Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air
Chul-Sik Kee, Jae-Eun Kim, and Hae Yong Park
Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea
Received 25 July 1997
We show that an absolute photonic band gap exists in a two-dimensional square lattice of square dielectric
rods in air. The index of refraction of the square dielectric rods must be larger than 2.65 to have the band gap.
This structure is a good candidate for two-dimensional photonic crystals in the IR or the visible to near-IR
wavelength range, and can be applied to high efficiency optoelectronic devices. S1063-651X9750312-6
PACS numbers: 42.70.Qs, 41.90.+e
It has been well known that periodic dielectric structures
photonic crystals can possess a frequency region in which
electromagnetic EM waves cannot propagate in any direc-
tion 1,2. This frequency region is called the photonic band
gap PBG, which is analogous to the electronic band gap
due to the spatial periodic electrostatic potential in natural
crystals. The absence of normal modes of EM waves inside
PBG can give rise to unusual physical phenomena, such as
the suppressed dipole-dipole interaction between atoms 3
and the photon-atom bound states 4. The enhancement of
the density of states of EM waves near PBG edge can im-
prove the performance of optoelectronic devices 5. Thus,
the search for photonic crystals generating PBG in two or
three dimensions has attracted a lot of attention 6,7. Pho-
tonic crystals have very useful and attractive properties
which semiconductor crystals do not. Defects in photonic
crystals are easily created by either adding other dielectric
materials to or removing dielectric material from a chosen
unit cell in the periodic lattice 8. This defect can create a
local mode of EM wave in PBG and act like a microcavity.
Thus, it is possible to tune the defect modes to any frequency
in PBG by designing the size, the shape, and the dielectric
constant of the defect.
Two-dimensional 2D photonic crystals can be fabricated
more easily than three-dimensional ones in the infrared IR
or the visible to near-IR wavelength ranges. For this reason,
attention may have been drawn towards 2D photonic crys-
tals. They have been mainly investigated for square and hex-
agonal lattices of air rods and dielectric rods with various
cross sections. It was reported that square and hexagonal
lattices of circular air rods can give rise to absolute PBG’s
9. A hexagonal lattice of single-circular dielectric rods in
air do not give rise to absolute PBG’s. On the other hand, it
has been recently known that a hexagonal lattice of two- and
three-basis circular dielectric rods in air can give rise to large
absolute PBG’s 10,11. In a square lattice, while a symme-
try breaking by changing the shape of square air rods to
rectangular reduces the width of the absolute PBG’s 9,a
symmetry reduction by placing an air rod of smaller diameter
at the center of each square unit cell enlarges it 11.
It was reported that a square lattice of square and circular
dielectric rods in air do not give rise to absolute PBG’s
9,12. However, we discuss in this paper that a square lattice
of square dielectric rods in air can have a sizable absolute
PBG at higher frequencies of EM waves. The absolute PBG
of a square lattice of square dielectric rods can occur when
the index of refraction of dielectric square rods is larger than
2.65, which is smaller than those of useful semiconductors.
The absolute PBG of a square lattice of square dielectric rods
lies in a higher frequency region than those of both a square
lattice of square circular air rods and a hexagonal lattice of
circular air rods, and exists over a wide range of filling frac-
tions.
We calculate the photonic band structure for the electric
field ( E polarization and the magnetic field ( H polarization
by the plane-wave method, where E ( H ) polarization means
the field parallel to the rod axis. The eigenvalue equations
are as follows:
det
A K, K' -
2
c
2
=0, 1
where
A K, K' =| K|| K'|
-1
K-K' 2
for the E polarization, and
A K, K' =K• K'
-1
K-K' 3
for the H polarization. Here, K=k+G, K' =k+G' , where
k is the wave vector in the first Brillouin zone, and G, G' the
reciprocal vectors.
-1
( K-K') is the Fourier transform of
the inverse of dielectric constant ( r). We use 797 plane
waves in our calculations. When the number of plane waves
was increased to 1297, the difference in the results was less
than 0.5%. Thus, we believe that the results are well con-
verged within at least 1% of their true value.
Figure 1 depicts the photonic band structure for a square
lattice of square dielectric rods in air. The index of refraction
of the rod is 3.5, which corresponds to that of GaAs in the IR
wavelength range, and a filling fraction of the dielectric rods
f =( d / a )
2
=41%, where d is the width of the square dielec-
tric rods and a the lattice constant. An absolute PBG occurs
where E
8
and H
6
gaps overlap, where E
i
and H
i
denotes the
gap that occurs between the i th and ( i +1)th bands for the
corresponding polarization. The shaded area represents the
absolute PBG. The midgap frequency
mid
=0.6696(2 c / a )
and the gap size =0.0370(2 c / a ). While the first gap of
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