improving the coupled-mode method by means of step functions: application to partial-height...

The antenna radiates predominantly the E� component of field withthe cross polarization level below 13 dB.

Next, radiation-pattern data are computed for the proposedfeed. A half-wave dipole feed is also modeled for comparison. Theparameters of the reflector are S � 0.21�, L � 0.21�, and � �0.65�. Computed horizontal-plane (xy plane) patterns for bothfeeds are shown in Figure 3. It is clear that the front-to-back (F/B)ratio for the MLBT feed is better than the dipole feed. Thedirectivity for the MLBT feed is 7.6 dBi, whereas that for thedipole feed is 6.9 dBi. The new feed results in some small crosspolarization (suppressed below 18 dB).

The second example given in Figure 4 has the same front–to-back ratio for both feeds. To achieve this the height of the reflector,H is adjusted for each feed, and L is fixed. For the same F/B ratio,the reflector height for the MLBT feed is 34% shorter (H � 0.65�)than that for the dipole feed (H � 0.98�).

The final model is generated by adjusting the length L andheight H of the reflector for fixed F/B ratio, directivity, andbeamwidth for both feeds. The pattern data for this case is shownin Figure 5. For fixed F/B ratio, directivity, and beamwidth of 15.5dB, 7.6 dBi, and 100o, the reflector dimension for the MLBT feedis 46% smaller than the dipole feed. The elevation-plane patternsfor both feeds have been found to be almost identical.

CONCLUSIONS

A novel primary feed antenna is introduced for application withplane sheet metallic reflectors. Such a feed, when compared to aconventional half-wave dipole feed, can reduce the reflector di-mension by almost half for fixed directivity, beamwidth, andfront-to-back ratio. These antennas can also be used in conjunctionwith corner reflectors or other types of reflectors.

REFERENCES

1. T.S. Rappaport, Wireless communications principles & practice. Pren-tice Hall, Englewood Cliffs, NJ, 1996.

2. K.R. Fujimoto and J.R. James, Mobile antenna systems handbook.Artech House, Dedham, MA, 1994.

3. M. Ali, S.S. Stuchly, and K. Caputa, A compact flat reflector antenna forpotential base station applications. Microwave Opt Technol Lett 18(1998), 319–320.

4. G.J. Burke and A.J. Poggio, Numerical electromagnetic code (NEC)–Method of moments, Parts I and III. Tech Document No. 116, NavalOcean Systems Center, 1980.

© 2002 Wiley Periodicals, Inc.

IMPROVING THE COUPLED-MODEMETHOD BY MEANS OF STEPFUNCTIONS: APPLICATION TOPARTIAL-HEIGHT ISOTROPIC ORANISOTROPIC DIELECTRIC PARALLEL-PLATE WAVEGUIDES

Alvaro Gomez, Juan S. Ipina, Miguel A. Solano, Andres Prieto,and Angel VegasDepartamento de Ingenierıa de ComunicacionesUniversity of CantabriaAvenida de los Castros s/n39005 Santander, Spain

Received 3 December 2001

ABSTRACT: The main drawback of the coupled-mode method is thepoor convergence and accuracy of the discontinuous field components.The greater the discontinuity, the worse these problems become. ThisLetter presents an improvement of the method, allowing this drawbackto be minimized. © 2002 Wiley Periodicals, Inc. Microwave Opt

Figure 3 Horizontal-plane patterns in front of a plane sheet reflector

Figure 4 Horizontal-plane patterns for same F/B ratio

Figure 5 Horizontal-plane patterns for fixed beamwidth, directivity, andF/B ratio

408 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 33, No. 6, June 20 2002

Technol Lett 33: 408–414, 2002; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.10337

Key words: coupled-mode method; microwave propagation; anisotropicmedia

1. INTRODUCTION

The coupled mode method is a moment method that is useful forobtaining the electromagnetic field inside cylindrical waveguidesthat are partially filled with isotropic and anisotropic media [1–5].Moreover, this method can easily be combined with the mode-matching method to characterize discontinuities, which results inconsiderable numerical benefits [6,7]. Its main advantage is itsversatility, because it is not necessary to apply boundary condi-tions to the dielectric interfaces. As a result, the formulation doesnot have to be redone if the geometry of the problem changes.However, the source of its versatility is also the cause of its maindrawback. If a field component is discontinuous in a transversedirection, the magnitude of the discontinuity will increase as thedifference between the permittivities of the two media becomesgreater. Because each field component is approximated by the sumof continuous sinusoidal functions, a large number of functionsmay be needed to describe the discontinuity adequately. Thisobviously will have negative effects on both the CPU time and thenumerical efficiency of the method.

To alleviate this problem, an alternative technique has beendeveloped [8] for waveguides with two transverse discontinuities,which consists of using as basis functions the functions corre-sponding to the LSE and LSM modes of a waveguide with a singletransverse discontinuity. This makes the formulation of the prob-lem considerably more complicated than the usual case, in whichthe basis functions used correspond to the modes of an emptywaveguide with the same external dimensions as the waveguide tobe studied.

Another alternative is to use step functions in the developmentof the formulation of the coupled-mode method. The strategyconsists of imposing, a priori, the discontinuity on those compo-nents of the electromagnetic field that present an abrupt jump(discontinuity) due to a change in the characteristics of the dielec-tric medium inside the waveguide. This can be done easily bymultiplying the usual expansion of the component in question byan appropriate sum of step functions. In this way, the discontinu-ous behavior of the field can be described more accurately withoutsacrificing too much of the formulation, because the basis func-tions are still the usual ones.

2. THEORY

In order to simplify the mathematical development, the formula-tion of the method for a case where there is discontinuity in onlyone of the transverse directions will be considered—for example aparallel-plate waveguide partially filled with a ferrite magnetizedin the Y direction (see Figure 1). The waveguide with isotropicdielectric will be considered as a particular case. In both structures,the electromagnetic field only varies in one transverse direction,along the y coordinate. The basis functions are chosen to be equalto the fields corresponding to the modes of an empty parallel-platewaveguide (basis modes). These are the TE and TM modes, aswell as a TEM mode, which is the fundamental mode.

2.1. TE Modes.The scalar potential T[n](y) is defined as

T�n�� y� ��2

kc�n��acos�n�

ay�, n � 1, 2, . . . , �, (1)

where kc[n] � n�/a is the cutoff wave number for TE modes. Then,the normalized transverse field components are [1]

et�n� � �az � �tT�n��y�, (2a)

h�n� � �tT�n��y�. (2b)

Note that the wave admittance of the TE mode has not beenincluded in the magnetic field; it will be taken into account later inthe modal expansion.

2.2. TM Modes.The scalar potential T(n)(y) is defined as

T�n� ��2

kc�n��asin�n�

ay�, n � 1, 2, . . . , �, (3)

where kc(n) � n�/a is the cutoff wave number for TM modes. Thenthe normalized transverse field components are [1]

et�n� � �tT�n��y�, (4a)

ht�n� � az � �tT�n��y�, (4b)

where, once again, the wave admittance in the magnetic field hasnot been taken into account.

2.3. TEM Modes.In addition to the TE and TM modes in a parallel-plate waveguide,there is a TEM mode, which is the fundamental mode. The scalarpotential for the TEM mode is

T�0� �1

�ay, (5)

and the expressions for the fields are the same as for the TMmodes. As can be seen [9, pp. 114–117], this mode corresponds tothe TM0 mode; that is, its fields do not vary with the Y coordinateand there is no cutoff frequency. Hence, the formulation for theTM modes can be extended to the TEM mode, simply by adding0 to the subindices, so that the scalar potential T(0) will be givenby (5).

2.4. Field in the Partially Filled Waveguide.The transverse components of the electromagnetic field of a uni-form parallel-plate waveguide partially filled with a slab of mag-

Figure 1 Parallel-plate waveguide partially filled with isotropic or aniso-tropic dielectric slabs (P.E.C. means perfect electric conductor)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 33, No. 6, June 20 2002 409

netized ferrite can be expanded in terms of the former TE, TM, andTEM basis modes as

E�y, z� � i�0

�

V�i��z�et�i��y� � j�1

�

V� j��z�et� j��y�, (6)

H�y, z� � i�0

�

I�i��z�ht�i��y� � j�1

�

I� j��z�ht� j��y�, (7)

where V(i)(z), V[j](z), I(i)(z), I[j](z) are the coefficients of the ex-pansion, whose dependence on the z coordinate for uniformwaveguides have the form V(i) exp(�k) with k � 1,2, . . . ,� withV(i) being a constant and �k the propagation constant for the modescorresponding to the partially filled waveguide (they will be calledproper modes). Analogous expressions could be written for theother coefficients. In the notation employed, parentheses refer toTM modes and brackets to TE modes of the empty waveguide. TheTEM mode will be taken into account as TM0 mode.

If Eqs. (2) and (4) are substituted into (6) and (7), the transversecomponents of the electromagnetic field in terms of the scalarpotentials are obtained:

Ex� y, z� � j�1

�

V� j�� z��T� j�� y�

� y, (8)

Ey� y, z� � i�0

�

V�i�� z��T�i�� y�

� y, (9)

Hx� y, z� � � i�0

�

I�i��z��T�i��y�

�y, (10)

Hy� y, z� � j�1

�

I� j�� z��T� j�� y�

� y. (11)

In the same manner, the longitudinal components of the electro-magnetic field can be expressed as [6]

Ez� y, z� � i�1

�

Kc�i�V�i�z � z�T�i�� y�, (12)

Hz� y, z� � j�1

�

Kc� j�I� j�z � z�T� j�� y� � H0� z�, (13)

where again the dependence of Vz(i)(z), I z

[j](z), and H0(z) on the zcoordinate for uniform waveguides has the same form as men-tioned before for V(i)(z), and kc(i) and kc[j] are introduced in orderto maintain dimensional coherence. The term H0(z) has to beintroduced because of the presence of magnetic media inside thewaveguide [4]. This term comes from the static solution (n � 0)for the TE modes [10]. For computer implementation, the infinitenumber of modes is truncated to NTE number of TE modes andNTM number of TM modes.

2.5. Use of the Step Function.Upon inspection of the above equations, it is seen that all thecomponents of the electromagnetic field are sinusoidal functions.

Figure 2 Convergence results for the phase constant normalized to vacuum wave number versus the number of basis modes introduced in the expansion.Dimensions in millimeters: a � 10, h � 3, �r � 15.3, F � 11 GHz


So, numerically, they are continuous functions. For the structureshown in Figure 1, the only discontinuous field at y � h is the Ey

component, and it has to be reproduced by means of continuoussinusoidal functions. This situation is analogous to the one shownin [11] in analyzing chirowaveguides and bianisotropicwaveguides. Figure 1 of [11] shows how a square function can beexpanded in a Fourier series. However, it is reasonable to supposethat the bigger the size of the step, the larger the number offunctions that must be used to adequately represent it. If thissituation is applied to the present problem, it is easy to see that inorder to adequately describe the behavior of a discontinuous fieldcomponent, a very large number of T functions would have to beused. This will impair the convergence and accuracy of the elec-tromagnetic field and the propagation constants, especially, for thehigher modes. For structures like that shown in Figure 1, withdiscontinuities in the y direction, the development given in Eq. (9)for the Ey component is more adequate for the Dy component,because Dy is a continuous function in the y direction, whereas Ey

is not. Then, if one introduces a new function Fstep1/� to describe the

discontinuous geometric behavior of the structure of Figure 1,given by

Fstep1/� �y� �

k�1

Nslabs 1

�rk

k�y�, (14)

where Nslabs is the number of slabs inside the guide, �rk is therelative dielectric constant of the rth-slab and k(y) is the pulsefunction given in terms of the step function U(y)

k� y� � U� y � hk� � U� y � hk�1� � 1, if hk � y � hk�1,0, elsewhere,

(15)

the Ey component can be expanded as

Ey� y, z� � Fstep1/� �y�

i�0

�

V�i��z��T�i��y�

�y. (16)

In the present case, the constitutive relations for the slab of ferritemagnetized to saturation in the y direction are [10]

TABLE 1 Relative Errors for Three Modes Corresponding to the Structure of Figure 2 with Three Different Permittivities

Error (%)

TM(1) TM(2) TM(9)

Schelkunoff’sFormulation [1]

PresentFormulation


PresentFormulation


PresentFormulation

30 50 30 50 30 50 30 50 30 50 30 50

�r � 2.5 0.05 — 0.0002 — 0.07 0.04 0.001 — 0.13 0.08 0.0123 —�r � 10 0.29 — 0.0019 — 0.93 0.55 0.0033 — 0.89 0.52 0.0901 —�r � 30 0.90 0.54 0.0026 — 1.59 0.95 0.0081 — 1.61 1.11 1.0516 0.0495

Figure 3 Ey component normalized to its maximum value as a function of the y coordinate for the fundamental TM1 mode for the same structure as Figure 2


� � �0�r, �� 0� � 0 jk0 �0 0

�jk 0 �� . (17)

From this point on, the process can be seen in detail, for arectangular waveguide, in [6], with the only change being theexpansion for the Ey component. Thus, the eigenvalue systemobtained is

�

� z �V�n�

V�m�

I�n�

I�m�

��

0 T �n�� j�v Z�n��i�

T 00 0 0 j�0U�m�� j�

j�0U�n��i� 0 0 00 Y�m�� j�

T T �m��i�i 0

��V�i�

V� j�

I�i�

I� j�

� (18)

Figure 4 Ey component normalized to its maximum value as a function of the y coordinate for the evanescent TM9 mode for the same structure as Figure 2

Figure 5 Dispersion diagram for a parallel-plate waveguide totally filledwith a ferrite magnetized to saturation. Dimensions in millimeters: a � 10,�r � 15.3, Ms � 1790 Gauss, H0 � 1000 Oersteds (Ms is the saturationmagnetization of the ferrite and H0 is the intensity of the applied externalmagnetic field)

Figure 6 Dispersion diagram for a parallel-plate waveguide partiallyfilled with three slabs of ferrite magnetized to saturation. Dimensions inmillimeters: a � 10, h1 � 2, h2 � 4, h3 � 6, h4 � 8, �r � 15.3, Ms � 1790Gauss, H0 � 1000 Oersteds (Ms is the saturation magnetization of theferrite and H0 is the intensity of the applied external magnetic field)


where the expressions for the matrices are shown in Appendix 1.The subindices n and i vary from 0 to NTM (0 is the index for theTEM mode) and the subindices j and m vary from 1 to NTE. TheU matrix is the identity matrix. The eigenvalues of the matrix in(19) are the propagation constants and the eigenvectors are thecoefficients of the development of Eqs. (8)–(13).

3. RESULTS

To check the behavior of the formulation with the use of the stepfunction proposed here, several structures that have an analyticsolution have been analyzed. First a parallel-plate waveguide withisotropic dielectric slab having the characteristics and dimensionsindicated in Figure 2 are considered. For this structure, the Ey

component is zero for the TE modes, and they are not influencedby the new development for this component. Furthermore, the TEand TM modes are not coupled; and only results for the TM modesthat have an Ey component that is not zero will be presented. Thefigure shows the convergence of the results obtained with theSchelkunoff formulation [1] and with the present method, for thephase constant normalized to the free-space wave number corre-sponding to the fundamental mode as a function of the number ofTM basis modes introduced in the expansion. Both formulationsare compared with the exact solution obtained by solving thecharacteristic equation for the TM modes. It can be seen that withonly 10 modes the formulation using a step function provides avalue very close to the exact solution. The Schelkunoff formula-tion, on the other hand, fails to achieve such good accuracy evenwith 50 modes. Table 1 summarizes the results obtained for othermodes and permittivities with the same waveguide. To be specific,the table shows the errors obtained with respect to the exact valuesof the phase constant when these two formulations are used toanalyze the TM1, TM2, and TM9 modes for three different per-mittivities. The cells of the table with a line indicate that, even with30 basis modes, the errors are already negligible and therefore arenot included in the table. It can be seen that for a given mode, thegreater the permittivity of the dielectric, the larger the errors. Also,for a given permittivity, the errors become greater as attempts aremade to describe the behavior of modes with a large attenuationconstant. In any case, the errors obtained with the present formu-lation are always two or more orders of magnitude lower thanthose obtained with the Schelkunoff formulation.

With the same configuration under consideration, Figure 3compares the results of the two formulations to the exact value ofthe Ey component normalized to its maximum value as a functionof the y coordinate, with the use of 30 basis modes correspondingto the TM1 mode. It can be seen that the formulation that makesuse of step functions provides excellent accuracy. The accuracy ofthe Schelkunoff formulation, on the other hand, is worse than thatobtained by our formulation. Furthermore, if the number of basismodes is raised, there is no significant improvement and theovershooting at y � 3 mm increases, which is where there is adiscontinuity in the component Ey.

The results shown in Figure 4, which are for the TM9 mode, areanalogous to those of Figure 3. It can be seen that even for aprofoundly cutoff mode (i.e., the mode with a large attenuationconstant) the discontinuous field component is reproduced fairlyaccurately.

After the present formulation demonstrated good performancefor isotropic media, testing it with anisotropic media was under-taken. Figure 5 shows the phase constants for the first threepropagating modes and one evanescent mode corresponding to aparallel-plate waveguide totally filled with a ferrite magnetized tosaturation, with the characteristics and dimensions indicated in the

figure. It can be seen that there is an excellent agreement with theanalytic results obtained from the analysis shown in [12].

Next a more complicated structure, corresponding to the oneshown in Figure 1 with three ferrite slabs, was analyzed. Themodes of this structure are hybrid modes with the six componentsof the electromagnetic field. Figure 6 shows the results for the firstthree modes and for the tenth mode (evanescent mode), with thedimensions indicated in the figure.

Finally, Figure 7 shows the three components of the electricfield normalized to their maximum value as a function of the ycoordinate, for the fundamental mode for the waveguide of Figure6. It can be seen that the Ey component presents abrupt changes atthe interfaces between the ferrite slabs and free space. The othertwo components of the electric field, on the other hand, have acontinuous behavior at these interfaces.

APPENDIX

The expressions for the matrices that appear in Eq. (18) are

�Z�n��i�� inv�n��l �TM ��S�l ��i�

Dtotal � �1

j��inv�l ��i��

� �S�l �� j�Etotal ��Snv� j��m�

Btotal ��S�m��i�Ctotal � , (A-1)

�Y�m�� j�� j�0��m�� j�TE � � �Snv�m�� j�

Btotal �, (A-2)

�T�n�� j�V � � ��inv�n��i�

TM ��S�i��m�Etotal ��Snv�m�� j�

Btotal �, (A-3)

�T�m��i�I � � �U�m�� j��Snv� j��m�

Btotal ��S�m��i�Ctotal �, (A-4)

�S� j��m�Btotal � � ��S� j��m�

B � �1

��S� j�

A ��S�m�A �t, (A-5)

�Snv�m�� j�Btotal � � �S� j��m�

Btotal ��1, (A-6)

�S�m��i�Ctotal � � �S�m��i�

Ctrans � �1

��S�m�

A ��SS�i�C �t, (A-7)

�S�l ��i�Dtotal� � ��l ��i�

TM � �1

��SS�l �

C ��SS�i�C �t, (A-8)

Figure 7 The three components of the electric field normalized to theirmaximum values as a function of the y coordinate, for the fundamentalmode of the structure referred to in Figure 6


�S�l �� j�Etotal � � �S�l �� j�

C � �1

��SS�l �

C ��S� j�A �t, (A-9)

�SS�i�C � � �0 �

0

a

k�T�i�

� yd y, (A-10)

� � �j�0 �0

a

� dy, (A-11)

�S� j�A � � j�0 �

0

a

�T� j� d y, (A-12)

�S� j��m�B � � j�0 �

0

a

�T�m�T� j� d y, (A-13)

�S�i��m�C � � �0 �

0

a

kT�m�

�T�i�

� yd y, (A-14)

�S�m��i�Ctrans� � �S�i��m�

C �t, (A-15)

��i��n�TM � � j�0 �

0

a

��T�n�

�y

�T�i�

�ydy, (A-16)

��i��n�TM � ��

0

a

Fstep1/�

�T�n�

�y

�T�i�

�ydy, (A-17)

��inv�n��i�TM � � ��i��n�

TM ��1, (A-18)

��m�� j�TE � ��

0

a

FSTEP�

�T�m�

�y

�T� j�

�ydy, (A-19)

��i��l �� 0

a

Fstep� T�l �T�i� dy, (A-20)

��inv�l ��i�� i��l ��1, (A-21)

where, analogously to the definition of expression (15), the func-tion

Fstep� �y� �

k�1

Nslabs

�rkyk�y�. (A-22)

has been introduced.The indices for the TE modes vary from 1 to NTE (total

number of TE basis modes), and the indices for the TM modesvary 0 to NTM (total number of TM basis modes).

REFERENCES

1. S.A. Schelkunoff, Generalized telegraphist’s equations forwaveguides. Bell Syst Tech J 3 (1952). 784–801.

2. K. Ogusu, Numerical analysis of rectangular dielectric waveguide and

its modifications. IEEE Trans Microwave Theory Tech MTT-25(1977), 441–446.

3. I. Awai and T. Itoh, Coupled-mode theory analysis of distributednonreciprocal structures. IEEE Trans Microwave Theory TechMTT-29 (1981), 1077–1086.

4. M.A. Solano, A. Vegas, and A. Prieto, Modelling multiple disconti-nuities in rectangular waveguide partially filled with non-reciprocalferrites. IEEE Trans Microwave Theory Tech MTT-41 (1993), 797–802.

5. Y. Xu and TR.G. Bosisio, An efficient method for study of generalbi-anisotropic waveguides. IEEE Trans Microwave Theory TechMTT-43 (1995), 873–879.

6. M.A. Solano, A. Vegas, and A. Prieto, Numerical analysis of discon-tinuities in a rectangular waveguide loaded with isotropic or anisotro-pic obstacles by means of the coupled-mode method and the mode-matching method. Int J Numer Model Electron Networks DevicesFields 7 (1994), 433–452.

7. M.A. Solano, A. Prieto, and A. Vegas, Theoretical and experimentalstudy of two ports structures composed of different size rectangularwaveguides partially filled with isotropic dielectrics. Int J Electron 84(1998), 521–528.

8. J.W. Tao, J. Atechian, R. Ratovondrahanta, and H. Baudrand, Trans-verse operator study of a large class of multidielectric waveguide. IEEProc Pt H 137 (1990), 311–317.

9. D.M. Pozar, Microwave engineering, 2nd ed. Wiley, New York, 1998.10. H. Chaloupka, A coupled-line model for the scattering by dielectric

and ferrimagnetic obstacles in waveguides. Arch Elektronik Ubertra-gun 34 (1980), 145–151.

11. Y. Xu and R.G. Bosisio, A study on the solutions of chirowaveguidesand bianisotropic waveguides with the use of coupled-mode analysis.Microwave Opt Technol Lett 14 (1997), 308–311.

12. H. Unz, Propagation in arbitrarily magnetized ferrites between twoconducting parallel planes. IEEE Trans Microwave Theory Tech 1963,204–210.

© 2002 Wiley Periodicals, Inc.

ON THE PROPAGATION OFELECTROMAGNETIC WAVESTHROUGH PARABOLICCYLINDRICAL CHIROGUIDESWITH SMALL FLARE ANGLES

P. K. ChoudhurySVBLFaculty of EngineeringGunma UniversityKiryu, Gunma, Japan

Received 6 December 2001

ABSTRACT: With the use of Maxwell’s field equations, an analyticalinvestigation of the propagation of electromagnetic (EM) waves througha three-layer parabolic cylindrical chirowaveguide is presented; differ-ent regions of the guide have different values of chirality admittance.The analytical approaches to derive the dispersion relations and theeigenvalue equations for cutoff are discussed under the assumption ofsmall flare angles of the parabolic cylindrical boundaries. However,more emphasis is given on the modal field cutoffs. It is observed that thebunching tendency of field cutoffs, which is seen in the cases of para-bolic cylindrical dielectric guides, is greatly affected in this case, whichis attributed to the presence of chirality in the medium. © 2002 WileyPeriodicals, Inc. Microwave Opt Technol Lett 33: 414–419, 2002; Pub-lished online in Wiley InterScience (www.interscience.wiley.com). DOI10.1002/mop.10338

Key words: optical waveguides; chiroguides; EM wave propagation


improving the coupled-mode method by means of step functions: application to partial-height...

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