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

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Chapter 7

Periodic Structures and

the Coupling of Modes

In microwave band, there are two types of charged-particle-ﬁeld interaction

devices, ampliﬁers and particle accelerators. In the ampliﬁer, kinetic energy

or potential energy of charged-particles is converted into the energy of the

ﬁeld, and the wave is strengthened. On the contrary, in the accelerator, the

energy of the ﬁeld is converted into the kinetic energy of charged-particles,

and the particle is accelerated.

During the operations of traveling wave interaction devices, such as

traveling-wave ampliﬁers or linear accelerators, the electron or ion beam is

to interact with an electromagnetic wave eﬀectively. For this purpose, the

charged particles (electrons or ions) need to be kept in phase with a retarding

ﬁeld in the ampliﬁers’ case or an accelerating ﬁeld in the accelerators’ case

over a long distance. This means that the phase velocity of the wave need

to be roughly equal to the average velocity of the charged particles, for the

phase velocity is the velocity with which an observer would have to move so

as to always be able to remain in the same phase of the wave. Since electrons

and ions can be accelerated only to velocities less than the velocity of light,

we need to look for electromagnetic structures capable of sustaining waves

propagating with phase velocities less than that of a plane wave in free space,

i.e., the speed of light. Such waves are called slow waves and the structures

capable of having slow waves propagating along them are called slow-wave

structures or slow-wave systems.

The slow waves are also used in many devices in which the electromagnetic

wave is to interact with a surface acoustic wave (SAW), a magnetostatic wave

(MSW), and those waves with phase velocities less than the speed of light.

In dielectric waveguides, the longitudinal phase velocity for the guided

mode is larger than the plane-wave phase velocity in the dielectric material

the core is made of, so one has fast waves in the core. For fast waves, the ﬁelds

402 7. Periodic Structures and the Coupling of Modes

are standing waves in the transverse direction. Although the longitudinal

phase velocity in the cladding is equal to that in the core, the permittivity

of the medium of which the cladding is made is less than that of the core,

so that the plane-wave phase velocity in the cladding is larger than that in

the core; consequently, the guided mode in the cladding for which the ﬁelds

are decaying ﬁelds in the transverse direction is a slow wave. Therefore, all

the dielectric waveguides, dielectric coated metallic planes, dielectric coated

metallic rods, and metallic waveguides with dielectric coating on the inner

walls can properly be viewed as slow-wave structures.

In this chapter, we will discuss the slow-wave structures with metallic

boundaries, which suit application in many devices, especially high-power

devices, for which small attenuation and high-power capacity are demanded.

In Chapter 4, we have seen that a system bounded by short-circuit or

open-circuit boundaries, for example uniform smooth conductors can sup-

port only TEM-wave and fast-wave modes, because the ﬁelds conﬁned by ho-

mogeneous boundary conditions must be Laplacian ﬁelds or standing waves.

Therefore, the boundaries of a system in which the slow waves can be sup-

ported must be impedance boundaries. Consequently, the metallic slow-wave

structures needs to be constructed with nonuniform or periodic boundaries

and is known as a periodic structure or periodic system. The structure can

be analyzed approximately as a uniform system when the spatial period is

much less than the guided wavelength, which means that the phase shift in

a period is inﬁnitesimally small. On the contrary, if the spatial period is

comparable to or larger than the guided wavelength, ﬁeld theory must be

developed for periodic systems.

Much of the mathematics and arguments employed in studying periodic

transmission structures is the same as used in studying the phenomena of

light (including x-rays) or electrons passing through a crystal lattice and the

artiﬁcial photonic crystals.

In the remainder of this chapter, a coupled-mode formalism in space is

given and, as examples, waveguide couplers and distributed feedback struc-

tures (DFB) are discussed. The coupled-mode theory is used not only for

treating electromagnetic wave modes, but also for studying all the phenom-

ena involving interaction of waves.

7.1 Characteristics of Slow Waves

7.1.1 Dispersion Characteristics

The relation between phase velocity v

and frequency f is known as dispersion

Characteristics or dispersion relations of the transmission system.

Alternative expressions for the dispersion characteristics are the ω–β di-

agram and the k–β diagram.

We know that the slope of the straight line connecting the origin and a

7.1 Characteristics of Slow Waves 403

Figure 7.1: Dispersion curves, phase velocity, and group velocity for FW (a)

and BW (b) of guided wave structures.

certain point on the ω–β curve represents the phase velocity v

, and that the

slope of the line tangential to the curve at that point represents the group

velocity v

. The corresponding slopes deﬁned on the k–β diagram represent

the slow-wave ratios v

/c and v

/c, respectively.

Two typical k–β diagrams describing the characteristics of slow-wave

structures are shown in Fig. 7.1. It is easily seen that for the wave shown in

Figure 7.1(a), the phase velocity points in the same direction as the group

velocity. This kind of wave is known as a forward wave, denoted by FW.

All guided modes in common transmission lines, metallic waveguides, and

dielectric waveguides discussed in the previous chapters are forward waves.

On the contrary, in Problem 3.7, we found that for the transmission line

of high-pass ﬁlter type, which consists of distributed series capacitance and

shunt inductance, the phase coeﬃcient β decreases with frequency. That is

to say, the slope of the line tangential to the ω–β curve is negative, so that

the group velocity and the phase velocity are in opposite directions. This

kind of wave is known as a backward wave, denoted by BW; see Fig. 7.1(b).

Generally speaking, in a guided wave system, some of the modes are forward

waves and some are backward waves.

The k–β curve in the upper half of the plane above 45

◦

line, i.e., k = β

line, or area covered by k > β, represents a fast wave, and that in the lower

half of the plane below 45

◦

line, or area covered by k < β, represents a slow

wave.

7.1.2 Interaction Impedance

In traveling-wave ampliﬁers, particle accelerators and other active devices,

the charged particle (electron or ion) beam is to interact with the longitu-

dinal component of the electric ﬁeld. The eﬀectiveness of the interaction

404 7. Periodic Structures and the Coupling of Modes

is described by a parameter called the coupling impedance or interaction

impedance, which is a measure of the strength of the electric ﬁeld, usually

the z component, of a given mode referred to the total power ﬂow carried by

the mode.

K =

where U denotes the amplitude of the eﬀective voltage across two points

separated by a quarter wavelength along the longitudinal direction:

U =

dz =

sin

2πz

dz =

By substituting the expression for U into the expression for K, we have

K =

2β

, (7.1)

where E

is the amplitude of the longitudinal electric ﬁeld, P is the total

power ﬂow carried by the given mode and W is the energy of the mode stored

in the system of unit length.

7.2 A Corrugated Conducting Surface

as a Uniform System

The dielectric coated conducting plane introduced in Section 6.3 and Fig. 6.2b

is a typical uniform planar slow-wave structure. In the case of the metallic

structure, a corrugated conducting surface is used instead of the dielectric

coating; see Fig. 7.2. If the spatial period of the structure is much less than

the longitudinal wavelength, p ¿ λ

, so that the phase shift in one period

is inﬁnitesimally small, βp ¿ 2π, the system can be analyzed approximately

by means of the uniform system model.

7.2.1 Unbounded Structure

In the unbounded structure or open structure shown in Fig. 7.2(a), the ﬁeld

extends to inﬁnity in the +x direction. We are interested in the TM modes

with ﬁelds uniform in the y direction; V = 0 and U(x, z) 6= 0.

In region 1, the upper half-space, 0 ≤ x ≤ ∞, For a slow wave, U function

is required to be an exponentially decaying function with respect to x, e

−τx

and have a traveling wave form in z, e

−jβz

. Consequently

U = Ae

−τx

−jβz

, (7.2)

and

= −τ

U = −τ

−τx

−jβz

, (7.3)

7.2 A Corrugated Conducting Surface as a Uniform System 405

Figure 7.2: Corrugated conducting surface, (a) unbounded structure, (b)

bounded structure.

= −jβ

∂U

∂x

= jβτ Ae

−τx

−jβz

, (7.4)

y 1

= −jω²

∂U

∂x

= jω²

τAe

−τx

, (7.5)

where

− τ

= k

= ω

. (7.6)

In region 2, the corrugated region, −h ≤ x ≤ 0, the space between each

pair of plates can be viewed as a parallel-plate transmission line with a short-

circuit plane at x = −h. Since p ¿ λ

, βp ¿ 2π, the only mode that has

to be considered is the TEM mode propagating in the ±x directions, and

the phase shift along z can be considered as an approximately continuous

function, e

−jβz

. Hence we have

= B

sin k(x + h)

sin kh

−jβz

, (7.7)

y 2

= −j

cos k(x + h)

sin kh

−jβz

. (7.8)

The boundary conditions on the surface, x = 0, are

(0) = E

(0) → B = −τ

A, (7.9)

y 1

(0) = H

y 2

(0) → −j

cos kh

sin kh

= jω²

τA. (7.10)

From (7.9) and (7.10) we get the eigenvalue equation

τh = kh tan kh. (7.11)

Substituting (7.6) into the above equation, we have

βh = ±

cos kh

. (7.12)

406 7. Periodic Structures and the Coupling of Modes

Figure 7.3: The k–β diagram of the open corrugated conducting surface

structure (a) and the closed structure (b) as uniform systems.

The k–β curves resulting from this dispersion equation are plotted in

Fig. 7.3(a).

It is evident from the dispersion curves in Fig. 7.3(a) that there are pass

bands of TM modes in the range nπ ≤ kh ≤ (n + 1/2)π, n = 0, 1, 2, ···,

and that in between the pass bands there are stop bands. In the pass band,

the length of the shorted parallel-plate line is longer than zero or a certain

even multiple of quarter wavelengths and is shorter than the next-nearest odd

multiple of quarter wavelengths. In this case, the input impedance at x = 0

is an inductance. This is just the requirement that the surface impedance

should support a TM slow wave.

From the eigenvalue equation (7.11), we may argue that if τ is imaginary

and β ≤ k, the root of the equation does not exist in the real frequency

domain. Hence all the dispersion curves of an unbounded corrugated con-

ducting plane are located in the region of β ≥ k and τ must be real. In

the region β ≤ k, τ becomes imaginary; then there are radiation ﬁelds in

x direction. In this case, the modes become radiation modes. As a result

of this fact, the region β ≥ k in the k–β diagram is a forbidden region, see

Fig. 7.3(a), and no fast wave can exist in unbounded systems.

The characteristics of the corrugated conducting surface under the condi-

tion p ¿ λ

are the same as those of the dielectric coated conducting plane

given in Section 6.3. Thus this kind of metallic structure is known as an

artiﬁcial dielectric.

7.2.2 Bounded Structure

If we put a conducting plate over the corrugated conducting surface, it must

become the bounded or closed structure shown in Fig. 7.2(b). We again

consider the TM modes with ﬁelds uniform in the y direction.

In region 1, 0 ≤ x ≤ b. For U to be zero at x = b, it must b e a hyperbolic

7.3 A Disk-Loaded Waveguide as a Uniform System 407

sine function, sinh τ (x−b), rather than an exponential function as in an open

system. From this argument we have

U = A sinh τ (x − b)e

−jβz

, (7.13)

= −τ

A sinh τ (x − b)e

−jβz

, (7.14)

= −jβτ A cosh τ(x − b)e

−jβz

, (7.15)

y 1

= −jω²

τA cosh τ (x − b)e

−τx

. (7.16)

In region 2, the ﬁeld components here are the same as those in the corre-

sponding region in an open system, (7.7) and (7.8).

By virtue of the same boundary conditions at x = 0 as in an open system,

(0) = E

(0) and H

y 1

(0) = H

y 2

(0), we have the following eigenvalue

equation

τb tanh τ b =

kh tan kh. (7.17)

The solution of this equation along with (7.6) gives the dispersion curves

shown in Fig. 7.3(b). According to this equation, if τ is imaginary then

tanh τ b is also imaginary so as to allow the roots to still exist in the real

frequency domain. Hence, there is no forbidden region in the k–β diagram,

and fast waves do exist in bounded systems.

7.3 A Disk-Loaded Waveguide as a

Uniform System

The disk-loaded waveguide is a circular metallic waveguide loaded with

equally spaced metallic disks as shown in Fig. 7.4(a) for center coupling

hole structure, and Fig. 7.4(b) for edge coupling hole structure. The disk-

loaded waveguide is also known as a coupled-cavity-chain structure. Various

slow-wave systems used in traveling wave ampliﬁers are modiﬁed disk-loaded

waveguides or coupled-cavity structures.

7.3.1 Disk-Loaded Waveguide with Center Coupling

Hole

The disk-loaded waveguide with center coupling holes shown in Fig. 7.4(a)

is the guided-wave system used in linear accelerators. At SLAC (Stanford

Linear Accelerator Center), the length of the accelerator extends for 2 miles

(≈ 3 km). The modiﬁed disk-loaded waveguides are used in high-power

traveling-wave ampliﬁers.

We are interested in the azimuthal invariant TM modes, in which the

longitudinal electric ﬁeld component at the center is suitable for the purpose

of ﬁeld–electron interaction. The coupling region or slow-wave region ρ ≤ b

is called region 1 and the disk region or cavity region b ≤ ρ ≤ a is called

408 7. Periodic Structures and the Coupling of Modes

Figure 7.4: Disk-loaded waveguide with coupling holes at the center (a) and

at the edge (b).

region 2. Suppose that the spatial period is much less than the longitudinal

wavelength, p ¿ λ

, and that the structure can be analyzed as a uniform

system.

In region 1, ρ ≤ b, the azimuthal invariant slow-wave solution in cylin-

drical coordinates is a modiﬁed Bessel function of the ﬁrst kind, I

(τρ), with

the coeﬃcient of K

(τρ) being zero to avoid singularity on the axis ρ = 0.

Then the U

function and ﬁeld components are given by

= AI

(τρ)e

−jβz

, (7.18)

ρ1

= −jβ

∂U

∂ρ

= −jβτ AI

(τρ)e

−jβz

, (7.19)

= −τ

(τρ)e

−jβz

, (7.20)

φ1

= −jω²

∂U

∂ρ

= −jω²

τAI

(τρ)e

−jβz

, (7.21)

where

− τ

= k

= ω

. (7.22)

In region 2, b ≤ ρ ≤ a, the space between the disks can be viewed as

a radial line with a short-circuit surface at ρ = a or a cylindrical cavity of

radius a. Under the condition that p ¿ λ

, i.e., βp ¿ 2π, only TEM

(ρ)

modes, in other words TM

(z)

0m0

modes, exist in the shorted radial line or

cylindrical cavity. For the same reason as given in the last section, the phase

shift along z can still be considered as an approximately continuous function,

−jβz

. Hence we have the solution with the standing wave in ρ as follows

= [B

(kρ) + B

(kρ)]e

−jβz

In satisfying the short-circuit boundary condition at ρ = a, we have

(ka) + B

(ka) = 0,

(ka)

= −

(ka)

= B.

7.3 A Disk-Loaded Waveguide as a Uniform System 409

Figure 7.5: The dispersion curves of disk-loaded waveguide with coupling

holes at the center (a) and at the edge (b).

Then U

and the ﬁeld components become

= B[N

(ka)J

(kρ) − J

(ka)N

(kρ)]e

−jβz

, (7.23)

= k

B[N

(ka)J

(kρ) − J

(ka)N

(kρ)]e

−jβz

, (7.24)

φ2

= −jω²

∂U

∂ρ

= jω²

kB[N

(ka)J

(kρ) − J

(ka)N

(kρ)]e

−jβz

, (7.25)

In order to satisfy the continuous conditions of tangential ﬁeld components

on the cylindrical boundary ρ = b, E

(b) = E

(b), and H

φ1

(b) = H

φ2

(b),

we must have

−τ

(τb) = k

B[N

(ka)J

(kb) − J

(ka)N

(kb)], (7.26)

−jω²

τAI

(τb) = jω²

kB[N

(ka)J

(kb) − J

(ka)N

(kb)]. (7.27)

Dividing the second of the two expressions by the ﬁrst one, we obtain the

eigenvalue equation

(τb)

τbI

(τb)

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (7.28)

The dispersion relation is determined by this equation and the equation

of β and τ , (7.22). The k–β diagram of the dominant TM mode in a disk-

loaded waveguide with coupling holes at the center is given in Fig. 7.5(a).

We see that for a disk-loaded waveguide, there is no forbidden region in the

k–β diagram because it is a b ounded structure.

The lower cutoﬀ frequency of a center-holed disk-loaded waveguide is

equal to the cutoﬀ frequency of the TM

mode in the circular waveguide

with radius a, which is the transverse resonant frequency and the longitudinal

phase coeﬃcient becomes zero. This cutoﬀ frequency is independent of the

radius b of the hole, and ka = x

= 2.405.

410 7. Periodic Structures and the Coupling of Modes

The higher cutoﬀ frequency of the system is determined by the radius b

of the hole. If b = 0, the system becomes a chain of isolated cavities and the

pass-band width becomes zero, i.e., the upper cutoﬀ frequency is equal to

the lower one. If b = a, the system becomes a hollow circular waveguide, the

upper cutoﬀ frequency becomes inﬁnity, and the system becomes a high-pass

ﬁlter. When the radius of the hole b is smaller than the radius of the cavity a

and larger than zero, the system becomes a band-pass ﬁlter. The larger the

radius b of the hole, the higher the upper cutoﬀ frequency.

The dominant TM mode in a center-hole disk-loaded waveguide is a for-

ward wave, for which the phase velocity is in the same direction as the group

velocity.

The disk-loaded waveguide can also be considered as a chain of resonant

cavities mutually coupled by the coupling hole. The larger the coupling

coeﬃcient the broader the bandwidth.

7.3.2 Disk-Loaded Waveguide with Edge Coupling Hole

The disk-loaded waveguide with edge coupling holes is shown in Fig. 7.4(b).

In practice, for the passing through of a electron beam, there are also center

holes. The diameter of the center hole is rather small so that the coupling

eﬀect of the center hole can be neglected. We are also interested in the

azimuthal invariant TM modes.

In region 1, b ≤ ρ ≤ a, the azimuthal invariant slow-wave solution are

both the modiﬁed Bessel function of the ﬁrst and the second kinds, I

(τρ)

and K

(τρ), for the axis ρ = 0 is outside the region. The U

function and

ﬁeld components are given by

= [A

(τρ) + A

(τρ)]e

−jβz

For satisfying the boundary condition on the wall of the waveguide r = a, we

must have

(a) = 0, A

(τa) + A

(τa) = 0

i.e.,

(τa)

= −

(τa)

= A.

Function U

and ﬁeld components become

= A[K

(τa)I

(τρ) − I

(τa)K

(τρ)]e

−jβz

, (7.29)

ρ1

= −jβτ A[K

(τa)I

(τρ) + I

(τa)K

(τρ)]e

−jβz

, (7.30)

= −τ

A[K

(τa)I

(τρ) − I

(τa)K

(τρ)]e

−jβz

, (7.31)

φ1

= −jω²

τA[K

(τa)I

(τρ) + I

(τa)K

(τρ)]e

−jβz

, (7.32)

where

− τ

= k

= ω

. (7.33)