Using the Jones matrix to model light propagation in anisotropic media
I. V. Valyukh* and A. V. Slobodyanyuk
Taras Shevchenko Kiev National University, Kiev, Ukraine
S. I. Valyukh
Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine
J. Osterman and K. Skarp
Dalarna University, Borla ¨nge, Sweden
Submitted December 11, 2002
Opticheski Zhurnal 70, 24–28 July 2003
Using a twisted nematic liquid crystal as an example, this paper discusses the use of the Jones
matrix to find the parameters of a light wave propagating in an inhomogeneous anisotropic
medium, which is represented for the numerical calculations by a sequence of thin homogeneous
layers. The accuracy of the result is analyzed as a function of the parameters of the
medium and the number of layers into which it is broken up. A criterion is proposed for finding
the optimum number of layers. © 2003 Optical Society of America
Numerous optical devices are currently being developed
and successfully operated in which inhomogeneous aniso-
tropic or anisotropic–hypotropic media are used as the key
element. This involves various interference filters, electroop-
tic modulators, polarizers, or liquid-crystal LC methods of
displaying information in which the desired intensity distri-
bution and state of polarization is achieved by specially
choosing the anisotropic plates or by the character of the
optical inhomogeneity of a complex anisotropic medium.
Computer engineering is ordinarily used in planning and
developing the designs of the devices mentioned above.
However, despite the relatively high speed of modern com-
puters, there are a number of problems for which it is crucial
to shorten the calculation time. This is usually the case for
inverse problems and optimization problems, in which the
values of several parameters of the optical system must be
varied simultaneously.
This paper will discuss light propagation in a twisted
nematic liquid crystal TNLC. From the viewpoint of optics,
a TNLC is a continuous nonabsorbing anisotropic medium
the orientation of whose optic axis smoothly varies along
some fixed direction. The matrix methods of Jones
1
or
Berreman
2
are ordinarily used to find the parameters of the
light wave transmitted through such a liquid crystal. It is
convenient in this case to break up an inhomogeneous aniso-
tropic medium into a set of layers of equal thickness, assum-
ing that the medium is homogeneous within each of them
Fig. 1. The accuracy of the results and the calculation time
depend on the number of such layers.
3,4
In this paper, we
shall investigate the criteria for determining the optimum
number of partitions for which sufficiently accurate compu-
tations can be obtained in the minimum time.
Let the orientation of the optic axis be specified by some
unit vector c. Let us introduce a coordinate system in such a
way that the Z axis is along the normal to the cell filled with
the TNLC. Then the coordinates of vector c( Z ) can be given
by means of the polar angle and the azimuthal angle :
c z =
cos z cos z
cos z sin z
sin z
. 1
Let
i
be the phase lag of the i th anisotropic layer. Ex-
pressing
i
in terms of the refractive index and thickness of
the layer, we can write
i
=
2
N
n
e
i
* -n
o
d , 2
where d is the thickness of the LC layer, is the wavelength,
n
o
and n
e
i
* are the refractive indices for the ordinary and
extraordinary waves, and N is the number of partitions. For
normal incidence along the Z axis in our case, n
e
i
* is de-
termined by
n
e
i
* =
n
e
1 +
n
e
2
n
o
2
-1
sin
2
i
, 3
where n
e
is the principal refractive index for the extraordi-
nary wave, and
i
is the angle between the propagation di-
rection and the optic axis. FIG. 1. TNLC in the form of a sequence of anisotropic plates.
470 470 J. Opt. Technol. 70 (7), July 2003 1070-9762/2003/070470-04$20.00 © 2003 The Optical Society of America