IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 7, JULY 2007
1963
A Robust Method for the Computation of Green’s Functions in Stratified Media Athanasios G. Polimeridis, Student Member, IEEE, Traianos V. Yioultsis, and Theodoros D. Tsiboukis, Senior Member, IEEE
Abstract—Closed-form Green’s functions for unbounded planar stratified media are derived in terms of cylindrical and spherical waves. The methodology is based on a two-level approximation of the spectral-domain representation of Green’s functions. This robust, efficient, and fully numerical approach does not call for an analytical extraction of “problematic” behaviors, such as the quasi-static terms and the surface wave poles, prior to the spectrum fitting. Instead, the spatial-domain Green’s functions derived in this paper provide an accurate description of both the near-field singularity in the vicinity of the source and the far-field dominant behavior of the surface waves. Index Terms—Closed-form Green’s functions, discrete complex image method (DCIM), quasi-static terms, rational function fitting method (RFFM).
I. INTRODUCTION HE analysis of printed structures, embedded in planar stratified media that consist of an arbitrary number of layers, constitutes one of the most widely studied topics in applied electromagnetics. Among the currently used models, methods based on integral equation techniques, formulated in the space domain, are particularly attractive, since they provide excellent accuracy and fast computation. Moreover, the mixedpotential form [1], [2] of the electric field integral equation (EFIE) is preferable to several other possible variants of the EFIE because it requires only the potential forms of the Green’s functions. These are less singular than their derivatives, which are involved in other forms of the EFIE [3]. The solution of the governing integral equation via the method of moments (MoM) [4] calls for an accurate and fast computation of the vector and scalar potentials in the spatial domain, where they are represented as oscillatory integrals, called Sommerfeld integrals (SIs). Generally, the analytical solution of the SIs is not available, and the numerical integration is time consuming, since the integrands are both highly oscillating and slowly decaying. Among the several methods that have been proposed in order to tackle this problem stands the discrete complex image method (DCIM) [5]–[7]. One of the most important and cumbersome steps in its application is the analytic extraction of the surface wave poles, along with the slowly decaying part (quasi-static
T
Manuscript received September 18, 2006; revised January 26, 2007. This work was supported by the Greek General Secretariat of Research and Technology under Grant PENED 03ED936. The authors are with the Telecommunications Division, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAP.2007.900258
part) of the spectrum of the Green’s function, prior to the complex exponential fitting via the generalized pencil-of-function (GPOF) method [8]. Although extraction of the slowly decaying part of the spectrum can be avoided by using the two-level approach [9], extraction of the surface wave poles remains essential [10], when Green’s function is to be accurately computed far from the source at distances of several wavelengths. The reason for such an extraction lies in the fact that the approximating functions represent spherical waves with complex distances, referred to as complex images, and that dominant wave constituents of the fields of a dipole source are spherical in nature. However, in a layered structure, there are other wave constituents due to a dipole source, like cylindrical and lateral waves, which may not be approximated in terms of complex images unless their contributions are explicitly accounted for. It has been shown [10] that if the surface wave poles are not extracted prior to the exponential fitting, the DCIM approximation seems to deteriorate violently for moderate distances from the source. Another approach was proposed recently for evaluating the Green’s functions of the potentials in planar stratified media [11], [12]. By solving the Helmholtz equation using the finitedifference (FD) method, this technique represents the spectraldomain Green’s function in terms of a pole-residue form. Implementation of this new approach provides closed-form expressions for the space-domain counterparts in terms of Hankel functions. Hence, without the extraction of any singularities, this approach is direct and convenient to implement. However, the accuracy of this method in the near field is not satisfactory, even if a large number of Hankel functions are employed. The physical explanation is that, in the near field, the quasi-static components dominate and their description in terms of cylindrical waves becomes mathematically inadequate. Recently, a novel approach was proposed that provides an accurate approximation of the Green’s functions in terms of a finite sum of spherical and cylindrical waves [13]. The main idea behind this method is to fit the spectrum of the Green’s function via the rational function fitting method (RFFM), based on the vector fitting algorithm (VECTFIT) presented in [14], after the analytical extraction of the quasi-static part of the spectrum. Although the RFFM was first presented in [15] and [16], the authors in [13] realized that this specific method experienced two major shortcomings, related to the sampling path and the restrictions of the VECTFIT algorithm. The above shortcomings prompted them to explore the possibility of improving the efficiency and accuracy of RFFM by sampling the spectrum off the real axis and by lifting the restrictions that the original VECTFIT posed on the poles and residues of the rational
0018-926X/$25.00 © 2007 IEEE
1964
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 7, JULY 2007
Fig. 2. Deformed sampling path for the two-level approximation.
select the vector and scalar Green’s functions due to a horizontally oriented electric source for the layer of the source, and more specifically the expressions derived in [18] Fig. 1. Typical case of a PEC backed stratified medium with a HED in layer i.
(1) function fit. This was done by ultimately introducing the modified RFFM (MRFFM), the key feature being that the surface wave poles and the associated residues are readily available from the process of fitting of the dynamic part of the spectrum with great accuracy, thus eliminating the need for a sophisticated pole-search algorithm. The only drawback of the MRFFM seems to be the analytical determination and extraction of the quasi-static part of the spectrum, a procedure that is both not unique and can become very tedious, as it is clearly stated by the authors [13]. On the other hand, the fact that the space-domain counterparts, derived via the Sommerfeld identity [17], are space waves is of significant physical importance, since the 1 singularity is recovered. Hence, it is obvious that in order to get a robust, fully automatic algorithm for the computation of Green’s functions, it is of paramount importance to incorporate a rigorous algorithm for the arithmetic extraction of the quasi-static part to the existing MRFFM. This can be done efficiently, as will be shown in this paper, by embodying the DCIM in the first path of the fitting procedure. More specifically, a spectrum fitting via exponential functions is performed for the accurate approximation of the quasi-static terms, in addition to some dynamic terms, while the vector fitting algorithm is employed for the approximation of the remaining spectral-domain function. In Section II, after a brief review of the spectral-domain Green’s functions in planar stratified media, the issue of whether the quasi-static terms can be accurately evaluated in a robust and fully numerical way by the approximation over the first path of the DCIM is clarified. In Section III, the formulation of the new approach based on a two-level approximation is explained in detail. Numerical results are provided in Section IV for the validation of the proposed methodology. II. SPECTRAL-DOMAIN GREEN’S FUNCTIONS AND QUASI-STATIC TERMS EVALUATION A general planar stratified medium is shown in Fig. 1, where the source is embedded in layer . Each layer can have different and thickness . electric and magnetic properties As typical examples of spectral-domain Green’s functions, we
(2) and
(3)
where
and
,
,
, and
are functions
[17] and the of the generalized reflection coefficients source location (for simplicity the source plane is chosen as the ). The time-harmonic variation reference plane is assumed and is suppressed. The spatial-domain Green’s functions are obtained either by using a two-dimensional inverse Fourier transform or by employing a one-dimensional Hankel (Fourier–Bessel) transform of the corresponding spectral-domain Green’s functions (4) where “SIP” stands for the Sommerfeld integration path. The first-level approximation, performed over the path (Fig. 2) in the two-level DCIM approach, is designed for the extraction of the quasi-static terms of the spectrum, as well as for enabling two different frequencies of sampling in the GPOF method [10]. It is the first one of those characteristics of the two-level DCIM that prompted us to explore the possible incorporation between this method and the recently proposed MRFFM, since the extraction of the quasi-static terms seems to be a part of utmost importance for the accuracy and efficiency of the second one.
POLIMERIDIS et al.: COMPUTATION OF GREEN’S FUNCTIONS IN STRATIFIED MEDIA
1965
cm; layer 2: , , cm; layer 3: free space; horizontal electric dipole (HED) is at interface between layer 3 and layer 2; the observation plane coincides with . The scalar potential Green’s function the source plane, in spectral-domain for such a geometry can be cast into the following form:
(5) Fig. 3. Four-layer structure with the HED at the interface between layer 3 (air) and layer 2.
To begin with the mathematical and physical reasoning of the quasi-static extraction procedure, we mention that in the first VECTFIT-based methods to appear in the literature [15], [16], this issue was not really treated in a general way. More specifically, in [15] and for the case of a dielectric slab above a perfectly conducting plane, the authors obtained closed-form Green’s functions only for expressions of the spectral-domain counterparts with primary wave terms of pure exponential nature. It was also mentioned that an analytical extraction of the quasi-static terms could improve the accuracy of the proposed method, but there was no further elaboration. Equally, only a simple single-layered structure was considered in [16], and as is already known, the analytic evaluation of the quasi-static terms for such a geometry is quite a simple task [5]. On the other hand, in the context of the MRFFM [13], a relatively simple scheme was employed for the determination of the high spatial frequency part of the spectrum, which is responsible for the near-field behavior of the space-domain Green’s functions. Moreover, the quasi-static part of the spectrum was defined in such a way that its space-domain counterpart could be evaluated analytically, in terms of spherical waves. Hence, the key feature of the MRFFM is that it yields closed-form Green’s functions, for any number of layers, in terms of both spherical and cylindrical waves. Those of spherical nature represent the source and quasi-static terms and provide an accurate description of the singularity and the near-field physics in the vicinity of the source, while the cylindrical ones capture correctly the surface wave behavior that dominates the far field. In order to alleviate the necessity of the explicit extraction of the quasi-static terms prior to the rational function fitting, we propose the application of the first level of the DCIM. As is comprehensively explained in [10], the approximation of the specvia complex extral-domain Green’s functions on the path ponentials can be considered an extraction of quasi-static terms, in addition to some dynamic terms as well. In other words, instead of the quite bothersome analytical extraction, which determines the limiting terms as , once the approximation over the path is subtracted from the spectrum, it not only can make the spectral-domain Green’s functions smoothly converge to zero as but also makes them zero beyond a predefined value of . To demonstrate the effectiveness of the first-level approximation, regarding the quasi-static terms of the spectrum, we choose GHz; layer a four-layer geometry, as shown in Fig. 3: , , 0: perfect electric conductor (PEC); layer 1:
where ( is the wavenumber of air) and the branch of the square root is chosen to satisfy the radiation condition. By performing the first-level approximation over the path , which of the spectral quantity is described by the following parametric equation: (6) where the parameters , are chosen according to [9], we obtain a finite sum of exponentials
(7)
over the path , enThe smooth behavior of ables us to relax the tolerance in the singular value decomposition analysis of the GPOF method and get only three exponentials for an accurate approximation. More specifically, a concan be used to adaptively control the accustant parameter is racy of the GPOF method. The number of exponentials chosen by analyzing the singular values [8]. Hence, we set the and discard the small , which are , tolerance where is the largest singular value. A typical choice for the is because most approximation over the path singular values are zero on this path. Extracting the approximating exponentials makes the spectral-domain function converge to zero, with a convergence rate that gets faster as the number of exponentials increases, which is clearly shown in Figs. 4 and 5. Hence, it is made clear from is superior to the analysis that approximation over the path analytical extraction of the quasi-static terms, because it is robust, applicable for any geometry no matter how complex it is, and fully automatic [10]. In other words, it exhibits all the required characteristics for the implementation in a closed-form Green’s function method. III. TWO-LEVEL APPROACH FOR APPROXIMATING THE SPECTRAL-DOMAIN GREEN’S FUNCTIONS AND THE CORRESPONDING SPACE-DOMAIN COUNTERPARTS In this section, we present the details of the proposed twolevel approach for the approximation of the spectral-domain Green’s functions. For the sake of illustration, we consider the general planar stratified geometry (Fig. 1) and the scalar Green’s function (3) as a representative case. Moreover, to demonstrate
1966
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 7, JULY 2007
Next, we subtract the approximating expansion for the range from the original funcof , will tion. The remaining function, divided by the term 2 be nonzero over a small range of and can be evaluated by an accurate fitting, in terms of rational functions according to [13]
(10)
Fig. 4. Real part of the remainder of the spectral-domain function f~ (k ; z; z ) sampled over C after the extraction of the first-level approximation over (quasi-static terms). C
Fig. 5. Imaginary part of the remainder of the spectral-domain function after the extraction of the first-level approxif~ (k ; z; z ) sampled over C (quasi-static terms). mation over C
the steps of getting an accurate spectral approximation and, finally, a closed-form Green’s function in terms of spherical and cylindrical waves, we use the following more compact form: (8) The first part of the approximation is performed along the path , where and , , via the GPOF method
. Since the first level of the proposed where method is identical with the first level of the DCIM and the second level with the first level of the MRFFM, as it is implemented in [13], it is clear that this method could be considered as a hybrid one. The details of the way the VECTFIT and the GPOF algorithms are used in order to fit the spectrum are omitted, since in [13], [9], and [10] all issues regarding the sampling paths, number of samples, and starting poles are successfully clarified. After the approximation of the spectral-domain Green’s function is concluded, the space-domain counterpart can be obtained with use of (4) and, by making use of the approximations (9) (10), we obtain
(11) or in a more compact form (12) beNote that the third integral is evaluated along the path but the second integral cause the integrand is negligible on . Therefore, the standard integral is evaluated along identities can be applied to the integrals in (11). More specifiis obtained via cally, the space-domain function the Sommerfeld identity [6] and the space-domain counterpart of (10) can be obtained with the use of the following integral identity [19], [20]:
(9)
(13)
As it is thoroughly explained in the previous section, the approximating exponentials represent the quasi-static terms in addition to some dynamic terms.
It is intuitively meaningful to mention that the rational function fitting procedure captures the surface wave poles and the
POLIMERIDIS et al.: COMPUTATION OF GREEN’S FUNCTIONS IN STRATIFIED MEDIA
1967
associated residues with great accuracy [13], providing the desired behavior in the far-field region. Finally, the space-domain Green’s function for a general planar stratified medium is cast into a finite sum of spherical and cylindrical waves
(14) . where To reveal the main contribution of the proposed method, it is essential that we proceed to a comparison with both the DCIM and the MRFFM. The DCIM, for instance, requires a pole-search algorithm for the surface wave extraction prior to the exponential fitting in order to maintain its accuracy for moderate to large distances from the source. The MRFFM, on the other hand, calls for an analytical determination of the quasi-static part of the spectrum prior to the rational function fitting in order to capture the near-field physics in the vicinity of the source. Hence, the method proposed in this paper succeeds in dealing with both issues simultaneously. To be more specific, both the quasi-static terms and the surface wave poles are evaluated in a robust and fully computational way within the framework of the accurate fitting of the spectral-domain Green’s functions. Hence, it alleviates the necessity for any special treatment and provides the means for the closed-form Green’s functions evaluation in a straightforward manner.
Fig. 6. Magnitude of the vector Green’s function G (; z; z ) versus k for the geometry given in Fig. 3. The solid lines correspond to results obtained with use of numerical integration while the marks correspond to results obtained with use of the proposed methodology.
IV. NUMERICAL RESULTS To demonstrate the validity and accuracy of the proposed methodology, the geometry in Fig. 3 was investigated. The observation plane was taken to coincide with the source plane . The two-level algorithm presented in this paper was applied to the approximation of the , elements of the dyadic Green’s function. The reference solution is constructed via numerical integration of the spectrum, according to the procedures presented in [21]. To be more specific, the integrals were evaluated to machine precision on the deformed path of Fig. 2 by an adaptive quadrature based on Patterson’s formulas. In addition, the algorithm of Shanks transformation was implemented for the convergence acceleration of the tail integral over the real axis. , where The total number of samples is and are the number of samples on and respectively, assuming uniform sampling of the real variable . In approximating the spectral-domain functions, the following parameters were used: , ; • number of samples: , ; • sampling range parameters: • number of exponentials: ; . • number of rational functions: Figs. 6 and 7 demonstrate the calculated magnitude of the at 1 and 30 GHz corresponding Green’s functions versus . The solid lines are the results obtained using numerical integration, while the results obtained with the use of
Fig. 7. Magnitude of the scalar Green’s function G (; z; z ) versus k for the geometry given in Fig. 3. The solid lines correspond to results obtained with use of numerical integration while the marks correspond to results obtained with use of the proposed methodology.
the proposed algorithm are shown by the marks in the plots. In addition, Figs. 8 and 9 demonstrate the calculated magnitude at 30 GHz, of the corresponding Green’s functions for the case where the observation plane does not coversus incide with the source plane ( mm, mm). As is obvious, the results are in very good agreement and the relative error remains below 0.5% for all values of distances between the source and observation point. In Fig. 10, a typical example of the behavior of the relative error of the magnitude versus for the geometry given in Fig. 3 at 30 GHz is shown. The proposed method is quite general and is applied, without modification, in the case of lossy media as well. More specifically, we consider the three-layer geometry (Fig. 11) with a loss tangent equal to 0.08. A nonzero loss tangent will introduce
1968
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 7, JULY 2007
Fig. 8. Magnitude of the vector Green’s function G (; z; z ) versus k for the geometry given in Fig. 3 at 30 GHz with z = 0:5 mm. The solid lines correspond to results obtained with use of numerical integration while the marks correspond to results obtained with use of the proposed methodology.
Fig. 10. Relative error of the magnitude jG (; z; z )j versus k for the geometry given in Fig. 3 at 30 GHz.
Fig. 11. Three-layer medium-loss (tan ( ) = 0:08) structure with the HED at the interface between layer 2 (air) and layer 1.
Fig. 9. Magnitude of the scalar Green’s function G (; z; z ) versus k for the geometry given in Fig. 3 at 30 GHz with z = 0:5 mm. The solid lines correspond to results obtained with use of numerical integration while the marks correspond to results obtained with use of the proposed methodology.
a small exponentially decreasing behavior in the Hankel functions of the corresponding surface waves, as is clearly shown in Fig. 12 for the case of the scalar potential at a frequency of 10 GHz. As a concluding remark, we notice that the proposed methodology requires as much computational time as the two-level DCIM with surface wave extraction and less computational time compared to the MRFFM, since the first level of approximation is performed via the GPOF algorithm, which is less computationally expensive, compared to the iterative VECTFIT algorithm. More specifically, the total amount of computation time for the accurate evaluation of the spatial domain Green’s function was less than 1 s for all cases, with the algorithm programmed in MATLAB and running on a Pentium IV 2.4 GHz processor.
Fig. 12. Magnitude of the scalar Green’s function G (; z; z ) versus k for the geometry given in Fig. 10 at 10 GHz. The solid lines correspond to results obtained with use of numerical integration while the marks correspond to results obtained with use of the proposed methodology.
V. CONCLUSION An efficient and fully numerical methodology for the computation of Green’s functions for unbounded planar stratified media has been presented. This technique combines the DCIM
POLIMERIDIS et al.: COMPUTATION OF GREEN’S FUNCTIONS IN STRATIFIED MEDIA
together with the MRFFM for casting the spectral-domain Green’s functions in a complex-exponential and a pole-residue form, respectively. Hence, the closed-form Green’s functions are cast into terms of cylindrical and spherical waves, thus enabling the accurate description of both the near-field and far-field physics. The key feature of the proposed methodology is that, due to its fully automatic nature, there is no need for analytical determination and extraction of any “problematic” behaviors of the spectrum prior to the fitting.
1969
[17] W. C. Chew, Waves and Fields in Inhomogeneous Media, ser. Electromagnetic Waves. New York: IEEE Press, 1995. [18] G. Dural and M. I. Aksun, “Closed-form Green’s functions for general sources and stratified media,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 1545–1552, Jul. 1995. [19] I. M. Vrancken, “Full wave integral equation based electromagnetic modelling of 3D metallic structures in planar stratified media,” Ph. D. dissertation, Leuven, Belgium, 2002. [20] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products. New York: Academic, 2000. [21] K. A. Michalski, “Extrapolation methods for Sommerfeld integral tails,” IEEE Trans. Antennas Propag., vol. 46, pp. 1405–1418, Oct. 1998.
REFERENCES [1] J. R. Mosig, “Arbitrarily shaped microstrip structures and their analysis with a mixed potential integral equation,” IEEE Trans. Microwave Theory Tech., vol. 36, pp. 314–323, Feb. 1988. [2] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory,” IEEE Trans. Antennas Propag., vol. 38, pp. 335–335, Mar. 1990. [3] A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag., vol. AP-28, pp. 593–603, Sep. 1980. [4] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [5] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed-form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theory Tech., vol. 39, pp. 588–592, Mar. 1991. [6] M. I. Aksun and R. Mittra, “Derivation of closed-form Green’s functions for a general microstrip geometry,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2055–2062, Nov. 1992. [7] N. V. Shuley, R. R. Boix, F. Medina, and M. Horno, “On the fast approximation of Green’s functions in MPIE formulations for planar layered media,” IEEE Trans. Microwave Theory Tech., vol. 50, pp. 2185–2192, Sep. 2002. [8] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, pp. 229–234, Feb. 1989. [9] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s functions,” IEEE Trans. Microwave Theory Tech., vol. 44, pp. 651–658, May 1996. [10] M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag., vol. 53, pp. 3644–3653, Nov. 2005. [11] A. C. Cangellaris and V. I. Okhmatovski, “New closed-form Green’s function in shielded planar layered media,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 2225–2232, Dec. 2000. [12] V. I. Okhmatovski and A. C. Cangellaris, “A new technique for the derivation of closed-form electromagnetic Green’s functions for unbounded planar layered media,” IEEE Trans. Antennas Propag., vol. 50, pp. 1005–1016, Jul. 2002. [13] V. N. Kourkoulos and A. C. Cangellaris, “Accurate approximation of Green’s functions in planar stratified media in terms of a finite sum of spherical and cylindrical waves,” IEEE Trans. Antennas Propag., vol. 54, pp. 1568–1576, May 2006. [14] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery, vol. 14, pp. 1568–1576, Jul. 1999. [15] V. I. Okhmatovski and A. C. Cangellaris, “Evaluation of layered media Green’s functions via rational function fitting,” IEEE Microw. Compon. Lett., vol. 14, pp. 22–24, Jan. 2004. [16] H. Chen, Z. Shen, and E. Li, “A simple approach to closed-form Green’s functions of microstrip structures,” Microw. Opt. Technol. Lett., vol. 45, no. 5, pp. 435–438, Jun. 2005.
Athanasios G. Polimeridis (S’01) was born in Thessaloniki, Greece, in 1980. He received the diploma degree in electrical and computer engineering from Aristotle University of Thessaloniki, Greece, in 2003, where he is currently pursuing the Ph.D. degree. His research interests include computational electromagnetics, with emphasis on the development and implementation of integral-equation based algorithms.
Traianos V. Yioultsis was born in Yiannitsa, Greece, in 1969. He received the diploma degree (with honors) in electrical engineering and the Ph.D. degree in electrical and computer engineering from Aristotle University of Thessaloniki, Greece, in 1992 and 1998, respectively. From 1993 to 1998, he was a Research and Teaching Assistant in the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki. From 2001 to 2002, he as a Postdoctoral Research Associate with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign. Since 2002, he has been a Lecturer in the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki. His current interests include the analysis and design of microwave circuits and antennas with fast computational and optimization techniques and the modeling of complex wave propagation problems.
Theodoros D. Tsiboukis (S’79–M’81–SM’99) received the Diploma degree in electrical and mechanical engineering from the National Technical University of Athens, Athens, Greece, in 1971 and the Ph.D. degree from the Aristotle University of Thessaloniki (AUTH), Greece, in 1981. From 1981 to 1982, he was with the Electrical Engineering Department, University of Southampton, Southampton, U.K., as a Senior Research Fellow. Since 1982, he has been with the Department of Electrical and Computer Engineering (DECE), AUTH, where he is currently a Professor. He has served in numerous administrative positions, including Director of the Division of Telecommunications, DECE (1993—1997) and Chairman, DECE (1997–2001). He is also Head of the Advanced and Computational Electromagnetics Laboratory, DECE. He has authored or coauthored eight books and textbooks including Higher-Order FDTD Schemes for Waveguide and Antenna Structures (Morgan & Claypool, 2006). He has authored or coauthored more than 125 refereed journal papers and more than 100 international conference papers. He was Guest Editor of a special issue of International Journal of Theoretical Electrotechnics (1996). His main research interests include electromagnetic-field analysis by energy methods, computational electromagnetics (finite-element method, boundary-element method, vector finite elements, method of moments, finite difference time-domain method (FDTD), ADI-FDTD method, integral equations, and ABCs), metamaterials, photonic crystals, and inverse and electromagnetic compatibility problems. Prof. Tsiboukis is a member of various societies, associations, chambers, and institutions. He was Chairman of the local organizing committee of the 8th International Symposium on Theoretical Electrical Engineering in 1995. He has received several awards and distinctions.
