Abstract:

This paper studies the property possessed in general by anisotropic
materials, namely that of splitting the transverse electric (TE)
polarized component from the transverse magnetic (TM) polarized
component of the waves that go through these materials. The aim of this
study is to classify the anisotropic materials according to the
splitting angle they can provide, this feature being extremely important
for the field of optical communications. The study undertaken in this
paper is also important from a theoretical point of view, because each
of the groups resulted from this classification is characterized by
arguments of mathematical nature.

Key words: anisotropic materials, electric permittivity and magnetic permeability, dispersion equations of TE and TH polarized waves, poynting vector, polarized beam splitting devices

Key words: anisotropic materials, electric permittivity and magnetic permeability, dispersion equations of TE and TH polarized waves, poynting vector, polarized beam splitting devices

Article Type:

Report

Subject:

Materials
(Electric properties)

Materials (Magnetic properties)

Anisotropy (Research)

Waves (Properties)

Materials (Magnetic properties)

Anisotropy (Research)

Waves (Properties)

Authors:

Sandru, Ovidiu Ilie

Sandru, Alexandra

Sandru, Alexandra

Pub Date:

01/01/2009

Publication:

Name: Annals of DAAAM & Proceedings Publisher: DAAAM International Vienna Audience: Academic Format: Magazine/Journal Subject: Engineering and manufacturing industries Copyright: COPYRIGHT 2009 DAAAM International Vienna ISSN: 1726-9679

Issue:

Date: Annual, 2009

Topic:

Event Code: 310 Science & research

Geographic:

Geographic Scope: Austria Geographic Code: 4EUAU Austria

Accession Number:

224713004

Full Text:

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

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

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