THE UNIVERSITY OF MICHIGAN 5825-1-F ON THE ANTENNA RADIATION THROUGH A PLASMA SHEATH A. Olte and Y. Hayashi June 1964 NASA Grant NsG-472 National Aeronautics and Space Administration Langley Research Center Lanle FilVrii 213365 -

THE UNIVERSITY OF MICHIGAN 5825-1 -F TABLE OF CONTENTS ABSTRA CT V ACKNOWLEDGMENTS vii I INTRODUCTION 1 References for I 6 II RADIATION OF AN ELEMENTARY CYLINDER ANTENNA THROUGH A SLOTTED ENCLOSURE 7 2. 1 Introduction7 2.2 Reduction of the Boundary Value Problem to a Fredholm Integral Equation of the First Kind 9 2. 3 The Appearance of the Singular Integral Equation 16 2. 4 Solution of the Fredholm Integral Equation for the Case of a Narrow Slot 18 2. 5 The Results 31 2. 6 Discussion 4 2. 7 Conclusions 5 References for II 55 III THEORY OF ELECTROMAGNETIC FIELDS FOR COAXIAL CIRCULAR CYLINDERS WITH A SLOT AND MAGNETIC LINE SOURCE 56 3. 1 Introduction 56 3. 2 Reduction of the Problem to a Singular Integral Equation 56 3. 3 Solution of the Singular Integral Equation 62 3. 4 A Note on Solution of Non-Singular Integral Equations 67 3. 5 Electromagnetic Field for a Narrow Slot 69 References for III 73 IV CONSIDERATION OF ENERGY STORAGE AND DISSIPATION IN A CAVITY BACKED SLOT ANTENNA AS IT AFFECTS INPUT IMPEDANCE AND RADIATED POWER 74 4.1i Introduction 74 4.2 The Energy Theorem 75 4. 3 The Circuit Representation 81 4.4 The Broad-band and the Narrow-band Antenna Problem 87 4. 5 Discussion 95 References for IV 100 V FUTURE WORK 101

THE UNIVERSITY OF MICHIGAN 5825-1 -F ABSTRA CT The influence of the plasma sheath on the slot antenna properties is considered. It is shown that opening a slot in a thin overdense plasma sheath re-establishes in many cases the radiation of an elementary cylinder antenna. The changes in the impedance of a cavity backed slot antenna with the onset of a plasma sheath are formulated on the basis of an energy theorem.

THE UNIVERSITY OF MICHIGAN 5825-1-F ACKNOWLEDGMENTS The authors express gratitude to NASA for supporting this work under Grant No. 472, and in particular we thank Mr. T. E. Sims and Mr. Win. Croswell and their colleagues of the Langley Research Center for the interest and encouragement shown. The authors have benefited from many discussions with Drs. T. B. A. Senior, R. E. Kleinman and R. F. Goodrich of the Radiation Laboratory. For the computations., we thank Mr. H. E. Hunter and Mr. P. Sikri.

THE UNIVERSITY OF MICHIGAN 5825-1-F I. INTRODUCTION The difficulty in maintaining communications with the aerospace vehicle during those phases of the flight when a plasma sheath forms on the antenna surfaces is well known (Rotman and Meltz, 1960; Ellis, 1962). 1The necessity of keeping radio contact with the vehicle at these times is equally clear (Huber and Nelson, 1962). It appears that a contribution to the solution of this problem will come from a quantitative understanding of the role the plasma sheath plays in modifying the absolute gain and the radiation pattern of the aerospace vehicle antennas. This requires developing analytical solutions and physical concepts to explain the effect on the far-zone antenna characteristics caused by the presence of reactive and lossy plasma sheath on the antenna surfaces. Once these interaction mechanisms are understood., a purposeful control of the plasma configurations by aerodynamic shaping of the vehicle., and perhaps by opening slots in the plasma sheath (Cuddihy, et al, 1963) or some other means will hopefully reduce the undesirable antenna pattern changes and signal attenuation to levels that are tolerable. A simplified version of an aerospace vehicle is shown in Fig. 1 -la and 1 -lb. It is a composite body consisting of a hemisphere, a section of a cone., and a section of a cylinder. Some sections of the body may be coated with a dielectric layer. A problem of current interest involves a radiating slot on the cone surface when the

Plasma Sheath- Dielectric A o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i tQ I~~~~~~~~~~~~~~~~~ FIG. 1 -la Section A-A 0 0<0 <180 FIG. 1-1: AEROSPACE VEHICLE FIG. 1 -lb

THE UNIVERSITY OF MICHIGAN 5825-1-F vehicle is surrounded by a plasma sheath. We would like to find the changes in the antenna impedance., in the radiation pattern, and in the absolute gain with the onset of the plasma sheath. The plasma sheath may have an axial slot. The plasma slot may be displaced circumferentially from the cone slot. The arrangement of the various layers and the slots are shown in the cross section drawing, Fig. 1 -lb. It is clear that the configuration of this simple aerospace vehicle is such that the vector wave equation is not separable, and therefore the theoretical treatment of the antenna problem is extremely difficult. It appears that one approach to this problem should be that of first solving an appropriate canonical antenna problem and then extending the results of this analysis, using some physical insight and perhaps an experiment,, to the vehicle shapes of interest. For the canonical problem it is advisable to choose the cylindrical geometry where the possibilities of a solutio are favorable. The cross section view of the cylinder antenna would be the same as in Fig. 1 -lb. A narrow axial slot of infinite length is cut in a perfectly conducting., infinitely long cylinder that is enclosed by a uniform dielectric layer. On the surface of the dielectric layer is a uniform plasma sheath. A narrow axial slot of infinite length is maintained in the plasma sheath. The slot of the plasma sheath may be above the cylinder slot., or displaced circumferentially from it. It is proposed that for a given tangential electric field in the cylinder slot the solution of the electromagnetic boundary value problem be obtained. In the first attack on this problem we

THE UNIVERSITY OF MICHIGAN 582 5-1 -F to the wavelength and highly overdense, we can replace it by a vanishingly thin, perfectly conducting shell. This reduces the configuration to a two-region problem. Furthermore, we take the dielectric under the shell to be the same as outside the shell. This problem has been formulated in Chapter II as a Fredholm integral equation of the first kind. It has been solved for the case of the narrow slot, and from this solution we derive the effect of the slotted shell on the radiation properties of the cylinder antenna. The cylinder diameters considered range from a small fraction of the wavelength to 1. 8/ w wavelength. The spacing between the cylinder and the shell is a small fraction of the wavelength. The principal conclusion is that opening a slot in the shell re-establishes the cylinder antenna radiation, except for certain angular separations of the cylinder slot from the shell slot. For details one should refer to the radiation curves in Chapter IL. In Chapter III we have an interesting new approach to the solution of the Fredholm integral equation that appeared in the previous chapter. The approach is to construct a solution to the Fredholm integral equation from a solution of a singular integral equation. The singular integral equation is derived by differentiating both sides of the Fredholm integral equation of the first kind. We hope that this solution will be of considerable value, especially for the discussion of the wide shell slot, in the future. In Chapter IV we discuss a more general problem than those of II and III,

THE UNIVERSITY OF MICHIGAN 5825-1-F and backed by a cavity. Using the Energy Theorem we express the impedance in terms of certain integrals over the electromagnetic fields. From this analysis it appears that the antenna will be least unfavorably affected by the plasma sheath when it is tuned primarily by internal resonances, and the electric reactive energy is kept at a minimum on the antenna surfaces. It is doubtful if making the cavity lossy alone would be a desirable feature in devising plasma-proof antennas. In Chapter V we discuss some suggestions for future work.

THE UNIVERSITY OF MICHIGAN 5825-1 -F REFERENCES FOR CHAPTER I Cuddihy, W. F.,2 I. E. Beckwith and L. C. Schroeder (1 963), "RAM B -2 Flight Test of A Method for Reducing Radio Attenuation During Hypersonic Re -entry.," NASA Report No. NASA-TM-X-902., October,. 1963 (CONFIDENTIAL). Ellis,, C. E. Jr. (1962), Proceedings of the Second Symposium on the Plasma Sheath, Boston., Mass. (SECRET) 10-12 April. Huber, P. W. and C. H. Nelson (1962), "Plasma Frequency and Radio Attenuation," NASA -University Conference on Science and Technology of Space Exploration Chicago, November 1-3, Vol. 2, pp. 347-360. Rotman., W. and G. Meltz (1961), Electromagnetic Effects of Re-entry,, Pergamon Press., New York.

THE UNIVERSITY OF MICHIGAN 5825-1-F II1. RADIATION OF AN ELEMENTARY CYLIINDER ANTENNA THROUGH A SLOTTE ENCLOSURE. A. Olte 2. 1 Introduction In this chapter we treat the radiation problems of a cylinder antenna shrouded by a concentric axially sloted shell. The source of the cylinder antenna is an infinite axial slot uniformly excited, with the electric field in the circumferential direction. In the following five sections we present the analyses of the problem. In the first section we reduce the boundary value problem of the antenna by employing a conventional series representation of the fields to a Fredholm integral equation of the first kind. The integral equation uniquely determines the tangential electric field of the slotted shell for the given source on the cylinder surface. The kernel of the integral equation is complex, non-Hermitian, and has a logarithmic singularity. In the second section we briefly discuss the physical aspects of a singular integral equation which follows by differentiation of the Fredholm integral equation of the first kind. Hayashi was the first to derive the singular integral equation in a similar problem. This is discussed in detail in Chapter III of this report. In the third section we report a solution of the Fredholm integral equation for the case of a narrow shell slot. Recognizing that in a narrow slot the field distribution is dominated by the edge singularity we are led to a slot field representation by a Fourier series of a kind where the first term is the dominant one. This idea has

THE UNIVERSITY OF MICHIGAN 5825-1 -F an electromagnetic wave normally incident on an axially slotted, perfectly conducting cylindrical shell. In the fourth section of this chapter we report the numerical calculations which are based on the above solution of the Fredholm integral equation. The purpose of the calculations is to exhibit the influence of the slotted shell on the radiation of the cylinder antenna. The cylinder diameters considered are, in wavelength, from 0. 2 / wr to 1. 8/ 7r. The radial spacings between the cylinder and the shell are 0. 1 /-t, 0. 05/itr, and 0. 025/i-t. It will be obvious that for the parameter values considered the slotted shell does not significantly modify the form of the cylinder antenna radiation pattern. However., the pattern is rotated by the angle between the source on the cylinder surface and the shell slot, although for the cylinder diameters considered the radiation is nearly omnidirectional anyway. Even for the largest diameter antenna considered the radiation field is omnidirectional to within +25 percent (Wait, 1959). Therefore, we have chosen to report the ratio of the radiated power with the slotted shell and without it, as a function of the source and the shell slot separation angle. In the fifth section of this chapter we discuss the accuracy of the approximate solution of the integral equation, and the error reflected in the power radiated. We also discuss some of the physical implications of the solution. In the last section we briefly discuss and summarize the main features of the results. The slotted shell p~revents radiation only for certain parameter

THE UNIVERSITY OF MICHIGAN 5825-1 -F combinations; for most others, the radiation remains the same and for some it is even enhanced. 2.2 Reduction of the Boundary Value Problem to a Fredhoim kInte gral EqCluation of the Fi r st K ihAJ. We consider a wedge waveguide of width 20 0feeding in the lowest order transverse electric mode a perfectly conducting circular cylinder of radius a, as shown in Fig. 2-1. The cylinder is concentrically shrouded by a vanish ingly thin perfectly conducting shell of radius b. The shell has an axial slot of wvidth 20 The I0 center-to-center circumferential displacement of the shell slot and the cylinder slot is indicated by the angle 0. We employ a right-hand circular cylindrical coordinate system (r, P., z) for which P~ is measured counter-clockwise from the center of the shell slot and z is along the axis of the cylinder. Thie constitutive parameters E and A are assumed to be real. The rational MKS system of units is used and the time dependence of e jct is implied for all field quantities. We have a two-dimensional problem, since both the antenna structure and the source is independent of z. Furthermore, it is a three region problem: r< a; a,<,r,<b; r.>,b. For this particular study we limit it to a two region problem by considering the tangential electric field of the cylinder slot as given. We set out to find the fields in the coaxial region and in the free space. The fields are given (Stratton, 1941) in a series form. For a( r ), b 2OD N~r

THE UNIVERSITY OF MICHIGAN 5825-1I-F 10~~~~~~

THE UNIVERSITY OF MICHIGAN 582 5-1 -F E0 j W~jkZ{B JtI (kr) +C N I (kr))eJ~ (2.2) n n n n E r r n BnJ (kr) +C N (kr) e n,(2.3) and for r >, b 00 2 (2) -jno H=zi k A H (kr) e (2.4) fl — 00 E= (krl -je Ek kjwp A H (2 r)T (2.5) vi ~~nfl ni = -00 O0 E= z:A H~2 (kr) e (2.6) r ~r n n n =o-0 where k =Li~A~~ We regard as given and we seek to find We require that jwp k n~oBJ I(kb)~+Cn N' (kb) e j E() =0 < 7 (2. 7) 11~~~~~~

THE UNIVERSITY OF MICHIGAN 5825-1 -F jwpk ] J I (ka) + C N I (ka)le~n =f(0),10-0 (P<0+0 nin nf J 0 0 =0,1 0+0 (Vi(27r+(0-0) 0 0 (2.8) co ~~~~(2)' jn > jwopkA nHn (kb)e < 0 0 029 Because of the orthogonality of the circular functions we obtain from (2. 7)., (2.8) and (2. 9) respectively jiopkjBJ1 J(kb)+C11Nk (kb) 12 jr E(P') e do ' (2. 10) 0+0 0 -0i 0 jwpkA H(2)' jno' j k An Hn (kb) 27 x E(Pt) e do, (2. 12) - 00 where ~ and Ti are dummy variables of integration. From (2. 10) and (2. 11) we obtain

THE UNIVERSITY OF MICHIGAN 5825-1 -F 0+ P~rw Pk D(ka, kb) B N' (kb) ~ __ dr 0 00 ~ 0+ -00~~~ 0-0 0 11 0 where D (ka., kb) =J' (ka)N' (kb) - J (kb) N' (ka). (2.15) fl n n n n We observe that (2.12), (2. 13) and (2. 14) give us the Fourier coefficients of the fields for both regions once the electric slot fields are known. However., we do not know the shell slot field. We seek to find it by enforcing the remaining boundary con dition: the continuity of the tangential magnetic field through the shell slot., i. e. Z A H (kb)e =tZ LnJn(kb) +CnNn(kbj e n =-oon= o (2.16)

THE UNIVERSITY OF MICHIGAN 5825-1 -F We eliminate the coefficients in (2. 16), and after factoring and transposing of terms obtain LHn2(k:: 1 ~ O E~)jnpd? -jngo n=-o) H Kn)(ka, kb) _ ~ )en jen 0 5 0(2. 18)0 0~~~~~~~~~ co (2)'~ ~ ~ ~ ~ ~~~~~~2.7 equation-Zof1 the form 00~~~~~~~+ 0~~~~~~~~~~ Itiseare ta 2 8 saFehl nerl qaino h is idwt comple non-Hrmitia kernel i.e. K(P Is 0) n ~14eno 0

THE UNIVERSITY OF MICHIGAN 582 5-1 -F The kernel has also a logarithmic singularity. Interchanging integration with summation in the representation of the known function, we have 0 +0 F (a) g(045)J f (77) (ri, dr (2.20) 0 - 0 where (a) \71 jnn) K (ri46) D (ka., kb)e n=-oo n The kernel K ()(n, 0) is a continuous function of ri and 9.We see that the known function of the integral equation (2. 18) is obtained by transforming the electric field of the cylinder slot according to (2. 20). We may further simplify the two kernels by observing that D (ka, kb) = D (ka, kb) H() (ka) H2) (ka) -n -n (2)' (2)' H- (kb) H (kb) and introducing a factor En such that n~~~~ E 2, n01%2, n We have then

THE UNIVERSITY OF MICHIGAN 5825-1-F U(2)' Go- H (ka) E K(0'I, 0) n nCos______ )(2.21) n=o H2) (kb) Dnkak) n and ',a) 00 K a)(D~) ~ D(ka-, kb) Cos n (v(2.22) n0O n We have reduced the boundary value problem to a Fredholm integral equation of the first kind with a non-Hermatian kernel. This is a unique statement of the original problem and no additional conditions need be imposed. 2.3 The Appearance of the Singular Integral Equation It is well known that the normal component of the electric field must be continuous through an aperture. It is also well known that the continuity of the tangential magnetic field through the aperture automatically insures the continuity of the normal electrical field component as well. The reverse., however., is not true and therefore the continuity of the normal component of the electric field through the aperture is necessary, but not sufficient for a boundary condition. The continuity condition of the normal component of the electric field through the shell aperture in our case corresponds to differentiating the Fredholm integral equation of the first kind with respect to 5.We obtain o +o0 pl E('K(',)0 (a)' P ~~~~~~f(n)K (nid,~ <0 (2.23) a~0 -

THE UNIVERSITY OF MICHIGAN 582 5-1 -F where the primes indicate that the kernels are to be differentiated with respect toj6 The integral of this type in a similar problem was first obtained by Hayashi (see Chapter III of this report). The singularity of the kernel K(P I, 0) by differentiation has been increased to a Cauchy type singularity and therefore we have to take the integral in the principal value sense which we explicityly indicate by P in front of the integration sign. Whereas the Fredholm integral equation exists as + - 0 0~~~~~~ K (a) ~)is continuous and the right hand side of (2. 2 3) exists even as O-*+~ 0 Evidently the Cauchy integral equation for this particular problem admits a set of solutions. Somehow one has to choose a solution that satisfies the Fredholm integral equation. One may possibly choose from the set a solution that satisfies the well known condition of the edge singularity at 0 = +0. This solution is a sum of a certain particular integral and a certain general solution of the homogeneous equation of (2. 23), both parts satisfying the edge singularity condition independent of each other. The amplitude of the general solution of the homogeneous equation clearly remains arbitrary. If finding the proper amplitude of this part of the solution gives us the unique slot field, then it may be determined by substituting the total solution i the Fredholm integral equation. These appear to be the physical aspects of the singular integral equation method used by Hayashi in constructing a formal solution to a similar boundary value problem in the subsequent chapter.

THE UNIVERSITY OF MICHIGAN 5825-1 -F 2.4 Solution of the Fredhoim Integral Equation for the Case of a Narrow Slot. The Fredhoim integral equation of the first kind (2. 18) was derived for the cylinder slot of arbitrary width 20.0 We simplify the problem, but retain its essential features by letting 0 -*0-, because then we can let the slot field assume a 0 6 -function distribution, 1. e. V f(P) - 6 )Oas 0 -> 0, (2.24) a o where V 15i the slot voltage. In this case then 0+ 0 6(0 -M) (a) 1 (a) g(0, ) vj K (ri,)dn=-V K (O,~) (2.25) a 0 -0 a a a 0 From (2.21) and (2.22) we observe that K(P I, y0) and K()(0, 0) may be divided into even and odd parts with respect to both variables, i. e. e o K(a) -O0)= (a) (O )K(a) (O.0(2.27) where 00H2) (ka) E K (O' 0)L n n cos(no ')cos(nP) (2.26a) e no= 0(2)? kb D n(ka, kb) n

THE UNIVERSITY OF MICHIGAN 5825-1 -F ODH(2)' ka K n0 ka s in(no') s in(no 2.26)b K n ~'H (2)(kb) D n(ka.,kb) (.6b n~~ r~a locOD En Ke (O.)=k (a b cos (no )cos(no) (2. 2 7-a) K0(0.95 ZD (ka., kb) sin(nO )sin tno) (2. 2 7-b) Since the unknown slot field may also be represented by an even and an odd part, we have E(P) = E e(0) + E 0(0) ~~~~~~~(2.28) and the Fredholm integral equation becomes E (0')K (0IAd E (0')K (A ', )dp =v ()( )(, -00 e e o aae -.0 '<-,0 -<' 00(2.29) We observe that the first integral is an even function in 4,while the second integral is an odd function, therefore we have that 0 ~~~~1 (a) (2.30) E ()K(')d'-V K (0,PP e e a a e 0'~<~~ 00 o a ~~~~~~a o 0 ~ 0

THE UNIVERSITY OF MICHIGAN 5825-i -F We have succeeded in breaking the Fredhoim integral equation into two integral equations of the same kind. The first one determines the even part of the shell slot field and the second one, the odd part. Stating the problem in this form means that we invert two small matrices instead of one large one in order to obtain the same accuracy in the solution. Electromagnetic fields cannot have large spatial variations over distances that are small compared to the wavelength., except in the vicinity of the sources., at the discontinuities in the medium constitutive parameters., and at sharp conducting edges. The slot field of a narrow slot is therefore dominated by the edge singularity. We separate this out in the first term of a Fourier representation of the even and odd parts of the slot field, i. e. a co E( ~+2 a o (2. 32) e ~ 22 q1 q 0 and + b s n(2 3 0 ~212 q =2 q (233 7 oP 0 We are guided in selecting these particular forms of the field singularities by the work of Sommerfeld (1964) and Millar (1960) on the diffraction by an infinite slit in a vanishingly thin perfectly conducting plane screen.

THE UNIVERSITY OF MICHIGAN 582 5-1 -F K (0',y0)do' 0 7 a ~~e + d Pf,0 o 2 2 a K 0 o 0 For the first term we have after using the series representation for the kernel and by interchanging summation with integration (2)' OD H (ka) a E n cos(nO ) cos(n0 ')d0' a LIJ (2)' b D (kakb) n =o Hn(b n 2r0 00 H (2)? (ka) J (n0) a Z En (2? (kb cos(n0) 0n=onH (2t(ka) Dn~k. b n Interchanging integration with summation in the second term we have to evaluate 00 which after using the series expansion for the kernel and again interchanging integration with summation becomes Go H (2)1(ka) PO q~ E n cos(no) ____?)os d n E n H(2)' kb D n(ka, kb) co00)o ~ OD H~~(2)' (ka) s in(nO AM2 >\ /Xq+1 H1 11 o

THE UNIVERSITY OF MICHIGAN 5825-1 -F By this the integral equation (2. 30) has been converted into the form OD H (2)? (ka) J (nop) 0a I E n 2' ( 0kb cos(no) + 0n =o nH (21(kb) Dn kkb n Go OD H ~~~~(2)1 k)nsnn) (20 )2 ~1(-1) q+1 a v Hn 1snn0)- cos(nO = '0 q q n=1 H (2)? (b D n(ka.,kb) q) 2 _n 2 q1 ~~~~~~~n o aae Integrating (2. 34), term by term on the interval -0 (0 A we have co (2)? Z71 Hn (ka) J 0(no ) sin(no ) a0 Ec H(2)? 1 Dn(ka.,kb) no + n =o n Qw co H(2)' k) i 2 n 40 (_)+1 H (a1 asin(0) (01q q In 21(b Dn(ka, kb) (qirij2 _(noS)2 q n1 n co0 Cn sin(no) C n).(.5 Multiplying (2. 34) by cos (mxp / y 0) and integrating term by term on the same interval we have

THE UNIVERSITY OF MICHIGAN 582 5-1 -F OD H (2'(ka) J(no) n sin(nO n I 0 00 + a0 ( H2)1 D (ka, kb) 2 2 n 0 OD OD0 H (21(ka) 2I(no )2 sin2n >~ ~1~q+1 1 0I% Sif Go s in(no ) v n 0~~~ 2Cos (nO) (2.36) a an1 D n'k. b (mir) 2-_(not 0 We may summarize (2. 35) and (2. 36) in the matrix notation as [hz] aq]- Vke] (2.37) where (e) H (21(ka) J (no sin(noi he =n 0 0 0 00 n =o n H2' (kb) Dn(ka..kb) no n (e) q+ 1 ~~0 H (21(ka) 1sin 2(no) h~e 40 (-1) (2)' D0k k)q oq 0 n =1 H (2(kb) Dn (k,) (qir) 2 (nt)2 n Go H (2)' (ka) J (no n sin (nyo (e) n 0 00 h = mo (21(b D (ka, kb) 2 2' > nl n (k)n (mir) -(no) 000 h(e) =22 (1q+1 H~2 (ka) 1n2 si (nt) mq 0 n =1 H-~(b Dk~b -q)2(n) F(.i2(n — ()1- D(k. k)m. >1. q 2>12

THE UNIVERSITY OF MICHIGAN 5825-1 -F (e)= o En sin(no )] (e) ~ nsnn) cos(no),m ID(kakb) 2n 2 m n n' (m7r) -(no~) Turning to the derivation of the odd part of the slot field, we observe that the left hand side of the integral equation (2. 31), after the substitution of(2. 33) becomes 0 b 0~ qo q Representing the kernel by the series (2.26-b), interchanging summation with integration and integrating once by parts the first term becomes b n ~(a) 2 sin(n0 2 ZI7~~ (2)? D (ka~~~~kb) 1-(- -)Cos xdx, nl H (21(kb)D n~k. 3b r n 0 and the second term after interchanging summation with integration twice becomes 00 00 72) 40 (-1) q+1 FH(2 (ka) 2 rsi~P 0q(()q+1kbq D (ka, kb) q2 si 2~) sin n) q =2 nl1 LH (qir) _n The integral equation (2. 31) now takes on the form

THE UNIVERSITY OF MICHIGAN 582 5-1 -F n V2ka) 2 20 ii S ] b1 j(2)? D (ka, kb) t 1- xn 2 cosxdxl sin(nO)+ n1 n (kb) n J 00 00 2 0 q q (2)? D (ka) ____ qwsin 2 sn ~n) n 1 ~~-i~ bq L (kb) ~~~~ VK -(n0 (2.38) Multiplying (2. 38) by sin(mrnw- / 0 ) and integrating onteinterval 4e 0 ~ ~ ~~~~~~~0 obtain an infinite set of algebraic equations. We summarize them in the matrix form as h() b1 -V V I, (2.39) mq qj a a Am where OD (2)? no 0 (o) 40 V' H n(ka) 1 s in (no ) 2 h - ~ j1-( ---)Cos xdx, ml 7t nl1 (2)? kb D n(ka, kb) mr)2 _(o2 n n o 0 h =40(-1~q n1 n (2? 1 0l mq 0 n=1Hn2 (kb) Dn (k, b) ir) 2-(nV 2) [mir)2-(no 2] (0) nD(k k)2 2 (o) = 2)' (ka) 2 sin~~~~~n~) sin (no) m>1 m ~~~~21(kb) n (mit)) 2(i n n ~~~~~~~~0

THE UNIVERSITY OF MICHIGAN 5825-1 -F We have reduced the problem of finding the shell slot electric field to an inversion of two complex matrices. By the inclusion of the edge singularity in the first term for each part of the slot field we have enhanced the first column at the expense of the rest of the matrix. The enhancement is particularly large for a narrow slot. Thus the problem has been set up in a way so as to lead to the narrow slot approximation'. However, before we delve into the numerical details, we should indicate the physical quantities we want to find. We are interested in the narrow slot antennas. Further., we restrict the radial separation of the shell from the cylinder to a small fraction of the wavelength X, i. e. kb-ka <K< 1. Under these conditions the form of the radiation pattern of the cylinder with the shell will be essentially the same as for the cylinder alone. However, the power radiated into the free space will depend most probably on the orientation of the shell. Therefore we want to show how the radiated power depends on the slot field in the next few paragraphs. The power radiated per unit length by the shell slot is given by 2wRi E xH.ardo 2 ejr 0 2w- 00 (2kOH(D = 'R C' 71 jwpkA H/(kr)e~n~h nA Hb (kr Jn(-Id 2 ej n-oD n = oD 0 n 00

THE UNIVERSITY OF MICHIGAN 5825-1-F When kr~> 1 and n ~<kr, we may represent the Hankel function and its derivative;, respectively, by H~~2~kr)~~ 2 ek -l+2n] (241 Hn (k) rkr e.1 (2)' 2 i~-jkr - + ] H (21(kr)-., -j e, 4(2.42) n -J7rk and taking the limit of the general term of the series as r -> o, before we perform the summation on n., we obtain the exact result 2 7 2wpk A A (2.43) n n n =o-0 Using (2. 12) we rewrite this as 2 ______ j;~~0jn 00 ~E(O)e~ do 1 7 0 2wr Li c H2) (kb)2 n 1 jn en 2~~~~~E P~os5 2d0 27rL n o H(2)' k) Ee(cond + -~~ 0 27

THE UNIVERSITY OF MICHIGAN 5825-1-F Substituting (2. 32) and (2. 33) in the preceding equation, we have, after the interchange of integration with summation 1(0 )' Jo(n ao_200 ~~~~~~~~~~~~~~2o 1=) 1 nTI (no 2 _____ _ 2 7r2 LJP n = oH(2)1T(kb) 2 'n 020 sn I (qff)2_(no)2 n Go (_1)q 2, -1qq~r bq + 2 s in(n ) -l (2.45) o 0 ql (qir)2 2 q = 7 _ (q~r)-(no ) 0 From the behavior of the denominator in the general term of the series we conclude that only 2kb terms need be considered independent of the slot width. A computation will show that for a narrow slot the second part of the numerator is negligible compared to a even for a comparable b. Only when b becomes substantially q larger than a do we have to take, in some special cases, the second part into consideration. Also in the first part of the numerator the aterm is negligible compared to the a of the narrow slot. Thus we have that for a narrow slot, 2 ]2 a2 2kb E J (no n n o 0 P (ka, kb, 0 )/ 2 Z(2)1, 2, (2.46 The power radiated by the cylinder without the shell is given by P(ka)^- 1 n21 (2.47) 27rfw p n~o H (ka)i

THE UNIVERSITY OF MICHIGAN 5825-1-F We define the voltage of the slotted shell as Vb~b J E(O5)do (2. 48) and it follows that V ba.(2. 49) b o The ratio of equations (2. 46) and (2. 47) we may write as 2 P~~ka~~kbO) bF a2 kb,(2. 50) P(ka) Va (b F0ak,) where F~ka., kb., n o 2 no ()En 2 -1 (2. 51) 0 n= 0H ()1(kb) 2 no H 2) )(ka)2 For further discussion we elect to keep Va = a volts, whether or not the cylinder is enclosed by the shell. Then (2. 50) takes the form P~ka, b, 0) a 2 (kaRb,(2. 50 -a) P(ka) F0a b,) and a we regard as a dimensionless quantity. The last formula gives the enhancement (or depression) of radiation when the cylinder is enclosed by the shell. Aside from the essentially geometric factor F, all depends on the amplitude of the a term in the shell slot field expression.

THE UNIVERSITY OF MICHIGAN 5825-1 -F Since Maxwell's equations are linear the result (2. 50) is independent of the amplitude of the source voltage V a; thus the right hand s ide s of (2. 5 0) and (2. 50Oa) must be equal, and we obtain V b k b a 0(2. 51 -a) V k~ia K a The last formula expresses the transformer properties of the shell. We may regard (kb/ ka) af0 as the transformer turns ratio. We are fortunate that for a narrow slot the a term is also dominant in the 0 slot field representation. This assertion is expected to hold when << 7r ~~~~~~~~~~~~~~(2. 52 -a) 2byi ~X< (2. 5 2-b) b-a) 2b (2. 52 -c) From (2.37) with the understanding that V a= a., we have X(e) a C:: 0(2.53) 0 e) 00 When D n(ka, kb) ->0, then (2. 53) takes on the particularly simple form H (21(kb) n cos(nO) a 0 (2. (2. 53-a) 0H (k) (no) n o 'o Theloes6ode root - oftD (ka, kbxccrv(rel,143 hn kbhe2-ka, kra (1,

THE UNIVERSITY OF MICHIGAN 5825-1 -F lowest order root of D2(ka, kb) occurs when kb -~ 4-ka, ka 2. Of particular interest are also the roots of D 0(ka, kb), because then a 0does not depend on the angle 00 We list the lowest order root of D 0(ka, kb) (Jahnke, Emde., Losch, 1960) for some parameter values of possible interest in Table Il-i. TABLE II-1; Parameters for Lowest Order Root of D 0(ka.,kb) ka kb/ ka 31. 427 1.1 15. 7275 1.2 10.4993 1. 3 7. 8875 1.4 6. 2702 1.5 2. 5 The Results We are left with the task of computing a. The series expansion of the numerator and denominator of a involve Bessel functions and Neumann functions. 0 In view of the intended applications of the results of this study, we restrict the r adius, of the shell to kb < 2. We use the exact series for computing the cylindrical functions of order 0, 1, 2, 3, 4 and 5. Recursion relations are used to compute orders 6 and 7. For orders greater than 7 we use the small argument (Jahnke-Emde-Losch, 1960) approximation to obtain 1 __ 7r (kb)2 ka) (2.4-a D (ka.,kb) [ ka 2i 'kb 254-a L kb J

THE UNIVERSITY OF MICHIGAN 5825-1 -F H() (ka) kbn+1 (2) ka (2. 5 4-b) H 2) (kb) k n We sum 90 terms in the series determining h and!.We retain only five 00 terms in the F(ka, kb, 0 0) series. The computations were performed on a digital computer for the slot width of 0. 06 radians, and ka in steps of 0. 2 from 0. 2 to 1. 8, for each step, kb-ka = 0. 05, 0.10, 0.20. We elect to present the results in the following form. 10lgP(ka,, kb_, 0) 10 logP(ka) we choose as the ordinate and 0 as the abscissa in Figures 2 -2 through 2 -1 0. A zero on the ordinate axis means that the source radiates the same amount of power into the free space with and without the slotted shell. A negative number, say -20, means that introduction of the slotted shell decreases the radiated power to one hundredth of the previous value, while +20 means that the radiated power is increased by a factor on one hundred. The origin on the ordinate axis (0 00 ) means that the cylinder slot is under the shell slot, and+ +0 = 1800 means that the slots are on the opposite sides of the cylinder. The radiation curves are even functions in 0.I

H 20 -kb-ka - [_.~~~ *, -, - -_ -.. -. -_ - - *-.20 *~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ---.20,~lo~ 10 ii.05 C~ H co a 0 _ - 0 '0 ~ t 20~ 40 ~ 60 ~ 800 100~ 1200 1 60 180 0,-40 0+ -10 2: 0.03 FIG. 2-2: RADIATION THROUGH THE SLOTTED SHELL FOR ka = 0.20 - 0

30 20 10 0-0~~~~~~~~~~~~~~~~~~~ 2& 0~~~~~~~~~~2 LCCLI~~~~~~~~~~~~~~~~~~~~~~~~~~C FIG 40R 600 T S C~~o lo~00 1200 1400 10 / P~~~~~~~~~~~~~~~~~~~~~~~'~~~~0. 03

10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 0 ~~~~~o0~ ~ ~ ~ ~ ~~ dow~~~~~~~~~~C FI. -A P6DAToN HOG H LTE HL O a=06

10H 200 40~ 60~ 800~ 100~ 12 0~ 140~ 160 kb-ka ~0 J ~~~+0.- -.0 -s 2~~~~~~10nXS,~~~~ P ~ ~ ~ ~ ~ ~=0.03 m \10 -c~ - C~ ~ A C ~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 1 FIG.0 ~ 20 CIS ~~~~1 -30 -C)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C -40 0 FIG. 2-5: RADIATION THROUGH THE SLOTTED SHELL FOR ka - 0.80

I 0~~~~~~~~~~~~~~~ 0~~~~~~~~~~~~~~~~~~~~2 140 -1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c o 16 -20~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-40~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~FIG.2 -: " D'A ION THU UGH HE LOT ED R Lka0

20 -ka -ka ~~ ~ ~ '* '0.~~' ~ 20 10. 0 o. -40- FIG. 2-7: RADIATION THROUGH THE SLOTTED SHELL FOR ka =1.2 00 Cd~ ~ ~ ~ ~ ~ ~~~~ -2~~~~0~ - -3~~~~~00 0 o ~~. 'o 6 o ~o \ '. + e~~~~~~~~el 6,0 ~ X 40~ FIG. 2 -7: RADIATION THROUGH THE SLOTTED SHELL FOR0.03.I

20 kb - ka 1.0 201T ~0~~~~~~~~~~~w 40 '0m" ~ ' * oz 20~ 0o - 0. 03 j:)~~~~~~~~~~~~~C -30 I -40 FIG. 2 -8: RADIATION THROUGH THE SLOTTED SHELL FOR ka=l. 4

lo-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ k~~~~~~~l.Now ago~~~~~~~~~~U 0 cow &*low0 "fts 0 1 0 0 GOOH -1 0-~~~~~~~~~~~~~~~~ -30 --40 il FIG. 2 -9: RADIATION THROUGH THE SLOTTED SHELL FOR ka =1. 6

10 0 0 0 0 0 0 ~~~~~~~~~~~~~~~~~~kb -k 200 400 600 800 1000 1200 1400 1600.05 0.10~~~~~~~~0 ~..* +....2 -1 _,,~, 1..0o -1 0- ~ ~ ~ ~ ~ ~ ~ ~~ // 00. 03 C,~~~~,~ W~~~~~~~~~~~~~~ -20 --30-~~~~~~~~~~ 'I I. ~~~~~~~~~~~~~~~~~~~~I -40 — FIG 2-10: RADIATION THROUGH THE SLOTTED SHELL FOR ka = 1.8

THE UNIVERSITY OF MICHIGAN 5825-1 -F All along we assume that the source amplitude on the cylinder remains unchanged, i. e. the voltage of the cylinder slot remains at la' volts. Thus these calculations do not include the de-tuning of the antenna that must arise in most cases when it is surrounded by a perfectly conducting slotted shell. However, these calculations do show the effect the slotted shell has on the coupling between the line source of the cylindrical antenna and the radiation field in the free space. In Fig. 2-2., the antenna diameter is 7 X (ka = 0. 2), where X is the free space wavelength. Enclosing this antenna by the slotted shell increases the radiated power, the bigger the separation between the antenna and the shell, the more radiation we get, which is only weakly dependent on the source and the slot separation angle 0. For kb-ka = 0. 05, 0.10, 0.20 the radiated power is increased b1w 3, 7.5 and 13db, respectively. The maximum increase is when 0 = 180 0and the minimum when 0= 0. The difference between maximum and minimum for a given kb -ka is only about 2 db. In Fig. 2-3, we have increased the antenna diameter to ka = 0. 40 and the closer the shell is to the antenna the more radiation we get, which is just the opposite situation we had when ka = 0. 2 in the preceding figure. When kb-ka 0. 05 0 the radiation is increased by 11. 5db when 0 = 0., and increased to 23. 5db when 0 4180.When we increase kb -ka to 0. 10 the radiation very significantly decreases; when 0 00 we have only -1db, however the radiation increases to 13db 0 l_ I1

THE UNIVERSITY OF MICHIGAN 5825-1-F at 0 =00we have -18db and the radiation increases only to +2db as 0->-1800 In Fig. 2 -4, we have ka =0. 6 and the farther the shell is from the antenna,, the more it depresses the radiation'. When kb-ka = 0. 05 the radiation is -3db at 0 =0: but decreases to a deep minimum (the approximation gives zero radiated power., but this is not expected to be true) at 0 = 360, and from then rapidly rises to +6db as O — >1 80.Increasing kb -ka to 0. 10 depresses radiation at 0 = 00to -5db and the deep minima moves to 0 = 41; as 0 ->1 80 the radiation recovers to +2. 5db,. As kb-ka --->0. 2 the curve is depressed all along below zero., and the deep minima moves further to the right. In Fig. 2-5 the increase of ka to 0. 80 has brought, maintaining the same order., all three curves closer together, and minima have moved further to the right. When 0 = 00 the radiation is a few db below zero; at 1800 either just above or below zero. In Fig. 2-6 we have ka = 1. 0. The minima have moved farther to the right and for the three curves occur between 90 0 and 1000. The order of the curves has been reversed: the larger the kb-ka value, the more radiation we get into the free space. At 0 = 0 0, and 1800, kb -ka = 0. 05 give s 0 db, kb -ka = 0. 10 give s + 0. 5db and kb -ka = 0. 20 gives 2 db. In Fig. 2 -7 we have ka=1. 2 and the minima are occurring between 105 0 0 and 115.The curve ordering remains the same as in the preceding figure, but

THE UNIVERSITY OF MICHIGAN 5825-1 -F this curve the radiation at 0 = O 0 and 180 0 is +13db and +16db, respectively. In Fig. 2 -8 we have increased the antenna radius to ka =1. 4,, and the minimas now occur between 115 0and 1200. The curves have started to reverse the order. The kb-ka=O. 20 curve has dropped about 15db below the other curves and is approximately where the other two curves were in the preceding figure., For the high curves the maximum radiation is 18 db. All three curves show a second minimum starting to form at 0 = 0 0. In Fig. 2-9 we have increased ka to 1. 6, and the reversal of the curve order has been completed, i. e. increased kb-ka decreases the radiation. Also two sets of 0 0 minima have formed: the new set is between 10 to 20 and the old set has moved to 1250 to 130 0. The radiation is increased only by 6. 5 db at the maximum. In Fig. 2 -1 0 we have ka =1. 8., and all three curves have moved very closely together. The first minima occur at about 33 and the second at about 130. At 0 the radiation is -1db; at 800 and 1800, it is + 1 db. 2.6 Discussion In the figures presented, we have increased the radius of the cylinder., a, in nine equal steps from 0. 1/n XrA to 0. 9/ir X.. In all nine cases we have shown the effect of the slotted shell on the radiation when the radial separation between the cylinder and the shell is 0. 025/n-r X., 0. 05/nz- X., and 0. 10/n-r X. The shell increases radiation independently of 0 when a < 0. 1/ 7r X. A very substantial increase in

— THE UNIVERSITY OF MICHIGAN 5825-1-F indicates that for those angles the slotted shell decouples the cylindrical antenna from the free space. When a is 0. 4/ r X, or 0. 5/ ir X the slotted shell leaves the antenna radiation largely unaffected for extensive ranges of 0, except when the slot is in the 900range from the source where then deep minima occur. As a is increased to 0. 6/ wr X the antenna radiation is enhanced, and also becomes sensitive to the cylinder and the slotted shell separation distance. The same thing remains true as a is further increased to 0. 7/ wr X, except that a new minimum appears to form at 0 = 0. As a is further increased to 0. 8/ wr X, both the radiation enhancement and the sensitivity of the radiation enhancement on the cylindrical antenna and the slotted shell separation markedly decrease for all 0 values. Two deep minima have formed as well. Increasing a further to 0. 9/7 A.X reduces the radiation enhancement practically to zero, and the radiation becomes independent of the antenna and the slotted shell radial separation. This also occurred at a = 0. 5/ ir X.. This phenomenon appears to be associated with the 'resonances' in the coaxial cavity formed by the cylindrical antenna and the coaxial shell. The 'resonances' occur when D n(ka, kb) = 0. For our range of variables the first 'resonance' appears when ka,-~-'1, the second when ka,-' —2, i. e. a, —j0.5/ r X, and 1/w XrA, respectively. In equation (2. 50-a) essentially a geometrical factor relates the radiated power per unit length to the square of the amplitude of the a coefficient. This factor we denoted by F, and in particular cases considered here, the infinite series

THE UNIVERSITY OF MICHIGAN 5825-1-F 5 2 rA5~ F(ka., kb, (2 =no. > n(k) o ~ (2)' k)2 = (2)' ka21(.5 We plot F(ka., kb, 0 0) in Fig. 2 -1 1 for the same ranges of variables as appeared in the preceding figures. Since the shell is close to the cylinder in the three cases considered we have that the factor F is close to unity when ka > 1. Only for ka < 1 do we have a substantial increase of the factor above unity. Using this factor we can very simply obtain the ratio of the slot voltages from the preceding figures. From (2. 50) we have 101lo b _0 lgP(ka, kb,, 0) F10lo ka () F(ka, kbi,7 (2. 56) og -0lg P(ka) -0lgkb aL We notice that in the second term on the right hand side the argument of the log is close to unity. Thus for most parameter configurations the Figures 2-2 through 2-10 give also directly the ratio of the slot voltages as a function of 0. We may add that in this approximation the phase of V ais constant between the minima, and it suffers a 180 0 change as one goes through a minimum. The slot voltage phase is given by the phase of 1 /h e)and the 180 0phase change comes 00 from the sign reversal of (e on going through the minimum. 0 We have presented the approximate solution of the integral equation (2. 30), and discussed to some extent the physical consequences of this solution. The method of approximation sugg-ested itself from the results of the narrow slot in

5. 0 -5- o-~ ~ ~~~ 5 E J(n0) ~ 4 \, ~~~~n o 0 IH(2) (kb) 2 n 4.0- F( F =n= I n kb -ka n H(2)' (ka) 12 0b 005 4 ~1 n O 0. 05 Z I ~~~~~~\ o~El 0.10 3.0' A 0.20 F I, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~O 2. 0 -1. 0 0.i 0.5 1. 0 1.5 ka FIG. 2-11: COMPARATIVE RADIATION EFFICIENCY OF CYLINDERS

THE UNIVERSITY OF MICHIGAN 5825-1 -F a plane screen. The use of the same leading terms in the slot field representation is justified largely on the physical grounds and it leads to the geometrical restrictions (2. 52). Although we feel the approximation is a good one for the range of parameters discussed, a quantitative statement of the slot voltage approximation would be very desirable. We may truncate the matrix in (2. 3 7) and invert it. This procedure is laborious, the results may not be conclusive as to the error in any case, and particularly so when the original approximation is a good one. We choose to go back to the integral equation itself. We rewrite it in the form [K (a)(,0 1j Ee(P t)Ke(PI.,P)dP I -1 = 0. (2. 57) 0 When we use the approximate solution, the left hand side of the above equation will not be quite zero. This difference we denote by Aze (P ), i. e. 0 (2. 58) where the prime on the field indicates that it is the approximate solution we have obtained., i. e. (e) 1O Et(p?) = 0 (2.59) e h(e) 22 00 Wr 04

THE UNIVERSITY OF MICHIGAN 5825-1-F (e)( -1 ZO H() (ka) e J (n (e) o (a() n no o os(n)/\A (0)- h(e) K( n =o H(2)'(kb) D (ka,kb) C 00 fn (2. 60) In Table II-2 we present some of the calculations from (2. 60)for the parameters of 0(e) (a) Figs. 2-2, 2-5, 2-6 and 2-9. The A (0) and the K (0, 0) series have been O e summed to 90 terms, the other two series to 200 terms. The calculations showed that the approximate solution (2. 59) 'satisfies' the integral equation (2. 57) essentially (e) independent of the angle 0. In the table we have shown how A (0) depends on the field point coordinate 0. We have carried out the computations for + 0 = 0, 0. 015, and 0. 030, i.e. at the slot center, half-way to either end, and at either slot edge. TABLE II-2: VALUES OF Ae)(e ) (e) A for 0 = 0. 030 ka kb-ka I ~ 0=+ 0.015 =+ 0.030 I 0.40 0.05 0. 030-jO. 013 -0. 063+j 0 032 0. 2 0O+j 0. 10 0.40 0.10 -0. 015-jO. 0045 0. 037+jO. 0097 -0. 12-jO. 030 0.40 0.20 -0. 010-j0. 0017 0.022+jO. 0035 -0. 072-jO. 011 1.0 0.05 0. 0004-jO. 0000002 -0. 0004+j 0. 000001 0. 0014-jO.000004 1.0 0.10 0. 0010-j0. 000008 -0. 0016+j0. 00002 0. 005-jO.00006 1.0 0.20 0. 0040-jO. 00023 -0. 0077+jO. 0005 0. 024-jO. 0016 1.2 0.05 0. 0052-j0. 00001 -0. 004+j 0. 0002 0. 017-jO. 00067 1.2 0. 10 O. 0082-j 0. 00055 -0.013+j jO.0015 0. 043-j0. 0048 1.2 0.20 O. 024-jO. 041 -0. 047+jO. 090 0. 14-jO. 27 1.8 0.05 -0. 0060+jO. 000027 0. 0033+jO. 000085 0. 013-jO. 00033 1.8 0. 10 -0. 0050-jO. 000037 0. 0046+j O. 00018 -0. 01 5-jO. 00062 1.8 0.20 -0. 0036-jO. 0001 0. 0053+jO. 0002( 0. 01 7-jO. 00083 49

THE UNIVERSITY OF MICHIGAN 582 5-1 -F The approximate solution 'satisfies' the integral equation better at the slot center and the worst at the slot edges. Also it appears that the integral equation is 'satisfied' better when the radiation is insensitive to the cylinder and the shell radial separation, i. e. when we are closer to some particular co-axial 'resonance' than when in between them. In order to be able to make a quantitative estimate of the error we rewrite (2. 5 7) in the form j {E (PI)-E'(o')] Ke (P' 0)do' K (a) 0 A3e() (2. 61) Taking the absolute values we have ~~~O EfA1)K K(045) (e) j~ ~e'~' ~ K(~'A) do A a (e)) (2. 62) Since Kea (0,O is a real function we may argue that at the maximum E( )E'( e( E(~ (2.63) and hence =l [- A ~e(A) jE() (2. 64) From Table 11-2 and the last formula we compute that the maximum possible error in the slot field is 22 percent for ka= 0. 40, 2. 4 percent for ka= 1. 0, 32 percent for ka = 1. 2 and 1. 7 percent for ka 1. 8. For the radiation this corresponds to

THE UNIVERSITY OF MICHIGAN 5825-1-F maximum possible error in the data plotted with the exception of the radiation minima, should be a few tenths of a db when the curves are close together, and a few db when they are separated, i. e. when we are far from a particular co-axial resonance. The nature of (2. 45) indicates the minima in the radiation coupling curve for the slot width considered should be at least 25 db deep. Some further work is necessary to establish the depth of the minima. We have computed the radiation through the shell for three different shell spacings from the cylinder. For the parameters we have selected it would appear that bringing the shell closer to the cylinder in most cases gives stronger radiation than letting the shell be farther away from the cylinder. However., the type of the problems we have does not allow us to extrapolate these results in either direction. Further calculations are necessary to establish this behavior. However, we may offer some comments based on simple physical arguments. It would appear that the radiation goes to zero as the slotted shell coalesces with the cylinder for < 0 < 2w -0, i.e. the source is not under the shell slot, because then the source is enclosed by a perfectly conducting medium which precludes radiation. When the source is under the shell slot (-?0 <0 < 0 ) as b —a, then of course we have the familiar situation of a source on the perfectly conducting cylinder., These comments are also supported by some additional calculations for the case of Fig. 2 -3 (ka = 0. 40). These were done for kb-ka=O. 025, 0. 010, 0. 005, and the

THE UNIVERSITY OF MICHIGAN 5825-1-F limit, that of increasing kb-ka to infinity for a given ka has no practical significance. Increasing kb -ka beyond 0. 20 we may expect the radiation alternatively to increase and decrease. We may also ask the physical question: How does the radiation get out when the source is not directly visible through the slot? In an attempt to read some physics into the mathematics we may compute the Fourier coefficient of theE component of the electric field (2.3) in the co-axial space. Taking the cylinder slot voltage at ' a' volts, we have that jwp kF J'I (kr) +C N'I(kr]' 1i{E(kr)N' (kb)-N' (kr)J' (kb)l ejn0 Ln n n n J 2ir D (ka.,kb) _ n nJ n I [J(kr) N' (ka) - N' (kr) J1 Jkaa,n (2.65) where the approximate sign enters because we use the narrow slot approximation for the shell slot field. The first few coefficients we cannot discuss without some numerical computations. However, when ka < 2 and n ~ kb, we may use the small argument approximation of the cylindrical functions, and obtain krBJ~fk~C N' (kU 1/fF~n-1 ka n+1(kr)n-1] n Ln n~ n n r2w L kr/(kb) kbe +FIkrx)nl - 1 jk n+ 1 (k n-l a n kb <2, n >>kb. (2. 6 5-a) Lkb kr kb 1a o oJ From (2. 65-a) we have that on the cylinder surface 1 WU k[B J'(ka) +C N' (eane (2. 6 5-b)

THE UNIVERSITY OF MICHIGAN 5825-1-F and on the shell surface jwpkF JI (kb)+ C N? (kb) a aJ(no) (2. 6 5-c) L nn n nJ27r 0 0 0 It is clear that many coefficients are necessary to approximate reasonably well the E0 component in the co-axial space. The same statement applies to the other two field components. No Fourier coefficient alone under any conditions dominates the field ccomponents in the co-axial region, even in the case when for some particular coefficient Dn(ka,, kb)-*- 0. From this behavior of the Fourier coefficients we may conclude that no simple model may be devised to explain the transfer of power through the co-axial region. The power transfer results from the interference of very many Fourier coefficients. 2. 7 Conclusions We briefly summarize the salient features of the numerical results of this study. With a fixed magnetic line source on the cylinder the addition of a relatively close fitting slotted shell: 1) enhances radiation for all source and shell slot separation angles 9 when the cylinder diameter < 0. 5/ IrX., 2) leaves radiation roughly unchanged when 00 5/ ir X.< the cylinder diameter < 1. 1/ ir A, except for a deep minima in the vicinity of e&-, + 900, 3) enhances radiation when 1. 1/i7r X < the cylinder diameter < 1. 5/ 7 A. except for a deep minmainth viint of O-. 1100

THE UNIVERSITY OF MICHIGAN 5825-1 -F 4) leaves radiation roughly unchanged when 1. 5/ ir A. < the cylinder diameter <2 /A., except for a deep minima in the vicinity of,-,-,+ 450,+130. The deep minima in case 2) is associated with the lowest root of D I(ka., kb) =O, and in case 4) with the lowest root of D2(ka, kb)0O. It appears that radiation will have no deep minima when D 0(ka, kb)0O, or very close to. zero. Some of the antenna parameters for which this will occur are shown in Table II-i. We also note that the theoretical work on the problem has been carried to the point where the. numerical calculations can be carried out for wider slots, if we so desire. In this particular theoretical work the solution of the Fredholm integral equation of the first kind is based on the truncation of two infinite matrices. It would appear that this particular method of solving the fundamental integral equation of the problem has been put in the most favorable form for carrying out further computations.

THE UNIVERSITY OF MICHIGAN 5825-1 -F REFERENCES FOR CHAPTER II Jahnke-Emde-Losch (1960), Tables of Higher Functions. McGraw-Hill, New York Millar, R. F. (1960), "A Note on Diffraction by an Infinite Slit," Can. J. Phys., 38, No. 1, 38-47. Morse, P.M. and H. Feshbach (1953), Methods of Theoretical Physics, Part II, McGraw-Hill, New York. Sommerfeld, A. (1964), Optics, Lectures on Theoretical Physics, Vol. IV, Academic Press, New York and London, pp 273-289. Stratton, J. A. (1941), Electromagnetic Theory, McGraw-Hill, New York, p. 361. Truell, R. (1943), "Concerning the Roots of J' (x) N' (kx) - J' (kx) N' (x) = 0," t n n n n J. Appl. Phys., 14, pp. 350-352. Wait, J. R. (1959), Electromagnetic Radiation from Cylindrical Structures, Pergamon Press, New York, p. 30. 55

THE UNIVERSITY OF MICHIGAN 582 5-1 -F Ill. THEORY OF ELECTROMAGNETIC FIELDS FOR COAXIAL CIRCULAR CYLINDERS WITH A SLOT AND MAGNETIC LINE SOURCE. Y. Hayashi 3. 1 Introduction In this chapter it is shown how to solve the Maxwell's equations for coaxial circular cylinders with an axial slot of arbitrary width and an axial magnetic line source. This is a simplified version of the solution for coaxial circular cylinders with arbitrary number of slots and arbitrary number of axial electric and magnetic line sources (Hayashi, 1964a). The result obtained here is exact and rigorous, because it satisfies the Maxwell?'s equations in the domain, the boundary condition on the walls of the cylinders,, the radiation condition at infinity, the edge condition at the edges of the slot and the continuity condition at the slot. The result is also true for any values of wave number and radii of the cylinders. First of all., the field is represented by Fourier series with unknown coefficients. Then a singular integral equation is derived for the unknown coefficients. The singular integral equation is solved exactly by the theory developed by the author. As an example, the interesting case of a narrow slot is presented in a form ready for a numerical calculation. 3.2 Reduction of the Problem to a Singular Integral Equation Suppose that the structure of the perfectly conducting cylindrical shells are

THE UNIVERSITY OF MICHIGAN 5825-1-F FIG. 3 -1 Wall: raO 5~2r and, b a< ~2wr -a Slot: r b, -a< < a Assume that the axial line source is located at Q: r=a., 0 In this case, as is well known, Maxwell's equations are reduced to the twodimensional wave equation, i. e. A~u +k uO0 (3.1) where u = H =axial magnetic field and k2 =2 w The rational MKS units are z iW t employed and el time dependence is suppressed throughout. The requirements on u = H are as follows, The radiation condition lim r1/2{ aDu +i k u =0. (3.2) The boundary condition an 57nte al (.3

THE UNIVERSITY OF MICHIGAN 582 5 -1 -F The continuity condition., i. e. u and auare continuous through the slot. (3. 4) The edge condition u= O(pl-/2) (3. 5) where p is the distance from an edge of the slot. As a solution of (3. 1) which satisfies (3. 2)., u is represented as uu = AnHn(kr)e (b< r) n = n- c u U. BJ(kr) +C H(kr)e + f H(kR)., 1 4~~~n n n n 0 (a,<,r <b) (3.6) where Jn and Hn are the Bessel function and the Hankel function of the second kind, f is an arbitrary source constant, and R is the distance from the source to the point of observation. It is easy to see that (3. 3) and (3. 4) are equivalent to Al =0 ( r a., 0< < 2wr) (3. 7) ar a U au. e - 1 (r b, 0<(< 2ir) (3.8) ar dr e 0 (r b, a< < 2wf- a) (3.9) ar~ ~~~~5

THE UNIVERSITY OF MICHIGAN 5825-1-F e= i r=b <O<)310 On substituting (3. 6) into (3. 7) and (3. 8) and making use of the orthogonality of finK, simultaneous linear equations for An, B and C are derived with which n n n B nand Cn are determined in terms of An as follows, B HHn Ib ~H(ka) A + fn nT (a, b) ni r ka n J I(ka)F Cn n H? (kb)A -f{J nk) nkb n k)H(ka) n T(a.,b)L n (ka)H kb)ntnkb) JTI (kb) n where -ink f f e 0 n T (a, b) = JTI (ka)HI (kb) - JTI (kb)HI (ka), n n n n n and (C) means the derivative with respect to the arguments. Eqs. (3. 9) turns out, by the substitution of (3. 6), to be AZ Ann(kb)ein =f0;c<2r- (3.12) n =oD 2r(~) -oK5<o~. The upper line of the right hand side of (3. 12) is the direct consequence of (3. 9), whileT'r(o) in the lower line is an unknown quantity which is proportional to E in

THE UNIVERSITY OF MICHIGAN 582 5-1 -F Similarly, from (3. 1 0), we have T~a~b) H{ H(ka) A + fJ 0 -a< <a ~ (3.13) n=-0O n) (3. 12) and (3.13) are the "Dual series equations" with respect to An0 It is easy to see that (3. 12) is equivalent to An = j T ino e dO (3.14) n L here L stands for -a< < a. Then the substitution of (3. 14) into (3. 13) gives nooSU- T(&)ei 0dO -iw7(kb)2 f(o) (3.15) here H'I (ka) n T (a.,b)H'I (kb)' n n AO ) =2 0en~o 2 ( ZII2T (a.,b) 7T (ka)(b n= oo n Interchanging integration with summation in (3. 15) we obtain an integral equation with respect to the unknown function s If we solve the integral equation, then A' s are n defined by (3. 14) in terms Of i-, Bn's and C Is are determined by (3. 11), in terms of An's and finally, ue and u. are obtained by (3. 6). It is not difficult to prove that u,, obtained by this way, satisfies all the requirements (3. 1), (3. 2), (3. 3)., (3. 4), and (3. 5)i h ouin' a iglaiya h depiti.

THE UNIVERSITY OF MICHIGAN 5825-1-F T=O(p-1/2) (3. 16) Therefore, the original problem is equivalent to solving (3. 15) with respect to Tr which satisfies edge condition (3. 16). According to the formulas due to the author (Hayashi, 1964b) the Bessel and Hankel functions are represented as follows: ()1nfn - 1 n n. 2 jn ~ dnY 2(n-1)'. 2n H (P = -( n hHI w 2P {1+hI (3.17) n? n n(2r 2. 2 here j, j h and h' are proved to be quantities of order-2-. Hence, for a n n n n 4n ositive number N which is sufficiently large, Sn is estimated, when n > N, as 2~~~~~~~ he re s is of order p / 4n. For negative n (3. 18) is also true, because we can n prove that S -n= S nwith the help of well-known formulas iJ = (-1)n Jn H n(_1)n H n n n -n By virtue of (3. 18) and the formula OD cosn 0 n 2n 2 -2cos 0 "r' andinterchanging integration with summation,, (3. 15) is converted into ~, I-T T()(iAOndO=f( - (3.19) In

THE UNIVERSITY OF MICHIGAN 5825-1-F where MO., 0 2-= cos9 +ncos no 2 (3.20) 2-2 os0 n1 nit(kb)2 Differentiating (3. 19) with respect to iwe have sin0 dO' J 7(0) - 229 - dO = f?(3.21) L (2 2 -2 oso where N inS k(o, 0) i 1j+ n 'W~2,(in9 -e n9) (3.22) Equation (3. 21) is a singular integral equation with a Cauchy kernel. We will solve it in the following section by the application of the general theory on the singular integral equations developed by the author. 3. 3 Solution, of the Singular Integral Equation io Suppose that z = r e is a point in the complex z-plane,, then t =belO t =be io -a< 0, ~<a (3.23) P 0. correspond to points on L., i. e.,, in the slot., and Cbe, c- =be correspond to the edge points of the slot. Then it is not difficult to see that (3.21) is equivalent to A 1 f' -ktt

THE UNIVERSITY OF MICHIGAN 5825-1-F where T t) and h (t ) are functions which correspond to ilrT(O) and -, o 2 respectively under the transformation (3.23). k(t, t) is a kernel corresponding to k (0, O )and is N tn k(t t) k 0 (3. 25) 0 'i'n n+1 where 7rE (kb)2n > n 2 n -1 + -in sj n< 0 The solution T(t) of (3. 24) must have, because of (3. 16), a singularity at 1that is limT~t)0(1/ + (3.26) t ->c1 Suppose that M is a set of functions, the element / (z) of which satisfies the following conditions (i)^-,'(v), (i 0 (z is holomorphic in z -plane, except on L, at 0 and oD (iii) lz = 0(z _N z 40

THE UNIVERSITY OF MICHIG 582 5-1 -F (v) (t )+~ (t) h(t ), t eAL here (t )(V( means the limiting value of q (z) when z tends to a point t EL from the left (right) hand side of L. Then we can see (Muskhelishvili, 1953a) fWz)= 1 = 5 1 -k(z, t) Tk (t)dt atisfies if T is a solution of (3.24) which satisfies the requirement (3.26) T( tt -f(t ()3.27) On the other hand, it is not difficult to see that the general expression of a function (z) which satisfies (i) —(v) is given by N+1 x(z) \\i 1 -~(z, t)nh(t) =27T= i XW t z -X(z, t) dt+x(z) n 2 i Jt-z jx(t) n- H N n Nwhere p(n -N, N+ 1) are constants, X (z, t) is a function similar to k(z,

THE UNIVERSITY OF MICHIGAN 582 5-1 -F where 'Xh(t ) = 0 h~t2 h(t) x~~~t) ~ ~ t dt o w9I 0 x (t) N+1 lo0 0 N no0 Conversely, on substituting (3. 28) into (3. 24), we have (Muskhelishvili, 1953b kT Kh (3.29) where L 0 K (t, t) = '~'x(~)~' -k(t~?{ -x(., t)~ d~ (3. 30) Because (3. 29) is of the same form as (3. 24), we can see that Tis given by A ~~ ~~~~~~~~~~A 2 T1 XK h + T2 whereT T is again a solution of kT2 = K h. Repeating this process, we have Am T=XK h+T- 3 1 m m + 1 (1 where K is an operator with k-th iterated kernel K (t, t) of K(t, t). By the mo0 0 addition of (3. 3 1) from m =0 to m = m, we have m T A K nh+ T m+1 no If Xischoen so41 that K <1,the n covre, nbcus ehv

THE UNIVERSITY OF MICHIGAN 5826-1 -F lrn Kn h = 0 as the necessary condition for the convergence, then n — >oo (0 = urn T must be a solution of n -cOD A (0) kT = 0 (3. 32) Thus we have obtained the solution of (3. 24), that is nD (0) TX K h+T (3. 33) n =0 Especially, if K = 0, then (3. 33) is reduced to (0)~ ~ ~ ~ ~~~~ 0 T(0 has the same form as (3. 28), i.e. N+1 T(0) (t)Xt n (3. 35) 0 0 w n o n-N must be a solution of (3. 32). In other words, p n(n = -N,..., N+ 1) must be de - termined so that (3. 35) satisfies (3. 32). On substituting (3. 35) into (3. 32), we have N+1 N+ 1 \X(t)tnt tn Zj tn t dt =p Sk(t t) x(t dt (3. 36) n N L 0 n=-N L which is the necessary and sufficient condition for pn with which (3. 35) satisfies (3. 32). After manipulations., we can reduce (3. 36) to

THE UNIVERSITY OF MICHIGAN 582 5-1 -F N+1 -N ZkrnnrP = p-Nm1 m n-m n m~~~~Z ur-n n n=-N ~~~~~n =mr N+1 N+1 n krn rn Z pnnn O<rn<N (3.37) where -a — X ~dt (3.38) n-rn iri L and i n and y are constants defined by rn- rn-n nX(Z) zn (3.39) 0D (3. 3 7) are 2 N + 1 conditions for 2 N + 2 constants p(n N N.* N+ 1) which are equivalent to (3. 36). Thus., pn(n=-N... N+ 1) are deterrnined up to a constant actor..4 A Note On Solution of Non-Singular Integral Equations Although we have solved (3. 24), or (3.Q21) a solution of (3.21) does not necessarily satisfy the original equation (3. 19) because (3.31) has been derived by the differentiation of (3. 19) with respect to f6. However., we can pick the unique solution of (3. 19) frorn the general solution of (3. 21),, by taking appropriate values of

THE UNIVERSITY OF MICHIGAN 582 5-1 -F pn (n =.N.. N+1). This will be discussed in the following. Suppose that a set of pnsatisfies (3. 37), then (o X(0 ) cZc n=-N n where c =bp nx(OV n Pn bY(e -e )(e - e )3 0 satisfies kT =0, T(O) (N'G 0) d0 = 0 (3. 41) L where N1(0, 0) is the kernel in the left hand side of (3.21). Furthermore, T(0) = T(0) +T (O) ( p where T~ P(0) is the particular solution XA f, satisfies (3. 21), or p~~~~~~~ LS ~ ~T (O9)N N(O.,0)dO = f I(O(.2 From(3.1) ad (.42) wehav

THE UNIVERSITY OF MICHIGAN 5825-1-F or, on integrating the last equation with respect to 0, we have whereg isa knwncnsta t. On the o othe had, f(3.4) turns ot to3b L where /3 is a known constant corresponding to particular values of n If we change {P by {., or what is the same thing., (o) (o) if we change T r0 by T(0, then T + T(0 satisfies (3. 19). In fact, p / T (O )N(0,O0)dO + T (O)N(O6, O)d6EP f(o) -g + /f().5 Electromagnetic Field for a Narrow Slot As an example of the application of the general theory obtained in the previous sections, we will solve the field of a narrow slot. We assume that the slot is narrow in the sense that 2 (.4 and also that Ikal Jkbj<1. (3.45) Then, s, which is of the order (kb) 2/ 4n, is negligible for n > 2. Hence, we can asuetan 1 n ehv

THE UNIVERSITY OF MICHIGAN 582 5-1 -F 1 - k(t,t) 15 0 + s sin l t -t o b L 2-2 cos 0 1 J 0 where 0= - BecauseI00IIOI~2 aI<<1 sinO and cosO can be expanded in power series with respect to O., and we have * i ~~~~2 1 -k(t, t)(-t ) 1+- 0 -)+0 (a ),(3.46) 2 IfIs is so large that~ I a 2 ---- 1o then we have to take one more term in (3. 46). Here, we have assumed that ~s a2 < 1] On taking X (t, t) to be equal to k(t 0o t), one has K~t t g~1 0to1d _ 0 2 w x~t)(t -t 0 where L By virtue of (3. 46), we have g(to t, ) 1. (-0)~0(a 2 0 ~~~2 which is independent of ~ if we neglect terms of O(a ). Hence., K(t, t) will be g(t,t) Q ( 1 K(t ot) = 0X(Q d~ = 0 o 72 X((-t -t0 -t 0 L he re g(t, t)O g(t,t, ) Thus we have proved, under the assumptions (3. 44), (3. 45) and 0( 2)=Ota K(t, t) = 0. Therefore., by virtue of (3. 34), the solution T of (3. 21), or the tangen0 ial component of the electric field E in the slot is obtained as

THE UNIVERSITY OF MICHIGAN 5825-1-F T(O) W E E (0) 2w7k 0 a OD n ninbn i ) -fx(0) 1 llf i 74(k)( 2 1 x(0)(0 -0) nT n(a, b) -a +x(0) n b mO (3.47) (3. 47) is rewritten., by virtue of 1 - -ir (kb) 2 (a n+1 10(kb)2 (.8 T (a, b) n b +04n J3 8 -TO E (0) - 2'rk 0 ifx(0) (a n+1 7r ka j (V)(o-V}) Z b sinV-0~ 2 n mnO +X(0) pnb e (3.49) where M(O 1 io 2 2 b e a -0 [ote that if (3. 48) is not true for n = 1, then the first term of (3. 49) should be modified]

THE UNIVERSITY OF MICHIGAN 5825-1 -F (ka + 'y) P1 +k_ Y p +k1 p' + k1 - p 0 ka P- +k p + (k a +1 )p +(k a +3)= o 1 0 0 0o -1 1 0 2 + -2) 2 k1a-2 p1+k 1a1 p + k1 a~p1+ (k1 a1 + 013) = 0 (3. 50) where a bn~ x(O )e ndO. II ir %J With the knowledge of the distribution of E 0in the slot obtained by (3. 49), the fields are given by (3. 14), (3. 11) and (3. 6).

THE UNIVERSITY OF MICHIGAN 5825-1-F REFERENCES FOR CHAPTER III Hayashi, Y. (1964a), "Electromagnetic Field for Circular Boundaries with Slots, 1 Proc'. Jap. Academy of Scil., submitted for publication. Hayashi, Y. (1964b), "Electromagnetic Field Solution for Circular Boundaries with Slots Using a Singular Integral Equation Approach."1 To be Published. Muskhelishvili, N. L. (1 953) Singular Integral Equations.," P. Noordhoff, N. V. Groningen-Holland, a) p. 2 9; b) p. 2 3.

THE UNIVERSITY OF MICHIGAN 5825-1-F IV. CONSIDERATION OF ENERGY STORAGE AND DISSIPATION IN A CAVITY BACKED SLOT ANTENNA AS IT AFFECTS INPUT IMPEDANCE AND RADIATION POWER. A. Olte 4. 1 Introduction In this chapter we consider the interaction of a cavity backed slot antenna with the plasma sheath. The slot is small with respect to the wavelength and hence the antenna is very narrow band. With the onset of the over-dense plasma sheath, the antenna becomes detuned and this in addition to the usual overdense plasma interference with the radio waves strongly limits transmission. One could broad-band the antenna at the expense of the antenna's efficiency by introducing losses in the cavity. Would this help in maintaining transmissions with the onset of the plasma sheath? Can we in any other way limit the undesirable influence of the plasma sheath? We undertook this study in an attempt to search out and define some favorable modes of interaction. This work is reported in four sections. In the first section we derive energy theorem for the cavity backed slot antenna. In the second section we use the energy theorem to express the antenna impedance in terms of certain integrals. In the third section we derive expressions for the antenna impedance with and without the plasma sheath both for the loss-less and the lossy cavity. In the fourth section we discuss and compare these expressions. The cavity losses do not appear to play any useful role. It appears that the plasma sheat interference will be kept at a minimum if the antenna resonance does not depend on

THE UNIVERSITY OF MICHIGAN 582 5-1 -F magnetic type. To the extent these conditions can be brought about by inside tuning is of continuing interest. 4.2 The Energy Theorem We consider antenna structure as shown in Figure 4-1. A cavity is cut in a erfectly conducting body of a finite size. The cavity is connected to the free space y a slot which is center-fed by a co-axial line. We select the input terminals of the antenna at a reference plane A in the co-axial line. The volume of the co-axial mne from the reference plane A to the opening at the slot we denote by V1. the cavity volume by V 2, and the plasma volume by V 3. The free space volume bounded y the plasma surface and the geometrical surface S4 is denoted by V4. The olowing electromagnetic parameters are assumed for the respective volumes: V 1 E1 (r eal), P 2 2 2 2 o V E El Elf PCt~ 3 3 3 3 V4 4 o0 o 1 -7 We have that C - 2 where p0 4irxl, 0 and c is the velocit 0 1 f light in vacuum.

THE UNIVERSITY OF MICHIGAN 5825-1-F I I1 I I~~~2I L I _ _ _ _ I I A 4~ C ~~~~~~~~ The Antenna Structure Figure 4-1

THE UNIVERSITY OF MICHIGAN 5825-1 -F We start our analysis with the vector identity V(E xH)=HV'xE-E-VxH, (4.1) where E and H are the complex amplitudes of the electric and magnetic jw.t fields, respectively. The time dependence is taken as e. Assuming that no sources are present in the regions enumerated, or on their boundaries, we obtain from (4. 1) after substituting the appropriate Maxwell's equations, V (E xH )jw LE E E-0H H(4.2) We multiply Eq. (4.2) by the differential volume dv and integrate in each region. We add the results together for the four regions. After transforming the left hand side by the Divergence Theorem, only two surface integrals remain. The others either vanish or cancel each other because of the usual electro-magnetic boundary conditions. The end result of this operation is SE1 x H Nds + JE x H nds SA S4 4~~~~~~~~ zjW jJE E ) H *H Jdv, i where n is the outward normal and the superscripts identify the fields of the respective regions as shown in Figure 4-1. rThe, time0 aveJ0roiag power rad~iated by theX 01a ntenna, Pi

THE UNIVERSITY OF MICHIGAN 5825-1 -F 1 (4)*~(4 2-R j E x H *nds (4.4) S4 When the surface S is sufficiently far removed from the antenna, then the power flow is radially outward and the electromagnetic fields locally assume a plane wave character, and 10 P 2~ E(4 x H nds (4. 5) S4 In order not to have the question always come up as to how far we should take S4.from the antenna, we let S — > S, the surface at infinity. 4 G We assume that the radial dimenions of the co-axial line are such that only the TEM mode is propagating. We select the reference plane A sufficiently far removed from the co-ax opening (a fraction of the wavelength is sufficient) to have the higher order mode fields essentially zero. At the reference plane the fields are then the sum of the incident and the reflected TEM modes. The amplitudes of the electric and the magnetic fields may be associated with equivalent electric voltage Vand current I, epciey One of the defining expressions is taken as VI ~ E x H n ds (4.6) SA The minus sign is selected to conform to the convention that the positive direc

THE UNIVERSITY OF MICHIGAN 5825-1 -F transmission line voltages and currents may be made unique by taking the characteristic impedance of the line, Z0, as z = 1n -,(4. 7) 0 2w r where r. and r are inside and outside cylinder diameters., respectively. Substituting (4.5) and (4.6) in (4.3) we have VI =2P +j PH H -E.(Ec E )dv., (4. 8) r (i)_ V. Because we have let S4 recede to infinity,, the volume V4 is infinite. However, this volume integral is finite because the fields assume the plane wave character in the far-zone and as a result the integrand vanishes there. Substituting the respective values of permittivity, in Eq. (4. 8), and separating the right hand side of (4. 8) into real and imaginary parts, VI =2P +LiEEi). E~i dv] + i= V.LH + j pH IsH -E4w d o H (i) Ei 79 H 4w

THE UNIVERSITY OF MICHIGAN 5825-1 -F We recognize that wHG and w E) is the time average energy density of the magnetic and electric fields., respectively, in the i-th region. With these definitions equation (4.9) may be rewritten as 3- - VI =P+ElE?WE dv + 2 r 2 ~i =2 V. 1 4 ~~~~~~~~~~~(4.11) 2jLo iwG) w~i dv i=1 V-H E The right-hand side of (4. 11) consists of a real part and an imaginary part. The real part consists of the first two expressions. The first expression is the average radiated power of the antenna. We identify the physical meaning of the terms in the second expression by observing that the divergence of the instantaneous PoyntingIs vector is equal to the negative instantaneous dissipated power density in the medium. From this., one may show (Landau and Lifshitz,, 1960) that the time average density of the power dissipated in the permittivity losses is given by kU wLw ()(4.12) 2 C!E and consequently the average power losses in the cavity and the plasma sheath P 2 and P3,9 respectively,, are given by the terms of the second expression on the right hand side of Eq. (4-11). The last set of terms on the right hand side we interpret

THE UNIVERSITY OF MICHIGAN 5825-1 -F 4. 3 The Circuit Representation We define the antenna impedance at the reference plane A by Z =R + jX =(4.13) Substituting this in Eq. (4-11) and separating the real and imaginary parts, we obtain R R + R 2+R (4.14) where 1 ~~(4) _() Rj E xH *iids r II R 2' - w(2)dv V2 R 3Jw () dv; 3 and4 X = X, (4.15) where II Kw (i) wE dv V. 1

THE UNIVERSITY OF MICHIGAN 5825-1 -F the "radiation resistance," the "cavity resistance," and the plasma "sheath resistance." The "radiation resistance" is given by twice the average power radiated divided by the antenna input current amplitude squared. The "cavity resistance" and the "plasma sheath resistance" is given by twice the average power dissipated in the cavity and in the plasma sheath, respectively, divided by the square of the antenna input current. From Eq. (4-15) we see that the reactance of the antenna consists of four reactances in series: the "co-axial termination reactance," the "cavity reactance," the "plasma sheath reactance," and the "free space reactance," X1, X2, X3, and X4, respectively. Each reactance is given by the difference in the magnetic and the 4o electric energies stored in the respective region multiplied byi.| In this formulation of the equivalent circuit problem the current I is the independent variable. The equivalent antenna circuit may be represented by an ideal current generator that is connected to the impedance Z = R + R2 + R3 + r 2 3 +X1+X2 + X3+X 4 as shown in Figure 4-2. R2 R3 Rr?~f! l Il 2 3 jX4 jX3 jX2 Figure 4-2. 8- 82 -

THE UNIVERSITY OF MICHIGAN 5825-1 -F Specifying the input current in the one-port network corresponds to specifying the tangential magnetic field at the input terminals of the antenna. In the four re - gions we must then solve for the electromagnetic fields that satisfy the vector wave equation and the boundary conditions; a formidable problem for the antenna structure under discussion here. If we had specified the voltage V at the input terminals, then this would correspond to specifying the tangential electric field (in case of the TEM mode the total electric field) at the reference plane A and the appropriate circuit representation would involve an ideal voltage generator connected to an admittance which represents the antenna. The admittance Y is given by Z - The above energy balance frame -work does not give any outright answers to the antenna problem. It does provide a form which suggests possible mechanisms to account for and explain some of the observed antenna behaviour. Before we take up the discussion of antenna configurations of interest we should make some observation on the realizability of the equivalent circuit representation as well as discuss some of the circuit aspects of the problem. The formulation of the antenna impedance representation pre-supposes the availability of an ideal current generator, 1. e. a device which produces a specified current that is independent of the load. The voltage across the load then is a variable and depends for a given current generator on the load connected to it. For quasi-static situations we have devices that approximate reasonably well the ideal current generator. For high frequencies such devices are not ava7,ilable, and the internal imnpedance of the geinerator enters- the problem.

THE UNIVERSITY OF MICHIGAN 582 5-1 -F physically approximate the ideal power generator: a device that sends a fixed amount of power on the load independent of the load and absorbs any reflected power. When the antenna and its surroundings are composed of a linear medium, and such a case is assumed for this discussion, then the ratio of the reflected power to the incident power is independent of the incident power amplitude, and this ratio then is of some use in describing the antenna properties. The power of a particular incident waveguide mode determines the electromagnetic field amplitude. The other main feature of the mode is its phase. Therefore, the boundary value problem of an ideal power generator source may be visualized as a waveguide mode incident on the reference plane A (in our case the TEM mode of the co-ax) for which we seek to find the amplitude and the phase of the reflected mode. Again we recall the assumption that the TEM mode only may propagate in the co-ax., and thus the incident and the reflected TEM modes constitute in our case the total field at the reference plane A., for all practical purposes. We see that in the ideal power generator boundary value problem we specify neither the tangential electric field nor the tangential magnetic field at A, but a partial field of each., namely, the electromagnetic field of the incident mode. The boundary value problem consists of finding the electromagnetic field in each region subject to the local boundary conditions and the provision that in the coax in addition to the incident TEM mode there exists an outward travelling TEM mode., i.e., the reflected TEM mode. In this case the total tangential electric and magnetic field at the antenna input terminals, depends on

THE UNIVERSITY OF MICHIGAN 582 5-1 -F the load, i.e. on any changes in the antenna structure. As before, we may associate equivalent current and voltage at the input terminals, but none of these are now independent of the load. However, formulas (4. 14) and (4. 15) are valid for computing the antenna resistance and reactance. The characterization of the antenna that naturally fits the ideal power generator representation is the reflection coefficient, F1. It is defined for any mode, as a ratio of the transverse electric field amplitudes of the reflected and incident modes, taken at the reference plane A. The same amount of physical information about the antenna is contained in the reflection coefficient representation as in the impedance representation. A well known simple transformation connects both representations: 7 0 ~~~~~~~~~(4.16) Z+ Z 0 The amplitude of the reflection coefficient squared is the fraction of the incident power reflected from the antenna. One often represents the antenna in terms of the voltage standing wave ratio (VSWR) given by VS +R= W! (4. 16a) which is the ratio of the electric field maximum to minimum amplitudes in the waveguide feeding the antenna.

THE UNIVERSITY OF MICHIGAN 5825-1 -F Whenever the circuit dimensions are small compared to the wavelength simple circuits are common in which the resistance and reactance may be varied independently from each other. In some cases the antenna reactance may possibly be varied without significantly affecting the radiation resistances. But this is not to be expected as a matter of course. Let us assume that in Eq. (4. 16) the real part of Z is fixed and let us vary the imaginary part so as to obtain the minimum power reflections from the load for a given Z and R. The fraction of the reflected power from (4.16) is 2 (R-Z ) 2+X2 17I= 2 2 ~~~~~~~~~(4.17) (R+Z) +X The extremum of 1FIunder the conditions discussed above is aLri]= (4.18) It leads to the condition that XRZ =0 (4.19) 0 and this can only be satisfied when X = 0 (4.20) Clearly this extremum point gives the minimum condition for the reflected power. And from Eq. (4. 16a), we see that whenever I I is minimum then also VSWR is minimum. The absolute minimum (7 F 0, and VSWR = 1) occurs when ZR=Z We then say that the antenna is matched to the feeding transmission line. In circuit theory1 we17 sayX that the circulit is in resonancen0 w,,hen the reactive, pa-rt of Zva n is-hes.

THE UNIVERSITY OF MICHIGAN 5825-1 -F Similarly we say that the antenna is at resonance when X = 0. This condition implies from Eq. (4.15) that the magnetic energy storage is equal to the electric energy storage in the antenna. The same condition as in the RLC circuit resonance. 4.4 The Broad-Band and the Narrow-Band Antenna Problem We want to discuss specific antenna configurations and possibly infer some -mechanism to explain the reported antenna behavior. We will use the current generator representation, since it contains the same amount of information as any other one. Case a. The cavity dielectric is loss-free: P2 = 0, and no plasma sheath on antenna surfaces, i. e. E3 E We have then from(4. 14)and (4.15) that a 1 lxeH n ds (4.21) S and X rr w~~~E dv (4.22) V. 1 We have used the subscript a in Eq. (4.21) and (4.22) to specifically indicate those quantities which will depend on the antenna structure changes initiated in the later discussions, and the antenna configuration will be identified by them as well.

THE UNIVERSITY OF MICHIGAN 5825-1 -F made on the wall opposite the cavity slot, so as to keep the spatial character of the slot field approximately the same. The antenna is tuned when Xa = 0 in Eq. (4. 22). We then have a specific value of the antenna resistance which we denote by R ae We desire to make the antenna resistances equal to the characteristic impedance of the feeder line as the optimum condition of the operation. The next question of interest is to determine what causes Xa to equal zero. Does each integral in Eq. (4.22) vanish separately, or do they cancel each other in some manner from which we may infer some physical meaning? The first integral is over the volume of the co-ax feeder section. This is normally a small volume and the difference in the average energies stored is expected to be small compared with the other energies to be discussed. For the time being we leave it out of the discussion. Since the cavity volume is considerable, a significant amount of average energy may be stored there: electric energy will predominate when the cavity is below resonance while magnetic energies predominates above resonance. In the present case of no plasma sheath the common boundary between volumes V3 and V4 is only a geometrical boundary and the two volumes should be discussed together. Although V 4 is an infinite vol - ume, the integral will exist because the far-zone fields assume a plane wave character which implies that the instantaneous energy storage in the magnetic and electric fields are equal, and hence the difference in the time average energy dens ities is zero as well i. e. (4) (4)

THE UNIVERSITY OF MICHIGAN 5825-1 -F where R is a sufficiently large number. Throughout volume V we will get a o 1 3 significant contribution to the integral, this volume being close to the slot and also in the immediate vicinity of the cylinder. To the extent of neglecting the energy storage in the co-axial feeder section, one possibility of obtaining antenna resonance, physically a very appealing one, is to consider that the cavity difference energy storage is compensated by the difference energy storage outside the antenna, i. e. ~' [(2)w(] dv 17 -KWi(i) dv, (4.24) Ha Ea i3-*L Ha Ea2 1. In this picture the slot couples the two predominant energy storage systems: the cavity and the near-zone of the antenna. The excess energy surges back and forth through the slot alternating directions each half cycle. This would be expected to enhance the slot field amplitude, and thus to increase the radiated power of the antenna. Unfortunately, we have no analytic means at our disposal to make this picture of the antenna resonance unique for a configuration as in Fig. 4-1. We may also see from a slightly different viewpoint that the picture of resonance presented is a physically reasonable one. Let us assume that the cavity slot is closed, and the cavity is excited by a small probe on the opposite side. We select the cavity dimensions such that it is in the lowest mode of resonance at the exciting frequency Wo. Now let us open a small slot. Energy from the cavity will leak through the slot into the outside space. The nature of the cavity fields close to the slot will also change radically, because of the changed boundary conditions. The cavity is no longer in

THE UNIVERSITY OF MICHIGAN 5825-1 -F new system which involves not only the cavity, but also the free space outside the body of the antenna. We want to find again the lowest mode of resonance of this new system. The simplest one and therefore possibly the lowest mode of resonance appears when we change the cavity dimensions just sufficient to make the sum of the difference energy storages in the cavity and the free space equal to zero. Only two possibilities are open, either the difference energy storage in th6 cavity and the free space vanish simultaneously, or they cancel each other. The possibility of the first situation cannot be excluded for all cases. However., the latter possibility appears to be physically much more probable than the first one. In our discussion, for most part., we will consider only the most probable case. When the coupling slot is small compared to the wavelength in both dimension, only a small fraction of the cavity energy will leak out, and consequently we will claim that the 'IQ" of the new system is just slightly lower than for the no slot cavity. In such a situation in order to obtain a significant amount of radiation from the antenna, the system must be brought to a resonance. That implies large field amplitudes in the cavity, large surface currents over the conducting walls, and also large electric fields in the slot, even when it is small compared to wavelengths. ith the proper transmission line coupling into the cavity and the system at resonance, we should be able to radiate all the power from the feed line., no matter how small the slot is. Such an antenna has a very narrow band-width and any changes in configuraiorfn will de -tune it marked-nly.v Thei factri that- ini orii ncase the, feeder exdcite

THE UNIVERSITY OF MICHIGAN 5825-1-F the slot directly does not change the substance of the discussion so far presented, it only makes the slot field more complex and thus less amenable to a theoretical treatment. From this discussion we see that it would be of considerable value to establish for some antenna shape the magnitude of the coupling provided by the slot for inside and outside reactive energies. The antenna configuration should be such that the exact solution of the boundary value problem can be carried out. At this point we should make a digression to observe that the impedance of a narrow slot in a perfectly conducting ground plane in free-space is related by Babinets' principle to the impedance of the thin electric antenna. (A thin wire cut in half and excited at the center by a voltage generator, also known as the electric dipole). The electric dipole is the most common and most studied antenna and its impedance is readily available. If Ze is the electric dipole impedance and Z the thin slot impedance then from Babinets I principle we have that z P_ 1- (4.25) M C Z e A thin slot in a perfectly conducting ground plane is an example of the physical realization of the magnetic dipole antenna. The magnetic dipole impedance may be applicable directly to antennas where the radii of curvature are large with respect to wavelength.,

THE UNIVERSITY OF MICHIGAN 5825-1-F Case b. We leave everything as in case a), except we add the plasma sheath. The permittivity of plasma we take as C = (4.26) where wo is the plasma frequency, and v the momentum transfer collision frequency. In case a) we had the resistance of the antenna given by (4-21) and the cavity was tuned such that the total reactance was zero. Maintaining the same current I at the input terminals we introduced the plasma sheath in V3. Both the resistance and the reactance will change. We denote the new quantities by subscript b., 1 ~(4) _4) _( Vw2 (3)_() R ~~E x H n ds + cc 2w2E.j~ dv b II, b b o W i+V S V ~~~~~m (4.27) -- ~~w -w dv (4.28) The antenna resistance now consists of two terms., The first term comes., as before., from the radiated power,. The second term is new and it represents the power absorbed in the plasma sheath,. The reactance, as before, consists of four

THE UNIVERSITY OF MICHIGAN 5825-1 -F integrals. We would like very much to make a statement about how much Rb differs from Ra, and Xb from zero. Unfortunately for this we have to calculate the fields first. In that case we would also find the tangential electric field at the reference plane A, and hence the voltage V for the specified I. There would be no need for this integral representation. The main value of the above expressions are in that they provide a qualitative picture of the components of the resistance and reactance, and furthermore give a frame of reference for making comparisons if reasonable approximations can be made in the fields. The best possible thing would be to solve some canonical problems whose results could then be extended to the antenna shape of interest by physical argument. We have not done that as yet and thus we have to rely only on our physical feelings for the problem in order to make some further comments. Conditions have not changed in the cavity, thus the fields would tend to maintain themselves there at the previous values. Let us assume therefore that the difference average energy storage in cavity remains approximately the same. What happens to the fields in the plasma sheath? Let us consider a case where w +v2 < w, then the real part of E is negative and hence w 15 i negative, and m p 3 Eb the difference energy storage in the plasma sheath is the sum of the average electric energy and average magnetic energy; always a positive quantity. The implication of this is that plasma beyond cut-off appears always inductive, no matter in what con

THE UNIVERSITY OF MICHIGAN 5825-1 -F difference average energy stored outside the plasma sheath. However, one over-all effect of the plasma sheath is obvious: the antenna will be de-tuned; probably Xb > 0, i. e. the antenna will appear to be inductive. This is the effect that the plasma sheath will play on the imaginary part of the antenna impedance. The real part of the antenna impedance consists of two parts., as pointed out before, the radiation resistance and the plasma sheath resistance. The radiation resistance with the plasma sheath beyond cut-off is definitely smaller than Ra because the sheath will impede radiation. The magnitude of the plasma sheath resistance will depend in large measure on the magnitude of the electric fields in the plasma sheath and the ratio. It may be comparable to the radiation resistance. Case c. The antenna has the same conditions as in case a), except the permittivity in the cavity is complex. We tune up the antenna,. i. e. make Xc = 0., and we have a corresponding resistance R. In terms of the fields for this case we have 1 () 4) -C2 (2) R ~ { E x H nds +w _ d_ (4.29) and 4~ ~~9

THE UNIVERSITY OF MICHIGAN 5825-1 -F This antenna will have a wider band-width than in case a) because the losses are greater and hence the antenna Q is lower. The antenna Q is defined in the same manner as for circuits. Case d) The antenna has the same conditions as in case c), except we add the plasma sheath and do not re-tune the antenna. The elements of the antenna impedanc we now express as 1 ~(4) _(4) (N, Ell(2) R E ~~x H n ds+wL -w dv + d IF, d d J Ed Go 2 OD 2~~~~ Li (3) _(3), Li ~~~E E dv (4.31) 0 Li 2+ 2 d d S m X d 1I w -Hd wi dv (4.32) V. The same comments can be made about this case as in case b). Things are changed only to the extent that we have energy dissipation in the cavity and this entails the change of fields in every other region to some extent. 4. 5 Discussion If the slot is narrow and less than half a wavelength long, then the coupling between thep cavityv and the freep space is notf a maximum asQ would4 be epected~r w~hen thec

THE UNIVERSITY OF MICHIGAN 5825-1-F slot is approximately one half-wavelength long. In order to obtain a significant amount of radiation the antenna must be brought to a resonance and the shorter the slot the more narrow the antenna bandwidth will be. Thus when the slot is short as in Case a) we definitely have a narrow band antenna. In our further discussion we will refer to this case as the "narrow band" antenna, and the Case b) as the narrow band" antenna with the plasma sheath. If we make the cavity dielectric lossy, we have Case c). The lossy cavity dielectric decreases the antenna Q and hence broadbands the antenna to some extent, causing a loss, of course, in the radiation efficiency, The Case c) we will denote in our discussion as the "broad band" antenna. And Case d) as the "broad band" antenna with the plasma sheath. We have represented the equivalent antenna impedance in terms of the fields in the appropriate regions. Each region is characterized by a resistance and a reactance. For the total effect they add in series. Now we want to discuss the power radiated in each case and make some comparisons. It is significant to discuss the antenna perrmance taking into account the internal impedance of the source. We select the oltage generator as the source and we consider a circuit as shown in Figure 4-3. s1 z Vgrt 9 Ak~~~~~9

THE UNIVERSITY OF MICHIGAN 5825-1 -F An ideal transmission line of length s, phase constant 13 and a characreristic impedance Z 0connects the antenna to the equivalent voltage generator which represents the transmitter. The length of the transmission line is from the reference plane A to the transmitter terminals. The transmitter is specified by voltage V and internal impedance Z. The current I through the impedance Z is given by V 1= 9( + )[Cos(13s) -iz sin (13s)] (4. 33) 0 g 5 where ~~ ~ Z+j Z tan(/3s) s o Z +jiZ tan(13s) 434 0 The above formulas result from elementary transmission line theory. The radiated power of the antenna is given by 1 2 P r I=2 R (4. 35) The maximum power will be extracted from the generator and transferred by the transmission line to the load when the load impedance as seen at the generator terminals is equal to the conjugate of the generator impedance., a well known condition, i.e. z =Z (4. 36) 5 g Furthermore, when the generator is not matched to the line, the reflection

THE UNIVERSITY OF MICHIGAN 5825-1 -F antenna with the antenna impedance changes. This is because the reflection coefficient is simply the reflected wave amplitude divided by the incident wave amplitude, and the incident wave amplitude changes as the load changes when the generator is not matched to the line, i.e., Z9 $ Z0. A good way to show that the generator is matched to the line is to monitor with a directional coupler the incident power. If it remains constant as the load is changed, then the generator is indeed matched to the line, and the reflection coefficient changes permit one to obtain a measure of. changes in the power absorbed by the antenna. In the subsequent discussion we will assume that the generator is matched to the transmission line. The main question before us is the experimental observation that the plasma sheath impeded the radiation field considerably more in the case of the "narrowband" antenna than in the "broad-band" antenna case. Even in spite of the fact that the narrow band antenna is a more efficient radiator when in resonance than the "broad band" antenna, the former gave a weaker far-zone field with the plasma present. This observation is very difficult to explain. On the basis of the impedance representation we can readily explain the antenna de-tuning when the plasma sheath appears in the "narrow band" antenna case. The sheath destroys the resonance condition and this introduces an appreciable reactance in the antenna impedance which decreases the antenna current and hence the radiation power. Part of the power taken by the antenna is dissipated in the plasma resistance and the remainder

THE UNIVERSITY OF MICHIGAN 5825-1 -F In case of the "broad band"? antenna similar comments apply, with the exception that the cavity is now lossy, and in addition to the plasma resistance and radiation resistance we add cavity resistance., Also we are told the resonance experimentally occured under different conditions. It was necessary to introduce a stub tuner in the transmission line between the reference plane A and the co-ax opening in order to obtain a better match between the antenna and the co-ax. In fact this creates an energy storage volume that no longer can be neglected in the over-all balance. One must consider the fact that this may change the resonance pattern, and either just reduce the reactive fields outside in the vicinity of the slot without unduly decreasing the slot field itself, or it may cause outside reactive energy to shift more into the magnetic field. The appearance of the plasma sheath under these conditions is expected to decrease the radiation field to a lesser degree than in the "?narrow band?? antenna case. The experimental geometry of the problem makes it impossible to consider an exact theoretical solution. We are left with considering a resonant structure where a solution is possible and to study the outside reactive fields and their control by changes in the inside energy storages with a view of making the above arguments more definite and unique than they are now. One of the main questions to be answered is: to what extent the energy storage outside is coupled to the inside energy storages, and to what extent it can be controlled by inside tuning? Can the

THE UNIVERSITY OF MICHIGAN 582 5-1 -F large reactive fields outside? In these areas there is very little theoretical information available. It must be added that for a resonant antenna of the type we are discussing the "fplane wave" approximation to predict the loss through the plasma sheath does appear to be very poor indeed. This type of approximation is expected to hold much better for apertures that are comparable with wavelength in both dimensions, such as an open waveguide radiator. One must also be aware of the fact that the plasma sheath not only impedes the transfer of power into the radiation field, but also changes the amplitude of the slot field (usually decreasing it) and thus compounding the influence of the plasma sheath. The real antenna problem is not equivalent to some prescribed source on the antenna surface, because in reality the true source is all the way back in the electronic package. The slot field can be regarded as an "equivalent" source for the field outside the antenna. However,, it lacks the requirements of a true source: the energy in the "equivalent source" is finite, while in a true source it must be infinite. REFERENCES Landau,, L. D. and Lifshitz, E. M., (1960), Electrodynamics of Continuous Media, Pergamon Press,, New York, p. 254.

THE UNIVERSITY OF MICHIGAN 5825-1 -F V. FUTURE WORK Some obvious extensions of the theoretical work accomplished so far include consideration of a dielectric under the shell that is different from that on the outside and the application of the method of Chapter II to the case of the narrow slot. On the basis of this study one would then be in a position to choose a dielectric constant ratio such that for a narrow frequency band of interest the cylinder antenna would radiate through the slotted shell for all angular positions of the shell slot. Another extension is to consider a finite thickness, partially transparent plasma sheath with an infinite axial slot. The radiation then would depend not only on the energy that leaks out through the slot, but also on the leakage through the plasma sheath itself. This is a new boundary value problem and it is expected to be somewhat more difficult than the one discussed in Chapters II and III. Another problem of some potential interest is to consider that the cylinder slot is excited by a wedge waveguide which has a magnetic line source at the origin. This in fact is a complete antenna problem. In order to find the slot fields of the cylinder slot and the sheath slot we have to solve two simultaneous integral equations. The solution of this problem would also show the extent to which the shell depresses the cylinder slot field. The further problem of relating these solutions to practical antenna configurations is also of continuing importance. For this, physical intuition and understanding of the canonical problem may not be sufficient, and some judicious experiments may