THE UNIVERSITY OF MICHIGAN 6633-1-T Derivation of Aerospace Antenna Coupling-Factor Interference Prediction Techniques Interim Report No. 1 1 June through 30 November 1964 Approved....A. M. Ly Professor. Electrical Engineering December 1964 Contract No. AF-33(615)-1761 Proj. 4357, Task 435705 Air Force Avionics Laboratory, AVWC Research and Technology Division, AFSC Wright-Patterson Air Force Base, Ohio 45433

THE UNIVERSITY OF MICHIGAN 6633-1-T FOREWORD This report was prepared by The University of Michigan under USAF Contract No. AF-33(615)-1761L. Task 435705, Project 4357. The work was administered under the direction of the Air Force Avionics Laboratory, Research and Technology Division, Air Force Systems Command, E. M. Turner, Technical Manager; Olin E. Horton, Project Engineer. This Report covers work conducted from June through November 1964.

THE UNIVERSITY OF MICHIGAN 6633-1-T TABLE OF CONTENTS ABSTRACT iv I INTRODUCTION 1 II THEORETICAL DERIVATION OF COUPLING 4 2. 1 E-Sectoral Horns 4 2. 2 H-Sectoral Horns 12 2. 3 Conical Horns 18 2. 4 General Analysis of Sectoral Horn Coupling 26 III EXPERIMENTAL COUPLING DATA 44 3. 1 E- and H-Sectoral Horns 44 3.2 Conical and Pyramidal Horns 47 IV CONCLUSIONS 52 V FUTURE WORK 53 REFERENCES 55 111i

THE UNIVERSITY OF MICHIGAN 6633-1-T ABSTRACT The progress which has been made in the determination of power interference between two similar antennas is described and both theoretical and experimental results are given. Mathematical methods are presented showing the coupling between one rectangular aperture and another. The methods are also extended to the case of the coupling between two conical horns. Good confirmation has been achieved between experimental and theoretical results. In some situations, the coupling will be below the -75 db value which has been given by the sponsor to indicate the range of interest for the measurements. Studies have been undertaken to determine the influence of scattering by auxiliary bodies or protuberances. Methods of data presentation involving nomographs and computer programs are discussed under future work. iv

THE UNIVERSITY OF MICHIGAN 6633-1-T I INTRODUCTION The present investigation of the coupling between horn antennas is a continuation of work performed on an earlier contract for the same sponsor. The previous work, summarized by Khan et al (1964), was largely concerned with coupling problems involving slot and spiral antennas. Some attention was also given to dielectric rod and horn antennas. This report presents a more detailed investigation of horn coupling behavior. Theoretical and experimental work has been performed to predict the coupling between identical horns of the E- and H-sectoral, pyramidal and conical types. Pairs of these horns were constructed and experimental patterns were measured. Due to the large number of possible designs of these types of horns, one cannot expect to cover the complete coupling problem with an experimental program. Consequently, much emphasis was placed on the construction of theoretical models which could be used to provide coupling data., The experimental coupling patterns were then used to spot-check the theoretical data. Preliminary checks of the theoretical coupling patterns indicate a high correlation with experimental data. It is hoped that the mathematical models can be used to provide coupling data for a wide range of operating frequencies for each of the four horn types mentioned with any specific set of design parameters. In the following sections the theoretical models are derived and the resulting coupling patterns are compared with experimentally measured patterns. In addition, many experimental coupling patterns are presented and analyzed. Throughout the following sections, reference is made to E- and H-plane coupling. E-plane coupling refers to the situation in which the E-field directions in the two horns are colinear at the beginning of the measurement (0 = 0). One antenna is then rotated as the coupling pattern is taken. Similarly, the term 1

THE UNIVERSITY OF MICHIGAN 6633-1-T H-plane coupling signifies an orientation in which the H-fields in the two horns are colinear when 0 = 900 (Fig. 1). In Sections 2. 1 and 2. 2 expressions for the far field coupling between E- and H-sectoral horns are derived and theoretical and experimental results are compared In Section 2. 3 a theoretical expression for the far field coupling between conical horns is derived. In Section 2. 4, the coupling between two sectoral horns is considered using a more unified approach to provide expressions which are valid in both the near and far regions. In Section III experimental coupling results are presented for E- and H-sectoral horns, for conical horns and for pyramidal horns - each for three X-band frequencies. In Section IV we present the conclusions of our study up to the report period and Section V discusses the work planned for the remainder of this contract. 2

THE UNIVERSITY OF MICHIGAN 6633-1-T (a) (b) E H E-plane Coupling H-plane Coupling FIG. 1: DIFFERENTIATION BETWEEN E- AND H-PLANE COUPLING.

THE UNIVERSITY OF MICHIGAN 6633-1-T II THEORETICAL DERIVATION OF COUPLING 2. 1 E-sectoral Horns In this section, an expression is derived for the far field coupling between Esectoral horns. The horns are flush mounted in an infinitely conducting ground plane of infinite extent and operate in the fundamental TE10 mode. One proceeds by first calculating the radiation fields on the ground plane produced by the transmitting antenna with the receiving antenna absent. The coordinate system is shown in Figure 2. According to the fundamental existence theorem of electromagnetic theory, a uniquely determined electromagnetic field exists provided the tangential component of either the electric, or the magnetic field is specified at each point (including discontinuity points) of a surface bounding a given region. For the present problem, the tangential electric field E x n vanishes on the ground plane and along the infinite hemisphere centered about the origin, and it has the value M in the aperture. The field in the aperture is equivalent to a magnetic surface cur rent distribution on a conductor of density M. By employing the duality of electric current and magnetic current, we obtain the Hertzian magnetic vector in the form (Stratton, 1941), * = 2w 3 M (r) dS', (1) aperture where a factor of 2 resulting from imaging effect is also taken into account (Lewin, 1951). Using this magnetic Hertzian potential, the electric and magnetic fields in the half space are given by (Stratton, 1941) E = -jt pVx (2)

THE UNIVERSITY OF MICHIGAN 6633-1-T I I I~Rig' /~ I/ I I a -~~~~~ -a- el J ~ ~ ~ ~ ~ -9 RI' R

THE UNIVERSITY OF MICHIGAN 6633-1-T H= V (V ~ 7 ) + k2 (3) where k2= e2 ~. At this point it was necessary to assume a distribution for the electric field in the aperture of the horn. The distribution chosen was -,7ry' g Ax2+R'2 A E =E cos e X, (4) X O b where kg is the waveguide propagation factor and the other variables are shown in Figure 2. This distribution is the result of the assumption that the wave fronts in the horn are circular and that the basic cosine variation of the field with y' in the waveguide is transmitted to the horn. The magnetic current M is then given by M=Exn=E x z=-E From (1) the magnetic Hertzian vector becomes A b/2 a/2 -j (kr+k x,2+ R,2) -y E g 7r 7ry 2r Los b r dx' dy' (5) Y 27rjW A b r -b/2 -aA where r = rI = IR -r In order to derive the electric fields, the curl of the Hertzian vector must be evaluated:

THE UNIVERSITY OF MICHIGAN 6633-1-T a7r a7T Vx 7r''=Vx 7 y=- Y x + Y (6) For coupling purposes, we are interested only in the fields on the ground plane. The x component of V x "'is, therefore, neglected leaving a' (V x I"' Y (7) a x Applying (7) to (5), the partial derivative with respect to x may be taken inside the integrals to give -E a (Vxi = ~y eCo jk J -2+R2 l. z 27rjqt c -a/2 b/2 a e-Jkr a ( r ) dx' dy' (8) If the differentiation is carried out, one obtains ( e-jkr (x-x)e-jkr 1 (jk + ax r r2 r Therefore, equation (8) becomes a/E r 2 r b/2 -j*r +k gx'2 +R 2')...[ y' e (x7rx = 2 - jk (x-x')cOs b r dx'dy -a2 -b/2 p eb/2 -j (kr+kg4x'2+R'2) 7 r -, a,, /

THE UNIVERSITY OF MICHIGAN 6633-1-T The first integral in the brackets is proportional to l/r2 while the second is proportional to 1/r3. Hence, for an approximation to the far fields of the horn, we neglect the second integral. The analysis was previously restricted to the calculation of the fields on the ground plane only. It is, therefore, possible to approximate r for far field calcula tions by the relation r=R- x'-xY y = R-px'-qy' R x R where x p = = cos 0 q = sin Applying the above indicated approximations to (9) and substituting the result into equation (2), produces the following expression for the electric field on the ground plane. E = -jwP3 (Vx 7r* ) jkE e -jkR (xa/ bxy j(kpx'+kqy'-k gix2+R2) E =- 2 (x-x')cos- e dx'dy' z 27rR b jkEe-_ b/2 y xa/2 j(kpx'-k' +R'2') - R cos j-e kqy'dy? (x-x') edx 27r- R e -b/2 A -jkR zjkE e E =- ~ I1I2 (10) z 2r8R 12 i, ~~~~~~~~~~8

THE UNIVERSITY OF MICHIGAN 6633-1-T where I1 = integral over y', and I2 = integral over x' 1 may be integrated in a straightforward manner to give kb bi( I kb ir b in (2 sine+22) sin( 2 sin e- I1= 2(- (-sin, + 2 ) + s (11) s A method for integrating I2 in an exact manner was not available. Therefore, the integration was performed on the IBM 7090 computer. Equation (10O) is an expression for the far electric field on the ground plane produced by the E-sectoral horn. The resulting component of Poynting vector along the ground plane is -Ekj -jkR 2 EokEk2 2 2 122 = wR 8 I) (IA(0)i TA) I (12) The directivity of the horn for any direction 0 along the ground plane may be obtained from the expression D()= N(0)(47R2 /2) - N(~)(2TR2) (13) w W where W = total power radiated from the antenna. A computation of W could be performed by solving for the Poynting vector at all positions on a large hemispherical surface centered at the origin and then integrating it over that surface. Several alternative methods may be used to find an approximation expression for W. 9.

THE UNIVERSITY OF MICHIGAN 6633-1-T A method which has been found to be satisfactory for present purposes is to integrate the Poynting vector over a constant phase wave front at the horn aperture. The Poynting vector on the cylindrical surface, defined in Fig. 2 by R", -0 <0'<0 b3 b0 and - < y'< 2' is 1NE 2 E 2 1S,2 27ry' 2 1o b Hence, the total power radiated is given by = NRd'dy= E2R cos2 d0dy cos d'dy'-b S b/2 0o b The integrations may be performed in a straightforward manner, giving 0 if E2R"b W= 0 0 2 With the directivity defined in (13) the power coupling between two such horns mounted in the common ground plane is D(O1)D(02)X N(Oi)N(02)R2 2 C=12- )2 _ 1 2 (14) where 01 and 02 are the orientation angles of the transmitting and receiving horns. Thus, by using (14), the coupling vs orientation pattern may be obtained theoretically A comparison of experimental and theoretical E-plane coupling curves for a pair of these horns operating at 10 Gc is shown in Fig. 3. The horns are fed with Xband wave guide and have a flare angle of 230. The aperture is 0.9x3.23 inches, and the center-to-center spacing between antennas is 14.5 inches. These curves compare favorably in many respects. The only major deviations are in the relative depths of the nulls. The nulls in the experimental curve are not as deep as the corresponding nulls in the theoretical pattern. This result may be attributed to neglecting all but far field terms in the analysis. 10

THE UNIVERSITY OF MICHIGAN 6633-1-T -50 _ll Experimental Theoretical f 10 Gc eo8 23 70-0 0__ _____ _ _____ / I I I I 0 11

THE UNIVERSITY OF MICHIGAN 6633-1-T Additional coupling patterns were calculated for E-sectoral horns with the same aperture size (.9x3.23") and spacing but with different flare angles. These patterns are shown in Figs. 4a through 4d, It should be noted that the lobes on either side of the positions 0= ~ 900 become smaller as the flare angle increases. For a flare angle of 400 these lobes have vanished altogether. It is also interesting that maximum coupling does not occur when 0 = 0~ but instead, when 0 = 240. These patterns also indicate the positions of many nulls. These nulls may prove useful to decouple the antennas. At present, mechanical devices are being built and tested which will permit experimental checks of the validity of the theoretical model for a variety of horn design parameters. 2.2 H-Sectoral Horns The far field coupling between H-sectoral horns was derived in a manner similar to that used for the E-sectoral horns. The aperture field distribution was assumed to be E =E cos — r e y o a where k is the free space propagation factor and the other parameters are defined in Fig. 5. The resulting far field expression for the electric field on the ground plane is a b/2 jE|e=- 1x' j(kpx'-kx'2+R'2) k(y-y') jkqy d E cos - e dx dy z 27rR a R a /2 -b/2 jEoe-jkR - 2R I1I2 ' p ==cos q= sin 0; I integral over x', and I2= integral over y' 12 may be integrated easily. The result is _ ky 2 kq 1 kqb 2. kqb 22kq 2 jRq 2 kq 2 12

THE UNIVERSITY OF MICHIGAN ' n.. 6633-1-T 13-Eo 13 Ooln~~~~~~._______;P I C, 1 PI~~~~~' ~~~~~~~~~~~~~~~~~~~o_ ___..__._.___I-~~~~~~~~~~~~~~t LI

(db) 4 —8 (db) -48 f a 10 Gc fI,-GcH 190 r. 300 60o= 40o -38 <\ | n / Y \ /\ | A / \ -58 Z lI.~~~~~~~~~~~~~~~~~~~~~~~~~~ I — 78 - -78 - C) -IlK -g0 0 +90 +e 1s0 -0 00 +0C.t0 Z _ _ _ _ _ _ _ _ _ _ I I J I (c) (d) FIGS. 4c-4d: THEORETICAL E-PLANE COUPLING PATTERNS FOR E-SECTORAL HORNS

THE UNIVERSITY OF MICHIGAN 6633-1-T z r b& 15 kx~~~~*e

THE UNIVERSITY OF MICHIGAN 6633-1-T The computer was used to calculate I1. The resulting Poynting vector for the 0 direction along the ground plane is N(O)= E I =(0) I(0) I I(0) (15)| An approximate value for W, the total power radiated, was calculated in a manner similar to that used for the E-sectoral horn. Referring to Fig. 5, the Poynting vector on the constant phase surface at the horn aperture was approximated by N= IE - E 2 2 Cos2 (270' Thus. W becomes S -b/2 0~ 2W=R"d'dy' b0~R"o E~ a- (16) | Equations (15) and (16) may be used in (13) and (14) to calculate the directivity and power coupling for H-sectoral antennas. A comparison of theoretical and experimental E-plane coupling patterns for two H-sectoral horns is shown in Fig. 6. The horns used have an aperture size of 0.4x3.25 inches and a flare angle of 00=330. The center-to-center spacing is 14.5" and the frequency is 10 Gc. These patterns compare favorably in many respects. The coupling levels are comparable and the general shapes of the patterns are similar. Unlike the E-sectoral horn, this horn is flared in the plane of the H-field. Hence, it is possible for the higher order TEno modes to exist at the aperture. The presence of one or more of these modes could account for the widening of the experimental pattern below the -50 db level as shown in Fig. 6, The effects of higher order modes will be studied in future work. 16

THE UNIVERSITY OF MICHIGAN 6633-1-T -30 / \ ~~ I Experimental --— Theoretical 0 J/ O50 '4 '4 4-~I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' ',,1 \ // r -60 ISO~ I+ I I~~~~~~~~~~~~~~~~~~~~~~~~~~ I t I Q I \ \~~~~~~ I I t I B I Pi'~~~~~~~~~~~~~~~~~~ I 0 -50 II I \ I I t I I I r ~~~~~~~~~~~~~~~~~~~~~~~~~~~I I -~o -60I ~ o l I \ ~~~~~~~~~~~~~~~~~~~~~~I \ r ~~~~~~~~~~~~~~~~~~~~~~~~~~I I, ~~~~~~~~~~~~~~~~~~~~~~I It8...... iI I I I I 8 I~~~~~~~~~~~~~~~ FIG 6: THEORETICAL AND EXPERIMENTAL H-SECTORAL HORN COUPLING PATTERNS. 17

THE UNIVERSITY OF MICHIGAN 6633-1-T 2. 3 Conical Horns For the conical horn a theoretical derivation of the far-field coupling can be made. By postulating a tangential E-field, Ep and Eo in the aperture of the conical horn, we can derive the Hertzian magnetic vector as: 27,U -jk R-pj 7T (R)= 21j __E(P_)xn __I dA'. (17) The factor in front of the integral already includes the imaging effect of the ground plane. Ex n, the tangential E-field, vanishes in the ground plane. Therefore, the only contribution to the integral conies from the horn aperture. From the Hertzian magnetic vector, we can obtain the fields as: E= -jwt V x VT (18) H=V(V' r'')+k2 '. (19) Using a spherical coordinate system as indicated in Fig. 7, in the ground plane (0= /2) we will have only E0 nonvanishing. The field component E0 is given by (Silver, 1949), 0(= jke2 [Ncoso+NysinB ] (20) where N and N represent the integrals in ' and 7r x y x y Nx=f (Et)xe jk(xcos+Ysin0)dA (21) N r= J (Et) eJk(x'cos0+Y'sin0)dA (22) Everywhere primed coordinates designate variables in the source aperture and unprimed coordinates, those at a field point. Using the relations x'=p'coso,' y'=p'sin ' (see Fig. 8), we get: 18

THE UNIVERSITY OF MICHIGAN 6633-1-T z FIG. 7: COORDINATE SYSTEM FOR CONICAL HORN R I XFIG. 8: COORDINATE SYSTEM IN GROUND PLANE FOR CONICAL FIG. 8: COORDINATE SYSTEM IN GROUND PLANE FOR CONICAL HORNS 19

THE UNIVERSITY OF MICHIGAN 6633-1-T N-= (Et) eJkp'cos(O-0 ')p'dp'do' (23) N ff (Et) ejkpcos(-0) p'dp1d'. (24) In order to evaluate the far field pattern, the tangential aperture field E t has to be derived or postulated. For the present case, it was assumed that in the aperture the tangential E-field could be represented as a combination of circular waveguide modes TEll and TM11, both of them having their phase centers at the point where the extended walls of the conical horn would intersect (see Fig. 9). There is experimental evidence available (Potter, 1963) that a mixture of these two modes actually exists in the aperture of even moderate flare angle conical horns. The aperture fields assumed to exist are given by: TE11: jwpK' F siny'Jl(K'hP') ~~~~~~~~~~~~~~~E ~~~~(25) Ep,: -jwo/ KjIF cos-'J{ (Kl1P') (26) where F exppi j74L2+p2 (27) (for the definition of L, see Fig. 9), and where (Klla) is the first root of J;l J1(K' a) = 0 (28) TMl Ep, 2jwiKilF sin0'Jl(Kllp') (29) 2jw/Kj1F cos I'Jl(K'p(P') E,= K1.p' (30) 20

THE UNIVERSITY OF MICHIGAN 6633-1-T Lr L - I 0 12a FIG. 9: CROSS SECTIONAL VIEW OF CONICAL HORN a = 4.6cm, d=2.54cm, L=26.4cm, 00=9.90. 21

THE UNIVERSITY OF MICHIGAN 6633-1-T where (K la) is defined as the first zero of J1(K la) = 0; (31) and F is defined above. In order to evaluate Nx and Ny (eqns. 23 and 24), we have to decompose the aperture fields into their rectangular components, giving: TE E, jwK F [JJ(Kp ')sin20l (32) jtK2 11F 2 11 E, =2jwLuKl1F[J (KI1p -J (Kt1pt)cos2] (33) TM11: EXT =-jCWpK11F J2(K11P )sin20' (34) Y'EjwltK11F {JO(K 1P,)+J2(K llP)cos20t1 (35) For the TE11 mode we obtain: Nx=+AjS F 2sine' kpc(0 i J2(Kl p')p'dp'do' (36) 1 1 11 Ny =A jKl F [Jo(K1 P') os2J2(K1 jkplcP)os(0 t) ptdptd (37) where A=2 jW kK Using the expansion jxcOsy=J (x)+ 2jnJn(X)cos ny, (38) these become: Nx=-2 wAsin2 0 r2(K 1) (39) 2 11 N =2zA Fo (K' )+r(K' )cos2~ (40) here rN)=-J Fp' J( p')Jo(kpt)dp' (41) 0 2 22

THE UNIVERSITY OF MICHIGAN 6633-1-T i2()- Fp'J2(y ') J2(kp')dp'. (42) For the TM 1 mode we obtain Nx=-2AfF sin 2'eijkpos ( - ')J(K p')P'dp'd't (43) =2Af FJ(KP ')+9+J2(K1 p')cos20i ejkp'C~s(-~0 )p'dp'do' (44) which becomes N =47rAsin20 (K 1) (45) x 2 11 (45) N =47rA [ (K 1 )-q(K )cos20] (46) Combining the contributions due to the x and y polarizations according to (20), we have _ jkA _ )]~r7 (47) EaTE jkA (KI _-(KI )]sin4 OTE1 R o 1 1 2 1) 11 EOTM 2jkjr [r )+(K)+(K)] sino (48) TM11 R 1 1 or, equivalently, wpK1 k E = r (K )-r2(K 1I sino (49) OTE R 11 11 11 R r(Kll )+r- (Kl) sin (50) EOTM R o 11 2 R si (50) The integrals P and r (eqns. 41 and 42) cannot be evaluated in closed form because of the factor F. However, in the limiting case of an aperture fed by a circular waveguide of the same size, this factor becomes equal to one. In this caser and/v can o 2 be evaluated exactly using the relationship (Silver, 1949); yJ (ay)J (f3y)dy= 2 [J n(aa)fJ'(3a)-Jn(1a)aJ n(aa) (51)....__ __ ___ __ ___ ___ __ __ _ 023

THE UNIVERSITY OF MICHIGAN 6633-1-T Now, using the boundary condition J1(K a)=O for the TE11 mode, and J (Kla)=O for the TM11 mode, we finally derive: 11 r (K' )= a J (Ka)[K?2aJ (ka)-kJ(ka] (52) 2(K. JK'k t a) FkJ(ka)-Kt2aJ(ka (53) 11 r r( ) -ka J (K ' a)J (ka) (54) iK k2 o 1 (Kl )KLL)=- KZ J,_k a( a)J1(() [D(K' J (K'I a) J (ka) (55) 2 11 K'2 -k0 ( o 11 1 11 Therefore, the particular combinations we need for deriving the normal E-field, equations (49) and (50), are: (K)-r(K )=- K a)J (ka) (56) 011 2 k J 1 1 1 1 d6 r(K 2ka o 1 (K2 -k 2 J (K.a)J1(ka) (57) The directivity of an antenna is given as. P()2 eR2 (58) where P(0) is the power density in the v-direction, and W is the total radiated power by the transmitting antenna. Then, 2 I~ 1E01 27R2 D - A2trR2(59) 1 To calculate W, evaluate Re(EtxHt) over the aperture, giving: 2 w: (E + jIE I )1P'dp'd'. (60) 24

THE UNIVERSITY OF MICHIGAN 6633-1-T There results for the TEl1 and TM11 modes, respectively: VoltK, 2 W0/ a TE 11 11 2 VO TE11 4 (K a2-1)J1 (Kl a) (61) where 2,2 11: I (62) k2 K2 TM116 lw 1 J (K a) (63) TMl ' nl oi where P11= k2-K2 (64) The total radiated power by the conical horn then will be the sum of that contained in the TE 1and TM modes, i.e. W =R2W +R 2 W (65) total 1 TE11 2 TM11 11 11 where R1 and R2 are the amplitudes with which the TE1l and TM11 modes are excited, respectively. Using the directivity of the conical horn, we can derive the coupling by the formula: D 2 D1D 2 c:47rR2 (66) where D = directivity of transmitting horn in the direction of the receiving horn, D2= directivity of the receiving horn in the direction of the transmitting horn 25

THE UNIVERSITY OF MICHIGAN 6633-1-T Now, for identical transmitting and receivirg horns, D=D1 D2 (67) and the coupling becomes C - D2R2 (68) While we can derive a closed form expression for F=1 forf" and r (equations 41 o 2 and 42), these equations can be evaluated very easily numerically for any F. A computer program was written for this purpose, evaluating the integrals by using Simpson's formula. A thorough study of the effects of having present both the TE11 mode and the TM mode is under way. Figure 10 shows the theoretical maximum coupling value obtained as a function of frequency with a pair of horns of radius a=4.6 cm, center-to-center spacing 36.6 cm, and length L=26.4cm, R=.9, R 2.1 Only a plot of the maximum coupling is needed in order to deduce the coupling for any orientation of the conical horns, since the directivity in the ground plane varies as sin9. 2. 4 General Analysis of Sectoral Horn Coupling A horn excited by a waveguide supporting a TEl0 mode is to be considered. Inside the horn the TEl0 mode from the waveguide will be decomposed into an infinite number of modes compatible with the geometry of the horn. If the length of the horn is sufficiently large, the pseudo TEl0 mode will be dominant in the region far from A and B (see Fig. 11). Similar observations can be made with a receiving horn and its associated structures. The legitimate approach to this problem is, of course, to match the coefficients of modes in each of the five regions (Fig. 11) in terms of the incident TE10 mode. The difficulties in this approach arise due to the fact that the geometry of a horn is not orthogonal in Cartesian coordinate systems. In other words, it is impossible to find one coordinate system which preserves the orthogonality in all regions, thus leading to the infinite mode transmission matrix. When flare angles of the horn are 26

- 60 -' -80 8.0 9.0 10.0 Gc 11.0 12.0 FIG. 10: COMPUTED COUPLING LEVELS FOR CONICAL HORNS AS A FUNCTION OF FREQUENCY. R1. 9, R2=. 1, a=4.6cm, s= center-to-center spacing=36.6cm. z

THE UNIVERSITY OF MICHIGAN 6633-1-T TE I Receiving Horn 10 I A' B TElo I 0 —Transmitting Horn B FIG. 11: CONFIGURATION OF TWO HORN SYSTEMS 28

THE UNIVERSITY OF MICHIGAN 6633-1-T small, the modes in the horn can be approximated in terms of the Cartesian coordinate system by using WKB methods. Even in this case, the amount of calculation is enormous. To avoid these difficulties, it is assumed that the aperture field of the horn con sists only of the pseudo TE10 mode, thus eliminating the necessity of considering many characteristics of waveguide and horn structures. In addition to the above, the following will be assumed. 1) No reflection occurs at A and A'. 2) Aperture field over the transmitting horn has the same distribution as the dominant mode in the horn. 3) Secondary scattering effects will be ignored. The above approach is merely the extension of analysis of the coupling of waveguide terminated slot antennas. Therefore, in the subsequent analysis many steps will be omitted as they have already been reported. As in the Introduction, a few words seem to be in order about the relations between the present approach and the coupling formula using directivity concepts which have been treated in earlier sections. Generally, the directivity has a meaning only in the region far from the source where the field intensity is proportional to l/r. The usefulness of the directivity ceases if the field intensity is proportional to l/r2 or the far field component is comparable to the near field component. The present approach has, as one of its objectives, to seek a unified form which is applicable both in the near and far regions. The two-horn system, flush mounted in the perfectly conducting plane, is shown in Fig. 12. The system A is transmitting and B is receiving. The orientation of B with respect to A is arbitrary. For computational convenience the sizes of two systems are assumed to be identical. The coordinate systems with reference to the centers of the Horn A and horn B are (~, rl, ) and (x, y, z) respectively. The relation 29

THE UNIVERSITY OF MICHIGAN 6633-1-T b 30 a~~~~~~~~~~~~~~~~~~~~~~~~~~I

THE UNIVERSITY OF MICHIGAN 6633-1-T between (E, rl, C) and (x, y, z) is = r cos a+x cos l-y sinf3 r7 = r sin a+ x sin f3+y cos / (69) ~ =Z. The coordinate system is shown in Fig. 13. The point 0 is located at the middle of the intersection line of the flared sides. The coupling C is defined as C = W/Wt, (70) where Wr is the power received by the receiving system and Wt is the power transmitted. The analysis of coupling consists, therefore, in the calculation of Wr and Wt. First, field intensities in the half-space will be calculated. When the receiving horn is covered by a perfectly conducting thin sheet, the magnetic field Ho in the half-space can be written in terms of the aperture field in the transmitting horn as follows: HO(rrA){ IYn(rAPA).R(PA)X OldSA (71) A where -k rA-PA YnrA'PA)= 27rja (k2 +VV) i-A- (72) n A' 2wA)= jcqu Ir -P I PA(5', ri7', ') = source coordinate of the transmitting aperture, rA ( I', q, ) = position vector of the field point from the source point in A, I is the unit dyadic. To calculate the field transmitted into the receiving horn, the following conditions on the boundary constraint will be made. 31

THE UNIVERS3 Y1TOF MICHIGAN x(g) T MXO / Y(77) y(rl) FIG. 13: HORN AND ITS COORDINATES......... 32 _

THE UNIVERSITY OF MICHIGAN 6633-1-T 1) Ignore the presence of the transmitting system (i. e. the field over the transmitting system is fixed). 2) When the receiving aperture is covered with a conducting thin sheet, the fields satisfy the boundary conditions: H =HOt' H =0 atz=0 E 0= En = 0 atz=O (73) where t and n indicate tangential and normal components respectively. 3) On removing the conductor, fields in the receiving horn satisfy the boundary conditions: Hit= Ht+H atz = 0 E it= E0s at z = 0 (74) where s denotes the tangential component of a scattered field. If the receiving horn were a uniform rectangular (or cylindrical) structure, H0s would be equal to (-Hit), thus leading to the relation: - 1 -Hit= -H0t (75) In the case of actual horns, this relation does not hold. However, for the horn with a small flare angle, we adopt the above relation which is not unreasonable in view of the assumptions we have been making so far. 4) Hit thus obtained travels through the horn to the waveguide where only the dominant TEl0 component has meaning as far as measurement of the received power is concerned. As already mentioned, it is assumed that the pseudo TE10 mode in the horn will be converted to the TE10 mode in the guide without any loss of power during the process of conversion. It is admitted that this is a very drastic assumption, but this seems to be the only way to avoid solving an infinite determinental characteristic equation. 33

THE UNIVERSITY OF MICHIGAN 6633-1-T In view of the above decisions or assumptions (1) through (4), and using equations (71) and (75), we obtain the amplitude I of the TE10 component in the following form: I= 2 fi(x,y) HO(PB dSB d EpdSABxO (76) 2- B SA PB)YPAPB)' ( PA)X (76) where h(pB)=h(x,y) is the normalized, transverse pseudo TE10 vector mode. The determination of E(-A) involves an integral equation for which the exact solution is not available. To avoid this difficulty, as stated earlier, it is assumed that E(PA) can be represented approximately by the dominant pseudo TE10 mode of the transmitting horn. The pseudo TE10 mode is given as H"= 2H(2 'yp)cos — x x 0 a H1 a ap L[Ho ('yp) sin (x) E = jP/ aH (/p)] cos -- (77) 0 p O a where =2(7r2 If y p >> 1, the above modes can be written as: H 2 /T1=- YP e C1/4Cos — x X TwyP a H"=j air/2 P -jy j7r/4 wCIfx H -13 - e-7 Pe cosp ary p a El I=w. JI' eJPe ie4 cos -. (78) Ths 0 V yp a These equations can also be written in terms of the Cartesian coordinates. The

THE UNIVERSITY OF MICHIGAN 6633-1-T transverse components in the x-y plane are: -jTpY/+y2I t, 2/ 2' jr/4 e 7rx H y= e Cosx i ry(221/4 a -j'y/p~+ 772 rl ' Ty 12 /(p2 +y2) / H"-j' 7 e e j r/y 4 sin (79) a ry(2 +772)3/4 a -_-n jr /4 pe __________ EY 7r'y ejr (p/4 2xcos a If the flare angle of the horn is small, then co = Po - -1 sinO = rl _ r1 4~T~ ~Po 2 +y2)1/4- 1j2 (80) Under this approximation (79) can be rewritten as H"= Y2f'Jir4 - COs 'x X a H =j/ e ~ e sin x (81) y a p P a 22 'jkr/4 -J /Pxoiy2' E" -w/ y / e e cos 7X y 7r yp a....- 35

THE UNIVERSITY OF MICHIGAN 6633-1-T Identifying E(pA)=E" (,rI) and taking h as the normalized version of H=x H '7 o x + yoH", the equation forms: 0oy 1 =_ -~ ~l4Iuo e- a.. ii4 zW1YJy'II~d BI %H $B]H Yn cs [H I dSB+JHX. y Sn x ry BllsAX(7) X (Y o7 o-psin -Y- cos a JJ A 078 a a a nao a 0 Po l HIJ3Bl IH"SI S B] (82) where ~~-~~J'~~~~Y~ J~o~172~~ '(83) X(rl)= e If we can ignore HY compared with H" (82) will take the form: y X / b/2 I = K / dy drX(r')X (y)J(ry) (84) i-b/2 J-b/2 where Ko 2ab (85) o 2jab, a X Ak2I+VV]~(GcA) J(r7, Y) d=/ d d cos a cos x +V (86) -ikF B-PA G(PBPA)= I = G(x, yl,, 7) (87) 36

THE UNIVERSITY OF MICHIGAN 6633-1-T where PB is the position vector over the receiving aperture in terms of the coordinate system (x, y, z) while PA still keeps the meaning of equation (72). It is to be noted that the differentiation affects only PB' At this point it should be mentioned that though the contribution of H? to the coupling is usually small, this effect cannot be ignored when the near field is dominant. However, this contribution of H" will not be dealt with in this report. y In view of equation (69) and Fig. 12; O2 a2 a a x.(kZI+VV) - =k cos 3+ cos - -sin,- (88) The substitution of (88) into (86) leads to ra/2 a/2 2 a2 a2 J (y, rl )= 7rx ~ G(X=[-cos ( l+) a a a where 2 and can be removed by partial integration. Then ax2 ax 7r2 7r x 7r J(y,r)=cos3(k2- f dxd cos Cos G 2. 4a a a/2 -( ar.)CSi n %13Wk 3r X x= - a (Zr.)sin f xdf s7 -— cos - G (90) a jj a a ay where 37

THE UNIVERSITY OF MICHIGAN 6633-1-T To proceed further, some approximations of G are in order: -jk lx_-i)+(y- )2 -jkr' G(x, y;, ) e. e _ e ~- 1 x _Y i( )........._ $cos(-fI)rrs + - cosa+ --- sina expjkxcosos(a-) -jky sin (orf)+jk~ cos a +jkr7sina, r0 (91) where ro is the center-to-center distance of the two horn apertures. The above is obtained as follows. By (69), (x-~)2+(y-rp)2=(rocos a+x cos 3-y sin I3- )2+(rosin ae+x sin,3+y cos -r1_)2 [r+ x cos (a-3)+y sin(a-/3)-B cos a-r7 sirn2 (92) and __ 1 ) (1- cos(a-)- - sin(ae-)+ -cos a+ r sin a) (93) _____________2 r r r r r The substitution of (91) into (89) and straightforward integration gives: ka 2 e o r.-jkr~1(C (a40) C(a) r= exp jky sin(+-k)+jk na since s in(a-3) sina fy r7-(~ r r(ca) + (a.-) C(a) sin(ca-3)sin Esin - -sin(ac+j 2- sinacosa sin(Caf) C(a-/) (S(a) 44ka cos a ) S(a-f) C(a) k4 C(ao-) C(a) +sincasin(c3 )cos( a(l3) - () cos(a- 2) (cos - 2 cos 3cos(a-I3)j, (94) 38

THE UNIVERSITY OF MICHIGAN 6633-1-T where C(a-)=cos O Cos (a-)] S(a-3)=sinkacos(ak2a2 ~(ac-)=l- 2 cos2() 95) C(a), S(a) and 0(a) follow the same rule. From the above formulas for J(y,rl), a few conclusions valid for this physical problem can be extracted. 1) For large ro and reasonably large a, the first term in the bracket is dominant. 2) For a= 0, there is only a near-field effect. 3) For small a, a combination of near- and far-fields exist. In calculating the total power transmitted Wt, the presence of a receiving horn is again covered. Then, a real part of the complex Poynting vector integrated over the transmitting aperture will give the desired value, i. e., 1 A Wt= Ref dSA(E xH')~.o (96) where Ho is given by (71) and E is the assumed aperture field. Substituting all relevant values, there results: Wt= -2 Re A iSdSA [E(PA)X~ Y'n(PAP) [E(PA) o] =ReL 22 dSA IdSX( r )X(rn) cos — cos (k2+ G(, r/, ', r(97) a a ax 39

THE UNIVERSITY OF MICHIGAN 6633-1-T Partial integration and change of integration variables leads to b/2 b/2 2 exp - (X)dX~ep k2 7)sinX+(1 - ' '7r" +X dXIi ____ 2 x2 Sink2 a2 2 2a2 b+X2 =Re - y 2 ka 21 (98) j2 2 )a where b/2 f be P ~= dri/ d7'X (r)X(7) dX (i' + a )sinX exp[kb IL(X kz a2)kCos X 2 - (99) (x-o)2+X2 Since W is given by Wr= Y~ W I2, (100) r 2 y the coupling formulas can be written as r 2 Poa 1 C =Wt = 472rr Lazb7b2 F (101) 40

THE UNIVERSITY OF MICHIGAN 6633-1-T where J /b / 2 J dy X drLX(rT)X (y)J'(y, ri) Fg2 -b/2,) (102) Re (jP) where J' (y, rj) is the term inside the squarebracket of equation(94) and P is given by (99). The integration of (102) in closed form seems to be impossible. However, for two very important cases, the relative values of coupling can be obtained without further integration. For E-plane coupling (Fig. 1), i. e. a = r/2, and r large, 0 ka cos( — sin 1) c(a-1) C(a) 2 J'(yr/)-( --- cs / -B k2 a 2....cos j. (103) 0 (ce- cosa) k2a2 2 Therefore, fb/2 kaos. ( sino)cos-12 C=C(13)=C X | 1,(y)e-jky cos adyL k2a22 j (104) 1- — 72sin 1 where / b/2 X(r)e ik1 2 C -b2 24 ( 4 (105) o Re(j P) r 2 422 o yab CO is independent of rotation angle 1. In this case, as indicated previously, the coupling has 1/ro tendency. Thus, ar-field coupling is dominant. The 1-dependence of (104) is compared with experimental results in Fig. 14. 41

THE UNIVERSITY OF MICHIGAN 6633-1-T -58 -60 o -62.m -64 0 0 -66 0 ~30 3(degrees)60 * Theoretical o Experimental 3.221?1 0~p.4" Normalized at the highest peak Po = 3.78" f= 10 Go. *- 0.9"-' FIG. 14: ANGULAR DEPENDENCE OF COUPLING OF TWO HORNS -__ 42

THE UNIVERSITY OF MICHIGAN 6633-1-T For H-plane coupling (cr=0), a41 C(at- ) C(a) k2a2 2 J'(y' )=-J 2 cosO 02 (l- 2Cos @ 2r ka ()7r 0 ka cos (ka cos) 2 2 1 kr (k2a2 2 k2 2 cos (106) o (I-2cs2/) (1- Co b/2 2 Os(k CO CSy27) - s2 where /2bi 2 X(r))drl 2 poa 1 1 C=167T 2 4 22 k2 2 -b2. (108) 1k~ 0 74a2b2 (k —a-2 Re(jP) 7r In this case the coupling has l/r4 tendency. It should be noted at this point that the above relation (107) is an incomplete one. The present situation is exactly the case where the second term inside the bracket of the numerator of equation (82) plays an important role which will be dealt with in the next report. However, the |/r4 tendency does not change even when this new factor is considered. 43

THE UNIVERSITY OF MICHIGAN 6633-1-T III EXPERIMENTAL COUPLING DATA 3. 1 E- and H-Sectoral Horns Experimental E- and H-plane coupling patterns for E- and H-sectoral horns are shown in Figs. 15 and 16 respectively. Patterns were measured at 8, 10 and 12 Gc. According to the theory for sectoral horn antennas, the coupling patterns should be symmetric with respect to the 0 = 00, +900 and +1800 lines. The deviations shown in Figs. 15 and 16 are due to mechanical defects in the horns. The E-sectoral horns used to obtain the patterns in Fig. 15 have an aperture siz of 0.9x3.23" and a flare angle of 230 (see 00 in Fig. 2). The center-to-center spacing between the horns is 14.5". These horns were designed to have maximum gain at 8.8Gc near the center of the X-band source. The H-sectoral horns used to obtain the patterns in Fig. 16 have an aperture size of 0.4x3.25" and a flare angle of 330 (see 00 in Fig. 5). The center-to-center spacing is 14.5". As in the case of the E-sectoral horns, these horns were designed to have maximum gain at a frequency of 8.8Gc. It should be noted that the average levels of coupling decrease as the frequency increases. This is a result of the fact that the spacing between horns becomes larger n terms of wavelength as the frequency increases. It is interesting to note that the ocations of local maxima and minima in these patterns do not change appreciably ith changes in frequency. This effect is desirable wnen one is concerned with decoupling two such antennas. If the H-plane curves in Figs. 15a and 15b for 8 and 10 Ge are compared, one ees that the lobe at 0=900 in Fig. 15a has been transformed into three lobes in Fig. 5b. In Fig. 15c these new lobes become more pronounced. This increases in the 44

THE UNIVERSITY OF MICHIGAN 6633-1-T -50 -50 (a) f=8Ge -60 -60 - 70 -70 -80 -l -r1!t d i i> + -80 4 > 30 lt T f l t -180 -90 90 0 -50..-50 -50 50 (db) ----— +(b) db) (db) [ 2C (db) f=10Ge =12G, ll-1 -— 1.- illllllllll illll lllll! 11 Hil MllT! T I 60 -60 rr irln i r!cr Irnr I I I I I 1 I I I ri rl Iii i i imm~mt~t~c-~~i~ l — 6 -60 - 60 -60 it60il lllillllillliliill]]llliilizil ':[: i: -60 70 90 10 -70 070 1 = XlFfflil = cdT~T~ TmTST~T n Il 141- I Illlll[ll[ [i; 1':i!;I 8-0 80 -80 I-90 IL I 9 1 80 I'_~l I1.1, l '! [J Ilu i -180 -90 Q 0 90 I180 -180 -90 lI 0 90 180 FIG. 15: E-SECTORAL HORN COUPLING PATTERNS ( —)E-PLANE. ( ---) H-PLANE. 45

THE UNIVERSITY OF MICHIGAN 6633-1-T -3O -70 (db).. ( db) ~ -50 -90Ai tllil llittil 1; tt 11 -60 t -100 t~~~~~~Af_ ri iif ti fl C11 -3o~~~~~~ - ' —60 -3 IllllE... i~ '' mC -7,-50!t I.-90. i,,,,,,,,,,,,,,,,,,,,:, W-,o db),-60 - 4t (db);f 10 f 1Ti 1_ 40 1 I Iv l i 1 lLr77 I l 1 il { TT I!nl I i~ r nl i I III 1 II l I= CI E II ~ YIII I p d I ~,l llU {I,{, I,,,,,i l,~ 1,, ll,,, l{,,i0, 11' 1,, i, {. -~ {lo, l0. l ~:, i t I I /1{1111 All l I II (ll. (C)llllllllllllllllllllll.lllllllll 11. 1 C PI, c' Pr~~~~~~~~~~~~~~~~~~~~~I l!-f1G (dbW ulllllm llllllllllllll llIIl~4lll' 1lll{.: '.l.f _db,_. 1... _I.~'. -'- l ( d IV[[LLL]-H[,II]'ILLII 11111111111511 11111 llll l illllllIIItlI FII IllFll Til l]~1G 1i1111111['1 IIIILIIlI II11 111111111 L 111111 111 1111 111 11111,1IIIIIIIIZ IIIIII111II.{' I l II;1 I I~;I il 1.'l]Jii.d I_. - lilll~lal]Tl/llilll~ll/llll~lil~llllllllllllll{~ll{ ff}1111} Illl-lPlol ll{lllllll ~'~*l i!\'l!- ~{~-1'~.1;I I I _ -4o 70-40 -8 '-50,{_8~1i{ 11 1 II NI1 1111l?..~-0 -50 ill!,. ii! 1i -,II.I~flIIIH IITIlIIII/IlII'IItIlEII. JI.INIIIIIIIIlIIIIYIIIlII lll I TI:t.. i aI:._I~_!EIW~~ —,.- o -fi I I II 111;\11111:11 11/ 1III I 11111 1111 11111 I 11 p II 111. IllliL lllli l /ITklilll lTTlll.III!111RlltlilrIII/111lTIIII I~l fki ~!i i i ilS i; L/-Lilltl l' l l;- ifIll lillllTTllIIIII.l-lllllllllllilllil lltllllllll~lllllll 1i I I iIld11'1! -I i! i lt —D-l,i lil I! I Il E llllllt Hilllil!i lill lil llliiil I!l J '. S -6 Iltll~l~i~llilillillililili~lil{ 1illt~lilllllli/{1lllllT'T.9 - 0 ri~llIl~l~l'~ ~t~fl;-R -t- l-1- 1 ItllililiiliilllIllilllfTlllilllllllll 1illlilllllllL I I I I IIII!IIIIIIIIU|tIii{lilLL!i3~ -J4ll-l-111r,,,~ ~ ~ V........""~..........t''t -60 I I90 -60 -100 t 1 1 0:- I~~~~~~~~rX 1 1 i~~~~~~~~mt; tLI~~~~~~~tI.......... j....l 111 I [j } l~l l....!11t!!! l!1 111 111 I Il II! -! l l!_ _I_ii _............. l....lJ.Jl -180 -90; 0 90 180 -180 -90 0 90 180 FIG. 16: H-SECTORAL HORN COUPLING PATTERNS. ( —)E-PLANE, ( ---)H-PLANE. 46

THE UNIVERSITY OF MICHIGAN 6633-1-T variation of the coupling patterns as the frequency is increased is due to the increase in the size of the horn aperture in terms of wavelengths and the resulting increase in phase cancellation which occurs in the radiated field. The lobe splitting effect is also evident in the E-plane curves of Figs. 15a and 15b. Here the lobe at 0=0~ in Fig. 15a splits into lobes in Fig. 15b leaving a local minimum in the coupling pattern at 0=0. 3.2 Conical and Pyramidal Horns Experimental data were gathered for a pair of conical horns mounted in a ground plane with L=26.4 cm, a=4.6, Q0=9.9~ (see Fig. 9). The gain of these horns is 18 db at 9 Gc. The data were taken for the linearly polarized case at 8, 10 and 12 Go, at a center-to-center spacing of 36.6 cm. The two curves shown in each of Figs. 17a to 17c, labelled E-plane and H-plane coupling, are described by the geometry of Fig. 1. It is seen from these curves that, indeed, as predicted by the analysis the coupling levels vary as sine. The maxima of the H-plane coupling are down 6 to 16 db from the maxima obtained for the E-plane coupling. At this spacing of 36.6 cm, the lowest coupling levels obtainable for the linearly polarized pairs of conical horns is about -90 db. Figure 18 shows the E-plane coupling pattern as a function of rotation angle 0 (of the receiving horn) at 8 Gc with both horns circularly polarized. Because of the phase quadrature relationship between the two orthogonal components existing in the horns, the nulls of the coupling patterns disappear. The slight variations in coupling are due to the aperture fields being somewhat elliptically polarized. A pair of pyramidal horns mounted in a ground plane was also tested at the same center-to-center spacing of 36.6 cm. These horns were fed from X-band waveguide (l.01x2.28cm), which was flared out over a length of 10.1cm to an opening of 4. 77x6.04cm. The horns have a gain of about 12 db at 9 Gc. Figures 19a through 19c show both the E- and H-plane coupling curves. The coupling pattern for the E-plane 47

,LNV LcL —L' ---J V.L —,.L -- (caZIloa I VaI I) SSHzLdcl oNIcdnOa NX1OH IVonINao:LT 'DIA 081 06 0 ~ 06- 081- 081 06 0 06- 081 -7T ITT- F7YI T JRWll~;!il!l:ltlN il]. i IIT -'ii. I I- Tr] I I rT"I iT'~~~~~~~~~~~~~~~~~I km e 5T~~~~~~lN f | X 1a1 I X tt ~~~~~~~~~~~ttltl >: 1 1 t tI i rhte E E n eH,i lr 1 1' I.r!, 001- d t S f 4CIXF -S~t ttlf a~ 6 - t Wr:<< S|8Xil I 4 rq oo[ 06- 08- 6 8 -001 T I 8I~~~~~0 -4I~ftletZ I ~V-te~f~~l I I ].[L ' ] i11 11 IT l I l XX A-,Eg...;? ~,?,:/!;?,llklIll'tllllllllllldlilllt 1l [~"~.~'] ---'~'-[-["-""'1'"~'"1 I', fT/ T~~~~~ ~~~~~~~~~ThfITillili!111,,1II d ILS~ ~~~or 1: + -: 1H tlti 33Et~ii fT i iI 1: I I I t 9ti l0+h 1++<99Xr0++iF<~~~~Al 08- tit L 06- ~~~~~~"" 6 --- 11 J l, II1 1 I I I II I ' —'-t0 I t li:~fl*~i; lri '! itlli ll/11tlmmO-E hli +ir 2 l I~1 ZLLLL 14Th+-<tl I I T I - I I f I 11 3'[11 I T 1 I i 1? ii i:!; 1 i[T.*1i] i1 i vi I i i i i ii [._, OL- P, ------ -----— (P)t X ~ 06- ['":: L[l, ][]n[[0-0 Ii1;li;'711; ir 1 ' ' tl 4';;l:ltlli/ll- ll l~l;.il I ',i/, [i itTJ~l iI 1 l l ~il l l I l t I l f/ I [ itl IIIII!Ili- I+,,II 0 '1ii il-liII I k I 11t1t iPl.lllll.l1 I 0 1 1 11'11i TT F1117PT [TI F[TTTI I I i I i I I I I ll i - T T It iiOL-4..-.1-.99 N I —,I Hi I I 0 I I lt tl l I Il OL~~~~~~~~~~~~0- i.... OLT I~~~~~~~~~~~~~~~~~~~~~ 08- 7~ 09 -( p )..........-...p... 01.,- O ~ ~ ~ ~~kl

6t, NIKLJLVd cNYNIfIOD NUOH IVDJNOD CTZIllYIOff PV't9nrDID:8T DI0 08 --ucOL i-t~~~~~~~~~~09 -30 8 =J lNVO 3IHc -I1O ISI aA I NIf% H0i

THE UNIVERSITY OF MICHIGAN 6633-1-T -5 -70 (db) (db) IIIIII+HF 11HI- T 1 f=8Gc -60 -80 2 IH ~fiR -70 -Aoo -8O ~F~ttrt-t ~ ttmtc-100 -4 -60 -40 -60 (db) (db) (db (db) -50 ~t~~f~tt~ttt~t-70 -50"t~fPt~~i~~;-: 1~C;ft —70 -60 -80 -60 -80 L~~~~~~~ -70 -90 70F -9 -180 90 ~ 0 90LLI 18~i 0 -180 -90 1} 0 90 180 FIG. 19: PYRAMIDAL HORN COUPIdNG PATTERNS. ( —)E-PLANE, ( ---) H-PLANE. 50

THE UNIVERSITY OF MICHIGAN 6633-1-T case is more directional than for the conical horns because there is more tapering of the E-field in the pyramidal horn aperture. It is to be noted that this more selective coupling occurs even though the conical horns have a much larger aperture, and correspondingly, much higher gain. Due to the lower level of the 'sidelobes' in the ground plane for the pyramidal horns, an antenna designer might want to consider using a pyramidal horn instead of a conical horn to achieve lower coupling levels between adjacent horns. 51

THE UNIVERSITY OF MICHIGAN 6633-1-T IV CONCLUSIONS So far the coupling data on a variety of antennas have clearly demonstrated the existence of a wide range of coupling values. The coupling level of spiral antennas is on the order of -30 db while the coupling levels of some of the horn type antennas having the H lines co-linear is less than -75 db. Thus some of the antennas now under study will show levels below the -75 db level of interest stipulated in the contract, at least for some spacings of the antennas. The -75 db level is believed to be a realistic limit for coupling measurements in a free space environment. Evidence now indicates that in a real life situation, effects due to scattering will be predominant for antennas having an intrinsic coupling level of -75 db. It should be noted that the coupling values which have been observed for a variety of antennas studied to date are far below levels commonly reported in the literature. Theoretical expressions have been developed for the coupling between E and H-sectoral horns using an approach applicable for far field only and by a second approach valid for both near and far regions. Good experimental verification has been achieved for these results. Theoretical expressions have also been derived for far field coupling between conical horns. 52

THE UNIVERSITY OF MICHIGAN 6633-1-T V FUTURE WORK To date the required experimental and theoretical effort has largely been accomplished for rectangular slot antennas, spiral antennas, and horn type antennas. In the remaining time it is expected that experimental work will be performed on a trough waveguide-type of traveling wave antenna and on recessed conical spiral antennas. The latter, it is anticipated, will be handled very much like the earlier spiral antenna work. Finally, during this period it is expected that coupling of scimitar antennas will be studied. This latter study will be very interesting because of the broad pattern of the scimitar and the multipolarization situation which exists. It is planned to give more attention to the influence of independent scatterers on the coupling between antennas. It is believed that two horn antennas appropriately spaced and mounted on a common ground plane on an aerospace vehicle may not have sufficient direct coupling for significant interference, but that problems will occur due to scattering from one type of protuberance or another. The interference effects due to scattering from objects in the anechoic room during coupling measurements have helped to show the importance of considering their effects. The detailed organization of the data acquired so far is a major objective in future work. Our present plans call for the presentation of data on rectangular slot antennas as well as circular and square Archimedean spiral antennas in avariety of nomograph charts. A separate nomograph will be made for each type of antenna. A typical nomograph for the circular Archimedean spiral antenna will predict the coupling for a given spacing and orientation angle of the second antenna with respect to the first. The bar scales in the nomograph will be on a normalized basis whereever possible. In order that the nomograph can be used for all designs of circular Archimedean spirals, it is possible that a supplementary nomograph involving the

THE UNIVERSITY OF MICHIGAN 6633-1-T Nomographs for the square Archimedean spiral would, no doubt, be very similar to those for the circular Archimedean spiral. A nomograph for the coupling of rectangular slot antennas would probably be somewhat simpler to design than those for the Archimedean spirals. It now appears that one simple nomograph taking into account the actual dimensions of the slot as well as the frequency of operation would be sufficient. In the cases of the antennas described above, it is contemplated that it will not be necessary to do much smoothing or averaging of the experimental curves. However, it appears that it will be necessary to average the experimental curves obtained for circular or rectangular horns in order to avoid the use of a double-valued scale on the nomograph. Simplicity in the construction of a nomograph precludes multiple valued functions. In addition to the antennas already cited, it is contemplated that coupling data for monopole antennas erected perpendicularly to a common ground plane will be presented in nomograph form. This appears to be straightforward at the present time. One of the more useful methods for the presentation of the data may very well be by means of a computer program. The possibility of preparing a program for the coupling between various rectangular and conical horns will be investigated. It is believed that one program will be sufficient for any and all rectangular horns. This program will be written in a standard programming language. During the next few weeks, a decision will be made in cooperation with the sponsor regarding the choice of the programming language. 1 _ _ _ _ _ _ _ _ _ __ ~54 _

THE UNIVERSITY OF MICHIGAN 6633-1-T REFERENCES Khan, P. J., et al (1964), "Derivation of Aerospace Antenna Coupling-Factor Interference Prediction Techniques, " The University of Michigan Cooley Electronics Laboratory Report No. 4957-8-F. Lewin, L. (1951), Advanced Theory of Waveguides, Iliffe Publishing Company Company, London, p. 123. Potter, P. D. (1963), "A New Horn Antenna with Suppressed Sidelobes and Equal Beamwidths, " Microwave Journal, pp. 71-78, June 1963. Silver, S. (1949), Microwave Antenna Theory and Design, McGraw-Hill, New York, pp. 336-337. Stratton, J. A. (1941) Electromagnetic Theory, McGraw-Hill, New York, pp. 29 and 431. 55 _