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A note on the Ewald-Oseen extinction theorem
Volume
2, number
OPTICS COMMUNICATIONS
4
A NOTE
ON
THE
EWALD-OSEEN
September
EXTINCTION
1970
THEOREM*
J. J. SEIN **
Received
16 .July 1970
The Ewald-Oseen extinction theorem is derived in a new and general form that is applicable to any medium. The usual form of the extinction theorem appears as a special case of the general formulation, It is shown that the extinction theorem is a consequence of the Helmholtz-Kirchhoff integral theorem.
I.
INTRODUCTION
The rigorous classical theory of dispersion is customarily formulated for an isotropic, linear, non-magnetic medium, beginning with the integrodifferential equation [1,2] E’(r,t)
= E(i)( r,f) +!
In (1.1).
[7 x [ I7x !??+-?@j .ji’
r1
c) is the incident electric
dI”
. (1.1)
field,
E’( r,t) is the effective field, N is the number of molecules per unit volume, and cy is the mean polarizability of a molecule. The relation (1.1) states that the effective field at any one molecule, which is regarded as a point-dipole, consists of the incident field and of the field that is radiated by all the other dipoles constituting the medium. From eq. (1.1) the rigorous theory of dispersion derives two results: the Lorentz-Lorenz formula and the Ewald-Oseen extinction theorem. The latter is usually interpreted as expressing the extinction of the incident field within the medium by the field that originates from the dipoles on the surface of the medium [l-4]. Recently it has been shown that this customary interpretation of the extinction theorem is invalid [5,6]. It has also recently been observed [4] that in the limit No - 0 (absence of matter), the extinction theorem reduces to a statement of the * Work supported by a NSF Science Faculty Fellowship. ** This work is based in part on a Ph. D. thesis submitted to New York University (October. 1969). Present address: Physics Department, St. Peter’s College. Jersey City, N.,J.. 07306, USA.
170
Helmholtz-K irchhoff integral theorem for the incident field. In this note we demonstrate that the extinction theorem is a statement that is independent of the properties of the medium, and is rather a consequence of Maxwell’s equations in vacua. In section 2 we derive a new and general form of the extinction theorem that is applicable to any medium. The usual form [2], which applies to isotropic, linear, non-magnetic media, follows readily as a special case from our general formulation. In section 3 we show that the extinction theorem is a consequence of the Helmholtz-Kirchhoff integral theorem.
2. NEW DERIVATION AND GENERALIZATION OF THE EXTINCTION THEOREM We consider an arbitrary closed domain s2 of free space, bounded by some regular surface C. We allow for the presence of non-singular, but otherwise arbitrary, charges and currents within and outside of Q. We wish to arrive at a statement for time-harmonic electromagnetic fields, if any, within 52. The electric field of interest may then be expressed in the form E(r,t)
=E(r,w)
emiwt ,
(2.1)
with a similar notation for the magnetic field B(r,t). It is well known [7] that Maxwell’s equations in vacua permit a solution for the fields B!?(T,CLI)and B( f,o) through E(r,o)
=ki = vx
Z(r.w)
+ 17( v.Z(r,w)
v r: Z(r,w)
- Fj(r,w)
;
(2.2)
Volume 2, number 4
= -iko Px Z(r,w)
B(r,w)
with the Hertz vector V2 Z(t,o)
OPTICS COMMUNICATIONS
+ki
z( r,o)
Z(r,o)
,
(2.3)
satisfying
= -Fi(r,w,
.
(2.4)
In the preceding relations k. = w/‘c, the i( r, w) is the amplitude of the time-harmonic current density, and we employ the gaussian system of units. The general solution to eq. (2.4), for the point of observation r in any closed domain a, is given by Kirchhoff’s formula [8,9]: Z(r,w)
=f
,s dr’j(
r’,w)G(R)
-S(z;
r,w)
where G(R) = $ eikoR
,
the notation
S(z;r,w)
T
=&JdS’[Z(r’,w) c
aZcr’, w, ant
(2.6)
ia
= -S(z;r;w)
In view of (2.2) and (2.7), E(‘)(r,w)
Z(r,w)
.
(2.7)
we then have
+ V x V xS(z;r,w)
=0 ,
= E(r,w)/k~
.
for such a medium,
(2.9)
eq. (2.8) becomes
E(i)(r,w)
In (2.6), the integration extends over the surface C that bounds the domain 52, r lies in a, and a/an’ is the spatial derivative normal to C, taken as positive out of the volume 51. The first term (the volume integral) on the right side of (2.5) represents the contribution to Z( r,w) from sources inside a; the term -S( z ; r,w) in (2.5) is the contribution to Z(r,w) from sources outside of 52 [8]. Since the field in 52 that originates from sources outside of Sl is what is customarily referred to as the incident field, the term -Sf z;r,w) in (2.5) represents the Hertz vector Z(‘) r ,w) that is associated with the incident field: Z(l)(r,w)
specific form of the extinction theorem for any specific medium, as may be derived from a relation of the type (l.l), follows directly from (2.8) by the elimination of the Hertz vector Z( r’, w) in S( z ; r, w) in favor of some equivalent that is appropriate to the specific medium and the specific problem. To illustrate, let us assume that the medium in 52 supports transverse fields only. Such would be the case, for example, in the medium for which the extinction theorem is usually derived: a homogeneous, isotropic, linear, non-magnetic dielectric. If only transverse fields exist in 52, then it follows from (2.2) that throughout Q
Therefore,
R=lr-I-‘/
and where we have introduced
_ G(R)
, (2.5)
September 1970
(2.8)
for r in W. It should be noted that the statement (2.8) is perfectly general; that is to say, it applies to any non-singular distribution of sources (charges and currents) within and outside of the domain W. Thus (2.8) applies also in the case when C is the boundary of a medium that extends throughout the domain a. In such a case, the relation (2.8) may be properly regarded as the general form of the Ewald-Oseen extinction theorem that is applicable to any medium. Any
+-+ I7 x V xS(E;r,w) =0 . (2.10) ko The relation (2.10) is just the usual form [2] of the Ewald-Oseen extinction theorem. If one so chooses, the electric field vector in S(E; r, w) of (2.10) may be replaced with the polarization vector by introducing the dielectric constant or the refractive index of the medium. It is not difficult to show [5] in a similar fashion that other specific forms of the extinction theorem for other specific types of media, such as non-linear [lo] or spatially dispersive dielectrics, also follow directly from the general form (2.8). A specialization of (2.8) of rather general interest is obtained when the field in the (unspecified) medium is expressible as a superposition of plane-waves. In such a case we may write the Hertz vector Z( r, W) in the medium in the form (2.11) Z(r,w) = $l Zk( r,w) = 2 Z(k,w) eik *r , with a similar notation for the current density i( r,w) in the medium. Note that in (2.11) the Z(k,w) and the k are in general complex; note also that inhomogeneous waves, and transverse and/or longitudinal waves are allowed for by (2.11). From (2.4) it then follows, in view of an expansion of the type (2.11)) that Z(k,w) Substitution
E( f r,W)+*
W
=Lj(k,o) . (2.12) k2 -k; of (2.11) and (2.12) into (2.8) yields
VX V
X C-
1
k k2-k$
!?(jk;r,W)
= 0
(2.13) 171
Volume 2. number 4
OPTICS COMMUNICATIONS
as the then applicable form of the extinction theorem. When the medium is an isotropic, linear dielectric, then all the waves in the medium at any given frequency propagate with wavevectors of identical magnitude. In such a case, noting also that jk = - iW pk, the expression (2.13) reduces to the usual statement [2] of the extinction theorem.
3. RELATIONSHIP OF THE EXTINCTION THEOREM TO THE HELMHOLTZ-KIRCHHOFF INTEGRAL THEOREM The Huygens-Fresnel principle [2,p. 1321, which may be regarded as the basic postulate of the wave theory of light, finds its rigorous formulation in the so-called Helmholtz-Kirchhoff integral theorem [ 2, section 8.31. For the Hertz vector Z(‘)( r,w) in a closed domain D the Helmholtz-Kirchhoff integral theorem reads .I?$)
+ S(z(‘);r,w)
=0 ,
(3.1)
with the notation having been defined in section 2. The Hertz vector 2 (‘)( r, w ) that is due to sources residing in 52 satisfies the complementary theorem ]27 section 8.31 Helmholtz-Kirchhoff s ( Ad’
;r,w)
=0
,
(3.2)
with r in 52. The relations (3.1) and (3.2) are the Helmholtz-Kirchhoff theorems for sources outside and inside fi respectively. The relation (3.2) follows from Green’s theorem [2, section 8.31 with the usual assumption of no sources at infinity. It May also be obtained directly from (2.7) as the limit of that expression for vanishing sources outside of 52 (recall that (2.7) applies for arbitrary sources inside and outside of a). Since the total Hertz vector z( r, wj in Q is
172
September 1970
(i)
the sum of Z(r,w)
(4
and of Z(r,w),
one obtains,
by adding (3.1) and (3.2), the relation (2.7). In section 2 it was shown that the extinction theorem (2.8) follows directly from (2.7) in view of the relation (2.2) between the electric field vector and the Hertz vector. Thus the extinction theorem is a consequence of the relations (3.1) and (3.2), which are statements of the Helmholtz-Kirchhoff integral theorem for the Hertz vectors that are associated with sources residing respectively outside and inside of any closed domain 0.
ACKNOWLEDGMENT The author thanks Professor Emil Wolf for helpful correspondence relating to this work.
REFERENCES [l] L. Rosenfeld, Theory of electrons (North-Holland, Amsterdam, 1951) Ch. 6. [2] M. Born and E. Wolf, Principles of c.ptics, 4th Ed. (Pergamon Press, Oxford and New York, 1970) section 2.4. [3] R. K. Bullough. J. Phvs. A2 (1968) 409. [4] E. Lalor. O$.‘Commun. 1 (i969) ‘50. [5] J. J. Sein, Ph. D. thesis. New York Universitv (1969) unpublished. [6] E. Wolf, private communication. [7] W. K. H. Panofsky and M. Phillips, Classical electricity and magnetism, 2nd Ed. (Addison-Wesley, Reading, 1962) section 14.5. [S] J. A. Stratton, Electromagnetic theorv (McGrawHill, New York and London, 1941) section 8.1. [9] A. N. Tikhonov and A. A. Samarskii, Equations of mathematical physics (Pergamon Press. Oxford and New York, 1963) p.466. [lo] N. Bloembergen and P.S. Pershan, Phys. Rev. 128 (1962) 606.
2, number
OPTICS COMMUNICATIONS
4
A NOTE
ON
THE
EWALD-OSEEN
September
EXTINCTION
1970
THEOREM*
J. J. SEIN **
Received
16 .July 1970
The Ewald-Oseen extinction theorem is derived in a new and general form that is applicable to any medium. The usual form of the extinction theorem appears as a special case of the general formulation, It is shown that the extinction theorem is a consequence of the Helmholtz-Kirchhoff integral theorem.
I.
INTRODUCTION
The rigorous classical theory of dispersion is customarily formulated for an isotropic, linear, non-magnetic medium, beginning with the integrodifferential equation [1,2] E’(r,t)
= E(i)( r,f) +!
In (1.1).
[7 x [ I7x !??+-?@j .ji’
r1
c) is the incident electric
dI”
. (1.1)
field,
E’( r,t) is the effective field, N is the number of molecules per unit volume, and cy is the mean polarizability of a molecule. The relation (1.1) states that the effective field at any one molecule, which is regarded as a point-dipole, consists of the incident field and of the field that is radiated by all the other dipoles constituting the medium. From eq. (1.1) the rigorous theory of dispersion derives two results: the Lorentz-Lorenz formula and the Ewald-Oseen extinction theorem. The latter is usually interpreted as expressing the extinction of the incident field within the medium by the field that originates from the dipoles on the surface of the medium [l-4]. Recently it has been shown that this customary interpretation of the extinction theorem is invalid [5,6]. It has also recently been observed [4] that in the limit No - 0 (absence of matter), the extinction theorem reduces to a statement of the * Work supported by a NSF Science Faculty Fellowship. ** This work is based in part on a Ph. D. thesis submitted to New York University (October. 1969). Present address: Physics Department, St. Peter’s College. Jersey City, N.,J.. 07306, USA.
170
Helmholtz-K irchhoff integral theorem for the incident field. In this note we demonstrate that the extinction theorem is a statement that is independent of the properties of the medium, and is rather a consequence of Maxwell’s equations in vacua. In section 2 we derive a new and general form of the extinction theorem that is applicable to any medium. The usual form [2], which applies to isotropic, linear, non-magnetic media, follows readily as a special case from our general formulation. In section 3 we show that the extinction theorem is a consequence of the Helmholtz-Kirchhoff integral theorem.
2. NEW DERIVATION AND GENERALIZATION OF THE EXTINCTION THEOREM We consider an arbitrary closed domain s2 of free space, bounded by some regular surface C. We allow for the presence of non-singular, but otherwise arbitrary, charges and currents within and outside of Q. We wish to arrive at a statement for time-harmonic electromagnetic fields, if any, within 52. The electric field of interest may then be expressed in the form E(r,t)
=E(r,w)
emiwt ,
(2.1)
with a similar notation for the magnetic field B(r,t). It is well known [7] that Maxwell’s equations in vacua permit a solution for the fields B!?(T,CLI)and B( f,o) through E(r,o)
=ki = vx
Z(r.w)
+ 17( v.Z(r,w)
v r: Z(r,w)
- Fj(r,w)
;
(2.2)
Volume 2, number 4
= -iko Px Z(r,w)
B(r,w)
with the Hertz vector V2 Z(t,o)
OPTICS COMMUNICATIONS
+ki
z( r,o)
Z(r,o)
,
(2.3)
satisfying
= -Fi(r,w,
.
(2.4)
In the preceding relations k. = w/‘c, the i( r, w) is the amplitude of the time-harmonic current density, and we employ the gaussian system of units. The general solution to eq. (2.4), for the point of observation r in any closed domain a, is given by Kirchhoff’s formula [8,9]: Z(r,w)
=f
,s dr’j(
r’,w)G(R)
-S(z;
r,w)
where G(R) = $ eikoR
,
the notation
S(z;r,w)
T
=&JdS’[Z(r’,w) c
aZcr’, w, ant
(2.6)
ia
= -S(z;r;w)
In view of (2.2) and (2.7), E(‘)(r,w)
Z(r,w)
.
(2.7)
we then have
+ V x V xS(z;r,w)
=0 ,
= E(r,w)/k~
.
for such a medium,
(2.9)
eq. (2.8) becomes
E(i)(r,w)
In (2.6), the integration extends over the surface C that bounds the domain 52, r lies in a, and a/an’ is the spatial derivative normal to C, taken as positive out of the volume 51. The first term (the volume integral) on the right side of (2.5) represents the contribution to Z( r,w) from sources inside a; the term -S( z ; r,w) in (2.5) is the contribution to Z(r,w) from sources outside of 52 [8]. Since the field in 52 that originates from sources outside of Sl is what is customarily referred to as the incident field, the term -Sf z;r,w) in (2.5) represents the Hertz vector Z(‘) r ,w) that is associated with the incident field: Z(l)(r,w)
specific form of the extinction theorem for any specific medium, as may be derived from a relation of the type (l.l), follows directly from (2.8) by the elimination of the Hertz vector Z( r’, w) in S( z ; r, w) in favor of some equivalent that is appropriate to the specific medium and the specific problem. To illustrate, let us assume that the medium in 52 supports transverse fields only. Such would be the case, for example, in the medium for which the extinction theorem is usually derived: a homogeneous, isotropic, linear, non-magnetic dielectric. If only transverse fields exist in 52, then it follows from (2.2) that throughout Q
Therefore,
R=lr-I-‘/
and where we have introduced
_ G(R)
, (2.5)
September 1970
(2.8)
for r in W. It should be noted that the statement (2.8) is perfectly general; that is to say, it applies to any non-singular distribution of sources (charges and currents) within and outside of the domain W. Thus (2.8) applies also in the case when C is the boundary of a medium that extends throughout the domain a. In such a case, the relation (2.8) may be properly regarded as the general form of the Ewald-Oseen extinction theorem that is applicable to any medium. Any
+-+ I7 x V xS(E;r,w) =0 . (2.10) ko The relation (2.10) is just the usual form [2] of the Ewald-Oseen extinction theorem. If one so chooses, the electric field vector in S(E; r, w) of (2.10) may be replaced with the polarization vector by introducing the dielectric constant or the refractive index of the medium. It is not difficult to show [5] in a similar fashion that other specific forms of the extinction theorem for other specific types of media, such as non-linear [lo] or spatially dispersive dielectrics, also follow directly from the general form (2.8). A specialization of (2.8) of rather general interest is obtained when the field in the (unspecified) medium is expressible as a superposition of plane-waves. In such a case we may write the Hertz vector Z( r, W) in the medium in the form (2.11) Z(r,w) = $l Zk( r,w) = 2 Z(k,w) eik *r , with a similar notation for the current density i( r,w) in the medium. Note that in (2.11) the Z(k,w) and the k are in general complex; note also that inhomogeneous waves, and transverse and/or longitudinal waves are allowed for by (2.11). From (2.4) it then follows, in view of an expansion of the type (2.11)) that Z(k,w) Substitution
E( f r,W)+*
W
=Lj(k,o) . (2.12) k2 -k; of (2.11) and (2.12) into (2.8) yields
VX V
X C-
1
k k2-k$
!?(jk;r,W)
= 0
(2.13) 171
Volume 2. number 4
OPTICS COMMUNICATIONS
as the then applicable form of the extinction theorem. When the medium is an isotropic, linear dielectric, then all the waves in the medium at any given frequency propagate with wavevectors of identical magnitude. In such a case, noting also that jk = - iW pk, the expression (2.13) reduces to the usual statement [2] of the extinction theorem.
3. RELATIONSHIP OF THE EXTINCTION THEOREM TO THE HELMHOLTZ-KIRCHHOFF INTEGRAL THEOREM The Huygens-Fresnel principle [2,p. 1321, which may be regarded as the basic postulate of the wave theory of light, finds its rigorous formulation in the so-called Helmholtz-Kirchhoff integral theorem [ 2, section 8.31. For the Hertz vector Z(‘)( r,w) in a closed domain D the Helmholtz-Kirchhoff integral theorem reads .I?$)
+ S(z(‘);r,w)
=0 ,
(3.1)
with the notation having been defined in section 2. The Hertz vector 2 (‘)( r, w ) that is due to sources residing in 52 satisfies the complementary theorem ]27 section 8.31 Helmholtz-Kirchhoff s ( Ad’
;r,w)
=0
,
(3.2)
with r in 52. The relations (3.1) and (3.2) are the Helmholtz-Kirchhoff theorems for sources outside and inside fi respectively. The relation (3.2) follows from Green’s theorem [2, section 8.31 with the usual assumption of no sources at infinity. It May also be obtained directly from (2.7) as the limit of that expression for vanishing sources outside of 52 (recall that (2.7) applies for arbitrary sources inside and outside of a). Since the total Hertz vector z( r, wj in Q is
172
September 1970
(i)
the sum of Z(r,w)
(4
and of Z(r,w),
one obtains,
by adding (3.1) and (3.2), the relation (2.7). In section 2 it was shown that the extinction theorem (2.8) follows directly from (2.7) in view of the relation (2.2) between the electric field vector and the Hertz vector. Thus the extinction theorem is a consequence of the relations (3.1) and (3.2), which are statements of the Helmholtz-Kirchhoff integral theorem for the Hertz vectors that are associated with sources residing respectively outside and inside of any closed domain 0.
ACKNOWLEDGMENT The author thanks Professor Emil Wolf for helpful correspondence relating to this work.
REFERENCES [l] L. Rosenfeld, Theory of electrons (North-Holland, Amsterdam, 1951) Ch. 6. [2] M. Born and E. Wolf, Principles of c.ptics, 4th Ed. (Pergamon Press, Oxford and New York, 1970) section 2.4. [3] R. K. Bullough. J. Phvs. A2 (1968) 409. [4] E. Lalor. O$.‘Commun. 1 (i969) ‘50. [5] J. J. Sein, Ph. D. thesis. New York Universitv (1969) unpublished. [6] E. Wolf, private communication. [7] W. K. H. Panofsky and M. Phillips, Classical electricity and magnetism, 2nd Ed. (Addison-Wesley, Reading, 1962) section 14.5. [S] J. A. Stratton, Electromagnetic theorv (McGrawHill, New York and London, 1941) section 8.1. [9] A. N. Tikhonov and A. A. Samarskii, Equations of mathematical physics (Pergamon Press. Oxford and New York, 1963) p.466. [lo] N. Bloembergen and P.S. Pershan, Phys. Rev. 128 (1962) 606.