The tensor solution for the electromagnetic wave being in quadrature with the surface of the cholesteric liquid crystal under the influence of the external electric field along axis of swing of crystal, has been found. Description of the reflection, transmission of the electromagnetic wave in multilayer structures composed from cholesteric liquid crystals have been shown. Futher the dependence of the reflection spectrum on the polarization of the incident beam and magnitude of the external electric field are determined

Dung, Pham Thuy

Hang, Pho Thi Nguyet

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 19, No. 1 (2009), pp. 26-32
THE PROPAGATION OF THE ELECTROMAGNETIC WAVE
IN MULTILAYERS STRUCTURES COMPOSED FROM
CHOLESTERIC LIQUID CRYSTALS UNDER THE
INFLUENCE OF THE EXTERNAL ELECTRIC FIELD
PHO THI NGUYET HANG
Institute of Engineering Physics, Hanoi University of Technology
PHAM THUY DUNG
Vietnam Research Institute of Electronics, Informatics, Automation
Abstract. The tensor solution for the electromagnetic wave being in quadrature with the surface
of the cholesteric liquid crystal under the influence of the external electric field along axis of swing
of crystal, has been found. Description of the reflection, transmission of the electromagnetic wave
in multilayer structures composed from cholesteric liquid crystals have been shown. Futher the
dependence of the reflection spectrum on the polarization of the incident beam and magnitude of
the external electric field are determined.
I. INTRODUCTION
Recently, liquid crystal in general and cholesteric liquid crystal, in particular, had
attracted great interests from both theoreticians and experimenters since their physical
properties are enable to be changed under the influence of some external actions, such as
electromagnetic field, stresses, temperature, . . . This significant property has been widely
applied in various technological devices. A sery of recent works have studied the effects of
external field on twist step of cholesteric liquid crystal [1-4], however all these works had
not consider the external effects on propagative characterizes of electromagnetic wave on
liquid crystal.
Noticed that the cholesteric liquid crystal is stratified anisotropic material, so in
the general case of the oblique electromagnetic wave in the crystal plate, the exactly so-
lution of the Maxwell’s equations can not be found. Theories of the propagation of the
electromagnetic wave in the cholesteric liquid crystal had been shown in the works [3-9].
These authors had used various approximate methods such as the geometrical – optics
approximation, the methods of part layers (i.e separate the plate into thinner layers which
are assumed as homogenous plates, then used the method of Jonnes’ matrixs to survey
the propagation of the electromagnetic wave in multilayers structure, without considering
the repeated reflection on the space between layers). Thereby, the problem of the elec-
tromagnetic wave being in quadrature with the surface of the cholesteric liquid crystal
plate was studied and found out analytical solution in [3, 7, 8]. However, these works
do not mention the problem of multilayers structures composed of liquid crystal that in
these cases the calculation is too complex. For example, to solve the problem of three
THE PROPAGATION OF THE ELECTROMAGNETIC WAVE ... 27
layers structure composed from a cholesteric liquid crystal plate between two isotropic
homogenous dielectric layers, we have to solve 32 equations for 32 unknown [8].
In works on crystal optics [10, 11], F.I. Fedorov, L.M. Barkovsky, and G.N. Borzdov
had proposed a method called operation method which allows to describe anisotropy of
material promoted general tensors form without using concrete coordinate axies. There-
fore, in the problems of propagation of the electromagnetic wave in multilayer structures
composed from anisotropic materials, using operation method is very compactly and effec-
tively so that the field of reflection and refraction wave have been expressed through the
field of incident wave only by tensor transformations and interesting repeated reflection
on the space between layers.
In the present paper, by using the above mentioned operation method, we have
derived the exactly solution for the electromagnetic wave that is in quadrature with the
surface of the cholesteric liquid crystal under the influence of the external electric field
along the axis of swing of crystal. Consequently, the problems of reflection, transmis-
sion of the electromagnetic wave in multilayer structures composed from cholesteric liquid
crystals has been solved. Furthermore, the dependence of the reflection spectrum on the
polarization of the incident beam and magnitude of the external electric field is investi-
gated.
II. THE ELECTROMAGNETIC WAVE IN CHOLESTERIC
LIQUID CRYSTAL PLATE
Let us consider a monochromatic plane wave transmitting from the isotropic mate-
rial into the cholesteric liquid crystal perpendicularly. Assume that the swing of crystal is
perpendicular to interface. Chose the swing of crystal is Oz axis, z = 0 at interface. The
external electric field vector has intensity E, along the axis of swing of crystal. Optical
properties of cholesteric liquid crystal is characterized by permittivity tensor ε (z)
ε (z) = S(z)εS˜(z), (1)
where ε = ε(0) is permittivity tensor at z = 0, S(z) is matrix rotating around Oz axis,
S˜ij = Sji, S(z) is described by
S (z) = exp
[
ϕ(z)q×
]
= I cosϕ(z) + sinϕ(z)q× + ~q ⊗ ~q. (2)
with ~q is unit real vector, normal to the interface (Oz), I = 1− ~q ⊗ ~q , (~q ⊗ ~q)ij = qiqj, ~q×
is an antisymmetric tensor of second rank, dual to the vector ~q, ϕ (z) =
pi
L
z, L is step of
spiral, L depends on external electric field intensity [3]:
L = L0
[
1−
(
E
Ec
)3]
, (3)
L0 is step of spiral when external electric field is zero, Ec is the critical value of electric field
intensity. When external electric field increases to the critical value Ec, spiral structure of
28 PHO THI NGUYET HANG AND PHAM THUY DUNG
cholesteric liquid crystal will be destroy completely. Maxwell’s equations takes the form:
d
dz
(
~Ht
~q× ~E
)
= ikM(z)
(
~Ht
~q× ~E
)
, (4)
where M(z) =
(
0 B(z)
I 0
)
and B(z) =
Iε¯(z)I
εq(z)
,
k is wave number, ~Ht, ~Et are tangential components of electric field intensity vector ~E
and magnetic field intensity vectors ~H, ~Ht = I ~H, ~Et = I ~E, εq = ~qε~q, ε¯ is the mutual
tensor of ε. The solution of the equation (4) has been found in following form [12]:(
~Ht(z)
~q× ~E(z)
)
= P
(
~Ht(0)
~q× ~E(0)
)
, (5)
where P =
z∫
0
(E + ikM(z)dz).
P is matrizant (multiplicative integral, integral exponent) [12]. The specific nature
of the multiplicative integral has a connection with the noncommutation of the operators
M(z). If operators M(z,) and M(z,,) are permutable at two arbitrary points z, and z,,:
M(z,)M(z,,) = M(z,,)M(z,); z,, z,, ∈ [z0, z], then the multiplicative integral reduces to
the operator P = exp
{
z∫
z0
iM(z)dz
}
. In the general case, the multiplicative integral is
calculated following [12]:
P =
z∫
0
(E + ikM(z)dz) = E +
z∫
0
ikM(z)dz +
z∫
0
ikM(z)dz
z∫
0
ikM(z1)dz1 + ... (6)
When the electromagnetic wave being in quadrature with the surface of the cholesteric
liquid crystal, having thickness l, the analytical expression of multiplicative integral will
be determined.
Using formula of multiplicative derivative [12]
DzX =
dX
dz
X−1 (7)
we can rewritten the multiplicative integral (5) by following form:
P =
l∫
0

I +



 −piL~q× ikSB(0)S˜
ikI −pi
L
~q×

+Dz

 exp
(pi
L
l~q×
)
0
0 exp
(pi
L
l~q×
)



 dz


(8)
Using integrate of parts formula for multiplicative integral
t∫
t0
[E + (Q+DtX) dt] = X(t)
t∫
t0
[
E +X−1QXdt
]
X(t0) (9)
THE PROPAGATION OF THE ELECTROMAGNETIC WAVE ... 29
(8) becomes
P =

 exp
(pi
L
l~q×
)
0
0 exp
(pi
L
l~q×
)

 exp

 −piLl~q× iklB (0)
iklI −pi
L
l~q×

 (10)
P is a characteristic matrix of the crystal plate. With matrix P , we can derive the relation
among the field vectors on the first and the last boundaries of the plate. Besides, the
problems of reflection, transmission of the electromagnetic wave in multilayer structures
composed from cholesteric liquid crystals can been solved.
III. THE REFLECTION SPECTRUM OF THE ELECTROMAGNETIC
WAVE IN MULTILAYERS STRUCTURES COMPOSED FROM
CHOLESTERIC LIQUID CRYSTALS
Consider a multilayer structure that consists of sequentiality alternate two compo-
nents, one from cholesteric liquid crystal layers, other from isotropic dielectric layers under
an external electric magnetic field. The interested system is in a isotropic medium with re-
fractive index n0. The liquid crystal layers have thickness l1, characterized by permittivity
tensor ε (z) (1). The isotropic dielectric layers have thickness l2 and refractive index n. In
the case the electromagnetic wave being in quadrature with the surface of the interested
system, we obtain the following expression of reflection and transmission tensors [11]:
R =
[
(−n0I, I)P
(
I
−n0I
)]
−
[
(n0I,−I)P
(
I
n0I
)]
,
D = 2n0I
[
(n0I, I)P
−1
(
I
n0I
)]
−
. (11)
where P = (P1P2)
N , N is period of the system, P1 is characteristic matrix of the crystal
plate, given by the formula (11), P2 is characteristic matrix of the isotropic dielectric plate,
given by the following formula:
P2 = exp (ikl2M2) , M2 =
(
0 n2I
I 0
)
(12)
In practice, we are interested in intensity and polarization of the reflection and
transmission waves through the system, concretely we will determine intensity of the
transmission wave through the system consisting of polarizer, optical system and analyzer.
The polarizer is characterized by dyad (12)
∏
= ~P ⊗ ~P ∗,
∣∣∣ ~P ∣∣∣2 = 1, the analyzer is
characterized by dyad A = ~A⊗ ~A∗, |A|2 = 1. Assume that intensity of the incident wave
equal to unit then intensity of the reflection wave JR and the transmission wave JD are
in form [13]:
JR =
∣∣∣ ~P ∗R~P ∣∣∣ , JD = ∣∣∣ ~A∗D~P ∣∣∣ (13)
With ~P ∗ is conjugate complex of ~P . The formula (13) is using for elliptic polarizer and
analyzer arbitrarily. For linear polarizer, P and A are real vectors. ϕ1 is the angle
between vectors ~P and ~b0, where ~b0 is the unit vector in incident plane and in quadrature
30 PHO THI NGUYET HANG AND PHAM THUY DUNG
with ~q, then ~P = cosϕ1~b0 + sinϕ1~b
×
0
~q. ϕ2 is the angle between vectors ~P and ~A, ~A =
cosϕ2 ~P +sinϕ2 ~P
×~q. For circular polarizer, P and A are complex vectors ~P 2 = 0, ~A2 = 0,
we have ~A = ~P =
√
2
2
(
~b0 ± i~a0
)
,with + is correspond with waves polarizing right –
circular and - is correspond with waves polarizing left – circular.
To calculate, we use the parameters of system:
ε1 = 2, 290; ε2 = 2, 143; L0 = 20µm; l1 = 25µm, n=2, 417; l2 = 100µm, no = 1,
(14)
ε1, ε2 are main values of tensor ε of cholesteric liquid crystal plate.
ε = ε1~a0 ⊗~a0 + ε2~b0 ⊗~b0 + ε2~q ⊗ ~q, (15)
6.03 6.035 6.04 6.045 6.05
x 10
-7
0
0.2
0.4
0.6
0.8
1
Buoc song
C
u
o
n
g
d
o
s
o
n
g
p
h
a
n
x
a
E=0
E/Ec=0.55
E/Ec=0.60
R
ef
le
c
ti
o
n
in
te
n
si
ty
Wavelength
Fig. 1. The dependence of the reflection intensity on wavelength for the different
values of the magnitudes of the external electric field when the incident beam is
linear polarization with ϕ1 = 30
0, ϕ2 = 15
0, N = 7
Fig. 1 show the spectrum of the reflection wave’s magnitude for the different values
of the external electric field when the incident wave polarizing linearly is perpendicular
to the multilayer structure that consists of sequentiality alternate two components, one
from cholesteric liquid crystal layers, other from isotropic dielectric layers (the period
of the system N = 7). Fig. 2 show the dependence of the reflection wave’s magnitude
∆R = R− R0 on wavelength for the different values of the external electric field. Where
alternately R0 and R is the reflection wave’s magnitude when the external electric field is
zero and is not equal zero.
From figures, we have proved that the reflection intensity varies following the mag-
nitudes of the external magnetic field. The change is clearer when the magnitude of the
external electric field is up to 50 per cent of the critical value of electric field intensity
(Fig. 1). When E/Ec < 0.03, the deference between R and R0 is insignificant ∆R < 0.06.
(Fig. 2).
THE PROPAGATION OF THE ELECTROMAGNETIC WAVE ... 31
6.035 6.04 6.045 6.05 6.055 6.06 6.065
x 10
-7
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Buoc song
D
R
E/Ec=0.1
E/Ec=0.2
Wavelength
Fig. 2. The changes of the reflection intensity ∆R on wavelength for the different
values of the magnitudes of the external electric field when the incident beam is
linear polarization with ϕ1 = 30
0, ϕ2 = 15
0, N = 7
IV. STUDY ON THE POLARIZATION OF REFLECTION AND
TRANSMISSION WAVES
The reflection and transmission intensity ~HR, ~HD vectors are expressed by means
of the reflection and transmission tensors in following form:
~HR = R ~H0 ~H
D = D ~H0 (16)
The polarization of the reflection and transmission waves are described in [10]. For
reflection wave, we use the following form:
γR =
∣∣∣( ~HR)2∣∣∣∣∣∣( ~HR)∣∣∣2 , (17)
For transmission wave, polarization is described as
γD =
∣∣∣( ~HD)2∣∣∣∣∣∣( ~HD)∣∣∣2 (18)
When the incident wave is linear polarization, then γ = 1. If the incident wave is
elliptic polarization 0 < γ < 1
Fig. 3 show the dependence of the polarization of the reflection wave on wavelength
for the different values of the magnitudes of the external electric field when the incident
beam is in quadrature with the surface of the system consist of a liquid crystal plate on an
isotropic dielectric layer. The parameters of system is used in (14). From the figure, we
have proved that the more magnitude of the external magnetic field is, the more obviously
the polarization of reflection wave is.
32 PHO THI NGUYET HANG AND PHAM THUY DUNG
6.05 6.06 6.07 6.08 6.09 6.1
x 10
-7
0.7
0.75
0.8
0.85
0.9
0.95
1
Buoc song
D
o
p
h
a
n
c
u
c
c
u
a
s
o
n
g
p
h
a
n
x
a
E/Ec=0.1
E/Ec=0.4
Fig. 3. The dependence of the polarization of the reflection wave on wavelength
for the different values of the magnitudes of the external electric field when the
incident beam is linear polarization with ϕ1 = 30
0, ϕ2 = 15
0
V. CONCLUSION
In a summary, using the operation method, we have found the exactly solution
for the electromagnetic wave be in quadrature with the surface of the cholesteric liquid
crystal under the influence of the external electric field, along axis of swing of crystal. As
a result, we have solved the problems of reflection, transmission of the electromagnetic
wave in multilayer structures composed from cholesteric liquid crystals in analytical-tensor
form without any approximation explaining the specific of the repeated reflection on the
space between layers. Our results therefore can used for arbitrary multilayer structures
composed from different cholesteric liquid crystals.
REFERENCES
[1] V. A. Beliakov, V. E. Dmitrienco, Physika Tverdovo Tela, 17 (2) (1975)491-495.
[2] S. M. Osadchi, Crystallographia, 29 (5) (1984) 976-983.
[3] V. A. Beliakov, A. C. Conhin, Optika Cholestericheskix Judkix Crystal, Moscow: Nauka, 1982, 360 p.
[4] A. A. Gevorgan, J. Tekhn. Phys., 70 (9) (2000) 75-82.
[5] I. V. Valukh, A. V. Clobodanuk, X. I. Valukh, E. Osterman, K. Scary, Opt. J., 70 (7) (2003) 24-28.
[6] M. D Arutinhan., G. A. Vardanhan, A. A. Gevorgan, Opt. J., 74 (4) (2007) 16-23.
[7] G. A. Vardanhan, A. A. Gevorgan, Crystalographia, 42 (4) (1997) 790-797.
[8] A. A. Gevorgan, K. V. Papoian, O. V. Pikichan, Optika i Spectroskopia, 88 (2000) 647-655.
[9] Pho Thi Nguyet Hang, L. M. Barkovsky, Proc. Hanoi Univ. Tech., Vietnam (1996)11-18.
[10] F. I. Fedorov, Cheoria Gyrotropy, Minsk: Nauka i Tekhnica,1976, 456 p.
[11] L. M. Barkovsky, G. N. Borzdov, J. Phys. A. Math. Gen., 20 (1987) 1095-1106.
[12] F. R. Gantmacher, Theory of Matrices, Moscow: Nauka, 1988, 548 p.
[13] G. N. Borzdov,Vesci Academi Nauk BSSR, 3 (1977) 85-90.
Received 02 August 2008.


The Propagation of the Electromagnetic Wave in Multilayers Structures Composed from Cholesteric Liquid Crystals under the Influence of the External Electric Field

http://vjs.ac.vn/index.php/cip/article/download/235/pdf%28Vietnamese%29

The Propagation of the Electromagnetic Wave in Multilayers Structures Composed from Cholesteric Liquid Crystals under the Influence of the External Electric Field

Abstract

Similar works

Full text

Available Versions

Vietnam Academy of Science and Technology: Journals Online