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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
sources in unbounded
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
sources in unbounded
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
!<'rom Eq. (47) it follows that \7 . r(r,r')
= -
~2
\7o(r - r').
With the
aid of this relation, Eq. (48) becomes
2+ k
2
(\7
)
= - (u
r(r,r')
+ :2 \7\7) o(r
(49)
- r')
Clearly this equation is satisfied by
r(r,r')
= (
+
u
;2 \7\7)
(50)
G(r,r')
where G(r,r') in turn satisfies (\72
+ k )G(r,r') 2
= -
o(r - r')
To meet the radiation
G(r,r')
iklr-r'l = 4 e .1rr-r
(51)
condition, the solution of Eq. (51) must be
'I
l
(52)
Thus the desired dyadic Green's function is 1
r(r,r')
=
(u
)
+ k2 \7\7
eiklr-r'l
(53)
41r/r _ r'l
The fields E", H" satisfy Maxwell's equations \7 X H" = - iwtE"
and
\7 X
E"
=
-Jm
+ iw~H"
from which it follows that H" satisfies the vector Helmholtz with Jm as its source term: \7 X \7 X H" -
k2H"
=
iwd",
(54) equation
(55)
As before, if we write
II" 28
=
iwtfr(r,r')
. Jm(r')dV'
(56)
Monochromatic sources in unbounded regions then Eq. (55) and the radiation condition will be satisfied when r is given by expression (53). That is, the dyadic functions in the integrands of Eqs. (44) and (56) are identical. Since H' expressions
= ~ V' 2W~
= V' X fr(r,r')
E"
= - V' X fr(r,r')
(V' X r(r,r')].
= -
J:- V' X H" 2WE'
it follows from
(44) and (.'56) that
H'
But r(r,r')
X E' and E"
=
. Jm(r')dV'
(u + ~ J(r')
= f(V' X r(r,r')]
. J(r')dV'
=
= -
. J(r')dV'
f[V' X r(r,r')]
. Jm(r')dV'
(57) (58)
V'V') G(r,r') and consequently
V'G(r,r') X J(r')
In view of this, Eqs. (57) and (58) become
= fV'G(r,r') X J(r')dV'
(59)
E" = - fV'G(r,r') X Jm(r')dV'
(60)
H'
Combining we get
E
=
H
=
expressions
(44) and (56) with (60) and (59) respectively,
+ E" = iw~fr(r,r') H' + H" = iWEfr(r,r')
E'
. J(r')dV' . Jm(r')dV'
- fV'G(r,r')
+ fV'G(r,r')
X Jm(r')dV'
(61)
X J(r')dV'
(62)
These expressions are identical to expressions (40) and (41).
2.3 Radiated Power For the computation of the power radiated by a monochromatic electric current, the complex Poynting vector theorem (see Sec. 1.5) V' • S
= - HJ* . E
+ 2iw(w
m
-
We)
(63) 29
Theory of electromagnetic
wave propagation
can be used as a point of departure. The real part of this equation when integrated throughout a volume V bounded by a closed surface A, which completely encloses the volume Vo occupied by the current J, yields Re
Iv
V'. S dV
= -72
Re
Ivo J*. E dV
(64)
Converting the left side by Gauss' theorem to a surface integral over the closed surface A with unit outward normal n, we get Re
f
A
n' S dA
= -72 Re
f
Vo
(65)
j*. E dV
The right side gives the net time-average power available for radiation and the left side the time-average radiated power crossing A in an outward direction. In agreement with the conservation of power this relation is valid regardless of the size and shape of the closed surface A as long as it completely encloses Vo. Thus we see that the time-average radiated power can be computed by integrating - (72) Re (j* . E) throughout Vo or, alternatively, by integrating Re (n' S) over any closed surface A eIl:closing Vo• In one extreme case, A coincides with the boundary Ao of Vo; in the other, A coincides with the sphere at infinity, Aoo' The imaginary part of Eq. (63) when integrated throughout V yields 1m
f
A
n'
S dA
= -72
1m
f
Vo
J*. E dV
+
f
2", v (wm
-
w.)dV
As before, A is an arbitrary surface completely enclosing Vo• A coincides with Ao this equation becomes 1m
lAO n . S dA
=
-72
1m
I
Vo
j* • E dV
+ 2",(W
mint -
W.int)
(66) When
(67)
where
Wm int =
I
Vo
w m dV
denote the (internal) 30
W.int =
f
Vo
time-average
w. dV magnetic
(68) and electric energies
Monochromatic
stored inside Vo. 1m
J
A.
regions
When A coincides with Aoo it becomes
S dA
n'
sources in unbounded
= - ~ 1m
J
Vo
J*. E dV
where
W mex =
J
11)
V-Vo
dV
m
Wex = •
J
dV
11) V - Vo
(70)
•
denote the (external) time-average magnetic and electric energies stored outside Vo. In the far zone, S is purely real and consequently Eq. (69) reduces to
From this relation it is seen that the volume integral of - (~) 1m (J*. E) throughout Vo gives 2w times the difference between the time-average electric and magnetic energies stored in all space, i.e., inside Vo and outside Vo• A relation involving only the external energies is obtained by subtracting Eq. (71) from Eq. (67), viz., (72)
Now, in accord with the left side of relation (65), we shall find the time-average radiated power by integrating Re (n' S) over the sphere at infinity. As was shown in Sees. 2.1 and 2.2, the electric field E produced by a monochromatic current J is given by E(r)
=
iwj.£
J
Vo
r(r,r').
J(r')dV'
(73)
where r(r,r')
=
1
(u
+k
2
) eiklr-r'l VV 41rJr _ r'l
(74) 31
Theory of electromagnetic
wave propagation
Since eiklr-r'l
V
eiklr-r'l
. r'l -
411"lr-
-
V' ~._-411"lr-
(75)
r'l
we may write expression (74) in the form
r(r,r')
=
(
u
1)
eiklr-r'l
+ k2 V'V'
411"lr-
(76)
r'r
with the double gradient operating with respect to the primed coordinates only. In the far zone, which is defined by 1'»
where valid:
kr» 1
and
1"
l'
=
yr-=r and
r'=
(77)
yr'~,
the following approximation
is
(78)
ere
where = rlr) is the unit vector in the direction of r. In this approximation we may replace exp (iklr - r'/) by exp [ik(r - er • r')] and l/lr - r'l by 1/1'. Accordingly Eq. (76) reduces to
r(r r')
=
,
(u
+ -k1 v'v')
in the far zone. and since
2
~ikr
e-ike,'r'
(79)
411"1'
The double gradient V'V' operates on
e-ike,.r'
only,
(80)
we have (81)
With the aid of this relation, expression (79) for the far zone r(r,r')
=
(u -
eikr
erer) 4-
11"1'
32
e-ike,'r'
r
becomes (82)
Monochromatic sources in unbounded regions
Substituting this expression into Eq. (73) and using the vector identity (u - ercr)' J = J - cr(er• J) = -Cr X (cr X J) we obtain the following representation for the far-zone electric field:
E(r)
=
ikr eer
-iwJ.l.
[J
X
41Tr
X
Cr
]
voe-;ke,'r'J(r')dV'
(83)
The far-zone magnetic field is found by taking the curl of Eq. (83) in accord with the Maxwell equation H
H(r)
ikr
= ik -e
41Tr
[Je
r
=
1.wJ.l.
X
E.
Thus
]-
e-ike,.r'J(r')dV'
X
J:- yo
Vo
(84)
Comparing expressions (83) and (84) we see that the far-zone E and H are perpendicular to each other and to Cr, in agreement with the fact that any far-zone electromagnetic field is purely transverse to the direction of propagation, viz., in the far zone
H
or
=
r~(c
'1M
r
X
E)
(85)
is always valid. Expressions (83) and (84) yield the following expression for the far-zone Poynting vector:
(86) The notation ICl2 where C is any vector means C. C*. From expression (86) we see that S is purely real and purely radial, i.e., directed parallel to Cr' The element of area of the sphere over which S is to be integrated is r2 dn, where dn is an element of solid angle. Hence, the time-average radiated power P is given by P
=
JA~
n .S
dA
=
J =
Cr'
Sr2 dn
I~-,~
'1
f
321T
2
f dn I
c, X
f
2
e-ike,.r'J(r')dV'1
(87)
Vo
33
Theory of electromagnetic
wave propagation
This way of calculating the radiated power is called the "Poynting vector method." A formally different way of calculating the radiated power consists in integrating throughout Vo the quantity - (72) Re (J* . E) in which E is taken to be the radiative electric field as given by Eq. (73). This alternative procedure, which was proposed by Brillouin,l yields p
J
= -72 Re vo J*. E dV =
WIJ. 81T
J vo J
J*(r). VO
(u + .!- vv) sin Ir - r I
(klr ~ r'D . J(r')dV'
k2
dV
(88)
and is called the "emf method" since it makes use of the induced electromotive force (emf) of the radiative electric field. Although representations (87) and (88) of these two methods are apparently different, they nevertheless yield the same result for P and in this sense are consistent. To exemplify this we now apply these two methods to the relatively simple case of a thin straight-wire antenna. The antenna has a length 2l and lies along the z axis of a cartesian coordinate system with origin at the center of the wire. Since the wire is thin, the antenna current is closely approximated by the filamentary current
J = e.Ioo(x)o(y)f(z)
(89)
where lois the reference current, e. is the unit vector in the z direction, and fez) is generally a complex function of the real variable z. By use of this current we get
Jvo
e-ike,'r'
J (r')dV'
= e.lo
J
~l
e-ik.,
coe
ef(z')dz'
(90)
where (J is the colatitude in the spherical coordinate system (r,(J,It» defined by x = r sin (J cos cP, Y = r sin (J sin cP, and z = r cos (J. Denoting the unit vectors in the r, (J, and cP directions respectively by er, ee, and eq, and noting that er X e. = -eq, sin (J, we find from Eq. (90) and 1
34
L. Brillouin, Origin of Radiation
Resistance, Radioelectricite, April, 1922.
Monochromatic
sources in unbounded
regions
expression (87) that the Poynting vector method yields
p =
~ ~ 101* '\j"i 1611" 0
I I I -I
I -I
f(z')f*(z)dz
dz'
r'"
Jo
eik(z-z')
C08
8
sin3 0 dO
(91)
Moreover, by substituting the current as given in Eq. (89) into expression (88), we see that the emf method yields
p =
WJJ.
101*
811"
0
II II -I
f(z')f*(z)
+ ~~)
(1
k2 az2
-I
sin (klz - z'l) dzdz' Iz - z'l
(92)
To show that expressions (91) and (92) are equivalent, we invoke the following elementary results:
r,..
Jo
1
(
eik(z-z')
COB
8
sin3 0 dO
=
i. ~
(~~~:U - u) u
+ ! ~) sin (klz - z'\) _ Iz-z 'I k a 2
Z2
cos
2k (sin U2
'u _
U
cos
) U
where u = k(z - z'). With the aid of these results and the introduction of the new variables ~ = kz and 1] = kz' expressions (91) and (92) pass into the 'common form
p =~
~ 1 Iri '\j"i 0
411"
I I kl -kl
kl -kl
d1]d~ f(1])f*W (~ -
1])2
[sin (~ ~-
1]) 1]
cos (~-
)J
1]
(93) Thus we see that the Poynting vector method and the emf method ultimately lead to the same formula (93) for the time-average radiated power P and hence are consistent with each other. From a practical viewpoint, formula (93) as it stands is too clumsy to use, owing to the presence of the double integral. However, Bouwkamp by successive transformations succeeded in reducing the double integral to a repeated integral and then finally to an elegant form involving only single integrals. To demonstrate the capabilities of this form he applied it to several "classical" cases which had been 35
Theory of electromagnetic wave propagation
handled previously by the Poynting vector method. For details we refer the reader to his original paper.! The present discussion may be extended to the case of magnetic currents by using the duality transformations of Sec. 1.2. For example, if we replace J, E, H respectively by - vi EI IL Jm, - vi ILl E H, vi EI IL E in the far-zone field formulas (83) and (84), we obtain the corresponding formulas for the far-zone field of a monochromatic magnetic current density:
=
H
(f e-ike,'r'Jm(r')dV' ) e~( f e-ike,.r'Jm(r')dV' ) ikr
-iWE
E = -ik -
41rr e er X
41rr
er
er X
(94)
vo
(95)
X
Vo
The Poynting vector of this far-zone electromagnetic field is S = ;YzE X H* =
er
I"! ~321r r I e, X f "J 2 2
IL
2
Vo
e-ike,.r'Jm(r')dV'1
(96)
and consequeritly the time-average radiated power is P =
f.
A..
n' S dA =
= !
36
f
e. Sr2 dn
A•• r
I~~ "J IL 3271
2
f dn I e X f r
2
e-ike,.r'Jm(r')dV'1 Vo
C. J. Bouwkamp, Philips Res. Rept., 1: 65 (1946).
(97)
Radiation from wire antennas
As a practical source of monochromatic radiation the wire antenna plays an important role. The field radiated by such an antenna can be obtained from a knowledge of its current distribution by using the formulas derived in the previous chapter. Although the determination of the antenna current is a boundary-value problem of considerable complexity, a sufficiently accurate estimate of the current distribution can be obtained in the case of thin wires by assuming that the antenna current is a solution of the onedimensional Helmholtz equation and hence consists of an appropriate superposition of simple waves of current. This simplifying approximation yields satisfactory results for the far-zone field and for those quantities that depend on the far-zone field, e.g., radiation resistance and gain, because the far-zone field in almost all directions is insensitive to small deviations of the current from the exact current. The radiation properties of thin-wire antennas and their arrays are discussed in this chapter.
3.1 Simple Waves of Current We consider a straight-wire antenna lying along the z axis of a cartesian coordinate system with one end at z = -l and the other at z = l, as shown in Fig. 3.1. Since the wire is 37
3
Theory of electromagnetic
wave propagation
z
Fig. 3.1
Coordinate systemfora straightwire antenna extending from z = -Z to z = Q is observation point. Q' is projection of Q in x-y plane.
z.
y
x
thin and since we wish to calculate only the far-zone field it is a permissible mathematical idealization to assume that the antenna current density is the filamentary distribution
J
=
(1)
ezo(x)o(y)j(z)
The total current is the integral of this distribution section of the wire: I(z)
= ezI(z) = IJ dx dy = e.j(z)Jo(x)o(y)dx
dy
= ezj(z)
over the cross (2)
It is supposed that the wire is cut at some cross section z = 1'/ and a monochromatic emf is applied across the gap. The current is necessarily a continuous function of z, but the z derivative of the current may be discontinuous at the gap. The antenna is said to be "centerfed" when 1'/ = 0 and "asymmetrically fed" when 1'/ ;;c O. For a center-fed antenna, j(z) is a symmetrical function of z and satisfies the one-dimensional Helmholtz equation 1
crJ +kj dz 2
2
=
0
k
= wlc = 2'Tr/'A
(3)
as well as the end conditions feZ)
=
f( -l)
=
0
(4)
1 It appears that Pocklington was the first to show that the currents along straight or curved thin wires in a first approximation satisfy the Helmholtz equation. See H. C. Pocklington, Proc. Cambridge PhiZ. Soc., 9: 324 (1897).
38
Radiation
from wire antennas
The two independent solutions of Eq. (3) are the simple waves eikz and e-ikz• Accordingly a general form for the current is
= Aeikz
I(z)
+ Be-
(5)
ikz
where A and B are constants. of the antennas, we have I1(z)
= Aleikz
I2(z)
=
A2eikz
Writing this form for the two segments
+ Ble-
for 0 ::::;z ::::;l
+
for -l ::::; z ::::;0
ikz
B2e-ikz
(6)
When applied to these expressions,the end conditions (4) yield (7)
from which it follows that BIIAI these results, Eqs. (6) become I1(z)
= -2iA1eikl
I2(z)
=
2iA2e-ikl
sin k(lsin k(l
The continuity condition 11(0) related by
_e2ikl,
BdA2
=
_e-2ikl•
With
for 0::::;z ::::;l
z)
+ z)
=
(8)
for -l ::::; z ::::;0 =
12(0) requires that A I and A2 be
(9)
In view of this connection between Al and A2 it followsfrom Eqs. (8) that the current distribution, apart from an arbitrary multiplicative constant 10, is given by the standing wavel I(z)
=
10 sin k(l - jz/)
(10)
which, for several typical cases, is displayed in Fig. 3.2. This "sinusoidal approximation" is adequate for the purpose of computing the far-zone radiation pattern of a center-fed straight-wire antenna, provided the antenna is neither "too thick" nor "too long." A closer approximation to the true current may be obtained heuristically by adding to the sinusoidal current a quadrature current, which takes into 1
J. Labus, Z. Hochjrequenztechnik und Elektrokustik, 41: 17 (1933).
39
(a)
(b)
kl=1r
(c)
Fig.3.2
40
Radiation patterns of a center-driven thin-wire antenna of current distribution shown by dotted lines.
kl •• 7-rrj6
(d)
various lengths
shown by solid lines.
Assumed
sinusoidal
41
Theory of electromagnetic
wave propagation
account the reaction of the radiation and the ohmic losses on the current. 1 Since the antenna is center-fed, the principal alteration that such a quadrature current can make on the far-zone radiation pattern is the presently negligible one of relaxing the intermediate nulls of the pattern. 2 However, for an asymmetrically fed antenna (1] ;t. 0) the radiation pattern calculated solely on the basis of a simple standing wave of current can be in serious error due to the presence of traveling waves of current. The standing wave, which for arbitrary values of 1] has the form3 1(z)
= 10 sin k(l +
1])
sin k(l - z)
1(z)
=
10 sin k(l-
1])
sin k(l
+ z)
for
1] ::;
z ::; l (11)
for -l ::;z ::;
1]
always gives rise to a radiation pattern that is symmetrical about the plane (J = 7r /2. On the other hand, a traveling wave produces an asymmetrical radiation pattern, viz., a pattern tilted toward the direction of the traveling wave. Accordingly the traveling waves tend to tilt the lobes of the pattern and to change their size. Thus if the traveling waves are appreciable, marked changes in the shape of the radiation pattern can occur. Generally the problem of finding the radiation pattern of an asymmetrically driven antenna cannot be handled adequately within the framework of the simple wave theory, except in those cases where either the standing wave or the traveling waves dominate the pat.tern.
3.2 Radiation from Center-driven Antennas As indicated by Eqs. (83), (84), and (86) of Chap. 2, the calculation of the far-zone radiation emitted by a distribution of monochromatic Ronold King and C. W. Harrison, Jr., Proc. IRE, 31: 548 (1943). C. W. Harrison, Jr., and Ronold King, Proc. IRE, 31: 693 (1943). 3 See, for example, S. A. Schelkunoff and H. T. Friis, "Antennas: Theory and Practice," chap. 8, John Wiley and Sons, Inc., New York, 1952. 1 2
42
Radiation from wire antennas
current centers on the evaluation of the so-called radiation vector N defined by the integrall N
J
=
Vo
e-ike,'r'J(r')dV'
(12)
where er is the unit vector pointing from the origin to the point of observation and r' is the position vector extending from the origin to the volume element dV'. The required information on J can be obtained either by solving the boundary-value problem which the analytical determination of J poses or by choosing the current on empirical grounds. In the present case of a thin-wire antenna, the latter alternative is adopted, according to which it is alleged that a sufficiently accurate representation of the antenna current can be built from simple waves to agree with the results of measurement. Accordingly, let us consider the case of a center-driven thin-wire antenna lying along the z axis with one end at z = -l and the other at z = l. It is known a posteriori that the current distribution along such an antenna may be approximated, insofar as the far-zone radiation is concerned, by the sinusoidal filamentary current
= e.Ioo(x)o(y)
J(r)
Izl)
sin k(l -
(13)
Substituting this assumed current into definition (12) and performing the integrations with respect to x' and y', we get the one-dimensional integral N
=
ezlo
J
~l
e-ikz'
cos 9
sin k(l -
Iz'l)dz'
(14)
which by use of the integration formula
Je
aE
sin (b~
+ c)d~
=
a2
~E
+b
2
[a sin (b~
+ c) -
b cos (b~
+ c)]
(15)
yields N
= ez2Io cos (kl cos.O) - cos kl k sm2 0
1 S. A. Schelkunoff, (1939).
A General Radiation
(16) Formula,
Proc. IRE, 27: 660-666 43
Theory of electromagnetic
wave propagation
With the aid of this result and the vector relations er X e. = -eq, sin (), er X (er X e.) = eB sin (), it follows from Eqs. (83), (84), and (86) of Chap. 2 that the far-zone electric and magnetic fields are
'.IB
=
I~e
ikr
_ i
E
~
27l"r
E
1 cos (kl co~ () - cos kl 0
(17)
----s~m-()---
and
Hq, =
-2
. eikr I cos (kl cos ()) - cos kl 211'1' 0 sin ()
and that the radial component 8
r
I~-.!.L [cos
= ~f
r
811'2 2
of the Poynting
(kl co~ ()) - cos kl]2 sm ()
(18) vector is (19)
In these expressions, the common factor
F«()
=
cos (kl co~ () - cos kl
(20)
sm()
is the radiation pattern of the antenna. Since the radiation pattern is independent of q, it is said to be "omnidirectional." When the antenna is short compared to the wavelength (kl« 1) the radiation pattern reduces to!
F«()
=
}-2(kl)2 sin ()
(21)
From this we see that the radiation pattern of a short wire antenna consists of a single lobe that straddles the equatorial plane () = 11'/2 and exhibits nulls at the poles () = 0 and () = 11'. As kl increases up to kl = 11' the lobe becomes narrower and more directive. As kl exceeds kl = 11' and approaches kl = 311'/2, two side lobes appear, gradually growing in size and ultimately becoming larger than the central lobe itself. (See Fig. 3.2.) Since F«() is an even function of () - 11'/2 that vanishes at () = 0 and 1 In the case of a Hertzian dipole F«() = kl sin (). To show this, we recall that the current density of a Hertzian dipole of length 2l, located at the origin of coordinates and directed parallel to the z axis, is defined as J = e,loo(x)o(y), then note that for this current N = e,21I 0 and hence Sr = VPJE (I 02 j811'2r2) (kl sin ()2.
44
Radiation from wire antennas
=
1r,
F(8)
=
8
it may be expanded
..
L
b2n+l sin (2n
I
in a Fourier series of the form
+ 1)8
(22)
n=O
where b2n+l
= ~ (" F(8) sin (2n + 1)8 d8 1r
(23)
10
For small kl, all the higher-order coefficients are, to a good approximation, negligible compared to the first coefficient bl
2/" [cos (kl cos 8) -
= 1r-
0
cos kl]d8
= 2Jo(kl) - 2 cos kl
(24)
Although this simple approximation deteriorates as the length of the antenna increases, for a half-wave dipole (kl = 1r/2) it is stilI satisfactory and yields cos F(8)
=
2 cos
1r (
. 8
sm
8) ~ 0.94.5 sin 8
(25)
From a practical viewpoint this approximate representation of the distant field of a half-wave dipole provides a useful simplification. For example, it enables one to obtain a convenient expression for the radiation resistance of certain linear arrays of half-wave dipoles. 2
3.3 Radiation Due to Traveling Waves of Current, Cerenkov Radiation In the previous section we noted that the far-zone radiation field of a center-fed thin-wire antenna is determined with sufficient accuracy by 1 R. King, The Approximate Representation of the Distant Field of Linear Radiators, Proc. IRE, 29: 458-463 (1941); C. J. Bouwkamp, On the Effective Length of a Linear Transmitting Antenna, Philips Res. Rept., 4: 179-188
(1949). 2 C. H. Papas and Ronold King, The Radiation Resist,ance of End-fire and Collinear Arrays, Proc. IRE, 36: 736-741 (1948).
45
Theory of electromagnetic
wave propagation
using the standing-wave part of the antenna current and ignoring the traveling-wave part. In the present section we shall discuss the converse state of the antenna, wherein the traveling-wave part of the current is dominant and the standing-wave part is quite negligible. Such a state can be achieved by the proper excitation and termination of the antenna.l Accordingly we assume that the current distribution
along a thin-
wire antenna is the traveling wave J(r)
=
(26)
(-l~z~l)
ezIoo(x)o(y)eipkz
Here the index p is the ratio of the velocity of light to the velocity of the current wave along the antenna. This index, which is equal to or greater than unity, depends on the degree to which the antenna is loaded. If the antenna is unloaded, i.e., if the antenna wire is bare, p is approximately equal to unity. Then as the loading is increased2 there is a corresponding increase in p. Substituting expression (26) into definition (12) we get N
=
1.
1
ezIo
.
e-,kZCOB
-I
e,pkz dz
6
=
sin [kl(p - cos 0)] ez2Io ------k(p - cos 0)
•
(27)
This expression for the radiation vector, when introduced into Eqs. (83), (84), and (86) of Chap. 2, yields the following nonvanishing components of the far-zone fields and the Poynting vector:
E
_ ~H ~
_
6 -
Sr
--
~~-
-
--
1
~ 87r2r2
-2
I
0
2
.~~
eikr . -I osm ~ 27rr
o sin [kl(p ---=----
cos 0)] p - cos 0
(28)
2 • 20 sin [kl(p - cos 0)] sm -------'~-------' (p - cos 0) 2
From these expressions it follows that the radiation
(29) pattern
of the
1 A practical example of such an antenna is the "wave antenna" or "Beverage antenna." See H. H. Beverage, C. W. Rice, and E. W. Kellogg, The Wave Antenna, a New Type of Highly Directive Antenna, Trans. AlEE, 42: 215 (1923). 2 The loading may take the form of a dielectric coating or a corrugation of the surface.
46
~ Radiation from wire antennas
---
Direction of traveling current wave
Fig.3.3
traveling F(e)
Typical radiation pattern for traveling wave of current.
wave of current
(26) is
= sin e sin [kl(p - cos 0)]
(30)
p-cose When the antenna reduces to F(e)
= kl sin e
is short
(kl«
1), the radiation
pattern
(30)
(31)
Comparing radiation patterns (31) and (21) we see that for short antennas (kl « 1) the radiation pattern (21) of the standing wave of current (13) has the same form (sin e) as the radiation pattern (31) of the traveling wave of current (26). However, for longer antennas the patterns (30) and (20) differ markedly, the essence of the difference being that the pattern (20) of the standing wave is symmetrical with respect to the equatorial plane e = 17"/2 whereas the pattern (30) of the traveling wave is asymmetrical. The maxium radiation of the traveling wave appears as a cone in the forward direction, i.e., in the direction of travel of the current wave; the half-angle of the cone decreases as p increases or as kl increases (see Fig. 3.3). This type of conical beam radiation resembles the Cerenkov radiation 1 from fast electrons. 1 P. A. Cerenkov, Phys. Rev., 52: 378 (1937). 1. Frank and 1. Tamm, Comptes rendus de ['Acad. Sci. U.R.S.S., 14: 109 (1937). See also, J. V. Jelley, "Cerenkov Radiation and Its Applications," Pergamon Press, New York, 1958.
47
Theory of electromagnetic
wave propagation
3.4 Integral Relations between Antenna Current and Radiation Pattern Again we study the thin-wire antenna, but in this instance we do not specify the current distribution. That is, we restrict the current distribution only to the extent of postulating a monochromatic current of the form J(r)
=
(Izj ~ l)
e.6(x)6(y)f(z)
(32)
where the functionf(z) may be complex. From Eq. (83) of Chap. 2, it directly follows that the far-zone electric field of this current distribution is EB=
ikr
- iwp. -e
sin 0
411"T
JI e-ik•
COB
Bf(z )dz
(33)
-I
Since the O-dependent factor of this expression is, by definition, the radiation pattern, we have F(O)
=
sin 0 JI
-I
e-ikzcOBBf(z)dz
(34)
This integral relation shows that when f(z) is given in the interval (izi ~ l), the radiation pattern F(O) is uniquely determined for all real angles in the interval (0 ~ 0 ~ 11"). To proceed toward a relation that would yield f(z) from a knowledge of F(O), we cast Eq. (34) into the form of a Fourier integral and then find its mate. Accordingly, the finite limits on the integral in Eq. (34) are replaced with infinite ones by assuming thatf(z) vanishes identically outside the interval (jzl ~ l), i.e., j(z)
= 0
for
(35)
With f(z) so continued, Eq. (34) can be written as F. (0) = smO 48
J-'"'" e-ikzc08 Bj(z )dz
(36)
Radiation from wire antennas or, in terms of the new variable
7](
= k cos 8), as
7]::;
(-k::;
k)
(37)
Now the range of validity of Eq. (37) is extended from (-l~ ::; 1]::; k) to (- 00 ::; 1]::; (0) by letting 8 trace the contour C in the complex 8 plane (Fig. 3.4). Such an extension of Eq. (37) leads to the Fourier integral F(cos-I1]/k) .----.-~ =
VI -
1]2/k2
f'"
.
e-'~'f(z)dz
(38)
-'"
By the Fourier integral theorem, the mate of Eq. (38) is
(-
00 ::;
z::;
(0)
(39)
Transforming to the complex 8 plane (by use of 1] = k cos 8) and explicitly taking into account the requirement (35) that f(z) vanish for
I I I I
I Fig. 3.4
Trace of contour C in the complex 8 plane.
I I I I I
o
49
Theory of electromagnetic
Izi ~ fez)
wave propagation
l, we obtain the desired relation = ~ ~ 211"
c
F(B)eikzcoge
dB
(izi ~
l)
(40)
and the side condition
o=~ 211"
~
c
F(B)eikzco9
e dB
(izi ~
l)
(41)
From Eq. (40) it is clear that F(B) must be known along the entire contour C before fez) can be evaluated from it. Moreover, since F(B) must satisfy the side condition (41), it cannot be chosen arbitrarily. Nevertheless, it seems possible] to find an F(B) which satisfies Eq. (41) and closely approximates. a prescribed radiation pattern in the range of real values (0 ~ B ~ 11").
3.5 Pattern Synthesis by Hermite Polynomials In connection with the antenna of the previous section we now briefly sketch the approximation method of Bouwkamp and De Bruijn, ~which enables one to calculate a current distribution that will produce a prescribed radiation pattern, or, in other words, enables one to synthesize a given radiation pattern. The point of departure is the integral relation (34) connecting the radiation pattern F(B) to the current distribution fez). For convenience, however, we express this relation in terms of the dimensionless variables t = cos B, ~ = kz and the dimensionless constant a = kl. Thus, Eq. (34), apart from an ignorable constant, is written first as (-l~t~l) 1
(42)
For a heuristic discussion of such a possibility, see P. M. Woodward and
J. D. Lawson, The Theoretical Precision with which an Arbitrary Radiation Pattern may be obtained from a Source of Finite Size, J. Inst. Elec. Eng., 95 (part III); 363-370 (1948). 2 C. J. Bouwkamp and N. G. de Bruijn, The Problem of Optimum Antenna Current Distribution, Philips Res. Rept., 1: 135-138 (1946). 50
Radiation from wire antennas
and then, by use of the shorthand
G(t)
=
~
VI -
(43)
t2
as (-1
S t S 1)
(44)
Referring to this integral equation, we see that the synthesis problem consists in finding fW when G(t) is given in the interval -1 S t S 1. By virtue of a theorem due to Weierstrass, I we may approximate the given function G(t) by a polynomial pet) of sufficiently high degree N: G(t)
= pet) ==
'Yo
+ 'Ylt + ... + 'YNtN
(45)
Moreover, we may invoke unknown functions fn(~) such that (46) Substituting expressions (45) and (46) into the integral equation (44), we see that functions fn(~) for n = 0, 1, ... , N must satisfy (-1 To find the functionsfn(O, defined by2
S t S 1)
(47)
we introduce the Hermite polynomials Hn(u)
(n
=
0, 1,2, ...
)
(48)
From this formula, the following result can be verified by repeated I See, for example, R. Courant and D. Hilbert, "Methods of Mathematical Physics," vol. 1, p. 65, Interscience Publishers, Inc., New York, 1953. 2 This definition agrees with that of E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis," p. 350, Cambridge University Press, London, 1940. Hn(u) = e"'/4Dn(u), where Dn(u) is that given by Whittaker and Watson.
51
Theory of electromagnetic
wave propagation
partial integrations:
(-I~t~l) (49)
When the arbitrary positive constant A is large, the factor exp (-A2~2/2)Hn(A~) decreases rapidly to zero as I~I~ 00. Hence, the contribution of the integration beyond a certain range (say, I~I> 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)
Radiation from wire antennas
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Radiation from wire antennas
tion vector: N
= J e-ike,'r' J (r')dV'
=
e. J
e-ikY'sin
X J~le-ikz'COS6COSkz'dz'
J
6 sin
n-l
l
Ape-ikz'sin6cos
- xp)dx'
p=o
which upon integration reduces to
N
2 cos (7r2 cos 8)
=
e -. •k
sm28
n -1
~ A e-ikzp 1.. p p=o
sin 6 co••
(66)
Substituting this result into Eq. (86) of Chap. 2 and recalling the vector relation e, X e. = -e", sin 8, one finds that the far-zone Poynting vector has only a radial component given by (67)
where cos (~cos 8) F(8)
(68)
sin 8
is the radiation pattern of each dipole, and n-l
A(8,4»
=
L
Ape-ik"p.in6co
••
(69)
p=o
is the array factor. U(8,4»
=
1F(8)A(8,4»
The radiation pattern of the entire array is
I=
F(8) IA (8,4»1
(70)
If we let if; denote the angle between the x axis and the line of observa59
Theory of electromagnetic
tion (cos if!
r
=
sin
e cos tP),
wave propagation
the array factor (69) takes the form
n-l
A(if!)
=
Ape-ikxpcos{l
(71)
p=o
which is recognized as the canonical expression for the complex radiation pattern of a similar array of isotropic antennas. Thus the radiation pattern U(e,tP) of the actual array is equal to the radiation pattern F(e) of a dipole multiplied by the radiation pattern A (if!) of the similar array of isotropic radiators. More generally, expression (71) is valid for any linear array irrespective of the type of its member antennas. For example, if each half-wave dipole of the array were replaced by an antenna having a radiation pattern G«(J,tP), then the resulting radiation pattern would be given by U«(J,tP) = IG«(J,
=
pd
(72)
where d is the uniform spacing. Imposing this spatial restriction (72) on the array factor (71) and expressing Ap as the product (73) which explicitly exhibits through the factor exp (-ip'Y) the progressive phasing 'Y of the currents, we get
r
n-l
A(if!)
=
p=o
r
n-l
ape-ip(kdcosH1)
=
apeipa
where the shorthand a = - kd cos if! - 'Y has been used. introduce the complex variable ~ defined by
~ = eia
(74)
p=o
Then, if we
(75)
the array factor (74) takes the form of a polynomial of degree n - 1 in 60
Radiation from wire antennas
the complex variable ~: n-l
A(lf) =
2:
(76)
ap~P
p~o
Since the coefficients ap are arbitrary, some of them may be zero. When this occurs, the antennas which correspond to the vanishing coefficients are absent from the array and the remaining antennas do not necessarily constitute an equidistantly spaced array. Nevertheless, an incomplete array of this sort can be considered equidistantly spaced by regarding d as the "apparent spacing" and n as the "apparent number" of antennas. Thus we see that the polynomial (76) can be identified with any linear array having commensurable separations. The importance of this one-to-one correspondence between polynomial and array stems from the fact that it permits application of the highly developed algebraic theory of polynomials to the synthesis problem. A case in point is Schelkunoff's well-known synthesis procedure,! which ingeneously exploits certain algebraic properties of the polynomial (76). When the coefficients ap of the polynomial (76) are equal to a constant, which for the present may be taken as unity, the array is said to be "uniform." The array factor of such a uniform linear array with commensurable separations has the closed form n-l
A(lf)
IP
=
p=o
1 ~- 1
~n -
(77)
=--
which, with the aid of ~ = exp (ia), becomes A (If)
=
ei(n-l)a/2
si~ (na/2) sm (a/2)
(78)
Consequently the radiation pattern of a uniform linear array of equidistantly spaced isotropic sources is given by
IA (If) I =
I sin (na/2) I = I si~ [n(kd cos If + 1')/2] I sin (a/2)
sm [(kd cos If
+
1') /2]
(79)
It is sometimes convenient to divide A(f) by n and thus normalize its 1
S. A. Schelkunoff, A Mathematical
Theory of Linear Arrays, Bell System
Tech. J., 22: 80 (1943). 61
Theory of electromagnetic
wave propagation
maximum value to unity. The resulting function K(y.,) is the "normalized radiation characteristic" of the array and is given by K(y.,)
= ! I sin n
(n(kd cos y., sin (kd cos y.,
+ 1')/2] I + 1')/2]
(80)
If the sources are in phase with each other (I' = 0) and if the spacing is less than a wavelength (kd < 211"), the radiation characteristic K(f) consists of a single major lobe straddling the plane y., = 11"/2 and a number of secondary lobes or "side lobes." As long as the spacing remains less than a wavelength, the spacing has only a secondary effect upon the radiation pattern. Hence, if I' = 0 and if kd < 211", the radiation is cast principally in the broadside direction and the array operates as a "broadside array." However, if the spacing becomes greater tha'n a wavelength (kd > 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
Theory of electromagnetic
wave propagation
kd= 71r
kd=211"
4
kd=
kd=311"
511"
2
Fig. 3.8 64
Continued.
Radiation from wire antennas
kd=4'l1'
kd=81T
Fig. 3.8
Continued.
65
Theory of electromagnetic
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.
Radiation from wire antennas
kd= 'Y= 3 'IT '4
kd='Y~1r
Fig. 3.9
Continued. 67
Theory of electromagnetic
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( <1l') are fixed and 'Yis varied from a to kd, the major lobe rotates from the broadside direction to the end-fire direction. This suggests that by continuously varying the phase 'Y the beam can be made to sweep continuously over an entire sector. It is on this principle that electrical scanning antennas operate. 7 The phases of the antennas are controlled elec1 D. Barbiere, A l\lethod for Calculating the Current Distribution of Tchebyscheff Arrays, Proc. IRE, 40: 78 (1952). 2 G. J. van der Maas, A Simplified Calculation for Dolph-Tchebyscheff Arrays, J. Appl. Phys., 25: 121 (1954). 3 T. T. Taylor, Design of Line-source Antennas for Narrow Beamwidth and Low Side Lobes, IRE Trans. Antennas Propagation, AP-3 (1): 16 (1955). 4 D. D. King, R. F. Packard, and R. K. Thomas, Unequally-spaced Broadband Arrays, IRE Trans. Antennas Propagation, AP-8 (4): 380 (1960). • H. Unz, Linear Arrays with Arbitrarily Distributed Elements, Electronics Research Lab., series 60, issue 168, University of California, Berkeley, Nov. 2,
1956. 6 S. S. Sandler, Some Equivalences between Equally and Unequally Spaced Arrays, IRE Trans. Antennas Propagation, AP-8 (5): 496 (1960). 7 For a review of the scanning properties of such arrays see, for example, W. H. von Aulock, Properties of Phased Arrays, Proc. IRE, 4.8: 1715 (1960).
69
Theory of electromagnetic
wave propagation
trically by phase shifters which form an integral part of the feed system. Although in many operational radars the scanning is done mechanically; electrical scanning is used in the case of large array antennas because it provides scanning patterns and scanning rates that cannot be obtained by mechanical means. Without further calculation, we can deduce the radiation pattern of a rectangular array of dipoles. We do this by compounding the radiation pattern of a parallel arrayl with that of a collinear array. An expression for the radiation pattern of a collinear array of half-wave dipoles can be constructed from expressions (68) and (71). We note that the parallel array of Fig. 3.7 is transformed into a collinear array when the dipoles are rotated until their axes are aligned with the x axis. Clearly then, in view of expression (68), the radiation pattern of each rotated dipole is given by cos (; cos F(I{;)
I{;)
sin I{;
and the array factor remains the same as it was before the rotation, viz., n-l
A(I{;)
=
2: Ape-ikx•
CDS'"
p=o
Hence the radiation
U(I{;)
pattern
= 1F(I{;)A(I{;)j =
of the collinear array turns out to be
cos (7r2 cos I{;)
.
sml{;
n-l
'\' A e-ikx.CDS'"
~
p=o
p
(85)
It follows from this expression (by replacing I{; with 0) that the radiation pattern of a collinear array of dipoles lying along the z axis with dipole centers at the points Zp is given by cos (-rr2 cos 0 ) U(O)
=
.
sm 0
n-l
'\' ,~p
A e-ikz.CDS8
(86)
p=o
1 The linear array shown in Fig. 3.7 is called a "parallel array" whenever it becomes necessary to distinguish it from a collinear array.
70
Radiation from wire antennas
By substituting this expression for F(8) in Eq. (70) we get the radiation pattern of a rectangular array of half-wave dipoles which are parallel to the Z axis and have centers at the points x = xl" Z = Zq (p = 0, 1, ... , n - 1; q = 0, 1, ... , m - 1) in the xz plane. We can regard expression (86) as the radiation pattern of each element of the parallel array, i.e., we can replace F(8) of expression (70) with U(8) of expression (86), and thus obtain the following expression for the radiation pattern of a rectangular array of dipoles:1 cos (,: cos U(8,
s~n 8
8) ILL
n-1 m-l
1'=0
Apqe-ih.
sin
6 cos ~e-ikz'
cos
61
(87)
q=O
where Apq denotes the complex magnitude of the current in the dipole at x = xl" Z = Zq. If the magnitudes of the dipole currents are equal to a constant, say 10, and if the array constitutes a two-dimensional periodic lattice with uniform spacings dx and d. in the x and Z directions (Xl' = pdx, Zq = qd.), expression (87) reduces to
~(':~8) 2 I sin
U(8
0
[n(kdx sin 8 cos
sin 8
-.
(88)
We see that such a rectangular array can cast a narrow beam in the direction (8 = 7r/2,
or by
s - ~ R. (LxL.) r
-
'\J;- 87r2r02
dx2d.2
2
(90)
1 Although this expression was derived by considering a parallel array of similar collinear arrays, it is valid also for the more general case where the complex amplitudes Apq of the dipole currents are arbitrarily chosen.
71
Theory of electromagnetic
wave propagation
where Lz( = ndz) and L.( = md.) are by definition the effective dimensions of the array.l In view of expression (90), it appears that Sr increases quadratically with the area LzL. of the array. However, expression (89) is valid only for "small" or "moderately sized" arrays because as the array is enlarged the field at the fixed observation point (T = TO, 8 = 71"/2, q, = 71"/2) changes in nature from a far-zone, or Fraunhofer, field to a near-zone, or Fresnel, field. If we take L( = V Lz2 Ly2) as the 2 typical dimension of the array, the condition that the array be contained well within the first Fresnel zone is
+
(91) From this it follows that L2
< X2 - 16
+ XTO
Since X/TO
(92)
2
« 1, expression
(92) reduces to (93)
Thus we see that the critical value of L is Le = VXTo/2. If L < Le, the observation point is in the far zone and the previously derived formulas are valid. On the other hand, if L > Le, the observation point is in the near zone and to find the radiation one must take into account the fact that the field is now of the Fresnel type. Tetelbaum3 1 The effective dimensions so defined are the limiting values of the actual dimensions (L.)actual = (n - l)dz and (L.)actual = (m - l)d. + 2l as n,m~ 00. 2 Let To-I cos wt be the field at observation point due to the dipole at origin. Then (To2 + £2)-~2 cos (wt + q,) is the field at observation point due to the farthest dipole. Assuming that YTo2 + L2 ~ To in the denominator, we see that the resulting field is cos (wt) + cos (wt + q,) = A cos (wt + a). It follows that the modulus A is given by A 2 = 2 + 2 cos q,. The second term is positive as long as q, ~ 71"/2, with q, == (271"/X)(YTo2 + L2 - To). Hence we
have the condition (271"/X)(YTo2
+ L2
- To) ~ 71"/2, or YTo2
+ L2
- To ~ ~.
3 S. Tetelbaum, On Some Problems of the Theory of Highly-directive Arrays, J. Phys., Acad. Sci. U.S.S.R., 10: 285 (1946).
72
Radiation from wire antennas
has performed such a calculation for the case of a square array; his results show that as the array is made larger, S. at first increases in accord with Eq. (90) and then behaves in a manner dictated by Cornu's spiral of Fresnel diffraction theory. A similar calculation has been made by Polkl for the case of it uniformly illuminated rectangular aperture antenna.
3.7 Directivity Gain The directivity gain g of a directional antenna can be calculated from the relation (r ~
00)
(94)
where S.(r,8,t/» denotes the radial component of the far-zone Poynting vector, (S.) max the major-lobe maximum of S.(r,8,t/», dn( = sin 8 d8 dt/» the element of solid angle, and r the radius of a far-zone sphere. This relation directly yields g = 1 for an isotropic antenna and g > 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
Theory of electromagnetic
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
Radiation from wire antennas
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
Theory of electromagnetic
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
Radiation from wire antennas
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
J p'o(p'
- a)o(z')p'dp'det/dz'
=
ez'Tra2Io
(17)
As an example of a configuration having an electric quadrupole mom.ent, 1 The three-index symbol Eap-r has the following meaning: Eali-r = 0 when any two of the subscripts are the same, EaB-r = 1 when a, {3,'Yare all different and occur in the order 12312 . . . (even permutations of 123), and EaB-r = -1 when a, {3,'Yare all different and occur in the order 21321 ... (odd permutations of 123). That is, E123 = Em = E3l2 = 1 and Em = E132 = Em = -1.
84
Multipole expansion of the radiation
field
let us take two antiparallel short filaments of current separated by a. distance d:
J = ezlo[/l(x' - d/2)/l(y')
- /lex'
+ d/2)/l(y')]
(-l ::;z' ::; l)
(18)
Substituting in Eq. (8), we get
Q
= ~ 21Io(eze.
+ e.e
z)
J [x'/l(x'
- d/2) - x'/l(x'
+ d/2)]dx'
or
Q = (~ 21I
0) (eze. + c.ez)d
(19)
Since the dipole moment of each filament is given by Eq. (15) and d = de. is the vector separation of the two filaments, we can write the electric quadrupole moment as Q
=
pd
+ dp
(20)
If we choose the function f(r,r') in relation (3) to be the free-space Green's function, i.e., if we let eikjr-r'l
f(r,r')
(21)
= 41Tlr- r'l
then by virtue of the identity eiklr-r'l
'il'~---
Ir - r'l -
eiklr-r'l
(22)
-'il---
Ir - r'l
relation (3) yields eikjr-r'l
J per') 4;[r
_ r'l dV'
i = -: -;;;'il.
J J(r')
eikjr-r'l
41Tlr_ r'l dV'
(23)
The left side of this equation is the Helmholtz integral representation of Eq,(r); the integral on the right is the Helmholtz integral representa85
Theory of electromagnetic
wave propagation
tion of (1/ jL)A(r). Hence, Eq. (23) expresses the Lorentz condition coupling the vector potential A(r) with the scalar potential !fJ(r), viz., ~ • A(r) = iWEjL!fJ(r)
(24)
Thus we see that the Lorentz gauge is the one that is consistent with the conservation of charge.
4.2 Taylor Expansion of Potentials As in the previous section, let us start by assuming that in a bounded region of space we have an arbitrary distribution of monochromatic current density J(r') and a distribution of charge density p(r') derived from it by the equation of continuity. Then the scalar and vector potentials of the electromagnetic field produced by such a source are given by the Helmholtz integrals 1 -4~E
!fJ(r)
=
A(r)
= 4~
jL
J p(r') J
eiklr-r'l
Ir -- r'1 dV' eik!r-r'l
J(r')
Ir __ r'l dV'
(25) . (26)
To expand these potentials in Taylor series we need the three-dimensional generalization of the familiar one-dimensional Taylor series. We recall that the one-dimensional Taylor series expansion of a function f(x) is given by (27)
When x is replaced by r, h by some vector a, and hd/dx by a'~, the Taylor series (27) heuristically takes the ~hree-dimensional form f(r
+ a)
=
'"1 L ,n. (a . ~)nf(r)
n~O
86
(28)
Multipole expansion of the radiation field
= -
By letting a
r' we see that for any function of r - r' the expansion
is f(r - r')
= ~ ~ (-r' . v)nf(r)
(29)
1., n!
n=O
and hence we have r r
eikl - '! __
Ir - r'l
1 eikr = I'" -(-r'.v)nn=O
n!
(30)
r
With the aid of expansion (30) we develop the Helmholtz integral representation (25) of the scalar potential in the Taylor series If>(r)
= ~41rE
J p(r')[l
- r'. V
+ 72(r'.
eikr ']-dV'
V)2 -
(31)
r
Keeping only the first three terms, we get 41rElf>(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
Theory of electromagnetic
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
Theory of electromagnetic
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
;<;E
(73)
of
Helec.quad..
In spherical coordinates,
+ ereBQrB + Cre~Qr~ + eBerQBr + eBeBQBB + + + + e~erQ~r
Scalarly postmultiplying this expression by 'VG hand for (ik - l/r)G, we get the vector
e~eBQ~B
=
eBe~QB~ e~e~Q~~
erf, where
(74)
f is a short(75)
which, when substituted
i; [;0
(Q~r sin
iw[
f
(Hr)elec,quad.
=
(H
B)elec.quad.
= 2'
(H
~)elec.quad.
=
94
into Eq. (73), yields
iw[
0) -
af)cP QBr] r stu
f)
r sin 0 acP Qrr - Q~r If)
2' QBr r ar (rf) -
fa] r
rIf)] ar (rf)
f)O Qrr
0
(76) (77) (78)
Multipole expansion of the radiation
field
Since 1 a (f) rar r
1 =,ar
a [( ~r'k
ikr
1) e ] 471'r
ikr
-t-
k2 e 471'r=
_
k2G.
(79)
as r -t 00, the only parts of the magnetic field components that survive in the far zone are (H
iwk2
B)elec.quad, =
(H ~)elec,quad,
""2
(80)
Q~rG
iwk2
(81)
= - 2 QB,G
To represent these far-zone field components in terms of the cartesian components of Q, we note that in cartesian coordinates Q has the form
Q = exexQxx
+ exeuQXY + exe.Qx. + eyexQyX + eyeyQyy + eye.Qy. + e.exQ.x + e.eyQ.y + e.e.Q..
which when premultiplied by QOr
= eo.
-
+ Qx.
Q . e,
=
eB, e~
and postmultiplied by
Qxx cos (J sin (J cos2 cJ>
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
+ Qy.
cos cJ>cos (J (84) 95
Theory of electromagnetic
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 <;omponents of Eelec.quad. directly from Eq. (72). We have already calculated the quantity Q • VG which appears in the second term of Eq. (72). The result of this calculation is shown by Eq. (75). Hence, the only quantity we now must calculate is V(Q: VVG) for r -+ 00. By definition of the double scalar product, we have Q:VVG
=
r
Qij(ei'
V)(ej'
V)G
(89)
i,;
where ej denotes the unit vectors in spherical coordinates. function of r only, this definition yields the expression
Since G is a
i)2
Q : VVG = Qrr cr2 G 96
(90)
Multipole expansion of the radiation field
which, in the far zone, reduces to
(r ~
<Xl)
Taking the gradient of this quantity
(91) and keeping only its far-zone term,
we get (92) Substituting Eqs. (75) and (92) into Eq. (72), we see that in the far zone (Er)elec.qu.d. disappears an_d the other two components of Eelec.qu.d. are given by
(Ee)elec.qu.d.
ik3 = - 2; QerG
(93)
(E.p)elec.qu.d.
ik3 = - 2;Q.prG
(94)
With the aid of Eqs. (80) and (81) it is clear that this result agrees with Eqs. (87) and (88).
4.4 Expansion of Radiation Field in Spherical Waves There is an alternative type of multipole expansion which in certain instances is more natural than the one based on the Taylor series expansion. In this section we shall construct such an expansion by first developing the radiation in spherical E and H waves, then defining the E waves as electric multipoles and the H waves as magnetic multipoles, and finally calculating the expansion coefficients through Bouwkamp and Casimir's method.! Outside the bounded region V 0, which completely contains the monochromatic source currents, the electric and magnetic fields can be con-
Ie. J. Bouwkamp and H. B. G. Casimir, On Multipole Expansions in the Theory of Electromagnetic Radiation, Physica, 20: 539 (1954). 97
Theory of electromagnetic
wave propagation
veniently derived from two scalar functions by use of the expressions
+ iWILV
E
=
V X V X (rv)
H
=
V X V X (ru) -
X (ru)
(95)
(96)
iWEV X (rv)
The two scalar functions u and v are the Debye potentialsl which satisfy the scalar Helmholtz equation (97) and obey the Sommerfeld radiation condition. Such a representation of an electromagnetic field in terms of the Debye potentials is quite general. Indeed, it has been proved2 that every electromagnetic field in a source-free region between two concentric spheres can be represented by the Debye potentials; the proof rests on Hodge's decomposition theorem for vector fields defined on a sphere. 3 We choose a spherical coordinate system (r,e,q,) with center somewhere ~ithin Vo• Then the acceptable solutions of Eq. (97) are the spherical wave functions (n ;:::0, m
= 0, II, ...
, In) (98)
The radial functions hn(kr) are the spherical Hankel functions of the first kind, which satisfy the differential equation r2
~;2n
+ 2r a:lr + [k r n
2 2 -
n(n
+ l)jh
n
=
0
(99)
and obey the radiation condition. They are related to the fractionalorder cylindrical Hankel functions of the first kind by
I P. Debye, Dissertation, Munich, 1908; also, Der Lichtdruck auf Kugeln von beliebigen Material, Ann. Phys., 30: 57 (1909). 2 C. H. Wilcox, Debye Potentials, J. Math. Meeh., 6: 167 (1957). 3 P. Bidal and G. de Rham, Les formes differentielles harmoniques, Commentarii Mathematiei Helvetiei, 19: 1 (1956).
98
Multipole expansion of the radiation
field
The fact that hn can be expressed in terms of the exponential function is sometimes useful; for examplel
ho(kr)
i eikr
= -
kT
(100)
etc. The angular functions Y nm(O,q,) are the surface spherical harmonics of degree n and order m. They constitute a complete set of orthogonal functions on the surface of a sphere. Displaying explicitly the normalization constant, we write (-n
~ m ~ n) (101)
where Pnm(cos 0) are the associated Legendre polynomials of degree n and order m j then in view of [m( Jo Pn [
'If'
]2' _ 2 (n cosO) smOdO-2n+l(n_m)!
+ m)!
(-n
~ m ~ n) (102)
we see that this choice of normalization constant leads to the orthogonality relations
(103)
where 5ij = 1 for i = j and 0 for i ~j. Here t,he quantity Ynm* is the conjugate complex of Ynm and is simply related to Yn-m as follows: (104) 1 See P. M. Morse, "Vibration and Sound," 2d ed., pp. 316-317, McGrawHill Book Company, New York, 1948.
99
Theory of electromagnetic
wave propagation
This relation clearly follows from definition (101) when we recall p,,-m(cos
8)
=
+ m)' m) i Pnm(cos
(n -
(n
(_1)m
8)
The Debye potentials are linear superpositions functions (98). That is, 00
v(",8,cp)
=
(105) of these spherical wave
m=n
L L
anmlfnm
(106)
bnm!/lnm
(107)
n=Om=-n m=n
u(r,O,cP)
=
L'" L 11=0
m=-ll.
The expansion coefficients anm, bnm could be calculated from a knowledge of u and v on the surface of a sphere of radius r = To by using the property that the functions !/Inm are orthogonal over the surface of the sphere, viz.,
However, we shall not determine them in this way. Rather, we shall determine them from the radial components of the electric and magnetic fields, in accord with the method of Bouwkamp and Casimir.! Substituting expressions (106) and (107) for the Debye potentials into representations (95) and (96), we get the following expansion of the electromagnetic field in spherical wave functions: (109) n.m
(110) n,m
W e ~re free to consider this electromagnetic field as a superposition of two electromagnetic fields, one being an E type field (E, ~ 0, Hr = 0) and the other an H type field (Hr ~ 0, Er = 0). Accordingly we ! C. J. Bouwkamp and H. B. G. Casimir, On Multipole Expansions in the Theory of Electromagnetic Radiation, Physica, 20: 539 (1954). Also, H. B. G. Casimir, A Note on Multipole Radiation, Helv. Phys. Acta, 33: 849
(1960). 100
Multipole expansion of the radiation field
decompose the electromagnetic field E, H as follows:
+ E" H" + H'
E
= E'
(111)
H
=
(112)
where E/, H' denote the E type field and E", H" the H type field. Comparing Eqs. (109) and (110) with Eqs. (111) and (112) respectively, we see that the E type field is given by (113) n,m
H'
L anm'V X (rif;nm)
= -iWf
(114)
n,m
and the H type field by E"
= iwJ.l
L bnm'V X (rif;nm)
(115)
n,m
H"
L bnm'V X 'V X (rif;nm)
=
(116)
n,m
Moreover, if we let (rif;nm) } ~lectric multipoles of X (rif;nm) degree n and order m
E~m
= 'V X 'V X
(117)
H~m
= -iWf'V
(118)
and E~m H:.'m
= iwJ.l'V =
X (rif;nm)
'V X 'V X (rif;nm)
} magnetic multipo.les of
(119)
degree n and order m
(120)
then Eqs. (113) through (116) become (121) H'
=
L anmH~m L bnmE:.'m
(122)
n,m
E"
=
(123)
n,m
H"
= L., ~
b nm H" nm
(124)
n,m
101
Theory of electromagnetic
wave propagation
We define E~m, H~m to be the electromagnetic field of the electric multipole of degree n and order m, and E;,'m, H;,'m to be the electromagnetic field of the magnetic multi pole of degree n and order m, so that expressions (121) through (124) constitute the multipole expansion of the electromagnetic field. Thus a superposition of the terms n = 1, m = 0, :!: 1 yields a dipolar field, and a superposition of the terms n = 2, m = 0, :!: 1, :!: 2 yields a quadrupolar field, and so forth. As yet the expansion coefficients have not been fixed; before we start to calculate them, let us deduce the spherical components of the multipole fields. To find the spherical components of E~m, H~m and E;,'m, H;,'m, we make use of the following relations. We note that
= n(n
where the second equality follows from Eq. (99).
+ 1)J/;nm
(125)
We also note that (126)
Denoting the unit vectors in the 0 and q, directions by eg and e.p respectively, we obtain by vector analysis the angular components of V X (rJ/;nm) and V. X V X (rJ/;nm):
eg' V X V X
(rJ/;nm)
e.p • V X V X (rJ/;nm)
1a
=
1 -0 sm
eg • V X (rJ/;nm)
= -.
e.p • V X (rJ/;nm)
= -
a
= ;: ar ao (rJ/;nm)
(127)
-!-o aar a:'I' (rJ/;nm) SIn
(128)
a J/;nm aq,
(129)
r
aoa J/;nm
(130)
With the aid of relations (125) through (130) we see that the spherical components of the multipole fields (117) through (120) are given by 102
Multipole expansion of the radiation field
=
(E~m)r
n(n
(E~mh = ~ ; (E~m)4> = (H~m)r
=
(H~m)e
=
+ 1) h"Y"m
r
(rh,,) :()
Y"m
i:n () dd (rh")Y,,m r sm r 0 ~E()
sm
spherical components of electric multipole field of degree n and order m (n ~ 0, -n 5 m 5 n)
(131)
spherical magnetic of degree (n ~ 0,
(132)
h"Y"m
(H~m)4> = iWEh" :() Y"m and (HI!) nm
,
(H'.:m)e
=
n(n
= ~;
r+
1) h Y n-
m n
(rh,,) :() Y"m
(H'.:m)4> =
. i:n () dd (rh,,) Y"m r sm r
(E~:m)r
=
0
(E'.:m)e
= - sm ~WM()h"Y"m
(E'.:m)4> = -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
Theory of electromagnetic
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'
=
r •E
WE
,
G(r,r)
Vo
v'
41rlr _ r'l
=
(144)
X J(r')dV'
n="
eiklr-r'l =
X
ik '\' 411" 1.. (2n
+
., 1)Jn(kr )hn(kr)Pn(cos
'Y)
(145)
n=O
where in(kr') = (1I"/2kr')'tiJn+",,(kr') and cOS'Y= cos (J cos (J' + sin (J sin (J' cos (I/J - I/J'). It is also known that m=n
Pn(cos 'Y) =
L
(-1)mPnm(cos (J)P n-m(COS (J')eim(4>-4>')
(146)
m=-n
Recalling Ytnm
= hn(kr) Y nm«(J,I/J) = [(2n
+ 1) ~: ~
and introducing the functions
Xnm,
:~:
r
hn(kr)P nm(COS(J)eimq, (147)
which are defined by
(148) 105
Theory of electromagnetic
wave propagation
we find from expansions (145) and (146) that the free-space Green's function can be expressed as follows: (149)
Substituting r' H
'k
= ~
this expression into Eq. (142) we get
L
(-I)"iYnm(r)
J
Xn-m(r')r' . \7' X J(r')dV'
Vo
(150)
n,m
An integration by parts yields
Jv. Xn-m(r')r'.
\7' X J(r')dV'
=
Jv. J(r') .
\7' X (r'xn-m)dV'
(151)
and hence the expansion for r . H becomes r' H
=
1: L
(-I)"'lfnm(r)
Jvo J(r')
• \7' X (r'xn-m)dV'
(152)
n,m
From Eq. (144) it similarly follows that
r. E
= - ~
471"
&. ~
'\j~ n,m L
f
(-I)"iYnm(r)
Vo
J(r/).
\7'
X \7' X (r/Xn-m)dV'
(153)
Comparing Eq. (152) with Eq. (134) and Eq. (153) with Eq. (133), we finally obtain the desired formulas
anm = b nm --
-
ik 471"
1 471"
{j; (_l)m '\j;- n(n + 1)
f
J( ')..." r
Vo
+ 1) f v. J(')"'" r •
(_l)m
n(n
v
X
.
('
r
v
X
...,1 v
-m)dV' Xn
X
('
-m)dV'
r Xn
(154) (155)
which give anm and bnm in terms of the current. From the above analysis we see that a multipole expansi9n of the electromagnetic field E, H radiated by a monochromatic current J is 106
Multipole expansion of the radiation field
obtained by developing E and H in basic multipole fields E~m, H~m and E;.'m, H;.'m, that is, by writing E and H in the following form:
E
=
L an",E~m
+
n,m
H
=
L anmH~m
n,m
L bnmE;.'m
(156)
n,m
+
L bnmH~m
(157)
n.m
Here the basic multipole fields are given explicitly in terms of spherical wave functions by the definitions (117) through (120), and the expansion coefficients anm, bnm (which constitute a decomposition of the known current J into electric and magnetic multipoles superposed at the origin of coordinates) are deduced from J by evaluating the integrals (154) and (155). In the far zone (kr -7 00), the basic multipole fields can be expressed most conveniently in terms of the operator 1 L=-:-rXV
(158)
~
which in wave mechanics is known as the angular-momentum operator. To show this, we note that the asymptotic form of the spherical Hankel function is
(kr
-7
00)
(159)
From this form it follows that
(160)
and
(161) 107
Theory of electromagnetic
wave propagation
Hence in terms of the operator L we have (162) (163) Using expressions (162) and (163), we thus see that the basic multipole fields (117) through (120) in the far zone are (164) (165)
eikr r
(166)
E:.'m = -(-i)niVJ.l/E-LYnm H" nm
=
-(-i)ni
eikr - r eT X LY
(167)
m n
Substituting these multipole fields into expansions (156) and (157), we find that the far-zone electromagnetic field is given by
H
=
e r [-LanmVE/J.l ikr
_
(-i)n+II .•Ynm
+L
108
bnm(_i)n+le, X Lynm]
(169)
Radioastronomical antennas
Observational radio astronomy is concerned with the measurement of the radio waves that are emitted by cosmic radio sources.! With the apparently single exception of the monochromatic radiation at A = 21 centimeters, i.e., the "hydrogen line" emitted by interstellar hydrogen, cosmic radio waves are rapidly and irregularly varying functions of time, resembling noise. The measurable properties of cosmic radio waves are their direction of arrival, state of polarization, spectrum, and strength. For ground-level observations, radio astronomy is limited essentially to the band ranging approximately from 1 centimeter to 10 meters, because waves of wavelength greater than about 10 meters are unable to penetrate the earth's ionosphere and those of wavelength less than about 1 centimeter are absorbed by the earth's atmospheric gases. However, radio-astronomical observa! For a popular exposition on radio astronomy, see the delightful and informative monograph by F. G. Smith, "Radio Astronomy," Penguin Books, Inc., Baltimore, 1960. For a comprehensive treatment of the subject, see J. L. Pawsey and R. N. Bracewell, "Radio Astronomy," Oxford University Press, Fair Lawn, N.J., 1955; also 1. S. Shklovsky, "Cosmic Radio Waves," Harvard University Press, Cambridge, Mass., 1960. See also F. T. Haddock, Introduction to Radio Astronomy, Proc. IRE, 46: 3 (1958); and R. N. Bracewell, Radio Astronomy Techniques, Handbuch der Physik, LIV, Springer-Verlag OHG, Berlin. See also J. L. Steinberg and J. Lequeux, "Radio Astronomy," McGraw-HilI Book Company, New York, 1963.
109
5
Theory of electromagnetic
wave propagation
tions have also been made in the band ranging from about 3 millimeters to 1 centimeter. The instrument that is used to measure cosmic radio waves is the "radio telescope." It consists of three basic components operating in tandem, viz., a receiving antenna, a sensitive receiver, and a recording device. Functionally, the antenna collects the incident radiation and transmits it by means of a wave guide or coaxial line to the input terminals of a receiver; the receiver in turn amplifies and rectifies the input signal; and then the recording device, which is driven by the rectified output of the receiver, presents the data for analysis. The rectified output of the receiver is a measure of the power fed to the receiver by the antenna. Since the cosmic signals arriving at the input terminals of the receiver are noiselike and similar to the unwanted noise signals which are unavoidably generated by the receiver itself, the receiver must be able to distinguish the desired noise signal from the undesired one. This is a difficult requirement and is met by a "radiometer," which consists of a high-quality receiver and special noise-reducing circuitry. To reduce even further the effects of the receiver noise, radiometers sometimes are supplemented with a low-noise amplifier such as a maser! or a parametric amplifier operating in front of the receiver.2 The part of the radio telescope that we shall consider in this chapter is the antenna, and our presentation will cover only the radiation theory of such radio-astronomical antennas. The reader interested in the more practical and operational aspects of the subject is referred to the literature. 3 1 The first application of a maser (X band) to radio astronomy was made by Giordmaine, Alsop, Mayer, and Townes [J. A. Giordmaine, L. E. Alsop, C. H. Mayer, and C. H. Townes, Proc. IRE, 47: 1062 (1959)]. See also J. V. Jelley and B. F. C. Cooper, An Operational Ruby Maser for Observations at 21 Centimeters with a 60-Foot Radio Telescope, Rev. Sci. Instr., 32: 166 (1961). 2 See, for example, F. D. Drake, Radio-astronomy Radiometers and Their Calibration, chap. 12 in G. P. Kuiper and B. M. Middlehurst (eds.), "Telescopes," The University of Chicago Press, Chicago, 1960. 3 See, for example, J. G. Bolton, Radio Telescopes,chap. 11 in G. P. Kuiper and B. M. Middlehurst (eds.), "Telescopes," The University of Chicago Press, Chicago, 1960.
110
Radio-astronomical antennas
5.1 Spectral Flux Density Since an incoming cosmic radio wave is a plane transverse electromagnetic (TEM) wave, its field vectors E(r,t) and H(r,t) are perpendicular to each other and to the direction of propagation. Consequently the Poynting vector of the wave, viz., S(r,t) = E(r,t) X H(r,t), is parallel to the direction of propagation, and its magnitude is given by the quadratic quantity S(r,t)
=
E(r,t)
VfO/ILO
(1)
• E(r,t)
It is an observed fact that each component of the field vectors, insofar as its time dependence is concerned, has the character of "noise." That is, at any fixed position r = ro the field vectors are rapidly and irregularly varying functions of time, yet in their gross behavior they are essentially independent of the time and in particular do not vanish at t = :!: 00. They constitute what is known as a stationary random (or stochastic) process.l By virtue of this noiselike behavior of the wave, it is the spectral density of the time-average value of S(r,t), and not the instantaneous value of S(r,t) itself, that constitutes a meaningful measure of the strength of the incoming wave. In order to resolve the incoming signal into its Fourier components, we must introduce the truncated function ET(ro,t) defined by ET(ro,t)
= E(ro,t)
for
It I ~
T
ET(ro,t)
= 0
for
It I >
T
(2)
where 2T is a long interval of time. its Fourier transform
AT(w)
=
1 271"
Joo _
00
ET(ro,t)eiwt dt
Since ET(ro,t) vanishes at t
= :!: 00,
(3)
1 For general theory of stochastic (random) processes see, for example, S. O. Rice, Mathematical Analysis of Random Noise, Bell System Tech. J., 23: 282 (1944); 25: 46 (1945); S. Chandrasekhar, Stochastic Problems in Physics and Astronomy, Rev. Mod. Phys., 15 (1): 1 (1943).
111
Theory of electromagnetic
wave propagation
and its Fourier integral representation (4)
always exist as long as T is finite. We know from the theory of stochastic processes that the transform AT(w) increases without bound as T -> 00, whereas the quadratic quantity IAT(w)12/T tends to a definite limit,l i.e., lim
1
-T
T-••,
IAT(w)!2 =
(5)
finite limit
This fact suggests that a quadratic quantity such as the time-average Poynting vector be considered. According to Eqs. (1) and (4), the magnitude of Poynting's vector is
and its time-average value, defined as (S(ro,t»
1 = T-+oo, lim 2 T
I~ST(ro,t)dt
(7)
T
is given by (S(ro,t»
=
r; T--.oo lim 2 T 1 T [1 '\J""io -T-oo 1
00
AT(w')e-iw't
dw'
Since lim T--.oo
..!-.. 2T
1
T -T
e-i(w'+w")t dt
=
lim.!!:. o(w' T--.oo
T
+ w")
(9)
1 For rigorous mathematical theory see N. Wiener, Generalized Harmonic Analysis, Acta Math., 55: 117 (1930).
112
Radio-astronomical antennas where 5 is the Dirac delta function,
(S(ro,t)
1r G T...• lim -T II'" '\JJ;o oo
=
Ar(w').
Eq. (8) reduces to
+ w")dw'dw'
AT(w")5(w'
-'"
=
~ '\JJ;o
I
1r
lim -T
T...• oo
00
AT(w).
Ar( -w)dw
(10)
-00
Using the relation Ar(w) = A~( -w), which is a consequence of the fact that Er(ro,t) in Eq. (3) is real, we see that Eq. (10) may be written as the one-sided integral (S(ro,t»
r '" [ '\JJ;o G lim
=
Jo
T...•'"
2 1rAr(w) • A~(w)] T
dw
(11)
This expression gives the time-average power density of the incoming wave in watts meter-2; hence the quantity S.,(ro) ==
~
'\JJ;o
lim [2 T...• '"
1rAT(w)
T
• A~(w)]
(12)
which is known to be finite by virtue of Eq. (5), gives its spectral flux density I in watts meter-2 (cycles per second)-I. I An alternative definition of the spectral flux density S.,(ro) in terms of the electric field E(ro,t) is based on constructing the autocorrelation function
q,(q)
=
1 I~ Tlim 2 T I~E(t). ...• '" T
'\J}.Lo
E(t
+ q)dt
~
'\J;,~
(E(t) . E(t
+ q»
and then identifying its Fourier transform with the spectral flux density. That is,
If'"
S.,
= 1r
-'"
q,(q)e''''' dq
= 2fcoo 1r
0
q,(q) cos wq dq
If S., is to be the spectral density, the integral of S., over all frequencies must yield the time-average power. To show that this requirement is met by the above definition, we note that
h
O
o
O
s'"
dw
2h'"
= -
1r
0
dq q,(q)
hoo 0
cos wq dw = 2
hoo 0
dq
=
and recognize that q,(O) is indeed the time-average power. 113
Theory of electromagnetic
wave propagation
From a strictly mathematical viewpoint, to obtain the spectral flux density 8", one would have to observe the incoming signal for an infinitely long period of time. In practice, obviously, this is neither possible nor desirable. As a practical expediency, the definition is relaxed by choosing T large enough and yet not so large as to iron out significant temporal variations of 8",. The best that can be done is to observe the incoming signal for successive periods of time equal to the time constant T of the receiving system. What is observed then is the signal smoothed by successive averaging over finite periods of duration T. The finiteness of T produces fluctuations in the record, as does the finiteness of the bandwidth Aw of the receiving system. The period 1 of the fluctuations is approximately 1/ Aw; hence in a time interval T, the incoming signal effectively consists of n( = TAw) independent pulses, whose standard deviation is 1/ y"n = 1/~. In view of this we can write (13)
where oR is the standard deviation of the readings R, and K is a dimensionless constant whose value depends on the detailed structure of the receiving system. Thus we see that the finiteness of T and Aw produces an uncertainty, or spread, in the readings. Consequently, in order that an incoming signal be detectable, the deflection produced by it must be greater than the deflection oR produced by the inherent fluctuations of the receiving system. Since 8", is the power per unit area per unit bandwidth, the power P 1 To see this, we consider the Fourier integral representation (4) for the frequency band w - Aw/2 to w + Aw/2 over which AT(W) is assumed constant. Then
Hence the period of the envelope is 411'/ Aw. 114
Radio-astron.omical
flowing normally through an area A in the frequency range", to '"
+ !:.",/2 is given
antennas
- !:.",/2
by (14)
P = AS",!:.",
5.2 Spectral Intensity, Brightness, Brightness Temperature, Apparent Disk Temperature In the previous radiation. In useful measures One of these defined by dP",
= I",(n')du
section we defined the spectral flux density S", of cosmic this section we shall define in terms of S", some other of cosmic radiation. measures of cosmic radiation is the spectral intensity,
nl . n'
dn(n')
(15)
where dP", is the radiant power per unit bandwidth flowing through an element of area du into a solid angle dQ(n'), nl the unit vector normal to du, and n' the unit vector along the axis of the solid angle (see Fig. 5.1). The quantity I",(n') is the spectral intensity of the radiation n'
dO (n')""""
Fig. 5.1
Geometric construction for definition of spectral intensity. Radiation is emitted by source and passes radially outward through solid angle an. 115
Theory of electromagnetic
wave propagation
traveling in the direction n'. Another such measure is the spectral brightness, which is defined in the same way as the spectral intensity except that n"( = -n') is now the direction from which the radiation is corning (see Fig. 5.2). Accordingly, the power per unit bandwidth falling on the area drT from the solid angle dn(n") is given by dP.,
= b.,(n")drT n2 . n" dn(n")
(16)
where b.,(n") is the spectral brightness of the incoming radiation and n2 = -nl. Comparing expressions (15) and (16), we get the relation b.,(-n')dn(
-n')
=
(17)
I.,(n')dn(n')
which places in evidence the fact that brightness refers to radiation traveling toward drT and intensity refers to radiation traveling away from drT. The quantity dP.,j(drT nl' n') is the power per unit bandwidth per unit area normal to the direction of travel of the radiation and hence it is equal to dS.,. Thus we see that the spectral intensity of the D'
Fig.5.2
116
Geometric construction for definition of brightness. Radiation is emitted by distributed source in sky and passes radially inward through solid angle dn.
Radio-astronomical antennas
emitted radiation is the spectral flux density per unit solid angle, i.e., I
= dB", '"
(18)
dn
Similarly the spectral brightness of the received radiation is the spectral flux density per unit solid angle, i.e., b
=
'"
dB", dn
(19)
If the source of radiation is distributed over the sky, then a convenient measure of the amount of radiation that falls on a receiving antenna from a given direction is the spectral brightness in that direction. As shown in Figs. 5.1 and 5.2, it is most convenient to choose the origin of coordinates at the source for L, and at the receiver for b.,. The spectral brightness b"" like the spectral intensity I w, is a function of 8, q, but not of r. The units of I", and b", are the same since they are defined in the same way, except that in the former the radiation is traveling outward from the vertex of the solid angle where the source is located, and in the latter it is traveling in toward the vertex where the receiver is located. Specifically, the units of I", and b", are watts meter-2 (cycles per second)-l steradian-I. It is sometimes convenient to specify the radiation in terms of the temperature that a blackbody would require in order to produce the measured spectral brightness. According to Planck's law for blackbody radiation in free space, the spectral brightness B", of a blackbody at temperature T (in degrees Kelvin) is given by B",
= 2hc
1 X3 exp (he/kXT)
- 1
(20)
where h = Planck's constant k = Boltzmann's constant c = velocity of light X = wavelength 117
Theory of electromagnetic
wave propagation
But in radio-astronomical applications he « kAT and hence expression (20) may be replaced by the Rayleigh-Jeans approximation to Planck's law, viz., (21) The spectral brightness temperature T.,b of the radiation coming toward the receiving antenna along a direction 8, c/J is obtained by equating expression (21) to b.,. Thus the spectral brightness temperature T.,b is related to the spectral brightness by (22) Like b." the quantity T.,b is a function of 8, c/J only. In case the source subtends a solid angle flo at the receiver, it follows from Eqs. (19) and (22) that (23) Noting that the "apparent 2k
p.isk temperature"
T.,d is defined by
r
S., = };2 } o. T.,d dO
(24)
we see from Eq. (23) that T.,d is related to T.,b by (25) and in this sense constitutes a measure of the average value of the spectral brightness temperature.
5.3 Poincare Sphere, Stokes .Parameters By its very nature a monochromatic electromagnetic wave must be elliptically polarized, i.e., the end point of its electric vector at each 118
Radio-astronomical
antennas
point of space must trace out periodically an ellipse or one of its special forms, viz., a circle or straight line. On the other hand, a polychromatic electromagnetic wave can be in any state of polarization, ranging from the elliptically polarized state to the unpolarized state, wherein the end point of the electric vector moves quite irregularly. Cosmic radio waves are generally in neither of these two extreme states, but rather in an intermediate state containing both elliptically polarized and unpolarized parts. A wave in such an intermediate state is said to be "partially polarized" and is describable by four parameters introduced by Sir George Gabriel Stokes in 1852 in connection with his investigations of partially polarized light.! In this section we define these Stokes parameters and show that they serve as a complete measure of the state of polarization. As an exemplar, we consider the case of a plane monochromatic TEM wave. The electric vector E(r,t) of such a wave traveling in the direction of the unit vector n has the form E(r,t)
= Re {Eoei(k.r-wt) I
(26)
where k( = n2'n-jX) is the wave vector and Eo the complex vector amplitude. Because the wave is plane and TEM, vector Eo is a constant and lies in a plane perpendicular to n, that is, n' Eo = O. Since the polarization of the wave is governed by Eo, and since Eo is a constant, the state of polarization is the same everywhere. This constancy of polarization is peculiar to plane, homogeneous waves, the polarization of more general types of electromagnetic fields possibly being different at different points of space. For example, if the field were a wave generated by a source of finite spatial extent, the polarization would vary with radial distance from the source as well as with polar and azimuthal angles. Without sacrificing generality, we choose a cartesian coordinate system x, y, z such that the z axis is parallel to n. With respect to this system Eo can be written as (27) 1 G. G. Stokes, On the Composition and Resolution of Streams of Polarized Light from Different Sources, Trans. Cambridge Phil. Soc., 9: 399 (1852). Reprinted in his "Mathematical and Physical Papers," vol. III, pp. 233-258, Cambridge University Press, London.
119
Theory of electromagnetic
wave propagation
where ez, eu are unit vectors along the x and y axes respectively, and where the amplitudes az, au as well as the phases oz, Oy are real constants. Thus from Eqs. (26) and (27) it follows that the cartesian components of E(z,t) are given by the real expressions
=
Ez
az cos (t/>
+
E,
oz)
=
0
(28)
where for brevity the shorthand t/> = wt - kz has been used. nating t/> from these expressions, we get
(E)2 a
+ (E)2 -!!. au
-.!: z
-
2
Ea Eau cos -.!: -!!.
Elimi-
0 = sin 2 0
(29)
z
where (30) y
2c)'
"
-------2c Fig. 5.3
120
.•------
Polarization ellipse for right-handed orientation angle if;.
....• polarized
wave having
Radio-astronomical
antennas
Taking E", and Ey as coordinate axes, we see that Eq. (29) represents an ellipse whose center is at the origin E", = Ey = O. Geometrically this means that at each point of space the vector E rotates in a plane perpendicular to n and in so doing traces out an ellipse. As is evident from expressions (28), the rotation of E and the direction of propagation n form either a right-handed screw or a left-handed screw, depending on whether sin 0 < 0 or sin 0 > 0 respectively. Accordingly, in conformity to standard radio terminology, the polarization of a wave receding from the observer is called right-handed if the electric vector appears to be rotating clockwise and left-handed if it appears to be rotating counterclockwise. See Fig. 5.3. To determine the polarization ellipse of a monochromatic wave, a set of three independent quantities is needed. One such set obviously consists of the amplitudes a"" ay and the phase difference O. Another set is made up of the semimajor and semiminor axes of the ellipse, denoted by a and b respectively, and the orientation angle 1/; between the major axis of the ellipse and the x axis of the coordinate system. These two sets are related such that a, b, 1/; can be found from a"" ay, 0 and vice versa. The well-known connection relations are!
a2
+ b2 = a",2 + a/ 2a",ay
tan 2 1/;=
2
a", -
Moreover,
ay
2COSO
(31)
(0 :::; 1/;
< 11")
(32)
we have
(33)
where
X
is an auxiliary angle defined by b
tanx=+--a
(34)
The numerical value of tan x yields the reciprocal of the axial ratio alb of the ellipse, and the sign of x differentiates the two senses of polariza1 See, for example, M. Born and E. Wolf, "Principles of Optics," pp. 24-31, Pergamon Press, New York, 1959.
121
Theory of electromagnetic
wave propagation
tion, e.g., for left-handed polarization 0 < X ~ 71"/4and for right-handed polarization -71"/4 ~ X < o. The Stokes parameters for the monochromatic plane TEM wave (28) are the four quantities
(35) But since the quantities are related by the identity (36) only three of the four parameters are independent. Alternatively, the Stokes parameters can be written in terms of the orientation angle Vt and the ellipticity angle X as follows: 81
=
80
cos 2x cos 2if;
82
=
80
cos 2x sin 2if;
83
=
80
sin 2x
(37)
where 80 is proportional to the intensity of the wave. From these expressions we see that if 81, 82, 83 are interpreted as the cartesian coordinates of a point on a sphere of radius 80, known as the Poincare sphere, 1 the longitude and latitude of the point are 2if; and 2x respectively (see Fig. 5.4). Thus, there is a one-to-one correspondence between the points on the sphere and the states of polarization of the wave. In order that the wave be linearly polarized, the phase difference 0 must be zero or an integral multiple of 71", and consequently, according to Eq. (33), X must be zero. Thus we see that the points on the equator of the Poincare sphere correspond to linearly polarized waves. In order that the wave be circularly polarized, the amplitudes az and au must be equal and the phase difference 0 must be either 71"/2or -71"/2, depending on whether the sense of polarization is left-handed or right-handed respectively. Hence, from Eq. (33) it follows that for a left-handed circularly polarized wave 2x = 71"/2and for a right-handed circularly polarized wave 2x = -71"/2; that is, the north and south poles of the Poincare sphere correspond respectively to left-handed and righthanded circular polarization. The other points on the Poincare sphere 1 H. Poincare, "Theorie mathematique de la lumiere," vol. 2, chap. 12, Paris, 1892.
122
Radio-astronomical
Fig. 5.4
Poincare sphere is a sphere of radius has latitude 2X and longitude 2if;.
80 •.•
antennas
A point on sphere
represent elliptic polarization, right-handed in the southern hemisphere and left-handed in the northern hemisphere. Since E(z,t) has only two components Ex, Ey it can be represented, for any fixed value of z( = zo), as a vector in the complex plane whose real and imaginary axes are Ex and Ey respectively. That is, as a function of t, to each value of E(zo,t) there corresponds, a point Ex + iEy in the Argand diagram. I With the aid of this representation an elliptically polarized wave may be decomposed into a right-handed and a left-handed circularly polarized wave. We note that in the complex plane circularly polarized waves of opposite senses are given by the complex vectors PI exp (iwt) and P2 exp (-iwt + i'Y), the former being 1 See, for example, K. C. Westfold, New Analysis of the Polarization of Radiation and the Faraday Effect in Terms of Complex Vectors, J. Opt. Soc. Am., 49: 717 (1959).
123
Theory of electromagnetic
wave propagation
ri/!:ht-handed and the latter left-handed. Thus in terms of the moduli PI, P2 the semimajor and semiminor axes of the polarization ellipse of the wave consisting of the superposition of these two circularly polarized waves are P2 + PI and P2 - PI. The orientation angle I/; of the ellipse is given by 21/; = 1', where l' is the phase angle between the complex vectors at t = 0 (see Fig. 5.5). Since the axial ratio of the polarization ellipse is (P2 - PI)/(P2 + PI), the angle X is given by tan x = (P2 - PI)/(P2 + PI). From this it follows that •
sm 2x =
P22 P2
2
PI2
+ PI
2
cos 2x
=
2P2PI P2
2
+ PI
(38)
2
Substituting Eqs. (38) into expressions (37), recalling that 21/; = 1', and noting that so( = P22 + P12) is the intensity of the wave, we get the following expressions for the Stokes parameters in terms of the moduli P2, PI and the phase difference 1': 2(P22
80
=
83
= 2(.P22
+ P12) -
82
= 4 P2PI sin l'
P12)
(39) iEy
Fig. 5.5
124
Complex plane. Splitting of elliptical polarization oppositely polarized circular components.
into two
Radio-astronomical
antennas
The state of polarization can be measured in a number of different ways. For example, as is suggested by expressions (35), the state of polarization can be measured by using two linearly polarized receiving antennas in such a way that one yields ax, the other yields ay, and the phase difference between their responses yields o. Alternatively, from expressions (39) it is seen that the state of polarization also can be measured by using two circularly polarized antennas of opposite senses, one of the antennas yielding Pz, the other yielding Pi, and the phase difference of their responses yielding 'Y. The accuracy of these methods of measurement depends largely on how purely linear is the linearly polarized antenna and how purely circular is the circularly polarized antenna. The techniques of measuring the polarization of monochromatic waves are well known and will not be discussed here.l Using the monochromatic wave (26) as a prototype, we now examine the case of a plane polychromatic TEM wave, which by virtue of its polychromatic character can be elliptically polarized, or unpolarized, or partially polarized. We assume that the frequency spectrum of the wave is confined to a relatively narrow band of width dw so that the electric vector of the wave, in analogy with expression (26), may have the simple analytic representation E(z,t)
= Re IEo(t)ei(.wlc-wo I
(40)
where w now denotes sOll1eaverage value of the frequency. Because the bandwidth is narrow, Eo(t) may change by only a relatively small amount in the time interval 1/dw and in this sense is a slowly varying function of time. If the bandwidth were unrestricted, the moot question of representing a broadband signal in analytical form would arise and the problem would have to be reformulated.2 In practice, howl See, for example, H. G. Booker, V. H. Rumsey, G. A. Deschamps, M. L. Kales, and J. 1. Bohnert, Techniques for Handling Elliptically Polarized Waves with Special Reference to Antennas, Proe. IRE, 39: 533 (1951); D. D. King, "Measurements at Centimeter Wavelength," pp. 298-309, D. Van Nostrand Company, Inc., Princeton, N.J., 1950; J. D. Kraus, "Antennas," pp. 479-484, McGraw-Hill Book Company, New York, 1950. 2 See A. D. Jacobson, Theory of Noise-like Electromagnetic Fields of Arbitrary Spectral Width, Calteeh Antenna Lab. Report, No. 32, June, 1964. Also, Robert M. Lerner, Representation of Signals, chap. 10in E. J. Baghdady (ed.), "Lectures on Communication System Theory," McGraw-Hill Book Company, 1961.
126
Theory of electromagnetic
wave propagation
ever, this difficulty is compulsorily bypassed inasmuch as the instruments used in measuring polarization are inherently narrowband devices. Writing Eo(t) in the form Eo(t)
= e"a,,(t)e-wz(t)
'.~
+ eyauCt)e- .(t)
(41)
i6
where the amplitudes a,,(t), ay(t) and the phases o,,(t), Oy(t) are slowly varying functions of time, we see from Eq. (40) that the cartesian components of E(z,t) are given by E"
=
Ey
= ay(t)
E.
=
+ o~(t)] cos [ep + o,,(t) + o(t)]
a,,(t) cos [ep
(42)
0
where ep = wt - zw/c, o(t) = Oy(t) - o,,(t). Although the amplitudes and phases are irregularly varying functions of time, certain correlations may exist among them. .It is these correlations that determine the Stokes parameters and consequently the polarization of the wave. By definition, the Stokes parameters of the polychromatic wave (42) are the time-averaged quantities 80
= (a,,2(t)
82 =
+ (ay2(t)
2(a,,(t)auCt) cos o(t)
81 = 83
(a,,2(t) - (a/(t)
=
(43)
2(a,,(t)ay(t) sin o(t)
which are generalizations of the monochromatic Stokes parameters (35). It can be shown 1 that the polychromatic Stokes parameters satisfy the relation 2
80
2
;::: 81
+
2
82
+
..
(44)
2
83
where the equality sign holds only when the polychromatic wave is elliptically polarized. The polychromatic wave (42) is 'elliptically polarized when the ratio q of the amplitudes (q = ay/a,,) and the phase differences 0 are absolute 1 See, for example, S. Chandrasekhar, "Radiative Dover Publications, IhC., New York, 1960.
126
Transfer,"
pp. 24-34,
Radio-astronomical
antennas
constants. That is, when q and IJ are time-independent, the electric vector of the wave traces out an ellipse whose size continually varies at a rate controlled by the bandwidth ~w but whose shape, orientation, and sense of polarization do not change. To demonstrate this, we note that for an elliptically polarized wave the Stokes parameters (43) become (1 + q2)(az2(t»
80
=
82
= 2q(az2(t»
(1 - q2)(az2(t»
81 = 83 =
cos IJ
2q(az2(t»
(45)
sin IJ
+
+
2 2 Since these parameters satisfy the identity 802 = 812 82 83 , only three of them are independent. In analogy with Eqs. (37) we can write the Stokes parameters of an elliptically polarized polychromatic wave
in the form 81
=
80
cos 2x cos 2if;
Consequently
82
=
the orientation
80
cos 2x sin 2if;
83 = 80
angle if;of the polarization
sin 2x
(46)
ellipse is given
by tan2if;
82
= -
81
=
2q
--cos 1 - q2
and its ellipticity . 2
SIn X
angle
= -83 = -- 2q.
1 + q2
80
IJ
X
(47)
by
sm IJ
(48)
Since q and IJ are time-independent, it is clear from Eqs. (47) and (48) that if;and X are time-independent, in confirmation of the fact that the shape, orientation, and sense of polarization do not change. We return to the polychromatic wave (42) and now assume that the phase of Ey is shifted with respect to the phase of Ez by an arbitrary constant amount E. The cartesian components of such a polychromatic wave are given by Ez
=
az(t) cos
Ey
=
ait)
E.
= 0
[q,
cos [q,
+ IJz(t)J + IJ (t) + lJ(t) + EJ z
(49)
127
Theory of electrOlpagnetic
wave propagation
As is clear from Fig. 5.6, the component of the electric field along the x' axis making an angle ()with the x axis is Ez'«(),E) = Ez cos () + Ell sin () and its square is Ez,2«(),e) = Ez2 cos2 () + E/ sin2 () + 2EzEII cos ()sin (). Substituting expressions (49) into this quadratic form, we find that tM instantaneous value of Ez,2«(),e) is Ez,2«(),e)
= az2(t) cos2 T cos2
()
+ aIl2(t)
+ 2az(t)ay(t)
+ B(t) + sin cos T cos [T + B(t) + EJ cos fJ sin () cos2 [T
EJ
2 ()
where T = 4>+ Bz(t). Recalling that az(t), all(t), B(t) are slowly varying functions of time and that cos T = cos [4>+ Bz(t)J = cos [wt - zwjc
+ Bz(t)J
is a rapidly varying function of time, we find that the mean value of 2Ez,2«(),e), which we denote by I«(),e), has the following'representation: I«(),e)
=
2(Ez,2«(),e»
+ [(az(t)all(t)
=
(az2(t»
cos2
()
+ (aIl2(t»
cos B(t» cos e - (az(t)ay(t)
sin2
()
sin B(t» sin EJ sin 2() (50)
y
Fig. 5.6 :~~',
128
Linearly polarized antenna that picks up the component Ez of electric field along x' axis. Its response is proportiQnal to the mean-square value of Ez'.
Radio-astronomical
With the aid of definitions (43) this representation
antennas
leads directly to the
relation [(O,E)
=
~[80
+
81
cos 20
+
(82
cos E -
8a
sin E) sin 20]
(51)
which shows that [(O,E) is linearly related to the Stokes parameters. It is evident from relation (51) that the Stokes parameters can be determined by measuring [(O,E) for various values of 0 and E. If [(O,E) happens to be independent of 0 and E, the wave is said to be "unpolarized." In other words, an unpolarized wave is one that satisfies [(O,E)
(52)
= ~8o
independently of 0 and E, or, equivalently, the necessary and sufficient condition that the wave be unpolarized is 81
=
82
=
8a
= 0
(53)
If the polychromatic wave consists of a superposition of several physically independent waves, the intensity of the resulting wave is the sum of the intensities of the independent waves. That is, if [(n) denotes the intensity of the nth independent wave, the intensity [ of the composite wave is given by (54)
Moreover, since each of the independent we have for the nth independent wave
waves satisfies relation (51),
where 80(n), 81 (n), 82(n), 8a(n) are the corresponding Stokes parameters. Hence, from Eqs. (54) and (55) we get the expression
(56) 129
Theory of elec!romagnetic
wave propagation
which, when compared with expression (51), shows that each of the Stokes parameters of the composite wave is the sum of the respective Stokes parameters of the independent waves. That is, the Stokes parameters are additive in the sense that (57)
where
are the Stokes parameters of the composite wave and (n = 1,2,3, ... )aretheStokesparametersofthe independent waves into which the composite wave can be decomposed. With the aid of this additivity of the Stokes parameters we can show that a polychromatic wave is decomposable uniquely into an unpolarized part and an elliptically polarized part, the two parts being mutually independent. To do this, we denote the Stokes parameters of the composite wave by (80,81,82,83), those of the unpolarized part by (80(1) ,0,0,0), and those of the polarized part by (80(2),81(2),82(2),83(2». Then by the additivity relation (57) we have 80, 81, 82, 83
80(n>, 81(n),
82(n),
83(n)
(58)
The degree of polarization m is defined as the ratio of the intensity of the polarized part to the intensity of the composite wave, i.e., by definition 80(2)
m=-
(59)
80
From relation (44) we know that the Stokes parameters of the polarized part are connected by the relation (60) which, with the aid of the last three equalities of Eq. (58), can be written as (61)
It follows from definition (59) and relation (61) that in terms of the Stokes parameters of the composite wave the degree of polarization is 130
Radio-astronomical
antennas
given by (62)
Furthermore, the orientation of the polarization ellipse is given by (63)
and its ellipticity by (64)
where use has been made of Eqs. (58) and (61). Thus we see that when the Stokes parameters 80, 81, 82, 83 of a partially polarized wave are known we can calculate the degree of polarization from Eq. (62), and the properties of the polarization ellipse of the polarized part of the wave from Eqs. (63) and (64). Since X is restricted to the interval -7r/4 ~ X ~ 11'/4, Eq. (64) unambiguously yields a single value for x. However, Eq. (63) can be satisfied by two values of t/I differing by 11'/2, the restriction that t/llie in the interval 0 :S t/I ~ 11' not being sufficient to fix t/I unambiguously. But from the first two of Eqs. (46) we see that t/I must be chosen such that 81 and 82 have the proper signs. Consequently, t/I is determined by Eq. (63) and by the requirement that the appropriate part of the Poincare sphere be used. Another way of decomposing a polychromatic wave is to express it as the superposition of two oppositely polarized independent waves. Two waves are said to be "oppositely polarized" if the orientation and ellipticity angles t/l1, XI of one of the waves are related as follows to the orientation and ellipticity angles t/l2, X2 of the other wave: (6.5)
This means that the major axes of the polarization ellipses of oppositely polarized waves are perpendicular to each other, that the axial ratios of the ellipses are equal, and that the senses of polarization are opposite. 131
Theory of electromagnetic
wave propagation
Let (80,81,82,83) denote the Stokes parameters of the polychromatic waves, and let (80(1) ,81(1) ,82(1) ,83(1» and (80(2),81(2),82(2) ,83(2» denote the Stokes parameters of the two independent and oppositely polarized waves. By the additivity of the Stokes parameters we have (66)
This relation is satisfied if we choose 80(1)
=
~80
-
80(2)
=
~80
+ ex
ex
(67)
where ex is an unknown quantity. oppositely polarized waves are (~80 - ex)
Then the Stokes parameters of the
(~80 - ex) cos 2Xl cos 21/11
(~80 - ex) cos 2Xl sin 21/11
O/z80 - ex) sin 2Xl
(68)
and (~80
+ ex)
(~80.
+ ex) cos 2X2 (~80
cos 21/12
+ ex) sin
(69) 2X2
Since these waves are oppositely polarized, we have cos 2Xl
= cos 2X2
sin 2Xl = - sin 2X2 (70)
In view of these relations we see that if the additivity theorem is applied to the Stokes parameters of the original wave and to the Stokes parameters (68) and (69) of the two oppositely polarized waves, the following relations are obtained: - 2ex
cos 2Xl cos 21/11 =
81
- 2ex
cos 2Xl sin 21/11 =
82
-2ex
sin 2Xl =
132
83
(71)
Radio-astronomical antennas
Squaring and adding Eqs. (71), we obtain (72)
From Eqs. (67) and (72) it followsthat the intensities of the two oppositely polarized waves are 80(1) = 7280
80(2)
=
7280
-
72
+
72
v'812 + 822 + 832 v'812 + 822 + 832
From Eqs. (71) we see that
1/;1
and
(73)
Xl
are given by (74)
Thus we see that a polychromatic wave whose Stokes parameters are (80,81,82,83) can be decomposed into two polarized waves having the intensities (72)80 :!: (72)(812 + 822 + 832)~ and being in the opposite states of polarization (x,1/;) and ( -x, 1/;
+ ;)
where X and 1/; are given
by Eqs. (74). Since we have used a fixed cartesian system of coordinates (x,y,z) to describe the Stokes parameters, the question of how these parameters change under a rotation of the axes naturally arises. To find the law of transformation, we need to consider only an elliptically polarized wave. This follows from the fact that a partially polarized wave always can be decomposed into two oppositely polarized independent waves. Let (80,81,82,83) denote the Stokes parameters of one of the elliptically polarized waves when referred to the original system, and let (8~,8i,8~,8;) denote these parameters when referred to the rotated system. The rotation consistsof a clockwisetwisting of the coordinates about the z axis and through an angle cf>. By virtue of the fact that the wave is elliptically polarized, we can write the Stokes parameters (80,81,82,83) in terms of the ellipticity angle x and orientation angle 1/;, as follows: 80
80
cos 2x cos 21/;
80
cos 2x sin 21/;
80
sin 2x 133
Theory of electromagnetic
Obviously, become So
So
wave propagation
when referred to the rotated
cos 2x cos 2(if; - f/J)
So
system these parameters
cos 2x sin 2(if; - f/J)
So
sin 2x
Clearly then, the Stokes parameters referred to rotated coordinates are given by S~
=
So
+
S~ = So
cos 2x cos 2(if; - f/J)
= SI
cos 2f/J
s~ = So
cos 2x sin 2("" -
= S2
cos 2f/J -
S;
=
f/J)
S2
sin 2f/J
SI
sin 2f/J
(n))
S3
where (SO,SI,S2,S3) and (s~,s~,s~,s;) are the Stokes parameters in, respectively, the original and rotated coordinates. The parameters So and S3 are invariant under the rotation, i.e., the intensity and the ellipticity of the wave do not change when the axes are rotated. On the other hand, SI and S2 do not remain the same and hence the orientation angle if; changes when the axes are rotated.
5.4 Coherency Matrices In the previous section it was demonstrated that the state of polarization of a narrowband polychromatic wave is specified completely by the four Stokes parameters So, SI, S2, S3. In this section we shall show that the state of polarization can be specified alternatively by means of a 2 X 2 matrix whose elements characterize the state of coherency between the transverse components of the wave. Let us again consider a plane TEM narrowband (quasi-monochromatic) polychromatic wave traveling in the z direction. In accord with Eqs. (42) the cartesian components of such a wave are E",
= Re {a",(t)eikze-io'e-iwt}
Ell = Re {all(t)eikze-io.e-iwt} Ez
= 0
134
(76)
Radio-astronOlnical
In terms of the complex vector A, whose components
A. the electric field components
=
antennas
are given by (77)
0
(76) may be written in the form E.
=
(78)
0
which shows that A is the phasor of the electric vector of the wave. Unlike the phasor of a monochromatic wave, A is time-dependent. The elements J pq of the coherency matrix J are defined by
(p,q
If Ap and Aq are physically independent obvious from definition (79) that
=
(79)
x,y)
then (ApA:)
=
O.
It is
(80) and hence the coherency matrix
(81)
is hermitian. To find the connection between the Stokes parameters and the coherency matrix, we note that when expressions (77) are substituted into definition (79) we get
= (ax2(t» J yy = (ay2(t» J xy = (ax(t)ay(t)ei
JyX
= (ax(t)ayc-i
cos o(t»
+ i(a.(t)ay(t)
cos oCt»~ - i(ax(t)ay(t)
sin o(t»
(82)
sin o(t»
where oCt) == Oy(t) - ox(t). Comparing expressions (82) with expressions (43), we find that the Stokes parameters are related to the ele135
Theory of electromagnetic
wave propagation
ments of the coherency matrix as follows:
82
+ JyZ 72(80 + 81)
= JZy
Jzz
=
83
= i(Jyz - JZY) (83)
= 72(80 - 81)
J1I1I
These relations show that the Stokes parameters and the elements of the coherency matrix are linearly related and that a specification of the wave in terms of the latter is in all respects equivalent to its specification in terms of the former.1 Since the additivity theorem applies to the Stokes parameters, it must, in view of the linear relations (83), also apply to the coherency matrix, in the sense that if JW, J(2), • • • , J(N) are the coherency matrices of N independent waves traveling in the same direction, then the coherency matrix J of the resulting wave is the sum of the coherency matrices of the independent waves, viz., N
L
J =
(84)
J(n)
n=1
To show this, we let Az(n), Ay(n) be the cartesian components of the phasor of the nth independent wave. Then by superposition the cartesian components of the phasor of the resulting wave are N
N
Az
=
L
Ay
A",(n)
L Ali(n)
=
n=1
(85)
n=1
The elements of the coherency matrix of the resulting wave are N
Jpq
= (A~:)
=
N
L L
(Ap(n)Aq(m)*)
n=1 m=1 N
=
L n=l
(Ap(n)Aq(n)*)
+
L
(Ap(n)Aq(m)*)
(86)
n"'m
1 Compare with E. Wolf, Coherence Properties of Partially Polarized Electromagnetic Radiation, Nuovo Cirrumto, 13: 1165 (1959).
136
Radio-astronomical
Each term of the last summation is zero since n ~ m are independent. Hence we have
Ap(n)
and
antennas
Aq(m)
for
N
J pq
=
L
J pq (n)
(87)
n=l
where J pq(n) denotes the elements of the coherency matrix of the nth independent wave. Thus the additivity theorem (84) is verified. From Schwarz's inequality, which is expressed by
.and fWll;l definition (79) it followstllp,t (88)
or, because of Eq. (80), that (89)
'The equality sign in these expressions obtains only when Api Aq is constant, which in turn means that the determinant of the coherency matrix vanishes only if the wave is elliptically pola~ized. If the determinant does not vanish, then the wave is partially polarized.
Thatis, det
J
0
for elliptic polarization
det
J>0
for partial polarization
=;=
(90)
We know from our study of the Stokes parameters that for an unpolarized wave 80 ~ o and 81 = 82 = 83 = O. Casting this into the l!1nguageof the coherency matrix, we see from Eqs. (83) that J zz = J yy = (72)80. Thus we find that the coherency matrix of an unpolarized wave has the form (91) 137
Theory of electromagnetic
wave propagation
Moreover, from expressions (46) and Eqs. (83) we see that the coherency matrix of an elliptically polarized wave has the form
J = ~ [ (1 + cos 2x cos 2if;) 2 (cos 2x sin 2if; - i sin 2x)
+
(cos 2x sin 2if; i sin 2x)] (1 - cos 2x cos 2if;)
(92)
where if; is the orientation angle of the polarization ellipse and X is its ellipticity angle. To see what this matrix looks like for certain simple states of polarization, we recall that for linear polarization X = 0, for right-handed circular polarization x = -71"/4, and for left-handed circular polarization X = 71"/4. Hence from expression (92) we find that sin"2if; ] J = ~ [1 + cos 2if; 2 sin 2if; 1 - cos 2if;
(93)
is the coherency matrix of a linearly polarized wave making an angle if; with the x axis; (94) is the coherency matrix for right-circular polarization; and (95) is the coherency matrix for left-circular polarization. It follows from relations (83) that the coherency matrix can be expanded in terms of the Stokes parameters and certain elementary matrices which in wave mechanics are called the Pauli spin matrices.l That is, 1
J
=
2"
l 3
(96)
spOp
p=o
where
8p
(p
=
0, 1, 2, 3) are the Stokes parameters,
do is the unit
I See, for example, U. Fano, A Stokes-parameter Technique for the Treatment of Polarization in Quantum Mechanics, Phys. Rev., 93: 121 (1954).
138
Radio-astronomical
antennas
matrix (97) and
are the Pauli spin matrices
dl, d2, da
(98) From Eq. (91) we see that do represents an unpolarized wave. more, using the decompositions
dl = [~
d2
~IJ
= [~
J
= [~ ~ = 72 [~
da =
[~i
~J
=
72
Further-
~J- [~ ~J ~J
-
[~i
72 [ ~
1
1 ~ ]
(99)
;J - ~~n ~iJ
and recalling the states of polarization that the matrices (93), (94), and (95) express, we see that dl characterizes the excess of a linearly polarized wave making an angle"" = 0 over a linearly polarized wave making an angle"" = 7r/2; d2 the excess of a linearly polarized wave making an angle"" = 7r/ 4 over a linearly polarized wave making an angle"" = 37r / 4; and da the excess of a wave polarized circularly to the left over one polarized circularly to the right. If we decompose the wave into an unpolarized part and an elliptically polarized part, then the ratio of the intensity of the polarized part to the intensity of the original wave is the degree of polarization m of the wave. The quantity (100) is the trace (or spur) 'of the matrix and represents the intensity of the original wave. The degree of polarization is given by the expression m
=
VI -
4 det J/(Tr J)2
(101) 139
Theory of electromagnetic
wave propagation
which can be derived from Eqs. (62) and (83). Since this expression involves only the rotational invariants det J and Tr J, the degree of polarization does not change with a rotation of the coordinate axis. From Eqs. (63) and (83) it follows that the orientation of the polarization ellipse of the polarized part of the wave is given by tan 21/1
=
JXY Jxx
+ JJ
yX
-
(102)
yy
and from Eqs. (64) and (83) that its ellipticity is given by . 2
sm
JyX
.
-
JXY
X = ~-=============
y(Tr
J)2 - 4 det J
(103)
Under rotation X does not change because the denominator of Eq. (103) is a rotational invariant, as is the numerator i(Jyx - JXY). However, 1/1 does change under rotation, as might have been expected. Thus we see that m and X are independent of the choice of orientation of the coordinate axes, while 1/1 is not. The quantities Tr J and det J do not change when the coherency matrix is transposed; on the other hand, the quantity J yx - J xy simply changes in sign. Therefore, from Eq. (103) we see that X simply changes in sign when the coherency matrix is transposed. Since the sign of X determines the sense of polarization, this means that if a coherency matrix describes a wave with a certain sense of polarization, then the transpose of the matrix describes a wave traveling in the same direction but with the opposite sense of polarization; or if a coherency matrix describes a wave traveling in a certain direction, the same matrix also describes a wave traveling in the opposite direction with opposite polarization.
5.5 Reception of Partially Polarized Waves In this section we shall calculate how much power a given antenna can extract from an incident polychromatic wave. We shall carry out the calculation by recalling the results of the conventional case, where the 140
Radio-astronomical antennas incoming wave is monochromatic, and then generalizing these results to the case where the incoming wave is polychromatic. This method of analyzing the problem, which uses the monochromatic theory of antennas as a point of departure, appears to be the most tractable, because it takes advantage of the fact that the receiving properties of an antenna are most conveniently expressed in terms of its monochromatic behavior as a transmitter. Hence, for the present we confine our attention to the conventional monochromatic theory of receiving antennas. According to this theory an antenna, actually or effectively, has two circuit terminals and with respect to these terminals its behavior is as follows: When the antenna is driven by a monochromatic voltage source applied to its terminals and no radiation is incident, the source "sees" an impedance, namely, the input impedance Zi of the antenna; on the other hand, when a monochromatic wave of the same frequency is incident on the antenna and the terminals are open-circuited, a voltage appears across the terminals, namely, the open-circuit voltage Vo• Then, in accord with Thevenin's theorem of circuit theory, when the antenna operates as a receiving antenna having a load impedance Zz connected to its terminals, the equivalent circuit of the antenna consists of the voltage Vo in series with Zi and Zz. From this equivalent circuit it is clear that the power absorbed by the load is a maximum when Zi and Zz are conjugate-matched, i.e., Zi = Zi. Under this condition of optimum power transfer, the power generated by Vo is divided equally between the power absorbed by Zi and the power absorbed by Zz. Physically, the power absorbed by Zi consists of the (reversible) power that is carried away from the antenna by the scattered, or reradiated, portion of the incident power and the (irreversible) power that goes into ohmic losses, i.e., into the heating of the antenna structure. In the hypothetical case where the conjugate-matched antenna is free of ohmic losses, one-half of the applied incident power is scattered into space and the other half is absorbed by the load. The power that an incident monochromatic wave delivers to the conjugate-matched load of a receiving antenna is related to the behavior of the antenna as a transmitter. To present this relation, let us suppose that the antenna in question is driven as a transmitter by a monochromatic voltage applied to its terminals. Let us also suppose that the antenna is located at the origin of a spherical coordinate system 141
Theory of electromagnetic
wave propagation
(r,8,e/». Then if the electric vector (actually phasor) of the far-zone field radiated by the antenna is Erad, the radial component of the Poynting vector of this field is (104) the field polarization vector is Erad -1..) - ---,-,=== Pr.d(8 ,'I' - vErad. Er.d'
(105)
and the gain function is 47rr2Srad(r,8,e/»
g(8,e/» fo
41r
Srad(r,8,e/»r2
(106)
dn
where dn( = sin 8 d8 de/» is an element of solid angle. Alternatively, let us now suppose that the antenna is operated as a receiving antenna with a conjugate-matched load attached to its terminals, and that a plane monochromatic wave is incident on it from a direction 8 = 80, e/> = e/>o. If the electric vector of the incident wave is Einc, the radial component of the Poynting vector of the incident wave is (107)
and the field polarization vector is (108) Then, in compliance with the reciprocity theorem, 1the power absorbed by the load is given by the relation (109) 1 S. A. Schelkunoff and H. T. Friis, "Antennas: Theory and Practice," pp. 390-394, John Wiley & Sons, Inc., New York, 1952.
142
Radio-astronomical
antennas
where the quantities g(B,q,) and prad(B,q,) describe the behavior of the antenna in transmission and the quantities Sinc(B,q,) and pinc(B,q,) describe the incident wave in reception. The polarization loss factor (110)
which appears in relation (109) can take on any value in the range o ::; K ::; 1, depending on how closely the polarization of the wave radiated in a direction (O,q,) is matched to the polarization of the incident wave falling on the antenna from the same direction. When (111)
the radiated wave and the incident wave are matched completely and K = 1. If the field polarization vector of the incident wave is conjugate-matched in this sense to the field polarization vector of the radiated wave, the power absorbed by the conjugate-matched load is a maximum and, according to Eq. (109), has the value! (112)
By definition the ratio (Pabs)max/Sinc(B,q,) is the effective area A (B,q,) of the receiving antenna,2 and consequently the effective area of the antenna in reception is proportional to the gain function of the antenna in transmission, i.e., A (B,q,)
=
A2
47r
g(O,q,)
(113)
With the aid of this result and definition (110) we can write Eq. (109) in the alternative form Pab•
= A(B,q,)Sinc(B,q,)K(B,q,)
(114)
! Y.-C. Yeh, The Received Power of a Receiving Antenna and the Criteria for Its Design, Proc. IRE, 37: 155 (1949). 2 Compare C. T. Tai, On the Definition of the Effective Aperture of Antennas, IRE Trans. Antennas Propagation, AP-9: 224-225 (March, 1961).
143
Theory of electromagnetic
wave propagation
which explicitly displays the dependence of the absorbed power on the effective area of the antenna and on the polarization loss factor. To generalize the above discussion to the case where the incident wave is partially polarized and polychromatic, we write Eq. (114) in the equivalent form (115)
where
pradprad.
is the dyadic associated with the wave radiated in a
----
direction (8,1/», and pincpinc. denotes the transpose of the dyadic pincpinc. associated with the wave incident from the same direction (8,1/». Moreover, we can in turn write Eq. (115) as (116)
Now if the incident wave happens to be a polychromatic wave and if over the entire spectrum of the wave the antenna is conjugate-matched to the load, then Eq. (116) remains valid for each frequency of the spectrum. Assuming that the antenna and load are so matched, we get the total absorbed power by integrating Eq. (116) over all fre~ quencies or, as mentioned in Sec. 5.1, by averaging with respect to time. Such an integration would require a knowledge of the frequency dependence of prad and A. However, we shall assume that prad, and hence A, is independent of frequency over the spectrum of the polychromatic wave and shall thus obtain the following expression for the total absorbed power (117)
Let us now consider the case where the incident polychromatic wave is narrowband and has the form (118)
Here the complex components Eg(t) and E",(t) are slowly varying functions of time, w is a mean frequency, and k = wlc. For such an inci144
Radio-astronomical
dent wave the matrix of the components the dyadic EincEinc* is
(i
antennas
of the time-average
=
1,2; j
=
1,2)
value of
(119)
The coherency matrix of the incident wave in Eq. (117)is the transpose [4] of [h]. The matrix of the components of the dyadic A (pradprad*) , that is, (i
= 1,2; j = 1,2)
(120)
is the effective-area matrix of the antenna. In terms of the effectivearea matrix [Ai;] of the receiving antenna and the coherency matrix [l;] of the incident wave, the power absorbed in the conjugatematched load of the receiving antenna is given by the compact relation (i
=
1,2; j
=
1,2)
(121)
which follows directly from Eq. (116) and definitions (119), (120). We can divide the incident wave into two mutually independent parts, viz., an unpolarized part and a polarized part. We do this by splitting [I;;] into (122) and noting that the first matrix on the right represents the unpolarized part and the second matrix on the right represents the polarized part. Taking the trace of this matrix equation, we obtain the expression (123) whose left side represents the average value (Sinc) of the incident power density, and whose right side represents the power density 2a of its unpolarized part plus the power density (3(qll + q22) of its polarized part. By definition the degree of polarization m is the ratio of the power density of the polarized part to the total power density; hence in 145
Theory of electromagnetic
wave propagation
terms of m Eq. (123) can be written as (124) From this it follows that (125) Since we are free to choose
{j,
we make the choice (126)
on the grounds of convenience. In view of expressions (125) and (126) we see that Eq. (122) can be written in terms of the time-average power density (Sine) of the incident wave and its degree m of polarization:
As a consequence of choice (126) we have (128) and because [qij] represents
a completely polarized wave, we have (129)
Moreover, by virtue of the fact that [[;;] is hermitian,
we also have (130)
From conditions (128), (129), and (130) we see that the components of qij may be written in the following way in terms of the orientation angle 1f/ and the ellipticity X of the polarization ellipse of the polarized part of the incident wave falling on the antenna from a direction (8,cP): qU [ q21
+
q12] _ ~ [ 1 cos 2x cos 21f/ q22 2 cos 2x sin 21f/- i sin 2x
+
cos 2x sin 21f/ i sin 2x] 1 - cos 2X cos 21f/ (131)
146
Radio-astronomical
antennas
Similarly, since [Aij] represents a completely polarized wave, viz., the wave the antenna would radiate if it were used as a transmitter, we can write it in terms of the orientation angle 1/;' and ellipticity angle x' of the wave radiated in a direction (8,4»: A-. -
1
A 8
[
[ ,,] - % (,4»
+
+
cos 2x' sin 21/;' i sin 2x' ] 1 - cos 2x' cos 21/;'
1 cos 2x' cos 21/;' cos 2x' sin 21/;' - i sin 2x'
(132)
Substituting Pab•
Eq. (127) into Eq. (121), we get
Tr [Aij][i-;;]
=
=
+ A )(Sinc) + A12q12 + A q21 + A
~'2(1 - m)(All
+ m(Allqll
22
21
q22)(Sinc)
(133)
22
and then, using Eqs. (131) and (132), we find that the time-average power absorbed by the conjugate-matched load is given by) Pab•
=
H(1 - m)A(8,4»(Sinc(8,4>))
+ mA(8,4»(Sinc(8,4>))
cos2
~
(134)
where cos'Y
== cos 2x' cos 2x cos (21/;' - 21/;) - sin 2x' sin 2x
(135)
On the Poincare sphere, 'Y is the angle between the point (21/;,-2x) describing the polarization ellipse of the incident wave and the point (21/;',2x') describing the polarization ellipse of the radiated wave. When 1/;' = 1/; and x' = - x, that is, the two points coincide and 'Y = 0, the polarizations of the radiated and incident waves are conjugate-matched and there is no polarization loss. This, of course, means that the two polarization ellipses have the same orientation in space and the same axial ratio. It also means that the sense of rotation of the incident wave is the same as the sense of rotation of the radiated wave if the former is viewed from infinity and the latter from the antenna. If viewed from some fixed position, the senses of rotation would appear to be opposite. ) H. C. Ko, Theoretical Techniques for Handling Partially Polarized Radio Waves with Special Reference to Antennas, Proc. IRE, 49: 1446 (1961). 147
Theory of electromagnetic
wave propagation
The first term on the right of Eq. (134) represents the contribution to
P ab. of the unpolarized part of the incident wave, whereas the second represents the contribution of the polarized part. If the polarization of the antenna in a direction (O,e/» is conjugate-matched to the incident wave coming from the same direction, then l' = 0 and the power absorbed in the conjugate-matched load resistance is a maximum, i.e., (136) Moreover, if the incident wave is completely polarized, we have m and hence
=
1
(137) On the other hand, if the incident wave is completelyunpolarized, have m = 0 and hence
we
(138) In this case there is no question of matching.
5.6 Antenna Temperature and Integral Equation for Brightness Temperature From the discussion in the previous section we know that if a plane unpolarized polychromatic wave is incident from a direction 0, cfJ on a lossless receiving antenna located at the origin of a spherical coordinate system (r,O,cfJ), the power absorbed by the matched load (the receiver) is given by (139) where Sine is the spectral flux density of the incident wave, A.w the bandwidth of the receiver, and A the effective area of the antenna. The validity of this expression rests on the assumption that A and Sine are independent of frequency within the relatively narrow bandwidth 148
Radio-astronomical
antennas
To find the absorbed power for the case where the source is distributed over the sky, we note that the elemental contribution to the absorbed power of the radiation falling within a cone of solid angle dn and within a bandwidth ~w can be expressed as
~w.
(140) where, in view of Eq. (23), dSinc is related to the brightness temperature Tb of the sky by (141) The subscript w has been dropped from Tb for simplicity. Then we assume that the radiation falling on the antenna from any direction is incoherent with respect to the radiation from the other directions. By virtue of this assumption, the total absorbed power is the sum of the elemental powers delivered to the antenna by various incident rays. In other words, if the incident rays are physically independent, the total absorbed noiselike power may be calculated by integrating expression (140) over the solid angle subtended by the distributed source: (142)
The quantity in the brackets has the dimension of temperature and is known as the antenna temperature. It provides a convenient measure for the noiselike power picked up by the antenna in a bandwidth ~w. Thus antenna temperature T" is defined by (143)
or, in terms of the gain function g(8,q,), by (144) 149
Theory of electromagnetic
wave propagation
According to this definition, one possible physical interpretation of Ta is as follows: If the antenna is completely enclosed by a surface which radiates as a blackbody at temperature Ta, then the antenna will absorb in its load resistance the power kTa ~w. Alternatively, Ta may be regarded as the temperature to which the effective input resistance of the receiver (which, if matched, equals the radiation resistance of the antenna) must be raised so that the noise power, produced by the thermal motion of the electrons and delivered to the receiver through a lossless line, would equal P
(145)
This is the integral equation for the brightness temperature Tb(n'). By changing the orientation n of the antenna so that its radiation pattern effectively scans the sky, we can measure Ta as a function of n. Moreover, by measuring the radiation pattern or by predicting it theoretically, we can deduce the gain function. Accordingly, we regard Ta(n) and g(n,n') as known quantities, and find the brightness temperature Tb(n') of the sky in terms of Ta(n) and g(n,n') by solving the integral equation. A practical way of solving the integral equation is by successive approximations.! To show what the scheme of the method is, let us write Eq. (145) in operator form (146) 1 See J. G. Bolton and K. C. Westfold, Galactic Radiation quencies, Au;tralian J. Sci. Res., 3: 19 (1950).
150
at Radio Fre-
Radio-astronomical
antennas
where K(n,n') is the integral operator defined by K(n,n')f(n')
==
4~
J g(n,n')f(n')dn(n')
(147)
the quantity fen') being a typical function of n'. Also, for simplicity, we do not bother to write explicitly the arguments n, n'. Thus in operator form Eq. (145) becomes
or equivalently (148) Suppose as a zero-order approximation tion T a and then take '1'1
= Ta
+ (1 -
to '1'b we choose the known func-
(149)
K)Ta
as the first-order approximation to Tb• By applying the same procedure to '1'1, we obtain the second-order approximation to Tb: (150) Clearly for the nth approximation
Tn = Ta
+ (1
to Tb we have (Hil)
- K)Tn-1
or in terms of T a
Tn = Ta
+ (1 -
K)Ta
+ (1 -
K)2Ta
+ ...
(1 - K)nTa
(152)
5.7 Elementary Theory of the Twoelement Radio Interferometer To attain high resolving power, antenna arrays having multilobe receiving patterns are used. The high resolving power of such arrays stems 151
Theory of electromagnetic
wave propagation
Direction of incident radiation
I
t-- Axial
plane
I
I I I I I
Antenna
I I
Antenna
Transmission line
Fig. 5.7
Two-element interferometer. Receiver is connected to two identical and similarly oriented antennas. Direction of incident radiation makes angle'" with base line and angle a with axial plane. Separation of antennas is l. Receiver is at electrical center of transmission line.
from the fact that each lobe of the multilobe pattern becomes narrower and hence more resolvent as the spacing between adjacent antennas is increased. The simplest array that exhibits a multilobe receiving pattern is the two-element radio interferometer,! consisting of two identical and similarly oriented receiving antennas separated by a distance l and connected to a single tuned2 receiver by a transmission line (Fig ..5.7). To find the receiving pattern of such an interferometer, we note that by lOne of the first applications of the two-element radio interferometer, which we recognize as the radio analog of Michelson's optical interferometer, was made by L. L. McReady,J. L. Pawsey, and R. Payne-Scott, Solar Radiation at Radio Frequencies and Its Relation to Sunspots, Proc. Roy. Soc., (A) 190: 357 (1947). 2 Because the receiver is sharply tuned we can use a monochromatic theory in most of the analysis. 152
Radio-astronomical
antennas
the reciprocity theorem its receiving and radiation patterns must be the same and we recall from Sec. 3.5 that its radiation pattern must be the product of the radiation pattern F of one antenna and the array factor A of the two antennas. Thus it follows from the reciprocity and multiplication theorems that the receiving pattern of the interferometer is IFAI and that the power fed to the receiver is proportional to IFA12. In a typical two-element interferometer F has one main lobe and the factor A has numerous lobes; these are called "grating" lobes. Consequently, the array factor A is responsible for the multilobe structure (fringes) of the receiving pattern and F gives the pattern's slowly vary.iog envelope (Fig. 5.8). Since we are interested in the resolving properties of the interferometer and since they depend chiefly on A, we may, insofar as radiation falling within the central portion of the main lobe of F is concerned, set the factor F equal to unity and thus assume that the receiving pattern of the interferometer is given by [AI alone. Accordingly, from
Grating lobes
Fig: 5.8
Pola,. plot of typical receiving pattern .jerome,ter.
of two-element
inter-
153
Theory of electromagnetic
wave propagation
Eq. (79) of Sec. 3.6 we see that the receiving pattern of the two-element interferometer is given byl IA(lf)1
=
2 cos ("72kl cos 11')
(153)
where 11' is the angle between the direction of the incident wave and the base line, i.e., the straight line joining the two antennas. In terms of the complementary angle a( = 7rj2 - 11'), i.e., the angle the direction of the source makes with the plane perpendicular to the base line (axial plane), the radiation pattern is IA (a) I
=
(V54)
2 cos ("72kl sin a)
The power P fed to the receiver of the interferometer IA(a)12 and hence pea)
=
2Po cos2 (>~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
Radio-astronomical antennas
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
Theory of electromagnetic
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
Radio-astronomical antennas
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
Theory of electromagnetic
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
Radio-astronomical antennas
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Electromagnetic waves in a plasma
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.<;ycles ill Electromagnetic Wave Propagation," in.M. Desi~ rant and ,T. 4 Mi<;hiel~ (eds.), Apade.mi(l Pre~s Inc., New Yor~, 1960.
1(;19
6
Theory of electromagnetic
wave propagation
the atmosphere,l and the propagation of microwaves in laboratory plasmas.2 In these applications the medium through which the electromagnetic wave must travel is formally the same: it is a plasma, or more descriptively, a macroscopically neutral ionized gas consisting principally of free electrons, free ions, and neutral atoms or molecules. This means that from one application to another the nature of the problem does not change essentially, despite the large variations the medium may undergo in, say, its degree of ionization and its temperature. However, in the presence of a beam of charged particles interacting with the plasma, an electromagnetic wave does acquire characteristics which differ qualitatively from those in a beam-free plasma. One such characteristic is, for example, wave amplification by beamgenerated plasma instabilities. 3 Accordingly, phenomena of this kind have to be treated separately and for this reason are excluded from the present discussion. In this chapter we shall analyze the problem of electromagnetic wave propagation in a plasma medium by calculating the constitutive parameters of the plasma and then treating the problem as a conventional problem in the theory of electromagnetic wave propagation in a continuous medium.
6.1 Alternative Descriptions of Continuous Media We recall from electromagnetic theory that for a continuous medium at rest Maxwell's equations can be written in the following elementary 1 Proc. Symp. Plasma Sheath, vol. 1, U.S. Air Force, Cambridge Research Center, December, 1959. 2 V. E. Goland, Microwave Plasma Diagnostic Techniques, J. Tech. Phys., U.S.S.R., 30: 1265 (1960). 3 R. A. Demirkhanov, A. K. Gevorkov, and A. F. Popov, The Interaction of a Beam of Charged Particles with a Plasma, Proc. Fourth Intern. Conf. on Ionization Phen. in Gases, vol. 2, p. 665, North Holland Publishing Company, Amsterdam, August, 1959.
170
Electromagnetic
waves in a plasma
form,l
1
= Jt
IJo V' X
B
V' X E
= -
E
= Pt
EOV"
:t
{J
+ at E
(1)
B
(2)
EO
(3) (4)
V'.B=O
which describes the macroscopic electromagnetic field in the medium by the two vector fields E and B and characterizes the medium by the total macroscopic charge density Pt and total macroscopic current density Jt. The constants IJo and EO denote respectively the permeability and dielectric constant of the vacuum. The total charge density Pt consists of the free charge density P and bound charge density Pb; similarly the total current density Jt consists of the free current density J and bound current density Jb, that is, Pt = P
+ Pb
(5) (6)
The free charge is that part of the total charge which exists independently of the field. On the other hand, the bound charge is an attribute of the multipoles that are induced in the medium by the electromagnetic field. Indeed, Pb and Jb are given by the series2 Pb
= - V' • P
Jb
=
{J
at P
-
+ 7~V'V':Q + ... L
a
2 at
V' • Q
+v
X M
(7)
+
(8)
1 See, for example, R. W. P. King, "Electromagnetic Engineering," McGrawHill Book Company, New York, 1945; also, L. Rosenfeld, "Theory of Electrons," North Holland Publishing Company, Amsterdam, 1951. 2 We keep only the leading terms. When the series are terminated at a certain degree of approximation, the number of electric multipoles exceeds the number of magnetic multi poles by one. In compliance with ihis rule we have kept two electric multi poles P and Q and one magnetic multi pole M.
171
Theory of electromagnetic
wave propagation
where P, M, Q denote respectively the volume densities of the electric dipoles, magnetic dipoles, and electric quadrupoles that are produced by the action of the electromagnetic field on the neutral molecules of the medium. In other words, P, M, Q are functionals of E and B. In view of the series (7) and (8), Maxwell's equations (1), (2), (3), (4) become
1
;;; v
ata
a
x B = J + EO E + at P -
VXE = -
Ia
2 at v . Q + v X M +
ft B
(10)
EoV.E=p-V,P+~VV:Q+ V. B
=
=
(11)
0
(12)
If we define the electric displacement V.D
(9)
D by (13)
p
then on comparing this relation with Eq. (11) we see that this definition leads to
D = EoE+ P - ~V . Q + ... Moreover,
(14)
if we define the vector H by
1 H=-B-M
(15)
/Joo
then Eq. (9) leads to (16)
VXH=J+ftD
Hence, when D is defined by Eq. (13) and H by Eq. (15), the Maxwell equations (9), (10), (11), (12) assume their conventional form:
a
VxH=J+-D VxE=--B
= p V.B = 0 V.D
172
a at
at
(17) (18) (19) (20)
Electromagnetic waves in a plasma
To apply these considerations to the case of an electromagnetic wave passing through a plasma medium, we note that the wave, in principle, interacts with all three components of the plasma, viz., the free electrons, the free ions, and the neutral molecules. However, the interaction of the wave with the neutral particles is so feeble in comparison to the interaction between the wave and the charged particles that it can be neglected. This means that P, M, Q, which constitute a measure of the interaction between the wave and the neutral particles, can be set equal to zero. Moreover, since the ions are much more massive than the electrons, the velocity imparted to the ions by the wave is negligibly small compared to the velocity given to the electrons. That is, when an electromagnetic wave passes through a sufficiently ionized plasma only the free electrons of the plasma influence appreciably the transmission of the wave. The interaction between the wave and the electrons is introduced into Maxwell's equations through the current density term J. As will be shown subsequently (see Eq. 43), the electronic current density J produced in the plasma by the wave is related in the steady state to the electric vector E of the wave by a linear relation of the form
J = aE
+ iwbE
(a, b
= positive real)
(21)
unless E exceeds a value at which nonlinearities come into play. It therefore follows that when an electromagnetic wave whose electric vector E lies within the bound of linearity passes through a sufficiently ionized plasma, the Maxwell equations for the phenomenon in the steady state become
v
X H
V X E
+ iwbE
= aE =
- iwEoE
(22) (23)
iWJLoH
Let us write Eq. (22) in the form VXH
+ iwb)E
= (a
- iWEOE
(24)
where (a + iwb)E appears as a conduction current and -iWEOE as a vacuum displacement current. This form suggests that we think of the complex factor (a + iwb) as a complex conductivity given by (To
=
(T,
+ iu
i =
a
+ iwb
(25) 173
Theory of electromagnetic
wave propagation
and thus describe the plasma as a conductor having a dielectric constant EO, and a complex conductivity shall not use this mode of description here. Instead, the term iwbE of Eq. (22) as a polarization current the plasma as a lossy dielectric. To do this, we recall that for a lossy dielectric by
v
XH
= qE
-
a permeability p'o, qc. However, we we shall interpret and thus consider definition we have (26)
iwP - iWEoE
where q is the conductivity of the dielectric and P is the polarization of the neutral molecules of the dielectric. Also for a dielectric we have
P
(27)
= xcE
where Xc, the electric susceptibility of the dielectric, is always positive. Since the relation D
=
EoE
+P=
(28)
EE
defines the dielectric constant E of the dielectric, it follows that the dielectric constant of the dielectric is given by E
=
EO
+ Xc
(29)
Clearly, for a true dielectric E is always greater than EO because Xc ~ O. If we are to describe the plasma as a lossy dielectric, we must identify Eq. (22) with Eq. (26), setting aE = qE and iwbE = -iwP = -iwx.E. This means that the conductivity q of the dielectric must equal a and its electric susceptibility X. must equal -b, that is, q = a and Xc = -b. Since b is positive, x. must be negative. Thus, if the effect of the motion of the electrons is to be accounted for by a conductivity and a polarization, then we must think of the plasma as a lossy dielectric whose electric susceptibility is negative. The constitutive parameters of the dielectric are then given by q=a
J.l = J.lo
E = EO -
b
(30)
Here we note that in contrast to an actual dielectric E is less than EO. Also we may combine the conductivity with the dielectric constant 174
Electromagnetic
waves in a plasma
and thus obtain a complex dielectric constant Ec• If this is done, the plasma is described by the constitutive parameters
Jl
=
Ec
jlO
=
a - •.... +
(31)
b
EO -
UAI
6.2 Constitutive Parameters of a Plasma When a high-frequency electromagnetic wave passes through a plasma, only the interaction between the wave and the free electrons need be considered. Therefore, from a statistical point of view the macroscopic state of the plasma can be described in terms of a single distribution function f(r,w,t), which determines the probable number of electrons that at the time t lie within the spatial volume dx dy dz centered at r and have velocities within the intervals dwx, dwu, dw. centered at w. This function of the position vector r, the velocity vector w, and the time t must satisfy the Boltzmann (or kinetic) equation
-df == -af + w.
dt
at
"ilf
+ (ddt" - w)
.
"il
f
= C
(32)
where "il••f is the gradient of f in velocity space, "ilf is the gradient of f in coordinate space, and C is the temporal rate of change inf caused by collisions. The acceleration dw/dt is related to E and B of the wave in accord with the Lorentz force equation m~ w
=
q(E
+w
(33)
X B)
where q and m denote respectively the charge and mass of the electron. Substituting expression (33) into Eq. (32), we obtain aj
at
+ w.
"ilf
+ mq (E + w
X B) . "il••f =
C
(34)
which shows explicitly that the driving force is the macroscopic electromagnetic field E, B. Multiplying this equation by mw and 175
ihtegrating over all motion
ve16cltYe-s, we
6btMn1 themalcro~Cbpi'ceql1s,tibnof
(35)
this equation the particle density n(r,l) and the macroscopic velocity v(r,t) are defined respectively by tfi
n(r,t)
=
11"f
j(r,w,t)dwJi,wtliw.
Iii
v(r,t) == ~
wj(r, w,t)dWxdwudw.
(36)
(37)
The dyadic S is the stress, defined by
S ==
mil I "
(w .•...v)(w - v)j(r,w,t)dwi:dwydW.
and the vector G is the net gain of momentum due to collisions. In the present case all the nonlinear terms as well as the v X .a term are dropped from Eq. (35), and thus the equation of motion is reduced, in the steady state, to the following simple form: -'- iwnmv = nqE
+G
(3\)
Moreov~r, since G is the net gain in momentum per unit Volume per unit time, we may write G
(40)
= -nmvWefl
where the proportionality constant We!! is the collision frequency and measures the numoor of effective collisions an electron makes per unit time. Furthermore, the density of electronic current J and the plasma 1 See, for example, L. Spit~er, Jt., "Physics of Fully Ionized Gases," Inter~ science Publishers, Inc., New York, 1956.
176
---~--"._ •.-
~~~._
..'~---~
Electromagnetic
frequency
waves in a plasma
are defined by
Wp
J=nqv
(41)
and by (42) Hence, from the equation of motion (39) and the expressions (40), (41), and (42) we find that the electronic current density J is related to E as follows: 2
J =
E =
EOWp
-iw
+
Welf
2 EOWelfWp 2 2 W Wefl
+
E
+ iw
2
E
EOWp W
2
+
2
(43)
Wel1
Comparing expression (43) with Eq. (21) of the previous section, we determine the coefficients a and b; and then by using relations (30) of the previous section, we find the constitutive parameters of the plasma. Accordingly, if we think of the plasma as a lossy dielectric, its con. ductivity is given by
(44) its dielectric constant by
E
=
EO
(1 - ~p2 2) 2
W
(45)
Well.
and its permeability by I/o =
1/00
(46)
The elementary derivation of the constitutive parameters given above makes use of the collision frequency merely as an unknown parameter, without providing any information about its value. To evaluate WeI/' the microprocesses which the plasma particles undergo must be taken into account explicitly. This has been done elsewhere 177
Theory of electromagnetic
wave propagation
by kinetic theory and the results show that Well is not constant at all. Nevertheless, expressions (44) and (45) with Well taken to be constant adequately describe the plasma for our present purposes.
6.3 Energy Density in Dispersive Media Using Maxwell's equations for a lossless medium, we can write
a
a
V' • Set) = - E(t) • - D(t)
(47)
- H(t) • - B(t) at
at
where (48)
= E(t) X H(t)
Set)
is the Poynting vector. The quantity V'. Set) represents the rate of change of the -electromagnetic energy density wet), that is, V' •
set)
= -
a . wet) at
(49)
-
From Eqs. (47) and (49) we see that aw at
=
a E(t) • - D(t) at
+ H(t)
a • - B(t) at
(.50)
For a simple, nondispersive, lossless dielectric is equal to f.LO; hence D(t)
=
EE(t)
B(t)
=
E
is a real constant and
f.L
(51)
f.LoH(t)
and relation (50) reduces to
a at- wet)
a
= !I
vt
[7~EE(t) • E(t)
+ ~f.LoH(t)
which shows that the electromagnetic 178
• H(t»)
(52)
energy density for a simple,
Electromagnetic
waves in a plasma
nondispersive, lossless dielectric is given by w(t)
=
%eE(t)
• E(t)
+ %/loH(t)
• H(t)
(53)
The first term on the right is the electric energy density w. and the second term is the magnetic energy density wm: w.(t) wm(t)
= %eE(t) • E(t) =
%/loH(t)
(54) (55)
• H(t)
In the case of harmonic time dependence, where E(t) = Re {Ee-iwt} and H(t) = Re {He-iwt}, the time-average energy densities may be written in terms of the phasors E, H as follows: Wm = '/1). =
7.:t:/loH. H* 7.:t:eE. E*
(.56) (.57)
To define the electric and magnetic energy densities of an electromagnetic wave in a plasma, we must assume that the plasma is lossless, because it is only for a lossless medium that electromagnetic energy can be rationally defined as a thermodynamic quantity. For this reason we must limit our consideration to situations where the collision frequency w.!! is so small that we may set it equal to zero, In keeping with this restriction, we consider a plasma whose collision frequency is zero and note that its constitutive parameters are
/l
=
/lO
0'=0
(58)
as can be seen by setting w.!! equal to zero in Eqs. (44) and (45). Since /l is a constant, the magnetic energy density can be evaluated by means of relation (54) or (.56). However, since e is a function of frequency, the medium is dispersive and relations (.'55)and (57) no longer can be used to evaluate the electric energy density. For example, if we use relation (.57) we obtain the expression IV. =
7.:t:eo(1 -
:;22) E . E*
(59) 179
Theory of electromagnetic
wave propagation
which predicts that w. < 0 when w < Wp, in contradiction to the fact that w. must always be positive-definite. Since the plasma is dispersive we cannot compute the electric energy density on a monochromatic basis. The reason for this is that since ow./at = E(t) • aD(t)/at, the expression for the electric energy density, viz., w.(t) = JE(t) • aD(t)/at dt + C, contains the integration constant C, whose value depends on how the field is established. To determine C, we assume that the wave is quasi-monochromatic; then for t we have E( - 00) = 0, w.( - 00) = 0, and hence C = O. That is, for a quasi-monochromatic wave that starts in the remote past from value zero and builds up gradually, the integration constant is zero and w.(t) is fully determined. A high-frequency wave whose amplitude is slowly modulated is a simple type of wave that builds up gradually in time and thus serves well in calculating electric energy density. Accordingly, we assume that the time dependence of the electric vector in the lossless plasma has the form -Y
=
E(t)
72Eo[COS
(w
+ Ilw)t
-
cos
(w -
Ilwt
sin
where Eo is a constant vector and Ilw is small compared to D = E(w)E, the resulting displacement vector is
=
72EO[E(W
+ Ilw)
cos (w
+ Ilw)t
00
Ilw)t]
- Eo sin
D(t)
-
-
wt
(60)
w.
Since
E(W - Ilw) cos (w - Ilw)t]
(61)
and the resulting displacement current is
a
at
D(t)
-72Eo[(W + IlW)E(W + Ilw) -
(w -
sin
(w
Ilw)E(w -
+ Ilw)t Ilw)
sin
(w -
Ilw)t]
(62)
Expanding (w + IlW)E(W + Ilw) and (w - IlW)E(W - Ilw) in a Taylor series and retaining only the first two terms, we get the approximate expressions (w
+ Ilw)E(w + Ilw)
(w 180
IlW)E(W -
Ilw)
=
WE
+ Ilw awa (EW) +
= WE -
a
Ilw aw (EW)
+
(63) (64)
Electromagnetic
waves in a plasma
which when substituted into expression (62) lead to the following expression for the displacement current: ft D(t) = - Eo [ WE sin t:.wt cos wt
+ t:.w :w
(WE) cos t:.wt sin wtJ
(65)
We see from Eq. (50) that the rate of change of the electric energy density is
ata w.
a
= E(t) • at D(t)
(66)
and hence the energy gained during the time interval tt - to is given by W.(tl) - w.(to)
a
(II
= J to E(t) • at D(t)dt
(67)
From expression (60) it is evident that E(t) is zero when t = 0 and has the form of a high-frequency carrier sin wt whose modulation envelope sin t:.wt increases slowly with time. The time required for E(t) to build up from zero to its maximum value is t:.wt = 71"/2 or t = 71"/2t:.w. The energy gained during the time interval to = 0 to tt = 71" /2t:.w is given by w. =
(,,/2/),I»
J0
E(t) •
ata D(t)dt
(68)
Substituting expressions (60) and (65) into Eq. (68), we get ("/2/),,,,.
W. = EO' EOWE Jo
+ Eo • E
.
sm2 t:.wt sm wt cos wt dt
0
a (WE ) J("/2/),,,, 0
t:.w aw
.,
d
sm 2 wt sm t:.wt cos t:.wt t
(69)
The first integral on the right is negligibly small compared to the second. In the second integral we may replace sin2 wt by ~ and thus approximate the integral by ("/2/)',,,
~ Jo
•
sm t:.wt cos t:.wt dt
1
= 4t:.w
(70) 181
Theory of electromagnetic
wave propagation
It follows that the time-average electric energy density is given by
(71)
If instead of the form (60) for E(t) we take E(t) = Re {Eo(t)e-U.l}, where Eo(t) is a slowly varying function, we would get again (72) Since E
=
EO(l -
w1}
/
w2),
expression (72) leads to (73)
which shows that We is the sum of two terms, the first representing the energy in the vacuum and the second representing the kinetic energy of the electrons.l To demonstrate that the second term does equal the time-average kinetic energy of the electrons, we recall from Eq. (39) of the previous section that for a lossless plasma
-iwnmv
=
(74)
nqE
The time-average kinetic energy density is, therefore, given by _ K = %nmv'v*
=
I nq2 --E.E*
4 w2m
(75)
Using definition (42) of the plasma frequency, we get (76) which is identical with the second term of expression (73). Thus we see that' for a lossless plasma the time-average
electro-
1 Formally, this result can also be obtained from the energy theorem of Chap. 1; see F. Borgnis, Zur e!ektromagnetischen Energiedichte in Medien mit Dispersion, Z. Physik, 159: 1-6 (1960).
182
Electromagnetic
waves in a plasma
magnetic energy density is given byl
tV
=
~loIoH
•
H*
+ !i. (w~)E 4 ow
• E*
(77)
6.4 Propagation of Transverse Waves in Homogeneous Isotropic Plasma To determine the propagation properties of transverse electromagnetic waves in a homogeneous isotropic plasma, we consider a linearly polarized plane transverse wave whose electric vector E(l) has the form (78) where Eo(t) is a slowly varying function of time, w is the real mean angular frequency, and k is the propagation constant or mean wave number, which may be complex. In a medium whose constitutive parameters are ~, 1010, u, the electric vector must satisfy
(79)
Since in the present case E(t) ~s transverse, Le., perpendicular to the direction of propagation, the quantity V X V X E may be replaced by - V2E. Moreover, since Eo(t) is a slowly varying function in comparision to e-u..l, we may replace iJ/ot by -iw and 02/ot2 by -w2. Thus when expression (78) is substituted in Eq. (79) we find that the propagation constant is given by
(80)
Since w is assumed real, it is clear from Eq. (80) that k is generally 1 See, for example, L. Brillouin, Congr. intern. elee./ Paris, pp. 739-788, Gauthier-Villars, Paris, 1933.
1932, vol. 2,
183
Theory of electromagnetic
complex.
k
= {:3
wave propagation
Accordingly, we write k in the form
+ ia
= ~ 71
c
+ ia
(a, (:3,
71
= positive-definite)
(81)
which displays as real quantities the phase factor {:3,the attenuation factor a, and the index of refraction 71. To obtain explicit expressions for these factors in terms of the constitutive parameters, we substitute relations (81) into Eq. (80). Thus we find that
{:3=WV~[~+~GY + (2SYT a =
W
V~ [- ~+
1 [€ V;;; 2-+
_ 71--
(82)
~GY+ (;wyr
(83)
~(t)2- + (U)2]~' 2
(84)
2w
Applying expressions (82), (83), and (84) to a lossless (nonabsorptive) plasma whose constitutive parameters are € = €o(l - wp2jw2), J.I. = J.l.o, u = 0, we get
a=O for w
{:3=O {:3=0
71=0
a=O
71
= 0
for
w =
for w Wp
< Wp
>
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
Theory of electromagnetic
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
Electromagnetic waves in a plasma
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
Theory of electromagnetic
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
Theory of electromagnetic
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
,
Eik
=
Eik
+i
- rTik W
(119)
so that Eik and rTik are hermitian. When the frequency of the electromagnetic waves that are passing through a magnetically biased plasma is very low, the motion of the plasma ions must be included in the analysis. We can find the dielectric constant in this low-frequency case by calculating the convection current as the sum of the ionic current and the previously determined 190
Electromagnetic waves in a plasma
electronic current, and by finding t from a knowledge of J through the use of relation (105). To proceed with the calculation, we note that the equation of motion for the ions is formally the same as the equation of motion for the electrons. Accordingly, since we have (in the loss-free case) -inmwv
=
+v
nq(E
X Bo)
(120)
as the equation of motion for the electrons, then for the ions the equation of motion must be -inimiWVi = niqi(E
+
Vi
X Bo)
(121)
Here mi denotes the ionic mass, qi the ionic charge, ni the ionic population density, and Vi the macroscopic velocity of the ions. We know from previous calculation that the electronic convection current nqv is given by expression (103). Hence, it follows from the similarity of Eqs. (120) and (121) that the ionic convection current niqivi is given by the same expression (103) but with Wp replaced by the ionic plasma frequency Wpi and Wg replaced by the ionic gyrofrequency, where Wgi
qi mi
= - Bo
(122)
Superposing nqv and niq,v" we get J, that is,
J = nqv
+ niqivi
(123)
and then substituting this J into relation (105), we find that the nonzero components of t for a loss-free magnetically biased plasma are given byl (124) (125) (126) 1
See, for example, E. Astrom, On Waves in an Ionized Gas, Arkiv Fysik, 2:
443 (1950). 191
Theory of electromagnetic
wave propagation
These components are in accord with the generalized symmetry relation (111)~nd with the hermiticity condition (112). The hermitian property of the dielectric tensor is a consequence of the assuinpt~on that the plasma is loss-free. To show that the hermiticity of the tensor is preserved under a rotation of the coordinate system, we introduce another cartesian system x', y', z', which is obtained from the original cartesian system x, y, z by a pure rotation. Let ai, with i = x', y', z', denote the unit vectors along the axes of the primed system; and as before let Ci, with i = x, y, z, denote the unit vectors along the axes of the unprimed system. In the unprimed system the dielectric tensor is given by (i, k
= x,
y, z)
(127)
and in the primed system it must have the form t
=
~aiakEik
(i, k
(a)
=
x', y', z')
(128)
where Eik(a) denote the components of t with respect to the primed system. Since ai • ak = Oik, it follows from expression (128) that Emn (a)
= am.
Substituting relation
£ •
an
(129)
expression (127) In expression (129), we obtain the
(130) which, by means of the shorthand 'Yik
==
ai'
ek
(131)
can be written as. (132) Similarly we obtain (133) 192
Electromagnetic
waves in a plasma
Since Eik is hermitian, it follows from Eqs. (132) and (133) that also hermitian, i.e.,
Enm(l»
is
(134) Thus we see that hermiticity is preserved under a rotation of the axes. Although we have found it convenient to express the constitutive relation of a magnetically biased plasma by means of a single tensor, it is simpler in certain considerations to deal instead with the elementary vector operations that carry E into D. To determine these operations, we assume for simplicity that the plasma is loss-free. Consequently, its dielectric tensor has the form -ig
a
o
0)
(135)
0 b
where a, b, and g are real quantities. a 0 0 a OOa
E=
(
0) 0
+ (00 00
and then substituting obtain D
= aE
+ (b -
0 0
Splitting this matrix as follows,
~)+(~
b - a
it into the constitutive
a)e.(e.' E)
(136)
0
+ ige. X E
relation D
= I: •
E, we
(137)
as an alternative statement of the constitutive relation. In the case where the motion of the ions can be neglected, i.e., in the case where a, b, and ig are given respectively by expressions (107), (109), and (108), we have
When the biasing field is weak or when the frequency is high, the ratio is small compared to unity and relation (138) to first order in
wg/w
193
Theory of electromagnetic
wave propagation
o/ w becomes
W
D
EE
=
W 2
+
iEO
-T w
(,)0
XE
(139)
where E = Eo(l - Wp2/(2) is the dielectric constant of an isotropic plasma. Returning to the dielectric tensor (135), we ask whether there is a special coordinate system with respect to which the dielectric tensor is diagonal. The ansy;C! to this is that since the tensor is hermitian its matrix can be diagonalized by a unitary transformation which amounts to a complex rotation in Hilbert space.l More simply, however, we observe that when the dielectric tensor (135) is substituted into the constitutive relation D = t. E, we obtain
Dz = aEz - igEy
(140)
Dy
= igEz + aEy
(141)
Dz
= bEz
(142)
With the aid of the following combinations o( expressions (140) and (141), Dz
+ iDy
Dz
-
iDII
= (a - g)(Ez
+ iEy)
(143)
+ g)(Ez
- iEy)
(144)
=
(a
we get the matrix equation
Dz Dz
(
+ iDy) -
Dz
iDy.
=
(a - g 0 0
0
a
+
g
(145)
0
which displays the dieiectric constant as a diagonal matrix. Since D . (cz ::!: icy) = (czDz eyDy ezDz) • (ez ::!: icy) = Dz ::!: iDy, the component Dz + iDy is the projection of D on the vector Cz + icy and
+
+
1 See, for example, Hermann Weyl, "The Theory of Groups and Quantum Mechanics," chap. 1, Dover Publications, Inc., 1931. Translated from the German by H. P. Robertson.
194
Electromagnetic
waves in a plasma
Dx, - iDy is the projection of D on the vector eX, - iey• Thus we see that the elements of the matrices in Eq. (145) are referred to a coordinate system x' = I/V2 (x + iy), y' = 1/0 (x - iy), z' = z, whose unit vectors are eX,' = 1/0 (ex, + iey), ey' = 1/0 (ex, - ie)y, ez' = ez• The vectors eX,', ey', ez', which are unit orthogonal vectors in the hermitian sense, that is, ei 'ek* = Oik where i, k = x', y', z', constitute the principal axes of the dielectric tensor.l
6.6 Plane Wave in Magnetically Biased Plasma In this section we shall study the propagation and polarization properties of a plane monochromatic wave in a magnetically biased homogeneous plasma which for simplicity is assumed to be lossless. We regard the plasma as a continuous medium whose conductivity is zero, whose permeability is equal to the vacuum permeability /-'0, and whose dielectric constant is the tensor £ given by Eqs. (107), (108), and (109) of the previous section. By definition, the electric vector of a plane monochromatic wave has the form E(r)
=
Eoeik•r
(146)
where Eo is a constant vector, k is the vector wave number, and r is the position vector. We may write k as k
= n~
v
(147)
where n is the unit vector in the direction of propagation and v is the phase velocity of the wave. The problem is to determine the vector k, which describes the propagation of the wave, and the vector Eo, which describes the polarization of the wave. 1 For an exhaustive discussion, see G. Lange-Hesse, Vergleich der Doppelbrechung in Kristall und in der Ionosphare, Archiv der Elektrischen Vbertragung, 6: 149-158 (1952).
195
Theory of electromagnetic
wave propagation
The vector E must satisfy the Helmholtz equation
v
X VX E
=
w2}J.o£
•
E
(148)
as can be seen from the Maxwell equations V X E
= iW}J.oH
VXH
=
-i~£. E
(149)
by taking the curl of the first and then using the second to eliminate H. Substituting expression (146) into Eq. (148), and using relation (147), we obtain Eo -
n(n
• Eo)
1v2
= - - £. Eo EO
(150)
c2
wherec = I/V }J.OEOis the vacuum velocity of light. Without loss of generality we choose a cartesian system of coordinates so oriented that the z axis is parallel to Bo and the yz plane contains n. As shown in Fig. 6.1, the angle between nand Bo is denoted by 8. Accordingly, the x, y, z components of the vector equation (150) are given by Eox
(1 - ~ EXX) c2 Eo
Eox (-
~ EYX) C EO
o + EOy( -
EOy (~ 2 EXY) c
EO
+ EOy (cos
2
0 0
+ =
8 - ~ EYY)
C EO
+ Eo.(
cos 8 sin 8) + Eo. (sin2 8 - ~~) C
EO
- cos (J sin 8) = 0
(151)
=0
where Eox, Eoy, Eo. are the cartesian components of Eo. Since these three simultaneous equations are homogeneous, they yield a nontrivial solution only when
o -sin8cos(J
o 196
-sin8cos8
2
• 2 (J sm --- V2 C
=0 En EO
(152)
Electromagnetic
waves in a plasma
%
Fig. 6.1
Arbitrary direction n of wave propagation in plasma with applied magnetostatic field Bo•
D
y
With the aid of the quantities
E}
= -EEoxz -
• fzu tEO
E2
=
E",,,, Eo
EI, E2, Ea,
+i
E"," Eo
which are defined by Ezz
fa = Eo
(153)
we find that Eq. (152) can be written
(154)
This equation determines two values of v2/e2 for each value of 9. In the case where the propagation is parallel to Bo, we have 9 = 0; accordingly, Eq. (154) yields the two solutions v2
1
- = -EI =
e2
1
1 Exx
• Exy
--tEO
Eo
1 ---1
X
(155)
+Y
and v2
1
-e2 = - = E2
1
1 E",,,, EO
+i
E",u EO
X 1--1- Y
(156)
197
Theory of electromagnetic
wave propagation
where X = (Wp/W)2 and Y = -wo/w.! From these expressions it follows that the propagation constants of the two waves that travel parallel to Bo are given by
(157) and
(158)
Moreover, when the propagation is along the y axis, i.e., perpendicular to Bo, f) is equal to 7r/2 and in this case the two solutions of Eq. (154) are
(159)
and
1 1- 1For the propagation
k' /2 "c
= w- VI--- -
X
X P/(1
(160) - X)
constants of the corresponding two waves, we have
= w~ -C
1-
W
2
-.!!-
w2
(161)
and
(162) ! Since q in the case of electrons is a negative quantity, then wo, which is given by (q/m)Bo, is also a negative quantity. We wish Y to be a positive quantity and therefore we include a minus sign in the definition.
198
Electromagnetic
waves in a plasma
In general, when 8 is arbitrary we have the two solutions.
~ =
[1- --1
-Y-T2----,X 1 - 2 1 _ X :!:
1=1 =Y=T=4===2]-1 \}"4 (1 _ X)2 YL
(163)
+
and hence (164)
(165)
where YT = Y sin 8 and YL = Y cos 8. Thus we see that there are two waves traveling in any arbitrary direction 8, and that one of them has a propagation constant k~ given by expression (164) while the other has a propagation constant k': given by expression (165). Since as a function of X the propagation constant k~ resembles the propagation constant of a wave in an isotropic plasma more closely than k': does, the wave whose propagation constant is k~ is sometimes referred to as the ordinary wave and the wave whose propagation constant is k': as the extraordinary wave. Indeed k~/2' the value k~ has when 8 = 71'/2, is identically equal to the propagation constant of a wave in an isotropic plasma. Each of the field vectors of a wave is proportional to exp Uk. r). Therefore the Maxwell equations yo X E = iw~H, yo X H = -iwD reduce to the relations 2k X E
=
iw~oH
2k X H
=
-iwD
(166)
which clearly indicate that the vectors k, E, D lie in a plane perpendicular to H (Fig. 6.2). Since H is necessarily perpendicular to k, the wave cannot be an H wave (also known as a TE wave). In general the wave must be an E wave (also known as a TM wave), but in certain special directions the wave is a TEM wave. The Poynting vector 199
Theory of electromagnetic
wave propagation
E
Fig. 6.2
The
vectors
k,
E, D, S lie in the plane of the paper, and H is perpendicular to it. D and H are perpendicular to k. S is perpendicular to E and H. S is generally not parallel to k.
s S(= ~ Ex H*)of the wave is not parallel to k except in those directions of travel where the wave is TEM.l Let us again consider the special case where the propagation is parallel to Bo• In this case, (J = 0 and Eqs. (151) reduce to E Oz
(1 - ~
e2
E
oz (_
Ezz) Eo
~ Eyz) e2 EO
Eo. (-
~2 e
E0
-
Y
+ Eoy
EZZ) =
(~EZY) e2 Eo
(1 _ ~
e2
=
EW) EO
0 =
0
(167)
0
EO
with v2/e2 given by Eq. (155) and by Eq. (156). From the third of these equations, we see that Eo. is zero. Consequently, the two waves that travel parallel to Bo are TEM waves. When v2/e2 is given by Eq. (155), the first or second of Eqs. (167) yields Eoz
.
-=t EOy
(168)
and when v2/e2 is given by Eq. (156), we find that Eoz EOy
=-t
.
(169)
1 However, it has been shown by S. M. Rytov, J. ExpU. Theoret. Phys., U.S.S.R., 17: 930 (1947), that the time-average Poynting vector is parallel to
the group velocity. 200
Electromagnetic
waves in a plasma
Therefore, the electric vectors of the two waves traveling parallel to Bo can be written as (170)
and E" = (ex
+ ieu)Ceiko"x
(171)
where A and C are arbitrary amplitudes. Clearly E' is a left-handed circularly polarized wave, whereas E" is a right-handed circularly polarized wave.l The sum of these two waves yields the composite wave
To study the polarization of this composite wave, we consider theratio E E From (172) we obtain x/
y•
+ (C/A) -Ex = 1,.1--~--~~---Ey
exp [i(k~' - k~)z] 1- (C/A)exp[i(k~' - k~)z]
(173)
If the waves E' and E" are chosen to have equal amplitudes, then the constants A and C become equal. As a consequence of this choice, Eq. (17:3)reduces to
Ex -_ cot (k~ -2 k~' z ) y
(174)
E
Since this relation is real, the composite wave at any position z is linearly polarized; however, the orientation angle of its plane of polarization (the plane containing E and k) depends on z and rotates as z 1 A geometric interpretation may be obtained by considering the real vectors He E'e-iwt and Re E"e-iwt. Setting A = C = 1, we obtain from Eqs. (170) and (171) the expressions
Re E'e-iwt = e. cos (k~z - wt)
+ ey
sin (k~z - wt)
Re E"e-iwt = e. cos (k~'z - wt) - ey sin (k~'z - wt) Clearly, at any fixed time the locus of the tip of the vector Re E'e-iwt is a right-handed helix. As time increases this helix rotates counter-clockwise. On the other hand, the locus of the tip of the vectorRe E"e-iwt is a left-handed helix, which rotates clockwise. 201
Theory of electromagnetic
wave propagation
increases or decreases. In other words, the composite wave undergoes Faraday rotation. The angle T through which the resultant vector E rotates as the wave travels a unit distance is given by
=
T
k~ - k~' 2
(175)
The rotation is clockwise because k~ > k~' always. With the aid of expressions (157) and (158), we see that T can be written in the form1 (176) which displays the dependence of the Faraday rotation T on frequency. We note that if a wave travels parallel to Bo it undergoes a clockwise Faraday rotation. On the other hand, if a wave travels antiparallel to Bo it undergoes a Faraday rotation of the opposite sense. That is, on reversing the direction of propagation, a clockwise wave becomes counterclockwise, and vice versa. This means that if the plane of polarization of a wave traveling parallel to Bo is rotated through a certain angle, then upon reflection it will be rotated still further, the rotation for the round trip being double the rotation for a single crossing. For weak biasing fields the Faraday rotation depends linearly on Bo• To deduce this fact from expression (176), which in terms of the parameters X = (wp/w)2 and Y = -wo/w can be written as
T
= ~ ~ (~1
-
1:
Y-
~1 -
1 ~
Y)
(177)
we expand the square roots and retain only the first two terms in accord with the assumption that X« 1 and Y« 1. Thus we obtain the relation T
lw
= --
2c
XY
=
_ 1.. (w )2 W p
2c
w
(178)
0
which shows that the Faraday rotation T for weak biasing fields (Y « 1) and high frequencies (X « 1) is linearly proportional to Wo and hence 1
Recall that
202
Wo
is a negative quantity.
Electromagnetic
waves in a plasma
linearly proportional to Bo• Since Wo is negative for electrons, we again see that T is positive (clockwise rotation) in the case of parallel propagation. In the other special case, propagation being perpendicular to Bo, that is, along the y axis, we have () = 7rj2, and Eqs. (151) reduce to
Eox
(1 - ~ EX.:) -
Eox
(_
c2
c2
Eo. (1 -
+ Eoy
EYX)
~
Eoy
EO
EO
~E •• ) 2 EO
c
=
(_
(~EXll) c 2
~
EO
c2
EYY)
=
0
=
0
(179) (180)
EO
0
(181)
When in accord with Eq. (159) we choose (182) then from Eqs. (179), (180), and (181) it follows that Eox and Eoy are identically zero, and the only surviving component of the electric vector is Eo.. Thus we see that one of the two waves traveling in the y direction is a linearly polarized TEM wave whose electric vector is parallel to Bo and has the form E'
= e.A eik',,'211
(183)
where A is an arbitrary constant. Since the propagation constant as given by Eq. (161) is independent of Bo and equal to the propagation constant of a wave in an isotropic plasma, this TEM wave (the ordinary wave) is independent of Bo in its propagation properties and behaves as though it were a TEM wave in an isotropic plasma. To obtain the extraordinary wave propagating perpendicular to Bo, the other possible value of v2/ c2 as given by Eq. (160) is used. That is, k~/2
v2
C2 =
Exxj EO (Exx/EO)2
is substituted
+
(184)
(EXy/EO)2
into Eqs. (179), (180), and (181). ,Thus it is found that 203
Theory of electromagnetic
wave propagation
Eo. vanishes identically and that Eo" EOy
.1-X-P XY
E
=
_..!!!!.=~----Ell"
(185)
Therefore the electric vector of this extraordinary wave has the form (186) where C is an arbitrary constant. The magnetic vector H" is obtained by substituting E" into the first of Eqs. (166). Thus H"
. k':'21 - X - Y2 Ceik'.;,'lI = -~e.-----WfJoo XY
(187)
From expressions (186) and (187), we see that the extraordinary wave traveling perpendicular to Bo is an E wave (TM wave) with its mag~ netic vector parallel to Bo• For propagation in an arbitrary direction (J, it follows from Eqs. (151) that the ratio p of the electric vector components perpendicular to n is given byl (188)
for the ordinary wave whose propagation constant is k~ and by
P
" _ E'; _ - E~ -
-
i YL
[12
Y T2
1- X
+
~14
4
YT
(1 - X)2
+ Y 2J
(189)
L
for the extraordinary wave whose propagation constant is k~. Here E, is the component of E in the direction of the unit vector ee, which is defined bye" X e, = n. That is, E, = -E. sin (J + Ell cos (J. The ratio E"jE, is a measure of the polarization of the part of E that is transverse to the direction of propagation n and is sometimes referred to as the polarization factor. The projection of the tip of E on a plane 1
Without loss of generality, we still take n to lie in the
204
zy
plane.
Electromagnetic
Fig. 6.3
waves in a plasma
Polarization ellipses of ordinary and extraordinary waves traveling into the plane of the paper. Ordinary wave is counterclockwise. Extraordinary wave is clockwise.
transverse to n sweeps out an ellipse and, accordingly, the wave is said to he elliptically polarized. We note that p' p" = 1 and consequently the ordinary and extraordinary waves are oppositely polarized. In the case of the ordinary wave the sense of polarization is counterclockwise and in the case of the extraordinary wave it is clockwise. See Fig. 6.3.
6.7 Antenna Radiation in Isotropic Plasma So far we have been concerned with only the plane wave solutions of Maxwell's equations for a homogeneous plasma medium. Now, as a generalization to a case that involves spherical waves, we consider the 205
Theory of electromagnetic
wave propagation
far-zone radiation field of a primary source in an unbounded plasma. For simplicity, the primary source is taken to be a thin, center-driven, straight-wire antenna of length 2l, and the ambient plasma is assumed to be homogeneous and isotropic. The antenna is driven monochromatically at an angular frequency wand the time-average power fed into its input terminals is Pi. The problem is to find for fixed Pi and w the far-zone radiation field of the antenna as a function of X( = Wp2/(2). Actually the basic part of the calculation has already been made in Chap. 3. Indeed, all we are required to do is to replace E by Eo(1 - X) and k by (w/e) Vi - X in expressions (17), (18), and (19) of Sec. 3.2. However, since these expressions are valid only in the far zone, we must be careful not to violate the condition (w/e) Vi - X r »1. Clearly this condition can be met for the range 0 :::; X < 1 by making r, the distance from the center of the antenna to the observation point, sufficiently large; but for X = 1 (plasma resonance) the condition is violated. Moreover, at X = 1 we have cutoff, i.e., no wave propagation can occur, and the power fed into the antenna goes into heating the plasma. As in Sec. 3.2, we place the antenna along the z axis of a cartesian coordinate system, with one end of the antenna at z = -l and the other end at z = l. With respect to the concentric spherical coordinate system (r,O,cf» shown in Fig. 3.1, we see that the far-zone field components of the antenna immersed in a homogeneous isotropic plasma medium are
Ee
=
1
yl- Xr
iei(w/c)
H~
(190)
. ~H~
Vi - x'1~
= -
2
rrr
(191)
Io(X)F(O,X)
and the radial component far zone is
of the time-average
Poynting
vector in the
(192)
This follows from Eqs. (17), (18), and (19) of Chap. 3 when 206
E
is replaced
Electromagnetic
waves in a plasma
V'l=X.
The radiation pat-
by Eo(1- X) and k is replaced by (wle) tern F«(),X) of the antenna is given by F«() X)
== cos [(wle) y1=X l cos.()j
,
-
cos [(wle) y1=X lj
(193)
SIn ()
To find how 10, the magnitude of the current at the driving point, depends on the time-average real power Pi fed into the antenna's input terminals and on the parameter X( = wp2lw2) which completely describes the plasma medium into which the antenna radiates, we note that since the plasma is assumed to be lossless, the time-average power P radiated by the antenna must be equal to Pi. Substituting expression (192) into the definition P.
=
S
(2" (" 10 10
rr2
.
(194)
sm ()d() dq,
and equating P to Pi, we find that lois related to Pi as follows: Pi
=.
I~ I 2(X)
_1__
yl-
(" F2«(),X)
0
xVEo
411"
10
sin ()d()
(195)
More conveniently, we write this relation in the form (196)
where the new parameter Rrad, the so-called radiation resistance of the antenna, has the representation Rrad(X)
= y_l__
l-X'\J~
~
21 (" F2«(),X) sin ()d()
(197)
11"10
By substituting expression (193) into the integral and performing the operations that led to Eq. (101) of Chap. 3,we obtain Rrad(X)
=
y 1 1-
+ sin22a (Si 4a
X
1 '\JI~ 2 EO
11"
- 2Si 2a)
[c +
In 2a -
Ci 2a
+ co~ 2a (C + In a + Ci 4a
- 2Ci 2a) ] (198) 207
Theory of electromagnetic
wave propagation
whereC(= 0.5722) is Euler's constant and a == (wle) VI - Xl. Thus we see from Eq. (196) that 10 depends on Pi and Rrad as follows,
(199)
and from expression (198) that Rrad(X) can be calculated for any X in the range 0 S X < 1. In view of relation (199) the far-zone field expressions (190) and (191) can be written as
(200)
(201)
These are the desired forms because they show how the far-zone fields Eo, Hq, depend on X. In the special case where X = 0 they reduce, as they should, to the conventional expressions for the far-zone fields of a straight-wire antenna in vacuum. In the other special case where Pi, w, and 1 are fixed and X is made to approach unity, we find that F(O,X) Rrad(X)
-+
21 (w)2 C
-+
1.~ (~)4 [4(1_ 67r '\j;;; e
Consequently, ~
Eo
"'J
'\j;;;
[2(1 - X) sin 0
as X
_ /sin 0 V Pi (1 _ X)l4
r _
Hq,"'J----vP;(I-
r
(203)
1, expressions (200) and (201) reduce to
-+
ei(w/c)Vl-Xr
ei(w/c)Vl-Xr
X)%
(202)
X)l4sinO
(204)
(205)
This shows that as X -+ 1, the antenna's radiation pattern approaches the radiation pattern of a Hertzian dipole. It also shows that the 208
Electromagnetic
waves in a plasma
wave impedance Z, which is given by (206)
increases without bound as X ~ 1.
6.8 Dipole Radiation in Anisotropic Plasma As was shown in Chap. 2, the radiation field of a monochromatic source in an unbounded homogeneous isotropic medium can be calculated by either the method of potentials or the method of the dyadic Green's function. As long as the medium is homogeneous and isotropic these two methods are equally convenient. However, in the case where the surrounding medium is anisotropic, the method of potentials1 leads to difficulties in the early stages of the calculation and the Green's function method becomes the more fruitful of the two. Indeed, Bunkin,2 Kogelnik,3 and Kuehl4 used the Green's function method with considerable success to analyze various aspects of the problem of a primary source in an anisotropic medium. Recalling some of their results, we shall now show how one proceeds in the Green's function method to find the radiation field of a dipole immersed in an unbounded homogeneous anisotropic plasma. The electric field E of a monochromatic source J immersed in an 1 A. Nisbet, Electromagnetic Potentials in a Heterogeneous Non-Conducting Medium, Proc. Royal Soc. (London), (4) 240: 375-381 (1957). 2 F. V. Bunkin, On Radiation in Anisotropic Media, J. Exptl. Theoret. Phys., U.S.S.R., 32: 338-346 (1957); also Soviet Physics JETP, 5: 277-283 (1957). 8 H. Kogelnik, The Radiation Resistance of an Elementary Dipole in Anisotropic Plasmas, Proc. Fourth Intern. Conf. on Ionization Phen. in Gases (Uppsala, 1959), pp. 721-725, North Holland Publishing Company, Amsterdam, 1960. Also J. Res. Natl. Bur. Std., 64D (5): 515-523 (1960). 4 H. Kuehl, Radiation from an Electric Dipole in an Anisotropic Cold Plasma, Caltech Antenna Lab. Rept. 24, October, 1960; also Phys. Fluids, 5: 1095--1103(1962).
209
Theory of electromagnetic
wave propagation
unbounded anisotropic plasma medium must satisfy (207) Moreover, E must have the form of a wave traveling away from the source. Hence, we are required to find the particular integral of Eq. (207) that satisfies the radiation condition. By virtue of the linearity of Eq. (207), the desired solution may be expressed in the form
=
E(r)
(208)
iW/loJr(r,r') . J(r')dV'
where the integration extends throughout the region of finite extent occupied by the current. If this form is to be the solution of Eq. (207), the dyadic Green's function r(r,r') must satisfy
v
X V X r(r,r')
-
r(r,r')
W2/lo£'
=
u8(r - r')
(209)
or
vv.
r(r,r')
- V2r(r,r')
-
W2/lo£ •
r(r,r')
=
u8(r - r')
(210)
where u is the unit dyadic and 8(r - r') is the three-dimensional Dirac delta function. To facilitate the construction of the dyadic Green's function, we express it as a Fourier integral. That is, we write r(r r')
= ~
,
811"8
J
A(k)eik.(r-r'l dk
00
-
(211)
00
and by so doing transform the problem of finding r into one of first finding the dyadic function A(k) and then evaluating the integral in k space. Substituting expression (211) into Eq. (210) and recalling the integral representation 8(r - r')
=
_1_ 811"8
J
00
-
e'ik'(r-r'l dk
(212)
00
we see that A(k) is determined by V(k) • A(k) 210
= u
(213)
Electromagnetic
waves in a plasma
where V(k)
=
+ku
-kk
2
-
(214)
W2~O£
With the aid of the theory of matrices, Eq. (213) yields for A(k) the expression A(k)
=
adj V(k) det V(k)
(215)
Here det V(k) stands for the determinant of the matrix of V(k) and - adj V(k) represents the dyadic whose matrix is the adjoint of the matrix of V(k).l It therefore follows from Eqs. (211) and (215) that the integral form of the dyadic Green's function is r(r r') ,
= ~ 811"3
f
00
-
00
adj V(k) det V(k)
eik'(r-r')
dk
(216)
This form obeys the radiation condition and hence constitutes the only solution of Eq. (211) that leads to a physically acceptable result. Since the source of radiation in the present instance is an oscillating electric dipole, we write the current distribution as J(r')
= -iwplJ(r')
(217)
where p denotes the electric dipole moment. Substituting the current (217) and the Green's function (216) into the form (208), we obtain the integral representation E(r)
=
W2f..lo 811"3
f
00 -00
[adj V(I{)] . P eik'r dk det V(k)
(218)
which is the desired expression for the electric field E of the dipole p. Thus we see that in the Green's function method the problem of calculating the field of a dipole in a homogeneous anisotropic medium splits into an algebraic part, which consists in finding the adjoint and the determinant of the matrix components of the dyadic V(k), and into 1 See, for example, H. Margenau and G. M. l\1urphy, "The Mathematics of Physics and Chemistry," p. 295, D. Van Nostrand Company, Inc., Princeton, N.J., 1943.
211
Theory of electromagnetic
wave propagation
an analytic part, which requires the evaluation of the integral in expression (218). According to Kuehl, when the dipole oscillates at a high frequency, i.e., when X = Wp2/W2« 1 and Y2 = w//w2« 1, the dipole's far~zone electric field in the spherical coordinates r, 8, t!> 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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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
Theory of electromagnetic
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)
This is the equation into which Eq. (72) is transformed when the frame of reference is changed from K' to K. In three-dimensional form, Eq. (86) leads to the following equations for the vector potential A(r,t) and the scalar potential
Theory of electromagnetic
wave propagation
reference frame K:
[
V'2 -
1 82
"& at2
-
K'Y2
(8at +
v. V'
)2]
A(r,t) (87)
[
1 82
"& 8t2
V'2 -
-
K'Y2
(8.at +
v • V'
)2J
q,(r,t)
, =
-J.l.'C2p
-
~,~ 'YC2('YJ' v -
'YC2p)
(88)
where, as before, K = (C2E'J.I.' - 1)jc2 = (n'2 - 1)jc2. With aknowledge of these equations, we can find the vector and scalar potentials of a source surrounded by a homogeneous isotropic medium moving at a velocity v with respect to the source. Moreover, these equations enable one to calculate the electric vector E = -V'q, - (8j8t)A and the magnetic vector B = V' X A of the source in the presence of a wind. The above discussion is based on the transformation of the inhomogeneous wave equation from the K' frame to the K frame. Actually, the same results can be achieved by using the tensor form of Maxwell's equations as the point of departure. 1 To show this, we recall that Maxwell's equations can be written as follows: (89)
8Ga(3 8X(3
=
J
(90) a
These tensor equations hold in all Lorentz frames, and in particular they hold in K' and K. In K' the constitutive relations are D'
=
E'E'
and
H' = .; H'
(91)
J.I.
1 K. S. H. Lee, On the Doppler Effect in a Medium, Antenna Lab. Rept. 29, California Institute of Technology, December, 1963. See also, J. M. Jauch and K. M. Watson, Phenomenological Quantum-electrodynamics, Phys. Rev., 74: 950, 1485 (1948).
236
The Doppler effect
Expressing D', E', H', B' in terms of D, E, H, B of the reference frame K, we find with the aid of the Eqs. (24), (25), (26), and (27) that the constitutive relations in K are 1 D +"2 v X H = l(E + v X B)
(92)
C
H -
vXD
=
.!. (B _.!.v c 2
JJ.'
XE)
(93)
When written in tensor form, these constitutive relations become Ga{JV{J
= c2lFapVp
Ga{JVp
+ G{JpVa + GpaV{J
(94)
= JJ..,1 (Fa{JVp
+ F{JpVa + FpaV{J)
(95)
where as before V denotes the velocity 4-vector ('Yv,i'Yc). To express the field tensor Ga{J explicitly in terms of the field tensor Fa{J, we multiply Eq. (95) by VP' Noting that p
VpVp
= -c2
we thus find that
By virtue of the constitutive relations (94), we have (97)
Hence, it follows from Eq. (96) that (98)
Substituting expression (98) into the Maxwell equation (90), we find that oFa{J oX{J
+ ua u • of{Jp OX~ K
u
_ K
p
U oFa• {J OX{J -
'J JJ.
••
(99) 237
Theory of electromagnetic
wave propagation
However, from Eqs. (90) and (94) we see that
(100)
Therefore, Eq. (99) becomes (101) Now we have two equations for the field tensor Fa/3, one being the Maxwell equation (89) and the other being equation (101). If we write the field tensor Fa/3 in terms of the 4-potential A,
=
(A,i, ~}
that is, if we write (102)
then Eq. (89) is satisfied. Substituting expression (102) into Eq. (101), we obtain the following equation for the 4-potential:
Rearranging terms, we get
Since the 4-potential is not completely determined by Eq. (102), we are free to impose on it the following additional condition, (105) 238
The Doppler effect
which is called the "generalized Lorentz condition" for the 4-potential. When this condition is satisfied, Eq. (104) reduces to (106)
This equation is identical to Eq. (86) and, in three-dimensional form, amounts to Eqs. (87) and (88). To show that Eq. (106) can be used to find the index ofrefraction of a moving media, we assume that A(r,t) has the form of a plane wave: (107) Substituting this expression into Eq. (87), with the right side set equal to zero, we find (108) On solving this equation the index of refraction.
for n
= ek/w, we are led to relation (61) for
239
Name Index Alsop, L. E., 110 Andrade, E. N. Da C., 217n. Astrom, E., 191n. Aulock, W. H. von, 69n. Baghdady, E. J., 125n. Barbiere, D., 69 Beverage, H. H., 46n. Bidal, P., 98n. Bladel, J. van, 26n. Blumenthal, 0., 9n. Bohnert, J. I., 125n. Bolton, J. G., 110n., 150n., 158n. Bontsch-Bruewitsch, M. A., 73n. Booker, H. G., 125n. Bopp, F., 189n. Borgnis, F., 182n. Born, M., 121n., 156n. Bouwkamp, C., 9, 35-36, 45n., 50n., 57n., 97, 100, 104n. Bracewell, R. N., 109n. Bramley, E. N., 167n. Brillouin, L., 34, 183n. Brouwer, L. E. J., 57n. Brown, G. H., 57n. Brown, R. H., 159 Bruckmann, H., 57n. Bruijn, N. G. de, 50n. Budden, K. G., 169n., 214n. Bunkin, F. V., 209 Campbell, G. A., 57n. Carter, P. S., 57n. Casimir, H. B. G., 57n., 97, 100, 104n. Cerenkov, P. A., 47 Chandrasekhar, S., 111n., 126n. Chu, L. J., 80n. Cooper, B. F. C., 110n. Courant, R, 21n., 51n. Debye, P., 98 Demirkhanov, R. A., 170n. Deschamps, G. A., 125n. Desirant, M., 169n. Dirac, P. A. M., 21n. Dolph, C. L., 68 Doppler, Ch., 217 Drake, F. D., 110n. Einstein, A., 218n. Emde, F., 16, 75n.
Fano, D., 138n. Fay, W. J., 169n. Feshbach, H., 11n. Fock, V., 218n. Frank, I., 47n., 228n. Franz, K., 77n. Friis, H. T., 42n., 57n., 142n.
Middlehurst, B. M., 110n., 158n. Minkowski, H., 218n. Ml'Sller, C., 218n. Morse, P. M., 11n., 99n. Muller, C., 56n. Murphy, G. M., 211n. Murray, F. H., 57n.
Geverkov, A. K., 170n. Ginzburg, V. L., 169n. Giordmaine, J. A., 110n. Goland, V. E., 170n.
Nisbet, A., 209n.
Haddock, F. T., 109n. Hansen, W. W., 62 Harrison, C. W., Jr., 42n. Helmholtz, H. von, 9 Heritage, J. L., 169n. Hedofson, N., 169n. Hilbert, D., 21n., 51n. Hodge, W. V. D., 98 Jacobson, A. D., 125n. Jahnke, E., 75n. Jauch, J. M., 236n. Jelley, J. V., 47n., 110n. Kales, M. L., 125n. Kellogg, E. W., 46n. King, D. D., 69, 125n. King, R: W. P., 42n., 45, 57n., 76n., 171n. Knudsen, H. L., 57n. Ko, H. C., 147n. Kogelnik, H., 209 Kraus, J. D., 57n., 125n. Kuehl, H., 209 Kuiper, G. P., 110n., 158n. Labus, J., 39n. Lange-Hesse, G., 195n. Lassen, H., 77n. Lawson, J. D., 50n. Lee, K. S. H., 225n., 236n. Lequeux, J., 109n. Lerner, R. M., 125n. Lorentz, H. A., 213n., 218. Maas, G. J. van der, 69 McReady, L. L., 152n. Margenau, H., 211n. Mathis, H. F., 57n. Mayer, C. H., 110n. Meixner, J., 9, 189 Michiels, J. L., 169n.
Packard, R F., 69 Papas, C. H., 45n., 76n., 187n., 225n. Pauli, W., 218n. Pawsey, J. L., 109n., 152n. Payne-Scott, R, 152n. Pistolkors, A. A., 57n., 74n. Pock1ington, H. C., 38n. Poincare, H., 122, 218n. Poincelot, P., 213n. Pokrovskii, V. L., 68 Polk, C., 73 Popov, A. F., 170n. Ratcliffe, J. A., 167n., 169n. Rayleigh, Lord, 213n. Rham, G. de, 98n. Riblet, H. J., 68, 80n. Rice, C. W., 46n. Rice, S. 0., 111n. Robertson, H. P., 194n. Rosenfeld, L., 171n. Rumsey, V. H., 125n. Rydbeck, O. E. H., 228 Rytov, S. M., 200n. Sandler, S. S., 69 Schelkunoff, S. A., 7n., 42n., 43n., 57n., 61, 80n., 142n. Schwartz, L., 21n. Shk1ovsky, I. S., 109n., 169n. Smith, F. G., 109n. Smythe, W. R, 11n. Sommerfeld, A., 20n., 62n., 189n., 218n. Sonine, N. J., 77 Spitzer, L., Jr., 176n. Steinberg, J. L., 109n. Stokes, G. R, 9n., 119 Stone, J. S., 68n. Stratton, J. A., In. Tai, C. T., 143n. Tamm, I., 47n.
241
Theory of electromagnetic Taylor, T. T., 69, SOn. Tetelbaum, S., 72 Thomas, R. K., 69 Toraldo, G. de Francia, 80n.
Townes, C. R., 110n. Twiss, R. Q., 159
wave propagation
Watson, G. N., 51n. Watson, K. M., 236n. Weisbrod, S., 169n. Westfold, K. C., 123n., 150n.
Weyl, R., 194n. Whittaker, E. T.; 5In., 21Sn.
Unz, R., 69
242
Wiener, N., 112n.
Wilcox, C. R., 98n. Wilmotte, R. M., 80n. Wolf, E., 121n., 136n., I56n.
Woodward, P. M., 50n. Woodyard, J. R.., 62 Yeh, Y.-C., 143n. Zernike, :F., 159n.
Subject Index Aberration, 226 Angular-momentum operator, 107 Antenna, dipole, 44, 208 isotropic, 57 radio-astronomical, 109110 scanning, 69-70 straight wire, 37-56 current in, 37-42 integral relation for, 48-50 pattern synthesis, 50-
56 radiation from, 42-47 Antenna temperature, 149151 Antipotentials, 13-14, 23 Apparent disk temperature, 118 Area, effective, in matrix form, 145 of receiving antenna, 143 Argand diagram, 123 Array factor, 57, 59, 60 Arrays, binomial, 62-68 broadside, 62, 68, 69 Chebyshev, 68-69 collinear, 70 end-fire, 62, 69 linear, 57-70 parallel, 70 rectangular, 71 superdirective, 80 uniform, 61 Attenuation factor, 184, 186 Autocorrelation function, 113 Axial ratio, 121 Binomial theorem, 53 Blackbody spectral brightness, 117 Boltzmann equation, 175 Boundary conditions, 8-9 Brightness temperature, 150 Brown and Twiss interferometer, 159, 167168 Cerenkov radiation, 47 Coherence, degree of, 160168 Coherency matrix, 135140, 145 Collision frequency, 176, 177,178
Complex dielectric constant, 175 Conjugate matching, 141, 143 Constitutive parameters, of anisotropic plasma, 189-191 of isotropic plasma, 174, 177 of lossy dielectric, 173174 of simple media, 2, 5-6 transformation of, 192 Cornu spiral, 73 Correlation coefficient, 166-167 Correlation function, 113 Correlation interferometer, 159-168 Coulomb gauge, 11 Covariance of Maxwell's equations, 223 Current 4-vector, 222 Debye potentials, 97-98 Degree, of coherence, 159161, 165-166 of polarization, 130-131, 139-140, 145-146 Dipole (see Electric dipole; Magnetic dipole) Dirac delta function, 21 Directivity gain, definition, 73 full-wave dipole, 76 half-wave dipole, 76 rectangular array, 78-80 short dipole, 74 uniform parallel array, 76-77 Dispersion, 185 Distribution function, 175 Doppler effect, 217, 226227 complex, 228-229 Duality, 6-8 Dyadic Green's function, 19, 26-29, 210-211 E wave, 81, 92, 100-101, 204 Electric dipole, 82, 89-93 field of, 90-93, 102 short filament of current, 84 Electric energy density, in dispersive media, 178183 instantaneous, 14
Electric energy density, time-average, 15":17 Electric potential, scalar, 10 vector, 13, 14 Electric quadrupole, 82, 83 fields of, 94-97 two antiparallel filaments,85 Electric wall, 9 Electrostatic wave, 185 EMF method, 34, 74 Energy theorem, 17 Evanescent wave, 184 Extraordinary wave, 199, 204-205 Far zone, definition, 32 of multipoles, 108 of rectangular array, 71-72 Faraday rotation, 202, 212 Field tensors, 220, 238 Four-potential, 238-239 Four-tensor, 219 Four-vector, 219 Fraunhofer field, 72 Fresnel drag formula, 232 Fresnel field, 72 Gain (see Directivity gain) Gain function, 142 Gauge, Coulomb, 11 Lorentz, 11 Giorgi system of units, 1 Grating lobes, 153 Green's function, dyadic, 26-29, 32, 210-211 scalar, 20-29, 89, 104106 Gyrofrequency, 188, 191 Hwave, 81, 93,100-101 Hankel function, spherical,98-99 Helmholtz equation, scalar, 11, 12, 19,21-23, 38, 98 vector, 104, 196 Helmholtz integral, 22, 23, 85,87 Helmholtz's partition theorem, 9 Hermite polynomials, 5154 Hermiticity,
of coherency
matrix, 135 of dielectric tensor, 189193
243
Theory of electromagnetic Hertzian dipole, 44, 208 Hilbert space, 194 Hodge's decomposition theorem, 98 Hydrogen line, 109 Impedance of antenna, 141 Inertial frame of reference, 217-218 Intensity, polychromatic wave, 129 spectral, 115-117 Interferometer, correlation, 159-168 two-element, 151-159 Irrotational vector, 9 Isotropic antenna, 57 Kronecker delta, 24 Legendre polynomials, associated, 99 Lorentz condition for fourpotential, 238-239 Lorentz force, 2, 175 Lorentz gauge, 11, 13, 19, 23, 86 Lorentz transformation, 217-223 Magnetic dipole, 83, 93 field of, 93 loop of current, 84 Magnetic energy, 14-17 Magnetic potential, scalar, 13 vector, 10 Magnetic wall, 9 Maxwell's equations, 1, 4, 171-174 in tensor form, 220, 236 Modulation index, 156 Multipolar fields, 101-108 Newton's law, 4 Noise, 111 Ordinary wave, 199, 203205 Orientation angle, 121124, 127, 131, 133, 138, 140, 146-147 Pauli spin matrices, 139 Phase invariance, 223-225 Phasor, 4, 135, 136 Planck's law, 117 Plasma, 170 anisotropic, dielectric tensor of, 187-195 dipole radiation in, 209-212 plane waves in, 195205 reciprocity relation for, 212-215
244
wave propagation
Plasma frequency, 177 Poincare sphere, 122, 131, 147 Poisson's equation, 12 Polarization, 109, 118-134 degree of, 130-131, 139140, 146 measurement of, 125 sense of, 121-123, 131 Polarization loss factor, 143 Polarization vector, 171, 172,174 Polarized wave, circularly, 122-124, 138-139 elliptically, 118-134, 137-139 linearly, 122, 123, 138139 oppositely, 131-133 partially, 119, 125, 133, 137, 140-148 Potentials, 9-12, 19-24 in spherical wave functions, 100 Taylor expansion of, 87-88 Power, absorbed, 142 radiated, 29-34 Poynting's vector, 15-16, 29-34, 111-113, 142143, 178, 185 of center-driven antenna, 44, 206 of linear array, 59 of monochromatic source, 33, 36 of rectangular array, 71 of traveling wave of current, 46 Poynting's vector theorem, 16, 178 QuadrupOle (8ee Electric quadrupole) Radiation characteristic, normalized, 62 Radiation condition, 20, 27-29, 98, 104 Radiation pattern, of antenna in plasma, 207, 208 of center-driven antenna, 44-50 of collinear array, 70-71 of linear array, 59 of monochromatic current source, 48 of rectangular array, 71 of traveling wave of current, 47 of two-element interferometer, 153 Radiation resistance, 37, 45, 207
Radio astronomy, 109 Radio telescope, 109-110 Radiometer, 110 Random (stochastic) process, 111-112 Rayleigh-Jeans law, 118 Reciprocity, 212-215 Reciprocity theorem, 142 Refraction, index of, 184186,226-227,230-233 Reversibility, 215 Schelkunoff's synthesis method,61 Schwarz's inequality, 137 Sea interferometer, 158159 Sommerfeld's radiation condition, 20, 98 Spectral brightness, 115118 Spectral flux density, 113117 Spectral intensity, 115-116 Spherical wave expansion, 97 Spur, 139 Stationary random process, 111 Stochastic process, 111 Stokes parameters, VS. coherency matrix, 135-136, 138-139 for monochromatic wave, 122-124 for polychromatic wave, 126-133 under rotation, 133-134 Stress dyadic, 176 Superdirectivity,80 Synthesis of radiation patterns, 48-56 Taylor's series, 86-88, 180 Thevinin's theorem, 141 Trace of matrix, 139 Truncated function, 111 Unilateral end-fire array, 62 Unit dyadic, 24 Unitary transformation, 194 Unpolarized wave, 129 Variance, 166 Velocity, energy transport, 185 group, 185-187 phase, 185-186, 226, 232 Visibility factor, 156, 161, 166 Wave four-vector, 224-226 Wave impedance, 209
A CATALOG
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ED
A CATALOG
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BOOKS IN ALL FIELDS OF INTEREST DRAWINGS OF REMBRANDT, edited by Seymour Slive. Updated Lippmann, Hofstede de Groot edition, with definitive scholarly apparatus. All portraits, biblical sketches, landscapes, nudes. Oriental figures, classical studies, together with selection of work by followers. 550 illustrations. Total of 630pp. 9~ x 12\( 21485-0,21486-9 Pa., Two-vol. set $25.00 GHOST AND HORROR STORIES OF AMBROSE BIERCE, Ambrose Bierce. 24 tales vividly imagined, strangely prophetic, and decades ahead of their time in technical skill: "The Damned Thing," "An Inhabitant of Carcosa, " "The Eyes of the Panther," "Moxon's Master," and 20 more. 199pp. 5%x 8~. 20767.6 Pa. $3.95 ETHICAL WRITINGS OF MAIMONIDES, Maimonides. Most significant ethical works of great medieval sage, newly translated for utmost precision, readability. Laws Concerning Character Traits, Eight Chapters, more. 192pp. 5%x 8~. 24522-5 Pa. $4.50 THE EXPLORATION OF THE COLORADO RIVER AND ITS CANYONS, J. W. Powell. Full text of Powell's I ,OOO-mileexpedition down the£abled Colorado in 1869. Superb account of terrain, geology, vegetation, Indians, famine, mutiny, treacherous rapids, mighty canyons, during exploration of last unknown part of continental U.S. 400pp. 5%x 8~. 20094-9 Pa. $6.95 HISTOR Y OF PHILOSOPHY, Julian Marias. Clearest one-volume history on the market. Every major philosopher and dozens of others, to Existentialism and later. 505pp. 5%x 8~. 21739-6 Pa. $8.50 ALL ABOUT LIGHTNING, Martin A. Uman. Highly readable non-technical survey of nature and causes of lightning, thunderstorms, ball lightning, St. Elmo's Fire, much more. Illustrated. 192pp. 5%x 8~. 25237-X Pa. $5.95 SAILING ALONE AROUND THE WORLD, Captain Joshua Slocum. First man to sail around the world, alone, in small boat. One of great feats of seamanship told in delig-htful manner. 67 illustrations. 294pp. 5%x 8~. 20326-3 Pa. $4.95 LETTERS AND NOTES ON THE MANNERS, CUSTOMS AND CONDI. TIONS OF THE NORTH AMERICAN INDIANS, George Catlin. Classic account of life among Plains Indians: ceremonies, hunt, warfare, etc. 312 plates. 572pp. of text. 6~ x 9\4. 22118-0,22119.9 Pa. Two-vol. set $15.90 ALASKA: The Harriman Expedition, 1899, John Burroughs, John Muir, et al. Informative, engrossing accounts of two-mon!!}, 9,000-mile expedition. Native peoples, wildlife, forests, geography, salmon industry, glaciers, more. Profusely illustrated. 240 black-and-white line drawings. 124 black-and-white photographs. 3 maps. Index. 576pp. 5%x 8~. 25109-8 Pa. $11.95
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THE BOOK OF BEASTS: Being a Translation from a Latin Bestiary of the Twelfth Century, T. H. White. Wonderful catalog real and fanciful beasts: manticore, griffin, phoenix, amphivius, jaculus, many more. White's witty erudite commentary on scientific, historical aspects. Fascinating glimpse of medieval mind. Illustrated. 296pp. 5%x 8\4.(Available in U.S. only) 24609-4 Pa. $5.95 FRANK LLOYD WRIGHT: ARCHITECTURE AND NATURE With 160 Illustrations, Donald Hoffmann. Profusely illustrated study of influence of nature-especially prairie-on Wright's designs for Fallingwater, Robie House, Guggenheim Museum, other masterpieces. 96pp. 9\4x 101<\. 25098-9 Pa. $7.95 FRANK LLOYD WRIGHT'S FALLINGWATER, Donald Hoffmann. Wright's famous waterfall house: planning and construction of organic idea. History of site, owners, Wright's personal involvement. Photographs of various stages of building. Preface by Edgar Kaufmann, Jr. 100 illustrations. 112pp. 9\4x 10. 23671-4 Pa. $7.95 YEARS WITH FRANK LLOYD WRIGHT: Apprentice to Genius, Edgar Tafel. Insightful memoir by a former apprentice presents a revealing portrait of Wright the man, the inspired teacher, the greatest American architect. 372black-and-white illustrations. Preface. Index. vi + 228pp. 8\4x II. 24801-1 Pa. $9.95 THE STORY OF KING ARTHUR AND HIS KNIGHTS, Howard Pyle. Enchanting version of King Arthur fable has delighted generations with imaginative narratives of exciting adventures and unforgettable illustrations by the author. 41 illustrations. xviii + 313pp. 6%x 9\4. 21445-1 Pa. $5.95 THE GODS OF THE EGYPTIANS, E. A. Wallis Budge. Thorough coverage of numerous gods of ancient Egypt by foremost Egyptologist. Information on evolution of cults, rites and gods; the cult of Osiris; the Book of the Dead and its rites; the sacred animals and birds; Heaven and Hell; and more. 956pp. 6li x 9\4. 22055-9,22056-7 Pa., Two-vol. set $21.90 A THEOLOGICO-POLITICAL TREATISE, Benedict Spinoza. Also contains unfinished Political Treatise. Great classic on religious liberty, theory of government on common consent. R. Elwes translation. Total of 421pp. 5%x 8~. 20249-6 Pa. $6.95 INCIDENTS OF TRAVEL IN CENTRAL AMERICA, CHIAPAS, AND YUCATAN, John L. Stephens. Almost single-handed discovery of Maya culture; exploration of ruined cities, monuments, temples; customs of Indians. 115 drawings. 892pp. 5%x 8~. 22404-X, 22405-8Pa., Two-vol. set $15.90 LOS CAPRICHOS, Francisco Goya. 80 plates of wild, grotesque monsters and caricatures. Prado manuscript included. 183pp. 6%x 9%. 22384-1 Pa. $4.95 AUTOBIOGRAPHY: The Story of My Experiments with Truth, Mohandas K. Gandhi. Not hagiography, but Gandhi in his own words. Boyhood, legal stu~ies, purification, the growth of the Satyagraha (nonviolent protest) movement. Cnucal, inspiring work of the man who freed India. 480pp. 5%x 8~. (AvaIlable In U.S. only) 24593-4 Pa. $6.95
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ILLUSTRATED DICTIONARY OF HISTORIC ARCHITECTURE, edited by Cyril M. Harris. Extraordinary compendium of clear, concise definitions for over 5,000 important architectural terms complemented by over 2,000 Ime drawmgs. Covers full spectrum of architecture from ancient ruins to 20th-century Modermsm. Preface. 592pp. 7Y, x 9%. 24444-X Pa. $14.95 THE NIGHT from original 4% x 6.
BEFORE CHRISTMAS, Clement Moore. Full text, and woodcuts 1848 book. Also critical, historical material. 19 illustrations. 40pp. 22797-9 Pa. $2.50
THE LESSON OF JAPANESE ARCHITECTURE: 165 Photographs, Jiro Harada. Memorable gallery of 165 photographs taken in the 1930's of exquisite Japanese homes of the well-to-do and historic buildings. 13 line diagrams. 192pp. 8%x lilt 24778-3 Pa. $8.95 THE AUTOBIOGRAPHY OF CHARLES DARWIN AND SELECTED LETTERS, edited by Francis Darwin. The fascinating life of eccentric genius composed of an intimate memoir by Darwin (intended for his children); commentary by his son, Francis; hundreds of fragments from notebooks, journals, papers; and letters to and from Lyell, Hooker, Huxley, Wallace and Henslow. xi + 365pp. 5%x 8. 20479-0 Pa. $5.95 WONDERS OF THE SKY: Observing Rainbows, Comets, Eclipses, the Stars and Other Phenomena, Fred Schaaf. Charming, easy-to-read poetic guide to all manner of celestial events visible to the naked eye. Mock suns, glories, Belt of Venus, more. Illustrated. 299pp. 51i x 81i. 24402-4 Pa. $7.95 BURNHAM'S CELESTIAL HANDBOOK, Robert Burnham, Jr. Thorough guide to the stars beyond our solar system. Exhaustive treatment. Alphabetical by constellation: Andromeda to Cetus in Vol. I; Chamaeleon to Orion in Vol. 2; and Pavo to Vulpecula in Vol. 3. Hundreds of illustrations. Index in Vol. 3. 2,OOOpp. 6li x 9\4. 23567-X, 23568-8, 23673-0 Pa., Three-vol. set $37.85 STAR NAMES: Their Lore and Meaning, Richard Hinckley Allen. Fascinating history of names various cultures have given to constellations and literary and folkloristic uses that have been made of stars. Indexes to subjects. Arabic and Greek names. Biblical references. Bibliography. 563pp. 5%x 8Y,. 21079-0 Pa. $7.95 THIRTY YEARS THAT SHOOK PHYSICS: The Story of Quantum Theory, George Gamow. Lucid, accessible introduction to influential theory of energy and matter. Careful explanations of Dirac's anti-particles, Bohr's model of the atom, much more. 12 plates. Numerous drawings. 240pp. 5%x 8Y,. 24895-X Pa. $4.95 CHINESE DOMESTIC FURNITURE IN PHOTOGRAPHS AND MEASURED DRAWINGS, Gustav Ecke. A rare volume, now affordably priced for antique collectors, furniture buffs and art historians. Detailed review of styles ranging from early Shang to late Ming. Unabridged republication. 161 black-and-white drawings, photos. Total of 224pp. 8%x II Ii. (Available in U.S. only) 25171-3 Pa. $12.95 VINCENT VAN GOGH: A Biography, Julius Meier-Graefe. Dynamic, penetrating study of artist's life, relationship with brother, Thea, painting techniques, travels, more. Readable, engrossing. 160pp. 5%x 8Y,. (Available in U.S. only) 25253-1 Pa. $3.95
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HOW TO WRITE, Gertrude Stein. Gertrude Stein claimed anyone could ~nderst~nd her unconventional writing-here are clues to help. Fascinating ImprovISatiOns, language expenments, explanations illuminate Stein's craft and the art of writing. Total of 414pp. 4%x 6%. 23144.5 Pa. $5.95 ADVENTURES AT SEA IN THE GREAT AGE OF SAIL: Five Firsthand Narratives, edited by Elliot Snow. Rare true accounts of exploration, whaling, shipwreck, fierce natives, trade, shipboard life, more. 33 illustrations. Introduction. 353pp. 5%x 8lj!. 25177.2 Pa. $7.95 THE HERBAL OR GENERAL HISTORY OF PLANTS, John Gerard. Classic descriptions of about 2,850 plants-with over 2,700 illustrations-includes Latin and English names, physical descriptions, varieties, time and place of growth, more. 2,706 illustrations. xlv + 1,678pp. 8lj!x 12'4. 23147.X Cloth. $75.00 DOROTHY AND THE WIZARD IN OZ, L. Frank Baum. Dorothy and the Wizard visit the center of the Earth, where people are vegetables, glass houses grow and Oz characters reappear. Classic sequel to Wizard of Oz. 256pp. 5%x 8. 24714.7 Pa. $4.95 SONGS OF EXPERIENCE: Facsimile Reproduction with 26 Plates in Full Color, William Blake. This facsimile of Blake's original "Illuminated Book" reproduces 26 full.color plates from a rare 1826edition. Includes "The Tyger," "London," "Holy Thursday," and other immortal poems. 26 color plates. Printed text of poems. 48pp. 5'4x 7. 24636.1 Pa. $3.50 SONGS OF INNOCENCE, William Blake. The first and most popular of Blake's famous "Illuminated Books," in a facsimile edition reproducing all 31 brightly colored plates. Additional printed text of each poem. 64pp. 5'4x 7. 22764.2 Pa. $3.50 PRECIOUS STONES, Max Bauer. Classic, thorough study of diamonds, rubies, emeralds, garnets, etc.: physical character, occurrence, properties, use, similar topics. 20 plates, 8 in color. 94 figures. 659pp. 6~ x 9'4. 21910.0,21911.9 Pa., Two.vol. set $15.90 ENCYCLOPEDIA OF VICTORIAN NEEDLEWORK, S. F. A. Caulfeild and Blanche Saward. Full, precise descriptions of stitches, techniques for dozens of needlecrafts-most exhaustive reference of its kind. Over 800 figures. Total of 679pp. 8~ x 11.Two volumes. Vol. I 22800.2 Pa. $11.95 Vol. 2 22801.0 Pa. $11.95 THE MARVELOUS LAND OF OZ, L. Frank Baum. Second Oz book, the Scarecrow and Tin Woodman are back with hero named Tip, Oz magic. 136 illustrations. 287pp. 5%x 8lj!. 20692.0 Pa. $5.95 WILD FOWL DECOYS, Joel Barber. Basic book on the subject, by foremost authority and collector. Reveals history of decoy making and rigging, place in American culture, different kinds of decoys, how to make them, and how to use them. 140plates. 156pp. 7Y. x 10'4. 20011.6 Pa. $8.95 HISTORY OF LACE, Mrs. Bury Palliser. Definitive, profusely illustrated chron. icle of lace from earliest times to late 19th century. Laces of Italy, Greece, England, France, Belgium, etc. Landmark of needlework scholarship. 266 illustrations. 672pp. 6~ x 9'4. 24742.2 Pa. $14.95
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ILLUSTRATED GUIDE TO SHAKER FURNITURE, Robert Meader. All furniture and appurtenances, with much on unknown local styles. 235 photos. 146pp. 9 x 12. 22819-3Pa. $7.95 WHALE SHIPS AND WHALING: A Pictorial Survey, George Francis Dow. Over 200 vintage engravings, drawings, photographs of barks, brigs, cutters, other vessels.Also harpoons, lances, whaling guns, many other artifacts. Comprehensive text by foremost authority. 207 black-and-white illustrations. 288pp. 6 x 9. 24808-9Pa. $8.95 THE BERTRAMS, Anthony Trollope. Powerful portrayal of blind self-will and thwarted ambition includes one of Trollope's most heartrending love stories. 497pp. 5%x 8!-l. 25119-5Pa. $8.95 ADVENTURES WITH A HAND LENS, Richard Headstrom. Clearly written guide to observing and studying flowers and grasses, fish scales, moth and insect wings, egg cases, buds, feathers, seeds, leaf scars, moss, molds, ferns, common crystals, etc.-all with an ordinary, inexpensive magnifying glass. 209 exact line drawings aid in your discoveries. 220pp. 5%x 8!-l. 23330-8Pa. $4.50 RODIN ON ART AND ARTISTS, Auguste Rodin. Great sculptor's candid, wideranging comments on meaning of art; great artists; relation of sculpture to poetry, painting, music; philosophy of life, more. 76 superb black-and-white illustrations of Rodin's sculpture, drawings and prints. 119pp. 8%x 11\4. 24487-3Pa. $6.95 FIFTY CLASSIC FRENCH FILMS, 1912-1982: A Pictorial Record, Anthony Slide. Memorable stills from Grand Illusion, Beauty and the Beast, Hiroshima, Mon Amour, many more. Credits, plot synopses, reviews, etc. 160pp. 8\4x 11. 25256-6Pa. $11.95 THE PRINCIPLES OF PSYCHOLOGY, William James. Famous long course complete, unabridged. Stream of thought, time perception, memory, experimental methods; great work decades ahead of its time. 94 figures. 1,39Ipp. 5%x 8!-l. 20381-6,20382-4Pa., Two-vol. set $19.90 BODIES IN A BOOKSHOP, R. T. Campbell. Challenging mystery of blackmail and murder with ingenious plot and superbly drawn characters. In the best tradition of British suspense fiction. 192pp. 5%x 8!-l. 24720-1Pa. $3.95 CALLAS: PORTRAIT OF A PRIMA DONNA, George Jellinek. Renowned commentator on the musical scene chronicles incredible career and life of the most controversial, fascinating, influential operatic personality of our time. 64 blackand-white photographs. 416pp. 5%x 8\4. 25047-4Pa. $7.95 GEOMETRY, RELATIVITY AND THE FOURTH DIMENSION, Rudolph Rucker. Exposition of fourth dimension, concepts of relativity as Flatland characters continue adventures. Popular, easily followed yet accurate, profound. 141 illustrations. 133pp. 5%x 8!-l. 23400-2Pa. $3.50 HOUSEHOLD STORIES BY THE BROTHERS GRIMM, with pictures by Walter Crane. 53 classic stories-Rumpelstiltskin, Rapunzel, Hansel and Gretel, the Fisherman and his Wife, Snow White, Tom Thumb, Sleeping Beauty, Cinderella, and so much more-lavishly illustrated with original 19th century drawings. 114illustrations. x + 269pp. 5%x 8!-l. 21080-4Pa. $4.50
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~UNDIALS, AI~ert Waugh. Far and away the best, most thorough coverage of Ideas, mathematIcs concerned, types, construction, adjusting anywhere. Over 100 illustrations. 230pp. 5%x 8~. 22947-5 Pa. $4.50 PICTURE HISTORY OF THE NORMAN DIE: With 190Illustrations, Frank O. Braynard. Full story of legendary French ocean liner: Art Deco interiors, design innovations, furnishings, celebrities, maiden voyage, tragic fire, much more. Extensive text. 144pp. 8%x lilt 25257-4 Pa. $9.95 THE FIRST AMERICAN COOKBOOK: A Facsimile of "American Cookery," 1796, Amelia Simmons. Facsimile of the first American-written cookbook published in the United States contains authentic recipes for colonial favoritespumpkin pudding, winter squash pudding, spruce beer, Indian slapjacks, and more. Introductory Essay and Glossary of colonial cooking terms. 80pp. 5%x 8~. 24710-4 Pa. $3.50 101 PUZZLES IN THOUGHT AND LOGIC, C. R. Wylie, Jr. Solve murders and robberies, find out which fishermen are liars, how a blind man could possibly identify a color-purely by your own reasoning! 107pp. 5%x 8~. 20367-0 Pa. $2.50 THE BOOK OF WORLD.FAMOUS MUSIC-CLASSICAL, POPULAR AND FOLK, James J. Fuld. Revised and enlarged republication of landmark work in musico-bibliography. Full information about nearly 1,000songs and compositions including first lines of music and lyrics. New supplement. Index. 800pp. 5%x 8\( 24857-7Pa. $14.95 ANTHROPOLOGY AND MODERN LIFE, Franz Boas. Great anthropologist's classic treatise on race and culture. Introduction by Ruth Bunzel. Only inexpensive paperback edition. 255pp. 5%x 8~. 25245-0 Pa. $5.95 THE TALE OF PETER RABBIT, Beatrix Potter. The inimitable Peter's terrifying adventure in Mr. McGregor's garden, with all 27 wonderful, full-color Potter illustrations. 55pp. 4\4x 5~. (Available in U.S. only) 22827-4 Pa. $1.75 THREE PROPHETIC SCIENCE FICTION NOVELS, H. G. Wells. When the Sleeper Wakes, A Story of the Days to Come and The Time Machine (full version). 335pp. 5%x 8~. (Available in U.S. only) 20605-X Pa. $5.95 APICIUS COOKERY AND DINING IN IMPERIAL ROME, edited and translated by Joseph Dommers Vehling. Oldest known cookbook in existence offers readers a clear picture of what foods Romans ate, how they prepared them, etc. 49 illustrations. 301pp. 6li x 9\4. 23563-7 Pa. $6.50 SHAKESPEARE LEXICON AND QUOTATION DICTIONARY, Alexander Schmidt. Full definitions, locations, shades of meaning of every word in plays and poems. More than 50,000exact quotations. 1,485pp. 6~ x 9%. 22726-X, 22727-8Pa., Two-vol. set $27.90 THE WORLD'S GREAT SPEECHES, edited by Lewis Copeland and Lawrence W. Lamm. Vast collection of 278 speeches from Greeks to 1970. Powerful and effective models; unique look at history. 842pp. 5%x 8~. 20468-5 Pa. $11.95
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THE BLUE FAIRY BOOK, Andrew Lang. The first, most famous collection, with many familiar tales: Little Red Riding Hood, Aladdin and th.e Wonderful Lamp, Puss in Boots, Sleeping Beauty, Hansel and Gretel, Rumpelsultskm; 37 mall. 138 illustrations. 390pp. 5%x 8!1. 21437-0 Pa. $5.95 THE STORY OF THE CHAMPIONS OF THE ROUND TABLE, Howard Pyle. Sir Launcelot, Sir Tristram and Sir Percival in spirited adventures of love and triumph retold in Pyle's inimitable style. 50 drawings, 31 full-page. xviii + 329pp. 6!1x 9'4. 21883-X Pa. $6.95 AUDUBON AND HIS JOURNALS, Maria I\udubon. Unmatched two-volume portrait of the great artist, naturalist and author contains his journals, an excellent biography by his granddaughter, expert annotations by the noted ornithologist, Dr. Elliott Coues, and 37 superb illustrations. Total of 1,200pp. 5%x 8. Vol. 125143-8 Pa. $8.95 Vol. II 25144-6 Pa. $8.95 GREAT DINOSAUR HUNTERS AND THEIR DISCOVERIES, Edwin H. Colbert. Fascinating, lavishly illustrated chronicle of dinosaur research, 1820's to 1960. Achievements of Cope, Marsh, Brown, Buckland, Mantell, Huxley, many others. 384pp. 5'4 x 8'4. 24701-5 Pa. $6.95 THE T ASTEMAKERS, Russell Lynes. Informal, illustrated social history of American taste 1850's-1950's. First popularized categories Highbrow, Lowbrow, Middlebrow. 129 illustrations. New (1979) afterword. 384pp. 6 x 9. 23993-4 Pa. $6.95 DOUBLE CROSS PURPOSES, Ronald A. Knox. A treasure hunt in the Scottish Highlands, an old map, unidentified corpse, surprise discoveries keep reader guessing in this cleverly intricate tale of financial skullduggery. 2 black-and-white maps. 320pp. 5%x 8!1.(Available in U.S. only) 25032-6 Pa. $5.95 AUTHENTIC VICTORIAN DECORATION AND ORNAMENTATION IN FULL COLOR: 46 Plates from "Studies in Design," Christopher Dresser. Superb full-color lithographs reproduced from rare original portfolio of a major Victorian designer. 48pp. 9~.x 12'4. 25083-0 Pa. $7.95 PRIMITIVE ART, Franz Boas. Remains the best text ever prepared on subject, thoroughly discussing Indian, African, Asian, Australian, and, especially, Northern American primitive art. Over 950 illustrations show ceramics, masks, totem poles, weapons, textiles, paintings, much more. 376pp. 5%x 8. 20025-6 Pa. $6.95 SIDELIGHTS ON RELATIVITY, Albert Einstein. Unabridged republication of two lectures delivered by the great physicist in 1920-21. Ether and Relativity and Geometry and Experience. Elega ••••• ideas in non-mathematical form, accessible to intelligent layman. vi + 56pp. 5%x 8!1. 24511-X Pa. $2.95 THE WIT AND HUMOR OF OSCAR WILDE, edited by Alvin Redman. More than 1,000ripostes, paradoxes, wisecracks: Work is the curse of the drinking classes, I can resist everything except temptation, etc. 258pp. 5%x 8!1. 20602-5 Pa. $4.50 ADVENTURES WITH A MICROSCOPE, Richard Headstrom. 59 adventures with clothing fibers, protozoa, ferns and lichens, roots and leaves, much more. 142 illustrations. 232pp. 5%x 8!1. 23471-1 Pa. $3.95
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PLANTS OF THE BIBLE, Harold N. Moldenke and Alma L. Moldenke. Standard reference to all 230 plants mentioned in Scriptures. Latin name, biblical reference, uses, modern identity, much more. Unsurpassed encyclopedic resource for scholars, botanists, nature lovers, students of Bible. Bibliography. Indexes. 123black-andwhite illustrations. 384pp. 6 x 9. 25069-5 Pa. $8.95 FAMOUS AMERICAN WOMEN: A Biographical Dictionary from Colonial Times to the Present, Robert McHenry, ed. From Pocahontas to Rosa Parks, 1,035 distinguished American women documented in separate biographical entries. Accurate, up-to-date data, ••umerous categories, spans 400 years. Indices. 493pp. 6!1x 9\4. 24523-3 Pa. $9.95 THE FABULOUS INTERIORS OF THE GREAT OCEAN LINERS IN HISTORIC PHOTOGRAPHS, William H. Miller, Jr. Some 200 superb photographs . capture exquisite interiors of world's great "floating palaces"-1890's to 1980's: Titanic, Ile de France, Queen Elizabeth, United States, Europa, more. Approx. 200 black-and-white photographs. Captions. Text. Introduction. 160pp. 8li!x 11K 24756-2 Pa. $9.95 THE GREAT LUXURY LINERS, 1927-1954:A Photographic Record, William H. Miller, Jr. Nostalgic tribute to heyday of ocean liners. 186 photos of lie de France, Normandie, Leviathan, Queen Elizabeth, United States, many others. Interior and exterior views. Introduction. Captions. 160pp. 9 x 12. 24056-8 Pa. $9.95 A NATURAL HISTORY OF THE DUCKS, John Charles Phillips. Great landmark of ornithology offers complete detailed coverage of nearly 200species and subspecies of ducks: gadwall, sheldrake, merganser, pintail, many more. 74 fullcolor plates, 102black-and.white. Bibliography. Total of 1,920pp. 811x 11\4. 25141-1,25142-X Cloth. Two-vol. set $100.00 THE SEAWEED HANDBOOK: An Illustrated Guide to Seaweeds from North Carolina to Canada, Thomas F. Lee. Concise reference covers 78 species. Scientific and common names, habitat, distribution, more. Finding keys for easy identification. 224pp. 511x 8!1. 25215-9 Pa. $5.95 THE TEN BOOKS OF ARCHITECTURE: The 1755Leoni Edition, Leon Battista Alberti. Rare classic helped introduce the glories of ancient architecture to the Renaissance. 68 black-and-white plates. 336pp. 811x 11\4. 25239-6 Pa. $14.95 MISS MACKENZIE, Anthony Trollope. Minor masterpieces by Victorian master unmasks many truths about life in 19th-century England. First inexpensive edition in years. 392pp. 511x 8!1. 25201-9 Pa. $7.95 THE RIME OF THE ANCIENT MARINER, Gustave Dore, Samuel Taylor Coleridge. Dramatic engravings considered by many to be his greatest work. The terrifying space of the open sea, the storms and whirlpools of an unknown ocean, the ice of Antarctica, more-all rendered in a powerful, chilling manner. Full text. 38 plates. 77pp. 9\4x 12. 22305-1 Pa. $4.95 THE EXPEDITIONS OF ZEBULON MONTGOMERY PIKE, Zebulon Montgomery Pike. Fascinating first-hand accounts (1805-6) of exploration of Missis. sippi River, Indian wars, capture by Spanish dragoons, much more. 1,088pp. 5%x 8!1. 25254-X, 25255-8 Pa. Two-vol. set $23.90
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A CONCISE HISTOR Y OF PHOTOGRAPHY: Third Revised Edition, Helmut Gernsheim. Best one-volume history-camera obscura, photochemistry, daguerreotypes, evolution of cameras, film, more. Also artistic aspects-landscape, portraits, fine art, etc. 281 black-and-white photographs. 26 in color. 176pp. 8lp 11\i. 25128-4 Pa. $12.95 THE DORE BIBLE ILLUSTRATIONS, Gustave Dore. 241 detailed plates from the Bible: the Creation scenes, Adam and Eve, Flood, Babylon, battle sequences, life of Jesus, etc. Each plate is accompanied by the verses from the King James version of the Bible. 241pp. 9 x 12. 23004-X Pa. $8.95 HUGGER-MUGGER IN THE LOUVRE, Elliot Paul. Second Homer Evans mystery-comedy. Theft at the Louvre involves sleuth in hilarious, madcap caper. "A knockout. "-Books. 336pp. 5% x 8~. 25185-3 Pa. $5.95 FLATLAND, E. A. Abbott. Intriguing and enormously popular science-fiction classic explores the complexities of trying to survive as a two-dimensional being in a three-dimensional world. Amusingly illustrated by the author. 16 illustrations. 103pp. 5% x 8~. 20001-9 Pa. $2.25 THE HISTORY OF THE LEWIS AND CLARK EXPEDITION"Meriwether Lewis and William Clark, edited by Elliott Coues. Classic edition of Lewis and Clark's day-by-day journals that later became the basis for U.S. claims to Oregon and the West. Accurate and invaluable geographical, botanical, biological, meteorological and anthropological material. Total of 1,508pp. 5% x 8~. 21268-8,21269-6, 21270-X Pa. Three-vol. set $25.50 LANGUAGE, TRUTH AND LOGIC, Alfred J. Ayer. Famous, clear introduction to Vienna, Cambridge schools of Logical Positivism. Role of philosophy, elimination of metaphysics, nature of analysis, etc. 160pp. 5% x 8~. (Available in U.S. and Canada only) 20010-8 Pa. $2.95 MATHEMATICS FOR THE NON MATHEMATICIAN, Morris Kline. Detailed, college-level treatment of mathematics in cultural and historical context, w.ith numerous exercises. For liberal arts students. Preface. Recommended Readmg Lists. Tables. Index. Numerous black-and-white figures. xvi + 641pp. 5% x 8~. 24823-2 Pa. $11.95 28 SCIENCE
FICTION
STORIES,
H. G. Wells. Novels, Slar Begotten and Men of the Ants," "A Story ofthe Stone Age," "In the Abyss," etc. 915pp. 5%x 8~. (Available in U.S. only) 20265-8 Cloth. $10.95
Like Gods, plus 26 short stories: "Empire "The Stolen Bacillus,"
HANDBOOK OF PICTORIAL SYMBOLS, Rudolph Modley. 3,250 signs and symbols, many systems in full; official or heavy commercial use. Arranged by subject. Most in Pictorial Archive series. 143pp. 8~ x II. 23357-X Pa. $05.95 INCIDENTS OF TRAVEL IN YUCATAN, John L. Stephens. Classic (1843) exploration of jungles of Yucatan, looking for evidences of Maya civilization. Travel adventures, Mexican and Indian culture, etc. Total of 669pp. 5% x 8~. 20926-1, 20927-X Pa., Two-vol. set $9.90
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DEGAS: An Intimate Portrait, Ambroise Vollard. Charming, anecdotal memoir by famous art dealer of one of the greatest 19th-century French painters. 14black-andwhite illustrations. Introduction by Harold L. Van Doren. 96pp. 5%x 8l-!. 25131-4 Pa. $3.95 PERSONAL NARRATIVE OF A PILGRIMAGE TO ALMANDINAH AND MECCAH, Richard Burton. Great travel classic by remarkably colorful personality. Burton, disguised as a Moroccan, visited sacred shrines of Islam, narrowly escaping death. 47 illustrations. 959pp. 5%x 8l-!. 21217-3,21218-1 Pa., Two-vol. set $17.90 PHRASE AND WORD ORIGINS, A. H. Holt. Entertaining, reliable, modern study of more than 1,200 colorful words, phrases, origins and histories. Much unexpected information. 254pp. 5%x 8l-!. 20758-7 Pa. $5.95 THE RED THUMB MARK, R. Austin Freeman. In this first Dr. Thorndyke case, the great scientific detective draws fascinating conclusions from the nature of a single fingerprint. Exciting story, authentic science. 320pp. 5%x 8l-!.(Available in U.S. only) 25210-8 Pa. $5.95 AN EGYPTIAN HIEROGLYPHIC DICTIONARY, E. A. Wallis Budge. Monumental work containing about 25,000 words or terms that occur in texts ranging from 3000 B.C. to 600 A.D. Each entry consists of a transliteration of the word, the word in hieroglyphs, and the meaning in English. 1,314pp. 6%x 10. 23615-3,23616-1 Pa., Two-vol. set $27.90 THE COMPLEAT STRATEGYST: Being a Primer on the Theory of Games of Strategy, J. D. Williams. Highly entertaining classic describes, with many illustrated examples, how to select best strategies in conflict situations. Prefaces. Appendices. xvi + 268pp. 5%x 8l-!. 25101-2 Pa. $5.95 THE ROAD TO OZ, L. Frank Baum. Dorothy meets the Shaggy Man, little Button-Bright and the Rainbow's beautiful daughter in this delightful trip to the magical Land of Oz. 272pp. 5%x 8. 25208-6 Pa. $4.95 POINT AND LINE TO PLANE, Wassily Kandinsky. Seminal exposition of role of point, line, other elements in non-objective painting. Essential to understanding 20th-century art. 127 illustrations. 192pp. 6l-!x 9\4. 23808-3 Pa. $4.50 LADY ANNA, Anthony Trollope. Moving chronicle of Countess Lovel's bitter struggle to win for herself and daughter Anna their rightful rank and fortuneperhaps at cost of sanity itself. 384pp. 5%x 8l-!. 24669-8 Pa. $6.95 EGYPTIAN MAGIC, 1<..A. Wallis Budge. Sums up all that is known about magic in Ancient Egypt: the role of magic in controlling the gods, powerful amulets that warded off evil spirits, scarabs of immortality, use of wax images, formulas and spells, the secret name, much more. 253pp. 5%x 8l-!. 22681-6 Pa. $4.50 THE DANCE OF SIVA, Ananda Coomaraswamy. Preeminent authority unfolds the vast metaphysic of India: the revelation of her art, conception of the universe, social organization, etc. 27 reproductions of art masterpieces. 192pp. 5%x 8l-!. 24817-8 Pa. $5.95
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CHRISTMAS CUSTOMS AND TRADITIONS, Clement A. Miles. Origin, evolution, significance of religious, secular practices. Caroling, gifts, yule logs, much more. Full, scholarly yet fascinating; non-sectarian. 400pp .. ~% x 8l\. 23354-5 Pa. $6.50 THE HUMAN FIGURE IN MOTION, Eadweard Muybridge. More than 4,500 stopped-action photos, in action series, showing undraped men, women, children jumping, lying down, throwing, sitting, wrestling, carrying, etc. 390pp. 7'1.x 10%. 20204-6 Cloth. $19.95 THE MAN WHO WAS THURSDAY, Gilbert Keith Chesterton. Witty, fast-paced novel about a club of anarchists in turn-of-the-century London. Brilliant social, religious, philosophical speculations. 128pp. 5Jiix 8l\. 25121-7 Pa. $3.95 A CEZANNE SKETCHBOOK: Figures, Portraits, Landscapes and Still Lifes, Paul Cezanne. Great artist experiments with tonal effects, light, mass, other qualities in over 100 drawings. A revealing view of developing master painter, precursor of Cubism. 102 black-and-white illustrations. 144pp. 8%x 6Jii. 24790-2 Pa. $5.95 AN ENCYCLOPEDIA OF BATTLES: Accounts of Over 1,560 Battles from 1479 D.C. to the Present, David Eggenberger. Presents essential details of every major battle in recorded history, from the first battle of Megiddo in 1479 ilL to Grenada in 1984. List of Battle Maps. New Appendix covering the years 1967-1984. Index. 99 illustrations. 544pp. 6l\x 9\4. 24913-1 Pa. $14.95 AN ETYMOLOGICAL DICTIONARY OF MODERN ENGLISH, Ernest Weekley. Richest, fullest work, by foremost British lexicographer. Detailed word histories. Inexhaustible. Total of 856pp. 6l\ x 9\4. 21873-2,21874-0 Pa., Two-vol. set $17.00 WEBSTER'S AMERICAN MILITARY BIOGRAPHIES, edited by Robert McHenry. Over 1,000 figures who shaped 3 centuries of American military history. Detailed biographies of Nathan Hale, Douglas MacArthur, Mary Hallaren, others. Chronologies of engagements, more. Introduction. Addenda. 1,033 entries in alphabetical order. xi + 548pp. 6l\ x 9\4. (Available in U.S. only) 24758-9 Pa. $11.95 LIFE IN ANCIENT EG YPT, Adolf Erman. Detailed older account, with much not in more recent books: domestic life, religion, magic, medicine, commerce, and whatever else needed for complete picture. Many illustrations. 597pp. 5Jiix 8l\. 22632-8 Pa. $8.95 HISTORIC COSTUME IN PICTURES, Braun & Schneider. Over 1,450costumed figures shown, covering a wide variety of peoples: kings, emperors, nobles, priests, servants, soldiers, scholars, townsfolk, peasants, merchants, courtiers, cavaliers, and more. 256pp. 8Jiix 11\4. 231.~0-X Pa. $7.95 THE NOTEBOOKS OF LEONARDO DA VINCI, edited by J. P. Richter. Extracts from manuscripts reveal great genius; on painting, sculpture, anatomy, sciences, geography, etc. Both Italian and English. 186 ms. pages reproduced, plus 500 additional drawings, including studies for Last Supper, Sforza monument, etc. 860pp. 7Ysx 10%. (Available in U.S. only) 22572-0, 22573-9 Pa., Two-vol. set $25.90
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THE ART NOUVEAU STYLE BOOK OF ALPHONSE MUCHA: All 72 Plates from "Documents Decoratifs" in Original Color, Alphonse Mucha. Rare copyright-free design portfolio by high priest of Art Nouveau. Jewelry, wallpaper, stained glass, furniture, figure studies, plant and animal motifs, etc. Only complete one-volume edition. 80pp. 9%x 12\4. 24044-4 Pa. $8.95 ANIMALS: 1,419 COPYRIGHT-FREE ILLUSTRATIONS OF MAMMALS, BIRDS, FISH, INSECTS, ETC., edited by Jim Harter. Clear wood engravings present, in extremely lifelike poses, over 1,000 species of animals. One of the most extensive pictorial sourcebooks of its kind. Captions. Index. 284pp. 9 x 12. 23766-4 Pa. $9.95 OBELISTS FL Y HIGH, C. Daly King. Masterpiece of American detective fiction, long out of print, involves murder on a 1935 transcontinental flight-"a very thrilling story"-NY Times. Unabridged and unaltered republication of the edition published by William Collins Sons & Co. Ltd., London, 1935. 288pp. 5%x 8~. (Available in U.S. only) 25036-9 Pa. $4.95 VICTORIAN AND EDWARDIAN FASHION: A Photographic Survey, Alison Gernsheim. First fashion history completely illustrated by contemporary photographs. Full text plus 235 photos, 1840-1914, in which many celebrities appear. 240pp. 6l~x 9\4. 24205.6 Pa. $6.00 THE ART OF THE FRENCH ILLUSTRATED BOOK, 1700-1914, Gordon N. Ray. Over 630 superb book illustrations by Fragonard, Delacroix, Daumier, Dore, Grandville, Manet, Mucha, Steinlen, Toulouse-Lautrec and many others. Preface. Introduction. 633 halftones. Indices of artists, authors & tiLles, binders and provenances. Appendices. Bibliography. 608pp. 8%x II \4. 25086-5 Pa. $24.95 THE WONDERFUL WIZARD OF OZ, L. Frank Baum. Facsimile in full color of America's finest children's classic. 143 illustrations by W. W. Denslow. 267pp. 5%x 8~. 20691-2 Pa. $5.95 FRONTIERS OF MODERN PHYSICS: New Perspectives on Cosmology, Relativity, Black Holes and Extraterrestrial Intelligence, Tony Rothman, et al. For the intelligent layman. Subjects include: cosmological models of the universe; black holes; the neutrino; the search for extraterrestrial intelligence. Introduction. 46 black-and-white illustrations. 192pp. 5%x 8~. 24587-X Pa. $6.95 THE FRIENDLY STARS, Martha Evans Martin & Donald Howard Menzel. Classic text marshalls the stars together in an engaging, non-technical survey, presenting them as sources of beauty in night sky. 23 illustrations. Foreword. 2 star charts. Index. 147pp. 5%x 8~. 21099-5 Pa. $3.50 FADS AND FALLACIES IN THE NAME OF SCIENCE, Martin Gardner. Fair, willy appraisal of cranks, quacks, and quackeries of science and pseudoscience: hollow earth, Velikovsky, orgone energy, Dianetics, flying saucers, Bridey Murphy, food and medical fads, etc. Revised, expanded In the Name of Science. "A very able and even-tempered presentation."The New Yorker. 363pp. 5%x 8. 20394.8 Pa. $6.50 ANCIENT EG YPT: ITS CULTURE AND HISTOR Y, J. E Manchip White. From pre-dynastics through Ptolemies: society, history, political structure, religion, daily life, literature, cultural heritage. 48 plates. 217pp. 5%x 8~. 22548-8 Pa. $4.95
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SIR HARRY HOTSPUR OF HUMBLETHWAITE, Anthony Trollope. Incisive, unconventional psychological study of a conflict between a wealthy baronet, his idealistic daughter, and their scapegrace cousin. The 1870 novel in its first inexpensive edition in years. 250pp. 5%x 8l>. 24953-0 Pa. $5.95 LASERS AND HOLOGRAPHY, Winston E. Kock. Sound introduction to burgeoning field, expanded (1981) for second edition. Wave patterns, coherence, lasers, diffraction, zone plates, properties of holograms, recent advances. 84 illustrations. 160pp. 5* x 8\4.(Except in United Kingdom) 24041-X Pa. $3.50 INTRODUCTION TO ARTIFICIAL INTELLIGENCE: SECOND, ENLARGED EDITION, Philip C. Jackson, Jr. Comprehensive survey of artificial intelligence-the study of how machines (computers) can be made to act intelligently. Includes introductory and advanced material. Extensive notes updating the main text. 132 black-and-white illustrations. 512pp. 5%x 8l>. 24864-X Pa. $8.95 HISTORY OF INDIAN AND INDONESIAN ART, Ananda K. Coomaraswamy. Over 400 illustrations illuminate classic study of Indian art from earliest Hdrappa finds to early 20th century. Provides philosophical, religious and social insights. 304pp. 6%x 9%. 25005-9 Pa. $8.95 THE GOLEM, Gustav Meyrink. Most famous supernatural novel in modern European literature, set in Ghetto of Old Prague around 1890.Compelling story of mystical experiences, strange transformations, profound terror. 13black-and-white illustrations. 224pp. 5%x 8l>.(Available in U.S. only) 25025-3 Pa. $5.95 ARMADALE, Wilkie Collins. Third great mystery novel by the author of The Woman in White and The Moonstone. Original magazine version with 40 illustrations. 597pp. 5%x 8l>. 23429-0 Pa. $9.95 PICTORIAL ENCYCLOPEDIA OF HISTORIC ARCHITECTURAL PLANS, DETAILS AND ELEMENTS: With 1,880 Line Drawings of Arches, Domes, Doorways, Facades, Gables, Windows, etc., John Theodore Haneman. Sourcebook of inspiration for architects, designers, others. Bibliography. Captions. 141pp. 9 x 12. 24605-1 Pa. $6.95 BENCHLEY LOST AND FOUND, Robert Benchley. Finest humor from early 30's, about pet peeves, child psychologists, post office and others. Mostly unavailable elsewhere. 73 illustrations by Peter Arno and others. 183pp. 5%x 8l>. 22410-4 Pa. $3.95 ERTE GRAPHICS, Erte. Collection of striking color graphics: Seasons, Alphabet, Aces and Precious Stones. 50 plates, including 4 on covers. 48pp. 9%x 12\4. 23580-7 Pa. $6.95
Numerals,
THE JOURNAL OF HENRY D. THOREAU, edited by Bradford Torrey, F. H. Allen. Complete reprinting of 14 volumes, 1837-61, over two million words; the sourcebooks for Walden, etc. Definitive. All original sketches, plus 75 photographs. 1,804pp. 8l>x 12\4. 20312-3,20313-1 Cloth., Two-vol. set $80.00 CASTLES: THEIR CONSTRUCTION AND HISTORY, Sidney Toy. Traces castle development from ancient roots. Nearly 200 photographs and drawings illustrate moats, keeps, baileys, many other features. Caernarvon, Dover Castles, Hadrian's Wall, Tower of London, dozens more. 256pp. 5* x 8\4. 24898-4 Pa. $5.95
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AMERICAN CLIPPER SHIPS: 1833-1858, Octavius T. Howe & Frederick C. Matthews. Fully-illustrated, encyclopedic review of 352 clipper ships from the period of America's greatest maritime supremacy. Introduction. 109 halftones. 5 black-and-white line illustrations. Index. Total of 928pp. 5%x 8\>. 25115-2,25116-0 Pa., Two-vol. set $17.90 TOW ARDS A NEW ARCHITECTURE, Le Corbusier. Pioneering manifesto by great architect, near legendary founder of "International School." Technical and aesthetic theories, views on industry, economics, relation of form to function, "mass-production spirit," much more. Profusely illustrated. Unabridged translation of 13th French edition. Introduction by Frederick Etchells. 320pp. 6ii x 9\4. (Available in U.S. only) 25023-7 Pa. $8.95 THE BOOK OF KELLS, edited by Blanche Cirker. Inexpensive collection of 32 full-color, full-page plates from the greatest illuminated manuscript of the Middle Ages, painstakingly reproduced from rare facsimile edition. Publisher's Note. Captions. 32pp. 9%x 12\4. 24345-1 Pa. $4.95 BEST SCIENCE
FICTION
STORIES
OF H. G. WELLS, H. G. Wells. Full novel Island," etc. 303pp. 5%x 8\>.(Available in U.S. only) 21531-8 Pa. $4.95
The Invisible Man, plus 17 short stories: "The Crystal Egg," "Aepyornis "The Strange Orchid,"
AMERICAN SAILING SHIPS: Their Plans and History, Charles G. Davis. Photos, construction details of schooners, frigates, clippers, other sailcraft of 18th to early 20th centuries-plus entertaining discourse on design, rigging, nautical lore, much more. 137 black-and-white illustrations. 240pp. 6ii x 9\4. 246.~8-2 Pa. $5.95 ENTERTAINING MATHEMATICAL PUZZLES, Martin Gardner. Selection of author's favorite conundrums involving arithmetic, money, speed, etc., with lively commentary. Complete solutions. 112pp. 5%x 8\>. 25211-6 Pa. $2.95 THE WILL TO BELIEVE, HUMAN IMMORTALITY, William James. Two books bound together. Effect of irrational on logical, and arguments for human immortality. 402pp. 5%x 8\>. 20291-7 Pa. $7.50 THE HAUNTED MONASTERY and THE CHINESE MAZE MURDERS, Robert Van Gulik. 2 full novels by Van Gulik continue adventures of Judge Dee and his companions. An evil Taoist monastery, seemingly supernatural events; overgrown topiary maze that hides strange crimes. Set in 7th-centurv China. 27 illustrations. 328pp. 5%x 8\>. 23502-5 Pa. $5.95 CELEBRATED CASES OF JUDGE DEE (DEE GOONG AN), translated by Robert Van Gulik. Authentic 18th-century Chinese detective novel; Dee and associates solve three interlocked cases. Led to Van Gulik's own stories with same characters. Extensive introduction. 9 illustrations. 237pp. 5%x 8\>. 23337-5 Pa. $4.95
Prices subject to change without notice. Available at your book dealer or write for free catalog to Dept. GI, Dover Publications, Inc., 31 East 2nd St., Mineola, N.Y. 11501. Dover publishes more than 175 books each year on science, elementary and advanced mathematics, biology, music, art, literary history, social sciences and other areas.
