Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium
Influence of small periodic perturbation in periodic dielectric medium on propagation of ТМ-polarized waves is investigated by the Krylov-Bogolyubov averaging method. As a result of periodic perturbation inside the allowed bands of unperturbed medium forbidden bands appear. The appearance of forbidd...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Zitieren: | Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium / K.Yu. Kramarenko, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2001. — № 1. — С. 94-95. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-785212015-03-19T03:02:28Z Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium Kramarenko, K.Yu. Khizhnyak, N.A. Theory and technics of particle acceleration Influence of small periodic perturbation in periodic dielectric medium on propagation of ТМ-polarized waves is investigated by the Krylov-Bogolyubov averaging method. As a result of periodic perturbation inside the allowed bands of unperturbed medium forbidden bands appear. The appearance of forbidden bands is conditioned by a parametric resonance between spatial harmonics of perturbation and plane waves, on which the solution of the wave equation in periodic medium is decomposed. The location, number and width of these bands are determined. 2001 Article Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium / K.Yu. Kramarenko, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2001. — № 1. — С. 94-95. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS 41.20. http://dspace.nbuv.gov.ua/handle/123456789/78521 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Theory and technics of particle acceleration Theory and technics of particle acceleration |
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Theory and technics of particle acceleration Theory and technics of particle acceleration Kramarenko, K.Yu. Khizhnyak, N.A. Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium Вопросы атомной науки и техники |
| description |
Influence of small periodic perturbation in periodic dielectric medium on propagation of ТМ-polarized waves is investigated by the Krylov-Bogolyubov averaging method. As a result of periodic perturbation inside the allowed bands of unperturbed medium forbidden bands appear. The appearance of forbidden bands is conditioned by a parametric resonance between spatial harmonics of perturbation and plane waves, on which the solution of the wave equation in periodic medium is decomposed. The location, number and width of these bands are determined. |
| format |
Article |
| author |
Kramarenko, K.Yu. Khizhnyak, N.A. |
| author_facet |
Kramarenko, K.Yu. Khizhnyak, N.A. |
| author_sort |
Kramarenko, K.Yu. |
| title |
Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium |
| title_short |
Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium |
| title_full |
Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium |
| title_fullStr |
Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium |
| title_full_unstemmed |
Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium |
| title_sort |
application of krylov-bogolyubov averaging method to the problem of periodic perturbations in the layered medium |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Theory and technics of particle acceleration |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78521 |
| citation_txt |
Application of Krylov-Bogolyubov averaging method to the problem of periodic perturbations in the layered medium / K.Yu. Kramarenko, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2001. — № 1. — С. 94-95. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT kramarenkokyu applicationofkrylovbogolyubovaveragingmethodtotheproblemofperiodicperturbationsinthelayeredmedium AT khizhnyakna applicationofkrylovbogolyubovaveragingmethodtotheproblemofperiodicperturbationsinthelayeredmedium |
| first_indexed |
2025-07-06T02:35:32Z |
| last_indexed |
2025-07-06T02:35:32Z |
| _version_ |
1836863280287580160 |
| fulltext |
APPLICATION OF KRYLOV-BOGOLYUBOV AVERAGING METHOD
TO THE PROBLEM OF PERIODIC PERTURBATIONS
IN THE LAYERED MEDIUM
K.Yu. Kramarenko, N.A. Khizhnyak
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
Influence of small periodic perturbation in periodic dielectric medium on propagation of ТМ−polarized waves is
investigated by the Krylov−Bogolyubov averaging method. As a result of periodic perturbation inside the allowed
bands of unperturbed medium forbidden bands appear. The appearance of forbidden bands is conditioned by a
parametric resonance between spatial harmonics of perturbation and plane waves, on which the solution of the wave
equation in periodic medium is decomposed. The location, number and width of these bands are determined.
PACS 41.20.
1. INTRODUCTION
Periodic structures with varying parameters can
prove useful in many applications. In paper [1] the
theory of wave propagation in periodic structures with
smoothly varying parameters is developed with the help
of Wannier-function expansion. In particular, it is
shown that the double periodic structure possesses a
miniband structure. Such a spectrum has been observed
in direct numerical simulations, described in paper [2].
The Krylov−Bogolyubov averaging method
was developed for the investigation of non−linear
differential equations. However, it is successfully
applied for solving linear equations with small
parameter in the theory of cyclic accelerators [3]. It is
also possible to apply such method for investigation of
TM−polarized wave propagation in a periodic medium
with small periodic perturbation at an arbitrary period.
The purpose of this paper is to describe the effects
occurring when an electromagnetic wave propagates in
a periodic medium with small periodic perturbation.
2. KRYLOV−BOGOLYUBOV AVERAGING
METHOD
At first, we consider the propagation of a TM
polarized wave in the periodic medium. The periodicity
exists only in one direction, and propagation of the
electromagnetic wave in that direction is governed by
the equation:
( ) ,)( 0DkzkL
dz
dD1
dz
d 222 =−+
⊥ε
ε
ε (1)
where
λ
π2k = is the wave number, ⊥k - the transverse
component of wave vector, L is the period of
undisturbed medium, z - the undimensional variable
and )()( z1z εε =+ . By replacing
11 , 2 dDU D U
dzε
= =
from the linear differential equation of the second order
we come to the system of two equations of the first
order:
1 ( ) 2 0,
2 ( ) 1 0,
dU z U
dz
dU z U
dz
ε − =
+ Ω =
(2)
where ( )22
2
kzk
z
Lz ⊥−=Ω )(
)(
)( ε
ε
. The solution of the
above system of linear homogeneous differential
equations with periodic coefficients can be written in
the Floquet form:
),exp()exp( zifCziCf1U 11 ψψ −+= ∗∗
(3)
),exp()exp( zifCziCf2U 22 ψψ −+= ∗∗
where ψ,, 21 ff are determined by the fundamental
solutions of the equation (1). Functions 21 ff , are
periodic ( )()( zf1zf 11 =+ , )()( zf1zf 22 =+ ),
normalized by the following way: i2ffff 1221 −=− ∗∗ .
In the case of periodic perturbation
)()(),()()( zzzzz εεεεε ∆=Λ+∆∆+→ the set of
equations (2) can be considered as inhomogeneous:
1 ( ) 2 ( ) 2,
2 ( ) 1 1,
dU z U z U
dz
dU z U U
dz
ε ε − = ∆
+ Ω = − ∆ Ω
(4)
where
2k ⊥∆+
∆=∆ Ω
)( εεε
ε
. The solution of
inhomogeneous set of equations is as follows:
),exp()()exp()( zifzCzifzC1U 11 ψψ −+= ∗∗
(5)
),exp()()exp()( zifzCzifzC2U 22 ψψ −+= ∗∗
94 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 1.
Series: Nuclear Physics Investigations (37), p. 94-95.
where )(),( zCzC ∗ are the solutions of the system of
differential equations:
( )[ +∆ Ω+∆= 2
1
2
2 ffC
2
i
dz
dC ε (6a)
( ) ( )( )2 2
2 1 exp( 2 )],C f f i zε ψ∗ ∗ ∗+ ∆ + ∆ Ω −
( )[ +∆ Ω+∆−= ∗
∗
2
1
2
2 ffC
2
i
dz
dC ε
( )2 2
2 1 exp(2 )].C f f i zε ψ+ ∆ + ∆ Ω (6b)
Application of the averaging method to equations (6)
becomes possible if we assume the following
parameters ∆ Ω∆ ,ε are small. The application of the
above method is based on the idea that if the derivatives
are small, than values of functions can be naturally seen
as the superposition of slowly varying part C and small
rapidly oscillating terms. Considering, that these terms
cause only small oscillations of real function about its
mean part, it can be neglected in zero-order
approximation: CC = . The right-hand part of equations
(6) is averaged on explicitly contained variable z :
( ) ( ) ,)exp( ∗∗∗ −
∆ Ω+∆+
+
∆ Ω+∆=
Czi2ff
2
i
Cff
2
1i
dz
dC
2
1
2
2
2
1
2
2
ψε
ε
( ) ,)exp( Czi2ff
2
i
Cff
2
1i
dz
dC
2
1
2
2
2
1
2
2
ψε
ε
∆ Ω+∆+
+
∆ Ω+∆−= ∗
∗
(7)
where ∫
∞→
=
T
0T T
1 ...... lim
. At first, consider the second
items in the equations (7). The periodic functions can be
decomposed in Fourier series. After the carrying out the
averaging operation, only the items satisfying the
following condition remain:
,, ppnl
ln +=+
+
Λ
= ψπψ (8)
where 0, 1, 2,...l = ± ± , 0, 1, 2,...n = ± ± , 1p < < . By
introducing the following descriptions
( )2
1
2
2 ff
2
1 ∆ Ω+∆= εδ ψ ,
( ) ( ) )exp( , zi2ff
2
i
ln
2
1
2
2 ψεψ −
∆ Ω+∆=∆ ∗∗ one can
receive such equations:
),exp( ipz2CCi
dz
dC −∆+= ∗ψδ ψ (9)
).exp( ipz2CCi
dz
dC ∗∗
∗
∆+−= ψδ ψ
These equations (9) describe the phenomenon of a
parametric resonance [3]. In the considered medium the
resonance is conditioned by the interaction of spatial
harmonics of periodic perturbation and plane waves, on
which the solution of wave equation in periodic medium
is decomposed. The resonance bands correspond to
forbidden bands of electromagnetic waves. As all
previous presumptions were made only for objective ψ ,
the forbidden bands occur inside the allowed bands of
unperturbed periodic medium. The position of the
forbidden bands is given by the formula:
,, δ ψψδ ψπψ +=+
+
Λ
= lnnl (10)
where 0, 1, 2,...l = ± ± ; n=0, ±1, ±2, ... . The width of
forbidden band is as follows:
( )
( )
.)exp( , zi2
f
f
i2 ln2
1
2
2 ψ
ε
ψ −
∆ Ω
+∆
=∆
∗
∗ (11)
The number of appearing bands depends on Λ . If
Λ is integer ( 1>Λ ), 1−Λ forbidden bands appear in
each allowed band of undisturbed medium. m-1
forbidden bands appear if Λ is a rational number, Λ=m/
j. Such a forbidden bands structure for layered medium
with meander perturbation was observed in [4].
3. CONCLUSION
Influence of small periodic perturbation in periodic
dielectric medium on propagation of ТМ−polarized
waves is investigated by the Krylov−Bogolyubov
averaging method. As a result of periodic perturbation
inside the allowed bands of undisturbed medium
forbidden bands appear. The location, number and
width of these bands are determined.
By using the averaging method, it is possible to
evaluate the influence of the sum of perturbations on
periodic medium dispersion properties. For example, for
two perturbations with periods Λ1=m1/j1 , Λ2=m2/j2 the
number of forbidden bands is determined according to
the formula: ( )1M − , where M is the least aliquot for
1m and 2m .
REFERENCES
1. V.V. Konotop. On wave propagation in periodic
structures with smoothly varying parameters // J.
Opt. Soc. Am., B.1997, v. 14, №2, p. 364-369.
2. V.M. Agranovich, S.A. Kiselev, D.L. Mills. Optical
multistability in nonlinear superlattices with very
thin layers // Physical Review, B. 1991, v. 44, №9,
p. 10917-10920.
3. A.A. Kolomensky, A.N. Lebedev. The theory of
cyclic accelerators. Moscow, 1962, p. 352 (in
Russian).
4. N.A. Khizhnyak, E.Yu. Kramarenko. Propagation of
electromagnetic waves in space-periodic structures
with dual periodicity // Ukr. Phys. Journal. 1997,
v. 42, №10, p. 1256-1259 (in Ukrainian).
95
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
REFERENCES
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