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. 21
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 22
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: 23
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 transverse fields become the Maxwell s curl equations for The transverse fields are irrotational vector functions of the transverse coordinates may be expressed as 2D-gradients of scalar potentials :, In a source free region, the fields are solenoidal vector functions so that and 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 24
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 and 4.8) 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. 25
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 26
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 [ ] 27
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 28
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-β n curve is simply the ω-β 0 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. 29
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, [ )] [ ] 30
) 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 ) and all the field components may be represented in infinite series of the form 4.21) where 4.21) 4.22) 31