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

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5.3 Waveguides and Cavities in Rectangular Coordinates 261

Figure 5.17: Mode spectrum of a rectangular resonant cavity.

mnp

√

µ²

√

µ²

. (5.109)

It can be seen from (5.102)–(5.107) that one of m and n is allowed to be

zero but p is not allowed to be zero. So the TE

101

mode is the lowest TE

mode when a > b. The distribution of the natural wavelength is shown in

Figure 5.17.

(2) TM Modes

The function V is zero and

U(x, y, z) = X(x)Y (y)Z(z)

= (A sin k

x + B cos k

x)(C sin k

y + D cos k

F e

+ Ge

−jk

.(5.110)

The boundary conditions on the walls at x = 0, x = a, y = 0, and y = b

are given in (5.47)–(5.50). The boundary conditions of function U on the

short-circuit surface at z = 0 and z = l are given by

∂U

∂z

z=0

= 0 −→ jk

F − jk

G = 0, G = F, (5.111)

∂U

∂z

z=l

= 0 −→ jk

− e

−jk

= −2k

sin k

l = 0, k

pπ

, (5.112)

where p is an integer. Substituting (5.47)–(5.50) and (5.111) and (5.112) into

(5.110), let U

= 2ACF , gives

U(x, y, z) = U

sin(k

x) sin(k

y) cos(k

z), (5.113)

The ﬁeld components are given by (4.141)–(4.146):

∂

∂x ∂z

= −k

cos(k

x) sin(k

y) sin(k

z), (5.114)

262 5. Metallic Waveguides and Resonant Cavities

Figure 5.18: Field maps in a rectangular resonant cavity.

∂

∂y ∂z

= −k

sin(k

x) cos(k

y) sin(k

z), (5.115)

= T

U =

+ k

sin(k

x) sin(k

y) cos(k

z), (5.116)

= jω²

∂U

∂y

= jω²k

sin(k

x) cos(k

y) cos(k

z), (5.117)

= −jω²

∂U

∂x

= −jω²k

cos(k

x) sin(k

y) cos(k

z), (5.118)

= T

V = 0, (5.119)

where k

= mπ/a, k

= nπ/b and k

= pπ/l. They are the same as those for

TE modes. Then the natural angular wave number and the natural angular

frequency are also the same as those for TE modes given in (5.108) and

(5.109). So the TE

mnp

and TM

mnp

modes are degenerate modes. It can be

seen from (5.114)–(5.119) that neither m nor n is allowed to be zero but p is

allowed to be zero. So the TM

110

mode is the lowest TM mode.

A typical mode spectrum of a rectangular cavity is shown in Figure 5.17.

The ﬁeld maps of some low-order modes are given in Fig. 5.18.

5.3 Waveguides and Cavities in Rectangular Coordinates 263

Figure 5.19: Instantaneous ﬁelds of the TE

101

mode in a rectangular cavity.

(3) The Dominant Mode

For a cavity with a > b and l > b, the TE

101

is the lowest mo de or dominant

mode, and the natural frequency and the natural wavelength are

101

√

µ²

, (5.120)

101

√

µ²

1/a

+ 1/l

. (5.121)

Putting m = 1, n = 0, p = 1 into (5.102)–(5.107), we have the ﬁeld compo-

nents of the TE

101

mode

= E

sin

, (5.122)

= −j

ωµl

sin

cos

, (5.123)

= j

ωµa

cos

sin

. (5.124)

The instantaneous ﬁelds of the TE

101

mode in a rectangular cavity with

respect to time are shown in Figure 5.19. It can be seen that the diﬀerence

between the phases of the electric and magnetic ﬁelds is π/2. The energy is

stored in the electric ﬁeld and the magnetic ﬁeld alternatively.

If we consider y as the longitudinal axis, the dominant mode becomes

110

264 5. Metallic Waveguides and Resonant Cavities

Figure 5.20: (a) Sectorial cavity and (b) sectorial waveguide.

5.4 Waveguides and Cavities in Circular

Cylindrical Coordinates

The waveguides and cavities in the circular cylindrical coordinates include

the circular waveguide and circular cylindrical cavity, coaxial line and coaxial

cavity, sectorial waveguide and sectorial cavity, radial transmission line, and

cylindrical horn waveguide.

5.4.1 Sectorial Cavities

The most general metallic electromagnetic structure in circular cylindrical

system is the sectorial cross-sectional cylindrical cavity, shown in Fig. 5.20(a).

(1) TM Modes or E Modes

In circular cylindrical coordinates, for TM modes, V (ρ, φ, z) = 0, and

U (ρ, φ, z) = R(ρ)Φ(φ)Z(z)

= [AJ

(T ρ) + BN

(T ρ)]

jνφ

+ De

−jνφ

¢¡

F e

jβz

+ Ge

−jβz

, (5.125)

where

β = k

, β

+ T

= k

= ω

µ². (5.126)

The boundary condition for function U on the short-circuit surfaces are

ρ=a

= 0 −→ R(a) = AJ

(T a) + BN

(T a) = 0, (5.127)

ρ=b

= 0 −→ R(b) = AJ

(T b) + BN

(T b) = 0, (5.128)

φ=0

= 0 −→ Φ(0) = C + D = 0, D = −C, (5.129)

φ=α

= 0 −→ Φ(α) = Ce

jνα

+ De

−jνα

= 0, (5.130)

∂U

∂z

z=0

= 0 −→ Z

(0) = jβ(F − G) = 0, G = F, (5.131)

∂U

∂z

z=l

= 0 −→ Z

(l) = jβ

F e

jβl

− Ge

−jβl

= 0. (5.132)

5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 265

From (5.129) and (5.130), we have

jνα

− e

−jνα

= 0, sin να = 0, ν =

nπ

, n is an integer. (5.133)

From (5.131) and (5.132), we have

jβl

− e

−jβl

= 0, sin βl = 0, β =

pπ

, p is an integer. (5.134)

Finally, from (5.127) and (5.128), we have

B = −A

(T a)

, (5.135)

and

(T b)N

(T a) − J

(T a)N

(T b) = 0. (5.136)

This is the eigenvalue equation or characteristic equation of the TM mode in

a sectorial cavity, which is a transcendental equation of the Bessel functions

of the νth order and has inﬁnite discrete roots. The mth root of the equation

of the νth order is denoted by T

νm

The natural angular wave numbers and the natural angular frequencies

can then be found from the resonant conditions (5.133), (5.134), and (5.136)

as follows:

νmp

+ T

νm

, ω

νmp

√

µ²

+ T

νm

. (5.137)

Applying the above results into (5.125) gives

U(ρ, φ, z) = U

(T a)J

(T ρ) − J

(T a)N

(T ρ)] sin(νφ) cos(βz), (5.138)

where U

= 4jACF/N

(T a).

Substituting (5.138) into the expressions for the ﬁeld components in cir-

cular cylindrical coordinates in terms of Borgnis’ potentials, (4.190)–(4.195),

we have

= −βT U

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(νφ) sin(βz), (5.139)

= −

βν

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(νφ) sin(βz), (5.140)

= T

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(νφ) cos(βz), (5.141)

jω²ν

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(νφ) cos(βz), (5.142)

= −jω²T U

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(νφ) cos(βz),(5.143)

= 0. (5.144)

It can be seen from (5.133) that ν is not necessarily an integer, so the

Bessel functions in the ﬁeld expressions are not necessarily with integer order.

The TM

νm

modes are also denoted by E

νm

modes, because only electric

ﬁelds have longitudinal components in the waneguide.

266 5. Metallic Waveguides and Resonant Cavities

(2) TE Modes or H Modes

For TE modes, U(ρ, φ, z) = 0, and

V (ρ, φ, z) = R(ρ)Φ(φ)Z(z)

= [AJ

(T ρ) + BN

(T ρ)]

jνφ

+ De

−jνφ

¢¡

F e

jβz

+ Ge

−jβz

, (5.145)

where

β = k

, β

+ T

= k

= ω

µ². (5.146)

The boundary conditions for function V on the short-circuit surfaces are

as follows:

∂V

∂ρ

ρ=a

= 0 −→ R

(a) = AJ

(T a) + BN

(T a) = 0, (5.147)

∂V

∂ρ

ρ=b

= 0 −→ R

(b) = AJ

(T b) + BN

(T b) = 0, (5.148)

∂V

∂φ

φ=0

= 0 −→ Φ

(0) = C − D = 0, D = C, (5.149)

∂V

∂φ

φ=α

= 0 −→ Φ

(α) = Ce

jνα

− De

−jνα

= 0, (5.150)

V |

z=0

= 0 −→ Z(0) = F + G = 0, G = −F, (5.151)

V |

z=l

= 0 −→ Z(l) = F e

jβl

+ Ge

−jβl

= 0. (5.152)

From the above boundary equations, we have

ν =

nπ

, n is an integer, β =

pπ

, p is an integer, (5.153)

B = −A

(T a)

, (5.154)

and

(T b)N

(T a) − J

(T a)N

(T b) = 0. (5.155)

This is the eigenvalue equation of the TE modes in a sectorial cavity; The

mth root of the equation of νth order is denoted by T

νm

The natural angular wave numbers and the natural angular frequencies

can then be found from the resonant conditions (5.153) and (5.155) as follows

νmp

+ T

νm

, ω

νmp

√

µ²

+ T

νm

. (5.156)

Substituting the above results into (5.145) gives

V (ρ, φ, z) = V

(T a)J

(T ρ) − J

(T a)N

(T ρ)] cos(νφ) sin(βz), (5.157)

where U

= 4jACF/N

(T a).

5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 267

Substituting (5.157) into the expressions for the ﬁeld components in cir-

cular cylindrical coordinates (4.190)–(4.195), we have

jωµν

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(νφ) cos(βz), (5.158)

= jωµT V

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(νφ) sin(βz), (5.159)

= 0, (5.160)

= βT V

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(νφ) cos(βz), (5.161)

= −

βν

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(νφ) cos(βz), (5.162)

= T

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(νφ) sin(βz). (5.163)

The TE

νm

modes are also denoted by H

νm

modes, because only magnetic

ﬁelds have longitudinal components in the waneguide.

The ﬁelds in a sectorial cavity are standing waves in three directions, the

ρ dependence is a cylindrical standing wave represented by Bessel functions.

The sectorial cavity is the general electromagnetic structure in circular

cylindrical coordinates. The other waveguides and cavities of cylindrical ge-

ometry can be developed from it by giving special dimensions and angles.

5.4.2 Sectorial Waveguides

If the system is unbounded in the longitudinal direction, z, it becomes a sec-

torial waveguide, see Fig. 5.20(b). The boundary equations (5.131), (5.132),

(5.151), and (5.152) at z = 0 and z = l are no longer valid, β becomes a

continuous value. Considering the traveling wave along +z in an inﬁnitely

long waveguide; we have the function U of the TM modes,

U(ρ, φ, z) = U

(T a)J

(T ρ) − J

(T a)N

(T ρ)] sin(νφ)e

−jβz

, (5.164)

and the function V of the TE modes,

V (ρ, φ, z) = V

(T a)J

(T ρ) − J

(T a)N

(T ρ)] cos(νφ)e

−jβz

. (5.165)

The ﬁeld components can then be obtained by substituting one of the above

two expressions into (4.196)–(4.201).

The eigenvalue equations for sectorial waveguides are the same as those

for sectorial cavities, (5.136) for TM modes and (5.155) for TE modes. The

eigenvalue equation or characteristic equation is also known as the dispersion

equation for the transmission system. The mth root of (5.136) for the νth

order is the cutoﬀ angular wave number for the TM

νm

mode, denoted by

νm

, and the mth root of (5.155) for the νth order is the cutoﬀ angular

wave number for the TE

νm

mode, denoted by T

νm

. The longitudinal phase

coeﬃcient is then given by

νm

−T

νm

= ω

µ² −T

νm

, (5.166)

268 5. Metallic Waveguides and Resonant Cavities

Figure 5.21: Field maps in a sectorial waveguide.

νm

−T

νm

= ω

µ² −T

νm

. (5.167)

The ﬁeld maps for some lower-order modes in the cross section of the

sectorial waveguide are shown in Fig. 5.21. The ﬁeld distributions are similar

to those for rectangular waveguides, especially when a − b ¿ a and α ¿ 2π.

The lowest mode is TE

when α is small and a − b is large, on the

contrary, the lowest mode becomes TE

when α is large and a − b is small.

5.4.3 Coaxial Lines and Coaxial Cavities

If the sectorial angle of the sectorial waveguide is enlarged to α = 2π, then ν

becomes

ν =

nπ

This is not a coaxial line but a coaxial line with a conducting plate in a plane

of equal φ, as shown in Fig. 5.22(a).

In a coaxial line shown in Fig. 5.22(b), the ﬁeld must be continuous around

the whole circle, and ν must satisfy

ν(φ + α) = νφ + 2nπ, ν =

2nπ

= n, n = 0, 1, 2 ···.

The radial functions of the ﬁelds become Bessel functions with integer order.

and the angular functions of the ﬁelds become cos nφ or sin nφ or the linear

combination of them, which depends upon the choice of the initial orientation

of the coordinate φ.

(1) TM and TE Modes in Coaxial Lines

In problems including the whole circumference, the orientation of φ = 0 can

be chosen arbitrarily. Here, the orientation of φ = 0 is chosen so that the

5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 269

Figure 5.22: (a) Sectorial waveguide with α = 2π, and (b) a coaxial line.

Borgnis’ function becomes an even function with respect to φ. Considering

ν becomes integer n, and the short-circuit boundary condition on ρ = a, the

expression for U function (5.164) for TM modes becomes

U(ρ, φ, z) = U

(T a)J

(T ρ) − J

(T a)N

(T ρ)] cos(nφ)e

−jβz

. (5.168)

Substituting (5.168) into the expressions for the ﬁeld components of the

traveling-wave solution in circular cylindrical coordinates (4.196)–(4.201), we

have the ﬁeld-component expressions for the TM modes:

= −jβT U

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(nφ)e

−jβz

, (5.169)

= j

βn

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(nφ)e

−jβz

, (5.170)

= T

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(nφ)e

−jβz

, (5.171)

= −

jω²n

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(nφ)e

−jβz

, (5.172)

= −jω²T U

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(nφ)e

−jβz

, (5.173)

= 0. (5.174)

The dispersion equation for the TM mode in a coaxial line is just that for

a sectorial waveguide and sectorial cavity (5.136) but the order of the Bessel

functions becomes an integer,

(T b)N

(T a) − J

(T a)N

(T b) = 0. (5.175)

Similarly, the function V for TE mode is given by

V (ρ, φ, z) = V

(T a)J

(T ρ) − J

(T a)N

(T ρ)] cos(nφ)e

−jβz

. (5.176)

Substituting (5.176) into (4.196)–(4.201), we have the ﬁeld-component ex-

pressions of the TE modes:

jωµn

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(nφ)e

−jβz

, (5.177)

270 5. Metallic Waveguides and Resonant Cavities

= jωµT V

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(nφ)e

−jβz

, (5.178)

= 0, (5.179)

= −jβT V

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(nφ)e

−jβz

, (5.180)

jβn

(T a)J

(T ρ)−J

(T a)N

(T ρ)] sin(nφ)e

−jβz

, (5.181)

= T

(T a)J

(T ρ)−J

(T a)N

(T ρ)] cos(nφ)e

−jβz

. (5.182)

The dispersion equation for the TE mode in a coaxial line is also similar

to that for sectorial waveguide and is given by

(T b)N

(T a) − J

(T a)N

(T b) = 0. (5.183)

Dispersion equations (5.175) and (5.183) are two transcendental equations

of the Bessel functions of the nth order and both have inﬁnite discrete roots.

Let

x = T a, or y = T a, c =

Equations (5.175) and (5.183) become

(x)N

(cx) − J

(cx)N

(x) = 0, (5.184)

(y)N

(cy) − J

(cy)N

(y) = 0. (5.185)

The mth root of (5.184) for the nth order is denoted by x

and the mth root

of (5.185) for the nth order is denoted by y

. They are given in Table 5.1

[49, 67].

Table 5.1

The roots of J

(x)N

(cx) − J

(cx)N

(x) = 0

(c − 1)x

c nm = 01 11 21 02 12 22

1.2 3.140 3.146 3.161 6.282 6.285 6.293

1.5

3.135 3.161 3.237 6.280 6.293 6.332

2.0

3.123 3.197 3.400 6.273 6.312 6.430

3.0

3.097 3.271 6.258 6.357

4.0 3.073 3.336 6.243 6.403

The roots of J

(y)N

(cy) − J

(cy)N

(y) = 0

(c + 1)y

(c − 1)y

c nm = 11 21 01 12 22 02

1.2 2.002 4.006 3.145 3.151 3.167 6.285

1.5 2.013 4.020 3.161 3.188 3.270 6.293

2.0 2.031 4.023 3.197 3.282 3.500 6.312

3.0

2.056 3.908 3.271 3.516 6.357

4.0

2.055 3.760 3.336 3.753 6.403