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

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5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 281

Figure 5.31: Field patterns of some modes in a circular cylindrical cavity.

The ﬁeld patterns of some low-order modes in a circular cylindrical cavity

are given in Fig. 5.31.

The Q factor of the TM and TE modes in a circular cylindrical cavity

due to surface loss may be evaluated by substituting the ﬁeld-component

expressions into (5.25). The ﬁnal result for the Q of TE modes is

nmp

1 − n

+ (pπa/l)

+ (2a/l)(pπa/l)

+ (1 − 2a/l)(npπa/y

, (5.223)

and the Q of TM modes can be evaluated to give

nmp

+ (pπa/l)

1 + δ

a/l

, (5.224)

where η =

µ/², R

ωµ/2σ = 1/σδ, η/2R

= λ/2πδ, and

1 p = 0,

2 p 6= 0.

An interesting mode in circular cylindrical cavity is the TE

0mp

mode. It

has only circumferential currents in both the cylindrical wall and the end

plates. Thus, if a cavity for such a mode is tuned by moving the end plate,

a good contact between the end plate and the cylindrical wall is not needed

since no current ﬂows across the boundary line. Furthermore, the Q factor

282 5. Metallic Waveguides and Resonant Cavities

of TE

0mp

mode is considerably higher than that of the other modes. So

the TE

0mp

modes, especially the lowest-order one, TE

011

mode, are widely

used as wave meter, echo box, frequency standard and other high-Q resonant

devices.

The other interesting mode in a circular cylindrical cavity is the TM

010

mode. It is the simplest mode, analogous to the TE

011

or TM

110

mode in

a rectangular cavity. We can see from Fig. 5.31 that the longitudinal elec-

tric ﬁeld has a maximum at the center of the cavity and a circumferential

magnetic ﬁeld surrounds the electric ﬁeld. Neither ﬁeld varies in the ax-

ial or circumferential direction. This mode may be considered as a TM

mode in a circular waveguide operating at cutoﬀ, or it may be thought of

as the standing wave produced by inward and outward radial propagating

waves of the cylindrical TEM mo de in the radial transmission line, refer to

Section 5.4.6. This cavity and its deformation, the reentrant or small-gap

cavity, are usually used in electron devices in which the carriers, i.e., charges

or holes interact with the longitudinal electric ﬁeld at the center of the cavity,

refer to Section 5.6.

5.4.5 Cylindrical Horn Waveguides and

Inclined-Plate Lines

(1) Cylindrical Waves in Cylindrical Horn Waveguides

If there is no boundary along the ρ direction in the sectorial structure de-

scribed in Section 5.4.1, it becomes a cylindrical horn waveguide, shown in

Fig. 5.32(a). The expressions for U of TM modes and V of TE modes are

the same as those for sectorial cavities, (5.125) and (5.145), respectively. The

boundary conditions in the φ and z directions, (5.129)–(5.132) and (5.149)–

(5.152) are still valid but the boundary conditions in the ρ direction, (5.127)

and (5.128), are no longer valid, and the transverse angular wave number T

becomes continuous. The functions U and V become

U(ρ, φ, z) = [AJ

(T ρ) + BN

(T ρ)] sin(νφ) cos(βz), (5.225)

V (ρ, φ, z) = [AJ

(T ρ) + BN

(T ρ)] cos(νφ) sin(βz), (5.226)

where

ν = nπ/α, β = k

= pπ/l , T

= k

− β

This two expressions may also be written in terms of Hankel functions

U(ρ, φ, z) =

−

(1)

(T ρ) + U

(2)

(T ρ)

sin(νφ) cos(βz), (5.227)

V (ρ, φ, z) =

−

(1)

(T ρ) + V

(2)

(T ρ)

cos(νφ) sin(βz), (5.228)

where

= (A+jB)/2, U

−

= (A−jB)/2, V

= (A+jB)/2, V

= (A−jB)/2.

5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 283

Figure 5.32: (a) Cylindrical horn waveguide and (b) an inclined-plate line.

In the above expressions, the function H

(1)

(T ρ) represents the cylindrical

wave along the −ρ direction and the function H

(2)

(T ρ) represents the cylin-

drical wave along the +ρ direction.

Substituting (5.227) or (5.228) into the ﬁeld-component expressions in

terms of Borgnis’ functions, (4.190)–(4.195), we may have the ﬁelds of the

TM and TE modes in a cylindrical horn waveguide.

In a cylindrical horn waveguide, the cylindrical traveling wave propagates

along the ±ρ direction with angular wave numb er T , and β becomes the

cutoﬀ angular wave number. When k > β, T is real and the dependence of

the ﬁeld along ρ becomes a Hankel function. This is the wave-propagation

state. When k < β, T is imaginary and the dependence of the ﬁeld along ρ

becomes a modiﬁed Bessel function. This is the cutoﬀ state.

(2) Cylindrical Waves in Inclined-Plate Lines

If there is no boundary along both the ρ and the z direction in the sectorial

structure, it becomes an inclined-plate line, shown in Fig. 5.32(b). We are

interested in the waves with a uniform ﬁeld in the z direction, in which

β = 0, T = k.

The expressions for U and V , (5.227) and (5.228), become

U(ρ, φ, z) =

−

(1)

(kρ) + U

(2)

(kρ)

sin(νφ), (5.229)

V (ρ, φ, z) =

−

(1)

(kρ) + V

(2)

(kρ)

cos(νφ). (5.230)

The ﬁeld components of the wave with V = 0 become

= k

−

(1)

(kρ) + U

(2)

(kρ)

sin(νφ), (5.231)

jω²ν

−

(1)

(kρ) + U

(2)

(kρ)

cos(νφ), (5.232)

= −jω²k

−

(1)

(kρ) + U

(2)

(kρ)

sin(νφ), (5.233)

284 5. Metallic Waveguides and Resonant Cavities

Figure 5.33: The ﬁelds of the TEM

(ρ)

mode in an inclined-plate line.

= 0, E

= 0, H

= 0.

The modes of this class are known as TM or E modes with respect to z as

the propagation direction. Now, the propagation direction, i.e., longitudinal

direction becomes ρ. The modes of this class with E

= 0 and H

6= 0 become

TE or H modes with respect to ρ as the propagation direction, denoted by

(ρ)

or H

(ρ)

modes. With this point of view, the TM or E modes with

respect to z as the propagation direction are denoted by TM

(z)

or E

(z)

modes.

The ﬁeld components of the wave with U = 0 become

jωµν

−

(1)

(kρ) + UV + H

(2)

(kρ)

sin(νφ), (5.234)

= jωµk

−

(1)

(kρ) + V

(2)

(kρ)

cos(νφ), (5.235)

= k

−

(1)

(kρ) + V

(2)

(kρ)

cos(νφ), (5.236)

= 0, H

= 0.

The mo des of this class are known as the TE or H modes with respect to z,

but they become TM or E modes with respect to ρ, denoted by TM

(ρ)

or E

(ρ)

modes. The TE or H modes with respect to z as the propagation direction

are denoted by TE

(z)

or H

(z)

modes.

If ν = 0, all ﬁeld components of the TE

(ρ)

mode become zero, but the

ﬁeld components of the TM

(ρ)

mode exist

= −jk

−

(1)

(kρ) + V

(2)

(kρ)

, (5.237)

= k

−

(1)

(kρ) + V

(2)

(kρ)

. (5.238)

This is the φ-polarized uniform cylindrical TEM wave or TEM

(ρ)

mode. The

ﬁeld map of this mode is shown in Fig. 5.33.

The theory for cylindrical horn waveguides and inclined-plate lines can be

used in the analysis of the transition between two waveguides with diﬀerent

cross-sections.

5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 285

Figure 5.34: (a) Radial line and (b) radial line cavity.

5.4.6 Radial Transmission Lines and Radial-Line

Cavities

If in the cylindrical horn waveguide, α = 2π and the conducting plate at the

constant-φ plane is removed, one has a pair of circular parallel plates with

a separation l, which is known as the radial transmission line or simply the

radial line as shown in Fig. 5.34(a). There are cylindrical traveling waves in

the +ρ and −ρ directions in the radial line. If the radial line is shorted at a

constant-ρ surface, it becomes a radial line cavity, shown in Fig. 5.34(b).

(1) TE and TM Modes in Radial Lines

In the expressions for U and V of the cylindrical horn waveguide, (5.227) and

(5.228), if ν is an integer, ν = n, we have the expressions for U and V of the

radial line in the form of traveling-waves as follows:

U(ρ, φ, z) =

−

(1)

(T ρ) + U

(2)

(T ρ)

cos(nφ) cos(β

z), (5.239)

V (ρ, φ, z) =

−

(1)

(T ρ) + V

(2)

(T ρ)

cos(nφ) sin(β

z), (5.240)

where β

= pπ/l becomes the two-dimensional eigenvalue or cutoﬀ angular

wave number of the radial guided wave. The cutoﬀ wavelength of the radial

line becomes

2π

√

. (5.241)

The phase coeﬃcient of the cylindrical traveling wave becomes

T =

− β

− (pπ/l)

. (5.242)

Function H

(1)

(T ρ) represents the inward cylindrical wave in the −ρ direction

and H

(2)

(T ρ) represents the outward cylindrical wave in the +ρ direction.

286 5. Metallic Waveguides and Resonant Cavities

Substituting (5.239) or (5.240) into the expressions (4.190)–(4.195), we

have the ﬁeld components of the modes of V = 0 or modes of U = 0 in the

radial line. For example, the ﬁeld components of the circumference uniform

modes of V = 0, n = 0, are given by

U =

−

(1)

(T ρ) + U

(2)

(T ρ)

cos(βz), (5.243)

= T

−

(1)

(T ρ) + U

(2)

(T ρ)

cos(βz), (5.244)

= T β

−

(1)

(T ρ) + U

(2)

(T ρ)

sin(βz), (5.245)

= jω²T

−

(1)

(T ρ) + U

(2)

(T ρ)

cos(βz). (5.246)

The functions U and V in the radial line may also be written in the form

of standing wave as

U(ρ, φ, z) = [AJ

(T ρ) + BN

(T ρ)] cos(nφ) cos(βz), (5.247)

V (ρ, φ, z) = [AJ

(T ρ) + BN

(T ρ)] cos(nφ) sin(βz). (5.248)

The ﬁeld components of the circumference uniform modes of V = 0 in the

form of standing waves become

U = [AJ

(T ρ) + BN

(T ρ)] cos(βz), (5.249)

= T

[AJ

(T ρ) + BN

(T ρ)] cos(βz), (5.250)

= T β[AJ

(T ρ) + BN

(T ρ)] sin(βz), (5.251)

= jω²T [AJ

(T ρ) + BN

(T ρ)] cos(βz), (5.252)

= 0, H

= 0.

These modes of V = 0 are TM modes with respect to both z and ρ since

= 0 and H

= 0.

The modes of U = 0 can also derived by the similar procedure.

(2) Cylindrical TEM Mode in Radial Lines

In the expressions for the modes of V = 0 in radial lines, if n = 0 and p = 0,

i.e., β = 0, T = k, expressions (5.243)–(5.252) become

U = AJ

(kρ) + BN

(kρ) = U

−

(1)

(kρ) + U

(2)

(kρ), (5.253)

= k

[AJ

(kρ) + BN

(kρ)] = k

−

(1)

(kρ) + U

(2)

(kρ)

, (5.254)

[AJ

(kρ) + BN

(kρ)] =

−

(1)

(kρ) + U

(2)

(kρ)

. (5.255)

= 0, E

= 0, H

= 0.

This mode is the TM or H mode with respect to z but the TEM mode with

respect to ρ, denoted by TEM

(ρ)

mode. There is no longitudinal electric ﬁeld

5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 287

Figure 5.35: Fields of the TEM mode in a radial line.

component nor longitudinal magnetic ﬁeld component, i.e., no ﬁelds in the ρ

direction. It is a cylindrical TEM mode.

The other cylindrical TEM mode is the TEM

(ρ)

mode in an inclined-plate

line, given in (5.237) and (5.238). The TEM

(ρ)

mode in a radial line, (5.254)

and (5.254), and the TEM

(ρ)

mode in an inclined-plate line, (5.237) and

(5.238), are dual modes. The ﬁeld map of the TEM

(ρ)

mode in a radial line

is given in Fig. 5.35.

The condition for single-TEM-mode transmission in a radial line can be

found from (5.241):

l <

√

. (5.256)

(3) Radial Line Cavities

A section of a radial line closed by short-circuit surfaces at ρ = b and ρ = a,

b < a, becomes a radial line cavity or toroidal cavity. In fact, it is the same

as a coaxial cavity with short-circuit end plates.

The radial line cavity with b = 0 becomes a circular cylindrical cavity,

in which the axis ρ = 0 is included in the ﬁeld region and the coeﬃcients

of N

(T ρ) become zero. For example, according to (5.254) and (5.255), the

ﬁelds of the TEM

mode in a radial line cavity are given by

= k

(kρ) = E

(kρ), (5.257)

288 5. Metallic Waveguides and Resonant Cavities

(kρ) =

(kρ). (5.258)

Applying the boundary condition at ρ = a, we have

(ka) = 0, k =

, (5.259)

where x

is the mth root of the Bessel function of the zeroth order.

We can have the same result from the expression for the circular cylin-

drical cavity (5.217) of Section 5.4.4 by letting n = 0 and β = 0. So the

TEM

mode in a radial line cavity is just the same as the TM

0m0

or E

0m0

mode in a circular cylindrical cavity. The ﬁeld pattern of the TM

010

mode

in a circular cylindrical cavity, i.e., the TEM

mode in a radial line cavity is

given in Fig. 5.31.

5.5 Waveguides and Cavities in Spherical

Coordinates

The waveguides and cavities in spherical coordinates include the spherical

cavity and spherical radial waveguides such as the biconical line, coaxial

biconical line, wedge line, and spherical horn. Their boundary coincide with

the coordinate surfaces of the spherical coordinate system.

The solutions of the Helmholtz equations in spherical coordinates were

given by (4.231) and (4.232). The ﬁelds in spherical coordinates may be

classiﬁed into TM and TE modes according to the following criterion.

TM or E mode: H

= 0, E

6= 0, i.e., V = 0, U 6= 0.

TE or H mode: E

= 0, H

6= 0, i.e., U = 0, V 6= 0.

In spherical coordinates, r becomes the longitudinal direction, θ and φ

become the transverse directions.

5.5.1 Spherical Cavities

The ﬁeld region of a spherical cavity includes the polar axes, θ = 0 and θ = π,

so the co eﬃcient of the function Q

(cos θ) in the solution must be zero. It

also includes the origin, r = 0, so the coeﬃcient of the function N

n+1/2

(kr)

must also be zero. The ﬁeld region of a spherical cavity includes the whole

circumference in φ, so the orientation of φ = 0 may be chosen arbitrarily,

and m must be an integer. Here, the orientation of φ = 0 is chosen such

that Borgnis’ function becomes an even function respect to φ. The resulting

solutions U and V , (4.231) and (4.232), for a spherical cavity become

U =a

(kr)P

(cos θ) cos(mφ) = A

√

n+1/2

(kr)P

(cos θ) cos(mφ),

(5.260)

V = b

(kr)P

(cos θ) cos(mφ) = B

√

n+1/2

(kr)P

(cos θ) cos(mφ).

(5.261)

5.5 Waveguides and Cavities in Spherical Coordinates 289

Substituting (5.260) or (5.261) into (4.233)–(4.238), we have the ﬁeld com-

ponents of TM or TE modes in the spherical cavity.

For a spherical cavity enclosed by short-circuit surface at r = a, the

boundary conditions are

r=a

= 0 and E

r=a

= 0. (5.262)

For TE or H modes, the above b oundary conditions are satisﬁed when

V |

r=a

= 0, i.e.,

(ka) = 0, k

, (5.263)

where x

is the pth root of the eigenvalue equation

(x) = 0, refer to the

upper half of Table 5.3. [37]

Table 5.3

The roots of

(x) = 0

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

p = 1 4.493 5.763 6.988 8.183 9.356 10.513

p = 2

7.725 9.095 10.417 11.705 12.967 14.207

p = 3

10.904 12.323 13.698 15.040 16.355 17.648

p = 4 14.066 15.515 16.924 18.301 19.653 20.983

The roots of

(y) = 0

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

p = 1 2.744 3.870 4.973 6.062 7.140 8.211

p = 2

6.117 7.443 8.722 9.968 11.189 12.391

p = 3

9.317 10.713 12.064 13.380 14.670 15.939

p = 4 12.486 13.921 15.314 16.674 18.009 19.321

The natural angular frequency and the natural wavelength of TE

nmp

modes are

nmp

√

µ²

√

µ²

, λ

nmp

2πc

nmp

2πa

√

. (5.264)

For TM or E modes, the boundary conditions are

∂U

∂r

r=a

= 0, i.e.,

(ka) = 0, k

, (5.265)

where y

is the pth root of the eigenvalue equation

(y) = 0, refer to the

lower half of Table 4.3.

The natural angular frequency and the natural wavelength of TM

nmp

modes are

nmp

√

µ²

√

µ²

, λ

nmp

2πc

nmp

2πa

√

. (5.266)

290 5. Metallic Waveguides and Resonant Cavities

The natural frequency of the TE

nmp

and TM

nmp

modes are independent

of m, the azimuthal variation of the ﬁelds, so the modes in a spherical cavity

are always nth-order degenerate, since m ≤ n. On the other hand, there are

even and odd degenerate modes when m 6= 0. Furthermore, the orientation

of the polar axis is arbitrary settled, so the modes of diﬀerent orientations

of polarization are also degenerate. In conclusion, the mo des in a spherical

cavity are highly degenerate, because of the highly symmetrical conﬁguration

of the cavity.

The lowest TM mode in a spherical cavity is the TM

101

mode, in which

n = 1, m = 0 and p = 1, and the function U becomes

U = A

√

3/2

(kr)P

(cos θ).

Since

3/2

(kr) =

πkr

sin kr

− cos kr

and

(cos θ) = cos θ,

we have

U = U

sin kr

− cos kr

cos θ, (5.267)

where U

= A

2/πk. Substituting (5.267) into (4.233)–(4.238), we have the

ﬁeld components

∂

∂r

+ k

U = 2k

sin kr

−

cos kr

cos θ, (5.268)

∂

∂θ ∂r

= k

sin kr

−

cos kr

−

sin kr

sin θ, (5.269)

= −

jω²

∂U

∂θ

sin kr

−

cos kr

sin θ, (5.270)

where

k =

2.744

. (5.271)

The lowest TE mode in a spherical cavity is the TE

101

mode, in which

n = 1, m = 0 and p = 1, and the function V becomes

V = B

√

3/2

(kr)P

(cos θ) = V

sin kr

− cos kr

cos θ, (5.272)

where V

= B

2/πk. Substituting (5.272) into (4.233)–(4.238), we have the

ﬁeld components

∂

∂r

+ k

V = 2k

sin kr

−

cos kr

cos θ, (5.273)