ZONE STRUCTURE AND POLARIZATION PROPERTIES OF THE STACK OF A METAMATERIAL-BASED CHOLESTERIC LIQUID CRYSTAL AND ISOTROPIC MEDIUM LAYERS
A. H. Gevorgyan"*, G. K. Matinyanh
" Department of Physics, Yerevan State University 0025. Yerevan. Armenia
bArmenian Agrarian State University 009, Yerevan, Armenia
Received May 26, 2013
The optical properties of a stack of metamaterial-based cholesteric liquid crystal (CLC) layers and isotropic medium layers are investigated. The problem is solved by a modification of Ambartsumian's layer addition method. CLCs with two types of chiral nihility are defined. The peculiarities of the reflection spectra of this system are investigated and it is shown that the reflection spectra of the stacks of CLC layers of these two types differ from each other. Besides, in contrast to the single CLC layer case, these systems have multiple photonic band gaps. There are two types of such gaps: those selective with respect to polarization of the incident light and nonselective ones. It is shown that the system eigenpolarizations mainly coincide with the quasi-orthogonal, quasi-circular polarizations for normally incident light, except the regions of diffraction reflection selective with respect to the polarization of incident light. The influence of the CLC sublayer thicknesses, the incidence angle, the local dielectric (magnetic) anisotropy of the CLC layers, and the refractive indices and thicknesses of the isotropic media layers on the reflection spectra and other optical characteristics of the system is investigated.
DOI: 10.7868/S0044451014050115
1. INTRODUCTION
Material science was energetically developing recently, and its part concerning the optical materials was developing even more energetically. In particular, metamaterials are of great interest. Metamaterials are artificial composites containing sublongwave structures and exhibiting new linear and nonlinear optical properties such as negative refraction, reverse Doppler effect, electromagnetic energy propagation in the direction opposite to the wave vector, and so on fl 9]. They have surprising applications to perfect lenses [10], invisible cloaks fll 16], perfect absorbers [17], etc.
Investigations of photonic crystals (PCs) are still of great interest both for their wide application in science and techniques and for developing the modern technology of creating new media. They have a photonic band gap (PBG) in their transmittance spectrum that can be changed either by external fields or byE-mail: agevorgyan'&ysu.am
the changes in the crystal internal structure [IS 20]. The optical devices based on PCs have such properties as multifunctionality and tunability, compactness and low energetic losses, high reliability and good compatibility with other optical devices. Cholesteric liquid crystals (CLCs) are known as PCs with easily tunable parameters (their parameters can be tuned by external electric, magnetic, and strong light fields, thermal gradients, or UV radiation, etc.) A CLC is a self-assembled PC formed by rod-like molecules, including chiral molecules that arrange themselves in a helical fashion. The CLC has a single PBG and an associated one-color reflection band for circularly polarized light with the same handedness as the CLC helix (at normal light incidence). On the other hand, PCs with multiple (polychromatic) PBGs are attracting much attention recently. They find wide application, in particular, in display industry.
Multiple PBGs of one-dimensional structures containing CLC and isotropic layers were reported in some theoretical and experimental works [21 23]. Analoguos investigations of one-dimensional multilayer structures
containing CLC and anisotropic layers wore carried out in [24]. In [25], quasi-periodic systems described by the Fibonacci sequence and containing CLC layers were investigated. The multiple PBGs are also formed in a stack containing right- and left-hand CLC layers [26 30]. In [31], the reflection and polarization peculiarities of stacks of CLC and isotropic media layers were investigated.
Recently, the chiral nihility media have become interesting. The concept of chiral nihility in electromag-netism was introduced by Lakhtakia [32], as a medium in which both dielectric and magnetic permittivities are zero. The nihility concept was then applied to isotropic chiral metamaterials in [33], whose peculiarities have recently been energetically investigated in [34 37] (also see the references in them).
In this paper the concept of nihility is generalized to structurally chiral media (such as CLC) and, using it, the peculiarities of a stack formed by CLC layers with nihility and isotropic media layers are investigated.
2. THE METHOD OF ANALYSIS
The problem is solved by a modification of Ambart-sumian's layer addition method [26,38]. According to [26,38], if there is a system containing two layers, A and B, stacked up "from left to right", then the reflection and transmission matrices of the system A li. denoted by Ra+b and Ta+b< are defined by the analogous matrices of the separate layers as follows:
11\.........H = 11\ \ l \ 11 n II — 11 \ It n l\ .
; -1 W
/ I.........H = I ji \I — 11 \ 11 n l \ ,
where J is the unit matrix and the tilde denotes the reflection and transmission matrices of the reverse light propagation. The same matrices for the reverse propagation of light are defined by the matrix equations
Ra+b = Rb + TgRa \ I ~ RbRa\ T g.
(2)
- r- - - { 1
l\.........g — l \ \I — Rg Ra T g.
In the case where the subject layer borders the same medium on its both sides, the reflection and transmission matrices for the incidence "from right to left" are related by
T = F^TF, R = F^RF, (3)
where F =
tions and F =
for linear base polariza-for circular base polariza-
tions. The exact reflection and transmission matrices for a finite CLC layer (for normal light incidence) and an isotropic layer of a finite thickness are well known [39,40].
Transmission/reflection through a stack of CLC layers and isotropic medium layers is calculated using matrix equations (1) by successively applying them to the new layers added to the stack; the stack was considered as layer A and the added layer as layer B. Hence, to organize the calculations more conveniently, system (1) is presented in the form of difference matrix equations
I'J '-j-tjl'j i [I-fjRj-1 0
IJ 'J
(4)
with R0 = 0 and T0 = /. Here, Rj, Tj, Rj-i, and are the reflection and transmission matrices for the systems with j and j — 1 sublayers respectively, and fj and tj are the reflection and transmission matrices for the jth sublayer and 0 is the zero matrix.
It is to be noted that in [41,42], a new method for solving the problem of light propagation through a one-dimensional layer system was described.
We now pass to the eigenpolarizations (EPs) and eigenvalues of the amplitude. As is known, EPs are the two polarizations of the incident wave that do not change when passed through the system, and the eigenvalues are the amplitude coefficients of reflection and transmission for the incident light with the EPs [38,40]. The EPs and eigenvalues deliver much information about the interaction of light with the system; therefore, their calculation is important for every optical system. It follows from the definition of EPs that they must be connected with the polarizations of the internal waves (eigenmodes) excited in the medium (they mainly coincide with the polarizations of eigenmodes). Our investigations show, in particular, that in homogeneous media and CLC (for the normal incidence) for which the exact solution is known and hence the polarizations of the eigenmodes are known, the EPs practically coincide with the polarizations of eigenmodes. As is known (in particular, for normal incidence), the EPs of CLCs or gyrotropic media practically coincide with the orthogonal circular polarizations, whereas for nongyrotropic media, they coincide with the orthogonal linear polarizations. It follows from the foregoing that the investigation of the EP peculiarities is especially important in the case of nonhomogeneous media, for which, in general, the exact solution of the problem is not known.
We let the ratio of the field complex amplitude components at the entrance of the system be denoted by \, (xa = E- /Ef) and that at the exit of the system by \i (xi. = Ef / Ef), and take into account that
Ef ' Tu T12 ' ' E? '
Ef . T21 T22 E?
The ellipticity e1:-2 and the azimuth of the EPs
are expressed in terms of xj,2 as
('i,2 = tg I - arcsm
1
1 - I \ 1,2 I
2 I m \
1
hi
(7)
to obtain their relation:
Xt —
'!'■>■> \, + T-21 I \ > \, + Tu
(5)
The function, \, = /(\,). is called the polarization transfer function [40]; it carries information about the transformation of the polarization ellipse as light passes through the system. Every optical system has two EPs obtained from the definition of EPs: \, = \t. Hence, according to (5), the \ i and \ > EPs are given by
A: 1,2 =
T-22 — Tu i \/(T22 — Ti
11)
■ 4Ti2?21
2Ti
12
(6)
3. RESULTS AND DISCUSSION
Now we analyze the spectra peculiarities of the multilayer structure that is a stack of CLC layers with nihility and an isotropic medium layers (Fig. 1). We first discuss some properties of a single CLC layer and the possibility of generalizing the chiral nihility concept for CLCs. We assume that the electromagnetic wavelength is longer than the characteristic lengths of the subject metamaterial structure elements (of which the medium is composed), which allows considering the medium continuous and characterizing it in terms of the dielectric and magnetic permittivity matrices of the form
Ae A "
cos(2uz) sin(2uz
Hz) =
Ae
sin (2« î
0
Ae
cos(2uz) 0
0
= 2 J
A/; A//. \
Ii w. + — cos(2ac) — sm(2ac) 0
(8)
m =
A// .
sm ( 2a î
0
/'2 /
where
€1 + £•2 fl 1 + fl 2
£m = 2 ' tl,n = 2 '
£ 1 — £2 . /'1 — fl-2
£1 and £2 are the principal values of the dielectric tensor, f¡i and
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