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

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7.7 The Helix 431

Figure 7.16: The k–β diagrams of the dominant mode (a) and some lower

modes (b) in a disk-loaded waveguide.

The two functions J

(x/2) and sinc(x/2) are similar when x is not large, see

Fig 7.15(b).

The k–β diagram of the dominant mode, i.e., the azimuthal invariant TM

modes in a disk-loaded waveguide with center coupling holes as a periodic

system is given in Fig. 7.16(a). The k–β diagram of some lower modes are

shown in Fig. 7.16(b).

7.7 The Helix

The earliest slow-wave structure used in a traveling-wave tube invented by

R.Kompfner is the helix. It is employed in all low- and medium-power

traveling-wave ampliﬁers and backward-wave oscillators. It is also used in

highly directive broad-band antennas and in high-frequency delay lines.

The helix is made of metallic wire as shown in Fig. 7.17 (a). The average

radius of the helix is a, the pitch of the winding, i.e., the period is p, and the

diameter of the wire is δ. The expanded view of a helix is shown in Fig. 7.17

(b). The ratio of the pitch p to the circumference 2πa is the tangent of the

pitch angle ψ,

tan ψ =

2πa

. (7.103)

Initially, people proposed that an electromagnetic wave propagates along the

helical wire with the speed of light c, and that the velocity of the wave in

the direction of z is equal to the ratio of the pitch of the winding p to the

circumference 2πa/ cos ψ times the speed of light:

2πa/ cos ψ

c = sin ψ. (7.104)

This is the simplest model of a helix and is known as the helical wave model.

According to this model, v

is independent of the frequency. That is to say

432 7. Periodic Structures and the Coupling of Modes

Figure 7.17: The helix (a) and its expanded view (b).

that the system is non-dispersive. This conclusion is correct in the high-

frequency regime but in the low-frequency regime, a practical helix does

become a dispersive system.

The boundary conditions of helix are rather complicated and diﬃcult to

deal with for ﬁnding the ﬁeld solution. Hence a number of physical models

have been put forward by diﬀerent authors. Among them the simplest and

most successful models are the sheath helix and tape helix.

7.7.1 The Sheath Helix

A physical abstraction, known as the sheath helix, given by J. R. Pierce [79]

yields solutions to Maxwell’s equations which show many of the properties of

an actual helix. The sheath helix is a cylindrical surface, i.e., a inﬁnitesimally

thin cylinder, conducting only in the helical direction, as shown in Fig 7.18.

The sheath is perfectly conducting in a direction making an angle ψ with the

plane perpendicular to the axis, but it is nonconducting in the direction nor-

mal to the direction of conduction. A physical approximation to this model

could be reasonably made by winding a ﬂat tape of width p consisting of a

large number of ﬁne wires all insulated with each other on a cylindrical form

of radius a, with all the windings being wound side by side. This structure

would be a perfect sheath helix if the diameter of the wires approaches zero

and the number of wires in the tape approaches inﬁnity. The sheath model

is found to be a good approximation to the actual helix at frequencies and

for modes for which there are many turns per guided wavelength,

À p, i.e., βp ¿ π.

This is exactly the condition required by the approach in which the validity

of the uniform system is assumed.

7.7 The Helix 433

Figure 7.18: The sheath helix.

(1) General Solutions

Because of the skew and anisotropic boundary conditions it is necessary to

have both TE and TM ﬁelds present in a certain mode, so neither U nor V

can be zero. Since we expect slow waves rather than fast waves, the ﬁeld

components are written in the slow-wave form in both region 1 and region

2. For a slow wave in circular cylindrical coordinates, the functions in ρ

must be modiﬁed Bessel functions. The functions in φ are e

jnφ

, where n = 0

for axially symmetric modes or meridional waves, n is a positive integer for

counterclockwise skew-wave modes and n is a negative integer for clockwise

skew-wave modes.

Region 1 (ρ ≤ a): The axis, ρ = 0, is included in the region, hence the

coeﬃcients of functions K

must be zero in order to avoid singularity, then

= A

(τρ)e

jnφ

−jβz

, V

= B

(τρ)e

jnφ

−jβz

and the ﬁeld components are

= −τ

(τρ)e

jnφ

−jβz

, (7.105)

ρ1

−jβτA

(τρ) +

ωµn

(τρ)

jnφ

−jβz

, (7.106)

φ1

nβ

(τρ) + jωµτ B

(τρ)

jnφ

−jβz

, (7.107)

= −τ

(τρ)e

jnφ

−jβz

, (7.108)

ρ1

−

ω²n

(τρ) − jβτB

(τρ)

jnφ

−jβz

, (7.109)

φ1

−jω²τA

(τρ) +

nβ

(τρ)

jnφ

−jβz

. (7.110)

Region 2 (ρ ≥ a): Inﬁnity, ρ → ∞, is included in the region, hence the

coeﬃcients of functions I

must be zero in order to avoid singularity, then

= C

(τρ)e

jnφ

−jβz

, V

= D

(τρ)e

jnφ

−jβz

434 7. Periodic Structures and the Coupling of Modes

and

= −τ

(τρ)e

jnφ

−jβz

, (7.111)

ρ2

−jβτC

(τρ) +

ωµn

(τρ)

jnφ

−jβz

, (7.112)

φ2

nβ

(τρ) + jωµτ D

(τρ)

jnφ

−jβz

, (7.113)

= −τ

(τρ)e

jnφ

−jβz

, (7.114)

ρ2

−

ω²n

(τρ) − jβτD

(τρ)

jnφ

−jβz

, (7.115)

φ2

−jω²τC

(τρ) +

nβ

(τρ)

jnφ

−jβz

. (7.116)

The relation among β, τ, and k is given by

− τ

= k

= ω

µ². (7.117)

The boundary conditions at ρ = a are as follows. The electric ﬁeld com-

ponents in the direction of conduction must be zero

(a) = 0, E

(a) = 0, (7.118)

and those normal to the direction of conduction must be continuous

1⊥

(a) = E

2⊥

(a). (7.119)

The magnetic ﬁeld components in the direction of conduction must be con-

tinuous, because there is no surface current normal to the magnetic ﬁelds,

(a) = H

(a). (7.120)

The subscript k denotes the components parallel to the direction of the helical

wire and ⊥ denotes the components normal to the direction of the helical

wire. The relations among the k, ⊥ and the z, φ components are shown in

Fig. 7.19.

= E

sin ψ + E

cos ψ, E

⊥

= E

cos ψ − E

sin ψ,

= H

sin ψ + H

cos ψ, H

⊥

= H

cos ψ − H

sin ψ.

Then the boundary conditions (7.118) may be written in terms of the ﬁeld

components of (7.105)–(7.116) as follows:

sin ψ + E

φ1

cos ψ = 0, (7.121)

sin ψ + E

φ2

cos ψ = 0, (7.122)

cos ψ − E

φ1

sin ψ = E

cos ψ − E

φ2

sin ψ, (7.123)

sin ψ + H

φ1

cos ψ = H

sin ψ + H

φ2

cos ψ. (7.124)

7.7 The Helix 435

Figure 7.19: The expanded view of a sheath helix and the helical coordinates.

Substituting (7.105) through (7.116) into them, yields

−τ

sin ψ +

nβ

cos ψ

(τa)A

+ jωµτ I

(τa) cos ψB

= 0, (7.125)

−τ

sin ψ +

nβ

cos ψ

(τa)C

+ jωµτ K

(τa) cos ψD

= 0, (7.126)

−τ

cos ψ −

nβ

sin ψ

(τa)A

− jωµτ I

(τa) sin ψB

cos ψ +

nβ

sin ψ

(τa)C

+ jωµτ K

(τa) sin ψD

= 0, (7.127)

−jω²τI

(τa) cos ψA

−τ

sin ψ +

nβ

cos ψ

(τa)B

+jω²τK

(τa) cos ψC

sin ψ −

nβ

cos ψ

(τa)D

= 0. (7.128)

This is a set of homogeneous linear equations with variables A

, B

, C

, and

. The homogeneous equations are satisﬁed simultaneously by nontrivial

solutions only when the determinant of the coeﬃcients vanishes.

−Ψ

(τa) Φ

(τa) 0 0

0 0 −Ψ

(τa) Φ

(τa)

−Ψ

(τa) −Φ

(τa) Ψ

(τa) Φ

(τa)

−Φ

(τa) −Ψ

(τa) Φ

(τa) Ψ

(τa),

= 0,

where

= τ

sin ψ −

nβ

cos ψ, Ψ

= τ

sin ψ +

nβ

cos ψ.

436 7. Periodic Structures and the Coupling of Modes

= jωµτ cos ψ, Φ

= jωµτ sin ψ, Φ

= jω²τ cos ψ.

After going through a lot of algebra, we obtain the eigenvalue equation:

(τa)K

(τa)

(τa)K

(τa)

= −

cot

(τ

− nβa cot ψ)

. (7.129)

Following this eigenvalue equation and equation (7.117), the propagation

characteristics of the sheath helix for diﬀerent modes are given.

The relations of the coeﬃcients in the ﬁeld-component expressions are

also obtained from (7.125)–(7.128) as follows:

− nβa cot ψ

jωµτa

cot ψ

(τa)

, (7.130)

(τa)

, (7.131)

(τa)

− nβa cot ψ

jωµτa

cot ψ

(τa)

. (7.132)

Substituting these into (7.105)–(7.116), we have the ﬁnal expressions of the

ﬁeld components inside and outside the helical sheath. Only one coeﬃcient

remains in the expressions, which is determined by the amplitude of the

wave of the nth mode propagating in the guided-wave system.

(2) The Dominant Mode

The dominant mode in the helix is the azimuthal uniform mode, i.e., the

mode for which n = 0. The phase velocity of the n = 0 mode is nearly equal

to the group velocity and is nearly constant for relatively high frequencies,

which is approximately equal to the value we estimated earlier by means of

the helical wave mo del (7.104). This mode is the commonly used as slow-wave

structure in traveling-wave ampliﬁers and delay lines.

For n = 0, the co eﬃcients in the ﬁeld comp onent expressions (7.130)–

(7.132) become

B =

jωµ cot ψ

(τa)

A, (7.133)

C =

(τa)

A, (7.134)

D = −

jωµ cot ψ

(τa)

A, (7.135)

and the ﬁeld components inside and outside the sheath become

= −τ

(τρ)e

−jβz

, (7.136)

ρ1

= −jβτ A

(τρ)e

−jβz

, (7.137)

7.7 The Helix 437

φ1

cot ψ

(τa)

(τρ)e

−jβz

, (7.138)

= −

jωµ cot ψ

(τa)

(τρ)e

−jβz

, (7.139)

ρ1

= −

βτ

ωµ cot ψ

(τa)

(τρ)e

−jβz

, (7.140)

φ1

= −jω²τA

(τρ)e

−jβz

. (7.141)

= −τ

(τa)

(τρ)e

−jβz

, (7.142)

ρ2

= jβτ

(τa)

(τρ)e

−jβz

, (7.143)

φ2

cot ψ

(τa)

(τρ)e

−jβz

, (7.144)

jωµ cot ψ

(τa)

(τρ)e

−jβz

, (7.145)

ρ2

= −

βτ

ωµ cot ψ

(τa)

(τρ)e

−jβz

, (7.146)

φ2

= jω²τ

(τa)

(τρ)e

−jβz

. (7.147)

For n = 0, The eigenvalue equation (7.129) becomes

(τa)K

(τa)

(τa)K

(τa)

cot ψ

, or (τa)

(τa)K

(τa)

(τa)K

(τa)

= (ka cot ψ)

. (7.148)

These equations are solved by solving for (k/τ ) cot ψ and ka cot ψ in terms

of the value of τ a. The solutions are plotted in Fig. 7.20 and are known as

the normalized disp ersion curves of the dominant mode on sheath helix and

are independent of the dimensions of the helix.

The actual dispersion curves may be plotted when the pitch angle ψ or

the average diameter of the helix a and the pitch p is given. The dep endencies

of the slow-wave ratio v

/c = ka/βa and normalized frequency ωa/c = ka

are plotted for several pitch angles in Fig. 7.21.

When the frequency is suﬃciently high, i.e., ka cot ψ is suﬃciently large,

we see from equation (7.148), Fig. 7.20 and Fig. 7.21 that (k /τ ) cot ψ → 1

and the longitudinal phase constant β and the slow-wave ratio become

β ≈ k

1 + cot

ψ,

≈ sin ψ. (7.149)

This is just the value we estimated earlier by means of the helical wave model

(7.104).

When the frequency is low, i.e., ka cot ψ is small, the slow-wave ratio is

larger than sin ψ. Moreover, when ω → 0 or ψ → π/2, i.e., ka cot ψ → 0, we

438 7. Periodic Structures and the Coupling of Modes

Figure 7.20: Normalized dispersion curves of the dominant mode on a sheath

helix.

see from equation (7.148), Fig. 7.20 and Fig. 7.21 that

cot ψ → ∞, τ ¿ k, β =

+ τ

≈ k,

≈ 1. (7.150)

The longitudinal phase velocity tends to approach the speed of light at very

low frequencies.

A plot of the ﬁeld conﬁguration within a longitudinal section of the sheath

helix is given in Fig. 7.22. When the frequency is suﬃciently high, the elec-

tric and magnetic ﬂux lines are inclined approximately by an angle ψ and

are approximately conﬁned on the plane normal to the helical direction of

conduction. The Poynting vector is then in the helical direction. This is

Figure 7.21: Dispersion curves of the dominant mode on a sheath helix.

7.7 The Helix 439

Figure 7.22: Field conﬁguration of the dominant mode on a sheath helix.

nothing but a nonuniform plane wave or TEM wave propagating in the heli-

cal direction. When the frequency is low, the angle of inclination of the ﬂux

lines is no longer ψ, the direction of the Poynting vector is oriented towards

the longitudinal axis z and the longitudinal phase velocity tends to c.

All the ﬁelds inside and outside the helix are traveling waves in z and

decaying ﬁelds in ρ. Such waves are surface waves. The ﬁeld strength at the

axis is smaller than that at the sheath surface. See Fig. 7.23.

The interaction impedance of the dominant mode on a sheath helix is

obtained by substituting the ﬁeld components (7.136)–(7.147) into (7.1), in

which

P =

ρ1

∗

φ1

−E

φ1

∗

ρ1

2πρ dρ+

∞

ρ2

∗

φ2

−E

φ2

∗

ρ2

2πρ dρ

The interaction impedance on the axis is then given by

K(0) =

(0)

2β

F (τa), (7.151)

where

F (τa) =

(

π(τa)

−I

−K

−1

Applying the recurrence formulas (C.14), (C.15) and Wronskian (C.28), we

reduces F (τ a) to

F (τa) =

πτa

·µ

−

τa

¸¾

−1

440 7. Periodic Structures and the Coupling of Modes

Figure 7.23: The ρ dependence of the ﬁeld components of the dominant mode

on a sheath helix.

All the arguments of the modiﬁed Bessel functions are τa.

The oﬀ-axis interaction impedance at ρ = b is

K(b) = K(0)I

(τb), (7.152)

which is higher than that on the axis.

The average interaction impedance in an electron beam of radius b is given

K =

πb

πK(0)I

(τρ)ρ dφ dρ = K(0)

(τb) − I

(τb)

. (7.153)

See Fig. 7.24.

The interaction impedance of an actual helix is lower than that predicated

for the ideal sheath helix because of the inﬂuences of the space harmonics

and the dielectric holder of the wire helix.

(3) The Higher Modes

Rewrite the transcendental eigenvalue equation for sheath helix (7.129),

(τa)K

(τa)

(τa)K

(τa)

= −

cot

(τ

− nβa cot ψ)

The solutions to the eigenvalue equation for general cases are shown in

Fig. 7.25 as the solid lines. The shaded region in the ﬁgure is a forbid-

den region because the helix is an unbounded structure. The dominant mode