Theory of electromagnetic wave propagation

THEORY OF ELECTROMAGNETIC WAVE PROPAGATION CHARLES HERACH PAPAS PROFESSOR OF ELECTRICAL ENGINEERING CALIFORNIA INSTITUT...

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THEORY OF ELECTROMAGNETIC WAVE PROPAGATION

CHARLES HERACH PAPAS PROFESSOR OF ELECTRICAL ENGINEERING CALIFORNIA INSTITUTE OF TECHNOLOGY

DOVER PUBLICATIONS,

INC., NEW YORK

Copyright @ 1965,1988 by Charles Herach Papas. All rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 30 Lesmill 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 and corrected republication of the work first published by the McGrawHill Book Company, New York, 1965, in its Physical and Quantum Electronics Series. For this Dover edition, the author has written a new preface. Manufactured in the United States of America Dover Publications, Inc., 31 East2nd Street, Mineola, N.Y. 11501 Library

of Congress Cataloging-in-Publication

Data

Papas, 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 530.1'41-dcI9 88-12291 CIP "i'

To RONOLDWYETH PERCIVAL KING Gordon McKay Professor of Appl1e~ Physics, Harvard University Outstanding

Scientist, Inspiring

Teacher,

and Dear Friend

"""', .~ ,"

Preface This book represents the substance of a course of lectures I gave during the winter of 1964 at the California Institute of Technology. In these lectures I expounded a number of newly important topics in the theory of electromagnetic wave propagation and antennas, with the purpose of presenting a coherent account of the subject in a way that would reveal the inherent simplicity of the basic ideas and would place in evidence their logical development from the Maxwell field equations. So enthusiastically were the lectures received that I was encouraged to put them into book form and thus make them available to a wider audience. The scope of the book is as follows: Chapter 1 provides the reader with a brief introduction to Maxwell's field equations and those parts of electromagnetic field theory which he will need to understand the rest of the book. Chapter 2 presents the dyadic Green's function and shows how it can be used to compute the radiation from monochromatic sources. In Chapter 3 the problem of radiation emitted by wire antennas and by antenna arrays is treated from the viewpoint of analysis and synthesis. In Chapter 4 two methods of expanding a radiation field in multipoles are given, one based on the Taylor expansion of the Helmholtz integrals and the other on 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, coherency 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 Green's function, the behavior of an antenna immersed in such a medium. Chapter 7 is concerned with the covariance of Maxwell's vii

Preface

equations in material media and its application to phenomena such as the Doppler effect and aberration in dispersive media. The approach of the book is theoretical in the sense that the subject matter is developed step by step from the Maxwell field equations. The advantage of such an approach is that it tends to unify the various topics under the single mantle of electromagnetic theory and serves the didactic purpose of making the contents of the book easy to learn and convenient to teach. The text contains many results that can be found only in the research literature of the Caltech Antenna Laboratory 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 textbook or a manual of self-instruction for researchers. My grateful thanks are due to Professor W. R. Smythe of the California Institute of Technology, Professor Z. A. Kaprielian of the University of Southern California, and Dr. K. S. H. Lee of the California 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 Edition Except for the correction of minor errors and misprints, this edition of the book is an unchanged reproduction of the original. My thanks are due to my graduate students, past and present, for the vigilance they exercised in the compilation of the list of corrections, and to Dover Publications for making the book readily available once again. Charles Herach Papas

viii

Contents

Preface

vii

Preface to the Dover Edition

viii

The electromagnetic

1 1.1

Maxwell's Equations in Simple Media

1.2

Duality

1.3 1.4

Boundary Conditions 8 The Field Potentials and Antipotentials

1.5

Energy Relations

2

field

1

6 9

14

Radiation from monochromatic sources in unbounded regions

2.1

The Helmholtz Integrals

2.2

Free-space Dyadic Green's Function

2.3

Radiated Power

3

1

19

19 26

29

Radiation from wire antennas

3.1

Simple Waves of Current

3.2

Radiation from Center-driven Antennas

3.3

Radiation Due to Traveling Waves of Current,

3.4

Cerenkov Radiation 45 Integral Relations between Antenna Current

3.5

and Radiation Pattern 48 Pattern Synthesis by Hermite Polynomials

3.6

General Remarks on Linear Arrays

3.7

Directivity Gain

73

37

56

42

50

37

Multipole expansion of the radiated field

4

4.1

Dipole and Quadrupole Moments

4.2

Taylor Expansion of Potentials

4.3

Dipole and Quadrupole Radiation

4.4

Expansion of Radiation Field in Spherical Waves

5

81

81 86 89 97

Radio-astronomical antennas

5.1

Spectral Flux Density

5.2

Spectral Intensity, Brightness, Brightness Temperature,

109

111

Apparent Disk Temperature

115

5.3 Poincare Sphere, Stokes Parameters

118

5.4

Coherency Matrices

5.5

Reception of Partially Polarized Waves

134 140

5.6 Antenna Temperature and Integral Equation for Brightness Temperature 5.7

Radio Interferometer 5.8

148

Elementary Theory of the Two-element 151

Correlation Interferometer

6

159

Electromagnetic waves in a plasma

6.1

Alternative Descriptions of Continuous Media

6.2

Constitutive Parameters of a Plasma

170

175

6.3

Energy Density in Dispersive Media

6.4

Propagation of Transverse Waves in Homogeneous Isotropic Plasma 183

178

6.5

Dielectric Tensor of Magnetically Biased Plasma

6.6 Plane Wave in Magnetically Biased Plasma 195 6.7 Antenna Radiation in Isotropic Plasma 205 6.8 Dipole Radiation in Anisotropic Plasma 6.9 Reciprocity

212

209

187

169

7

The Doppler effect

7.1

Covariance of Maxwell's Equations

7.2

Phase Invariance

7.3

Doppler Effect and Aberration

7.4

Doppler Effect in Homogeneous

7.5

Index of Refraetion

7.6

Wave Equation

and Wave 4-vector

217

218 223

225 Dispersive Media

of a Moving Homogeneous

for Moving Homogeneous

227

Medium

Isotropie Media

Index

230 233

24i

The

1

electromagnetic field

In this introductory chapter some basic relations and concepts of the classic electromagnetic field are briefly reviewed for the sake of easy reference and to make clear the significance of the symbols.

1.1 Maxwell's Equations in Simple Media In the mks, or Giorgi, system of units, which we shall use throughout this book, Maxwell's field equationsl are

a v x E(r,t) = - iii B(r,t) vx

R(r,t)

V. B(r,t)

J(r,t)

=

where E(r,t)

R(r,t)

(2) (3)

= 0

v . D(r,t) =

+ iiia D(r,t)

(1)

p(r,t)

(4)

= electric field intensity vector, volts per meter

=

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

Theory of electromagnetic

wave propagation

= electric displacement vector, coulombs per meter2

D(r,t)

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, seconds The equation of continuity

= -

V' • J(r,t)

a

at p(r,t)

(5)

which expresses the conservation of charge is a corollary of Eq. (4) and the divergence of Eq. (2). The quantities E(r,t) and B(r,t) are defined in a given frame of reference by the density of force f(r,t) in newtons per meter3 acting on the charge and current density in accord with the Lorentz force equation f(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) by constitutive parameters which characterize the electromagnetic nature of the material medium involved. For a homogeneous isotropic linear medium, viz., a "simple" medium, the constitutive relations are D(r,t)

= EE(r,t)

H(r,t)

= - B(r,t)

(7)

1

(8)

IJ.

where the constitutive parameters E in farads per meter and IJ. in henrys per meter are respectively the dielectric constant and the permeability of the medium. In simple media, Maxwell's equations reduce to V' X E(r,t)

= - J.L

V' X H(r,t)

=

2

ata H(r,t)

J(r,t)

+

E

a

at E(r,t)

(9) (10)

The electromagnetic V. H(r,t)

=

V • E(r,t)

= ~ p(r,t)

field (11)

0

(12)

E

The curl of Eq. (9) taken simultaneously

+

V X V X E(r,t) Alternatively,

ILE

02 ot2 E(r,t)

=

-IL

with Eq. (10) leads to

at0 J(r,t)

(13)

the curl of Eq. (10) with the aid of Eq. (9) yields

V X V X H(r,t)

+

ILE

02 i)t2 H(r,t)

= V X J(r,t)

(14)

The vector wave equations (13) and (14) serve to determine E(r,t) and H(r,t) respectively when the source quantity J(r,t) is specified and when the field quantities are required to satisfy certain prescribed boundary and radiation conditions. Thus it is seen that in the case of simple media, Maxwell's equations determine the electromagnetic field 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 the relations connecting B(r,t) to H(r,t) and D(r,t) to E(r,t) are linear, be it anisotropic, inhomogeneous, or both. To form a complete field theory an additional relation connecting J(r,t) to the field quantities is necessary. If J(r,t) is purely an ohmic conduction current in a medium of conductivity u in mhos per meter, then Ohm's law J(r,t)

=

uE(r,t)

(Hi)

applies and provides the necessary relation. On the other hand, J(r,t) is purely a convection current density, given by J(r,t)

= p(r,t)v(r,t)

if

(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


wave propagation

the total force F(r,t) acting on a charge carrier by first integrating the force 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 of inertia in accord with Newton's law of motion

F(r,t)

=

d

dt [mv(r,t)]

(18)

where m is the mass of the charge carrier in kilograms. In the case where the convection current is a charged fluid in motion (continuous case), the force density f(r,t) is entered directly into the equation of motion of the fluid. Because Maxwell's 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, by Fourier's theorem, any linear field of arbitrary time dependence can be synthesized from a knowledge of the monochromatic field. To reduce the system to the monochromatic state we choose exp (- iwt) for the time dependence and adopt the convention G(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 concomitant complex function of position {sometimes called a "phasor"), which depends parametrically on the frequency f( = w/27r) in cycles per second, and Re is shorthand for "real part of." Application of this convention to the quantities entering the field equations (1) through (4) yields the monochromatic form of Maxwell's equations:

v

X E",(r)

V

X

H.,(r)

= iwB",(r) =

J",(r) - i~D",(r)

(20) (21)

V • B",(r)

= 0

(22)

V. D",(r)

= p",(r)

(23)

4

The electromagnetic

field

In a similar manner the monochromatic form of the equation of continuity V' • J",(r) = iwp",(r)

(24)

is derived from Eq. (5). The divergence of Eq. (20) yields Eq. (22), and the divergence of Eq. (21) in conjunction with Eq. (24) leads to Eq. (23). We infer from this that of the four monochromatic ~Iaxwell equations only the two curl relations are independent. Since there are only two independent vectorial equations, viz., Eqs. (20) and (21), for the determination of the five vectorial quantities E",(r), H",(r), D",(r), B",(r) , and J",(r) , the monochromatic Maxwell equations form an underdetermined system of first-order differential equations. If the system is to be made determinate, linear constitutive relations involving the constitutive parameters must be invoked. One way of doing this is first to assume 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 with this assumption the system is determinate and possesses solutions involving 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)

=

(27)

u",E.,(r)

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 mono5


wave propagation

chromatic case, and we shall suppress the argument r in the timedependent case. For example, E(t) will mean E(r,t) and E will mean Ew(r). Accordingly, the monochromatic form of Maxwell's equations in simple media is

v

X E

VXH

= iW,llH

(28)

J - iweE

=

(29) (30)

V.H=O V.E=-p

1

(31)

e

1.2 Duality (J = 0), Maxwell's equations possess a By this we mean that if two new vectors

In a region free of current certain duality in E and H. E' and H' are defined by and

H'

=

+

iE

then as a consequence of Maxwell's equations VXH

= -iweE

V.H=O

V X E

(32) (source-free)

= iW,llH

it follows that E' and H' likewise satisfy Maxwell's equations free) V X H' V. E'

(33)

V.E=O

=

0

= -iweE'

(source-

(34)

V. H' = 0

and thereby constitute an electromagnetic field E', H' which is the "dual" of the original field. This duality can be extended to regions containing current by employing the mathematical artifice of magnetic charge and magnetic 6

The electromagnetic field

current.l

In such regions Maxwell's equations are

v

VXH=J-iwEE

= iWIlH (35)

1

V.E=-p

V.H=O

XE

E

and under the transformation (32) they become

v

X

E'

= :t ~ J

+ iWIlH'

V X

H' = -iWEE' (36)

V.

E'

=

0

= :t! ~p Il '\j;

V • H'

Formally these relations are Maxwell's equations for an electromagnetic field E', H' produced by the "magnetic current" +: vi III E J and the "magnetic charge" :t vi p.1 E p. These considerations suggest that complete duality is achieved by generalizing Maxwell's equations as follows: V X H = J - iWEE.

+ iwp.H

V X E = -Jm

V.E=-p

(37)

1 E

where Jm and Pm are the magnetic current and charge densities. Indeed, under the duality transformation H' =

E' = :t~H J;" =

+:

V X E' = -J;" V.

E'

p'

~~J

= !p' E

+:

V.

H'

(38)

= :t ~~

+ iWIlH'

J' = :t ~~ Jm

~~E

V X H'

1, =-p p.

p;"

Pm =

= + -~ ;P

J' - iWEE' (39)

m

1 See, for example, S. A. Schelkunoff, "Electromagnetic D. Van Nostrand Company, Inc., Princeton, N.J., 1943.

Waves," chap. 4,

7


wave propagation

Thus to every electromagnetic field E, H produced by electric current J there is a dual field H', E' produced by a fictive magnetic current J~.

1.3 Boundary Conditions The electromagnetic field at a point on one side of a smooth interface between two simple media, 1 and 2, is related to the field at the neighboring point on the opposite side of the interface by boundary conditions which are direct consequences of Maxwell's equations. We denote by n a unit vector which is normal to the interface and directed from medium 1 into medium 2, and we distinguish quantities in medium 1 from those in medium 2 by labeling them with the subscripts 1 and 2 respectively. From an application of Gauss' divergence theorem to Maxwell's divergence equations, V'. B = Pm and V'. D = P, it follows that the normal components of Band D are respectively discontinuous by an amount equal to the magnetic surface-charge density 7/m and the electric surface-charge density 7/ in coulombs per meter2: (40) From an application of Stokes' theorem to Maxwell's curl equations, V' X E = -Jm + iWJ.lH and V' X H = J - iweE, it follows that the tangential components of E and H are respectively discontinuous by an amount equal to the magnetic surface-current density Km and the electric 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 sheets are mathematical abstractions which can be simulated by limiting forms of electromagnetic objects. For example, if medium 1 is a perfect conductor and medium 2 a perfect dielectric, Le., if 0"1 = 00 and 0"2 = 0, then all the field vectors in medium I as well as 7/m and Km vanish identically and the boundary conditions reduce to (42)

s

The electromagnetic

field

A surface having these boundary conditions is said to be an "electric wall." By duality a surface displaying the boundary conditions Bz

n'

=

7]m

n

X H2

= 0 (43)

is said to be a "magnetic wall." At sharp edges the field vectors may become infinite. However, the order of this singularity is restricted by the Bouwkamp-Meixner1 edge condition. According to this condition, the energy density must be integrable over any finite domain even if this domain happens to include field singularities, i.e., the energy in any finite region of space must be finite. For example, when applied to a perfectly conducting sharp edge, this condition states that the singular components of the electric and magnetic vectors are of the order o-~, where /) is the distance from the edge, whereas the parallel components are always finite.

1.4 The Field Potentials and Antipotentials According to Helmholtz's partition theorem2 any well-behaved vector field can be split into an irrotational part and a solenoidal part, or, equivalently, a vector field is determined by a knowledge of its curl and divergence. To partition an electromagnetic field generated by a current J and a charge p, we recall Maxwell's equations V' X H =

J - iwD

V' X E = iwB

(44) (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). This theorem was proved earlier in less complete form by G. B. Stokes in his paper On the Dynamical Theory of Diffraction, Trans. Cambridge Phil. Soc., 9: 1 (1849). For a mathematically rigorous proof, see O. Blumenthal, Uber die Zerlegung unendlicher Vektorfelder, Math. Ann., 61: 235 (1905).

9


wave propagation

V.D=p

(46)

V.B=O

(47)

and the constitutive relations for a simple medium D

= eE

(48)

B

= ,uH

(49)

From the solenoidal nature of B, which is displayed by Eq. (47), it followsthat B is derivable from a magnetic vector potential A: (50)

B=VxA

This relation involves only the curl of A and leaves free the divergence of A. That is, V . A is not restricted and may be chosen arbitrarily to suit the needs of calculation. Inserting Eq. (50) into Eq. (45) we see that E - iwA is irrotational and hence derivable from a scalar electric potential et>: E

= -Vet>

+ iwA

(51)

This expression does not necessarily constitute a complete partition of the electric field because A itself may possess both irrotational and solenoidal parts. Only when A is purely solenoidal is the electric field completely partitioned into an irrotational part Vet> and a solenoidal part A. The magnetic field need not be partitioned intentionally because it is always purely solenoidal. By virtue of their form, expressions (50) and (51) satisfy the two Maxwell equations (45) and (47). But in addition they must also satisfy the other two Maxwell equations, which, with the aid of the constitutive 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, the 10

The electromagnetic

field

following simultaneous differential equations are obtained,l relating cP and A to the source quantities J and p: V2cP -

V2A

= -p/E

iwV. A

+kA 2

=

-jLJ

(53)

+ V(V. A -

iWEjLcP)

(54)

where k2 = W2jLE. Here V . A is not yet specified and may be chosen to suit our convenience. Clearly a prudent choice is one that uncouples the equations, Le., reduces the system to an equation involving cP alone and an equation involving A alone. Accordingly, we choose V • A = iWEjLcP or V • A = O. If we choose the Lorentz gauge V •A

= iWEjLcP

(55)

then Eqs. (53) and (54) reduce to the Helmholtz equations

+ k2cP V2A + k A

V2cP

2

= =

p/E

-jLJ

(56) (57)

The Lorentz gauge is the conventional one, but in this gauge the electric field is not completely partitioned. If complete partition is desired, we must choose the Coulomb gauge2 V.A

= 0

(58)

1 Also the vector identity V X V X A = V(V. A) - V2A is used. The quantity'V2A is defined by the identity itself or by the formal operation

V2A =

L V2(eiAi), where the Ai

are the components of A and the ei are the

i

unit base vectors of the coordinate system. The Laplacian V2 operates on not 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 = eiV2Ai. See, for example, P. M. Morse and H. Feshbach,

L i

"Methods of Theoretical Physics," part I, pp. 51-52, McGraw-Hill Book Company, 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


wave propagation

which reduces Eqs. (53) and (.54)to V2q, = -

1

- P

(59)

E

(60)

We note that Eq. (59) is Poisson's equation and can be reduced no further. However, Eq. (60) may be simplified by partitioning J into an irrotational part Ji and a solenoidal part J., and by noting that the irrotational part just cancels the term involving the gradient. To show this, J is split up as follows: J = Ji + J., where by definition V X Ji = 0 and V. J. = O. Since Ji is irrotational, it is derivable from a scalar function !J;, viz., Ji = V!J;. The divergence of this relation, V. Ji = V2!J;, when combined with the continuity equation V • J = V. (Ji + J.) =V. Ji = iwp, leads to V2!J;= iwp. A comparison of this result with Eq. (59) shows that !J; = -iWEq, and hence Ji = 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 part J. of the current distribution and q, by its irrotational part Ji. Since q, satisfies Poisson's equation, its spatial distribution resembles that of an electrostatic potential and therefore contributes predominantly to the near-zone electric field. It is like an electrostatic field only in its space dependence; its time dependence is harmonic. In regions free of current (J = 0) and charge (p = 0) we may supplement the gauge V . A = 0 by taking q, == O. Then Eq. (53) is trivially satisfied and Eq. (54) reduces to the homogeneous Helmholtz equation (62)

In this case the electromagnetic field is derived from the vector potential A alone. Let us now partition the electromagnetic field generated by a magnetic current Jm and a magnetic charge Pm. We recall that Maxwell's 12

The electromagnetic field

equations for such a field are V' X H = -iwD

(63)

= -Jm +iwB

V' xE

(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 from an electric vector potential A.: D

= -V' X A.

(67)

In turn it follows from Eq. (63) that H - iwA. is irrotational and hence equal to - V'cPm, where cPm is a magnetic scalar potential: H

= -V'cPm

+ iwA.

(68)

Substituting expressions (67) and (68) into Eqs. (64) and (66), we get, with the aid of the constitutive relations, the following differential equations for A. and cPm: V'2cPm- iwV' • A. V'2A.

+ k A. 2

=

~-Pm p.

= -

+ V'(V' • A.

-elm

(69)

- iwp.EcPm)

If we choose the conventional gauge V' •

A.

= iwp.EcPm

(70)

then cPm and A. satisfy V'2cPm

+ k cPm

V'2A.

+ 1c2A. =

2

= -

1

p.

Pm

-EJm

In this gauge cPm and A. are called "antipotentials."

(71) (72)

Clearly we may 13


wave propagation

also choose the gauge V'. Ae = 0 which leads to V'2cf>m

= -

1

- Pm /L

(73)

where JmB is the solenoidal part of the magnetic current; this gauge leads 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 currents and charges, then the field for the conventional gauge is given in terms of the potentials A, cf> and the antipotentials Ae, cf>m by E

= -

V'cf>

+ iwA

B = V' X A -

-

/LV'cf>m

!V' X Ae

(75~

+ iW/LAe

(76)

t

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

where E(t) stands for E(r,t), D(t) for D(r,t), etc. instance these expressions reduce to We =

~tE(t)

. E(t)

and

Wm =

~ILH(t) • H(t)

(77)

In the present

(78)

Both We and Wm are measured in joules per meter3• To transform these quadratic quantities into the monochromatic domain we recall 14

The electromagnetic

field

that E(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 always be written as E = E1 + 'iE2, where E1 andE2 are respectively the real and imaginary parts of E, the first of Eqs. (79) is equivalent to E(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 electric energy density

where the bar denotes the time average.

Since

where E* is the conjugate complex of E, we can express 'II'. in the equivalent form We =

%eE. E*

(83)

By a similar procedure it follows from the second of Eqs. (78) and the second of Eqs. (79) that the time-average magnetic energy is given by Win

= %J.tH.H*

(84)

The instantaneous Poynting vect~r S(t) is defined by S(t)

=

E(t)

X H(t)

where S(t) stands for S(r,t) and is measured in watts per meter2•

(85)

With 15


wave propagation

the aid of expressions (79), the time average of Eq. (85) leads to the following expression for the complex Poynting vector: S

=

~E

(86)

X H*

If from the scalar product of H* and V' X E = iWJLH the scalar product of E and V' X H* = J* iweE*(e is assumed to be real) is subtracted, and if use is made of the vector identity

+

V' . (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 monochromatic form of Poynting's vector theorem! V'. S

= -~J*. E

+ 2iw(w

m

-

(88)

1V.)

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 correspondingly the one on the left an outflow (when positive) or an inflow (when negative). In Poynting's vector theorem (88) a term involving the difference wm - W. appears. To obtain an energy relation (for the monochromatic 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' .

1

16

(oEow

X H*

+ E*

X

oH) ow

F. Emde, Elektrotech. M aschinenbau, 27: 112 (1909).

(90)

The electromagnetic

field

- E* . V' X oH

(91)

is identically equal to H* • V' X oE - oE . V' X H*

ow

ow

+~!:! .V' X E* ow

ow

From Maxwell's equations V' X E = iWJLH and V' X H = J - iwEE it follows that

V' X oH = ~ (V' X H) = ~. (J - iWEE) = oj _ iEE _ iWEoE _ iwE OE

ow

ow

ow

ow

and

V' X H*

= J*

ow

ow

+ iWEE*

Substituting these relations into expression (91) we obtain the desired energy relation V' . (oE

oW

X H*

+ E*

X

~!!) ow =

i [o(WJL) H • H*

ow

+ O(WE) E ow

. E*J

_ oE. 1* _ E* . oj

ow

ow

(92)

which we call the "energy theorem." Here we interpret as the timeaverage electric and magnetic energy densities the quantities (93)

which reduce respectively to expressions (83) and (84) when the medium is nondispersive, i.e., when OE/OW = 0 and OJL/ow = o.

17

Radiation from monochromatic

sources in unbounded regions The problem of determining the electromagnetic field radiated by a given monochromatic source in a simple, unbounded medium is usually handled by first finding the potentials 'of the source and then calculating the field from a knowledge of these potentials. However, this is not the only method of determining the field. There is an alternative method, that of the dyadic Green's function, which yields the field directly in terms of the source current. In this chapter these two methods are discussed.

2.1 The Helmholtz Integrals We wish to find the vector potential A and the scalar potential cP of a monochromatic current J, which is confined to a region of finite spatial extent and completely surrounded by a simple, lossless, unbounded medium. For this purpose it is convenient to choose the Lorentz gauge V' • A

= iWEfJ.cP

(1)

In this gauge, cP and A must satisfy the Helmholtz equations (see Sec. 1.4)

+ k2cP(r) V'2A(r) + k2A(r) V'2et>(r)

= - -1 per)

(2)

E

=

-fJ.J(r)

(3) 19

2


wave propagation

Since the medium is unbounded, q, and A must also satisfy the radiation condition. In physical terms this means that q, and A in the far zone must have the form of outwardly traveling spherical (but not necessarily isotropic) waves, the sphericity of the waves being a consequence of 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 the theory of the scalar Helmholtz equation that q, is uniquely determined by Eq. (2) and by the radiation condition 1 lim

l'

r....• '"

(uq, - ikq,) ur

=

0

(4)

where r = (yr' r) is the radial coordinate of a spherical coordinate system r, e, 1/;. To deduce from this radiation condition the explicit behavior of q, on the sphere at infinity, we note that the scalar Helmholtz 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 the angular coordinates and u is a function of l' only. Clearly the radiation condition (4) is satisfied by u(1') = (1/1') exp (ik1') and accordingly at great distances from the source the behavior of q, must be in accord with lim q,(r)

=

T-+OO

eikr r

f(e,1/;) -

(.5)

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 Sommerfeld's "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 complementary solution. In the present instance, however, the radiation condition requires that the complementary solution vanish identically.

20

Monochromatic

sources in unbounded

regions

where G(r,r') is a function of the coordinates of the observation point r and of the source point r', and where the integration with respect to the primed coordinates extends throughout the volume V occupied by p. The unknown function G is determined by making expression (6) satisfy Eq. (2) and condition (5). Substituting expression (6) into Eq. (2) we get

f p(r')(V2

+ lc )G(r,r')dV' 2

= -

(7)

p(r)

where the Laplacian operator operates with respect to the unprimed coordinates only. Then with the aid of the Dirac 0 functionl which permits p to be represented as the volume integral p(r)

= fp(r')o(r

(r in V)

- r')dV'

(8)

we see that Eq. (7) can be written as

f p(r')[(V2 + k2)G(r,r') + o(r

- r')]dV'

(9)"

= 0

From this it follows that G must satisfy the scalar Helmholtz V2G(r,r')

+ k2G(r,r')

= -

equation

o(r - r')

(10)

Since G satisfies Eq. (2) with its source term replaced by a 0 function, G is said to be a Green's function2 of Eq. (2). The appropriate solution of Eq. (10) for r ~ r' is eiklr-r'l

G(r,r') 1

= a

Ir-r 'I

(11)

The 0 function has the following definitive properties: oCr - r')

r ~ r' and =

00

for r = r'; /.f(r)o(r

- r')dV

= fer') forr'

= 0

for

in V and = 0

for r' outside of V wherefis any well-behaved function. See P. A. M. Dirac, "The Principles of Quantum Mechanics," pp. 58-61, Oxford University Press, 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 Mathematical Physics," vol. 1, pp. 351-388, Interscience Publishers, Inc., New York, 1953.

21


wave propagation

where a is a constant. It becomes clear that this solution is compatible with the requirement that the form (6) satisfy condition (5) when we recall the geometric relation Ir - r'l

= =

where r2 form

lim G(r,r')

+

vr2

r'2 - 2r' r'

r . rand r'2

=

r

VI +

(r' /r)2 - 2r' . r/r2

(12)

= r' . r/, and from this relation find the limiting

eiklr-r'l

Ir

lim a r~c:o

,-....+00

=

- r

/1 ~

eikr exp (-ikr' . r/r) r

a-

(13)

To determine the constant a, expression (11) is substituted into Eq. (10) and the resulting equation is integrated throughout a small spherical volume centered on the point r = r". It turns out that a must be equal to 7i7l", and hence the Green's function is eiklr-r'l

G(r,r')

=

(14)

4 7I"r I - r '1

Therefore, since the form (6) satisfies Eq. (2) and condition (5) when G is given by expression (14), the desired solution of Eq. (2) can be written as the Helmholtz integral 1 q,(r) = E

J p(r')

eiklr-r'l 4

I

7I"r - r

'1 dV'

(15)

Now the related problem of finding A can be easily handled. Clearly, the appropriate solution of Eq. (3) must be the Helmholtz integral A(r)

=

p.

J J(r')

eiklr-r'l

(16)

471"Ir_ r/I dV'

because it has the proper behavior on the sphere at infinity and it satisfies 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 Green's 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 antipotentials q,m and Ae should be iJ?-voked. The magnetic scalar potential cPm and the electric vector potential Ae satisfy the Helmholtz equations

+

22

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 the one we used in obtaining the Helmholtz integrals for !Pand A leads to the following Helmholtz integrals for !Pm and A.: 1

!PmCr)

= p.-

A.(r)

=

E

f Pm(r')

f Jm(r')

eiklr-r'l

4 11'1'-1' I

'I dV'

(19)

411'11' _ 1"1dV'

(20)

eikjr-r'l

From a knowledge of !P,A, !Pm, A. the radiated electric and magnetic fields can be derived by use of the relations (see Sec. 1.4) E

= - V'!p + iwA -

H

=

! V' X A p.

!V' X A.

(21)

E

- V'!pm + iwA.

(22)

It is sometimes desirable to eliminate !P and !Pmfrom these relations and thereby express E and H in terms of A and A. only. This can be done with the aid of -V'!p

=

t

and

-V'(V'.A)

WEP.

-V'!pm

= -

i

WEP.

V'(V. A.)

(23)

which follow from the gradients of the Lorentz conditions V'. A = iWEP.!P and V' • A. = iwp.!Pm' Thus relations (21) and (22) may be written as follows: E

= iw [ A

H = ; V'

X

+

b

V'(V' • A) ] ....• ~ V'

A

+

iw [ A.

+

b

X

A.

V'(V' • A.) ]

(24) (25) 23


To enable us to cast A

wave propagation

+ ~ V(V.

A) and A.

+

k

V(V. A.) into the

form of an operator operating on A and A., we introduce the unit dyadic u and the double-gradient dyadic VV which in a cartesian system of coordinates are expressed by m,=3n=3

I I

U =

(26)

emenOmn

m,=ln=l m=3n=3

vv =

\' \'

L. L.

emen

m,=ln=l

a a ax:' aX

(27) n

where Xi (i = 1, 2, 3) are the'cartesian coordinates, ei (i = 1, 2, 3) are the unit base vectors, and the symbol omn is the Kronecker delta, which is 1 for m = nand 0 for m ;;e n. The properties of u and \7V that we will need are u. C = C and (VV) . C = V(V. C), where C is any vector function. These properties can be demonstrated by writing C in component form and then carrying out the calculation. Thus

=

LLL

eme" . epCpOmn

=

mn.p

where e" . ep

L L L emOnpCpOmn mll.p

=

=

L epCp

=

C

(28)

P

Onp, and

With the aid of these results, relations (24) and (25) become E

=

24

iw

(u + ~ k-

VV) . A -

1 V' X €

A.

(30)

Monochromatic


regions

(31)

Using the Helmholtz integrals (16) and (20) and taking the curl oper-

E

=

J(

iwp,

u

1

iz

+

ator and the operator u )

+ k2 V'V'

[

•

V'V'under the integral sign, we get

eiklr-r'I] 41l"lr _ r'l

J(r')

J

-

H =

J

V' X

[

J(r')

eiklr-r'I] 41l"lr _ r'l

+ iWE J

(

V' X

[

eiklr-r'l ] Jm(r') 41l"jr _ r'l

dV'

)

[

eiklr-r'l ] Jm(r') 41l"tr _ r'l

dV'

(32)

dV' 1

U

dV'

+ k2 V'V'

•

To reduce these expressions we invoke the following considerations. If a is a vector function of the primed coordinates only and w is a scalar function of the primed and unprimed coordinates, then

= (,

)'

'-' L. m

n

e e m

~-~"!!-). a

-n

= (V'V'w) .

a

(34)

OXmOXn

and V' X (aw)

=

(I

em

m

In view of identities

o~J

X aw =

(I

em :~)

X a = V'w X a

m

t35)

.

(34) and (35), expressions (32) and (33) reduce to

the following:

E = iwp,

J [(

1

U

)

+ k2 V'V'

eiklr-r'l ] 41l"lr _ r'l • J(r')dV'

~

eiklr-r'l) r'l

J ( V' 41l"lr _

X Jm(r')dV'

(36)

25

Theory

of electromagnetic

J(

H =

eiklr-r'l) V 41J"lr_ r'l

wave propagation

X J(r')dV'

+ iWEJ

[(

1

U

+k

2

)

eiklr-r'l ]

VV 41J"lr_ r'l

• Jm(r')d.V'

(37)

Since the quantity eiklr-r'l

== 41J"r I

G(r,r')

-

'I

r

(38)

is known as the free-space scalar Green's function, it is appropriate refer to the quantity

== (

r(r,r')

U

+

b

VV)

4:1:lr=r'~'1 == (

U

+

b

VV) G(r,r')

to

(39)

as the free-space dyadic Green's function. Using (38) and (39) we can write expressions (36) and (37) as follows: E(r)

= iwp.fr(r,r')

H(r)

=

fVG(r,r')

• J(r')dV' X J(r')dV'

- fVG(r,r')

X Jm(r')dV'

(40)

+ iWEfr(r,r')

. Jm(r')dV'

(41)

These relations formally express the radiated fields E, H in terms of the source currents J and Jm.1

2.2 Free-space Dyadic Green's Function In the previous section we derived the free-space dyadic Green's function using the potentials and antipotentials as an intermediary. In this section we shall derive it directly from Maxwell's equations. We denote the fields of the electric current by E', H' and those of the magnetic 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 - r'l ~ 0 everywhere and the integrals are proper. On the other hand, if r lies within the region of the source, then Ir - r'l = 0 at one point in the region and there the integrals diverge. This improper behavior arises from interchanging the order of integration and differentiation. See, for example, J. Van Bladel, Some Remarks on Green's Dyadic for Infinite Space, IRE Trans. Antennas Propagation, AP-9 (6): 563-566 (1961).

26

Monochromatic


regions

by superposition, i.e., E = E' + E" and H = H' + H". Let us consider first the fields E', H' which satisfy Maxwell's equations V' X H'

=

J - iWEE'

and

V' X E'

=

iWJLH'

(42)

From these equations it follows that E' satisfies the vector Helmholtz equation with J as its source term: (43) In this equation, E' is linearly related to J; on the strength of this linearity we may write E'(r)

= iWJLfr(r,r') . J(r')dV'

(44)

where r is an unknown dyadic function of l' and 1". To deduce the differential equation that r must satisfy we substitute this expression into the vector Helmholtz equation. Thus we obtain V' 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 integral sign and observing that V' X V' X (r. J) = (V' X V' X r) . J, we get the following equation:

f[V' X V' X r(r,l")

- k2r(r,r')

- uo(r - 1")]. J(r')dV'

Since this equation holds for any current distribution that r(r,r/) must satisfy (curl curl - k2)r(r,r')

=

ull(r -

1")

=

0

(46)

J(r'), it follows

(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 - V'2 and write Eq. (47) in the form (V'2

+ k2)r(r,r')

= -

ull(r - 1")

+ V'V'• r(r,r')

(48) 27

Theory of electromagn~tic wave propagation

! a) is negligible. Also, when A is large, the factor exp ( - t2/2A 2) approaches unity. Accordingly, if we choose A sufficiently large, then Eq. (49) closely approximates

(-I~t~l)

Comparing Eqs. (50) and (47), we see that the functionsfnW

(50)

are given

by

(51)

Substituting this result into Eq. (46) We get the formal solution of integral equation (44):

(A large)

(52)

As an application of the above method we now synthesize the radiation pattern F(O)

=

Since t G(t) 52

(53)

sin2NH 0 =

cos 0, then F(t)

= (1 - t2)N

=

VI -

t2 (1 - t2)N and hence (54)


By the binomial theorem, we have

(55)

where the binomial coefficients are given by

(N) = N! l

(N - l)!l!

Comparing expansions (55) and (45) we see that

-Y21

= (~)

(56)

(_i)21

and substituting

these values into Eq. (52) we get

(57)

The arbitrary positive constant A is chosen such that

A=-

{1 a

(58)

where (3 is greater than the largest root of H 21(U) = o. With the use of expression (58), the current distribution (57) is transformed to

(59)

Thus, corresponding to N = 0, N = 2, N = 4, we have the radiation patterns F=sine

F = sin5

e

F = sin9

e

(60) 53


wave propagation

a

-1.00 Fig. 3.5

Current distributions along antenna for N = 0 and N Length of antenna is approximately quarter wave.

=

2.

and the respective current distributions that produce them: (61)

(62)

(63) 54

Radiation

from wire antennas

According to the calculations of Bouwkamp and De Bruijn, these distributions. are explicitly given by fo(0.8,~,4) f2(%A,6)

= =

2.394rI2.oEI 5.175 X 104e-aW(1 -

= 1.4208 X

f4(7r/4,~,9)

lQlle-6oP(1

+ 1379.59~4) 526P + 34,559~4 - 605,706~6 + 2,843,678~8)

128.663p -

In Figs. 3.5 and 3.6, curves of foWlfo(O), f2Wlh(0), and f4(~)lf4(0) versus ~ are plotted. From these curves, we see that as N increases the number of oscillations increases. These spatial oscillations cause

650

o

a

-0.50

-1.00 Fig. 3.6

Current distribution along antenna for N = 4. antenna is quarter wave.

Length

of

55


wave propagation

the far-zone waves to interfere destructively in every direction except the equatorial one, where they add constructively and thus produce a sharp omnidirectional beam straddling the equatorial plane.

3.6 General Remarks on Linear Arrays A great variety of radiation patterns can be realized by arranging in space a set of antennas operating at the same frequency. The fields radiated by the separate antennas interfere constructively in certain directions and destructively in others, and thus produce a directional radiation pattern. A knowledge of each antenna's location, orientation, and current distribution, being tantamount to a complete description of the monochromatic source currents, uniquely determines the resultant radiation pattern. Once the vector currents are known, the radiation pattern can be calculated in a straightforward manner by the methods described in Chap. 2. On the other hand, the converse problem of finding a set of antennas that would produce a specified radiation pattern has no unique solution. For this reason, the problem of synthesizing a set of antennas to achieve a prescribed radiation pattern is considerably more challenging than the one of analyzing a prescribed set of antennas for its resultant radiation pattern. Actually, the indeterminacy of the synthesis problem is circumvented by imposing at the start certain constraints on the set which reduce sufficiently its generality and then by specifying the desired radiation pattern with that degree of completeness which would make the problem determinate.l Although any arrangement of antennas can be analyzed for its radiation pattern when the vector current distribution along each of the antennas is known, a synthesis procedure is possible only for certain sets. An important example of such a set is the configuration called the array, which by definition is composed of a finite number of identical antennas, identically oriented, and excited in such a manner that the current distributions on the separate antennas are the same in form but may differ in phase and amplitude. It follows from this definition that 1 See, for example, Claus Muller, Electromagnetic Radiation Patterns and Sources, IRE Trans. Antennas Propagation, AP-4 (3): 224-232 (1956).

56

Radiation from wire antenna8

the radiation pattern of an array is always the product of two functions, one representing the radiation pattern of a single antenna in the array and the other, called the array factor or space factor, being interpretable as the radiation pattern of a similar array of nondirective (isotropic) antennas. This separability simplifies the problems of analysis and synthesis to the extent that it permits the actual array to be replaced by a similar array of isotropic antennas.! Of all possible arrays, the linear array is the simplest to handle mathematically and hence constitutes a natural basis for a discussion of antenna arrays. Here we shall limit our attention to linear arrays. 2 Let us consider then a linear array which for definiteness is assumed to consist of n center-driven half-wave dipoles oriented parallel to the z axis with centers at the points xp(p = 0, 1, ... , n - 1) on the x axis (see Fig. 3.7). Each dipole is independently fed, has a length 2l, and is resonant (kl = 11'/2). Under the simplifying approximation that the proximity of the dipoles does not modify the dipole currents or, equivalently, that the dipoles do not interact with each other, 3 the cur! An isotropic antenna is no more than a conceptual convenience. Actually a system of coherent currents radiating isotropic ally in all directions of free space is a physical impossibility. This was proved by Mathis using a theorem due to L. E. .J. Brouwer concerning continuous vector distributions on surfaces. See H. F. Mathis, A Short Proof that an Isotropic Antenna is Impossible, Proc. IRE, 39: 970 (1951). For another proof see C. J. Bouwkamp and H. B. G. Casimir, On Multipole Expansions in the Theory of Electromagnetic Radiation, Physica, 20: 539 (1954). 2 For comprehensive accounts of antenna arrays we refer the reader to the excellent literature on the subject. See, for example, G. A. Campbell, "Collected Papers," American Tel. and Tel. Co., New York, 1937; Ronold King, "Theory of Linear Antennas," Harvard University Press, Cambridge, Mass., 1956; S. A. Schelkunoff and H. T. Friis, "Antennas: Theory and Practice," John Wiley & Sons, Inc., New York, 1952; H. Bruckmann, "Antennen ihre Theorie und Technik," S. Hirzel Verlag KG, Stuttgart, 1939; J. D. Kraus, "Antennas," McGraw-Hill Book Company, New York, 1950; H. L. Knudsen, "Bidrag til teorien fjilrantennesystemer med hel eller delvis rotationssymmetri," I Kommission hos Teknick Forlag, Copenhagen, 1952. 3 In practice, one would take into account this mutual interaction or coupling by invoking the concept of mutual impedance. See, for example, P. S. Carter, Circuit Relations in Radiating Systems and Application to Antenna Problems, Proc. IRE, 20: 1004 (1932); G. H. Brown, Directional Antennas, Proc. IRE, 25: 78 (1937); A. A. Pistolkors, The Radiation Resistance of Beam Antennas. Proc. IRE, 17: 562 (1929); F. H. Murray, Mutual Impedance of Two Skew Antenna Wires, Proc. IRE, 21: 154 (1933). 157


wave propagation

rent density along thepth dipole is taken to be that of an isolated dipole: (64)

(-1 ~ z ~ 1)

where Ap denotes the complex magnitude of the current. resulting current density for the entire array is the sum

.r

n-l

J =

Hence the

n-!

J(p)

p=o

= e,o(y) cos kz

L

Apo(x - xp)

(65)

p=o

This current density gives rise to the following expression for the radia-

z Q

Fig. 3.7

58

Linear array of half-wave dipoles at points xo, along X axis. Each dipole is parallel to z axis. observation. Q' is projection of Q on xy plane.

Xl,

• , Xn-l

OQ is line of


tion vector: N

= J e-ike,'r' J (r')dV'

=

e. J

e-ikY'sin

X J~le-ikz'COS6COSkz'dz'

J

6 sin

211"), the radiation characteristic K(y.,) changes markedly; it develops a multilobe structure consisting of "grating lobes" which collectively resemble the diffraction pattern of a linear optical grating. I On the other hand, if the sources are phased progressively such that kd = -I' or kd = 1',the radiation is cast principally in the direction of the line of sources and the array operates as an "end-fire array." If the spacing is less than a half wavelength (kd < 7T), there is a single end-fire lobe in the direction l/J = 7T when kd = 'Y. But if the spacing is equal to a half wavelength (kd = 7T), two end-fire lobes exist simultaneously, one along y., = 0 and the other along y., = 11". Hence, when kd < 11"the array is a "unilateral end-fire array" and when kd = 11" it is a "bilateral end-fire array." An increase in the directivity of a unilateral end-fire array is realized when the condition of Hansen and Woodyard is satisfied, viz., I' = -(kd 1I"/n) or I' = (kd 1I"/n).2 If one desires the major lobe to point in' some arbitrary direction y., = y.,1, then the phase I' and the spacing d must be chosen such that kd cos y.,1 I' = o. When the coefficients ap of the polynomial (76) are smoothly tapered in accord with the binomial coefficients

+

+

+

n ap = ( 1

p

1)

I)! 1 _ p)!p!

(n = (n -

(81)

See, for example, A. Sommerfeld, "Optics," pp. 180-185, Academic Press

Inc., New York, 1954. 2 W. W. Hansen and J. R. Woodyard, A New Principle in Directional Antenna Design, Proc. IRE, 26: 333 (1938).

62

Radiation

from wire antennas

kd= 311' 4

kd=lr Fig.3.8

kd= 511' .4

kd=31r

. 2

Radiation characteristic K(t/t) of a uniform linear array for various spacings. Calculated from Eq. (80) with 'Y = 0 and n = 12. Broadside array. Grating lobes. 63


wave propagation

kd= 71r

kd=211"

4

kd=

kd=311"

511"

2

Fig. 3.8 64

Continued.


kd=4'l1'

kd=81T

Fig. 3.8

Continued.

65


wave propagation

kd='Y=JI.. 4

kd='Y=!!:. 2

Fig.3.9

66

Radiation characteristic K(~) of auniform linear array cal. culated from Eq. (80) with n = 12 for various values oj kd = 'Y Unilateral end-fire array. Bilateral end-fire array.


kd= 'Y= 3 'IT '4

kd='Y~1r

Fig. 3.9

Continued. 67


wave propagation

the array factor becomes

(82) Hence, the radiation IA(~-)I

=

pattern

2n-1lcosn-1 (a/2)

I

=

of such a "binomial 2n-1Icosn-1 [(kd cos

array" 1/1

is given by

+ 1')/2]1

(83)

For I' = 0 and kd = 11", the binomial array yields the following broadside pattern, which is distinguished by the fact that it is free of side lobes:! (84) Comparing a uniform broadside array with a binomial broadside array having the same number of radiators, we see from the above examples that the broadside lobe of the former is narrower than the broadside lobe of the latter. Thus by tapering the strengths of the radiators we reduce the side lobes, but in so doing we broaden the broadside lobe. However, it is possible to choose the coefficients ap such that the width of the broadside lobe is minimized for a fixed side-lobe level, or conversely, the side-lobe level is minimized for a fixed width of the broadside lobe. Indeed, Dolph2 demonstrated that for the case in which the number of sources in the array is even and d ~ X/2, such an optimum pattern can be achieved by matching the antenna polynomial (76) to a Chebyshev polynomial. Then Riblet3 extended the discussion to the case in which the number of sources is odd and d < X/2. And finally Pokrovskii,4 through the use of the so-called Chebyshev-Akhiezer polynomials, which constitute a natural extension of the Chebyshev polynomials, succeeded in handling the general case where d ~ X/2 or d < )0../2. Certain simplifications in the practical calculation of such 1

J. S. Stone, U.S. Patents 1,643,323 and 1,715,433.

C. L. Dolph, Current Distribution for Broadside Arrays which Optimize the Relationship between Beam Width and Side-lobe Level, Proc. IRE, 34: 335 (1946). 3 H. J. Riblet, Discussion on Dolph's Paper, Proc. IRE, 35: 489 (1947). 4 V. L. Pokrovskii, On Optimum Linear Antennas, Radiotekhn. i ElektrQrt" I: 593 (1956). 2

68

Radiation fronl wire antennas

Chebyshev arrays were made by Barbiere1 and by Van der Maas.2 For a continuous distribution of isotropic radiators along a straight line, i.e., for a line source, the problem of an optimum broadside pattern (narrow beam width and low side lobes) was solved by T. T. Taylor. 3 When the sources are incommensurably spaced, the point of departure is no longer the polynomial (76) but the more general expression (71). Clearly expression (71) is considerably more difficult to handle than expression (76), especially when the number of sources becomes large; but with the use of a computer, numerical results can be obtained in a straightforward manner. An unequally spaced array is generally more "broadband" than an equally spaced array, in the sense that its radiation pattern remains essentially unaltered over a broader band of operating frequencies. King, Packard, and Thomas4 studied this attribute of unequally spaced arrays by numerically evaluating the radiation pattern for Ap = 1 and Xp chosen according to various spacing schemes. A general discussion of unequally spaced linear arrays has been reported by Unz,. and certain equivalences between equally and unequally spaced arrays have been noted by Sandler. 6 Returning to the case of a uniform array whose radiation characteristic is given by expression (80), we see that if nand kd( 1 for all other antennas. Unless the antenna happens to be a short dipole or some other equally simple antenna, the problem of calculating directivity gain is complicated by the fact that the integral representing the time-average power P radiated by the antenna, viz., P

=

(4..

10 ST(r,8,t/»r

2

dn

(r ~

00)

(95)

cannot be evaluated by elementary means. The same difficulty arises in connection with the calculation of the radiation resistance R of an antenna,2 because to find R from the definition R = 2P/12, where I is 1 C. Polk, Optical Fresnel-zone Gain of a Rectangular Aperture, IRE Trans. Antennas Propagation, AP-4 (1): 65-69 (1956). 2 M. A. Bontsch-Bruewitsch, Die Strahlung cler komplizierten rechtwinkeligen Antennen mit gleichbeschaffenen Vibratoren, Ann. Phys., 81: 425

(1926). 73


wave propagation

an arbitrary reference current, one is again faced with the task of calculating P. As an alternative, it is always possible to calculate P by Brillouin's emf method, 1 but the integral to which this method leads is generally as difficult to evaluate as the integral (95) posed by Poynting's vector method. The situation is eased considerably when the antenna is highly directional, for then ST(r,8,q,) may be approximated by a function that simplifies the evaluation of the integral (95). Let us first consider the simple case of a short wire antenna. From Eq. (19) we see that for kl « 1, the far-zone radial component of the Poynting vector has the form

= ~r

ST(r,8,q,)

(96)

sin2 8

where K is a constant that will drop out of the calculation due to the homogeneity of relation (94). The maximum of ST(r,8,q,) occurs at 8 = 7r/2 and has the value (97) Substituting expressions (96) and (97) into definition (94), we find that the gain of a short dipole is given by g =

2

10"

sina

3

(98)

=-

8 d8

2

As the antenna is lengthened, its gain increases moderately. To show this, we recall from Eq. (19) that for a center-driven antenna of arbitrary length the far-zone radial component of the Poynting vector has the form ST(r 8 q,) , ,

=

!i r 2

[cos (kl co~ 8) - cos kl]2 SIn

8

(99)

where, now, K = Vp./Elo2/87r2. Substituting this expression into Eq. (95), we obtain the following integral representation for the time1 A. A. Pistolkors, The Radiation Resistance of Beam Antennas, Proc. IRE, 17: 562 (1929).

74


average power radiated by the antenna: P

=

27rK (1r [cos (kl COS.8) - cos klJ2 d8 10 sm 8

(100)

To evaluate this integral we introduce the new variables u( = kl cos and v( = kl - u). Thus ( r

8)

[cos (kl co~ 8) - cos klJ2 d8

10

sm 8

_IJkl - 2"

-kl

(cos U

cos kl)

-

2( kl _1 u + kl +1)u

= J_klkl (cos ~l-=- c~s kl)2 du = J 2kl 0

[(1 + cos

. 2kl ( sm . v - 2"1 sm . 2)v - sm

2kl)(1

du - cos v)

v

cos 2kl ( 1 - cos 2v )] dv - -2-

and hence P

=

27rK [ C

+ In 2kl

- Ci 2kl

+ COS22kl

+ sin22kl

(C

(Si 4kl - 2Si 2kl)

+ In kl + Ci 4kl

- 2Ci 2kl)]

(101)

where Si x

=

(z

10

sin ~ d~

~

is the sine integral Ci x = -

f'" z

cos ~ d~ = C ~

+ In x

_

(z

10

1 - cos ~ d~

~

is the cosine integral, and C( = 0.5722 ... ) is Euler's constant. With the aid of a table of sine arid cosine integrals, I P can be easily I See, for example, E. Jahnke and F. Emde, "Tables of Functions," Dover Publications, Inc., New York, 1943.

75


wave propagation

computed from expression (101). In the case of a half-wave dipole (kl = 7r/2) it follows from Eqs. (99) and (101) that (Sr)max = K/r2 and P = 27r(1.22)K. Inserting these results into Eq. (94), we find that the gain of a half-wave dipolel is g = 1.64. Similarly, in the case of a full-wave dipole we would find that g = 2.53. These examples illustrate that the gain of a linear antenna increases rather slowly with length, and to get really high gains from thin-wire antennas one must operate them in multielement arrays. As an antenna of high-gain capabilities, let us now consider a uniform parallel array whose far-zone Poynting vector, in accord with Eqs. (67) and (79), has the radial component

S

_ r -

~ _1_ "\J; 87r2r2

cos (~cos ()) sin (n(kd sin ()cos cP + ")')/2] 2 sin () sin ((kd sin ()cos cP + ")')/2]

(102)

By virtue of approximation (25) we can write this expression in the simpler form 2

S r

_ I~(0.945)2 \ . ()sin (n(kd sin ()cos cP + ")')/2] 1 -"\JE 87r2r2 sm sin ((kd sin ()cos cP + ")')/2]

(103)

It is clear that the maximum of Sr occurs at () = 7r/2 and cP = cPo, where cPo is fixed by kd cos cPo + ")'= O. Thus (104) Substituting (103) and (104) into definition (94), we get 47rn2

g

=

{2" { •.. 3 () sin2 (n(kd sin ()cos cP + ")')/2] d()dcP Jo Jo sm sin2 ((kd sin ()cos cP + ")')/2]

(105)

The integral in this expression can be evaluated exactly2 through the 1 The relative gain gr of an antenna is its gain over a half-wave dipole. That is, gr = g/1.64. 2 C. H. Papas and R. King, The Radiation Resistance of End-fire and Collinear Arrays, Proc. IRE, 36: 736 (1948).

76


use of Bonine's first integral theorem 1 (2"

Jo

f" . 3 1:1 sin2 [n(kd sin 1:1 cos cP + 'Y)/2j dl:1dcP Jo sm sin2 [(kd sin 1:1 cos cP + 'Y)/2j

nI-l (n-q ) cos ()q'Y

8~n + 8~

=-

3

(sin ----- u U

q=l

sin u u3

+ --cos u) u2

(106)

where u = qM. Hence the gain (105) of the array can be expressed in terms of the finite series (106): (J

4~2

=

n-l

8~n

3

+ 8~L \' ( n

-

)

--u -

()

(sin u

- q cos q'Y

sin u US

+ cos U2u)

(107)

q=l

This expression is convenient for numerical calculation, especially when the number n of dipoles is small. When n is very large and the array is operating as a broadside array ('Y = 0, d ::;; X), we have the simple limiting form 4nd

(J =-

(n ~ 00)

X

(108)

which may be obtained2 by comparing the denominator of expression (107) with the Fourier expansions of the functions x, x2, and x3 for the interval (0,2~). Let us now calculate the gain of a large uniform rectangular array. We recall from Eq. (88) that its radiation pattern is given by cos (~ cos U(1:1 )

,cP

=

I

~

0

1:1).I

3m 1:1

.

mf31

na Sl~ sm a sm f3

Sl~l

(109)

where a = (kdx/2) sin 1:1 cos cP and f3 = (kd./2) cos 1:1. The spacings dx and dz are assumed to be less than a wavelength (dx < x, d. < X) and 1 N. J. Sonine, Recherches sur les fonctions cylindriques et Ie developpement des fonctions continues en series, llfath. Ann., 16: 1 (1880). 2 See, for example, K. Franz and H. Lassen, "Antennen und Ausbreitung," p. 255, Springer-Verlag OHG, Berlin, 1956.

77


wave propagation

hence the radiation pattern consists of two broadside beams, one in the direction 0 = 1r/2, q, = 1r/2 and the other on the opposite side of the array in the direction 0 = 1r/2, q, = 31r/2, Each of these beams has the maximum value (110)

Umax = Ionm

With the aid of expressions (109) and (110) and the fact that Sr is proportional to U2/r2, definition (94) leads to the following expression for the gain: ' 47rn2m2

g=

r

2•.

r"

cos (~cos

J0 J0

2

?

sm

0) .

Sl~2

0

nex

sm2

ex

.

Sl~2

(111) m{3 dOdq,

sm2/3

Since the array is large, most of its radiation is concentrated in the two narrow broadside beams. Because of the symmetry of the radiation pattern, the q, integration may be restricted to the beam lying in the interval (0,1r), and because of the sharpness of the beam, the following approximations obtain: cos q, "'" 1r/2 - q, dex

= (kdx/2) (cos 0 cos q, dO - sin 0 sin q, dq,) "'" - (kdx/2)dq,

d/3

= - (kdz/2)

sin 0 dO "'" - (kdz/2)dO

Applying these approximations to the integral in Eq. (111), we get (112)

Here the actual limits have been replaced by infinite ones on the ground that the two factors in the integrand rapidly decrease as ex and /3 depart from zero. Since the ex integration yields n1r and the {3integration m1r, expression (112) yields the following limiting value for the gain of the 78

Radiation from wire antennas rectangular array:1 (n, m~

00)

(113)

If the array were backed by a reflector, which eliminates one of the beams and concentrates all the energy in the other, the limiting value of the gain would be twice as large, viz., (n, m~

00)

(114)

In terms of the effective dimensions of the array Lz( = ndz) and mdz) and the effective area of the array A (= LzLz), the limiting values (113) and (114) respectively become

Lz( =

(115)

1 If the limits are chosen to include only the broadside beam, then the integral in Eq. (112) must be replaced by

Integral

=

f"ln --2 sin2ndsin2m{3 f" f" - -{32 dOld{3 = nm f"lm -,,1m -"In 01

-"

-..

sin2xsin2y -2- --2- dxdy X

Y

Since

then sin2 ~ f"_"~d~

=

- sin2 ~ ~

I"

_"

sin 2~ + f"_,,-~-d~

The first term on the right vanishes and the second term is equal to 2Si (211"). Using this result, we get Integral = 4nm[Si (211"))2.Hence the corresponding . f h .. 211"(ndz)(mdz) [ 11" J2 Th.IS resu It agrees expressIOn or t e gam IS g = -y2 2Si (211")' with Eq. (113) since [1I"/2Si(211"))2 is approximately equal to 1. 79


wave propagation

for the array without a reflector, and (116)

for the array with a reflector. It is clear from the above results that the gain of an array can be increased by increasing its size. However, it is also possible in principle to achieve very high gain, i.e., supergain, with an array of limited dimensions.l Since the elements of such superdirective arrays are closely spaced, their mutual interactions playa determining role. These interactions have the effect of storing reactive energy in the neighborhood of the elements and of thus making narrow the bandwidth of the array. 2 Moreover, the large currents that superdirectivity demands lead to high ohmic losses and consequently to reductions in operating efficiency. 3 In addition to narrow bandwidths and low efficiencies, superdirective antennas are burdened with the requirement that the amplitudes and phases of the currents be maintained with a relatively high degree of precision. Superdirective arrays are useful in those cases where a very sharp narrow beam is desired regardless of the cost in bandwidth, efficiency, and critical tolerances. Some aspects of the supergain phenomenon are closely related to the problem of optical resolving power. 4 I S. A. Schelkunoff, A Mathematical Theory of Linear Arrays, Bell System Tech. J., 22: 80 (1943). 2 L. J. Chu, Physical Limitations of Omni-directional Antennas, J. Appl. Phys., 19: 1163 (1948). 3 R. M. Wilmotte, Note on Practical Limitations in the Directivity of Antennas, Proc. IRE, 36: 878 (1948); T. T. Taylor, A Discussion on the Maximum Directivity of an Antenna, Proc. IRE, 36: 1135 (1948); H. J. Riblet, Note on Maximum Directivity of an Antenna, Proc. IRE, 36: 620 (1948). 4 G. Toraldo di Francia, Directivity, Super-gain and Information Theory, IRE Trans. Antennas Propagation, AP-4 (3): 473 (1956).

80

Multipole expansion of the

radiation field One method of expanding a radiation field in multipoles is to develop in Taylor series the Helmholtz integral representations of the scalar and vector potentials and then to identify the terms of the series with formal generalizations of the conventional multi poles of electrostatics and magnetostatics. Another method of expansion consists in developing the radiation field in spherical E and H waves and defining the E waves as electric multipole fields and the H waves as magnetic multi pole fields. In this chapter a brief account is given of these two methods.

4.1 Dipole and Quadrupole Moments We assume that a monochromatic current density J(r') is distributed throughout some bounded region of space. Then by virtue of the conservation of charge there also exists in the region a monochromatic charge density p(r') given by \7' •

J(r') = iwp(r')

(1)

To deduce a relation which we will use for defining the moments of the charge in terms of the current and for framing the gauge of the potentials produced by the charge and 81

4


wave propagation

current, we multiply this equation of continuity by an arbitrary function f(r,r') and then integrate with respect to the primed coordinates. Thus from Eq. (1) we obtain

J p(r')f(r,r')dV'

=

L

jf(r,r')V'

(2)

• J(r')dV'

Here the region of integration includes the entire space occupied by the current and charge, and the normal component of the current is zero on the surface which bounds the region. Using the identity v' . (Jf) = f"V' • J J . V'f, and noting that the term fV'. (Jf)dV' disappears because by the divergence theorem it equals the surface integral ffn' J dS' whose integrand disappears, we see that Eq. (2) leads to the desired relation

+

J p(r')f(r,r')dV'

=

£ J J (r') . V'f(r,r')dV'

(3)

On the proper selection of f(r,r'), the left side of this relation becomes a moment of the charge and the right side becomes an equivalent representation of the moment in terms of the current. When f = 1, relation (3) reduces to f p(r')dV'

=

(4)

0

and we thus see that the total charge is zero. Moreover, when we denote the cartesian components of r' and J by x~ and J a (ex = 1,2,3), and when we successively assume thatf = x~andf == x~x~(ex, (3 = 1,2,3), relation (3) gives rise to the first and second moments of the ,charge, which define respectively the cartesian components pa and Qafj of the electric dipole moment p and the electric quadrupole moment Q:

J p(r')x~ dV' J J",(r')dV' Q",fj = J p(r')x~x/l dV' = ~ J [J a(r')x~ + J fj(r')x~ dV' ] pa

82

=

= ~

(ex

=

1, 2, 3)

(ex, (3 = 1, 2, 3)

(5)

(6)

Multipole expansion of the radiation field It is clear from these expressions that p is a vector and Q is a dyadic whose components constitute a symmetrical matrix, QafJ = QfJa. In vector form, Eqs. (5) and (6) are

J p(r')r' dV' = ~ J J (r')dV' Q = J p(r')r'r' dV' = ~ J (J(r')r' p

(7)

=

+ r'J(r')JdV'

(8)

These relations show how p and Q can be calculated from a knowledge of either the charge or the current. Associated with the electric charge and electric current are their magnetic counterparts, the magnetic charge density Pm (1") and magnetic current density Jm(r'). These conceptual entities serve the purpose of establishing a formal duality between electric and magnetic quantities. The magnetic current density is defined by Jm(r') = (w/2i)r' X J(r') and the magnetic charge density is deduced in turn from Jm(r') by the conservation law 'V'. Jm(r') = iWPm(r'). Since Jm(r') and Pm(r') obey the conservation law, it follows that they also obey a relation which is formally the same as Eq. (3), viz.,

J Pm(r')f(r,r')dV'

=~

J Jm(r') . 'V'f(r,r')dV'

(9)

Choosing f to be successively the cartesian components of 1", and recalling the definition of Jm(r'), we get the following expression for the first moment of the magnetic charge density:

J Pm(r')r' dV'

= ~

J Jm(r')dV'

= ~

J

1"

X J(r')dV'

(10)

Since the first moment of Pm(r') is by definition the magnetic dipole moment m, this equation gives the following expression for m in terms of the electric current: m = ~Jr'

X

J(r')dV'

(11)

Clearly m can be regarded as a pseudo vector whose cartesian com83


wave propagation

ponents arel 3

m-r

I J x~J

= ~

Eali-r

('Y

B(r')dV'

=

(12)

1, 2, 3)

a,B=l

or as an antisymmetrical

dyadic

m

having the cartesian (a, (3 = 1, 2, 3)

components (13)

The simplest current configuration that possesses an electric dipole moment is the short filament of current

J = e.Ioo(x')o(y')

(-l

(14)

~ z' ~ l)

Substituting this expression into Eqs. (7), (8), (11) we find that the electric dipole moment is given by

p

i

(Hi)

= e. - 21Io w

and the electric quadrupole and magnetic dipole moments are zero. The dual of this configuration is the small filamentary loop of current

J = eq,Jq, = eq,Ioo(p' - a)o(z')

(16)

where a is the radius of the loop. Substituting into Eqs. (7), (8), (11) we find in this case that the only nonzero moment is the magnetic dipole moment given by

m

=1 20 ep

X e(r) =

[ J p(r')dV' ] r - J p(r')r'dV'. ] V r eikr

ikr

[

e

+ 72

[J

e:

r

p(r')r/r/

dV' ] :VV

(32)

where in the third term (r' . V)2 has been written as the double scalar product r/r/: VV of the dyadics r'r' and VV. The first term is zero by virtue of Eq. (4). The second and third terms involve respectively the electric dipole and quadrupole moments, as defined by Eqs. (7) and (8). Thus the leading terms of the Taylor expansion of If>(r) can be written in the following concise form: ikr

If>(r)

= - 41rE - 1 (eP'

ikr

V-

r

- 72Q:VV-

e

r

+ ..

-)

(33)

where p is the electric dipole moment and Q the electric quadrupole moment. Similarly, the Taylor series development of the HelmholtZ integral 87


representation

41l-A(r)

=

(471-/p)A(r)

(26) for the vector potential

I-lfJ(r')[l

Considering

wave propagation

- r'. V

+ 72(r'.

is ikr

V)2 -

.

'J~dV' r

(34)

dll' ] . V e;'

(35)

only the first two terms, we get

=

[J

[J

J(r')dV' ] e;' -

J(r')r'

The first integral by Eq. (7) is equal to -iwp. To express the second integral in terms of Q and m, we decompose the dyadic J(r')r' into its symmetric and antisymmetric parts: J(r')r'

=

+ J(r')r'J

%[r'J(r')

- %[r'J(r')

- J(r')r'J

When integrated, the first symmetric part yields by Eq. (8) the sym~ metric dyadic (-iw/2)Q and the second anti symmetric part yields by Eq. (13) the antisymmetric dyadic m. Thus the expansion for the vector potential up to the second term turns out to be 41l'A(r) If m m of

=

.

-tWI-lP-

eikr

. r

+ iWI-l -2 Q'

eikr

V-

r

-

eikr I-lm . Vr

(36)

one prefers to think of m as a pseudo vector, then the operator . V, where m is an antisymmetric dyadic, has to be replaced by X V, where m is a pseudo vector. Accordingly, an alternative form expression (36) is (37)

Hence, for a source that can be described as a superposition of an electric dipole, a magnetic dipole, and an electric quadrupole, the scalar and vector potentials are given by Eqs. (33) and (37). If a source is such that poles of higher multiplicity are required, it becomes more convenient to calculate q,(r) and A(r) by evaluating directly the Helmholtz integrals than to use the method of multipole expansion., 88

Multipole expansion of the radiation

field

4.3 Dipole and Quadrupole Radiation The electromagnetic field of a monochromatic source can be found by substituting the Taylor expansions of the scalar and vector potentials, viz.,

et>(r)

=

_! (p'

VG - ;YzQ:VVG

E

A(r)

=

-iwJL(pG -

where G

E

+ ... )

;YzQ • VG - !:. m X VG + W

(38)

... )

(39)

= eikr/47rr, into the relations

= -Vet>

+ iwA

(40)

which yield the electromagnetic field E, H. By virtue of the linearity of the system, the resulting electromagnetic field may be thought of as the vector sum of the individual electromagnetic fields of the various poles. Since each multipole radiates a spherical wave and the most natural coordinate system for a mathematical description of the radiation is a spherical one centered on the multipoles, we assume that the multipoles are located at the origin of a spherical coordinate system (r,8,et» defined in terms of the cartesian system (x,y,z) by x = r sin 8 cos et>, y = r sin 8 sin et>,z = r cos 8. A consequence of this assumption is that the free-space Green's function G = eikr / 47rr which appears in expressions (38) and (39) is a function of the radial coordinate r only. From expressions (38) and (39) we see that the potentials of the electric~dipole part of the source are

et>elec.dip. =

1

- -E

p . VG

Aelec.dip. = -iwJLpG

(41) (42)

Applying relations (40) to these potentials and using the identities V(p' VG) = (p • V)VG and V.X (pG) = -p X VG, which follow from vector a.-nalysisand the constancy of p, we obtain the electric and mag89


wave propagation

netic field of the electric dipole: 1

Eelee.dip.

= -E

Helee.dip.

=

[(p' V)Va

iwp

X

+ k pG]

(43)

2

va

(44)

Since the gradient operator in spherical coordinates

v .=

a

la

va

the vector

(45)

r sm 9 at/>

which appears in Eqs. (43) and (44) is given by

1) a

ikr

va = v -e41l'r = Moreover,

a

1

er-+er--+e-'-ar r a9

is

(46)

er ( ik - -

r

in spherical coordinates,

p has the form (47)

+

+

where P = VP' P = VPr2 ps2 p2is the strength dipole. Hence the scalar product of p and V yields

of the electric

a + p .•.---r sin1 9 at/>

(48)

When this operator acts on the vector (ajar)e, = 0, (aja9)er = es, and (ajat/»er

(46) and it is recalled that = e sin 9, we get

P •v

=

a P , ar

(p' V)Va

=

+ ps

la - r a9.

erpr ( -k2

'I'

-

2ik - .. r

2)

+ r- a + esps 2

1) a

(ik - - -2 r r

'k + ep", (7 -

1) a

1=2

(49)

Using this result and expression (47), we easily obtain from Eq. (43) the spherical component of Eelee.dip. in terms of the spherical components 90

Multipole expansion of the radiation field of p.

Thus ( - 2ik -

+ 2)

(Er)elec.dip.

= -1 pr Err

(ES)clcc.dip.

=! Ps (ik - ~ + k2) G Err

(E4»clcc.dip.

= -1 P4> Err

(ik -

-

2

1

+

2

(50)

G

(51)

k2 ) G

(52)

In cartesian coordinates, p has the form p

=

e",p",

+ eypy + e.p.

(53)

S calarly multiplying this expression by

Cr,

es, e4>in succession, noting

that pr

=

PS

P' er

=

P' es

P4>

= p'e4>

and recalling that er ' e", = sin eS ' e", =

e cos q, cos e cos q,

er

ey = sin

'

es ' Cy =

e sin q, er ' e. = cos e cos e sin q, es ' e. = ~

e4>'e",=

-sinq,

we get the following connection between the cartesian components of p:

~=~~e~q,+~~e~q,+~~e ps = p", cos e cos q, + py P4>

=

-p", sin

sin

e

e4>'ey = cosq,

and spherical

(54) cos

q, + py cos q,

e sin q, -

p. sin

e

(55) (56)

Substituting these expressions into Eqs. (50) through (52), we find that the spherical components of E.lcc.dip. in terms of the cartesian compo91

Tlteory of electromagnetic

wave propagation

nents of p are given by (Er)elee.dip.

=

! (pz sin

(Ee)elee.dip.

=

! (Pz cos e cos q, + Pu cos e sin

f

e cos q,

+ Pu sin

e sin q, +

f

p. cos

e)

q, - p. sin e)

x (~! + k )G l' 1'2

(.58)

+' k' 2) G:r

(.59)

2

(E '") elec.dip. --

-1 f

( - pz SIll .

(ik

1 q, + Pu cos q,) -l' - -2 l'

To find the spherical components of Helee.dip. in terms of the spherical components of p, we substitute expressions (46) and (47) into Eq. (44) and thus obtain (60)

(H e)elee.dip.

=

iwp", (ik - ~) G

(61)

(H"')elec.dip.

= -iwpe (ik - ~) G

(62)

With the aid of Eqs. (55) and (56) we can express these spherical components of (H)eleo.dip. in terms of the cartesian components of p: (H

e)elee.dip. =

(H",)elee.dip.

iw( -pz sin q,

+ Pu cos q,) (ik

-

0

(63)

G

= -iw(pz cos e cos q, + Pu cos e sin q, - p. sin e) X

(ik - ~) G

(64)

Since (H r)elec.dip. is identically zero and (Er)elee.dip. is not, the radiation field of the electric dipole is an E wave or, equivalently, a TM wave. But as kr is increased, (Er)elee.dip. becomes negligibly small compared to (Ee)elee.dip. and (E",)elee.dip., and therefore in the far zone the radiation 92

Multipole expansion of the radiation field of a TEM

field has the structure er X

(E)elec.dip.

= ~

(H)elec.dip.

wave, and the

is valid there.

simple relation

As kr is decreased, the

magnetic field becomes negligible compared to the electric field, and this electric field approaches the electric field of an electrostatic dipole. As can be seen from expressions (38) and (39), the potentials of the magnetic dipole are cPmag.dip.

= 0

Amag.dip.

= -

(6.5) (66)

I'm X 'VG

When substituted into the second of relations (40), this vector potential yieldl:l the magnetic field of the magnetic dipole. That is, Hmag.dip.

= -

'V X

(m

X

'VG)

(67)

Since m is a constant vector, it follows from vector analysis that 'V X (m x'VG) = m'V2G - (m. 'V)'VG. But 'V2G = -k2G for r > O. Hence the magnetic field (67) of the magnetic dipole may be written alternatively Hmag.dip.

=

as -[(m. 'V)'VG

+

k2mG]

The electric field of the magnetic dipole is found by substituting sions (65) and (66) into the first of relations (40). Thus

(68) expres-

(69) :~ On comparing Eq. (43) with Eq. (68) and Eq. (44) with Eq. (69), we see that the electromagnetic field of an electric dipole, except for certain multiplicative factors, is formally equivalent to the electromagnetic field of a magnetic dipole, with electric and magnetic fields interchanged. Hence, we can obtain the components of the magnetic dipole by simply applying a duality transformation to the already obtained field components of the electric dipole. Since the radiation field of an electric dipole is an E wave and the dual of an E wave is an H wave, the radiation field of a magnetic dipole thus must be an H wave or, equivalently, a TE wave. 93

Them..y of electromagnetic

wave propagation

According to expressions (38) and (39), the potentials quadrupole are

cPelec.quad. =

of the electric

I

(70)

2E Q : 'V'VG iwp,

Aelec.quad.

=

2 Q . "VG

(71)

With the aid of relations (40), these potentials yield the following expressions for the electric and magnetic fields of the electric quadrupole:

Eelec.quad.

= -

I

+ k (Q.

,,['V(Q:'V'VG)

iw

Heleo.quad,

= 2'

ererQrr

(72)

'V X (Q . 'VG)

First let us find the components Q can be written as

Q=

'VG)]

2

;

Q •• sin (J cos (J

+

(QXY

+

+ Qyy

er

(82)

yields

cos (J sin (J sin2 cJ>

Qyx) cos (J sin (J cos cJ>sin cJ>

cos2 (J cos cJ>- Qzx sin (J cos cJ>+ Qy. cos2 (J sin cJ>- Q.y sin2 (J sin cJ> 2

and Q~r

= e~ . Q • e, =

(Qyy

.+ QyX

-

Qxx) sin (J sin cJ>cos cJ>-

QXY sin (J sin2 cJ>

sin (J eos2 cJ>- Qx. sin cJ>cos (J

+ Qy.

cos cJ>cos (J

Invoking the symmetry of Q and using some simple trigonometric identities, we reduce these results to QBr

= ~ sin 2(J(Qxx cos2 cJ> + Quv sin2 cJ>- Q •• + QXY sin 2cJ»

+ cos 2(J(Qx. Q~r

= ~(Qyy

-

Qxx) sin (J sin 2cJ> -

+ Qyx

cos cJ>+ Qy. sin cJ»

(83)

sin (J cos 2cJ>

Qx. sin
cos (J (84) 95


wave propagation

Substituting Eqs. (84) and (83) into Eqs. (80) and (81), we obtain the following expressions for the spherical components of the far-zone magnetic field of an electric quadrupole in terms of the cartesian components of the quadrupole moment: (H8)elec.quad.

=

iwk2 eikr 41l"r [~(Qyu

2

+ QyZ sin (H

_ 4»elec.quad. -

-

- Qzz

(J cos 2et> -

.. - Qzz) sm (J sm 2et> Qzz sin et>cos (J

+

QyZ cos et>cos (J]

(85)

iwk2 eikr 1 • 2 • 2 2 41l"r [~ sm 2(J( Qzz cos et> + Qyy sm et>

+ QZY sin 2et» + cos 2(J(Qzz

In the far zone the relation

er X .

Eelec.quad.

cos et>

=

+ QyZ sin et»]

f!!: ~;

Hele •.quad.

(86)

is valid.

Consequently the spherical components of the far-zone electric field of the electric quadrupole are derivable from expressions (85) and (86) by use of the following simple connections:

(87)

(88) Alternatively one may calculate the far-zone = -iWMh"

components of multi pole field n and order m -n 5 m 5 n)

:() Y"m

With the aid of relations (125) and (126) it follows from Eqs. (109) and (110) that r . E and r . H can be written as1 r' E

= ~

(133)

a"mn(n + l)lf"m

n,m

(134) n,'n

By virtue of the orthogonality of the functions If,,m over the surface of a sphere, it is obvious from expansions (133) and (134) that the coeffi1 Comparing these expressions for r' E and r . H with expressions (106) and (107) for v and u we see that apart from the factor n(n 1) they are respectively the same.

+

103


wave propagation

cients anm and bnm can be determined from a knowledge of the scalar functions r . E and r . H over the surface of a sphere. Accordingly we now shall find the expansion coefficients anm, bnm from the currents within V 0 by first calculating r . E and r . H in terms of the currents and then incorporating these results with expansions (133) and (134). The validity of this procedure is assured by the theorem 1 that any electromagnetic field in the empty space between two concentric spheres is completely determined by the radial components Er, Hr' Our task now is to obtain r . E and r . H in terms of the currents within Vo. We recall (see Sec. 1.1) that the E and H produced by a monochromatic J must satisfy the Helmholtz equations

v X V X H - k2H V X V X E -

=

k2E

VX

=

J

(135) (136)

iwp.J

When these equations are scalarly multiplied by the position vector r, we get the relations r . V X V X H - k2r . H r •V X V X

E - k2r • E

=

J

r. V X

(137)

= iwp.r • J

(138)

which by vector analysis reduce2 to (V2

+k

(V2

+k

2)

2)

(r . H)

(r .

E

= -

r .V X

+ i.-

r .J)

WE

J

(139)

r.

= .;... "'WE

VXVX

J

(140)

In terms of the free-space Green's function ,

G(r,r)

eiklr-r'l

= 41rr-r I '1

(141)

Bouwkamp and Casimir, loco cit. We use the vector identity (V2 + k2)(r' C) = 2V. C + r' V(V. C) r . V X V X C + k2r • C, where C is an arbitrary vector field. To obtain Eq. (139) we let C = H. To obtain Eq. (140) we let C = E and C = J successively. 1

2

104

Multipole expansion of the radiation

field

the solutions of the scalar Helmholtz equations (139) and (140) whioh satisfy the Sommerfeld radiation condition are r' H r'

E

=

J

=

J:-

Vo

v'

G(r,r')r'.

J -

r'

1-WE

J:- J 1-WE

(142)

X J(r')dV'

Vo

G(r,r')r'.

V' X V' X J(r')dV'

(143)

These relations are valid for r inside and outside Vo• For r outside V 0, the first term on the right side of Eq. (143) is identically zero and we have

.i J

G(r,r')r'.

v'

It is known that for r'

~kl sin a)

=

Po!1

+ cos (kl

is proportional

to

(155)

sin a)]

where Po denotes the power fed to the receiver by a single antenna. As a point source of radiation sweeps across the sky, the angle a changes and P oscillates between the limits 0 and 2Po• This is strictly true for small a only. Actually, when a becomes large, the power fed to the receiver is no longer given by expression (155) alone, but by the product of expressio)1 (155) and IFj2. The factor IFI2 has the effect of tapering off the oscillations (Fig. 5.9). The nulls of the receiving pattern occur where kl sin a

= (2n

+ 1)7r

(n

=

0, 1,2, ...

(156)

)

and the maxima occur where kl sin a

=

2n7r

(n

=

0, 1, 2, ...

(157)

)

For small values of a, i.e., for values of a such that sin a

=

a, the width

We obtain this expression from Eq. (79) of Chap. 3 by setting n = 2 and The fact that the receiver of the interferometer is located at the electrical center of the transmission line connecting the two antennas requires that'Y = O. 1

l' = O.

154


of each grating lobe is given by the simple relation 211"

Aa

= kf =

X

(158)

r

which shows that as the spacing l is increased the width of each grating lobe is decreased. It also shows that for a fixed spacing the width of each grating lobe is decreased as the wavelength Xto which the receiver is tuned is decreased. In the derivation of formula (155) it was tacitly assumed that the incident radiation comes from a point source. We now shed this restriction and consider the more realistic case where the source has angular extent. In this case the received power is given by P(ao)

= I[1

+ cos (kl

sin a)Jf(a

(159)

- ao)da

where f(a - ao) is the distribution across the incoherent source and ao is the angle that the mean direction of the source makes with the axial plane. If the width of the source is 2w, the limits of int~gration are a = ao - wand a = ao + w. We assume that ao and 2w are small, i.e., we assume that the source is narrow and near the axial plane. Expression (159) is a generalization of expression (155) and reduces to it when f(a - ao) is the Dirac delta function Il(a - ao).

cx __ Fig. 5.9

Rectangular plot of receiving pattern of two-element interferometer for point source. The minima are zero. The maxima are tapered, by virtue of the fact that IFI2 is not equal to one for all values of a. Actually, IFI2 behaves in a manner indicated by the envelope. 155


Fig. 5.10

wave propagation

Rectangular plot of receiving pattern having uniform distribution.

ao--for a narrow source

However, if f(a - ao) has a narrow rectangular shape, i.e., if = Po/2w for ao - w ::::;; a ::::;; ao + wand f(a - ao) = 0 for all other values of a, expression (159) leads to

f(a - ao)

P(ao) = f(1

+

cos kla)f(a

- ao)da = Po(l

+

V cos klao)

(160)

where the quantity V defined by V

= sin klw klw

(161)

is the "visibility factor," a term borrowed from optics.l As the rectangular distribution sweeps across the sky, ao changes and P(ao) oscillates sinusoidally between Po(l - V) and Po(1 + V). The ratio of the minimum value to the maximum value is the modulation index M given by 1-

M=I+V

V

(162)

From this we see that if the distribution function is rectangular the width of the source can be determined by measuring M and then computing w from Eqs. (161) and (162). See Fig. 5.10. 1 See, for example, M. Born and E. Wolf, "Principles of Optics," pp. 264-267, Pergamon Press, New York, 1959.

156


More generally, if the source is narrow but otherwise arbitrary, it follows from Eq. (159) that for small values of ao the received power is given by

+ cos (kl sin a)J.f(a - ao)da = f[1 + cos (kla)]f(a - ao)da = fj(a - ao)da + f cos (kla)f(a -

= J[l

P(ao)

ao)da

(163)

The first term on the right is Po, the power fed to the receiver by one antenna; the second term on the right we denote by Pl. Accordingly, we write (164)

where PI = f cos (kla)f(a

(165)

- ao)da

If we let u = a - ao and note that f(u) == 0 for lui> w, then Pi can be cast in a form that explicitly displays its amplitu4e and phase, viz., Pi =

J cos [kl( u + ao) ]f( u )du

= Re eiklao

= Re eiklaoQ(kl)ei~(kl)

J_"'",eik1uf( u )du

= Q(kl) cos [klao + cf>(kl)] (166)

Here the amplitude Q(kl) and the phase cf>(kl) are defined by . (167) The inverse Fourier transform of Eq. (167) yields the relation f(u)

1J:'"

= -

11" 0

Q(kl) cos [klu -

cf>(kl)]d(kl)

(168)

which expressesj(u) in terms of the amplitude and phase of the observed quantity Pi, viz., Q(kl) and cf>(kl). Relation (168) shows that it is possible, in principle, to find the distribution by measuring the amplitude 157


wave propagation

and phase with different base lines. However, the measurement of phase sometimes presents difficulties. Unfortunately, it is not possible to determine uniquely the distribution from a knowledge of the amplitude alone, unless some information is available beforehand about the general shape of the distribution function. As an example of a two-element radio interferometer with a horizontal base line we mention the one in Owens Valley, California, which is operated by the California Institute of Technology~ Each element of the interferometer is a steerable parabolic reflector antenna, 90 feet in diameter, placed on the ground. It is used for the measurement of angular diameters at centimeter and decimeter wavelengths, and for positional work. 1 A two-element interferometer having a vertical base line can be effected by placing a single horizontally beamed antenna on a cliff of height l/2 overlooking the sea. The surface of the sea acts as an image plane. Thus the elevated antenna and its image constitute a twoelement interferometer. 2 The elevated antenna is horizontally polarized to take advantage of the fact that the surface of the sea approximates a perfect reflector most closely for horizontal polarization. The image antenna is out of time-phase with respect to the elevated antenna and hence the power received from a point source is given by P(OI)

=

(169)

2Po[1 - cos (kl sin 01)]

where 01 is the angle the direction of the source makes with the axial plane, i.e., the surface of the sea, and Po is the power the elevated antenna would receive if it were not operating as an interferometer. In this case the nulls of the receiving pattern occur where kl sin

01

=

2n1l'

(n

=

(170)

0, 1,2, ... )

and the maxima occur where kl sin

01

=

(2n

+ 1)11'

(n = 0, 1, 2, ...

)

(171)

1 For details the reader is referred to J. G. Bolton, Radio Telescopes, chap. 11 in G. P. Kuiper and B. M. Middlehurst (eds.), "Telescopes," The University of Chicago Press, Chicago, 1960. 2 An interferometer of this type is called a "sea interferometer," a "cliff interferometer," or a "Lloyd's mirror" after its optical analog.

158


From Eq. (169) we see that as the source rises above the horizon and cuts through the grating lobes of the interferometer, the received power increases from zero and oscillates in characteristic fashion. Then as the source rises above and out of the beam, the received power gradually tapers to zero, an effect which would have been displayed by Eq. (169) had it been multiplied by IFj2.

5.8 Correlation Interferometer The two-element interferometer discussed in the previous section behaves as though the incident radiation were monochromatic because the receiver of the interferometer is sharply tuned and accepts only a very narrow band of the incident radiation's broad spectrum. Since the energy residing outside this band is rejected and thus wasted, the sensitivity of the interferometer is limited by the bandwidth of the receiver. Increasing the bandwidth would increase the sensitivity but would also deteriorate the multilobe pattern and hence decrease the precision of the system. This means that in a phase-comparison type of interferometer the bandwidth is necessarily narrow and the sensitivity is limited by the bandwidth. In addition to this inherent limitation on the sensitivity there is a practical limitation on the resolving power. As the antennas are moved farther apart for the purpose of increasing the resolving power, it becomes more difficult to compare accurately the phases of the antenna outputs. The awkwardness of measuring the phases of two widely separated signals places a practical limitation on the antenna separation and this in turn places a limitation on the resolving power. Because of these and other limitations, the two-element phase-comparison interferometer has been superseded in certain applications by more sophisticated systems. In this section we shall discuss one such system, namely, the correlation interferometer of Brown and Twiss.l But before we do this, let us discuss the concept of degree of coherence2 upon which it is based. 1 R. H. Brown and R. Q. Twiss, A New Type of Interferometer for Use in Radio Astronomy, Phil. Mag., 45: 663 (1954). 2 F. Zernike, The Concept of Degree of Coherence and Its Application to Optical Problems, Physica, 5: 785 (1938).

159


wave propagation

To measure the degree of coherence of the polychromatic radiation from an extended source we use two identical and similarly oriented antennas. These antennas receive the incident radiation and consequently develop at their respective output terminals the voltages VI(t) and V2(t), which for mathematical convenience are assumed to have the form of an analytical signal. The resulting voltages are fed into a receiver whose output is the time-average power given by P

=

([VI(t)

+ V (t)][Vt(t) + VW))) = (VI(t) Vi (t» + (V (t)VW» + 2 Re (VI(t)vt(t» 2

2

(172)

The first term on the right is the time-average power output of one antenna operating singly, and the second term is the time-average power of the other antenna operating singly. Hence, the third term is the only one that involves the mutual effects or mutual coherence of the incident radiation. Accordingly, as a quantitative measure of the mutual coherence of the incident radiation, we choose the complex quantity l' defined by

l' =

(V I(t) Viet»~ v(VI(t)ViCt»(V2(t)vi(t)

(173)

and referred to as the complex degree of coherence. The modulus h'l of l' is known as the degree of coherence. By the Schwau inequa.lity it can be shown that 11'1s:; 1

(174)

When 11'1= 0 the incident radiation is incoherent; when 11'1= 1 the incident radiation is coherent; and when 0 < 11'1< 1 the incident radiation is partially coherent. In terms of 1', expression (172) for the power output of the receiver becomes P

=

(VI(t)Vi(t»

+ (V (t)Vi(t» + 2 V(VI(t) Vi(t»(V (t) 2

2

Vi(t»

11'1cos (arg'Y)

(175)

where arg'Y is the phase of 1', that is, l' = 11'1exp (i arg 1'). For simplicity we assume that the power outputs of the antennas when 160

Radio-astronomical

antennas

operated separately are equal; that is, we assume that (176) With the aid of this assumption expression (175) reduces to the relation P = 2Po[1

+ h'l cos (arg'Y)]

(177)

which clearly indicates that the degree of coherence 11'1 of the incident radiation is measured by the visibility. Expression (177) provides an operational definition o(the degree of coherence. To show how I' is related to data that specify the source and to the spacing of the antennas we proceed as follows. We choose a cartesian coordinate system with origin 0 in order that the antennas be located along the x axis at the points x = =+= 1/2. For simplicity the source is assumed to be a line source lying along the 1; axis of a parallel cartesian system with origin 0'. The distance between 0 and 0' is R. See Fig. 5.11. We think of the source as being divided into elements of length dh, db dl;a, . . . , and we denote the respective antenna output voltages due to the radiation from the mth element by the analytic signals tr ml(t) and Vm2(t). The respective antenna output voltages due to the radiation from the entire source are given by the sums (178)

We assume that each element of the source is an isotropic radiator. Consequently the radiation from the mth element produces the voltage

V ml(t) = Am

(t -

Rml)

-

C

e-iw(t-Rmdc)

R

(179)

ml

in one antenna and the voltage

R V m2(t) = Am ( t - ~ C

2) .

e-iw(t-Rm,/c)

R .

(180)

m2

in the other. Here Rm1 and Rm2 are the distances from the mth element to the antennas, C is the velocity of light, w is the mean frequency 161


wave propagation

Line source

Rm2

Anten na No. 1

Antenna NO.2

\

/

x--1/2

Fig. 5.11

Arrangementfor the measurement of degree of coherence. Two identical and similarly oriented antennas are exposed to the polychromatic radiation from line source. R is distance from 0 to 0'. The angle that the line connecting 0 with the center of the line source makes with the axial line 00' is ao. The angle that the line from 0 to the element d~m makes with the line 00' is am. The distances from the element d~m to the antennas are Rm1 and Rm2.

of the incident radiation, and Am is the complex amplitude It follows from expressions (178) that (V1(t)vt(t»

= ~ m

(Vml(t)V;::l(t»+

function.

(181)

~~ (Vml(t)V:1(t» fn;Jlin

However, the isotropic radiators that make up the source are assumed to be statistically independent and to have a mean value of zero, Le.,

when m 162

:;6

n

(182)

Radio-astronomical

and consequently

the cross-product

antennas

terms of Eq. (181) vanish.

Thus

we get

(VI(t)vt(t»

=

L (Vml(t)V~I(t»

(183)

m

Similarly, we obtain

(V 2(t)VW»

=

L (V m2(t)V~2(t»

(184)

m

and (VI(t) VW»

=

L (V ml(t) V~2(t»

(185)

m

Substituting expressions (179) and (180) into Eqs. (183) and (184) respectively, and noting that Am is stationary, we see that

(VI(t)Vt(t»

= ~

(t _ Rml) A * (t _ Rml)\

_1_ / A

';;:RmI2

\

m

C

m

=

C

/

l R~12 (Am(t)A~(t»

(186)

m

and

(V2(t)VW»

=

f

R~22 (Am (t - R;2) A~ (t ~ R;2))

=

l R~22 (Am(t)A~(t»

(187)

m

Since Rmi and Rm2 are approximately equal, these two expressions in this approximation are equal to each other and to Po. That is, in agreement with assumption (176) we have (188)

Substituting

expressions

(179) and (180) into Eq. (185), we obtain

163


wave propagation

where k = wle. Since Rm1 and Rm2 are approximately equal, and since Am is stationary, Eq. (189) reduces to (190)

To cast this expression into the form of an integral, we introduce the following geometric considerations. From Fig. 5.11 it is clear that (191)

Since R»

(~m + J) and R» (~m- J) it follows from Eqs. (191)

that Rm2 - Rm1 =

_ ~l

and

(192)

Using these approximations, we see that Eq. (190) becomes (V1(t)VW»

=

L

~2

(Am(t)A;::(t»eiklt ••!R

(193)

m

Moreover, from Fig. 5.11 it is clear that tan am small we have the simpler relation

= ~mIR, but since am is (194)

With the aid of relation (194) we may cast Eq. (193) in the following form: (V1(t)VW»

=

l

~2

(Am(t)A;::(t»eikla..

(195)

m

which suggests that the sum may be written as an integral.

164

If we let

Radio-astroIiomical antennas

then Eq. (195) in the limit becomes (196)

where ao is the angle that the line connecting 0 with the center of the source makes with the axial plane. Substituting Eqs. (188) and (196) into expression (173), we find that the complex degree of coherence is related to the source distribution function f(a - ao) and to the antenna spacing kl by the relation 'Y

= -l Po

J f(a

- ao)eik1a da

(197)

Since Po

(198)

= ff(a - ao)da

we may also write

'Y

in the homogeneous form

ff(a - ao)eik1a da ff(a - ao)da

'Y=-~----

(199)

If in accord with the notation of the previous section we denote the amplitude and phase of the integral appearing in Eq. (196) by Q(kl) and q,(kl) respectively, then we may write - ao)eik1a da

ff(a

= Q(kl)eikla'ei4>(kl)

(200)

and from Eqs. (199) and (200) note that the degree of coherence is given by

I I= 'Y

Q(kl)

(201)

Q(O)

and the phase of 'Y by arg

'Y

= klao

+ q,(kl)

(202)

Substituting expressions (201) and (202) into Eq. (177), we obtain the 165


wave propagation

expression

= 2Po { 1

P

Q(kl) + Q(O)

cos [klao

+ q,(kl)] }

(203)

which places in evidence the equivalence of visibility and degree of coherence. In the special case where f(a - ao) is a rectangular function of width 2w, that is, f(a - ao) = Po/2w for ao - w ~ a ~ ao + wand f(a - ao) = 0 for all other values of a, we have

J f(a

.

- ao)e'kla da

Po fa.+w 2w a.-w

= -

Hence for such a rectangular given by

I I= 'Y

Q(kl) Q(O)

=

. da e,kla

=

distribution

sin klw klw

sin klw e,kla. . Po -klw

(204)

the degree of coherence is

(205)

and the phase of 'Yby arg'Y

=

(206)

klao

Thus we see that for a uniform source the magnitude 'Yis related by Eq. (20.5) to the width of the source and the phase arg'Y is related by Eq. (206) to the angle between the axial plane and the line running from the origin to the center of the source. The correlation coefficient p(V1,V2) of VI(t) and V2(t) by definition is (207) where (V1)

= «VI - (V1»(vt - (Vi»)

(V2)

= «V2

2

U

2

U

-

are the variances.

(V2»(V:

(208)

- (v:m

Since VI(t) and V2(t) have zero mean value, i.e., (209)

166

Radio-astronomical

the expression for

p

antennas

reduces to (210)

Comparing expressions (173) and (210), we see that the complex degree of coherence 'Y is equal to the correlation coefficient p(V 1,V 2)' Hence what is actually measured in the above arrangement is the amplitude and phase of the correlation coefficient p( V 1, V 2). Now let us suppose that the circuits are changed (to a Brown and Twiss system) so that we can measure the correlation coefficient of the square of the moduli Ml(t) and M2(t) of Vl(t) and V2(t) respectively. The correlation coefficient p(M 12 ,M 22) by definition is p(M 12,M 22)

=

«M12 - (MI2»(M22 - (M22») u(M !2)u(M 22)

(211)

where u2(M

12)

=

«M

12 -

(M 12»2)

u2(M22) = «M22 - (M22»2)

(212)

Under the assumption that the receiver noise is negligible compared to the desired signal, it can be shown by statistical calculations! that (213)

But

and hence (214) 1 E. N. Bramley, Diversity Effects in Spaced-aerial Reception of Ionospheric Waves, Proc. Inst. Elec. Engrs., 98 (3): 9-25 (1951); also, J. A. Ratcliffe, Some Aspects of Diffraction Theory and their Application to the Ionosphere, Rept. Prog. Phys., 19: 188-267 (1956).

167


wave propagation

This means that the correlation coefficient of the squares of the moduli of the antenna voltages is equal to the square of the degree of coherence of the incident radiation. In the case where the source is a rectangular distribution of width 2w, we see from Eqs. (205) and (214) that M

p( .

2 1,

M 22)

= I sinklwklw

2

1

(215)

With the aid of this result w can easily be computed from a knowledge of the correlation coefficient p(M 12,M 22). Although p(M 12,M 22) yields information about I'YI only, and p(V1, V2) yields information about I'YI as well as arg 'Y, the former is easier to measure, as no phase-preserving link between the antennas is required. The correlation interferometer of Brown and Twiss may be defined as an interferometer that measures p(M 12,M 22). It differs from a conventional interferometer, which measures p(V1, V2). Since no radio~ frequency phase-preserving link is necessary in the measurement of p(M 12,M 22), the antennas can be separated greatly and thus high resolving powers can be realized.

168


In recent yea-Faconsiqerable attention has beeil focused on the theory of electl'Omagnetic wave propagation in a plasma medium. In large measure this interest in the theory has been stimulated by its applicability to current problems in radio communications, radio' astronomy, and controlled thermonuclear fusion. For example, the theory has been invoked to m'plain such phenomena as the propagation of radio Waves in the ionosphere,! the propagation of cosmic radio waves in the flolar atmosphere, in nebulae, and iil interstellar and interplanetary space, 2 the reflection of radio waveS frpm meteor tra.ils3 and from the envelope of ionized g!ts that s\lrrOl,l,m:Js a !SPacecraft as it pen~trates 1 K. G.Budden, "Radio Waves in the Ionosphere," Cambridge University Press, New York, 1961; also, J. A. Ratcliffe, "The Magneto-ionic Theory," Cambridge University Press, New York, 1961. 2 V. L. Ginzburg, "Propagation of Electromagnetic Waves in Plasma," Gord{)n l!tod Breach, Science Publishers, Inc., New York, 1961; also, I. S.Shldovsky, "Cosmic Radio Wlleyes," Harvard Uni~ versity Press, Cambridge, Mass., 1960. 3 N. Herlofson, Plasma Resonance in Ionospheric Irregularities, ArkjlJ Fysik, 3: 247 (1951); also, J. L. Heritage, S. Weisbrod, and W. J. Fay, "Experimental Studies of Meteor Echoes at 200 M~gll.

Wp

(85) (86) (87)

These expressions show the marked difference in behavior between a wave whose operating freq~ency is greater than the plasma frequency and a wave whose operating frequency is less than the plasma frequency. When w > Wp, the wave travels without attenuation at a phase velocity greater than that of light in vacuum. On the other hand, when w < Wp the wave is evanescent (nonabsorptively damped) and carries no power. At w = Wp the wave is cut off; the magnetic 184

Elect.romagnetic waves in a plasma

field is zero and the electric field must satisfy V' X E(t) = o. Hence, at cutoff a transverse electromagnetic wave cannot exist. However, a longitudinal electrical wave, sometimes called a~'plasma wave" or "electrostatic wave," can exist. To examine the properties of such a wave, spatial dispersion must be taken into account. There are three types of velocity that pertain to the transverse wave: the phase velocity Vph, whose value can be found from a knowledge of 71 by using the relation Vph = c/'T/; the group velocity Vg, which by definition is iJw/iJ{3; and the velocity of energy transport Ven, which is defined by the ratio S./7IJ. Again restricting the discussion to a lossless plasma, we see from expressions (85) that the phase and group velocities are given by (88)

(89)

Since an increase of wavelength (or, equivalently, a decrease of frequency) results in an increase in phase velocity, the dispersion is s~id to be "normal." To find Ven, we note that the time-average value of the Poynting vector of the wave is z directed and has the value -

S.

= 72 Re e.' (E

X

H*)

= 72 Re ~ -JJ.oE Eo'

Et

(90)

Moreover, we note that the time-average energy density (77) in this case reduces to (91)

Therefore, the velocity of energy transport assumes the form V~n =

S. -=- = w

(7~)

(72)E

Re

V~

+ (7::i)W

ilE/ilw

(92) 185


wave propagation

Substituting E = Eo(l - wp2 / w2) into this form, we find that the velocity of energy transport in a lossless plasma is given by Ven

=

W2

1-

C

w2

~

(93)

(W ~ Wp)

..2...

which is identical to expression (89) for the group velocity. Let us consider now a plasma with small losses. In the limiting case where lEI» u/w, the losses are incidental and expressions (82), (83), and (84) reduce to

{3=wV;;

(W ~ wp)

(94)

Using relations (44) and (45), i.e., E

=

Eo

(1 -

2 ~2

W

Well

2)

(95)

we see that expressions (94) yield (96) (97)

(98)

and the corresponding phase and group velocities are given by C Vph

= ~

2

1

(99)

.Wp

-

w

2

+

Weu2

(100)

Comparing expression (88) with expression (99), we see that the phase velocity is decreased by the presence of loss. On the other hand, comparing expression (89) with expression (100), we see that the group 186


velocity is increased by the presence of loss. The interpretation of group velocity as the velocity of energy transport breaks down when the medium is dissipative.

6.5 Dielectric Tensor of Magnetically Biased Plasma When a magnetostatic field Bo is applied to a plasma, the plasma becomes electrically anisotropic for electromagnetic waves. That is, the permeability of the plasma remains equal to the vacuum permeability p.o, whereas the dielectric constant of the plasma is transformed into a tensor! (or dyadic) quantity £. To derive the dielectric tensor of a magnetically biased plasma, which for simplicity is assumed for the present to be lossless, we use the macroscopic equation of motion (35). In the present instance this equation reduces to - inmwv

= nq(E

+v

(101)

X Bo)

and yields the following expression for the macroscopic velocity of the plasma electrons: v=

-w2(q/m)E

- iW(q2/m2)E -iw

[(~

X Bo

Bo) .(~

+ (q3/m3)(E.

Bo)Bo

Bo) - w2]

(102)

Since the density of the electronic convection current J by definition is equal to nqv, it follows from expression (102) that J is given by

where Wp is the plasma frequency (wp2 = nq2/mEo)

and where the

1 See, for example, C. H. Papas, A Note Concerning a Gyroelectric Medium, Calteeh Tech. Rept. 4, prepared for the Office of Naval Research, May, 1954.

187


amplitude1

(,)Q

WQ

wave propagation

of the vector

== !I Bo

(104)

m

represents the gyrofrequency of the electrons. From a knowledge of J we can find the dielectric constant of the plasma by noting that the total current density is the sum of the convection current density J and the vacuum displacement current density -iWEOE, and then by regarding this total current density as a displacement current in a dielectric medium whose dielectric constant I: is fixed by the relation

J - iWEoE = -iwl:'

(105)

E

According to expression (103), it appears that J is generally not parallel to E; the quantity I: must be a tensor or dyadic to take this into account. Since, by definition, the displacement vector D is calculated from D = I:.E

(106)

the tensor character of I: also means that D is not generally parallel to E. Although a tensor is independent of coordinates, its components are not. If we are given the components of a tensor with respect to one coordinate system, we can find its components with respect to any other coordinate system by applying the transformation law connecting the coordinates of one system with those of the other. Therefore we are free to choose any coordinate system without risking loss of generality. In the present instance, for simplicity, we choose a cartesian system of coordinates (x,y,z) whose z axis is parallel to Bo, that is, Bo = ezBo; Cz is the z-directed unit vector. When Bo > 0, the vector Bo is parallel to the z axis; and when Bo < 0, the vector Bo is antiparallel to the z axis. The components of I: in this cartesian system are denoted by Eik, with i, k = x, y, z. Substituting expression (103) into Eq. (105) leads to the following expressions for the components Eik of I: in the cartesian system whose 1

Wg

This means that

= (-lql/m)Bo.

ISS

WQ

= (q/m)Bo.

For electrons

q

is negative and hence

Electromagnetic

waves in a plasma

z axis is parallel to Do: Ezz

=

EO

(1 -

2 Wp2 W

-

2)

= Euu

(107)

Wg

(108) (109) The remaining components Ezz, Ezz, Euz, Ezu are identically zero. We note that when the magnetostatic field Bo vanishes, Wg vanishes and the diagonal terms become equal to each other, i.e., (110) and the off-diagonal terms disappear. That is, when Bo = 0, the plasma becomes isotropic as it should. Also we note that when Bo is replaced by - Bo, the gyrofrequency Wg changes sign and, consequently, the components satisfy the generalized symmetry relation (111) as must the components of the dielectric constant of any medium whose anisotropy is due to an externally applied magnetostatic field.l In addition we see that the components constitute a hermitian matrix, i.e., (112)

The hermitian nature of the dielectric tensor results from the assumption that the plasma is lossless. Expressions (107), (108), and (109) for the components of the dielectric tensor may be easily generalized to take into account collision losses. For the case where the collision losses are appreciable, we must add to the right side of Eq. (101) a collision term. Thus for the equal See, for example, A. Sommerfeld, "Lectures on Theoretical Physics," vol. 5, "Thermodynamics and Statistical Mechanics," p. 163, Academic Press Inc., New York, 1956.

189


wave propagation

tion of motion of the electrons we get

=

-inmwv

where form

Well

Bo) ~

iWell)V

=

+v

nq(E

(113)

nmVWefl

is the collision frequency.

+

-inm(w

+ v'x

nq(E

Rewriting this equation in the

X Bo)

(114)

and comparing with Eq. (101), we see that the resulting expression for J is the same as expression (103), with W replaced by W iWell' It then follows from Eq. (105) that the cartesian components of the dielectric tensor of a lossy dielectric are given by

+

, = (1

Exx

EO

-

Wp2(W

W

+ iWell)

[( W +')2~Well

E~. = Eo

)

Wg

2]

,

(115)

= Ew

,

Wp2Wg

•

-~EO

-

+

W(W

[1 _ (

ww

iWel1

+

~2.

Wg)(W

)]

+

iWel1

-

Wg)

-£1/%

(116)

(117)

~Well

where the prime is used to distinguish the lossy components from the loss-free ones. As in the lossless case, we again have (118)

but, unlike the lossless case, the components is given by E

= -

W)2

-

( C

ei(w/c)(I-X/2)r

p. sin 8

4

1I"Eor

(eg cos {3r -

(219)

e", sin (3r)

for a z-directed dipole of moment p. parallel to the biasing field Bo, and by W)2

E

= ( C-

p. Vi

-

ei(w/c)(I-X/2)r

sin2 8 cos2 t!>

471"

[ee cos ({3r

Eor

-

+ a)

e", sin ({3r

+ a)]

(220)

for an x-directed dipole of moment p. perpendicular to the biasing field Bo• Here {3 = k (~)XY cos 8 and a = tan-I (tan t!>/cos 8). Comparing these expressions with the corresponding ones for a dipole in an isotropic plasma, we see that in the case of high frequencies the anisotropy does not change the amplitude VE . E* of the radiated field E but does change its state of polarization: it causes the field to undergo Faraday rotation.

6.9 Reciprocity Let EI, HI be the electromagnetic field occupying a finite volume V I and let E2, H2 radiated by a current J 2 occupying another source currents oscillate monochromatically the medium occupying the space V 3 outside and may be inhomogeneous. Clearly EI, HI are related to JI and E2, equations

v

X HI

212

= JI - iwt. EI

radiated by a current Jl be the electromagnetic field finite volume V 2. The two at the same frequency and of VI and V 2 is anisotropic H2 are related to

J2

by the

(221)

Electromagnetic

waves in a plasma

Multiplying the first one by E2 and the second one by El, and then subtracting the resulting equations, we get E2• yo X HI - El• yo X H2

= E2•

Jl

-

E1•

- iwE2

J2

• t •

El

+ iwE

l

• t •

E2

(222)

With the aid of yo X El = iw~oHl and yo X E2 = iw~oH2 we write the left side of Eq. (222) as a divergence and thus obtain yo • (El X H2

-

E2 X HI)

= E2

• J 1 -

E1

- iwE2

• J 2

• t •

E1

+ iwE

I

• t •

E2

(223)

Integrating this relation throughout all space and converting the left side of the resulting equation to a surface integral which vanishes by virtue of the behavior of the fields over the sphere at infinity, we are led to the expression (224) where (225)

When U is zero, Eq. (224) yields the relation (226)

which defines what we usually mean by reciprocity. 1 That is, two monochromatic sources are said to be reciprocal when the source cur1 The reciprocity theorem for electromagnetic waves is a generalization of Rayleigh's reciprocity theorem for sound waves (see Lord Rayleigh, "Theory of Sound," 2d ed., vol. II, pp. 145-148, Dover Publications, Inc., New York, 1945) and stems from the work of Lorentz [see H. A. Lorentz, Amsterdammer Akademie van Wetenschappen, 4: 176 (1895-1896)]. For a detailed discussion .see P. Poincelot, "Pr6cis d'6lectromagn6tisme th6orique," chap. 18, Dunod, Paris, 1963.

213


wave propagation

rents and their radiated electric fields satidy relation (226) or, equivalently, when the quantity U vanishes. Clearly, U vanishes when the dielectric constant of the medium is symmetric (Elk = EM). However, for a magnetically biased plasma the dielectric constant is hermitian and hence U does not necessarily vanish. This means that in the case of an anisotropic plasma reciprocity does not necessarily hold. Nevertheless, the concept of reciprocity can be generalized, at least formally, to include the case of an anisotropic plasma.! Such a generalization is based on the fact that the dielectric tensor of a magnetically biased plasma is symmetrical under a reversal of the biasing magnetostatic field, i.e., (227)

or t(Be)

=

(228)

t( - Be)

where the tilde indicates the transposed dyadic. When the biasing field is Be, we have for the fields produced by Jl the Maxwell equation (229) Moreover, when the biasing field is -Be, we have for the fields produced by J2 the Maxwell equation V' X H2(

-

Be)

=

J2

-

iwt(- Be) . E2(

-

Be)

(230)

which, in view of the symmetry relation (228), assumes the form V' X H2( -Be)

=

J2

-

iwt(Be) • E2( -Be)

(231)

Proceeding as before, we find from Eqs. (229) and (231) the relation (232) ! For application to ionospheric propagation see K. G. Budden, A Reciprocity Theorem on the Propagation of Radio Waves via the Ionosphere, Proc. Cambridge Phil. Soc., 50: 604 (1954).

214

Electromagnetic

waves in a plasma

which is the desired generalization of the reciprocity theorem to the case of an anisotropic plasma.1 If J 2 is such that E2( - Bo) = E2(Bo), or if J1 is such that E1(Bo) = E1( -Bo), this relation reduces to the usual reciprocal relation (226). 1 Reciprocity and reversibility are not unrelated properties. If the current density J transforms into -J' when t is replaced by -t', the Maxwell equations can be made invariant under time reversal by replacing D by D', H by -H", B by -B', and E by E'. However, in a lossy medium the presence of a conduction current term q E makes it impossible for the Maxwell equations to be invariant under time reversal.

215

The Doppler effect

If a source of monochromatic radiation is in motion relative to an observer, the observed frequency of radiation will increase as the source and observer approach each other and will decrease as they get farther apart. This principle, enunciated by Christian Dopplerl in 1843, is called the "Doppler principle" or the "Doppler effect." Basically the Doppler effect is a consequence of the covariance of Maxwell's equations under the Lorentz transformation. For the usual case where .the source and observer are in free space, the exact relativistic formulation of the Doppler effect is well known. But in the presence of material media the Doppler effect is more intricate and involves questions which as yet have not been completely settled. In this chapter the problem of calculating the Doppler effect in material media is discussed. It is shown that for homogeneous media the calculation can be made by using the principle of phase invariance, whereas for inhomogeneous media. a more elementary point of departure is required. 1 Ch. Doppler, nber das farbige Licht der Doppelsterne, Abhandlungen der Koniglichen Bohmischen Gesellschaft der W issenschaften, 1843. See also E. N. Da C. Andrade, Doppler and the Doppler Effect, Endeavor, vol. 18, no. 69, January, 1959.

217

7


wave propagation

7.1 Covariance of Maxwell's Equations According to the theory of relativity, the Maxwell equations must have the same form in all inertial frames of reference, i.e., they must be covariant under the Lorentz transformation.l This means that if we write the Maxwell equations in an inertial frame K and then by a proper Lorentz transformation pass from the coordinates x, y, z, t of K to the coordinates x', V', Zl, t' of another inertial frame K' which is moving at a uniform velocity with respect to K, the dependent functions, i.e., the four field vectors, the current density vector, and the charge density, must transform in such a way that the transformed equations have the same formal appearance as the original equations. The Lorentz transformations can be considered a consequence of the postulate that the velocity of light in vacuum has the same value c in all frames of reference. To show this, we make the spatial origins of K and K' coincident at t = t' = 0 and introduce the convenient notation Xl = x, X2 = y, Xa = Z, X4 = ict, x~ = x', x~ = V', x~ = x', x~ = ict'. Then, in this notation, the postulate demands that the condition (1) be satisfied. Here and in analogous cases we suppress the summation sign and use the convention that repeated indices are summed from 1 to 4. This condition in turn leads to the requirement that the coordi1 The covariance of the Maxwell equations under the Lorentz transformation was proved by Lorentz and Poincare, and physically interpreted by Einstein. Their work, however, was intentionally restricted to the Maxwell equations of electron theory, Le., to the so-called microscopic Maxwell-Lorentz equations, and said nothing of material media. The required generalization of the theory to the case of material media was finally worked out by Minkowski from the postulate that the macroscopic Maxwell equations are covariant under the Lorentz transformation. See, for example, W. Pauli, "Theory of Relativity," Pergamon Press, New York, 1958; A. Sommerfeld, "Electrodynamics," Academic Press Inc., New York, 1952; V. Fock, "The Theory of Space Time and Gravitation," Pergamon Press, New York, 1952; E. Whittaker, "A History of the Theories of Aether and Electricity," vol. II, Harper & Row, Publishers, Incorporated, New York, 1953; C. M~ller, "The Theory of Relativity," Oxford University Press, Fair Lawn, N.J., 1952.

218

The Doppler effect

nates x; and xl' be related by the linear transformations (2) whose coefficients aI" obey the side conditions

=

for

JI

for

JI~A

A

(3)

These linear transformations constitute the complete Lorentz group of transformations. Since the determinant lal',1 may equal 1or -1, this complete group splits naturally into the positive transformations for which lal',1 = 1 and the negative transformations for which lal',1 = -1. From these the positive transformations are selected because they include the identity transformation

+

x;

=

XI'

(J.L

= 1, 2, 3, 4)

(4)

The positive transformations, which can be thought of as a rotation in four-dimensional space or, equivalently, as six rotations in the XIX2, XIX3, XIX4, X2X3, X2X4, X3X4 planes, contain not only the proper Lorentz transformations but also extraneous transformations involving the reversal of two or four axes. Therefore, when these extraneous transformations are excluded, those that remain of the positive transformations constitute the proper Lorentz transformations. Assuming that the coordinates undergo a proper Lorentz transformation, we define a 4-vector as a set of four quantities AI' (J.L = 1, 2, 3,4) that transform like the coordinates: (5) Moreover, we define a 4-tensor AI" of rank 2 as a set of 42 quantities that obey the transformation law (6) and a 4-tensor AI">" of rank 3 as a set of 43 quantities transformation law

that obey the

(7) 219


wave propagation

In terms of the quantities are given by

FafJ, GafJ,

FafJ

0

B.

-B.

0

~E", iE c c

afJ

G

--E

0

-~E

H. 0 -H",

icD", icD"

icD.

H"

(8)

•

0

eE. -H" H2O 0

[ -H. 0

=

c

i

"

2O

i c "

B20

-B2O

(a, (3 = 1,2,3,4), whose values

i

--E c

-B"

=

B"

J'a

-icDo] -icD"

(9)

-icD. 0

J.~[n

(10)

wp

the two Maxwell equations V.B

=

0

VxE=--B

a at

(ll)

become (a, f3, ~

= 1,2,3,4)

(12)

and the other two Maxwell equations

a vxH--D=J at

V.D

=

p

(13)

become (a 220

= 1, 2, 3, 4)

(14)

The Doppler effect From the postulate that Maxwell's equations are covariant under a proper Lorentz transformation of the coordinates, viz., that the fourdimensional forms (12) and (14) are covariant, it follows that F a{3 and Ga{3 are 4-tensors of rank 2 and J a is a 4-vector. This means that when the coordinates undergo a proper Lorentz transformation (/.I the quantities

J a (the 4-current) (/.I

and

(15)

= 1, 2, 3, 4)

=

transform

like the coordinates: (16)

1, 2, 3, 4)

the field tensors

Fa{3,

Ga{3 transform

like the product

of the

coordinates:

F;. =

apaa.{3F

a{3

(J,l,

G;.

apaa.{3Ga{3

(J,l,

=

= II =

II

1, 2, 3, 4)

(17)

1, 2, 3, 4)

(18)

So far the only restrictions we have placed on the reference frames are that their spatial origins be coincident at t = t' = 0 and that their relative velocity v be uniform. Now we shall place an additional restriction on the reference frames, namely, that they have the same orientation. With the velocity and orientation specified, the coefficients ap• can be uniquely determined from Eqs. (2) and (3) and the condition lap.1 = 1. One can show that if the two inertial frames K and K' have the same orientation, and if their relative velocity is v, then the coefficients ap• are given by V 2

(oy _ 1) vxvy v2

1+ (oy -I)-=-2 v (oy _ 1) VyV. v2 (oy _ 1) v,vx 02

1

+ (oy

-

(oy _ 1)

(oy _ 1)

V 2

(oy _ 1) vI/v, v2

1)-.!'-

v2

V.Vy v2

V"'V, v2

1

+

(oy -

(19)

V 2

1) -!-

v2

'Y

Using these values of the coefficients and expressing the results in three-dimensional form, we find that the transformation law (15) for 221


the position 4-vector

,=

r

,

t

r - -yvt

=-y

+ ('Y

-

wave propagation

which can be written as (r,iet), becomes

XI"

~.~

1) --

(20)

V

v2

( t-7r. v)

(21)

where 1

VI -

=

-y

{3=~

{32

e

and that the transformation law (16) for the 4-vector (J,iep) assumes the form

,

J =J-

'YVP

+

J.v

1) -

('Y -

v2

(22)

V

(23)

Also, we find that the transformation law (17) leads to

+v X

+ (1

E'

=

'Y(E

B'

=

-y B - C2v X E

(

B)

1)

-

'Y)

E.v

+ (1 -

V

-y)

(24)

v

-2

B.v V2

v

(25)

and that the transformation law (18) yields D' = H'

'Y

( D + C21)

= -y(H -

v X

v X D)

H

+ (1 -

+ (1

-

'Y)

'Y)

D.v

V2 v

H.v -2-

V

V

(26) (27)

Clearly Eqs. (22) and (23) follow from Eqs. (20) and (21) by replacing r by J and iet by iep. Also Eqs. (26) and (27) follow from Eqs. (24) and (25) by replacing E by eD and B by Hie. 222

The Doppler effect

Thus we see that when the coordinates and time undergo the proper Lorentz transformations expressed by Eqs. (20) and (21), the Maxwell equations with respect to K, viz.,

a

vxE=--B

VxH=J+-D at

a at V.D=p

V. B

=

0

(28)

transform into the Maxwell equations with respect to K', viz.,

v'

X H'

=

J'

+ ~,D'

v'

X E'

= - ~at' V'. D'

H' = p'

v' . B' = 0 (29)

provided the primed quantities are related to the unprimed quantities by relations (20) through (27).

7.2 Phase Invariance and Wave 4-Vector If a reference frame K is at rest with respect to a homogeneous medium, the Maxwell equations in K admit solutions of the form E(r,t) B(r,t)

= Re = Re

Eoei(k.r-",O

(30)

Boei(k.r-..O

(31)

where Eo is a constant and Bo, which is related to Eo by Bo = (1/ w)k X Eo, is likewise a constant. Expressions (30) and (31) represent in K the electric and magnetic vectors of a plane homogeneous wave of angular frequency wand wave vector k. To see what form this plane wave takes in a reference frame K' moving at uniform velocity v with respect to K, we first substitute expressions (30) and (31) into the transformation law (24) and thus obtain the expression E'(r,t)

=

Re

E~ei(k.r-",t)

(32) 223


wave propagation

where E~ is a constant given by

= -y(Eo

E~

+ v X Bo) + (1 .

-y) Eo' v v v2

(33)

Then we transform the coordinates rand t into the coordinates r' and t' of K' by means of the proper Lorentz transformation r

r'

=

+ -yvt'. + (-y

-

r'. v 1) -- 2 V

(34)

v

t = (t + r'c: v) i

-y

(35)

Applying this transformation to expression (32), we see that the electric vector of the wave in K' takes the form E' (r' ,t')

= Re

E~ei(k'.r'-""t')

(36)

where k

,w

w'

+ (-y

=

k - -y - 2 v c

=

-y(w - v. k)

-

k. v 1) - 2 v

(37)

V

(38)

This shows that in going from K to K' the plane wave (30) is transformed into the plane wave (36). By the mann~r in which k' and w' appear in expression (36), we are led to the interpretation that k' is the wave vector of the wave in K' and w' is its frequency. Accordingly, we regard relation!! (37) and (38) as t~e transformation laws for the wave vector and the frequency. Comparing these relations with Eqs. (20) and (21), we see that

(k,i~) transforms like the 4-vector

is a 4-vector. 224

(r,ict).

It is called the wave 4-vector.

Hence'

The Doppler effect

The phase cP of the wave in K is defined by cP

=

k.

(40)

wt

r -

and in terms of kl' and

XI'

it takes the form (41)

Since kl' and XI' are 4-vectors, it follows from Eq. (41) that cP is invariant. What we have shown above is that the phase cP of a uniform plane wave in a homogeneous medium remains invariant under a proper Lorentz transformation of the coordinates. This invariance of the phase, sometimes referred to as the principle of phase invariance, applies not only to waves in vacuum but also to waves in homogeneous media, even if these homogeneous media be anisotropic and dispersive. However, in the case of inhomogeneous media the Maxwell equations do not admit uniform plane wave solutions and hence preclude the possibility of devising an invariant phase. 1

7.3 Doppler Effect and Aberration As in the previous section, we consider a plane monochromatic wave traveling in a homogeneous medium. We recall that if k and ware respectively the wave vector and angular frequency of the wave in the reference frame K, which is at rest with respect to the medium, then the wave vector k' and the angular frequency w' of the 'wave, as observed in a reference frame K' moving with uniform velocity v with respect to K, are given by k'

=

k -

w'

=

'Y(w -

w

c2 v

'Y -

+

v • k)

(

'Y -

k.v 1) - v2

V

(42) (43)

1 K. S. H. Lee and C. H. Papas, Doppler Effects in Inhomogeneous Anisotropic Ionized Gases, J. Math. Phys., 42 (3): 189-199 (September, 1963).

225


wave propagation

where

1

'Y =

VI _ {32

From Eq. (42) we "~n calculate the angle between the directions of k' and k and thus obtain the aberration of the wave vector due to the relative motion of the reference frames. Also, from (43) we can calculate the difference between w' and w, which gives the corresponding Doppler shift in frequency. To derive the aberration formula, we note that the spatial axes of K and K' are similarly oriented, i.e., the x', y', z' axes are parallel respectively to the x, y, z axes, and we assume that v is parallel to the x axis and hence to the x' axis. Since v = ezv, it follows from the scalar multiplication of Eq. (42) by the unit vectors ez and ell that k' cos 8'

= -yk cos 8 -

k'sin8'

=

w

'Y 2

c

v

(44)

ksin8

(45)

where 8' is the angle between k' and v, and 8 is the angle between k and v. Dividing Eq. (45) by Eq. (44) and using the relations k = wlvph, where Vph is the phase velocity in K, n = clvph, where n is the index of refraction in K, and (3 = vic, we get the aberration formula tan 8'

=! 'Y

sin 8 cos 8 -

1

tan 8

fi = ~ 1 - fi sec 8

n

(46)

n

In vacuum, we have n = 1 and, accordingly, Eq. (46) reduces to the familiar relativistic formula for aberration. The formula (43) for the Doppler effect can be written as w' = 'Y(w -

vk cos 8) = 'Yw(I -

(3n cos 8)

(47)

where 8 is the angle between the wave vector k and the relative velocity v. From this equation we see that a wave of angular frequency w in reference frame K appears to have a different frequency w' when 226

The Doppler effect

observed from the moving frame K'. The Doppler shift in frequency, viz., the quantity w' - w, is a maximum when (J = 0 and is a minimum when (J = 7rj2. In the latter case we have the relation w'

=

')'w

(48)

which expresses the so-called "transverse Doppler effect."

7.4 Doppler Effect in Homogeneous Dispersive Media We shall now apply the Doppler formula to the situation in which a monochromatic source and an observer are in a homogeneous dispersive medium. We shall limit the discussion to two cases: in one the source is fixed with respect to the medium and in the other the observer is fixed with respect to the medium. The observer is assumed to be in the far field of the source so that, to a good approximation, the waves incident upon the observer are plane. In the case where the source is fixed with respect to the medium, we choose the reference frame K to be at rest with respect to the medium and the source, and the reference frame K' to be moving with the observer at velocity v with respect to K. Hence, from Eq. (47) we see that w'

=

')'w[1 - (3n(w) cos (J]

(49)

where w is the source frequency in K, and w' is the frequency observed in the moving frame K'. The index of refraction n(w) is evaluated in K. Since n(w) ~ 0, it follows from expression (49) that when the observer is moving toward the source «(J = 7r), w' is greater than w, as in a vacuum. However, when the observer is moving away from the source «(J = 0), w' is not necessarily less than w. Under special circumstances (for example, when the medium is a nearly resonant plasma), n(w) could be so small that w' would be greater than w, in contradistinction to the corresponding phenomenon in a vacuum, where w' would necessarily have to be less than w. 227


wave propagation

In the case where the observer is fixed with respect to the medium, we choose K to be at rest with respect to the medium and the observer, and K' to be moving with the source at velocity v with respect to K. Accordingly we again have W'

= 'Yw[1 - [3n(w) cos OJ

(50)

but now w' is the source frequency and w the observed frequency. When 0 = 1r the source is moving away from the observer, and when o = 0 it is moving toward the observer. Due to the dispersive nature of n(w), expression (50) is not, in general, monotonic between w' and w. Therefore, a given value of w' may yield more than one value of w. This means that the radiation incident upon the observer may appear to have several spectral components even though the source is oscillating at a single frequency. This splitting of the emitted monochromatic radiation into several modes is called the complex Doppler effect. This effect has been studied by Frankl in connection with the problem of determining the radiation of an oscillating dipole moving through a refractive medium. If the medium were nondispersive, expression (.50) would, of course, yield a monotonic relation between w' and w, and hence no complex Doppler modes would be generated. As an illustrative example, let us examine the complex Doppler effect in the special instance where the medium is a homogeneous plasma. For such a medium, Eq. (50) becomes W'

=

'Y(w - [3yw2

-

wp2 cos 0)

(51)

where Wp is the plasma frequency. A plot of w' versus w is shown in Fig. 7.1. The curve has two branches, one given by the solid line and the other by the broken line. The broken line represents Eq. (51) for o = 1r (source receding from the observer), and the solid line represents Eq. (51) for 0 = 0 (source approaching the observer). The two branches join at point A, where w = w" and w' = 'YW". The solid branch is a minimum at point B, where w = 'YW" and w' = Wp• The asymptotes make with the axes an angle If which depends on the relative 11. M. Frank, Doppler Effect in a Refractive Medium, J. Phys. U.S.S.R., 7 (2): 49-67 (1943). See also, O. E. H. Rydbeck, Chalmers Res. Rept. 10, 1960. 228

The Doppler effect

Fig. 7.1

A sketch of the source frequency w' versus the observedfrequency w, in the case where the observer is at rest with respect to a homogeneous isotropic plasma medium and the source is moving through the medium at relative velocity {3c.

velocity v according to the relation tan if; = VI - {3/v'f+I3. From the curve, we see that for a given value w~ of w' greater than 'YWp, we get a single value w, of w when the source is receding, and a single value Wa of w when the source is approaching. We also see that if w~ is less than 'YWp but greater than Wp, the wave due to the receding source is beyond cutoff, and the wave due to the approaching source splits into two, thus yielding two values of Wa instead of only one. One of these two frequencies is always greater than w~, while the other may be greater or less than w~ depending on how close w~ is to -ywp' Finally, we note that if w~ is less than Wp, even the wave due to the approaching source is beyond cutoff. 229


wave propagation

7.5 Index of Refraction of a Moving Homogeneous Medium To compute the index of refraction of a homogeneous medium moving at velocity v with respect to a reference frame K, we choose a frame K' that is at rest with respect to the medium, and we assume that in K' there is a monochromatic plane wave having wave vector k' and frequency w'. In K the wave is perceived as a plane wave of wave vector k and frequency w. The index of refraction of the medium is defined by n' = ck'/w' in K' and by n = ek/w in K. As a point of departure for the calculation, we use the transformations

+ "I w' - v + ("I e

k

=

k'

w

=

'Y(w'

2

+ v.

-

k'.v

1)-- 2 v

(52)

V

k')

(53)

From Eq. (52) we find that k is given by

(54)

Dividing Eq. (54) by Eq. (53) and noting that k' . v obtain

=

k'v cos 0', we

k

(55)

w

Since by definition n that ek

n

= -;; =

vn'2

=

ek/w and n'

=

+ 2"(2n'(3 cos 0' + ("(2 "((1

ek'/w', it follows from Eq. (55)

-

l)n'2 cos2 0'

+ n'(3 cos 0')

+ "(2(32

(56)

Although this relation relates n to n', it is not yet the relation we want, because it involves the angle 0'. To obtain the desired relation, we must eliminate 0' in favor of the angle 0 between k and v. Accordingly, 230

The Doppler effect

we invoke the aberration relations

-yen' cos (J'

cos

(J

cos

(J

+ fJ)

= -:=. V n'2==.============ sm2 (J' + -y2(n' cos (J' + {j)2

(57)

-y(n cos (J - (j) sin (J -y2(n cos

(58)

'

= --:============= 2

V n2

+

which follow from Eq. (46). led to

n cos

(J

=

(J -

{j)2

Combining Eqs. (56) and (57), we are

+

n' cos (J' {3 1 n' {3cos (J'

+

(59)

which, with the aid of Eq. (58), yields the following quadratic equation for n:

(60) Solving this equation and choosing the root that yields n v = 0, we obtain the desired relation: n

=

VI + -y2(n'2

-

1) (1 - {32cos2 (J) - (3-y2(n'2 1 - -y2(n'2 - 1){32 cos2 (J

1) cos (J

= n' for

(61)

Here n' is the index of refraction of the medium in the K' frame, which is at rest with respect to the medium, n is the index of refraction in the K frame;with respect to which the medium is moving at velocity v, and (J is the angle between v and the wave vector k. We see from Eq. (61) that the index of refraction n of a moving medium depends on the velocity v( = (3c) of the medium and on the angle (J between k and v. When {32 « 1, Eq. (61) reduces to the following equation, n

= n' - (n'2 - 1){3cos (J

which is valid for dispersive as well as nondispersive media.

(62)

In the 231


wave propagation

case where the direction of k is parallel (8 = 0) or antiparallel to v and the medium is nondispersive, Eq. (62) yields

i: v

Vph = ~

n'

(1 - ~)

(8

= 11")

(63)

n'2

where Vph( = c/n) is the phase velocity of the wave in K. This is the well-known formula of Fresnel. The coefficient (1 - 1/n'2) is called the Fresnel drag coefficient. The Fresnel formula was verified experimentally by Fizeau who used streaming water as the moving medium. For a dispersive medium Eq. (63) has to be modified. To find what this modification is, we note that in Eq. (62) the index of refraction n' is a function of w'. Since the Doppler formula (47) for low velocities «32 « 1) yields w' = w =+= (3nw, where the upper sign is for 8 = 0 and the lower one is for 8 = 11", we see that n'(w')

=

n'(w

Expanding we get n '( w')

=

= n'

n '( w)

=+=

(3nw)

(64)

this relation about wand keeping only the first two terms,

Substituting n

=+=

=+=

(3nw ~an'(w)

(65)

this expansion into the equation (66)

(n'2 - 1)(3

which follows from Eq. (62) when 8 terms in (32, we get n

= n'(w)

=+=

[n'2(w) - 1](3

=+=

0 and 8

=

, an'(w) (3wn (w) ~

11",

and neglecting

(67)

= c/n, we then deduce from Eq. (67) that

Since

Vph

Vph =

c [ n'(w) i: v 1 - n'2(w)

232

=

1]

w an'(w) i: v n'(w) ~

(68)

The Doppler effect

This is the form that Eq. (63) takes for a dispersive medium. We see that the dispersive nature of the medium is accounted for by the last term on the right side. This term is sometimes referred to as the "Lorentz term". It was verified experimentally by Zeeman.

7.6 Wave Equation for Moving Homogeneous Isotropic Media In a frame of reference K' which is at rest with respect to a homogeneous isotropic medium, the vector potential A' (r' ,t') and the scalar potential cfJ'(r',t') due to a current density J'(r',t') and a charge density p'(r',t') clearly must obey the inhomogeneous wave equations \7'2

-

'2 [ \7

-

[

n'2 ~] c2 i)t'2

A'(r' ' t')

'J'(r' t')

(69)

n'2 i)2]

, , , _ 1", cfJ (r ,t) - - ;: p (r ,t )

(70)

C2 at'2

= -

/-l,

where /-l' and e' are the permeability and the dielectric constant of the medium and n' is the index of refraction. With the aid of the 4-vectors and A:, whose values are given by

J:

J' a

=

[~t]

A: = [~~]

J~

(71)

-: cfJ' c

icp'

these equations can be combined to give (-1-

KC2(4)

a

( \7'2

-

n'2 i)2) A' - C2 i)t'2 a

= ,.. ,,'J'a

(72)

where K = /-l'e' - (1/c2) = (n'2 - 1)/c2 and 1l4a is the Kronecker delta. We wish to transform Eq. (72) to reference frame K, with respect to which the medium is moving at velocity v. Since and are 4-vec-

A:

J:

233


wave propagation

tors, they transform as follows: (73)

Here All and Jil are 4-vectors in K, and the a"l1 are the coefficients of the proper Lorentz transformation that carries K' into K. To transform the differential operator that appears in Eq. (72), we write (74)

The first two terms on the right side constitute an invariant operator, and hence (75)

By means of the transformations (76)

r

=

r'

+ 'Yvt' + ('Y

-

r'. v 1) -2- V

(77)

V

it can be shown that n'2 -

c2

1

iJ2 at'2 = K'Y2

(aat + )2

(78)

v •V

Thus from relations (75) and (78) we see that the operator (74) transforms as follows: V'2 -

n'2 a2 = - c at'2 2

V2 -

1 a2 c at2

-2 -

K'Y2

(a + at -

V •

V

)2

(79)

Now, with the aid of the transformations (73) and (79), it becomes evident that equation (72) in K' transforms into the following equation 234

The Doppler effect

in K: (80) where the operator L is defined by L

==

1 iP 2

V'2 -

-2 -

c

Multiplying relation

at

(aat + )2

K"'/ -

v • V'

(81)

Eq. (80) by aav, summing on a, and using the orthogonality

(3), we find that (82)

For a

= 4, Eq. (80) yields (83)

Therefore we can cast Eq. (82) in the form (84)

Using Eq. (19), we see that (85) where Uv is the velocity 4-vector ("(v,i"(c). Eq. (84) yields

With the aid of this result,

, LAv = -p.'Jv

-

~,~

U".lfJUfJ

(86)

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