mtt48-4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 4, APRIL 2000 589

Analysis of Inhomogeneously Filled WaveguidesUsing a Bi-Orthonormal-Basis Method

Enrique Silvestre, Miguel Angel Abián, Benito Gimeno, Albert Ferrando, Miguel V. Andrés, Member, IEEE, andVicente E. Boria

Abstract—A general theoretical formulation to analyze inhomo-geneously filled waveguides with lossy dielectrics is presented inthis paper. The wave equations for the tranverse-field componentsare written in terms of a nonself-adjoint linear operator and itsadjoint. The eigenvectors of this pair of linear operators define abiorthonormal-basis, allowing for a matrix representation of thewave equations in the basis of an auxiliary waveguide. Thus, theproblem of solving a system of differential equations is transformedinto a linear matrix eigenvalue problem. This formulation is ap-plied to rectangular waveguides loaded with an arbitrary numberof dielectric slabs centered at arbitrary points. The comparisonwith theoretical results available in the literature gives good agree-ment.

Index Terms—Dielectric-loaded waveguides, eigenvalues, eigen-vectors, numerical analysis, waveguide theory.

I. INTRODUCTION

I NHOMOGENIOUSLY filled waveguides have receivedconsiderable attention in the last decades because of their

applications in a variety of waveguide components. The modesof propagation of such waveguides are not, in general, TM orTE modes, but hybrid modes. The boundary value method hasbeen used to calculate the modal solutions for concentric [1],[2] and eccentric [3] dielectric-loaded circular guides, as wellas for rectangular guides filled with dielectric slabs [4], [5]. Inthat method, the electromagnetic field is expanded in terms ofanalytical functions in the relevant regions of the waveguide,and a linear eigenvalue problem is obtained after imposing theboundary conditions in the corresponding interfaces. When theguide is filled with two or more dielectrics, the determinationof the propagation constants and of the mode fields becomesdifficult because of the transcendental equations involved.Alternatively, a variational method is used in [6] to calculatethe eigenvalues in rectangular waveguides loaded with losslessdielectric slabs. The finite-element method has been extensivelyapplied to find the eigenvalues and modal fields in dielectric

Manuscript received January 12, 1999. This work was supported by the Gen-eralitat Valenciana, Valencia, Spain, under Grant GV96-D-CN-05141, by theDirección General de Investigación Científica y Técnica under Grant TIC97-1153, and by the Ministerio de Educación y Ciencia, Spain.

E. Silvestre and A. Ferrando are with the Departamento de Óptica, Univer-sidad de Valencia, 46100 Burjassot, Valencia, Spain.

M. A. Abián, B. Gimeno, and M. V. Andrés are with the Departamento deFísica Aplicada–ICMUV, Universidad de Valencia, 46100 Burjassot, Valencia,Spain.

V. E. Boria is with the Departamento de Comunicaciones, UniversidadPolitécnica de Valencia, 46022 Valencia, Spain.

Publisher Item Identifier S 0018-9480(00)02780-0.

loaded guides [7], [8]–[10]. There are, as well, a numberof matrix formulation methods to analyze inhomogeneouslyfilled waveguides in which the fields are expanded in a set ofbasis functions [11]–[16], and based on Laplace and Fouriertransforms techniques [17].

In this paper, we develop a rigorous and computationallyefficient method to obtain the modal spectrum in inhomoge-neously filled waveguides with lossy dielectric of arbitraryprofiles. Starting with the differential equations governingthe propagation of the tranverse electric and magnetic fields,we identify a pair of linear nonself-adjoint operators, whoseeigenvectors satisfy a biorthogonality relationship. The key el-ement of our approach is to transform the system of differentialequations into a linear matrix eigenvalue problem by meansof the Galerkin method, using the eigenvectors of an auxiliaryproblem. From a computational point-of-view, this methodis very efficient because the integrals involved in the matrixelements are, in principle, frequency independent, so they haveto be evaluated only once to obtain the dispersion curves, thusgenerating a robust and efficient code. This method has beenapplied to study open dielectric waveguides, as reported in [18]and [19]. Comparisons between our results and the availablenumerical published data fully validate the theory presentedhere.

II. THEORETICAL FORMULATION

Our starting point are Maxwell’s equations for uniform cross-section waveguides partially or totally filled with a lossy di-electric media defined by its dielectric permittivity

. We assume that the considered media does not havemagnetic properties . Thus, the solution of the problemcan be obtained as a superposition of fields with explicit har-monic dependence on(we assume that the time dependenceis always implicit and has a harmonic form for all vectorfields)

(1)

where is the propagation constant andand represent thetransverse-dependent (depending onand only) three-dimen-sional vector amplitudes of the electric and magnetic fields, re-spectively. We are interested in rewriting Maxwell’s equations

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590 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 4, APRIL 2000

in terms of the transverse components of the electric and mag-netic fields and . Following [20], we canobtain a set of equations for them as follows:

(2a)

(2b)

where is the relative dielectric permittivity, is thefree-space wavenumber ( ), and the operatoris the transverse gradient operator. The axial componentsand are determined by and through constraint relationsgiven by Maxwell’s equations.

For our purposes, it is more interesting to rewrite (2) in adifferent way. We will express the two previous equations in atwo-dimensional matrix form. Both and are two-dimen-sional vector fields that are represented by two component vec-tors. Thus, the differential operators acting on them can be ex-pressed as 2 2 matrices. Equation (2b) involves tranverse two-dimensional vectors only. However, (2a) includes the three-di-mensional axial vector ( ) on its right-hand side. On theone hand, it is easy to check that the double three-dimensionalvector product in (2a) can be rewritten in terms of a 22 matrixacting on (using, e.g., the completely antisymmetric tensor intwo-dimensions ). On the other hand, for reasons that willbecome clear later, it is more convenient to rewrite (2b) notin terms of , but in terms of the closely related vector field

( is the conjugation operation). After manipu-lating (2b) in a suitable way, one can obtain the following equiv-alent set of equations in matrix form:

(3)

where and are 2 2 matrix differential operators given by

(4a)

(4b)

where , and the derivative ap-pearing in the second matrix of acts both on the componentsof and on the vector field . The operator is called the

formal adjointto the linear operator , and is defined as follows(see [21]):

(5)

where and are two-dimensional vector complex func-tions defined on a two-dimensional closed and bounded region

, with boundary . They are members of a Hilbert spacewith inner product

(6)

for all , . When , we say that is formallyself-adjoint.

The set of (3) is a system of eigenvalue equations for the non-self-adjoint operator and its adjoint

(7)

where and .The eigenvectors of a nonself-adjoint operator do not satisfy

a orthogonality relation. The same applies to the eigenvectorsof its adjoint. In our case, this means that and

, being the Kronecker delta symbol.Apparently, the impossibility of using the standard orthog-onality relations associated to a self-adjoint operator wouldprevent us from expanding arbitrary functions in terms of itseigenvectors. However, since we are considering asystem, this is not so because we can take advantage of what itis called thebiorthogonality relation[22]

(8)

The biorthogonality relations were successfully used by Paivaand Barbosa to analyze inhomogeneous biisotropic planarguides [23]. Despite its apparently formal character, thisrelation has a very clear physical meaning. If we write theinner product in its integral from and restore the originalthree-dimensional notation, (8) reads

(9)

where represents the unitary vector along the-direction. Inthe waveguide literature, the relation (9) is known as the orthog-onality condition for the waveguide modes [20].

The previous relation allows us to expand any vector functionof in terms of either the eigenvectors, , or those of

its adjoint ,

(10)

where the complex expansion coefficients are given by the innerproducts and . Notice the ’s and’s coefficients are not trivially related, unless whenis self-

adjoint, in that case, and .

SILVESTREet al.: ANALYSIS OF INHOMOGENEOUSLY FILLED WAVEGUIDES 591

The previous results define the framework in which ourmethod is developed. Our aim is to find the propagationmodes of a realistic waveguide characterized by a complexrelative dielectric permittivity . As we have proven, theelectromagnetic propagation in this waveguide is described bythe system of eigenvalue (7). Let be the matrix differentialoperator (4a) representing the waveguide, we are interestedin and , the eigenmodes of and , respectively, and

, the propagation constant of theth mode. The system ofequations describing this waveguide and that we want to solveis then

(11)

Now we define anauxiliary problemas a waveguide charac-terized by a relative dielectric permittivity and with thesame boundary conditions as the waveguide described by (11).The eigenmodes of the auxiliary problem and, thus,their respective propagation constants, are supposed to be per-fectly known. The equations describing the auxiliary problemconstitute another system

(12)

and, consequently, all the properties of a biorthogonal-basisapply to the set. In particular, the modescan be used as a basis to represent any arbitrary vector. Thus,following (10)

(13)

We are certainly concerned with the matrix representation ofa linear operator of the real problem. The matrix elements ofthe operator in the basis will then easily be obtainedby applying the standard Galerkin moment method [21]. By in-serting the first equation of (13) into the first equation of (11),and applying the linear properties of, we find

(14)

The next step in the application of the Galerkin procedure is tochoose a set of weighting functions , and to take the innerproduct for each yielding

(15)

The above system can be written in matrix form as

......

...

......

......

...

...

...

...(16)

where the elements of the matrix are obtained as

(17)

For practical purposes, it is convenient to introduce the dif-ference operator

(18)

Thus, the elements of the operator in the auxiliarybasis are trivially obtained by means of (17)

(19)

where the first term is diagonal because the operatoris ex-pressed in its own biorthogonal basis. At this point, it is impor-tant to remark that we have tranformed the differential operatorsystem (11) into a linear matrix eigenvalue problem defined in(16). An analog equation for its adjoint matrix

can also be derived. Thus, the information con-tained in the above matrix equations is the same as in the differ-ential equations for the and operators (11). Diagonaliza-tion of the matrix yields the squared of theth-mode propa-gation constant—theth eigenvalue of —and also its trans-verse magnetic amplitude through the knowledge of thetheigenvector (recall that its components constitute the ex-pansion coefficients of the unknown mode in terms of theauxiliary modes ). It is important to note that the diago-nalization of not only provides us with the propagation con-stants and tranverse magnetic amplitudes of the modes, but alsowith their whole three-dimensional magnetic- and electric-fieldstructure. Both the axial component of the magnetic field and thetransverse and axial components of the electric field are relatedto through constraints given by Maxwell’s equations [20].This fact is very important from a computational point-of-viewbecause only the diagonalization process for thematrix isrequested in the numerical implementation of this method.

However, the matrix is infinitely dimensional. In order todevelop a realistic method, we have to work with a finite set ofauxiliary fields. Unfortunately, there are no general conditionsthat guarantee the convergence of the expansions. This conver-gence will depend on both the nature of theoperator and theauxiliary problem chosen to define the biorthogonal basis. Ingeneral, we observe that the real modes are better describedby increasing the number of auxiliary modes. In the same way,auxiliary basis encompassing the most relevant features of thereal problem produce faster convergence. In any case, numer-ical convergence tests must be done by sweeping the number ofauxiliary modes over meaningful ranges and studying the sta-bility of the solutions.

The method we have just presented involves no restriction onthe vector character of the electromagnetic field. The key eigen-value (7) are completely general and involve the nonself adjointoperator . It is remarkable that the nonself-adjoint characterof is present even when the medium is lossless ( isa real function)] showing the inherent nonself-adjoint characterof electromagnetic propagation. It is also interesting to note thatit is precisely the nonself-adjoint part of, the second matrix in(4a), the one responsible for polarization mixing. The diagonal


Fig. 1. Cross section of an inhomogeneously filled rectangular waveguide witharbitrarily placed lossy dielectric slabs.

Fig. 2. Rectangular waveguide loaded with a dielectric slab along the sidewall.

TABLE INORMALIZED PROPAGATION CONSTANT

�=k OF A RECTANGULAR WAVEGUIDE LOADED WITH A DIELECTRIC SLAB

ALONG THE SIDEWALL

and, thus, nonpolarization mixing, part is self-adjoint in loss-less media. This fact makes evident the close relation betweenthe nonself-adjoint character ofand the full-vector descriptiongiven by its eigenvectors. Indeed, a cylindrical waveguide uni-formly loaded with a homogeneous dielectric is described by thefirst matrix of (4a) (the second matrix is zero), and only TM andTE modes appear. However, when it is filled with an inhomo-geneous dielectric, due to the second matrix of (4a), most of themodes are hybrid. To end, we emphasize the unambiguous andrigorous character of the matrix construction and of the modeexpansions presented here, which are based on the biorthogo-nality property (8) satisfied by the auxiliary basis.

III. N UMERICAL RESULTS

The present method can be applied to a large variety ofwaveguides. In fact, its suitability has already been demon-

Fig. 3. Rectangular waveguide loaded with a centered dielectric slab.

TABLE IINORMALIZED PROPAGATION CONSTANT �a OF A RECTANGULAR WAVEGUIDE

LOADED WITH A CENTEREDDIELECTRIC SLAB

strated to deal with open guides like optical fibers [18],[19]. We will now focus on dielectric-loaded rectangularwaveguides. We have developed aFORTRAN code to analyze arectangular waveguide partially loaded with arbitrarily placedlossy dielectrics of rectangular cross section, as can be shownin Fig. 1. Thus, the relative dielectric permittivity ofthis guide can be expressed as follows:

(20)


(a) (b)

(c) (d)

Fig. 4. (a) Rectangular waveguide loaded with three dielectric slabs. Plots of the real part of the Poynting vector (z-component) for the: (b)TE , (c)TE ,and (d)TE modes, respectively.

where is the Heaviside function, is the number of di-electric slabs, and theth dielectric is centered at point ,its size being by . In our case, the auxiliary problem ischosen to be a homogeneously filled rectangular waveguide,characterized by , whose eigenvalues and eigen-vectors are well known (see, e.g., [24]). Following the theoret-ical formulation, the matrix elements of , derived from (19),are given by

(21)

After some algebraic manipulations, these integrals have beenanalytically calculated. As a consequence, only a numerical di-agonalization process has to be performed for each frequencypoint, thus resulting in fast code implementation.

We have compared the results of our approach with existingones for five different dielectric-slab-loaded rectangular guides.The first case is a rectangular waveguide of widthand height

, with a dielectric slab along the sidewall, as shown in Fig. 2.Half of the waveguide is filled with dielectric material whoserelative permittivity is and the other half is vacuum.This case is particularly interesting because the analytical solu-tion does exist [20]. To solve this simple case, we have properly

located one dielectric rectangular slab, and we have taken thevector mode functions of an empty rectangular guide as the aux-iliary basis. In Table I, we compare our results with the exact so-lution, and also with results calculated in [9] using the finite-el-ement method. For the comparison, we present the normalizedpropagation constant of the first modes , calculated for theworking frequency given by . Only 200 auxiliary modeshave been used, taking 31 s on a CrayOrigin2000 machine. Theresults agree with each other accurately.

The second case is again a rectangular waveguide of widthand height with a centered dielectric slab, as shown in

Fig. 3. In Table II, we present the normalized propagation con-stant for the LSE and LSM modes calculated for the oper-ating frequency defined by . We compare our resultswith those provided by a variational approach in [6]. The dielec-tric region size is by with relative permittivity .As in the previous case, the auxiliary problem is an empty rect-angular guide. The results of Table II have been obtained using400 basis functions, the computation time being 103 s on a Cray-Origin2000 machine. The agreement between both methods isgood.

The third example is a rectangular waveguide loaded withthree dielectric slabs [see Fig. 4(a)]. This structure is used tomodel a microwave cure applicator in [5]. In this problem, welocate a dielectric slab of relative permittivity at thecenter of a standard WR-340 guide, which was homogeneouslyfilled with a dielectric of relative permittivity , whosemodes are the auxiliary basis. In Table III, we give the cutoff


TABLE IIICUTOFFFREQUENCIES(GHz) OF A RECTANGULAR WAVEGUIDE LOADED WITH

THREEDIELECTRIC SLABS FORDIFFERENTVALUES OFTHE RELATIVE WIDTH

OF THE CENTRAL DIELECTRIC REGION (c=a)

frequencies for four different widths of the central dielectric re-gion, and we compare our results with those obtained by solvinga transcendental equation [5]. The number of basis functionsemployed in this case is 300. The distribution of the electro-magnetic field is important in the applications of this kind ofstructures in order to focus the energy in the central region ofthe guide. Our method provides the propagation constants andthe fields of the modes. As an example, we show the Poyntingvector profiles for the first propagating modes in Fig. 4(b)–(d).In these plots, the operating frequency is 5 GHz, and the rel-ative width of the central dielectric region is . Oneof the most attractive features of our method is the versatilityand flexibility for the analysis and design of complex dielectricstructures filling a rectangular guide, thus becoming a powerfuland effective computer-aided design (CAD) tool. That is to say,the inclusion of other slabs inside the guide is a very simple taskemploying this algorithm, while other techniques require to re-calculate the full problem in order to find the new transcendentalequation.

The fourth example is a shielded rectangular dielectric wave-guide. In Fig. 5, we present the dispersion curves for the firsttwo modes, comparing our results with those obtained with a fi-nite-difference method [25]. The relative permittivity of the coreis , and the dimensions of the rectangular rod areand , . The dielectric rectangular rod is shieldedby a metallic rectangular guide of dimensions and

Fig. 5. Dispersion curves for theEH and HE modes of a shieldedrectangular dielectric waveguide. Comparison between our results (solid lines)and the results obtained with the finite-difference method [25] (dots).

Fig. 6. Dispersion curves for the fundamental mode of a rectangular guideloaded with a centered inhomogeneous dielectric slab, as a function of thethickness: (1)c=a = 0:5, (2) c=a = 0:2, and (3)c=a = 0:1. Comparisonbetween our results (solid lines) and the results obtained with a technique basedon a variational formulation [6] (dots).

. Only 200 basis functions are necessary to obtain thefirst modes. The computation time required to obtain the resultsshown in Fig. 5 is 12 s per frequency point on a CrayOrigin2000machine. We find a good agreement with previous results.

Finally, we want to show how our method can also be appliedto analyze waveguides loaded with inhomogeneous dielectricslabs. The fifth example is a rectangular waveguide loaded witha centered inhomogeneous dielectric slab. The relative permit-tivity of the inhomogeneous slab depends on the coordinatein the form (see Fig. 6)

(22)

with and where the origin of the-axis is atthe center of the waveguide. We have computed the dispersioncurves of the fundamental mode for three different values of


, using 300 auxiliary modes. In Fig. 6, we compare our re-sults with those presented in [6] using a variational formulation.

IV. CONCLUSIONS

In this paper, we have developed a method for the analysisof inhomogeneously filled waveguides with lossy dielectrics.Once Maxwell’s equations are written in terms of the transversecomponents of the fields of guided modes, we have shown thatthey can be rewritten as a system of eigenvalue equations for anonself-adjoint operator and its adjointand , respectively.The eigenvectors of the system define a biorthonormalbasis and allow to transform the differential operator system intoa linear matrix eigenvalue problem, using the eigenvectors of anauxiliary problem to expand the modes of the original problem.

We have developed aFORTRANcode to obtain the modal spec-trum of rectangular waveguides filled with dielectric slabs. Inprinciple, our program can deal with any number of lossy dielec-tric slabs with arbitrary size and location within the rectangularwaveguide. We have tested it by comparison with theoretical re-sults found in the technical literature. We demonstrate that it canbe used to work out the modal spectrum of a large variety of di-electric guides.

Finally, we showed that our method can deal with inhomoge-neous dielectrics. Furthermore, our method can be easily usedto analyze dielectric rods with nonrectangular cross section andinhomogeneously magnetic-media-filled waveguides.

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[3] H. A. Ragheb, A. Sebak, and L. Shafai, “Cutoff frequencies of circularwaveguide loaded with eccentric dielectric cylinder,”Proc. Inst. Elect.Eng., vol. 144, no. 1, pp. 7–12, Feb. 1997.

[4] J. Strube and F. Arndt, “Rigurous hybrid-mode analysis of the transitionfrom rectangular waveguide to shielded dielectric image guide,”IEEETrans. Microwave Theory Tech., vol. MTT-33, pp. 391–401, May 1985.

[5] L. Outifa, M. Delmotte, and H. Jullien, “Dielectric and geometric de-pendence of electric field and power distribution in a waveguide hetero-geneously filled with lossy dielectrics,”IEEE Trans. Microwave TheoryTech., vol. 45, pp. 1154–1161, Aug. 1997.

[6] C.-C. Yu and T.-H. Chu, “Analysis of dielectric-loaded waveguide,”IEEE Trans. Microwave Theory Tech., vol. 38, pp. 1333–1338, Sept.1990.

[7] B. M. Azizur and J. B. Davies, “Penalty function improvement ofwaveguide solution by finite elements,”IEEE Trans. Microwave TheoryTech., vol. MTT-32, pp. 922–928, Aug. 1984.

[8] T. Angkaew, M. Matsuhara, and N. Kumagai, “Finite-element analysisof waveguide modes: A novel approach that eliminates spurious modes,”IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp. 117–123, Feb.1987.

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[10] L. Nuño, J. V. Barbastre, and H. Castañé, “Analysis of general lossy in-homogeneous ans anisotropic waveguides by the finite-element method(FEM) using edge elements,”IEEE Trans. Microwave Theory Tech., vol.45, pp. 446–449, Mar. 1997.

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[18] E. Silvestre, M. V. Andrés, and P. Andrés, “Biorthonormal-basis methodfor vector description of optical-fiber modes,”J. Ligthwave Technol.,vol. 16, pp. 923–928, May 1998.

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[20] R. E. Collin,Field Theory of Guided Waves, 2nd ed. New York: IEEEPress, 1991.

[21] D. G. Dudley,Mathematical Foundations for Electromagnetic Theory,1st ed. Piscataway, NJ: IEEE Press, 1994.

[22] P. M. Morse and H. Feshbach,Methods of Theoretical Physics, 1sted. New York: McGraw-Hill, 1953.

[23] C. R. Paiva and A. M. Barbosa, “A linear-operator formalism for theanalysis of inhomogeneous biisotropic planar waveguides,”IEEE Trans.Microwave Theory Tech., vol. 40, pp. 672–678, Apr. 1992.

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Enrique Silvestrewas born in Valencia, Spain, in 1962. He received the Licen-ciado degree in physics, the Master’s degree in theoretical physics, the Master’sdegree in optics, and the Ph.D. degree in optics from the Universitat de Valencia(UV), Valencia, Spain, in 1986, 1989, 1997, and 1999, respectively.

Since 1997, he has been an Assistant Professor in the Department of Optics,UV. He is currently a Visiting Scientist in the Department of Physics, Universityof Bath, Bath, U.K. His research interests are in the area of modal techniques forelectromagnetic-wave propagation in nonsymmetrical and periodic structures.

Miguel Angel Abián was born in Soria, Spain, onSeptember 26, 1972. He received the Licenciado de-gree in physics from the Universidad de Valencia, Va-lencia, Spain, in 1995, and is currently working to-ward the Ph.D. degree at the Universidad de Valencia.

He is currently with the Universidad de Valencia.He is also involved with the Technical Unit forPackaging, Transport and Environment of AIDIMA(Instituto Tecnologico del Mueble y Afines),Paterna, Spain. His current research interestsinclude computer-aided techniques for analysis of

waveguide structures and photonic crystals.


Benito Gimenowas born in Valencia, Spain, on Jan-uary 29, 1964. He received the Licenciado degree inphysics and the Ph.D. degree from the Universidadde Valencia, Valencia, Spain, in 1987 and 1992, re-spectively.

From 1987 to 1990, he was a Fellow at the Univer-sidad de Valencia. From 1990 to 1996, he was an As-sistant Professor in the Departmento de Física Apli-cada, Universidad de Valencia, and became an Asso-ciate Professor in 1997. From 1994 to 1995, he was aResearch Fellow with the European Space Research

and Technology Centre (ESTEC), European Space Agency. His current researchinterests include the areas of computer-aided techniques for analysis of mi-crowave passive components, waveguide structures, and photonic crystals.

Albert Ferrando was born in Valencia, Spain, in1963. He received the Licenciado degree, and theM.S., and Ph.D degree in theoretical physics fromthe Universitat de Valencia (UV), Valencia, Spain,in 1985, 1986, and 1991, respectively.

His research work was performed at UV, theUniversidad Autónoma de Madrid, Madrid, Spain,as an Assistant Professor, the University of Bern,Bern, Switzerland, as a Post-Doctoral Fellow, theNiels Bohr Institute of Copenhagen, Copenhagen,Denmark, and the Massachusetts Institute of Tech-

nology, Cambridge, as a Visiting Scientist. In 1996, he joined the Departamentof Optics, UV, where he is currently an Assistant Professor. His researchinterests are in the area of electromagnetic propagation in optical waveguidesand fiber devices, as well as in nonlinear and quantum effects in optical fibers.

Miguel V. Andrés (M’91) was born in Valencia,Spain, in 1957. He received the Licenciado enFísica and the Doctor en Física degrees from theUniversidad de Valencia (UV), Valencia, Spain, in1979 and 1985, respectively.

Since 1983, he has served successively as an As-sistant Professor and Lecturer in the Departamentode Física Aplicada, UV. From 1984 to 1987, hewas with the Department of Physics, University ofSurrey, U.K., as a Visiting Research Fellow. Until1984, he was engaged in research on microwave

surface waveguides. His current research interests are optical fiber devicesand systems for signal processing and sensor applications, and electromag-netic-wave propagation in microwave and optical waveguides and devices.

Dr. Andrés is a member of the Institute of Physics (IOP).

Vicente E. Boria was born in Valencia, Spain,on May 18, 1970. He received the Ingeniero deTelecomunicación degree (first-class honors) andthe Doctor Ingeniero de Telecomunicación degreefrom the Universidad Politécnica de Valencia (UV),Valencia, Spain, in 1993 and 1997, respectively.

In 1993, he joined the Departamento de Comunica-ciones, UV, where he was an Assistant Lecturer from1993 to 1995, a Lecturer from 1996 to 1997, and anAssociate Professor since 1998. In 1995 and 1996, hehad a Spanish Trainee position at ESTEC-ESA, No-

ordwijk, The Netherlands, where he was involved in the area of electromagneticanalysis and design of waveguide devices. His current research interests includenumerical methods for the analysis of waveguide and scattering structures, au-tomated design of waveguide components, radiating systems, and measurementtechniques.

Dr. Boria received the First National Prize of Telecommunication Engi-neering Studies for his outstanding student record in 1993 presented by theSpanish Ministerio de Educación y Ciencia.

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