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- 1. THEORY OFELECTROMAGNETICWAVE PROPAGATION CHARLES HERACH PAPAS PROFESSOR OF ELECTRICAL ENGINEERING CALIFORNIA INSTITUTE OF TECHNOLOGYDOVER PUBLICATIONS, INC., NEW YORK
- 2. Copyright @ 1965,1988 by Charles Herach Papas. All rights reserved under Pan American and InternationalCopyright Conventions. Published in Canada by General Publishing Company, Ltd., 30Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company,Ltd., 10 Orange Street, London WC2H 7EG. This Dover edition, first published in 1988, is an unabridged andcorrected republication of the work first published by the McGraw-Hill Book Company, New York, 1965, in its Physical and QuantumElectronics Series. For this Dover edition, the author has written anew preface. Manufactured in the United States of America Dover Publications, Inc., 31 East2nd Street, Mineola, N.Y. 11501 Library of Congress Cataloging-in-Publication DataPapas, Charles Herach. Theory of electromagnetic wave propagation / Charles Herach Papas. p. em. Reprint. Originally published: New York : McGraw-Hill, cl965. (McGraw-Hill physical and quantum electronics series) With new pref. Includes index. ISBN 0-486-65678-0 (pbk.) 1. Electromagnetic waves.~. Title. QC661.P29 1988 "i530.141-dcI9 88-12291 CIP
- 3. To RONOLDWYETH PERCIVAL KINGGordon McKay Professor of Appl1e~ Physics, Harvard University Outstanding Scientist, Inspiring Teacher, and Dear Friend """, ," .~
- 4. PrefaceThis book represents the substance of a course of lectures Igave during the winter of 1964 at the California Institute ofTechnology. In these lectures I expounded a number ofnewly important topics in the theory of electromagnetic wavepropagation and antennas, with the purpose of presenting acoherent account of the subject in a way that would revealthe inherent simplicity of the basic ideas and would place inevidence their logical development from the Maxwell fieldequations. So enthusiastically were the lectures receivedthat I was encouraged to put them into book form and thusmake them available to a wider audience. The scope of the book is as follows: Chapter 1 provides thereader with a brief introduction to Maxwells field equationsand those parts of electromagnetic field theory which he willneed to understand the rest of the book. Chapter 2 presentsthe dyadic Greens function and shows how it can be used tocompute the radiation from monochromatic sources. InChapter 3 the problem of radiation emitted by wire antennasand by antenna arrays is treated from the viewpoint of anal-ysis and synthesis. In Chapter 4 two methods of expandinga radiation field in multipoles are given, one based on theTaylor expansion of the Helmholtz integrals and the otheron an expansion in spherical waves. Chapter 5 deals with the wave aspects of radio-astronomical antenna theory and explains the Poincare sphere, the Stokes parameters, coher- ency matrices, the reception of partially polarized radiation, the two-element radio interferometer, and the correlation coefficients in interferometry. Chapter 6 gives the theory of electromagnetic wave propagation in a plasma medium and describes, with the aid of the dyadic Greens function, the behavior of an antenna immersed in such a medium. Chapter 7 is concerned with the covariance of Maxwells vii
- 5. Prefaceequations in material media and its application to phenomena such asthe Doppler effect and aberration in dispersive media. The approach of the book is theoretical in the sense that the subjectmatter is developed step by step from the Maxwell field equations.The advantage of such an approach is that it tends to unify the varioustopics under the single mantle of electromagnetic theory and servesthe didactic purpose of making the contents of the book easy to learnand convenient to teach. The text contains many results that can be found only in the research literature of the Caltech AntennaLaboratory and similar laboratories in the U.S.A., the U.S.S.R., and Europe. Accordingly, the book can be used as a graduate-level text- book or a manual of self-instruction for researchers. My grateful thanks are due to Professor W. R. Smythe of the Cali- fornia Institute of Technology, Professor Z. A. Kaprielian of the University of Southern California, and Dr. K. S. H. Lee of the Cali- fornia Institute of Technology for their advice, encouragement, and generous help. I also wish to thank Mrs. Ruth Stratton for her unstinting aid in the preparation of the entire typescript. Charles Herach Papas Preface to the Dover EditionExcept for the correction of minor errors and misprints, this edition ofthe book is an unchanged reproduction of the original. My thanks are due to my graduate students, past and present, for thevigilance they exercised in the compilation of the list of corrections, andto Dover Publications for making the book readily available once again. Charles Herach Papasviii
- 6. ContentsPreface viiPreface to the Dover Edition viii 1 The electromagnetic field 11.1 Maxwells Equations in Simple Media 11.2 Duality 61.3 Boundary Conditions 81.4 The Field Potentials and Antipotentials 91.5 Energy Relations 14 2 Radiation from monochromatic sources in unbounded regions 192.1 The Helmholtz Integrals 192.2 Free-space Dyadic Greens Function 262.3 Radiated Power 29 3 Radiation from wire antennas 373.1 Simple Waves of Current 373.2 Radiation from Center-driven Antennas 423.3 Radiation Due to Traveling Waves of Current, Cerenkov Radiation 453.4 Integral Relations between Antenna Current and Radiation Pattern 483.5 Pattern Synthesis by Hermite Polynomials 503.6 General Remarks on Linear Arrays 563.7 Directivity Gain 73
- 7. 4 Multipole expansion of the radiated field 814.1 Dipole and Quadrupole Moments 814.2 Taylor Expansion of Potentials 864.3 Dipole and Quadrupole Radiation 894.4 Expansion of Radiation Field in Spherical Waves 97 5 Radio-astronomical antennas 1095.1 Spectral Flux Density 1115.2 Spectral Intensity, Brightness, Brightness Temperature, Apparent Disk Temperature 1155.3 Poincare Sphere, Stokes Parameters 1185.4 Coherency Matrices 1345.5 Reception of Partially Polarized Waves 1405.6 Antenna Temperature and Integral Equation for Brightness Temperature 1485.7 Elementary Theory of the Two-element Radio Interferometer 1515.8 Correlation Interferometer 159 6 Electromagnetic waves in a plasma 1696.1 Alternative Descriptions of Continuous Media 1706.2 Constitutive Parameters of a Plasma 1756.3 Energy Density in Dispersive Media 1786.4 Propagation of Transverse Waves in Homogeneous Isotropic Plasma 1836.5 Dielectric Tensor of Magnetically Biased Plasma 1876.6 Plane Wave in Magnetically Biased Plasma 1956.7 Antenna Radiation in Isotropic Plasma 2056.8 Dipole Radiation in Anisotropic Plasma 2096.9 Reciprocity 212
- 8. 7 The Doppler effect 2177.1 Covariance of Maxwells Equations 2187.2 Phase Invariance and Wave 4-vector 2237.3 Doppler Effect and Aberration 2257.4 Doppler Effect in Homogeneous Dispersive Media 2277.5 Index of Refraetion of a Moving Homogeneous Medium 2307.6 Wave Equation for Moving Homogeneous Isotropie Media 233 Index 24i
- 9. The electromagnetic 1 fieldIn this introductory chapter some basic relations and con-cepts of the classic electromagnetic field are briefly reviewedfor the sake of easy reference and to make clear the signifi-cance of the symbols. 1.1 Maxwells Equations in Simple MediaIn the mks, or Giorgi, system of units, which we shall usethroughout this book, Maxwells field equationsl are a (1)v x E(r,t) = - iii B(r,t) avx R(r,t) = J(r,t) + iii D(r,t) (2)V. B(r,t) = 0 (3)v . D(r,t) = p(r,t) (4)where E(r,t) = electric field intensity vector, volts per meter R(r,t) = magnetic field intensity vector, ainperes per meter 1 See, for example, J. A. Stratton, "Electromagnetic Theory,"chap. 1, McGraw-Hill Book Company, New York, 1941. 1
- 10. Theory of electromagnetic wave propagation D(r,t) = electric displacement vector, coulombs per meter2 B(r,t) = magnetic induction vector, webers per meter2 J(r,t) = current-density vector, amperes per meter2 p(r,t) = volume density of charge, coulombs per meter3 r = position vector, meters t = time, secondsThe equation of continuityV • J(r,t) = - a at p(r,t) (5)which expresses the conservation of charge is a corollary of Eq. (4) andthe divergence of Eq. (2). The quantities E(r,t) and B(r,t) are defined in a given frame ofreference by the density of force f(r,t) in newtons per meter3 acting onthe charge and current density in accord with the Lorentz forceequationf(r,t) = p(r,t)E(r,t) + J(r,t) X B(r,t) (6)In turn D(r,t) and H(r,t) are related respectively to E(r,t) and B(r,t) byconstitutive parameters which characterize the electromagnetic natureof the material medium involved. For a homogeneous isotropiclinear medium, viz., a "simple" medium, the constitutive relations areD(r,t) = EE(r,t) (7) 1H(r,t) = - B(r,t) (8) IJ.where the constitutive parameters E in farads per meter and IJ. in henrysper meter are respectively the dielectric constant and the permeabilityof the medium. In simple media, Maxwells equations reduce toV X E(r,t) = - J.L a at H(r,t) (9)V X H(r,t) = J(r,t) + a at E(r,t) E (10)2
- 11. The electromagnetic fieldV. H(r,t) = 0 (11)V • E(r,t) = ~ p(r,t) (12) EThe curl of Eq. (9) taken simultaneously with Eq. (10) leads to 02 0V X V X E(r,t) + ILE ot2 E(r,t) = -IL at J(r,t) (13)Alternatively, the curl of Eq. (10) with the aid of Eq. (9) yields 02V X V X H(r,t) + ILE i)t2 H(r,t) = V X J(r,t) (14)The vector wave equations (13) and (14) serve to determine E(r,t) andH(r,t) respectively when the source quantity J(r,t) is specified andwhen the field quantities are required to satisfy certain prescribedboundary and radiation conditions. Thus it is seen that in the case ofsimple media, Maxwells equations determine the electromagneticfield when the current density J(r,t) is a given quantity. Moreover,this is true for any linear medium, i.e., any medium for which therelations connecting B(r,t) to H(r,t) and D(r,t) to E(r,t) are linear, beit anisotropic, inhomogeneous, or both. To form a complete field theory an additional relation connectingJ(r,t) to the field quantities is necessary. If J(r,t) is purely an ohmicconduction current in a medium of conductivity u in mhos per meter,then Ohms lawJ(r,t) = uE(r,t) (Hi) applies and provides the necessary relation. On the other hand, if J(r,t) is purely a convection current density, given by J(r,t) = p(r,t)v(r,t) (16) where v(r,t) is the velocity of the charge density in meters per second, the necessary relation is one that connects the velocity with the field. To find such a connection in the case where the convection current is made up of charge carriers in motion (discrete case), we must calculate 3
- 12. Theory of electromagnetic wave propagationthe total force F(r,t) acting on a charge carrier by first integrating theforce density f(r,t) throughout the volume occupied by the carrier, i.e.,F(r,t) = ff(r + r,t)dV = q[E(r,t) + v(r,t) X B(r,t)J (17)where q is the total charge, and then equating this force to the force ofinertia in accord with Newtons law of motion dF(r,t) = dt [mv(r,t)] (18)where m is the mass of the charge carrier in kilograms. In the casewhere the convection current is a charged fluid in motion (continuouscase), the force density f(r,t) is entered directly into the equation ofmotion of the fluid. Because Maxwells equations in simple media form a linear system,no generality is lost by considering the "monochromatic" or "steady"state, in which all quantities are simply periodic in time. Indeed, byFouriers theorem, any linear field of arbitrary time dependence can besynthesized from a knowledge of the monochromatic field. To reducethe system to the monochromatic state we choose exp (- iwt) for thetime dependence and adopt the conventionG(r,t) = Re {G",(r)e-i",t} (19)where G(r,t) is any real function of space and time, G",(r) is the con-comitant complex function of position {sometimes called a "phasor"),which depends parametrically on the frequency f( = w/27r) in cyclesper second, and Re is shorthand for "real part of." Application of thisconvention to the quantities entering the field equations (1) through(4) yields the monochromatic form of Maxwells equations:v X E",(r) = iwB",(r) (20)V X H.,(r) = J",(r) - i~D",(r) (21)V • B",(r) = 0 (22)V. D",(r) = p",(r) (23)4
- 13. The electromagnetic fieldIn a similar manner the monochromatic form of the equation ofcontinuityV • J",(r) = iwp",(r) (24)is derived from Eq. (5). The divergence of Eq. (20) yields Eq. (22), and the divergence ofEq. (21) in conjunction with Eq. (24) leads to Eq. (23). We inferfrom this that of the four monochromatic ~Iaxwell equations only thetwo curl relations are independent. Since there are only two inde-pendent vectorial equations, viz., Eqs. (20) and (21), for the deter-mination of the five vectorial quantities E",(r), H",(r), D",(r), B",(r) ,and J",(r) , the monochromatic Maxwell equations form an under-determined system of first-order differential equations. If the system isto be made determinate, linear constitutive relations involving the con-stitutive parameters must be invoked. One way of doing this is first toassume that in a given medium the linear relations B",(r) = aH",(r),D",(r) = iSE",(r), and J",(r) = YE",(r) are valid, then to note that withthis assumption the system is determinate and possesses solutionsinvolving the unknown constants a, is, and Y, and finally to choose the values of these constants so that the mathematical solutions agree with the observations of experiment. These appropriately chosen values are said to be the monochromatic permeability /-l"" dielectric constant E"" and conductivity u" of the medium. Another way of defining the constitutive parameters is to resort to the microscopic point of view, according to which the entire system consists of free and bound charges interacting with the two vector fields E",(r) and B",(r) only. For simple media the constitutive relations are B",(r) = /-l",H",(r) (25) D",(r) = E",E",(r) (26) J",(r) = u",E.,(r) (27) In media showing microscopicinertial or relaxation effects, one or more of these parameters may be complex frequency-dependent quantities. For the sake of notational simplicity, in most of what follows we shall drop the subscriptw and omit the argument r in the mono- 5
- 14. Theory of electromagnetic wave propagationchromatic case, and we shall suppress the argument r in the time-dependent case. For example, E(t) will mean E(r,t) and E will meanEw(r). Accordingly, the monochromatic form of Maxwells equationsin simple media isv X E = iW,llH (28)VXH = J - iweE (29)V.H=O (30) 1V.E=-p (31) e1.2 DualityIn a region free of current (J = 0), Maxwells equations possess acertain duality in E and H. By this we mean that if two new vectorsE and H are defined by and H = + iE (32)then as a consequence of Maxwells equations (source-free)VXH = -iweE V X E = iW,llH (33)V.H=O V.E=Oit follows that E and H likewise satisfy Maxwells equations (source-free) V X H = -iweE (34)V. E = 0 V. H = 0and thereby constitute an electromagnetic field E, H which is the"dual" of the original field. This duality can be extended to regions containing current byemploying the mathematical artifice of magnetic charge and magnetic6
- 15. The electromagnetic fieldcurrent.l In such regions Maxwells equations areVXH=J-iwEE v XE = iWIlH (35) 1V.H=O V.E=-p Eand under the transformation (32) they becomev X E = :t ~ J + iWIlH V X H = -iWEE (36)V. E = 0 V • H = :t!j; p Il ~Formally these relations are Maxwells equations for an electro-magnetic field E, H produced by the "magnetic current" +: vi III E Jand the "magnetic charge" :t vi p.1 E p. These considerations suggestthat complete duality is achieved by generalizing Maxwells equationsas follows:V X H = J - iWEE. V X E = -Jm + iwp.H (37) 1 V.E=-p Ewhere Jm and Pm are the magnetic current and charge densities.Indeed, under the duality transformationE = :t~H H = +: ~~E J = :t ~~ Jm (38)J;" = +: ~~J p = :t ~~ Pm p;" = -~ ;P +V X E = -J;" + iWIlH V X H = J - iWEE (39)V. E = !p V. H 1, =-p p. m E 1 See, for example, S. A. Schelkunoff, "Electromagnetic Waves," chap. 4,D. Van Nostrand Company, Inc., Princeton, N.J., 1943. 7
- 16. Theory of electromagnetic wave propagationThus to every electromagnetic field E, H produced by electric current Jthere is a dual field H, E produced by a fictive magnetic current J~.1.3 Boundary ConditionsThe electromagnetic field at a point on one side of a smooth interfacebetween two simple media, 1 and 2, is related to the field at the neigh-boring point on the opposite side of the interface by boundary condi-tions which are direct consequences of Maxwells equations. We denote by n a unit vector which is normal to the interface anddirected from medium 1 into medium 2, and we distinguish quantitiesin medium 1 from those in medium 2 by labeling them with the sub-scripts 1 and 2 respectively. From an application of Gauss divergencetheorem to Maxwells divergence equations, V. B = Pm and V. D = P,it follows that the normal components of Band D are respectively dis-continuous by an amount equal to the magnetic surface-charge density7/m and the electric surface-charge density 7/ in coulombs per meter2: (40)From an application of Stokes theorem to Maxwells curl equations,V X E = -Jm + iWJ.lH and V X H = J - iweE, it follows that thetangential components of E and H are respectively discontinuous byan amount equal to the magnetic surface-current density Km and theelectric surface-current density K in amperes per meter: (41)In these relations Km and K are magnetic and electric "current sheets"carrying charge densities 7/m and 7/ respectively. Such current sheetsare mathematical abstractions which can be simulated by limitingforms of electromagnetic objects. For example, if medium 1 is aperfect conductor and medium 2 a perfect dielectric, Le., if 0"1 = 00and 0"2 = 0, then all the field vectors in medium I as well as 7/m and Kmvanish identically and the boundary conditions reduce to (42)s
- 17. The electromagnetic fieldA surface having these boundary conditions is said to be an "electricwall." By duality a surface displaying the boundary conditions n Bz = 7]m n X H2 = 0 (43)is said to be a "magnetic wall." At sharp edges the field vectors may become infinite. However, theorder of this singularity is restricted by the Bouwkamp-Meixner1 edgecondition. According to this condition, the energy density must beintegrable over any finite domain even if this domain happens toinclude field singularities, i.e., the energy in any finite region of spacemust be finite. For example, when applied to a perfectly conductingsharp edge, this condition states that the singular components of theelectric and magnetic vectors are of the order o-~, where /) is the dis-tance from the edge, whereas the parallel components are alwaysfinite. 1.4 The Field Potentials and AntipotentialsAccording to Helmholtzs partition theorem2 any well-behaved vectorfield can be split into an irrotational part and a solenoidal part, or,equivalently, a vector field is determined by a knowledge of its curl anddivergence. To partition an electromagnetic field generated by a cur-rent J and a charge p, we recall Maxwells equationsV X H = J - iwD (44)V X E = iwB (45) 1 C. Bouwkamp, Physica, 12: 467 (1946); J. Meixner, Ann. Phys., (6) 6: 1(1949). 2 H. von Helmholtz, Uber Integrale der hydrodynamischen Gleichungen,welche den Wirbelbewegungen entsprechen, Crelles J., 55: 25 (1858). Thistheorem was proved earlier in less complete form by G. B. Stokes in his paperOn the Dynamical Theory of Diffraction, Trans. Cambridge Phil. Soc., 9: 1(1849). For a mathematically rigorous proof, see O. Blumenthal, Uber dieZerlegung unendlicher Vektorfelder, Math. Ann., 61: 235 (1905). 9
- 18. Theory of electromagnetic wave propagationV.D=p (46)V.B=O (47)and the constitutive relations for a simple mediumD = eE (48)B = ,uH (49) From the solenoidal nature of B, which is displayed by Eq. (47), itfollowsthat B is derivable from a magnetic vector potential A:B=VxA (50)This relation involves only the curl of A and leaves free the divergenceof A. That is, V . A is not restricted and may be chosen arbitrarily tosuit the needs of calculation. Inserting Eq. (50) into Eq. (45) we seethat E - iwA is irrotational and hence derivable from a scalar electricpotential et>:E = -Vet> + iwA (51)This expression does not necessarily constitute a complete partition ofthe electric field because A itself may possess both irrotational andsolenoidal parts. Only when A is purely solenoidal is the electricfield completely partitioned into an irrotational part Vet> and a sole-noidal part A. The magnetic field need not be partitioned inten-tionally because it is always purely solenoidal. By virtue of their form, expressions (50) and (51) satisfy the twoMaxwell equations (45) and (47). But in addition they must alsosatisfy the other two Maxwell equations, which, with the aid of theconstitutive relations (48) and (49), become!V,u X B = J- iweE and V.E = pie (52)When relations (50) and (51) are substituted into these equations, the10
- 19. The electromagnetic fieldfollowing simultaneous differential equations are obtained,l relating cPand A to the source quantities J and p:V2cP - iwV. A = -p/E (53)V2A +kA 2 = -jLJ + V(V. A - iWEjLcP) (54)where k2 = W2jLE. Here V . A is not yet specified and may be chosen tosuit our convenience. Clearly a prudent choice is one that uncouplesthe equations, Le., reduces the system to an equation involving cPalone and an equation involving A alone. Accordingly, we chooseV • A = iWEjLcP or V • A = O. If we choose the Lorentz gaugeV •A = iWEjLcP (55)then Eqs. (53) and (54) reduce to the Helmholtz equationsV2cP+ k2cP = - p/E (56)V2A + k A 2 = -jLJ (57)The Lorentz gauge is the conventional one, but in this gauge the elec-tric field is not completely partitioned. If complete partition isdesired, we must choose the Coulomb gauge2V.A = 0 (58) 1 Also the vector identity V X V X A = V(V. A) - V2A is used. ThequantityV2A is defined by the identity itself or by the formal operationV2A = L V2(eiAi), where the Ai i are the components of A and the ei are theunit base vectors of the coordinate system. The Laplacian V2 operates onnot only the Ai but also the ei In the special case of cartesian coordinates,the base vectors are constant; hence the Laplacian operates on only the Ai,that is, V2A = L eiV2Ai. See, for example, P. M. Morse and H. Feshbach, i"Methods of Theoretical Physics," part I, pp. 51-52, McGraw-Hill BookCompany, New York, 1953. 2See, for example, W. R. Smythe, "Static and Dynamic Electricity,"2d ed., p. 469, McGraw-Hill Book Company, New York, 1950. 11
- 20. Theory of electromagnetic wave propagationwhich reduces Eqs. (53) and (.54)to 1 (59)V2q, = - - P E (60)We note that Eq. (59) is Poissons equation and can be reduced nofurther. However, Eq. (60) may be simplified by partitioning J intoan irrotational part Ji and a solenoidal part J., and by noting thatthe irrotational part just cancels the term involving the gradient. Toshow this, J is split up as follows: J = Ji + J., where by definitionV X Ji = 0 and V. J. = O. Since Ji is irrotational, it is derivablefrom a scalar function !J;, viz., Ji = V!J;. The divergence of this rela-tion, V. Ji = V2!J;, when combined with the continuity equationV • J = V. (Ji + J.) =V. Ji = iwp, leads to V2!J;= iwp. A com-parison of this result with Eq. (59) shows that !J; = -iWEq, and henceJi = V!J; = -iWEVq,. From this expression it therefore follows that - IJoJi - iWEIJoVq, vanishes and consequently Eq. (60) reduces to (61)Thus we see that in this gauge, A is determined by the solenoidal partJ. of the current distribution and q, by its irrotational part Ji. Sinceq, satisfies Poissons equation, its spatial distribution resembles that ofan electrostatic potential and therefore contributes predominantly tothe near-zone electric field. It is like an electrostatic field only in itsspace dependence; its time dependence is harmonic. In regions free of current (J = 0) and charge (p = 0) we maysupplement the gauge V . A = 0 by taking q, == O. Then Eq. (53) istrivially satisfied and Eq. (54) reduces to the homogeneous Helmholtzequation (62)In this case the electromagnetic field is derived from the vector poten-tial A alone. Let us now partition the electromagnetic field generated by a mag-netic current Jm and a magnetic charge Pm. We recall that Maxwells12
- 21. The electromagnetic fieldequations for such a field areV X H = -iwD (63)V xE = -Jm +iwB (64)V.D=O (65)V.B=Pm (66)and, as before, the constitutive relations (48) and (49) are valid.From Eq. (65) it follows that D is solenoidal and hence derivable froman electric vector potential A.:D = -V X A. (67)In turn it follows from Eq. (63) that H - iwA. is irrotational and henceequal to - VcPm, where cPm is a magnetic scalar potential:H = -VcPm + iwA. (68)Substituting expressions (67) and (68) into Eqs. (64) and (66), we get,with the aid of the constitutive relations, the following differentialequations for A. and cPm:V2cPm- iwV • A. = - ~- Pm p. (69)V2A. + k A. 2 = -elm + V(V • A. - iwp.EcPm)If we choose the conventional gaugeV • A. = iwp.EcPm (70)then cPm and A. satisfy 1V2cPm + k cPm 2 = - - p. Pm (71)V2A. + 1c2A. = -EJm (72)In this gauge cPm and A. are called "antipotentials." Clearly we may 13
- 22. Theory of electromagnetic wave propagationalso choose the gauge V. Ae = 0 which leads to 1V2cf>m = - - Pm /L (73)where JmB is the solenoidal part of the magnetic current; this gaugeleads also to cf>m = 0 and (74)for regions where Jm = 0 and Pm = O. If the electromagnetic field is due to magnetic as well as electric cur-rents and charges, then the field for the conventional gauge is given interms of the potentials A, cf> and the antipotentials Ae, cf>m byE = - Vcf> + iwA - !V X Ae t (75~ B = V X A - /LVcf>m + iW/LAe (76) 1.5 Energy Relations The instantaneous electric and magnetic energy densities for a losslese medium are defined respectively by We = J E(t) . ft D(t)dt and Wm = J H(t) . ft B(t)dt (77) where E(t) stands for E(r,t), D(t) for D(r,t), etc. In the present instance these expressions reduce to We = ~tE(t) . E(t) and Wm = ~ILH(t) • H(t) (78) Both We and Wm are measured in joules per meter3• To transform these quadratic quantities into the monochromatic domain we recall 14
- 23. The electromagnetic fieldthatE(t) = Re {Ee-u.t} and H(t) = Re {He-i.,t} (79)where E is shorthand for E.,(r) and H for H.,(r). Since E can alwaysbe written as E = E1 + iE2, where E1 andE2 are respectively the realand imaginary parts of E, the first of Eqs. (79) is equivalent toE(t) = E1 cos wt + E2 sin wt (80)Inserting this representation into the first of Eqs. (78) we obtain (81)which, when averaged over a period, yields the time-average electricenergy densitywhere the bar denotes the time average. Sincewhere E* is the conjugate complex of E, we can express II. in the equiv-alent formWe = %eE. E* (83)By a similar procedure it follows from the second of Eqs. (78) and thesecond of Eqs. (79) that the time-average magnetic energy is given byWin = %J.tH.H* (84) The instantaneous Poynting vect~r S(t) is defined byS(t) = E(t) X H(t) (85)where S(t) stands for S(r,t) and is measured in watts per meter2• With 15
- 24. Theory of electromagnetic wave propagationthe aid of expressions (79), the time average of Eq. (85) leads to thefollowing expression for the complex Poynting vector:S = ~E X H* (86) If from the scalar product of H* and V X E = iWJLH the scalar prod-uct of E and V X H* = J* + iweE*(e is assumed to be real) is sub-tracted, and if use is made of the vector identityV . (E X H*) = H* .V X E - E . V X H*the following equation is obtained:V . (E X H*) = - J* . E + iW(JLH • H* - eE . E*) (87)which, with the aid of definitions (83), (84), and (86), yields the mono-chromatic form of Poyntings vector theorem! V. S = -~J*. E + 2iw(w m - 1V.) (88) The real part of this relation, i.e., V • (Re S) = Re ( - ~J* . E) (89) expresses the conservation of time-average power, the term on the right representing a source (when positive) or a sink (when negative) and cor- respondingly the one on the left an outflow (when positive) or an inflow (when negative). In Poyntings vector theorem (88) a term involving the difference wm - W. appears. To obtain an energy relation (for the monochro- matic state) which contains the sum Wm + lb. instead of the difference wm - W. we proceed as follows. From vector analysis we recall that. the quantity V . (oE ow X H* + E* X oH) ow (90) 1 F. Emde, Elektrotech. M aschinenbau, 27: 112 (1909). 16
- 25. The electromagnetic fieldis identically equal toH* • V X oE - oE . V X H* +~!:!V X E* . - E* . V X oH (91) ow ow ow owFrom Maxwells equations V X E = iWJLH and V X H = J - iwEE itfollows thatV X oH = ~ (V X H) = ~. (J - iWEE) = oj _ iEE _ iWEoE _ iwE OE ow ow ow ow ow ow and V X H* = J* + iWEE*Substituting these relations into expression (91) we obtain the desiredenergy relationV . (oE oW X H* + E* X ~!!) ow = i [o(WJL) H • H* ow + O(WE) E ow . E*J _ oE. 1* _ E* . oj (92) ow owwhich we call the "energy theorem." Here we interpret as the time-average electric and magnetic energy densities the quantities (93)which reduce respectively to expressions (83) and (84) when the mediumis nondispersive, i.e., when OE/OW = 0 and OJL/ow = o. 17
- 26. Radiation from monochromatic sources in unbounded regions 2 The problem of determining the electromagnetic field radi-ated by a given monochromatic source in a simple,unbounded medium is usually handled by first finding thepotentials of the source and then calculating the field froma knowledge of these potentials. However, this is not theonly method of determining the field. There is an alter-native method, that of the dyadic Greens function, whichyields the field directly in terms of the source current. Inthis chapter these two methods are discussed. 2.1 The Helmholtz IntegralsWe wish to find the vector potential A and the scalarpotential cP of a monochromatic current J, which is confinedto a region of finite spatial extent and completely surroundedby a simple, lossless, unbounded medium. For this pur-pose it is convenient to choose the Lorentz gaugeV • A = iWEfJ.cP (1)In this gauge, cP and A must satisfy the Helmholtz equations(see Sec. 1.4) 1 + k2cP(r)V2et>(r) = - - per) E (2)V2A(r) + k2A(r) = -fJ.J(r) (3) 19
- 27. Theory of electromagnetic wave propagationSince the medium is unbounded, q, and A must also satisfy the radiationcondition. In physical terms this means that q, and A in the far zonemust have the form of outwardly traveling spherical (but not neces-sarily isotropic) waves, the sphericity of the waves being a consequenceof the confinement of the sources p and J to a finite part of space. Let us first consider the problem of finding q,. We recall from thetheory of the scalar Helmholtz equation that q, is uniquely determinedby Eq. (2) and by the radiation condition 1limr....• " l (uq, - ikq,) ur = 0 (4)where r = (yr r) is the radial coordinate of a spherical coordinatesystem r, e, 1/;. To deduce from this radiation condition the explicitbehavior of q, on the sphere at infinity, we note that the scalar Helm-holtz equation is separable in spherical coordinates and then write q,in the separated form q,(r) = f(e,1/;) u(1), where f is a function of theangular coordinates and u is a function of l only. Clearly the radiationcondition (4) is satisfied by u(1) = (1/1) exp (ik1) and accordingly atgreat distances from the source the behavior of q, must be in accord with eikr (.5) lim q,(r) = f(e,1/;) - T-+OO r That is, the solution of Eq. (2) that we are seeking is the one that has the far-zone behavior (5). Since the scalar Helmholtz equation (2) is linear, we may write q, in the form2 q,(r) = ~ f p(r)G(r,r)dV (6) 1 This is Sommerfelds "Ausstrahlungbedingung"; see A. Sommerfeld, Die Greensche Funktion del Schwingungsgleichung, JahTesbericht d. D. Math. VeT., 21: 309 (1912). 2 From the point of view of the theory of differential equations, the solution of Eq. (2) consists of not only the particular integral (6) but also a comple- mentary solution. In the present instance, however, the radiation condition requires that the complementary solution vanish identically. 20
- 28. Monochromatic sources in unbounded regionswhere G(r,r) is a function of the coordinates of the observation pointr and of the source point r, and where the integration with respect tothe primed coordinates extends throughout the volume V occupied byp. The unknown function G is determined by making expression (6)satisfy Eq. (2) and condition (5). Substituting expression (6) into Eq.(2) we getf p(r)(V2 + lc )G(r,r)dV 2 = - p(r) (7)where the Laplacian operator operates with respect to the unprimedcoordinates only. Then with the aid of the Dirac 0 functionl whichpermits p to be represented as the volume integralp(r) = fp(r)o(r - r)dV (r in V) (8)we see that Eq. (7) can be written asf p(r)[(V2 + k2)G(r,r) + o(r - r)]dV = 0 (9)"From this it follows that G must satisfy the scalar Helmholtz equationV2G(r,r) + k2G(r,r) = - o(r - r) (10)Since G satisfies Eq. (2) with its source term replaced by a 0 function,G is said to be a Greens function2 of Eq. (2). The appropriate solution of Eq. (10) for r ~ r is eiklr-rlG(r,r) = a Ir-r I (11) 1 The 0 function has the following definitive properties: oCr - r) = 0 forr ~ r and = 00 for r = r; /.f(r)o(r - r)dV = fer) forr in V and = 0for r outside of V wherefis any well-behaved function. See P. A. M. Dirac,"The Principles of Quantum Mechanics," pp. 58-61, Oxford UniversityPress, London, 1947. See also L. Schwartz, TMorie des distributions,Actualites scientijiques et industrielles, 1091 and 1122, Hermann et Cie, Paris,1950-51. 2 See, for example, R. Courant and D. Hilbert, "Methods of MathematicalPhysics," vol. 1, pp. 351-388, Interscience Publishers, Inc., New York, 1953. 21
- 29. Theory of electromagnetic wave propagationwhere a is a constant. It becomes clear that this solution is compatiblewith the requirement that the form (6) satisfy condition (5) when werecall the geometric relationIr - rl = vr2 + r2 - 2r r = r VI + (r /r)2 - 2r . r/r2 (12)where r2 = r . rand r2 = r . r/, and from this relation find the limitingform eiklr-rl eikrlim G(r,r) = lim a Ir /1 ~ a- exp (-ikr . r/r) (13),-....+00 r~c:o - r rTo determine the constant a, expression (11) is substituted into Eq. (10)and the resulting equation is integrated throughout a small sphericalvolume centered on the point r = r". It turns out that a must beequal to 7i7l", and hence the Greens function is eiklr-rlG(r,r) = 4 7I"r - r 1 I (14)Therefore, since the form (6) satisfies Eq. (2) and condition (5) whenG is given by expression (14), the desired solution of Eq. (2) can bewritten as the Helmholtz integral 1 q,(r) = - J p(r) 4 eiklr-rl I 1 dV (15) E 7I"r - r Now the related problem of finding A can be easily handled. Clearly,the appropriate solution of Eq. (3) must be the Helmholtz integral eiklr-rlA(r) = p. J J(r) 471"Ir r/I dV _ (16) because it has the proper behavior on the sphere at infinity and it sat- isfies Eq. (3). To show that it satisfies Eq. (3), one only has to operate on Eq. (16) with the operator (V + k2) and note that J(r) depends on the primed coordinates alone and that the Greens function (14) obeys Eq. (10). When in addition to the electric current J there is a monochromatic magnetic current distribution Jm of nnite spatial extent, the antipoten- tials q,m and Ae should be iJ?-voked. The magnetic scalar potential cPm and the electric vector potential Ae satisfy the Helmholtz equations 22
- 30. Monochromatic sources in unhounded regions(see Sec. 1.4) (17) (18)where V • A. = iWP.E!Pm and V. Jm = iwPm. A procedure similar to theone we used in obtaining the Helmholtz integrals for !Pand A leads tothe following Helmholtz integrals for !Pm and A.:!PmCr) = p. - 1 f Pm(r) eiklr-rl I 4 111-1 I dV (19) eikjr-rlA.(r) = E f Jm(r) 41111 _ 1"1dV (20) From a knowledge of !P,A, !Pm, A. the radiated electric and magneticfields can be derived by use of the relations (see Sec. 1.4)E = - V!p + iwA - !V X A. (21) EH = !V X A p. - V!pm + iwA. (22)It is sometimes desirable to eliminate !P and !Pmfrom these relations andthereby express E and H in terms of A and A. only. This can be donewith the aid of t i-V!p = -V(V.A) and -V!pm = - V(V. A.) (23) WEP. WEP.which follow from the gradients of the Lorentz conditions V. A = iWEP.!Pand V • A. = iwp.!Pm Thus relations (21) and (22) may be written asfollows:E = iw [ A + b V(V • A) ] ....• ~ V X A. (24)H = ; V X A + iw [ A. + b V(V • A.) ] (25) 23
- 31. Theory of electromagnetic wave propagation To enable us to cast A + ~ V(V. A) and A. + k V(V. A.) into theform of an operator operating on A and A., we introduce the unit dyadicu and the double-gradient dyadic VV which in a cartesian system of coordinates are expressed by m,=3n=3U = I I m,=ln=l emenOmn (26) m=3n=3 a a (27)vv = L. L. emen ax: aXn m,=ln=lwhere Xi (i = 1, 2, 3) are thecartesian coordinates, ei (i = 1, 2, 3) arethe unit base vectors, and the symbol omn is the Kronecker delta, whichis 1 for m = nand 0 for m ;;e n. The properties of u and 7V that wewill need are u. C = C and (VV) . C = V(V. C), where C is any vectorfunction. These properties can be demonstrated by writing C in com-ponent form and then carrying out the calculation. Thus = LLL mn.p eme" . epCpOmn = L L L emOnpCpOmn mll.p = L epCp P = C (28)where e" . ep = Onp, andWith the aid of these results, relations (24) and (25) becomeE = iw (u + ~k- VV) . A - 1 V X € A. (30)24
- 32. Monochromatic sources in unbounded regions (31)Using the Helmholtz integrals (16) and (20) and taking the curl oper-ator and the operator u + iz VV nder the integral sign, we get u J( 1 ) [ eiklr-rI]E = iwp, u + k2 VV • J(r) 41l"lr _ rl dV eiklr-rl ] - J V X [ Jm(r) 41l"jr _ rl dV (32) eiklr-rI]H = J V X [ J(r) 41l"lr _ rl dV + iWE J ( U 1 + k2 VV ) • [ eiklr-rl ] Jm(r) 41l"tr _ rl dVTo reduce these expressions we invoke the following considerations.If a is a vector function of the primed coordinates only and w is a scalarfunction of the primed and unprimed coordinates, then = (, - L. ) e e m -n ~-~"!!-). OXmOXn a = (VVw) . a (34) m nand V X (aw) = (I m em o~J X aw = (I m em :~) X a = Vw X a . t35) In view of identities (34) and (35), expressions (32) and (33) reduce to the following: 1 ) eiklr-rl ] E = iwp, J [( U + k2 VV 41l"lr _ rl • J(r)dV eiklr-rl) ~ J ( V 41l"lr _ rl X Jm(r)dV (36) 25
- 33. Theory of electromagnetic wave propagation J( eiklr-rl)H = V 41J"lr_ rl X J(r)dV 1 ) + iWEJ [( U +k2 eiklr-rl ] VV 41J"lr_ rl • Jm(r)d.V (37)Since the quantity eiklr-rlG(r,r) I == 41J"r - r I (38)is known as the free-space scalar Greens function, it is appropriate torefer to the quantityr(r,r) == ( U + b VV) 4:1:lr=r~1 == ( U + b VV) G(r,r) (39)as the free-space dyadic Greens function. Using (38) and (39) we canwrite expressions (36) and (37) as follows:E(r) = iwp.fr(r,r) • J(r)dV - fVG(r,r) X Jm(r)dV (40)H(r) = fVG(r,r) X J(r)dV + iWEfr(r,r) . Jm(r)dV (41)These relations formally express the radiated fields E, H in terms of thesource currents J and Jm.12.2 Free-space Dyadic GreensFunctionIn the previous section we derived the free-space dyadic Greens func-tion using the potentials and antipotentials as an intermediary. Inthis section we shall derive it directly from Maxwells equations. We denote the fields of the electric current by E, H and those of themagnetic current by E", H". The resultant fields E, H are obtained 1 If the point of observation r lies outside the region occupied by the source(which is the case of interest here), then Ir - rl ~ 0 everywhere and theintegrals are proper. On the other hand, if r lies within the region of thesource, then Ir - rl = 0 at one point in the region and there the integralsdiverge. This improper behavior arises from interchanging the order ofintegration and differentiation. See, for example, J. Van Bladel, SomeRemarks on Greens Dyadic for Infinite Space, IRE Trans. Antennas Propa-gation, AP-9 (6): 563-566 (1961).26
- 34. Monochromatic sources in unbounded regionsby superposition, i.e., E = E + E" and H = H + H". Let us con-sider first the fields E, H which satisfy Maxwells equationsV X H = J - iWEE and V X E = iWJLH (42)From these equations it follows that E satisfies the vector Helmholtzequation with J as its source term: (43)In this equation, E is linearly related to J; on the strength of this lin-earity we may writeE(r) = iWJLfr(r,r) . J(r)dV (44)where r is an unknown dyadic function of l and 1". To deduce thedifferential equation that r must satisfy we substitute this expressioninto the vector Helmholtz equation. Thus we obtainV X V X fr(r,r) . J(r)dV - k2 fr(r,r). J(r)dV = fu.J(r)Il(r - r)dV (45)Noting that the double curl operator may be taken under the integralsign and observing that V X V X (r. J) = (V X V X r) . J, we getthe following equation: f[V X V X r(r,l") - k2r(r,r) - uo(r - 1")]. J(r)dV = 0 (46) Since this equation holds for any current distribution J(r), it follows that r(r,r/) must satisfy (curl curl - k2)r(r,r) = ull(r - 1") (47) Now we construct a dyadic function r such that Eq. (47) will be satisfied and expression (44) will have the proper behavior on the sphere at infinity. One way of doing this is to use the identity curl curl = grad div - V2 and write Eq. (47) in the form (V2 + k2)r(r,r) = - ull(r - 1") + VV• r(r,r) (48) 27
- 35. Theory of electromagn~tic wave propagation!<rom Eq. (47) it follows that 7 . r(r,r) = - ~2 7o(r - r). With theaid of this relation, Eq. (48) becomes(72+ k 2 ) r(r,r) = - (u + :2 77) o(r - r) (49)Clearly this equation is satisfied byr(r,r) = ( u + ;2 77) G(r,r) (50)where G(r,r) in turn satisfies(72 + k )G(r,r) 2 = - o(r - r) (51)To meet the radiation condition, the solution of Eq. (51) must be iklr-rlG(r,r) = 4e .1rr-r l I (52)Thus the desired dyadic Greens function is 1 ) eiklr-rlr(r,r) = (u + k2 77 41r/r _ rl (53) The fields E", H" satisfy Maxwells equations7 X H" = - iwtE" and 7 X E" = -Jm + iw~H" (54)from which it follows that H" satisfies the vector Helmholtz equationwith Jm as its source term:7 X 7 X H" - k2H" = iwd", (55)As before, if we writeII" = iwtfr(r,r) . Jm(r)dV (56)28
- 36. Monochromatic sources in unbounded regionsthen Eq. (55) and the radiation condition will be satisfied when r isgiven by expression (53). That is, the dyadic functions in the inte-grands of Eqs. (44) and (56) are identical. Since H = ~ V 2W~ X E and E" = - J:- V X H" 2WE it follows fromexpressions (44) and (.56) thatH = V X fr(r,r) . J(r)dV = f(V X r(r,r)] . J(r)dV (57)E" = - V X fr(r,r) . Jm(r)dV = - f[V X r(r,r)] . Jm(r)dV (58)But r(r,r) = (u + ~ VV G(r,r) and consequently )(V X r(r,r)]. J(r) = VG(r,r) X J(r)In view of this, Eqs. (57) and (58) becomeH = fVG(r,r) X J(r)dV (59)E" = - fVG(r,r) X Jm(r)dV (60)Combining expressions (44) and (56) with (60) and (59) respectively,we getE = E + E" = iw~fr(r,r) . J(r)dV - fVG(r,r) X Jm(r)dV (61)H = H + H" = iWEfr(r,r) . Jm(r)dV + fVG(r,r) X J(r)dV (62)These expressions are identical to expressions (40) and (41). 2.3 Radiated PowerFor the computation of the power radiated by a monochromatic elec-tric current, the complex Poynting vector theorem (see Sec. 1.5)V • S = - HJ* . E + 2iw(w m - We) (63) 29
- 37. Theory of electromagnetic wave propagationcan be used as a point of departure. The real part of this equationwhen integrated throughout a volume V bounded by a closed surfaceA, which completely encloses the volume Vo occupied by the current J,yieldsRe Iv V. S dV = -72 Re Ivo J*. E dV (64)Converting the left side by Gauss theorem to a surface integral overthe closed surface A with unit outward normal n, we getRe f A n S dA = -72 Re f Vo j*. E dV (65)The right side gives the net time-average power available for radiationand the left side the time-average radiated power crossing A in an out-ward direction. In agreement with the conservation of power this rela-tion is valid regardless of the size and shape of the closed surface A aslong as it completely encloses Vo. Thus we see that the time-averageradiated power can be computed by integrating - (72) Re (j* . E)throughout Vo or, alternatively, by integrating Re (n S) over anyclosed surface A eIl:closing Vo• In one extreme case, A coincides withthe boundary Ao of Vo; in the other, A coincides with the sphere atinfinity, Aoo The imaginary part of Eq. (63) when integrated throughout V yields1m f A n S dA = -72 1m f Vo J*. E dV + f 2", v (wm - w.)dV (66)As before, A is an arbitrary surface completely enclosing Vo• WhenA coincides with Ao this equation becomes1m lAO n . S dA = -72 1m IVo j* • E dV + 2",(W mint - W.int) (67)whereWm int = I Vo w m dV W.int = f Vo w. dV (68)denote the (internal) time-average magnetic and electric energies30
- 38. Monochromatic sources in unbounded regionsstored inside Vo. When A coincides with Aoo it becomes1m J A. n S dA = - ~ 1m J Vo J*. E dVwhereW mex = J V-Vo 11) m dV Wex = • J V - Vo 11) • dV (70)denote the (external) time-average magnetic and electric energiesstored outside Vo. In the far zone, S is purely real and consequentlyEq. (69) reduces toFrom this relation it is seen that the volume integral of - (~) 1m (J*. E)throughout Vo gives 2w times the difference between the time-averageelectric and magnetic energies stored in all space, i.e., inside Vo and out-side Vo• A relation involving only the external energies is obtained bysubtracting Eq. (71) from Eq. (67), viz., (72) Now, in accord with the left side of relation (65), we shall find thetime-average radiated power by integrating Re (n S) over the sphereat infinity. As was shown in Sees. 2.1 and 2.2, the electric field Eproduced by a monochromatic current J is given byE(r) = iwj.£ J Vo r(r,r). J(r)dV (73)where 1 ) eiklr-rlr(r,r) = (u +k 2 VV 41rJr _ rl (74) 31
- 39. Theory of electromagnetic wave propagationSince eiklr-rl eiklr-rlV . - V ~._-- (75) 411"lr- rl - - 411"lr- rlwe may write expression (74) in the formr(r,r) = ( u 1) + k2 VV eiklr-rl 411"lr- rr (76)with the double gradient operating with respect to the primed coordi-nates only. In the far zone, which is defined by1» 1" and kr» 1 (77)where l = yr-=r and r= yr~, the following approximation isvalid: (78)where ere = rlr) is the unit vector in the direction of r. In this approxi-mation we may replace exp (iklr - r/) by exp [ik(r - er • r)] andl/lr - rl by 1/1. Accordingly Eq. (76) reduces tor(r r) , = (u + -1 vv) k 2 ~ikr 411"1 e-ike,r (79)in the far zone. The double gradient VV operates on e-ike,.r only,and since (80)we have (81)With the aid of this relation, expression (79) for the far zone r becomes eikrr(r,r) = (u - erer) 4- e-ike,r (82) 11"132
- 40. Monochromatic sources in unbounded regionsSubstituting this expression into Eq. (73) and using the vector identity(u - ercr) J = J - cr(er• J) = -Cr X (cr X J) we obtain the follow-ing representation for the far-zone electric field: ikrE(r) = -iwJ.l. 41Tr eer X [J Cr X voe-;ke,rJ(r)dV ] (83)The far-zone magnetic field is found by taking the curl of Eq. (83) inaccord with the Maxwell equation H = J:- yo 1.wJ.l. X E. ThusH(r) = ik -e 41Tr ikr [Jer X Vo e-ike,.rJ(r)dV ]- (84)Comparing expressions (83) and (84) we see that the far-zone E and Hare perpendicular to each other and to Cr, in agreement with the factthat any far-zone electromagnetic field is purely transverse to thedirection of propagation, viz., in the far zone or H = r~(c 1M r X E) (85)is always valid. Expressions (83) and (84) yield the following expres-sion for the far-zone Poynting vector: (86)The notation ICl2 where C is any vector means C. C*. From expres-sion (86) we see that S is purely real and purely radial, i.e., directedparallel to Cr The element of area of the sphere over which S is to beintegrated is r2 dn, where dn is an element of solid angle. Hence, thetime-average radiated power P is given byP = JA~ n .S dA = J Cr Sr2 dn = 1 I~-,~ f 321T 2 f dn I c, X f Vo e-ike,.rJ(r)dV1 2 (87) 33
- 41. Theory of electromagnetic wave propagationThis way of calculating the radiated power is called the "Poyntingvector method." A formally different way of calculating the radiated power consists inintegrating throughout Vo the quantity - (72) Re (J* . E) in which Eis taken to be the radiative electric field as given by Eq. (73). Thisalternative procedure, which was proposed by Brillouin,l yieldsp = -72 Re vo J*. E dV J = WIJ. 81T J vo J VO J*(r). (u + .!- vv) sin Ir - r I k2 (klr ~ rD . J(r)dV dV (88)and is called the "emf method" since it makes use of the inducedelectromotive force (emf) of the radiative electric field. Although representations (87) and (88) of these two methods areapparently different, they nevertheless yield the same result for P andin this sense are consistent. To exemplify this we now apply these twomethods to the relatively simple case of a thin straight-wire antenna.The antenna has a length 2l and lies along the z axis of a cartesiancoordinate system with origin at the center of the wire. Since the wireis thin, the antenna current is closely approximated by the filamentarycurrentJ = e.Ioo(x)o(y)f(z) (89)where lois the reference current, e. is the unit vector in the z direction,and fez) is generally a complex function of the real variable z. By useof this current we getJvo e-ike,r J (r)dV = e.lo J ~l e-ik., coe ef(z)dz (90)where (J is the colatitude in the spherical coordinate system (r,(J,It»defined by x = r sin (J cos cP, Y = r sin (J sin cP, and z = r cos (J. Denot-ing the unit vectors in the r, (J, and cP directions respectively by er, ee,and eq, and noting that er X e. = -eq, sin (J, we find from Eq. (90) and 1 L. Brillouin, Origin of Radiation Resistance, Radioelectricite, April, 1922.34
- 42. Monochromatic sources in unbounded regionsexpression (87) that the Poynting vector method yieldsp = ~ ~ 101* j"i 1611" 0 I I I -I I -I f(z)f*(z)dz dz r" Jo eik(z-z) C08 8 sin3 0 dO (91)Moreover, by substituting the current as given in Eq. (89) into expres-sion (88), we see that the emf method yieldsp = WJJ. 811" 101* 0 II II -I -I f(z)f*(z) (1 + ~~) k2 az2 sin (klz - zl) dzdz Iz - zl (92)To show that expressions (91) and (92) are equivalent, we invoke thefollowing elementary results: r,..Jo eik(z-z) COB 8 sin3 0 dO = i. ~ (~~~:U u) u - cos 1 + ! ~) I(klz - Iz) _ sin - 2k (sin u _ )( k a2 z-z Z2 U2 U cos Uwhere u = k(z - z). With the aid of these results and the introduc-tion of the new variables ~ = kz and 1] = kz expressions (91) and (92)pass into the common formp =~ 411" ~ 1 Iri j"i 0 I I kl -kl kl -kl d1]d~ f(1])f*W (~ - 1])2 [sin (~ - ~- 1] 1]) - cos (~- )J 1] (93) Thus we see that the Poynting vector method and the emf method ultimately lead to the same formula (93) for the time-average radiated power P and hence are consistent with each other. From a practical viewpoint, formula (93) as it stands is too clumsy to use, owing to the presence of the double integral. However, Bouwkamp by successive transformations succeeded in reducing the double integral to a repeated integral and then finally to an elegant form involving only single integrals. To demonstrate the capabilities of this form he applied it to several "classical" cases which had been 35
- 43. Theory of electromagnetic wave propagationhandled previously by the Poynting vector method. For details werefer the reader to his original paper.! The present discussion may be extended to the case of magnetic cur-rents by using the duality transformations of Sec. 1.2. For example,if we replace J, E, H respectively by - vi EI IL Jm, - vi ILl E H, vi EI IL Ein the far-zone field formulas (83) and (84), we obtain the correspondingformulas for the far-zone field of a monochromatic magnetic currentdensity:H = -iWE (f e-ike,rJm(r)dV ) ikr 41rr er X e er X vo (94)E = -ik - e~( f e-ike,.rJm(r)dV ) 41rr er X Vo (95)The Poynting vector of this far-zone electromagnetic field is I"! ~ r I e, X f 2S = ;YzE X H* = er e-ike,.rJm(r)dV1 (96) "J IL 321r 2 2 Voand consequeritly the time-average radiated power isP = f. A.. n S dA = f A•• r e. Sr2 dn = I~ ~ "J IL 3271 2 f dn I e X f r Vo e-ike,.rJm(r)dV1 2 (97) ! C. J. Bouwkamp, Philips Res. Rept., 1: 65 (1946).36
- 44. Radiation from wire 3 antennasAs a practical source of monochromatic radiation the wireantenna plays an important role. The field radiated bysuch an antenna can be obtained from a knowledge of itscurrent distribution by using the formulas derived in theprevious chapter. Although the determination of theantenna current is a boundary-value problem of considerablecomplexity, a sufficiently accurate estimate of the currentdistribution can be obtained in the case of thin wires byassuming that the antenna current is a solution of the one-dimensional Helmholtz equation and hence consists of anappropriate superposition of simple waves of current. Thissimplifying approximation yields satisfactory results for thefar-zone field and for those quantities that depend on thefar-zone field, e.g., radiation resistance and gain, becausethe far-zone field in almost all directions is insensitive tosmall deviations of the current from the exact current. Theradiation properties of thin-wire antennas and their arraysare discussed in this chapter. 3.1 Simple Waves of CurrentWe consider a straight-wire antenna lying along the z axisof a cartesian coordinate system with one end at z = -l andthe other at z = l, as shown in Fig. 3.1. Since the wire is 37
- 45. Theory of electromagnetic wave propagation z Fig. 3.1 Coordinate sys- temfora straight- wire antenna ex- tending from z = -Z to z = z. Q is observation point. Q is pro- y jection of Q in x-y plane.xthin and since we wish to calculate only the far-zone field it is apermissible mathematical idealization to assume that the antennacurrent density is the filamentary distributionJ = ezo(x)o(y)j(z) (1)The total current is the integral of this distribution over the crosssection of the wire:I(z) = ezI(z) = IJ dx dy = e.j(z)Jo(x)o(y)dx dy = ezj(z) (2)It is supposed that the wire is cut at some cross section z = 1/ and amonochromatic emf is applied across the gap. The current is neces-sarily a continuous function of z, but the z derivative of the currentmay be discontinuous at the gap. The antenna is said to be "center-fed" when 1/ = 0 and "asymmetrically fed" when 1/ ;;c O. For a center-fed antenna, j(z) is a symmetrical function of z andsatisfies the one-dimensional Helmholtz equation 1crJ + k jdz 2 2 = 0 k = wlc = 2Tr/A (3)as well as the end conditionsfeZ) = f( -l) = 0 (4) 1 It appears that Pocklington was the first to show that the currents alongstraight or curved thin wires in a first approximation satisfy the Helmholtzequation. See H. C. Pocklington, Proc. Cambridge PhiZ. Soc., 9: 324 (1897).38
- 46. Radiation from wire antennasThe two independent solutions of Eq. (3) are the simple waves eikz ande-ikz• Accordingly a general form for the current isI(z) = Aeikz + Be- ikz (5)where A and B are constants. Writing this form for the two segmentsof the antennas, we haveI1(z) = Aleikz + Ble- ikz for 0 ::::; ::::;l z (6)I2(z) = A2eikz + B2e-ikz for -l ::::; 0 z ::::;When applied to these expressions,the end conditions (4) yield (7)from which it follows that BIIAI = _e2ikl, BdA2 = _e-2ikl• Withthese results, Eqs. (6) becomeI1(z) = -2iA1eikl sin k(l- z) for 0::::;z ::::;l (8)I2(z) = 2iA2e-ikl sin k(l + z) for -l ::::; ::::;0 zThe continuity condition 11(0) = 12(0) requires that A I and A2 berelated by (9)In view of this connection between Al and A2 it followsfrom Eqs. (8)that the current distribution, apart from an arbitrary multiplicativeconstant 10, is given by the standing wavelI(z) = 10 sin k(l - jz/) (10)which, for several typical cases, is displayed in Fig. 3.2. This "sinus-oidal approximation" is adequate for the purpose of computing thefar-zone radiation pattern of a center-fed straight-wire antenna,provided the antenna is neither "too thick" nor "too long." A closerapproximation to the true current may be obtained heuristically byadding to the sinusoidal current a quadrature current, which takes into 1 J. Labus, Z. Hochjrequenztechnik und Elektrokustik, 41: 17 (1933). 39
- 47. (a) (b) kl=1r (c)Fig.3.2 Radiation patterns of a center-driven thin-wire antenna of current distribution shown by dotted lines.40
- 48. kl •• 7-rrj6 (d)various lengths shown by solid lines. Assumed sinusoidal 41
- 49. Theory of electromagnetic wave propagationaccount the reaction of the radiation and the ohmic losses on thecurrent. 1 Since the antenna is center-fed, the principal alteration thatsuch a quadrature current can make on the far-zone radiation pattern isthe presently negligible one of relaxing the intermediate nulls of thepattern. 2 However, for an asymmetrically fed antenna (1] ;t. 0) the radiationpattern calculated solely on the basis of a simple standing wave ofcurrent can be in serious error due to the presence of traveling waves ofcurrent. The standing wave, which for arbitrary values of 1] has theform31(z) = 10 sin k(l + 1]) sin k(l - z) for 1] ::; z ::; l (11)1(z) = 10 sin k(l- 1]) sin k(l + z) for -l ::; ::; z 1]always gives rise to a radiation pattern that is symmetrical about theplane (J = 7r /2. On the other hand, a traveling wave produces anasymmetrical radiation pattern, viz., a pattern tilted toward the direc-tion of the traveling wave. Accordingly the traveling waves tend totilt the lobes of the pattern and to change their size. Thus if thetraveling waves are appreciable, marked changes in the shape of theradiation pattern can occur. Generally the problem of finding theradiation pattern of an asymmetrically driven antenna cannot behandled adequately within the framework of the simple wave theory,except in those cases where either the standing wave or the travelingwaves dominate the pat.tern. 3.2 Radiation from Center-driven Antennas As indicated by Eqs. (83), (84), and (86) of Chap. 2, the calculation of the far-zone radiation emitted by a distribution of monochromatic 1 Ronold King and C. W. Harrison, Jr., Proc. IRE, 31: 548 (1943). 2 C. W. Harrison, Jr., and Ronold King, Proc. IRE, 31: 693 (1943). 3 See, for example, S. A. Schelkunoff and H. T. Friis, "Antennas: Theory and Practice," chap. 8, John Wiley and Sons, Inc., New York, 1952. 42
- 50. Radiation from wire antennascurrent centers on the evaluation of the so-called radiation vector Ndefined by the integrallN = J Vo e-ike,rJ(r)dV (12)where er is the unit vector pointing from the origin to the point ofobservation and r is the position vector extending from the origin tothe volume element dV. The required information on J can beobtained either by solving the boundary-value problem which theanalytical determination of J poses or by choosing the current onempirical grounds. In the present case of a thin-wire antenna, thelatter alternative is adopted, according to which it is alleged that asufficiently accurate representation of the antenna current can bebuilt from simple waves to agree with the results of measurement. Accordingly, let us consider the case of a center-driven thin-wireantenna lying along the z axis with one end at z = -l and the other atz = l. It is known a posteriori that the current distribution alongsuch an antenna may be approximated, insofar as the far-zone radiationis concerned, by the sinusoidal filamentary currentJ(r) = e.Ioo(x)o(y) sin k(l - Izl) (13)Substituting this assumed current into definition (12) and performingthe integrations with respect to x and y, we get the one-dimensionalintegralN = ezlo J ~l e-ikz cos 9 sin k(l - Izl)dz (14)which by use of the integration formula ~EJe aE sin (b~ + c)d~ = a2 +b 2 [a sin (b~ + c) - b cos (b~ + c)] (15)yieldsN = ez2Io cos (kl cos.O) - cos kl (16) k sm2 0 1 S. A. Schelkunoff, A General Radiation Formula, Proc. IRE, 27: 660-666(1939). 43

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