Elsevier

Optics Communications

Volume 283, Issue 14, 15 July 2010, Pages 2835-2840
Optics Communications

Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method

https://doi.org/10.1016/j.optcom.2010.03.057Get rights and content

Abstract

A full-vectorial mode solver in terms of the transverse magnetic field components for optical waveguides with transverse anisotropy is described by using the multidomain spectral collocation method based on Chebyshev polynomials. The waveguide cross section surrounded by the perfectly matched layers is divided into suitable number of homogeneous rectangles, and then connected with by imposing the continuities of the longitudinal field components at the dielectric interfaces shared by the adjacent rectangles, resulting in a generalized matrix eigenvalue problem. To validate the established method, results of an anisotropic square waveguide and a magnetooptic rib waveguide are presented and compared with those from the full-vectorial finite difference method, full-vectorial beam propagation method, and the experimental data.

Introduction

Anisotropic optical waveguides like poling-induced polymer waveguides and magnetooptic waveguides have been widely applied to form various kinds of integrated optical devices, including optical modulators/switches, polarization converters/splitters, and optical isolators [1], [2], in photonic integrated circuits or planar lightwave circuits (PICs/PLCs). An efficient and accurate full-vectorial mode solver is indispensable for the analysis and design of these devices. It is almost impossible, however, to get the analytical solutions. Therefore, the use of numerical (or approximate) analysis becomes necessary.

Vectorial coupled mode theory (V-CMT) [2], [3] is often used to analyze the magnetooptic waveguides. In V-CMT, the imaginary off-diagonal terms of the relative permittivity tensor are treated by perturbation theory, which is only valid for the waveguide structures where the off-diagonal terms are small compared to the relative permittivity of the corresponding isotropic waveguides [2], [3]. Over the years, more rigorously numerical techniques, e.g., finite difference (FD) method [4], [5], [6], [7], finite element (FE) method [8], [9], [10], [11], [12], [13], and FD- [14], [15] and FE-based [16], [17], [18], [19], [20] beam propagation method (BPM), have been proposed and successfully applied to analyze the guide modes or the propagation characteristics of the anisotropic dielectric waveguides. However, FD and FE method are often based on the low-order basis functions, normally leading to large matrix.

Alternatively, the series expansion method, e.g., the Galerkin method (GM) and spectral collocation method (SCM), expresses the unknown fields by a complete set of orthogonal basis functions, resulting in a small matrix. The authors have developed a full-vectorial mode solver for optical waveguides with step-index profiles by using the GM with variable transformation technique [21], [22]. Although the order of the resulted matrix is small compared to the FD and FE method, it is also time-consuming since laborious integral procedure is required for calculating the matrix elements. In SCM, the unknown fields are imposed to be satisfied with the wave equation at a set of collocation points so that the integral procedure is avoided. As a result, the calculation is more efficient. So far, several kinds of SCM have been proposed and successfully applied in studying the optical waveguide problems. Sharma and Banerjee [23] applied the SCM with single domain to analyze the guided modes and the propagation characteristics of optical waveguides. However, for inhomogeneous waveguide structures, the solutions show the Gibbs phenomenon, resulting in poor convergent behavior. To overcome this problem, Huang et al. [24], [25], [26], [27] applied the domain decomposition (DD) technique to the SCM, the so-called multidomain SCM (MSCM), in which the whole interest domain is divided into several homogeneous subdomains, and then connected with by imposing the continuities of the longitudinal field components at the dielectric interfaces. The results [24], [25], [26], [27] show that such treatment greatly improves the numerical stability and accuracy. However, the scaling factors of the Laguerre-Gauss basis functions used in the semi-infinite subsomains, which strongly affect the computational accuracy, should be carefully chosen. More recently, Chiang et al. [28], [29] proposed a more versatile MSCM, in which a curvilinear coordinate mapping technique is used to transform each curvilinear quadrilateral subdomain into a square one. With the help of this technique, the MSCM can be applied to solve the guided modes of the waveguide structures with curved dielectric interfaces, e.g., circular optical fibers and fused fiber couplers, and compute the band diagrams of the 2-D photonic crystals. Although the MSCM has proved to be powerful and versatile, the recent reported studies only focus on the isotropic waveguide structures, in which the anisotropy of the constituent material is completely neglected.

We here describe a full-vectorial mode solver for optical waveguides with transverse anisotropy by using the MSCM in terms of transverse magnetic field components. To avoid the nonphysical reflection from the computational window edges, the robust perfectly matched layer (PML) absorbing boundary conditions [30] are incorporated into the present method. Chebyshev polynomials are chosen as the basis functions for each subdomain because of the nonperiodicity of the waveguide structures [31]. Moreover, additional efforts for choosing the optimum scaling factor as used in [24], [25], [26] are avoided since only one kind of polynomials is utilized in our formulation. In order to test the validity and utility of the established method, an anisotropic square waveguide and a magnetooptic rib waveguide are analyzed, and the results are compared with those from the full-vectorial FD method, full-vectorial BPM, and the experimental data.

Section snippets

Description of the method

Assuming monochromatic electromagnetic fields with angular frequency ω propagating along the z-direction and using the complex coordinate stretching technique [30], the curl Maxwell's equations can be written as˜×E=jωμ0H˜×H=jωε0εˆEwhere ε0 and µ0 are the electric permittivity and the magnetic permeability in free space, respectively, and the operator ∇̃ is defined as˜αxxxˆ+αyyyˆ+αzzzˆwhere αx, αy, and αz are the complex PML parameters. The parameter αz is set to be unity since

Numerical results

The validation of the present method is first performed for the anisotropic square waveguide with high index contrast between the core and cladding. The waveguide cross section is surrounded by the PMLs and then divided into nine homogeneous rectangles, as shown in Fig.1. The cladding is set to be isotropic with the index ns = 1.0 (air), whereas the core is set to be anisotropic, a poling-induced polymer. For the sake of simplicity, the core is assumed to be homogeneously poled at an angle θ = 45°

Conclusion

We have developed a full-vectorial mode solver for optical waveguides with transverse anisotropy by using the multidomain spectral collocation method based upon the Chebyshev polynomials. The PML absorption boundary conditions via the complex coordinate stretching technique are incorporated into the present method. The numerical results for an anisotropic square waveguide and a magnetooptic rib waveguide indicate that the established method show superior convergent behavior to the FD method,

Acknowledgements

This work was supported by the Natural Science Foundation of China under Grant 60978005, in part by the Natural Science Foundation of Jiangsu Province of China under Grant BK2007102, in part by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant 20060286042, and in part by the Hi-Tech Program of Jiangsu Province of China under Grant BG2007042.

References (34)

  • J. Xiao

    Opt. Commun.

    (2006)
  • J.W. Wu

    J. Opt. Soc. Am. B

    (1991)
  • H. Dotsch

    J. Opt. Soc. Am. B

    (2005)
  • M. Loymeyer

    Opt. Commun.

    (1998)
  • C.L. da Silva Souza Sobrinho

    IEE Proc.-H

    (1993)
  • P. Lusse

    Electron. Lett.

    (1996)
  • A.B. Fallahkhair

    J. Lightwave Technol.

    (2008)
  • M.Y. Chen

    Opt. Express

    (2009)
  • M. Koshiba

    J. Lightwave Technol.

    (1986)
  • W.C. Chew

    IEEE Trans. Microwave. Theory Tech.

    (1989)
  • K. Hayata

    IEEE Trans. Microwave. Theory Tech.

    (1989)
  • Y. Liu

    IEEE Trans. Microwave. Theory Tech.

    (1993)
  • F.A. Kastriku

    J. Lightwave Technol.

    (1996)
  • V. Schulz

    IEEE Trans. Microwave. Theory Tech.

    (2003)
  • C.L. Xu

    J. Lightwave Technol

    (1994)
  • L.D.S. Alcantara

    J. Lightwave Technol.

    (2005)
  • Y. Tsuji

    J. Lightwave Technol.

    (1999)
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