ISSN 1068–3372, Journal of Contemporary Physics (Armenian Academy of Sciences), 2015, Vol. 50, No. 2, pp. 137–143. © Allerton Press, Inc., 2015. Original Russian Text © R.B. Alaverdyan, A.L. Aslanyan, L.S. Aslanyan, G.S. Gevorgyan, S.Ts. Nersisyan, 2015, published in Izvestiya NAN Armenii, Fizika, 2015, Vol. 50, No. 2, pp. 185–193.
Polarized Light in Anisotropic Medium with Twisting R. B. Alaverdyan, A. L. Aslanyan, L. S. Aslanyan, G. S. Gevorgyan*, and S. Ts. Nersisyan Yerevan State University, Yerevan, Armenia *
[email protected] Received November 6, 2014
Abstract⎯The evolution of state of light polarization in a smoothly inhomogeneous anisotropic medium is analyzed both theoretically and experimentally. The analytical expressions describing the polarization state in such a medium are obtained. In particular, the results of analysis are applicable to a nematic liquid crystal with twist orientation. It is shown that although the polarization state, when light propagates in such a medium, undergoes oscillations, spatial frequency of which depends on the thickness of the sample, nevertheless, the phenomenon of adiabatic tracking is observed at the exit in the case of both input e and o waves. The experiments confirm the results of theoretical analysis. DOI: 10.3103/S106833721502005X Keywords: nematic liquid crystal, polarized light, anisotropic medium
The analysis of light propagation in anisotropic media continues to attract the attention of researchers, despite the large number of publications (see, for example, [1–7]). This is caused by the possibilities of wide application in LCD and spatial light modulators. Despite the variety of optical phenomena in anisotropic and circularly anisotropic media, in photonic and liquid crystals the presence of inhomogeneity of anisotropy (both natural and induced) significantly enhances the range of possible phenomena and their applications (see [8] and references cited therein). This explains the necessity to create universal methods of analysis of optical phenomena in such media. The analytical solution of wave equation in such a media is related to certain difficulties even if take account of the slow change of parameters of medium and the possibility of applying of approximate methods. However, the analysis of behavior of the state of light polarization in smoothly inhomogeneous media is very important and actual task, since, firstly, the polarization of light wave after interaction contains an essential information about the medium itself and, secondly, the control of the medium parameters is the basis for the controlling of light polarization state by way of inducing one or another inhomogeneity with use of external influence and creation of controlled achromatic compensators. The propagation of plane monochromatic wave in media with spatial inhomogeneity of anisotropy is described mathematically by the same equations as the behavior of two-level system in nonstationary fields [9, 10]. Such an analogy enables us to use the well-developed methods of theory of quasi-resonant radiation interaction with the two-level atom. The aim of the present work is the theoretical and experimental investigation of evolution of light polarization in smoothly inhomogeneous media, in particular, in the nematic liquid crystals (NLC) with twist-orientation. Let the plane monochromatic wave propagates in anisotropic medium with inhomogeneous azimuthal angle of optical axis ψ(z ) (see Fig. 1). We represent the permittivity tensor of smoothly inhomogeneous media ε ij (z) in the following form [11] 137
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Fig. 1. Geometry of the problem.
ε ij ( z ) = ε ⊥ δij + ε a mi ( z ) m j ( z ) .
(1)
Here mi ( z ) is the unit vector (director), describing the local orientation of optical axis, δij the Kronecker symbol, ε a = ε& − ε ⊥ the anisotropy of LC. The absorption is neglected and the magnetic permeability is equal to unity. In the case of normal incidence on the medium we can easily obtain from the Maxwell's equations the following system for two-dimensional Jones vector E = ( Ex E y ) , describing the state of Т
polarization of light wave [12, 13]: d 2E ( z ) d z2
+
ω2 εˆ ( z ) E ( z ) = 0 , c2
(2)
where i, j = x, y and z is the coordinate along the light propagation. In the case of smallness of change of angle ψ( z ) with the wavelength, which is justified for LC, the geometrical optics method can be used [12–15]. We seek a solution in the form
⎧ ω ⎫ E ( z ) = E0 ( z ) exp ⎨i ∫ n0 ( z ) dz ⎬ , ⎩ c ⎭
(3)
where n0 ( z ) =
ε xx ( z ) + ε yy ( z ) 2
.
(3a)
This enables us to separate in the wave the fast oscillations of field and the relatively slow changes of the medium parameters and related parameters of wave. Taking into account the slowness of change of complex amplitude E0 ( z ) , we obtain the following vector equation [12, 13]: dJ ( z ) dz
=i
ω ˆ H (z)J(z) , c
(4)
where
{
}
Hˆ ( z ) = εˆ ( z ) − n 2 ( z ) Iˆ 2n ( z ) ,
(4a)
J ( z ) = n ( z )E0 ( z ) .
(4b)
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Expression (4) is a Schrödinger-type equation, which describes the evolution of polarized light state, and the matrix Hˆ ( z ) is an analogue of the Hamiltonian and one characterizes the properties of the medium. In the case of propagation of polarized light in the NLC with twist-orientation, for the matrix Hˆ ( z ) we have ε ⎛ cos2ψ ( z ) sin2ψ ( z ) ⎞ Hˆ ( z ) = a ⎜ ⎟. 4n0 ⎜⎝ sin2ψ ( z ) −cos2ψ ( z ) ⎟⎠
(5)
The solution of equation (4) is simplified essentially and it is possible to obtain the analytical solution of the problem if we change the coordinates and transform them to the rotating coordinate system by means of the following relation [12, 16] J ( z ) = Rˆ −1 A ( z ) , Hˆ ( z ) = Rˆ −1 Hˆ 0 Rˆ ,
(6)
Т where A = ( Aξ Aη ) is the Jones vector in the rotating coordinate system, Rˆ ( ψ ) is the following matrix of rotation of coordinate axes
⎛ cosψ ( z ) sinψ ( z ) ⎞ Rˆ ( ψ ) = ⎜ ⎟. ⎜ −sinψ ( z ) cosψ ( z ) ⎟ ⎝ ⎠
Here Hˆ 0 is the “Hamiltonian” of inhomogeneous anisotropic media in the local coordinate system. After simple but cumbersome transformations, we obtain the following coupled equations for Аξ,η [12]: dАξ ( z ) dz dАη ( z ) dz
= iγАξ ( z ) +
=−
dψ ( z ) dz
dψ ( z ) dz
Аη ( z ) ,
(7a)
Аξ ( z ) − iγАη ( z ) ,
(7b)
where the following notation is used Γ=
πε a . 2λn0
(7c)
In the case of uniform change of orientation angle ψ(z) = αz of director, the equations (7a,b) represent by itself a system of equations with constant coefficients, the solution of which is well known [17]. Based on this solution and after the inverse transformation into the laboratory coordinate system the final analytical solution can be represented in the form Eμ ( z ) =
2π i n0 z i ⎡⎣ Aμ ( z ) e −i Ωz + Bμ ( z ) ei Ωz ⎤⎦ e λ , 2Ω
(8)
where μ = x, y. The notations introduced here have the following form: ⎛ ⎞ α Ax ( z ) = ⎜ B0 − i A0 ⎟ ⎣⎡αcosψ ( z ) + i ( γ + Ω ) sinψ ( z ) ⎤⎦ , γ+Ω ⎠ ⎝ ⎡ ⎤ α Bx ( z ) = ⎡− ⎣ i ( γ + Ω ) A0 − αB0 ⎦⎤ ⎢ cosψ ( z ) + i γ + Ω sinψ ( z ) ⎥ , ⎣ ⎦ ⎛ ⎞ α Ay ( z ) = ⎜ B0 − i A0 ⎟ ⎣⎡ αsinψ ( z ) − i ( γ + Ω ) cosψ ( z ) ⎤⎦ , γ+Ω ⎠ ⎝ JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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⎡ ⎤ α By ( z ) = ⎡− ⎣ i ( γ + Ω ) A0 − αB0 ⎦⎤ ⎢sinψ ( z ) + i γ + Ω cosψ ( z ) ⎥ , ⎣ ⎦ Ω 2 = γ 2 + α 2 , A0 = Ex ( z = 0 ) , B0 = E y ( z = 0 ) .
The quantities A0 and B0 can be complex ones (in the case of elliptically polarized input wave). It is easy to verify that in the limit α → 0 the resulting solution (8) transforms into the well-known expression [2, 17]. Thus, the analytical expressions allow one to describe the spatial dynamics of the state of light polarization in the inhomogeneous anisotropic media. As an example, we consider the NLC 5CB, the parameters of which are well known [18]. Let an input wave is polarized linearly along the x-axis (in the case of normalized intensity A0 = 1, B0 = 0). As can be seen from Fig. 2a, the output wave is completely polarized along the y-axis according to the phenomenon of adiabatic following [2, 6]. When the input wave is polarized linearly along the y -axis, the output wave is linearly polarized along the x -axis (see Fig. 2b) (it was mentioned qualitatively in [6, 18]).
(b)
(a) 2
⏐Ex(z)⏐ + ⏐Ey(z)⏐
2
2
2
⏐Ex(z)⏐ + ⏐Ey(z)⏐
⏐E(z)⏐
⏐E(z)⏐
2
2
2
⏐Ey(z)⏐
2
2
⏐Ex(z)⏐
2
⏐Ex(z)⏐
⏐Ey(z)⏐
z(μ)
z(μ)
Fig. 2. Evolution of state of light polarization in the twist-oriented NLC 5CB. The values of parameters are the following: ε 0 = n02 = 2.31, ε a = 0.58, λ = 0.5145, the thickness of NLC layer is 30 µm. (a) Input wave is polarized along the x-axis. (b) Input wave is polarized along the y-axis.
Thus, our analysis shows that if at the input to the medium the incident wave is polarized linearly along one of the normal waves, the polarization vector of light wave traces the rotation of principal axes provided the coefficient of torsion is small. However, the state evolution of light polarization in the layer of twist-oriented NLC depends on the sample thickness. At small thicknesses (of order of several µm) this is a sufficiently smooth transition through the intermediate elliptically polarized states (this also was mentioned in [6]). At the thicknesses of the order of tens of microns such a transition has an oscillatory behavior (see Fig. 2a, b). For greater clarity, the spatial dynamics of the state of light polarization can be represented on the Poincaré sphere. To do so, we determine the Stokes parameters by the relation
Sˆ = E+ σˆ E,
(9)
where σˆ i is the Pauli matrices [6,13]: ⎛ 1 0⎞ ⎛1 0 ⎞ σˆ 0 = ⎜ ⎟ , σˆ 1 = ⎜ ⎟, ⎝0 1⎠ ⎝ 0 −1⎠
⎛0 1⎞ ⎛ 0 −i ⎞ σˆ 2 = ⎜ ⎟ , σˆ 3 = ⎜ ⎟. ⎝1 0⎠ ⎝i 0 ⎠
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It is easy to verify that 2
2
2
2
S0 = Ex + E y , S1 = E x − E y ,
(10a)
S2 = Ex E y + Ex E y , S3 = i ( Ex E y − E x E y ) .
(10b)
Figure 3 represents the spatial dynamics of the state of light polarization on the Poincaré sphere drown with use of relations (8) and (10). The following regularity is observed: with the increase in sample thickness, the amplitude of spatial oscillations is decreasing and the Stokes vector is rotating practically in the equatorial plane. This means that the input linearly polarized wave follows the rotation of NLC director adiabatically and remains linearly polarized.
(a)
(b)
(c)
Fig. 3. Evolution of state of light polarization in the twist-oriented nematic 5CB on the Poincaré sphere. The values of parameters for the NLC 5CB are the following ε 0 = n02 = 2.31, ε a = 0.58, λ = 0.5145 µm, the thickness of LC layer: (a) 2.5 µm, (b) 10 µm, and (c) 30 µm.
Quite often there is a situation, especially in the twisted NLC, when the quantity γ, describing the phase delay, is essentially greater than the specific rotation α. As an example, we consider the twist oriented NLC 5CB, which has γ α .Using the definition (7b) and taking into account that αl = π / 2 , we obtain γ εa l = . α n0 λ This means that at l λ the condition γ α is met. At γ α , neglecting the terms with α, we obtain E x ( z ) = ( A0 cosψ ( z ) e iγz − B0 sinψ ( z ) e − iγz ) e
i
E y ( z ) = ( B0 cosψ ( z ) e − iγz + A0 sinψ ( z ) eiγz ) e
2π n0 z λ
i
2π n0z λ
,
.
Consequently Ey ( z ) Ex ( z )
=
B0 cosψ ( z ) e − iγz + A0sinψ ( z ) eiγz A0 cosψ ( z ) eiγz − B0sinψ ( z ) e − iγz
.
(11)
Consider the previous particular cases. Let A0 = 1 and B0 = 0, i.e. the input wave is linearly polarized along the x-axis. It follows from (11) that Ey ( z ) Ex ( z )
= tan ψ ( z ) .
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Since ψ ( l ) = π / 2, the output wave is polarized along the y-axis. Similarly, when the input wave is linearly polarized along the y -axis, then A0 = 0 and B0 = 1 . It follows from Eq. (11) that Ey ( z ) Ex ( z )
= − cot ψ ( z ) .
And at ψ(l ) = π 2 the output wave is linearly polarized along the x-axis. To investigate the changes of polarization of initially linearly polarized laser beam by transition through the twist-oriented NLC cell, an experimental setup was assembled, the scheme of which is shown in Fig. 4. The laser beam of He-Ne laser (1) with the wave length 0.63 µm passes initially through the polarizer (2), with aid of it the laser beam becomes linearly polarized. The beam passed through a polarizer is incident on the twist NLC cell (3). The state of polarization of passed light is investigated with the use of analyzer (4) and power meter (5). The polarization of a beam passed through the twist NLC cell is determined by way of rotation of analyzer and by measuring the power of laser beam. During the experiment, the normal incidence of beam on the cell with the thickness of 100 µm has been ensured. The LC 5CB has been used. The experiment was carried out at the room temperature.
Fig. 4. Scheme of experimental setup: 1 – He-Ne laser, 2 – polarizer, 3 – twist cell of NLC (the thickness 100 µm), 4 – analyzer, 5 – power meter.
It is known that if the polarization of laser beam is parallel to the direction of director at the front wall of the cell, the polarization of beam rotates (remaining parallel to the orientation of molecules) and at the output of the cell the polarization direction becomes perpendicular to the initial polarization [18, 19]. The aim of this experiment was to determine the change of polarization of initially linearly polarized laser radiation under condition that this polarization is perpendicular to the director at the front wall of the cell. The investigations show that the polarization of linearly polarized beam after passing through the twist oriented NLC cell is changed in such a way that at the output of the cell it obtains polarization, the direction of which is perpendicular to the initial polarization. Thus, when passing through the twist NLC cell the polarization of linearly polarized laser beam rotates by 90о regardless of whether the polarization of beam is parallel or perpendicular to the director direction at the front wall of the cell. Particular emphasis should be given to the frequency independence (achromaticity) of such rotation. The experimental results confirm the foregoing theoretical results. ACKNOWLEDGMENT This work was financially supported by the SCS RA MES within the scientific project SCS 13-1C240. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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