Polarization gaps and negative group velocity in chiral phononic crystals:
Layer multiple scattering method
Huanyang Chen,
1,2
Kin Hung Fung,
1
Hongru Ma,
2
and C. T. Chan
1
1
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
2
Institute of Theoretical Physics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Received 18 January 2008; revised manuscript received 21 May 2008; published 20 June 2008
We designed chiral phononic crystals and studied their properties by using the layer multiple scattering
method. The transmittance curves and the corresponding band structures show that this kind of structure
possesses significant polarization gaps. The chiral structures break the symmetry so that the degenerate trans-
verse modes split into a pair of right-hand polarized mode and left-hand polarized mode. The polarization
splitting in the low-frequency range is enhanced in systems of large filling ratios. We also demonstrate that
chiral structures containing strongly resonant units can induce negative group velocity in elastic waves, and
stronger resonance brings a wider band of negative group velocity.
DOI: 10.1103/PhysRevB.77.224304 PACS numbers: 63.20.-e, 43.35.+d
I. INTRODUCTION
Studies on the properties of photonic and acoustic mate-
rials with chiral structures attracted growing interest for their
special characteristics, such as negative refraction
1,2
and po-
larization gaps.
3–7
Since the symmetry between the right-
hand and left-hand circularly polarized waves is broken by
chiral structures, the two polarizations travel with different
speeds and one of the circular polarizations can potentially
be blocked by the polarization gaps or chiral Bragg gaps.
Significant polarization gaps for circularly polarized electro-
magnetic EM waves in spiral photonic crystals have been
observed.
3
Such chiral structures can give exotic thermal ra-
diation properties.
8
Similar results about polarization gaps of
the circularly polarized transverse CPT elastic waves were
also observed in one kind of materials called piezoelectric
continuously twisted structurally chiral medium PCTSCM
proposed by Lakhtakia.
9
Recently, a kind of acoustic
metamaterials
10
has been demonstrated to realize the nega-
tive refraction in acoustic waves, which is similar to the
original idea of negative refraction in EM waves proposed by
Veselago.
11
This kind of metamaterials is named the “double-
negative acoustic metamaterials” due to their simultaneously
negative effective bulk modulus and mass density. Such kind
of metamaterials possesses a passband of negative group ve-
locity that can lead to negative refraction. In addition to
double negativity, there are various mechanisms that can
cause negative group velocity and/or negative refraction,
such as Bragg scattering and chiral structures.
1,7,12
The nega-
tive refraction induced by Bragg scattering has already real-
ized in acoustic waves.
13
However, neither negative group
velocity nor negative refraction induced by chiral structures
has been explicitly demonstrated. There are very few studies,
especially experimental ones, which focus on CPT elastic
waves. Given that a technique to generate CPT elastic waves
was proposed as early as 1964 by Einspruch,
14
the study of
the CPT elastic waves should be given more consideration
both experimentally and theoretically, in view of the surge in
the interest of artificial wave-functional structures such as
phononic crystals. In this paper, we show the existence of
polarization gaps circular Bragg gaps for CPT elastic
waves in a layer-by-layer chiral phononic crystal and dem-
onstrate that there exists a passband of negative group veloc-
ity with controllable bandwidth when some locally reso-
nant units
15
are included in the structure. We then have a
more complete picture that links special structures to wave
properties. We note that an artificial helical structure with
one sphere per two-dimensional unit cell might also be able
to generate negative refraction, as suggested in EM waves.
7
Such a structure was originally suggested by Karathanos et
al.
16
as an artificial helical material that can be used for
rotating the plane of polarization of linearly polarized light.
Here, we propose a chiral structure with two spheres per
two-dimensional unit cell so that more flexibility is allowed
to generate the proposed results in elastic waves. The paper
is arranged as follows. In Sec. II, we will describe the struc-
ture of the chiral phononic crystal. In Sec. III, we will show
the existence of polarization gaps in such structures. In Sec.
IV, we will demonstrate that the negative group-velocity
bands can be found in the chiral structure embedded with
resonators. In Sec. V, we will give the conclusions.
II. CHIRAL PHONONIC CRYSTAL
The method of calculation is a layer multiple scattering
theory MST, which was formulated by Sainidou et al.
17
and extended by Chen et al.
18
The method was used to cal-
culate transmission properties and band structures of
phononic crystals made of layers that consist of two kinds of
nonoverlapping spheres in the two-dimensional unit cell of
each layer.
18,19
Here we consider a phononic crystal gener-
ated by layers of spheres arranged in a hexagonal lattice Fig.
1a with a basis of two spheres. We start with the primitive
vectors a
1
= 1,0,0a
0
and a
2
= 1 / 2,
3 / 2,0a
0
, where a
0
is
the lattice constant of the structure. The two identical spheres
of a basis are positioned as 0, 0, 0 and 0.5a
0
,0,0 and this
forms one layer. The successive layers are formed using the
operation |R|T, where the rotation R is an anticlockwise
120° rotation and T = 0,0, d. The structure repeats itself af-
ter every three layers in the vertical direction, and the struc-
ture of the first, second, and third layers are depicted in Figs.
1b–1d, respectively. The chiral phononic crystal is de-
PHYSICAL REVIEW B 77, 224304 2008
1098-0121/2008/7722/2243045 ©2008 The American Physical Society 224304-1