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Ultra-Near Fields of Bodies ofTranslation With Discontinuities:ω-κ RepresentationsL.N. Medgyesi-Mitschang a , P.G. Moore b , D.L. Smith c &S.G. Lambrakos da Naval Research Laboratory, Code 5050 4555 OverlookAve., SW, Washington, D.C. 20375-5320, USAb Naval Research Laboratory, Code 5050 4555 OverlookAve., SW, Washington, D.C. 20375-5320, USAc Naval Research Laboratory, Code 5050 4555 OverlookAve., SW, Washington, D.C. 20375-5320, USAd Naval Research Laboratory, Code 5050 4555 OverlookAve., SW, Washington, D.C. 20375-5320, USAPublished online: 03 Apr 2012.

To cite this article: L.N. Medgyesi-Mitschang , P.G. Moore , D.L. Smith & S.G.Lambrakos (2001) Ultra-Near Fields of Bodies of Translation With Discontinuities: ω-κRepresentations, Journal of Electromagnetic Waves and Applications, 15:12, 1701-1723,DOI: 10.1163/156939301X00238

To link to this article: http://dx.doi.org/10.1163/156939301X00238

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ULTRA-NEAR FIELDS OF BODIES OF TRANSLATION WITH DISCONTINUITIES: ω-κ REPRESENTATIONS

L. N. Medgyesi-Mitschang, P. G. Moore, D. L. Smith, and S. G. Lambrakos

Naval Research Laboratory, Code 5050 4555 Overlook Ave., SW, Washington, D.C. 20375-5320, USA

Abstract-Ultra-near fields scattered from bodies of translation

(BOTs) with surface and gap discontinuities are analyzed using frequency wave vector (w-k) representations. The scattering surfaces are assumed to be perfectly electrically conducting (p.e.c.) and can be electrically large. The theoretical formulation is based on first

principles. The electric field integral equation (EFIE) for the BOT scatterer is solved by the method of moments (MM). Next the scattered fields in the operator form of the EFIE are Fourier transformed over the spatial domain. The resulting ω-κ representation delineates the

propagated and evanescent waves near the scattering surface. The

analytical results are compared with experimental data.

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1. INTRODUCTION

Following the seminal work of Brillouin and subsequent investigators, w-k representations have been used to study wave phenomena in var- ious settings such as periodic structures, plasma sheaths, and an- tenna applications to name a few. [1-3] Recent investigations in near- field acoustic holography [4] have stimulated a re-examination of these

representations in the electromagnetics setting [5]. As in acoustics, cv-k

representations provide deeper insights into the scattering physics by delineating the propagating and evanescent fields near an object.

The mapping of near fields into the w-x domain can be carried out in two ways. In one approach, designated here as the indirect method, the near fields in the vicinity of the target are obtained numerically or

experimentally at a finite set of spatial points. These are then FFT to the w-x domain. Since the near field sample is finite, the truncation of the spatial data along the propagation direction can lead to significant errors in the w-x results unless a very large spatial sample is calculated

or measured. In addition to the errors incurred, the computational resources (time and memory) required with the indirect approach in numerical simulations are considerable.

An alternate, direct approach is possible. The currents on the

target are obtained using a numerical solution procedure such as the MM technique. Next, the scattered electric field, expressed in

operator form, is Fourier transformed over the spatial domain along the desired propagation direction. The resulting transformed field

maps directly into the space delineating the propagating and evanescent regions in the proximity of the target. The computational resource requirements with the direct method are significantly less when compared with the earlier indirect method with no loss of

accuracy. We illustrate this approach for an arbitrary 2-D geometry, often referred to as a body of translation (BOT). In this discussion, the BOT is restricted to be p.e.c.

2. MM FORMULATION FOR BOTS

Representative formulations for BOTs based on the MM technique are given in [6-12]. The present formulation is adapted from [9]. Referring to Fig. la, the p.e.c. BOT surface is formed by translating the generating curve C(t) along the z-axis. The curve is parametrized by t and may have an arbitrary number of discontinuities as shown. The curve can be partially convex or concave, and either open or closed.

Assuming a time harmonic incident plane wave Ei (ejúJt is

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Figure 1. Generic p.e.c. body of translation. (a) Coordinate

geometry. (b) Subdomain representation of the BOT generating curve used in the MM implementation.

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assumed), the total electric field Et outside the body is:

where the scattered field Es is given by

Z/[ ] is the integro-differential operator over the unknown surface currents J(t) on the BOT. The integral is interpreted in the principal value sense. The 2-D free space Green's function kernel -4? is given by

where H¿2) denotes the zeroth order Hankel function of the second

kinds = 27r/A; R = ((z - x')2 + (y - and 1-£0 and eo are the permeability and permittivity of free space, respectively. Noting that the tangential component of the total electric field on a perfect conductor vanishes, i.e.,

Equation (2) is the well known the electric field integral equation

(EFIE).t This equation can be solved for the unknown currents. If we

apply the MM technique, the BOT generating curve C(t) is discretized into N sub-segments Ci parallel to the z-axis. The unknown surface currents on these sub-segments are expanded as

with a = t for TE polarization and a = z for TM polarization and where _ r/1 I - ,-,., v I- 1

ict is the tangent vector on C(t), i.e., ut = Üx cos v' + Üy sin v' and v' is the angle subtended by the BOT generating curve and the x-axis. Here the TE and TM polarization denote the case where the incident

t The present discussion implicitly excludes cases where internal resonances occur. In such cases a combined (CFIE) formulation is needed as discussed in Ref. 12, p. 97ff. The present approach can be generalized to the CFIE formulation.

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electric field is perpendicular or parallel to the z-axis, respectively. For

simplicity in the present formulation we let the discretized segments Cj be straight lines. The basis functions fj(t) are chosen to be overlapping triangle functions spanning Cj (see Fig. 1b). They are defined as

A non-vanishing half-triangle at an edge of a gap discontinuity is chosen for f) (t) . (For alternate representations of currents on a BOT see [10] and [11].)

Following the usual MM (Galerkin) procedure, the current

expansions in Eq. (3) are substituted into Eq. (2) to form the inner

products of the integral operator L[ ] with the testing functions The asterisk denotes the conjugate operation. This yields

a system of algebraic equations for the unknown current expansion coefficients h :

and where Wt and lj are the i-th testing and j-th current basis

functions on C(t), respectively. The term f7i' is the incident electric field tested on the BOT segment spanning the i-th triangle function, i.e.,

For ease of reference and completeness, the defining expressions for the Z's and V's in Eq. (4) adapted from [9], are summarized in the

Appendix A. The unknown current coefficients in Eq. (4) are obtained

using standard matrix solution methods. ,

3. w-x TRANSFORMATION

Knowing the MM derived currents, we carry out the Fourier transform

directly on the scattered electric field given by the operator in Eq. (la). Formally, the transformed electric field over x for the

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propagation constant kz is given by:

In the foregoing the implicit assumption is made that the interchange of the order of integrations (i.e., over x and t) does not affect the value of the result. This is justified by the convergence of the reordered infinite integral over x. The transform in Eq. (5) can be carried out

analytically as detailed in Appendix B. The transformed electric field in Eq. (5) can be evaluated numeri-

cally using the Galerkin discretization used earlier on the L [ ] operator in computing the impedance elements in Eq. (4). Using the current coefficients obtained from the solution of Eq. (4) we obtain the desired

expressions for the transformed fields for the respective polarizations as:

- J Note the summation of the integrals in the above expressions is over the discretized BOT segments Cj, spanned by the j-th triangle function

(t). The terms Aj and are the discretized forms of A and B in

Eqs. (B3) and (B5), respectively, where x' and the

angle v' in the sine and cosine terms. The numerical integrations in Eqs. (6) and (7) are carried out using standard techniques, i.e., the

Simpson method or Gaussian quadrature.

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4. RESULTS

The foregoing formulation was numerically implemented by modifying the CARLOS code. This code is a general purpose electromagnetics code based on the MM technique for solving complex 2-D and 3-D

scattering problems [13, 14]. In the computed results presented here, we restrict ourselves to the Galerkin expansion option in the code using overlapping triangle functions. This provides the needed flexibility to handle efficiently a broad range of surface discontinuities and gaps. These can be electrically large or small compared to a wavelength. We use a discretization density of ten triangle functions per wavelength in the computations.

The electric ultra-near fields and w-x maps were computed for a number of BOT geometries including strips and dihedrals with and without gaps. As a subcase of the latter, dihedrals with sharp (right angle) and rounded corners were also considered. To illustrate the

generality of the present formulation, the w-x maps were also generated for finite arrays of electrically narrow strips.

The numerical results were compared with ultra-near field measurements made in the Naval Research Laboratory's (NRL) Ultra- Near Field Facility [15]. The principal elements of the experimental set-up are summarized in Figure 2. The facility consists of a very stable 152.4 cm square (5 x 5 feet) planar scanner specially adapted by ORBIT/FR for this application. Using a heavy duty rotating stage,

cylindrical scans up to 106.7 cm (3.5 feetl' in radius can be generated. For sampling the near and ultra-near fields, a variety of probes were used including electrically small monopoles and dipoles. An X-band version of the latter with a small etched balun proved most effective for the present investigations. This probe has a bandwidth approaching 3 GHz and a polarization selectivity of 20 dB. The balanced pick- up of the dipole minimized deleterious ground loop currents in the measurements. The probe data was fed into a Hewlett Packard network

analyzer. Specialized processing software provided for automated data reduction.

For the experimental phase of this investigation a number of

targets were measured in the NRL facility. The ultra-near fields were measured bistatically with the object under test illuminated by a

Singer broadband horn. The position of the horn was adjusted to minimize interference with the probe. Generally, planar scans were made 7.62 to 15.24cm (3 to 6 inches)beyond the target surface to

negate truncation effects when the measured near field data was FFT to the w-x domain.

I The English units are given for reference in the experimental phase of the work.

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Figure 2. Diagram of the principal elements of the NRL Ultra-Near Field Facility.

The calculated ultra-near electric fields scattered from an infinite

strip are shown in Fig. 3. Two cases are considered: a strip with and without a bisecting gap. The strip lies on the ?-axis having a width (w) of 45.72 cm (18 inches). The strip is illuminated bistatically for 0 = 167° measured from the x-axis. This angle approximates the Peter's lobe angle at 10 GHz where the maximum traveling wave scattering occurs for slender electrically extended conducting bodies. [16] The ultra-near fields for TE and TM polarization are computed at y = 0.254cm (0.1 inches). For TE

polarization, traveling waves are induced on the strip. As expected near grazing, the y component of the field is much larger than the x

component. The resultant fields near the surface mirror the well known

high trailing edge currents for this polarization. A gap introduces a strong discontinuity in the ultra-near field magnitudes. With the

gap a complex interference pattern is produced by two traveling waves associated with each segment of the bisected strip. For TM

polarization, no traveling waves are generated and the discontinuity in the field magnitudes is less pronounced than before.

Figure 4 depicts the w-x maps corresponding to the cases given in Fig. 3. It is customary to plot these maps in terms of frequency f

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Figure 3. Computed ultra-near field magnitudes (V/m) from a p.e.c. strip with and without gaps illuminated at = 167°. The strip width w = 45.72 cm. The gap width (g) is referenced to 10 GHz. The fields are sampled along y = 0.254 cm (0.1 in.) from the strip surface.

versus the propagation constant We follow this convention here. In these plots the propagating region is defined as the locus between two lines having slopes of +c/27r and -c/27r where c is the speed of light. Often these are denoted as the "light" lines. The region outside these is the non-propagating or evanescent region. As can be seen in Fig. 4, a V-shaped feature appears prominently in both polarizations. Taking the case of the y component of the transformed electric field, the line

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Figure 4. Computed w-x representations for a p.e.c. strip with and without gaps, illuminated at = 167°. The gap widths are referenced to 10 GHz. The strip width and the field sampling are as in Figure 3. The intensity scale is in 20 loglo of the magnitude of the transformed scattered fields; likewise in the other figures.

sloping to the right has a slope of c/[27r cos(13°)] = 1.03c/27r. It is associated with the forward scattered and reflected transformed field intensities along a surface parallel to the strip, i.e., at y = 0.254 cm. Since these are approximately coincident with the "light" line, these fields are propagating along the positive x direction with a velocity approaching c. The line sloping to the left is coincident with the light line with slope -

c/27r. These are associated with (backscattered)

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Figure 5. Measured w-x representations for a p.e.c. plate without a

gap, illuminated at = 167°. The width of the plate corresponds to that of the strip in Fig. 3. The field sampling is as in Figure 3. The intensity scale for the measured fields are those of the total field; likewise for the measured results in the other figures.

waves propagating in the negative 2; direction. These fields are much weaker. As the strip width w - oo, these fields vanish altogether. The gap in the strip suppresses the y component of secondary fields within the propagating region. As noted before at grazing illumination, the x component of the electric field and its transform are very small. For TM polarization evanescent fields are present. They appear to be insensitive to the presence of gaps.

The computed results for the strip were compared with measurements of the ultra-near fields for a rectangular aluminum plate. Its dimensions were 22.86 x 45.72 cm (9 x 18 in.) with a thickness of 0.159cm (1/16in.). The edges were unleveled. Thinner plates were also measured showing that the data was relatively insensitive to these effects. To correlate with the calculated strip results, the plate was centered in the z-z plane with the longer edge aligned parallel to the z-axis.

Figure 5 depicts the measured w-x results for the bistatically illuminated plate. These results were obtained by sampling the total electric field along a line y = 0.254 cm. above the plate. As in the calculated results, the forward scattered electric field intensities are

approximately coincident with the light line. The variation in the forward scattered field intensity is an experimental artifact. It arises

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from quasi-resonant probe interactions with the plate. The line in the evanescent region with a slope of approximately twice the light line, is also a probe induced effect. As was noted with the computed results in Figure 4, the backscattered fields are much weaker. Consequently, they were difficult to resolve experimentally. This is also true of the fine details seen in the computed secondary propagating and evanescent fields in Figure 4.

Figure 6 compares the computed w-x results for a strip without a gap, when illuminated at broadside and at oblique angles. For TE

polarization, the dominant electric field intensities for the x component occur at kz = 0, indicative of broadside backscattering. For the y component, the finite width of the strip also results in weak forward and backscattered fields even at broadside. Finally, we note that for

oi = 135°, the dominant forward scattered field intensity occurs at

kx = k cos(7r - 135°) in conformity with Snell's law of reflection. The finite width of the strip causes a spreading of the intensities over a number of kx values. This corresponds to edge diffraction from the

strip. For TM polarization, the fields are also dominant for kx = 0 at broadside. In addition, weak fields propagating along the strip are

present at broadside. In this polarization, evanescent fields appear at all illumination angles.

Figure 7 compares the computed w-x maps for a dihedral where one of its sides has a bisecting gap. Two cases are shown. In one, the

gap is in the illuminated region; in the other it is shadowed. For the latter case in TE polarization, there appear bands of cancellation of the x components of the forward and backscattered fields. These bands are approximately centered at 7 and 15 GHz. The dominant scattering occurs for kz s5 k. As in the case of the strip for TM polarization, evanescent fields are present in the forward and backscatter directions for both sets of illumination.

Figure 8 compares the computed w-x maps for two right angle dihedrals. One has a squared corner; the other has a rounded corner. In both cases the gaps are in the shadowed region. In both cases there appears to be cancellation of the ultra-near fields. This is most pronounced for the x component of fields. Furthermore in TM

polarization, these is substantial difference in the evanescent fields in the backscatter direction.

Figure 9 shows the measured and computed field intensities for a finite 14 element array of narrow thin strips aligned parallel to the z-axis. In the experimental implementation, we used an array of thin aluminum rods in place of narrow strips. The rods were 30.48 cm

(12in.) long with a diameter of O.159cm (1/8in.). The centers of the elements were spaced 2.29 cm (0.9 in.) apart. The array was illuminated

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r igure o. comparison 01 compuieu w-x represemaiions ror a p.e.c. strip without a gap at broadside and oblique illumination. The strip width and the field sampling are as in Figure 3.

in TM polarization with §1 = 145°. The total electric fields were

sampled at y = 0.254 cm above the array. The resulting measured w-x plots in Fig. 9a show that the forward

propagating fields are dominant. Their intensity forms a line inclined to the right with a slope equal to c/27r cos(35°) = 1.22c/27r. This line is difficult to resolve in the experimental data from that of the light line. In the calculated results (Fig. 9b), these two lines are distinct. In the

propagating region, the parallel lines of lower intensity are the Bragg

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Figure 7. Comparison of computed w-x representations for a p.e.c. right angle dihedral where the gap is either in the shadowed or illuminated region. The gap width is 0.77 cm (0.3 in.). The sides of the dihedral are 45.72 cm (18 in.) wide. The fields are sampled along y = 0.254 cm (0.1 in).

modes, characteristic of a periodic structure. The lines associated with the Bragg modes are separated along x-direction by Akx = 27rn/s, where s is the spacing of the array elements and n is the

Bragg mode

number. In the present case for n = 1, 2.74 cm . For

frequencies above 7 GHz, this is a propagating mode; for frequencies above 14 GHz, the next order mode n = 2 is also propagating. The

negative hx in these cases indicates that the fields associated with these modes are moving in a direction opposite to the incident field. There is also a very weak evanescent field component as evidenced by the line

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Figure 8. Comparison of computed w-x representations for p.e.c. dihedrals with right angle and rounded corners. The illumination is at

§1 = 200°. The other parameters are as in Figure 7.

to the right of the forward scattered fields. The corresponding calculated w-x plots for the scattered fields are

given in Fig. 9b. These results show the same general features observed in the experimental data. The relative intensity levels for the measured and calculated results are different since the latter are the bistatically scattered (not total) field intensities.

Finally, Fig. 10 depicts the measured and calculated w-x results for a finite 10 element array of elements. The elements are arranged with two different spacings sl and s2. In the infinite case, this corresponds to a doubly periodic structure. For the measurements again aluminum rods were used with the earlier dimensions. For greater clarity a gray scale is used for the calculated data. The measured and calculated w-x

plots show two sets of intensity lines corresponding to the two spacings.

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Figure 9. representations for a 14 element evenly spaced array: (a) measured, (b) computed. The incident field is TM polarized and

oi = 145°, s = 2.29 cm (0.9 in.). The electric field is sampled along y = 0.254cm (0.1 in.) above the array. The gray lines refer to the

positive and negative light lines. The black lines are associated with the illumination angle.

One is associated with Sl replicating the results in Fig. 9. The second set is associated with s2. In both the measured and computed plots, higher order propagating Bragg modes are visible. In addition, weak evanescent fields appear outside the propagating region. Overall there is good correlation between the measured and computed results.

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Figure 10. w-x representations for a 10 element evenly spaced array: (a) measured, (b) computed. The incident field is TM polarized and

oi = 145°, sl = 2.29 cm (0.9 in.) and s2 = 4.58 cm (1.8 in.). The fields are sampled along y = 0.254 cm (0.1 in.) above the array. The gray lines refer to the positive and negative light lines. The black lines are associated with the illumination angle.

5. SUMMARY

Expressions for the frequency wave vector (w-k) representations for

p.e.c. BOTs were developed. They are applicable to BOTs with

arbitrary surface and gap discontinuities that can be electrically large or small. The results were implemented numerically within the framework of the MM (Galerkin) technique. The ω-k representations allow delineation and visualization of the propagating and evanescent

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fields ultra-near to a target. Representative results were given for

strips and dihedrals with various discontinuities. The case of finite, evenly spaced array of scatterers was also considered. Experiments were carried out in the NRL Ultra-Near Field Facility to correlate with the analytical results.

ACKNOWLEDGMENT

The authors acknowledge the generous assistance of Mr. John M. Putnam of the Boeing Company in the adaptation of the CARLOS software for this investigation. We also thank Drs. Donald Forester of NRL and Ronald Radlinski of ONR for their support and

encouragement of this project. This research was sponsored in part by the Office of Naval

Research under Contract N00014-97-AF00002 and in part by the Naval Research Laboratory.

APPENDIX A.

The general expression for the impedance elements in Eq. (4a) is given by:

/ where a = t or z. Explicit expressions for the impedance elements

in Eq. (4) can be derived from the finite BOT results in [9] as

follows. Since in the present case the BOT is infinite along the z-axis, the expansion and testing functions do not have any z-dependence, i.e., this corresponds to the n = 0 mode result in Eqs. (8) and (11) of [9]. Thus

where the divergence of a vector A on the BOT is V - A = + Q and where v and v' are the angles between the BOT generating curve

C(t) and x-axis at the points (x, y) and (x', y'), respectively. Since in

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the present case the BOT is infinite along the z-axis, there are no t- directed edges on the surface and thus there are no impedance elements

cross-coupling the t- and z-directed currents, i.e., terms corresponding to Eqs. (9) and (10) in [9] are zero.

It is customary when using overlapping triangle functions as basis and testing functions to approximate these fj and their derivatives

dfj / dt as four pulses, denoted by Tp and Tp respectively, i.e., Tp =

( 1 /4, 3 /4, 3 /4, 1 /4) and Tp = Thus each segment of

C(t) is approximated by 4 straight line segments associated with these

pulses. Using a centroid approximation for the integrations over t and

t', yields

where q is the free space wave impedance. The expression for the excitation vector in Eq. (4) is given by

where for TE and TM the respective expressions are

and 1jJp = xp cos + yp sin

APPENDIX B.

The transform in Eq. (5) can be carried out as follows. Shifting the coordinates 7 = z - x' and 11 = y - y' yields

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where R = x The integrand with the sine term is odd and hence its contribution is zero, thus yielding

Expressing the Hankel function in terms of the zeroth order Bessel and

Neumann functions, i.e., = Jo - j No and invoking the integral identities [17], we obtain for y > 0,

where Kl = and Explicitly written, the

B (scalar potential) term is

For the first part of Eq. (B4) we invoke the theorem in [18]. For the second part we use the result in Eq. (B2), yielding

The terms in Eqs. (B3) and (B5) with the imaginary exponential

represent the propagating parts of A and B, respectively. The others denote the evanescent parts. To obtain corresponding results for the w-

ky domain, we substitute ky for kx and y for z in the above expressions.

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3. Mayes, P. E., G. A. Deschamps, and W. T. Patton, "Backward- wave radiation from periodic structures and application to the

design of frequency independent antennas," Proc. IRE, Vol. 49, 962, 1961.

4. Williams, E. G., "Supersonic acoustic intensity," J. Acoust. Soc.

Am., Vol. 97, 121-127, 1995.

5. Taylor, D., P. Loschialpo, E. Williams, and W. Pala, "Dispersion analysis of surface waves in the near field," Proc. PIERS, 146, Cambridge, MA, July 7-11, 1997.

6. Andreasen, M. G., "Scattering from parallel metallic cylinders with arbitrary cross sections," IEEE Trans. Antennas Propagat., Vol. AP-12, 746-754, 1964.

7. Wallenberg, R. F. and R. F. Harrington, "Radiation from

apertures in conducting cylinders of arbitrary cross section," IEEE Trans. Antennas Propagat., Vol. AP-17, 56-62, 1969.

8. Wilton, D. R. and R. Mittra, "A new numerical approach to the calculation of EM scattering properties of two-dimensional bodies of arbitrary cross section," IEEE Trans. Antennas Propagat., Vol. AP-20, 310-317, 1972.

9. Medgyesi-Mitschang, L. N. and J. M. Putnam, "Scattering from finite bodies of translation: plates, curved surfaces, and noncircular cylinders," IEEE Trans. Antennas Propagat., Vol. AP-

31, 847-852, 1983. 10. Medgyesi-Mitschang, L. N. and J. M. Putnam, "Electromagnetic

scattering from extended wires and two- and three-dimensional

surfaces," IEEE Trans. Antennas Propagat., Vol. AP-33, 1090-

1100, 1985. 11. Bornholdt, J. M. and L. N. Medgyesi-Mitschang, "Mixed-

domain Galerkin expansions in scattering problems," IEEE Trans. Antennas Propagat., Vol. AP-36, 216-227, 1988.

12. Miller, E. K. , L. N. Medgyesi-Mitschang, and E. H. Newman

(Eds.), Computational Electromagnetics: Frequency-Domain Me- thod of Moments, IEEE Press, Inc., New York, 1991.

13. Medgyesi-Mitschang, L. N., J. M. Putnam, and M. B. Gedera, "Generalized method of moments for three-dimensional penetrable scatterers," J. Opt. Soc. Am., Vol. 11, 1383-1398, 1994.

14. Putnam, J. M. and M. B. Gedera, "CARLOS-3D: A general purpose 3-D method of moments scattering code," IEEE Antennas

Propagat. Mag., Vol. 35, 69-71, 1993.

15. Smith, D., S. Browning, S. Love, M. Morgan, and M. Parent, "Experimental issues in ultra-near field measurements," AMTA

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'99 Symp., 8-12, Monterey, CA, Oct. 4-8, 1999.

16. Peters Jr., L., "End-fire echo of long thin bodies," IRE Trans. Antennas Propagat., Vol. AP-6, 133, 1958.

17. Gradshteyn, I. S. and I. M. Ryzhik, Table of Integral, Series, and

Products, #6.677.3-4, 736, Academic Press, New York, 1980.

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Louis N. Medgyesi-Mitschang received the B.S. (1961), M.S.

(1962), and D.Sc. (1967) degrees in Electrical Engineering from

Washington University, St. Louis. In 1967 he joined the McDonnell

Douglas Corporation (MDC) as a research scientist in the corporate laboratory where he developed new microwave, radiometric, and interferometric diagnostic techniques for high temperature plasmas. Subsequently, during the past two decades he pioneered new techniques in computational electromagnetics for scattering from complex targets, analysis of airborne arrays, and simulation of wave phenomena in

composite materials. He has been a principal investigator on numerous R&D programs for the US Tri-Services and national labs. He has served on special assignments for the MDC Corporate Office on technology integration, assessment and strategic planning. In 1991 he was elected a Senior Technical Fellow of MDC. With the merger of MDC with the Boeing Company, he was named Staff Director of the Computational Electromagnetics Group within Phantom Works, the Boeing advanced R&D organization. In 1999, he assumed the responsibilities of a Chief Scientist at the Naval Research

Laboratory. Dr. Medgyesi-Mitschang has authored numerous papers and chapter length contributions in the field of electromagnetics, antennas, numerical analysis and computer simulation/modeling. He is a co-editor of Computational Electromagnetics, IEEE Press. He served as Guest Editor of IEEE Proceedings, was Associate Editor of IEEE Transactions on Antennas and Propagation and ADCOM member of the IEEE MTT-S. He is a Fellow of the IEEE, Associate Fellow of the

AIAA, member of URSI (Commission B), ACES, Tau Beta Pi, Eta

Kappa Nu and Sigma Xi.

Peter G. Moore received the B.A. (1970), B.S. in Physics (1970), M.A. (1973), and Ph.D. (1975) degrees from the University of Texas at Austin, following graduate studies at the University of

Kaiserslautern, Germany. He was a NRC post-doctoral fellow at the Naval Research Laboratory from 1975 to 1977 when he joined the

Laboratory as a research physicist. He has been principal investigator

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for DoD research programs in acoustics, atomic and molecular physics, materials processing and characterization. He has published over 40 scientific and technical papers. His current research interests include the modeling and simulation of electromagnetic waves in

complex materials and structures, radar cross section analysis, and

antenna/array optimization. He is a member of ASM.

Douglas L. Smith received his Ph.D. in physics from the University of Illinois in 1989. He pursued postdoctoral studies at the Microwave

Technologies Division of the Naval Research Laboratory (NRL) investigating the development of electronic devices with large band-gap materials. Subsequently, he conducted research in the magnetostrictive properties of soft-ferromagnetic materials at microwave frequencies with the Optics Division of NRL. In 1994 he joined SFA, Inc.

engaged in materials research for antenna applications. In 1998 he

joined NRL as a research physicist responsible for the development of the NRL Ultra-Near Field Laboratory and associated measurement and analytical techniques used to characterize the electromagnetic scattering from complex targets

Samuel G. Lambrakos received the B.S. (1977), M.S. (1979), and

Ph.D, (1983) degrees in Physics from the Brooklyn Polytechnic. He has been a research physicist at the Naval Research Laboratory since 1984. He has published over 60 technical papers in mathematical

physics and related fields. This includes particle beam scattering, long- wavelength electromagnetics, optics, radar cross section analysis, pattern recognition, surface-structure spectroscopy and thin film

analysis. In the field of molecular dynamics, he has developed new constraint and tracking algorithms for large-number-of-particles systems. He has extended fluid dynamics to biological systems and has developed a fluid mechanics based theory for conventional and

deep penetration welding processes. During his tenure at NRL, Dr. Lambrakos has also served as an invited professor of physics at the

University of Burgundy, Dijon, France developing a new curriculum in

computational methods at the university. He is a member of SIAM.

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