Convergence phenomenon in the solution ofdichroic scattering problems by Galerkin's method

F.S. Johansson, MSc

Indexing terms: Antennas (reflectors), Electromagnetic theory, Mathematical techniques

Abstract: The convergence problem that ariseswhen the scattering problem of dichroic surfaces issolved numerically by means of Galerkin's methodand Floquet's theorem is investigated. A theoreti-cal discussion is outlined showing that a trunca-tion of the infinite Floquet spectrum is actuallyequivalent to modifying the used basis functions.Examples of modified basis functions due to differ-ent Floquet-mode truncations and the effect onthe numerical results are demonstrated for theparticular cases of arrays of thin dipoles andcrossed dipoles. Some useful truncation rules arealso presented, and the theories are verified bycomparing predicted values with experimentalresults obtained by waveguide simulator measure-ments.

1 Introduction

Periodic arrays of passive metallic elements, commonlyreferred to as dichroic surfaces or frequency selective sur-faces (FSS), have found many interesting applications inrecent years. For example, the use of dichroic surfaces assubreflectors in Cassegrain antenna systems to obtaintwo focus feed positions [1-3] has become an attractivemethod for increasing the capacity of communicationlinks.

As is well documented in the literature [3-5], the theo-retical analysis of dichroic surfaces is handled byassuming a plane infinite periodic array illuminated by aplane wave. Under these assumptions, the electromag-netic field is expanded in terms of an infinite discretespectrum of plane waves according to Floquet's theorem.From the boundary conditions, an integral equation forthe current on the metallic elements is established. Byexpanding the unknown current in a set of basis func-tions and applying Galerkin's method, the integral equa-tion is reduced to a system of linear equations.

A numerical treatment of the resulting equationsystem implies that the number of basis functions has tobe limited and that the infinite Floquet-mode spectrummust be truncated. The two degrees of freedom in select-ing the number of basis functions and Floquet modesexhibit a convergence problem that is sometimes referredto as the phenomenon of relative convergence. Althoughthis type of problem has been discussed in several papers[6-9], no general criterium has been presented as to

Paper 4954H (Ell), first received 2nd April and in revised form 29thJuly 1986

The author is with the Division of Network Theory, Chalmers Uni-versity of Technology, S-412 96 Gothenburg, Sweden

under which conditions a satisfactory solution may beobtained.

In this paper a different approach for considering theabove type of convergence problem is presented, withspecial emphasis on dichroic scattering problems. Theobjective is not to present a mathematically stringent sol-ution of the problem, but rather to explain and illustrateanalytically and numerically, the effect of numerical trun-cations. Nevertheless, the discussion results in someuseful truncation rules, which, when applied, give satis-factory results with good correspondence with measuredresults.

2 Theory

Consider the periodic array in Fig. 1. The surface isassumed to be infinite in the xy-plane and illuminated by

Fig. 1 Geometry of an arbitrary periodic array of thin conductingplates etched on a dielectric sheeta Front viewb Side view

a plane wave under the incidence angles (0, (f>). Byexpanding the scattered field in terms of Floquet modesand applying the boundary conditions, the followingintegral equation, with the induced surface current Js asunknown, is obtained:

Y bm(i + KcL^ "iv mo

m = l2 oo

m= 1 p, q= — oo'mpq

JAi

X I - W ^ M M dSQ>pq(r)KnJAs

(1)

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987 87

where

= k0 s m s m

2%y - pzxd2) (2)

r = xx + yy

The notations used are the same as in Reference 4, wherea detailed derivation of the integral equation and theremaining equations may also be found.

Let the unknown current be approximated by

(3)n = l

where hn(r) forms a set of linear independent basis func-tions. By weighting the resulting integral equationaccording to Galerkin's method and using an innerproduct defined as,

•JJ a b* dS

where A is the area of an periodic cell, the followingsystem of linear equations can be established:

+ gl{*Too)m = l

n = 1 m—lp, q~ — <x>mpq

X gn{kTpq)*mPq

Where L = 1, . . . , N a n d

gn(kTpq)= [ kJLrpjJLr) dSU{kTpq) = *»(JA

(4)

(5)

Eqn. 4 is a matrix equation for the coefficients of thecurrent expansion, and once the solution to the equationsystem is found, the scattered field can be obtained [3, 4].

Besides the unknown current coefficients Cn, thematrix equation involves the equivalent modal admit-

gn. pq (6)p, q = - co

For the numerical solution of the linear equation system,eqn. 4, the infinite Floquet-mode summation has to betruncated. Let DPQ be the set of indices of the Floquetmodes that are to be included. The truncation results inonly the part of the spectrum of the basis functions withindices belonging to DPQ being considered for thenumerical solution. This is equivalent to assuming basisfunctions with a truncated spectrum given by

gn(kTpq) = ^pqgni^Tpq)

when solving the matrix equation, where

for pq s DPQ

for pq $ DPQ

(7)

(8)

It follows from eqn. 6 that the basis functions corre-sponding to the truncated spectra are given by

pq= - c o

Zpq e DPQ

(9)

As is well known, the above relation can be written as aconvolution between the original basis function and afunction s{r) with a Floquet spectrum equal to the trunca-tion function, i.e.

(10)

tance rf£pq, the dielectric reflection coefficients Rsmoo, the T V / "

vectors #n> pq = gn{kTpq), and the unit vectors Kmpq whichrelate the scalar wave function <DTC(r) to the transversevectorial field components [5]. As seen from eqn. 5, thevectors gn pq are the vectorial coefficients for the expan-sion of the basis functions hn{r) in terms of the scalarwave functions OM(r). That is, eqn. 5 represents the spec-trum of the basis functions in the Floquet-mode domainand we have the following reversed relationship:

Kir) = s{r) * hn{r)

where * denotes the convolution and

S(r) = X" 1 X %q{r)pq e DPQ

Hence the convergence problem of numerically solvingthe integral equation, eqn. 1, by Galerkin's method canbe considered to be composed of two parts. The firstproblem is to select an appropriate set of basis functionsfor the current expansion to reduce the integralequation to a system of linear equations. The second con-cerns the proper truncation of the infinite Floquet sum-mation involved in the matrix equation. Since thiscorresponds to solving the linear equation system withmodified basis functions, a proper numerical solutionrequires that the truncation is carried out so that theresulting basis functions, due to the truncation, havegood correspondence with the original basis functions.

The following Section illustrates, by example, the effectdifferent Floquet-mode truncations can have on the basisfunctions and the numerical results.

3 Examples

Consider the particular case of the dipole array shown inFig. 2. Let the incident field have the £-vector in the yz-plane and assume that the dipoles are narrow so the

w

d2

n n nFig. 2 Single array of dipoles considered in the waveguide simulatormeasurements. The incident field is a TE-wave with the E-field in theyz-planeDimensions are: dl = 11.43 mm, d2 = 10.16 mm, L = 9.2 mm, W = 1.0 mm,t = 1.59 mm, er = 2.62, the waveguide cross-section is 22.86 x 10.16 mm2

current across the width of the dipoles can be neglected.A suitable set of basis functions is then

1 cos] fnnsin • • ~ ysin ]\L

for n odd

for n even (11)

Here the first factor is included in order to satisfy theedge condition by assuming that the transversal behav-iour is the same as in the static case [10].

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Due to the rectangular lattice geometry, the transverse

propagating vector takes the form

k0 sin 0 cos 0 + — p )x

« // 0

+ I k0 sin 6 sin cj> + — q )y\ "2 /

= kxpx + kyqy (12)

Thus the double integral in eqn. 5 can be separatedaccording to

Sn, pq y P n, q

where

•w/2

•tL/2 fcosj/mr

-L/2

exp [jkxpx~] dx

dy

(13)

(14a)

(146)

The integrals can be evaluated in closed form, and in Fig.3 the magnitudes are plotted for different values of n.

p= 3 13

Fig. 3a Magnitude of the integral in eqn. 14a as a function ofkxWArrows mark the levels of kxW for different P, when kx = 2nP/dlt dl = 11.43 mm,W = 1.0 mm

Pn=1

n = 3 n=5 n=7

, A A AIf «! '•' I

•l I II : 1

Q=

Fig. 3b Magnitudes for the integral in eqn. 14b as a function of kyLfor different nArrows mark the levels of kfL for different Q, when ky = 2nQ/d2, d2 = 10.16 mm,L = 9.2 mm

A straightforward truncation of the Floquet-modesummation in eqn. 4 consists in excluding all terms withindices outside the intervals p = { — P, P} andq — { — Q, Q}. Following the discussion in the previousSection, this corresponds to assuming basis functionsgiven by

[-jkxpx]J

x I In,q^Pl-Jkyqyl (15)q=~Q

when solving the matrix equation. In Fig. 4 a few exam-ples of the Fourier series expansion used for the basisfunctions in eqn. 11 are plotted, together with theresulting distributions in eqn. 15. As seen from Fig. 4, theresulting basis functions approximate the original dis-tributions well, except for the case when the spectrum ofthe function of order n — 7 is truncated at Q = 3. By con-sidering Fig. 3b, it is obvious that the large difference inthis case depends on the truncation at Q = 3 excludingthe dominant spectral components. Thus, if the resultingbasis functions are to represent the used Fourier seriesexpansion, it is realistic to set a minimum truncationlevel at the first zero after the highest mainlobe of thespectrum of the basis functions. That is, all Floquetmodes satisfying

where L is the dipole length and nmax, the highest usedFourier series term in eqn. 11, should at least be included.

Fig. 5 shows comparisons between predicted valuesand experimental results obtained by waveguide simula-tor measurements. The measured results correspond toTE-incidence on the infinite dipole array in Fig. 2, withthe £"-field directed along the dipoles and with incidenceangles between 6 = 30-55°, <f> = 0°. As no asymmetricalcurrents will be excited in this case, only the cosine func-tions are needed in the current expansion.

For the predicted values in Fig. 5a, four cosine func-tions are used for the longitudinal current expansion, i.e.n = 1, 3, 5 and 7 in eqn. 11. The truncation of theFloquet-mode spectrum is held constant in the/c^-direction while the truncation in the fc^-direction isvaried; that is only the part of the spectrum correspond-ing to the longitudinal current distribution is changed. Itis seen from the results that if the condition in eqn. 16 isnot satisfied, i.e. all main lobes of the spectrum of thecosine functions are not included, see Fig. 3b, the result-ant numerical solutions become incorrect.

Fig. 5b shows computed results when only two cosinefunctions (n = 1, 3) are used for the current expansion. Aspreviously, only the truncation level in the fcy-direction isvaried. In this case satisfactory numerical solutions areobtained for less Floquet modes than previously, butagain it is seen that if eqn. 16 is not satisfied, the pre-dicted results become erroneous. Using a larger trunca-tion level than Q = l has almost no effect on the pre-dicted transmission coefficients. This can be explained bythe fact that the major part of the Floquet spectrum forthe two cosine functions are already included, see Fig. 3b.

Truncation rules similar to the one given in eqn. 16have been suggested previously in the literature [2, 9]. Incontrast to what has been reported recently for the caseof solving the scattering problem of a single periodic stripgrating [9], the erroneous truncations of the Floquetspectrum in the cases treated here have not been indi-cated by the condition number of the resulting matrix

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987 89

equations. It should be emphasised that eqn. 16 is aminimum condition and that more Floquet modes can berequired if solutions with high accuracy are desired.

1.0

The example above has concentrated on the rectangu-lar array of dipoles in Fig. 2 because of the simplicity ofseparating the spectrum of the used basis functions and

1.0

0.5

Consider next the truncation of the part of the spec-trum in eqn. 13 that corresponds to the transversecurrent distribution. In Fig. 6 the transverse functionused for the basis functions in eqn. 11 is plotted togetherwith the resulting distribution in eqn. 15 for differenttruncation levels. As seen from Fig. 6, the sharp discon-tinuities and the fast variations of the transverse distribu-tion are smoothed out efficiently by the truncation of theFloquet spectrum. It is obvious that, for a good corre-spondence with the desired distribution, an unpracticallylarge number of Floquet modes would be required.

Fig. 5c shows predicted values when the truncation ofthe Floquet-mode spectrum is varied in the /e^-direction,while the truncation level in the fey-direction is held con-stant, i.e. only the part of the spectrum corresponding tothe transverse current distribution is changed. For thelongitudinal current expansion, two cosine functions areused. It is seen from the computed results that the trunca-tion level only has a minor effect if a sufficient number ofFloquet modes is included. By a closer study of varioustruncation levels for different cases it was found thatstable solutions for the far-field properties are obtained ifat least the part of the spectrum of the transversal dis-tribution up to roughly the first null is included, see Fig.3a.

That a transverse distribution of the form plotted atthe top in Fig. 6 is sufficient to give satisfactory numeri-cal results can be explained by the fact that the trans-versal behaviour of the current is not directly significantfor the resonance characteristic of the dipole array. Wealso assume a fixed transverse behaviour and expandonly the longitudinal current distribution in differentbasis functions. Actually it was found that a constanttransverse distribution can also be used for the currentexpansion in eqn. 11 without causing any large changesin the predicted values.

90

Fig. 4 Examples of the longitudinalcurrent expansion in eqn. 11 for thedipole array in Fig. 2 and the resultingdistributions due to truncations of theFloquet spectrum at different Q, seeeqn. 15

longitudinal current expansiontruncation of Floquet distribution

an = 1, Q = 7; b n = I, Q = 3

illustrating the effect of different Floquet-mode trunca-tions. However, the principles of the given truncationrules have also been used for the crossed dipole elementand for other lattice geometries, resulting in numericalsolutions with excellent agreement with experimentalresults [3]. The same ideas for the truncation of theFloquet-mode spectrum can be applied with minor exten-sions for other dichroic elements, if a similar type ofentire domain basis functions as in eqn. 11 is used.

Consider, for example, the more complicatedJerusalem-cross element [11]. This element can be con-sidered to consist of six arms of conducting rectangularstrips. If the strips are narrow, the same type of basisfunctions as in eqn. 11 can be used for the current expan-sion on each arm, except that, for the center arms, oddsine and even cosine terms also ought to be included[11]. Thus, with the arms of the Jerusalem cross directedalong the x- and y-axis, the spectrum of the basis func-tions, eqn. 5, will involve integrals of the type given ineqn. 14. For arbitrary lattice geometry, kxp and kyp ineqn. 14 have to be replaced by x • kTpq and y • kTpq. Theminimum condition of including all the main lobes of thespectrum means, in this case, that all Floquet modesshould at least be included that satisfy

(17)

Where n'xmax represents the order of that basis functionfor the x-directed strips that will produce the highestspectral mainlobe in the /cx-direction and L is the lengthof this strip, n'y max and L' are the corresponding values forthe y-directed strips.

Although erroneous Floquet-mode truncations,according to the minimum condition in eqn. 16, haveresulted in obviously incorrect numerical results for the

1EE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987

cases considered above, it is important to emphasise thatincorrect truncations are not always clearly seen withoutcomparing the predicted values with measured results. Asan example, consider Fig. 7, where comparison betweennumerical and experimental results for an array ofcrossed dipoles is shown. For the predicted values, thesame type of current expansion as in eqn. 11 has beenused, with two cosine functions and five sine functions forthe current expansion on each dipole arm. The dashedcurve corresponds to predicted values when the Floquetspectrum is truncated according to the principle of theminimum condition in eqn. 17. The dotted curve isobtained from computed transmission coefficients whenthe number of Floquet modes is truncated so the mainlobes of the spectrum of the highest used basis functions(n = 10) are excluded. It is seen from Fig. 7 that the lattercase gives a solution where the effect of the asymmetricalcurrent resonance [12] on the transmission response isvery different from the measured result. For the numeri-

-6

-12

-18

-24

-3080 9.0 10.0 11.0

frequency , GHz

• •

12.0

-6

-12

-18

-24

-308.0 9.0 10.0 11.0 12.0

frequency,GHzb

10.0 11.0frequency, GHz

12.0

Fig. 5 Measured and predicted transmission coefficients for the simu-lated dipole array in Fig. 2. For the computed results in (a) four cosinefunctions have been used in the current expansion, eqn. 11, and in (b), (c)two cosine functions. The various results correspond to a variation of thetruncation levels P and Q of the Floquet-mode spectrum, see Fig. 3

measuredfl>/> = 7,e = 3; OP = 7, 2=4; * P = 7,G = 11b M P = 7 , Q = l ; O/> = 7,e = 2; * P = 7, £) = 7c • P = 3, g = 7; O ? = 7,8 = 7; * P = 13, Q = 7

Fig. 6 Transversal current expansion in eqn. 11 for the dipole array inFig. 2 and the resulting distribution due to a truncation of the Floquetspectrum at different P, see eqn. 15

a P = l; b P = 3transversal current expansiontruncation of Floquet spectrum

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8.0 9.0 10.0 11.0frequency, GHz

12.0

Fig. 7 A Measured and predicted transmission coefficients for an arrayof crossed dipoles, simulated by waveguide simulator measurements

measuredComputed results using 180 Floquet modesComputed results using 86 Floquet modes

Fig. 7B Cross-section of the waveguide simulator

a = 22.86 mm; b= 10.16 mm;L = 10.0 mm; W = 1.0 mm; t = 1.59 mm; er = 2.62

cal results, when all mainlobes of the spectrum of thebasis functions are included, the correspondence with themeasured result is better, although the solution is some-what displaced in frequency. This frequency displacementcan be reduced if even more Floquet modes are includedwhen solving the resulting matrix equation.

4 Conclusions

The objective of this study has been to investigate theconvergence problem that arises when the integral equa-tion for the scattering problem of dichroic surfaces issolved numerically by means of Galerkin's method. Theconvergence problem has been considered to be com-posed of two parts, where the first problem is to select an

appropriate set of basis functions in order to reduce theintegral equation to a system of linear equations. Thesecond concerns the numerical truncation of the infiniteFloquet-mode summations involved in the matrix equa-tion. The discussion shows that the truncation of theFloquet spectrum is actually equivalent to modifying thebasis functions when solving the linear equation system.It has therefore been pointed out that if satisfactory solu-tions should be obtained, the truncation has to be doneso that the resulting basis functions have good corre-spondence with the original functions.

In order to study the effect different Floquet-modetruncations can have on the basis functions and thenumerical results, the particular case of a rectangulararray of dipoles has been considered, where an entiredomain Fourier-series type of expansion has been usedfor the current expansion. Minimum conditions for thetruncation of the Floquet spectrum have been establishedand the necessity of fulfilling the conditions has been con-firmed for arrays of dipoles and crossed dipoles, by com-paring predicted values with measured results.

5 Acknowledgments

The author acknowledges the support and encour-agement of Prof. E. Folke Bolinder, Head of the Divisionof Network Theory, Chalmers University of Technology,Gothenburg, Sweden. Special thanks are also due to Dr.Lars G. Josefsson for stimulating discussions on somepoints of theory.

6 References

1 AGRAWAL, V.D., and IMBRIAL, W.A.: 'Design of a dichroic cass-egrain reflector', IEEE Trans., 1979, AP-27, pp. 466-473

2 BIELLI, P., BRESCIANI, D., CONTU, S., FORLGO, D., andSAVINI, D.: 'Study of dichroic subreflectors for multifrequencyantennae, electrical design'. ESA report, contract 5355/83/NL/GM,1984

3 JOHANSSON, F.S.: 'Analysis and design of double-layerfrequency-selective surfaces', IEE Proc. H, Microwaves, Antennas &Propag., 1985,132, pp. 319-325

4 MONTGOMERY, J.P.: 'Scattering by an infinite periodic array ofthin conductors on a dielectric sheet', IEEE Trans., 1975, AP-23,pp. 70-75

5 CHEN, C.C.: 'Scattering by a two-dimensional periodic array ofconducting plates', ibid., 1970, AP-18, pp. 660-665

6 LEE, S.W., JONES, W.R., and CAMPBELL, J.J.: 'Convergence ofnumerical solutions of iris-type discontinuity problems', ibid., 1971,MTT-19, pp. 528-536

7 MITTRA, R., ITOH, T., and LI, T.S.: 'Analytical and numericalstudies of the relative convergence phenomenon arising in the solu-tion of an integral equation by moment method', ibid., 1972,MTT-20, pp. 96-104

8 LEROY, M.: 'On the convergence of numerical results in modalanalysis', ibid., 1983, AP-31, pp. 655-659

9 SHULEY, N.V.: 'Relative convergence for moment-method solu-tions of integral equation of the first kine as applied to dichroicproblems', Electron. Lett., 1985, 21, (3), pp. 95-97

10 DENLINGER, E.J.: 'A frequency dependent solution for microstriptransmission lines', IEEE Trans., 1971, MTT-19, pp. 30-39

11 PARKER, E.A., HAMDY, S.M.A., and LANGLEY, R.J.: 'Modes ofresonance of the Jerusalem cross in frequency-selective surfaces',IEE Proc. H, Microwaves, Opt. & Antennas, 1983,130, pp. 203-208

12 HAMDY, S.M.A., and PARKER, E.A.: 'Influence of latticegeometry on transmission of electromagnetic waves through arrayof crossed dipoles', ibid., 1982,129, pp. 7-10

92 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 1, FEBRUARY 1987