We employ a homogenization technique based on the Lorentz electronic theory
to show that planar chiral structures (PCSs) can be described by an effective
dielectric tensor similar to that of biaxial elliptically dichroic crystals.
Such a crystal is shown to behave like a PCS insofar as it exhibits its
characteristic optical properties, namely, co-rotating elliptical polarization
eigenstates and asymmetric, direction-dependent transmission for
left/right-handed incident wave polarization.Comment: 3 pages, version as accepted in Optics Letters but before final
  shortening

Andrey V. Novitsky

Borzdov

Decker

Drezet

Fedotov

Kuwata-Gonokami

Lyons

Papakostas

Plum

Reichelt

Rockstuhl

Sergei V. Zhukovsky

Shalaev

Spielman

Vladimir M. Galynsky

Zhukovsky

English

arXiv

Crossref

Elliptical dichroism: operating principle of planar chiral metamaterials

arxiv.org/abs/0903.4126We employ a homogenization technique based on the Lorentz electronic theory to show that planar chiral structures (PCSs) can be described by an effective dielectric tensor similar to that of biaxial elliptically dichroic crystals. Such a crystal is shown to behave like a PCS insofar as it exhibits its characteristic optical properties, namely, corotating elliptical polarization eigenstates and asymmetric, direction-dependent transmission for left- or right-handed incident wave polarization

Zhukovsky, Sergei V.

Novitsky, Andrey V.

Galynsky, Vladimir M.

Электронная библиотека БГУ

1988 OPTICS LETTERS / Vol. 34, No. 13 / July 1, 2009Elliptical dichroism: operating principle of planar
chiral metamaterials
Sergei V. Zhukovsky,1,* Andrey V. Novitsky,2 and Vladimir M. Galynsky2
1Theoretical Nano-Photonics, Institute of High-Frequency and Communication Technology, Faculty of Electrical,
Information and Media Engineering, University of Wuppertal, Rainer-Gruenter-Strasse 21,
D-42119 Wuppertal, Germany
2Department of Theoretical Physics, Belarusian State University, Nezavisimosti Avenue 4, 220030 Minsk, Belarus
*Corresponding author: sergei@uni-wuppertal.de
Received March 24, 2009; revised May 3, 2009; accepted May 15, 2009;
posted June 1, 2009 (Doc. ID 109206); published June 24, 2009
We employ a homogenization technique based on the Lorentz electronic theory to show that planar chiral
structures (PCSs) can be described by an effective dielectric tensor similar to that of biaxial elliptically di-
chroic crystals. Such a crystal is shown to behave like a PCS insofar as it exhibits its characteristic optical
properties, namely, corotating elliptical polarization eigenstates and asymmetric, direction-dependent trans-
mission for left- or right-handed incident wave polarization. © 2009 Optical Society of America
OCIS codes: 050.2065, 160.1245, 160.1585, 160.1190, 350.3618, 050.6624.Metamaterials show promise for a wide range of un-
usual physical phenomena rare or absent in nature
[1,2], including giant optical activity [2–6]. Following
a pioneering work by the group of Zheludev and co-
workers in 2006 [7], planar chiral structures (PCSs)
were recently introduced as a distinct class of
metamaterials. They consist of planar elements [7–9]
that have a sense of twist and cannot be superim-
posed on their in-plane mirror image. PCSs are
asymmetric in electromagnetic wave propagation for
right-handed (RH) versus left-handed (LH) circularly
polarized incident waves. Unlike 3D or bilayer chiral
metamaterials [4,6,10], which resemble bi-isotropic
or gyrotropic media, PCSs change their properties if
the wave propagation direction is reversed [7,11].
They also differ from Faraday media (also known to
have related asymmetry) because PCSs have corotat-
ing elliptical polarization eigenstates [7,12], while in
bi-isotropic or Faraday media the eigenstates always
come in pairs of RH and LH polarizations.
In this Letter, we interpret optical properties of
PCSs in terms of elliptical dichroism. The properties
of chiral metamaterials are usually investigated on
the level of resonant electromagnetic response in an
individual element [8–13]. This heavily depends on
the element shape, a wide variety of which were re-
ported lately [7–9,12,13]. Although a recent Letter
[13] offers a generalized explanation based on
polarization-sensitive excitation of electric and mag-
netic dipoles, such a treatment should still be per-
formed separately for different PCS designs. Thus,
there is a need for a macroscopic description that
would make planar chirality available for studies on
an abstract crystallographic level, similar to other
optical phenomena like birefringence or optical activ-
ity.
The idea of regarding elliptical dichroism as a
mechanism behind the optical manifestation of pla-
nar chirality is suggested by earlier experimental ac-
counts of circular dichroism in PCSs [7,13]. We con-
firm its existence by using the Lorentz-theory
homogenization scheme in chiral split-ring (CSR)
0146-9592/09/131988-3/$15.00 ©structures [13–15]. Moreover, we show that a bulk el-
liptically dichroic medium exhibits all the character-
istic properties of PCS, namely, (i) corotating ellipti-
cal polarization eigenstates, (ii) asymmetric trans-
mission for RH versus LH circular polarization, and
(iii) enantiomeric asymmetry and change of proper-
ties for different wave propagation direction.
We begin by considering a CSR unit cell [Fig. 1(b)].
For planar (as opposed to 3D) elements, the electric
response is dominant because the magnetic moment
is parallel to the wave propagation direction [7].
Hence, one can arrive at homogenized material prop-
erties by considering the response within the frame-
work of the electronic Lorentz theory by using ˆeffE
=E− 4 / ieNv. N and v are the electron concen-
tration and velocity [16], which are position-
dependent in a structured material. To derive the ef-
fective dielectric tensor ˆeff, one needs to average over
the unit cell. If dR, the electrons in the ring can be
mobile only in the azimuthal direction; so vm=R˙e
Fig. 1. (Color online) (a) Bilayer (3D) versus monolayer
(planar) chiral metamaterial. (b) Unit cell of a CSR struc-
ture [13].
2009 Optical Society of America
July 1, 2009 / Vol. 34, No. 13 / OPTICS LETTERS 1989and ¨+˙+0
2= e /Rmee ·Em. The electrons in the
dielectric substrate are bound, and their velocity vd
in response to the electric field is position indepen-
dent. The averaging is then performed as
Nv = L−2
metal
Nmvmd2r + 
diel
Ndvdd2r .
1
For the structure shown in Fig. 1, integration in Eq.
(1) results in a complex symmetric tensor ˆeff:
ˆeff = 
x xy 0
xy y 0
0 0 z
	 . 2
Its components depend on the dielectric constants of
metal m and dielectric d, as well as on the
geometrical parameters, e.g., the angles 1,2 and 1,2
(Fig. 1). Figure 2 shows the dependencies x,y,xy. If
the split ring is symmetric (2=0, 1=2, or 1=2),
then xy=0 and tensor (2) supports two linearly po-
larized eigenwaves that are different in their attenu-
ation (linear dichroism). Such CSRs display no pla-
nar chirality [14,15]. Otherwise, xy0 and the
dichroism is elliptical. We further rewrite the tensor
ˆeff in axial representation [17] as
ˆ = o + e − oc  c, c = c + i	c, 3
where the vectors c and c determine the optical
axes, and c c denotes the outer dyadic product c
 cij
cicj.
At any fixed frequency, the medium’s polarization
eigenstates can be recovered from its normal refrac-
tion tensor NH [18]. For nonmagnetic media and nor-
mal incidence
NH2 = − q
ˆq
 + qˆq−1q
ˆq  qˆq
, 4
where q is a unit vector normal to the plane and q

is defined as q
u
q
u. The eigenvectors H1,2 of
NH2 represent the polarization states of the field that
are preserved as the wave propagates along q. If the
polarization is circular or elliptical, the handedness
of Hj is determined by the sign of the product
q · Re Hj
Im Hj [17]. The condition for corotation of
polarization eigenstates then becomes
 = q · Re H1
 Im H1q · Re H2
 Im H2 0.
5
Fig. 2. (Color online) Frequency dependencies for (a) real
and (b) imaginary parts of ˆeff in Eq. (2) for L=15 mm, R
=6 mm, d=0.8 mm, 1=140°, 2=160°, 1=40°, 2=20°.For anisotropic media without absorption, =0
(the eigenstates have linear polarization). For bi-
isotropic chiral or Faraday media, it can be shown
that 0, so circular or elliptical polarization eigen-
states are counterrotating. However, for ˆ of Eq. (3),
where c
ex cos +ey sin  and c
ex cos 
+ey sin , Eqs. (4) and (5) result in
 =
4	2 sin2 − 
1 + 	22 − cos 2 + 	2 cos 22
 0. 6
Figure 3 shows the dependence of  on the orienta-
tion of the axes c and c. One can see that polariza-
tion eigenstates are indeed corotating. They reverse
their handedness at the line =. Including gyrot-
ropy in ˆ would result in a negative addition to ,
thus counteracting planar chirality [12].
Next, we consider the transmission spectra of an
elliptically dichroic layer, which can be obtained by
constructing the wave evolution operator in a gener-
alized transfer matrix formalism [18]. Figure 4 shows
the spectra for the LH and the RH circularly polar-
ized incident wave. The transmission of LH versus
RH polarization is clearly asymmetric. For higher
frequencies, the spectra become flat. This is caused
by the absence of material dispersion (the Lorentzian
resonance is assumed to be broad), so that spectrally
uniform elliptical dichroism results in spectrally uni-
form planar chiral behavior. Real PCSs are disper-
sive and thus display planar chirality only in a nar-
row spectral range close to resonance (see Fig. 2), in
accordance with earlier findings [7,13].
Fig. 3. (Color online) The dependence  , as in Eq. (6)
for 	=0.2 with polarization eigenstates given by Eq. (4) at
points (a)–(h).
1990 OPTICS LETTERS / Vol. 34, No. 13 / July 1, 2009If the material is replaced with its enantiomeric
counterpart →− ,→−, or if the wave propa-
gates in the opposite direction q→−q, the LH and
RH plots are exchanged (Fig. 4). This agrees with ex-
perimental results for PCSs [13] and with the Lor-
entz homogenization scheme.
When an initially circularly polarized wave travels
through the layer, the wave is seen to assume the po-
larization matching one of the eigenstates in orienta-
tion and both eigenstates in handedness. So, a wave
whose handedness does not match that of the eigen-
states undergoes polarization conversion, as reported
earlier [7]. This can be understood from the combined
action of anisotropy (creating LH↔RH polarization
mixing) and dichroism (diminishing the circular po-
larization component not matching the eigenstates in
handedness).
The results obtained are complementary to a re-
cent account [13] linking planar chirality with
polarization-sensitive excitation of resonant modes in
CSRs. There, e.g., LH and RH waves were shown to
excite an electric dipole (well coupled to the field) and
a magnetic dipole (poorly coupled to the field), respec-
tively. For the latter, the energy is trapped in the
resonant mode and dissipates, resulting in circular
dichroism. The range of PCS designs where dichro-
ism is brought about by this mechanism remains to
be determined. It is also interesting to note that in
high-Tc anyon superconductors circular dichroism
was reported [19], yet no sign of Faraday-like non-
reciprocity was found [20], similar to PCSs. This
striking similarity may be attributed to the layered
structure of such superconductors.
To summarize, we have shown that bulk media
with elliptical dichroism exhibit optical properties
characteristic for PCSs. First, the elliptical polariza-
tion eigenstates are corotating (Fig. 3). Second, trans-
mission for a LH–RH-polarized incident wave is
asymmetric and is exchanged if the material is re-
placed with its enantiomeric counterpart (Fig. 4). Fi-
nally, a wave propagating in such a material under-
goes circular polarization conversion so as to match
the eigenstates. The medium is described solely by a
Fig. 4. (Color online) Transmission spectra for an LH–RH
circularly polarized wave passing through a slab with
thickness a made of material as in Eq. (3) for = ±45°, 
= ±10°, o=1.1, e=1.0, 	=0.2, along with the incident (in)
and transmitted (out) wave polarization at a /2c=100.complex symmetric dielectric tensor whose structureis given by homogenization of a split-ring PCS. This
way, it is shown that a crystallographic approach is
possible in studying PCSs, offering a theoretical
starting point for analyzing planar chirality as an op-
tical phenomenon, e.g., in bianisotropic multilayer
solvers [21].
The authors acknowledge helpful suggestions from
V. Fedotov, E. Plum, and D. N. Chigrin. This work
was supported in part by the Deutsche Fouschungs-
gemeinschaft (DFS) (FOR 557) and the Basic Re-
search Foundation of Belarus (F08MS-006).
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