On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery)
After the basic Pekar papers on crystal optics with spatial dispersion (CSD) among a lot of others there was the “dielectric approximation” (DA) method. But soon it has been fully rejected because, as it was recognized by the main group of authors on the theme, it came in conflict with the law of co...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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Piskovoi, V.N. Venger, E.F. 2017-06-13T16:52:31Z 2017-06-13T16:52:31Z 2015 On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) / V.N. Piskovoi, E.F. Venger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 3. — С. 255-258. — Бібліогр.: 11 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.03.255 PACS 78.20.Bh, 71.35.Aa, 71.36.+c, 77.22.Ch https://nasplib.isofts.kiev.ua/handle/123456789/121209 After the basic Pekar papers on crystal optics with spatial dispersion (CSD) among a lot of others there was the “dielectric approximation” (DA) method. But soon it has been fully rejected because, as it was recognized by the main group of authors on the theme, it came in conflict with the law of conservation of energy flux through the ideal vacuum-crystal interface. The objective of this paper is to advance a way for rehabilitation of DA method by using indispensable broadening and essential generalization of the expression just for the energy flux density vector (EFDV) at the vacuum-medium interface (the so-called Poynting-Pekar vector in the additional light waves (ALW) theory). The authors would like to thank the members of Pekar’s seminar for helpful discussions on the theme and personally Dr. V. Korotyeyev for helpful discussions on the theme. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) Article published earlier |
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On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) |
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On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) Piskovoi, V.N. Venger, E.F. |
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On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) |
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On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) |
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On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) |
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On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) |
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on the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) |
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Piskovoi, V.N. Venger, E.F. |
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Piskovoi, V.N. Venger, E.F. |
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2015 |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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After the basic Pekar papers on crystal optics with spatial dispersion (CSD) among a lot of others there was the “dielectric approximation” (DA) method. But soon it has been fully rejected because, as it was recognized by the main group of authors on the theme, it came in conflict with the law of conservation of energy flux through the ideal vacuum-crystal interface. The objective of this paper is to advance a way for rehabilitation of DA method by using indispensable broadening and essential generalization of the expression just for the energy flux density vector (EFDV) at the vacuum-medium interface (the so-called Poynting-Pekar vector in the additional light waves (ALW) theory).
|
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1560-8034 |
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https://nasplib.isofts.kiev.ua/handle/123456789/121209 |
| citation_txt |
On the dielectric approximation in the additional light waves theory: an historical rehabilitation (recovery) / V.N. Piskovoi, E.F. Venger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 3. — С. 255-258. — Бібліогр.: 11 назв. — англ. |
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2025-11-26T13:13:23Z |
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2025-11-26T13:13:23Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 255-258.
doi: 10.15407/spqeo18.03.255
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
255
PACS 78.20.Bh, 71.35.Aa, 71.36.+c, 77.22.Ch
On the dielectric approximation in the additional light waves theory:
an historical rehabilitation (recovery)
V.N. Piskovoi, E.F. Venger
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03028 Kyiv, Ukraine
Abstract. After the basic Pekar papers on crystal optics with spatial dispersion (CSD)
among a lot of others there was the “dielectric approximation” (DA) method. But soon it
has been fully rejected because, as it was recognized by the main group of authors on the
theme, it came in conflict with the law of conservation of energy flux through the ideal
vacuum-crystal interface. The objective of this paper is to advance a way for
rehabilitation of DA method by using indispensable broadening and essential
generalization of the expression just for the energy flux density vector (EFDV) at the
vacuum-medium interface (the so-called Poynting-Pekar vector in the additional light
waves (ALW) theory).
Keywords: additional light waves theory, dielectric approximation.
Manuscript received 02.02.15; revised version received 22.05.15; accepted for
publication 03.09.15; published online 30.09.15.
For the sake of simplicity and clearness, we shall
restrict ourselves to a simple model of polarization
oscillators in a medium as well as the simplest
geometrical configuration of the light-medium system.
Also, we shall stand here in the framework of the
exciton effective mass (EEM) approximation (the
exciton effective mass M is taken positive for the sake of
definiteness).
Let us consider a case that a monochromatic plane
wave (with the frequency ω and wave vector k0 = ω/c,
where c is the speed of light in vacuum) is incident from
vacuum along the normal to the surface of a semi-
infinite optically uniaxial crystal (occupying the half-
space 0≥x ). We assume that this wave is in resonance
with some nondegenerate dipole-allowed excitonic state
to which a polarization vector ))((0,0, xPP corresponds.
The P vector, being directed along the crystal axis
C(Z), is parallel to that of the wave electric field
))((0,0, xEE and determines the partial contribution
from the state studied into the total crystal polarization.
The ideal Maxwell equation in a medium has the
standard form [1]
0=)4( 0
2
02
2
PEk
x
E
π+ε+
∂
∂ , (1)
where 0ε is the background permittivity.
As it was shown in our papers [2], the material
equations appropriate for the situation considered may
be written in one of the following forms:
1. In the pioneer Pekar form [3] that arises from
supplemented material Maxwell equation in the
)(0,∞ half-space to the ),( ∞−∞ space with further
reducing to zero the remainder part ,0)(−∞ in 0≥x
space (in a more generalized case, it corresponds to
1= −pU or 0=)(0+P in the point 4, see bellow).
2. In Tolpygo-Mashkevich form that is based on the
simultaneous use of the material and ideal
equations in the integral form [4]. We have
checked, of course, identicity of this approach to
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 255-258.
doi: 10.15407/spqeo18.03.255
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
256
the problem with that of the points 3, 4, and came
to conclusion that there is no reasons to base the
paper on the different formulations of the ideal
Maxwell equations.
3. One more historical approach is based on the
fundamental Hopfield-Thomas paper [5]. In this
approach, the Maxwell material equations for the
above presented configuration is determined by the
following material equation:
EPi
x
P
M ex π
Δε
−γ+ω−ω+
∂
∂
4
=)(
2
0
2
2h . (2)
Here, exω and γ are the resonance exciton
frequency and damping constant, respectively; the
parameter Δ is equal to the constant of the so-
called “longitudinal – transverse” splitting of
exciton line, which, in its turn, may be written in
several possible equivalent forms (see Eq. (9) in
4mob of 2004).
According to the studies of such class of
equations in mathematical physics, the linear in P
boundary condition for Eq. (2) is of the following
form
0=)(0)(0 ++
∂
∂
α+
x
PP , (3)
where α, in general, is some rather arbitrary
parameter that will be partly determined in the
point 4.
4. In the simplest integral form with some free enough
parameter 1)|(| ≤UU that determine the exciton
interaction with crystal surface
xdxExxUxxxP ′′ω′+χ+ω′−χ∫
∞
)()],(),([=)(
0
, (4)
where ),( ω′−χ xx is the polarization operator for
an infinite medium:
( ) .2= with
,||exp
4
=),(
2
1
0
−γ+ω−ω
⎟
⎠
⎞
⎜
⎝
⎛ ′−
π
Δε
−ω′−χ
iMr
xx
r
iMrixx
ex
h
h
(5)
It is obvious that all the above approaches to the
problem must be equivalent, and their parameters should
be closely connected to one another. Namely, parameter
1= −pU (6)
corresponds to 0=α in (3), i.e., is the main and the
most often used Pekar (or Dirichlet) condition with
0=)(0+P
1=gU (7)
that corresponds to the Ginzburg (or Neumann) case [6]
with 0=1−α in Eq. (3), and 0=)(0+
∂
∂
x
P
2
2
2
= with,
)(
21=
Md
C
Cr
diU s
h
−⎥
⎦
⎤
⎢
⎣
⎡
δ+
δ
+− , (8)
where d is the lattice constant in the x direction and δ is
the energy shift of a molecular level on the surface as
compared to that of the crystal bulk. The latter
corresponds to the Sugakov case [7] with
)(= δ+δα Cd in Eq. (3) – that was the first in ALW
theory representation of expression for P in the form of
Eq. (4).
All the above relations can be easily checked by
using Eq. (4) and its differentiation with respect to x, for
arbitrarily x as well as for x approaching to +0 .
In full compliance with the abovementioned, the
DA case was already considered in papers [8] with
0=daU . (9)
In terminology of the point 3, it determines α in
Eq. (3) uniquely:
irda −α = . (10)
At the same time, if the exciton effective mass
approximation is applied consistently, then the
expression for the energy flux density is also well
defined, and the possible (virtual) jump of time-averaged
normal components of the energy flux density 0SΔ at
the vacuum-medium interface is given by the expression
known in crystal optics with spatial dispersion:
+
∗
∗
Δ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
−
∂
∂
εΔ
ω
00
0 2
=
x
PP
x
PP
M
iS exh . (11)
For the most cases of Eqs. (6) to (8) type, the
necessary turning SΔ to zero was easily checked, so
from the very beginning this fact does not call any doubt
as to its general applicability ad hoc and, naturally, has
been omitted at the calculation of reflection and passing
of EM waves for all above situations, that is for all
possible meanings of parameter U, including also the
case 0=daU .
But soon a big enough group of authors (see, e.g.,
[3, 6, 9, 10]) who used these to simple calculations for
U = 0 has noted, that in this case the law of energy
conservation violates and 0SΔ has a discontinuity at +0 .
Namely, this fact results in total ignoring the DA method
in ALW theory as inevitable cost of scientific
production.
Now our aim is to reanimate it. To this end, we
shall take the generalized expression for SΔ (see the
papers [2]) including the undetermined for a moment
expression for the density of the surface polarization
current )(0,0, 0jj . In this case, SΔ takes the form:
0
*
00 ].[
4
1= SccjES ΔΔ ++−
π
(12)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 255-258.
doi: 10.15407/spqeo18.03.255
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
257
with 00 )(= Ej ωσ , and σ as the one more free but not
completely independent parameter, because it may be
related with the parameter U or α by the relation:
0=)(
0+Δ S . (13)
As it has been argued in [2], one can choose σ in
such a way that 0=)(Im σ in the absence of real surface
conductivity.
So, the boundary conditions for electric field and
magnetic induction ⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
− EB rot=1
tc
B , in the frame of
the taken above space configuration and the time
dependence of the fields, take the following form:
)(0=)(0 +− EE , (14)
0
04)(0=)(0 j
c
ik
x
E
x
E
π+
∂
∂
∂
∂ +− . (15)
Here, m0 designates approaching the crystal
boundary on the left and right to the origin of
coordinates, respectively.
So, step by step we come to the complete system of
consistent equations necessary for solving the assigned
problem. As usually, we seek its solution in the
framework of the plane waves, namely
)(exp),()(),( xikPExExP jjj→ with jj nkk 0= for
each possible value of the wave-vector. We obtain two
different roots for refraction, n1 and n2 in accordance
with the second order in x of two joint homogeneous
Eqs. (1) and (2) and with the requirement for absence of
any electromagnetic field sources at )(+∞ – for half-
space configuration.
These two roots for refraction are
( ) bnn +ε−μ±ε+μ≡ −+
2
00
2
,
2
1,2 4
1)(
2
1= (16)
where 2
0
0
2
0
2=),(2
k
Mbi
k
M
ex
hh
εΔ
γ+ω−ω≡μ . And, accor-
dingly, the relations
( ) jjj EnP 0
2
4
1= ε−
π
, (17)
)(0=)(0 0
++
∂
∂
jj
j Pnik
x
P
. (18)
As a result, we have the complete system of
equations for the problem under consideration. Being
written in terminology of Ei, ER, E1, E2 for amplitudes of
incident, reflected and two penetrating waves, it has the
following form
( )( )
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
α+ε−
σ
π
++−
≡++
∑ 0=1
4=
=
00
2
1,2=
02211
021
jjj
j
Ri
Ri
Enkin
E
c
EnEnEE
EEEEE
(19)
where ( ) 2
0210 Im
4
= −α++α
π
σ fkinnbkc (with
021
2
2
2
1= ε−++ nnnnf ). It follows from Eq. (13) by a
transformation from jP and
x
Pj
∂
∂
to 1,2E (see equations
(17), (18)) and then via 0E from the first and third
equations of the system (19).
This fact makes it possible to express all fields
arising in crystal and free space via the field 0E and to
construct the reflection amplitude coefficient (R) for the
situation considered:
eff
eff
i
R
n
n
E
ER
+
−
≡
1
1
= , (20)
where effn is the effective refractive index:
σ
π
+
α++
+α+ε+
cfkinn
nnnnkinnneff
4)(=
021
21210021 . (21)
It is evident that different parts of the expression
for neff may be combined in many ways, but here we take
those of them that can be easily compared with the
simplest and well known ones, namely:
the Pekar [3] result ( 0=)(00,=0,= +σα exP ) with
21
021=
nn
nnneff +
ε+ (22)
and the Ginzburg [6] result ( 0=)(00,=0,1 +−
∂
∂
σ→α
x
Pex )
with
1
2121 )(= −+ fnnnnneff . (23)
So, we fulfilled the main aim of the paper. First, we
have given the way for rehabilitation of DA and a group
of other similar cases in the ALW theory, in principle.
Second, we have given the correct expressions for the
coefficients of reflection and refraction of polariton
waves in DA for a specific “light – crystal” orientation
(see Eqs. (20) and (21) with ri =α , and 0≠σ , in
principle).
Some additional remarks.
After the first Pekar paper on the ALW problem, a
lot of publications on the subject appeared. Many of
them deal with different types of generalization (as to
the variety of exciton-like states, so for their degeneracy
or multiplicity, for oblique incidence with different ABC
for different polarization vector projections, etc.) and
can be found in the rather complete review [11], in the
cited above papers [2] and elsewhere. But here we cited
the really pioneer works only (see points 1-4 above)
which turned out enough for our aim.
The authors would like to thank the members of
Pekar’s seminar for helpful discussions on the theme and
personally Dr. V. Korotyeyev for helpful discussions on
the theme.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 255-258.
doi: 10.15407/spqeo18.03.255
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
258
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