On a simple model of the photonic or phononic crystal
A model is proposed for a one-dimensional dielectric or elastic superlattice (SL) that relatively simply describes the frequency spectrum of electromagnetic or acoustic waves. The band frequency spectrum is reduced to mini-bands contracting with increasing frequency. A procedure is suggested for obt...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | On a simple model of the photonic or phononic crystal / A.M. Kosevich // Вопросы атомной науки и техники. — 2001. — № 6. — С. 287-290. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859647675176058880 |
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| author | Kosevich, A.M. |
| author_facet | Kosevich, A.M. |
| citation_txt | On a simple model of the photonic or phononic crystal / A.M. Kosevich // Вопросы атомной науки и техники. — 2001. — № 6. — С. 287-290. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | A model is proposed for a one-dimensional dielectric or elastic superlattice (SL) that relatively simply describes the frequency spectrum of electromagnetic or acoustic waves. The band frequency spectrum is reduced to mini-bands contracting with increasing frequency. A procedure is suggested for obtaining local states near a defect in a SL, and the simplest of these states is described. Conditions for the initiation of Bloch oscillations of a wave packet in a SL are discussed.
|
| first_indexed | 2025-12-07T13:30:06Z |
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ON A SIMPLE MODEL OF THE PHOTONIC
OR PHONONIC CRYSTAL
A.M. Kosevich
B. Verkin Institute for Low Temperature Physics & Engineering
National Academy of Sciences of Ukraine, Kharkov, Ukraine
e-mail: kosevich@ilt.kharkov.ua
A model is proposed for a one-dimensional dielectric or elastic superlattice (SL) that relatively simply describes
the frequency spectrum of electromagnetic or acoustic waves. The band frequency spectrum is reduced to mini-
bands contracting with increasing frequency. A procedure is suggested for obtaining local states near a defect in a
SL, and the simplest of these states is described. Conditions for the initiation of Bloch oscillations of a wave packet
in a SL are discussed.
PACS: 43.20,+g; 62.65.+k; 63.20DJ; 68.35.Ja
1. By a photonic crystal is meant a macroscopic
periodic structure composed of two spatially alternating
dielectrics differing in dielectric constants (velocities of
electromagnetic waves) [1]. Analogously, by a
phononic crystal or acoustic superlattice (SL) is meant a
periodic structure composed of two alternating elastic
materials differing in elastic moduli and velocities of
sound (the general acoustics theory of layered media is
expounded in [2], and a useful bibliography on acoustic
SLs is given in one of the last publications [3]). A great
number of publications are devoted to studying the fre-
quency spectrum of SLs. It is clear that, in the general
case, this spectrum is extremely complicated and con-
tains a system of both a great number of eigenfrequency
bands and gaps corresponding to forbidden frequencies
of eigenmodes. In order to characterize such spectra
qualitatively and to illustrate their main quantitative
features, it would be appropriate to use simple models
that allow for these features. The well-known ID Kro-
nig-Penney model [4] may serve as an example of such
a model in the electronic theory of crystals. In this
work, a model of a SL is proposed that provides an ana-
lytical description of the high-frequency part of its
spectrum and suggests a possible implementation of an
interesting acoustic SL.
Consider a SL in the form of alternating plane-par-
allel layers of two materials differing in either elastic or
dielectric (depending on the implementation of interest)
characteristics. Denote the layer thicknesses by d1 and
d2; then, the SL period equals d = d1 + d2. The elastic or
electromagnetic field inside each material, which is
assumed to be isotropic, is described by the wave
equation
,2
2
2
2
2
x
uc
t
u
∂
∂−
∂
∂ α
α
α
α = 1,2, (1)
where cα is the wave velocity in the layer of the α type.
The velocity of light in a dielectric equals cα= c/ ε (c
is the velocity of light in free space), and that in an
elastic medium equals cα= αα ρµ / ; εα, μα and ρα
(α=1,2) are dielectric constants,1 elastic moduli, and
mass densities, respectively.
Consider a wave propagating along the X axis
perpendicular to the layers. In this case, waves of two
possible polarizations do not interact, and it is possible
to study scalar fields u(α) (α = 1, 2).
The standard boundary conditions will be
formulated as applied to the acoustic problem. The
displacements u(α) and stresses σα = μα(∂u(α)/ ∂x) at the
layer boundaries will be considered continuous. It is
known that, by virtue of the periodicity of a structure
with a period of d, eigenmodes can be characterized by
a quasi-wave number k, considering that the field in a
unit cell with the number n takes the form
un(x) = un(x-nd)eiknd. (2)
The dispersion equations in this problem were obtained
by Rytov for both electromagnetic field [5] and
acoustics [6]
cos(kd) = cos(k1d1) cos(k2d2)
2211
1
2
2
1
2
1 dksindksin
k
k
k
k
+− (3)
where k1 = ω/c1 and k2 = ω/c2 (ω is frequency).
Equation (3) determines the frequency as an implicit
function of the quasi-wave number. It allows the
1 Because I will be interested mainly in narrow frequency bands, the
frequency dispersion of e can be neglected, and e can be related to the
corresponding frequencies.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 287-290. 31
spectrum of long-wavelength vibrations (kd<< 1) to be
described readily, for which a sound spectrum with
averaged elastic moduli <µ> and density <ρ> is
naturally obtained. It was shown [6] that
<ρ>d = ρ1d1+ρ2d2; d/<µ> = d1/µ1+ d2/µ2. (4)
The relationship for <µ>, which contains only dα/µα
ratios, is curious. A limiting case that is commonly of
no interest in the dynamics of a quantum particle can be
considered based on this relationship. Consider the limit
d2→0 and µ2→0 at d2/µ2 = P = const.2
In this case, d1→d
and k2d2 = ωd2/c2 = 2222 µωρ dd →0
therefore, Eq. (3) is reduced to the following equation:
cos(kd) = cos(k1d-(1/2)P((ρ2µ1/ρ1d)(k1d))sink1d (5)
It is useful to note that the dispersion law (Eq. 5)
coresponds to an elastic SL composed of a chain of
regularly repeating elements of length d with parameters
µ1 and c1. The following boundary conditions are
fulfilled at their joints: (1) continuity of the normal
stresses [σ] +
− = 0, which is equivalent to [∂u/ ∂x] +
− = 0
and (2) occurrence of a jump of displacements at a soft
inter-layer determined by the stresses at the joint
[u] +
− = Mσ ≡ µ1M(∂u/ ∂x), (6)
where M = P(ρ1/ρ2).
A set of such boundary conditions at a fixed M is
used in describing capillary phenomena in solids [7] or
planar defects in crystals [8]. If the parameter M is
small, the system in hands is reduced to a periodic
sequence of elastic sections weakly bound together. A
chain of piezoelectric sections bound together by thin
vacuum interlayers may serve, for example, as a
possible implementation of such a system. Then, the
coupling of elastic vibrations in neighboring sections
would be accomplished through electromagnetic os-
cillations in vacuum gaps.
To illustrate the distribution of roots ω = ω(k) of Eq.
(5), this equation will be represented in the form
coskd = cosz-Qzsinz, (7)
where z = k1d = ωd/c1 and Q = P(ρ2µ1/2ρ1d). Consider
the graphical construction in Fig. 1. The figure shows a
plot of the right-hand side of Eq. (7). When it runs over
values between ±1, the roots of the equation run over
2 A more general case d2→0 and c2→0 at d2/c2= const could be
considered; however, no new results arise in this case.
values within intervals marked off on the abscissa axis.
Fig. 1. Graphical solution of Eq. (7).
Eigenfrequency bands are shown in heavy lines on the z
axis
Note that the allowed frequencies are localized in
contracting intervals at values k1d = ±mπ, where m is a
large integer, as z increases.3 Under the condition that
m2Q>>1, the dispersion laws in these intervals take the
form
ω = mω0+
Ω
),2/(cos
),2/(sin2
2
2
kd
kd
m ,12
,2
+=
=
pm
pm
(8)
where (ω0 = πc/d and Ω. = c/πQd. It is clear that Eq. (8)
gives the size quantization phonon spectrum in a layer
of thickness d, whose levels are split into minibands
because of low "transparency" of interlayer boundaries.
An attempt to analyze the character of the SL band
spectrum was made in [9], where the dispersion relation
(Eq. (3)) was derived once again. However, their
analysis is not satisfactory in a limiting case close to
that considered in this work, because it leads to the
conclusion that the miniband widths do not vary with
increasing frequency.
Consider Eq. (8) from another point of view: Eq. (8)
describes the spectrum of a pseudo-quantum particle for
which the Schrödinger equation within the tight-binding
model takes the form (for m = 2p)
i∂ψ/∂t = mω0ψn-(Ω/m)(2ψn-ψn+1-ψn-1) (9)
i∂ψ/∂t = mω0ψn+(Ω/2m)(2ψn+ψn+1+ψn-1) (9a)
Actually, Eqs. (9) and (9a) are equations for the
envelop curve of SL vibrations taken at discrete points
(at joints). As usual, the order of derivative with respect
to time decreases in such equations. These equations
describe analytically the dynamics of a wave packet
corresponding to the allowable high frequencies. Using
the explicit form of the dispersion laws (Eq. (8)) and
3 The contraction of bands with increasing frequency was also noted
previously; in particular, this was mentioned in [3].
32
simple Eqs. (8) and (9), the passage of wave packets
through the system under study can be described
readily, and explicit relationships can be proposed for
comparison with possible experimental results.
Nonlinear effects in optical SLs associated with the
dependence of the refraction coefficient (that is, the
velocity of light c and the parameter ω0) or the
characteristic of joints Q on the field strength ψn can be
readily taken into account using Eqs. (9) and (9a) much
as it was done in [10] when describing optical solitons
in such SLs.
The frequencies of forbidden bands correspond to
displacements of the type un~e±knd (when k = iκ) or un~(-
1)ne±knd (when k = iκ+π), which drop (grow) with
increasing displacement number n. The frequency
dependence of the parameter K for solutions of the first
type can be found from the relationship
coshκd = cosz-Qzsinz >1, (10)
and, for solutions of the second type, from the
relationship
-coshκd = cosz-Qzsinz<-1 (10a)
It is clear that such states have a physical meaning
only in the x semiaxis under the condition that a
solution vanishing at infinity and corresponding to
certain boundary conditions at the origin is selected.
Solutions of the first and the second types correspond to
frequencies in the intervals
(2p –1)π < z < 2pπ
and
2pπ < z < (2p + 1)π, respectively, (see Fig. 1).
The necessity of using exponentially decreasing
solutions arises in describing displacements in the
vicinity of a local SL defect.
2. Assume that the boundary conditions at one of the
joints (let its number n =0) differ from those described
above, or more specifically, these conditions differ in
the parameter M: M* ≠ M. The local vibration
frequency is essentially determined by the difference
M*-M = ξM Calculations show that the boundary
condition at the defect leads to the relationship
sinhκd = ξQzsinz (11)
which, along with Eq. (10) or (l0a) (depending on the
sign ξ) gives the local vibration frequency. The local
frequencies are determined by the intersection points of
plots of the right-hand sides of Eqs. (10) and (l0a) with
a plot of the function
f(z) = z2sinh1 + = [1 + (ξQz)2sin2z]1/2,
which is determined by Eq. (11). Because κ > 0, the
solutions correspond to the frequencies (values of z)
determined by the equation (see Fig. 2)
Fig. 2. Determining a series of roots of Eq.(12)
graphically: roots z1, z2, ... and z1, z3 ... correspond to
two types of vibrations
cosz-Qzsinz =
=sgn{ξQzsinz} zQz 22 sin)(1 ξ+ . (12)
The local vibration frequencies corresponding to
different signs of ξ are located in alternating intervals
between z =:2pπ and z = (2p +1)π (p = 0, 1, 2, ...):ω =
ωs., s = 1, 2, 3, .... The corresponding solutions can be
presented in the standard form un(x,t) = wn(x)exp{-iωt},
where wn(x) is an odd function w-n(-x) = wn(x) of the
following form (see Fig. 3):
Fig. 3. Coordinate dependence of displacements of
a SL in the vicinity of a defect for two types of
vibrations consistent with roots in Fig. 2: (a)
corresponds to z0, z2 ... roots (antiphase vibrations of
unit cells) and (b) corresponds to z1, z3, ... roots (in-
phase vibrations of unit cells)
33
),cos()( )(
1
)(
0 s
ss xkaxw θ−= 0<x< d;
,)cos()( )(
11
)( nd
s
ss
n exkaxw κθ −−= (13)
nd<x< (n+1) d, n≥1;
where θs is the constant phase corresponding to the
eigenfrequency ωs. The function w1(ξ) depends
harmonically on the argument and can be found easily.
In this case, the local vibrations for which
(2p-1)π<z<2pπ (points z2 and z4 in Fig. 2) are described
by a monotonic function decreasing with increasing
number of the unit cell, and the vibrations with
frequencies
(2p-1)π<z<2pπ
(points z1 and z3 in Fig. 2) are described by a function
proportional to (-l)ne-κnd. It is essential that a local
vibration may arise at any sign of the perturbation ξ.
A local vibration with an even eigenfunction cannot
arise at a defect localized at one boundary at any sign of
ξ. Assume that this is a joint with n = 0; at this joint,
[u] +
− = 0 and σ0 = 0;
therefore, an excitation in the form of a standing wave
with an even dependence on the x coordinate is not
sensitive to the value of the parameter Q at the joint n =
0 and does not differ from the vibration of the free SL
boundary passing along this joint.
The free SL boundary corresponds to a section
through the joint n = 0. This is equivalent to the
condition σ0 = 0, which is obtained in the given model
at ξ = ∞ (M* = ∞).
It follows from Eq. (13) that only uniform vibrations
(κ = 0) are possible in this case at frequencies
ω = (c/d)πm, m = 0, 1, 2, ....
Hence, no localized wave exists at the free SL end.
This means that vibrations of the even type are
impossible if the defect is lumped at one joint. Such
localized excitations arise upon variation (perturbation)
of the parameter M at least at two neighboring joints. As
in the case of an odd solution, the regions of occurrence
of such local vibrations with in-phase and antiphase
displacements of neighboring unit cells alternate,
depending on the sign of ξ, with the period ∆z = π.
3. It is interesting to discuss the possibility of
occurrence and experimental observation of Bloch
oscillations of a wave packet in the SL under
consideration. Bloch oscillations of an optical pulse in a
different situation were described and observed
experimentally [11,12]. Therefore, this discussion is not
groundless.
This work was partly supported by INTAS (INTAS
open call 1999, project №167).
REFERENCES
1. J.D. Joannopulos, R.D. Meade, J.N. Win.
Photonic Crystals. Princeton, Princeton University
Press, 1995.
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Media M.: "Nauka", 1973, 373 p. (Academic, New
York, 1980).
3. Pi-Gang Luan, Zhen Ye. Acoustic wave
propagation in an one-dimensional layered system
// Phys. Rev. E. 2000, v. 63, p. 066611.
4. R. de L. Kronig, W.G. Penni. Quantum
mechanics of electrons in crystal lattices // Proc.
Roy. Soc. London, Ser. A. 1931, v. 130, p. 499-513.
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laminated medium // Zh. Eksp. Teor. Fiz. 1955,
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p. 466].
6. S.M. Rytov. Properties of laminated
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Phys. Rev. Lett. 1992, v. 68, p. 3650-3653;
A.M. Kosevich, A.V. Tutov. Quasi-local surface
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waves” // Fiz. Nizk. Temp. 1993, v. 19, p. 1273-
1276 [Low Temp. Phys. 1993, v. 19, p. 905];
A.M. Kosevich, A.V. Tutov. Localized and
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defect in a crystal // Phys. Lett. A. 1996, v. 213,
p. 265-272.
9. A. Figotin, V. Gorentsveig. Localized
electromagnetic waves in a layered periodic
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10. A.A. Sukhorukov, Yu.S. Kivshar.
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Phys. Rev. Lett. 2001, v. 87, 083901-1 - 083901-4.
11. N. Petsch, P. Dannberg, W. Elflein et al.
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34
A.M. Kosevich
|
| id | nasplib_isofts_kiev_ua-123456789-80034 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:30:06Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kosevich, A.M. 2015-04-09T16:05:50Z 2015-04-09T16:05:50Z 2001 On a simple model of the photonic or phononic crystal / A.M. Kosevich // Вопросы атомной науки и техники. — 2001. — № 6. — С. 287-290. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 43.20,+g; 62.65.+k; 63.20DJ; 68.35.Ja https://nasplib.isofts.kiev.ua/handle/123456789/80034 A model is proposed for a one-dimensional dielectric or elastic superlattice (SL) that relatively simply describes the frequency spectrum of electromagnetic or acoustic waves. The band frequency spectrum is reduced to mini-bands contracting with increasing frequency. A procedure is suggested for obtaining local states near a defect in a SL, and the simplest of these states is described. Conditions for the initiation of Bloch oscillations of a wave packet in a SL are discussed. This work was partly supported by INTAS (INTAS open call 1999, project №167). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory On a simple model of the photonic or phononic crystal Простая модель фотонного или фононного кристаллa Article published earlier |
| spellingShingle | On a simple model of the photonic or phononic crystal Kosevich, A.M. Kinetic theory |
| title | On a simple model of the photonic or phononic crystal |
| title_alt | Простая модель фотонного или фононного кристаллa |
| title_full | On a simple model of the photonic or phononic crystal |
| title_fullStr | On a simple model of the photonic or phononic crystal |
| title_full_unstemmed | On a simple model of the photonic or phononic crystal |
| title_short | On a simple model of the photonic or phononic crystal |
| title_sort | on a simple model of the photonic or phononic crystal |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80034 |
| work_keys_str_mv | AT kosevicham onasimplemodelofthephotonicorphononiccrystal AT kosevicham prostaâmodelʹfotonnogoilifononnogokristalla |