1. INTRODUCTION
The problem of splitting the polarized components of the light
beams is important due to its numerous applications in many current
fields of interest, such as the field of optical communications. In
general, the splitters used to separate light beams' polarized
components are made from certain types of crystals or multilayer
transparent materials. The shortcoming of these devices consists in the
fact that they cannot offer splitting angles and splitting distances,
respectively, large enough as desired. Recent researches point out the
possibility to realize splitting devices using certain types of special
anisotropic materials such as the ones presented in (Luo et al., 2006;
Luo & Ren, 2008). Also, by extending the applicability range of the
geometrical transformation method (Ward & Pendry, 1996; Pendry,
2003), we have indicated in (Sandru et al., 2009) other types of
materials that have the property mentioned above.
In this paper we shall characterize from a mathematical point of
view which of the anisotropic media possess the property of separating
the TE polarized component from the TM polarized component of the waves
that go through them or not. Due to the practical applications of the
splitting phenomena of differently polarized light beam components, it
is important to know under which conditions a transparent medium offers
only splitting angles less than 90[degrees] and under which conditions
it offers splitting angles larger than 90[degrees].
2. THE BEHAVIOR OF ANISOTROPIC MATERIALS TOWARDS THE DIFFERENTLY
POLARIZED COMPONENTS OF LIGHT BEAMS
Anisotropic materials distinguish themselves through the tensorial
structure of the electric permittivity and magnetic permeability. Due to
the symmetry of these tensors for each of them there exists a
tri-orthogonal coordinate system in relation to which these tensors
admit a diagonal matricial representation. In general, the frame against
the electric permittivity admits the diagonal shape differs from the
frame against which the magnetic permeability admits the diagonal shape.
Anisotropic materials for which these frames coincide, build an special
subclass of anisotropic materials. In this paper we shall focus only on
studying this important subclass.
Let us assume the following experiment: we consider a
parallelepipedous slab ABCDA BC' D' made of an anisotropic
material for which there exists a tri-orthogonal frame (let this be Oxyz
) against which electric permittivity [epsilon] and magnetic
permeability [mu] (of slab ABCDABC'D') admit the expressions
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where [[epsilon].sub.1], [[epsilon].sub.2], [[epsilon].sub.3],
[[mu].sub.1], [[mu].sub.2], [[mu].sub.3], are real nonzero numbers.
Remark: In the anisotropy hypothesis we are using, it becomes
obvious that the elements [[epsilon].sub.i], [[mu].sub.i], i = 1,2,3, of
these two matrices cannot satisfy the relations [[epsilon].sub.1] =
[[epsilon].sub.2]= [[epsilon].sub.3], [[mu].sub.1] = [[mu].sub.2] =
[[mu].sub.3] simultaneously.
In relation with this coordinate system, we place the slab ABCDA B
C'D so that the conditions A (d ,0,0), B(d +1,0,0), C(d +1, h,0),
D(d, h,0), d > 0, l > 0, h > 0, are fulfilled. Regarding the
environment which surrounds this slab, we suppose that it possesses the
material parameters described through
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
with [[epsilon].sub.0] > 0, and [[mu].sub.0] > 0.
Upon the slab ABCDA'B'C'D' a plane beam w of
frequency w is sent out and intercepts the face ABCD of the slab in
point P (d, p,0), p < h, under the angle of incidence
[[theta].sub.I]. By [w.sup.E] we denote the component of beam w
transverse electric polarized (TE)--its electric field is perpendicular
to the plane of incidence, and by [w.sup.H], the component of beam w
transverse magnetic polarized (TM)--its magnetic field is perpendicular
to the plane of incidence. In the surrounding medium the time-averaged
Poynting vector corresponding to [w.sup.E] has the expression
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
where [E.sub.I] is a real constant that can be determined depending
on the conditions in which the experiment will be carried out, while
components [square root of [[epsilon].sub.0]/[[mu].sub.0]] cos
[[theta].sub.I]
and [square root of [square root of [[epsilon].sub.0]/[[mu].sub.0]
sin [[theta].sub.I where determined from the condition that the angle of
incidence of the [w.sup.E] component is equal to [[theta].sub.I], and
from the corresponding equation of dispersion.
If we suppose that there exists [[theta].sub.I] so that
[[epsilon].sub.3][[mu].sub.1][[mu].sub.2]--[[epsilon].sub.0]
[[mu].sub.0][[mu].sub.2] [sin.sup.2][[theta].sub.I]/[[mu].sub.1]] >
0, (1)
then, within the medium the ABCDA'B'C'D' slab
is made of, the time-averaged Poynting vector corresponding to the
[w.sup.E] component, has the expression
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where parameter ET, and components
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where determined from the conditions of continuity that wave
[w.sup.E] must verify on the interface x = d, from the equation of
dispersion and from the law of conservation of energy.
Analogously. in the surrounding medium the time-averaged Poynting
vector corresponding to [w.sup.H] has the expression
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where [H.sub.I] is a real constant that can be determined depending
on the conditions in which the experiment will be carried out, while
coefficients [square root of [[mu].sub.0]/[[epsilon].sub.0]] cos
[[theta].sub.1], and [square root of [mu].sub.0]/[[epsilon].sub.0] sin
[[theta].sub.I], where determined from the condition that the angle of
incidence of the [w.sup.H] component is equal to [[theta].sub.I] and
from the corresponding equation of dispersion.
If we suppose that there exists [[theta].sub.I], so that
[[epsilon].sub.1] [[epsilon].sub.2]
[[epsilon].sub.3]--[[epsilon].sub.0][[mu].sub.0]
[[epsilon].sub.2][sin.sup.2][[theta].sub.I]/[[epsilon].sub.I] > 0,
(2)
then, within the medium the ABCDA BC' D' slab is made of,
the time-averaged Poynting vector corresponding to the [w.sup.H]
component, has the expression
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where parameter [H.sub.T], and coefficients
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
can be determined from the conditions of continuity that the
[w.sup.H] wave must verify on the interface x = d, from the equation of
dispersion and from the energy conservation law.
For the reasoning that we are about to make, it is important to
observe that Re([E.sup.2.sub.T])Re([H.sup.2.sub.T]) > 0. Indeed, this
result can be easily obtained taking into account that the transmission
coefficients for TE and TM polarized waves are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
With these preparations, from the expression of vectors
[S.sup.E.sub.a], and [S.sup.H.sub.a], it can be easily noticed that for
those incidence angles [[theta].sub.I] for which the relation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
is satisfied, the time-averaged Poynting vectors [S.sup.E.sub.a],
and [S.sup.H.sub.a], are not collinear, fact that demonstrates that the
two components [w.sup.E] and [w.sup.H] will follow different
trajectories within the ABCDABC'D slab. Thus, anisotropic materials
that possess the property of splitting TE polarized components from TM
polarized components of the incident beams are characterized by the
relations (1), (2), and (3).
3. ANISOTROPIC MATERIALS THAT CAN SEPARATE COMPONENTS [w.sup.E],
[w.sup.H] OF THE w WAVE MORE THAN 90[degrees]
We begin this section with the following important observation. If
[[epsilon].sub.1][[mu].sub.1] < 0, (4)
then, under the hypothesis (1) and (2), components
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
of vectors [S.sup.E.sub.a] and [S.sup.H.sub.a] have opposed signs.
Under these conditions, if moreover the following relations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)
take place, then the angle between the Poynting vectors of
components [w.sup.E] and [w.sup.H] will be obtuse. Thus, those
anisotropic materials which ensure splitting angles higher than
90[degrees] are characterized through conditions (1), (2), (4) and (5).
Observations: 1) Conditions (1), (2) and (4), without condition (5),
ensure the fact that the material which possesses these properties might
provide splitting angles higher than 90[degrees], but does not guarantee
this thing. Fulfilling this property is ensured by the conditions (5).
2) In case that the dispersion equations of components [w.sup.E]
and [w.sup.H] which cross the ABCDA'BC'D slab provide
single-sheeted hyperboloids we obtain the results found in (Luo et al.,
2006).
4. ANISOTROPOIC MATERIALS THAT CAN NOT SEPARATE COMPONENTS
[w.sup.E], [w.sup.H] OF THE w WAVE UNDER ANGLES LARGER THAN 90[degrees]
If in the established relations (1), (2), (4) and (5) we replace
condition (4) by condition
[epsilon].sub.1][[mu].sub.1] > 0, (6)
and condition (5) by condition (3), then the components of the
non-collinear Poynting vectors [S.sup.E.sub.a] and [S.sup.H.sub.a]
corresponding to axis Oy, have the same sign. From this reason, the
angle between vectors [S.sup.E.sub.a] and [S.sup.H.sub.a] cannot be
larger than 90[degrees].
5. REFERENCES
Luo, H., Shu, W., Li, F. & Ren, Z. (2006). A polarized splitter
using an anisotropic medium slab, arXiv:physics 0605161vi
[physics.optics], pp 1-14
Luo, H. & Ren, Z. (2008). Polarization-sensitive in an
anisotropic metamaterial with double-sheeted hyperboloid dispersion
relation. Optics Communications, issue 4, (February 2008) pp 501-507,
ISSN: 0030-4018
Pendry, J. B. (2003). Perfect cylindrical lenses. Optics Express,
Vol. 11, No. 7, (April 2003) pp 755--760, ISSN 1094-4087
Sandru, O. I., Schiopu, P. & Sandru, A. (2009). Geometrical
ways of obtaining polarized beam splitting devices, International Spring
Seminar on Electronics Technology, Prasek, J. et al., pp 1-6, (CD-ROM)
ISBN 978-1-4244-4260-7, Brno Univ. of Technology, May 2009, IEEE, Brno
Ward, A. J. & Pendry, J. B. (1996). Refraction and geometry in
Maxwell's equations, Journal of Modern Optics, Vol. 43, No. 4,
(April 1996) pp 773--793, ISSN 0950-0340