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CHAPTER – 4

ELECTROMAGNETIC WAVES IN CYLINDRICAL SYSTEMS

The vector Helmholtz equations satisfied by the (phasor) electric and magnetic fields are

where . In low-loss media and for a high frequency, i.e.,

and are real, √ . Since the vector Laplacian does not in general separate in an

orthogonal curvilinear coordinate system other than the Cartesian, it is natural to look for some

methods of reducing the three-dimensional vector Helmholtz equations to scalar Helmholtz

equations at least under certain not too restrictive conditions. They are the method of Borgnis’

potentials, the method of Hertz’s vector potentials, and the method of longitudinal components.

All three methods depend on the choice of the coordinate system in which the equations are to be

solved.

4.1 Orthogonal Curvilinear Coordinate Systems

Orthogonal curvilinear coordinate system in may be defined in terms of three sets of

mutually orthogonal (curved ) surfaces as shown in Fig 4.1. A family of surfaces in three

dimensional space may be represented as ( , where is a parameter indexing the

family. Consider three families of curved surfaces, that are mutually orthogonal, defined by the

following equations

( ( (

The intersection of three of these surfaces, one from each family, defines a point in space, which

may be described by means of ( . Then, are defined as the orthogonal

curvilinear coordinates of that point.

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Figure 4.1 Orthogonal curvilinear coordinate system

In an arbitrary cylindrical coordinate system the coordinate corresponding to the Lame

coefficient, is usually taken as the axial coordinate . The transverse (curvilinear)

coordinates are then . For time-harmonic waves traveling along -axis of a cylindrical

coordinate system, the propagation factor is of the form ( and

. The

expressions for the axial components of the fields become [13]

( (

(4.1)

where and are the Borgnis’ functions satisfying the homogeneous scalar Helmholtz

equations

In an arbitrary cylindrical coordinate system, all of the functions satisfy the

same scalar Helmholtz equation

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(4.2)

We look for a solution in the separable form

( ( (

(4.3)

Substituting (4.3) into (4.2) and dividing by , we have

(4.4)

The first term is a function only of and , and the second term is a function only of z.

Therefore each of them must be separately equal to a constant, so that the sum of the two

constants will be equal to – . Let

and

. Then or

√ and

(4.5)

(4.6)

The general solution of (4.5) may be taken to be a superposition of two travelling waves

propagating along the direction:

(

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where √ is the propagation phase constant. The second equation (4.6) is a 2D-scalar

Helmholtz equation and is the transverse mode number to be fixed by the boundary

conditions.

Guided waves in bounded cylindrical systems are classified as follows according to the

value of the transverse mode number .

4.2 The TEM Mode

When √ and √ . Thus the phase speed of the TEM mode will be

equal to the speed of propagation of uniform plane waves in an unbounded medium with the

same electrical constants . From the relations , it is seen that

for a TEM mode. The transverse part of the Borgins’ potentials satisfy the 2D-

Laplace equation

Under the conditions the Maxwell’s curl equations for

the transverse fields become

The transverse fields are irrotational vector functions of the transverse coordinates and

may be expressed as 2D-gradients of scalar potentials ( ( :

( ( , ( (

In a source free region, the fields are solenoidal vector functions so that

Hence, we have

(

(

4.3 Fast Wave Modes

In fast wave modes, is real. The field configuration depends on relative values

of :

(i) If is real and . Since

√ we have

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√

√

√

√

(4.7)

which is the speed of light in the unbounded medium. We have already seen that there is no

problem with , since phase speed is not associated with the propagation of signal, energy

or wavefront.

Since is a constant, the group speed

√

√

√

(4.8)

and

(4.9)

(ii) If is imaginary. The corresponding field is not

associated with wave propagation and are called as evanescent modes.

(iii) If . This corresponds to the critical state of the waveguide

mode. The cut-off wave number of the waveguide is given as .

4.4 Slow Waves

When , then is imaginary and is real and | | . Thus

| |

√

√

√

√

(4.10)

and the phase speed along the direction is less than the phase speed of a uniform plane wave in

the unbounded medium. Hence this type of wave is called a slow wave..

For a slow wave, the eigenvalue is no longer constant with respect to frequency and

(4.8) and (4.9) for are no longer valid. The group speed of a slow wave is still smaller than the

speed of light in space. In some systems, two or even three types of waves can be supported

simultaneously.

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4.5 Periodic Structures

In a periodic structure, shape, size and material properties vary periodically along the

longitudinal axis. Wave propagation in a periodic structure is goverened by Floquet’s

theorem stated as “ 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 periodic. The proof of the theorem lies in the fact that when a periodic

structure (necessarily infinite in the axial direction) is displaced along its axis by one period or an

integer multiple of periods, it remains invariant.

4.5.1 Floquet theorem

Suppose that the spatial period of the system is , and the axial distance between the two

cross-sections is ( . Then the complex constant may be expressed as and the relation

between the phasor fields at the cross-sections separated by are related by

( ( (4.11)

In a periodic system, the distribution of the field on a transverse plane may vary with the

- coordinate (but only periodically). Therefore, the time-harmonic field at any location (

within the structure must have the form

( ( (4.12)

and accordingly the phasor field at is given by

( ( (4.13)

We can readily see that ( must be periodic function of with period to be consistent

with Floquet’s theorem. The phasor field at is

( ( (

If ( is a periodic function of with a period , then

( (

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Therefore

( (

(

as required by Floquet’s theorem. For lossless system , and we have

( (

4.5.2 Space Harmonics

The periodic function ( can be expanded in a Fourier series

( ∑ (

(4.14)

so that

( ∑ ( (

To evaluate the “Fourier Coefficients”, multiply equation (4.14) by ( and

integrate both sides from

∫ (

∑ ( ∫ (

( , since ∫ (

Thus

( ∫ (

∫ [ ( ] (

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∫ (

where

(4.15)

The field expression becomes

( ∑ (

The term of the above series is called the space harmonic which is associated to a phase

constant . Note that takes both positive and negative values. The space harmonic for

is called the fundamental space harmonic.

The phase speed of the space harmonic is given by

which is different for various values of and will be negative whenever . The group

speed of the space harmonic

(

)

is the same for all space harmonics. Since only an infinite-series expansion involving all the

space harmonics can satisfy all the boundary conditions in general, the phase speed of an

individual space harmonic does not have any physical significance; however, since all space

harmonics have the same group speed, the infinite-series representation of a propagating mode in

a periodic system will also have the same group speed at a given frequency. This is in contrast to

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the case of an individual fast-wave mode, which is capable of satisfying all the boundary

condition on its own, supported by a uniform system.

For a system with given , can be obtained by adding to it, this is to say that

the ω-βn curve is simply the ω-β0 curve shifted along the axis by . Therefore ω is a

periodic function of . It is apparent that ω is an even function of , 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 speed in direction is shown in Fig. 4.2. For the

wave with negative group speed, the phase constants of the space harmonic becomes

(

Figure 4.2 diagram for the positive and negative group velocity

4.6 Field expansion for an infinite helical structure

The invariance of the helical geometry under rotation and translation permit an infinite

series representation for the field components in terms of space harmonics. Hence the periodic

structure exhibits the following geometric invariance properties:

1. Being a periodic structure it is invariant under a translation in the axial direction by any

integer multiple of the pitch .

2. When it is rotated around its axis by any integer multiple of radians, it also remains

invariant.

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The periodicity in the axial direction implies, by Floquet’s theorem, that any phasor field

component has the representation

( (

(4.16)

where is periodic in with period ;

( (

(4.17)

The property of the axial symmetry implies that is also periodic in with period ;

( (

(4.18)

Thus admits the double Fourier-series representation

( ∑ ∑ (

(4.19)

3. When the helix is translated in the axial direction by an arbitrary distance and

simultaneously rotated around its axis by the single , it remains invariant.

This invariance implies that

(

) (

that is,

∑ ∑ ( [ (

)]

[

( ]

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∑ ∑

(

(

∑ ∑ ( (

)

(4.20)

For the last two infinite series to be equal for an arbitrary value of , it is necessary that

( whenever . Redenoting ( by ( , the double infinite series

representation of ( is seen to reduce to single series

( ∑ ( (

)

(4.21)

and all the field components may be represented in infinite series of the form

( ( ∑ (

(4.21)

where

(4.22)