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

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7.4 Periodic Systems 421

Figure 7.11: Field patterns in a periodic disk-loaded waveguide.

422 7. Periodic Structures and the Coupling of Modes

7.4.5 Two Theorems on Lossless Periodic Systems

There are two important theorems on lossless periodic transmission systems.

Their statements are as follows [107].

Theorem 1

The time-averaged electric stored energy per period is equal to the time-

averaged magnetic stored energy per period in the pass bands of a lossless

periodic transmission system.

The mathematical expression of Theorem 1 is

µH · H

∗

dV −

²E · E

∗

dV = 0, (7.58)

where V represents the volume of the periodic structure of one spatial period.

Notice that the time-averaged electric stored energy in an arbitrary length

is equal to the time-averaged magnetic stored energy in the same length in a

lossless uniform transmission system. In this case, V in (7.58) represents the

volume of the system of an arbitrary length.

Theorem 2

The time-averaged power ﬂow in the pass band of a lossless periodic trans-

mission system is equal to the group velocity times the time-averaged electric

and magnetic stored energy per period divided by the length of the period

The mathematical expression of Theorem 2 is given by

E × H

∗

· dS

= v

²E · E

∗

µH · H

∗

dV, (7.59)

where S represents the area of an arbitrary cross section of the system, V

represents the volume of the structure of one spatial period and p is the

spatial period of the system.

Notice that the time-averaged power ﬂow in a lossless uniform transmis-

sion system is equal to the group velocity times the time-averaged electric

and magnetic stored energy in an arbitrary length divided by the length.

The proofs of these two theorems are left to the reader as an exercise,

refer to Problem 7.11.

7.4.6 The Interaction Impedance for Periodic Systems

For a given mode in a periodic system, the phase velocities ascribed to dif-

ferent space harmonics are diﬀerent, so the charged particles with certain

velocity can interact with only that particular space harmonic with phase

velocity close to the velocity of the particles. The eﬀectiveness of the interac-

tion is determined by the interaction impedance of a speciﬁc space harmonic,

7.5 Corrugated Conducting Plane as a Periodic System 423

i.e., the strength of the electric ﬁeld of the speciﬁc space harmonic of the

given mode referred to the total power ﬂow carried by the mode:

znm

2β

znm

2β

, (7.60)

where E

znm

denotes the amplitude of the z component of the electric ﬁeld

speciﬁed by the nth space harmonic, P denotes the total power ﬂow carried

by the mode, and W denotes the energy of the given mo de stored in the

system of unit length.

7.5 Corrugated Conducting Plane

as a Periodic System

Consider the two-dimensional corrugated conducting plane shown in

Fig. 7.12(a) as a periodic system. The spatial period of the structure is

no longer much less than the longitudinal wavelength, so the phase shift in

one period cannot be made inﬁnitesimally small. In this case, the system

must be analyzed by means of the periodic system model [107]. We are again

interested in TM modes with no y-dependence, V = 0, U(x, z) 6= 0, and

∂/∂y = 0.

For the unbounded structure of Fig. 7.12(a), the ﬁeld extends to inﬁnity in

the +x direction. In region 1, 0 ≤ x ≤ ∞, the U function must be composed

of the complete set of space harmonics shown in (7.55) and each term of the

series must assume the form of (7.2), i.e., an exponential decaying function

in the x direction, e

−τ

, and a traveling wave in the z direction, e

−jβ

U =

∞

n=−∞

−τ

−jβ

, (7.61)

and

∞

n=−∞

−τ

−jβ

, (7.62)

∞

n=−∞

jβ

−τ

−jβ

, (7.63)

y 1

∞

n=−∞

jω²

−τ

−jβ

, (7.64)

where

− τ

= k

= ω

, (7.65)

and

= β

2πn

. (7.66)

424 7. Periodic Structures and the Coupling of Modes

Figure 7.12: Corrugated conducting surface as a periodic system, (a) un-

bounded structure, (b) bounded structure.

In region 2, −h ≤ x ≤ 0, the gap may as well be viewed as a parallel-plate

transmission line with a short-circuit plane at x = −h. For an exact solution,

each of the ﬁeld components in region 2 is also a complete set or inﬁnite series

consisting of all eigenfunctions that satisfy the boundary conditions. The

eigenvalue equation will include an inﬁnite-dimensional determinant and all

the coeﬃcients make up an inﬁnite series similar to those for the reentrant

cavity given in Section 5.6.1. We would rather ﬁnd the approximate solution

according to the method given in Section 4.11 than bother with the exact

solution. Although the spatial period of the structure is no longer much

less than the longitudinal wavelength p 6¿ λ

, it is still much less than the

wavelength in space, p ¿ λ, and the only mode we need to consider is the

TEM mode propagating in the ±x directions. Since p 6¿ λ

, the phase shift

along z must be considered as a discontinuous function e

−jβ

. We have

(

sin k(x + h)

sin kh

−jβ

, mp−d/2 <z < mp+d/2,

0 mp+d/2 < z <(m + 1)p−d/2,

(7.67)

y 2

(

−j

ω²

cos k(x+h)

sin kh

−jβ

, mp−d/2<z <mp+d/2,

0 mp+d/2<z <(m+1)p−d/2.

(7.68)

The electric ﬁeld in region 1 at the boundary x = 0 is given by

(0) =

∞

n=−∞

−τ

−jβ

(mp+z

)

∞

n=−∞

−τ

−jβ

, (7.69)

where z

denotes the z coordinate with respect to the center of the mth gap,

= z − mp. Note that both n and m are integers so that e

−j2πnm

= 1.

The electric ﬁeld in region 2 at the boundary x = 0 is given by

(0) =

−jβ

, −d/2 < z

< +d/2,

0 +d/2 < z

< p − d/2.

(7.70)

7.5 Corrugated Conducting Plane as a Periodic System 425

The ﬁeld-matching condition of the electric ﬁeld E

at the boundary be-

tween region 1 and region 2, x = 0, is then given by E

(0) = E

(0), i.e.,

∞

n=−∞

−τ

−jβ

B, −d/2 < z

< +d/2,

0 +d/2 < z

< p − d/2.

(7.71)

The coeﬃcient −τ

of the Fourier series is

−τ

d/2

−d/2

jβ

, (7.72)

The above integral is calculated as

d/2

−d/2

jβ

= d

sin(β

d/2)

d/2

= d sinc

. (7.73)

Then we have the ﬁnal expression of the coeﬃcient −τ

−τ

= B

sinc

. (7.74)

Substituting this into the ﬁeld-component expressions (7.62) and (7.64), we

have

∞

n=−∞

sinc

−τ

−jβ

, (7.75)

y 1

∞

n=−∞

−

jω²

sinc

−τ

−jβ

. (7.76)

The exact matching conditions for the electric ﬁeld E

and the magnetic

ﬁeld H

at the boundary x = 0 cannot be satisﬁed simultaneously, for they

are both trial functions but not actual ﬁelds in the gap region. Once the

electric ﬁeld E

is matched exactly at x = 0, the magnetic ﬁeld H

can only

be matched approximately by applying the average matching condition given

in Section 4.11, which is

d/2

−d/2

y 1

(0) dz

= H

y 2

(0) d. (7.77)

Hence we have

∞

n=−∞

sinc

d/2

−d/2

jβ

cot kh. (7.78)

Using the integral formula (7.73), we have the eigenvalue equation

∞

n=−∞

sinc

kh tan kh

. (7.79)

426 7. Periodic Structures and the Coupling of Modes

Figure 7.13: The k–β diagram of the open corrugated conducting surface

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

The diﬀerence between this equation and the eigenvalue equation of the same

structure based on a uniform system approach (7.11) is that the left-hand

side of the equation becomes a series. Furthermore, the factors p/d and

sinc(β

d/2) pertaining to details of the structure app ear in the eigenvalue

equation obtained by the periodic system approach.

Following this dispersion equation, we have the k–β curves plotted in

Fig. 7.13(a). It is shown by the dispersion curves in Fig. 7.13(a) and

Fig. 7.3(a) that when βp is small the two curves are close to each other

and when βp is large the periodic character of the system is obvious. The

dispersion curve becomes periodic and the forbidden regions appear periodi-

cally. This fact shows that as a guided wave system, for any space harmonic,

transverse radiation is never allowed.

The eigenvalue equation for the bounded structure shown in Fig. 7.12(b)

is obtained in a similar manner:

∞

n=−∞

h tan τ

sinc

kh tan kh

. (7.80)

The k–β curves are plotted in Fig. 7.13(b). Note that there is no forbidden

region in the k–β diagram for the bounded system, because the fast wave with

transverse standing-wave ﬁelds can be supported by an unbounded structure.

7.6 Disk-Loaded Waveguide as a Periodic

System

The disk-loaded waveguide with coupling holes at the center as a periodic

system is shown in Fig. 7.14. The thickness of the disk is t, the spacing

between disks is d, and the period of the system is p = t+d. We are interested

in the azimuthal invariant TM modes for which n = 0. The spatial period is

7.6 Disk Loaded Waveguide as a Periodic System 427

Figure 7.14: Disk-loaded waveguide as a p eriodic system.

no longer much less than the longitudinal wavelength and the structure must

be analyzed as a periodic system.

In region 1, ρ ≤ b, according to (7.55) and (7.18), the U function can be

expressed as the following series:

∞

n=−∞

(τ

ρ)e

−jβ

, (7.81)

where

− τ

= k

= ω

, β

= β

2πn

. (7.82)

The ﬁeld components become

ρ1

∞

n=−∞

−jβ

(τ

ρ)e

−jβ

, (7.83)

∞

n=−∞

−τ

(τ

ρ)e

−jβ

, (7.84)

φ1

∞

n=−∞

−jω²

(τ

ρ)e

−jβ

. (7.85)

In region 2, b ≤ ρ ≤ a, the region between two disks is a radial line

with short-circuit surface at ρ = a. The exact solution must be the following

series:

(

∞

l=0

ρ) + C

ρ)] cos β

)

−jβ

, (7.86)

428 7. Periodic Structures and the Coupling of Modes

where β

is the phase coeﬃcient inside the radial line, readers should distin-

guish it from the phase coeﬃcient of the system β

and β

. Since p 6¿ λ

the phase shift along z is considered as a discontinuous function e

−jβ

Although the spatial period of the structure is no longer much less than the

longitudinal wavelength, it is still much less than the wavelength in space,

p ¿ λ, and the only mode we need to consider is the TEM

(ρ)

mode propa-

gating in the ±ρ directions, i.e., β

= 0 and T = k. Hence we have

= [B

(kρ) + B

(kρ)] e

−jβ

. (7.87)

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

(ka)

= −

(ka)

= B.

Then the U

function and the ﬁeld components become

= B [N

(ka)J

(kρ) − J

(ka)N

(kρ)] e

−jβ

, (7.88)

= k

B [N

(ka)J

(kρ) − J

(ka)N

(kρ)] e

−jβ

, (7.89)

φ2

= jω²

kB [N

(ka)J

(kρ) − J

(ka)N

(kρ)] e

−jβ

. (7.90)

Suppose that the electric ﬁeld at the gap is uniform, i.e., on the boundary,

ρ = b, the electric ﬁeld is of the form

(b) =

, −d/2 < z

< +d/2,

0 +d/2 < z

< p − d/2.

(7.91)

where z

denotes the z coordinate with respect to the center of the mth gap,

= z − mp.

Let E

(b) = E

(b). We get

∞

n=−∞

−τ

(τ

b)e

−jβ

, −d/2 < z

< +d/2,

0 +d/2 < z

< p − d/2.

The nth coeﬃcient of the Fourier series is evaluated by

−τ

(τ

b) =

d/2

−d/2

jβ

= E

sinc

and

= −

(τ

sinc

. (7.92)

Let E

(b) = E

(b). We get

B [N

(ka)J

(kb) − J

(ka)N

(kb)] = E

7.6 Disk Loaded Waveguide as a Periodic System 429

and

B =

(ka)J

(kb) − J

(ka)N

(kb)]

. (7.93)

The magnetic ﬁeld H

can be matched approximately on the cylindri-

cal boundary ρ = b by applying the average matching condition given in

Section 3.10, which is

d/2

−d/2

φ1

(b) dz

d/2

−d/2

φ2

(b) dz

. (7.94)

Substituting (7.85) and (7.90) into this and applying the integral formula

(7.73), we have

∞

n=−∞

−τ

(τ

b) sinc

= kB[N

(ka)J

(kb) − J

(ka)N

(kb)]. (7.95)

Substituting the expressions for A

and B into this equation, we have the

eigenvalue equation as follows:

∞

n=−∞

(τ

sinc

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (7.96)

If we use the speciﬁc point matching condition instead of the average match-

ing condition for the magnetic ﬁeld, then the eigenvalue equation becomes

∞

n=−∞

(τ

sinc

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (7.97)

In fact, the ﬁelds in the gap are not uniform. The ﬁeld at the center of

the gap must be weaker than that at the edge. The alternative approach is

that the electric ﬁeld at the gap is assumed to be a quasi-static ﬁeld between

opposite knife-edges, refer to Fig 7.15(a). According to the result of conformal

mapping, the potential between the edges is given by

ϕ(z

) =

arcsin

and the electric ﬁeld is derived to be

(b) = −

dϕ

1 − (2z

/d)

, (7.98)

where E

= 2V/πd. Let E

(b) = E

(b). We get

∞

n=−∞

−τ

(τ

b)e

−jβ







1 − (2z

/d)

, −d/2 < z

< +d/2,

0 +d/2 < z

< p − d/2.

430 7. Periodic Structures and the Coupling of Modes

Figure 7.15: (a) Uniform ﬁeld between parallel plates and quasi-static ﬁeld

between opposite knife edges. (b) Functions J

(x/2) and sinc(x/2).

The nth coeﬃcient of the Fourier series satisﬁes the relation

−τ

(τ

b) =

d/2

−d/2

1 − (2z

/d)

jβ

= E

and the expression for A

takes the form

= −

(τ

. (7.99)

Let E

(b) = E

(b). We get

B[N

(ka)J

(kb) − J

(ka)N

(kb)] =

d/2

−d/2

1 − (2z

/d)

and

B =

(ka)J

(kb) − J

(ka)N

(kb)]

. (7.100)

Substituting these expressions for A

and B into (7.95), we have the eigen-

value equation

∞

n=−∞

(τ

sinc

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

(7.101)

If we use the speciﬁc point matching condition for a magnetic ﬁeld, then

the eigenvalue equation takes another form:

∞

n=−∞

(τ

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (7.102)