus inﬁnity, then only guided modes exist in

the guide; the same is true in metallic waveguides.

6.2.6 Fields in Symmetrical Planar Dielectric

Waveguides

The x dependencies of the transverse ﬁeld components of guided modes and

radiation modes in a symmetrical dielectric slab waveguide are illustrated

in Fig. 6.5. For guided modes, the ﬁelds are standing waves along x in the

core and are decaying ﬁelds along ±x in the cladding, whereas for radiation

modes, the ﬁelds are radiation waves along ±x in the cladding.

The ﬁeld maps of some low-order TM and TE guided modes are given in

Fig. 6.6 and Fig. 6.7, respectively. We easily see that the ﬁelds in the central

part of the slab are similar to those in the metallic parallel-plate line shown

in Figure 5.14, and the ﬁelds at the boundary and in the cladding are the

same as those for the total internal reﬂection shown in Fig. 2.20.

336 6. Dielectric Waveguides and Resonators

Figure 6.6: Field maps of some low-order guided TE modes in symmetrical

planar dielectric waveguide.

6.2 Symmetrical Planar Dielectric Waveguides 337

Figure 6.7: Field maps of some low-order guided TM modes in symmetrical

planar dielectric waveguide.

338 6. Dielectric Waveguides and Resonators

6.2.7 The Dominant Modes in Symmetrical Planar

Dielectric Waveguides

The lowest mode or dominant mode in a symmetrical planar dielectric waveg-

uide are TE or TM modes with m = 0. According to (6.97), the cutoﬀ

frequencies are consequently zero for these modes:

mπ

√

− µ

m=0

= 0.

These TM

and TE

modes remain guided modes down to zero frequency.

This feature of the lowest modes in a dielectric waveguide is shared by the

TEM mode in a two-conductor transmission line.

For TM

and TE

modes, if the operating frequency is so low that τ → 0

and T → 0, then from the ﬁeld-component expressions (6.51) to (6.53) and

(6.78) to (6.80), we notice that the longitudinal components are much smaller

than the transverse components. The ﬁeld expressions in the core reduce to

−jβz

, H

y 1

ω²

−jβz

, for TM

mode, (6.102)

y 1

−jβz

, H

=−

ωµ

−jβz

=−

−jβz

, for TE

mode. (6.103)

The ﬁelds of TM

and TE

modes in the core asymptotically approach those

of the uniform plane wave or the TEM mode in a parallel-plate transmission

line, as the frequency decreases to zero.

6.2.8 The Weekly Guiding Dielectric Waveguides

In typical dielectric waveguides for optical frequencies, the refractive index of

the core is only slightly larger than that of the cladding, (n

−n

)/n

¿ 1, so

that n

≈ n

. The diﬀerence between the indices of the core and the cladding

is less than a few percent. This is the weakly guiding condition, and this kind

of optical waveguide is known as the weakly guiding optical waveguide.

For weakly guiding optical waveguides, the critical angle on the boundary

is rather large. For guided modes, the angle of incidence must b e larger than

the critical angle and close to π/2, i.e. the wave vector of the incident wave

is almost parallel to the z axis. In this case, the longitudinal components of

the ﬁelds are much less than the transverse components, and the longitudinal

wave number is approximately equal to the wave number of a plane wave in

the core material, that is to say, the wave is close to the TEM mode. This

kind of modes are known as quasi-TEM modes.

For non-magnetic dielectric waveguides, µ

= µ

, the eigenvalue

equation for TE modes, (6.93) becomes

T h tan

T h −

mπ

= τh.

6.3 Dielectric Coated Conductor Plate 339

Figure 6.8: Dielectric coated conducting plate.

For weekly-guiding dielectric waveguides, ²

≈ ²

, the approximate eigenvalue

equation for TM modes, (6.72) becomes

T h tan

T h −

mπ

= τh,

We see that, for weekly guiding dielectric waveguides, the TE

mode and

mode are approximately degenerate modes.

6.3 Dielectric Coated Conductor Plate

The dielectric coated conducting plate shown in Fig. 6.8, is also known as

the dielectric image waveguide. The analysis of a dielectric coated conduct-

ing plate is similar to that of the symmetrical dielectric slab waveguide. By

imposing the short-circuit boundary condition at x = 0, we deduce that only

even TM modes and odd TE modes can exist in a dielectric coated conduct-

ing plate, i.e., TM

, TM

, ··· and TE

, TE

, ··· modes remain

nontrivial. The propagation characteristics and ﬁeld distribution for the di-

electric coated conducting plate are the same as those for the corresponding

modes in symmetrical dielectric slab waveguide.

The dielectric coated conducting plate can be used as a slow-wave struc-

ture, and the equivalent metallic structure will be introduced in the next

chapter.

6.4 Asymmetrical Planar Dielectric

Waveguides

In the optical waveband, especially in integrated optics applications, the com-

monly used waveguide is the asymmetrical planar dielectric waveguide shown

in Fig. 6.9, where the core, a planar ﬁlm of refractive index n

, is sandwiched

between a substrate of refractive index n

and a cladding or cover of refractive

index n

. In order to support guided modes it is necessary to make n

> n

340 6. Dielectric Waveguides and Resonators

Figure 6.9: Asymmetrical planar dielectric waveguide.

and n

> n

. Sometimes the cladding material is nothing but air, in which

case n

= 1. The dielectric waveguide is usually fabricated by growing thin

ﬁlm on substrate such as glass, crystal, or semiconductor material. The ﬁlm

can be deposited by evaporation or sputtering, or done by means of epitax-

ial growth techniques. Alternatively, dielectric optical waveguide fabrication

employs doping techniques including proton exchange, diﬀusion and ion im-

plantation techniques. Typical diﬀerences between the indices of the core

and the substrate range from 10

−3

to 10

−1

, and the typical ﬁlm thickness is

in the micrometer range. In the guided mode the wave is conﬁned in the core

or so-called guided layer by total internal reﬂection at the core–substrate and

core–cladding boundaries.

Suppose the permittivity and permeability are ²

, µ

for the material

of the core, region 1, ²

, µ

for the substrate, region 2, and ²

, µ

for the

cladding, region 3. The thickness of the core is d in the x direction and the

waveguide extends to inﬁnity in the y and z directions; refer to Fig. 6.9.

For guided modes, the ﬁeld dependence in the guided layer is a standing

wave in the x direction with phase coeﬃcient T , and the ﬁelds in the substrate

and the cladding are decaying ﬁelds in the transverse directions with decaying

coeﬃcients τ

and τ

, respectively. The longitudinal dependencies of the ﬁeld

in all the three regions are traveling waves with the same phase coeﬃcient

β, as required by the boundary conditions. The relations among the phase

coeﬃcients and decaying coeﬃcients are as follows:

=ω

, β

−τ

=ω

, β

−τ

=ω

which lead to

= k

− T

= k

+ τ

= k

+ τ

. (6.104)

This implies that

β < k

, β >k

, v

√

, v

√

, v

√

. (6.105)