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The “missing” electromagnetic wave and the additional boundary conditions for light in a crystal
Solid State Communications, Vol. 46, No. 4, pp. 363-365, 1983. Printed in Great Britain.
0038-1098/83 / 160363 - 0 3 $03.00/0 Pergamon Press Ltd
THE "MISSING" ELECTROMAGNETIC WAVE AND THE ADDITIONAL BOUNDARY CONDITIONS FOR LIGHT IN A CRYSTAL S.I. Pekar and V.I. Pipa Institute of Semiconductors, Academy of Science of the Ukrainian SSR, 252028, Prospect Nauki 115, Kiev, U.S.S.R.
(Received 7 January 1983 by E.A. Kaner) Light penetrating from vacuum into a crystal gives rise to a non-uniform wave in both media that has not been taken into account up to now. Its wavevector k normal projection is imaginary and k 2 = 0. The allowance for this wave makes it for the first time possible in the general case to satsify all the additional boundary conditions of the additional light wave theory. A PECULIAR FEATURE of the exciton resonance range is the existence of the additional light waves along with the two waves of the traditional birefringence [1 ]. To describe these, one has to supplement the Maxwell boundary conditions given at the crystal surface with the additional boundary conditions (ABC). The letters are determined unambiguously by the laws governing the exciton reflection from the crystal surface. Being different for various exciton models, the laws and hence the ABC are, however, of the same form for the three most widely exploited exciton modles - the Frenkel exciton, the optical phonon, and the Wannier-Mott exciton (with the radius of the electron-hole relative motion far shorter than the exciton wavelength) [2]. Our paper deals just with these ABC. The ABC were derived for the first time in 1957 for the Frenkel exciton. With the interaction between the adjacent and close crystal elementary cells only taken into account, the ABC are of the form [1, 3]
(kr is a real vector), the OZ-axis is directed inwards the crystal, Im kz i> 0. Inasmuch as any arbitrary vector Pex given in a semi-infinite crystal may be uniquely expanded in exponential waves (3) by means of the Laplace transformation, equation (4) determines operator I" applied to any Pex. Let s be the degeneracy multiplicity for the exciton under consideration that is in resonance with light (s ~ 3). Then P~x lies in s-dimensional subspace [2, 3], vector equation (1) is equivalent to s scalar equations, and the ABC are given by s + 1 equations including scalar equation (2). In paper [3], s additional light waves were taken into account and it was impossible to satisfy s + 1 ABC with s amplitudes of the waves mentioned. So, the problem concerning light reflection from and penetration through the crystal-vacuum interface has not been solved in the general case until very recently. The solution has been found only for the following three important special cases:
[Pex(r,t)]z=o = 0,
(1) The light incidence is normal, k r = 0 for all waves 7"u = 0, equation (2) is an identity, and only s equations (1) are to be satisfied. (2) The incident light wave and all penetrated ones are polarized perpendicular to the plane of incidence (s-polarization, u ± kr, uz = 0). Again, equation (2) becomes an identity. (3) The interaction between either distant or adjacent crystal cells may be neglected in the exciton problem. In this case one may consider respectively the ABC in the form (1) or (2) alone.
(1)
where z = 0 is the crystal surface plane, Pex is the partial exciton contribution in the specific crystal polarization P [2, 31. The contribution in the ABC due to the interaction between distant crystal cells was considered in 1978 [3] to obtain, along with equation (1), the following ABC [7"Pex(r,t)]z= o = O,
(2)
where the definition for the operator 7" applied to an exponential wave of the form U
=
Uo
e i(kr-wt)
(3)
is the following: Tu --- i(kr, u) - Ikrluz
Ikr I + ikz
(4)
Here kr is the wavevector k projection at the plane z = 0 363
In order to obtain the general solution of the light reflection and penetration problem the authors had to find the missing (s + 1)-st additional electromagnetic wave in the crystal. Let us make it clear, how can one miss waves when solving wave equations written in doubtless form (in particular, the Maxwell equations). A wave may be
THE "MISSING" ELECTROMAGNETIC WAVE
364
characterized by the electric field E, induction D, magnetic induction B, polarization P, and other quantities. These are connected by certain relations, hence, the wave equation may be written for any quantity. It is to be emphasized, however, that the dispersion equations (relating k and ~ ) obtained for each case are, generally speaking, inequivalent. Let us consider for example an isotropic spatially dispersive medium with scalar dielectric permittivity e(co, k2). If the wave equation is written for E, then the condition for nontrivial solutions to exist, i.e., the dispersion equation, is of the form [-~
--
6(03, k 2
)1
e(co, k 2)
] -- 1
0.
c
c
= 0
(I0)
c
(1 1)
1 •
E = ---A--V(p
(12)
B = rot A
(13)
c
where p' and P are expressed in terms of E by means of formulas (7) and (8). Let us seek the solution of equations (10) and (11) in the form usual for the problem o f light penetration through an interface between two media: A, ~ ~ e i(krrr-t°t) ,
= 0,
p' = -- div P,
(6)
(7)
where ~E,
AA-~A--1V~+4nP
(5)
which is not equivalent to equation (5): it describes the same transverse waves but fails to describe the longitudinal ones. Thus one "misses" longitudinal waves when solving the equation for D, the reason being, D = 0 for these waves, i.e., they are trivial solutions of the wave equation for D and do not obey the condition for the nontrivial solutions of equation (6) to exist. Having written the wave equation for the dielectric polarization fictitious charge density
P =
the context of the Maxwell equations, B = P = D = 0. To obtain this wave as a nontrivial solution of the wave equation, the latter is to be written for a quantity that is not equal to zero rather than for E (or B, P, D). Let us consider the vector and scalar potentials A and ~b and assume the gauge condition to be div A = 0. Then the Maxwell equations for a crystal without external charges and currents may be written as:
A~b = -- 4rip' =
The roots k~ of the equation [(c2k2/w 2) -- e] 2 = 0 are associated with the transverse waves. With the appropriate k-dependence of e, there may be more than two roots, i.e., there exist additional light waves (ALW) [1, 2]. The roots k] o f the equation e(w, k 2) = 0 are known to correspond to the longitudinal waves. But, the wave equation written for D yields a dispersion equation o f the form c2k2
Vol. 46, No. 4
K =
e--1 - 4n
AA = 0,
(14)
/,4 = o.
(15)
The solution of equations (14) and (15) is a non-uniform wave A = Ao
(9)
that is not equivalent to either equations (5) or (6) and only describes the longitudinal waves. This time all transverse waves are missing. Coming from the isotropic medium back to the general case of an anisotropic crystal, let us consider the peculiar features of the wave that could be missing in the papers dealing with the ALW theory developed in course of the last 25 years. Insofar as the wave equation has been written in these papers for E, the missing wave must possess the property E = 0, since it is to be a trivial solution of the wave equation made use of. Then, within
e itk°r-wt),
q5 =
- iCe
i0%r-t°t),
(16)
where ko -
(8)
one obtains the dispersion law e(uJ,k 2) = 0,
where co and k r are similar to the ones of the wave incident from vacuum. As it has been shown above, we have to find a partial solution associated with E = 0. Then equations (10) and (11) reduce to
{kr, kz},
kz -- ilkrl,
kg = 0.
(17)
For the right-hand side of equation (I 2) to be equal to zero, one has to require that ico --Ao
¢
=
koC.
(18)
The waves (16), (17) and (18) cannot exist in an infinite crystal since it would grow infinite as z ~ -- oo. It can exist, however, in a semi-infinite crystal that occupies the half-space 0 <~ z ~< oo. It is just the "missing" wave. The term Ep = - - ( 1 / c ) A in equation (12) is the whole rotational part of the electric field. Just Ep (rather than E) is the field that perturbs the crystal energy Schr6dinger operator and, in particular, causes the photransitions. Thus, the "missing" wave A is the perturbation field wave. It was obtained in [4] in a somewhat different manner.
Vol. 46, No. 4
THE "MISSING" ELECTROMAGNETIC WAVE
The allowance for the missing wave (16) does not give rise to any additional terms in the Maxwell boundary conditions since E = D = B = 0 for this wave. No additional terms appear as well in the ABC (1) because the contribution of the missing wave in Pex can be shown to be equal to zero. An additional term appears in the ABC (2) alone, since the relevant fraction denominator vanishes by virtue of equation (17). As was shown in [4], condition (2) can be satisfied once and for all with
2~r ~ (ko,KEj) C -
Ik,-I j=11krl + ikzj '
365
be valid only for the three special cases mentioned, when one could ignore the ABC (2). Now these results of [2] turn out to be valid in the general case as well: there is no need to disregard the ABC (2) which is satisfied provided one takes into account the missing wave with the amplitude given by equation (19). The exciton resonance ranges and the spatial dispersion are irrelevant to the existence of the missing light wave. The results ( I 0 ) - ( 1 9 ) are valid for the whole frequency spectrum. In particular, relation (19) may be derived [4] making no use of the ABC (2).
(19)
where j labels usually penetrated light waves witn nonzero electric field; Ei are relevant wave amplitudes. Thus, one has to satisfy s equations (1) for s usual ALW which is possible to carry out in the general case. This was done in Section 29 of [2] when obtaining the coefficients of light reflection from and penetration through the surface for crystals of various symmetries [see formulas (29.1)(29.63)]. But these results of [2] have been supposed to
REFERENCES 1. 2.
3. 4.
S.I. Pekar, Zh. Eksp. Teor. Fiz. 33, 1022 (1957). S.I. Pekar, Crystal Optics and Additional Light Waves, Naukova Dumka, Kiev (1982) (~,nglish translation to be published by Addison-Wesley, 1983). S.I. Pekar,Zh. Eksp. Teor. Fiz. 74, 1458 (1978). S.I. Pekar & V.I. Pipa, Fiz. Tverd. Tela (Leningrad) 25,266 (1983).
0038-1098/83 / 160363 - 0 3 $03.00/0 Pergamon Press Ltd
THE "MISSING" ELECTROMAGNETIC WAVE AND THE ADDITIONAL BOUNDARY CONDITIONS FOR LIGHT IN A CRYSTAL S.I. Pekar and V.I. Pipa Institute of Semiconductors, Academy of Science of the Ukrainian SSR, 252028, Prospect Nauki 115, Kiev, U.S.S.R.
(Received 7 January 1983 by E.A. Kaner) Light penetrating from vacuum into a crystal gives rise to a non-uniform wave in both media that has not been taken into account up to now. Its wavevector k normal projection is imaginary and k 2 = 0. The allowance for this wave makes it for the first time possible in the general case to satsify all the additional boundary conditions of the additional light wave theory. A PECULIAR FEATURE of the exciton resonance range is the existence of the additional light waves along with the two waves of the traditional birefringence [1 ]. To describe these, one has to supplement the Maxwell boundary conditions given at the crystal surface with the additional boundary conditions (ABC). The letters are determined unambiguously by the laws governing the exciton reflection from the crystal surface. Being different for various exciton models, the laws and hence the ABC are, however, of the same form for the three most widely exploited exciton modles - the Frenkel exciton, the optical phonon, and the Wannier-Mott exciton (with the radius of the electron-hole relative motion far shorter than the exciton wavelength) [2]. Our paper deals just with these ABC. The ABC were derived for the first time in 1957 for the Frenkel exciton. With the interaction between the adjacent and close crystal elementary cells only taken into account, the ABC are of the form [1, 3]
(kr is a real vector), the OZ-axis is directed inwards the crystal, Im kz i> 0. Inasmuch as any arbitrary vector Pex given in a semi-infinite crystal may be uniquely expanded in exponential waves (3) by means of the Laplace transformation, equation (4) determines operator I" applied to any Pex. Let s be the degeneracy multiplicity for the exciton under consideration that is in resonance with light (s ~ 3). Then P~x lies in s-dimensional subspace [2, 3], vector equation (1) is equivalent to s scalar equations, and the ABC are given by s + 1 equations including scalar equation (2). In paper [3], s additional light waves were taken into account and it was impossible to satisfy s + 1 ABC with s amplitudes of the waves mentioned. So, the problem concerning light reflection from and penetration through the crystal-vacuum interface has not been solved in the general case until very recently. The solution has been found only for the following three important special cases:
[Pex(r,t)]z=o = 0,
(1) The light incidence is normal, k r = 0 for all waves 7"u = 0, equation (2) is an identity, and only s equations (1) are to be satisfied. (2) The incident light wave and all penetrated ones are polarized perpendicular to the plane of incidence (s-polarization, u ± kr, uz = 0). Again, equation (2) becomes an identity. (3) The interaction between either distant or adjacent crystal cells may be neglected in the exciton problem. In this case one may consider respectively the ABC in the form (1) or (2) alone.
(1)
where z = 0 is the crystal surface plane, Pex is the partial exciton contribution in the specific crystal polarization P [2, 31. The contribution in the ABC due to the interaction between distant crystal cells was considered in 1978 [3] to obtain, along with equation (1), the following ABC [7"Pex(r,t)]z= o = O,
(2)
where the definition for the operator 7" applied to an exponential wave of the form U
=
Uo
e i(kr-wt)
(3)
is the following: Tu --- i(kr, u) - Ikrluz
Ikr I + ikz
(4)
Here kr is the wavevector k projection at the plane z = 0 363
In order to obtain the general solution of the light reflection and penetration problem the authors had to find the missing (s + 1)-st additional electromagnetic wave in the crystal. Let us make it clear, how can one miss waves when solving wave equations written in doubtless form (in particular, the Maxwell equations). A wave may be
THE "MISSING" ELECTROMAGNETIC WAVE
364
characterized by the electric field E, induction D, magnetic induction B, polarization P, and other quantities. These are connected by certain relations, hence, the wave equation may be written for any quantity. It is to be emphasized, however, that the dispersion equations (relating k and ~ ) obtained for each case are, generally speaking, inequivalent. Let us consider for example an isotropic spatially dispersive medium with scalar dielectric permittivity e(co, k2). If the wave equation is written for E, then the condition for nontrivial solutions to exist, i.e., the dispersion equation, is of the form [-~
--
6(03, k 2
)1
e(co, k 2)
] -- 1
0.
c
c
= 0
(I0)
c
(1 1)
1 •
E = ---A--V(p
(12)
B = rot A
(13)
c
where p' and P are expressed in terms of E by means of formulas (7) and (8). Let us seek the solution of equations (10) and (11) in the form usual for the problem o f light penetration through an interface between two media: A, ~ ~ e i(krrr-t°t) ,
= 0,
p' = -- div P,
(6)
(7)
where ~E,
AA-~A--1V~+4nP
(5)
which is not equivalent to equation (5): it describes the same transverse waves but fails to describe the longitudinal ones. Thus one "misses" longitudinal waves when solving the equation for D, the reason being, D = 0 for these waves, i.e., they are trivial solutions of the wave equation for D and do not obey the condition for the nontrivial solutions of equation (6) to exist. Having written the wave equation for the dielectric polarization fictitious charge density
P =
the context of the Maxwell equations, B = P = D = 0. To obtain this wave as a nontrivial solution of the wave equation, the latter is to be written for a quantity that is not equal to zero rather than for E (or B, P, D). Let us consider the vector and scalar potentials A and ~b and assume the gauge condition to be div A = 0. Then the Maxwell equations for a crystal without external charges and currents may be written as:
A~b = -- 4rip' =
The roots k~ of the equation [(c2k2/w 2) -- e] 2 = 0 are associated with the transverse waves. With the appropriate k-dependence of e, there may be more than two roots, i.e., there exist additional light waves (ALW) [1, 2]. The roots k] o f the equation e(w, k 2) = 0 are known to correspond to the longitudinal waves. But, the wave equation written for D yields a dispersion equation o f the form c2k2
Vol. 46, No. 4
K =
e--1 - 4n
AA = 0,
(14)
/,4 = o.
(15)
The solution of equations (14) and (15) is a non-uniform wave A = Ao
(9)
that is not equivalent to either equations (5) or (6) and only describes the longitudinal waves. This time all transverse waves are missing. Coming from the isotropic medium back to the general case of an anisotropic crystal, let us consider the peculiar features of the wave that could be missing in the papers dealing with the ALW theory developed in course of the last 25 years. Insofar as the wave equation has been written in these papers for E, the missing wave must possess the property E = 0, since it is to be a trivial solution of the wave equation made use of. Then, within
e itk°r-wt),
q5 =
- iCe
i0%r-t°t),
(16)
where ko -
(8)
one obtains the dispersion law e(uJ,k 2) = 0,
where co and k r are similar to the ones of the wave incident from vacuum. As it has been shown above, we have to find a partial solution associated with E = 0. Then equations (10) and (11) reduce to
{kr, kz},
kz -- ilkrl,
kg = 0.
(17)
For the right-hand side of equation (I 2) to be equal to zero, one has to require that ico --Ao
¢
=
koC.
(18)
The waves (16), (17) and (18) cannot exist in an infinite crystal since it would grow infinite as z ~ -- oo. It can exist, however, in a semi-infinite crystal that occupies the half-space 0 <~ z ~< oo. It is just the "missing" wave. The term Ep = - - ( 1 / c ) A in equation (12) is the whole rotational part of the electric field. Just Ep (rather than E) is the field that perturbs the crystal energy Schr6dinger operator and, in particular, causes the photransitions. Thus, the "missing" wave A is the perturbation field wave. It was obtained in [4] in a somewhat different manner.
Vol. 46, No. 4
THE "MISSING" ELECTROMAGNETIC WAVE
The allowance for the missing wave (16) does not give rise to any additional terms in the Maxwell boundary conditions since E = D = B = 0 for this wave. No additional terms appear as well in the ABC (1) because the contribution of the missing wave in Pex can be shown to be equal to zero. An additional term appears in the ABC (2) alone, since the relevant fraction denominator vanishes by virtue of equation (17). As was shown in [4], condition (2) can be satisfied once and for all with
2~r ~ (ko,KEj) C -
Ik,-I j=11krl + ikzj '
365
be valid only for the three special cases mentioned, when one could ignore the ABC (2). Now these results of [2] turn out to be valid in the general case as well: there is no need to disregard the ABC (2) which is satisfied provided one takes into account the missing wave with the amplitude given by equation (19). The exciton resonance ranges and the spatial dispersion are irrelevant to the existence of the missing light wave. The results ( I 0 ) - ( 1 9 ) are valid for the whole frequency spectrum. In particular, relation (19) may be derived [4] making no use of the ABC (2).
(19)
where j labels usually penetrated light waves witn nonzero electric field; Ei are relevant wave amplitudes. Thus, one has to satisfy s equations (1) for s usual ALW which is possible to carry out in the general case. This was done in Section 29 of [2] when obtaining the coefficients of light reflection from and penetration through the surface for crystals of various symmetries [see formulas (29.1)(29.63)]. But these results of [2] have been supposed to
REFERENCES 1. 2.
3. 4.
S.I. Pekar, Zh. Eksp. Teor. Fiz. 33, 1022 (1957). S.I. Pekar, Crystal Optics and Additional Light Waves, Naukova Dumka, Kiev (1982) (~,nglish translation to be published by Addison-Wesley, 1983). S.I. Pekar,Zh. Eksp. Teor. Fiz. 74, 1458 (1978). S.I. Pekar & V.I. Pipa, Fiz. Tverd. Tela (Leningrad) 25,266 (1983).