1.1 Introduction This chapter is concerned primarily with establishing formulas for the electromagnetic field vectors E and H in terms of all the sources causing these radiating fields, but at points far removed from the sources. The collection of sources is called an antenna and the formulas to be derived form the basis for what is generally referred to as antenna pattern analysis and synthesis. A natural division into two types of antennas will emerge as the analysis develops. There are radiators, such as dipoles and helices, on which the current dis- tribution can be hypothesized with good accuracy; for these, one set of formulas will prove useful. But there are other radiators, such as slots and horns, for which an estimation of the actual current distribution is exceedingly difficult, but for which the close-in fields can be described quite accurately. In such cases it is possible to replace the actual sources, for purposes of field calculation, with equivalent sources that properly terminate the close-in fields. This procedure leads to an alternate set of formulas, useful for antennas of this type. The chapter begins with a brief review of relevant electromagnetic theory, including an inductive establishment of the retarded potential functions. This is fol- lowed by a rigorous derivation of the Stratton-Chu integrals (based on a vector Green's theorem), which give the fields at any point within a volume Kin terms of the sources within Fand the field values on the surfaces S that bound V. This formulation possesses the virtue that it applies to either type of antenna, or to a hybrid mix of the two. Simplifications due to the remoteness of the field point from the antenna will lead to compact integral formulas, from which all the pattern characteristics of the different types of antennas can be deduced. A general derivation of the reciprocity theorem is presented; the result is used to demonstrate that the transmitting and receiving patterns of an antenna are identical. The concept of directivity of a radiation pattern is introduced and a connection is estab- 3 4 The Far-Field Integrals, Reciprocity, Directivity lished between the receiving cross section of an antenna and its directivity when trans- mitting. The chapter concludes with a discussion of the polarization of an antenna pattern. A. REVIEW OF RELEVANT ELECTROMAGNETIC THEORY1 It will generally be assumed that the reader of this text is already familiar with elec- tromagnetic theory at the intermediate level and possesses a knowledge of basic trans- mission line analysis (including the use of Smith charts) and of waveguide modal representations. What follows in the next several sections is a brief review of the pertinent field theory, primarily for the purposes of introducing the notation that will be adopted and highlighting some useful analogies.2 Throughout this text MKS rationalized units are used; the dimensions of the various source and field quantities introduced in the review are listed on the inside of the front cover. 1.2 Electrostatics and Magnetostatics in Free Space A time-independent charge distribution p(x,p(x,y,z) y, z) (1.1a(I-la) expressed in couloumbs per cubic meter, placed in what is otherwise free space, gives rise to an electrostatic field E(x, y9 z). Similarly, a time-independent current distribu- tion J(x,y,z)J(x, y, z) (Lib(Lib) expressed in amperes per square meter, produces a magnetostatic field B(x, y, z). To heighten the analogies between electrostatics and magnetostatics, it is sometimes useful to refer to the "reduced" source distributions P(x,p(x, y, z) J(x, y, z) ((1.2 2)) ?o Vol l in which eQ is the permittivity of free space and fa is the reciprocal of the perme- ability of free space. Coulomb's law can be introduced as the experimental postulate for electrostatics and described by the equations 2The reader who prefers to omit this review should begin with Section 1.7. 2The pairing of B with E (and thus of H with D), the use of //Q l, the introduction of reduced sources, and the parallel numbering of the early equations in this review all serve to emphasize the analogies that occur between electrostatics and magnetostatics. This is done in the belief that percep- tion of these analogies adds significantly to one's comprehension of the subject. See R. S. Elliott, "Some Useful Analogies in the Teaching of Electromagnetic Theory," IEEE Trans, on Education, E-22 (1979), 7-10. Reprinted with permission. 1.2 Electrostatics and Magnetostatics in Free Space 5 F = qEqE (1.3a) • P(Z,t1,OKdV E(x,y,E(x,y,z) z) == y 47I€QR3 3 (1.4a)) JV 4ne0R in which R is the directed distance from the source point (£, //, £) to the field point (x, y, z), and F is the force on a charge q placed at (x, y, z), due to its interaction with the source system /?(£, rj, Q. Similarly, the Biot-Savart law can be introduced as the experimental postulate for magnetostatics and is represented by the equations F = qYXBqv X B (1.3b(1.3b) f'J(£,n,OxRdV J(£,//,0 XKdV B(x{X ,y y, Z)z) == l 3 ((1.4bl Ab)) ' > IJV 4jiMoAnp^R*R One can show by performing the indicated vector operations on (1.4a) that VVXEEE X E =O 0 (1.5a(1.5a) ) V-V.E = -p^£ (1.5b) fo In like manner, the curl and divergence of (1.4b) yield J VxB--V X B = i 1 (1.5c) Mo1 V.BB = 00 (1.5d) Equations 1.5 are Maxwell's equations for static fields. Integration of (1.5b) and use of the divergence theorem gives Gauss' law, that is, E • dS = [T~)dV£ = total reduced charge enclosed (1.6a) tfo. s 1(£'V ) Similarly, integration of (1.5c) and use of Stokes' theorem yields Ampere's circuital law: B • dl = \~TT) • dS = total reduced current enclosed (1.6b) c In like manner, integration of (1.5a) and (1.5d), followed by the application of Stokes' theorem or the divergence theorem results in the following relations. j> E-dlssOE-rf/=0 (1.7a) c | B -• dSdS~0 = 0 (1.7b)) J ss 6 The Far-Field Integrals, Reciprocity, Directivity From (1.7a) it can be concluded that E (x, y, z) is a conservative field and that <f E • dl between any two points is independent of the path. Equation 1.7b permits the con- clusion that the flux lines of B are everywhere continuous. Equation 1.4a can be manipulated into the form E--VE- O-V O (1.8a) in which O(x, y, z) = p(Z,n,Odv (1.9a) ly 4ne0R is the electrostatic potential function. In like manner, Equation 1.4b can be rewritten in the form BB-VX = V XA A (1.8b) where mjkQdv A(x,y,z)-^A(x, y, z) = 4n/iolRl ((1.9b }) IV 4njHo R is the magnetostatic vector potential function. One can see that the reduced sources (1.2) play analogous roles in the integrands of the potential functions (1.8a) and (1.8b), as well as in the integrands of the field functions (1.4a) and (1.4b). There is no compelling reason to introduce either D or H until a discussion of dielectric and magnetic materials is undertaken, but if one wishes to do it at this earlier stage, where only primary sources in what is otherwise free space are being assumed, then it is suggestive to write Do = 660E (1.10a) l Ho - fa B (1.10b) with the subscripts on D and H denoting that the medium is free space. Then it fol- lows logically from (1.5) that V .• Do = />? VxHV X H0o = JJ (1.11) and from (1.6) that <p Do • dS = \ p dV = total charge encloseenclosed (1.12a) s Jy | Hoo •• dldl == \f JJ •• dSdS == totatotall currencurrentt enclosed (1.12b(1.12b)) c s Equations 1.12 are the forms in which one is more apt to find Gauss' law and Ampere's circuital law expressed. It is apparent from (1.12) that Do and Ho play analogous roles in the two laws. 1.3 The Introduction of Dielectric, Magnetic, and Conductive Materials 7 When flux maps are introduced, (1.12a) leads to the conclusion that the lines of Do start on positive charge and end on negative charge. If one chooses to defer the introduction of D and H until materials are present, a flux map interpretation of (1.6a) includes the idea that the lines of E start on reduced positive charge and end on reduced negative charge. It has already been noted in connection with equation (1.7b) that the flux lines of B are continuous. Since Ho differs from B only by a multiplicative constant, the flux lines of Ho are also continuous. 1.3 The Introduction of Dielectric, Magnetic, and Conductive Materials The electrostatic behavior of dielectric materials can be explained quite satisfactorily by imagining the dielectric to be composed of many dipole moments of the type p = lnqd, in which q is the positive charge of the oppositely charged pair, d is their separation, and ln is a unit vector drawn from — q to +q. If P(x, y9 z) is the volume density of these elementary dipole moments, one can show3 that their aggregated effect is to cause an electrostatic field given by ' pP . dSs , t(-V ^P)dV' E(x,y,z)=-V. n -^ 4. r (-Vj-p)<*rs i ((1.13a) sAneQR Jv 4neQK with S the dielectric surface and Fits volume.