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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3363–3378
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Solving the full anisotropic liquid crystal waveguides by using an iterative pseudospectral-based eigenvalue method

Chia-Chien Huang  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3363-3378 (2011)
http://dx.doi.org/10.1364/OE.19.003363


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Abstract

This study develops an efficient mode solver based on pseudospectral eigenvalue algorithm to analyze liquid crystal waveguides with full 3 × 3 anisotropic permittivity tensors. Present formulation yields a cubic eigenvalue matrix equation with an eigenvalue of the propagation constant, and they are solved using an iterative approach following the transformation of the matrix equation to a standard linear eigenvalue equation. The proposed scheme significantly reduces the memory storage and computational time by using only transverse magnetic field components. Although the proposed scheme requires an iterative procedure, the convergent eigenvalues are achieved after performing only four iterations. Therefore, for this scheme, computational efforts remain greatly lower than those for other reported schemes that used at least three field components. For solving the modes of nematic liquid crystal waveguides, the numerical results obtained by the proposed scheme are in good agreement with those calculated by using the finite-element and the finite-difference frequency-domain schemes, thus verifying the applicability of the proposed approach. Furthermore, the mode patterns of liquid crystal waveguides under arbitrary molecular orientations are also characterized in detail.

© 2011 OSA

1. Introduction

Dielectric waveguides are essential building blocks of diverse photonic integrated circuits such as polarizers, switches, filters, rotators, and modulators. To accomplish the desired characteristics, these functional devices are mainly designed by altering the material characteristics by using electro-optic, magneto-optic, and thermo-optic modulations. Recently, liquid crystals (LCs) have been reconsidered as versatile optoelectronic materials for integrated optics [1

1. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37(1-3), 95–106 (2005). [CrossRef]

11

11. D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010). [CrossRef]

] because of their low scattering loss at near-infrared wavelengths [12

12. M. Kawachi, N. Shibata, and T. Edahiro, “Possibility of use of liquid crystals as optical waveguide material for 1.3μm and 1.55μm bands,” Jpn. J. Appl. Phys. 21(Part 2, No. 3), L162–L164 (1982). [CrossRef]

] and low power requirement resulting from their appreciable birefringence. The birefringence can be controlled using an applied voltage from the designed electrode. For LCs with arbitrary molecular orientations, the nine elements of the permittivity tensor are all nonzero. In consequence, for accurately analyzing the LC device performance, it is indispensible to develop a numerical scheme that considers an arbitrary permittivity tensor.

For solving anisotropic LC waveguides with full 3 × 3 permittivity tensors by using full-wave techniques, a number of numerical schemes have been proposed. These approaches are the finite-element frequency-domain (FEFD)-based eigenvalue mode solvers [6

6. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009). [CrossRef]

,13

13. M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986). [CrossRef]

15

15. S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express 15(24), 15797–15811 (2007). [CrossRef] [PubMed]

], the finite-element beam propagation methods (FE-BPMs) [3

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(10), 733–748 (2008). [CrossRef]

,9

9. 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(13), 10895–10909 (2009). [CrossRef] [PubMed]

,16

16. S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000). [CrossRef]

,17

17. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19(3), 405–413 (2001). [CrossRef]

], the finite-difference beam propagation method (FD-BPM) [4

4. E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707–5714 (2000). [CrossRef]

], and the finite-difference frequency-domain (FDFD)-based eigenvalue mode solvers [10

10. M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009). [CrossRef] [PubMed]

]. In the FE-BPMs [3

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(10), 733–748 (2008). [CrossRef]

,16

16. S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000). [CrossRef]

,17

17. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19(3), 405–413 (2001). [CrossRef]

], three electric (or magnetic) field components with 3N unknowns, are used to result in a matrix size of 9N 2, where N is the number of grid points. To avoid introducing Pade approximation, the FE-BPM algorithm developed by Vanbrabant et al. [9

9. 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(13), 10895–10909 (2009). [CrossRef] [PubMed]

] used the transverse electric field components and their derivatives with respect to the propagation direction z to determine the second-order derivative term; however, this leads to an increase of up to 4N in the number of unknowns, resulting in a matrix size of 16N 2. In the FD-BPM [4

4. E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707–5714 (2000). [CrossRef]

], a system of coupled differential equations, which involves an electric and a magnetic field component with 2N unknowns (the matrix size is 4N 2), is proposed for solving the twisted nematic pixels found in microdisplays. The required computational effort is relatively low. These BPM-based schemes [3

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(10), 733–748 (2008). [CrossRef]

,4

4. E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707–5714 (2000). [CrossRef]

,9

9. 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(13), 10895–10909 (2009). [CrossRef] [PubMed]

,16

16. S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beam propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36(12), 1392–1401 (2000). [CrossRef]

,17

17. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19(3), 405–413 (2001). [CrossRef]

] can study field profiles along the longitudinally variant waveguides but they need to consider the extra convergence conditions involving the step size of propagation, the two-point recurrence scheme, and the reference refractive index while solving the mode problems. Moreover, the BPM-based schemes cannot determine higher-order modes unless they transform the real propagation axis to the imaginary axis as in the imaginary-distance BPMs [18

18. J. C. Chen and S. Jüngling, “Computation of high-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, 199–205 (1994). [CrossRef]

,19

19. T. Ando, H. Nakayama, S. Numata, J. Yamauchi, and H. Nakano, “Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface,” J. Lightwave Technol. 20(8), 1627–1634 (2002). [CrossRef]

]. Thus far, to the best of my knowledge, imaginary-distance BPM schemes have not been developed to solve full anisotropy waveguide problems. In contrast, mode solver algorithms based on eigenvalues, such as the FEFD [6

6. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009). [CrossRef]

,13

13. M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986). [CrossRef]

15

15. S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express 15(24), 15797–15811 (2007). [CrossRef] [PubMed]

] and FDFD [10

10. M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009). [CrossRef] [PubMed]

] schemes, can deal with problems including all high-order modes. In [13

13. M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986). [CrossRef]

15

15. S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express 15(24), 15797–15811 (2007). [CrossRef] [PubMed]

], the FEFD schemes are formulated in term of all three magnetic or electric components with 3N unknowns to study the gyrotropic rectangular waveguides [13

13. M. Koshiba, K. Hayata, and M. Suzuki, “Finite element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol. 4(2), 121–126 (1986). [CrossRef]

], tilted channel waveguides [14

14. G. Tartarini and H. Renner, “Efficient finite-element analysis of tilted open anisotropic optical channel waveguides,”, ” IEEE Microw. Guid. Wave Lett. 9(10), 389–391 (1999). [CrossRef]

], and 2D photonic crystals [15

15. S. M. Hsu and H. C. Chang, “Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy,” Opt. Express 15(24), 15797–15811 (2007). [CrossRef] [PubMed]

]. To maintain the symmetry of matrices in the FEFD method, Beeckman et al. [6

6. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009). [CrossRef]

] used 6N unknowns (the matrix size is 36N 2) including three electric field components (3N) and the product (3N) of three electric field components and the wavevector component along z. To formulate a standard eigenvalue matrix equation in the FDFD [10

10. M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009). [CrossRef] [PubMed]

], the authors adopted 4N unknowns including two transverse electric field and two transverse magnetic field components to study LC waveguides with arbitrary molecular director orientations.

2. Computational schemes

A. Mathematical formulations

Assuming a monochromatic electromagnetic wave that has a time dependence of exp(jωt) and is propagating along the z direction in a medium with the permittivity tensor [ε]. The vector wave equation based on the magnetic field vector H is given by
×([ε]1×H)ω2μ0H=0,
(1)
where ω is the angular frequency and μ 0 is the permeability in vacuum. In this work, the permittivity tensor [ε] with the general form is considered as follows:
[ε]=ε0[εr]=ε0[εxxεxyεxzεyxεyyεyzεzxεzyεzz],
(2)
where ε 0 is the permittivity in free space and [εr] is the relative permittivity tensor. Considering a z-invariant waveguide structure, all electromagnetic field components are assumed to have a z dependence of exp(−jßz), where the propagation constant is β = k 0 neff, the wave number in free space is k 0 = ω 2 μ 0 ε 0, and neff is the effective refractive index. In the study, the governing wave equations based on transverse magnetic field components Hx and Hy are developed, and the magnetic field component in the z-direction Hz is thus replaced by transverse magnetic field components Hx and Hy through applying the divergence-free of the magnetic field vector. Under the consideration of full anisotropy of [ε], a cubic eigenvalue matrix equation is formulated. By transforming the cubic eigenvalue matrix equation to a linear eigenvalue equation with an eigenvalue of β 2, the full vector eigenvalue equation represented using the transverse magnetic field components Hx and Hy is obtained as follows:
[Pxx(β)Pxy(β)Pyx(β)Pyy(β)][HxHy]=β2[ηyyηyxηxyηxx][HxHy],
(3)
where the differential operators Pxx(β), Pxy(β), Pyx(β), and Pyy(β) are defined as a function of β by the following.
Pxx(β)Hx=ηyy2Hxx2+ηzz2Hxy2ηyx2Hxyx+k02Hx+jβ(ηyz+ηzy)Hxy...                  +jβ(ηzxyηzyx)2Hxyx,
(4a)
Pxy(β)Hy=(ηyyηzz)2Hyxyηyx2Hyy2jβ(ηyzHyx+ηzxHyy)...                   +jβ(ηzxyηzyx)2Hyy2,
(4b)
Pyx(β)Hx=(ηxxηzz)2Hxyxηxy2Hxx2jβ(ηxzHxy+ηzyHxx)...                   +jβ(ηzyxηzxy)2Hxx2,
(4c)
Pyy(β)Hy=ηzz2Hyx2+ηxx2Hyy2+ηxy2Hyxy+k02Hy+jβ(ηxz+ηzx)Hyx...                   +jβ(ηzyxηzxy)2Hyxy,
(4d)
where
ηxx=(εyyεzzεyzεzy)/Δ,ηxy=(εxzεzyεxyεzz)/Δ,ηxz=(εxyεyzεxzεyy)/Δ,
ηyx=(εyzεzxεyxεzz)/Δ,ηyy=(εxxεzzεxzεzx)/Δ,ηyz=(εxzεyxεxxεyz)/Δ,
ηzx=(εyxεzyεyyεzx)/Δ,ηzy=(εxyεzxεxxεzy)/Δ,ηzz=(εxxεyyεxyεyx)/Δ,
and

Δ=εxxεyyεzz+εxyεyzεzx+εxzεyxεzyεxzεyyεzxεxyεyxεzzεxxεyzεzy.
(5)

Compared with that in the transverse anisotropic media [27

27. C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

], the operators in Eqs. (4)a) to (4d) including the eigenvalue β lead to a nonlinear eigenvalue matrix equation rather than a simple linear one as shown in [27

27. C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

]. Therefore, an approach for solving a nonlinear eigenvalue problem is required. In the proposed scheme, the computational window is first divided at interfaces between different materials into several subdomains having uniform refractive index profiles and is then assembled using the physical interfacial conditions between these subdomains. Interfacial conditions include continuous normal and tangential components of the magnetic fields Hx and Hy at each intra-element boundary, and the other two interfacial conditions can be derived from the divergence condition of the magnetic field vector and from the Ampere’s law. In the divergence condition of the magnetic fieldH=0, the longitudinal component Hz is expressed by
Hz=1jβ(Hxx+Hyy),
(6)
and that the Ez is expressed by
Ez=ηzx[(1β)y(Hxx+Hyy)+βHy]+ηzy[(1β)x(Hxx+Hyy)βHx]+jηzz(HxyHyx),
(7)
through the Ampere’s law×H= [ε]E. Accordingly, Eqs. (6) and (7) are used as the two interfacial conditions because of the continuities of Hz and Ez at interfaces between different materials. Note that the first three continuous interface conditions, the normal and tangential components of the magnetic fields (Hx and Hy) and Hz, are the same as those applied in the transverse anisotropic media [27

27. C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

]. The main difference of interface conditions between the full 3 × 3 anisotropic here and the transverse anisotropic media [27

27. C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

] is appeared in Eq. (7). Equation (7) involving the eigenvalue β makes the proposed work significantly complex while patching the subdomains than that in [27

27. C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

] which has only the last term of Eq. (7). In the proposed approach, only horizontal and vertical interfaces are considered because the aim of this study is to develop an efficient waveguide mode solver for anisotropic waveguides with arbitrary permittivity tensors. If the waveguides, such as optical fibers, tapered waveguides, or photonic crystals, with curved boundaries are considered, the transfinite interpolation schemes with bending functions [29

29. W. J. Gordon and C. A. Hall, “Transfinite element methods: blending function interpolation over arbitrary curved element domains,” Numer. Math. 21(2), 109–129 (1973). [CrossRef]

] are the most often used approaches in various numerical schemes such as the FD, FE, and pseudospectral methods [30

30. D. A. Kopriva, Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers (Springer–Verlag, 2009).

]. The schemes use linear interpolations to map a reference orthogonal square in computational space to an arbitrarily curvilinear quadrilateral region that represent the physical space. Under the mapping, the Eqs. (3), (6), and (7) are transformed by the result of chain rule. The implementations of the transfinite interpolation schemes in the pseudospectral approach for analyzing the optical fibers and photonic crystals with curved interfaces, readers are referred to references [22

22. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2 Pt 2), 026703 (2007). [CrossRef] [PubMed]

,23

23. P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, and H.- Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44(1), 56–66 (2008). [CrossRef]

].

B. Numerical approach

In the pseudospectral scheme [20

20. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Springer–Verlag, 2nd edition, 2001).

], the computational window is divided at interfaces between different materials into several subdomains having uniform refractive index profiles. For an arbitrary interior rectangle subdomain r, it involves (nx + 1) × (ny + 1) collocation points in the corresponding (nx + 1) × (ny + 1) field unknowns as shown in Fig. 1(a)
Fig. 1 The mesh division of an arbitrary interior subdomain r
(where nx = ny = 10 is illustrated here).

In the subdomain r, the magnetic field components Hx and Hy are expanded using a set of suitable orthogonal basis functions as follows:
Hxr(x,y)=p=0nxq=0nyθpr(x)ψqr(y)Hx,pqr,
(8a)
Hyr(x,y)=p=0nxq=0nyθpr(x)ψqr(y)Hy,pqr,
(8b)
where θ(x) and ψ(y) are Lagrange-type basis functions, Hx,pq and Hy,pq are grid point values at (nx + 1) (ny + 1) collocation points, and θpr(xm)=δmp, ψpr(ym)=δmp, δmp denotes the Kronecker delta. Substituting Eqs. (8)a) and (8b) into Eq. (3), Eq. (3) is demanded to perfectly be satisfied at these (nx − 1) × (n y− 1) interior collocation points (namely, (x 1, yq), (x 2, yq),… (x 9, yq), as shown in Fig. 1, where q = 1 to 9). Equation (3) is thus converted to a system of linear equations to form a matrix eigenvalue equation with an eigenvalue of β 2 given by
[Pxxr(β)Pxyr(β)Pyxr(β)Pyyr(β)][HxrHyr]=β2[ηyyrηyxrηxyrηxxr][HxrHyr],
(9)
where the operatorsPxxr(β),Pxyr(β),Pyxr(β), andPyyr(β)are
Puvr(β)=[(D˜00nx1,1)uvr(D˜01nx1,1)uvr(D˜0nynx1,1)uvr(D˜00nx1,2)uvr(D˜01nx1,2)uvr(D˜0nynx1,2)uvr(D˜00nx1,ny1)uvr(D˜01nx1,2)uvr(D˜0nynx1,2)uvr],uv=xx,xy,yx,  and  yy,
(10a)
(D˜0snx1,t)uvr=[(D0s1,t)uvr(D1s1,t)uvr(Dnxs1,t)uvr(D0s2,t)uvr(D1s2,t)uvr(Dnxs2,1)uvr(D0snx1,t)uvr(D1snx1,t)uvr   (Dnxsnx1,t)uvr],s=0,1,...,ny,t=0,1,...,nx,
(10b)
(Dpqi,j)xxr=ηyyrθpr(2)(xir)ψqr(yjr)+ηzzrθpr(xir)ψqr(2)(yjr)ηyxrθpr(1)(xir)ψqr(1)(yjr)+k02θpr(xir)ψqr(yjr)...              +jβ(ηyzr+ηzyr)θpr(xir)ψqr(1)(yjr)+jβ(ηzxrθpr(1)(xir)ψqr(2)(yjr)ηzyrθpr(2)(xir)ψqr(1)(yjr)),
(10c)
(Dpqi,j)xyr=(ηyyrηzzr)θpr(1)(xir)ψqr(1)(yjr)ηyxrθpr(xir)ψqr(2)(yjr)jβ(ηyzrθpr(1)(xir)ψqr(yjr)...               +ηzxrθpr(xir)ψqr(1)(yjr))+jβ(ηzxrθpr(xir)ψqr(3)(yjr)ηzyrθpr(1)(xir)ψqr(2)(yjr)),
(10d)
(Dpqi,j)yxr=(ηxxrηzzr)θpr(1)(xir)ψqr(1)(yjr)ηxyrθpr(2)(xir)ψqr(yjr)jβ(ηxzrθpr(xir)ψqr(1)(yjr)...               +ηzyrθpr(1)(xir)ψqr(yjr))+jβ(ηzyrθpr(3)(xir)ψqr(yjr)ηzxrθpr(2)(xir)ψqr(1)(yjr)),
(10e)
(Dpqi,j)yyr=ηzzrθpr(2)(xir)ψqr(yjr)+ηxxrθpr(xir)ψqr(2)(yjr)+ηxyrθpr(1)(xir)ψqr(1)(yjr)+k02θpr(xir)ψqr(yjr)...               +jβ(ηxzr+ηzxr)θpr(1)(xir)ψqr(yjr)+jβ(ηzyrθpr(2)(xir)ψqr(1)(yjr)ηzxrθpr(1)(xir)ψqr(2)(yjr)),
(10f)
where θp(h)(x)andψq(h)(y)represent the h-th order derivatives of θp(x) to x and ψq(y) to y, respectively. Once the matrix eigenvalue equation for each subdomain is obtained, a global matrix equation can be formed by assembling the matrix equations in all subdomains. Assuming there are m subdomains, the pattern of the matrix elements is given by
[Q1(β)0000Q2(β)00000000Qm(β)][H1H2Hm]=β2[η10000η200000000ηm][H1H2Hm],
(11)
where

Qr(β)=[Pxxr(β)Pxyr(β)Pyxr(β)Pyyr(β)],Ηr=[HxrHyr],   ηr=[ηyyrηyxrηxyrηxxr],(r=1,2,3...m).
(12)

Note that the interfacial conditions between subdomains have not yet been enforced to Eq. (11) to form the final matrix equation. The exponential convergence behavior can be preserved as the same as that in the single domain version by explicitly enforcing the physical interface conditions. First, the continuities of the normal and tangential components of the magnetic field vector H are implicitly imposed. In addition, the other two interface conditions are the continuities of Hz in Eq. (6) and Ez in Eq. (7). For a specific vertical interface as shown in Fig. 1, the interface collocation points (x 0, yj) and (xnx, yj) belong to the right (denoted by + ) and left (denoted by −) subdomains, respectively, where j = 1, 2, 3… ny. The continuity of Hz gives
[Ax+Ay+][Hx+Hy+]=[AxAy][HxHy],
(13)
where
Ak±=[Ak±(ψ0(y0±))Ak±(ψ1(y0±))Ak±(ψny(y0±))Ak±(ψ0(y1±))Ak±(ψ1(y1±))Ak±(ψny(y1±))Ak±(ψ0(yny±))Ak±(ψ1(yny±))Ak±(ψny(yny±))],k=x,y,
(14a)
Ax+(ψi(yj+))=[θ0(1)(x0+)ψi(yj+)θ1(1)(x0+)ψi(yj+)θnx(1)(x0+)ψi(yj+)],
(14b)
Ay+(ψi(yj+))=[θ0(x0+)ψi(1)(yj+)θ1(x0+)ψi(1)(yj+)θnx(x0+)ψi(1)(yj+)],
(14c)
Ax(ψi(yj))=[θ0(1)(xnx)ψi(yj)θ1(1)(xnx)ψi(yj)θnx(1)(xnx)ψi(yj)],
(14d)
Ay(ψi(yj))=[θ0(xnx)ψi(1)(yj)θ1(xnx)ψi(1)(yj)θnx(xnx)ψi(1)(yj)],
(14e)
and the continuity of Ez gives
[Bx+By+][Hx+Hy+]=[BxBy][HxHy],
(15)
where
Bk±=[Bk±(ψ0(y0±))Bk±(ψ1(y0±))   Bk±(ψny(y0±)) Bk±(ψny(y0±)) Bk±(ψ1(y1±))   Bk±(ψny(y1±))Bk±(ψ0(yny±))Bk±(ψ1(yny±))   Bk±(ψny(yny±))],k=x,y,
(16a)
Bx+(ψi(yj+))=[ηzx(1β)θ0(1)(x0+)ψi(1)(yj+)+ηzy(1β)θ0(2)(x0+)ψi(yj+)βηzyδij+jηzzθ0(x0+)ψi(1)(yj+)ηzx(1β)θ1(1)(x0+)ψi(1)(yj+)+ηzy(1β)θ1(2)(x0+)ψi(yj+)βηzyδij+jηzzθ1(x0+)ψi(1)(yj+)ηzx(1β)θnx(1)(x0+)ψi(1)(yj+)+ηzy(1β)θnx(2)(x0+)ψi(yj+)βηzyδij+jηzzθnx(x0+)ψi(1)(yj+)]T,
(16b)
By+(ψi(yj+))=[ηzx(1β)θ0(x0+)ψi(2)(yj+)+βηzxδij+ηzy(1β)θ0(1)(x0+)ψi(1)(yj+)jηzzθ0(1)(x0+)ψi(yj+)ηzx(1β)θ1(x0+)ψi(2)(yj+)+βηzxδij+ηzy(1β)θ1(1)(x0+)ψi(1)(yj+)jηzzθ1(1)(x0+)ψi(yj+)ηzx(1β)θnx(x0+)ψi(2)(yj+)+βηzxδij+ηzy(1β)θnx(1)(x0+)ψi(1)(yj+)jηzzθnx(1)(x0+)ψi(yj+)]T,
(16c)
Bx(ψi(yj))=[ηzx(1β)θ0(1)(xnx)ψi(1)(yj)+ηzy(1β)θ0(2)(xnx)ψi(yj)βηzyδij+jηzzθ0(xnx)ψi(1)(yj)ηzx(1β)θ1(1)(xnx)ψi(1)(yj)+ηzy(1β)θ1(2)(xnx)ψi(yj)βηzyδij+jηzzθ1(xnx)ψi(1)(yj)ηzx(1β)θnx(1)(xnx)ψi(1)(yj)+ηzy(1β)θnx(2)(xnx)ψi(yj)βηzyδij+jηzzθnx(xnx)ψi(1)(yj)]T,
(16d)
By(ψi(yj))=[ηzx(1β)θ0(xnx)ψi(2)(yj)+βηzxδij+ηzy(1β)θ0(1)(xnx)ψi(1)(yj)jηzzθ0(1)(xnx)ψi(yj)ηzx(1β)θ1(xnx)ψi(2)(yj)+βηzxδij+ηzy(1β)θ1(1)(xnx)ψi(1)(yj)jηzzθ1(1)(xnx)ψi(yj)ηzx(1β)θnx(xnx)ψi(2)(yj)+βηzxδij+ηzy(1β)θnx(1)(xnx)ψi(1)(yj)jηzzθnx(1)(xnx)ψi(yj)]T,
(16e)
where i = 0,1 2, …nx and T denotes the transpose of a matrix. The derived formulations are similar for a horizontal interface, if the vertical interface collocation points (x 0, yj) and (xnx, yj) are altered by the horizontal ones (xi, y 0) and (xi, yny). For brevity, these formulations are not shown here. Following the global matrix equation of Eq. (11), in addition to the continuous normal and tangential components of the magnetic fields at all interface points, the final matrix equation is formed by also imposing the interfacial conditions in Eqs. (13) and (15) to Eq. (11), which no longer has a block diagonal form but turns into a matrix with an approximate sparsity of 47%. A detailed description for clearly recognizing the coupling of fields between subdomains can be found in the appendix of the work [22

22. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2 Pt 2), 026703 (2007). [CrossRef] [PubMed]

], which solved the band structures of photonics crystals by scalar pseudospectral method. Notably, solving the full 3 × 3 anisotropic waveguides is approximately 8.5 times the memory required and 1.9 times the computational time of those for solving the transverse anisotropic ones [27

27. C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

] having the approximate sparsity of 38%. The dependent variables in the final matrix include only the interior collocation points of each subdomain; however, the dependent variables in interface points can be obtained once the solutions of the final matrix are solved. In addition, the computational boundary conditions are automatically satisfied because of the use of LGFs to represent the exterior subdomains.

4. Simulation results and discussion

In this section, a square LC waveguide with five nonzero elements of the permittivity tensor is first calculated to examine the convergence of the proposed mode solver compared with that by using the FEFD-based eigenvalue approach with higher-order elements [6

6. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009). [CrossRef]

]. Furthermore, to show the full capability of the proposed scheme, the second example considers a nematic LC waveguide with nine nonzero elements of the permittivity tensor. The calculated effective indices are then compared with those obtained using the FDFD-based eigenvalue approach [10

10. M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009). [CrossRef] [PubMed]

]. The mode patterns under various orientations of the LC director are also characterized in detail.

Figure 2(a)
Fig. 2 (a) Cross-sectional view of the square LC waveguide with the permittivity tensor [ε] in the core region and refractive index ns of the surrounding substrate. (b) Schematic diagram of the orientationn^of the LC director. (c) Division of the computational domain for the LC waveguide.
shows a cross-sectional view of the square LC waveguide surrounded by an isotropic cladding with refractive index ns = 1.5. The structure parameters of its core region are: width w = 1 μm and thickness t = 1 μm.

Assuming that the orientation of the square LC directorn^ tilts at an angle of θc = 45° with respect to the z-axis (the propagation direction) as shown in Fig. 2(b), and thus, the permittivity tensor [ε] of the LC is given by
[ε]=ε0[no2000no2+Δεsin2θcΔεsinθccosθc0Δεsinθccosθcno2+Δεcos2θc],
(20)
where no = 1.55, ne = 1.8, and Δε=ne2no2. In this example, the operating wavelength λ = 1μm in free space. To preserve the fast convergence of the pseudospectral method, the proposed scheme divides the computational window at material interfaces into 9 subdomains (Fig. 2(c)). The guided mode profiles in the interior subdomains having finite intervals are expanded using Chebyshev polynomials, and those in the exterior subdomains having semi-infinite intervals are expanded using LGFs. For example, the mode profiles in both the x- and y-directions of subdomains 1, 3, 7, and 9 are expanded using LGFs. In contrast, the Chebyshev polynomials expand the mode profiles in both the x- and y-directions of the subdomain 5. For subdomains 2 and 8, the mode profiles in the x-direction are expanded using Chebyshev polynomials, and those in the y-direction are expanded using LGFs.

For examining the convergence of the proposed scheme, Fig. 3
Fig. 3 Convergent effective index versus iteration time for different terms of the basis functions n.
shows the effective indices of the i-th iteration of the fundamental mode versus the iteration time under different number of terms of basis functions n. We note that equal number of terms of the basis function in both the x- and y- directions for all subdomains are used throughout the paper, and the chosen initial guess of the effective index is set as n guess = no = 1.55. In particular, the semi-infinite intervals are fixed at 10 terms of the basis functions due to the well-matched behaviors of the exponential decay fields of guided modes and the mathematical behavior of LGFs. In Fig. 3, the ns denote the number of terms required to expand the finite intervals. Thus, the total number of unknowns used are 2 × [(10 − 1) + (n − 2) + (10 − 1)]2.

The differences of the effective indices between the fourth and fifth iterations are achieved in the order of 10−8, and the calculated effective indices are 1.604108 and 1.604152 for n = 20 (the number of unknowns is 2592) and n = 25 (the number of unknowns is 3362) at the fourth iteration, respectively. Compared with the effective index neff = 1.6055 obtained using the FEFD-based eigenvalue approach with higher-order elements [6

6. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009). [CrossRef]

] (the degrees of freedom used are ~10000), the result neff = 1.6041 calculated using the proposed scheme shows good agreement. However, the proposed scheme has the advantage that it requires fewer degrees of freedom (less memory storage). In addition, Fig. 4
Fig. 4 Mode profiles of (a) |Hx| and (b) |Hy| of the fundamental mode of the square nematic LC waveguide.
shows the mode profiles of |Hx| and |Hy|.

The following example considers an LC dielectric waveguide comprising a core region filled with nematic LCs (5CB) [34

34. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons. Inc., New York, 1999).

], surrounded by a glass substrate having refractive index ng = 1.45 and air with refractive index na, (Fig. 5(a)
Fig. 5 (a) Cross-sectional view of the nematic LC-core channel waveguide with the core permittivity tensor [ε], substrate refractive index ng, and air refractive index na. (b) Schematic diagram of the twist angle φc and the tilt angle θc of the LC directorn^.
). In addition, this example investigates arbitrary orientations of the LC director shown in Fig. 5(b).

The permittivity tensor [ε] of the LC-core region at the operating wavelength λ = 1.55 μm is given by
[ε]=ε0[no2+Δεsin2θccos2φcΔεsin2θcsinφccosφcΔεsinθccosθccosφcΔεsin2θcsinφccosφcno2+Δεsin2θcsin2φcΔεsinθccosθcsinφcΔεsinθccosθccosφcΔεsinθccosθcsinφcno2+Δεcos2θc],
(21)
whereΔε=ne2no2, the ordinary refractive index is no = 1.5292, the extraordinary refractive index is ne = 1.7072, θc is the tilt angle between the LC director n^ and the z-axis, and φc is the twist angle between the projection of n^ on the xy plane and the x-axis. The width and thickness of the core are w = 3 μm and t = 3 μm, respectively. The orientation n^ of the LC director can be controlled using an applied voltage on the designed electrode. For the condition θc = 90°, the elements of [ε] under different angles of φc are reduced to those of a waveguide with transverse anisotropy [26

26. J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010). [CrossRef]

,27

27. C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

]. For the condition θc = 0°, nonzero elements of [ε] are further reduced to be only in diagonal positions. This study accordingly considers the conditions θc = 30° and θc = 60° in which nine elements of [ε] are all nonzero. The computational window is divided into nine subdomains, which is similar to that in Fig. 2(c). Furthermore, all calculations use n = 20, and results are obtained at the fourth iteration. The calculated effective indices for θc = 30° and θc = 60° under different φc’s along with the results obtained using the FDFD [10

10. M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009). [CrossRef] [PubMed]

] are shown in Figs. 6(a)
Fig. 6 Calculated effective indices versus different values of φc for tilt angles (a) θc = 30° and (b) θc = 60°.
and 6(b), respectively. On comparison with the calculated effective indices by the FDFD [10

10. M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009). [CrossRef] [PubMed]

] (indicated by circles in Fig. 6), the results obtained using the proposed scheme (indicated by crosses in Fig. 6) are found to be in good agreement.

For each individual value of the tilt angles θc, the differences in the effective indices of the fundamental guided mode (mode 1) are fairly small (of the order of 10−4) for different values of twist angles φc. This can be understood by the opposite variances of the diagonal elements εxx and εyy as well as the non-diagonal elements εxz and εyz as the twist angles φc alter. The major field patterns of mode 1 for four values of φc shown in Fig. 7
Fig. 7 |Hx| and |Hy| mode patterns of mode 1 for the tilt angle θc = 30°
(for θc = 30°) and Fig. 8
Fig. 8 |Hx| and |Hy| mode patterns of mode 1 for the tilt angle θc = 60°.
(for θc = 60°) also verify these results. The symbols |H1,x| and |H1,y| on the left sides of Figs. 7 and 8 denote the moduli of the Hx and Hy components of mode 1, respectively. The effects of altering φc emerge in the minor field patterns and the relative amplitude between |Hx| and |Hy|. In addition, Figs. 7 and 8 also show the effect of transforming the major field Hy to Hx beyond the twist angle φc = 45°.

For higher-order modes, the differences in the values of the calculated effective indices for different values of φc are greater than 0.001; thus, the major field patterns show modest variations. Figure 9
Fig. 9 |Hx| and |Hy| mode patterns of the higher-order modes for the tilt angle θc = 30°.
shows the |Hx| and |Hy| components of higher-order modes 2–4 for four values of φc at θc = 30°; Fig. 10
Fig. 10 |Hx| and |Hy| mode patterns of the higher-order modes for the tilt angle θc = 60°.
shows the same for modes 2−4 at θc = 60°.

As shown in Figs. 9 and 10, mode symmetries relative to the y-axis are observed only at φc = 0° and φc = 90°, and symmetric features also appear at θc = 45°. Particularly, the major field does not always have the same polarization for all modes for an individual value of φc. For instance, at the orientation of the LC director θc = 30° and φc = 90°, the major field is Hx polarization for the first three modes but turns into Hy polarization for mode 4. Contrary to the condition of φc = 90°, only mode 4 has major Hx field while the orientation is at φc = 0°. For the case of θc = 60° and φc = 90°, the major field Hy polarizations for the first six modes are converted to the Hx polarization for mode 7 (not shown here). Further observations were made of the higher-order modes; however, the reverse phenomenon appears merely in some modes. In addition, for examining the influence of θc at certain angles of φc in more detail, the calculated effective indices by the proposed approach versus different values of θc at φc = 30° and φc = 60° are shown in Fig. 11(a) and (b)
Fig. 11 Calculated effective indices versus different values of θc for twisted angles (a) φc = 30° and (b) φc = 60°.
, respectively. The effective index increases monotonously as the angle θc increases. For a specific angle of φc, it can be realized that the elements εxx, εyy, and εxy increase as the angle of θc increases.

5. Conclusion

This study has developed an efficient pseudospectral mode solver for solving dielectric waveguides with full 3 × 3 anisotropy. The proposed scheme was formulated as an eigenvalue matrix equation by applying only the transverse magnetic field components with 2N unknowns (matrix size of 4N 2), where N is the number of unknowns. Thus, it requires fewer grid points than both the finite-element frequency-domain (FEFD)-based eigenvalue scheme with 3N unknowns (matrix size of 9N 2) and the finite-difference frequency-domain (FDFD)-based eigenvalue scheme with 4N unknowns (matrix size of 16N 2). In consequence, the computational time and memory required for the proposed scheme are significantly reduced. Having to treat a cubic eigenvalue equation creates a computational penalty because of the complicated wave equations and the interfacial conditions; however, convergence of the effective indices can be achieved by performing only four iterations. As a result, the computational efforts for the proposed scheme are still much lesser than those required for reported FEFD and FDFD approaches. The first example considered a square LC waveguide with the orientation of the director at an tilt angle of θc = 45° with respect to the propagation direction (having five nonzero elements in the permittivity tensor), and the calculated results of the proposed scheme are compared with those obtained using the FEFD scheme. A good agreement of the effective index of the fundamental mode is obtained to validate the numerical accuracy of the proposed scheme. To further demonstrate the full capability of the proposed scheme, in this study, I analyzed a nematic liquid crystal with arbitrary orientations of the director embedded in a glass substrate. Good agreement was also shown for the calculated results when compared with those from the FDFD scheme. In addition, mode profiles at arbitrary orientations of the LC director were presented. On examining these mode profiles, the proposed scheme was shown to offer not only an accurate numerical scheme but also a solution method that is more efficient than the previously reported approaches. Finally, the proposed mode solver can be used in the future to investigate and design more complex photonic devices while using arbitrary anisotropic materials with nine nonzero elements of the permittivity tensor.

Acknowledgements

The author would like to thank the National Science Council of China, Taiwan, for financially supporting this research under Contract No. NSC 99-2112-M-005-005-MY3.

References and links

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J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37(1-3), 95–106 (2005). [CrossRef]

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A. D’Alessandro, B. D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006). [CrossRef]

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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(10), 733–748 (2008). [CrossRef]

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E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707–5714 (2000). [CrossRef]

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B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009). [CrossRef]

6.

J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009). [CrossRef]

7.

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(13), 10895–10909 (2009). [CrossRef] [PubMed]

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M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009). [CrossRef]

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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(13), 10895–10909 (2009). [CrossRef] [PubMed]

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M. Y. Chen, S. M. Hsu, and H. C. Chang, “A finite-difference frequency-domain method for full-vectorial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor,” Opt. Express 17(8), 5965–5979 (2009). [CrossRef] [PubMed]

11.

D. Donisi, B. Bellini, R. Beccherelli, R. Asquini, G. Gilardi, M. Trotta, and A. Dálessandro, “A switchable liquid-crystal optical channel waveguide on silicon,” IEEE J. Quantum Electron. 46(5), 762–768 (2010). [CrossRef]

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M. Kawachi, N. Shibata, and T. Edahiro, “Possibility of use of liquid crystals as optical waveguide material for 1.3μm and 1.55μm bands,” Jpn. J. Appl. Phys. 21(Part 2, No. 3), L162–L164 (1982). [CrossRef]

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J. B. Xiao and X. H. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010). [CrossRef]

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C. C. Huang, “Modeling mode characteristics of transverse anisotropic waveguides using a vector pseudospectral approach,” Opt. Express 18(25), 26583–26599 (2010). [CrossRef] [PubMed]

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OCIS Codes
(130.2790) Integrated optics : Guided waves
(230.3720) Optical devices : Liquid-crystal devices
(230.7380) Optical devices : Waveguides, channeled

ToC Category:
Integrated Optics

History
Original Manuscript: December 7, 2010
Revised Manuscript: January 25, 2011
Manuscript Accepted: January 28, 2011
Published: February 4, 2011

Citation
Chia-Chien Huang, "Solving the full anisotropic liquid crystal waveguides by using an iterative pseudospectral-based eigenvalue method," Opt. Express 19, 3363-3378 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3363


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References

  1. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron. 37(1-3), 95–106 (2005). [CrossRef]
  2. A. D’Alessandro, B. D. Donisi, R. Beccherelli, and R. Asquini, “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006). [CrossRef]
  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(10), 733–748 (2008). [CrossRef]
  4. E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707–5714 (2000). [CrossRef]
  5. B. Bellini and R. Beccherelli, “Modeling, design and analysis of liquid crystal waveguides in preferentially etched silicon grooves,” J. Phys. D Appl. Phys. 42(4), 045111 (2009). [CrossRef]
  6. J. Beeckman, R. James, F. A. Í. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” J. Lightwave Technol. 27(17), 3812–3819 (2009). [CrossRef]
  7. 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(13), 10895–10909 (2009). [CrossRef] [PubMed]
  8. M. F. O. Hameed, S. S. A. Obayya, K. Al-Begain, M. I. Abo el Maaty, and A. M. Nasr, “Modal properties of an index guiding nematic liquid crystal based photonic crystal fiber,” J. Lightwave Technol. 27(21), 4754–4762 (2009). [CrossRef]
  9. 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(13), 10895–10909 (2009). [CrossRef] [PubMed]
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