Propagation analysis of chirowaveguides using the finite-element method

1488 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 10, OCTOBER 1990

Propagation Analysis of Chirowaveguides Using the Finite-Element Method

Abstract -Propagation characteristics of chirowaveguides, i.e., waveguides including chiral media, have been calculated using a generalization of the full vector finite-element formulation in terms of the electric and magnetic fields. The described formulation permits numerous inbo- mogeneous waveguide structures of arbitrary linear composition including chiral media to be analyzed without any nonphysical, spurious modes. In the proposed formulation both the necessary conditions on the tangential field components and the additional conditions on the normal field components are automatically satisfied by the trial functions. In this way the dimension of the resulting sparse generalized eigenvalue problem is significantly reduced. The straightforward extension to the novel class of chirowaveguides, which exhibit a number of interesting new features, demonstrates the versatility of the utilized formulation. The previously noted advantageous numerical properties have thus been preserved. Numerical examples on both metal and dielectric chirowaveguides are given. The finite-element results are com- pared with exact solutions, which are also reported, and the correspondence is found to be excellent.

I. INTRODUCTION HIROWAVEGUIDES constitute a recently pro- C posed [l], [21 class of waveguides characterized by

the incorporation of chiral media. The existence of chiral, or optically active, media has been known since the begin- ning of the 19th century and has attained a renewed interest due to the possibility of manufacturing such materials not only for the optic region but also for the microwave and millimeter-wave domains [3]. As a waveguide filled with a chiral medium demonstrates a number of new and interesting features, it has been suggested [2] that such waveguides will give rise to a variety of new applications in, e.g., integrated optical devices and optical communication systems as well as in the microwave and millimeter-wave fields. For such applications a method to accurately predict propagation characteristics and modal fields will be of great concern, as exact solutions are feasible only for a strictly limited number of structures.

The two-dimensional finite-element method (FEM) has been widely used during the last two decades in the analysis of waveguide components [4]-[ 121. With this method propagation characteristics of arbitrarily shaped waveguides are easily attainable. A drawback which ini- tially limited the use of the method is the appearance of nonphysical, spurious solutions [41, [51, [71, [8] in the

Manuscript received December 6, 1989; revised May 3, 1990. The author is with the Department of Information Technology

(FOM), Swedish Defence Research Establishment, P.O. Box 1165, S-581 11 Linkoping, Sweden.

IEEE Log Number 9037693.

earlier vector formulations. The spurious solutions satisfy the finite-element equations but do not even approxi- mately satisfy the original boundary-value problem. A reported origin [5] of the spurious solutions is that some of the tangential boundary conditions, which are necessary to unambiguously define the boundary-value problem, are not explicitly satisfied.

During the past few years finite-element formulations have been presented that are purported to avoid the spurious solutions [9]-[ll]. Of these the formulation in [ l l ] is the only one applicable to arbitrary linear media. It is a formulation in terms of both the electric and magnetic fields, E and H , and in which both the necessary conditions on n X E and n X H and the additional conditions on n . B and n . D are automatically satisfied by the test and expansion functions. The enforcement of normal conditions reduces the final number of variables to or below the level of, e.g., the formulations given in [9] and

In this paper the formulation in [ l l ] is extended to include chirowaveguides. The straightforward extension and successful application to the novel class of chirowaveguides demonstrate the versatility of the adopted formulation. Previously observed [ 111, [ 121 numerical properties of the resulting generalized eigenvalue problem, such as high accuracy and convergence rate, sparsity of the matrices, and the possibility of treating either the complex propagation constant or the angular frequency as the eigenvalue, are thus conserved.

Some canonical examples, a circular metal chirowaveguide and a circular dielectric chirowaveguide, are analyzed, and the finite-element results are found to be in excellent correspondence with the exact solutions, which are also given in the form of characteristic equations. Typical chirowaveguide features, e.g. the phenomenon of mode splitting and the absence of pure TE and TM modes, are observed. No spurious modes are found.

11. THE FINITE-ELEMENT FORMULATION Consider the arbitrary chirowaveguide shown in Fig. 1,

which may consist of a number of different linear achiral or isotropic lossless chiral media. By taking the constitutive relations to be of the form

[101.

D = E ~ [ E ] B + j [ , B

H = jt ,E + [ P ~ P I - ~ B (1)

( 2)

0018-9480/90/ 1000- 1488$01 .00 0 1990 IEEE

SVEDIN: PROPAGATION ANALYSIS OF CHIROWAVEGUIDES 1489

... where E , H , D , B, eo, po, [E], [ p ] , and 5, are, respectively, the electric field, the magnetic field, the electric displacement, the magnetic induction, the permittivity of vacuum, the permeability of vacuum, the relative permittivity 3 X 3 matrix, the relative permeability 3 X 3 matrix, and the chirality admittance, we may treat both arbitrary linear achiral media (5, = 0) as in [ l l ] and [12] and lossless isotropic chiral media (5, f O), for which it has been shown [131 that the constitutive relations are of the

By analogy with the derivation in [I11 for achiral waveguides, we base the weak finite-element formulation on the classical Maxwell curl equations:

form (1) and ( 2 ) with scalar [E] and [p]. Y ~ i ~ . 1. The arbitrary waveguide structure considered, which may be

composed of arbitrary linear achiral media and isotropic lossless chiral media.

V X E = j w B = j w p o [ p ] ( H - j 5 , E ) (3)

= - j w [ ( ~ ~ [ ~ I + t ~ ~ ~ [ ~ L l ) ~ + j t , ~ ~ [ ~ C L l ~ l (4)

where 2, = d G is the intrinsic impedance of vacuum, and

V X H = - j w D

where the harmonic time dependence e-’”‘ has been assumed, and (1) and (2) have been solved for B and D in order to replace these quantities in (3) and (4). Proceed- ing with the analogy, we require that for admissible test functions Etest and Htest,

where the appropriate interelement and boundary conditions are handled during the assembly Le over all elements 2141. Here, both the necessary continuity re- quirements on n x E and n X H and the additional re- quirements on n - B and n . D are, as described below, explicitly enforced on the test and expansion functions.

In this paper the six components of the electric and magnetic test and expansion fields are approximated on each element in terms of the values at each of the nodal points according to

The real k x 1 column vector { N } is the element shape function vector [15], where k is the number of nodal points on each element, {O} is a k x 1 null vector, and the superscript T denotes a matrix transposition. The column vectors {E,}, {E,}, {E,}, {H,}, {H,}, and {H,} are k X 1 complex field vectors representing the nodal point values of, respectively, E, / Z o , E , / Z o , - jE , / Z o , H,, H,, and - jH, on each element.

Using the standard Galerkin procedure with the expansion (6)-(8), one obtains the following generalized eigenvalue problem:

(9)

where the column vector is composed of all the nodal point variables used to represent E and H throughout the waveguide cross section. Note that both the complex propagation constant p and the real angular frequency o may be treated as the eigenvalue depending on which is numerically most advantageous or of primary interest.

where p = p’+ j p ” is the complex propagation constant, with p’, p” E 9,

[ N I = (7)


By expanding (5) in component form, the quadratic sparse matrices [PI , [Ql, and [RI are found to be

where

[AI = W ) ( W (13)

[B] = ( N ) a ( N ) T / a x (14)

[Cl = w ) a ( N ) T / a Y . (15)

In the present analysis the constitutive parameters [ E ] ,

[ p ] , 5, are allowed to vary between elements, and in (10)-(12), E , , , E , , , etc., represent the a, xy, etc., components of [ E ] , with the components of [ p l represented in analogous fashion.

The continuity of n X E , n X H , n. B , and n . D , where n is the unit normal vector perpendicular to an element side in the transversal x - y plane, is enforced. As an example consider a portion of the sample third-order triangular mesh illustrated in Fig. 2. In order to satisfy the

interelement conditions, it is sufficient to enforce at each nodal point along each internal side

(16)

(17)

n X ( E , - E, ) = 0

n X ( H , - H I ) = 0 and

n . ( ( ( €01 €,I + E: Po[ PCL, 1) E, + jt,PO [ P l 1 ) - ( ( ~ o [ ~ , ~ + ~ ~ ~ o [ ~ I ~ ) ~ , + J ~ , , P J ~ ~ I H , ) ) = O (18)

n. ([ cuc I ( H , - j t , , ~ , ) - [ CL, I ( H , - jt,,E, 1) = 0 (19) where E,, H , , E,, and HI denote unconnected field vectors in adjacent elements, i and j . To satisfy the external boundary conditions, it is sufficient to enforce at each nodal point along each external side

n X E , = O (20)

n . ( [ ~ , l ( ~ , +, ,E l ) ) = O (21)


, . , . , . , , _,' which is self-adjoint because I .~

(26b)

( 26c)

Fig. 2. A portion of a sample third-order mesh. Interelement or boundary conditions are in general enforced at each nodal point along each element side w.r.t. the normal vector n.

and hence

[RI = [RI'.

on electric walls and

n X H , = O

The curve integral along the boundary r in (26b) vanishes owing to the explicit enforcement of the conditions on n X E and n x H. When lossy structures are considered (22)

on magnetic (symmetry) walls. Other examples, which are not considered in this paper, are surface impedance walls and phase-shift walls. From (16)-(23), it is easily con- cluded that the total number of unknowns is < 6Np, where Np is the number of nodal points.

For nondissipative structures the matrices [PI, [ e l , and [RI are all self-adjoint. From (5 ) and (9) we note that [PI corresponds to the zeroth-order operator

which is easily shown to be self-adjoint, i.e. L, =Lip, employing the fact that [E], [p], and 6, are all self-adjoint for nondissipative media. Consequently,

[ P ] = [ P ] ' . (24b) Because -[PI is related to the longitudinal energy density, [PI must be negative definite. The matrix [Ql is associated with the zeroth-order operator

1: 0 0 - 1 0 0

1 0 0

which indeed is self-adjoint and thus

[el = [ e l ' . (25b) The matrix [RI corresponds to the first-order operator

L R = j ( -:tlx 7;)

[PI will no longer be self-adjoint. The most important property of the eigenvalue prob-

lem (9) is the 0(1 /N) density of the matrices [PI, [Ql, and [RI. This property ensures that the maximum number of nontrivial matrix elements on each row is independent of the dimension of the matrices. Thus an upper bound of the densities becomes p = NNz/N2 a 0(1/N), where N, is the maximum number of nonzero elements on each row and N is the matrix dimension. Hence, for large-size problems, sparse eigenvalue codes may be used to save significant amounts of computer time and memory, as has been pointed out elsewhere [71, [161.

111. NUMERICAL EXAMPLES In this section we consider some chirowaveguides in

order to confirm the validity of the presented finite-element formulation. To make possible a comparison of finite-element results with solutions of known accuracy, only waveguide structures for which exact solutions are obtainable are considered in this paper.

The complex generalized eigenvalue problem (9) for chirowaveguides has been solved with p treated as the eigenvalue using the (dense) QZ algorithm, as imple- mented in the double precision version of the NAG [171 routine F02GJF. The computations were performed on a 1Mflop/l6MB computer. In order to use more elements than in the examples below, a sparse eigenvalue code would have been necessary. With p being treated as the eigenvalue, the eigenvalue problem is of a type for which, to the author's knowledge, there do not seem to exist any well-established sparse codes. On the other hand, for lossless problems with w treated as the eigenvalue, the problem is of the type [A]{x} = h [ B ] { x ) , where [A] and [B] are self-adjoint and [B] is positive definite. For that case efficient sparse eigenvalue codes are available [161, [171.


A

0.0

1.5 2.0 k,R 2.5 3.0

order EH,,, and HE,,, modes in the circular metal chirowaveguide. Fig. 4. Dispersion relations for the fundamental HE,,, and higher

treatment is necessary because of the undefined normal vector at the vertex points.

The resulting propagation diagram is plotted in Fig. 4, where crosses denote FEM results and solid lines denote exact solutions (derived in the Appendix). Considering

2R

v the relatively crude mesh that has been employed the

(b) Fig. 3. (a) Cross section of the circular metal chirowaveguide, which is

bounded by an electric wall at p = R. (b) Finite-element model of the chirowaveguide shown in Fig. 3(a). The field components E,, E,, and B, are forced to vanish at vertex points on the external boundary at p = R.

A. The Circular Metal Chirowaueguide This example consists of a waveguide of radius R ,

depicted in Fig. 3(a), homogeneously filled with a chiral medium of relative scalar permittivity E , = 1, relative scalar permeability p, = 1, and chiral admittance 5, = 1 mS and bounded by an assumed perfect electric conductor. The finite-element mesh shown in Fig. 3(b) was employed to calculate the dispersion relations for the fundamental HE,,, and higher order EH,,, and HE:,, modes.

As all modes in a metal chirowaveguide are hybrid [l], [2], the descriptor EH,,, is here used for chiral modes originating in achiral TM,,, modes, and HE,,, is used for chiral modes stemming from achiral TE,,, modes. Here, n and m denote the azimuthal and radial quantum numbers, of which n implies an azimuthal variation e'"@' and m implies that the mode is number m when ordered after increasing cutoff frequency for given n.

The applied FEM mesh consists of 80 first-order triangular elements, which corresponds to 246 complex variables. On the external (electric wall) boundary at p = R , (20) and (21) were enforced w.r.t. n = p, i.e., not w.r.t. the triangle side normals at the vertex points. This special

degraded near points where a p / a w 4 W.

Some special features of the metal chirowaveguide that have been accurately modeled using the FEM are as follows. The first is the absence of pure TE and TM modes; i.e., all modes are hybrid. The second is that each chiral mode is split into two branches with different propagation constants depending on the sign of n. The cutoff frequencies for the two branches are, however, equal. For n = 0 only one branch is obtained. By reversing the direction of propagation, the magnitude of the propagation constants are retained if one makes the change n -+ - n. The chirowaveguide is consequently reciprocal. By reflecting the dispersion diagram in the k , R axis and making the substitution n + - n, the whole dispersion diagram can be obtained. It is thus only the relation between the propagation direction and the sign of n that is important. The third feature is that there exists a backward region (where the phase and group velocities have opposite signs) as for a transversely magnetized ferrite-loaded waveguide (which is nonreciprocal). Finally, at large frequencies, for 6, > 0 (6, < 01, the propagation constant approaches k + ( k - ) (see the Appendix), which is the bulk propagation constant for right-handed circularly polarized (left-handed circularly polarized) waves in an unbounded chiral medium [2]. This feature has also been reported in [2] for parallel-plate waveguides and in 131 for circular metal chirowaveguides. All modes become TU-(TU+) [3] at high frequencies, where U , and U - are functions used to decouple the wave equations according


to (Al)-(A4). The dotted line in Fig. 4 indicates the position of k , / k , = 1.445 for the chirowaveguide being discussed.

B. The Circular Dielectric Chirowaveguide This waveguide, as illustrated in Fig. %a), consists of a

circular chiral core of radius R with relative scalar permittivity = 1.1, relative scalar permeability pl = 1, and chiral admittance & = 1 mS surrounded by an achiral medium with constitutive parameters E , = p2 = 1.

The finite-element mesh shown in Fig. 5(b) was used to calculate dispersion relations for the fundamental HE1: and higher order modes. The descriptor HE,,, is here used for a hybrid mode with azimuthal and radial quantum numbers n and m, respectively. The mesh used comprises 108 first-order triangular elements, correspond- ing to 330 complex variables.

The infinite exterior region, p > R, has been modeled by a finite region extending to p = 5R that was termi- nated with an electric wall (this position was not opti- mized). With this, one cannot analyze the waveguide below p / k , = 1, k , = w&/co (when it starts acting as an antenna), but with increasing frequency the approxi- mation will be increasingly good.

In Fig. 6, the resulting dispersion diagram is plotted for p / k 2 > 1, where crosses denote FEM results and solid lines represent exact solutions, as given in the Appendix. Again, by reflecting the dispersion diagram in the k , R axis and letting n + - n, the whole diagram can be obtained for Ip l /k , > 1. For this chirowaveguide it was numerically confirmed using (A331 that p + k + ( k - 1 when w + m for 6, > 0 (5 , < 0). The dotted line in Fig. 6 indicates the position of k , / k 2 = 1.491 for the chirowaveguide in question.

The correspondence between FEM results and exact solutions is again very satisfying in view of the crude mesh that has been utilized. Of the studied modes, the relative error in p is largest for the HE- 1, branch, which has the most rapid variation in the radial direction. However, as only five points were used for this direction, the relative error, which is always less than 2%, must be considered quite satisfactory.

For the studied examples, no spurious modes have been observed. As was pointed out in [ll], the relative error in p2 increases rapidly with the mode indices, with the result that some higher order modes are predicted to have a propagation constant in the same region as the fundamental and first higher order modes. However, by calculating the residual of the solutions and only accept- ing modes with a residual smaller than a suitable limit, this problem is diminished. To calculate a dispersion diagram it is thus easy to follow a certain mode along the w (or p ) axis.

IV. CONCLUSIONS A finite-element procedure to analyze inhomogeneous

waveguide structures of arbitrary linear composition in- cludine lossless isotronic chiral media has been DrODOSed.

z Lx (b) (a) Cross section of the circular dielectric chirowaveguide. (b)

Finite element model of the chirowaveguide shown in Fig. Xa). The field components E+, E,, and B, are forced to vanish at vertex points on the artificial boundary at p = 5R.

Fig. 5.

0 5 k 2 R 10 15

Fig. 6. Dispersion relations for the fundamental HE,,, and higher order HE,, , mode in the circular dielectric chirowaveguide.

a 1 I Y . .

1494 IEEE TRANSACXIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 10, OCTOBER 1990

The procedure combines a generalization of the full vector finite-element formulation [ l l ] in terms of both the electric and magnetic fields with an explicit enforcement of the conditions on both the tangential and the normal field components.

With the presented procedure, advantages such as the absence of spurious modes, the sparsity of the resulting generalized eigenvalue problem, and the possibility of treating both the propagation constant and the eigenfre- quency as the eigenvalue result.

The validity of the presented procedure has been confirmed by applying it to both metal and dielectric circular chirowaveguides. The correspondence between FEM results and exact solutions has been found to be excellent. Characteristic equations for the waveguides considered have also been presented.

APPENDIX The exact solutions plotted in Fig. 4 and Fig. 6 have

been obtained by solving the appropriate characteristic equations. As derived elsewhere 111, [2] for a chirowaveguide, the longitudinal components E, and H, can be expressed in terms of functions U , and U - according to

E , = p , U + + p - U - (AI)

H, = q + U , + q - U - (A21

v;u, + p + U , = 0 (A31

y2u- + p - U - = 0 (A41

with

range p < R :

U + = A,J , (Jp ,p)e j”+ (Al l )

U - = ~ , ~ , ( & p ) e j , , (A121

where A , and A , are constants to be determined. Here, J, is the Bessel function of the first kind of order n.

For the dielectric chirowaveguide we must also consider the exterior region, p > R (here assumed achiral), where the longitudinal components obey 1181

y2E , + ( k 2 - P2)E, = 0

y’H, + (k’ - p ) H , = 0.

(A131

(A141

Owing to the assumed time dependence, the solutions have to be of the form

E, = A , Hi1)( d-p)ejn4

H, = A , Hi1)( d-p)ein4

(A15)

(A16)

where Hi’) is the Hankel function of the first kind and order n.

A. Characteristic Equation for the Circular Metal Chirowaveguide

By enforcing the boundary condition

O = p x ( E , p + E,++ E z ~ ) l , = ~ = ( E , z - E , + ) ~ , = R

(A171

and constants [2]

P + = (k: - P ’ ) p - = ( k 2 - p’)

on the assumed perfect electric conductor at p = R (see Fig. 3(a)) and using (Al)-(A16) together with the expres- sion for E, as a function of E, and H, given in [l], the following system of equations is obtained:

( ~ 5 )

(A6)

where

q+ = (k: - k l ) ~ , / ( 4 j ~ ’ ~ ’ t , ) (A7) a=P([(k:+k!)/2] - P ’ ) / [ ( P ’ - k : ) ( P ’ - k ? ) ]

k = +_ w p t , + J k 2 + ( wpt,)’

q - = -(k:- k2)p-/(4ju2p2tC) (A8) (A191

(A9) b = w p ( p 2 - k2)/[(p2 - k : ) ( p 2 - k ! ) ] ( M O )

k=w\ICLE P=POP, E = E ~ E , . (A10) p = 2 W 2 ~ ’ t c P / [ ( P 2 - - k : ) ( p 2 - k l ) ]

Here, k, are the wavenumbers for right-handed and left-handed circularly polarized plane waves, respectively. The assumed harmonic time and z dependence has been suppressed in this Appendix.

For the circular chirowaveguides considered in this paper, the use of cylindrical coordinates ( p , 4 , z ) yields the following solutions for U , and U - in the interior

= Wptc(p2 + k2)/[ (p2 - k : ) ( p 2 - k ? ) ] ( ~ 2 2 )

are constants .[1]. In order to accomplish a nontrivial solution of (A18), we must require

det[A] = 0 (A231

where [A] is the coefficient matrix in (A18).


at the interface between the interior chiral and the exterior achiral region (see Fig. 5(a)), the following system of equations results by using (Al)-(A16) and expressions for E, and H4 as functions of E, and H, given in [l] and [HI:

where

and

Note that whereas the submatrices [ B , , ] and [ B , , ] depend on constitutive parameters E , , p , (and &), the submatrices [ B,,] and [ B,,] depend on parameters E , and p,. Again, in order to have nontrivial solutions, we must demand

det [ B ] = 0 (A331

where [ B ] is the coefficient matrix in (A26).

ACKNOWLEDGMENT

The author would like to thank Dr. L. E. Pettersson and Dr. L. 0. Pettersson at the Swedish Defence Re- search Establishment for very helpful discussions.

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vol. 14, no. 11, pp. 593-595, June 1989. [2] N. Engheta and P. Pelet, “Guided-wave structures filled with

chiral materials,” in h o c . 1989 URSI Int. Symp. Electromagnetic Theory (Stockholm), Aug. 1989, pp. 277-279. C. Eftimiu and L. W. Pearson, “Guided electromagnetic waves in chiral media,” Radio Sci., vol. 24, no. 3, pp. 351-359, May-June 1989.

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[7] B. M. A. Rahman and J. B. Davies, “Penalty function improve- ment of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp. 922-928, Aug. 1984.

[8] M. Hano, “Finite-element analysis of dielectric-loaded waveguides,” IEEE Trans. Microwaue Theory Tech., vol. MTT-32, pp.

[3]

1275-1279, Oct. 1984.

[9] T. Angkaew, M. Matsuhara, and N. Kumagai, “Finite-element analysis of waveguide modes: A novel approach that eliminates spurious modes,” IEEE Trans. Microwaue Theory Tech., vol. MTT- 35, DD. 117-123, Feb. 1987. (A31)

[lo] K. Hayata, K. Miura, and M. Koshiba, “Finite-element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech., vol. 36, pp. 268-275, Feb. 1988. (A32)


J. Svedin, “A numerically efficient finite-element formulation for the general waveguide problem without spurious modes,” IEEE Trans. Microwaue Theory Tech., vol. 37, pp. 1708-1715, Nov. 1989. J. Svedin, “A fast converging finite-element procedure without spurious modes for waveguide propagation analysis,” in Proc. 1989 URSI Int. Symp. Electromagnetic Theory (Stockholm) Aug.

D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., vol. 18, pp. 211-216, 1979. P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers. R. Wait and A. R. Mitchell, Finite Element Analysis and Applica- tions. Chichester: John Wiley, 1985. I. S. Duff, “Survey of sparse matrix research,” Proc. IEEE, vol. 65, pp. 500-535, Apr. 1977. NAG Fortran Library, Numerical Algorithms Groups Ltd., Oxford, England.

1989, pp. 46-48.

Cambridge: Cambridge University Press, 1983.

[18] R. E. Collin, Field Theory of Guided Waues. New York: McGraw-Hill, 1960.

rite control componen Ph.D. degree in theoi nolou..

Jan A. M. Svedin (S’89) was born in Skelleftei, Sweden, on November 4, 1962. He received the M.Sc. degree in applied physics and electrical engineering in 1986 from the Linkoping Insti- tute of Technology, Linkoping, Sweden.

Since 1987, he has been a research officer in the Division of Microwave Technology, Depart- ment of Information Technology, Swedish De- fence Research Establishment, working on the field analysis and design of dielectric resonator oscillators, microstrip antenna arrays, and fer-

its. In addition, he is currently working toward the retical physics at the Linkoping Institute of Tech-

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Propagation analysis of chirowaveguides using the finite-element method

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