10
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 12, DECEMBER 1988 1769 Complex Pole Patterns of the Scattering Amplitude for Conducting Spheroids and Finite-Length Cylinders BARBARA L. MERCHANT, MEMBER, IEEE, PHILIP J. MOSER, SENIOR MEMBER, IEEE, ANTON NAGL, MEMBER, IEEE, AND HERBERT UBERALL, MEMBER, IEEE Abstract-The positions in the complex frequency plane of the poles in the electromagnetic scattering amplitude for perfectly conducting prolate spheroids and finite length cylinders with length-to-diameter ratios of up to 5.0 are presented. Using our modification of the Waterman T-matrix method, these poles were obtained for both classes of electromagnetic eigenvibrations of the scatterer and for various values of the azimuthal mode number. I. INTRODUCTION T IS WELL KNOWN that in the scattering of electromag- I netic waves from dielectric or conducting objects, reso- nances appear in the scattering amplitude. Mathematically, these resonances take the form of poles, which occur at the complex eigenfrequencies of the objects. Obtaining information on the complex resonance frequency pattern of perfectly conducting objects is of fundamental importance from the standpoint of basic scientific knowledge, since this provides us with an understanding of the scattering mechanism in terms of surface waves [l]. In such a model, surface waves are generated during the scattering process. The resonances then originate from a phase matching of these surface waves as they repeatedly circumnavigate the scattering object. Scatterers of elongated shape, such as prolate spheroids or finite-length cylinders, are of special interest here, since for these objects the surface wave paths are geodesics (helices for the case of cylinders [2]-[4]) which can be readily determined. The set of allowed pitch angles of these geodesics then gives rise to different sublayers of complex poles. These sublayers correspond to different values of the azimuthal mode number m that appears in the azimuthal component exp (imqi) of the surface wave. Such a splitting of the resonance frequencies according to their m values was first pointed out by Moser and Uberall [5], [6]. Previous calculations of complex pole patterns for conducting spheroids [7] and finite cylinders [8] had been restricted to m = 0 poles only. Manuscript received July 29, 1986; revised November 6, 1987. This work was supported in part by the Sperry Corporation, Corporate Technology Center, Reston, VA, and by the Office of Naval Research. B. L. Merchant was with the Radar Division, Naval Research Laboratory, Washington, DC. She is now with Harry Diamond Laboratories, 2800 Powder Mill Road, Adelphi, MD 20783. P. J. Moser is with AVTEC Systems, Inc., 3025 Hamaker Court, Fairfax, VA 2203 1. A. Nag1 and H. Uberall are with the Department of Physics, Catholic University of America, Washington, DC 20064. IEEE Log Number 8823646. In a spherical coordinate system, the multipole modal expansion of the field of a perfectly conducting sphere [9, pp. 769-7701 contains two distinct types of vector-multipole basis functions, known as transverse magnetic (TM) or “electric” and as transverse electric (TE) or “magnetic” multipole types, which differ from each other in their parity [9, pp. 745- 7461. They have the property that for the TM (TE) portions of the field, no radial magnetic (electric) field component is present, which accounts for the terminology (TM, TE) used for them. (“Transverse” here refers to transversality of the fields relative to the surface normal of the sphere.) The complex eigenfrequencies or pole positions are obtained by letting the modal expansion coefficients of the TM (TE) multipole functions tend to infinity. Accordingly, the sphere poles also fall into two corresponding classes labeled TM (TE) 1101. If the conducting sphere is modified, e.g., by applying a dielectric coating of constant thickness (the case of [lo]), or by deforming it into a conducting spheroid (the present case), the two pole families of TM or TE type still remain different and distinct, as can be seen by considering a smooth continuous variation starting from the conducting sphere to end up with a coated sphere or with a conducting spheroid. In both cases, a smooth motion of the poles along continuous trajectories in the complex frequency plane takes place. (In the case of the spheroid, there is an rn-fold multiplicity of trajectories all branching out from each given sphere pole, as mentioned before.) In view of this continuous motion, the distinction between the two pole types remains intact, and we shall continue to label them as being of “TM” or “TE” type, respectively. While for the coated-sphere case, the fields remain trans- verse with respect to the surface normal due to the spherical symmetry, such is no longer the case for the conducting spheroid. Therefore, in this paper we shall put “TM” and “TE” in quotation marks to indicate this effect. (Analogously, for a conducting finite cylinder [3], the poles also separate into “TM” and “TE” types, see also below.) In addition to the loss of the geometrical feature of the spheroidal field transver- sality with respect to the surface normal (since the surface normal no longer coincides with the radius vector), the poles in general can no longer be expected to retain their pure “electric” or “magnetic” character, but they become hybrid mixtures. This is most easily seen in the framework of the T- 0018-926X/88/1200-1769$01.00 O 1988 IEEE

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Page 1: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 12, DECEMBER 1988 1769

Complex Pole Patterns of the Scattering Amplitude for Conducting Spheroids and Finite-Length

Cylinders BARBARA L. MERCHANT, MEMBER, IEEE, PHILIP J. MOSER, SENIOR MEMBER, IEEE, ANTON NAGL, MEMBER, IEEE, AND

HERBERT UBERALL, MEMBER, IEEE

Abstract-The positions in the complex frequency plane of the poles in the electromagnetic scattering amplitude for perfectly conducting prolate spheroids and finite length cylinders with length-to-diameter ratios of up to 5.0 are presented. Using our modification of the Waterman T-matrix method, these poles were obtained for both classes of electromagnetic eigenvibrations of the scatterer and for various values of the azimuthal mode number.

I. INTRODUCTION T IS WELL KNOWN that in the scattering of electromag- I netic waves from dielectric or conducting objects, reso-

nances appear in the scattering amplitude. Mathematically, these resonances take the form of poles, which occur at the complex eigenfrequencies of the objects.

Obtaining information on the complex resonance frequency pattern of perfectly conducting objects is of fundamental importance from the standpoint of basic scientific knowledge, since this provides us with an understanding of the scattering mechanism in terms of surface waves [l]. In such a model, surface waves are generated during the scattering process. The resonances then originate from a phase matching of these surface waves as they repeatedly circumnavigate the scattering object.

Scatterers of elongated shape, such as prolate spheroids or finite-length cylinders, are of special interest here, since for these objects the surface wave paths are geodesics (helices for the case of cylinders [2]-[4]) which can be readily determined. The set of allowed pitch angles of these geodesics then gives rise to different sublayers of complex poles. These sublayers correspond to different values of the azimuthal mode number m that appears in the azimuthal component exp (imqi) of the surface wave. Such a splitting of the resonance frequencies according to their m values was first pointed out by Moser and Uberall [5], [6]. Previous calculations of complex pole patterns for conducting spheroids [7] and finite cylinders [8] had been restricted to m = 0 poles only.

Manuscript received July 29, 1986; revised November 6, 1987. This work was supported in part by the Sperry Corporation, Corporate Technology Center, Reston, VA, and by the Office of Naval Research.

B. L. Merchant was with the Radar Division, Naval Research Laboratory, Washington, DC. She is now with Harry Diamond Laboratories, 2800 Powder Mill Road, Adelphi, MD 20783.

P. J . Moser is with AVTEC Systems, Inc., 3025 Hamaker Court, Fairfax, VA 2203 1.

A. Nag1 and H. Uberall are with the Department of Physics, Catholic University of America, Washington, DC 20064.

IEEE Log Number 8823646.

In a spherical coordinate system, the multipole modal expansion of the field of a perfectly conducting sphere [9, pp. 769-7701 contains two distinct types of vector-multipole basis functions, known as transverse magnetic (TM) or “electric” and as transverse electric (TE) or “magnetic” multipole types, which differ from each other in their parity [9, pp. 745- 7461. They have the property that for the TM (TE) portions of the field, no radial magnetic (electric) field component is present, which accounts for the terminology (TM, TE) used for them. (“Transverse” here refers to transversality of the fields relative to the surface normal of the sphere.) The complex eigenfrequencies or pole positions are obtained by letting the modal expansion coefficients of the TM (TE) multipole functions tend to infinity. Accordingly, the sphere poles also fall into two corresponding classes labeled TM (TE) 1101.

If the conducting sphere is modified, e.g., by applying a dielectric coating of constant thickness (the case of [lo]), or by deforming it into a conducting spheroid (the present case), the two pole families of TM or TE type still remain different and distinct, as can be seen by considering a smooth continuous variation starting from the conducting sphere to end up with a coated sphere or with a conducting spheroid. In both cases, a smooth motion of the poles along continuous trajectories in the complex frequency plane takes place. (In the case of the spheroid, there is an rn-fold multiplicity of trajectories all branching out from each given sphere pole, as mentioned before.) In view of this continuous motion, the distinction between the two pole types remains intact, and we shall continue to label them as being of “TM” or “TE” type, respectively.

While for the coated-sphere case, the fields remain trans- verse with respect to the surface normal due to the spherical symmetry, such is no longer the case for the conducting spheroid. Therefore, in this paper we shall put “TM” and “TE” in quotation marks to indicate this effect. (Analogously, for a conducting finite cylinder [3], the poles also separate into “TM” and “TE” types, see also below.) In addition to the loss of the geometrical feature of the spheroidal field transver- sality with respect to the surface normal (since the surface normal no longer coincides with the radius vector), the poles in general can no longer be expected to retain their pure “electric” or “magnetic” character, but they become hybrid mixtures. This is most easily seen in the framework of the T-

0018-926X/88/1200-1769$01.00 O 1988 IEEE

Page 2: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

1770 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 12, DECEMBER 1988

Re kb -

SPHERE 0 3:l SPHEROID v 5:l SPHEROID Jt 1.5:l SPHEROID + 3.5:l SPHEROID 0 1O:l SPHEROID A 2:l SPHEROID 4:l SPHEROID 0 2.5:l SPHEROID A 4 5 1 SPHEROID

Fig. 1. Pole pattern in complex frequency plane (of “TM” type) for perfectly conducting prolate spheroids with semimajor axis b (Poles on imaginary axis have been dispersed horizontally for better visibility.)

matrix expansion of the fields or, alternately, the surface currents. Instead of letting the expansion coefficients of the TM or TE basis functions tend to infinity separately and individually, we obtained our poles through the T-matrix

predictions of a helical surface wave model [3], [4]. For the prolate-spheroid case, an analogous model based on the phase matching of surface waves propagating along geodesics is now being developed [ 121.

method [3]-[6] by setting to zero the determinant of a matrix that couples the two types of expansion coefficients. Nonethe- less, as shown by the continuity argument, the two classes of “TM” and “TE” type poles remain distinct. Our continued use of the corresponding labels then must be interpreted such that the electromagnetic eigenfunctions corresponding to these complex pole eigenvalues tend continuously toward the geometrically transverse magnetic or electric eigenfunctions of the sphere as the spheroid reverts back to a sphere.

In this paper, we present comprehensive patterns of the complex-frequency poles that appear in the scattering ampli- tude of perfectly conducting prolate spheroids and finite- length cylinders for both “TM” and “TE” eigenvibration modes. We trace the pole migration for the cases of both m = 0 and m > 0 through the complex plane as the values of the length-to-diameter or major-to-minor axis ratios are increased up to values of 5.0. We obtained our numerical results by using a modified version of Waterman’s T-matrix code for the electromagnetic scattering amplitudes of axisymmetric con- ducting bodies [ll] which also furnishes the poles of the scattering amplitudes 151, [6]. For the finite-length cylinder case, the mentioned surface wave explanation of the pole pattern has already been confirmed by its agreement with the

11. DETERMINATION OF COMPLEX-FREQUENCY POLES We obtained the complex-frequency poles of the scattering

amplitude from a modification of the Waterman T-matrix (or extended boundary condition) method. The Waterman T- matrix code calculates only the electromagnetic scattering amplitude from axially symmetric conducting objects [ 1 11.

We have modified the original T-matrix code in two ways [5], [6]. First, the resonance poles of the scattering object are calculated as the complex zeros of a secular determinant. Second, a test to determine the order and type (“TE” or “TM”) of the poles has been added.

Our search for the poles begins by selecting a rectangle around a suspected pole location in the complex frequency plane. A grid of frequencies is laid out on the rectangle and the value of the complex secular determinant is calculated with our computer code at each frequency on the grid. A pole is located at that frequency where both the real and imaginary parts of the value of the determinant go to zero. We found that the use of an exponent of (IEEE G-FLOATING) instead of lok3* (IEEE F-FLOATING) extended the area of the complex-frequency plane over which we could search for pole locations.

Page 3: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

1771 MERCHANT AND MOSER: COMPLEX POLE PATTERNS OF SCATTERING AMPLITUDE

Re kb - 3 4 5 6 7 0 1 2

I I I I I I

D - 3 b

Y -4 E

-7 c 0 3: l SPHEROID Pp SPHERE + 3.5:l SPHEROID * 1.5:l SPHEROID

A 2:l SPHEROID 4:l SPHEROID

0 2.5:l SPHEROID A 4.5:l SPHEROID

v 5: l SPHEROID 0 1O:l SPHEROID

Fig. 2. Pole pattern in complex frequency plane (of “TE” type) for perfectly conducting prolate spheroids with semimajor axis b (Poles on imaginary axis have been dispersed horizontally for better visibility .)

111. RESULTS FOR PROLATE SPHEROIDS is deformed continously into a longer and longer spheroid.

Figs. 1 and 2 are plots of our calculated spheroidal complex- resonance poles for the “TM” and “TE” modes, respec- tively, as well as the pole values for a sphere. The plots are located in the fourth quadrant of the complex frequency plane. Both axes are scaled in terms of kb, where k = 2a/X and b is the semimajor axis of the spheroid. The “TM” and “TE” poles have been placed on two different plots for clarity.

In both Figs. 1 and 2, the locations of the sphere poles are indicated by the squares containing an x. The TE sphere poles lie at the zeros of the nth spherical Hankel function of the first kind hy). The TM sphere poles are given by

[xh ?’(x)] ’ = 0 (1)

where the prime indicates differentiation with respect to the argument x [13]. The labels n on the plots indicate the nth spherical Hankel function of the first kind A!,‘). These functions have n multiple roots. The first layer of sphere poles in these figures corresponds to the first root of each h?). Similarly, the second layer corresponds to the second root of h t ) . The layers of the two types alternate with the first TM layer closer to the real axis than the first TE layer.

The trajectories originating from both the TE and TM sphere poles represent the change in pole location as the sphere

The spheroid pole locations were calculated for axial ratios b/a (semimajor to semiminor axis ratio) of 2.0, 3.0, 4.0, and 5.0. In some cases, a pole was calculated for axial ratios of 1.5, 2 . 5 , 3.5, or 4.5 also. For every order n, azimuthal mode number rn, and type (“TE” or “TM”) considered, the spheroid pole locations move away from the sphere pole location such that the more elongated the spheroid, the further is the spheroid pole location from the sphere pole. The pole locations for the “TM” (“TE”) type spheroid oscillations lie along a smooth line originating at the TM (TE) sphere pole. This supports the view that the spheroid hybrid modes correspond to a progressive relaxation of the pure transversal- ity of the sphere modes, and of the use of the terminology “TM” and “TE.”

From each sphere pole location, a number of different trajectories are formed upon deformation of the sphere. For b > a, the sphere pole is split into several distinct, different poles which correspond to different values of m, where rn is the azimuthal mode number. Each trajectory originating at a sphere pole corresponds to one of the allowed values of rn. Additionally, the more elongated the spheroid, the further apart are the poles for the various allowed values of rn for a given sphere pole. For each sphere pole of order n, the allowed integer values of rn are 0, 1, * * * , n for a total of n +

Page 4: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

1772 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 12, DECEMBER 1988

TABLE I "TM" POLES FOR SPHEROID'

m = l m = 2 m = 3 m = 4

a) b / a = 2.0

1.350-0.9901 1 1

2

3

4

2 2

3

1,150-0.4521'

2.374-0.7001

3.6 10-0.8841

4.86 -1.04i

0.000-2.1241'

1.21 -2.871'

2.40 -3.381'

3.59 -3.81i

4.80 -4.16i

(1.152-0.4531)

(2.380-0.703 1)

(3.623-0.8901)

(4.876-1.0421)

(0.000-2.1301)

(1.210-2.87911

(2.410-3.4081)

(3,609-3.83Of)

(4.8 10-4.1 87 0.000-3.9701 1,204-4.7641 2.42 -5.39i

2.334-1.4721 3.10 -1.60i

3.474-1.3541 4.06 -2.04i

4.42 -0.81i 5.02 -1.92i

5.02 -2.01i

5.88 --2.50i 7.00 -2.2901

O.OO0-2.1281 0.00 -2.751'

1.22 -2.973 1.12 -3.20i

2.37 -3.51i 2.48 -3.67i

1.52 -3.88i

2.50 -4.20; 3.10 -4.80i 4

5 3.59 -3.881 3.64 -4.14i

4.82 -4.241 4.84 -4.42i

0.000-3.992i 0.00 -4.271 1.21 -4.821 1.18 -4.981 2.40 -5.44i 2.42 -5.60i

b) b / a = 3.0

1.62 -1.425i 2.385-1.891 4.02 -2.621 3.81 -1.515i 4.635-3.0151 5.06 -1.16i 5.91 -2.461

1.18 -3.88; 2.66 -4.05i 3.00 -4.275i

4.22 -4.90i

3.74 -4.40i

5.04 -4.761'

0.00 -4.51i 1.30 -5.30i 2.46 -5.881

4.12 -5.081

5.20 -5.221

0.00 -5.431 1.24 -5.74i 2.60 -6.20i

6

3 4 5 6

1 1

6.855-3.541 7.275-3.93 i 1.87 -5.261 2.72 -5.51; 4.17 -5.421 5.75 -5.821'

9.825-4.245 i 2 1.43 -3.191

C) b / a = 4.0

1.776-1.7841 2.44 -2.18i 4.72 -3.74i 4.10 -1.78i 5.08 -3.861' 5.40 -1.72i 6.74 -3.28i 1.64 -3.84i 1.16 -4.40i 2.82 -4.40i 3.40 -4.661 4.44 -4.60i 4.70 -5.421

1.64 -6.04i 1.40 -6.40i

d) b/a = 5.0

1.86 -2.09i

2.47 -2.371 5.222-4.65 i

4.32 -2.05i 5.51 -4.66i

1 1.332-0.361 2.732-0.5561 4.148-0.7 12i 5.568-0.8401 1.56 -3.401 3.00 -3.96i 4.44 -4.361'

8.420-5.081 8.72 -5.42i 1.96 -6.30i 2.70 -6.381' 4.30 -6.341' 5.88 -6.521'

12.3 -6.02i 2

1.60 -5.841

1.38 -0.33i

2.81 -0.53i

4.28 -0.651

5.73 -0.78i (5.745-0.773i') 0.00 -2.663

(0.000-2.672i) 1.63 -3.521'

(1.374-0.3361)

(2.817-0.5 160

(4.277-0.6551)

(1.652-3.522n 3.17 -4.08;

(3.195-4.070r) 4.67 -4.481'

6.15 -4.82i

1.68 -6.05i

(4.694-4.491 11

(6.174-4.8391)

1

2

3

4

2

3

4

5

6

5

10.10 -6.70i

10.22 -6.951' 5.62 -2.30i 7.25 -4.281'

0.00 -2.71i

1.81 -4.18i 1.19 -4.821

14.8 -7.80i

2

1.96 -7.02i

2.75 -7.06i

4.33 -7.05i

5.88 -7.08i

2.97 -4.681 3.53 -4.91i

4.73 -4.88i 5.27 -5.891'

6.04 -5.381'

e) b /a = 10.0

1

2

1.44 -0.2651'

2.96 -0.393i (1.458-0.26511

(2.977-0.400i)

Page 5: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

1773 MERCHANT AND MOSER: COMPLEX POLE PATTERNS OF SCATTERING AMPLITUDE

2

~~ ~~

3 4.46 -0.485i (4.5 10-0.497 1’)

2 0.00 -2.961 (O.oo0-2.969i’)

3 1.88 -3.73i

4 3.54 -4.24i

5 5.15 -4.621’

6 6.72 -4.931’

(1.888-3.776i’)

(3.575-4.278i’)

(5.194-4.67Oi)

(6.782-4.975iJ

a The numbers in the parentheses are reproduced for comparison from Marin [7]

1 allowed values. (Actually, there are 2n + 1 allowed values of rn for a given n, but the eigenfrequencies are degenerate for f r n due to reflection symmetry of the spheroid about the plane formed by the minor axes.) The poles for a sphere coincide (or are degenerate) for all the allowed values of m due to symmetry of the sphere. Previous calculations of these spheroid poles [7] had considered only the rn = 0 case, corresponding to axial eigenvibrations.

In the first “TM” layer, Fig. 1, the rn = 0 poles tend toward integer multiples of 7r/2 as the spheroid elongation increases. In other words, the imaginary parts decrease and go to zero for an infinitely long spheroid. In contrast, the rn > 0 poles in the first “TM” layer split away from the rn = 0 poles and move toward larger imaginary parts with increasing elongation.

In the second “TM” layer, Fig. 1, the rn = 0 pole behavior is different from that in the first “TM” layer. Here the rn = 0 poles tend to lie parallel to the real axis along with the rn = 1 poles. The higher rn poles, rn = 2 and 3, tend to move toward increasingly negative imaginary parts with increasing elonga- tion.

The first “TE” layer, Fig. 2, looks similar to the second “TM” layer. Again, the trajectories for rn = 0 and rn = 1 tend to intertwine and lie approximately parallel to the real axis. The m = 2 and m = 3 poles move to larger imaginary frequencies.

Tables I and I1 contain our calculated “TM” and “TE” prolate spheroid complex-resonance poles of those axial ratios b/a for which we have the most complete sets. In each of the tables, the column labeled I gives the pole’s layer (correspond- ing to the Ith root of hjl])), while the column labeled n gives the pole’s order. The columns labeled with various values of rn correspond to the allowed azimuthal mode number m for the given n. The pole values are scaled in terms of kb, where b is the semimajor axis of the spheroid. For comparison, Table I- a), d), and e) also contain the poles for rn = O calculated with a different method by Marin [7].

Our interpretation of the rn = 0 spheroidal poles in terms of the phase matching of surface waves propagating along a meridional geodesic [ 121 has demonstrated the accuracy of the poles calculated using our T-matrix code, up to axial ratios of

5.0. In fact, this phase matching method constitutes the most reliable test for T-matrix calculations known to us. A similar interpretation of the m > 0 poles in terms of the phase matching of surface waves propagating along closed quasi- helical geodesics is now in progress.

IV. RESULTS FOR FINITE-LENGTH CYLINDERS

Figs. 3 and 4 show plots of our calculated complex poles for finite-length cylinders of “TM” and “TE” type, respectively. For reference, the sphere poles are shown in these figures as well as the “TM” poles for L/2a = 100.0 found by Baum and Singaraju [14]. The axes in Figs. 3 and 4 are scaled in terms of kL/2, where L is the length of the cylinder. (For the sphere, the pole value is scaled equivalently in terms of ka, where a is the radius.)

In many respects, the cylinder pole plots in Figs. 3 and 4 are similar to the ones in Figs. 1 and 2 for spheroids. Trajectories of pole locations are formed as the length-to-diameter ratio is increased from 1.0 up to 5.0. Different trajectories exist for each of the n + 1 allowed values of the azimuthal mode number m for the family of nth-order trajectories that originate around the sphere pole of order n. Again the poles tend to lie in layers approximately parallel to the real axis with the first “TM” layer closest to the real axis and the “TE” and “TM” layers alternating as the imaginary part of the frequency becomes increasingly negative.

Some major differences exist between the cylinder and spheroid pole plots. First, the pole trajectories for the various allowed rn values, formed by varying the length-to-diameter ratio, do not merge into the sphere poles. Secondly, the cylinder pole positions, even for a 1 .O cylinder, do not become degenerate in rn due to a lack of spherical symmetry.

The values of our calculated cylinder “TM” and “TE” complex poles, shown in Figs. 3 and 4, are tabulated in Tables I11 and IV for those length-to-diameter ratios for which we have the most complete sets. The columns are labeled the same as in Tables I and 11. The pole values are scaled in terms of kL/2, where L is the cylinder length.

We attribute the physical origin of the complex poles to the phase matching of multiply circumnavigating surface waves generated, e.g., during a scattering process. We have shown

Page 6: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

1774 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 12, DECEMBER 1988

TABLE I1 "TE" POLES FOR SPHEROID

I n

1 1 2 3 4 5

2 3 4 5 6

3 5

1 2 3 4 5

2 4 5 6

1 2 3 4 5

2 4 5 6

1 1 2 3 4 5

2 4 5 6

1 1 2 3 4 5 6

m = O m = l m = 2 m = 3 m = 4

0.00 -1.6921 1.222-2.1741 2.420-2.526 1 3.618-2.8241 4.82 -3.06; 0.00 -3.2401 1.21 -3.971 2.41 -4.531 3.60 -5.00; 0.000-4.9821

1.42 -2.64; 2.77 -2.951 4.12 -3.221 5.46 -3.43; 1.47 -4.61; 2.80 -5.18; 4.18 -5.661'

1.52 -2.963 3.00 -3.281 4.44 -3.521 5.80 -3.641 1.60 -5.041

4.56 -6.041 3.12 -5.643

0.00 -3.011 1.63 -3.281 3.13 -3.531 4.60 -3.733 6.08 -3.901 1.82 -5.401 3.30 -5.921 4.85 -6.351

0.00 -4.081 1.80 -4.201 3.37 -4.361 4.97 -4.50; 6.53 -4.62; 8.07 -4.72;

a) b/a = 2.0

0.00 -1.4921 1.036-2.1041 1.40-2.541 2.520-2.5541 2.40-2.96; 3.664-2.9481 3.62-3.081 4.84 -3.16; 4.96-3.38; 0.00 -3.2621 0.00-3.331 1.17 -3.981' 1.26-4.16; 2.43 -4.561 2.40-4.74 i 3.61 -5.07; 3.62-5.2Oi 0.000-5.032 i 0.00-5.13 i

b) b/a = 3.0

1.08 -2.401 1.74-3.403 2.94 -2.791 2.58-3.761 4.20 -3.40; 4.08-3.611 5.49 -3.43; 5.58-4.123 1.31 -4.54; 1.58-5.041 2.86 -5.201 2.74-5.621' 4.16 -5.791 4.24-5.961

c) b/a = 4.0

1.08 -2.601 1.92-4.081 3.24 -2.961 2.66-4.34 1 4.52 -3.601 4.30-4.161 6.02 -3.541 5.78-4.681 1.40 -4.921' 1.88-5.721 3.16 -5.621 2.96-6.261 4.52 -6.241

d) b/a = 5.0

0.00 -2.411 1.11 -2.771 1.94-4.64; 3.43 -2.931 2.73-4.771 4.92 -3.611 4.37-4.67;

1.43 -5.19; 2.08-6.401 3.43 -5.871' 3.12-6.821

6.46 -3.591 5.75-5.30;

4.82 -6.483

e) b/a = 10.0

3.04-3.34i 4.02-3.721 5.10-3.781 0.00-4.09 1 1.18-4.501 2.50-4.98 i 3.72-5.48i 0.00-5.471

3.91-4.931 4.78-5.141 5.94-4.901 1.20-5.56i 3.21-5.941' 4.4 1-6.64;

5.10-6.261 6.56-6.101 1.20-6.421

4.80-8.281'

5.11-7.031 6.70-7.011 1.19-7.083

4.78-4.01 5.76-4.301'

1.54-5.241' 2.56-5.561 3.84-5.821 0.00-5.771'

Page 7: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

MERCHANT AND MOSER: COMPLEX POLE PATTERNS OF SCATTERING AMPLITUDE

Re kL/2 -

-7

1775

- , 1 1 I I

-1

-2 0

t -3 cv Y 2 -4

E -5

-

-6

-7

SPHERE A 2:l CYLINDER Jt 1OO:l CYLINDER 0 1:l CYLINDER 0 3:l CYLINDER 0 1.51 CYLINDER v 5 1 CYLINDER

Fig. 3. Pole pattern in complex frequency plane (of “TM” type) for perfectly conducting circular cylinders of length L . (Poles on imaginary axis have been dispersed horizontally for better visibility.)

w

7 0

n=l

n=3

Re kL/2 - 1 2 3 4 5 6

0

I I I I I I

0 SPHERE 0 1.51 CYLINDER v 5:l CYLINDER 0 1:l CYLINDER A 2:l CYLINDER

Fig. 4. Pole pattern in complex frequency plane (of “TE” type) for perfectly conducting circular cylinders of length L. (Poles on imaginary axis have been dispersed horizontally for better visibility.)

Page 8: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

1776 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 12, DECEMBER 1988

TABLE 111 "TM" POLES FOR CYLINDER

~~~

-~ ~ ~~~~~ -

I n m = O m = l m = 2 m = 3 m = 4

1 2 3 4 2 3 4 5 4 5

1 2 3 4 2 3 4 5 4 5

1 2 3 4 2 3 4 5 4 5

0.722-0.361 i 1 S64-0.7791 2.3 10-0.885 1 3.08 -0.72; 0.000-1.3871 0.734- 1.860; 1.487-2.1711 2.257-2.4521 0.000-2.494; 0.743-3.0141

0.93 -0.32; 2.10 -0.681 3.22 -1.071 4.13 -1.311 0.00 -1.831 1.06 -2.531 2.05 -2.961 3.04 -3.251 0.00 -3.441 1.03 -4.09;

1.15 -0.245; 2.49 -0.465; 3.88 -0.67i 5.25 -0.851 0.000-2.5451 1.51 -3.16;

4.20 -4.15; 2.91 -3.71;

0.000-4.625 i 1.55 -5.50;

a) L/2u = 1.0

0.753-0.431 1 1.435-0.413; 1.60 -0.651 2.303-0.7 18; 2.23 -0.55; 3.06 -1.35; 2.99 -0.65; 0.000-1.2931' 0.00 -1.391

1.382-2.195 i 1.45 -2.101 2.1 84-2.478 i 2.10 -2.461 0.000-2.4421' 0.00 -2.411 0.721-2.9861 0.68 -2.901

0.713-1.8041 0.69 -1.73;

b) L/2u = 2.0

1.12 -0.871 1.82 -1.121 2.47 -1.581 2.98 -0.85i 2.94 -1.731 4.22 -0.921 4.08 -1.24;

1.05 -2.50; 0.83 -2.51; 1.96 -2.961' 2.20 -2.971 3.11 -3.33; 3.00 -3.401 0.00 -3.341 0.00 -3.551 1.03 -4.151 0.99 -4.091

0.00 -1.68; 0.00 -2.231'

c) L/2u = 5.0

1.47 -1.7451 2.045-1.8751 3.49 -3.48; 3.365-1.5 10; 4.10 -3.42; 4.865-1.6551 6.23 -3.08i 0.005-2.1501 1.53 -3.6051' 0.933-3.825 i 2.60 -4.00; 2.865-3.6251'

0.000-4.4151' 0.000-5.545 1 1.25 -5.85;

2.47-0.871' 3.10-0.751

0.76-1.911 1.38-2.091 2.18-2.341 0.00-2.341 0.72-2.901

4.08-2.281 4.33-2.88;

1.18-3.281 1.79-3.321

0 .OO-5.04 1 3.30-3.301

1.15-4.42;

5.95-4.95 1 7.32-4.751

2.07-5.451 2.67-5.351' 5.27-5.05 i

1.50-5.461

3.39-1.07;

1.53-2.321 2.09-2..42; 0.00-2.621 0.66-2.851

3 39-3.921

2.30-4.191 2.85-4.161 0 .OO-5.5 3 i 0.80-4.491

TABLE IV "TE" POLES FOR CYLINDER

I n m = O m = l m = 2 m = 3 m = 4

1 1 2 3 4 5

2 3 4 5 6

3 5 6

1 1 2 3 4 5

2 3 4 5 6

3 5 6

0.000-0.859 i 0.685-1.1891

2.288-1.823; 1.486-1.5321

3.06 -1.97; 0.000-1.963; 0.743-2.4941 1.497-2.865 1 2.258-3.1541 O.OO0-3.121 i 0.747-3.6171

0.00 -1.42; 0.96 -1.711 2.03 -2.011 3.13 -2.30; 4.22 -2.60; 0.00 -2.741

2.10 -3.911 3.10 -4.32; 0.00 -4.31; 1.04 -4.981

1.06 -3.41;

a) L/2a = 1.0

0.752-1.3261 0.74 -1.311 1.443-1.5701 1.44 -1.59; 2.037-1.7901 2.22 -1.80; 2.96 -1.96; 2.81 -2.021 0.000-1.9271 0.00 -1.92; 0.655-2.308 i 0.72 -2.431 1.455-2.8031 1.36 -2.721

0.000-3.051; 0.00 -2.941 0.727-3.598; 0.72 -3.50;

b) L/2u = 2.0

O.OO0-0.855;

2.213-3.183; 2.13 -3.06;

0.00 -1.28; 0.87 -1.79; 1.17 -2.17; 2.14 -2.24; 1.81 -2.381 2.97 -2.50; 3.19 -2.381 4.06 -2.541' 4.16 -2.83i 0.00 -2.80; 0.00 -2.631 0.94 -3.291 1.09 -3.561 2.08 -3.821 1.97 -3.961 3.07 -4.291 3.1 -4.3; 0.00 -4.231 0.00 -4.241 1.00 -4.901 1.04 -4.911

1.53-1.641 2.14- 1.85 1 2.34-1.941 2.96- 1.95 1 3.02-2.041 0.00-2.04 i

1.44-2.801 1.36-2.781 2.08-3.07; 2.17-3.08; 0.00-3.02i 0.00-2.951

0.72-3.511 0.68-3.451

0.67-2.381 0.77-2.58;

2.38-3 ,001 2.86-3.111 3.43-4.261 4.07-2.801 4.37-3.73; 0.00-4.61;

1.22-4.361 0.82-3.51; 2.32-4.141 1.79-4.391 3.0 -4.6;

0.00-6.021 0.00-5.921 0.99-5.06;

Page 9: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

MERCHANT AND MOSER COMPLEX POLE PATTERNS OF SCATTERING AMPLITUDE

TABLE IV (Continued) ~ ~~~ ~- ~

__

I n m = O m = l m = 2 m = 3 m = 4 __ ~-

c) L/2a = 5.0

1 2 3 4 5 3 4 5 5 .-

O.OO0-2.3 15i 1.316-2.6501 0,962-2.3151 2.62 -2.781’ 3.36 -2.18i 3.96 -2.92i 4.70 -2.741’ 5.35 -3.05i 6.26 -1.44i O.OO0-4.1451 O.Oo0-4.0151 1.52 -4.70i 1.265-4.57i

that on a finite cylinder the poles with nonvanishing azimuthal components, that is rn > 0, can be explained by the phase matching of helical waves on the cylinder [3], [4].

V. CONCLUSION

In summary, using a modification of the Waterman T- matrix code, we have calculated extensive sets of pole locations in the complex frequency plane for perfectly conducting prolate spheroids and finite-length cylinders. We calculated poles for both hybrid mode types (“TM” and “TE”).

The spheroid poles were calculated for axial ratios up to 5.0. The plots in Figs. 1 and 2 show that the spheroid poles for increasing elongation lie along a smooth curve originating at the sphere poles. The “TM” poles lie along a trajectory which originates at the TM sphere pole, and similarly in the “TE” case.

In addition, we calculated poles for various values of the allowed values of m, the azimuthal mode number. The plots in Figs. 1 and 2 show that the sphere pole is split into several rn poles as soon as the sphere is deformed into a spheroid.

Plots of our calculated cylinder poles show features similar to the spheroid poles. That is, two hybrid modes, smooth trajectories for increasing elongation, and rn-splitting of the poles.

Surface wave interpretations of these results are now in progress, including phase matching along closed quasi- helicoidal geodesics, and the evaluation of surface-wave phase and group velocities as functions of frequency.

ACKNOWLEDGMENT

We are grateful to Dr. J . Diarmuid Murphy for having provided us with the pole values for spheres.

REFERENCES [ l ] J . D. Murphy, P . J . Moser, A. Nagl, and H. Uberall, “A surface wave

interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propagat., vol. AP-28, pp. 924-927, 1980. A. Nagl, H. Uberall, P. P. Delsanto, J . D. Alemar, and E. Rosario, “Refraction effects in the generation of helical surface waves on a cylindrical obstacle,” Wave Motion, vol. 5 , pp. 235-247, 1983.

[3] P . J . Moser et al., “Complex eigenfrequencies of axisymmetric objects: Physical interpretation in terms of resonances,” Proc. IEEE,

H. Uberall et al., “Complex acoustic and electromagnetic frequencies of prolate spheroids and related elongated objects and their physical interpretation,” .I. Appl. Phys., vol. 5 8 , pp. 2109-2124, 1985.

[5] P. J . Moser, “The isolation, identification, and interpretation of

[2]

v01.,72, pp. 1652-1653, 1984. (41

1.517-3.701 2.14 -3.721 3.95-5.251 4.811-3.4371 4.62-5.151’ 5.60 -3.17i 6.63-4.853

0.91-5.511 1.65 -5.60i 3.35-5.3i

0.00 -6.10i

1777

~

resonances in the radar scattering cross section for conducting bodies of finite general shape,” Ph.D. dissertation, Catholic Univ. of America, Washington, DC, 1982. P. J . Moser and H. Uberall, “Complex eigenfrequencies of axisym- metric, perfectly conducting bodies: Radar spectroscopy,” Proc.

L. Marin, “Natural-mode representation of transient scattering from rotationally symmetric bodies,” IEEE Trans. Antennas Propagat.,

F. M. Tesche, “On the analysis of scattering and antenna problems using the singularity expansion technique,” IEEE Trans. Antennas Propagat., vol. AP-21, pp. 53-62, 1973. J . D. Jackson, Classical Electrodynamics, 2nd ed. New York: Wiley, 1975. W. E. Howell and H. Uberall, “Complex frequency poles of radar scattering from coated conducting spheres,” ZEEE Trans. Antennas Propagat., vol. AP-32, pp. 624-627, 1984. P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” in Computer Techniques f o r Electromagnetics, vol. 7 , R. Mittra, Ed. New York: Pergamon, 1973, ch. 3 . B. L. Merchant, A. Nagl, and H. Uberall, “Resonance frequencies of conducting spheroids and the phase matching of surface waves,” IEEE Trans. Antennas Propagat., vol. AP-34, pp. 1464-1467, 1986. J . A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941, p. 558. C. E. Baum and B. K. Singaraju, “The singularity and eigenmode expansion methods with application to equivalent circuits and related topics,” in Acoustic, Electromagnetic, and Elastic Wave Scatter- ing-Focus on the T-matrix Approach, V. K. Varadan and V. V. Varadan. Eds.

IEEE, vol. 71, pp. 171-172, 1983.

vol. AP-22, pp. 266-274, 1974.

New York: Pergamon, 1980, pp. 431-452.

Barbara L. Merchant (M’W) was born in Balti- more, MD, on February 13, 1949. She received the B.S. degree in mathematics from the University of Maryland, College Park, and the M.S. and Ph.D. degrees in physics from the Catholic University of America, Washington, DC, in 1971, 1982, and 1987, respectively.

She was a Physicist at the Naval Research Laboratory, Washington, DC, from 1971 until 1988, where she worked in the areas of radar countermeasures and electromagnetic scattering.

Since 1988 she has been with Harry Diamond Laboratories, Adelphi, MD. Her present research interests include resonance effects and surface waves, and numerical solution of electromagnetic scattering problems.

Philip J. Moser (M’gl-SM’87) was born in Rich- mond, VA, on June 5 , 1947. He received the B S degree in physics from Virginia Polytechnic Insti- tute. Blacksburg, and the M S and Ph.D. degrees in physics from the Catholic University of America, Washington, DC, in 1970, 1979, and 1982, respec- tively

From 1970 to 1974 he fulfilled a rmlitary obliga- tion in the U.S. Air Force From 1974 to 1984 he was employed as a Physicist with the Naval Re- search Laboratory, Washington, DC. From 1984 to

1987 he was a Member of the Technical Staff at the Corporate Technology Center for Signal Processing and Artificial Intelligence at Sperry Corporation,

Page 10: Complex pole patterns of the scattering amplitude for conducting spheroids and finite-length cylinders

1778 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 12, DECEMBER 1988

Reston, VA. Since 1987 he has been Program Manager for Signal Processing at AVTEC Systems, Inc. in Fairfax, VA. Also, since 1985 he has been a Lecturer in the Department of Electrical Engineering at the Catholic University of America and has taught courses in signal processing and electromagnetic scattering. His current research interests include adaptive signal processing techniques and target identification using radar scattering resonance theory.

Dr. Moser is an industrial representative to the Institute for Information Technology of Virginia’s Center for Innovative Technology and a member of the American Geophysical Union, the American Physical Society and American Association for the Advancement of Science.

Anton Nag1 (M’87) was born in Munich, West Germany He received the M.S. degree in electrical engineering from the Technical University, Mu- nich, and the Ph D degree in nuclear physics from the Catholic University of America, Washington, DC, in 1961, and 1983, respectively

From 1964 to 1971 he was with the Naval Weapons Laboratory (now Naval Surface Weapons Center), Dahlgren, VA, where he worked on computer software and computer simulation pro- jects for submarine-based and airborne attact sys-

tems. He then joined the Catholic University where he has been working on pion-nuclear physics, underwater sound propagation, electromagnetic and

acoustic resonance scattering, channeling radiation, and ionospheric propaga- tion of electromagnetic waves.

Dr. Nag1 is a member of the Acoustical Society of America.

Herbert Uberall (M’80) was born in Neunhrchen, Austria He received the Ph.D degree from the University of Vienna, Austria, in 1953, and the Ph D. degree from Cornell University, Ithaca, NY, in 1953 and 1956, respectively, both in theoretical physics, as well as an honorary Doctorate of Science from the University of Le Havre, France, in 1987

He was Assistant Professor of Physics at the University of Michigan from 1960 to 1964, and has been with the Catholic University of America,

Washington, DC, since 1965 as Associate Professor and then as Professor of Physics. He has been Consultant to the Naval Research Laboratory since 1966 and Consultant to the Naval Surface Weapons Center during 1976-1981, and he was a Visiting Professor at the University of Paris (Ecole Normale) in 1984-1985 His interests include theoretical nuclear physics in which he authored and coauthored several books, as well as theoretical acoustics and electromagnetic theory. In these latter fields, he has been instrumental, together with several collaborators, in developing the theory of resonance scattering of acoustic, elastic, and electromagnetic waves, and in establishing the connection between these resonance phenomena and the surface waves generated in the scattering process, utilizing the theory of nuclear resonance scattering for such purposes

Dr. Uberall is a Fellow of the American Physical Society, the Acoustical Society of America, and the Washington Academy of Sciences He is a member of the International Union of Radio Science and the American Association of University Professors