Fifty years of high frequency diffraction - Wiley Online Library

Fifty Years of High Frequency Diffraction Constantine A. Balanis,1 Levent Sevgi,2,3 Pyotr Ya. Ufimtsev4 1 2 3 4

School of Electrical, Computer and Energy Engineering, Arizona State University, Temple, AZ Department of Electronics and Communications Engineering, Dogus University, Istanbul, Turkey Department of Electrical and Computer Engineering, UMASS Lowell (UML), MA 01854 EM Consulting, LA, CA 90025

Received 1 December 2012; accepted 1 February 2013

ABSTRACT: The study of high frequency diffraction phenomena mostly associated with geometrical theory of diffraction, uniform theory of diffraction, physical theory of diffraction, and their alternatives is reviewed. Several examples illustrate these theories and their C 2013 Wiley Periodicals, Inc. Int J RF and applications in antennas and scattering problems. V Microwave CAE 23:394–402, 2013.

Keywords: electromagnetics; diffraction; high frequency asymptotics; GO; GTD; UTD; PO; PTD

1940s. Milestone events happened in 1957 and 1962 with publications of papers on GTD [1, 2] and PTD [9, 10]. Both of these techniques have been developed and applied to a number of various problems. For example, a collection of articles associated with GTD is presented in Ref. 3. Among papers related to applications and extensions of GTD, most developed for practical applications, appeared to be the uniform theory of diffraction (UTD) [4, 5]. The modern form of PTD is presented in Refs. 11, 12. The most impressive application of PTD relates to design of US low observable (stealth) aircraft F-117 and B-2 [13–15]. One may believe that these techniques are different and independent. However, a basic connection exists between them; they have a common foundation. They are based on the localization principle that in the high frequency limit (k ¼ 2p f =c ! 1), the diffracted field in vicinity of a scattering object depends on its local properties. However, this principle is utilized in PTD and GTD in different ways. In PTD, it is applied to the field (currents) on the object surface while in GTD to the diffracted rays, i.e. only to the ray-part of the field radiated by these currents. Thus, already on this fundamental level it is understood that GTD can be interpreted as the asymptotic ray form of PTD. That is why it is not surprising that the GTD edge diffracted rays are derived from the PTD field integrals by asymptotic evaluation [11, 16, 17]. Basic ideas of these techniques and some illustrative examples of their applications are briefly introduced in this paper and elsewhere [15–23]. The next two sections illustrate details about GTD/UTD and PTD. Section IV presents two test problems with GTD/UTD and PTD comparisons. The 2D wedge scattering [25–28] and Flat-Earth

I. INTRODUCTION

The treatment of scattering characteristics from electromagnetic (EM) wave–object interactions using modal solutions is limited to structures whose surfaces can be described by orthogonal curvilinear coordinates. Moreover, most of the solutions are in the form of infinite series, which are poorly convergent when the dimensions of the object are greater than a few wavelengths. These limitations, therefore, exclude closed-form analyses of many practical scattering systems. When the dimensions of the scattering object are many wavelengths, high frequency asymptotic (HFA) techniques can be used to analyze many problems that are otherwise mathematically intractable. Two such techniques, which have received considerable attention in the past five decades, are the ray-type geometrical theory of diffraction (GTD) and the wave- (i.e., induced-source)-based physical theory of diffraction (PTD) [1–23]. Usually the diffraction phenomenon is understood as a deviation in a wave behavior from the geometrical optics (GO) laws. Just in this sense this term was introduced by Grimaldi who was the first to observe and describe the diffraction of light almost three hundred fifty years ago [24]. Since then this phenomenon has been continuously investigated. An extensive study of HFA diffraction was stimulated by invention and development of radar in the

Correspondence to: L. Sevgi; e-mail: [email protected] or [email protected] DOI 10.1002/mmce.20725 Published online 8 April 2013 in Wiley Online Library (wileyonlinelibrary.com). C 2013 Wiley Periodicals, Inc. V

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Fifty Years of High Frequency Diffraction

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Figure 2 EM field–object interaction. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 1 Radiation pattern of a vertical monopole above a PEC ground (f ¼ 1GHz) [8]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

with double knife-edge obstacles [29] scenarios are used for this purpose. Finally, conclusions are summarized in Section V. II. GEOMETRICAL/UNIFORM THEORY OF DIFFRACTION

The GTD, originated by Keller [1, 2], and extended to UTD by Kouyoumjian and Pathak [3, 4] to remove the discontinuities along the incident and reflection shadow boundaries (ISB, RSB), is an extension of the classical GO, and it overcomes some of the limitations of GO by introducing a diffraction mechanism [5, 6]. At high frequencies, diffraction—like reflection and refraction—is a local phenomenon and it depends on the geometry of the object at the point of diffraction (e.g., edge, vertex, curved surface) and the amplitude, phase, and polarization of the incident field at the point of diffraction. A field is associated with each diffracted ray, and the total field at a point is the sum of all the rays at that point. Some of the diffracted rays enter the shadow regions and account for the field intensity there. The diffracted field, which is determined by a generalization of Fermat’s Principle [2, 3], is initiated at points on the surface of the object that create a discontinuity in the incident GO field (ISB and RSB). The phase of the field on a ray is assumed to be equal to the product of the optical/geometrical length of the ray from some reference point and the wave number of the medium. Appropriate phase jumps must be added as rays pass through caustics. The amplitude is assumed to vary in accordance with the principle of conservation of energy in a narrow tube of rays. The initial value of the field on a diffracted ray is determined from the incident field with the aid of an appropriate diffraction coefficient that is a dyadic for EM fields. This is analogous to the manner reflected fields are

determined using the reflection coefficient. The rays also follow paths that make the optical distance from the source to the observation point an extremum (usually a minimum). This leads to straight-line propagation within homogeneous media and along geodesics (surface extrema) on smooth surfaces. The field intensity also attenuates exponentially as it travels along surface geodesics. The diffraction and attenuation coefficients are usually determined from the asymptotic solutions of the simplest boundary-value problems, which have the same local geometry at the points of diffraction on the object at the points of interest. Geometries of this type are referred to as canonical problems. One of the simplest geometries that will be discussed in this paper is a conducting wedge. The primary objective in using GTD is to resolve each problem to smaller components, each representing a canonical geometry with a known solution. The ultimate solution is a superposition of the contributions from each canonical problem. The contribution of diffraction is illustrated with the example presented in Figure 1. Here, a quarter wavelength, bottom-fed vertical monopole antenna is erected above a finite-size square perfectly electrical conductor (PEC) ground plane. The finite ground plane introduces not only reflections but also significant diffractions as shown in the figure. III. PHYSICAL THOEORY OF DIFFRACTION

PTD is extension of the physical optics (PO) approximation introduced by Kirchhoff for scalar waves [18] and by Macdonald for EM waves [19]. Here, we expose PO and PTD for scalar/acoustic waves and utilize acoustic terminology. A scattering object is considered perfectly reflecting with soft (Dirichlet) or hard (Neumann) boundary conditions. According to PO, the total field on the illuminated side of scattering object is assumed to be the same as on the infinite tangential plane (see Fig. 2). The object (in Fig. 2) is convex; its radii of curvature are large compared to the wavelength. On the shadow side, the total field is zero. This approximation is a powerful tool widely used in acoustics, optics, and EMs. In particular, it is extensively applied in the design of

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

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Balanis, Sevgi, and Ufimtsev

Figure 3

Directivity patterns of f ð1Þ , gð1Þ [9–11]. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

reflector antennas and in calculations of scattering from objects. The PO approximation describes properly both all reflected rays away from the GO boundaries and the diffracted field near these boundaries, as well as near foci and caustics. The PO drawbacks are the following. First, it is not self-consistent: When the observation point approaches the scattering surface, the PO integrals do not reproduce the initial values for the surface field. Second, the PO field does not satisfy rigorously the boundary conditions and the reciprocity principle. The reason for these shortcomings is the GO approximation for the surface field, which does not include its diffraction components. The PO shortcomings are overcome by PTD, which improves PO by taking into account the diffracted surface field. According to PTD, the field scattered by an object is considered as the radiation generated by induced surface sources/currents. These sources are separated into two components j ¼ jð0Þ þ jð1Þ

(1)

The first one,jð0Þ , is the PO component and it is determined by GO and represents the ray component. The field radiated by this component is the ordinary PO field. The second term, jð1Þ , is the diffracted component and it is caused by any deviation of the object surface from a plane tangential to the object. In the case of large convex objects with edges, this component in the vicinity of edges can be asymptotically approximated by the surface sources induced on a tangential wedge. The field radiated by this component is referred to as the fringe field. Its directivity patterns are shown in Figure 3 in the case of diffraction of a plane wave by an infinite wedge.

Figure 4 Scattering from a trihedral soft cylinder. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Functions f ð1Þ and gð1Þ represent the directivity patterns of fringe waves scattered from soft and hard wedges, respectively. PTD actually consists of asymptotic calculation of fringe waves [11, 12]. A typical soft trihedral cylinder is displayed in Figure 4. Backscattering from this trihedral soft cylinder is presented in Figure 5 and shows significant influence of fringe waves on the sidelobes. The modern form of PTD is based on the concept of elementary edge waves (EEW). A total edge wave scattered from convex-curved edge L can be considered as a linear combination of EEW (see Fig. 6). uðPÞ ¼

Z

duðfÞ

(2)

L

Here, du(f) is the elementary edge wave. It is assumed that the curvature radius of the edge L is large in terms of the wavelength (q >> k) and it can slowly change along the edge.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 23, No. 4, July 2013


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Figure 7 Cone diffraction. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 5 Backscattering from a soft trihedral cylinder [11].

Figure 8 Wedge scattering problem. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 6 Convex-curved edge scattering. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

In the directions of the diffraction cone (see Fig. 7) and underexpðixtÞtime-dependence, the PTD fringe EEW have the following asymptotic form [11]: ð1Þ

dus ð1Þ duh

df ¼ u ðfÞ 2p inc

f ð1Þ ð/; /0 ; nÞ gð1Þ ð/; /0 ; nÞ

eikR : R

(3)

Functions f ð1Þ and gð1Þ relate to the soft and hard boundary conditions, respectively, and they are represented by simple trigonometric functions f ð1Þ ¼ f f ð0Þ ; gð1Þ ¼ g gð0Þ

(4)

reason why GTD fails in the directions of incident and reflected rays (/ ¼ p þ /0 , / ¼ p /0 ); functions f, g are infinite in these directions. In contrast, the PTD fringe functions f ð1Þ ; gð1Þ are finite along those directions because the singularities of functions f, g are compensated by the singularities of the functions f ð0Þ ; gð0Þ related to the PO waves. This property of the functions f ð1Þ ; gð1Þ is the basic advantage of PTD compared to GTD and other techniques. The following examples demonstrate the PTD technique. Application of the stationary phase method to the integral (2) for the fringe waves leads to their following ray form [11]: p

1 ei 4 uts;h ¼ uinc ðfst Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sin c pffiffiffiffiffiffiffiffi 0 2p k R 1 þ Rq

f ð1Þ ð/; /0 ; nÞ ikR e gð1Þ ð/; /0 ; nÞ (7)

where (n ¼ a=p); f ð/; /0 ; nÞ sinðp=nÞ ¼ n gð/; /0 ; nÞ 8 9 < = 1 1 6 :cos p cos //0 0 ; cos pn cos /þ/ n n n sinð/0 Þ ; gð0Þ ð/; /0 Þ cosð/Þ þ cosð/0 Þ sinð/0 Þ ¼ : cosð/Þ þ cosð/0 Þ

(5)

Following the same procedure, one obtains the ray asymptotics for the field (2) generated by the total surface sources (1): p

1 ei 4 inc ffi utot s;h ¼ u ðfst Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin c pffiffiffiffiffiffiffiffi 0 2p k R 1 þ Rq

f ð0Þ ð/; /0 Þ ¼

f ð/; /0 ; nÞ ikR e gð/; /0 ; nÞ (8)

(6)

The angles/; /0 and a are shown in Figure 8. Functions f, g relate to the waves generated by the total scattering sources (1). These functions, (5) and (6), determine the edge-diffracted rays in GTD, and this is the

These are exactly the equations postulated in GTD. In this way, the PTD concept of EEW validates GTD. Notice that the functions f and g or f ð1Þ and gð1Þ , representing the directivity patterns of EEW (3), can be considered as the differential diffraction coefficients in contrast to the factors:


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Figure 9 A finite cone. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 11

RCS vs. vertex angle of the cone [11].

Figure 12 Multiple edge-waves. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 10

where uð1Þ is of the same order in magnitude as the GOreflected rays. Its amplitude does not depend on the frequency due to focusing. Its RCS vs. length and RCS vs. vertex angle are displayed in Figures 10 and 11, respectively. A big difference between SBC and HBC data is indicated; it is about 40 dB for narrow cones.

RCS vs. length of the cone [11].

f ð1Þ ð/; /0 ; nÞ ð1Þ g ð/; /0 ; nÞ p ei 4 f ð/; /0 ; nÞ pffiffiffiffiffiffiffiffi sin c0 2pk gð/; /0 ; nÞ p

ei 4 pffiffiffiffiffiffiffiffi sin c0 2pk

(9) (10)

which can be interpreted as the integral diffraction coefficients. Thus, the above derivation of the ray asymptotics sheds insight on the nature of diffraction coefficients introduced in GTD. The following examples illustrate the PTD technique. A. Backscattering from Cones 2 The normalized scattering cross-section rnorm s;h ¼ rs;h pa of the cone shown in Figure 9 is calculated as a function of two variables; the length l and the vertex angle x. Contribution from non-uniform/fringe edge currents are [11]: uð1Þ s ¼ u0

ð1Þ

uh ¼ u0

1 eikR 2p R 1 eikR 2p R

Z

2p

f ð1Þ ðwÞadw0 ¼ u0 af ð1Þ ð/; /0 ; nÞ

0

Z

2p 0

gð1Þ ðwÞadw0 ¼ u0 agð1Þ ð/; /0 ; nÞ

eikR R (11) eikR R (12)

B. Multiple Edge Waves Diffracted at a Hard Disk The scenario of this problem is displayed in Figure 12. This problem is more complicated because the wave traveling along one face of the disk generates (due to diffraction at the edge) higher-order waves not only on the same face but also on the other face. The scattered field is investigated at the points P on the z-axis, which is the focal line of the edge-diffracted waves. Taking into account all multiple edge waves, one can show that the PTD scattering cross-section of the disk can be represented by [11] ( rPTD h

2

¼ 2pa

1 2X sin½mð2ka p=4Þ 1þ ð1Þm ka m¼1 2m1 ðpkaÞm=2

) (13)

Comparison of rh with the exact asymptotic solution [30], which contains the first six terms for the total cross-section, confirms that PTD correctly predicts the first asymptotic term for every multiple edge wave; this is the unique result. Notice also some alternative or similar to PTD approaches where the EEW were studied and presented in different forms: Bateman (1955), Rubinowicz (1965), Wolf (1967), Mitzner (1974), Michaeli (1986), Breinberg (1992), Tiberio and Maci (1994), Johanson (1996), Syed



Figure 13 WedgeGUI diffraction tool presented in Ref. 27. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Figure 15 Flat-earth and double knife-edge problem. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 14 Total and diffracted fields around the wedge calculated via exact mode summation, UTD and PTD methods. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

and Volakis (1996). Exact references to these and some other related techniques are given in Ref. 11. IV. TEST PROBLEMS AND GTD/UTD-PTD COMPARISONS

The GTD/UTD and PTD models are compared on two canonical diffraction problems. A. Wedge Scattering Problem The wedge scattering problem illustrated in Figure 8 presents a canonical problem where all HFA techniques can be investigated and comparisons can be performed. A series of papers published recently [25–28] both review all the HFA techniques and present an attractive virtual diffraction tool that can be used for teaching/training as well as research. Figures 13 and 14 belong to this virtual tool. One of the main interests of diffraction by wedges is that engineers and scientists have investigated how the shape and material properties of complex structures affect the backscattering characteristics. The attraction in this area is primarily aimed toward designs of low-profile (stealth) technology by using appropriate shaping along with lossy or coated materials to reduce the radar visibility, as repre-

Figure 16 3D field maps for edges at 3 and 5 km ranges (Line source at 100 m height, HBC, f ¼ 300 MHz) [29]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

sented by radar cross-section (RCS), of complex radar targets, such as aircraft, spacecraft, and missiles [8–10, 31]. B. Flat-Earth and Double Knife-Edge Problem The scenario of this problem is shown Figure 15 showing possible rays at different observation points (A, B, and C). This is a complicated problem which includes extraction of GO eigenrays (source-emanating rays which reach/pass the receiver) plus calculation of reflected and diffracted fields for the specified source and observer locations [29]. Typical simulations are performed with GO plus both GTD/UTD and PTD with 150 m- and 175 m-high double knife-edges at 3 km and 5 km ranges, respectively. The line source is 100 m above the ground with HBC. The frequency is 300 MHz. Results are illustrated in Figures 16 and 17.


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• Can be utilized to develop efficient hybrid techniques (in combination with numerical methods) for the calculation of complex diffraction problems. • Is flexible for various modifications and extensions.

REFERENCES

Figure 17 Propagation factor vs. range (Line source at 100 m height, HBC, f ¼ 300 MHz) [29]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

V. CONCLUSIONS

HFA methods have long been used in diffraction modeling. The GO can model reflections and refractions but fails to account for the field intensity in shadow regions. GTD describes diffraction everywhere except at and near incidence and reflection shadow transitions; UTD removes the discontinuities along these shadow boundaries. However, GO/GTD/UTD fails near caustics. The PTD supplements PO to provide corrections that are due to diffractions at edges of conducting surfaces. Ufimtsev suggested the existence of nonuniform (fringe) edge currents in addition to the uniform PO surface currents. Some characteristics of each method follows: 1. GTD/UTD: • Is simple to apply. • Can be used to solve complicated problems that do not have exact solutions. • Provides physical insight into the radiation and scattering mechanisms from the various parts of the structure. • Yields accurate results that compare quite well with measurement and other methods. • Can be combined with other techniques, such as the method of moments (MoM), to form a hybrid method. 2. PO/PTD: • Provides correctly only the first asymptotic terms for main components of the scattered field in 3D problems. • Is the maximum of what modern asymptotic theories can do for complex objects and it is a main need in engineering. High-order terms are usually not necessary, but they may be of interest to test approximate techniques. • Allows constructing relatively simple solutions of various practical problems. • Provides uniform asymptotics for the scattered field which are valid both in the ray regions and in the vicinity of foci and caustics. • Clarifies the physical structure of the scattered field. • Establishes the diffraction limit of reduction of scattering by absorbing coatings.

1. J.B. Keller, Diffraction by an aperture, J Appl Phys 28 (1957), 426–444. 2. J.B. Keller, Geometrical theory of diffraction, J Opt Soc Am 52 (1962), 116–130. 3. R.G. Kouyoumjian, Asymptotic high frequency methods, Proc IEEE 53 (1965), 864–876. 4. R.G. Kouyoumjian and P.H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc IEEE 62 (1974), 1448–1461. 5. R.C. Hansen, Ed, Geometric theory of diffraction, IEEE Press, New York, 1981. 6. G.L. James, Geometrical theory of diffraction for electromagnetic waves, 3rd ed Revised, Peregrinus, London, 1986. 7. D.A. McNamara, C.W.I. Pistorius, and J.A.G. Malherbe, Introduction to the uniform geometrical theory of diffraction, Artech House, Boston–London, 1990. 8. C.A. Balanis, Advanced engineering electromagnetics, 2nd ed., Wiley, NJ, 2012. 9. P. Ya. Ufimtsev, Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies, Sov Phys—Tech Phys (1957), 1708–1718. 10. P. Ya. Ufimtsev, Method of edge waves in the physical theory of diffraction, Soviet Radio, Moscow, 1962; translated into English by the US Air Force, Foreign Technology Division (National Air Intelligence Center), Wright-Patterson Air Force Base, OH, 1971, Technical Report AD 733203, Defense Technical Information Center of USA, Cameron Station, Alexandria, VA, 22304-6145, USA. 11. P. Ya. Ufimtsev, Fundamentals of the physical theory of diffraction, Wiley, Hoboken, New Jersey, 2007. 12. P. Ya. Ufimtsev, Theory of edge diffraction in electromagnetics: Origination and validation of the physical theory of diffraction, SciTech Publishing Inc., Raleigh, NC, 2009. 13. M.W. Browne, Two rival designers led the way to stealthy warplanes, New York Times, Science Times Section, May 14, 1991. 14. M.W. Browne, Lockheed credits soviet theory in design of F-117, Aviation Week Space Technology, December 1991, p.27. 15. B. Rich and L. Janos, Skunk Works, Little Brown, Boston, 1994. 16. P. Ya. Ufimtsev, Theory of acoustical edge waves, J Acoust Soc Am 86 (1989), 463–474. 17. P. Ya. Ufimtsev, Elementary edge waves and the physical theory of diffraction, Electromagnetics 11 (1991), 125–160. 18. G.R.Kirchhoff, Zur Theorie der Lichtstrahlen, Annalen der Physik 18 (1883), 663–695. 19. H.M. Macdonald, The effect produced by an obstacle on a train of electric waves, Phil Trans R Soc London Ser A 212 (1912), 299–337. 20. C.A. Balanis and L. Peters Jr, Analysis of aperture radiation from an axially slotted circular conducting cylinder using geometrical theory of diffraction, IEEE Trans Antennas Propagat 17 (1969), 93–97.


Fifty Years of High Frequency Diffraction 21. C.A. Balanis and L. Peters Jr, Equatorial plane pattern of an axial-TEM slot on a finite size ground plane, IEEE Trans Antennas Propagat 17 (1969), 351–353. 22. C.A. Balanis, Radiation characteristics of current elements near a finite length cylinder, IEEE Trans Antennas Propagat 18 (1970), 352–359. 23. C.A. Balanis, Analysis of an array of line sources above a finite ground plane, IEEE Trans Antennas Propagat 19 (1971), 181–185. 24. F.M. Grimaldi, Physico-Mathesis de Lumine, Coloribus at Iride, Bonomiae, 1665. 25. M.A. Uslu and L. Sevgi, Matlab-based virtual wedge scattering tool for the comparison of high frequency asymptotics and FDTD method, ACES, Int J Appl Comput Electromagnet 27 (2012), 697–705. 26. F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, Electromagnetic wave scattering from a wedge with perfectly reflecting boundaries: Analysis of asymptotic techniques, IEEE Antennas Propagat Mag 53 (2011), 232–253.

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27. F. Hacivelioglu, M.A. Uslu, and L. Sevgi, A Matlab-based simulator for the electromagnetic wave scattering from a wedge with perfectly reflecting boundaries, IEEE Antennas Propagat Mag 53 (2011), 234–243. 28. G. Çakir, L. Sevgi, and P. Ya. Ufimtsev, FDTD modeling of electromagnetic wave scattering from a wedge with perfectly reflecting boundaries: Comparisons against analytical models and calibration, IEEE Trans Antennas Propagat 60 (2012), 3336–3342. 29. O. Ozgun and L. Sevgi, Comparative study of analytical and numerical techniques in modeling electromagnetic scattering from single and double knife-edge in 2D ground wave propagation problems, ACES Int J Appl Comput Electromagnet 27 (2012), 376–388. 30. H.H. Witte and K. Westphal, ‘‘Hochfrequente Schallbeugung an der Kreisblende: numerische Ergebnisse,’’ Annalen der Physik, Folge 7, Band 25, Heft 4, (1970), pp.375–382. 31. P. Ya Ufimtsev, Comments on diffraction principles and limitations of RCS reduction techniques, Proc IEEE 84 (1996), 1830–1851.

BIOGRAPHIES

Constantine A. Balanis received the BSEE from Virginia Tech, Blacksburg, VA, in 1964, the MEE from the University of Virginia, Charlottesville, VA, in 1966, and the PhD in Electrical Engineering from Ohio State University, Columbus, OH, in l969. From 1964 to 1970 he was with NASA Langley Research Center, Hampton VA, and from 1970 to 1983 he was with the Department of Electrical Engineering, West Virginia University, Morgantown, WV. Since 1983 he has been with the School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, where he is Regents’ Professor. His research interests are in computational electromagnetics, flexible antennas and high impedance surfaces, smart antennas, and multipath propagation. He received in 2004 a Honorary Doctorate from the Aristotle University of Thessaloniki, the 2012 Distinguished Achievement Award of the IEEE Antennas and Propagation Society, the 2005 Chen-To Tai Distinguished Educator Award of the IEEE Antennas and Propagation Society, the 2000 IEEE Millennium Award, the 1996 Graduate Mentor Award of Arizona State University, the 1992 Special Professionalism Award of the IEEE Phoenix Section, the 1989 Individual Achievement Award of the IEEE Region 6, and the 1987–1988 Graduate Teaching Excellence Award, School of Engineering, Arizona State University. Dr. Balanis is a Life Fellow of the IEEE. He has served as Associate Editor of the IEEE Transactions on Antennas and Propagation (1974–1977) and the IEEE Transactions on Geoscience and Remote Sensing (1981–1984), as Editor of the Newsletter for the IEEE Geoscience and Remote Sensing Society (1982–1983), as Second Vice-President (1984) and member of the Administrative Committee (1984–1985) of the IEEE Geoscience and

Remote Sensing Society, and as Distinguished Lecturer (2003–2005), Chairman of the Distinguished Lecturer Program (1988–1991) and member of the AdCom (1992– 1995, 1997–1999) of the IEEE Antennas and Propagation Society. He is the author of Antenna Theory: Analysis and Design (Wiley, 2005, 1997, 1982), Advanced Engineering Electromagnetics (Wiley, 2012, 1989) and Introduction to Smart Antennas (Morgan and Claypool, 2007), and editor of Modern Antenna Handbook (Wiley, 2008) and for the Morgan & Claypool Publishers, series on Antennas and Propagation series, and series on Computational Electromagnetics. Levent Sevgi received the PhD degree from Istanbul Technical University, Istanbul, Turkey, and Polytechnic Institute of New York University, Brooklyn, in 1990. Prof. Leo Felsen was his advisor. He was with Istanbul Technical University from 1991 to 1998; TUBITAK-MRC, Information Technologies Research Institute, Gebze/ Kocaeli, Turkey, from 1999 to 2000; Weber Research Institute/Polytechnic University of New York, from 1988 to 1990; Scientific Research Group of Raytheon Systems, Canada, from 1998 to 1999; and the Center for Defense Studies, ITUVSAM, from 1993 to 1998 and from 2000 to 2002. Since 2001, he has been with Dogus University, Istanbul. Since September 2012, he has been with the University of Massachusetts, Lowell (UML) on his sabbatical leave. He has been involved with complex electromagnetic problems and systems for more than two decades. Prof. Sevgi is an IEEE Fellow, the Writer/Editor of the ‘‘Testing ourselves’’ Column in the IEEE Antennas and Propagation Magazine and a member of the IEEE Antennas and Propagation Society Education Committee.


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Pyotr Ya. Ufimtsev received his PhD degree in EE from the Central Research Radio Engineering Institute of the Defense Ministry (Russia, Moscow. 1959) and PhD degree in theoretical and mathematical physics from St. Petersburg University (Russia, St. Petersburg, 1970). He was affiliated with a number of research and academic institutions, including the Institute of Radio Engineering and Electronics of the Academy of Sciences (Moscow), the Moscow Aviation Institute, Northrop-Grumman Corp. (California, USA), the University of California at Los Angeles and Irvine. As a Guest Professor he taught courses on the diffraction theory at Singapore National University (1993), Air Force Institute of Technology (1994, 1999, Dayton, Ohio, USA), Moscow State University (2007), Sienna University (2008, Italy), and Dogus University (2011, 2012 Istanbul/Turkey). Dr. Ufimtsev is working on electromagnetic and acoustic wave theory and its applications. Among his fundamental results are the theory of scattering from black objects, the Physical Theory of Diffraction, and the discovery of new physical phenomena related to surface waves in absorbing materials. His Physical Theory of Diffraction is used worldwide in the design of microwave antennas and in calculations of radar

cross-section of scattering objects. In particular, it was used in the design of American aircraft and ships invisible to radar (stealth objects). This theory played a critical role in the design of American stealth-aircraft F-117 and B-2. He is the author of the Method of Edge Waves in the Physical Theory of Diffraction (Sovetskoe Radio, Moscow, 1962) translated into English by the US Air Force (Foreign Technology Division, Wright-Patterson AFB, OH, 1971), Theory of Edge Diffraction in Electromagnetics (Tech Science Press, Encino, California, 2003) translated into Russian (BINOM, Moscow, 2007), Theory of Edge Diffraction in Electromagnetics: Origination and Validation of the Physical Theory of Diffraction, Revised Printing (SciTech Publishing, Inc. Raleigh, NC, 2009), Fundamentals of the Physical Theory of Diffraction (Wiley & Sons. Inc., New Jersey. 2007) translated into Russian (BINOM, Moscow, 2009), and Theory of Edge Diffraction in Electromagnetics: Introduction into the Physical Theory of Diffraction, 2nd edition (in Russian, BINOM, Moscow, 2012). Among Dr. Ufimtsev’s many honors and awards are the USSR State Prize (Moscow, 1990) and the Leroy Randle Grumman Medal (New York, 1991). He was elected a Member of the Electromagnetics Academy (MIT, 1989), Life Fellow of IEEE (2012), and Associate Fellow of AIAA (1992). He is listed in Who’s Who in the World, Who’s Who in America, Who’s Who in Science and Engineering.
