Abstract

The exact molecular reorientation model for nematic liquid crystals taking into account all diagonal Frank elastic constants and using two angles to describe director orientation is presented. Solutions and simplified equations are shown for the most common planar and chiral configurations. Gaussian beam propagation simulated using fully vectorial Beam Propagation Method in nonlinear case is also provided. Detailed comparison between exact solutions and single Frank constant approximation is made. However, no significant differences between these two models were found neither in beam propagation nor in polarization distribution, some difficulties may occur in choosing single Frank constant especially when it comes to quantitative results. Presented results correspond to a propagation of a beam of the Gaussian or topologically similar shapes.

© 2012 Optical Society of America

1. Introduction

Nematic liquid crystals are unique materials with many applications, like stress and temperature sensors, tunable filters, lasers and the most important and widely used - displays [13] where the nematic cell is controlled by the external electric field. Liquid crystals may be also controlled by the optical signals due to the nonlinear properties of these materials. Light beam launched into the cell can become non-diffractive and form a nematicon (i.e. a soliton in nematic liquid crystals) [46]. This phenomenon was observed in many experiments [7, 8] as well modeled numerically [912]. To simulate light propagation in a nematic cell it is needed to model electromagnetic field propagation as well as reorientation of liquid crystals molecules, because in nematics, nonlinearity mainly comes from the reorientation of the molecules. One of the methods is to use Beam Propagation Method (BPM) [13,14] for field propagation and Frank-Osseen molecular model to describe director orientation (i.e. mean direction of molecules). The most common approach to date is to use a single Frank constant approximation and one [15, 16] or rarer two angles [1719] to describe director orientation. When it comes to chiral nematics (i.e. cholesterics) there is an additional need to use vectorial BPM [20].

In this paper we present exact equations describing molecular reorientation in chiral and non-chiral nematic liquid crystals based on the Frank-Osseen theory. All Frank elastic constants are kept and two angles to describe director orientation are used. Comparison between this model and one using a single constant approximation is shown. At last we present results of beam propagation obtained with vectorial BPM combined with molecular reorientation model.

At first we deal with molecular reorientation in chiral and non-chiral nematic liquid crystals. The setup and coordinate system are shown in Fig. 1. Two angles to describe director orientation are used so n⃗ = [cosθ, sinθ sinϕ, sinθ cosϕ]. To derive equations describing molecular reorientation we start up with Frank-Oseen equation on free energy density [2123]:

f=12K11(n)2+12K22(n(×n)G)2+12K33(n×(×n))212Δεε0(nE)2
where Δε - electrical anisotropy, K11,K22, K33 - Frank elastic constants, G=2πp - parameter describing chirality, p - pitch, E⃗ - electric field. Subsequently n⃗ is substituted to Eq. (1) and then Euler-Lagrange equations of the following form are used:
xfϕx+yfϕy+zfϕzfϕ=0xfθx+yfθy+zfθzfθ=0
where ϕx=ϕx, ϕy=ϕy, ϕz=ϕz, θx=θx, θy=θy, θz=θz. We are searching for the extrema of the free energy. Equations (2) give two following equations describing molecular reorientation:
sin2θ(K33cos2θ+K22sin2θ)2ϕx2++sin2θ(K11cos2ϕ+sin2ϕ(K22cos2θ+K33sin2θ))2ϕy2++sin2θ(K11sin2ϕ+cos2ϕ(K22cos2θ+K33sin2θ))2ϕz2++sin2θ(K11K22cos2θK33sin2θ)××[((ϕz)2(ϕy)222ϕyz)12sin2ϕcos2ϕϕzϕy]++12sin2θ(K22K11)[(2θz22θy2)12sin2ϕcos2ϕ2θyz]++sin2θ(K33+K112K22)[((θz)2(θy)2)12sin2ϕcos2ϕθzθy]++sin2θ(K22K11)[2θxycosϕ2θxzsinϕ]++2cosθsin3θ(K33K22)[cosϕ(2ϕxz+12ϕyϕx)+sinϕ(2ϕxy12ϕzϕx)]++sin2θ(sin2θ3cos2θ)(K22K33)××[cosϕ(θzϕx+ϕzθx)+sinϕ(θyϕx+ϕyθx)]++sin2θ[sin2θ(2K22K33)+K33cos2θ]ϕxθx++12sin2θ(2K22K11K33)[cosϕθxθysinϕθzθx]++sin2θ(K11sin2ϕ+cos2ϕ(K22cos2θ+(2K33K22)sin2θ))ϕzθz++sin2θ(K11cos2ϕ+sin2ϕ(K22cos2θ+(2K33K22)sin2θ))ϕyθy++12sin2θsin2ϕ(K22cos2θ+(2K33K22)sin2θK11)(ϕzθy+ϕyθz)+GK22(sin2θθx+2sin2θsinϕθy+2sin2θcosϕϕz)+f(E)=0
where
f(E)=Δεε02(2EyEzsin2θcos2ϕ+sin2θ(ExEycosϕExEzsinϕ)++sin2θsin2ϕ(|Ey|2|Ez|2))
(K11sin2θ+K33cos2θ)2θx2+(K22cos2ϕ+sin2ϕ(K11cos2θ+K33sin2θ))2θy2++(K22sin2ϕ+cos2ϕ(K11cos2θ+K33sin2θ))2θz2++12sin2θ(K22K11)[(2ϕz22ϕy2)12sin2ϕcos2ϕ2ϕyz]++12sin2θ(K33K11)[(θz)2cos2ϕ+(θy)2(θx)2+sin2ϕθzθy+sinϕ(θzϕx+ϕzθx22θxy)+cosϕ(θyϕx+ϕyθx+22θxz)]++12sin2θ[(2sin2θ(K22K33)K11)cos2ϕK22sin2ϕ](ϕz)2++12sin2θ[(2sin2θ(K22K33)K11)sin2ϕK22cos2ϕ](ϕy)2++(K11cos2θ+K33sin2θK22)[sin2ϕ(2θyz+ϕyθyϕzθz)++cos2ϕ(ϕyθz+ϕzθy)]+12sin2θ[(K332K22)sin2θK33cos2θ](ϕx)2++cos2θ(K33K11)[sinϕθyθxcosϕθzθx]++sin2θ[K11+(3cos2θsin2θ)(K22K33)K22](sinϕϕyϕx+cosϕϕzϕx)+12sin2ϕsin2θ[2sin2θ(K22K33)+K22K11]ϕzϕy++sin2θ(K11K22)(2ϕxzsinϕ2ϕxycosϕ)++GK22(sin2θϕx+2sin2θsinϕϕy+2sin2θcosϕϕz)+ξ(E)=0
where
ξ(E)=Δεε02(EzEysin2θsin2ϕ+2cos2θ(ExEysinϕ++ExEzcosϕ)+sin2θ(|Ez|2cos2ϕ+|Ey|2sin2ϕ|Ex|2))
where EiEj=Ei*Ej+EiEj* and i, j ∈ {x,y,z}. Equations (3) and (5) can be simplified by neglecting derivatives with respect to z. Such assumption means that intensity and polarization slowly varies along the direction of propagation i.e. we use slowly varying envelope approximation. The following equations are obtained:
sin2θ(K33cos2θ+K22sin2θ)2ϕx2++sin2θ(K11cos2ϕ+sin2ϕ(K22cos2θ+K33sin2θ)2ϕy2+12sin2ϕsin2θ(K11K22cos2θK33sin2θ)(ϕy)2+14sin2θsin2ϕ(K22K11)2θy212sin2ϕsin2θ(K33+K112K22)(θy)2++sin2θcosϕ(K22K11)2θxy++2cosθsin3θ(K33K22)[sinϕ2ϕxy+12cosϕϕyϕx]++sin2θ(sin2θ3cos2θ)sinϕ(K22K33)[θyϕx+ϕyθx]++sin2θ(sin2θ(2K22K33)+cos2θK33)ϕxθx++12sin2θ(2K22K11K33)cosϕθxθy++sin2θ[K11cos2ϕ+sin2ϕ(K22cos2θ+(2K33K22)sin2θ)]ϕyθy+GK22(sin2θθx+2sin2θsinϕθy)+f(E)=0
(K11sin2θ+K33cos2θ)2θx2+(K22cos2ϕ+sin2ϕ(K11cos2θ+K33sin2θ))2θy2+14sin2θsin2ϕ(K22K11)2ϕy2++12sin2θ(K33K11)[(θy)2(θx)2+cosϕ(θyϕx+ϕyθx)+2sinϕ2θxy]++12sin2θ[(2sin2θ(K22K33)K11)sin2ϕK22cos2ϕ](ϕy)2++sin2ϕ(K11cos2θ+K33sin2θK22)ϕyθy++12sin2θ[(K332K22)sin2θK33cos2θ](ϕx)2++cos2θsinϕ(K33K11)θyθx++sin2θsinϕ[K11+(3cos2θsin2θ)(K22K33)K22]ϕyϕx+sin2θcosϕ(K11K22)2ϕxy++GK22(sin2θϕx+2sin2θsinϕϕy)+ξ(E)=0

 figure: Fig. 1

Fig. 1 The analyzed setup and coordinate system.

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Another significant simplification can be made by using single Frank constant approximation. We preserve all derivatives with respect to x, y and z directions and assume K11 = K22 = K33 = K. In this case Eqs. (3) and (5) simplify to the already known form [1719]:

sin2θ2ϕ+sin2θ(ϕxθx+ϕyθy+ϕzθz)+G(sin2θθx+2sin2θsinϕθy+2sin2θcosϕθz)+f(E)K=0
2θ12sin2θ((ϕx)2+(ϕy)2+(ϕz)2)++G(sin2θϕx+2sin2θsinϕϕy+2sin2θcosϕϕz)+ξ(E)K=0
When we assume that θ = 90° and Ex = 0, which means that the reorientation only occurs in yz plane, Eq. (7) simplifies to the following form:
K222ϕx2+(K11cos2ϕ+K33sin2ϕ)2ϕy212sin2ϕ(K11K33)(ϕy)2++Δεε02(2EyEzcos2ϕ+sin2ϕ(|Ey|2|Ez|2))=0
However, both equations have to be satisfied. Equation (8) is fulfilled for ϕy=0. Moreover, for planar configuration (G = 0) Eq. (8) is fulfilled for ϕy=0 or for ϕx=0. It means that Eq. (11) can only be used for homogeneous field along the y direction or (in planar case) along the x or y direction. In a single constant approximation Eq. (8) vanishes so Eq. (11) can be used without additional assumptions.

In the following part we will discuss only planar configuration (G = 0). When we assume ϕ = 0° and Ey = 0 so the reorientation occurs in xz plane Eq. (8) simplifies to the form:

(K11sin2θ+K33cos2θ)2θx2+K222θy2+12sin2θ(K33K11)[(θy)2(θx)2]++Δεε02(2cos2θExEz+sin2θ(|Ez|2|Ex|2))=0
while Eq. (7) if fulfilled for θx=0 or for θy=0 i.e. for homogeneous fields along either x or y direction.

As it was stated, the above Eqs. (11) and (12) can be used for homogeneous fields only. For finite beams reorientation will occur not only in a single plane and will lead to both ϕ and θ reorientations. Such molecular reorientation occurs mainly at the edges of the beam, where the field evanesce and it gives the lowest energy of orientation. In other words reorientation of molecules in a single plane will not lead to the lowest energy state and thus ϕ and θ reorientation have to occur.

Assuming that ϕ = 90° and Ez = 0 i.e. the reorientation occurs in xy plane, Eq. (8) simplifies to:

(K11sin2θ+K33cos2θ)2θx2+(K11cos2θ+K33sin2θ)2θy2++12sin2θ(K33K11)((θy)2(θx)2+22θxy)+cos2θ(K33K11)θyθx++Δεε02(2cos2θExEy+sin2θ(|Ey|2|Ex|2))=0
and while Eq. (7) vanishes for G = 0 the above equation needs no additional assumptions.

2. Numerical results

In this section numerical simulations of molecular reorientation and beam propagation in nematic liquid crystals are presented. For simulations Gaussian beam containing the electric (E⃗) and magnetic (H⃗) field is used:

Ex=E0xexp((x0x)2(y0y)22w02)sinαHx=ε22(x,y)Z0EyEy=E0yexp((x0x)2+(y0y)22w02)cosαHy=ε11(x,y)Z0ExEz=0Hz=0
where Z0=μ0ε0 - vacuum impedance, w0 - beam waist, α - polarization angle, E0 - amplitude Ex,y,z - electric field components, Hx,y,z - magnetic field components, E0-amplitude, ε - electrical permittivity tensor defined as:
ε=[ε±+Δεcos2θΔεsinθcosθsinϕΔεsinθcosθcosϕΔεsinθcosθsinϕε+Δεsin2ϕsin2θΔεsinϕcosϕsin2θΔεsinθcosθcosϕΔεsinϕcosϕsin2θε+Δεsin2θcos2ϕ]

For modeling molecular reorientation only electric field components of Gaussian beam are used. To obtain desired power (P) of the beam the amplitudes E0x and E0y were chosen to fulfill ∬Szdxdy = P were Sz - is a z component of Poynting vector.

2.1. Molecular reorientation

Calculations of molecular reorientation are made for a cell of L = 25μm in height and W = 50μm in width on a 2D grid of 50x100. Electric field amplitude corresponds to a Gaussian beam of a power P = 100mW. Optical field waist is equal ω0 = 3μm (FWHM≈ 7μm). Parameters of liquid crystals correspond to 6CHBT liquid crystals [24, 25] with anisotropy Δε = 0.47 and Frank elastic constants K11 = 8.8pN, K22 = 3.7pN and K33 = 9.5pN. Equations (7) and (8) are solved using SOR (Succesive Over-Relaxation) iterative method [26, 27], with relaxation parameter Ω = 1.8 and resolution Δx = Δy = Δh = 0.5μm if not otherwise stated. Typically convergence is obtained after no more than 300 iterations which is discussed later. Initial and boundary conditions depend on liquid crystal configuration. We consider chiral and non-chiral (planar) configurations. In planar configuration initial conditions are as follows θ(x,y)=π2 and ϕ(x,y) = ϕ0. In many simulations reorientation occurs with Freedericksz threshold [28] therefore molecules (except boundary conditions) where realigned in relation to the initial conditions as follows Δϕ = +0.01° and Δθ = −0.01°. For instance for Ex ≠ 0 and Ey = 0, Ez = 0, terms f(E⃗) and ξ(E⃗) are equal 0 (vanish) so no reorientation can occur. Realigning the molecules makes terms f (E⃗) and ξ(E⃗) non-zero and calculations with a threshold are possible. In chiral configuration initial conditions are ϕ(x,y) = ϕ0 +Gx where G=2πp and θ(x,y)=π2. In calculations pitch of a cholesterics p and pitch induced by the boundary conditions p′ are equal to the cell height so p = p′ = L and G′ = G. For each configuration (chiral and planar) at the edges of the system initial conditions are fixed and constitute the boundary conditions. In the next paragraphs we will discuss solutions of Eqs. (7) and (8) and compare them to the solutions of Eqs. (9) and (10) assuming single Frank constant approximation. We will also present empirical equations which can help to determine value of an elastic constant K when using single constant approximation.

Planar configuration

At first let us consider planar configuration where molecules initially lie in the yz plane (θ(x,y) = const. = 90°) and are rotated through some angle ϕ0 corresponding to the z axis. Solutions for ϕ0 = 0° and electric field applied along the y-axis (α = 0°) are presented in Fig. 2. Assuming reorientation only in yz plane Eq. (7) simplifies to Eq. (11) but, as it was stated before, it can be only used for homogeneous field along either x or y direction, because Eq. (8) is satisfied only for homogeneous fields. In fact there is some θ reorientation up to ±2°. Such reorientation can be neglected so the simplified Eq. (11) can be used. Moreover in a single constant approximation there is no θ reorientation (it is below 10−5 deg. which is caused by numerical error). It is consistent with our previous conclusions made while describing Eq. (11). However simplified equation can be used it is difficult to determine a proper value of K in a single constant approximation. In the considered case reorientation occurs only in yz plane. Due to our simulations performed for different liquid crystals and setups empirical equation can be used to determine elastic constant:

K=23K22+13(K11+K332)
Using (16) gives well reproduction of ϕ reorientation not only in discussed setup but in all the following configurations including chiral one. However, maximum values of ϕ reorientation are comparable between both sets of equations, analyzing results obtained with exact Eqs. (7) and (8) shows that the shape of ϕ reorientation is elliptical in contrast to the results obtained using Eqs. (9) and (10) where the reorientation is rather circular. Elliptical shape is caused by the differences between Frank constants corresponding to different deformations (i.e. K11 - splay, K22 - twist and K33 - bend). It was also checked that the elliptical shape of reorientation is neither caused by the boundary conditions nor by using two angles to describe director orientation. Even in one-angle model while using all Frank elastic constants the reorientation has elliptical shape.

 figure: Fig. 2

Fig. 2 Results of molecular reorientation for planar configuration ϕ0 = 0° and Ex = 0, Ey ≠ 0 (y-polarized light). On the left solutions obtained using Eqs. (9) and (10), on the right solutions obtained using Eqs. (7) and (8). Form top to bottom: ϕ(x,y) distribution, θ(x,y) distribution and visualization of molecules orientation from two orthogonal points of view.

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When assuming initial conditions ϕ0 = 0° and θ(x,y) = 90° as in the previous case and an electric field applied along the x-axis (x-polarized α = 90°), reorientation will mainly occur in the xz plane (ϕ = const.). Solutions of this configuration are presented in Fig. 3. In this case Eq. (8) simplifies to Eq. (12). However, Eq. (7) has solutions only for homogeneous fields along either x or y directions, it appears that ϕ reorientation is relatively weak (up to ±5°), corresponding to θ reorientation (more than 70°) so the simplified Eq. (12) may be used. However, maximum θ reorientation obtained using Eqs. (7)(8) and Eqs. (9)(10) is the same, the shape of reorientation in a single constant approximation is elliptical in contrast to the circular shape obtained with full equations. To determine K in this configuration we propose an equation:

K=K22+K332

 figure: Fig. 3

Fig. 3 Results of molecular reorientation for planar configuration ϕ0 = 0° and Ey = 0, Ex ≠ 0 (x-polarized light).

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Consider another planar configuration where the molecules lie in yz plane (θ(x,y) = 90°) and are aligned along the y-axis (ϕ0 = 90°) and electric field is applied along the x-axis (i.e. is x-polarized α = 90°). Numerical results are shown in Fig.4. The reorientation is expected to occur in xy plane only so Eq. (8) can be simplified to Eq. (13). For planar configuration Eq. (7) vanishes so Eq. (13) is an exact one. In fact, results given by exact Eqs. (7) and (8) as well as simplified Eqs. (9) and (10) show that there is no reorientation in yz plane (ϕ reorientation is below 10−5 deg.). To determine value of K for this configuration we propose the following equation:

K=K11+K332
In Fig. 4 it is visible that not only maximum values of reorientation but also a shape is the same for both (exact and simplified) sets of equations.

 figure: Fig. 4

Fig. 4 Results of molecular reorientation for planar configuration ϕ0 = 90° and Ez = 0, Ex ≠ 0 (x-polarized light).

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In the previously analyzed, particular planar configurations, equations could be simplified and differences between results obtained using single constant approximation and full equations were not so significant. Moreover, it was quite easy to determine proper K value as the reorientation occurred in a single plane. For now, let us consider more general case, where molecules initially lie in the yz plane and are rotated through an angle ϕ0 = 45° and the applied field is polarized at α = 45° (Ex = Ey). In such case both ϕ and θ reorientations are important and need be taken into account. Solutions are presented in Fig. 5. However, maximum values of reorientation obtained with full Eqs. (7) and (8) and equations using single constant approximation (9) and (10) are comparable, the shape of ϕ reorientation is not only elliptical but also the long axis of ellipse does not lie along the y direction. This rotation of the long axis vanishes in one-angle model, while using only angle ϕ to describe director orientation.

 figure: Fig. 5

Fig. 5 Results of molecular reorientation for planar configuration ϕ0 = 45° and Ez = 0, Ex = Ey ≠ 0 (α = 45°).

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In this configuration, where as well ϕ as θ reorientation is important, it is very difficult to chose proper K value.

Chiral configuration

Another simulations were made for chiral configuration where molecules are rotated through an angle of 360°. Optical field is polarized at α = 45°. Solutions are presented in Fig. 6. Results obtained with full equations and simplified equations (using single constant approximation) are nearly the same. Total reorientation and the shape of reorientation are comparable. To determine proper K value we propose to use Eq. (16).

 figure: Fig. 6

Fig. 6 Results of molecular reorientation for chiral configuration where molecules are rotated through an angle of 360°. Optical field is polarized at α = 45°, (Ex = Ey, Ez = 0).

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Convergence, timing and numerical uncertainties

To determine convergence of Eqs. (7) and (8) we have performed simulations of planar configuration where ϕ0 = 45°, w0 = 3μm, α = 45°, P = 100mW, Ω = 1.8, K11 = 8.8pN, K22 = 3.7pN, K33 = 9.5pN. Calculations were performed as previously for a cell of L = 25μm in height and W = 50μm in width. We have measure maximum positive ϕ reorientation and maximum negative θ reorientation as only these occur in such configuration. Reorientation versus number of iterations is presented in Fig. 7. Convergence of the analyzed equations is very well, but what has to be stressed it strongly depends on resolution. For instance for Δh = 0.5μm (grid of 100x50 points) convergence is obtain after approximately 50 iterations, but for Δh = 0.1μm convergence is not obtained even for 300 iterations. We have also compared convergence of Eqs. (7) and (8) with Eqs. (9) and (10) (see Fig. 7). Convergence of both sets of equations is comparable.

 figure: Fig. 7

Fig. 7 Maximum molecular reorientation versus number of iterations for planar configuration. Results for different resolution are presented on the left. On the right comparison between exact Eqs. (7) and (8) and simplified ones (9) and (10) is shown.

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Calculations were made on PC with an Intel Pentium IV clocked at 2.66 GHz and 1.5GB of RAM. Computation time of the above planar configuration on a grid of 50x100 points with 300 iterations was up to 7 sec. for full equations and 6.5 sec. for simplified ones. For a grid of 100x200 points timing was 28 sec. and 26.5 sec. respectively.

To estimate the numerical uncertainty calculations were made for a planar configuration with no external field applied and a noise of amplitude Anoise = 45° added to the initial conditions. Uncertainty was estimated to be not grater than 0.3°.

To show the changes in the results due to different numerical steps, calculations for planar configuration were performed. Results of 3000 iterations performed on a grid of 50x100 (Δh = 0.5μm), 100x200 (Δh = 0.25μm) and 250x500 (Δh = 0.1μm) points using both models were compared. Differences in maximum and minimum reorientation between models did not exceed 2°. Differences were mainly caused by choosing slightly inappropriate K value. When it comes to results for different resolution there was no change between the results (with accuracy ± 0.1°).

2.2. Beam propagation

In this section we describe light propagation in nematic liquid crystal cell. To model molecular reorientation Eqs. (7) and (8) have been used. For modeling optical field propagation we use Full Vector Beam Propagation Method (FVBPM) [9, 20] derived directly from Maxwell equations with no paraxial approximation. Nevertheless, paraxial approximation is indirectly preset as we neglected derivatives with respect to z in molecular reorientation model. The following equations were used to describe light propagation:

HzyHyz=iωε0(ε11Ex+ε12Ey+ε13Ez)Hx=1iμ0ω(EzyEyz)HxzHzx=iωε0(ε21Ex+ε22Ey+ε23Ez)Hy=1iμ0ω(ExzEzx)HyxHxy=iωε0(ε31Ex+ε32Ey+ε33Ez)Hz=1iμ0ω(EyxExy)

Equations (19) were rewritten using finite differences and solved using Runge-Kutta 4th order method. Calculations were performed on a 2D transverse grid with resolution Δx = Δy = 0.5μm. The same resolution was taken to calculate molecular reorientation. Longitudinal resolution was set to Δz = 0.01μm. Molecular reorientation was recalculated after each 100nm of propagation. At the boundaries Dirichlet conditions were used. Simulations were performed for two different liquid crystals 6CHBT (K11 = 8.8pN K22 = 3.7pN K33 = 9.5pN, Δε = 0.44@793nm) and 1110 (K11 = 21pN K22 = 8pN K33 = 21pN Δε = 0.13@1064nm). Gaussian beam described by (14) was launched into the system. Light was propagated along the z axis. Simulations showing beam propagation in a cell filled with liquid crystals in a planar configuration are shown in Fig. 9, 10 and 11. The beam is polarized at α = 45°. Comparison between results obtained using full Eqs. (7) and (8) and results obtained using single constant approximation (Eqs. (9) and (10)) is also made. There are no significant differences neither in beam propagation nor in its polarization distribution between these two models. However, it has to be stressed that it is very important to choose appropriate K value in a single constant approximation. In Fig. 9 and 10 it appears that for inappropriate value of K = 19pN (right) light distribution and beam polarization differs from the exact solution (left). In the xz plane the beam ”breathes” stronger in a single constant approximation. It is caused by the fact that in a single constant approximation ϕ reorientation along the x and y direction is comparable and nearly symmetrical in contrast to the non-symmetrical reorientation appearing when using exact equations (see Fig. 5). Moreover for inappropriate K value the portion of the x-polarized light propagated nearly on axis is larger, and the x-polarized light guided by the y-polarized soliton propagates on a greater distance (see. Fig. 9). It has to be stressed that in a single constant approximation changing K value leads to the same results as changing power. So in fact, choosing appropriate K value in a single constant approximation is about finding the right power values. In Fig. 9 and 10 at the end of propagation the extraordinary part of light slightly changes direction of propagation. It is caused by the boundary which is very close to the propagated beam and has fixed anchoring conditions. Simulations performed for wider numerical grids (in y direction) show no such effect. To compare exact equations with single constant approximation walk-off angle versus input power was also examined (Fig. 8). Both models give comparable results with differences not greater than 0.3°.

 figure: Fig. 8

Fig. 8 Walk-off angle versus input power, for 1110 liquid crystals in planar configuration. Marked uncertainties are resulting from error in reading walk-off angle (≈ 0.11°) from simulations and do not take into account numerical errors of calculations.

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 figure: Fig. 9

Fig. 9 Beam propagation in 1110 liquid crystal cell (50μm × 100μm) of a planar configuration. Beam launched into the system was polarized at α = 45° with a FWHM=9μm and power P = 25mW. Simulations were carried out using exact equations with all Frank constants (left) and using single constant approximation (middle and right). From top to bottom: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction) and polarization distribution of a cross-section for x = 25μm. The x-polarized light is marked as red and the y-polarized light as blue.

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 figure: Fig. 10

Fig. 10 Beam propagation for the same parameters as in Fig. 9 with power P=50mW.

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 figure: Fig. 11

Fig. 11 Beam propagation in 6CHBT liquid crystals cell (50μm ×125μm) of a planar configuration. Beam launched into the system was polarized at α = 45° with a FWHM=7μm and power P = 1mW. Simulations were carried out using exact equations with all Frank constants (top) and using single constant approximation (bottom). From left to right: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction) and polarization distribution of a cross-section for x = 25μm. The x-polarized light is marked as red and the y-polarized light as blue.

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When it comes to higher-birefringence liquid crystals 6CHBT, the results obtained with both models are comparable when using correct K value. However, there are some differences comparing to the low birefringence liquid crystals 1110. Obviously, the minimum power needed to form a soliton is significantly lower (about P ≈ 1mW) but there is also a change in polarization propagation. For 1110 some part of the x-polarized is propagated nearly on axis and the rest is guided by the y-polarized soliton. In 6CHBT whole x-polarized light is propagated straight ahead and only the y-polarized forms a soliton.

Simulations of light propagation in a cholesteric configuration are shown in Fig. 12. Cholesterics is rotated through an angle of 360°. The beam is y-polarized. Equations and results are much more stable for y-polarized light in comparison to α = 45°. It may be connected with paraxial approximation which is better fulfilled for the y-polarized light because there is mostly ϕ reorientation and there are nearly no changes in polarization along the z axis. It is visible that for appropriately chosen K the result obtained with both models are comparable. However, diffraction of the beam is confined in yz plane, in the xz plane some ”breathing” appears. It is caused by the presence of the layers in cholesterics. Light reflects from the edge of the layer and since there is no distinguished direction, intensity distribution is symmetric with respect to the center of the cell.

 figure: Fig. 12

Fig. 12 Beam propagation in a cell filled with 1110 liquid crystals (50μm × 50μm) in a cholesteric configuration. Beam launched into the system was y-polarized with a FWHM=9μm and power P = 200mW. Simulations were carried out using exact equations with all Frank constants (left) and using single constant approximation (right). From top to bottom: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction).

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3. Conclusions

Summarizing, we have presented exact equations describing molecular reorientation in nematic liquid crystals. We have taken into account all derivatives with respect to all directions x, y and z and all diagonal Frank constants K11,K22, K33. To the best of our knowledge it is the first time these equations have been explicitly written. We have shown similarities and differences between molecular reorientation obtained using full equations and equations with single Frank constant approximation. Most of the results obtained using these two models are comparable, but it is crucial to choose an appropriate K value. We also proposed a simple, empirical equation which facilitates calculating K in some simple cases. Full equations neglecting only derivative with respect to z were combined with FV-BPM to provide beam propagation in a liquid crystal cell. Calculations were performed for planar and chiral configurations. No significant differences were observed between results obtained using all Frank constants in comparison to single constant approximation. However, in many cases it is very difficult to choose a correct K value. In such cases we propose to recalculate molecular reorientation using full equations and then recalculate reorientation for a few different K values (using single constant approximation) typically fulfilling K22 < K < K33 and found one which gives results close to the results obtained using full equations. Such K value can be used to calculate molecular reorientation and be combined with FV-BPM. Such approach will significantly shorten computation time as the Eqs. (9) and (10) are much simpler than full equations (7) and (8). Since qualitative results obtained using single constant approximation are similar to the results obtained using all Frank elastic constants, choosing appropriate K value is a matter of scaling the power at which particular phenomena occurs. In other words changing K is the same as changing power P so for qualitative results it is not so crucial and nearly any K value can be assumed. Finally it has to be stressed that presented results and conclusions correspond to the beams of a Gaussian or topologically similar shapes. For other cases, for instance vortexes or other topological defects, results may be different but this is out of scope of our analysis.

Acknowledgments

This work was financially supported by the National Science Centre. Filip Sala was supported by a scholarship from the Center for Advanced Studies.

References and links

1. D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980). [CrossRef]  

2. S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007). [CrossRef]  

3. B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific Publishing Co. Pte. Ltd., 1995).

4. G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003). [CrossRef]  

5. A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010). [CrossRef]  

6. G. Assanto and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009). [CrossRef]  

7. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011). [CrossRef]   [PubMed]  

8. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003). [CrossRef]   [PubMed]  

9. F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photon. Lett. Pol. 1, 163–165 (2009).

10. A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007). [CrossRef]  

11. P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17, 10895–10909 (2009). [CrossRef]   [PubMed]  

12. J. Beeckman, K. Neyts, P. Vanbrabant, R. James, and F. Fernandez, “Finding exact spatial soliton profiles in nematic liquid crystals,” Opt. Express 18, 3311–3321 (2010). [CrossRef]   [PubMed]  

13. G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008). [CrossRef]  

14. J. A. Fleck Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976). [CrossRef]  

15. B. Y. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985). [CrossRef]  

16. U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008). [CrossRef]  

17. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006). [CrossRef]  

18. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008). [CrossRef]   [PubMed]  

19. F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

20. F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 558, 176–183 (2012) [CrossRef]  

21. I.-C. Khoo, Liquid Crystals (John Wiley and Sons, 2007). [CrossRef]  

22. F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958). [CrossRef]  

23. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933). [CrossRef]  

24. W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985). [CrossRef]  

25. R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985). [CrossRef]  

26. D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” PhD thesis, Harvard University (1950).

27. L. Hageman and D. Young, Applied Iterative Methods (Academic Press, 1981).

28. P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

References

  • View by:

  1. D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
    [Crossref]
  2. S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
    [Crossref]
  3. B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific Publishing Co. Pte. Ltd., 1995).
  4. G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003).
    [Crossref]
  5. A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
    [Crossref]
  6. G. Assanto and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
    [Crossref]
  7. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011).
    [Crossref] [PubMed]
  8. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
    [Crossref] [PubMed]
  9. F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photon. Lett. Pol. 1, 163–165 (2009).
  10. A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
    [Crossref]
  11. P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17, 10895–10909 (2009).
    [Crossref] [PubMed]
  12. J. Beeckman, K. Neyts, P. Vanbrabant, R. James, and F. Fernandez, “Finding exact spatial soliton profiles in nematic liquid crystals,” Opt. Express 18, 3311–3321 (2010).
    [Crossref] [PubMed]
  13. G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
    [Crossref]
  14. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [Crossref]
  15. B. Y. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
    [Crossref]
  16. U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
    [Crossref]
  17. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
    [Crossref]
  18. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
    [Crossref] [PubMed]
  19. F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).
  20. F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 558, 176–183 (2012)
    [Crossref]
  21. I.-C. Khoo, Liquid Crystals (John Wiley and Sons, 2007).
    [Crossref]
  22. F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
    [Crossref]
  23. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
    [Crossref]
  24. W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
    [Crossref]
  25. R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
    [Crossref]
  26. D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” PhD thesis, Harvard University (1950).
  27. L. Hageman and D. Young, Applied Iterative Methods (Academic Press, 1981).
  28. P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

2012 (2)

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 558, 176–183 (2012)
[Crossref]

2011 (1)

2010 (2)

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[Crossref]

J. Beeckman, K. Neyts, P. Vanbrabant, R. James, and F. Fernandez, “Finding exact spatial soliton profiles in nematic liquid crystals,” Opt. Express 18, 3311–3321 (2010).
[Crossref] [PubMed]

2009 (3)

P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17, 10895–10909 (2009).
[Crossref] [PubMed]

G. Assanto and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[Crossref]

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photon. Lett. Pol. 1, 163–165 (2009).

2008 (3)

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[Crossref]

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[Crossref]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[Crossref] [PubMed]

2007 (2)

A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
[Crossref]

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[Crossref]

2006 (1)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[Crossref]

2003 (2)

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

1985 (3)

B. Y. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[Crossref]

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[Crossref]

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[Crossref]

1980 (1)

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[Crossref]

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

1958 (1)

F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[Crossref]

1933 (1)

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[Crossref]

Alberucci, A.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[Crossref]

A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
[Crossref]

Assanto, G.

Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011).
[Crossref] [PubMed]

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[Crossref]

G. Assanto and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[Crossref]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[Crossref] [PubMed]

A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
[Crossref]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003).
[Crossref]

Bahadur, B.

B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific Publishing Co. Pte. Ltd., 1995).

Baran, W.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[Crossref]

Beeckman, J.

Berreman, D. W.

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[Crossref]

Chien, L.-C.

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[Crossref]

Conti, C.

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

Dabrowski, R.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[Crossref]

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[Crossref]

de Gennes, P.

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

Desyatnikov, A. S.

Dyadyusha, A.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[Crossref] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[Crossref]

Dziaduszek, J.

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[Crossref]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Fernandez, F.

Fernandez, F. A.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Frank, F. C.

F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[Crossref]

Hageman, L.

L. Hageman and D. Young, Applied Iterative Methods (Academic Press, 1981).

Heffner, W. R.

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[Crossref]

Izdebskaya, Y. V.

James, R.

Jaworowicz, K.

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[Crossref]

Kaczmarek, M.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[Crossref]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[Crossref] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[Crossref]

Karpierz, M. A.

F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 558, 176–183 (2012)
[Crossref]

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

G. Assanto and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[Crossref]

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photon. Lett. Pol. 1, 163–165 (2009).

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[Crossref]

Kedzierski, J.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[Crossref]

Khoo, I.-C.

I.-C. Khoo, Liquid Crystals (John Wiley and Sons, 2007).
[Crossref]

Kivshar, Y. S.

Kriezis, E. E.

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[Crossref]

Laudyn, U. A.

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[Crossref]

Lu, S.-Y.

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[Crossref]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Neyts, K.

Oseen, C. W.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[Crossref]

Peccianti, M.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[Crossref]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[Crossref] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[Crossref]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

Piccardi, A.

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[Crossref]

Prost, J.

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

Raszewski, Z.

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[Crossref]

Sala, F. A.

F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 558, 176–183 (2012)
[Crossref]

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photon. Lett. Pol. 1, 163–165 (2009).

Szczucinski, T.

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[Crossref]

Tabiryan, N. V.

B. Y. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[Crossref]

Vanbrabant, P.

Vanbrabant, P. J. M.

Young, D.

L. Hageman and D. Young, Applied Iterative Methods (Academic Press, 1981).

Young, D. M.

D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” PhD thesis, Harvard University (1950).

Zel’dovich, B. Y.

B. Y. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[Crossref]

Ziogos, G. D.

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[Crossref]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Appl. Phys. Lett. (2)

D. W. Berreman and W. R. Heffner, “New bistable cholesteric liquid-crystal display,” Appl. Phys. Lett. 37, 109–111 (1980).
[Crossref]

S.-Y. Lu and L.-C. Chien, “A polymer-stabilized single-layer color cholesteric liquid crystal display with anisotropic reflection,” Appl. Phys. Lett. 91, 131119 (2007).
[Crossref]

Discuss. Faraday Soc. (1)

F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19–28 (1958).
[Crossref]

J. Opt. Soc. Am. B (2)

A. Alberucci and G. Assanto, “Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions,” J. Opt. Soc. Am. B 24, 2314–2320 (2007).
[Crossref]

F. A. Sala and M. A. Karpierz, “Chiral and non-chiral nematic liquid crystal reorientation induced by inhomogeneous electric fields,” J. Opt. Soc. Am. B 29 (submitted) (2012).

Liq. Cryst. (1)

G. Assanto and M. A. Karpierz, “Nematicons: self-localised beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009).
[Crossref]

Mol. Cryst. Liq. Cryst. (4)

F. A. Sala and M. A. Karpierz, “Modeling of nonlinear beam propagation in chiral nematic liquid crystals,” Mol. Cryst. Liq. Cryst. 558, 176–183 (2012)
[Crossref]

U. A. Laudyn, K. Jaworowicz, and M. A. Karpierz, “Spatial solitons in chiral nematics,” Mol. Cryst. Liq. Cryst. 489, 214–221 (2008).
[Crossref]

W. Baran, Z. Raszewski, R. Dabrowski, and J. Kedzierski, “Some physical properties of mesogenic 4-(trans-4’-n-alkylcyclohexyl) isothiocyanatobenzenes,” Mol. Cryst. Liq. Cryst. 123, 237–245 (1985).
[Crossref]

R. Dabrowski, J. Dziaduszek, and T. Szczucinski, “Mesomorphic characteristics of some new homologous series with the isothiocyanato terminal group,” Mol. Cryst. Liq. Cryst. 124, 241–257 (1985).
[Crossref]

Nat. Phys. (1)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. 2, 737–742 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Opt. Photonics News (1)

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical spatial solitons in nematic liquid crystals,” Opt. Photonics News 14, 44–48 (2003).
[Crossref]

Opt. Quantum Electron. (1)

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40, 733–748 (2008).
[Crossref]

Photon. Lett. Pol. (1)

F. A. Sala and M. A. Karpierz, “Numerical simulation of beam propagation in a layer filled with chiral nematic liquid crystals,” Photon. Lett. Pol. 1, 163–165 (2009).

Phys. Rev. A (1)

A. Alberucci, A. Piccardi, M. Peccianti, M. Kaczmarek, and G. Assanto, “Propagation of spatial optical solitons in a dielectric with adjustable nonlinearity,” Phys. Rev. A 82, 023806 (2010).
[Crossref]

Phys. Rev. Lett. (2)

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Escaping solitons from a trapping potential,” Phys. Rev. Lett. 101, 153902 (2008).
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

B. Y. Zel’dovich and N. V. Tabiryan, “Orientational optical nonlinearity of liquid crystals,” Sov. Phys. Usp. 28, 1059–1083 (1985).
[Crossref]

Trans. Faraday Soc. (1)

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[Crossref]

Other (5)

D. M. Young, “Iterative methods for solving partial difference equations of elliptical type,” PhD thesis, Harvard University (1950).

L. Hageman and D. Young, Applied Iterative Methods (Academic Press, 1981).

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

I.-C. Khoo, Liquid Crystals (John Wiley and Sons, 2007).
[Crossref]

B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific Publishing Co. Pte. Ltd., 1995).

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Figures (12)

Fig. 1
Fig. 1 The analyzed setup and coordinate system.
Fig. 2
Fig. 2 Results of molecular reorientation for planar configuration ϕ0 = 0° and Ex = 0, Ey ≠ 0 (y-polarized light). On the left solutions obtained using Eqs. (9) and (10), on the right solutions obtained using Eqs. (7) and (8). Form top to bottom: ϕ(x,y) distribution, θ(x,y) distribution and visualization of molecules orientation from two orthogonal points of view.
Fig. 3
Fig. 3 Results of molecular reorientation for planar configuration ϕ0 = 0° and Ey = 0, Ex ≠ 0 (x-polarized light).
Fig. 4
Fig. 4 Results of molecular reorientation for planar configuration ϕ0 = 90° and Ez = 0, Ex ≠ 0 (x-polarized light).
Fig. 5
Fig. 5 Results of molecular reorientation for planar configuration ϕ0 = 45° and Ez = 0, Ex = Ey ≠ 0 (α = 45°).
Fig. 6
Fig. 6 Results of molecular reorientation for chiral configuration where molecules are rotated through an angle of 360°. Optical field is polarized at α = 45°, (Ex = Ey, Ez = 0).
Fig. 7
Fig. 7 Maximum molecular reorientation versus number of iterations for planar configuration. Results for different resolution are presented on the left. On the right comparison between exact Eqs. (7) and (8) and simplified ones (9) and (10) is shown.
Fig. 8
Fig. 8 Walk-off angle versus input power, for 1110 liquid crystals in planar configuration. Marked uncertainties are resulting from error in reading walk-off angle (≈ 0.11°) from simulations and do not take into account numerical errors of calculations.
Fig. 9
Fig. 9 Beam propagation in 1110 liquid crystal cell (50μm × 100μm) of a planar configuration. Beam launched into the system was polarized at α = 45° with a FWHM=9μm and power P = 25mW. Simulations were carried out using exact equations with all Frank constants (left) and using single constant approximation (middle and right). From top to bottom: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction) and polarization distribution of a cross-section for x = 25μm. The x-polarized light is marked as red and the y-polarized light as blue.
Fig. 10
Fig. 10 Beam propagation for the same parameters as in Fig. 9 with power P=50mW.
Fig. 11
Fig. 11 Beam propagation in 6CHBT liquid crystals cell (50μm ×125μm) of a planar configuration. Beam launched into the system was polarized at α = 45° with a FWHM=7μm and power P = 1mW. Simulations were carried out using exact equations with all Frank constants (top) and using single constant approximation (bottom). From left to right: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction) and polarization distribution of a cross-section for x = 25μm. The x-polarized light is marked as red and the y-polarized light as blue.
Fig. 12
Fig. 12 Beam propagation in a cell filled with 1110 liquid crystals (50μm × 50μm) in a cholesteric configuration. Beam launched into the system was y-polarized with a FWHM=9μm and power P = 200mW. Simulations were carried out using exact equations with all Frank constants (left) and using single constant approximation (right). From top to bottom: light intensity distribution in the xz plane (integrated along the y direction), light intensity distribution in the yz plane (integrated along the x direction).

Equations (19)

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f = 1 2 K 11 ( n ) 2 + 1 2 K 22 ( n ( × n ) G ) 2 + 1 2 K 33 ( n × ( × n ) ) 2 1 2 Δ ε ε 0 ( n E ) 2
x f ϕ x + y f ϕ y + z f ϕ z f ϕ = 0 x f θ x + y f θ y + z f θ z f θ = 0
sin 2 θ ( K 33 cos 2 θ + K 22 sin 2 θ ) 2 ϕ x 2 + + sin 2 θ ( K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + K 33 sin 2 θ ) ) 2 ϕ y 2 + + sin 2 θ ( K 11 sin 2 ϕ + cos 2 ϕ ( K 22 cos 2 θ + K 33 sin 2 θ ) ) 2 ϕ z 2 + + sin 2 θ ( K 11 K 22 cos 2 θ K 33 sin 2 θ ) × × [ ( ( ϕ z ) 2 ( ϕ y ) 2 2 2 ϕ y z ) 1 2 sin 2 ϕ cos 2 ϕ ϕ z ϕ y ] + + 1 2 sin 2 θ ( K 22 K 11 ) [ ( 2 θ z 2 2 θ y 2 ) 1 2 sin 2 ϕ cos 2 ϕ 2 θ y z ] + + sin 2 θ ( K 33 + K 11 2 K 22 ) [ ( ( θ z ) 2 ( θ y ) 2 ) 1 2 sin 2 ϕ cos 2 ϕ θ z θ y ] + + sin 2 θ ( K 22 K 11 ) [ 2 θ x y cos ϕ 2 θ x z sin ϕ ] + + 2 cos θ sin 3 θ ( K 33 K 22 ) [ cos ϕ ( 2 ϕ x z + 1 2 ϕ y ϕ x ) + sin ϕ ( 2 ϕ x y 1 2 ϕ z ϕ x ) ] + + sin 2 θ ( sin 2 θ 3 cos 2 θ ) ( K 22 K 33 ) × × [ cos ϕ ( θ z ϕ x + ϕ z θ x ) + sin ϕ ( θ y ϕ x + ϕ y θ x ) ] + + sin 2 θ [ sin 2 θ ( 2 K 22 K 33 ) + K 33 cos 2 θ ] ϕ x θ x + + 1 2 sin 2 θ ( 2 K 22 K 11 K 33 ) [ cos ϕ θ x θ y sin ϕ θ z θ x ] + + sin 2 θ ( K 11 sin 2 ϕ + cos 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ ) ) ϕ z θ z + + sin 2 θ ( K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ ) ) ϕ y θ y + + 1 2 sin 2 θ sin 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ K 11 ) ( ϕ z θ y + ϕ y θ z ) + G K 22 ( sin 2 θ θ x + 2 sin 2 θ sin ϕ θ y + 2 sin 2 θ cos ϕ ϕ z ) + f ( E ) = 0
f ( E ) = Δ ε ε 0 2 ( 2 E y E z sin 2 θ cos 2 ϕ + sin 2 θ ( E x E y cos ϕ E x E z sin ϕ ) + + sin 2 θ sin 2 ϕ ( | E y | 2 | E z | 2 ) )
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + ( K 22 cos 2 ϕ + sin 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ ) ) 2 θ y 2 + + ( K 22 sin 2 ϕ + cos 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ ) ) 2 θ z 2 + + 1 2 sin 2 θ ( K 22 K 11 ) [ ( 2 ϕ z 2 2 ϕ y 2 ) 1 2 sin 2 ϕ cos 2 ϕ 2 ϕ y z ] + + 1 2 sin 2 θ ( K 33 K 11 ) [ ( θ z ) 2 cos 2 ϕ + ( θ y ) 2 ( θ x ) 2 + sin 2 ϕ θ z θ y + sin ϕ ( θ z ϕ x + ϕ z θ x 2 2 θ x y ) + cos ϕ ( θ y ϕ x + ϕ y θ x + 2 2 θ x z ) ] + + 1 2 sin 2 θ [ ( 2 sin 2 θ ( K 22 K 33 ) K 11 ) cos 2 ϕ K 22 sin 2 ϕ ] ( ϕ z ) 2 + + 1 2 sin 2 θ [ ( 2 sin 2 θ ( K 22 K 33 ) K 11 ) sin 2 ϕ K 22 cos 2 ϕ ] ( ϕ y ) 2 + + ( K 11 cos 2 θ + K 33 sin 2 θ K 22 ) [ sin 2 ϕ ( 2 θ y z + ϕ y θ y ϕ z θ z ) + + cos 2 ϕ ( ϕ y θ z + ϕ z θ y ) ] + 1 2 sin 2 θ [ ( K 33 2 K 22 ) sin 2 θ K 33 cos 2 θ ] ( ϕ x ) 2 + + cos 2 θ ( K 33 K 11 ) [ sin ϕ θ y θ x cos ϕ θ z θ x ] + + sin 2 θ [ K 11 + ( 3 cos 2 θ sin 2 θ ) ( K 22 K 33 ) K 22 ] ( sin ϕ ϕ y ϕ x + cos ϕ ϕ z ϕ x ) + 1 2 sin 2 ϕ sin 2 θ [ 2 sin 2 θ ( K 22 K 33 ) + K 22 K 11 ] ϕ z ϕ y + + sin 2 θ ( K 11 K 22 ) ( 2 ϕ x z sin ϕ 2 ϕ x y cos ϕ ) + + G K 22 ( sin 2 θ ϕ x + 2 sin 2 θ sin ϕ ϕ y + 2 sin 2 θ cos ϕ ϕ z ) + ξ ( E ) = 0
ξ ( E ) = Δ ε ε 0 2 ( E z E y sin 2 θ sin 2 ϕ + 2 cos 2 θ ( E x E y sin ϕ + + E x E z cos ϕ ) + sin 2 θ ( | E z | 2 cos 2 ϕ + | E y | 2 sin 2 ϕ | E x | 2 ) )
sin 2 θ ( K 33 cos 2 θ + K 22 sin 2 θ ) 2 ϕ x 2 + + sin 2 θ ( K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + K 33 sin 2 θ ) 2 ϕ y 2 + 1 2 sin 2 ϕ sin 2 θ ( K 11 K 22 cos 2 θ K 33 sin 2 θ ) ( ϕ y ) 2 + 1 4 sin 2 θ sin 2 ϕ ( K 22 K 11 ) 2 θ y 2 1 2 sin 2 ϕ sin 2 θ ( K 33 + K 11 2 K 22 ) ( θ y ) 2 + + sin 2 θ cos ϕ ( K 22 K 11 ) 2 θ x y + + 2 cos θ sin 3 θ ( K 33 K 22 ) [ sin ϕ 2 ϕ x y + 1 2 cos ϕ ϕ y ϕ x ] + + sin 2 θ ( sin 2 θ 3 cos 2 θ ) sin ϕ ( K 22 K 33 ) [ θ y ϕ x + ϕ y θ x ] + + sin 2 θ ( sin 2 θ ( 2 K 22 K 33 ) + cos 2 θ K 33 ) ϕ x θ x + + 1 2 sin 2 θ ( 2 K 22 K 11 K 33 ) cos ϕ θ x θ y + + sin 2 θ [ K 11 cos 2 ϕ + sin 2 ϕ ( K 22 cos 2 θ + ( 2 K 33 K 22 ) sin 2 θ ) ] ϕ y θ y + G K 22 ( sin 2 θ θ x + 2 sin 2 θ sin ϕ θ y ) + f ( E ) = 0
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + ( K 22 cos 2 ϕ + sin 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ ) ) 2 θ y 2 + 1 4 sin 2 θ sin 2 ϕ ( K 22 K 11 ) 2 ϕ y 2 + + 1 2 sin 2 θ ( K 33 K 11 ) [ ( θ y ) 2 ( θ x ) 2 + cos ϕ ( θ y ϕ x + ϕ y θ x ) + 2 sin ϕ 2 θ x y ] + + 1 2 sin 2 θ [ ( 2 sin 2 θ ( K 22 K 33 ) K 11 ) sin 2 ϕ K 22 cos 2 ϕ ] ( ϕ y ) 2 + + sin 2 ϕ ( K 11 cos 2 θ + K 33 sin 2 θ K 22 ) ϕ y θ y + + 1 2 sin 2 θ [ ( K 33 2 K 22 ) sin 2 θ K 33 cos 2 θ ] ( ϕ x ) 2 + + cos 2 θ sin ϕ ( K 33 K 11 ) θ y θ x + + sin 2 θ sin ϕ [ K 11 + ( 3 cos 2 θ sin 2 θ ) ( K 22 K 33 ) K 22 ] ϕ y ϕ x + sin 2 θ cos ϕ ( K 11 K 22 ) 2 ϕ x y + + G K 22 ( sin 2 θ ϕ x + 2 sin 2 θ sin ϕ ϕ y ) + ξ ( E ) = 0
sin 2 θ 2 ϕ + sin 2 θ ( ϕ x θ x + ϕ y θ y + ϕ z θ z ) + G ( sin 2 θ θ x + 2 sin 2 θ sin ϕ θ y + 2 sin 2 θ cos ϕ θ z ) + f ( E ) K = 0
2 θ 1 2 sin 2 θ ( ( ϕ x ) 2 + ( ϕ y ) 2 + ( ϕ z ) 2 ) + + G ( sin 2 θ ϕ x + 2 sin 2 θ sin ϕ ϕ y + 2 sin 2 θ cos ϕ ϕ z ) + ξ ( E ) K = 0
K 22 2 ϕ x 2 + ( K 11 cos 2 ϕ + K 33 sin 2 ϕ ) 2 ϕ y 2 1 2 sin 2 ϕ ( K 11 K 33 ) ( ϕ y ) 2 + + Δ ε ε 0 2 ( 2 E y E z cos 2 ϕ + sin 2 ϕ ( | E y | 2 | E z | 2 ) ) = 0
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + K 22 2 θ y 2 + 1 2 sin 2 θ ( K 33 K 11 ) [ ( θ y ) 2 ( θ x ) 2 ] + + Δ ε ε 0 2 ( 2 cos 2 θ E x E z + sin 2 θ ( | E z | 2 | E x | 2 ) ) = 0
( K 11 sin 2 θ + K 33 cos 2 θ ) 2 θ x 2 + ( K 11 cos 2 θ + K 33 sin 2 θ ) 2 θ y 2 + + 1 2 sin 2 θ ( K 33 K 11 ) ( ( θ y ) 2 ( θ x ) 2 + 2 2 θ x y ) + cos 2 θ ( K 33 K 11 ) θ y θ x + + Δ ε ε 0 2 ( 2 cos 2 θ E x E y + sin 2 θ ( | E y | 2 | E x | 2 ) ) = 0
E x = E 0 x exp ( ( x 0 x ) 2 ( y 0 y ) 2 2 w 0 2 ) sin α H x = ε 22 ( x , y ) Z 0 E y E y = E 0 y exp ( ( x 0 x ) 2 + ( y 0 y ) 2 2 w 0 2 ) cos α H y = ε 11 ( x , y ) Z 0 E x E z = 0 H z = 0
ε = [ ε ± + Δ ε cos 2 θ Δ ε sin θ cos θ sin ϕ Δ ε sin θ cos θ cos ϕ Δ ε sin θ cos θ sin ϕ ε + Δ ε sin 2 ϕ sin 2 θ Δ ε sin ϕ cos ϕ sin 2 θ Δ ε sin θ cos θ cos ϕ Δ ε sin ϕ cos ϕ sin 2 θ ε + Δ ε sin 2 θ cos 2 ϕ ]
K = 2 3 K 22 + 1 3 ( K 11 + K 33 2 )
K = K 22 + K 33 2
K = K 11 + K 33 2
H z y H y z = i ω ε 0 ( ε 11 E x + ε 12 E y + ε 13 E z ) H x = 1 i μ 0 ω ( E z y E y z ) H x z H z x = i ω ε 0 ( ε 21 E x + ε 22 E y + ε 23 E z ) H y = 1 i μ 0 ω ( E x z E z x ) H y x H x y = i ω ε 0 ( ε 31 E x + ε 32 E y + ε 33 E z ) H z = 1 i μ 0 ω ( E y x E x y )

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