3667 -6-P OPR No. 3667-6-P = RL-2108 DEPARTMENT OF ELECTRICAL ENGINEERING COOLEY ELECTRONICS LABORATORY Quarterly Progrss Report No. 6 Study and Investigation of a UHF-VHF Antenna Period Covering July 1, 1961 to October 1, 1961 Prepared by: J. C. PALAIS A. 1. SIMANYI R. M. KALAFUS A. T. ADAMS Approved by: J. A. M. LYON. EL Ec,0 C 4,R ~ - I ERAt '4 80 R Al Under Contract With: Air Research and Development Command United States Air Force Contract No. AF 33(616)-7180 Administered through: November 1961

THE UNIVERSITY OF MI CHI GAN COLLEGE OF ENGINEERING Department of Electrical Engineering Colley Electronics Laboratory Quarterly Progress Report No. 6 Period Covering July 1, 1961 to October 1, 1961 STUDY AND INVESTIGATION OF A UHF-VHF ANTENNA A. T. Adams R. M. Kalafus J. C. Palais A. I. Simanyi Approved by: ORA Project 03667 under contract with: UNITED STATES AIR FORCE AIR FORCE SYSTEMS COMMAND AERONAUTICAL SYSTEMS DIVISION CONTRACT NO. AF 33(616)-7180 WRIGHT-PATTERSON AIR FORCE BASE, OHIO administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November 1961

TABLE OF CONTENTS Page LIST OF FIGURES iv ABSTRACT v 1. REPORTS, TRAVEL, AND VISITORS 1 2. FACTUAL DATA 1 2.1 Plane Wave Incident on a Material Sphere in Which Is Imbedded a Lossy Inner-Sphere 1 2.1.1 General Discussion 1 2.1.2 Mathematical Analysis 2 2.2 Scattering of a Plane Wave by a Long Ferrite Cylinder 11 2.2.1 Mathematical Analysis 11 2.2.2 Computer Results-Interpretation of the Resonance Properties Shown by the Total Power Flow Through a Unit Length of Cylindrical Ferrite and Dielectric Material (TM Case) 24 2.3 Scattering of a Plane Wave by a Ferrite Prolate Spheriod 32 2.3.1 Scattered Field 33 2.3.2 Boundary Conditions 37 2.4 Radiation from a Material-Filled Rectangular-Waveguide HPlane Sectoral Horn 40 2.5 Radiation from Horns Flared in Two Dimensions and Filled with Ferrite Material 45 2.5.1 Radiation Field of the Horn 52 3. ACTIVITIES FOR THE NEXT PERIOD 59 4. SUMMARY 60 REFERENCES 62 DISTRIBUTION LIST 63 iii

LIST OF FIGURES Fi gure 1. Field coefficients, real part. 2. Field coefficients, imaginary part. 3. Power absorption ratio. 4. Plane wave incident on cylinder, electric to cylinder axis. 5 Plane wave incident on cylinder, electric dicular to cylinder axis. 6. Power flow through the cross section of a cylinder. 7. Power flow through the cross section of a cylinder. 8. Power flow through the cross section of a tric cylinder. 9. Power flow through the cross section of a cylinder. 1C. Prolate spheroidal coordinates. 11. Incident electromagnetic wave. 12. Coordinate systems Lsed for H-plane horn. 13. Double-taper ferrite-filled horn. Page 3 4 5 field parallel field perpenlong ferrite long ferrite long dieleclong ferrite 17 23 25 26 27 28 34 >)5 41 46 iv

ABSTRACT Several theoretical and experimental problems were studied during this period. The study of diffraction of a plane wave by a ferrite sphere was extended to include a small lossy inner sphere enclosed within and concentric with the original ferrite sphere. Computer results show that the total power absorbed by the inner sphere rises to peak values of about 500 times the power incident upon a cross-section area equal to that of the large ferrite sphere. A summary of the mathematical results has been compiled for the theoretical study of the diffraction of an incident plane wave by a long ferrite cylinder. The total power passing through the cylinder has been evaluated on the computer and compared with the power incident upon a cross-section area equal to that of the cylinder in free space. The ratio rises to peaks greater than ten. A study of the scattering of a plane wave by a ferrite spheroid was initiated, but, because of the need for more experimental activity in other areas, temporarily discontinued. The basic analytical steps are outlined. The radiation from a material-filled rectangular waveguide H plane sectoral horn has been analyzed. Computer results will be given in the next quari terly report. The problem of the radiation from a ferrite-filled double taper horn fed by a rectangular waveguide has been analyzed mathematically. v

1. REPORTS, TRAVEL, AND VISITORS During this period no reports were issued and no one visited the project. A trip was made to Wright-Patterson Air Force Base on September 5 to discuss project matters. CEL personnel present were CEL Director B.F. Barton, Professor John A.M. Lyon, A.T. Adams, R.M. Kalafus, and J.C. Palais. A number of VHF and microwave antenna problems of interest to the Aeronautical Systems Division were discussed in connection with possible extensions of the contract. 2. FACTUAL DATA 2.1 PLANE WAVE INCIDENT ON A MATERIAL SPHERE IN WHICH IS IMBEDDED A LOSSY INNER- SPHERE 2.1.1 General Discussion. -The sphere problem has been further modified to include an inner material sphere of complex i and E. There are two reasons for solving this problem. First, the integration of the Poynting vector over the inner sphere will yield the power absorbed by it. This can be compared to the amount of power incident upon an area in free space equal to the crosssection area of the large sphere, or to the power absorbed by the same inner sphere in free space. The second reason is the following: in an actual antenna the energy is carried away from the antenna into the receiver through the feed; the waves are not reflected exactly as they would be if the antenna were shorted at the terminals. By postulating a loss over the volume of the 1

inner, poorly conducting sphere, one can approximate the feed power by a distributed heat loss. This then yields information about the effect of the feed on the field distributions in and around the sphere. The mathematics of the derivation is given in the section entitled "Mathematical Analysis." The interior field oefficients are given in Figs. 1 and 2 as a function of outer radius with an ini er radius of.01 \o =.I ' A lossy conductor will have the parameters 1 = o, e = E'-J(e" + 2). If the propagation constant (k = w 47p) is allowed ro be complex, the radial functions must be likewise complex, so that there is a net real part of the power expression greater than zero. The problem was programmed for the IBM704 computer using the MAD compiler. The expression l'or the rati of power absorbed to the power incident (gra2 x the incident power density) is plotted as a function of normalized radius in Fig. 5. This shows that the power absorbed can be as much as 500 times the power which would pass through the large circular area in free space. Thus the energy must be concentrated toward the sphere near these resonant points. It is felt that the approach used here may help solve many antenna problems: investigation of the energy distribution of the scattered field can be used to interpret antenna problems and provide some design criteria. The results will be extended and a more complete discussion and interpretation given later. 2.1,2 Mathematical Analysis.-A concentric sphere interior to the original material sphere is postulated, and specified to be of a lossy material. As with an inner lossless metallic sphere, the field solutions in the inter 2

100 I0 0.1 0.01.001 -,-.0001 0.. 2. 3.4.5 a/Xm Fig. 1. Field coefficients, real part 1.0 5

o o 0 o 0 COEFFICIENTS 0 0 0 0 0 * I. ' C >3a 3 Fig. 2. Field coefficients, imaginary part.

1000 100 I0 INNER BOUNDARY AT:-r b:z O. Ikm) o ic0 4 z 0 I.CL 0.I 0 U) 4o 0 -ri 0P 02 to.01 ABSORPTION RATIO * POWER ABSORBED IN DISSIPATING SPHERE INCIDENT WAVE POWER DENSITY X 7ro 2 * 0011 INNER SPHERES I ~ 61 a doJ INTERMEDIATE REGION' P a Eun I0 10 0/Xm -o 5

mediate medium must be composed of Bessel and Neumann functions of half order. The inner sphere has only terms involving Bessel functions. This gives rise to eight simultaneous equations, allowing solution of any set of coefficients. The losses in the inner sphere can be treated by allowing p. and e to become complex. One can then integrate the Poynting vector over the sphere to find the power absorbed. The intermediate medium fields can be expressed by the same set of equations as for an inner conducting sphere. t= Eot Jn 2n+l t + ct (2) t() t- 2) EMt= Eo e E J Mn - enein " JdnRln n=l n(n+l) L. - jwt jn 2n+l bt( 1) t-(2) t-( ) t( 2) Ht =- i Eoe Jtn + djnMn n+ J annen + JCnNoln n=l n(n+l) (1) where: the superscript (2) denotes Bessel functions of the second kind. The interior fields are represented by the coefficients an and bn. _ Pn2n+l!-~.(1) ~,l(z)- i = Eoeje t z jn 2n+l l n=l n (n+l) Hijn 2n+l Ibj4 + j^(l(2) H1 =n=l n(n+l) nMeln + a, (ol2) where: E2 and p.2 are the complex permittivity and permeability of the inner sphere. There are eight equations resulting from equating the tangential E and H vectors at r = a and r = b. 6

4iSn(k~a) + 4SAI(kia) + K~ASn( k2b) ~.L L4iS( k2b) ctCn i)- Kla'R 2) (k a) =KSn(koa) cCACI(k~a) - pjeasRl(2) (koa) = SAk) = K~a4Sn(kib) + K2cnCn(k~b) = L + p2C4fSA(k~b) +2cC I(kib) (3) bnSn(..kla) + dnCn(k~a) - ~ibsRii2(koa) =~inka bt SA(k1a) + dtCA(kia) - Klb sRA(2) (koa) =KESA(koa) ribjnS(k2b) = 12bnSn(k~b) + n ndC(k lb) KlbASA (k2b) -KbtSAklb) + K~d CAI(k lb) (4) are the relative permeabilities and Kj = ki/ko. The solutions interior coefficients are: Here ~Lj 12 for the six I J~ aK~ an = p1jL2K14(a - 1K2*_b - I~L201*,c + 4~K* bI - J~.1jj~2KK2 n,K 2a - ~12K&1(b - ~IK2\*c + I~LlJI2K1d (5) 7

t = l [Kl42Sn(k2b) C(klb) - eq- eq.... K2iSn((k2b)Cn(kb) ] n - n bn=b 42K2 [IlK2Sn(k2b) Cn(klb) - 2KlSn(k2b)Cn(klb)] 42K2 [Ki42Sn(k2b)Sn(kjb) - K2iSni(k2b)Sn(kb) ] 42Kb [4lK2Sn(k2b) SA(klb) - 42K1SA(k2b) Sn(klb) ] ct n = a n dt = b VI L v (6) 42K2 where: Ia = R(2) (koa)Sn(k2b) [Cn(klb) Sn(ka) - Sn(kib)Cn(kla)] = R(2) (kOa) Sn(kb) [Cn(klb)Sn(kia) - Sn(klb)Cn(kla)] *c = R() (koa) Sn(k2b) [CI(klb) Sn(kza) - SA(kzb)Cn(kla)] *d = Rg2) (koa) S(kb) [Cn( kb) S(kia) - Sn( kb) C(kia)] (7) It can be shown from the propagation equation k2 = U24e - jwC4 (8) that losses due to dipole friction in a dielectric (which can be represented by the imaginary part of a complex permittivity) have the same effect on the propagation constant as conductive losses at a given frequency. Thus conductive losses can be simulated by allowing c to become complex; one can then allow the inner sphere to be a lossy conducting surface by letting e = c'-e". 8

Then k2 = W \/O(e'-je")o One can then integrate over the inner surface and find the total power absorbed. This can be calculated by integrating the normal component of the Poynting vector over the surface. Let r denote the surface of the sphere. The Poynting vector is defined by ' = 1/2 Re (E x H*); the surface integration is ff Sn da = b2 f f2 (f. n)sin 9OdOd (9) r 9=0 0=0 where: S n 1/2 Re (E x H*). ar = - 1/2 Re (EgHA - E^Ht)~ (10) where the asterisk denotes the complex conjugate quantity. Substituting the field equations, ff (EgeH}da = JfI Eot I t 2n+l 2m+l n=l m=l n(n+)' m(m+l) + laitl Sn(k2b) S!(k2b)* f Pn(cos e)Pm(1os ) dG (11) k2b k2b* =0 sin 9 '( 2 Sm(k2b) S (k2b) T rP1(os 9) 0 -Ibn ~.~. *.C0P(cos ) sind k2b k2b* 9=0 inG d The terms involving atn and b have an integration involving Jro sin y cos d~ = O, so that these cross-terms do not contribute. fJJ fEHaIda = jf E 2n+l 2m+l VI2 0n=l m=l n(n+l) m(m+l) 9

t I Sn(k2b) Sm(k2b)*. p 1 pm ~ II f- ~ sin GdG C k2b k2b* 9=0 as aq + 22 (k2b) Sm(k2b)* I P(cos )Pm(cos ) d (12) +I bnj f n dO (12) k2b k2b* 9=0 sin 9 J The Legendre functions have orthogonality properties which greatly simplify the expressions (Ref. 1, p. 417). ( ^ -. n1 P sn d \\ 2[n(n+l) ]2 O oe \ 8 sin2G/ 2n+l = 0, m n (13) Thus ff (S )da = b2 - f2 Re{[E - - EH)sin Qded^ r 2 9= —O =0 r 2~ = -j1 E Ei Z (2n+l)[-| 1a|2S(k2b)Sn(k2b)* 2 Ik2 12 n=l + btnl S(k2b) Sn(k2b)*] (14) The power in the incident wave over an area ra2 corresponds to the power passing through the area of the larger sphere if both spherical regions were composed of free space. The reference power is found by integrating the incident wave power density over a circular area ia2. P = * ira2 Re{Ei x ~). a' = i- E2 (15) 2 o 0o If we let the inner medium be a lossy conductor, E2/o = e'-je", L2 = 0o; the ratio of the absorbed power to the reference power is 10

e Ei E (2nfl)[|b| n(k2b)Sn(k2b)* - an Sn(k2b) S(kpb)*] (16) n=l jk2a12 n=l 2.2 SCATTERING OF A PLANE WAVE BY A LONG FERRITE CYLINDER 2.2.1 Mathematical Antlysis.-The basic solution of this problem was outlined in Quarterly Report No. 4. Since then, the expressions for the, Poynting vector inside and outside the cylinder, for TM and TE cases, have been formulated. In addition, expressions have been derived for the total power passing through a unit length of the cylinder. The fields and Poynting vector at the center of the cylinder have been evaluated on the IBM-704 and resonances noted for e = - = 10. The total power through the cylinder has been evaluated on the IBM-704 and resonances noted for several different values of 4 and e. The basic formulation of the problem is as follows. TM Case (Fig. 4) In addition to the expressions for the fields derived in Quarterly Progress Report No. 4, repeated below for convenience, expressions were derived for the Poynting vector and total power through the cylinder. Fields (Ref. 1, p. 360) Fields Inside Cylinder 00 EZ = k2 bne Jn(kr) n=-oo k2 ij Hr = Z ~ nbne Jn(kr) (17) 4wr n=-oo tik3 n00 H = -- bneinGJ(kr) 4w n=-oo 11

Fields Outside Cylinder 00 Ez = k2 Z. 0 n=-oo ane GH$1l (ko) + A,0 Z Jn,(kor) e n=-oo Hr = 7 na e r n flo = -oo inG (41) (k r) 4- AO —,,E 00 Cos G Z. Jn(kor)e ing n=-oo ik3 00 ing H9 Q Z8.ne HAJ') (k r) -A.E \Ii' 00 sinG Z fl=-oo in m JnI~kor) e (18) Poynting Vectors PXInside Cylinder at Point (r,,G) = - -Re(Ez 1 Rk2 Z 2 n=-oo bnemn Jn(kr)I xK-4 00 n=-oo b*e- minG m mkr 1 ik5 00 00 nm = - Re-i z z bnbe Jn~rnm/kr 2 4w L=-oo m=-00 k5 00 00 b + b. i~= - ~~c n= E =0 Jn~krl)Um\Lk) I(nrbmr+ nibmi sn~-)G + (bimr - nrbmi)cos(n-m)G41 f..Inside Cylinder at Point (r,G) (19) Pe= - Re (E x Hr) 2 -~00 = I Re k27Z 2 n o bne 11Jn(kr, a) 00 Z mb~eim M= -00 kr) _1 2 4 00 00 4wr L=W0 m=-00 mbnb -ei(n-m) GJn(kr) Jm(kr) I j k5 00 - xZF 2~i6&(kr) n=-o 00 ~ Z mJn(kr) Jm(kr) )(bnrbmr + bnibmi) cos(n-rn)G9 - =-00 0o M: (bnibmr - bnrbm.4) sin(n-m) 9 (20) l L.

POutside at (r,9) a Pr-I- 2 Re (Ez x H9) 2 - AO s~ sin 9 ~-L0 I= - -ReEk2 00 1 z Jm(kor)e-"'io Mfl-00 aneinH 1)(kor) + A0 E Jn(kor) e jx fl=-oo LA E qe t(1) *(kor) I. LoCi) m=-0o = 1ReE-0 00 z fl=-oo 88anHI'q (kor) H11(' (kor) ei~m M=-00 Aoik3 I -,i r A2Pow o 00 z:1=-CO 00 z a*Jn(kor)H1(l()*(kor)ei (n-m)g AokQ.2 U sin 9 M=-m 001 l=-.oo 00 anJm(kor) Fj~ (k~r) ei~m M=-00 00 sin 9 Z n=-oo 00 z M=-00 Jn(kor) Jm(kor) ei(n-m) 9 substituting: an =anr + iani H = Jn + in~ where N. = Neumann function k -9 p = - - 0 r 2iow n=00 m=E {Bnranr + aniami) (N(kor) Jlh(kor) - NA~(kor) Jn(kor) + (aniam - anrami) (J(kor) JAh(kor) + Nn(kor) Nih(kor)) cos(n-m)9 + (anramar + aniami) (J(kor) J i(kor) + Nn (kor) NMI(kor)) Aoko3 - (aniamr - anrami) (n(kor) JA,(kor) - Nm(kor) JAl(korj sin(n.-m)Qp @- Pc 00 z, n=-oo 00 Z ja~.J~h(k,,r) Mf= 00 - amjNA~(kor)~ s in( n-rn) 9 x Jn(kor) - (amjJI(kor) + am~rNA,(kor) c os(n-m) 9 x Jn(k ri ok / /2 V~ 00 s in 9 Z n=-oo 00 z M=-00 (21) [?nrJn(kor) - aniNn(ko-r.'cos(n-m) 9 +A sin @ Z 2 11o n=-oo 00 z M=-00 Jn (kor) Jm(kor) c os (n-m) J

P9 Outside Cylinder at, (Trjl) iing kZ JEk r)-ikn Pe iRe(E x IT,) Re)k2 - ane noJL)(kor) ~AO __ eiojJ xae H kr 2 2 =-o n=c* CWD Mf=-oo - 0 4 00 00 0c~(-) k2~ini.m 0 0 + AO Co Z JmJ(kor)em (k Re A0 co 9a anm% kr) m1(kor)Hi e (o~ n=-c M=- n=-co m==o-m=oon 0oo Ao 0 00 00.ii Ji 00 + COS 9, Za ZJ Jnkor r)J,(kor)e knJ+ AO ko Co ZJ9 ~ ~ an 11~kor) (n" n(kor)Jm o ILo a n=-w =-co -co n=-oo m= -wL~ + Nn(kor)Nm(kor) -(aniamrn- amianr) (Jm(kor)Nn(kor) - Jri(kor)Nm(kor)) x cos(n-m)Q - Laniamnr - amianr) (Jn krJ~kr n(kor)Nm(ko) + (anramr + a'niami) (J(kor)Nn(kor) - Jn(kor)Nm(kor)) x sin(n-m)9J 2 Dr fl0c m Lam O~~k - aiiNm(kor) conm, 9 + L1ium~kor) + 8amrNm(korj sin(n-n)G Jn(kor) Aok2!Eo 00 00[j fl-00 Cos anr~Jn(kor) - n~~o cos(n-m) 9 - a kr + 11 ~ m-0 0o ni~n'kor) anr~dn~or/)sinkn-m/yGJ~mkfor/ A2 00 00 + y- j-Ii c"os 9 Z Jn(kor)Jm(kor)cos(n-m)9 (22) 2 ~ 0 fn-00 mfl-0

Total Power Passing Through Unit Length of Cylinder = / lr aPrdGd~ = a JfPrdd (radius of cylinder = a) 00 0 r Total power -k5a 24o 00 o n=-oo oL (bnrbmr + m —~0 -bnib (bnib bnibmi) sin(n-m) + (bnibmr - bnrbmi) Cos(n-m)~ Jn(kr)Jm(kr) x dQ k5a 2p= 00 n=-oo 00 f-I Z Jn(ka)Jm(ka) |(bnrbmr + bnibmi) f sin(n-m) @d + (bnibmr - bnrb k 5a 2~iw 00 E n=-oo 00 E m= -o )mi) fi cos(n-m) d7 I i Jn(ka)Jm(ka)(bnrbmr + bnibmi) x - (contribution n-m _ only when n+m is odd) Changing Summation to (O to 0o) x m Jn(ka)J(ka)(bnrbmr+bnibmi)n (for m even n odd) E0) 0o Ei o "n2 - m2 4 o (25) alt5 00 E 0 00 0 EonEom Jn(ka) J (ka) (bnrbmr+bnibmi)m (for n even, m odd) n2 - m2 Power through same area in free space,- -- a = A! -s x a I M-o (24) Dividing Eq. (23) by (24) gives the ratio of power through the cylinder to the power through the same cross-sectional area. Eon = 1 (n = 0) = 2 (n / 0) 15

In te abve, nr, ~j, nr, i for 'the TM case are defined as follows: an = anr + iani bn = bnr + ibni AO [cxJn(ka) JAl(koa) - Jn(koa) JAl(ka) ] [cxHj' ) (k a) Jn(ka) - 1)' (k a) JAl(ka)]I (25) CAo [ H Ij') (k a) Jn(koa) - H)(ka) J'l(ko) ii k2 (1) ) k[ciHA I(koa) Jn(ka) - Hn(k a) JAn(ka) ] where I= 7, — ol = ko = 0) —0-0 k = W ~f HA (koa)Jn(koa) - H~)(koa)JA(koa) 2i (by Wronskian relationship). Primes denote differentiation with respect to argument. anr A0 ko A0 [ciJn(ka)JAf(koa) - Jn(k a) JA(ka) ][ciJA(koa) Jn(ka) - Jn(koa)JAl(ka)] Dn [ciJn (ka) JA (koa) - Jn (koa) JAIn(ka) ] [CINAI(k a) Jn (ka) - NnkaJAa) kD2 bnr irk 2k a [Nn0 koa) JAI(ka) - cZNAn(koa) Jn(ka) b 2cxA0 [Jn(koa) JAl(ka) - aJA(k Oa) Jn(ka) ] bni =- - 2 1 Trk koa Dn where Dn (TM case) = [oiJA(koa)Jn(ka) - Jn(koa)JA'(ke) ]2 (26) + [ctNn' (koa) Jn(ka) + [~NAkoaJn~a)- Nn(ko~a)JAf(ka) ]2 16

2Q i — INFINITELY LONG -" CYLINDER i Ez I Hi x Fig. 4. Plane wave incident on cylinder, electric field parallel to cylinder axis. 17

TE Case (Fig. 5) As in the TM case, expressions were derived for fields, Poynting vector, and total power through the cylinder. Fields (Ref. 1, p. 360) Fields Inside Cylinder 00 HZ = k2 bnein Jn(kr) n=-oo Er = - --, nbne inJn(kr) n=-oo (27) 00 = - ikui ZF n=-oo bne in (kr) n n\, Fields Outside Cylinder HZ = k 2 Z aneinQHl)(koa) + AoIC - Z. einQJn(kor) n=-o io n=-00 E ow w neing(1) ) Er nane H (koa) r n=-oo 00 + AO cos Z n=-oo ein Jn(kor) (28) 00 Eg = - ikoioC Z anein(1 (lkoa) n=-oo 00 - Ao sin 9 Z einjkor) n=-oo Poynting Vectors Pr Inside Cylinder at Point (r,G) Pr = Re(E x Iz) 2 1 Re -ik~pc 2 00 Z n=-oo bneiJ =oo imo ej(kr = i Re - ik3~ 2 n=-oo m= 00 z n= -00 00 i n-nm) Z bn~b ei( Jm)(kr)Jm(kr m= -oo.00 + (bnibmr - bnrbmi)cos(n-m)e x Jm(kr)Jn(kr) 18 (29)

ERInside 2ylinder at 'Point(rQ Re G Gon *i(n-m)Q0 ReHEHZ - 20 nbne'n Jn(k L, b11e i=J kRe 1 2 nbnbrne Jn(kr) Jm(kr) 2 2 Lr nm-cook~[ =c 2 ~o3 -l EnJn(kr) Jmkr), rm bnibrni) cos(n-m) 9 - (bnibmr bnrbumj) sin( n-rn) -2r n=-oo mn —o Pr Outside Cylinder at Point(r) i F mG i)n~ ing 2 c = Re (Fx HZ) =-ReI-i~Lc-nko 2 ane HA(' (kor) - AO sinG L e Jnkor) x 1k2 a4eH-'n((kor) 2r 2 Ln=-oo n=-oo Jn LAnc + AO5. 2 - ei"'Jrn(k Re- i~io~kg n=r-oo m=-oro i (n-m) 0 V o rn=-cx 2korj - Ak2Osin9 2 2 a4Jn(kor) Jl1)*(kor) ei(n-m)G1 A~dk nzkr kr n=-co m=-oo goi nm-co mm-coa.w o)IA (~~ 22 0 i (n-m)T - AS4. sin 00 co n~o)m~o go~ ~ n=-oo m=-cood1io~ ~I&re 3 00 00 2 n=-co m=-co mr+niami) x (J Alkor) Jm(kor) + N,(kor)Nn(kor)) (aniarnr - an.rani) (Ni(kor) J3(kor) -Nrn(kor) Jn~kor) sin( n-rn) 9 + [(ani amr - anrami) (JA(kor) J3( kor) + NAC kor) Nm(kor)) + (anram.r + aniarni) (NA~kor) Jm(kor) -1m( k~r)JA( kor)I cos (n-rn) Aok2sing oAk2 C w 22E I(&aJm(kor) -ajmkrconmQ (ai,'kr)erI ko)sin(n-m)G] Jn(kor) + A~0 00 7-(nrn J3(k-r)m00w00 n=-co rn=-cx ar~ ( an~lk~r))sin~n-m, G + (aniJn~or + nN~~)c n-m)~ -m r - sin 0 2 2, Jn(kor) Jm(kor)COB(n-m)GI anNC i 'J~~)2 go nm-co 3mw-co (5')

POutside Cylinder at Point (r,@) P Re1 (E = R nane H~ (kor) + AO Cos 9 Z einJn (k ri x k.4 i (k)* Ao 1 0ck *i(n-m)G9 (1) - Oe (kr enaname Hn(k r) Hm' *(k r) ~1o m=-m im02 r n=-c m=- wo Z *i~(n-m)G I ~oP~ 00 inm am' kr)e Jn( ) O ~ unane )Hl)(kr)Jm(kr) n=-cx m=-m Un~~o,- o0V r n=0 m=-oo 00 00 - g- CO 9Jn(kor)Jm(kor)ei~m@ I'o n=-oo m= —oo 0 Z2 Z [inam + 8aniami)(J-n(kOr)Jm(kOr) + Nn(kor)Nm(kor)) -aiar- amianr)(Jm(kOr)Nnk) =j 2r n=-oo m=-oo Lnarkoanir -Jn(kor)Nm(kor)Z cos(n-m)g - Lanraitxr + aniami)(Jm(kOr)Nn(kOr) - JI(kor)Nm(kor)) + aim - 8mianr) Aok 2COSQ 00 00 (Jn(kor)Jxn(kor) + N~o)mkr sin(n-m)g n - 2 2 (ajnrJm(kor) - amiNm(kOr))cos(n-m)G 2 n=-oo m=-00 -(amiJm(kor) + amrNm(k r) )s in(n-m) 9]Jn(kcor) - m=-00 (anrJn(kor) - aniNn~o)csn) 2r n=Jno m=r)Jm(kor))cos(nm)~ (5 -(aniJn(kor) + anrNn(kor) )sin(n-m) 9Jnim(kor) - AgY~ eCOS n=-oo m=kr-00 Jm)G(3

Total Power Passing Through Unit Length of Cylinder Total power = 1 aPr dGd~ (a = radius of cylinder) 00 Z. Jm(ka)Jn(ka) (bnrbmr + bnibmi) f. sin(n-m)9de m=-oo k 2 00 E n=- - + (bnibmr - bnrbmi) fo cos(n-m) Qdj ki2 2 00 n=-oo m= Jm(ka) Jn(ka) (bnrbmr + bnibmi) [(-1) ] m= —0 n-m (contribution only when n+m odd) Changing Summation to (0 to oo) Total power - 2 n0 Jn (ka) J(ka) (bnrbmr+bnibmi)n n=O m=O n2 m2 2 00 00 E Z n=O m=O Jn(ka) J( ka) (bnrbmr+bnibmi) m n2 - inm2 ConEom (for m even, n odd) EonEom (for n even, m odd) (533) Power through same area in free space = 2 ~ = AO xa. (5533a) Dividing Eq. (33) by (33a) gives the ratio of power through cylinder to the power through the same area in free space. In the above, anr, amr, bnr, bni for the TE case are defined as follows: 21

.AO -, [aJn(koa) JA(ka) + Jn(ka) JAt(koa)] n kS [a~n(koa) J n(ka) - HIA(koa) Jn(ka) A.0 r-: [n(koa) JA(koa) + Jn(koa) HA(koa)]() n [caJn(koa) JA~ka) + JA(koa) JA(kva) I[ZAk)J(~)-J(a,~) A [axjn(koa) JA(ka) + Jn(ka) JA(koa)]I[aJA(ka)Nn(koa) - Jn(ka)NA(koa)] ro r A0 RJ [2J;(koa) JA(k~a) I [cza)JA(ka)oka) - JWA(ka)NA(koa)I - [CXJaA(koa) J~ = [cxJAka)JJnk~k)a-Jn(koa)][aJAkka) 2+cJn(koa)N-n(koa)-Jn(koa)N [A'jkoa)lgn k0a) + Jn(koa) NA(koa) I [atJA(ka) Nn(koa) - Jn(ka) NA~(koa) ] Ik a) -Jn(ka) JAn(koa) ] [JAl(koa) Nn(koa) + Jn(koa) NAn(koa)]1 (35)

i ~Hz INFINITELY LONG fCYLINDER \-i -Y P(r,9) on cylinder, electric field perpendicular i Ex Fig. 5. Plane wave incident to cylinder axis. 23

2.2.2 Computer Results-Interpretation of the Resonance Properties Shown by the Total Power Flow Through a Unit Length of Cylindrical Ferrite and Dielectric Material (TM Case) -The results of a computer study showing the normalized power flowing through a unit length of a lossless ferrite or dielectric cylinder with an incident plane wave appear in Figs. 6-9. The normalized power is defined as the ratio of the power flowing through the front surface of the cylinder (subtending an arc of 180~ at the center of the circular cross section) with the material medium present to the power that would pass through the sam cross-sectional area in free space. Therefore, whenever the magnitude of the normalized power level exceeds one, there is more energy flowing into or out of a unit length of the cylinder than would pass through the same area in free space. Before the graphs are described in detail, some remarks on their general behavior might be in order. The expectation entertained before any numerical computations were performed was that the cylinder should exhibit transverse model resonances at well-defined frequencies, depending on the propagation characteristics. Since, in the building of any practical antenna structure utilizing the energy densities available inside the cylinder, it will be important to know the total energy flow into the cylinder rather than that computed on the basis of a few modes, it was decided to compute several final plots for the total normalized power flow into the cylinder. It was hoped that these plots of power flow would still show an improvement or resonance condition even though taking the integral over the front surface of the cylinder involves an averaging process which should 24

1.9 /L=3 3 I.I 1.5 a0.5 I - z PERCENTAGES. BANOWIDTH FOR INORMALIZED POERI> CENTER FREQUENCY OF RESONANCE a/Xf0.0 I I 1,, I 0.1.2.3.4.5.6 7.8.9 1.0 1. 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Fig. 6. Power flow through the cross section of a long ferrite cylinder.

10 1,5 e =10.56 nr' LJ 0 z m._J rr' 5 4 3 2 0 - I -2.78 83~A 1.00 r( K K / -3 13%0/J 14% T 3.53 ABOVE IMPROVEMENT THRESHOLD: 35% a/ Xf..17 IQ 2 I I., I,. I, I..I.2.3.4.5.6.7.8.9 I.u 1.1 c. I.o i1. 1.c i.o Fig. 7. Power flow through the cross section of a long ferrite cylinder.!. ( 1.0 i. e-.u

9 8 =100 7 6C 5S! 38 4 r\) -, o rr 0.. -J ro 0 z 3S 8% 21 A.62 A.88 A 1.12 i 1.38 A 1.62 A 1.88 A 0 -1 120/ J 3% 2% 2%. 6\ o ABOVE IMPROVEMENT THRESHOLD: 15%.36 a/Xf-! ' I! I I I I. --------- ~ ~~ ~ ~~ ~ ~~~~~~~ -06 --------------------------------- f A.1.2.3.4.5.6.7.8.9 1.0 I.I L2 1.3 1.4 1.5 1. I1. Fig. 8. Power flow through the cross section of a long dielectric cylinder. 1.8 1.9 Z'U

+ 1.0 -1.0 Lui N) 3. co o.J cr 0 z -2D 0.1.2.3.4.5.6.7.8.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Fig. 9. Power flow through the cross section of a long ferrite cylinder. 1.7 1.8 1.9 20

cut down the resonance peaks considerably. In addition to this averaging, there is a large amount of mismatch that will confront the incoming plane wave whenever the characteristic impedance of the cylinder is very much different from that of the surrounding free space, i.e., when V47/ is much different from 577 ohms. In two figures, Figs. 6 and 7, ZO is that of free space. In Figs. 8 and 9 ZO is 57.7 and 5770 ohms, respectively. Both of these impedances would give rise to a reflection coefficient of +.8 for a plane wave meeting a plane boundary. While the effect for a curved boundary is harder to predict, it is safe to say that this change of characteristic impedance will introduce a mismatch in fields at the boundary leading to a loss in the energy densities penetrating inside the body. Because of the difference in characteristic impedance it is natural to discuss the four graphs in two groups. Figures 6 and 7, for E = e = 5 and = e = 10, respectively, have ZO = 1. Figures 8 and 9,. = 1, e = 100, and = 100, e = 1, have ZO =.1 and ZO = 10, respectively. In the first two figures it can be observed that the value of the permeability and the permittivity has a profound influence upon the shape of the power resonance curve. In the first place, the low frequency value of the total power in the cylinder, i.e., the value for a thin cylinder, reaches.5 for Fig. 6 (i = e = 5) and.2 for Fig. 7 (k = e = 10). In case I the first resonance is reached at approximately a =.25\ferrite. All the resonances evident, up to the one at a = 1.6Xf occur spaced.21-.24 ferrite wavelengths apart. This is quite an amazing regularity, considering that terms from which the coefficients of the separate modes are derived 29

exhibit resonances due to the difference in the denominator of properly weighted products of Bessel and Hankel functions. Therefore, for the individual modes we would certainly not expect this regularity evident in the average power flow into the cylinder. All the resonances occurring are fairly broad, reaching bandwidths between 26% (at.5wf) and 12% (at l.l\f) of the center frequency. For all the following discussion, bandwidth will be defined as that region of frequency over which the absolute value of the normalized power flow into the cylinder reaches a value larger than one. This bandwidth is indicated for each peak on the graphs. All the resonances which occur are of moderate height, with the highest peak reached equal to 1.8. A feature which distinguishes case I from all the other cases is quite significant. For p = c = 3, the only resonances obtained are positive values, i.e., the normalized power oscillates about the +1 line, never reaching -1 (or any negative value). This is significant because it means that, up to a radius a = 2kf (at least), power always flows into the front half of the cylinder; never does power flow out of it. For all the other computed cases there occur both plus and minus resonances, signifying that there is a net power flow both into and out of the front half of the cylinder. Figure 7, p = e = 10, presents quite a different picture. Note the change of scale which was necessary to portray accurately the shape of each of the four graphs. This is the only case in which there is not a large amount of periodicity in the resonance peaks. One phenomenon typical of Figs. 8 and 9 is demonstrated by the twin resonance at.53 and.56kf (of magnitude 3 and 5, respectively). Between these two points, with a 6% change in fre 30

quency, the flow of power is reversed in direction. This change in the direction of power flow could be used in an antenna system to obtain some quite selective filtering of two slightly different frequencies. For this case peaks of magnitude 2 and bandwidths of over 10% are reached several times (at.78%f and 1.02\f)., Narrower twin resonances occur in the neighborhood of.55Xf and 1.27\f. A narrow resonance of 35% bandwidth and peak value of 12 is found at 1.5Xf. This is the highest power concentration found in the entire graph, and oddly enough it does not have minus resonance point associated with it. Figure 8 depicts the case L = 1, e = 100. The first impression the viewer gets from this graph is one of resonance peaks rising at quite regular intervals. Indeed, all the resonances from.62Xf on up occur at.25-wavelength intervals. These resonances are quite narrow (2-5%), reaching peaks between 1.5 and 2.35. All peaks are positive. The lower frequency behavior is quite interesting. At.36 and.358 a narrow twin resonance occurs, 6% and 12% in bandwidth, with the positive peak reaching up to 9. At the very low frequency end, it is seen that, in this particular case, there is resonance right from the lowest frequency until the radius of the rod reaches.12Xf. The concentration of power is not very outstanding; however, both bandwidth of operation and the physical shape of the setup (we require only a thin rod, instead of the heavier ones needed for higher-order resonances) are quite attractive. For Fig. 9, i = 100, and e = 1, the computer increments in a/xf were not small enough. Even though a computation was made for every.02 increment in a/Xf, it is apparent that most of the resonance peaks were missed. The 351

neighborhood of the resonances which appear will have to be explored with more care. Even though there is only one actual computed point which definitely shows resonance (at a =.60f), it still can be seen that there are other resonances also. All of them appear quite narrow, with bandwidths of 2% or less. Without any computer points of high value, it is of course impossible to say how high the peak values of the resonance curves are. Once more, several of these resonances are spaced about.25Xf apart. One disadvantage of this high permeability case would be presented by its low-frequency behavior; for thin cylinders up to about. 5Xf in radius, the normalized power is much less than one. From a practical standpoint, then, this configuration would be less desirable than case III since a much bulkier cylinder would be required to obtain resonance. In all the graphs shown, the peaks have not been examined in detail and could conceivably rise to higher values than those shown on the graphs. 2.3 SCATTERING OF A PLANE WAVE BY A FERRITE PROLATE SPHEROID An investigation of plane-wave scattering by a prolate spheroid was initiated. This study was planned as an extension of the ferrite-cylinder and ferrite-sphere problems currently under investigation. Early in the study, it became apparent that the problem, although solvable in theory, was quite complex. The results expected as an extension of the sphere and cylinder problems seem desirable. However, great urgency of other parts of the project has deferred work in this area. Accordingly, only the first step of the solution was carried out and the method of solution indicated. But work will continue on the related problem of the radiation properties of a constant cur 32

rent antenna embedded in a ferrite prolate spheroid. Assume an incident plane wave propagating in the negative z direction. The electric vector has a magnitude Eo and points in the positive y direction; the magnetic vector points in the positive x direction. The ferrite prolate spheroid is located at the origin. The orientation of the spheroid and the incident wave striking nose-on are shown in Figs. 10 and 11. Expanding the incident plane wave in spheroidal vector wave functions2 -.E (kz-t) -jEoek 20 Ei = ayEoe A.OeX-Moe - Eo J(kz-ot) _ Eoe- z (j()) Hi = 'Ex E 2 E Ao X (36) where 2( j) ' o) - N Z (dn ) N-o n=0,l oor 2 N = (d. ) (37) N n=O,l 2(n+l) The prime on the summation indicates that the summation is to be over even values of n if 2 is even, and over odd values of n if I is odd. The symbols dnm are numerical coefficients tabulated by Stratton, Morse, Chu, and Hutner.3 The expressions for the incident wave have the property that they are finite at all points in space. 2.3.1 Scattered Field.-At least eight sets of vector functions are available, the individual terms of each set satisfying the vector Helmholtz equation and having zero divergence.2 33

x PROLATE SPHEROID n=O 'ICHARACTERISTICS: II.01.01.00 %.00 k I,AO of o f' F / C9 kF k 2r 7 Fig. 10. Prolate spheroidal coordinates.

x DIRECTION OF PROPAGATION IZ Y Fig. 11. Incident electromagnetic wave. 35

In choosing two of the eight sets of vector functions to represent the scattered wave, it is highly desirable to choose functions of each of the three components of which vary with A in the same way as do the corresponding components of the incident wave. The vector functions which have the proper variation with p are: M) M(i) YNi) rM(i) and rN). Of these M(4 /M011Mo -o 0 Y andO ) -1j Y = i ) zM(4) were chosen for reasons of simplicity. However, the order (4) was -li chosen since M tends to - ( - const.) as tends to o, as is the - r case for the scattered wave. Now the expressions for the scattered wave are: -jEoe-jc~t 2 x (4) = ok =0 L+j E rEo e ) + ( 1Z Ss = ok2 No 5 8) where coas, s are unknown coefficients to be determined from the boundary conditions. The expressions for M, N are given in Ref. 2. The expressions for the transmitted field are: -JEoe- j Wt x(1) t z 1 2=0 -g IwEoe jWt ' ( ' where xM(l) M(1) were chosen since they are regular (no singularity) along the line g = 1 (which is the degenerate spheroid), and since each of the components varies with d in the same way as do the corresponding components of the incident wave. 36

2.3.2 Boundary Conditions.i x (Ei + E) = IS x t at 0 = o (0) i, x (Hi + Hs) = i x xt This results in four equations in the four unknowns aoQ, co t, o, and 0s,. These equations are: sk t k k, _t s ~BL '- tCL- + %ao DL - k 1 E = k = F 00 kt t ILI + k k 00 1' ' 3 20 __ LL M + — NLkv ~ Li 2=0 B_ + 3 Q L 2 1k k L - 3 VL k3 o L ( which are four infinite simultaneous equations. However, the constants BL, CL,...WLi are not simple expressions. Each one of these constants is an infinite series containing very complicated integrals. 37

These constants are: 2 BLA = R j~(4) f'd S 1)(CO)S (l)(r)~1T T) C1= Ri ()) -1 S (~)SQ)(1~) (T(1-Ti2) -id-rj DL1 = RIj')( x)(-1) f is 'I S q - 01 0 -1"o EL, = RI (S)s (TI) dyI (1)toft2s(l~r)s FL 01 Qj '(0)-) -1 01 O GL 001)R(4) 1 (i) (1) (2) (4 ) (1) (rI)rI(ilI) r - 1 2, S (.q \T qT )d- (E2-1)R1_T12 f1s T1), =o to 1L Ci)1,q (1)(to) i) 1 (1I) (i) 1 E ),(o -1, (TI) soL() (i-T2) 2dT1 - (t21)2 R1, (to) fsj (q)s (T1)T1(- TI2) 2dTj 1L2_ =R (i) 1 (1) + 1 11 T)(_q (i)4h( )'C)()S )CI) TjdTI + ER6 (to) f~ S4 1) (To S 4(O~I) dTl KL = (t2_i)%1( (E0)A01 f ) ) (T1)S (TI)TjdT + EOi)(t )AO, St o(T)71) soL(T)1_T1) dy (42)

or KL1 = AL11L, I~~R(3 ~)s3) (-r2~j Lo~L~(4)] F- 220 i.1 o l ()( )-y + 0~ — Rifj (to) f~ 1 )(1 ) TIc3OL2~ r N31= 3 sw 7) )csm Jiori toL (3- (toj d1q+-fs1 T L(I - NLI01 OL )t E_.2) —II t21) 5- ) (I ) 9(idT - 3 0- (3-) SO L )t (2_2)B + f3 SI ) 41 L T, t(to~ R~4)( o)dJ f3s 0W(1 4)l Ro'(4)(to) drl 03- = ~- 022 0 R('2) (E) PL1 A= - to if_ (o)___Si__ 3- ')'s d-q''d RI ~(4)(t) f3 1 2L dii -L f2_j1 OL- IO+10 3- 03 S3- z SO (YI) dii + R1i' (to)L3- S3. ~(TI) SOL~ 2L di =R$61 ( t ) 3 (-) d 2)i 4- T Et2_1) R3)(] is$()SOL 2) A f( Tl S, - + - 0 l(1 t) 6 C 0 VL1 = SoUL d E~22 s43- () iij] dii + -d [2_i)2 R' ('to) f s1) (q)SQ,() (1_1i2)~ dii

In Eq. (41) both I, L go from zero to infinity. The prime on R and S indicates differentiation with respect to the argument. The procedure'used to evaluate the above integrals is to express S1)(ri) and its derivatives in terms of P1m+n() and its derivatives. The difficulty in solving this problem comes from the fact that the vector wave functions in spheroidal coordinates are not orthogonal as in spherical and cylindrical coordinates. In spherical or cylindrical coordinates, the vector wave functions, under certain conditions, satisfy ff Vn mds = 0O s n / m (43) ff Un U mdS= O1 s and fJ Vn Unds = 0 s This orthogonality makes it possible in the sphere and cylinder to get four equations only. Unfortunately, these relations do not hold in the spheroidal case, but they do for the scalar wave functions from which the vector wave functions are derived.5 The four infinite simultaneous equations can be solved by taking a finite number of terms (for example, two terms), and then solving the resulting equations 2 L RADIATION FROM A MATERIAL-FILLED RECTANGULAR-WAVEGUIDE H-PLANE SECTORAL HORN The H-plane sectoral horn, shown in Fig 12, has been evaluated the 40

z r2 y RECTANGULAR COORD I NATES x y SPHERICAL COORDI NATES x z y CYLINDRICAL COORDINATES x Fig. 12. Coordinate systems used for H-plane horn. 41

oretically. The horn and waveguide are assumed to be filled with a material of relative permeability or and relative permittivity Er, The derivation for the radiated fields is similar to that done by Silver for an air-filled horn. Huygens' principle is used and the source replaced by the fields in the aperture. To simplify the solution, the current distribution over the exterior surface of the waveguide is neglected. It is also assumed that only the TEo mode exists in the rectangular waveguide feed and that only one mode propagates in the sectoral guide due to the dimensions of the horn throat and the excitation. Referring to Fig. 12, the only fields present in the horn are Ey, Hr, and A, because these fields match with the TE1o components in the rectangular waveguide feed. To determine the space dependence of the fields in the horn, Maxwell's equations are solved in cylindrical coordinates. By applying the boundary conditions and solving, the following fields are obtained. Ey = cos pj 2)(klr) + aH4 (klr) Hr = si 2)(k1r) r+ 4l) (klr (44) H kl cos pG [ (2) (k1r) + aH()(klr) where: H) (klr) = the Hankel function of order two, representing an outward traveling wave. H1) Hp) (klr) = the Hankel function of order one, representing an inward traveling wave. 42

P = -2, usually not an integer 29o 0o = the flare angle = reflection coefficient from the horn to free space. Primes represent derivatives with respect to k.r. k = p o mir E = permeability of the material inside the horn E = permittivity of the material inside the horn w = frequency of operation. To determine the radiated fields, the aperture diffraction method is used. The sources are replaced by the fields at the aperture of the horn and the far-field pattern is derived from this distribution. As shown by Silver (Ref. 6, p. 161), the radiated far field is given by: -_ - Ep = - r e x r e fpo x p t xe o ds (45) = 1 e NOxfaperture E - o ( pe d o5) where: Ro = a unit vector from the origin to the field point n = outward normal to the aperture, in this case the cylindrical F unit vector. E,H = the fields at the aperture of the horn, r = r2 k = I NIo = free space propagation factor o, Eo = permeability and permittivity of free space = vector from the origin to the element ds of the aperture area R = spherical coordinate of the field point. 43

Carrying out the calcula~tions, -I~he following fields are found: (a) for the yz-plane', or E-plane, = 9C', kV2 jkR ~b G ~jk(V12 cos Geos 9 + y sin 9) 0 Cos Ey osGHZI d~dy (b) for the xZ-plane, or H-plane, 0 = = jkr2ejkR fb fGO ekr2(sin Gsin 9 + cos Gcos 9) 4irR 0 G Los(G-G)Fy - -odd where: (46) (47) H9 Then., substituting found to be: = unit vector in 9 direction = unit vector in ~ direction = from Eq. (1) with r = r2 = from Eq. (35) with r = r2. Eqs. (1) and (35) into Eqs. (3) and (6), the far fields are (a) E-plane, 0 = 900 = kr2ejkR 1be jk sin %y JGO jkr2(COS Gcos 9) ~Cos Gcos o G k2 (kV2) + cHl(klV7 -Dcs 9 cos Pe jH~ )(klv2) + cxH (kor2)r de 'p p j (48) 44

(b) H-plane, A = 0~ kVe-jkR b G0 jkr2(sin Gsin G + cos Gcos G) = j f e 4irR o -O icos(G-9) cos pI(j2)(klV2) + aH(1)(klr2 - o,1,(2) fl) ~ cos pG Hp (klV2) + aH' (klV2)' dGdy (49) The integrals in Eqs. (7) and (8) can be evaluated easily using numerical integration on a computer. A program has been written, and results of the calculations will be given in the next Quarterly. 2.5 RADIATION FROM HORNS FLARED IN TWO DIMENSIONS AND FILLED WITH FERRITE MATERIAL The procedure followed is to solve the vector wave equation in spherical coordinates such that the boundary conditions are satisfied on the surface of the horn. Then, assuming that the horn supports only one mode, which corresponds to the TElo mode of the exciting rectangular guide, the fields at the aperture are evaluated and used in the modified Kirchoff formula to obtain the radiation field. The horn is defined by the cones G = Q1, 9 = t - 01, the planes 0 = 0, Z = 1 and the spheres r = r1 r = r2 as shown in Fig. 13. The vector wave equation VC+k2 = 0 has two solutions M, N which have zero divergence. These two solutions could be obtained from the solution of the scalar wave equation V24+k21 = 0. The general solution of the scalar wave equation in spherical coordinates is 45

z A i \ X\ x Fig. 15. Double-taper ferrite-filled horn. b\ "I, 46

Cr'- cs ) tm (1) *n (Am cos m~ + Bm sin m~)!C:"~o )+D~nMP~ncos 7] I~hn (kr) + Fnh$~2(kr), (50) (1) (2) hn, hn have been taken to represent traveling and reflected waves. n,m are not integers because the region 9 = 0, 9 =it is excluded and there are boundaries in the O-direction. Ml= 0 sin 9 (-mAm, sin mo ~ csm) 'C P(cos 9)+DrfP~n(-cos 9 sin 9 Enq kr)+ Fnlqj2(kr7] C..(Cos 9) P"(cos 9) = (A(~cos mo +3msin mo) Cm + D~mn ~~(l) + h(2)k (1 Enhn (kr) Fnhn (r(51 N1 n(n+i)*n kr n(n+1) -j + Bisin m) Q) F4~2) (kr)I kcr COm co, B i [cflmlP(COB ) DrmnF(FCos +n N2=kr ~r~o = (Am cos m$+ m sin e) n P(COS +D &n(-COS 9) kh L)' (kr + Fn (2~) N3 1 kcr sin 9 + 4~2) (k)i -j (52) J47

E = z n,m (Mn + bnNn) H = (Nn + bnMn) jwy n,m (53) Now the boundary conditions to be satisfied are: E1 = E2 = 0 on = 0, 0 = i2 r 1 < r < r2 E1 = E3 = 0 on @ = @1, Q = -0 ' where: the subscripts 1, 2, 3 refer to r, 9 and ~ components, respectively. E1 = Z b nn+ n,m kr,Enh(l)(kr) + = 0 at < = 0, (Am cos mO Fnh2) (kr) P = 2 - F —. + Bm s in mO) jCnMFr( c 0s e) + Dnmn(-cos e) The only way to satisfy this condition and the others is to take bn = 0. Therefore E -= 0 From the condition on E2 we get Bm = 0, mOl = ~,r, I = 1, 2, 3,... (54) From the conditions on E3 we get C Pn(cos ~1) rmu P -c bel Pn(-cos ~1) C6nm bel;Pn(-cos ~1) + Dnm- -_ + Dm, MP(cos ~1) D1 — a +Dmi1 = 0 = 0 48

Therefore 6P'n(cos 91) (Cos 1 2 - + P (-c0sQ) Therefore (33) the characteristic Eq. (55) determines n. If I is taken to be 1, then the field components will be Er - E3 s ing 4 — ~P(-cos @) - ~jsin 9 $1 B(2) kr+ nhl(l) (krjJ Ap"(-cos 91) C., n LPn"( c 0 s 91 9) I 'm/= Eocos i [~P'(cos @1) 6Pmn(-COS 9) -.~Pm(-COs @1) 6P~n(COS @7 L,C'G G IQ A+ Ynh (n j H r Eokn( n+l),jwy nr P(cos el) mn (-cs9 P(cs9 m~o h(l)(kj) + Y ~2(kr) H9 = Cos jcny FP' (Cos G1) L mn I'P"(-Cos 9) n( _ AlP,"( cs 9 Ikr ~r rh$12) (k?1 kr ~r Ln i G- Pmn(-cos Ho E=ki s in f. =JwD $jsin 9 0 9) - ~P4(-Cos 91) P4(cos 97J Lkr6rL k r 7r- Jj (56) )49

where::Tr n is given by the characteristic Eq. (53).If 01 is taken to be 500thnm=6 Then: 2'(X where: =C-() (Y+M). 2M(2-m)!m~ (X2- ) m/2 F(m -y m +y + 1 m + 1 -x) (Ref.- 7, p. 428) F is hypergeometric function. Therefore P'm(Cos @1) 2m(7Y-m) Mo F(m -, y, m + y + 1,y m + 1, 1COS G @) 2 F(m - y., m +7~y+ 1, m + 1, 2-o 91 = Ft (m- 7.KL)!(m+7). 20 (1-cos G1 K2) pm(c __Z Os Gl) G 2-m(7y M)! isin@1,F 1+ Lo mcos G1sn - G9F2 ___1 m! sin @1 z (m1+I (m+1+1+,e) 1+ 1-cos after getting the corresponding expression for F, substituting in Eq. (35) and manipulating, we get: 00 I( I+1)si2@ z: (m-7+I)!(m+Z+1+1)! s i 9 -I1=0 ifl(m+1+l)! 2 I fjCos @1 (37) - 2/ I + m Cos el ( -Cos @ I2+1 1+Cos G-I I, 2 2+1 = 0 50

Before specific values for 91 are taken, the dimensions of the feeding guide should be considered. Assuming the rectangular guide supports only TElo mode, the dimensions of the guide must be a < x < 2a, 2b < x Therefore b < a a = 2rl sin 91 sin 2 b = 2rl sin (A - 201) Therefore sin(ir-2@1) < sin @1 sin ~ 2 i.e., cos e1 < sin (58) 2 2 for i = 3(0~ therefore cos 91 < 0.1295 therefore 1 > 82.5~ therefore the angle of the cone is less than 150, and we notice from Eq. (58) that, as 01 increases, the minimum value of the angle of the horn in the 9 -direction increases. Equation (10) is solved approximately by taking the first ten terms, for example, instead of the infinite summation, calculating the value of the summation for different values of y, y = 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, plotting 51

the summation vs. y, and then taking the value of y for which the summation is zero. To evaluate negative factorials, the definitions (-x) = —; (x-l) sinjrx (-_1) = are used. In solving Eq. (57) there may be more than one value 2 of y satisfying the equation, the lowest value which satisfies the condition 7 > m is taken. Then this value is substituted in the field equations (56) to get the lowest mode in the horn, which corresponds to the TE1o in the rectangular waveguide. 6,8 2.5.1 Radiation Field of the Horn. ' -The field at the point P is given by Ep = A X f X - 1 R1 x (H x - (ds (59) The origin is the center of the sphere of which the horn is a part; r2, 3, A define a point on the aperture; R, 0', i' define the field point; R1 is a unit vector in direction of R; p is the vector from the origin to the element ds on the surface of the aperture; nT is the outward normal to the aperture. x E = arx [Egag + Ea ] = E6a - x H = ar x [Har + HG + Ha] = p = r2sin 9 cos Or + r2sin 9 sin aKy + R1 = sin 0' cos O'5^ + sin 9' sin i'd HKao - l-a r2cos 0Kz + cos 9'Ez From Fig. 13 it is seen that ay = az ax = - sin L ax + cos ay 2= cos2+ Bb = Cos 'Ex+ sin k 22 T3r (60) (61) 52

Note that the field point is defined with respect to the axes x', y', z'; the axis z' passes by the center of the aperture, while the x axis passes by the the center of the boundary 0 = 0, as shown in Fig. 13. From Eqs. (60) and (61) we get p. Ri = - r2sin a cos b sin O' cos f' sin L + r2sin @ cos cos G' cos d 2 2 + r2sin 0 sin ^ sin @' cos ^' cos L + r2sin 0 sin ^ cos 0' sin r1 2 2 + r2cos @ sin @' sin 0' (62) For the plane -' =, Eq. (62) reduces to i5 R1 = r2sin @ cos 0 cos G' cos il + r2sin @ sin 0 cos 1' sin L + r2cos @ 2 2 sin @' = r2[sin @ cos 9' cos(o - ^) + cos ~ sin @'] (63) Also Eq. (60) reduces to R1 = sin 9'ay + cos ' z (64) Therefore Eq. (59) reduces to jklr2e jkR - - ) '...R x f f (E Ea9) _ (i)2 R x (He - 7 jklr2[sin 9 cos 9' cos(o - i1) + cos 9 sin ' ] H2a) e 2 sin Od9do (65) Now let us put the integrand in terms of x', y' and z' components. Referring to Fig. 13, E = - sin ax + cos zay ag = - sin Gaz + cos 9 cos 0ax + cos 9 sin ray (66) ar = cos ~9az + sin 9 cos 0ax + sin 9 sin pay 53

x (E9-a - Eq -(sin G'Ijr + Cos G'-') x [E9(- sinl ~- + Cos ry -Eo(- sin @-az + cos G Cos 9Tax + cos @ sin ray) (67) and let us express 'E, 7~y 'd in terms of i~ 4,~ =-sin~~+ o~ ~ = co ~'+ sinA~g - 2 X2 Substituting from Eq. (68) in Eq. (67),, Therefore (sin 9- + cos G'a'Z) x g(sin ~ 2i sx-sn Cs a + Cos0 Co -9xI+ ao +scsn sin cos Cs o ~snL Cos G o o 2 + OosS CnOSs 2 ' + cos @ sin sin 2 X -AZ~~+ B-J+ C- (69) where: A =sin G' Egsin(L - ) -Eocos Q o(41 - -EO cos @' sin Q B =cos @' F-Cos(m - 0) + W~cos Q sin(O )j(0 C = siG' cos(L - ~)+ Eocos G sin(02 - 54

R, x [R1 x (Ha - lag) ] = (sin G'ay + cos ''az) x (A'tax + B'ay + C'az) where A', B', and C' are obtained from A, B, and C, respectively, by putting H instead of E. Therefore RI x [Rx x ( Ha - H-o)] = ax(sin I'C' - cos Q'B') + ei cos O'A' + Rz(-sin I'A') (71) Substituting from Eqs. (70) and (71) into Eq. (65), we get for the plane ~' = r/2 E T) - -kR 91 sin 9 1[Esin( - - E~cos 9 cos(l~ - ) ] - cos e' sin GEO + (-)2 [Hcos(i - ) + Hicos 9 sin( - ~) ] x exp jklr2[sin 9 cos 9@ cos(j -,) + cos 9 sin 1 ] sinjOdQ9 (72) (9 - - -jklR f 91 cs 9 [Eecos(4 - ) + Ecsin(4 - )cos 9] - (e)2 sin 9' cos T' [Hsin(L - /) ^ 1 2 - HIcos 9 cos(i - _) ] + ( )2 cos2T' sin 9H1J expIjklr2[sin 9 cos 9' cos(O - 2) + cos 9 sin 9]} x sin G9d9d (75) 55

E(-') = l- j1 e -1 -sin 9' [Eecos( - / E-t 4R 2o 2 + E cos 9 sin( - Z) + (1)2 sin2G' [Hsin( - -) - HWcos G cos( - ) ] - ()2 sin 9' cos 9' sin GE~ expljklr2[sin Q cos G' cos( - 1 + cos ~ sin ' ]!sinGdd Equations (72), (73), and terms of the field at the by putting r = r2. For the plane f' = 0 Therefore p * Ri = - r2sin 0 cos 0 + r2sin 0 sin f (74) (74) give the radiation field as a function of 0' in aperture E9 and Eg which are obtained from Eq. (56) substitute in Eq. (62). sin 0' sin L + r2sin 9 cos 0 cos 0' cos L 2 2 sin 0' cos 1 + r2sin 0 sin 0 cos 0' sin 2 2 = r2sin cos( - - - - 0') 2 R1 = sin 4a + cos O'az Therefore jr2e-jkR k n-01 - -(9) j e R i f 1 [EGa - Ea) - (i ) R1 x (He1 - H^) ] x exp-jklr2sin 'cos(- 2 - - ')sin 9Gdgd ~-2 R1 x (Eao - E-ag) = (sin 9'a x + cos 9'az) x [E0(-sin ax + cos Oay) - EO(-sin Gaz + cos 9 cos ax + cos 9 sin ay)] (75) (76) (77) (78) 56

as before -Ex= -Sin -1 d' + Cos 81 g' 2 2 'E = co5 + s in z' 2 2 "9Z= (79) Therefore R x EGO- E$_a) (sin GT~, + cos G'ld) x [-sin OEG(-sin + os 2 a 2 2 + os OEG(cos + sin - cs) + sin cos5- cos 9 cos OEB (-sin e~ ~ak + si~n S ~) 2 2 =9k(-cos G' sin GEO) + t[Egcos(G - L+ E cos 9 sin(G' - + 2) + 'E sin &1 sin GEO (80) x [R1 x (H %Ua - H~-NG) (sin G'9i6 + cos @'89) x (FaL + Ta- + HII) where F, G, H are the coefficients of -, -, i, respectively, in Eq. (80) but E is replaced by H. Therefore 71 x [Rj x (H14a - Hae)] [- C[Hecos 9' cos(G' H cos 9I cos 9 sin(9' - + 2 2 2 ~o 'cs8sn8 s r(sin 9 W + a[%sin 9' cos( ) + H1sin 9' cos 9 sin(9' - + Substituting in Eq. (77), we get for the plane ~' = 0

Therefore E(e') = jklr~e- JkiR i,r-9i '_ -jkR J-91 -cos 9 sin 9E + ( )1 rR 0El iHlcos ' cos(9' - - ) + HBcos 91 cos 9 sin(9' - Z + 21) exp Ijkr2 x sin 9 cos(9' - ^ + ^2 sin Qdgdo L- 2t ' c 1'- E ejR f [cos(' - +A) 4irR 0 G1 (82) E(Q') = + cos 9 sin(9' - ~ + 2)EE + ( )2 sin Q9H] 2 El Ez(e') = z exp jklr2sin 9 cos('9 - + ) sin @dQdd 2 -jk1R jklrr2e -jkiR9 l -i sin '9 sin GE^ - (1) - rR 0o 9l H5sin 9' cos(e' - + ) + Hsin '9 cos 9 sin(e' - + 2 exp jklrin s( xsin cos('- + x sin d (83) (84) Equations (82), (83), and (84) give the radiation field in terms of the fields EQ and E~ at the aperture. The integrations occurring in the radiation field expressions are of the form: o1 sin 1 sin ( - )eb cos( - )d The integrand is a trigonomentric function multiplied by an exponential function. The integrals can be evaluated by computer methods. 58

3. ACTIVITIES FOR THE NEXT PERIOD Computer facilities will not be available until October 15 while the IBM704 is being replaced by an IBM-709 computer. Computer programs presently in use can be used directly on the nex computer. As soon as the new computer is in operation, the computer results will be obtained on the details of power flow near resonance in the studies of sphere and cylinder diffraction. In addition, computer results will be obtained on the diffraction by a magnetized ferrite cylinder. Experimental work is presently being conducted on a biconical antenna, a ferrite-filled cavity backed slot antenna, a ferrite-filled disk antenna, a shielded ferrite-loaded balanced loop antenna, and a coaxial fed ferriteloaded disk antenna. In addition, a ridged ferrite-loaded cavity-backed slot antenna is being constructed. Work on approximately half of these antennas in nearing completion and will be summarized in the next quarterly report. The computer results on the sectoral horn analysis are being studied at present and will be given in the next quarterly report. During the next period several new theoretical studies will be initiated, including a variational solution of the radiation from a ferrite-loaded cavity backed slot antenna; a study of the radiation from a constant current antenna surrounded by a ferrite prolate spheroid; and studies of antennas immersed in ion plasmas. 59

4. SUMMARY The theoretical study of plane wave diffraction by a ferrite sphere has been extended to treat a lossy sphere enclosed within the ferrite sphere. Computer results show that the power absorbed by the lossy sphere rises to values greater than 500 times the power incident upon a cross-sectional area in free space equal to that of the large ferrite sphere. Computer results have been obtained for the total power flow through a long ferrite cylinder with an incident plane wave. The total power exhibits resonances where the total power rises to values greater than ten times the power that would pass through a cross-sectional area in free space equal to that of the cylinder. A study of plane wave diffraction by a ferrite prolate spheroid has been initiated and the preliminary analytical steps have been carried out. Because of the urgencies of other parts of the project effort, the study has been discontinued and will not be started again unless developments in other areas indicate that the results would be significant. The radiation from a material-filled rectangular waveguide sectoral horn has been analyzed and computer results are being obtained. The radiation from a ferrite-filled double taper horn fed by a rectangular waveguide has been analyzed mathematically. This problem is also quite complex and, unless further developments indicate a significance, probably will not be evaluated on the computer. Results are already being obtained on the single taper sectoral horn. The nC,

efforts of project members are now being directed to practical antennas utilizing ferrite materials and which are capable of relatively precise experimental evaluation. The bibliography has been omitted from this quarterly report in the interest of conserving space. A new bibliography will be compiled and published towards the end of the present contract. 61

REFERENCES 1. J. A. Stratton, Electromagnetic Theory, New York, McGraw-Hill Book Co., Inc., 1941. 2. F. V. Schultz, "Scattering by a Prolate Spheroid," Aeronautical Research Center, The University of Michigan, External Memorandum, No. UMM-42. 3. J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner, Elliptic Cylinder and Spheroidal Wave Functions, John Wiley and Sons, New York, 1941. 4. Carson Flammer, Spheroidal Wave Functions, Stanford University Press, Stanford, California, 1957. 5. "The Theory of Electromagnetic Waves," A symposium sponsored by NYU and AFCRC, Interscience Publishers Inc., New York, 1951. 6. S. Silver, "Microwave Antenna Theory and Design," Rad. Lab. Series, Vol. 12, McGraw-Hill, New York, 1949. 7. S. Schelkunoff, Applied Mathematics for Engineers and Scientists, D. Van Nostrand Publishing Co. 8. L. J. Chu, "Calculation of Radiation Properties of Hollow Pipes and Horns," Journal of Applied Physics, pp. 603-610, September, 1940. 62