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

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7.7 The Helix 441

Figure 7.24: The average interaction impedance of the dominant mode on a

sheath helix.

of n = 0 is shown in the diagram as a forward-wave mode, so the helix is a

forward fundamental system. It is also indicated in the diagram that only for

n = 0 mode and for τ a À 1, is the phase velocity nearly constant and nearly

equal to the group velocity. The modes for which n 6= 0 are forward-wave

modes in a certain frequency range and are backward-wave modes for rest of

the frequency range.

From the eigenvalue equation (7.129) and the k −β diagram Fig 7.25 we

see that, other from metallic waveguide, dielectric waveguide and disk-loaded

waveguide, for helix, the cutoﬀ conditions and the dispersion relations are dif-

ferent for +n and −n, i.e., τ

−n

6= τ

and β

−n

6= β

. So the clockwise and

the counterclockwise skew waves with functions e

−jnφ

−jβ

−n

and e

jnφ

−jβ

propagate with diﬀerent longitudinal phase coeﬃcients and cannot be com-

posed into standing wave ﬁelds with stationary polarization direction. This

is because of the skew and anisotropic nature of the boundary.

A right-handed helix is not fundamentally diﬀerent from a left-handed

helix. If we have the solution for a right-handed helix of the form

(ρ)e

jnφ

−jβz

, then a solution F

−n

(ρ)e

−jnφ

−jβz

exists for the left-handed

helix. This is so because n occurs only in conjunction with cot ψ and func-

tions I

and K

are even functions of n. Thus, if we change the sign of n and

the sign of cot ψ, we get the same propagation constant β.

The dominant mode with n = 0 of a helix is a forward-wave mode, which

is usually used as slow-wave structure for traveling wave ampliﬁer. The

backward-wave mode with n = −1 is used as the slow-wave structure for

backward-wave oscillator, which is a wide-band electronic-tuned oscillator.

442 7. Periodic Structures and the Coupling of Modes

Figure 7.25: k–β diagram of the sheath helix.

7.7.2 The Tape Helix

A more representative physical model of the wire helix is the tape helix [87]

shown in Fig. 7.26. The tape is of width δ and of zero thickness and considered

to be perfectly conducting. The radius of the helix is a, the pitch is p and

cot ψ = 2πa/p.

We have seen in the last subsection that the sheath helix is a uniform

helical structure. When a sheath helix of inﬁnite length is displaced along

its z axis by an arbitrary distance, it remains invariant and when the sheath

helix is rotated in φ by an arbitrary angle it again remains invariant.

Both the wire helix and the tape helix are periodic helical structures

rather than uniform-helical structures. The features of a periodic helical

structure are as follows:

(1) When the helix is moved a distance p in the z direction it remains

invariant. This is the feature of a p eriodic structure and according to (7.50)

and (7.48) we have

E (ρ, φ, z) = F (ρ, φ, z) e

−jβ

, (7.154)

F (ρ, φ, z + mp) = F (ρ, φ, z). (7.155)

(2) When the single-wire helix is rotated in φ by 2π it also remains in-

variant. This gives

F (ρ, φ + 2π, z) = F (ρ, φ, z). (7.156)

In satisfying the two conditions above, the function F (ρ, φ, z) must be of the

7.7 The Helix 443

Figure 7.26: The tape helix.

following form

F (ρ, φ, z) =

∞

ν=−∞

∞

n=−∞

νn

(ρ)e

jνφ

exp

− j

2πn

. (7.157)

(3) When the helix is moved an arbitrary distance ∆z and rotated in φ

by ∆φ = 2π∆z/p it again remains invariant. This gives

ρ, φ + 2π

∆z

, z + ∆z

= F (ρ, φ, z), (7.158)

i.e.,

∞

ν=−∞

∞

n=−∞

νn

(ρ) exp

jν

φ + 2π

∆z

¶¸

exp

−j

2πn

(z + ∆z)

∞

ν=−∞

∞

n=−∞

νn

(ρ)e

jνφ

exp

− j

2πn

exp

−j

2πn

(n − ν)∆z

∞

ν=−∞

∞

n=−∞

νn

(ρ)e

jνφ

exp

− j

2πn

. (7.159)

To ensure this, it is necessary to demand ν = n, i.e.,

νn

(ρ) 6= 0, for ν = n, and F

νn

(ρ) = 0, for ν 6= n.

So F (ρ, φ, z) must be in the following expanded series

F (ρ, φ, z) =

∞

n=−∞

(ρ)e

jnφ

exp

− j

2πn

, (7.160)

and the ﬁeld components must be the form

E (ρ, φ, z) = F (ρ, φ, z) e

−jβ

∞

n=−∞

(ρ)e

jnφ

−jβ

, (7.161)

444 7. Periodic Structures and the Coupling of Modes

where

= β

2πn

. (7.162)

Each term in the above series must satisfy Helmholtz’s equation in cylin-

drical co ordinates. Hence the wave functions for slow-wave solutions must be

series of I

(τρ) in region 1 (ρ ≤ a) and series of K

(τρ) in region 2 (ρ ≥ a):

∞

n=−∞

(τ

ρ)e

jnφ

−jβ

, V

∞

n=−∞

(τ

ρ)e

jnφ

−jβ

, (7.163)

∞

n=−∞

(τ

ρ)e

jnφ

−jβ

, V

∞

n=−∞

(τ

ρ)e

jnφ

−jβ

, (7.164)

where

− τ

= k

= ω

µ². (7.165)

It should be noted that, for sheath helix, n is the angular order of the mode.

Each mode represents a space-sinusoidal wave which alone can satisfy the

uniform-system boundary condition of the sheath helix. Here, for tape helix,

n is the longitudinal order as well as the angular order of the space harmonic.

The non-sinusoidal wave composed by all the space harmonics can satisfy the

periodic-system b oundary condition of the tap e helix.

The ﬁeld components in the two regions are given by

∞

n=−∞

−τ

(τ

ρ)e

jnφ

−jβ

, (7.166)

ρ1

∞

n=−∞

−jβ

(τ

ρ) +

ωµn

(τ

ρ)

jnφ

−jβ

, (7.167)

φ1

∞

n=−∞

nβ

(τ

ρ) + jωµτ

(τ

ρ)

jnφ

−jβ

, (7.168)

∞

n=−∞

−τ

(τ

ρ)e

jnφ

−jβ

, (7.169)

ρ1

∞

n=−∞

−

ω²n

(τ

ρ) − jβ

(τ

ρ)

jnφ

−jβ

, (7.170)

φ1

∞

n=−∞

−jω²τ

(τ

ρ) +

nβ

(τ

ρ)

jnφ

−jβ

. (7.171)

∞

n=−∞

−τ

(τ

ρ)e

jnφ

−jβ

, (7.172)

ρ2

∞

n=−∞

−jβ

(τ

ρ) +

ωµn

(τ

ρ)

jnφ

−jβ

, (7.173)

7.7 The Helix 445

φ2

∞

n=−∞

nβ

(τ

ρ) + jωµτ

(τ

ρ)

jnφ

−jβ

, (7.174)

∞

n=−∞

−τ

(τ

ρ)e

jnφ

−jβ

, (7.175)

ρ2

∞

n=−∞

−

ω²n

(τ

ρ) − jβ

(τ

ρ)

jnφ

−jβ

, (7.176)

φ2

∞

n=−∞

−jω²τ

(τ

ρ) +

nβ

(τ

ρ)

jnφ

−jβ

. (7.177)

The boundary conditions at ρ = a for the perfect conducting tape helix

are as follows.

1. The tangential electric ﬁeld is continuous for all φ and z.

2. The discontinuity in a tangential magnetic ﬁeld is equal to the surface

current density perpendicular to the magnetic ﬁeld.

3. The tangential electric ﬁeld is equal to zero on the tape surface.

The mathematical expressions of the above conditions are as follows.

(a) = E

(a), (7.178)

φ1

(a) = E

φ2

(a), (7.179)

(a) − H

(a) = −J

sφ

(a), (7.180)

φ2

(a) − H

φ1

(a) = J

(a) (7.181)

and

(a) = 0, for

pφ

2π

−

< z <

pφ

2π

, (7.182)

where E

(a) denotes the tangential electric ﬁeld on the surface ρ = a, J

sφ

(a)

and J

(a) are the surface current densities on ρ = a in the φ and z directions,

respectively. They are also in series of space harmonics

sφ

(a) =

∞

n=−∞

φn

jnφ

−jβ

, (7.183)

(a) =

∞

n=−∞

jnφ

−jβ

, (7.184)

where j

φn

and j

are the complex Fourier amplitudes of the current densities

associated with the nth space harmonic.

An exact solution should be obtained in principle if one tries to insert the

appropriate ﬁeld component quantities in the boundary equations (7.178) to

446 7. Periodic Structures and the Coupling of Modes

Figure 7.27: The expanded view of a tape helix.

(7.181). This results in a set of inﬁnite-by-inﬁnite simultaneous equations

which can be solved for the unknown propagation constant β

and the co-

eﬃcients A

to D

in expressions of ﬁeld components. This procedure is

similar to that used for the reentrant cavity given in Section 5.6.1 and adds

little physical insight to the problem. Therefore, only approximate methods

are considered here in which a reasonable current distribution in the tape is

assumed.

For slow waves and for regions smaller than the wavelength in space, the

r.f. ﬁeld solutions are like static ﬁeld solutions. We assume, therefore, that

the current in a thin, narrow tape ﬂows only in the tape direction, i.e.,

⊥

= 0, (7.185)

∞

n=−∞

jnφ

−jβ

for

pφ

2π

−

< z <

pφ

2π

, (7.186)

and

= 0, elsewhere.

The relations for j

φn

, j

, and j

are

φn

= j

cos ψ, j

= j

sin ψ. (7.187)

See the ﬁgure of the expanded view of a tape helix, Fig. 7.27.

For the amplitude variation on the tape, there are two possible assump-

tions.

1. The current density is constant over the width of the tape.

2. The quasi-static current density distribution in an isolated inﬁnitesi-

mally thin conducting tape, which can be obtained by means of con-

formal mapping is similar to the distribution of electric ﬁeld between

opposite knife-edges (7.98).

7.7 The Helix 447

For the phase variation on the tape, there are three possible assumptions.

1. The phase varies in the z direction.

2. The phase varies in the φ direction.

3. The phase varies in the helical direction of the tape.

Finally, the current density J

is expressed as follows:

1 − ξ

2(z + pφ/2π)

exp

−j

pφ

2π

+ β

z −

pφ

2π

¶¸¾

, (7.188)

for

pφ

2π

−

< z <

pφ

2π

and

= 0, elsewhere,

where

ξ = 0 means constant current density over the width of the tape;

ξ = 1 indicates the quasi-static current density in an isolated thin tape;

= β

means the phase varies in z, the phase factor becomes e

−jβ

;

= 0 means the phase varies in φ, the phase factor becomes

exp

−jβ

pφ

2π

;

= β

sin

ψ means the phase varies in the helical direction and the

phase factor becomes exp

−jβ

z sin

ψ +

pφ

2π

1 − sin

The Fourier coeﬃcients of (7.186), j

, are then obtained by equating it

to the given function (7.188) and performing proper integration:

pφ

2π

pφ

2π

−

1−ξ

2(z+pφ/2π)

exp

−j

pφ

2π

+β

z−

pφ

2π

¶¸¾

jβ

−jnφ

dz.

(7.189)

The result of the integration is

= JR

, (7.190)

where











sinc(nπδ/p), for ξ = 0, β

= β

;

(nπδ/p), for ξ = 1, β

= β

;

sinc(β

δ/2), for ξ = 0, β

= 0;

(β

δ/2), for ξ = 1, β

= 0;

(β

/β

)sinc(β

δ/2), for ξ = 0, β

= β

sin

ψ;

(β

δ/2)/J

(β

δ/2), for ξ = 1, β

= β

sin

ψ.

448 7. Periodic Structures and the Coupling of Modes

Substituting the ﬁeld-component expressions (7.166)–(7.177) and the sur-

face current density expressions (7.183), (7.184), (7.187), and (7.190) into the

boundary equations (7.178)–(7.181), we have the following set of simultane-

ous linear equations

−τ

+ τ

= 0, (7.191)

nβ

+jωµτ

−

nβ

−jωµτ

= 0, (7.192)

−τ

+ τ

= JR

cos φ, (7.193)

jω²τ

−

nβ

−jω²τ

nβ

= JR

sin φ, (7.194)

where I

, I

, K

, and K

represent I

(τ

a), I

(τ

a), K

(τ

a), and

(τ

a), respectively.

The solutions of this set of equations give the following expressions for

the ﬁeld coeﬃcients

−a sin ψ + (nβ

/τ

) cos ψ

jω²

(τ

a)JR

, (7.195)

−a cos ψ

(τ

a)JR

, (7.196)

−a sin ψ + (nβ

/τ

) cos ψ

jω²

(τ

a)JR

, (7.197)

−a cos ψ

(τ

a)JR

. (7.198)

Substituting these coeﬃcients into (7.166)–(7.177), we have all the ﬁeld com-

ponents inside and outside the tape helix. The only unknown quantity is J,

which is determined by the amplitude of traveling wave propagating along

the longitudinal direction z.

The eigenvalue equation is then obtained by applying the condition of

(7.182). Because the current density distribution on the tape is not the true

value, the condition of (7.182) cannot be satisﬁed strictly everywhere, instead,

the following approximate condition is used:

a, φ,

pφ

2π

= 0, (7.199)

which means that the electric ﬁeld parallel to the helical direction of the tape

is equal to zero on the central line of the tape. The relation of E

and E

is given by

= E

sin ψ + E

cos ψ. (7.200)

Substituting the corresponding ﬁeld-component expressions (7.166), (7.168)

or (7.172), (7.174) and the coeﬃcient expressions (7.195), (7.196) or (7.197),

7.7 The Helix 449

Figure 7.28: k–β diagram of the tape helix.

(7.198) into the above equations, we have

a, φ,

pφ

2π

= j

J sin

ω²a

jβ

∞

n=−∞

(τ

− 2nβ

a cot ψ +

(β

(τ

cot

×I

(τ

a)K

(τ

a)+(ka)

cot

ψ I

(τ

a)K

(τ

=0. (7.201)

It gives the approximate eigenvalue equation as follows

∞

n=−∞

(τ

−2nβ

a cot ψ +

(β

(τ

cot

(τ

a)K

(τ

+ (ka)

cot

ψ I

(τ

a)K

(τ

=0. (7.202)

This equation includes a series of modiﬁed Bessel functions which converge

slowly. If each term of the series in (7.202) is independently set to zero, it

reduces to the eigenvalue equation for the nth mode in a sheath helix, shown

in (7.129).

Equation (7.202) is solved for a speciﬁc value of cot ψ. A complete k–β

diagram of a tape helix is shown in Fig. 7.28 for ψ = 10

◦

and πδ/p = 0.1.

Comparison of Fig. 7.28 with Figure 7.25 shows that the forbidden regions

appear periodically at βp = 2nπ and the k–β curves are consistent with the

regulations for periodic systems. It is remarkable that the space harmonic

components on the tape helix correspond to individual modes on the sheath

helix and that the sheath approach is invalid in forbidden regions.

The interaction impedance of the tape helix can be obtained from its

ﬁeld components and the calculated value is lower than that of the sheath

approach due to the inﬂuences of space harmonics.

450 7. Periodic Structures and the Coupling of Modes

7.8 Coupling of Modes

The guided modes that we studied in the preceding sections propagate along

guided-wave systems undisturbed and free of mutual coupling provided that

the waveguide is uniform and free of irregularities or discontinuities. Some-

times, there are material inhomogeneities or slight changes in the boundaries

on the waveguide. These imperfections cause the modes in the waveguide

to couple among them. If a single mode is excited at the beginning of a

waveguide, some of the power may be transferred to other guided modes,

cutoﬀ modes, or radiation modes by means of the coupling. Furthermore,

the guided mo des in diﬀerent waveguides may also be coupled with each

other if there are some coupling mechanisms between the waveguides. In pe-

riodic systems the mode coupling can happen to diﬀerent space harmonics of

diﬀerent modes. When mode coupling occurs, the propagation constants will

be diﬀerent from those of the individual modes, which leads to an increase

or decrease of phase velocity or the growth or decay of the wave.

A great many phenomena occurring in physics or engineering can be quite

naturally viewed as coupled-mode processes and studied by the coupled-mode

theory. For example, the directional couplers in microwave and light-wave

technologies, the scattering loss due to waveguide irregularities, the inter-

action between electron beams and slow-wave structures in traveling-wave

ampliﬁers and backward-wave oscillators, the distributed feedback (DFB)

structures, and the scattering of light by gratings and by acoustic waves,

etc. The coupled-mode formalism is a perturbation analysis developed for

weak coupling, it includes the coupling-in-time formalism and the coupling-

in-space formalism. The former is applied to coupled oscillating modes and

the latter to coupled propagation modes. In this book, we deal with the

coupling-in-space formalism only. The coupling-in-time equations are analo-

gous to the coupling-in-space equations. Time in oscillating elements plays

the role of distance in propagating structures. The frequency plays the role

of the propagation constant, whereas the counterpart of power ﬂow in the

transmission system is energy in the oscillator. [38, 61, 80, 117]

7.8.1 Coupling of Modes in Space

Recall from Section 3.4 that in an arbitrary uniform lossless guided-wave sys-

tem any mode can be simulated by an equivalent ideal transmission line. For

the wave with time dependence e

jωt

, the basic equations for ideal transmis-

sion line are given in Section 3.1.1 as follows:

= jωLI,

= jωCU. (7.203)

The solutions to these equations are in the form e

−jβz

in which β is positive

or negative for the wave with phase velocity in the +z or the −z direction,

respectively, and β =

√

LC.