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2802 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 6, JUNE 2012

Nystrm Solution to Oblique Scattering of ArbitrarilyPolarized Waves by Dielectric-Filled Slotted

CylindersJohn L. Tsalamengas and Chrysostomos V. Nanakos

AbstractWe investigate oblique diffraction of arbitrarilypolarized waves by axially slotted, dielectric-filled, perfectlyconducting circular cylinders with infinitesimally thin walls. The

relevant 2 2 system of coupled singular integral-integrodiffer-

ential equations is solved by a highly accurate Nystrm method.For enhanced efficiency, several asymptotic terms are extractedfrom the kernels. This helps isolate the inherent singularitiesand, thus, transform the initial slowly converging Green-functionexpansions into rapidly converging series. As a consequence, all

elements of the resulting Nystrm matrix take exact expressions

in the form of very rapidly converging series of elementary terms.Numerical examples and case studies are presented that validatethe algorithm, illustrate its rapid convergence, and bring to lightthe effect of changing several physical and geometrical parameters

on the reflection, transmission, and absorption characteristics ofthe structure.

Index TermsAperture coupling, diffraction, integral equa-tions, Nystrm method, slotlines.

I. INTRODUCTION

T HERE has been considerable interest in problems of wavepenetration into conducting cylindrical cavities throughaxial slots owing to the importance of such problems in testingsystems and subsystems to high-level electromagnetic fieldsand in studying reflector antennas, frequency-selective surfacereflectors, remote sensing, etc. In addition, dielectric-filledslotted cylinders (cylindrical microslotlines) are practical openwaveguiding devices. Finally, periodic arrays of such resonatorsare potentially useful in building two-dimensional photoniccrystals. The relevant boundary-value problems have beentreated by several techniques such as the method of moments[1][6], the method of analytical regularization (MAR) [7], thecombined boundary conditions method [8], the method of sin-gular integral equations [9], [10], and the method of dual seriesequations in the framework of the so-called Riemann-HilbertProblem [11][16].

Manuscript received April 19, 2011; revised November 08, 2011; acceptedDecember 19, 2011. Date of publication April 12, 2012; date of current versionMay 29, 2012. This work was supported in part by the THALES ProjectANEMOS, funded by the ESPA Program of the Ministry of Education ofGreece.

The authors arewith theDepartmentof Electrical and Computer Engineering,National Technical University of Athens, Greece, GR, 15773 Zografou, Athens(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAP.2012.2194649

The present study deals with wave diffraction by an axiallyslotted, dielectric-filled, conducting circular cylinder. The pri-mary excitation is taken to be an obliquely incident, arbitrarily

polarized plane wave; this stands in contrast with previousrelated works which only concern the definitely simpler two-di-mensional case of a -independent primary excitation wherein

and waves decouple and can be treated separately.Our main objective here is the highly accurate evaluation of

the field both inside and around the cavity. As an interestingby-product, the resulting algorithms enable one to convenientlyinvestigate the characteristics (propagation constants, modalfields, etc.) of the hybrid modes supported by a cylindricalmicroslotline, too; this may be done at no additional analyticalor computational cost by simply solving the correspondinghomogeneous problem.

Using field equivalence principles, the relevant boundary-value problem will be formulated via a 2 2 set of coupledsingular integral-integrodifferential equations, having the com-

ponents of the equivalent surface magnetic current across theslot as unknowns, with the kernels being the so-defined Green

functions (GF). It is known that conventional expansions of theGreen functions for cylinders in terms of infinite series of or-thogonal eigenfunctions do not converge uniformly in their re-gion of validity, exhibiting a slow and conditional convergencein the following two cases: 1) near the source (singular) point,due to the singularity of the GF at source points, and 2) when thesource point is very near the surface of the cylinder, due to the

presence of a nearby image source which strongly affects thebehavior of the GF. As a result, such expansions are unsuitedfor the exact solution of singular integral equations, in whichvalues of the GF at the source point do appear inside the inte-gral. In [17], the aforesaid inadequacies were remedied by firstextracting the singular behavior of the solution in a closed-form

(Hankel) term out of the eigenfunction series. Next, an addi-tional simple image term was also extracted to improve the con-vergence of the expansion of the remaining, non-singular partof the GF. The so-obtained new eigenfunctions expansion con-verges uniformly over the whole region of its validity.

In the present paper, for enhanced efficiency, alternative ex-pansions withimproved convergence are proposed for the Greenfunctions. The main idea is to extract several asymptotic termsout of the kernels that help isolate the inherent singularities and,thus, transform the initial eigenfunction expansions into veryrapidly converging series. Using the final (improved) forms ofthe Green functions, the pertinent integral equations are then

solved in the framework of the Nystrm method of [18]. De-

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TSALAMENGAS AND NANAKOS: NYSTRM SOLUTION TO OBLIQUE SCATTERING OF ARBITRARILY POLARIZED WAVES 2803

tailed numerical results and case studies demonstrate the effi-ciency and accuracy of the proposed algorithm and bring to lightthe effect of changing several physical and geometrical param-eters on the reflection, transmission, and absorption character-istics of the structure.

It is emphasized that the analysis herein only concerns thephysical problem just described. Arbitrarily shaped cylindersmay be treated by extending alternative formulation methodslike those outlined in [3] and [4] which, however, necessitatethe construction of new boundary integral equations. Moreover,application of Nystrm techniques to the solution of such gener-alized structures demands development of new quadratures thatwill enable one to fully account for the singular nature of thesolution at pertinent singular points (edges, corners, etc.) of thegeometrical configuration. Clearly, such an extension, thoughfeasible, is outside the scope of the present paper.

II. FORMULATION

A. Preliminaries

Fig. 1 shows an infinitely long, dielectric-filled, slotted cir-cular cylinder ( , , ) of radius (region 1),surrounded by the space ( , , ) (region 2); theslot extends from to parallel to the axis. The

primary excitation is an obliquely incident plane wave (the as-sumed time dependence is suppressed)

(1)

originating from region 2 and propa-gating in the direction of

(2)

as shown in Fig. 1. The polarization of the incident wave isdescribed by the unit vector

(3)

where

(4)

which is normal to , with the particular direction specifiedby the polarization angle . The special cases (or )and (or ) correspond to the and

polarizations, respectively.Let and denote

the -components of the total field at , where, 2 designates the region of space, and let

and be the transverse (to a xis) field components;

Fig. 1. Geometry of the problem and details of the incident wave.

and can be expressed in terms of andvia the relations

(5)

(6)

where

(7)

B. Field Representations

In the following, and willdenote the -components of the known electric and magnetic

fields which are excited at by auxiliary electricand, respectively, magnetic phased line sources

( , ) impressed at ( ,2) in presence of the complete conducting cylinder, i.e., withthe slot short-circuited. Here, the phase factor has beenselected both f or and , in contrast t o the phasefactor relevant to the primary plane wave excitation; this choicefacilitates obtaining the pertinent integral equations as will be

seen shortly.Let , where

, be the equivalent surface magnetic current densityacross the slot. Following field equivalence principles, the totalfield in region 1 is identified with the field excited by the source

impressed at across the slot but withthe slot short-circuited. By twice applying the reaction theorem[19], first in the form

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with , and then in theform

we obtain the following integral representations for the -com-ponents of the total field in region 1 (see, also, section 2.6 of[20]):

(8)

(9)

In the last step in each of (8) and (9) use was made of (6), butwith in place of , to express and in terms of

and , respectively. The transverse components of the

total field in region 1 can be obtained in terms of andvia (5) and (6).

Let now be the known field excitedin region 2 by the incident plane wave in presence of the con-ducting cylinder with the slot short-circuited, and let

designate the scattered fieldin region 2. Followingfield equivalence principles, the scatteredfield is identified with the field excited by thesourceimpressed at across the slot but with the slot short-cir-cuited. By twice applying the reaction theorem in the form

we get for the -components of the scattered field the represen-tations

(10)

(11)

The transverse components of the field in region 2 can beobtained from (5) and (6) in terms of the -components.

III. INTEGRAL EQUATIONS

By enforcing the continuity conditionsand when ,

we obtain

(12)

(13)

Equations (12) and (13) form a 2 2 system of coupledfirst-kind integral equations with unknowns and .The efficient solution of this system, outlined in Section IV,

prerequisites the highly accurate computation of the quantitiesand

(14)

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( , 2) encountered inthe kernelfunctions; this taskis carriedout as outlined below.

A. Magnetic-Type Green Functions ( 1, 2)

1) The Original Form: Straightforward application of sepa-ration of variables yields

(15)

where is the Neumanns factor ( is the Kro-necker delta) and

(16)

For large , one may obtain the asymptotic expansions

(17)

where the expansion coefficients are given by

(18)

in terms of the dimensionless parameters

Equation (17) shows that the series in (15) converges as, i.e., conditionally, and that its derivative

with respect to diverges; hence, as it stands, (15) is uselessfor solving (12) and (13). Nevertheless, (15) can be recast intoa rapidly converging series as follows.

2) The New Form: Let be a prescribed integer parameter. We firstly subtract the -term asymptotic series

(19)

out of in (15), then add it back, and use the result

(20)

to obtain

(21)

In (20), is the polylogarithm function which can becomputed to arbitrary precision, e.g., by the software of [21].

Note: Alternative exponentially converging series expan-sions for which allow rapid computationare summarized in the Appendix. As will be made clear inSection III-C, these expansions play an essential role in car-rying out our Nystrm solution, because they form the basisfor proper handling (decomposition into a singular part and aregular part) of the kernel functions.

In view of (17) and (19), varies proportionally tofor large . Consequently, the infinite series encoun-

tered in(21) converges as , i.e., as rapidly as one desiresby properly selecting the integer parameter . The remainingfinite series, over , can be computed to arbitrary accuracy ei-ther from (20) or as in the Appendix. Therefore, (21) allows veryrapid computation of the Green function.

B. Electric-Type Green Functions: Computation of

1) The Original Form: Simple application of separation ofvariables yields

(22)

with defined in (16). For large , one may obtainthe asymp-totic expansions

(23)

where the expansion coefficients are given by

(24)

in terms of the dimensionless parameters ( , 2).Equation (23) shows that the series in (22) diverges. Never-

theless, (22) can be recast into a rapidly converging series ex-pansion as follows.

2) The New Form: Let be a prescribed integer parameter. We firstly subtract the -term asymptotic series

(25)

out of in (22), then add it back, and use (20) to obtain

(26)

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As seen, the infinite series encountered in (26) converges as, i.e., as rapidly as one desires by properly selecting .

Remark 1: It is instructive here to see how our (21) and (26)compare from the viewpoint of convergence withcorrespondingresults reported in [17]. Following the outline in [17], one ob-tains

(27)

(28)

where , withdenoting the image of ,

(29)

(30)

whereas and resultfrom(29)and (30)after 1)replacingby and 2) interchanging the Bessel and Hankel functions.A detailed asymptotic analysis, omitted here for brevity,

yields for large the asymptotic expressions

(31)

(32)

(the upper sign in the exponent pertains to the casewherein and , so that ; the lower sign

pertains to the case wherein and , so that). In terms of the dimensionless parameters

(33)

( , 2), the expansion coeffi

cients are given by

(34)

(35)

and so on. Evidently, (27) and (28) converge exponentially pro-vided that not both and are too close to . However, thestrong exponential convergence is lost when both and arevery near .

Fig. 2. for , versus , as obtained via(21), for several values of , and (27). The parameter values are: ,

, , .

For , (34) and (35) yield ,

. Hence, the series in (27) and (28) converge as

; even for , which is the case of interestin (21) and (26), the convergence is uniform .In summary, (27) for and (21) for have

the same convergence rate. As goes beyond 3, however, theconvergence of (21) becomes more and more rapidand, thus, (21) outperforms (27). This same conclusion appliesalso to (26) which outperforms its counterpart, derived from(28), when .

As an example that corroborates the above remarks,Fig. 2 shows the logarithm (base 10) of the relative error

versus, as obtained 1) from (21) for several values of and 2)

from (27). Here, is the number of terms retained in the

infinite series in each of (21) and (27) whereasis the final value to which settles down for sufficiently large ; this asymptotic value can be treated asthe exact value. As seen, (21) is both very rapidly convergentand highly accurate; in this numerical example the error is lessthan , i.e., comparable with the errors due to round off,when for (or when for ).The curve labelled is indistinguishable from the coloredcurve pertaining to (27), as expected. The superiority of (21)over (27) when becomes evident.

Remark 2: For sufficiently large values of , andare close to each other, and this renders the computation of the

infinite series in (21) and (26) susceptible to loss of accuracy.It is stressed, however, that so large values of (which wouldleadto loss of accuracy and possibly problematic behavior of thealgorithm) are not necessary: Because of the rapid convergence,excellent accuracy can be obtained with small values of . Forinstance, in the above example the relative error is less than

when and . (As our computationsshow, the relative error is only when and

).

C. Kernel Decomposition

For reasons explained in [18], proper decomposition of thekernel functions into a singular part anda regular part is a critical

prerequisite as far as obtaining a rapidly converging Nystrm

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solution of the system of IEs (12) and (13) is concerned. Thistask is carried out as follows:

Set

(36)

Appendix (82)(88) show that en-

countered both in (21) and (26) are logarithmically singular at, when is odd, and polynomial, i.e., analytic

functions when is even. Thus, (82)(88) suggest the decom-position

(37)

where, for ,

(38)

(39)

The first term in the right-hand side of (37) contains a loga-rithmic singularity. The second term, , is analyticwith limiting values

(40)

where denotes the Riemann zeta function.Using (37), we arrive at the required decompositions

(41)

(42)

where

(43)

(44)

(45)

(46)

D. Final Form of the Integral Equations

With the help of (41) and (42), the integral equations (12) and(13) take the form

(47)

(48)

Here, for simplicity in notation, and areidentified with and , respectively.

For or , and for or , the kernelfunctions in (47) and (48) are expressed as

(49a)

(49b)

where, for or ,

(50)

(51)

(52)

(53)

(54)

(55)

The singular integral equations (47) and (48) are amenable toan accurate solution by the Nystrm method as outlined below.

IV. SOLUTION BY THE NYSTRM METHOD

In conformity with the edge condition, and will besought in the form

(56)

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We substitute from (56) into (47) and (48). Working in theframework of the Nystrm method of [18] we then evaluateany singular integrals encountered via quadratures obtainedin [22], and any regular integrals by applying the standardGaussChebyshev rules [23]

(57)

(58)

where

(59)

(The integer parameter is selected as high as needed to assureany prescribed accuracy, as specified in Section V). Finally, wesatisfy the resulting two equations, respectively, at the colloca-tion points and , . This yieldsthe following discrete counterparts of (47) and (48):

(60)

(61)

For or , and for or , the matrixelements are expressed as

(62)

where, for or ,

(63)

(64)

(65)

(66)

with and given by ( and are Chebyshevpolynomials) [22]:

(67)

(68)

From the solution of the linear system of algebraic equa-tions (60) and (61) the quantities and ,

, are obtained. In terms of them, thefield can be com-puted everywhere using (8)(11) and (5), (6); all integrals en-countered in (8)(11) may be computed via (57), (58); for theGreen functions, expansions (27) and (28) may be used. Explicitexpressions for the farfield are given below.

A. Far Scattered Field and Radiated Power

The -components of the far scattered field are given by

(69)

(70)

where

(71)

(72)

The remaining field components may be obtained via (5), (6).The per unit length of -axis radiated power may be

obtained by integrating the complex Poynting vector over thesurface of a cylinder of infinite radius. The final result is

(73a)

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(73b)

Equation (73b) results by first substituting from (69) and (70)

into (73a) and, then, using the Parsevals equation.

B. Absorption and Transmission Efficiencies

The per unit length of -axis complex powerentering the interior of the cylinder results by integrating

the complex Poynting vector over the surface of the cylinderwith radius . In compact form the final result is

(74)

where , , and are the following vectors of length L:

(75a)

(75b)

(75c)

(75d)

whereas for , are the submatrices

(76)

( ; ) whose elements are givenin (63)(66).

amounts to the per unit length power dissipated in theinterior of the cavity, if any ( if the dielectric insidethe cavity is lossless), and amounts to the per unit lengthreactive power stored in the interior of the cavity. The ratio

(77)

where is the per unit length of -axis inci-dent plane-wave power intercepted by the width , designatesthe absorption efficiency of the cavity. In a similar way, the ratio

(78)

designates the transmission efficiency of the configuration.Finally, it can be shown that the per unit length of -axis

power associated with the equivalent surface source is givenby

(79)

TABLE IFOR INCREASING MATRIX SIZES WHEN , ,

, , , ( CASE), AND

TABLE IIFOR INCREASING MATRIX SIZES WHEN , ,

, , ( CASE), , AND OR

The energy conservation principle imposes the equality

(80)

Numerical validation of (80) has been extensively used as a par-tial correctness test of our algorithm.

V. NUMERICAL RESULTS

In the following it is taken, unless otherwise specified, thatregion 2 has the properties of vacuum, i.e., ,and that . Moreover, it is assumed that

.

A. Validation and Convergence

Correctness and accuracy tests are presented in Tables IIII,where comparisons with published results are, also, carried out.

Table I shows for increasing when( -case), , , , , . Asseen, the convergence is very rapid and stable. The final valuescoincide within 15 digits with those obtained by the highly ac-curate method presented in [24].

For increasing , Table II shows when(TE-case), , , and for tworesonant values of ( and ).Again, the convergence is very rapid and stable. Thefinal valuesare in excellent agreement with those obtained by the highlyaccurate method presented in [25].

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TABLE III1) WHEN , ( CASE), AND 2)

WHEN , ( CASE) FOR INCREASING MATRIX SIZES. THEPARAMETERVALUES ARE: , , , AND

TABLE IV1) WHEN ( CASE) AND 2) WHEN( CASE) FOR INCREASING MATRIX SIZES. THE PARAMETER VALUES ARE:

, , , , AND

The results for in Table III show that the proposedalgorithms yield accurate results for electrically large cylinders,too. Ourresultsare in agreement with [24] and [25].(Note, how-ever, that the label in the pertinent Table III of [25] is amissprint and should be corrected to ).

Table IV shows the convergence when the cylinder is filledby a lossy dielectric . Finally, Fig. 3 shows therelative errors of and versus for and for several slot openings ( , 10 , 30 , 60 , 90 ). Asseen, accurate results can be obtained even for large w.

B. Other Results

Fig. 4 shows the frequency dependence of the transmissionefficiency , defined in (78), when , , ,

, ( -case). The peaks correspond to theresonances, wherein the field is strongly localized in the cavity.Our results are in excellent agreement with those of Fig. 5 in [10](taking into account the time dependence assumedin [10]). Analogous results pertaining to the -case

are shown in Fig. 5 for and for .

The dependence of versus both in the andcases is shown in Fig. 6 for and for .

Fig. 3. Relative error versus , (a) of when ( -case),(b) of when ( -case). The other parameter values are:

, , , and , 10 ,30 , 60 , 90 .

Fig. 4. Transmission efficiency versus when, , , , ( -case).

The frequency dependence of the absorption efficiency ,see (77), is shown in Fig. 7 both in the and caseswhen , , , .

Pertaining to the -case , Fig. 8 shows (a)the magnitude of the electric field and (b) a snapshotof the magnetic field at , at the resonant value

of the mode of the closed waveguide. The otherparameter values are: , , ,

. Analogous results pertaining to the resonant valueof the mode are shown in Fig. 9.

Fig. 10 shows snapshots of the electric field atwhen ( -case), , , and a) ,

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Fig. 5. Transmission efficiency versus when , ,, ( -case) for , 2.

Fig. 6. Transmission efficiency versus when , ,and or . (a) ( -case), (b)

( -case).

( resonant value); b) ,( resonant value); c) , ; d)

, . The angle is in (a)(c) and 0 in (d).In cases (a), (c), and (d), our results are in excellent agreementwith corresponding ones reported in [17].

VI. CONCLUSION

An efficient Nystrm method has been developed to inves-tigate the physical problem of oblique diffraction by axiallyslotted, dielectric-filled, circular cylinders. The proposed algo-rithm may be, also, directly used to study the propagation char-acteristics of cylindrical microslotlines.

The method, suitable extended, can be applied to compositeresonators containing dielectric or conducting cylindrical in-clusions. Such generalized configurations will be considered inforthcoming papers.

Fig. 7. Absorption efficiency versus when ,, , . (a) ( -case), (b) ( -case).

APPENDIXTHE FUNCTION ,

Evidently,

(81)

Consequently, suffice to consider only the case . Inthis case, , , can be expressed as

(82)

(83)

(84)

(85)

(86)

(87)

(88)

and so on, where, for ,

(89)

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Fig. 8. (a) Magnitude of electric field, (b) snapshot of magnetic field at , when , , , , ,( -case).

Fig. 9. (a) Magnitude of electric field, (b) snapshot of magnetic field at , when , , , , ,( -case).

Fig. 10. Snapshots of electric field at when ( -case),, and a) , , (

resonant value); b) , , ( resonantvalue); c) , , ; d) , , .

The series in (89) converges very strongly (exponentially)when . For odd, contain a logarithmic

singularity at . For even, is a polynomial, i.e.,an analytic function of .

ACKNOWLEDGMENT

The authors would like to thank three anonymous Reviewersfor helpful comments and suggestions.

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[12] W. A. Johnson and R. W. Ziolkowski, The scattering of an H-polar-ized plane wave from an axially slotted infinite cylinder: A dual seriesapproach, Radio Sci., vol. 19, pp. 275291, 1984.

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[16] A. Y. Svezhentsev, Simulation of waves in coupled cylindrical slotlinesThe Riemann-Hilbert method model, J. Electromagn. Waves

Applicat., vol. 12, pp. 15811596, 1998.[17] D. M. Pozar, A useful decomposition for the Greens functions ofcylinders and spheres, Radio Sci., vol. 17, no. 5, pp. 11921198, Sep.

Oct. 1982.[18] R. Kress, Linear Integr al Equations. Berlin: Springer-Verlang, 1989.[19] J. A. Kong, Electromagnetic Wave Theory. New York: Wiley-Inter-

science, 1986.[20] N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods

for Electromagnetics. Norwood, MA: Artech House, 1990.

[21] S. Wolfram, Mathematica. A System for Doing Mathematics by Com-puter. New York: Addison-Wesley, 1998.

[22] J. L. Tsalamengas, Exponentially converging Nystrms methods forsystems of singular integral equations with applications to open/closedstrip- or slot-loaded 2D structures, IEEE Trans. Antennas Propag.,vol. 54, no. 5, pp. 15491558, May 2006.

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[24] J. L. Tsalamengas, Exponentially converging Nystrm methods inscattering from infinite curved smooth strips-part 1: TM-case, IEEETrans. Antennas Propag., vol. 58, no. 10, pp. 32653274, Oct. 2010.

[25] J. L. Tsalamengas, Exponentially converging Nystrm methods inscattering from infinite curved smooth strips-part 2: TE-case, IEEETrans. Antennas Propag., vol. 58, no. 10, pp. 32753281, Oct. 2010.

John L. Tsalamengas was born in Karditsa, Greece in 1953. He received theDiploma of Electrical and Mechanical Engineering and the doctors degreein Electrical Engineering from the National Technical University of Athens(NTUA), Greece, in 1977 and 1983, respectively.

From 1983 to 1984, he was with the Hellenic Aerospace Academy. He thenjoined NTUA, where he has been a Professor of electrical engineering sinceNovember 1995. His fields of interest include problems of wave propagation,radiation and scattering in the presence of complex media, computational elec-tromagnetics, and applied mathematics.

Chrysostomos V. Nanakos was born in Athens, Greece in April 8, 1979. Hereceived the Diplomaof Electrical and ComputerEngineering fromthe NationalTechnical University of Athens (NTUA) where he is currently working towardsthe Ph.D. degree.

His research interests include computational and applied electromagnetics aswell as applied mathematics.

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