1580
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 5, MAY 2011
Mixed-Impedance Boundary Conditions Henrik Wallén, Ismo V. Lindell, Life Fellow, IEEE, and Ari Sihvola, Fellow, IEEE
Abstract—A novel set of isotropic electromagnetic boundary conditions generalizing some of the recently introduced boundary conditions is introduced. For a planar boundary it is required that the boundary appears as an impedance boundary for the TE and TM components of the field with different surface impedances in the general case. Thus, such conditions can be dubbed as mixed-impedance boundary conditions. It is shown that the conditions can be alternatively expressed in terms of normal components of the fields at the boundary and in this form they can be defined for non-planar boundaries as well. The conditions generalize those of the isotropic impedance boundary and the more recently considered DB and D B boundaries. As a numerical example, scattering from a mixed-impedance sphere is considered, and it is found that the sphere could be useful as a cloaking device. Index Terms—Boundary impedance.
conditions,
scattering,
surface
• Perfect electromagnetic conductor (PEMC) [4]; • Self-dual impedance boundary with [5]; [6], [7]. • Soft-and-hard surface, previous case with Recently, boundary conditions in terms of field components normal to the boundary have been considered [8]–[10] and two conditions dubbed as DB-boundary and D B -boundary conditions were respectively defined for the planar boundary with as (2) and (3)
I. INTRODUCTION HE impedance boundary condition defines a linear relation between time-harmonic electric and magnetic field components tangential to the boundary surface. The basic form with scalar surface impedance was introduced by Shchukin [1] and Leontovich [2] in the 1940’s [3]. Denoting the outer normal unit vector by , its general form can be expressed as
T
(1) Here the subscript denotes component tangential to the sur, while is the two-dimensional surfaceface, impedance dyadic. Actually, (1) can be interpreted as two-dimensional extension of Ohm’s law if we replace the tangential magnetic field in terms of the effective surface current density . The most general impedance condition involves four scalar in any coordiparameters, the components of the dyadic nate system on the boundary. As special cases of the impedance boundary we may list the following. ; • Isotropic impedance boundary ; • Perfect electric conductor (PEC) ; • Perfect magnetic conductor (PMC)
Manuscript received May 05, 2010; revised August 30, 2010; accepted November 01, 2010. Date of publication March 07, 2011; date of current version May 04, 2011. This work was supported in part by the Academy of Finland. The authors are with the Department of Radio Science and Engineering, Aalto University School of Electrical Engineering, Espoo, Finland (e-mail: henrik.
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2123064
The DB conditions (2) were introduced already in 1959 by Rumsey [11] but applications were suggested only very recently [12]–[18]. It was shown that the planar DB boundary acts as a PEC plane for fields polarized TE with respect to the normal of the plane and PMC plane for the TM field [8], [11]. Similarly, the planar D B boundary acts as PMC and PEC planes for the TE and TM fields, respectively [10]. The purpose of this paper is to study a novel more general set of boundary conditions which follow by assigning separate impedance boundary conditions for the TE and TM components of the field. II. TE/TM DECOMPOSITION It is well known that, in a homogeneous and isotropic medium, outside the sources, any field can be decomposed in partial fields polarized TE and TM with respect to a fixed direction in space [19]. Defining the direction by , we can write (4) (5) The decomposed fields satisfy individually the Maxwell equations (6) (7) (8) and the divergences of the four vectors vanish. From these we obtain the following relations:
0018-926X/$26.00 © 2011 IEEE
(9) (10)
WALLÉN et al.: MIXED-IMPEDANCE BOUNDARY CONDITIONS
1581
where one should pay attention that, in the last terms, the total fields are involved.
TABLE I SOME SPECIAL CASES OF MIXED-IMPEDANCE BOUNDARY CONDITIONS
III. MIXED-IMPEDANCE BOUNDARY CONDITIONS Let us now consider a new type of electromagnetic boundary on which the impedance boundary conditions are valid for the , the TE and TM fields separately. At the planar boundary are required field components in an isotropic half space to satisfy the conditions (11) (12) and are two surface impedances. Obviously, where the boundary appears as a regfor the case ular isotropic impedance boundary with the surface impedance . Equation (12) can also be written as (13) Operating (11) by
ever a physical realization for the mathematical boundary can be found. Finding a realization for the MI boundary must be left as a topic of future work. However, it is known that localized sources cannot create TEM fields so that the problem of uniqueness is limited to sources of infinite extent. IV. POWER CONSIDERATIONS An isotropic impedance boundary with surface impedance , where the surface resistance and surare real, is passive when and lossless face reactance . Therefore, it seems reasonable to assume that an when MI boundary with surface impedances (17)
and applying (9) we obtain is passive when both surface resistances are non-negative
(18)
(14) . Similar where at the last step we have applied steps for (13) yield a similar result. In this way the following boundary conditions are obtained for the normal components of the total fields (15) (16) which are valid when each of the TE and TM components sees the boundary as an impedance boundary. To prove the equivalence between the two pairs of conditions we also need to show that (15) and (16) implies (11) and (12), which seems difficult in the general case. It is, however, straightforward to show the equivalence between the conditions for arbitrary TE and TM polarized plane waves. Since in the form (15), (16) the boundary conditions do not depend on the TE/TM decomposition of the fields, they appear more universal and we can call them mixed-impedance (MI) boundary conditions. The MI boundary is an isotropic boundary condition, since there is no preferred direction along the boundary, but this isotropy should not be confused with the isotropy of an ordinary impedance surface. Since the introduction was based on the TE and TM decomposition the conditions (15), (16) are not meaningful for fields which cannot be uniquely decomposed in TE and TM parts with respect to the given direction, i.e., fields which are TEM with respect to the axis. A plane wave incident normally to is an example of such a field. This the boundary plane means that, just like in the case of the DB or D B boundary [10], for such special fields the boundary conditions are not sufficient and require an additional condition which emerges when-
and lossless when . To show that this is indeed the case is straightforward for a plane wave reflecting from a planar MI boundary, and the Mie-computations below also confirm this assumption for the plane-wave scattering from an MI sphere. However, for general fields and boundary shapes the problem appears more complicated, and we leave the question whether (18) is a sufficient condition for passivity for a future study. V. SPECIAL CASES It was already mentioned that for the MI boundary becomes an ordinary isotropic impedance boundary. and PMC This includes the limiting cases PEC . Also, the DB and D B boundaries [8]–[10] are special cases of the MI boundary, as summarized in Table I. In [20] a generalization for the DB and D B boundary conditions (2), (3), dubbed as the generalized DB (GDB) boundary conditions, was introduced. The GDB boundary conditions were expressed in the form (19) (20) where is a parameter. From this we can see that the DB condiwhile the D B conditions tions (2) are obtained for . are obtained for The GDB boundary conditions form a special case of the MI boundary conditions. In fact, (16) corresponds to (19) for
(21)
1582
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 5, MAY 2011
and (15) corresponds to (20) for
(22) Actually, one can note that when the two surface impedances satisfy the relation (23) the MI boundary becomes a GDB boundary, where the paramcan be complex. eter It was shown previously that the GDB boundary and its special cases, the DB and D B boundaries are self dual, i.e., there exists a duality transformation in term of which the boundary conditions are invariant [20]. One can show that the MI boundary is self dual when the condition (23) is , valid. In fact, making the duality transformation in (15) and (16) the pair of conditions is invariant only if (23) is valid. Thus, the GDB boundary can also be called as the self-dual mixed-impedance boundary. The two impedance parameters can be expressed in terms of as two other parameters (24) where can be called the self-dual surface-impedance parameter and the anti-self-dual surface-impedance parameter. In the impedances satisfy (23) while for a fact, for minus sign must be added to the right side of (23). The parameters and are real for a resistive MI boundary, and in that case the passivity condition (18) corresponds to (25) For a lossless MI boundary, both and are imaginary. The condition for an isotropic impedance boundary implies the condition (26) which is the equation of a hyperbola. In the resistive case, , we get the one-sided hyperbola shown in the upper part of Fig. 1 when we require the surface resistance to be non-negative , both the upper and (passive). In the reactive case, lower branches of the hyperbola shown in the lower part of Fig. 1 are equally reasonable. , Another interesting special case is when which corresponds to the hyperbolic condition (27) which is also plotted in the lower part of Fig. 1. This condition is reasonable only in the reactive case, since would require an active boundary for one of the polarizations. , while The DB boundary condition corresponds to the PEC, PMC, and D B conditions require infinite values of and/or as indicated in Fig. 1.
Fig. 1. Some interesting special cases of the MI boundary in terms of the selfdual and anti-self-dual impedance parameters s, a for the resistive case (above) and reactive case (below).
VI. NON-PLANAR BOUNDARIES It does not appear straightforward to generalize the conditions by , the unit (11), (13) to non-planar surfaces by replacing normal of the boundary surface. In fact, it requires that at the surface is associated to a coordinate system in the space outside the surface and that there exists a TE/TM decomposition of the field with respect to that coordinate. It is better to start from the definition (15), (16) which is not directly associated with a global TE/TM decomposition. Following [10] we should now write the conditions as (28) (29)
WALLÉN et al.: MIXED-IMPEDANCE BOUNDARY CONDITIONS
1583
However, at the boundary, and can be still associated to surface impedances when the field is locally approximated by reflecting plane waves decomposed in TE and TM parts with respect to the direction of the local normal. As a concrete example let us consider the region outside the . It is well known that any field can be spherical surface globally decomposed in TE and TM components with respect [21]. Thus, we can start from to the radial unit vector the MI boundary conditions of the form (11), (13) written as (30) (31) and show that they lead to (28) and (29). we now have Because of (32) (33) whence we can operate (30) as
[24], and applying the boundary conditions (37) and (38), we get the scattering amplitudes (39) (40)
where is the size parameter, but otherwise all formulas from [24] remain the same.1 Of particular interest are perhaps the absorption, total scattering and backscattering of the MI sphere, and their dependence on the parameters and . For this purpose, we computed , the absorption efficiency , the extinction efficiency the scattering efficiency , and the backscattering efficiency for a representative selection of parameters presented in Figs. 2–5. These efficiencies are defined as the corresponding cross section divided by the physical cross section, and extinction is the sum of scattering and absorption (41)
(34)
As predicted by the extinction paradox [24], the extinction efficiency approaches 2 for large spheres
by . After similar operations on and replace thus become (31), the boundary conditions at (35) (36) which equal (28) and (29) written in spherical coordinates. In [22] we considered modes in a spherical resonator with DB boundary and, in [20], [23], plane wave scattering from a spherical object with DB, D B and GDB boundaries. It was found that, due to their self-dual character, there is no backscattering from any of these objects. Obviously this should no longer be true for the MI sphere without the self-dual property, i.e., for . To verify this, some numerical tests were made. VII. SCATTERING FROM A MIXED-IMPEDANCE SPHERE Let us consider the scattering of a plane wave from an MI sphere with radius in free space. In terms of the self-dual and anti-self-dual parameters and , the MI conditions (35) and can be expressed as (36) at (37) (38) It is obvious that, making the duality transformation to the fields, , is equivalent with replacing i.e., interchanging and by . This scattering problem can be solved using ordinary Mieseries with very small modifications, as was done for the DB, D B and GDB spheres in [20], [23]. Following the analysis in
(42) and all efficiencies are small for small spheres. The most inis of the teresting effects happen when the size parameter order one, i.e., when the circumference of the sphere is near one free-space wavelength. Fig. 2 shows all four efficiencies for a resistive MI sphere and varying real parameters with size parameter . As expected, the backscattering is zero, , , and if and only if the anti-self-dual parameter vanishes, the absorption is positive when the passivity condition (25) is satisfied. The largest variations in scattering and , and this absorption happens for a self-dual MI sphere case is also plotted in Fig. 3. The absorption has the maximum when and vanishes when and , which correspond to the DB and D B conditions, respectively. , when both the impedThe absorption is zero, ances and the parameters are purely imaginary. The extinction and backscattering efficiencies, and , for such a lossless reactive MI sphere are plotted . in Fig. 4, where the size parameter is chosen to be As already mentioned, the backscattering is zero if and only if , but we can also see that both and appear to have large and sharp maxima for the reactive MI sphere in Fig. 4. To get a better quantitative view of the efficiencies, we present a few horizontal cuts from Fig. 4 in Fig. 5, where a small real part is added to slightly smoothen the curves without 1Notice, however, that we use the e time convention, which corresponds j compared with [24]. to the replacement i
!0
1584
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 5, MAY 2011
=1
05 < s; a < 5. = 0. The same
Fig. 2. Extinction, absorption, scattering and backscattering efficiencies for a resistive MI sphere with size parameter kr as a function of The dashed green line marks the boundary where a s and the black contour lines are drawn at integer values of Q, with a thicker line at Q color-scale is also used in all four sub-figures to make comparisons easier.
=6
( =
Fig. 3. Extinction, absorption and scattering efficiencies for a self-dual a resistive s MI sphere with size parameter kr as a function of the self-dual parameter s. (The backscattering is zero, Q .)
0)
(Im = 0)
=1 0
removing any one of the main peaks. The sharpest maxima in the figures are naturally very sensitive to losses, but it seems that a reactive MI sphere can exhibit fairly large extinction and backscattering efficiencies even when losses are included. It is interesting to notice that the efficiencies plotted in Figs. 2 and 4 are symmetric with respect to the anti-self-dual parameter , while the sign of the self-dual parameter is very important. In the reactive case in Fig. 4, the total scattering is stronger when the TE reactance is capacitive and the TM reactance is inductive
, , ) than in the opposite case. For ( , in Fig. 5 we have the instance, in the self-dual case, at , maximum extinction efficiency while the minimum is at . in Fig. 5 suggests that the MI The very low minimum boundary condition could be very interesting from a cloaking perspective. We could, theoretically, place the object to be cloaked inside a sphere with MI boundary conditions on the outside, and choose the MI parameters and so that the total extinction is minimized. Looking at the results presented above, , it seems that the and also testing various ways to minimize optimal choice is , , with depending of the sphere. In this case we have on the size parameter and no absorption , zero backscattering . The results in Fig. 6 show which implies that that the minimum extinction (and scattering) efficiency is very much smaller than for a PEC sphere of the same size is small, while the extinction paradox (42) implies when for very large spheres. In between, for that around unity, we can significantly reduce the total extinction compared with a PEC sphere, provided that the needed MI boundary condition is somehow realizable. The cloaking is never perfect, but the performance might be competitive with other proposed approaches [25]. The GDB sphere considered in [20] corresponds to the reacand using the tive self-dual MI sphere with present notation.
WALLÉN et al.: MIXED-IMPEDANCE BOUNDARY CONDITIONS
1585
Fig. 6. Minimum extinction efficiency Q for a self-dual (a = 0) lossless (Re s = 0) MI sphere as a function of the size parameter kr . The left curve shows Q relative to the extinction efficiency of a PEC sphere Q , and the right curve shows the corresponding optimal choice of Im s. For small size parameters, say kr < 1, the MI sphere looks promising as a cloaking device.
Fig. 4. Extinction and backscattering efficiencies for a reactive MI sphere (Re s = Re a = 0) with size parameter kr = 1:5 as a function of 5 < Im s; Im a < 5. The absorption is exactly zero, and the backscattering is zero only when a = 0.
0
the normal total field components in a form which does not depend on the TE/TM decomposition of the field. In this form they can be expressed for non-planar boundaries as well. The MI boundary is a generalization of the previously defined DB, D B , and GDB boundaries, but also the more common PEC, PMC and isotropic impedance boundaries can be expressed as special cases of the MI boundary. As an example, a spherical MI boundary was considered and its scattering properties for some impedance parameters were studied. In the ideal lossless case, the MI sphere can be either a very strong or very weak scatterer depending on the choice of MI parameters. Choosing optimal parameters, it appears that the MI sphere could have some potential for electromagnetic cloaking purposes. A theoretical implementation of the DB boundary was presented in [9] and some promising steps towards an experimental realization was recently presented in [26], but the realization of the more general MI boundary is still an open problem. REFERENCES
Fig. 5. Extinction and backscattering efficiencies for a reactive MI sphere with size parameter kr = 1:5 and either a = 0 or a = j as a function of Im s. A small real (resistive) part, Re s = 0:02, is added to make the curves smoother, without removing the main peaks.
VIII. CONCLUSION A novel set of electromagnetic boundary conditions was introduced under the name mixed-impedance (MI) boundary conditions. Expanding the field in TE and TM polarized components, a planar MI boundary appears as an impedance boundary for the TE field and for the with surface impedance TM field. It was shown that the conditions can be expressed for
[1] A. N. Shchukin, Propagation of Radio Waves. Moscow: Svyazizdat, 1940. [2] M. A. Leontovich, “Methods of solution for problems of electromagnetic waves propagation along the earth surface,” (in Russian) Bull. Acad. Sci. USSR, Phys. Ser., vol. 8, no. 1, p. 1622, 1944. [3] G. Pelosi and P. Y. Ufimtsev, “The impedance-boundary condition,” IEEE Antennas Propag. Mag., vol. 38, no. 1, pp. 31–35, Feb. 1996. [4] I. V. Lindell and A. H. Sihvola, “Perfect electromagnetic conductor,” J. Electromagn. Waves Appl., vol. 19, no. 7, pp. 861–869, 2005. [5] I. V. Lindell, Methods for Electromagnetic Field Analysis, 2nd ed. New York: IEEE Press, 1995. [6] P.-S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,” Electron. Lett., vol. 24, no. 3, pp. 168–170, 1988. [7] P.-S. Kildal and A. Kishk, “EM modeling of surfaces with stop or go characteristics—Artificial magnetic conductors and soft and hard surfaces,” ACES J., vol. 18, no. 1, pp. 32–40, Mar. 2003. [8] I. V. Lindell and A. H. Sihvola, “Uniaxial IB-medium interface and novel boundary conditions,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 694–700, Mar. 2009. [9] I. V. Lindell and A. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E, vol. 79, no. 2, p. 026604, 2009. [10] I. V. Lindell and A. Sihvola, “Electromagnetic boundary conditions defined in terms of normal field components,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1128–1135, Apr. 2010.
1586
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 5, MAY 2011
[11] V. H. Rumsey, “Some new forms of Huygens’ principle,” IRE Trans. Antennas Propag., vol. 7, no. Special supplement, pp. S103–S116, Dec. 1959. [12] B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett., vol. 100, no. 6, p. 063904, 2008. [13] A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys., vol. 10, p. 115022, Nov. 2008. [14] A. D. Yaghjian and S. Maci, “Corrigendum: Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys., vol. 11, p. 039802, Mar. 2009. [15] R. Weder, “The boundary conditions for point transformed electromagnetic invisibility cloaks,” J. Phys. A, vol. 41, no. 41, p. 415401, 2008. [16] I. V. Lindell, A. Sihvola, P. Ylä-Oijala, and H. Wallén, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2725–2731, Sep. 2009. [17] P.-S. Kildal, “Fundamental properties of canonical soft and hard surfaces, perfect magnetic conductors and the newly introduced DB surface and their relation to different practical applications including cloaking,” in Proc. ICEAA’09, Torino, Italy, Aug. 2009, pp. 607–610. [18] P.-S. Kildal, A. Kishk, and Z. Sipus, “Introduction to canonical sufaces in electromagnetic computations: PEC, PMC, PEC/PMC strip grid, DB surface,” in Proc. 26th Annual Review of Progress in Applied Computational Electromagnetics (ACES 2010), Tampere, Finland, Apr. 26–29, 2010, pp. 514–519. [19] D. S. Jones, The Theory of Electromagnetism. Oxford: Pergamon Press, 1964, p. 19. [20] I. V. Lindell, H. Wallén, and A. Sihvola, “General electromagnetic boundary conditions involving normal field components,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 877–880, 2009. [21] J. G. Van Bladel, Electromagnetic Fields. Hoboken-Piscataway, NJ: Wiley-IEEE Press, 2007, pp. 314–317. [22] I. V. Lindell and A. H. Sihvola, “Spherical resonator with DB-boundary conditions,” Prog. Electromag. Res. Lett., vol. 6, pp. 131–137, 2009. [23] A. Sihvola, H. Wallén, P. Ylä-Oijala, M. Taskinen, H. Kettunen, and I. V. Lindell, “Scattering by DB spheres,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 542–545, Jun. 2009. [24] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983, ch. 4. [25] P. Alitalo and S. Tretyakov, “Electromagnetic cloaking with metamaterials,” Mater. Today, vol. 12, no. 3, pp. 22–29, Mar. 2009. [26] S. Hrabar, D. Zaluski, D. Muha, and B. Okorn, “Towards experimental realization of DB metamaterial layer,” presented at the META’10, 2nd Int. Conf. on Metamaterials, Photonic Crystals and Plasmonics, Cairo, Egypt, Feb. 22–25, 2010, Poster C17. Henrik Wallén was born in 1975 in Helsinki, Finland. He received the M.Sc. (Tech.) and D.Sc. (Tech.) degrees in electrical engineering from Helsinki University of Technology (which is now part of the Aalto University), in 2000 and 2006, respectively. He is currently working as a Postdoctoral Researcher at the Department of Radio Science and Engineering, Aalto University School of Science and Electrical Engineering, Espoo, Finland. His research interests include electromagnetic theory, modeling of complex materials and computational electromagnetics. Dr. Wallén is Secretary of the Finnish National Committee of URSI (International Union of Radio Science).
Ismo V. Lindell (S’68–M’69–SM’83–F’90–LF’05) was born in 1939 in Viipuri, Finland. He received the degrees of Electrical Engineer (1963), Licentiate of Technology (1967), and Doctor of Technology (1971), from the Helsinki University of Technology (HUT), Espoo, Finland. In 1962, he joined the Electrical Engineering Department, HUT, since 1975 as Associate Professor of Radio Engineering and, since 1989, as Professor of Electromagnetic Theory at the Electromagnetics Laboratory which he founded in 1984. During a sabbatical leave in 1996–2001 he held the position of Professor of the Academy of Finland. Currently he is Professor Emeritus at the Department of Radio Science and Engineering, Aalto University School of Electrical Engineering. He utilized a Fulbright scholarship as a Visiting Scientist at the University of Illinois, Champaign-Urbana, in 1972–1973, and the Senior Scientist scholarship of the Academy of Finland at the Massachusetts Institute of Technology, Cambridge, in 1986–1987. He has authored and coauthored 265 refereed scientific papers and 12 books including, Methods for Electromagnetic Field Analysis (IEEE Press, New York 3rd printing 2002), Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood MA, 1994), Differential Forms in Electromagnetics (Wiley and IEEE Press, New York 2004), and Long History of Electricity (Gaudeamus, Helsinki, Finland 2009, in Finnish). Dr. Lindell received the IEEE S.A. Schelkunoff award (1987), the IEE Maxwell Premium (1997 and 1998) and the URSI van der Pol gold medal in 2005, as well as the State Award for Public Information (2010).
Ari Sihvola (S’80–M’87–SM’91–F’06) was born on October 6, 1957, in Valkeala, Finland. He received the degrees of Diploma Engineer in 1981, Licentiate of Technology in 1984, and Doctor of Technology in 1987, all in electrical engineering, from the Helsinki University of Technology (TKK), Finland. Besides working for TKK and the Academy of Finland, he was a Visiting Engineer in the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, in 1985–1986, and in 1990–1991, he worked as a visiting scientist at the Pennsylvania State University, State College. In 1996, he was a Visiting Scientist at Lund University, Sweden, and for the academic year 2000–2001, he was a Visiting Professor at the Electromagnetics and Acoustics Laboratory, Swiss Federal Institute of Technology, Lausanne. In summer 2008, he was a Visiting Professor at the University of Paris XI, France. Currently he is a Professor of electromagnetics at Aalto University School of Electrical Engineering (before 2010, known as the Helsinki University of Technology) with interest in electromagnetic theory, complex media, materials modelling, remote sensing, and radar applications. Dr. Sihvola is Chairman of the Finnish National Committee of URSI (International Union of Radio Science) and a Fellow of IEEE. He was awarded the five-year Finnish Academy Professor position starting August 2005. Since January 2008, he is Director of the Graduate School of Electronics, Telecommunications, and Automation (GETA).
