Materials Sciences and Applications, 2010, 1, 32-35
doi:10.4236/msa.2010.11006 Published Online April 2010 (http://www.SciRP.org/journal/msa)
Optical Rotation of Linearly Polarized Light
Propagating through a Nonideal 1D-Superlattice
Vladimir V. Rumyantsev, Stanislav A. Fedorov
Galkin Donetsk Institute for Physics and Engineering of National Academy of Sciences, Donetsk, Ukraine.
Received January 13th, 2010; revised January 28th, 2010; accepted February 1st, 2010.
ABSTRACT
The problem of finding polariton modes (necessary for calculating gyrotropic characteristics) in space-dispersed su-
perlattices is not yet solved. At the same time the specified quantities can be approximately evaluated if the widths of
layers comprising a multilayer material are much bigger then the characteristic scales of space dispersion. In such a
case the contribution of individual layers to gyrotropy can be regarded as independed. Thus the corresponding optical
quantities can be expressed through the layers’ gyrotropic characteristics. This approach is applied to calculate the
specific rotation angle of plane of polarization of light propagating through a nonideal 1D-superlattice, which varies in
composition as well as in layers’ width.
Keywords: Light Propagation, Nonideal 1D-Superlattice, Specific Rotation Angle
1. Introduction
At present, there are numerous papers [1-4] dealing with
the studies of the optical properties of perfect and imperf-
ect dielectric superlattices. Urgency of the research is co-
nditioned by electrical engineering and electronics needs
in layered structures, as well as by the progress in the de-
velopment of the theory of superlattice optical properties.
The procedures used to calculate the transmission coeffi-
cients and refractive indices for light developed in [5-7]
allow the frequency-concentration dependence is expo-
sed and turn out to be useful in simulation of composite
materials with preset parameters with no spatial disper-
sion taken into account. At the same time, the investiga-
tion of the gyrotropy of crystals is often the only way to
determine stereo- and crystallochemical characteristics as
well as the fine details in construction of respective
space-dispersing structures. Such investigations are even
more urgent as now there exists a large quantity of or-
ganic complexes and polymers, which are optically ac-
tive due to structure peculiarities or to the optical activity
of molecules they are composed of [8,9].
The problem of finding polariton modes (necessary for
calculating the gyrotropic characteristics) in space-dis-
persing superlattices has not been solved yet. At the same
time, with thickness of layers composing a multilayer
much larger than the characteristic scales of spatial dis-
persion, the specified quantities can be evaluated only
approximately. In such a case, the contribution of indi-
vidual layer to gyrotropy can be regarded independent.
Thus, the corresponding optical quantities can be expre-
ssed through the layers gyrotropic characteristics. In this
research, this approach is applied to calculate the specific
angle of the light polarization plane rotation, the light
being propagated in an imperfect 1D-superlattice with an
arbitrary number of heterogeneous layers varying com-
position or thickness.
Here we consider light propagation along layer’s opti-
cal axes perpendicular to their planes. An analytical ex-
pression is derived for the specific rotation angle as a
function of the impurity layer concentration.
2. Modeling
According to the above-mentioned approach, in an im-
perfect topologically ordered one dimensional superlat-
tice composed of N unit cells, the angle of the light
propagation plane rotation is described by the expression:
 
11
Nσ
nαnα
nα
ρωρ ωa

 (1)
We assume the quantity of N to be larger enough (to
neglect the effects from the influence of sample bounda-
ries). In (1)


nα
ρω and are the configura-
tion-dependent specific angle of the light propagation
plane rotation and thickness of the α-th layer of the n-th
unit cell; σ is the number of unit-cell layers.

nα
aω
According to the general principles of the physics of

Optical Rotation of Linearly Polarized Light Propagating through a Nonideal 1D-Superlattice 33
disordered systems, the rotation angle measured experi-
mentally should be equal to
 
ˆ
Pρω ρω, where
is the configuration averaging operator [6,7,10] in-
fluencing the configuration-dependent function
ˆ
P


ρω.
In the imperfect 1D-superlattice under consideration, the
disordering is of two types, consequently, there are two
types of configuration dependence. The first disorder is
due to heterogeneous (defective) layers present in the
superlattice, which differ from the perfect-system layers
in physico-chemical composition (the configuration-dep-
endent quantity is ). The second is due to layers-
defects present in the system and differing in thickness
from the perfect superlattice (the configuration-depend-
ent quantity is ). In what follows, we believe the dis-
ordering factors to be mutually independent. Quantities
and relate to configuration-dependent sto-
chastic variables

nα
ρω
αn
a
nα
a

ω
nα
ρ


μα
sα
η and


να
sα
η as

 


1
rα
μα μα
nααnα
μα
ρω ρη

 (2)
and
 


sα
να να
nααnα
να
aaη (3)
where


 


11
1, 1,
rαsα
μα να
nαnα
μα να
ηη





1
μα
nα
η, if the-th layer
of the n-th unit cell is the layer of
α


μα -type
(




1.2... )μα rαμ
sα
η and 0

- in any other case;


1
να
nα
η, if thickness of the -th layer of the n-th unit
cell equals
α


να
α
a

 (να and

1.2.. )sα0ην
sα

- in
any other case.



μα
α
ρω
α

μα
is the specific rotation angle
of the -th layer of type. Now and then index
μ
enumerates layers of variable composition, - of
variable thickness.
ν
By using (1)-(3) as well as the averaging rules [10] for

ρω , we have



 






 





  


111
11
1
1
1
[
].
rα
σ
μα μα
αααα α
αμα
sα
να να
ααα
να
rαsα
μαναμανα
αααα
μα νε
ρω Nρωaa ρωC
ρω aC
ρωaC C









(4)












11
,
μα μανανα
ααααα
ρρρaaa
α


as well as




,
μα να
αα
CC
are concentrations of defect layers which differ from lay-
ers of base substance in composition and/or thickness,
respectively. In expression (4), the fist summand corresp-
onds to the rotation angle for the light polarization plane
of perfect 1D-superlattice composed of layers of the (1)-
type (this substance is assumed to be basic). The second
summand stands for superlattice compositional disorder,
it goes to zero in the absence of composition variation.
The third summand corresponds to disordering in thick-
ness (with no disordering, the summand goes to zero).
And the last summand stands for superlattice disordering
in layer composition and thickness at a time. In the ab-
sence of any disordering, the fourth summand (4) goes to
zero. In expression (4) each summand has meaning of
rotation angle per unit cell. The angles, in contrast to




,μα να
nα
ρ (measured in deg/unit length) are measured in
degrees.
3. Results
For a more concrete results, let us consider the propaga-
tion of electromagnetic radiation in an imperfect alumin-
ium organic 1D-superlattice with two elements-layers in
a cell: the first is the aluminium menthylate Al(O-Ment)3
layer (


1
161, 2ρ ), the second—aluminium bornylate
Al(O-Born)3 layer (


1
236,7ρ ). Let the first sublattice
contain impurity layer l-MentOH (), the
second—l-BornOH (

2
145,5ρ


2
235, 4ρ ) ones. Here we use
values of specific rotation angles from [9]. In first and
second sublattices, the concentration and the thickness of
base-substance layer are denoted by



1
11
,Ca
1
and




1
2
,C1
2
a, respectively,



2
a
(2)
1
1, CT
C and





2
2
2, CT
C2 a are
those of the impurity (index С(Т) stands for variation of
impurity layers in composition or thickness). With for-
mula (4) and by simple transformations we obtain the
following concentration dependence of the specific angle
of rotation






22
12
,/
СТ СТ
ρρССρNd (5)
(here
 


 


11 212 21
1211 122TT
daaa aCaaC 2
2
is the averaged period of the cell in 1D-sublattice) for the
light polarization plane in two-sublattice imperfect 1D-
sublattice in the form:
 



  
 




 




1
221
112 212
121 2
111222
21 22
11 111
21 22
22 222
1[
1
1]
TT
TT
TC
TC
ρafC faCρa1
ρ
ρfC ρfaC
ρρ fC C
ρρ faC C

 


 
(6)

Optical Rotation of Linearly Polarized Light Propagating through a Nonideal 1D-Superlattice 34
In (6), there is the following designation:




11
21
/aa a, .
 

 

21 21
1112 22
/1, /faa faa1
The concentration dependence
of the specific angle of the light polarization plane rota-
tion in the studied imperfect superlattice is graphically
shown in Figures 1 and 2.





22
12
,
СТ СТ
ρρСС
Figure 1 illustrates the presence of impurity layers in
the first sublattice, which differ in composition, those in
the second sublattice differ in thickness. In Figure 2 the-
re is a variation of superlattice layers in thickness, Figu-
re 3 shows the case when the layers vary in composition.
4. Conclusions
The paper is devoted to numerical simulation of the con-
centration dependence of the rotation angle for the light
polarization plane in imperfect 1D-multilayer with an
arbitrary number of sublattices. The choice of aluminium
Specific rotation angle, ρ
1
0.5
0 0
0.8 1
Concentration
of defect layers,
)2(
1C
C
0.6
50
0
-50
-100
-150
-200
Concentration
of defect layers,
)2(
2T
C
0.2
0.4
Figure 1. Concentration dependence






2
2
2
1,ТС СС
 
2/ )1(
2
)2(
2a
,
where , , 1/ )1(
1
)2(
1aa 2.0/ )1( 1
)2(
1aa a
Concentration
of defect layers,
)2(
2
T
C
Specific rotation angle, ρ
1
0.5
0 0.2
Concentration
of defect layers,
)2(
1T
C
0.6
0
-10
-20
-30
-40
-50
1
00.4 0.8
Figure 2. Concentration dependence





2
2
2
1,TT СС
 
5
)1 
of
the specific angle of the light polarization plane rotation in
studied imperfect superlattice; /(
1
)2(
1aa
Concentration
of defect layers,
)2(
2C
C
40
20
0
20
40
-60
Specific rotation angle, ρ
1
0.5
000.2 0.4 0.8 1
Concentration
of defect layers,
)2(
1C
C
0.6
1
2
3
4
Figure 3. Concentration dependence






2
2
2
1,CССС
 
8
)1 /)1(
2
)2(
2aa
of
the specific angle of the light polarization plane rotation in
studied imperfect superlattice; , ,
being equal 8, 3, 1 and 0.1 for case 1, 2, 3 and 4
respectively
/(
1
)2(
1aa3
)1(
1
)1(
2/aa
terpene alcogolates as objects of investigation is due to
the wide application of these optically active materials in
the capacity of catalysts of different asymmetric synthe-
ses stimulating catalytic reactions of other optically ac-
tive compounds [9].
At present, there exist a great number of organic com-
plexes and polymers, which are optically active due to
peculiarities of structure or optical activity of constituent
molecules [8,9]. Studies of polymeric composite materi-
als [1,2,9] are urgent enough due to variety of properties
and a wide range of application. A high interest in mate-
rials of this class has arisen, on the one hand, from de-
mands of electrical engineering and electronics for thin
films and layered structures and, on the other hand, it
relates to the latest achievements in nanotechnologies
and photonics [11], as well as a substantial progress in
the theory of optical properties of perfect and imperfect
layered structures.
5. Acknowledgment
The authors are grateful to Prof. Yurii Pashkevich for
valuable discussions. This work is supported by Project
“Dynamical and static properties of complex low-di-
mensional systems in external fields” of National Acad-
emy of Sciences of Ukraine.
REFERENCES
[1] C. Zhang and D. E. Hirt, “Layer-By-Layer Self-Assembly
of Polyelectrolyte Multilayers on Cross-Section Surfaces
of Multilayer Polymer Films: A Step toward Nano-Pat-
terning Flexible Substrates,” Polymer, Vol. 48, 2007, pp.
6748-6754.
[2] A. Pucci, M. Bernabò and P. Elvati, “Photoinduced For-

Optical Rotation of Linearly Polarized Light Propagating through a Nonideal 1D-Superlattice 35
mation of Gold Nanoparticles into Vinyl Alcohol Based
Polymers,” Journal of Materials Chemistry, Vol. 16,
2006, pp. 1058-1066.
[3] A. Kondilis and P. Tzanetakis, “Numerical Calculations
on Optical Localization in Multilayer Structures with
Random-Thickness Layers,” Physical Review, Vol. B46,
1992, pp. 15426-15431.
[4] L. Lyubchanskii, N. N. Dadoenkova, M. L. Lyubchanskii,
E. A. Shapovalov, A. Lakhtakia and Th. Rasing, “One-
Dimensional Bigyrotropic Magnetic Photonic Crystals,”
Applied Physics Letters, Vol. 85, 2004, pp. 5932-5934.
[5] A. Yariv and P. Yeh “Optical Waves in Crystals,” John
Willey & Sons, New York, 1987.
[6] V. V. Rumyantsev, S. A. Fedorov and E. Ya. Shtaerman,
“Peculiarities of Photonic Band Gap Width Dependence
upon Concentration of the Admixture Layers Randomly
Included in Composite Material,” Functional Materials,
Vol. 15, 2008, pp. 223-227.
[7] V. V. Rumyantsev and S. A. Fedorov, “Propagation of
Light in a Quasi-Two-Dimensional Si/Sio2 Superlattice
with Variable Layer Thickness,” Optics and Spectroscopy,
Vol. 106, 2009, pp. 627-631.
[8] J. Ryan and R. A. L. Jones, “Polymers: The Quest for
Motility,” Materials Today, Vol. 11, 2008, pp. 21-23.
[9] M. L. Sierra, R. Kumar, V. S. J. de Mel and J. P. Oliver,
“Synthesis and Spectroscopic Investigations of Alkylalu-
minum Alkoxides Derived from Optically Active Alco-
hols. The First Structural Identification of an Optically
Active Organoaluminum Alkoxide,” Organometallics,
Vol. 11, 1992, pp. 206-214.
[10] M. Ziman, “Models of Disorder,” John Willey & Sons,
New York, 1979
[11] D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D.
Meade, Photonic Crystals, 2nd Edition, Princeton Uni-
versity Press, Princeton, 2008.