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

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7.4 Periodic Systems 411

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

line with a load at ρ = b. The coeﬃcient of function N

(kρ) must be zero

because the axis ρ = 0 is included in the region. Hence we have the solution

with the standing wave in ρ as follows

= BJ

(kρ)e

−jβz

, (7.34)

= k

(kρ)e

−jβz

, (7.35)

φ2

= jω²

kBJ

(kρ)e

−jβz

. (7.36)

The boundary conditions on the boundary ρ = b are given by E

(b) =

(b) and H

φ1

(b) = H

φ2

(b) as follows,

−τ

A[K

(τa)I

(τb) − I

(τa)K

(τb)] = k

(kb), (7.37)

−jω²

τA[K

(τa)I

(τb) + I

(τa)K

(τb)] = jω²

kBJ

(kb). (7.38)

We obtain the eigenvalue equation

τb

(τa)I

(τb) + I

(τa)K

(τb)

(τa)I

(τb) − I

(τa)K

(τb)

(kb)

kbJ

(kb)

. (7.39)

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 edge is given in Fig. 7.5(b).

The dominant TM mode in a disk-loaded waveguide with coupling holes at

the edge is a backward wave, for which the phase velocity is in the opposite

direction of the group velocity.

The disk-loaded waveguide can also be considered as a coupled-cavity

chain. The center-holed coupling disk-loaded waveguide is a electric ﬁeld

(or capacitance) coupled-cavity chain, whereas the edge-holed coupling disk-

loaded waveguide is a magnetic ﬁeld (or inductance) coupled-cavity chain.

We will see later that in periodic system, the fundamental harmonic in a

capacitance-coupled-cavity chain is a forward wave, whereas the fundamental

harmonic in a inductance-coupled-cavity chain is a backward wave.

The dispersion curves given in Fig. 7.3 and Fig. 7.5 are obtained by the

uniform system approach. They are good approximations only in the region

of small β. When β is large, we must consider the eﬀect of the periodic

boundaries.

7.4 Periodic Systems

In the previous sections, a periodic structure is analyzed approximately by the

approach of the uniform-system model. When the spatial period is much less

than the longitudinal guided wavelength, this model is a good approximation.

If, however, the spatial period is not small enough so that the phase shift along

z is no longer continuous, we must develop the theory for periodic structures

or periodic transmission systems [10, 107].

412 7. Periodic Structures and the Coupling of Modes

Figure 7.6: Uniform system (a) and periodic system (b).

7.4.1 Floquet’s Theorem and Space Harmonics

(1) Uniform System

A uniform transmission system has its shape, size, and material kept uniform

along the longitudinal direction z, as shown in Fig. 7.6(a). The principle

feature of a uniform system can be described as follows.

For a given mode of propagation at a given steady-state frequency the

ﬁelds at one cross section diﬀer from those an arbitrary distance away merely

by a complex constant which depends upon the distance only.

The proof lies on the fact that when a uniform system of inﬁnite length

is displaced along its z axis by an arbitrary distance, it is indistinguishable

from the original one.

Suppose that the complex constant can be written as

−γ(z

−z

)

= e

−γ∆z

, where z

= z

+ ∆z, γ = α + jβ.

Then the relation between ﬁelds distributed on the two cross sections at

points z

and z

is given by

E (x, y, z

, t) = E (x, y, z

, t) e

−γ∆z

. (7.40)

For satisfying the above relation, the harmonic ﬁeld at a given steady-state

frequency distributed on an arbitrary cross section at point z is given by

E (x, y, z, t) = F (x, y) e

−γz

jωt

, (7.41)

and the complex amplitude of the ﬁeld is

E (x, y, z) = F (x, y) e

−γz

, (7.42)

where F (x, y) is the distribution function of the ﬁeld on the transverse cross

section which is independent of z.

7.4 Periodic Systems 413

For lossless uniform systems, γ = jβ, and

E (x, y, z) = F (x, y) e

−jβz

. (7.43)

Therefore, we conclude that the ﬁeld of a guided mode in a uniform system

is a single spatial harmonic wave, i.e., a sinusoidal wave that satisﬁes the

uniform boundary condition.

(2) Periodic System

In a periodic transmission system, the shape, size, and constitutive material

vary periodically along its z axis, see Fig. 7.6(b). The basis for the study of

periodic systems is a theorem ascribed to the French mathematician Floquet,

which may be stated as follows.

In a periodic system, for a given mode of propagation at a given steady-

state frequency, the ﬁelds at one cross section diﬀer from those one period

(or an integer multiple of periods) away by only a complex constant.

This theorem is true whether or not the structure has losses so long as it

is p eriodic. The proof of the theorem lies in the fact that when a periodic

system having inﬁnite length is displaced along its axis by one period or an

integer multiple of periods, it cannot be distinguishable from its original self.

Suppose that the spatial period of the system is p, and the distance be-

tween the two cross sections is mp, m is an integer, then the complex constant

can be written as

−γ(z

−z

)

= e

−γ

, where z

− z

= ∆z = mp, γ

= α

+ jβ

The relation between the complex amplitudes of the ﬁelds on the two cross

sections at z

and z

= z

+ mp are given by

E (x, y, z + mp) = E (x, y, z) e

−γ

. (7.44)

This is the mathematical formulation of Floquet’s theorem.

In a periodic system, the distribution function of the ﬁeld on the trans-

verse cross section is dependent on z, so the time-harmonic ﬁeld at a given

steady-state frequency on an arbitrary cross section at z must be

E (x, y, z, t) = F (x, y, z) e

−γ

jωt

, (7.45)

and the complex amplitude of the ﬁeld at z is

E (x, y, z) = F (x, y, z) e

−γ

. (7.46)

We can readily prove that if the function F (x, y, z) is a periodic function of

z with period p, then the Floquet’s theorem (7.44) is followed.

The complex amplitudes of the ﬁelds on the cross sections at z + mp is

E (x, y, z + mp) = F (x, y, z + mp) e

−γ

(z+mp)

. (7.47)

414 7. Periodic Structures and the Coupling of Modes

If F (x, y, z) is a periodic function in z, we must have

F (x, y, z + mp) = F (x, y, z). (7.48)

Then (7.47) becomes

E (x, y, z + mp) = F (x, y, z) e

−γ

= E (x, y, z) e

−γ

, (7.49)

so we see that the Floquet’s theorem (7.44) is obeyed.

For lossless systems, γ

= jβ

, and (7.46) becomes

E (x, y, z) = F (x, y, z) e

−jβ

. (7.50)

(3) Space Harmonics

The periodic function F (x, y, z) can be expanded into a Fourier series of the

form

F (x, y, z) =

∞

n=−∞

(x, y) exp

− jn

2π

, (7.51)

and the ﬁeld expression (7.50) becomes

E (x, y, z) =

∞

n=−∞

(x, y) exp

− j

2πn

. (7.52)

To ﬁnd E

(x, y), multiply (7.51) by exp

2π

and then integrate both

sides from z

− p/2 to z

+ p/2:

+p/2

−p/2

F (x, y, z) exp

2π

dz =

∞

n=−∞

+p/2

−p/2

(x, y) exp

j(m−n)

2π

dz.

By orthogonality of the functions exp

2π

+p/2

−p/2

exp

j(m − n)

2π

dz =

0 m 6= n,

p m = n,

the above integration becomes

(x, y) =

+p/2

−p/2

F (x, y, z) exp

2π

+p/2

−p/2

F (x, y, z) e

−jβ

exp

2πn

dz.

Using (7.50), we obtain

(x, y) =

+p/2

−p/2

E (x, y, z) e

jβ

dz, (7.53)

7.4 Periodic Systems 415

where

= β

2πn

, (7.54)

and the ﬁeld expression (7.52) becomes

E (x, y, z) =

∞

n=−∞

(x, y) e

−jβ

. (7.55)

The nth term of the above series is called the nth space harmonic or Hartree

harmonic which is associated to a phase constant β

. Some of the phase

constants are positive and some of them are negative. The space harmonic

with n = 0 is called the fundamental harmonic.

The corresponding phase velocity of the nth space harmonic is

+ 2πn/p

1/v

+ 2πn/ωp

, (7.56)

which is diﬀerent for diﬀerent n and will b e negative whenever β

is negative.

The group velocity of the nth space harmonic is

dω

dβ

dω

d(β

+ 2πn/p)

dω

dβ

= v

, (7.57)

and is the same for all space harmonics.

As mentioned before, the wave with phase and group velocities in the

same direction is called a forward wave, denoted by FW, and the wave with

phase and group velocities in opposite directions is called a backward wave,

denoted by BW. The k–β diagrams with unique features for forward waves

and for backward waves are shown in Fig. 7.1.

We come to the conclusion that the ﬁeld of a single spatial harmonic

wave mode cannot adjust to the boundary condition of a periodic system.

The ﬁeld which does satisfy the periodic boundary condition has to be a

spatial anharmonic wave mode which can b e expanded into an inﬁnite series

of space harmonics with phase velocities diﬀerent but group velocities the

same. In a propagation mode, all the space harmonics with the same group

velocity must be present simultaneously in order for the total ﬁeld to satisfy

the boundary conditions. For a given periodic boundary, the proportions of

space harmonics of a mode remain constant. If there are energy exchanges

between the wave and certain outside source or load, e.g., a charged particle

beam, the amplitudes of all space harmonics will scale by a common factor.

This means that in a periodic structure, for a given mode of propagation at

any frequency there is no one common phase velocity but it is found to be an

inﬁnite number of individual phase velocities characterizing the mode. The

diﬀerence in the concepts mode and space harmonic is that the former can

match the boundary conditions alone but the later cannot.

Because the phase velocities for diﬀerent space harmonics are diﬀerent,

the phase relations of diﬀerent harmonics, hence the composed wave form,

will vary during propagation.

416 7. Periodic Structures and the Coupling of Modes

Figure 7.7: The ω–β diagram of a periodic system.

7.4.2 The ω–β Diagram of Periodic Systems

The relation between the phase constant of the nth space harmonic and that

of the fundamental harmonic is given by

= β

2πn

For a system with given β

, β

can be obtained by adding 2πn/p to it, this

is to say that the ω–β

curve is simply the ω–β

curve shifted along the β

axis by 2πn/p; see Fig. 7.7(a). Therefore ω is a periodic function of β

It is apparent that ω is an even function of β

, since for a reciprocal

system, reversing the structure in z cannot change the physical situation.

The ω–β diagram of a periodic system for the wave with group velocity in

−z direction is shown in Fig. 7.7(b). For the wave with negative group

velocity, the phase coeﬃcients of the space harmonics become

= −

2πn

The complete ω–β diagram of a typical periodic system is shown in

Fig. 7.8(a). In this diagram, the phase velocity of the fundamental harmonic

and the group velocity are in the same direction. The diagram represents a

system in which the fundamental harmonic is a forward wave. In such sys-

tems, all the space harmonics with positive n are forward waves and all the

space harmonics with negative n are backward waves.

The ω–β diagram of a system with a fundamental harmonic that is a

backward wave is shown in Fig. 7.8(b). In such systems, all the space har-

monics with positive n are backward waves and all the space harmonics with

negative n are forward waves.

7.4 Periodic Systems 417

Figure 7.8: The ω–β diagram of a periodic system with a forward fundamental

(a) and with a backward fundamental (b).

In practice, there is a complete set of modes in a periodic system. Each

mode corresponds to a pass band, some of the pass bands have a forward

fundamental and some of them have a backward fundamental. Pass bands

are separated by stop bands.

The ω–β diagram of a periodic system is also known as a Brillouin dia-

gram.

7.4.3 The Band-Pass Character of Periodic Systems

To understand the pass bands and stop bands in the ω–β diagram of a peri-

odic system, we investigate, for example, the disk-loaded waveguide that is a

circular waveguide with thin obstacles lined up periodically inside, as shown

in Fig. 7.4.

We are interested in a speciﬁc mode, for which the ﬁrst cutoﬀ frequency

is equal to the cutoﬀ frequency of the waveguide without obstacles. This

ﬁrst cutoﬀ frequency is obtained when the phase shift along the waveguide

is set to be zero, βp = 0, and is denoted by ω

. For example, the cutoﬀ

condition for the TM

mode is k

a = x

= 2.405, λ

= 2.6a. The ﬁrst

cutoﬀ frequency ω

is also called the transverse resonant frequency. When

the wave of this speciﬁc mode propagates in the disk-loaded waveguide, at

each obstacle there will be transmission and reﬂection. There will be a certain

frequency for which the reﬂections from the successive obstacles returning to

a speciﬁc point in the waveguide will add in phase and the wave of that mode

in the guide will be cut oﬀ again. This second cutoﬀ frequency will be equal

to the frequency for which the one-way phase shift between obstacles is nπ,

where n is any integer and is denoted by ω

. The second cutoﬀ frequency ω

418 7. Periodic Structures and the Coupling of Modes

Figure 7.9: ω–β diagrams of coupled-cavity chain or disk-loaded waveguide.

is also called the longitudinal resonant frequency.

The disk-loaded waveguide and other periodic loaded transmission sys-

tems can also be viewed as a coupled-cavity chain. The disk-loaded waveguide

with coupling holes at the center is an electric-ﬁeld or capacitance-coupled-

cavity chain and the disk-loaded waveguide with coupling holes at the edge

is a magnetic-ﬁeld or inductance-coupled-cavity chain. When the size of the

coupling hole is reduced to zero, this becomes a chain of uncoupled cavities

for which the ω–β diagram is a horizontal straight line, meaning that any

phase shift between sections is possible at the natural frequency of the cav-

ity; see curve 1 in Fig. 7.9. When the coupling hole is enlarged to the size

of the waveguide, this becomes an unloaded waveguide and the second cutoﬀ

frequency tends to inﬁnity; see curve 2 in Fig. 7.9.

The patterns of the ﬁelds in a circular waveguide, disk-loaded waveguide,

and uncoupled cavity chain at ω

(βp = 0) and ω

(βp = π) are shown in

Fig. 7.10.

From the ﬁeld pattern at ω

, we observe that the introduction of obstacles

does not disturb the ﬁeld at ω

, and hence, by no means changes the ﬁrst

cutoﬀ frequency ω

In the case of a center-hole coupled-cavity chain, the coupling from section

to section is predominately capacitive, since the coupling holes are located in

a region in which the electric ﬁeld is stronger than the magnetic ﬁeld. From

Fig. 7.10 we easily see that the electric ﬁeld strength near the coupling hole

at βp = π is smaller than that at βp = 0 and the magnetic ﬁeld remains

almost the same. To see that the eﬀective capacitance becomes smaller and

the natural frequency is higher at βp = π than at βp = 0, i.e., ω

> ω

;

see curve 3 in Fig. 7.9. In this case, the slope of the dispersion curve must

be positive, the group velocity and the phase velocity must be in the same

direction, and the fundamental harmonic must be a forward wave. So the

center-hole coupled-cavity chain is a forward fundamental system.

In the case of an edge-hole coupled-cavity chain the coupling from section

to section is predominately inductive, since the coupling holes are located in

7.4 Periodic Systems 419

Figure 7.10: Electric and magnetic ﬁelds in a disk-loaded waveguide.

a region in which the magnetic ﬁeld is stronger than the electric ﬁeld. From

Fig. 7.10 we easily see that the magnetic ﬁeld strength near the coupling

hole at βp = π is smaller than that at βp = 0 and the electric ﬁeld remains

almost the same. This tells us that the eﬀective inductance becomes larger

and the natural frequency is lower at βp = π than at βp = 0, i.e., ω

< ω

;

see curve 4 in Fig. 7.9. In this case, the slope of the dispersion curve must

be negative, the group velocity and the phase velocity must be in opposite

directions, and the fundamental harmonic must be a backward wave. So the

edge-hole coupled-cavity chain is a backward fundamental system.

In fact, the coupling holes are also resonant elements, so the character-

istics of the coupled-cavity chain is determined not only by the resonant

characteristics of the cavities but also by the resonant characteristics of the

coupling elements. For example, the cavity mode of an edge-hole coupled-

cavity chain is a backward fundamental mode when the resonant frequency

of the hole is higher than that of the cavity, i.e., the hole should be viewed

as an inductance; however, the mode becomes a forward fundamental mode

when the resonant frequency of the hole is lower than that of the cavity, i.e.,

420 7. Periodic Structures and the Coupling of Modes

the hole should b e viewed as a capacitance. In the former case, there is a

higher hole mode with a forward fundamental and in the latter case, there

is a lower hole mode with a backward fundamental. In the considerations of

the hole mode, the cylindrical cavity plays the part of a resonant coupling

element.

Various modiﬁed disk-loaded waveguides are developed for high-power

traveling-wave ampliﬁers, they are Hughes structure with inductive (edge)

coupling, i.e., a backward fundamental structure; clover-leaf structure and

centipede structure with negative inductive coupling, i.e., forward fundamen-

tal structures; and long-slot structure with resonant coupling, also a forward

fundamental structure.

The basic center-hole disk-loaded waveguide is a narrow-band structure,

which is usually used in linear accelerators. Various kinds of modiﬁed disk-

loaded waveguides are used in medium-power and high-power broad-band

traveling ampliﬁers.

7.4.4 Fields in Periodic Systems

The boundary condition of a periodic system cannot be satisﬁed by ﬁelds de-

scribed by any single space harmonic. In order to satisfy the periodic bound-

ary condition one must have a spatial non-simple-sinusoidal ﬁeld which can

be expanded into an inﬁnite series of space harmonics with diﬀerent phase

velocities but with the same group velocity. Because the phase velocities

for diﬀerent space harmonics are diﬀerent, the phase relations among diﬀer-

ent harmonics will change during the propagation, thus the composed wave

form will change during propagation. The ﬁeld patterns of the lowest TM

mode in the drift region of a periodic disk-loaded waveguide are illustrated

in Figure 7.11.

From Fig. 7.11, we notice the following

1. The waveform of the composed ﬁeld is anharmonic and changes during

propagation.

2. The waveform of the composed ﬁeld is very sharp near the periodic

surface of the structure, ρ = b, so it must be composed of high order

space harmonics of larger amplitudes. But the waveform is much more

smooth near the axis, i.e., away from the periodic surface, so it must be

composed of space harmonics of smaller amplitudes especially higher

order ones. The reason for this phenomenon is that the higher the order

of space harmonic the larger the transverse decaying factor.

3. The electric ﬁeld on the axis has a non-zero longitudinal component, but

its transverse component there vanishes. This ﬁeld structure is suitable

for the interaction between the ﬁeld and a longitudinal charged-particle

beam.