1 372 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994

An Absorber Tip Diffraction Coefficient

Philip J. Joseph, Albert D. Tyson, and Walter D. Burnside

.4bstract-The UTD corner diffraction solution for a perfectly conduct- ing corner is empirically modified for the case of a dielectric corner. The dielectric may be lossless or lossy, but is assumed to be homogeneous This modified solution is used to caldate the bistatic scattering from the tip of a dielectric pyramid. Sample calculations display some features of the scattering from a single lossy dielectric pyramid. To verify the solution, calculations are compared with backscatter measurements of a single pyramid that is cut from a homogeneous lossless dielectric (polyethylene). Calculations are then compared with measurements for the more pertinent case of bistatic scattering from a wall of pyramidal radar absorbing material.

I. INTRODUCTION This short paper is concerned with the construction of an approxi-

mate high-frequency solution for the diffraction of an electromagnetic wave incident on a dielectric comer surrounded by an isotropic homogeneous medium. This will allow us to calculate the diffraction from a pyramidal tip. Tip diffraction was shown to be the dominant scattering mechanism for pyramidal radar absorbing material when source and observer directions are away from nose-on [ 11, a case of great interest in anechoic chamber applications.

11. DIELECTRIC CORNER DIFFRACTION COEFFICIENT Kouyoumjian and Pathak [2] obtained a uniform asymptotic solu-

tion of Maxwell’s equations for the problem of electromagnetic waves obliquely incident on a curved edge formed by perfectly conducting curved or plane surfaces. Bumside, Wang, and Pelton [3] developed a comer diffraction coefficient to account for the termination of an edge in a finite perfectly conducting object. This coefficient was based on an asymptotic evaluation of the radiation integral for the equivalent currents that would exist in the absence of the comer. The asymptotic result was then empirically modified to yield their comer diffraction term.

In this paper, the comer diffraction formulation of [3] is modified for the case of a dielectric comer. The modification is an extension of the work of Bumside and Burgener [4] on the two-dimensional dielectric slab, and that of DeWitt [ l ] on the two-dimensional dielectric wedge. These efforts are reviewed before the dielectric comer diffraction solution is developed.

A. Two-Dimensional Dielectric Slab

In [4], Bumside and Burgener consider edge diffraction from a tu o-dimensional, semi-infinite, thin dielectric slab. They recognized that the edge-diffracted field of a conducting half-plane smooths the geometrical optics (GO) discontinuities at the reflection shadow

Manuscript received January 4, 1993; revised April 27, 1994. This work was supported in part by the Air Force Institute of Technology, Wright-Patterson AFB, OH, and by the National Aeronautics and Space Administration, Langley Research Center, Hampton, VA, under Grant NSG 1613 with the Ohio State University Research Foundation.

P. J. Joseph is with Dedicated Electronics, Inc., Chester, NH 03036 USA. .A. D. Tyson, IV is with Teledyne Brown Engineering, Huntsville, AL 3580

W. D. Bumside is with The Ohio State University ElectroScience Labora-

IEEE Log Number 9404493.

USA..

tory, Department of Electrical Engineering, Columbus OH 43212 USA.

Line Source .\ \ P’

Observer \\ .. .. .. ..

Fig. 1. slab.

Geometry for diffraction from conducting half-plane or thin dielectric

boundary (RSB) and the incident shadow boundary (ISB). For the dielectric slab, they note that these discontinuities are scaled by R and 1 - T , respectively, where R and T represent the total reflection and transmission coefficients of the thin slab. They thus proposed scaling the edge diffracted field in the same manner.

Recall that the diffracted field from a conducting half-plane (see Fig. 1) is given by [2]

where u i n C ( Q ~ ) is the field incident on the edge, and DI is the diffraction coefficient, which is given by

II

where

(3)

The Fresnel transition function, F [ .], and its arguments are defined in [2]. Note that for an electric (magnetic) line source, udif and u i n c ( Q ~ ) are scalar electric (magnetic) fields, and DI (Dll) is used. (The subscripts on D indicate the electric field polarization relative to the plane of incidence.) The first term of (2) is labeled ISB since it is responsible for smoothing the GO discontinuity at the ISB. The second term of (2) smooths the GO discontinuity at the RSB.

Implementing the scaling approach of [4], the edge-diffracted field of the dielectric slab is also given by (l), but with the diffraction coefficient written as

Note that for a perfect conductor

Ttotal I = 0 and Rtltal = q=1 II II

such that the proposed dielectric solution reduces to the correct result for the perfectly conducting case. The accuracy of this dielectric solution was verified by comparison with Moment Method results for a finite two-dimensional slab [4].

0018-9375/94$04.00 0 1994 JEEE

I

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994

Line Source Y ‘\, P’

\

Observer

F g. 2. Two-dimensional wedge geometry.

B Two-Dimensional Dielectric Wedge

The wedge geometry is shown in Fig. 2. The angles 4,4’ may be measured from either wedge face, which is then called the “0 face” (where o,df = 0); while, the other is called the “n face” (where Q. of = n r ) . The diffraction coefficient for the perfect electrical conductor (PEC) wedge has four terms [2 ] ; each term smooths the GO discontinuity at one of the four possible shadow boundaries (an ISB and RSB for either the 0 or n face).

DeWitt [ 11 sought to apply the approach of [4] to the more general problem of the two-dimensional lossy dielectric wedge. Since the total rcflected and transmitted fields are considerably more complicated for this geometry, DeWitt developed the following method which proved to work well. First, due to the vanishing wedge thickness at the edge, energy will pass through the wedge and tend to smooth the discon- tinuity at the incident shadow boundary. Thus the two ISB terms of the diffraction coefficient were considered negligible. Second, the RSB terms were scaled by the Fresnel reflection coefficient for the initial external reflection from the wedge. Finally, half of the angle bztween the incident and scatter directions was used as the incidence a igle in the reflection coefficient calculations. This forced the solution to satisfy the reciprocity principle. The effects of these choices are illustrated in [l].

The resulting formulation for the edge-diffracted field of a two- dimensional dielectric wedge is given by (l), but with the diffraction c >efficient written as

+ R I Cot [ 7T - (4 + ””1 F [kLa- (4 + 471). (5) II 2n

The reflection coefficients are given by

COSO‘ - J ( E ~ / E ~ ) - sin2 0 2

C O S O ~ + J ( E ~ / E ~ ) - sin2 8% RI =

and

u here 0‘ is half the angle between the incident and scatter directions, tl is the complex permittivity of the wedge (medium 2), and €1 is the permittivity of the lossless dielectric (medium 1) surrounding the u edge. Since the convention used for Oz is independent of the 0- and n -face surface normals, a single reflection coefficient scales both the 0- and n-face RSB terms of the diffraction coefficient.

This modified UTD approach was compared against a general integral equation for an infinite dielectric wedge derived by Rawlins [5]. Rawlins found a Neumann series solution of the integral equation tluough a perturbation technique. Taking the first term of this series, and using asymptotic methods, he found explicit expressions for

313

“‘ki Corner Diffracted \ \ !if I \ . Ravs

7 ISB

Edge Rays

Diffi racted

Fig. 3. Comer diffraction geometry.

the diffracted field of a right-angle dielectric wedge under certain constraints. The comparisons, which can be found in [l], show very good agreement between the two solutions.

C. Three-Dimensional Dielectric Wedge and Comer Diffraction

This section develops a dielectric comer diffraction coefficient based on the PEC comer diffraction coefficient of [3], and the dielectric modifications of [I], [4]. The comer geometry shown in Fig. 3 is a semi-infinite wedge with flat faces and straight edges. Incident, reflected, edge-diffracted, and comer-diffracted rays will exist. The edge-diffracted rays scatter along cones whose half angle is equal to the angle formed by the incident ray and the edge. (Note that edge-diffracted rays associated with one edge only are depicted.) Comer-diffracted rays scatter in all directions from the comer. The GO field will be discontinuous at each RSB and ISB, which are planar surfaces (only those associated with the vertical edge are depicted). The edge-diffracted field compensates for the GO discontinuities at the GO shadow boundaries. However, since each edge stops at the comer, each edge-diffracted field will stop abruptly and create a discontinuity at what is called an edge-diffraction shadow boundary (EDSB). It is the comer-diffracted field that eliminates the discontinuities at the edge diffraction shadow boundaries, thus accounting for the termination of the edge.

The dielectric comer diffraction coefficient will be found in two steps. First, the GO discontinuities for the dielectric case will be found and used to determine the dielectric edge-diffracted field. The discontinuities at the edge diffraction shadow boundary will then be known, and will be used to determine the dielectric-comer-diffracted field.

Let us begin with the edge-diffracted field of a perfectly conducting three-dimensional wedge with flat faces and a straight edge, as

774 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994

Observation Z . Point (01

Edge Of

/R Ordinary Plane

o f Incidence

Edge-fixed Plane of Reflection

(vtaw along reflactea ray touard 0,)

Fig. 4. Three-dimensional wedge diffraction geometry.

depicted in Fig. 4, which may be written as [2]

where A and B are matrices given by

L A L A

and -e-3"/4

2n&Z sin po D ( x ) =

x {Cot ( T ) F [ l C L U + ( Z ) ]

+ cot ( F ) F [ k L u - T - X (s)l}.

(9)

The matrices A and B account for the GO discontinuities at the ISB and RSB, respectively. The matrices A' and B' that properly scale the PEC solution for the dielectric case must be found. The discontinuities of the reflected field at the reflection shadow bound- aries are investigated first to find the matrix B'. Consider an incident ray reflecting off a wedge face, as shown in Fig. 5. The point of reflection is infinitesimally close to the edge, so that the reflected ray lies along a diffraction cone, on the lit side of the RSB. This reflected ray represents the discontinuity of interest.

Before continuing, several definitions are in order. The ordinary plane of incidence is defined as the plane formed by the incident ray and the surface normal. The incident ray and the edge define the edge-fixed plane of incidence; while, the reflected ray and the edge define the edge-fixed plane of reflection. Any diffracted ray and the edge define an edge-fixed plane of diffraction. (The edge-fixed plane

(View along lncldant r a y toward 0,)

Fig. 5. Pertinent field polarizations for reflection near wedge edge.

' E'

Fig. 6. Ray fixed coordinates for reflection.

of reflection coincides with one edge-fixed plane of diffraction since our reflected ray lies on the diffraction cone.) Further, let us define vz to be the angle between the ordinary and edge-fixed planes of incidence. (The subscript indicates whether the reflecting surface is the 0 or n face of the wedge.) The angle between the ordinary plane of incidence and the edge-fixed plane of reflection is also uz.

Since reflection is a local phenomenon at high frequencies, the fields reflected from the wedge faces are computed assuming plane- wave incidence and Fresnel reflection coefficients. The reflected field is most conveniently written as

where E'(QR) is the field incident at the reflection point, QR, s is the distance along the reflected ray from QR, and RII and RI are given in (6) and (7), respectively. Most significantly, though, notice that the incident and reflected fields are expressed in terms of ray- fixed component directions [2] (illustrated in Fig. 6) that are parallel

iEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994 315

m d perpendicular to the ordinary plane of incidence. On the other hand, the edge-diffracted field formulation expresses the incident field via edge-fixed components [2] parallel and perpendicular to the edge-fixed plane of incidence, and expresses the diffracted field via edge-fixed components parallel and perpendicular to the edge-fixed plane of diffraction.

To find the discontinuities in the reflected field in terms of these edge-fixed components, one can take the incident field expressed in its Pk, 4' components, transform it into its il, components, use ( 11) to obtain the reflected field in its 21. O i components, and then rransform the reflected field into its P o , 4 components. Doing so, one finds that the reflected field can be written as

[ 2$:))] RII cos' v o - RI sin' Y O

- (RII+Rl)s invo cosvo -RIIsin'vo + R l c o s ' v o (RII + R I ) sin Y 0 cos v o

n n

This result is valid for any incident direction which illuminates the wedge face. Since the reflected field is zero on the dark side of the KSB, the 2 x 2 matrix in (12) represents the GO discontinuities being sought. Thus the matrix B' is written as

R I I C O S ~ v o - R l sin2 Y O ./=[ n

(RII + R1)s invo cosvo

n n - ( ~ l l + ~ l ) s i n v o cos;o -~I Is in 'vo

(13) The coefficients of the matrix A' are set to zero since the ISB terms ;ire considered negligible per the approach of [l].

Using these results, the edge-diffracted fields of a three- dimensional dielectric wedge are given by

(14)

where

D, = ( R I I cos2 vn - RI sin2 v n ) D n ( 4 + 4' )

Db = (RII +RI)s inv ,cosv ,Dn(4+4 ' )

D c = -(R11+R1)sinv,cosv,D,(4+4')

Dd = (RI cos2 vn - Rll sin' Y , ) D , ( ~ + 4')

+ ( R I I cos' YO - RL sin' vo)D0(4 + 4') (15)

(16) + (RI I + R L ) sin vo cos v0Do (4 + 4')

- (RII +RI)s invocosvoDo(4+4 ' ) (17)

+ ( R l cos' vo - RII sin' vo)D0(4 + 4') (18)

and

An interesting feature of this approximate dielectric wedge solution i \ that the two field polarizations are coupled; whereas, they are uncoupled for the perfect conductor.

Now consider the comer-diffracted field associated with one comer and one edge of a finite perfectly conducting three-dimensional wedge in the near field with spherical wave incidence, which can be written as [61

Observer

Source

-7'

4̂'= : x p^, ; = 6 x F o , (not shown)

Fig. 7. Pyramidal tip diffraction geometry.

where A and B are defined in (9), the comer diffraction coefficient is given by

- e - j ~ / 4

2 n J G X s i n PO C ( X ) =

and the distance parameters are expressed as

S/S/' L = - sin2 PO

s' + SI'

and

Lc = scs s c + s'

The angles Pc, Po=, and the unit vectors bo=, 4, 4' are shown in Fig. 7. The comer diffraction coefficient C(z) is a modified version of the edge diffraction coefficient D ( z ) . The modification factor I F [ J w 4 4 ) / A ] 1

LL,a(.rr + P0c - P c )

is a heuristic function which helps to avoid the abrupt sign changes in C ( X ) as the observer passes through the geometrical-optics shadow boundaries of the edge.

The comer-diffracted field compensates for the discontinuities at the EDSB. A careful analysis of these discontinuities reveals that the

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994

matrices needed to properly scale the PEC solution for the dielectric case areA the same A: a n d B ' described earlier. (Note that at the EDSB, O,, = Po and BC = PA!) The comer-diffracted field associated with one comer and one edge of a finite three-dimensional dielectric wedge (flat faces, straight edge) in the near field with spherical wave incidence is then given by

sin pc sin POc e-3"/4

x - m (COSP,, - COSPc)

Y

where

and

This form will be used to calculate the bistatic scattering from the tip of an absorber pyramid. The solution is approximate, but it is seen to predict scattering characteristics of a single pyramid which agree with measurements and intuition (Section 111-B); the solution is also useful in predicting the general level of scattering from a large number of pyramids. For more details on the development presented in this section, the reader is referred to [7].

111. BISTATIC SCAlTERING FROM A DIELECTRIC PYRAMIDAL TIP

In this section, the application of the dielectric comer diffraction solution to the pyramidal tip is described briefly; detailed information is available in [7]. Sample calculations of pyramidal tip diffraction are then presented and discussed.

A. Calculation of fip Diffraction

The pyramidal tip is a comer in a three-dimensional wedge which has four planar surfaces and four edges intersecting at a common point. Recall that the comer diffraction solution gives the fields associated with one comer and one edge; thus it must be applied to the pyramidal tip four times.

To begin, consider the edge and face conventions illustrated in Fig. 8. The tip of the pyramid is placed at the origin of a Cartesian coordinate system. The four faces are oriented such that a vector normal to face 1 or face 3 has no y component; while, a vector normal to face 2 or face 4 has no 2 component. The pyramid's shape is characterized by a single angle, cy, which is the angle between my two adjacent edges. Consider the wedge angle, r , a variable that

Edge 1 0 Face Edge 4 N Face

(FACE 1)

Fig. 8. Conventions used for pyramid geometry.

depends only on the geometry of the pyramid. It can be shown that the wedge angle is given by

r = 2sin-l (T). This expression is of course valid for all four edges of the pyramid.

Next, an edge-fixed coordinate system is found for each pyramid edge. These are needed for the calculation of many quantities involved in the comer diffraction solution. The coordinate systems chosen are rectangular; each have an edge vector ( e ) directed along the edge toward the pyramid tip, a vector ( 6 ) which is normal to the 0 face of the corresponding edge, and a binormal vector ( e ' ) which lies along the 0 face and is defined by 6' = 6 x e .

The pyramid is illuminated by a spherical wave. Source and observation points are assumed to be in the far zone of the tip. Thus the field incident at the tip is appfoximately planar. The incident electric field is defined via its 8, 4 components and its incidence angles (P , 4*). The scattered field is determined in a direction specified by the scattering angles (es,4"). These angles are depicted in Fig. 8.

All required computations are explained in [7] and will not be included here. However, a few comments are in order. First, note that source and observer are assumed to recede into the far zone at an equal rate; that is, they are equidistant from the tip. Thus while the location of QE is not known, there is no ambiguity in the value of PA. Second, if & or dm is greater than n7r, then the diffracted fields associated with the mth edge are ignored. Although regions of space do exist where there is no diffracted field contribution from a given edge, each region of space has at least one edge contribution.

B. Sample Calculations

Sample calculations of monostatic and bistatic cross section are presented here. First consider the backscatter case where 4; = 4' = 0" and 0" < 8' = 8" < 90", as depicted by the solid curve in Fig. 9. The incident electric field is 8 polarized, the frequency is 5 GHz, and the pyramidal absorber is characterized by a = 21" and

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994 371

Fig. 9. Backscatter in the 4

Fig. 10.

20 h

E c n o 2 .- $ -20

x e

V

'2 -60

v

i V

-40

.a v1

-80

-inn ."" 0 IO 20 30 40 50 60 70 80 90

6 (degrees)

= O", 45" planes from a pyramidal absorber tip.

- 50 I

I I

- 55

3 -60 P s

h

g -65

2 6 -75

.- e

-70

U .- .a 2 -80 E

-85 _ _ 0 IO 20 30 40 50 60 70 80 90

8' (degrees)

Bistatic scattering in the t-z plane from a pyramidal absorber tip.

-,. = 1.45 - j0.58. This E,. is chosen as an estimate value over the irequency range 2-18 GHz, based on vendor data reported in [ 11. As 9' varies, the incident ray striking the tip sweeps through the 4 = 0" plane, which is perpendicular to face 1. As 8z increases through lo", a \mall discontinuity is seen; here face 3 drops into the shadow region, m d false shadow boundaries' exist for the 0 face of edge 2 and the I L face of edge 3. As 8' increases through 79", a singularity is seen because the source is broadside to face 1 (the observer is along both the GO RSBs of face 1 and the edge diffraction shadow boundaries of d g e s 1 and 4). Note that if one were to consider a pyramid of finite txtent (by including the diffraction from the base of the pyramid), this singularity would not exist. The singular component of the tip diffracted term would cancel with a like component from the base in that the structure would be finite as opposed to the infinite pyramid itudied here.

The second (dashed) curve in Fig. 9 is for the same case, except that (5' = 4' = 45". At 8' = 0", both curves have the same value since both represent nose-on incidence. This is also the lowest backscatter value on the plot, indicating that the absorber performs best at nose-on incidence, as expected. In this second case, as 8' varies, the incident

' This term is explained in [7].

ray striking the tip sweeps through the 4 = 45" plane, which contains edge 1. A discontinuity is seen at Oa = 14"; here faces 3 and 4 drop into the shadow region, false shadow boundaries exist for the 0 face of edge 2 and the n face of edge 4, and contributions from edge 3 are ignored since 43, &, > nT. This case has a singularity at OZ = 75", where the source direction is perpendicular to edge 1; in addition, it has another singularity at 8' = 90", where the source direction is perpendicular to both edges 2 and 4. In both instances, the source (observer) is along an edge diffraction shadow boundary. Measured data which support these calculations can be found in [8].

A bistatically scattered case is depicted in Fig. 10, where the incident direction is specified by 82 = 45", 4' = 0" and the scattered direction by 0" < 8" < 90", 4' = 180". Again, the remaining parameters are indicated in the figure. The first discontinuity, at about 8" = 30", is due to false shadow boundaries associated with the 0 face of edge 2 and the n face of edge 3. The second discontinuity, at about 8" = 55", is due to false shadow boundaries associated with the 0 face of edge 4 and the n face of edge 1.

Note that varying the frequency only shifts the plot up or down. This is true because only the tip contribution is considered, which is equivalent to the pyramid having infinite extent.

~

1

378 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994

4" 1 - t--) f)

0" 6 . 4"

* * 40"

Fig. 1 1 . Polyethylene test target (side and end views).

Azimuth Angle ( Degrees )

Fig. 12. Measured versus calculated tip scattering of polyethylene pyramid, 0 = 35".

-10

-LO

-30

-40

4 -50 T d v

9 -60 -70

-80

-90

-100 0 10 20 30 40 50 60 70 80

Azimuth Angle (Degrees)

Fig. 13. Measured versus calculated tip scattering of polyethylene pyramid, ( I = 27".

I V . EXPERIMENTAL VERIFICATION Calculations are compared against backscatter measurements of

a homogeneous dielectric (polyethylene) pyramid, and then against histatic measurements from a wall of pyramidal radar absorber. The polyethylene material used has a measured relative permittivity of 2.28 at 12 GHz, with no appreciable loss.

A. Monostatic Scattering from Polyethylene Pyramids

Polyethylene pyramids were constructed so that the dielectric comer diffraction solution could be tested against measurements of a homogeneous dielectric. Backscatter measurements were made by first orienting the pyramid for nose-on incidence, with the edges of the pyramid base in the vertical and horizontal directions. The frequency response was then measured from 6 to 18 GHz. The data were transformed to the time domain, where the tip term was isolated with a time gate (special care was taken in determining a valid procedure for choice of time-gate width and center [9]). The data were then transformed back to the frequency domain. The pyramid

WALL

TRANSMIT LOCATION

TYPICAL RECEIVER LOCAT ION

'TER

' NOTE : DISTANCE MEASURED TO VALLEY OF CONES

Fig. 14. Experimental setup of bistatic absorber wall scatter measurement 111.

T I M E I N NANOSECS

Fig. 15. Bandlimited impulse response of wall scatter measurement [l].

was then rotated 5" in azimuth and the process was repeated. Separate measurements were made on a 2-in sphere (rotated along the same arc) for calibration.

The results presented next were obtained from the test target shown in Fig. 11. The 6-in tall pyramid has an angle Q of 35"; while, the 8- in tall pyramid has an angle cy of 27". Fig. 12 shows measured versus calculated data for the 6-in tall polyethylene pyramid, horizontal polarization, 12 GHz, and azimuth angle varying from 0" to 70". (Nose-on corresponds to 0" .) Agreement between measured and calculated data is very good. For azimuth angles greater than 58", scattered energy from the pyramid base begins to enter the time gate used to isolate the tip. This may account for the rise in measured data relative to predictions in this range.

Fig. 13 shows measured versus calculated data for the 8-in tall polyethylene pyramid, horizontal polarization, 12 GHz, and azimuth angle varying from 0" to 75". The calculated curve predicts the trend of the measured data very well; however, predictions exceed measurements by up to 5 dB at certain azimuth angles. For azimuth

ll IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 4, NOVEMBER 1994

. . . . . . . . . . .

Incnherpnt A d d i t i o n

4 6 8 10 12 14 16 18 Frequency (GIIz)

379

Fig. 16. Comparison of frequency-domain data versus calculations, wall scatter experiment (experimental data from [I]).

angles greater than 66”, scattered energy from the pyramid base begins to enter the time gate used to isolate the tip. Again, this may account for the rise in measured data relative to predictions in this range.

A comparison of the above results shows better agreement for the shorter pyramid. This was verified with additional measurements not shown here. Measurements were also made for vertical polarization. Agreement between measured and calculated data was seen to depend slightly on polarization; however, pyramid height was a stronger factor.

H. Bistatic Scattering from a Wall of Pyramidal Absorber

In [l], an experiment is described in which two 3-ft-diameter parabolic dish antennas are used to measure the bistatic scattering from a wall of pyramidal absorber. The wall absorber consisted of 8-in tall pyramids. The two antennas had broadband TEM horn feeds at their focii. The transmitter produced a vertically polarized electric field. The experimental setup is depicted in Fig. 14. The transmitting antenna is positioned at an angle of 45” relative to the wall; while, the receiver is positioned at a variety of angles, and the wall was in the near field of both antennas. Thus an elliptical region (of about 10 ft’) of the wall is assumed to be illuminated by a plane wave. ’I he receiver location specified by 4’ = 45” is of most interest. At this angle, all illuminated pyramids have (ideally) identical incidence and scattering angles, and identical phase paths from the transmitting horn to the receiving horn. However, pyramids on a block of absorber have slightly varying heights, and blocks of absorber are not perfectly positioned on the wall. If all of the illuminated pyramids were perfectly positioned, one would expect coherent addition of the fields. Three cases are considered here. Fields of pyramids on the same block are assumed to add either coherently, incoherently, or via actual tip height measurements that were performed on sample absorber blocks. In addition, the fields of pyramids on different blocks are always a w m e d to add incoherently. (Of course, the pyramid fields could interfere and produce a total field less than the incoherent sum.)

Fig. 15 shows the bandlimited impulse response for the wall sc’atter. The location of the expected time of the tip and base returns is indicated. The tip retum is dominant (and would become even more so at greater bistatic angles). Fig. 16 shows the corresponding fi equency-domain data. The prediction based on typical tip-height

deviation agrees best with the measured data, as expected. The results show that this solution can be used to predict the levels of bistatic scattering from pyramidal absorber.

V. CONCLUSION A high-frequency dielectric comer diffraction solution has been

developed in this short paper. It is based on the perfectly conducting comer diffraction solutions of [3] and the dielectric modifications of [l], [4]. The validity of this solution was confirmed by constructing and measuring homogeneous dielectric (polyethylene) pyramids. The applicability of this solution to the bistatic scattering of pyramidal absorber with source and observer away from nose-on was then experimentally verified. Such bistatic scattering occurs in all types of measurement ranges; thus this solution can be used to predict the levels of absorber scattering into the test zone.

REFERENCES

[ 11 B. T. DeWitt, “Analysis and measurement of electromagnetic scattering by pyramidal and wedge absorber,” Ph.D. dissertation, The Ohio State Univ., Columbus, 1986.

[2] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly-conducting surface,” Proc. IEEE, vol. 62, no. 11, pp. 148-1461. Nov. 1974.

[3] W. D. Bumside, N. Wang, and E. L. Pelton, “Near-field pattem analysis of airborne antennas,” IEEE Trans. Antennas Propagat., vol. AF-28, no. 3, pp. 318-327, 1980.

[4] W. D. Bumside and K. W. Burgener, “High frequency scattering by a thin lossless dielectric slab,” IEEE Trans. Antennas Propagat., vol. AP-31, no. 1, pp. 104-111, 1983.

[5 ] A. D. Rawlins, “Diffraction by a dielectric wedge,” J. Inst. Math. Appl., vol. 19, pp. 261-279, 1977.

[6] P. H. Pathak, “Techniques for high-frequency problems,” in Antenna Handbook-Theory, Application and Design, Y. T. Lo and S. H. Lee, Eds.

[7] P. J. Joseph, “A UTD scattering analysis of pyramidal absorber for design of compact range chambers,” Ph.D. dissertation, The Ohio State Univ., Columbus, 1988.

[8] S. Brumley and D. Droste, “Evaluation of anechoic chamber absorbers for improved chamber designs and RCS performance,” in Proc. 1987 AMTA Symp. (Seattle, WA).

[9] A. D. Tyson, IV, “Electromagnetic scattering from dielectric pyramidal tips,” Masters Thesis, The Air Force Institute of Technology, 1991.

New York: Van Nostrand Reinhold, 1988, ch. 4.