## Solving the full anisotropic liquid crystal waveguides by using an iterative pseudospectral-based eigenvalue method |

Optics Express, Vol. 19, Issue 4, pp. 3363-3378 (2011)

http://dx.doi.org/10.1364/OE.19.003363

Acrobat PDF (1515 KB)

### 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

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. 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]

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]

## 2. Computational schemes

### A. Mathematical formulations

*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 bywhere

*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:where

*ε*

_{0}is the permittivity in free space and [

**ε**] is the relative permittivity tensor. Considering a

_{r}*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}

*n*, the wave number in free space is

_{eff}*k*

_{0}=

*ω*

^{2}

*μ*

_{0}

*ε*

_{0}, and

*n*is the effective refractive index. In the study, the governing wave equations based on transverse magnetic field components

_{eff}*H*and

_{x}*H*are developed, and the magnetic field component in the

_{y}*z*-direction

*H*is thus replaced by transverse magnetic field components

_{z}*H*and

_{x}*H*through applying the divergence-free of the magnetic field vector. Under the consideration of full anisotropy of [

_{y}**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

*H*and

_{x}*H*is obtained as follows:where the differential operators

_{y}*P*(

_{xx}*β*),

*P*(

_{xy}*β*),

*P*(

_{yx}*β*), and

*P*(

_{yy}*β*) are defined as a function of

*β*by the following.

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]

*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]

*H*and

_{x}*H*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 field

_{y}*H*is expressed byand that the

_{z}*E*is expressed bythrough the Ampere’s law

_{z}*jω*[

**ε**]

**E**. Accordingly, Eqs. (6) and (7) are used as the two interfacial conditions because of the continuities of

*H*and

_{z}*E*at interfaces between different materials. Note that the first three continuous interface conditions, the normal and tangential components of the magnetic fields (

_{z}*H*and

_{x}*H*) and

_{y}*H*, are the same as those applied in the transverse anisotropic media [27

_{z}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]

**18**(25), 26583–26599 (2010). [CrossRef] [PubMed]

*β*makes the proposed work significantly complex while patching the subdomains than that in [27

**18**(25), 26583–26599 (2010). [CrossRef] [PubMed]

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]

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. 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

*r*, it involves (

*n*+ 1) × (

_{x}*n*+ 1) collocation points in the corresponding (

_{y}*n*+ 1) × (

_{x}*n*+ 1) field unknowns as shown in Fig. 1(a) (where

_{y}*n*=

_{x}*n*= 10 is illustrated here).

_{y}*r*, the magnetic field components

*H*and

_{x}*H*are expanded using a set of suitable orthogonal basis functions as follows: where

_{y}*θ*(

*x*) and

*ψ*(

*y*) are Lagrange-type basis functions,

*H*and

_{x,pq}*H*are grid point values at (

_{y,pq}*n*+ 1) (

_{x}*n*+ 1) collocation points, and

_{y}*δ*denotes the Kronecker delta. Substituting Eqs. (8)a) and (8b) into Eq. (3), Eq. (3) is demanded to perfectly be satisfied at these (

_{mp}*n*− 1) × (

_{x}*n*− 1) interior collocation points (namely, (

_{y}*x*

_{1},

*y*), (

_{q}*x*

_{2},

*y*),… (

_{q}*x*

_{9},

*y*), as shown in Fig. 1, where

_{q}*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 bywhere the operators

*h-*th order derivatives of

*θ*(

_{p}*x*) to

*x*and

*ψ*(

_{q}*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 bywhere

**H**are implicitly imposed. In addition, the other two interface conditions are the continuities of

*H*in Eq. (6) and

_{z}*E*in Eq. (7). For a specific vertical interface as shown in Fig. 1, the interface collocation points (

_{z}*x*

_{0},

*y*) and (

_{j}*x*,

_{nx}*y*) belong to the right (denoted by + ) and left (denoted by −) subdomains, respectively, where

_{j}*j*= 1, 2, 3…

*n*. The continuity of

_{y}*H*giveswhere

_{z}*E*giveswhere

_{z}*i*= 0,1 2, …

*n*and

_{x}*T*denotes the transpose of a matrix. The derived formulations are similar for a horizontal interface, if the vertical interface collocation points (

*x*

_{0},

*y*) and (

_{j}*x*,

_{nx}*y*) are altered by the horizontal ones (

_{j}*x*,

_{i}*y*

_{0}) and (

*x*,

_{i}*y*). 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

_{ny}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]

**18**(25), 26583–26599 (2010). [CrossRef] [PubMed]

*H*and

_{x}*H*components. Chebyshev polynomials are used for expanding the optical fields in interior subdomains having finite intervals because of their mathematical robustness to non-periodic structures. In contrast, LGFs are used for expanding the exterior subdomains having semi-infinite intervals because the exponential decay characteristic of LGFs matches the guided field profiles. For Chebyshev polynomials, the explicit form of the Lagrange-type interpolation function is represented as follows [20]:where

_{y}*T*(

_{v}*x*) is the general Chebyshev polynomial of the order

*v*, and

*x*denotes the collocation points for the Chebyshev polynomials. For the LGFs, the explicit form is as follows [20]:where

_{p}*L*(

_{v}*αx*) is the Laguerre polynomial of order

*v*, and

*x*denotes the collocation points for the LGFs. The scaling factor

_{p}*α*in Eq. (18) significantly affects the numerical accuracy of the results obtained using LGFs. For a given number of terms of basis functions,

*α*is determined by the derivation of Tang [31

31. T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. **14**(3), 594–605 (1993). [CrossRef]

31. T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. **14**(3), 594–605 (1993). [CrossRef]

*α*is determined by the following formula:where

*M*is the spreading of the guided fields to be solved, and

*x*is the collocation point at position

_{i}*i*. Now, the determination of

*M*is become a main concern for various physical problems while using LGFs. For a light beam propagating in waveguides, the

*M*of guided modes can be computed by a simple EIM [32]. The detailed derivation of determining

*M*has been shown in the previous work [27

**18**(25), 26583–26599 (2010). [CrossRef] [PubMed]

*α*is the same for the full 3 × 3 anisotropic permittivity tensors. The resultant global matrix is a cubic eigenvalue equation with respect to

*β*, which results from the wave equations of Eq. (11) and the interfacial condition of Eq. (15). By transforming the final matrix to a linear eigenvalue equation with an eigenvalue of

*β*

^{2}, a simple iterative scheme such as the following process is used to discover the mode eigenvalues. (1) Assign an initially guessed value of

*n*by that of the substrate (in fact, this method should be followed only if a reasonable range of values of

_{eff}*n*is required). (2) Calculate the operators

_{eff}**P**

*(*

_{xx}*β*),

**P**

*(*

_{xy}*β*),

**P**

*(*

_{yx}*β*), and

**P**

*(*

_{yy}*β*). (3) A standard linear eigenvalue problem with an eigenvalue of

*β*

^{2}is obtained and solved using the Arnoldi iteration method [33

33. R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. **17**(4), 789–821 (1996). [CrossRef]

*n*in step (3), repeatedly keep executing the processes (2) and (3) to achieve the desired convergence (the difference of

_{eff}*n*between adjacent iterations is set at 10

_{eff}^{−8}in this study). Particularly, four iterations are needed to achieve the accepted convergence solutions due to the nonlinearity of the eigenvalue equation for the full anisotropy, and thus the whole computational time is 8 times of solving a transverse anisotropic problem [27

**18**(25), 26583–26599 (2010). [CrossRef] [PubMed]

## 4. Simulation results and discussion

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]

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]

*n*= 1.5. The structure parameters of its core region are: width

_{s}*w*= 1 μm and thickness

*t*= 1 μm.

*θ*= 45° with respect to the

_{c}*z*-axis (the propagation direction) as shown in Fig. 2(b), and thus, the permittivity tensor [

**ε**] of the LC is given bywhere

*n*= 1.55,

_{o}*n*= 1.8, and

_{e}*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.

*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}=

*n*= 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

_{o}*n*s 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}.

^{−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

*n*= 1.6055 obtained using the FEFD-based eigenvalue approach with higher-order elements [6

_{eff}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]

*n*= 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 shows the mode profiles of |

_{eff}*H*| and |

_{x}*H*|.

_{y}*n*= 1.45 and air with refractive index

_{g}*n*, (Fig. 5(a) ). In addition, this example investigates arbitrary orientations of the LC director shown in Fig. 5(b).

_{a}**ε**] of the LC-core region at the operating wavelength

*λ*= 1.55 μm is given by

*n*= 1.5292, the extraordinary refractive index is

_{o}*n*= 1.7072,

_{e}*θ*is the tilt angle between the LC director

_{c}*z*-axis, and

*φ*is the twist angle between the projection of

_{c}*xy*plane and the

*x*-axis. The width and thickness of the core are

*w*= 3 μm and

*t*= 3 μm, respectively. The orientation

*θ*= 90°, the elements of [

_{c}**ε**] under different angles of

*φ*are reduced to those of a waveguide with transverse anisotropy [26

_{c}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]

**18**(25), 26583–26599 (2010). [CrossRef] [PubMed]

*θ*= 0°, nonzero elements of [

_{c}**ε**] are further reduced to be only in diagonal positions. This study accordingly considers the conditions

*θ*= 30° and

_{c}*θ*= 60° in which nine elements of [

_{c}**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

*θ*= 30° and

_{c}*θ*= 60° under different

_{c}*φ*’s along with the results obtained using the FDFD [10

_{c}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]

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]

*θ*, the differences in the effective indices of the fundamental guided mode (mode 1) are fairly small (of the order of 10

_{c}^{−4}) for different values of twist angles

*φ*. This can be understood by the opposite variances of the diagonal elements

_{c}*ε*and

_{xx}*ε*as well as the non-diagonal elements

_{yy}*ε*and

_{xz}*ε*as the twist angles

_{yz}*φ*alter. The major field patterns of mode 1 for four values of

_{c}*φ*shown in Fig. 7 (for

_{c}*θ*= 30°) and Fig. 8 (for

_{c}*θ*= 60°) also verify these results. The symbols |

_{c}*H*| and |

_{1,x}*H*| on the left sides of Figs. 7 and 8 denote the moduli of the

_{1,y}*H*and

_{x}*H*components of mode 1, respectively. The effects of altering

_{y}*φ*emerge in the minor field patterns and the relative amplitude between |

_{c}*H*| and |

_{x}*H*|. In addition, Figs. 7 and 8 also show the effect of transforming the major field

_{y}*H*to

_{y}*H*beyond the twist angle

_{x}*φ*= 45°.

_{c}*φ*are greater than 0.001; thus, the major field patterns show modest variations. Figure 9 shows the |

_{c}*H*| and |

_{x}*H*| components of higher-order modes 2–4 for four values of

_{y}*φ*at

_{c}*θ*= 30°; Fig. 10 shows the same for modes 2−4 at

_{c}*θ*= 60°.

_{c}*y*-axis are observed only at

*φ*= 0° and

_{c}*φ*= 90°, and symmetric features also appear at

_{c}*θ*= 45°. Particularly, the major field does not always have the same polarization for all modes for an individual value of

_{c}*φ*. For instance, at the orientation of the LC director

_{c}*θ*= 30° and

_{c}*φ*= 90°, the major field is

_{c}*H*polarization for the first three modes but turns into

_{x}*H*polarization for mode 4. Contrary to the condition of

_{y}*φ*= 90°, only mode 4 has major

_{c}*H*field while the orientation is at

_{x}*φ*= 0°. For the case of

_{c}*θ*= 60° and

_{c}*φ*= 90°, the major field

_{c}*H*polarizations for the first six modes are converted to the

_{y}*H*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

_{x}*θ*at certain angles of

_{c}*φ*in more detail, the calculated effective indices by the proposed approach versus different values of

_{c}*θ*at

_{c}*φ*= 30° and

_{c}*φ*= 60° are shown in Fig. 11(a) and (b) , respectively. The effective index increases monotonously as the angle

_{c}*θ*increases. For a specific angle of

_{c}*φ*, it can be realized that the elements

_{c}*ε*,

_{xx}*ε*, and

_{yy}*ε*increase as the angle of

_{xy}*θ*increases.

_{c}## 5. Conclusion

*N*unknowns (matrix size of 4

*N*

^{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 3

*N*unknowns (matrix size of 9

*N*

^{2}) and the finite-difference frequency-domain (FDFD)-based eigenvalue scheme with 4

*N*unknowns (matrix size of 16

*N*

^{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

*θ*= 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.

_{c}## Acknowledgements

## References and links

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29. | W. J. Gordon and C. A. Hall, “Transfinite element methods: blending function interpolation over arbitrary curved element domains,” Numer. Math. |

30. | D. A. Kopriva, |

31. | T. Tang, “The Hermite spectral method for Gauss-type functions,” SIAM J. Sci. Comput. |

32. | T. Tamir, |

33. | R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. |

34. | P. Yeh and C. Gu, |

**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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