Studies in Radar UMM-106 RL-2014 Cross-Sections VI Cross- Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies by R.R.Bonkowski, C.R.Lubitz, and C. E. Schensted Project MIRO Contract No. AF 30 (602)-9 Willow Run Research Center Engineering Research Institute University of Michigan UMM-106 October, 1953

roR INTrA-UNlVLSrrlt COiMrZPONIsNCB UNIVERS" OF MICHIGAN ENGINERING RESEARCH INSTITUTE 21 February 1958 MEMO TO: Prof. K. M. Siegel FROM: Security Office SUBJECT: Project 2015;This is to advise you that all reports and memos generated on Project 2015 (AF30(602)9 are unclassified. Please see that all documents in your possession are marked in the manner illustrated below. Regrading markings should appear on front and back covers, title, first and last pages, with the previous classification lined through. CLASSIFICAT;ON CIH-IANGED To C AS PER AUTHORITY Ap#'O 46P-af-.t Bulk files of aforementioned reports need not be so marked ifmediately. Regrading markings can be affixed when the documents are being used or charged out or transmitted. However, if bulk files are in your possession, the change of classification should be indicated inside the file drawer or storage container. T. J. 8C Security Officer Cf c.c.: J. H. Richter

-—,- UNIVERSITY OF MICHIGAN A UMM-*I Ob Studies in Radar Cross-Sections - VI: Cross-Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies by R. R. Bonkowski, C. R. Lubitz, and C. E. Schensted (UMM-106, October 1953; Reprinted June 1954). Contract No. AF 30(602)-9. SECRET (UNCLASSIFIED when Appendix is removed). Errata in First Printing Pg. 11, Eq. 2.4-3 A A Replace k + 2(k.... by k A k - 2(k.... Pg. 16, Caption on the horizontal scale should be DEGREES r DEGREES pg. 39, Eq. 3.4-6 The last term should be (z+b ) dO Replace "For 0 =.... Pg. 41, line below Eq. 3.4-16 by "For 0 = 0.... Errata in Second Printing Pg. 39, Eq. 3.4-6 Pg. 41, line below Eq. 3.4-16 Addenda None Same as in 1st printing Same as in 1st printing -- I - -

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN ___________ UMM-106 TABLE OF CONTENTS NUMBER TITLE PAGE List of Figures ii Nomenclature iv Preface ix I Introduction and Summary 1 II The Corner Reflector 3 2.1 Analytical Method for Determining the Radar Cross-Section of the Corner Reflector 3 2.2 An Optical Model for Corner Reflectors 5 2.3 Monostatic Cross-Section of Square and Triangular Corner Reflectors 7 2.4 Bistatic Cross-Section of a Square Corner Reflector for the Symmetric Case 10 2.5 Effect of Constructional Errors, Compensation, and Truncation 21 III Other Multiple Scatterers 22 3.1 Formulas for Scattering from Curved Surfaces: Fock's Method 22 3.2 Scattering from Two Spheres 24 3.3 Formulas for Scattering from Curved Surfaces: The Method of Stationary Phase 31 3.4 The Biconical Reflector 38 IV Experimental Data on Multiple Scatterers 43 References 52 II I I I i

Iov

~T~Z

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ___________ UMM-106 NOMENCLATURE -. The symbol for a vector is an arrow over the letter, e.g., R. The unit vector parallel to t is written as R while R stands for the magnitude of R. Thus f = RR. The subscripts i and s stand for incident and scattered respectively. ei'ut time dependence is used throughout. The transpose of a matrix is represented by a tilde. Thus the transpose of M is M. SYMBOL MEANING SECTION FIRST USED IN 2.4 A a A Hi ^ ^ ^ x + ay iy + a A aH A b b1 b2 B c C 1,C2, C3 d D E E,F,G 8pq A 1 1 ^ aH = -x +J7 Iy Area of equivalent aperture Side length of a corner reflector Smallest radius of the cones in a biconical reflector Largest radius of the cones in a biconical reflector Radius of gyration Velocity of light Defined in Eq. (2.4-9) Distance between two spheres Cross-sectional area of a bundle of rays Electric field A A A A A k + R = E ix + F iy + G iz d0r 2 = guu du2 + 28uv dudv + 8vvdv2 and 2.4 2.1 2.3 3.4 3.4 2.1 2.4 2.4 3.2 3.1 3.1 2.4 gUV = gvu 3.1 iv I IIII I I

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN __UMM-106 I II1 8Pq Gpq guu guv UU V 1p q G = - - anx an - pq = p qq -P aq 9p 'guv guu_ az q 3.1 h H HN A A ^ AA/i pi The characteristic dimension of a body The magnetic field A unit vector parallel to the N-tuply scattered field Unit vectors along the x, y, and z axes I I =... Jeik? MS e d Je i Mtdgl....dt 2N - N k A k K,m,n M n A n A N P,q ro r k= 2. A E. x H. k = 1 1 Ei x H I.= Kbn Direction cosines between the direction to the transmitter and the axes of a corner reflector Defined in Eq. (3.3-8).= Kbn A A Unit outward normal to surface = nix + nyiy + nziz Unit normal on the j'th reflecting surface Number of reflecting surfaces May equal either u or v Radius of a sphere Radius vector from origin to integration point 3.1 2.1 2.1 3.3 2.4 3.3 2.1 2.4 IV 2.3 3.3 IV 2.1 3.3 3.3 3.1 3.2 2.1 v I II s I I I I I I [ I I I I

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN _UMM-106 rj 3J R R R RN R' S 81,82,83 t T pq TP q Radius vector from arbitrary reference point to a point on the j'th reflecting surface Distance from specular reflection point on upper cone or from left hand sphere A Vector from origin to field point = RR Unit vector from origin to field point = ^ A A Rxix + Ryi + Rziz Vector from the j'th reflection point to the (j + i) st reflection point Vector from the N'th reflection point to the field point Vector from integration point to field point Surface of scattering body Defined in Eq. (2.4-9) Time T =g -.q + R(q - cos G ) pq pq p q pq pq TP = gP T + gP T q uq vq Curvilinear coordinates on the scattering surface May equal either u or v Cartesian coordinates Cosine of the angle between the x. and x xj and Yj+' and xj+1, and y and yj+ axes respectively cos /= R z R tan ' =-Y R X 3.3 3.2 2.1 2.1 3.3 3.3 2.1 2.1 2.4 2.4 3.1 3.1 3.1 3.1 2.4 3.3 2.4 2.4 u, v w x,y,z jil j12' (jz1' ~(j2z vi. I

VWILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN ____ UMM-106 qw A1' 2' s1t & '~j 2 ij $. J /^ d r T Christoffel symbol of the second kind = gpu [qw;u] + gpv [qw;v] qw Angle between symmetry axis of corner reflector and direction to transmitter in degrees Measure of constructional error Angle of incidence Polar Angles Wavelength =1' 2' 3'2''' 2 N 2j jl and 2;j-1 ' x Radius of curvature Radar cross-section Element of arc length r b= L -- sin/-cos3] Angle between center of beam and half-power point Azimuth angles 211c ) = kc = ik f (uv) eik () is the phase factor for a wave incident on a surface np= -a ap 3.1 2.3 2.5 3.1 3.2 2.1 3.3 3.3 3.3 2.1 3.1 2.4 2.1 3.2 2.4 C 1 2 3.1 3.1 np vii

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGANI UMM-106 &< a42 - r - rV a_ 3.1 p 9q sq pq au pq av [p,q; w] Christoffel symbol of the first kind 3.1 agpw gqw agpq [pq;w] a + p cl q?p a ) viii.. Vlll I

W ILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 PREFACE This paper is the sixth in a series of reports growing out of studies of radar cross-sections at the University of Michigan's Willow Run Research Center. The primary aims of this program are: (1) To show that radar cross-sections can be determined analytically. (2) To'elaborate means for computing cross-sections of objects of military interest. (3) To demonstrate that these theoretical cross-sections are in agreement with experimentally determined values. Intermediate objectives are: (1) To compute the exact theoretical cross-sections of various simple bodies by solution of the appropriate boundary-value problems arising from the electromagnetic vector wave equation. (2) To examine the various approximations possible in this problem, and determine the limits of their validity and utility. (3) To find means of combining the simple body solutions in order to determine the cross-sections of composite bodies. (4) To tabulate various formulas and functions necessary to enable such computations to be done quickly for arbitrary objects. (5) To collect, summarize, and evaluate existing experimental data. Titles of the papers already published or presently in process of publication are listed on the back of the title page. K. M. Siegel ix

WILLOW/ RUN RESEARCH CENTER-UNIVERSITY OF MICHIG. _______ UMM-106 ___... FIG. 1 MULTIPLE SCATTERERS I IIIII I I I I x _ i III I I I I! II I

,LOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 I INTRODUCTION AND SUMMARY If a body is not convex, radiation incident on it may be reflected a:iumber of times from one part of the body to another before finally being reflected away from the body. These multiple reflections have an impor-:ant effect on the radar cross-section of a complicated body such as an airplane. Therefore, as part of Willow Run Research Center's program 3f determining radar cross-sections, this study of the radar crosssections of multiple scatterers at short wavelengths has been made. rhis paper presents a summary of known data on multiple scatterers,;ogether with a few new formulas for special cases. The best known and best understood example of a multiple scatterer is the corner reflector, which is widely used as a beacon and as a stand-.rd in experimental determinations of cross-section. A corner reflector consists of sections of three mutually orthogonal planes, and has the characteristic property of giving a large monostatic cross-section over i wide range of directions of incidence.* A simple approximation to the bistatic cross-section of a corner reflector is given in Equations (2.1-5) and (2.1-6). An optical model to 3e used in conjunction with Eq. (2.1-5) for determining the monostatic:ross-section of a corner reflector is described in Section 2.2. Explicit expressions for the monostatic cross-sections of square and triangular:orner reflectors are given in Section 2.3. A study of the bistatic crossiection of a square corner reflector with the transmitter on the axis of symmetry is made in Section 2.4. A discussion of the effects of conAtructional errors, compensation, and truncation is given in Section 2.5. When the multiple scatterer has surfaces which are curved the:ross-section may be obtained by applying Eq.(2.1-4). The application )f this formula involves the geometrical optics approximation to the ields on the scattering surface and this is given in Eq.(3.1-1). In the special case when the radii of curvature of the scattering body are finite It all of the reflection points the cross-section may be obtained by using aq.(3.3-10) and,(2.1-3). To illustrate the methods used, the crossiections of a biconical reflector and of a pair of spheres are obtained Sec. 3.2 and 3.4). Certain closely related configurations are also commonly referred to as corner reflectors... 1

W'ILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM- 106 A sampling of experimental data on corner reflectors is quoted in Section IV. The authors wish to express their appreciation for the kind permission of the Bell System Technical Journal to reproduce Figures 17-21, and of Dr. R. D. O'Neal to reproduce Figures 15 and 16 and the figures in the appendix. 2 - -

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-106 II THE CORNER REFLECTOR* 2.1 Analytical Method for Determining the Radar Cross-Section of the Corner Reflector Although the simplest method for obtaining radar cross-sections is the method of geometrical optics (References 1 and 2), this method is not directly applicable to corner reflectors because it predicts that the radar cross-section is infinite in the directions in which radiation is specularly reflected and zero elsewhere. More explicitly, for a scattering body consisting only of plane surfaces, geometric optics predicts that the incident radiation is scattered into a region which, at large distances from the body, subtends a vanishingly small solid angle. Actually the radiation must be spread, by diffraction, over a region of solid angle ( X / h)2 where X is the wavelength of the radiation and h is the characteristic dimension of the body. Near the body this objection no longer exists so that geometrical optics can be used to obtain the fields on the surface of the scatterer when A << h. When the magnetic field is known on the surface of a perfectly conducting body the following formula (Reference 3, page 466) can be used to obtain the scattered magnetic field at any point in space: 1 eikR' = (n x H)x dS (2.1-1) where H is the magnetic field on the surface of integration, Us is the scattered magnetic field, n is the outward unit normal to the surface, k = 2 (/. R/ is the distance between the field point and the integration point. Thbe taken over the entire surface, S, of the body. When the field point is at a large distance from the body, (2.1 -1) can be approximated by *Much of the material presented in this section appears in Ref. 4. i I I III I II. 3 I

9MISSIN

L WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN ___ UMM-106 where A is the area of the above-mentioned aperture. The angle, SY, between the beam direction and the direction in which the radar cross-section has decreased by a factor of two is approximately (Ref. 4) = 7.50~ /B........ (2.1-6) where B is the radius of gyration of the aperture taken about an axis through the center of gravity of the aperture and perpendicular to the plane in which the deviation from the center of the beam is taken. 2.2 An Optical Model for Corner Reflectors. One of the beams in which the scattered energy is concentrated is reflected back toward the transmitter. The value of A for this beam determines the monostatic cross-section through (2.1-5). The task of obtaining A analytically can be avoided by use of an optical model which, looked at from any direction, presents an aperture whose projected area is A. Such an optical model can be constructed by cutting appropriate openings in three mutually orthogonal opaque sheets (Ref. 4 and 5). For the corner reflector in Figure 2, the openings are as shown in Figure 3. Each of the three apertures shown in Figure 3 is obtained by cutting one of the faces of the corner reflector-out of each of the four quadrants so as to give a symmetrical figure. Figure 4 shows the optical model consisting of the three apertures of Figure 3. An optical model for any corner reflector can be constructed in precisely the same manner. Z t FIG. 2 A CORNER REFLECTOR I I ~ 5

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-106 x FIG. 3 APERTURES IN OPTICAL MODEL OF CORNER REFLECTOR z The two views form a stereo pair A three dimensional effect may be obtained by focusing the right eye on the right view, the left eye on the left view, and then superposing the images. Alternatively a standard stereoscopic viewer may be used. FIG. 4 OPTICAL MODEL OF CORNER REFLECTOR.. 6

WILLOWV RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN. UMM- 106 2.3 Monostatic Cross-Section of Square and Triangular Corner Reflectors. In Reference 4, the value of A for the beam reflected towards the transmitter has been determined analytically for both square and triangular corner reflectors (Fig. 5). i i Z_7 - b Square Corner Reflector Triangular Corner Reflector FIG. 5 SQUARE AND TRIANGULAR CORNER REFLECTORS The value of A is expressed most simply in terms of the cosines of the angles between the axes of the corner reflector and the direction to the transmitter. If these cosines are.< m _ n, then A is given by: For a Square Corner Reflector: A = 4m b2/n, (m ' n/2)......... * (2.3-1) A=.(4 - b2 b, (m > n/2) For a Triangular Corner Reflector: Im b2 (~+m'n) A= 4- b (+ m n) +( m T n......... (2.3-2) A = + m + n m + n+ m n) The transmitter direction making equal angles with the three axes is a symmetry axis for square and triangular corner reflectors. If Jc is the angle, in degrees, between this symmetry axis and the direction I 7,

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 ___ to the transmitter, then, for small S, A is given by: For a Square Corner Reflector: r. —. a. A _ \j 3 (1-0.0274~ )b'............ (2.3-3) For a Triangular Corner Reflector: A l (/-)(-0.00076 2)b2.............. (2.3-4) From these equations and from (2.1-5) it follows that the dimensionless quantity QY. /41b4 = A /b depends only on the direction to the 2 4 transmitter. Curves of constant A /b are plotted in Figure 6 for a A2 2 square corner reflector using the trilinear coordinates, m, and n As can be seen from (2.1-5), (2.3-3), and (2.3-4) the maximum values of or for square and triangular corner reflectors are 1211'b / 2 and 41b /3)2 respectively. 8 [

W - - inI..... -*TY OF MICHIGAN i, { I- - ^9WBHPII^''qBUV9 "H~iBBMBHH~ W~j~mc exW~in~winx w i~l~^JrKLTIJ A -I m2 0.7 N 0.8 0.9 1.0 ~& a W * 3.0 2.5 2.0 1.5 1.0 0.5 - 0.3 0.2 0.1 I -- I 0 I& I I f I 0 0.1 0.2 0.3 0.4 A5 nA 0.7 C n2 The curves drawn here are for V = constant 4 rb b ).8 0.9 1.0 FIG. 6 THE MONOSTATIC RADAR CROSS - SECTION OF A SQUARE CORNER REFLECTOR 9

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-106 2.4 Bistatic Cross-Section of a Square Corner Reflector for the Symmetric Case. The analytic methods described in Section 2.1 are applicable to both the monostatic and bistatic cross-section problems by a suitable choice of the radius vector from the body to the field point. To illustrate the procedure for computing the bistatic cross-section, consider the case of a square corner reflector of side length b. The orientation of the transmitter is as indicated in Figure 7. The receiver is restricted to the first octant (x 0, y 3 0, z D 0). Transmitter Direction of Incident Radiation._rc cos y Ax FIG. 7 TRANSMITTER ORIENTATION FOR THE SYMMETRIC BISTATIC CASE When the wavelength of the incident radiation is less than the side length b, the radar cross-section is determined almost entirely by the triply reflected radiation. Thus, to apply (2.1-1), it is only necessary to obtain the magnetic field, H, for the triply reflected rays. Consider a ray reflected first from the x-plane, then from the y-plane, and finally from the z-plane, and let the incident magnetic field be A,L " ^ -iC(t - ) H. = a e c /.......... (2.4-1) 1 A where a is a unit vector. Suppressing the time factor e-iJt, the magnetic field along the ray going from the x-plane to the y-plane is A ik -(k. i )i. r a - 2 (a. i )i e. (2.4-2) X X 10. II~~ I

VWILLOWV RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN __ UMM-106 where ix is the unit vector in the x direction. The magnetic field along the ray going from the y-plane to the z-plane is ^ A ( ] -ik k + 2 (k. i ). i-a+Z a. i )iz e....... (2.4-3) z z A A A._ On the z-plane r = xi + yi, so that i * r x y z to this surface, then A, A T-ik(k. nx H = - 2 (i xa) e In general, for triply reflected radiation, A = 0. If n is a unit vector normal........... (2.4-4) -ik. r) A - 2 A A nx H= - 2 (nx a) e........ (2.4-5) on the scattering surface. It is still necessary to determine how much of the corner is illuminated by such triply reflected radiation. A consideration of the optical model shows that the entire corner is illuminated for the transmitter orientation of Figure 7. For orientations of the transmitter other than that in Figure 7, the corner is not entirely illuminated. However, these orientations present no new problems, since the part of the corner that is illuminated in these cases may also be found from the optical model. a 11,

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN _____ UMM-106 r i From (2.1-4) and (2.4-5) the cross-section of the corner reflector is A A A Cr =- (i x a)x R e dxdy 4 1b 1b x A -ik (k + R)^. r O 0 b b ^ ^ -ik (k+ R). r + / r (i xa) x R e dzdx (2.4 J J Y A A% 2 b ^ ^ ^ -ik (k + R). r o O A A A A A A A A A A A Let R =R i +R i +R i,k+R = Ei +Fi +Gi,and a =a i + xx yy zz x y z xx A A a i +a i yy zz In this notation (2.4-6) becomes 4 A A b b -ik (Ex + Fy) = i +R a i - (R a +R a )i e d Z x zxx yy xx yy JO (2.4 A A A o b -ik (Fy + Gz) + R a i +Ra i -(R a +R a )i x / e dyd r A A fb pb -ik(Gz+Ex) — 6) lxdy ( 1-7) z I z + R a i y zz,.' + R a i -(R a + R a )i y xx z z xx y J J e 0 0 dzdx. 12

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN e pnUMM-106 After performing the integration, (2.4-7) becomes X2 01 = — 4.11'3 A R a i z xx A +R a i -(R a Z VV xx + Ra )i e v v z, -ikb(E + F ) -ikbE -ikbF -e -e +1 i i i iiii i ii ~1~__ L A + R a i + x yy r ^ - " J EF A ^ -ikb(F + G) -ikbG -ikbE R a i -(R a + R a )i i e. -e -e +1 xzz yy z zx F(248 FG (2.4-8),1 -ikb(G + E) -ikbG -ikbE 2 + Rai +Rai -(R a +R a ) y zz yxx z z xx y.R~~y fi I! I I l -e -e +1 GE To simplify (2.4-8), the following condensation symbols are introduced. Cl = c2 = 2 C = 3 8 = cos kb(F + G) -cos kbF -cos kbG -1 FG cos kb(G+ E) -cos kbG -cos kbE -1 GE cos kb(E + F) -cos kbE -cos kbF -1 EF sin kb(F + G) -sin kbF -sin kbG FG sin kb(G + E) -sin kbG -sin kbE GE sin kb(E + F) -sin kbE -sin kbF EF (2.4-9) s3 - - 13

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAT UMM-106 In this notation, the radar cross-section of the square corner reflector for the symmetric case is given by 2 1 yx22 = 3 (R + R a ) c + R a c +R a c + R a c, -(R a +R aZ) c + R a c3 + R a c +R a c -(R a +R a ) c x z yz 2 xx yy + Rxa -(R a +R a )s 2+R a s xx z xz 2 zy3j 12 + Rxa s + R as -(R a +R a ) s3 z 1 yz 2 xx yy This formula gives the radar cross-section for any polarization of the incident electromagnetic wave. To show how the bistatic radar crosssection varies as a function of receiver position for this symmetric case, (2.4-10) has been plotted in Figures 9, 10, 11, and 12 for a corner reflector of side length b = 25 cm., for three values of wavelength, and for the incident magnetic field vector parallel to one of the coordinate sur1 1 A X faces, that is aH= H +_ The polar angles designating receiver position are indicated in Figure 8. 14 IIIII I I I

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN _____ UMM-106 Receiving Antenna y x FIG. 8 POLAR ANGLES,f AND Y DESCRIBING RECEIVER POSITION Figure 9 shows the variation of C" with,c for/ = 54.74~ and for wavelengths of 3, 10, and 30 cm. The variation of 0" with/? for a wavelength of 3 cm. and, = 150, 30~, and 45~ is shown in Figure 10. The " = 45~ values were obtained at two degree intervals while the b = 15~ and = 30~ values were obtained at 10 degree intervals. Because the 10~ interval is too large to show the variation of O" with /3 accurately, curves have not been drawn for a wavelength of 10 cm. Figures 11 and 12 show the variation of 0' with/5 for r = 45~ and wavelengths of 10 and 30 cm. respectively. II 15

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN I UMM- 106 1 3.8 3.4 3.0 2.6 T I-v.. 2.2 1.8 1.4 1.0.6 TA3 cm =10 cm =30 cm,,,4S 25 35 45 DEGREES 55 65. = 54.74' X = Wavelength Direction of Incidence along Axis of Symmetry Incident Magnetic Field Parallel to x - y Plane FIG. 9 BISTATIC RADAR CROSS - SECTION OF A SQUARE CORNER REFLECTOR AS A FUNCTION OF Y 16,

I,1Iwbi, A 'I I I I I I A I M W EC I a-C IegI MepSS.PS.IP. tAn of Pn oo V- y4NW 3.0 07.p15. l 2.3 __ _ The Vokumfor 3, C, 0, werO abiainod from Equation (2.4 - 10) 2.0 -Equation (2.4-10) Equation (2.4.-11) 1.5 1.4 1.3 1.2 1.1 1.0.9.8.7.6'. —.5.4-.3.2 -El9_ _ _ _e I.1 0 0 10 20 30 40 50,e-DEGREES 60 70 80 90 FIG. 10 BISTATIC RADAR CROSS SECrION OF"A SQUARE CORNER REFLECTOR AS A FUNCTION OF 6 FOR A WAVELENGTH OF 3 CM 17 - -

WILLOW RUN R.ESEARCH CENTER UNIVERSITY OF MICHIGA UMM- 106 --- -- Sidelength of Reflctor = 25 cm Direction of Incidence: along Axis of Symmetry Incident Magnetic Field Parallel to x - y Plane ~ I I I I I Eu E I o $ 0 10 20 30 40 50 8 - DEGREES 60 70 80 90 FIG. 11 BISTATIC RADAR CROSS-SECTION OF A SQUARE CORNER REFLECTOR AS A FUNCTION OF 3 FOR A WAVELENGTH OF 10 CM i... 18 - - - --

W i r W Wo C m ama UMM.106 3.5 _ __ — __ — _il -.. 3.0 - Sdlength of RIefctor 25 cm Direction of Incidece: long Axis of Incidnt Mo ic Field IoroWl to x 2.5 Y 45' 2.0 - Equation (2.4-10) -- Equation (2.4-11) V I 1 I~~~~~ ~ ~ ~ -iawaL-w i-. -. It,adz - -Clw - -am:tr ~~iS; -k A 40 50 - DEGREES FIG. 12 BISTATIC RADAR CROSS-SECTION OF A SQUARE CORNER REFLECTOR AS A FUNCTION OF f3 FOR A WAVELENGTH OF 30 CM 19. I I I

SWILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-106 As was noted in section 2.1, the scattering pattern of a corner reflector is approximately the same as the diffraction pattern of an equivalent aperture. For the symmetric case considered, the diffracting aperture is hexagonal. For y = 450, the bistatic radar cross-section for this aperture is 1081b4 2 By 18, 4 (sinfsinr/3) (. 4-11),X I'71b [ 1 where b = --- sin{l - cos | The values of radar cross-section (2.4-11) as predicted by this equivalent aperture are also plotted in Figures 10, 11, and 12 for comparison with the values obtained from (2.4-10). It should be noted that the halfpower widths given by both (2.4-10) and (2.4-11) agree with the values predicted by (2.1-6). Although the geometric optics and physical optics approximations are based on the assumption that the wavelength is small compared to the characteristic dimension of the body, there is reason to believe that the error introduced by the use of these approximation techniques when b/A is approximately one is sometimes much less than an order of magnitude. Kouyoumjian (Ref. 6), for example, has found that the monostatic radar cross-section predicted by physical optics for a flat plate at normal incidence does not deviate from the exact electromagnetic solution by more than a factor of five for the range b/A between 0.8 and 5. Since it is not likely that exact computations will be made of the cross-section of corner reflectors in the near future, and since there is reason to believe that the approximation techniques do yield order of magnitude answers for the square corner reflector for X ~ b, these techniques have been applied for a wavelength of 30 cm. (i.e. )/b = 1.2). 20 II I II I

L WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 2.5 Effect of Constructional Errors, Compensation, and Truncation. Corner reflectors are generally used to direct a large signal back toward the transmitter. This large signal is reduced in intensity if the corner is not perfectly constructed. If the faces of a corner reflector do not meet at exactly 900 then the beam which would have been reflected back to the transmitter is divided into several beams, none of which, in general, are directed exactly toward the transmitter. As a result, there will be a reduction in signal received at the transmitter. In Reference 4 the magnitudes of the errors which reduce the signal returned by square or triangular corners (Fig. 5) to one half the maximum returned signal are calculated. This error,A, is determined as follows: if one of the faces of the corner is rotated about one of the coordinate axes through which it passes, then A is the distance which the part of the face farthest from the axis moves. These errors are independent of the size of the corner, and therefore are difficult to avoid for large corners and small wave lengths. For incidence along the axis of symmetry these errors are Square Corner: one error, A =.40 | three equal errors, A =.24l Triangular Corner: one error, A =.70 X three equal errors,AL =.35)X For some applications, such as a movable corner used as a beacon, it is desirable to sacrifice some of the strength of the returned signal in order to obtain a usable signal over a wider range of incidence angles on the corner. This flattening and widening of the monostatic response pattern can be accomplished by truncation or compensation (Ref. 5), i.e., the removal of some of the reflecting surface (see Sec. 4). - 21.

W ILLOYW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 III OTHER MULTIPLE SCATTERERS 3.1 Formulas for Scattering from Curved Surfaces: Fock's Method. In Section II only scattering from surfaces having infinite radii of curvature was considered. In this section multiple scattering from surfaces having finite radii of curvature will be considered. In Reference 7, formulas are developed for the scattering from curved surfaces. These formulas, which are useful for computing the cross-section of bodies with curved surfaces, are summarized in this section. The scattered electric and magnetic fields, as given by geometric optics, are A r AA D (O) ikR E = E. 2n x (E x n) e s D(R) (3.1-1) A l D (0) ikR H H -2 (n. H.)n e 1 1 D(R) where D(R) is the cross-sectional area of a bundle of rays at a distance R from the specular reflection point, and Ei, H. is the incident field at the specular reflection point. The area of the bundle of rays at a distance R is given by Tu TU U V D(R) = (3.1-2) TV Tv u v where TP is the symmetrical tensor q 22 -~

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ____ UMM-106 T = gpu T +g T q uq vq (3.1-3) and T =g - nq + R (q - cos G ). Pq Pq P q Pq G Pq (3.1-4) Here u and v are curvilinear coordinates on the scattering surface and g is the metric tensor given by Pq 2 2 2 d0 = g du +2g dudv +g dv2 uu uv vv (3.1 -5) where dOT is an element of arc on the surface. The gPq (3.1-3) are related to the g by that appear in uu uv g g vu vv g g 1 2 guu vv uv 1 (3.1-6) G is the curvature tensor of the surface given by pq an @n an axn Y 91 -z -G = xS + _y + z z pq 9p aq app q a P q (3.1-7) where n, n, and n are the components of the unit normal to the surx y z face at a point x, y, z of the surface. The angle ~ is the angle between the direction of incidence and the normal to the surface. -a is defined in terms of the phase of the incident wave on the scattering surface e ik(uv). f is the ordinary derivative of -1 with respect to p. p 23

XWILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN __ UMM-106 JP = p (3.1 -8) fL is the second covariant derivative Pq 2 1p apa pq opdq pu au upq u F v pq E)v (3.1-9) p is the Christoffel symbol of the second kind, qw w =gu [q,w;u] + gv [qw;v] (3.1-10) and [p,q;w] is the Christoffel symbol of the first kind p1 w qw g pa [p., g;w + - 2 9q -rp l w (3.1-11) 3.2 Scattering from Two Spheres As an example of the application of the formulas in Section 3.1, consider the back-scattering from two spheres of equal radius for an electric field '1 e-ikz incident perpendicular to the common axis of the spheres.. 24 - I-II

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 (Positive x-axis points into the paper) jA-R, z1 FIG. 13 REFLECTION GEOMETRY FOR TWO SPHERES First, consider the doubly reflected ray shown in Figure 13. For the reflection from the first sphere the coordinates u and v are u 1=6, v I > (3.2-1) where 1 and 1 are related to the Cartesian coordinates by x= r sin e1 cos, y= r sinel sinZl, (3.2-2) z= r cos91 o 1 The normal on the surface of the sphere is n = sin sinl i + cos 1 i (3.2-3) Thus, by (3.1-7) - G[;:: G:I: r 0 (3.2-4) G0l91 Gi 0 r sin2Ei m. 25

W/ILLOWX RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ___________ UMM- 106 The square of the element of arc on the sphere is dr2 = r 2 d62 +r 2 o 1 o sin2 1 d 12 so that (3.2-5) (3.2-6) L gel'I gel 5I 2 r o 0 2. 2 0 r sin 1 0 and 161 g p 1 g g g ~1 1 2 r o o 7 (3.2-7) 1 0 2. 2 r sin /I o 1 The phase factor is A1 (E,1, 1)= - z = - r0 cos ( (3.2-8) and the first derivatives of the phase factor are 1) el=r sine9 — 0 The Christoffel symbols of the first kind are The Christoffel symbols of the first kind are (3.2-9) [/ ';] = - r sin 1 cos 1 2 1 [/ '1; ] r sin E1 cos I3 1 't 1;I t 1 o 1 1; (3.2-10) = 0..I 26 II

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-106 ___, I L The Christoffel symbols of the second kind are H1 1 r-li ~IM = - sin01 cos 1 = cotel, Th6erl fy 1 I TI 'l =1l1e seo = ~ a Therefore, by (3.1-9) the second covariant derivatives are 1 e1 e1 1 1 1 iL1 1- 1 0 0 -Ql sin2 1 1 9 (3.2-11) (3.2-12) 4) is (3.2-13) m the (3.2-14) Since 1 = e1' and f1 r = -r cos 91 the symmetric tensor (3.1 - T11 Tel &To o, % 1 2 1 - 2 R11 0 T,.. Ta <. 0 /L 1 1i /t1 The cross-sectional area specular reflection point Tel TA (ro 2 - '2L ( 1 -2 (rs2 r rays at a distance R1 froi of a bundle of is 2 A - 2 R11\ 1' 2RI 2 r 0 0 0 D1 (R1) = Tel TAN' 2 Rfl 1 1 -- ro s ( s +R,) = cos", ose~ +~/ V '"Or/ 1 + 2 ro cos e1 II I I I I I I 27

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN ______ UMM-106 The electric field (3.1-1) scattered from the first sphere is cos 1 I ik (R1 - ro cos 61) -x Rj Rl (3.2-15) x (cos 1 + 2 ) (1 +2 cos ) ) r ~ ro 1 For the ray shown in Figure 13, 1 = 1r/4 so that the electric field incident on the second sphere is A eik(d - r) -i e ' o) d ~ (3.2-16) (2 f -3)( 2 r On the second sphere the coordinates u and v are taken to be u2 = Q2 and v2 = 2 These coordinates are related to the Cartesian coordinates by x = ro sin e2 cos 2 y = d +r sin62 sin2 (3 (3.2-17) z = ro cos 92 The metric and curvature tensors, and the Christoffel symbols, for the second sphere are obtained from those of the first sphere by replacing G and,1 by 2 and y2. 28 [ [ I

WVILLOWV RUN RESE.ARCH CENTER-,UNIVERSITY OF MICHIGAN _____ _____UMM-106_ The phase function on the second sphere is given by the simultaneous equations (ALz ~.~)sin 29cosI = r sinG2 cos2 ro sine(1 COS, 1 (I2 ~-1)sin ze(91 sin1= d + ro sinOG2 sin, 2 -ro sin 01 sin 1l.(3.218) (fl2 -Al) cos 201 = ro cose - ro cose1l where in this example 01= 17 " and 3 Hence, the phase factor and its derivatives for this example are AfL2 = d - ACL2 3 -~ro 7r A.,, 4 1,;Yl2 = 0 2 ~ i (3.2-19) The metric and curvature tensors at d A O 2 +2f72'm - =. 2 rc -1 2 z~ 02 = 17'/4 and,2 = 3 T2 are r 22 o 0 2 g09202 g e29 3.2-20) g 262 g9~ 29

WILLOVV RUN RESEARCH CENTER-UNIVERSITY OF MICHIGA] UMM-106 ___ i__ and G26 G 11 G e2 0 (3.2-21) 0 r 2 Since 2 = 1/4, the symmetrical tensor (3.1-4) is 22 T 2 ro 2 2 4d - 2frO T+ _ - 2 4TroR2 7?a- 'A 0 -N' ' C - - " O T T2 dr R2 T'22 0 L _ 2 + d The cross-sectional area of the bundle of rays at a distance R the specular reflection point is (3.2-22) from D (Rz)= 2 2 1 R2 4d - 24o r 2 ro 2Zd - 3ro ( R2 2d + ro 4-Td-ro (3.2-23) Therefore, at a large distance from the scatterer, for the doubly reflected ray shown in Figure 13, the scattered electric field is 2 ik (z + d - 2 r0) e ^ ix 4 dz ro 1 - I d (3.2-24) I 30 I Ill I I II III

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-106 There is a second doubly reflected ray which gives a contribution equal to (3.2-24) and there are two singly reflected rays each of which contribute to the back scattered electric field by an amount o ik(z - Zr) (3.2-25) ---— e o i (3.2-25) In addition there are back-scattered rays which are reflected more than twice. If the distance between the centers of the spheres is much larger than the radii of the spheres, the radiation which undergoes more than two reflections may be neglected and the total reflected field is approximately r eik(d-2 "r) eikzA - 2ikr o e A r e o ik + 0. (3.2-26) 3.3 Formulas for Scattering From Curved Surfaces: The Method of Stationary Phase. Another technique for finding the scattered fields when the wavelength is less than a characteristic dimension of the scatterer is the method of stationary phase. The field associated with a multiply reflected ray, as given by this method, depends upon the radii of curvature of the body at the specular reflection points. These radii of curvature are assumed to be finite. A Cartesian coordinate system is used at each reflection point. The z-axis is taken along the normal to the surface, and the x- and yaxes are chosen so that the x and y planes are the principal sections of the surface, that is sections in which the principle radii of curvature are obtained. In the vicinity of the reflection points the equations of the surfaces are, approximately, I 31 [ [ [ I -

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN _UMM-106 2 x. z. = - / 2j1 2 y j 27j2 (j = 1, 2, 3,..., N ) (3.3-1) where/1,j and/j2 are the principle radii of curvature of the j'th surface. Let9j and. represent the polar angles at the j'th reflection point, r represent the radius vector to a point on the j'th surface from a fixed arbitrary reference point. Assume the incident magnetic field to be A ik(kR) - ae where R is the radius vector from the reference point to an arbitrary point in space. From equation (2.1-1) the field scattered from the first surface is ik lIR-r =1 A )n A 7 1 H = i(n1 x a)xV _~~.R - r1 A A ikk r e 1 ds1 (3.3-2) and the multiply scattered field reflected from the N surfaces in succession is 1 H s (2r)N a A x A n nN x LN-1 x...X**x (2x iX (3.3-3) ik r3 -r X v -- - 03 ik IR-rN R -rN ikk ~ r1 ds 1 N s.dsN. 32 i iiiii

WILLOW RUN RESEARCH CENTER ~UNIVERSITY OF MICHIGAN __UMM-106 Assume the wavelength to be so short that k i. - > 1. For this case + 1k r| -? | ik I?. k-. V e jl ike + l - (3.3-4) (r. r.) r -' 2 ~ 'j+ 3 3 "r r+-r rij+l j In the integrand of (3.3-3) all of the quantities except the exponential factor can be replaced by their values at the specular reflection points. With this approximation, (3.3-3) becomes -N Nx Nx[flx[[ x x axxR ] 2 x... xRN H ' R1 R2 RN N 1 N (3.3-5) N ikk.r1 + R. X... [e j1 dS..dS where - A A rj +1 -rj =R. =R.R.andR N RN RN RN Let =1 x, =. 1 Y15 3 ' X' 4 *.' N-lN = 2N = YN' and expand the phase factor in Equation (3.3-5) in the.. The first order terms will vanish at the specular reflection points, leaving terms of second order as the leading terms in the expansion. Neglecting all but second order terms, (3.3-5) becomes N (kIN 0 oo Cos NA iKr + Z R (3.3-6) H co coe cosN j=1 J 1 2 NR m I 33

WftM^ WUN IWJCEM CTlMlfYi' 'i 01 MICHIGAN whoe I * * e d - dj.. 1 2N Go ^ A is a ZN-4imvn*sohal Fresnel integral, HN is a uilt vector giving the polariation of the scattered wave, and X and M are the matrices 2 2N'l ' 2' ' ' ' N 2 2N 34 ii I III l,~ --

UM~M.1O r — q M=tk= 2 t0 11I siaIai Z I 7 RI 2 R I 0 sin2e Isin#, C.41 a2a a RI IB n ~Ot me ain G cooS 2R 2 A1 sin49 co4, sin& min Cow &Z 2 z I /I - (l 2 I 2 z I I z R 2 +(I -sin 02 coo '2,)i-R + - I I AZ winilaii sLn& xin -01 MA 1 izz - 2 I I 2 R sin al, a' nA C, " A.!R-+ I I ficz 0 0 0 0 0 0 0 fine3 coo$ stn6 c'log c stI9 s5Ln sinAeE COSA -"c 3 A 2 212 * 0. 0.1J 0 4 - 0 0 0 2 2 /Z Z RN-1 R~N IR ~N-1 N.(3.3-8) 4 I iii i i i Jl ii i iii i ii _ i ii i i i 35

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN _____ UMM-106 In (3.3-8) C( jll is the cosine of the angle between the xj and the xj axes, cjl2 is the cosine of the angle between the xj and the Yj+ axes, j.1 is the cosine of the angle between the yj and the xj+l axes, and j21 j+l Lj22 is the cosine of the angle between the yj and yj+l axes. Evaluation of the integral I yields,Ve 2 1 =, (3.3-9) k / f where M is the determinant of M. Thus (3.3-6) becomes A. N ik(k r + Z R.) - 1 j=i 3 H =H - (3.3-10) s N M When the radii of curvature at the reflection points are finite, Equation (3.3-10) is equivalent to Equation (3.1-1). To illustrate this equivalence, the method of Stationary Phase will be applied to the problem of multiple scattering from two spheres treated by Fock's method in section 3.2, In this problem /11 /'=12 /i2z / r=' R1 d - Wr, R2= z - - (3,3-11) l i= 32= ="'-l A | r k. r and =i = 1 i = = = 111 ' 112 121 122 1 2 ' /2 2 ' iii 36

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN ____ UMM-106 r Assuming that d - - <4 z,the substitution of (3.3-11) into (3.3-8) gives (2d - -'r) d 8 r (d - ~'ro) The substitution of (3.3-11) and (3.3-12) into (3.3-10) gives r2 eik(z+d- 22r ) 0 Hs = HN — 4d(z - ) 1 -l If T'1 z >> r Equation (3.3-13) reduces to Equation (3.2-26). 0 (3.3-12) (3.3-13) 37 -

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ___________ UMM-106 3.4 The Biconical Reflector. In the examples considered thus far the radii of curvature of the scattering surfaces have either been all finite or all infinite. However, many problems that arise in practice involve both finite and infinite radii of curvature. Rather than attempt to give a general formula for all the cases that might arise, the scattering from a biconical reflector will be treated to illustrate the appropriate technique. A method of attack for this problem has already been given, namely the use of (2.1-4) with the value of H in the integrand given by the geometric optics formula (3.1-1). As with the examples which have already been treated, it is usually advantageous to make simplifying approximations in the evaluation of the integral appearing in (2.1-4). One approximation is to take into account only the current induced on the scattering surface by the last reflection of a multiply reflected ray. A second approximation is to use stationary phase in evaluating the integral whenever appropriate. In the following computation only the case of transmitter and receiver along the x-axis is considered (Figure 14). I 90, x K —2b2 -FIG. 14 THE BICONICAL REFLECTOR The equation of the upper cone is x +y =(z +b1)2 (3.4-1) while the equation of the lower cone is x +y2 = (z -b ). (3.4-2).. 38

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-106 Equation (3.1-1) can be used to obtain the field reflected from the upper cone to the lower cone. If the surface coordinates are z and, then x = (z +b1) cos, y = (z +b) sin Z, (3.4-3) z = z. The outward unit normal on the upper cone is A cos2 sin A n = - +- i \F2 x y Applying (3.1-7) yields. GZZ GZ_ O The ar element is given by The arc element is given by 1<' - -— z1b] z +b 4 —l (3.4-4) (3.4-5). 2 2 +( dO2 =2dz + ( z + b 1 )d%2 (3.4-6) so that zz gz The inverse of the metric tensor is gIZ gZ] gZ gI 1/2 goz g ~ 0 1= 0 (z+b1)2j 0 (3.4-7) (3.4-8) (3.4-9) (z+b1) The phase ( ) (z ) =(z+b) cos/. - 39,.

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN __UMM- 106 __ The derivatives of the phase factor are -/] = - Cos /, z =-co (3.4-10) T j=(z +bl) sin/. The Christoffel symbols of the first kind are [z z; z; =[z, z;^] -=[z;z]= /(3.4-11) [, /; -] = - +b =, z;/. The Christoffel symbols of the second kind are rz= r,= r- = (34o. rz =-(1/2)(z+bl), z z+ b so that so that z z+bl — QZZ az7 0 0 I p 0. 0 (1/2)( z + b)cos Since cos = i = cosn, zz z.2 R os T T sin C os cos 0 + / z. z+b1 z + b where R is the distance from the specular reflection point on the upper cone. 3.4-13) )c 3.4-14) 40 Ill

L WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 Thus D( l/2,R1 os~' cos). Cos - (\ COB t (3.4-15) A -ikx If the magnetic field incident on the upper cone is i e, then the magnetic field scattered from the upper cone is (-sin cosi x+ cos ii +sini ) e k 1(Z bl)C Y I.a 3.4-16) For = 6 the reflected magnetic field on the surface of the lower cone is 3z+bl 3 z + b1 ik(z -bl)A i (3.4-17) e y where z is the height at which the incident ray strikes the upper cone. If (3.4-16) is written in terms of the coordinates x,y,z on the lower cone with x = (b - z' ) cos' (3.4-18) it becomes b - z' b 3 -3z' y' = (bl -z') sin ' ik- b - z' +(1/2 e 2 + (/')] (3.4-19) where O(X) is a function for which lim O(x)/x = constant which is x — 0 neither zero nor infinity. If (3.4-19) is used in (2.1-2) and the integration over A' is carried out by the method of stationary phase, it is found that the doubly scattered field at a large distance x is given by 41

WILLOWV RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ______ UMM-106 _ ^ ik eik(x -Zbl)( y 2. x y 2f x oo (b +z)(b +2 ) Z 2 S ik — b~3d- ' bl+3z d dz e 1 d dz -C= (3.4-20) k ~\T 1 - 2 31'ri. b -b1 eik(x- 2bl) 2 1 4 e 2 y x b1 + z r, dz. - bbl+ 2z 0 1 Integrating the last expression with respect to z gives 31 i ik(x- 2b ) 3 - A -— x (b2 2b -b - b2 ) A i. y (3.4-21) Therefore, taking into account the radiation reflected from the lower cone to the upper cone, the radar cross-section of the biconical reflector is r b2b -bl - bl (3.4-22) Numerically, the cross-section given by Equation (3.4-22) is in excellent agreement with experimental results that appear in Reference 5. Furthermore, the dependence on wave length is in agreement with Robertson's experimental results. [I [ I I [1[ 42 II I I I I II I IIIII I I

L WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-106 IV EXPERIMENTAL DATA ON MULTIPLE SCATTERERS Many experimental measurements have been made of the radar cross-section of the corner reflector. Of the bodies considered in this paper, data are available for square and triangular corner reflectors (Ref. 8) and for biconical reflectors (Ref. 5). In addition, the effects of compensation (Sec. 2.5) are discussed in detail in Reference 5. The material in this section is taken from these two references. Theoretical curves and experimental points for the back-scattering from square and triangular corner reflectors are shown in Figures 15 and 16 respectively. The results are plotted so as to be independent of the size of the corner. The experimental dependence of the crosssection on the size of the reflector is shown in Table 1 for a square corner reflector. TABLE 1 Variation of Cross-Section with Corner Side Length b ( A= 9.1 cm) Value of n in Size of Reflector = Kbnl 6 inch 3.3 2 foot 4.0 3 foot 3.8 4 foot 3.8 A one foot corner reflector was used to obtain the constant K. For the 6 inch reflector, whose dimensions are of the same order of magnitude as the wavelength of the incident radiation, the cross-section deviated from that predicted by physical optics by a factor of approximately 1.6. The discrepancies between physical optics theory and experiment for the 3 and 4 foot reflectors can be attributed to non-perpendicularity of the reflector sides. - I 43 l

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN _ UMM-106 I____..I.... 0 5 10 15 20 25 30 35 I- DEGREES FIG. 15 RELATIVE INTENSITY OF REFLECTION FROM SQUARE CORNER REFLECTOR 44

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN - T UMM-106 a I 1.1 1.0 < E C,' 0 5 10 15 20 25 30 35 40 -V. DEGREES FIG. 16 RELATIVE INTENSITY OF REFLECTION FROM TRIANGULAR CORNER REFLECTOR -- 45 I I III I......... ~1~

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGA] UMM-106 Compensation, i.e., reduction of scattering surface, can be used to widen the response pattern of a corner reflector (Sec. 2.5). The response pattern of the compensated triangular reflector, shown in Figure 17, is compared with the response pattern from an uncompensated reflector of the same dimensions in Figure 18. A special case of the compensated corner reflector is the corner which has been modified so as to yield a minimum response along the axis of symmetry (which usually yields the maximum response). The response pattern from this corner (Fig. 19) is shown in Figure 20. The response pattern from a biconical reflector is shown in Figure 21. This response pattern is independent of azimuth since the biconical reflector is axially symmetric. In conclusion it is felt that the profusion of multiple scatterers and the widespread use of corner reflectors warrants theoretical investigations, even though the corner reflector may be a poorer standard than the sphere since its exact solution is not known. This paper shows that when A(t< h the cross-section of these bodies can be predicted within an order of magnitude. 46

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN ___UMM-106.___.__ Symmetry ~~. To Radar (a) Aspct = 0', o = 0o Shaded Region Represent Effective Area A (b) Aspct 8 =30', =o0 FIG. 17 COMPENSATED TRIANGULAR CORNER REFLECTOR - 47..

WILLOW RUN RESEARCH CENTER'- UNIVERSITY OF MICHIGAN I UMM-106 ______ 60 -I, i - 40 Ci 20 - 0 O 0u Au 20 -O. I I I I - _ I0 {L -- - - -- I.40.20 0 20 40 60.60 -40.20 0 20 40 60 40 — --- - - - 40 - 20........... —.... -—.....d =10* 0- -. I, _I 0 J. —L I- I Z -60 -40.20 0 20 40 60 -60.40.20 0 20 40 60 MA uz MA sO Cs 60I 06- I t a. I -r i 4 4U 1. i 1 i " 1 -- I I I20 $-.20* I a 0 V I U. — -..- - -60 -40 -20 0 20 40 60 -60.40.20 0 20 40 60 s -60.40 -20 0 20 40 60 -60.40 -20 0 20 40 60 6 IN DEGREES Compensated Reflector Uncompensated Reflector ----- FIG. 18 REFLECTION CHARACTERISTICS OF COMPENSATED TRIANGULAR CORNER REFLECTOR 48..... mlm~......lira.......................

'..., U,Vi.IJ. i t, 6 0 e Aspecdt =0, =0 Effective Area, A = 0 a Adect =30' #=0' Shoded Region tRepresents Effective Area A FlOG. 19 SPECIALLY COMPENSATED TRIANGULAR CORNER REFLECTOR DESIGNED TO PRODUCE MINIMUM BACK-SCATTERING ALONG AXIS OF SYMMETRY 49 - 4 II Ill [ II '

WVILLOWX RUN RESEARCH CIENTER -UNIVERSITY OF MICHIGAN UMM4106 60 40 I I..LUII 20 0 =W Experimental Curves - --- - - Extrapolation of Experimental Datw 0 0 1.60.40.20 0 20 40 60' ta u 0 60 40 -a 20 w 0 I.o % t.11110..60.40.20 0 20 40 60 60 40 20 0 I - -- 4 - L. a 60.40.20 0 20 40 60 0U us60 40 '20 0 60 40 20 -- I 400 dow 000 / OF -02+2(r — I it MMMMMWA I —. I -- - I I I - - - - I - AL.60.40.20 0 20 40 -60.60.40.20 0 20 40 60 0 I N DEG0R E E SI FIG. 20 RESPONSE PATTERN OF -CORNER REFLECTOR DESIGNED FOR MINIMUM ECHO ON THE AXIS 50

......, -,... UMM-106 -50 0 0 * FIG. 21 RESPONSE PATTERN OF BICONICAL REFLECTOR., 51 m.* -

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM- 106 REFERENCES 1. "Mathematical Theory of Optics" by R. K. Luneberg, Brown University, Providence, R.I. (1944). 2. "A Note on the Foundations of Geometrical Optics" by N. Arley, Det Kgl. DanskeVidenskabernes Selskab. Matematisk- fysiske Meddelser, Vol. 22, No. 8. (1945-46). 3. Electromagnetic Theory by J. A. Stratton, McGraw-Hill, New York, (1941). 4. ATI-5763, "Optical Theory of the Corner Reflector", by R. C. Spencer, MIT Radiation Laboratory. (1944). 5. "Targets for Microwave Radar Navigation" by S. D. Robertson, Bell System Technical Journal, 26, 852. (1947). 6. "Symposium on Microwave Optics", Eaton Electronics Research Laboratory, McGill University. (June 1953). 7. "Generalization of the Reflection Formulae to the Case of Reflection of an Arbitrary Wave from a Surface of Arbitrary Form", by V. A. Fock, Zhurnal Eksperimental' noi i Teoreticheskoi Fiziki, 20, 961, (1950). 8. RL-280, "The Application of Corner Reflectors to Radar (Experimental)" by R. D. O'Neal, M.I.T. Radiation Laboratory (1 July 1943).... 52 I I I I I I

WWILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN L UMM- 106 APPENDIX THE USE OF CORNER REFLECTORS FOR CAMOUFLAGE The corner reflector, because of its large back-scattering crosssection, has long been used as a device for creating false radar echoes. This technique has been employed primarily by bomber aircraft for confusing search radars and tracking radars used in air defense. Dr. R. D. O'Neal, Assistant Division Manager of the Fort Worth Division of Consolidated Vultee Aircraft Corp., and others** have suggested the use of corner reflectors for camouflaging geographic features (such as bodies of water, cities, terrain irregularities, and large military or industrial installations) against recognition by means of navigation and bomb- sight radars. To test the feasibility of such camouflage, an experiment was carried out recently by Consolidated Vultee in which an attempt was made to divide Eagle Mountain Lake, near Fort Worth, Texas, in half when seen by an AN/APS-23 X-band navigation and bombing radar. This experiment, which is described in Figures A-l through A-7 inclusive,* met with striking success. The results of this test show clearly that the use of corner reflectors for camouflage merits further investigation...b *These figures are presented here through the courtesy of Dr. O'Neal and the Consolidated Vultee Aircraft Corporation. **Dr. O'Neal points out that Mr. L. H. Moffatt of Consolidated Vultee Aircraft Corp. originated the idea independently at Consolidated Vultee Aircraft Corp. and had much to do with getting the program started there. I... 53 -

r4 FIG A - 1 45 -INCH DIAMETER CORNER REFLECTORS USED IN CAMOUFLAGE EXPERIMENTS USDI CAOFLG EXEIET A. J I t -j.iJ n 0 Z T) rt 2: -4

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM- 106 Tie Tube to Reflector K Slot A Dia. 12 Plywood (Typ.) Aluminum Foil Cemented on both Sides. Use Epon 6 Cement i Inner Tube 7:00 x 20 or Equivalent Dash No A -6 30 " -8 20 " -10 45 " 1 Screw and Glue Plywood Discs 2 Spray Aluminum Foil after Completion with Light Coat of Zinc Chromate Primer FIG. A-2 SCHEMATIC DRAWING OF CORNER REFLECTOR ASSEMBLY R... 55.

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM- 106 The first test runs in the camouflage tests were used to determine the optimum size corner reflector for confusing the AN/APS-23 radar. For this purpose 20, 30, and 45-inch diameter reflectors were placed in one corner of Eagle Mountain Lake and observed by an APS-23 flown at 40,000 feet. Two runs were made, one approaching from the north and one approaching from the west. The radar scope was photographed once per minute from a range of 150 miles to a range of 59 miles and four times a minute from 50 miles to zero miles. ( 6) 30" DIA. REFLECTORS (6 ) 20" DIA. REFLECTORS (6 ) 45" DIA. REFLECTORS Reflectors Spaced 500' 0 5,000 10,000 FEET FIG. A-3 GEOGRAPHY OF EAGLE MOUNTAIN LAKE SHOWING FIRST TEST ARRANGEMENT -

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-106 1 FIG. A-4 TYPICAL PPI PHOTO, FIRST TEST ARRANGEMENT. 57 -

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM- 106 FIG. A-5a SECTOR PPI PHOTO, FIRST TEST ARRANGEMENT 30-MILE RANGE NORTH-SOUTH RUN It can be seen from these photographs (Figs. A-5a and A-5b) that the 45-inch diameter reflectors are more effective than the 30-inch diameter reflectors and that the return obtained is independent of the direction of flight. -- 58 - --

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM- 106 FIG. A-5b SECTOR PPI PHOTO, FIRST TEST ARRANGEMENT 25-MILE RANGE WEST- EAST RUN 59 --

%WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN I UMM-106 1 I The procedure used in the second test was the same as in the first test, eighteen 45-inch diameter reflectors spaced 500 feet apart were placed across a narrow portion of the lake in an attempt to divide the lake in half. (18) 45" DIA. REFLECTORS Reflectors Spaced 500',J II Il 0 5,000 10,000 FEET FIG. A-6 SECOND TEST ARRANGEMENT 60

WILLOWV RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM- 106 FIG. A-7a SECTOR PPI PHOTO, SECOND TEST ARRANGEMENT 30-MILE RANGE NORTH-SOUTH RUN Both these photographs (Figs. A-7a and A-7b) show the rearrangement obtained by use of the corner reflectors. In both cases, Eagle Mountain Lake is seen as two separate lakes. g.. (1 I

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN I UMM-106 1 FIG. A-7b SECTOR PPI PHOTO, SECOND TEST ARRANGEMENT 17-MILE RANGE NORTH- SOUTH RUN 62