Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites
We consider the long-wavelength limit for a periodic arrangement of carbon nanotubes. Using the Fourier expansion method we develop an effective-medium theory for photonic crystal of aligned optically anisotropic cylinders. Exact analytical formulas for the effective dielectric constants for the E...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1175652017-05-25T03:03:30Z Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites Gumen, L.N. Krokhin, A.A. Carbon nanotubes, quantum wires and Luttinger liquid We consider the long-wavelength limit for a periodic arrangement of carbon nanotubes. Using the Fourier expansion method we develop an effective-medium theory for photonic crystal of aligned optically anisotropic cylinders. Exact analytical formulas for the effective dielectric constants for the E and H eigenmodes are obtained for arbitrary 2D Bravais lattice and arbitrary cross-section of anisotropic cylinders. It is shown that depending on the symmetry of the unit cell photonic crystal of anisotropic cylinders behaves in the low-frequency limit like uniaxial or biaxial optical crystal. The developed theory of homogenization is in a good agreement with existing experimental results for the dielectric tensor of photonic crystals of carbon nanotubes. 2008 Article Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites / L.N. Gumen, A.A. Krokhin // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1072–1080. — Бібліогр.: 47 назв. — англ. 0132-6414 PACS: 42.70.Qs;41.20.Jb;42.25.Lc;78.67.Ch http://dspace.nbuv.gov.ua/handle/123456789/117565 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid |
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Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid Gumen, L.N. Krokhin, A.A. Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites Физика низких температур |
| description |
We consider the long-wavelength limit for a periodic arrangement of carbon nanotubes. Using the Fourier
expansion method we develop an effective-medium theory for photonic crystal of aligned optically
anisotropic cylinders. Exact analytical formulas for the effective dielectric constants for the E and H
eigenmodes are obtained for arbitrary 2D Bravais lattice and arbitrary cross-section of anisotropic cylinders.
It is shown that depending on the symmetry of the unit cell photonic crystal of anisotropic cylinders behaves
in the low-frequency limit like uniaxial or biaxial optical crystal. The developed theory of homogenization
is in a good agreement with existing experimental results for the dielectric tensor of photonic crystals
of carbon nanotubes. |
| format |
Article |
| author |
Gumen, L.N. Krokhin, A.A. |
| author_facet |
Gumen, L.N. Krokhin, A.A. |
| author_sort |
Gumen, L.N. |
| title |
Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites |
| title_short |
Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites |
| title_full |
Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites |
| title_fullStr |
Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites |
| title_full_unstemmed |
Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites |
| title_sort |
index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Carbon nanotubes, quantum wires and Luttinger liquid |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117565 |
| citation_txt |
Index of refraction of a photonic crystal of carbon
nanotubes and homogenization of optically anisotropic
periodic composites
/ L.N. Gumen, A.A. Krokhin
// Физика низких температур. — 2008. — Т. 34, № 10. — С. 1072–1080. — Бібліогр.: 47 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT gumenln indexofrefractionofaphotoniccrystalofcarbonnanotubesandhomogenizationofopticallyanisotropicperiodiccomposites AT krokhinaa indexofrefractionofaphotoniccrystalofcarbonnanotubesandhomogenizationofopticallyanisotropicperiodiccomposites |
| first_indexed |
2025-07-08T12:28:34Z |
| last_indexed |
2025-07-08T12:28:34Z |
| _version_ |
1837081784363253760 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1072–1080
Index of refraction of a photonic crystal of carbon
nanotubes and homogenization of optically anisotropic
periodic composites
L.N. Gumen1 and A.A. Krokhin2
1
Universidad Popular Autonoma del Estado de Puebla, 21 Sur, #1103, 72160, Mexico
2
Department of Physics, University of North Texas, P.O. Box 311427, Denton, TX, 76203, USA
E-mail: arkady@unt.edu
Received January 31, 2008
We consider the long-wavelength limit for a periodic arrangement of carbon nanotubes. Using the Fou-
rier expansion method we develop an effective-medium theory for photonic crystal of aligned optically
anisotropic cylinders. Exact analytical formulas for the effective dielectric constants for the E and H
eigenmodes are obtained for arbitrary 2D Bravais lattice and arbitrary cross-section of anisotropic cylin-
ders. It is shown that depending on the symmetry of the unit cell photonic crystal of anisotropic cylinders be-
haves in the low-frequency limit like uniaxial or biaxial optical crystal. The developed theory of homogeni-
zation is in a good agreement with existing experimental results for the dielectric tensor of photonic crystals
of carbon nanotubes.
PACS: 42.70.Qs Photonic bandgap materials;
41.20.Jb Electromagnetic wave propagation; radiowave propagation;
42.25.Lc Birefringence;
78.67.Ch Nanotubes.
Keywords: photonic crystal, carbon nanotubes, dielectric constants.
Introduction
Photonic crystals have been introduced after the publi-
cation by Yablonovitch and Gmitter [1], where the pres-
ence of a photonic band gap was experimentally demon-
strated. Since that the photonic band gap materials have
found numerous applications in optoelectronics and pho-
tonics [2]. A photonic crystal is an artificial structure with
periodic arrangement of «atoms» — dielectric (or metal-
lic) objects of arbitrary form with dielectric constant � a ,
imbedded in a homogeneous background material with
dielectric constant � b . Due to diffraction of electromag-
netic waves at the boundaries between two constituents
the band gap (or gaps) may open for some types of unit
cells if the dielectric contrast | |� �a b� exceeds some criti-
cal value. This gives rise to distinct optical phenomena
such as inhibition of spontaneous emission [3], high-re-
flecting omni-directional mirrors [4] and low-loss-wave-
guiding [2,5]. Photonic crystals with specially engineered
nanostructures may exhibit optical properties that do not
exist for natural crystals. In this case the artificial pho-
tonic crystals are referred to as metamaterials. The most
known, so far, phenomena attributed to the metamaterials
are the anomalous Doppler effect [6], negative index of
refraction [7], and huge optical anisotropy [8–10]. Only
the anomalous Doppler effect manifests itself at the fre-
quencies close to the band edge. Unlike this, the other two
phenomena may be observed at much lower frequencies.
This means that the photonic crystals can be also em-
ployed in the frequency region well below the gap, where
the wavelength covers many periods of the structure.
Here the periodic medium behaves like a homogenous
one and its optical properties can be characterized by ef-
fective parameters, like, e.g., the effective index of re-
fraction. The mathematical theory of heterogenous struc-
tures in the long-wavelength limit is called the theory of
homogenization [11]. Numerous practical applications of
photonic crystals made strong impact to the theory of ho-
mogenization. During a short period of time the new pow-
erful methods have been developed and many new results
have been obtained [8,12–29].
© L.N. Gumen and A.A. Krokhin, 2008
Fabrication of a photonic crystal with the period com-
parable near-infrared or optical range is a challenging
technological problem [2,4]. Carbon nanotubes, being
high-quality dielectric nanomaterial with extraordinary
mechanical properties, is a promising constituent for fab-
rication of 2D photonic crystals [30–34]. Due to appropri-
ate relation between dielectric and conductive properties
of carbon nanotubes, a triangular lattice of aligned carbon
nanotubes exhibits the effect of negative index of refrac-
tion [34]. Due to this metamaterial property, photonic
crystals of carbon nanotubes may be used in engineering
of superlens — optical device that produce images with
subwavelength resolution [35]. There are also a lot of
possible applications of photonic crystals as traditio-
nal optical elements like polarizers, prisms and lenses
[2,8,12]. In the latter case the diffractive properties of a
photonic crystal are not explored and it serves as an artifi-
cial dielectric material with custom-tailored optical char-
acteristics. These characteristics may be quite different
from those of natural crystals and give rise, for instance,
to unusually large birefringence.
In order to calculate these effective characteristics, we
develop an analytic approach to the optical properties of
photonic crystals with cylindrical atoms. We are using the
term «optical» in the sense that the wavelength of the
propagating wave is much larger than the lattice period of
the crystal; for natural crystals this condition fits the spec-
tral region up to the ultraviolet [36]. For photonic crystals
the lattice constant is, of course, a variable quantity.
Therefore, the long-wavelength regime’s upper limit may
be anywhere between radio waves and the far infrared. In
practice, many photonic band-gap materials exhibit a lin-
ear dispersion law, for both propagating modes, almost up
to the gap frequency. This rather wide region (usually
wider than the band-gap) can be considered as the domain
of «photonic crystal optics». Due to the linearity of the
dispersion law, each mode is characterized by a unique
parameter — its effective dielectric constant. It appears in
the homogenized solution of Maxwell’s equations for the
periodic medium, which thus can be replaced by an effec-
tive homogeneous medium with effective permittivity
�
�eff ( � ) limk � �
�
�
�
�k
ck
0
2
. (1)
In the general case, this effective parameter depends on
the direction of propagation � /k k� k and has tensor struc-
ture. The latter property is emphasized for 2D PC’s,
which are anisotropic uniaxial or biaxial crystals, depend-
ing on the symmetry of the unit cell [8,21]. Unlike this,
3D PC’s may be isotropic [15].
Optical anisotropy of the photonic crystals studied in
Refs. 8, 14, 15, 18–21, 23, 25, 27 is determined by the ge-
ometry of the unit cell only. The constituents themselves
are considered to be isotropic dielectrics. This is not the
case, for example, for a structure of aligned carbon nano-
tubes arranged periodically in the plane perpendicular to
the tubes. Here anisotropy manifests itself at the «micro-
scopic» level, since the nanotubes («atoms» of the pho-
tonic crystal) are optically anisotropic. Anisotropy origi-
nates from the layered structure of the graphite crystal,
which has different dielectric constants along the c axis
and in the perpendicular plane. The static values of these
dielectric constants are � || .�1 8225 and �� � 5 226. [37].
The elongated topology and the natural anisotropy of
graphite cause the photonic crystals of carbon nanotubes
to exhibit large optical anisotropy [30–34]. Three-dimen-
sional PC’s with anisotropic dielectric spheres have been
studied in Refs. 16. It was shown, that, depending on the
symmetry of the unit cell, the anisotropy of the spheres
may be favorable for either broadening or narrowing the
band gaps.
High anisotropy of 2D photonic crystals may find in-
teresting applications in nanophotonics as it was recently
proposed by Artigas and Torner [10]. Namely, the surface
of an anisotropic 2D photonic crystal supports propaga-
tion of a surface wave predicted by Dyakonov [38]. The
surface mode does not radiate and is localized close to the
surface due to the interference between the ordinary and
extraordinary waves. In natural crystals, it can be hardly
observed because of the low anisotropy. Since it is a sur-
face wave with very low energy losses, the Dyakonov
wave may replace surface plasmons in the near-field op-
tics and integrated photonic circuits. At the same time
application of optically anisotropic substrates leads to es-
sential increase of the propagation length of surface plas-
mons, thus, giving rise to higher efficiency of plasmonic
devices [39]. These and other examples show that dielec-
tric materials with high optical anisotropy is on demand in
modern optoelectronics.
Here we develop a theory of homogenization for pe-
riodic heterogeneous dielectric medium with intrinsic
anisotropy and apply the results for calculation of the ef-
fective dielectric tensor of a photonic crystal of carbon
nanotubes.
2. Method of plane waves
We consider a 2D periodic structure of dielectric cylin-
ders with their axes parallel to z and whose cross section
can have an arbitrary shape (Fig. 1). The cylinders are im-
bedded in a dielectric matrix. A 2D PC supports propaga-
tion of two uncoupled modes with either E polarization
(where the vector E is parallel to the cylinders, represent-
ing the TE mode), or H polarization (in this case the vec-
tor H is parallel to the cylinders, representing the TM
mode). The background material is an isotropic dielectric
with permittivity � b and the cylinders are rolled up from
an anisotropic dielectric sheet characterized by a tensor
�
( )� a . For carbon nanotubes, this tensor has two different
Index of refraction of a photonic crystal of carbon nanotubes and homogenization
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1073
eigenvalues, and in cylindrical coordinates is represented
by a diagonal matrix with elements � � �
( ) ( )a
zz
a� � � and
� �rr
a( )
||� . As a whole, the periodic inhomogeneous dielec-
tric medium is characterized by the coordinate-dependent
block-matrix,
�( )
( )
( )
�
�
�
��
r
r
r
�
�
�
��
�
0
0 zz
. (2)
Here ���( )r is a 2 2� Hermitian matrix in the x y� plane.
Outside the cylinders it reduces to a scalar, � ���b
( , ,� � � x y) and inside the cylinders it is given by [40]
�
� � � �
� �
�� �
� �
�
� �
�
( ) / ( / )( )
( / )( ) (
|| ||
||
x y r xy r
xy r
2 2 2 2
2 y x r2 2 2� �|| ) /�
�
�
�
�
�
�
. (3)
The wave equations for the E- and H-polarized modes
with frequency � have the following form:
� �2
2
2
E
c
Ezz
�
� ( )r , (4)
�
�
�
�
�
�
�
�
�
� �
x
a
H
x c
H
�
��
�
�2
2
0 , x x x y� �, ,� . (5)
Here E E x y� ( , ) and H H x y� ( , ) are the amplitudes of
the E and H monochromatic eigenmodes, respectively,
and a�� is a 2 2� Hermitian matrix with determinant 1:
a��
��
��
�
�
( )
det || ( )||
( )
r
r
r
� . (6)
The determinant det ( )��� r can be written as a product of
two eigenvalues. Within the graphite wall of the cylinders
this product is � �|| � . Outside the wall it is either ( )( )� b 2
(for r being outside the cylinders) or 1 for the interior re-
gion of the hollow cylinder, which is free from the dielec-
tric material.
Since for the E polarization the electric field is parallel
to the boundaries separating the background from the cyl-
inders, depends only the zz component of the dielectric
tensor enters in Eq. (4). It means that for this mode the ef-
fective dielectric constant is insensitive to in-plane aniso-
tropy. Due to the continuity of the electric field E x y( , )
across the cylindrical surfaces, the static dielectric con-
stant of any arrangement of parallel cylinders (not neces-
sarily periodic) is given by a simple formula
� �
eff
( )E
zz� , (7)
where
� �zz
c
zz
A
A
d
c
� �
1
( )r r , (8)
is the average over the area Ac of the unit cell zz compo-
nent of the tensor (2). For a binary composite
� � � �
eff
( )
( )
E
zz bf f� � � �� 1 , (9)
where f is the filling fraction of the component a. This ef-
fective dielectric constant is thus independent of the di-
rection of propagation and it is simply the weighted aver-
age dielectric constant [41].
Anisotropy affects the H-polarized mode. To obtain
the long-wavelength limit for Eq. (5) we apply the method
of plane waves. Using the Bloch theorem and the period-
ici ty of the function a��( )r , we get the Fourier
expansions,
H i h i( ) exp ( ) ( ) exp ( )r k r G G rk
G
� � �� , (10)
a a i�� ��( ) ( ) exp ( )r G G r
G
� �� , (11)
where the Fourier coefficients a��( )G are given by
a
A
a i d
Ac
A
c
a
ac
�� �� ��� �
�( ) ( ) exp ( ) (
||
( )
G r G r r r� � � �� �
�
1 1
) exp ( ) exp ( ) exp ( )� � � � � � � ��i d
A
i d i d
c b b
G r r G r r G r r
�
�
�� 1
2
c
�
�
�
�
�
�
�
�
�
.
(12)
1074 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
L.N. Gumen and A.A. Krokhin
z
x
Fig. 1. Periodic arrangement of dielectric cylinders. We as-
sume an arbitrary cross-sectional form of anisotropic cylin-
ders, arbitrary Bravais lattice, and filling fraction of the cylin-
ders.
Here G gives the reciprocal-lattice vectors. Indices a, b,
and c at the integrals label the domains of integration —
within the graphite walls, within the dielectric matrix, and
within the interior of the hollow nanotube, respectively.
Substituting Eqs. (10) and (11) into Eq. (5) we get an
eigenvalue problem in G space:
a k G k G h c h�� � � �
�
� � � � � � � �
G
k kG G G G( )( ) ( ) ( ) ( / ) ( ) .2 2
(13)
3. The long-wavelength limit
Equation (13) is an infinite set of homogenous linear
equations for the eigenfunctions hk ( )G . The nontrivial
solution is obtained by requiring that the determinant of
the coefficients of hk ( )G vanishes. This gives rise to the
band structure � �� n (k), where n is the band index. Be-
ing an analytic function, �n (k) may be expanded in a
power series of k (for any direction of k) around k � 0. For
the lowest (acoustic) band of the spectrum �( )0 0� and
the expansion starts from the linear term, i.e. �( )k � k. In
the static limit �� 0 there can be no magnetic field
(H � 0). Therefore all Fourier coefficients hk (G ) must
vanish if k � 0. The rates that they approach zero are dif-
ferent: the Fourier coefficients hk ( )G 0 vanish faster
than the zero harmonic hk ( )G � 0 . To obtain the behavior
of hk ( )G we substitute G � 0 in both sides of Eq. (13), di-
vide the both sides by h h0 0� �k G( ) and take the limit
k � 0,
� �� � � �� � �
2 2
0
/ ( ) ( )*c a k k a k G h� � � � � �
�
�
G
kG G .
(14)
Here a a�� ��! �( )G 0 is the bulk average of the matrix
(6) and h h hk kG G
* ( ) ( ) /� 0 is the normalized Fourier co-
efficient. In the long-wavelength limit the coefficients of
hk G
* ( )� on the right-hand side are proportional to k. In or-
der to make the right-hand side quadratic with k, the
amplitudes of nonzero harmonics, hk G
* ( )� must be pro-
portional to k. Namely, h kAk c
* /( )G �0 1 2. Thus, the Bloch
wave (10) can be written as a linear expansion over k:
H i h h i( ) exp ( ) [ ( ) exp ( )]*
r k r G G rk
G
� � � �
�0
0
1 . (15)
Since the sum over G vanishes linearly with k, Eq. (15)
shows that the medium becomes homogeneous, i.e., the
solution of the wave equation (5) approaches a plane
wave.
Now, to calculate the effective dielectric constant (1),
we develop a theory of perturbation with respect to a
small parameter ka (a Ac" 1 2/ is a lattice period). In
Eq. (13) we keep the linear terms and obtain the following
relation:
a G k a G G hk�� � � �� � �( ) ( ) ( )*
G G G G
G
� � � � � �
�
�
0
0 .
(16)
The quadratic approximation is given by Eq. (14), which
gives another linear relation between the eigenvectors
hk G
* ( ). Note that this relation is obtained from the
eigenvalue problem Eq. (13) for G � 0 and the linear ap-
proximation Eq. (16) is obtained for G 0. The linear re-
lations, Eqs. (14) and (16), are the homogenized equa-
tions for the Fourier components of the magnetic field.
These equations are consistent, if the corresponding de-
terminant vanishes:
det [ ( )
G,G
G G
�
� � � �
0
#a G G�� � �
� � � � �a a G n n G�� $� � � $ �( ) ( ) ]G G 0 . (17)
Here n k� / k is the unit vector in the direction of propa-
gation and # � � �( )a n n�� � � �eff
1 . Since Eqs. (14) and (16)
are homogenous with respect to k, the dispersion equation
(17) depends only on the inverse effective dielectric con-
stant, ( / )� ck 2. This fact is a manifestation of a general
property: At low frequencies an electromagnetic wave
has a linear dispersion in a periodic dielectric medium.
Although Eq. (17) is an infinite-order polynomial equ-
ation in #, it turns out that it has only a unique nonzero so-
lution. The fact that the second term in the determinant
Eq. (17) is a product of two factors, one of which depends
only on G and the other only on G �, plays a crucial role.
Omitting the mathematical details, which can be found in
Refs. 21, 25, we give here the final answer for the inverse
effective dielectric constant obtained from Eq. (17) as:
1
�
�� � �
eff
( )
( � )
H
a n n
n
� �
� � � � � � �
�
��a a n G n G a G G�� $� � � $ � %& & %( ) ( ) [ ( ) ] .G G G G
G,G 0
1
(18)
Here [...]�1 implies the inverse matrix in G space. Equa-
tion (18) is valid for an arbitrary form of the unit cell, ge-
ometry of the cylindrical inclusions, material composi-
tion of the photonic crystal, and the direction of
propagation in the plane of periodicity. In the case when
a�� ��' �( ) ( )G G� Eq. (18) is reduced to the formula ob-
tained for isotropic cylinders in Refs. 8, 21 ('( )G is the
Fourier component of the inverse dielectric constant,
' �( / ( )r) r�1 ).
The effective dielectric constant Eq. (18) depends on
the direction of propagation in the x y� plane and on the
details of the photonic crystal structure. For propagation
in the plane of periodicity, Eqs. (7), (8), and (18) give a
complete solution for the effective dielectric constants of
any 2D photonic crystal in the low-frequency limit. In
what follows we will show how to calculate the principal
dielectric constants, which give the customary descrip-
tion for anisotropic media in crystal optics [36].
Index of refraction of a photonic crystal of carbon nanotubes and homogenization
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1075
4. Index ellipsoid
As any natural crystal, artificial PC in the long-wave-
length limit can be characterized by an index ellipsoid
[36]. Taking into account Eq. (7) the equation for this el-
lipsoid can be written as follows:
x y z
zz
0
2
1
0
2
2
0
2
1
� � �
� � � . (19)
Here x y z0 0 0, , are three mutually orthogonal directions
along which the vectors of the electric field, E, and of the
displacement, D, are parallel to each other. For the E mode
we have E |D z| | | �, i.e., the z 0 direction coincides with z
axis. In the x– y plane the cross section of the index ellip-
soid is given by Eq. (18), which can be rewritten in the ca-
nonical form as
1 2 2
� (
( (
eff
( )
( )
( ) cos ( ) sin
H xx xx yy yya A a A� � � � �
� �( ) sina Axy xy 2( .
(20)
Here
A a a G G a G G�� �$ �� $ � %& & %� � � ��
�
� � ��
G,G
G G G G
0
1( ) ( ) [ ( ) ] ,
� � $ � & %, , , , , ,� x y . (21)
Equation (20) describes a rotated ellipse in the polar coor-
dinates ( , )) ( . The radius ) ( � (( ) ( )
( )�
eff
H
gives the index
of refraction of H mode and the angle( is related to the di-
rection of propagation, � (cos ,sin )n � ( ( . The directions
x 0 and y0 coincide with the semiaxes of the ellipse given
by Eq. (20) and the in-plane indices of refraction �1,
� 2 are given by the lengths of the semiaxes:
�
1
2 2 12� � � � �( sin cos sin )a A A Axx xx yy xy , (22)
�
2
2 2 12� � � � �( cos sin sin )a A A Ayy xx yy xy . (23)
The angle of rotation
of the axes of the ellipse Eq. (20)
with respect to some (initially) arbitrary chosen axes x y,
is given by the relation
tan 2
2
�
�
A
A A
xy
yy xx
. (24)
It is the symmetry of the unit cell that determines
whether a photonic crystal is uniaxial (� �1 2� ) or biaxial
(� �1 2 ). Unlike 3D photonic crystals, 2D crystals cannot
be isotropic (� � �1 2 3� � ). This property is guaranteed by
the Wiener bounds (� �1 2 3, * ) valid at least for in-plane
isotropy, namely � �1 2� [42]. If the crystal possesses a
third- or higher-order rotational axis z, then any sec-
ond-rank symmetric tensor such as A�� Eq. (21), is re-
duced to a scalar [43], A Aik � ��� (and a a�� ���� ).
Then Eqs. (22) and (23) may be simplified as
� �1 2
1� � � ��( )a A
� � � � ��
+
�
� � ��a a a G G a G G G
1
2
0
1
�$ �� $ � %& & %
G,G
G G G( ) ( ) [ ( ) ],
-
.-
/
0
-
1-
�1
.
(25)
Here [...]�1 implies matrix inversion, while {...}�1 means
«reciprocal». This compact formula gives the principal
dielectric constant (associated with the plane of periodic-
ity) of a uniaxial photonic crystal. The optical axis coin-
cides with the axis z, which is to say that birefringence is
absent for a single direction of propagation — the direc-
tion parallel to the cylinders. For propagation in this di-
rection (with E z��) the phase velocities of the «ordinary»
and «extraordinary» waves are the same, � �/ /k c� 1,
with �1 given by Eq. (25). Of course, for any direction of
propagation the «ordinary wave» propagates with the
same speed, c / �1, by definition. Because this velocity is
always less than the velocity c zz/ � of the «extraordi-
nary wave» (with E|| z�) that propagates in the plane of pe-
riodicity, we may conclude that uniaxial 2D photonic
crystals are necessarily «positive» optically anisotropic
crystals.
5. Uniaxial and biaxial photonic crystals of solid
graphite cylinders
In this section, we study 2D photonic crystals of solid
graphite cylinders arranged in square and rectangular lat-
tices. In Cartesian coordinates the dielectric function of a
carbon cylinder is given by Eq. (3). For rectangular and
square lattices with circular cylinders the semiaxes of the
index ellipsoid are directed along the basic lattice vectors.
Because of the cylindrical symmetry of the inclusions, the
off-diagonal elements of the tensor a��(G) vanish [44].
The diagonal elements for hollow cylinders with outer
and inner radii R and $R, respectively (0 12 2$ ), have the
following form:
a
R
A
xx b
c
b( )
( )
( )||0
1
2
21
2 2
1 1 1� �
��
�
�
�
�
� �� �
�
� ��
3 $
� � � ,
a A G GRxx c b( ) ( / ( ) { ( )G � � ��
�
�2 2 1 13 � �
� � � � ��
�
�[ ( ) ( )] ( ) ( ) ( )}||$ $ � � $J GR J GR J GR J GR1 1
1 1
0 0 .
(26)
The diagonal element a yy (G) is obtained from Eq. (26) by
the replacement � �� 4 ||.
1076 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
L.N. Gumen and A.A. Krokhin
For solid cylinders, i.e. $ ! 0. The circles in Fig. 2
show the effective dielectric constant, given by Eq. (20),
of the H mode as a function of the filling fraction,
f R Ac� 3 2 / , for the uniaxial photonic crystal with a
square lattice. The number of G values (plane waves)
considered in this calculation was 1200, which provides a
good convergence in Eq. (25). In accordance with the
Wiener bounds, the dielectric constant Eq. (7) for the ex-
tra-ordinary wave (E mode) (shown by triangles in Fig. 2)
is always larger than that for the ordinary wave (H mode),
i.e., the effective medium is an uniaxial positive crystal.
For a long time, there have been extensive efforts to
construct effective medium theories for inhomogeneous
media. The well-known Maxwell-Garnett approximation
[45] gives good results for very small filling fractions
( f ** 1or 1 1� **f ) but it fails otherwise. It also does not
take into account the microstructure of the inhomoge-
neous medium. To check the validity of the Maxwell-
Garnett approximation, we plot in Fig. 2 (squares) the ef-
fective dielectric constant proposed in Ref. 40:
�
� �
� �MG
H f
f
( ) || ||
|| ||
( )
( )
�
� � �
� � �
5 5
5 5
. (27)
Here 5 � �� �|| / . One can see that for all filling frac-
tions the Maxwell-Garnett approximation gives overesti-
mated values for the effective dielectric constant. For a
very dilute system, f * 0 07. , the Maxwell-Garnett approx-
imation gives results that are practically indistinguishable
from the exact ones (see inset in Fig. 2). For the close-
packed array of cylinders the Maxwell-Garnett approxi-
mation overestimates the dielectric constant by about 25%.
In Fig. 3 we plot two principal dielectric constants for
the biaxial PC of solid carbon cylinders with a rectangular
unit cell. The ratio of the sides of the rectangle is 1:2. The
difference between the two dielectric constants increases
with the filling fraction, giving rise to a higher anisotropy
of the corresponding effective medium. The Maxwell-
Garnett approximation Eq. (27), which does not take into
account the anisotropy of the unit cell, gives the values
for �MG that lie between the two principal values,
� � �1 2* *MG .
6. Uniaxial photonic crystal of carbon nanotubes
In our model we consider the carbon nanotubes as hol-
low graphite cylinders. In the experimental study [30] of
the dielectric properties of carbon nanotubes the outer ra-
dius of the cylinders was approximately R � 5 nm. The
nanotubes formed a thin film and they were oriented
along a specific direction. Although the nanotubes were
not necessarily arranged periodically, one can assume
that they formed almost a regular lattice, since the nano-
tube density is about 0.6–0.7 which is near the value of
f c � 63 / .4 0 785 for a close-packed structure. Thus,
the separation between the nanotubes (the period of
the square lattice d) slightly exceeds 2R, and in Ref. 40 it
was estimated to be d �1015. nm. The inner radius $R �
� 0.25–2 nm was evaluated from the amount of electro-
Index of refraction of a photonic crystal of carbon nanotubes and homogenization
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1077
1.2
1.1
0.1 0.2 0.3 0.4 0.5 0.6
Filling fraction
4.0
3.5
3.0
2.5
2.0
1.5
1.0
� e
ff
0.05 0.10 0.150
0
Fig. 2. In-plane effective dielectric constant for the H mode
for uniaxial PC of solid graphite cylinders with � || .�1 8225 and
�� � 5 226. in air, �b �1 (circles). Straight line (triangles) is the
effective dielectric constant � zz for the E mode, Eq. (8). The
squares show the results of the Maxwell-Garnett approxima-
tion (27). Inset shows the region of very small filling fractions.
0.1 0.2 0.3 0.4 0.5 0.6
Filling fraction
4.0
3.5
3.0
2.5
2.0
1.5
1.0
� e
ff
0
Fig. 3. A plot of the principal effective dielectric constants for
the photonic crystal of solid graphite cylinders arranged in a
rectangular lattice. In this case the effective medium is a bi-
axial crystal with all three different principal dielectric con-
stants. The larger (smaller) in-plane dielectric constant �1 (�2)
corresponds to the direction of the vector E along the short
(long) side of the rectangle. The Maxwell-Garnett dielectric
constant is shown by the squares. The triangles show � zz.
magnetic absorption for the E-polarized light [40]. The
four parameters f , R, $, and A ac � 2 are not independent
but related by the formula
f R Ac� �3 $2 21( ) / . (28)
Substituting the aforementioned parameters of the square
unit cell into this formula allows one to check that they
are self-consistent. It is worthwhile to mention that the
background material in the experiment [30] is not air but
the host material Delrin or Teflon with � b 7 1. Since nei-
ther the density of the host material nor its dielectric con-
stant is known, one cannot expect very good agreement
between the experimental results [30] and theory. In all
theoretical considerations it was assumed that � b �1. Be-
cause of this lack of experimental data, the effective me-
dium theories [32,40,46] and the results shown in Fig. 2
give lower values for �eff than that observed in the experi-
ment [30].
It is obvious that the inner cavity reduces the per-
mittivity of an isolated nanotube as compared to a solid
graphite cylinder of the same size. It was argued [40] that
for a periodic arrangement the effect of the inner cavity is
less than that for a single cylinder and even can be ig-
nored, if the ratio between the inner and outer radii $ does
not exceed 0.4. This conclusion was supported by com-
paring the results of the Maxwell-Garnett approximation
Eq. (27) and numerical band structure calculations. In
Fig. 4 we plot the dielectric constant for a square lattice of
hollow carbon nanotubes and compare the exact results
obtained from Eqs. (20), (21), and (26) (shown by the cir-
cles) with the results given by the Maxwell-Garnett ap-
proximation (squares). One can see that, for the same
outer radius, the effective dielectric constant drops with
an increase of the inner radius. Thus, if the outer radius is
fixed, the dependence on the inner radius cannot be ig-
nored, even in the Maxwell-Garnett approximation. How-
ever, the effective dielectric constants exhibits much less
sensitivity to the internal radius if it is plotted against
filling fraction, Fig. 5.
In the Maxwell-Garnett approximation (27) there is no
dependence on the parameter $, therefore, this approxi-
mation is represented by a single curve in Fig. 5. Here,
only the total amount of the dielectric material is impor-
tant (i.e., the filling fraction of the carbon), but not the to-
pology of the cylinders. In our exact theory the effective
dielectric constant depends on the details of the micro-
structure of the photonic crystal, but as far as the filling
fraction is concerned, the topology plays a much less im-
portant role. Since the cylinder is uniquely determined by
either two parameters out of three, R, $, and f , the curves
in Fig. 5 may cross each other. This means that at the
crossing point the values of f and $ correspond to the
same hollow cylinder. This can be easily seen from
Eq. (28).
7. Conclusions
We calculated the low-frequency dielectric tensor for
2D photonic crystal of optically anisotropic parallel cy-
linders arranged in a periodic lattice in the perpendicular
plane. The exact analytical formula for the principal val-
ues of the dielectric tensor was obtained. The results are
1078 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
L.N. Gumen and A.A. Krokhin
2.2
2.0
1.8
1.6
1.4
1.2
1.0
� e
ff
0.20 0.25 0.30 0.35 0.40 0.45
R/a
$ = 0
$ = 0.1
$ = 0.3
$ = 0.5
$ = 0.7
$ = 0.9
Fig. 4. The plot of the effective dielectric constant for square
lattice of carbon nanotubes versus the outer radius for tubes
with different ratios of the inner and outer radii, $ � 0.1, 0.3,
0.5, 0.7, 0.9. The exact results are shown by circles and the re-
sults of the Maxwell-Garnett approximation are shown by
squares.
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Filling fraction
� e
ff
$ = 0
$ = 0.1
$ = 0.3
$ = 0.5
$ = 0.7
$ = 0.9
Maxwell-Garnett
Fig. 5. A graph of the effective dielectric constant for a square
lattice of carbon nanotubes versus filling fraction for tubes
with different ratios of the inner and outer radii, $ � 0.1, 0.3,
0.5, 0.7, 0.9. The exact results are shown by circles and the
Maxwell-Garnett approximation is shown by squares.
applied for the periodic arrangement of carbon nanotubes
which are rolled up from uniaxial graphite crystal with
static values of the dielectric tensor � || � 1.8225 and
�� � 5.226. It was shown that the interior (vacuum) re-
gion of the nanotubes has a small effect on the dielectric
properties of the photonic crystal and can be ignored. Al-
though we are interested in the static dielectric tensor, it is
clear that the developed long-wavelength limit approach
remains valid, even for optical frequencies since the pe-
riod of the lattice of carbon nanotubes d �10 nm is much
less than the optical wavelength % 6 500 nm. To calculate
the dynamic dielectric tensor, one has to substitute in the
general formula Eq. (18) the corresponding frequency-
dependent values for � || and �� . Of course at finite fre-
quencies Eq. (18) gives the real part of the dielectric func-
tion. Calculations of the imaginary part require a genera-
lization of the presented theory. This result will be
reported elsewhere.
The exact theory presented here allows a calculation of
the effective dielectric constant of carbon nanotubes im-
bedded in a gas. Due to high absorbability of nanotubes,
the concentration of gas in the interior region of the
nanotubes may be different from that in the atmosphere.
This leads to slightly different dielectric constants of the
material in the interior and exterior regions of the cylin-
ders. This effect can be registered by precise measure-
ments of the shift of the resonant frequency of a resonant
cavity [47]. Thus, the proposed theory may find applica-
tions in the microwave detection of Poisson gases in the
atmosphere.
One more interesting application of carbon nanotube
photonic crystal is related to its huge anisotropy of the ef-
fective dielectric constant. Recently Artigas and Torner
[11] demonstrated that the electromagnetic surface wave
(Dyakonov wave [38]) can propagate along the surface of
a photonic crystal with high optical anisotropy. This wave
propagates in a lossless dielectric medium and decays
much slower than surface plasmon-polariton. Since crys-
tals with huge optical anisotropy are rare in nature, car-
bon nanotube photonic crystals may be considered as a
promising material for integrated photonic circuits where
information is transmitted by surface modes.
Acknowledgment
This work is supported by the US Department of En-
ergy grant # DE-FG02-06ER46312.
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