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

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6.6 Circular Dielectric Waveguides and Optical Fibers 381

n−1

(T ρ)

sin (n −1)φ

−cos (n − 1)φ

¾¸

−jβz

, (6.292)

− k

(x)

≈ k

(T ρ)

cos nφ

sin nφ

−jβz

, (6.293)

∂

(x)

∂z ∂x

= j

βT

n+1

(T ρ)

cos (n + 1)φ

sin (n + 1)φ

−J

n−1

(T ρ)

cos (n − 1)φ

sin (n −1)φ

¾¸

−jβz

, (6.294)

In region 2, the cladding, to avoid singularity as ρ → ∞, the functions in

ρ must be modiﬁed Bessel functions of the second kind. The functions in φ

are the same as those in the core. So the function V

(x)

is given by

(x)

=BK

(τρ)

cos nφ

sin nφ

−jβz

. (6.295)

Substituting (6.295) into the ﬁeld component expressions and using the

recurrence formula (C.15) and diﬀerential formula (C.18), we obtain the ﬁeld

components of the linearly polarized modes in the cladding:

y 2

= −jωµ

∂V

(x)

∂z

= −ωµ

βBK

(τρ)

cos nφ

sin nφ

−jβz

, (6.296)

= jωµ

∂V

(x)

∂y

= −j

ωµ

n+1

(τρ)

sin (n + 1)φ

−cos (n + 1)φ

−K

n−1

(τρ)

sin (n −1)φ

−cos (n − 1)φ

¾¸

−jβz

, (6.297)

− k

(x)

≈ k

(τρ)

cos nφ

sin nφ

−jβz

, (6.298)

∂

(x)

∂z ∂x

= j

βτ

n+1

(τρ)

cos (n + 1)φ

sin (n + 1)φ

n−1

(τρ)

cos (n − 1)φ

sin (n −1)φ

¾¸

−jβz

, (6.299)

The boundary conditions for the tangential ﬁeld components are

y 1

(a) cos φ = E

y 2

(a) cos φ, H

(a) sin φ = H

(a) sin φ, (6.300)

(a, φ) = E

(a, φ), H

(a, φ) = H

(a, φ). (6.301)

The solutions of these boundary equations are

B =

(T a)

A (6.302)

382 6. Dielectric Waveguides and Resonators

Figure 6.24: Field patterns of some linearly polarized modes in the core of

the weakly guiding optical ﬁber.

and the eigenvalue equations

T aJ

n+1

(T a)

τaK

n+1

(τa)

T aJ

n−1

(T a)

= −

τaK

n−1

(τa)

. (6.303)

By using the recurrence formulas of Bessel functions and modiﬁed Bessel

functions, we can prove that the above two equations are identical to each

other. The resulting eigenvalue equations are the same as the universal eigen-

value equation for a weakly guiding waveguide (6.286).

Similarly, it takes only a little eﬀort to obtain also the solutions of the

modes with transverse electric ﬁeld in the x direction.

The linearly polarized modes in a weakly guiding optical ﬁber are denoted

by LP

, where m means that the mth root of the eigenvalue equation is

taken. The ﬁeld patterns of some lower-order linearly polarized modes are

given in Fig. 6.24. We easily see that the LP

mode is just the HE

mode,

the LP

mode is the superposition of HE

and TE

or TM

modes, and

the other LP modes are the superposition of HE

n+1,m

and EH

n−1,m

modes.

The linearly polarized modes are merely approximations in the weakly

guiding condition. In fact, the longitudinal wave numbers of the two modes

elliptically polarized in opposite senses are not exactly the same, so the com-

posed mode will not remain linearly polarized forever during the propagation.

Only the LP

modes, i.e., HE

modes, are truly linearly polarized modes.

6.6.5 Dominant Modes in Circular Dielectric

Waveguides

The lowest mode, i.e., dominant mode, in a circular dielectric waveguide is

the HE

mode with zero cutoﬀ frequency. The HE modes with n = 1 and

n = −1 represent two skew waves circularly polarized in opposite senses with

equal phase coeﬃcients. The superposition of these two waves with equal

amplitudes results in a linearly polarized wave, i.e., the LP

mode.

6.6 Circular Dielectric Waveguides and Optical Fibers 383

For a weekly guiding optical ﬁber made of nonmagnetic material, µ

= µ

, ²

≈ ²

, and β ≈ k

. Field components (6.218)–(6.223) become

n = +1, χ = −1 n = −1, χ = +1

ρ1

= −jk

T AJ

(T ρ)e

jφ

−jβz

, jk

T AJ

(T ρ)e

−jφ

−jβz

φ1

= k

T AJ

(T ρ)e

jφ

−jβz

, k

T AJ

(T ρ)e

−jφ

−jβz

= T

(T ρ)e

jφ

−jβz

, −T

(T ρ)e

−jφ

−jβz

ρ1

= −(k

/η

)T AJ

(T ρ)e

jφ

−jβz

, −(k

/η

)T AJ

(T ρ)e

−jφ

−jβz

φ1

= −j(k

/η

)T AJ

(T ρ)e

jφ

−jβz

, j(k

/η

)T AJ

(T ρ)e

−jφ

−jβz

= −j(1/η

AJJ

(T ρ)e

jφ

−j6βz

, −j(1/η

(T ρ)e

−jφ

−jβz

The superposition of the corresponding ﬁeld components with n = +1 and

n = −1 is done by using simple mathematics so that

ρ1

= 2k

T AJ

(T ρ) sinφ e

−jβz

, (6.304)

φ1

= 2k

T AJ

(T ρ) cosφ e

−jβz

, (6.305)

= j2T

(T ρ) sinφ e

−jβz

, (6.306)

ρ1

= −2

T AJ

(T ρ) cosφ e

−jβz

, (6.307)

φ1

= 2

T AJ

(T ρ) sinφ e

−jβz

, (6.308)

= −j2

(T ρ) cosφ e

−jβz

. (6.309)

With the coordinate transformation

ρA

cos φ −

φA

sin φ,

ρA

sin φ +

φA

cos φ,

the cartesian components of the ﬁelds can be expressed by

y 1

= E

(T ρ) e

−jβz

, (6.310)

= j

(T ρ) sin φ e

−jβz

, (6.311)

= −

(T ρ) e

−jβz

, (6.312)

= −j

ωµ

(T ρ) cos φ e

−jβz

, (6.313)

where E

= 2k

T A. The above expressions can be veriﬁed by checking them

against the ﬁeld components of linearly polarized modes (6.291)–(6.294).

In weakly guiding waveguides, the wave propagates in a direction almost

parallel to the axis z. Hence T ¿ β, τ ¿ β, and β ≈ k

. It follows that in

this case, E

¿ E

y 1

and H

¿ H

and the ﬁeld components become

y 1

= E

−jβz

, H

= −

−jβz

, and

y 1

= η

Evidently the ﬁelds in the core are similar to those of plane waves.

384 6. Dielectric Waveguides and Resonators

Figure 6.25: Measured loss spectrum of a germano-silicate single-mode opti-

cal ﬁber.

6.6.6 Low-Attenuation Optical Fiber

The present worldwide eﬀort devoted to optical ﬁber communications systems

stems largely from two papers [47] [109] published in 1966, proposing that

optical ﬁbers be used as dielectric waveguide in telecommunications to replace

coaxial line transmission systems. It is interesting to note how close the

optical ﬁber communication systems developed in recent years are to those

original proposals.

The key to the success in the application of optical ﬁber as the transmis-

sion medium in communication is to attain the low losses in light propagation.

Glass that is routinely used in optical instruments is far too lossy to be used

for optical ﬁbers suitable for long-distance transmission. The attenuation

coeﬃcient of optical ﬁbers made by such glasses is about 1 dB/m, i.e., 1000

dB/km. That is to say, the power is attenuated to 10

−100

of the initial value

after 1 km, which is too bad to be acceptable for optical communication

purposes.

In 1970s the weakly guiding ﬁber made from very low OH contents

germania-doped fused silica by the chemical vapor deposition technique was

developed. The attenuation co eﬃcient was made as low as 20 dB/km so that

optical ﬁber communication began to be practical.

Today, the single-mode germania-doped silicate optical ﬁber operated at

mode has become the most important transmission medium in world-

wide networks for long-distance communications. The curve in Fig. 6.25

6.7 Dielectric-Coated Conductor Cylinder 385

Figure 6.26: Dielectric-coated conducting cylinder.

shows the loss versus wavelength relationship of a mo dern optical ﬁber. The

curve shows that the loss peak around 1.4 µm is due to the residual OH con-

tamination in the fused silica. A low loss value of ∼ 0.5 dB/km is achieved

near λ = 1.3 µm and ∼ 0.2 dB/km is achieved near λ = 1.55 µm. Con-

sequently, these regions in the spectrum are now favored for long-distance

communications.

6.7 Dielectric-Coated Conductor Cylinder

The dielectric-coated conducting cylinder, shown in Fig. 6.26 is a typical

surface-wave transmission line that can also be used as a surface-wave radia-

tor. The radius of the conducting rod is a, the constitutional parameters of

the dielectric are ² and µ, the thickness of the dielectric coating is t, and the

outer radius of the dielectric is then b = a + t.

We are interested in the angular uniform modes, n = 0. In this case the

TE or TM mode alone can satisfy the boundary conditions.

(1) TM Modes, V = 0

Region 1. Inside the dielectric, a ≤ ρ ≤ b,

= [AJ

(T ρ) + BN

(T ρ)] e

−jβz

, (6.314)

ρ1

= −jβ

∂U

∂ρ

= jβT [AJ

(T ρ) + BN

(T ρ)] e

−jβz

, (6.315)

φ1

= −

jβ

∂U

∂φ

= 0, (6.316)

= T

[AJ

(T ρ) + BN

(T ρ)] e

−jβz

, (6.317)

ρ1

jω²

∂U

∂φ

= 0, (6.318)

φ1

= −jω²

∂U

∂ρ

= jω²T [AJ

(T ρ) + BN

(T ρ)] e

−jβz

, (6.319)

386 6. Dielectric Waveguides and Dielectric Resonators

where

+ T

= k

= ω

µ². (6.320)

Region 2. Outside the dielectric, b ≤ ρ ≤ ∞,

= CK

(τρ) e

−jβz

, (6.321)

ρ2

= −jβ

∂U

∂ρ

= jβτ CK

(τρ) e

−jβz

, (6.322)

φ2

= −

jβ

∂U

∂φ

= 0, (6.323)

= −τ

(τρ) e

−jβz

, (6.324)

ρ2

jω²

∂U

∂φ

= 0, (6.325)

φ2

= −jω²

∂U

∂ρ

= jω²

τCK

(τρ) e

−jβz

, (6.326)

where

− τ

= k

= ω

. (6.327)

The boundary conditions are known to be speciﬁed as,

(a) = 0 → B = −A

(T a)

, (6.328)

(b) = E

(b) → T

[AJ

(T b) + BN

(T b)] = −τ

(τb), (6.329)

φ1

(b) = H

φ2

(b) → ²T [AJ

(T b) + BN

(T b)] = −²

τCK

(τb). (6.330)

Then we have

(T a)J

(T b) − J

(T a)N

(T b)

(T a)J

(T b) − J

(T a)N

(T b)

= −

(τb)

. (6.331)

This is the eigenvalue equation of the dielectric-coated conducting cylinder.

From (6.320) and (6.327) we have

+ τ

= k

− k

= ω

(µ² − µ

). (6.332)

The transverse wave numbers T and τ are determined by these two equations,

(6.331) and (6.332). Further, the longitudinal wave number β is obtained by

(6.320) or (6.327).

(2) TE Modes, U = 0

By following a similar procedure, we obtain the eigenvalue equation for the

TE modes in the dielectric-coated conducting cylinder

(T a)J

(T b) − J

(T a)N

(T b)

(T a)J

(T b) − J

(T a)N

(T b)

= −

(τb)

. (6.333)

6.8 Dielectric Resonators 387

The ﬁelds outside the dielectric coating decay oﬀ down the transverse

direction ρ, and the phase velocity of the traveling wave in the longitudinal

direction z is less than the speed of light in free space. This kind of wave is

known as a slow wave or surface wave, which will be discussed in more detail

in the next chapter.

It can be shown that for axial asymmetrical, i.e., angular nonuniform

ﬁelds, the TE or TM modes alone cannot satisfy the dielectric boundary, and

the modes are HEM modes. Refer to problem 6.10.

6.8 Dielectric Resonators

Dielectric resonators are dielectric objects such as spheres, disks, cylinders,

or parallelepipeds of high permittivity, which can be used as energy storage

devices [11]. Dielectric resonators were ﬁrst prop osed in 1939 [85], but for

about 25 years the theoretical proposal failed to excite a constant interest

because the material with the required high permittivity and low loss was

unknown. In the 1960s, the introduction of new materials, such as rutile,

of high dielectric constant (²

≈ 100) renewed the interest in dielectric res-

onators. However, resulting from the high temperature coeﬃcient of rutile,

poor frequency stability temporarily prevented the development of devices

toward practical applications. In the 1970s, low-loss, high-permittivity and

temperature-stable ceramics, such as barium titanate and zirconium titanate,

were ﬁnally introduced and applications of such materials were made in the

design of high-performance microwave devices such as oscillators and ﬁlters.

Dielectric resonators are small, lightweight, high-Q, temperature-stable, and

low-cost devices, they are good for design and fabrication of hybrid and mono-

lithic microwave integrated circuits and are compatible with semiconductor

devices.

For the analysis of the dielectric resonator, three approximate approaches

are given:

1. The op en-circuit boundary approximation or perfect-magnetic-

conductor (PMC) wall approach,

2. The cutoﬀ-waveguide-terminal approach,

3. The cutoﬀ-waveguide, cutoﬀ-radial-line approach.

6.8.1 Perfect-Magnetic-Conductor Wall Approach

The dielectric constant of the material used in a dielectric resonator must be

large, usually 30 or larger. Under this condition, the dielectric–air bound-

ary acts almost like a perfect-electric-conductor (PEC) wall or short-circuit

boundary when looking from the air to the dielectric, and almost like an

open-circuit boundary when looking from the dielectric to the air, which

388 6. Dielectric Waveguides and Resonators

Figure 6.27: Dielectric resonator (left) and perfect-magnetic-conductor wall

approach (right).

causes total internal reﬂections resulting in the conﬁnement of energy in the

dielectric object. The reﬂection coeﬃcient of a plane wave normally incident

at the dielectric to air boundary is given by

Γ =

− η

+ η

√

− 1

√

+ 1

À1

≈ +1,

where η

and η denote the wave impedance of the air and the dielectric, re-

spectively. Hence the dielectric to air boundary can be approximated by a

hypothetical perfect magnetic conductor (PMC) surface, on which the tan-

gential components of the magnetic ﬁeld is required to vanish, i.e., an open-

circuit surface. Note that, the air to dielectric boundary, on the contrary,

can be approximated by a perfect electric conductor (PEC) surface, i.e., a

short-circuit surface.

Although the PMC wall model may not lead to the exact solution, it gives

a ﬁrst-order approximation to the problem and supplies reasonable results.

We choose the circular cylindrical dielectric resonator as an example, as

shown in Fig. 6.27. The permittivity of the material is large enough and the

permeability is µ

(1) TE Modes, U = 0

Dielectric cylinder including the axis at the center, the coeﬃcient of the

Neumann function must be zero. The angular dependence of the ﬁeld function

may either be even symmetrical or odd symmetrical, here we choose the even

symmetrical functions. The function V are given as follows

V = AJ

(T ρ) cos nφ cos(βz + θ), (6.334)

The relation between β and T is given by

+ T

= k

= ω

². (6.335)

6.8 Dielectric Resonators 389

According to the PMC wall model, the boundary conditions on the side

surface of the cylinder, ρ = a, are

(a) = T

V (a) = 0

(a) =

∂

∂φ ∂z

= 0







→ J

(T a) = 0, T

, (6.336)

where x

is the mth root of the Bessel function of the nth order. Similarly,

the boundary conditions on the end surfaces, z = ±l/2, are

(−l/2) =

∂

∂φ ∂z

−l/2

= 0

(−l/2) =

∂

∂ρ ∂z

−l/2

= 0











→ sin(−βl/2+θ) = −sin(βl/2−θ) = 0,

(l/2) =

∂

∂φ ∂z

l/2

= 0

(l/2) =

∂

∂ρ ∂z

l/2

= 0











→ sin(βl/2 + θ) = 0,

which are easily put in the form

sin(βl/2) cos θ − cos(βl/2) sin θ = 0, (6.337)

and

sin(βl/2) cos θ + cos(βl/2) sin θ = 0. (6.338)

From these it follows that

sin(βl/2) cos θ = 0, cos(βl/2) sin θ = 0. (6.339)

There are two ways to satisfy the above two equations simultaneously:

θ = 0, sin(βl/2) = 0, β = pπ/l, p = 0, 2, 4, ···, for even modes.

2. θ = π/2, cos(βl/2) = 0, β = pπ/l, p = 1 , 3, 5, ···, for odd modes.

Finally we have the natural angular frequency of the TE

nmp

mode

nmp

√

pπ

. (6.340)

The lowest TE mode is the TE

010

mode, where n = 0, m = 1, p = 0, β = 0

and the natural frequency is

010

√

2.405

√

. (6.341)

The ﬁeld components of TE

010

mode are as follows

= −jωµ

T AJ

(T ρ), H

= T

(T ρ). (6.342)

390 6. Dielectric Waveguides and Resonators

(2) TM Modes, V = 0

The function U and the ﬁeld components are written in the form

U =AJ

(T ρ) cos nφ cos(βz + θ), (6.343)

The relation between β and T is also given by (6.335)

The boundary condition on the side surface of the cylinder, ρ = a, is

(a) =

jω²

∂U

∂φ

= 0 → J

(T a) = 0, T

, (6.344)

where y

is the mth root of the derivative of the Bessel function of the nth

order. The boundary conditions on the end surfaces, z = ±l/2, are

(−l/2) = −jω²

∂U

∂ρ

−l/2

= 0

(−l/2) =

jω²

∂U

∂φ

−l/2

= 0











→ cos(βl/2 − θ) = 0,

(l/2) = −jω²

∂U

∂ρ

l/2

= 0

(l/2) =

jω²

∂U

∂φ

l/2

= 0











→ cos(βl/2 + θ) = 0,

which give

β = pπ/l,

p=0, 2, 4, ···, for even modes,

p=1, 3, 5, ···, for odd modes,

Finally we have the natural angular frequency of TM

nmp

mode

nmp

√

pπ

. (6.345)

The lowest TM mode is the TM

111

mode, which is not a circumferential

symmetrical mode, where n = 1, m = 1, p = 1 and the natural frequency is

111

√

1.841

. (6.346)

When the resonator has a small length-radius ratio, the TE

010

mode is the

dominant mode, whereas for the resonator with a large length:radius ratio,

the TM

111

mode is the dominant mode.

A comparison of the resonant frequencies and ﬁeld components of the di-

electric resonator modeled by a PMC wall with those of the metallic resonator

modeled by a PEC wall shows that the TE modes in one type of resonator

are the dual modes of the TM modes in the other, and vice versa. The two

types of boundary conditions, a PMC or open-circuit surface and a PEC or

short-circuit surface, are dual boundary conditions.