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A generalized extinction theorem for exciton polarization in spatially dispersive media
Volume 40A, number 4
PHYSICS LETTERS
31 July 1972
A GENERALIZED EXTINCTION THEOREM FOR EXCITON POLARIZATION IN SPATIALLY DISPERSIVE MEDIA* G.S. AGARWAL** Institut für Theoretische Physik, Universitilt Stuttgart, Germany
and D.N. PATTANAYAK and E. WOLF Department of Physics and Astronomy, University of Rochester, Rochester, N. Y. 14627, USA Received 5 June 1972 A generalized extinction theorem for exciton polarization is derived. It expresses a constraint that the exciton polarization must satisfy on the surface of the medium. The theorem provides so-called additional boundary conditions, whose exact form has been frequently argued about.
In the electrodynamics of spatially dispersive media, the question of the so-called additional boundary conditions has attracted a good deal of attention [1—7],since such conditions seem to be needed to solve, for example, problems of refraction and reflection on a medium of this kind. Sein [6, 7] derived an integral relation, called extinction theorem for exciton polarization which provided a set of additional boundary conditions. We recently analyzed the problem of refraction and reflection [8], without introducing any additional boundary conditions. Instead we made use of certain mode coupling conditions that we previously derived in an analysis relating to the mode structure of an electromagnetic field in a spatially dispersive medium [9]. We discussed the relationship between the additional boundary conditions, the mode coupling conditions and the extinction theorem in another recent publication [10]. In refs. [6—10] a spatially dispersive medium was considered, whose dielectric constant in the k, wspace (Fourier conjugate to space-time) has the following form appropriate to an isolated exciton transition: a w2
For the meaning of all the symbols see ref. [9] In the present note we derive an extinction theorem appropriate to spatially dispersive media of a much wider class. Let us consider first an infinite medium in which the part Pexc(r, w) of the macroscopic polarization at a point r and a frequency w, that can be attributed to exciton transitions is related to the electric field E(r, w) by a relation of the form .
LP
(r,w)=—E(r,w),
where L is a linear differential operator in the coordinates of r. The form of this operator must be deduced from a microscopic theory. Some examples may be found in refs. [1, 4, 51. For the sake of simplicity we only consider media that are macroscopically spatially stationaryt and isotropic. For such media L will be a function of the Laplacian operator V2 only, with constant coefficients. The appearance of spatial derivatives in L is responsible for a spatially non-local relationship between the electric displacement vector D(r, w) and the electric field E(r, w) viz.
D(r,
~)
e
3r’. 0(w)E(r, w)
ê(k,w)e 0(w)+
e —
+
2 iwl’ (1)
e
—
.
(2)
exc
+
fe1(r —r’, w)E(r’,w)d (3)
(flw/m~)k
* Supported by the Army Research Office (Durham). **On leave from the Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627, USA.
tBy spatial stationarity, we mean that the constitutive relation (3) between D and E involves r and r’ through the difference r — r’ only.
279
Volume 40A, number 4
PHYSICS LETTERS
Here e0(w) has the same physical meaning as in (I), and e1(R, w) represents the contribution to the (nonlocal) dielectric “constants” from all the exciton transitions. The integration extends over the whole infinite medium. The contribution from the exciton transitions to the macroscopic polarization is 3r. (4) 4~J I (r, w) = ~ Ce (r r’. w)E(r’, w) d Let us now apply the L-operator to both sides of (4). Recalling that L is a differential operator on the coordinates of r, we obtain, if we also interchange the order of operators on the right-hand side of(4) and compare the resulting equation with (2), cxc
P
~L~I
31 July 1972
~
=
vc{~,~).
(8)
where L~is the operator adjoint to L. Comparison of (6) and (7) shows that for every point r inside V. fn .C{1 (r
r’, w). [P
(r
dS.
(9)
S
We may call the relation (9) extinction theorem fir exciton polarization in a medium characterized by the operator L. In the special case when the four-dimensional Fourier transform of the dielectric constant is given by (1), one readily finds that (9) reduces to the extinction theorem derived in ref. [10. eq. (22)1 and is then equivalent to the extinction theorem of Sein
L 1 (r
—
r’, w) = —4ith(r
—
r’),
(5)
showing that El is a Green function of the L-operator. Of the possible solutions of(s), e~is that solution which will satisfy appropriate constraints at infinity, dictated by physical considerations, such as causality. Let us now assume that the medium under consideration occupies afinite volume V. If the linear dimensions of this volume are large compared with the effective wavelengths of the exciton inodes, one can assume that, except perhaps at points close to the boundary of V, the exciton polarization at any point r inside V will, to a good approximation, be given by the expression of the form (4), with the integration extending over the volume V only: 1e (r
Pexc(r, w)
= 4ir~ ~ 1’
I
—
r’, w)E(r’, w)d3r’.
(6)
Now from eqs. (5) and (2) and from Green’s theo-
rem generalized to the case of an arbitrary operator [11] L it follows that
[6,7]. From the manner in which our generalized extinction theorem, expressed by eq. (9), was derived, it is clear that it may be regarded to be a consistency requirement of the theory: Only those exciton polarization fields are admissible, whose values on the boundary surface ensure that (9) is satisfied. Thus the extinction theorem plays the role of additional boundary conditions. By analogy with the more restrictive case discussed in ref. [101, these additional boundary conditions will presumably be equivalent to appropriate mode coupling conditions on the modes that can be generated in the medium. We plan to return to this question in another publication. References 11 SI. Pekar, Soviet l’hys. JITP 6 (1958) 785. 121 SI. Pekar, Soviet Phys. Solid State 4 (1962) 953. 131 V.L. Ginzburg, Soviet Phys. JETP 7 (1958)1096. 141 V.M. Agranovich and V.L. Ginzburg, Spatial dispersion in crystal optics and the theory of excitons (Interscience, New York. 1966). J.J. 1-Iopfield and D.G. Thomas, Phys. Rev. 132 (1963) 563. 161 ii. Scm, Ph. 1). Dissertation. New York University (1969). 171 ii. Scm, Phys. Lett .32A (197(1)141.
151
[Pcxc (r, w)l f =
4~i..’ _!_ 1e I (r
—
r’, w)E~(r’,w) d3r’
+
v +
~1-f n ‘C{e~(r
4ir ~1 S
(j = 1,2, 3
—
r’, o4, ~Pcxc (r’, w] / dS .~
labelling Cartesian components), where
(7)
S
is the surface bounding the volume V, tzis the unit outward normal to S and ~ s~}is the bilinear concomitant associated with the operator L, defined by the relation 28))
181
G.S. Agarwal, D.N. Paitanayak and E. Wolf. Optics
4(1971)255. 191 Commun. G.S. Agarwal, D.N. Pattanayak and E. Wolf, Phys. Rev. Lett. 27(1971) 1022. 1101 CS. Agarwal, D.N. Pattanayak and F. Wolf. Optics (‘ommun. 4(1971) 260. 1111 P.M. Morse and Ii. Feshbach. Methods of theoretical physics (McGraw Hill. New York. 1953) Part 1, p.870.
PHYSICS LETTERS
31 July 1972
A GENERALIZED EXTINCTION THEOREM FOR EXCITON POLARIZATION IN SPATIALLY DISPERSIVE MEDIA* G.S. AGARWAL** Institut für Theoretische Physik, Universitilt Stuttgart, Germany
and D.N. PATTANAYAK and E. WOLF Department of Physics and Astronomy, University of Rochester, Rochester, N. Y. 14627, USA Received 5 June 1972 A generalized extinction theorem for exciton polarization is derived. It expresses a constraint that the exciton polarization must satisfy on the surface of the medium. The theorem provides so-called additional boundary conditions, whose exact form has been frequently argued about.
In the electrodynamics of spatially dispersive media, the question of the so-called additional boundary conditions has attracted a good deal of attention [1—7],since such conditions seem to be needed to solve, for example, problems of refraction and reflection on a medium of this kind. Sein [6, 7] derived an integral relation, called extinction theorem for exciton polarization which provided a set of additional boundary conditions. We recently analyzed the problem of refraction and reflection [8], without introducing any additional boundary conditions. Instead we made use of certain mode coupling conditions that we previously derived in an analysis relating to the mode structure of an electromagnetic field in a spatially dispersive medium [9]. We discussed the relationship between the additional boundary conditions, the mode coupling conditions and the extinction theorem in another recent publication [10]. In refs. [6—10] a spatially dispersive medium was considered, whose dielectric constant in the k, wspace (Fourier conjugate to space-time) has the following form appropriate to an isolated exciton transition: a w2
For the meaning of all the symbols see ref. [9] In the present note we derive an extinction theorem appropriate to spatially dispersive media of a much wider class. Let us consider first an infinite medium in which the part Pexc(r, w) of the macroscopic polarization at a point r and a frequency w, that can be attributed to exciton transitions is related to the electric field E(r, w) by a relation of the form .
LP
(r,w)=—E(r,w),
where L is a linear differential operator in the coordinates of r. The form of this operator must be deduced from a microscopic theory. Some examples may be found in refs. [1, 4, 51. For the sake of simplicity we only consider media that are macroscopically spatially stationaryt and isotropic. For such media L will be a function of the Laplacian operator V2 only, with constant coefficients. The appearance of spatial derivatives in L is responsible for a spatially non-local relationship between the electric displacement vector D(r, w) and the electric field E(r, w) viz.
D(r,
~)
e
3r’. 0(w)E(r, w)
ê(k,w)e 0(w)+
e —
+
2 iwl’ (1)
e
—
.
(2)
exc
+
fe1(r —r’, w)E(r’,w)d (3)
(flw/m~)k
* Supported by the Army Research Office (Durham). **On leave from the Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627, USA.
tBy spatial stationarity, we mean that the constitutive relation (3) between D and E involves r and r’ through the difference r — r’ only.
279
Volume 40A, number 4
PHYSICS LETTERS
Here e0(w) has the same physical meaning as in (I), and e1(R, w) represents the contribution to the (nonlocal) dielectric “constants” from all the exciton transitions. The integration extends over the whole infinite medium. The contribution from the exciton transitions to the macroscopic polarization is 3r. (4) 4~J I (r, w) = ~ Ce (r r’. w)E(r’, w) d Let us now apply the L-operator to both sides of (4). Recalling that L is a differential operator on the coordinates of r, we obtain, if we also interchange the order of operators on the right-hand side of(4) and compare the resulting equation with (2), cxc
P
~L~I
31 July 1972
~
=
vc{~,~).
(8)
where L~is the operator adjoint to L. Comparison of (6) and (7) shows that for every point r inside V. fn .C{1 (r
r’, w). [P
(r
dS.
(9)
S
We may call the relation (9) extinction theorem fir exciton polarization in a medium characterized by the operator L. In the special case when the four-dimensional Fourier transform of the dielectric constant is given by (1), one readily finds that (9) reduces to the extinction theorem derived in ref. [10. eq. (22)1 and is then equivalent to the extinction theorem of Sein
L 1 (r
—
r’, w) = —4ith(r
—
r’),
(5)
showing that El is a Green function of the L-operator. Of the possible solutions of(s), e~is that solution which will satisfy appropriate constraints at infinity, dictated by physical considerations, such as causality. Let us now assume that the medium under consideration occupies afinite volume V. If the linear dimensions of this volume are large compared with the effective wavelengths of the exciton inodes, one can assume that, except perhaps at points close to the boundary of V, the exciton polarization at any point r inside V will, to a good approximation, be given by the expression of the form (4), with the integration extending over the volume V only: 1e (r
Pexc(r, w)
= 4ir~ ~ 1’
I
—
r’, w)E(r’, w)d3r’.
(6)
Now from eqs. (5) and (2) and from Green’s theo-
rem generalized to the case of an arbitrary operator [11] L it follows that
[6,7]. From the manner in which our generalized extinction theorem, expressed by eq. (9), was derived, it is clear that it may be regarded to be a consistency requirement of the theory: Only those exciton polarization fields are admissible, whose values on the boundary surface ensure that (9) is satisfied. Thus the extinction theorem plays the role of additional boundary conditions. By analogy with the more restrictive case discussed in ref. [101, these additional boundary conditions will presumably be equivalent to appropriate mode coupling conditions on the modes that can be generated in the medium. We plan to return to this question in another publication. References 11 SI. Pekar, Soviet l’hys. JITP 6 (1958) 785. 121 SI. Pekar, Soviet Phys. Solid State 4 (1962) 953. 131 V.L. Ginzburg, Soviet Phys. JETP 7 (1958)1096. 141 V.M. Agranovich and V.L. Ginzburg, Spatial dispersion in crystal optics and the theory of excitons (Interscience, New York. 1966). J.J. 1-Iopfield and D.G. Thomas, Phys. Rev. 132 (1963) 563. 161 ii. Scm, Ph. 1). Dissertation. New York University (1969). 171 ii. Scm, Phys. Lett .32A (197(1)141.
151
[Pcxc (r, w)l f =
4~i..’ _!_ 1e I (r
—
r’, w)E~(r’,w) d3r’
+
v +
~1-f n ‘C{e~(r
4ir ~1 S
(j = 1,2, 3
—
r’, o4, ~Pcxc (r’, w] / dS .~
labelling Cartesian components), where
(7)
S
is the surface bounding the volume V, tzis the unit outward normal to S and ~ s~}is the bilinear concomitant associated with the operator L, defined by the relation 28))
181
G.S. Agarwal, D.N. Paitanayak and E. Wolf. Optics
4(1971)255. 191 Commun. G.S. Agarwal, D.N. Pattanayak and E. Wolf, Phys. Rev. Lett. 27(1971) 1022. 1101 CS. Agarwal, D.N. Pattanayak and F. Wolf. Optics (‘ommun. 4(1971) 260. 1111 P.M. Morse and Ii. Feshbach. Methods of theoretical physics (McGraw Hill. New York. 1953) Part 1, p.870.