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0018-926X/86/1200-1459$01.00 0 1986 IEEE

This ellipsoid is determined by its three semi-axes which can be statistically estimated from common radar tracking data. Obviously, such a ,characterization cannot be used to uniquely determine a target’s morphology-but it can be used in structural classification schemes precisely because it makes no assumptions about the statistics of the intensities of the individual scattering centers (other then their independence). This last feature may make this a very general target classifier.

REFERENCES

H. P. Baltes, Inverse Scattering Problems in Optics, Topics in Current Physics: H. P. Baltes. Ed. New York: Springer-Verlag, 1980. W-M. Boerner, Inverse Methods in Electromagnetic Imaging, NATO AS1 Series, Series C: Math. Phys. Sci., vol. 143. Dordrecht, Holland: Reidel, 1985. J . B. Keller, “The inverse scattering problem in geometrical optics and the design of reflectors.” IRE Trans. Antennas Propagat., vol. AP- 7, p. 146, 1959. J. J. Stoker, “On the uniqueness theorems for the embedding of convex surfaces in three-dimensional space,” Comnz. Pure Appl. Math., vol. 3, pp. 231-257, 1950. G . T. Ruck. D. E. Barrick. W. D. Stuart. and C. K. Krichbaum, Radar Cross Section Handbook, G. T. Ruck, Ed. New York: Plenum, 1970. B. V. Gnedenko, The Theory of Probability. New York: Chelsea, 1962. R. H. Delano, “A theory of target glint or angular scintillation in radar tracking,” Proc. IRE, vol. 41, p. 1778, 1953. M. Fisz, Probability Theory and Mathematical Statistics. Mala- bar, FL: Krieger: 1963. D. R. Rhodes, Introduction to Monopuke. New York: McGraw- Hill, 1959.

Physical Optics and the Direction of Maximization of the Far-Field Average Power

JOHN S. ASVESTAS, MEMBER, IEEE

Abstruct-For the problem of physical optics scattering by a perfectly conducting plate of finite dimensions and arbitrary shape, attention is drawn to the fact that the directions in which the far-field average power is maximized can be easily determined for H-polarization, while the same is not true for E-polarization. Moreover, it is shown by means of an example that the directions of maximization for E-polarization are not necessarily those for H-polarization.

I. INTRODUCTION

This communication is the result of some recent conversations the author has had on the subject of physical optics (PO) scattering by a perfectly conducting flat plate. The consensus was that, according to PO, an electromagnetic plane wave will bounce from a flat plate with the main beam directed according to Snell’s law for an infinite plane. It is shown here that this statement is true only for H-polarization

Manuscript received November 16, 1985; revised May 5, 1986. The author is with the Corporate Research Center, Grumman Corporation,

IEEE Log Number 8610047. Bethpage, NY 11714.

(magnetic field in the plane formed by the direction of incidence and the normal to the plate). For E-polarization, this is not always the case, and the fact is demonstrated by considering the far field of a square plate.

Due to reviewers’ comments, the author wishes to stress two points. First, although the PO theory has both physical and mathematical shortcomings [l], it is an effective means for comput- ing scattered far fields from large and complex structures, especially when supplemented by appropriate correction terms. Second, in no way is it implied here that in reality (or, in an exact sense), the direction of the main beam should be dictated by Snell’s law; in fact, the author is not aware of any proof to this effect.

u. APPROACH AND RESULTS

With respect to the rectangular coordinate system of Fig. 1, consider a bounded, perfectly conducting, plane surface S on the z = 0 plane and an harmonic (e-iur), plane wave impinging on it. The electric vector of this wave is given by

Ei = Eeikk^.P , E . E=o (1)

with E‘ a constant vector and the direction of propagation, given by

l = s i n eip+cos 0i2, -<0i<a. (2) a 2

According to PO, the far-field average power in the direction

i = P sin 0 cos Q+3 sin 0 sin Q + 2 cos 0,

0 < 0 < T, 0 < Q<2a (3)

is given by

Yk2 8a2 I SS l 2 P ( ? ) = - l i x [2x ( k x 8 ) ] I 2 exp { ik (E-? ) * p’) dxd~

(4)

where Y is the admittance of free space, and p’ is the integration vector on the z = 0 plane.

By the Cauchy-Schwarz inequality

1 ss I2 exp { i k ( E - i ) ,7) dxdy < A ~ ( s ) ( 5 )

where A ( S ) is the area of S, and equality is attained if, and only if

( E - ? ) $=O. (6)

From (2) and (3) this implies that

sin 0 cos Q = O (7)

sin 0i- sin Q sin 0 = 0. (8)

If sin 0 = 0, then sin 0; = 0. But ~ / 2 < Oi < a, so that sin Oi # 0. Thus, cos Q = 0, or Q = a12 or Q = 3a/2. If Q = 3 ~ / 2 , then sin Bi + sin 0 = 0, which implies that 0 = 0; f a. Neither of these values is in the range of 0. The only acceptable value of 6 is, then, 6 = a/2. This leads to 0 = Oi and 0 = a - Oi = $ j .

Thus, there are two directions that maximize the last factor in (4), namely

? = E , i=?,,=p sin o i -2 COS ei . (9)

1460 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-34, NO. 12, DECEMBER 1986

Z

f

X K

Fig. 1. Geometry of the problem (the vector k lies in the yz-plane).

The first is the forward scattering direction while the second is the direction of reflection as dictated by Snell’s law for the corresponding infinite plate. The maximization of P, however, depends also on the remaining factor in (4) that is a function of i. For H-polarization, and with

E= E2 (10)

this factor becomes

I ? x [ i x ( k ^ x f E ) ] 1 2 = ( E COS Oi)21ix2E(2 (11)

and is maximized for any i that lies in the yz-plane, its value there being constant. This result and the one in (9) imply that, for H- polarization, the power P is maximized along the directions given in (9).

For E-polarization, and with

,??=E~x~^=E (-cos Oig+sin e,?) (12)

the first factor in (4) becomes

I i x [ ? x ( k ^ x ~ ) ] I 2 = E 2 ( i x p 1 2 (13)

and is maximized for any i that lies on the xz-plane. Since the last factor in (4) is maximized on the yz-plane, no conclusion can be drawn as to the maximizing directions. The expression in (4) must, for E-polarization, be maximized in its totality rather than term-by- term, and this the author has been unable to accomplish. The notion, however, that these directions are given by (9) can be dispelled by considering the case of the square plate 1x1 < a, (yI < a. For E- polarization

f2[ka (sin ei- sin e sin 4)] (14)

where

f (O)= 1, f ( x ) = y , X # O . sin x

Even for this simple case, the conventional method of maximizing a function of two variables [2] can not be applied in a practical way. By plotting level curves of (14), it was ascertained that the maximizing directions occur when 4 = ~ / 2 . With this value of 4, and for various values of the side-to-wavelength ratio (2a/h), the angle Om that maximizes (14) was found through a computer search. The results are displayed in Fig. 2 and clearly show that 8, # $i, unless $i = 0. Moreover, the difference between $i and 8, increases as $i increases

Fig. 2. Main beam direction versus angle of incidence $t for various values of side of square-to-wavelength ratio (E-polarization).

or as 2a/h decreases. Besides Om, there is also the “forward scattering” direction .rr - 0, which gives the same maximum for (14).

ACKNOWLEDGMENT

The author wishes to thank Mr. Bernard Guarino of Gnunman Corporation for his programming work.

REFERENCES

El] J. S. Asvestas, “The physical optics method in electromagnetic

[2] W. Kaplan, Advanced Calculus, 2nd ed. Reading, MA: Addison- scattering,” J. Math. Phys., vol. 21, pp. 290-299, 1980.

Wesley, 1973.

Tree Attenuation at 869 M H z Derived from Remotely Piloted Aircraft Measurements

WOLFHARD J. VOGEL, MEMBER, IEEE, AND JULIUS GOLDHIRSH, SENIOR MEMBER, IEEE

Abstract-Tree attenuation results are described based on data acquired from an experiment employing UHF transmissions at 869 MHz between a remotely piloted aircraft and a stationary vehicle. The objective of the experiment was directed toward providing input to the land mobile satellite community where the extent of shadowing from roadside trees

Manuscript received February 4, 1986; revised May 8, 1986. This work was supported by the NASA Communications Division under Contract 00024-85-C-5301 and Jet Propulsion Laboratory Contract 956520.

W. J . Vogel is with the Electrical Engineering Research Laboratory, The University of Texas, 10100 Burnet Road, Austin, TX 78758.

J. Goldhirsh is with the Applied Physics Laboratory, The Johns Hopkins University, Johns Hopkins Road, Laurel, MD 20707.

IEEE Log Number 8610049.

0018-926X/86/1200-1460~01.00 0 1986 IEEE