Vector analysis of bending waveguides by using a modified finite-difference method in a local cylindrical coordinate system

Jinbiao Xiao; Xiaohan Sun

doi:10.1364/OE.20.021583

1. Introduction

Optical dielectric waveguide bends are fundamental building blocks for photonic integrated circuits (PICs) or planar lightwave circuits (PLCs) [1, 2], where they are used to change propagation directions or introduce lateral displacements, or connect with components integrated on the same chip [1, 2]. They also may be used as such individual components as polarization converters or rotators [3, 4]. Modal analysis of the waveguide bends is basic and perhaps indispensible tools in designing the PICs/PLCs or understanding the operating principle of some components, where the real (related to the phase variation) and imaginary (related to the pure bending loss) parts of the effective indexes for leaky modes can be simultaneously obtained.

To accurately determine the complex effective indexes and the corresponding field distributions of the leaky modes supported by the bending waveguides with a small radius and even with high refractive index contrast, a vector analysis is needed. Moreover, although the reduced equation using the equivalent straight waveguide (ESW) approximation looks, in form, like that of the straight waveguide and can be relatively easily solved by using numerical techniques, it is only valid for some special waveguide structures, for instance, waveguide bends with large radius. Therefore, solutions of the rigorous vector wave equation in cylindrical coordinate systems (CCS) are greatly required. In addition, to effectively demonstrate the leaky nature of the waveguide bends, it is necessary to use highly effective numerical boundary conditions for dealing with the mesh grids located at the edges of the computational window. Compared with the Dirichlet or transparent boundary conditions (TBC), the robust perfectly matched layers (PML) absorption boundary conditions, earlier used in the finite difference time-domain method (FDTD) [5] and later applied to the beam propagation method (BPM) [6], is more suited for the bending waveguides. In the past decades, a large number of numerical techniques, such as method of lines (MoL) [7, 8], film mode matching (FMM) method [9, 10], finite element (FE) method [11, 12], finite difference frequency-domain (FDFD) method [13], multidomain collocation method [14], beam propagation method (BPM) [15], finite difference (FD) method [16–20] have been proposed and successfully applied to calculate the scalar, semi-vector, and full-vector leaky modes of the bending waveguides. Among them, the FD method is attractive due to the simplicity of its implementation and the sparsity of its resulting matrix. However, the vector mode solver for bending waveguides using FD methods published earlier often uses central difference scheme to approximate the partial derivates that appear in the vector equation, where averaged index approximation approach is applied to deal with the grid points neighboring the dielectric interfaces [17–19], leading to poor convergence, especially for those waveguide bends with high index contrast. Moreover, they are often limited to uniform mesh grids. Although formulas with non-uniform mesh grids for straight waveguides are derived, there are few reports on those for bending waveguides. More recently, an improved FD mode solver for bending waveguides without above-mentioned limitations is proposed [20], its implementation, however, is complex. In addition, although all field components can be calculated from either electric (E-formulation) or magnetic formulation (H-formulation), the resulted field patterns do not always look exactly the same. As a general rule, if H-fields (E-fields) are desired, it is best to choose the H-formulation (E-formulation).

In this paper, a vector mode solver for optical dielectric waveguide bends by using a modified FD method in terms of transverse magnetic field components in a local CCS is developed. The formulation is based on the rigorous vector equation without the ESW approximation, and the PML absorbing boundary conditions via the complex coordinate stretching technique [21] are incorporated. The computational window (including the PMLs) is scanned with a suitable number of sub-regions with uniform refractive index profiles, and the discrete points where fields are sampled are located at the vertices of each sub-region. With the help of Taylor series expansion technique and the continuity condition of the longitudinal field components for each discrete point together with its four neighboring points (sub-regions), the governing equation results in a standard matrix eigenvalue equation without the averaged index approximation approach to deal with the those discrete points neighboring the dielectric interfaces [13, 18, 19]. Therefore, the convergent behaviors are improved and it is suited for the waveguide structures with high index contrast. Moreover, the present method can be applied to both the uniform and non-uniform mesh grids, and its implementation is relatively easy and the programming is simple since the FD coefficients are expressed in an explicit form, which is independent of specific structures of waveguide. A typical rib bending waveguide and a silicon wire bend are considered as the numerical examples to validate the effectiveness of the established method.

2. Mathematical formulation

To incorporate the robust PML absorbing boundary conditions into the present method, the following variable transformation is introduced into the Maxwell equations [18, 19, 21]

\tilde{ς} = \int_{0}^{ς} s_{ς} (ς^{'}) d ς^{'},

where

s_{ς} (ς)

represents the complex stretching variables, and ζ corresponds to the r, θ, or z in CCS. The typical definitions of the complex stretching variables can be expressed by [18, 21]

s_{ς} (ς) = {\begin{matrix} 1, in the non-PML region \\ 1 - \frac{1}{50 π Δ ς ω ε_{0}} \frac{{| ς - ς_{0} |}^{2}}{d^{2}}, in the PML region \end{matrix},

where Δζ is the mesh size along ζ direction, ζ₀ is the PML interface, and d is the thickness of the PML. Using Eq. (1), we can derive the vector wave equation in terms of the magnetic fields as follows

{\tilde{\nabla}}^{2} H + (\tilde{\nabla} \log ε) \times \tilde{\nabla} \times H + k_{0}^{2} ε H = 0,

with

\tilde{\nabla} \to {\tilde{r}}_{0} \frac{\partial}{\partial \tilde{r}} + {\tilde{θ}}_{0} \frac{1}{\tilde{r}} \frac{\partial}{\partial \tilde{θ}} + {\tilde{z}}_{0} \frac{\partial}{\partial \tilde{z}},

{\tilde{\nabla}}^{2} \to \frac{1}{\tilde{r}} \frac{\partial}{\partial \tilde{r}} (\tilde{r} \frac{\partial}{\tilde{r}}) + \frac{1}{{\tilde{r}}^{2}} \frac{\partial^{2}}{\partial {\tilde{θ}}^{2}} + \frac{\partial^{2}}{\partial {\tilde{z}}^{2}},

where

k_{0} = 2 π / λ

with λ being the wavelength in free space and

ε = n^{2}

with n being the refractive index profile of the guiding medium, and

{\tilde{r}}_{0}

,

{\tilde{θ}}_{0}

,and

{\tilde{z}}_{0}

are the unit vectors along the r, θ, and z directions, respectively. By introducing the local CCS where the transformation rules from the global cylindrical system to the local system are as follows (see Fig. 1 ) [8], [12, 13]:

\tilde{r} \to \tilde{x} + R

,

\tilde{θ} \to \tilde{z} / R

, and

\tilde{z} \to \tilde{y}

where R is the bending radius, we have

[\begin{matrix} P_{x x} & P_{x y} \\ P_{y x} & P_{y y} \end{matrix}] [\begin{matrix} H_{x} \\ H_{y} \end{matrix}] = β^{2} [\begin{matrix} H_{x} \\ H_{y} \end{matrix}],

with

P_{x x} H_{x} = \frac{\partial}{\partial \tilde{x}} [{\tilde{t}}_{x} \frac{\partial ({\tilde{t}}_{x} H_{x})}{\partial \tilde{x}}] + {\tilde{t}}_{x}^{2} n^{2} \frac{\partial}{\partial \tilde{y}} (\frac{1}{n^{2}} \frac{\partial H_{x}}{\partial \tilde{y}}) + {\tilde{t}}_{x}^{2} k_{0}^{2} n^{2} H_{x},

P_{x y} H_{y} = \frac{\partial}{\partial \tilde{x}} ({\tilde{t}}_{x}^{2} \frac{\partial H_{y}}{\partial \tilde{y}}) - {\tilde{t}}_{x}^{2} n^{2} \frac{\partial}{\partial \tilde{y}} (\frac{1}{n^{2}} \frac{\partial H_{y}}{\partial \tilde{x}}),

P_{y y} H_{y} = {\tilde{t}}_{x} n^{2} \frac{\partial}{\partial \tilde{x}} (\frac{{\tilde{t}}_{x}}{n^{2}} \frac{\partial H_{y}}{\partial \tilde{x}}) + {\tilde{t}}_{x}^{2} \frac{\partial^{2} H_{y}}{\partial {\tilde{y}}^{2}} + {\tilde{t}}_{x}^{2} k_{0}^{2} n^{2} H_{y},

P_{y x} H_{x} = {\tilde{t}}_{x}^{2} \frac{\partial^{2} H_{x}}{\partial \tilde{x} \partial \tilde{y}} - {\tilde{t}}_{x}^{2} n^{2} \frac{\partial}{\partial \tilde{x}} (\frac{1}{n^{2}} \frac{\partial H_{x}}{\partial \tilde{y}}),

where H_x and H_y are the transverse magnetic field components, β is the complex propagation constant, and

{\tilde{t}}_{x} = 1 + \tilde{x} / R

. The complex effective index n_eff is defined as

n_{eff} = β / k_{0} = n_{r} - j n_{i}

where n_r and n_i correspond to the real and imaginary parts, respectively.

Fig. 1 Cross section of a typical bending waveguide, where local cylindrical coordinate system is used and the radius of curvature R is defines as the radius of the center of the rib.

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For the guiding medium with uniform index profiles, Eq. (5) is reduced as follows:

{\tilde{r}}^{2} \frac{\partial^{2} H_{x}}{\partial {\tilde{x}}^{2}} + 3 \tilde{r} \frac{\partial H_{x}}{\partial \tilde{x}} + {\tilde{r}}^{2} \frac{\partial^{2} H_{x}}{\partial {\tilde{y}}^{2}} + H_{x} + {\tilde{r}}^{2} k_{0}^{2} ε H_{x} + 2 \tilde{r} \frac{\partial H_{y}}{\partial \tilde{y}} = R^{2} β^{2} H_{x},

{\tilde{r}}^{2} \frac{\partial^{2} H_{y}}{\partial {\tilde{x}}^{2}} + \tilde{r} \frac{\partial H_{y}}{\partial \tilde{x}} + {\tilde{r}}^{2} \frac{\partial^{2} H_{y}}{\partial {\tilde{y}}^{2}} + {\tilde{r}}^{2} k_{0}^{2} ε H_{y} = R^{2} β^{2} H_{y},

for piecewise uniform medium, they can be recovered to a set of coupled equations by incorporating interfacial conditions between neighboring grids.

A modified FD scheme is constructed for solving Eqs. (7a) and (7b). To do this, the cross-section of the waveguide structures enclosed with the PMLs at the edges is scanned with a suitable number of sub-regions with uniform refractive index profiles, and the discrete points where fields are sampled are located at the vertices of each sub-region, as shown in Fig. 1, where discrete points at the dielectric interfaces are marked with the solid circles. Figure 2 illustrates the general situation for an arbitrary grid point P with its neighboring points N, S, E, and W, where the grid field values for H_x and H_y are denoted as $H_{x}^{P}$ , $H_{x}^{N}$ , $H_{x}^{S}$ , $H_{x}^{E}$ , and $H_{x}^{W}$ , and $H_{y}^{P}$ , $H_{y}^{N}$ , $H_{y}^{S}$ , $H_{y}^{E}$ , and $H_{y}^{W}$ , respectively. It is noted that point P is shared by the four neighboring sub-regions 1, 2, 3, and 4 with the relative permittivity of ε₁, ε₂, ε₃, and ε₄, respectively. The grid size along x (y) direction is denoted as e or w (n or s) whose values are real and complex in non-PML and PML regions, respectively, so the present method can be applied to the non-uniform mesh grids. Considering the grid point P, and using second-order (first-order) Taylor series expansion to approximate the second-order (first-order) partial derivatives that appear in Eqs. (7a) and (7b) within each sub-region, as shown in Fig. 2, yields

\begin{array}{l} {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{W}}{w^{2}} - \frac{2 H_{x}^{P}}{w^{2}} + \frac{2}{w} {\frac{\partial H_{x}}{\partial \tilde{x}} |}_{w}) + 3 {\tilde{r}}_{P} {\frac{\partial H_{x}}{\partial x} |}_{w} + {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{N}}{n^{2}} - \frac{2 H_{x}^{P}}{n^{2}} - \frac{2}{n} {\frac{\partial H_{x}}{\partial \tilde{y}} |}_{n}) + \\ H_{x}^{P} + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{1} H_{x}^{P} + {2 {\tilde{r}}_{P} \frac{\partial H_{y}}{\partial y} |}_{n} = R^{2} β^{2} H_{x}^{P}, \end{array}

\begin{array}{l} {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{W}}{w^{2}} - \frac{2 H_{x}^{P}}{w^{2}} + \frac{2}{w} {\frac{\partial H_{x}}{\partial \tilde{x}} |}_{w}) + 3 {\tilde{r}}_{P} {\frac{\partial H_{x}}{\partial x} |}_{w} + {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{S}}{s^{2}} - \frac{2 H_{x}^{P}}{s^{2}} + \frac{2}{s} {\frac{\partial H_{x}}{\partial \tilde{y}} |}_{s}) + \\ H_{x}^{P} + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{2} H_{x}^{P} + {2 {\tilde{r}}_{P} \frac{\partial H_{y}}{\partial y} |}_{s} = R^{2} β^{2} H_{x}^{P}, \end{array}

\begin{array}{l} {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{E}}{e^{2}} - \frac{2 H_{x}^{P}}{e^{2}} - \frac{2}{e} {\frac{\partial H_{x}}{\partial \tilde{x}} |}_{e}) + 3 {\tilde{r}}_{P} {\frac{\partial H_{x}}{\partial x} |}_{e} + {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{S}}{s^{2}} - \frac{2 H_{x}^{P}}{s^{2}} + \frac{2}{s} {\frac{\partial H_{x}}{\partial \tilde{y}} |}_{s}) + \\ H_{x}^{P} + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{3} H_{x}^{P} + {2 {\tilde{r}}_{P} \frac{\partial H_{y}}{\partial y} |}_{s} = R^{2} β^{2} H_{x}^{P}, \end{array}

\begin{array}{l} {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{E}}{e^{2}} - \frac{2 H_{x}^{P}}{e^{2}} - \frac{2}{e} {\frac{\partial H_{x}}{\partial \tilde{x}} |}_{e}) + 3 {\tilde{r}}_{P} {\frac{\partial H_{x}}{\partial x} |}_{e} + {\tilde{r}}_{P}^{2} (\frac{2 H_{x}^{S}}{n^{2}} - \frac{2 H_{x}^{P}}{n^{2}} - \frac{2}{n} {\frac{\partial H_{x}}{\partial \tilde{y}} |}_{n}) + \\ H_{x}^{P} + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{4} H_{x}^{P} + {2 {\tilde{r}}_{P} \frac{\partial H_{y}}{\partial y} |}_{n} = R^{2} β^{2} H_{x}^{P}, \end{array}

for Eq. (7a), and

{\tilde{r}}_{P}^{2} (\frac{2 H_{y}^{W}}{w^{2}} - \frac{2 H_{y}^{P}}{w^{2}} + \frac{2}{w} {\frac{\partial H_{y}}{\partial \tilde{x}} |}_{w}) + {\tilde{r}}_{P} {\frac{\partial H_{y}}{\partial \tilde{x}} |}_{w} + {\tilde{r}}_{P}^{2} (\frac{2 H_{y}^{N}}{n^{2}} - \frac{2 H_{y}^{P}}{n^{2}} - \frac{2}{n} {\frac{\partial H_{y}}{\partial \tilde{y}} |}_{n}) + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{1} H_{y}^{P} = R^{2} β^{2} H_{y}^{P},

{\tilde{r}}_{P}^{2} (\frac{2 H_{y}^{W}}{w^{2}} - \frac{2 H_{y}^{P}}{w^{2}} + \frac{2}{w} {\frac{\partial H_{y}}{\partial \tilde{x}} |}_{w}) + {\tilde{r}}_{P} {\frac{\partial H_{y}}{\partial \tilde{x}} |}_{w} + {\tilde{r}}_{P}^{2} (\frac{2 H_{y}^{S}}{s^{2}} - \frac{2 H_{y}^{P}}{s^{2}} + \frac{2}{s} {\frac{\partial H_{y}}{\partial \tilde{y}} |}_{s}) + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{2} H_{y}^{P} = R^{2} β^{2} H_{y}^{P},

{\tilde{r}}_{P}^{2} (\frac{2 H_{y}^{E}}{e^{2}} - \frac{2 H_{y}^{P}}{e^{2}} - \frac{2}{e} {\frac{\partial H_{y}}{\partial \tilde{x}} |}_{e}) + r_{P} {\frac{\partial H_{y}}{\partial \tilde{x}} |}_{e} + {\tilde{r}}_{P}^{2} (\frac{2 H_{y}^{S}}{s^{2}} - \frac{2 H_{y}^{P}}{s^{2}} + \frac{2}{s} {\frac{\partial H_{y}}{\partial \tilde{y}} |}_{s}) + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{3} H_{y}^{P} = R^{2} β^{2} H_{y}^{P},

{\tilde{r}}_{P}^{2} (\frac{2 H_{y}^{E}}{e^{2}} - \frac{2 H_{y}^{P}}{e^{2}} - \frac{2}{e} {\frac{\partial H_{y}}{\partial \tilde{x}} |}_{e}) + {\tilde{r}}_{P} {\frac{\partial H_{y}}{\partial x} |}_{e} + r_{P}^{2} (\frac{2 H_{y}^{N}}{n^{2}} - \frac{2 H_{y}^{P}}{n^{2}} - \frac{2}{n} {\frac{\partial H_{y}}{\partial \tilde{y}} |}_{n}) + {\tilde{r}}_{P}^{2} k_{0}^{2} ε_{4} H_{y}^{P} = R^{2} β^{2} H_{y}^{P},

for Eq. (7b).

Fig. 2 Point P and its neighboring points and sub-regions.

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To obtain the desired coupled equations, continuity condition of the longitudinal field components is utilized. Considering the horizontal interfaces as show in Fig. 3(a) , the continuity of E_z gives

ε_{n} {\frac{\partial H_{x}}{\partial \tilde{y}} |}_{s} - ε_{s} {\frac{\partial H_{x}}{\partial \tilde{y}} |}_{n} = (ε_{n} - ε_{s}) \frac{\partial H_{y}}{\partial \tilde{x}},

and the continuity of H_z yields

{\frac{\partial H_{y}}{\partial \tilde{y}} |}_{n} = {\frac{\partial H_{y}}{\partial \tilde{y}} |}_{s},

where n and s denote the locations at the infinitesimally upper and lower side of the horizontal interface, respectively. Similarly, for a vertical interface as show in Fig. 3(b), we have

{ε_{e} \frac{\partial H_{y}}{\partial \tilde{x}} |}_{w} - {ε_{w} \frac{\partial H_{y}}{\partial \tilde{x}} |}_{e} = (ε_{e} - ε_{w}) \frac{\partial H_{x}}{\partial \tilde{y}},

and

{\frac{\partial H_{x}}{\partial \tilde{x}} |}_{e} = {\frac{\partial H_{x}}{\partial \tilde{x}} |}_{w},

where e and w denote the locations at the infinitesimally right and left of the vertical interface, respectively. With the help of above boundary conditions Eqs. (10a), (10b), (10c), and (10d), after some mathematical work similar to that described in [22], Eqs. (8a)-(8d) and (9a)-(9d) are combined into the coupled FD equations regarding point P for H_x and H_y and given as below

\begin{array}{l} a_{x x}^{N} H_{x}^{N} + a_{x x}^{S} H_{x}^{S} + a_{x x}^{E} H_{x}^{E} + a_{x x}^{W} H_{x}^{W} + a_{x x}^{P} H_{x}^{P} + \\ a_{x y}^{N} H_{y}^{N} + a_{x y}^{S} H_{y}^{S} + a_{x y}^{E} H_{y}^{E} + a_{x y}^{W} H_{y}^{W} + a_{x y}^{P} H_{y}^{P} = β^{2} H_{x}^{P}, \end{array}

\begin{array}{l} a_{y x}^{N} H_{x}^{N} + a_{y x}^{S} H_{x}^{S} + a_{y x}^{P} H_{x}^{P} + \\ a_{y y}^{N} H_{y}^{N} + a_{y y}^{S} H_{y}^{S} + a_{y y}^{E} H_{y}^{E} + a_{y y}^{W} H_{y}^{W} + a_{y y}^{P} H_{y}^{P} = β^{2} H_{y}^{P}, \end{array}

where the FD coefficients, e.g.,

a_{x x}^{N}, a_{x x}^{S}

, on the left hand sides are presented in the Appendix as Eqs. (13)-(30), respectively. After considering all the discrete points, we obtain the standard matrix eigenvalue equation as follows

[\begin{matrix} [A_{x x}] & [A_{x y}] \\ [A_{y x}] & [A_{y y}] \end{matrix}] [\begin{matrix} {H_{x}} \\ {H_{y}} \end{matrix}] = β^{2} [\begin{matrix} {H_{x}} \\ {H_{y}} \end{matrix}],

where [A_xx] and [A_xy], [A_yx], and [A_yy] are sub-matrices whose elements are determined from Eqs. (13)-(30), and {H_x} and {H_y} are column vectors related to the grid field values. We use MATLAB subroutines eigs to solve the resulting sparse matrix, in which the eigenvectors are related to the modal field distributions, while the corresponding eigenvalues are related to the complex propagation constants whose real and imaginary parts represent the phase variation and the pure bending loss, respectively.

Fig. 3 Horizontal (a) and vertical (b) dielectric interfaces.

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3. Numerical results

The first structure studied is of a typical rib bending waveguide whose cross section is shown in Fig. 1 [17–19]. The refractive indexes of the substrate, the core, the upper cladding are n_s = 3.17 (InP), n_f = 3.27 (InGaAsP), and n_c = 1.0 (air) at wavelength of 1.55μm. The total thickness of the core is h₁ = 1.3μm where thickness of the slab is t = 0.1μm. The thickness of the cap layer (InP) is h₂ = 0.5μm. The width of the rib is chosen as w = 3.0μm and the computational window is set to be 9.0 × 5.0μm, including the thickness of the PML layers, 1.0μm, imposed on the right and bottom edges of the computational window since the wave will leak towards these two sides.

The convergent behavior of the present method (method 1) is first examined by reducing the mesh grid size. Figures 4(a) and 4(b), respectively, show the real parts, n_r, and the imaginary parts, log₁₀(n_i), of the effective indexes of fundamental TE- and TM-like modes as a function of the mesh grid size (Δx = Δy), in which the bending radius R is set to be 100μm. For the sake of comparison, solutions from the conventional FD method (method 2) described in [18], where central difference scheme is used to approximate the partial derivates that appear in the coupled vector wave equations, and a four-point FD scheme is applied to approximate the cross-coupling terms (CCTs), and the improved one (method 3) described in [19], similar to that in [18] except that a six-point FD scheme is constructed to approximate the CCTs, are also plotted in the same figures. It can be observed that that, for both the real and imaginary parts in TE- and TM-like modes, the numerical results show good convergent behavior as the mesh grid size decreases. It also can be found that the convergence behavior of method 1 is superior to that of the method 2, and methods 1 and 3 have nearly similar convergent behaviors. For FD method 1 and 3, the mesh grid size, Δx = Δy = 0.025μm, is moderate since the convergent solutions can be obtained, while for FD method 2, the mesh grid size should be set to be less than 0.025μm for obtaining the convergent solutions. As a result, more CPU time and memory are required. It is noted that methods 2 and 3 are limited to the uniform mesh grid, while method 1 can be applied to both uniform and non-uniform mesh grids. We can use fine and coarse mesh grid size, respectively, to scan the interested regions (such as the neighboring areas of the dielectric interfaces) and other regions (such as claddings far away from the core). As a result, computational efficiency can be improved while keeping computational accuracy. Therefore, method 1 is more general and versatile.

Fig. 4 Effective indexes of the fundamental modes for a typical bending rib waveguide as a function of the mesh grid size by different FD method: real (a) and imaginary (b) parts.

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The modal characteristics of this bending waveguides are then studied in detail. Fine mesh grid size, Δx = Δy = 0.02μm, is used for a window 4.0 × 3.0μm, covering the core and its neighboring claddings, and coarse mesh grid size, Δx = Δy = 0.05μm, used for the rest regions including the PMLs. Figures 5(a) and 5(b), respectively, present the variation of the real and imaginary parts of the effective indexes of the fundamental TE- and TM-like modes with the bending radius. It is seen that computed results converge to those of the straight waveguide for both the real and imaginary parts in TE- and TM-like modes with the increment of the bending radius. When the bending radius is below 300μm, the values of the real parts in TM-like modes are greater than those in TE-like modes, but the difference decreases as the bending radius increases, while the results show the opposite behavior as the bending radius is over 300μm. Moreover, the bending loss of the TM-like modes is slightly lower than that of the TE-like modes as bending radius is below 200μm, but their values have the same order of magnitude at each bending radius. In order to examine the accuracy of the present method, results from the FMM [9, 10] method in CCS are also given in these figures. Our solutions agree well with those from the FMM, which is so far regarded as a benchmark results for comparing the accuracy. It is noted that the results from the FMM method in the present paper are calculated by the authors’ in-house software package, based on Ref [9, 10]. and programmed by the MATLAB subroutines, where the number of the TE/TM mode pairs for each slice is set to be 100. In [9], the bending waveguide structure studied is of low index contrast, so we do not use that example. Moreover, the calculations (including the following analysis) are performed on a personal computer with an Intel Core 2 Duo CPU, with 2 GB of physical RAM. The computational time to obtain the complex effect index for one polarization state (e.g., fundamental TE-like mode) of the present method, the FD method in Ref [18, 19]. (uniform mesh, i.e., Δx = Δy = 0.02μm), and the FMM method is of 5.43, 9.02, and 21.16, respectively. Therefore, the present method is more efficient.

Fig. 5 Effective indexes of the fundamental modes for a typical bending rib waveguide as a function of the bending radius: real (a) and imaginary (b) parts.

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Figure 6 gives the distributions of the field intensity of the real parts of the fundamental TE- and TM-like modes with R = 100μm, where the values for each component are normalized to its maximum value. It can be observed that the center of fields shift towards the bottom-right corner. Moreover, it also can be found that shape of the minor (H_x for TE-like modes and H_y for TM-like modes) and the major (H_y for TE-like modes and H_x for TM-like modes) components are very similar, especially for TM-like modes, and the ratio between the amplitudes of major and minor components is found to be below 20, much smaller than that of the straight waveguides. Therefore, we can use this waveguide bend to form the compact polarization converters or rotators [2, 3].

Fig. 6 Distributions of field intensity of the real parts of the fundamental modes for a typical bending waveguide: (a) H_x and (b) H_y for TE-like modes; (c) H_x and (d) H_y for TM-like modes.

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Next, modal characteristics of a silicon wire bend is analyzed using present method. This waveguide structure has submicron cross-section due to its high index contrast, and the bending radius can be reduced to under several micrometers while the bending losses can be nearly neglected [23]. So far, various kinds of compact functional photonic components using this silicon wire have been proposed or demonstrated [24]. And the importance of high-density on-chip integration of these components, i.e., PICs, has been widely recognized [1, 25, 26], where waveguide bends are indispensible elements.

The cross-section of this silicon wire bend is similar to that in Fig. 1 except the cap layer and the slab vanish, i.e., t = h₂ = 0. Under the following analysis, if not specified, the computational parameters are: n_s = 1.46(SiO₂), n_f = 3.48(Si), n_c = 1.0(air), w = 500nm, h₁ = 220nm, and λ = 1.55μm. The computational window is set to be 4.0μm × 4.0μm. Here again, the PML layers with the thickness of 1.0μm are imposed on the right and bottom edges of the computational window. Fine mesh grid size, Δx = Δy = 10nm, is used for a window 1.5 × 1.5μm, covering the core and its neighboring claddings, and coarse mesh grid size, Δx = Δy = 50nm, used for the rest regions including the PMLs.

Figure 7(a) presents the variation of the real parts of the effective indexes, n_r, for the fundamental modes with the bending radius. From this figure, it is seen that solutions, both for TE- and TM-like modes, decrease with the increment of the bending radius, and come close to those of the straight waveguides (i.e., bending radius approaches infinity). For TE-like modes, solutions are nearly similar to those of the straight counterparts when the bending radius is over 2.0μm, for TM-like modes; however, the bending radius must be over 5.0μm. It is also found that this bend shows strongly birefringence, where the values for TE-like modes are much larger than those for TM-like modes at each bending radius, and the difference between them is up to 0.75, slightly fluctuating dependent on the bending radius. Figure 7(b) shows the imaginary parts of the effective indexes, log₁₀(n_i), for the fundamental modes as a function of the bending radius. It is seen that the bending losses, for both TE- and TM-like modes, decrease as the bending radius increases, but the values for TE-like modes decrease faster than those for TM-like modes. Moreover, for TE-like modes, the values nearly linearly decrease when the bending radius is below (or over) 3.0μm, but they decrease faster as R is below 3.0μm. While for TM-like modes, the values always nearly linearly decrease as the bending radius is ranged from 1.0 to 5.0μm. It is also found that bending losses for TM-like modes are larger than those for TE-like modes at each bending radius. The value of log₁₀(n_i) for TE-like modes is up to −10 when the bending radius is of 3.0μm, which means that the pure bending losses are very small and can be neglected. While for TM-like modes, value is of −2 at the same radius, so the bending losses should be carefully taken into account. In addition, results from the FMM are also illustrated in Figs. 7(a) and 7(b), and the corresponding values for TE-like modes are summarized in Table 1 . Also, our results are good agreement with those from the FMM.

Fig. 7 Effective indexes of the fundamental modes for a silicon wire bend as a function of the bending radius: real (a) and imaginary (b) parts.

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Table 1. Effective Indexes of the Fundamental TE-like Modes for a Silicon Wire Bend Computed by Different Methods

View Table

We also investigate the modal characteristics of this silicon wire bend dependent on the operating wavelength. Figures 8(a) and 8(b), respectively, present the real and imaginary parts of the effective indexes for the fundamental modes as a function of the operating wavelength with R = 2.0μm. It is seen that real parts, n_r, both for TE- and TM-like modes, nearly linearly decrease with the increasing of the operating wavelength, while the imaginary parts, log₁₀(n_i,) nearly linearly increases. Therefore, as the operating wavelength increases, the phase variation of the leaky modes decreases, but the pure bending loss increases. From Fig. 8(b), it also can be seen that the pure bending loss for TM-like is much larger than that for TE-like mode at each wavelength, ranged from 1.45 to 1.65μm. However, the slopes of the two curves both in Figs. 8(a) and 8(b) are almost identical, which means that the phase variation and the pure bending loss for the two polarization states dependent on the operating wavelength are almost similar.

Fig. 8 Effective indexes of the fundamental modes for a silicon wire bend as a function of the operating wavelength: real (a) and imaginary (b) parts.

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Figure 9 gives the distributions of the field intensity of the real parts of the fundamental TE- and TM-like modes with R = 2.0μm and λ = 1.55μm. It is seen that the TE-like modes are strongly confined inside the core, not clearly showing leaky behaviors, but the center slightly shifts to the right side. TM-like modes, however, clearly leak towards the bottom-right-hand corner, and the center also shifts to the right side. Moreover, although the magnetic fields are continuous over the entire computational window, the derivatives of the magnetic field components with respect to some coordinates, e.g., x or y, are not continuous across the dielectric interfaces according to the Maxwell’s equations (see Eqs. (10a) and 19(c)), so the patterns of the fields are not rigorously smooth for some field components, especially for those waveguides with small core and high index contrast. As a result, the field patterns look like discontinuous, as illustrated in Figs. 9(a) and 9(c). For the present method, the sampled points are located at the dielectric interfaces, and the continuity condition of the longitudinal field components is utilized (transverse field components are naturally built-in).

Fig. 9 Distributions of field intensity of the real parts of the fundamental modes for a silicon wire bend: (a) H_x and (b) H_y for TE-like modes; (c) H_x and (d) H_y for TM-like modes.

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4. Conclusion

A full-vectorial mode solver for optical dielectric waveguide bends by using a modified FD method in terms of transverse magnetic field components in a local CCS is described, where the robust PML absorbing boundary conditions via the complex coordinate stretching technique are adopted for effectively demonstrating the leaky nature of the bending waveguides. With the help of the Taylor series expansion technique and the continuity condition of the longitudinal field components, a standard matrix eigenvalue equation is derived without the averaged index approximation technique for dealing with the discrete points located at or near the dielectric interfaces. The established method can be applied to both the uniform and non-uniform mesh grids. The leaky modes of a typical rib bending waveguide and a silicon wire bend are calculated, and the complex effective indexes and the field distributions are presented. Numerical results indicate that the present method show good convergent behaviors. And solutions are good agreement with those from the film mode matching method, which validates its effectiveness.

Appendix

Using Taylor series expansion technique and continuity condition of the longitudinal field components, Eq. (7) is converted into the FD Eq. (11), where the corresponding FD coefficients are given as follows:

a_{x x}^{N} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2}{n (e + w)} (\frac{w ε_{2}}{n ε_{2} + s ε_{1}} + \frac{e ε_{3}}{s ε_{4} + n ε_{3}}),

a_{x x}^{S} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2}{s (e + w)} (\frac{w ε_{1}}{n ε_{2} + s ε_{1}} + \frac{e ε_{4}}{s ε_{4} + n ε_{3}}),

a_{x x}^{E} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{1}{e + w} (\frac{2}{e} + \frac{3}{{\tilde{r}}_{P}}),

a_{x x}^{W} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{1}{e + w} (\frac{2}{w} - \frac{3}{{\tilde{r}}_{P}}),

a_{x x}^{P} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} [\frac{e (c_{1} + c_{2})}{(e + w) (s ε_{4} + n ε_{3})} + \frac{w (c_{3} + c_{4})}{(e + w) (n ε_{2} + s ε_{1})}],

with

c_{1} = s ε_{4} (k_{0}^{2} ε_{3} - \frac{2}{e^{2}} - \frac{2}{s^{2}} + \frac{1}{{\tilde{r}}_{P}^{2}} - \frac{3}{e {\tilde{r}}_{P}}),

c_{2} = n ε_{3} (k_{0}^{2} ε_{4} - \frac{2}{e^{2}} - \frac{2}{n^{2}} + \frac{1}{{\tilde{r}}_{P}^{2}} - \frac{3}{e {\tilde{r}}_{P}}),

c_{3} = n ε_{2} (k_{0}^{2} ε_{1} - \frac{2}{w^{2}} - \frac{2}{n^{2}} + \frac{1}{{\tilde{r}}_{p}^{2}} + \frac{3}{w {\tilde{r}}_{P}}),

c_{4} = s ε_{1} (k_{0}^{2} ε_{2} - \frac{2}{w^{2}} - \frac{2}{s^{2}} + \frac{1}{{\tilde{r}}_{P}^{2}} + \frac{3}{w {\tilde{r}}_{P}}),

for [A_xx],

a_{x y}^{N} = \frac{{\tilde{r}}_{P}}{R^{2}} \frac{2}{e + w} (\frac{w ε_{2}}{n ε_{2} + s ε_{1}} + \frac{e ε_{3}}{s ε_{4} + n ε_{3}}),

a_{x y}^{S} = - \frac{{\tilde{r}}_{P}}{R^{2}} \frac{2}{e + w} (\frac{w ε_{1}}{n ε_{2} + s ε_{1}} + \frac{e ε_{4}}{s ε_{4} + n ε_{3}}),

a_{x y}^{E} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2 w}{e {(e + w)}^{2}} [\frac{w (ε_{1} - ε_{2})}{n ε_{2} + s ε_{1}} + \frac{e (ε_{4} - ε_{3})}{s ε_{4} + n ε_{3}}],

a_{x y}^{W} = - \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2 e}{w {(e + w)}^{2}} [\frac{w (ε_{1} - ε_{2})}{n ε_{2} + s ε_{1}} + \frac{e (ε_{4} - ε_{3})}{s ε_{4} + n ε_{3}}],

a_{x y}^{P} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2}{e + w} (\frac{e - w}{e w} + \frac{1}{r_{P}}) [\frac{w (ε_{1} - ε_{2})}{n ε_{2} + s ε_{1}} + \frac{e (ε_{4} - ε_{3})}{s ε_{4} + n ε_{3}}],

for [A_xy],

a_{y x}^{N} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2 s}{n {(n + s)}^{2}} [\frac{s (ε_{3} - ε_{2})}{w ε_{3} + e ε_{2}} + \frac{n (ε_{4} - ε_{1})}{w ε_{4} + e ε_{1}}],

a_{y x}^{S} = - \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2 n}{s {(n + s)}^{2}} [\frac{s (ε_{3} - ε_{2})}{w ε_{3} + e ε_{2}} + \frac{n (ε_{4} - ε_{1})}{w ε_{4} + e ε_{1}}],

a_{y x}^{P} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2 (n - s)}{n s (n + s)} [\frac{s (ε_{3} - ε_{2})}{w ε_{3} + e ε_{2}} + \frac{n (ε_{4} - ε_{1})}{w ε_{4} + e ε_{1}}],

for [A_yx], and

a_{y y}^{N} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2}{n (n + s)},

a_{y y}^{S} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{2}{s (n + s)},

a_{y y}^{E} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{1}{n + s} (\frac{2}{e} + \frac{1}{{\tilde{r}}_{P}}) (\frac{s ε_{2}}{w ε_{3} + e ε_{2}} + \frac{n ε_{1}}{w ε_{4} + e ε_{1}}),

a_{y y}^{W} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} \frac{1}{n + s} (\frac{2}{w} - \frac{1}{{\tilde{r}}_{P}}) (\frac{s ε_{3}}{w ε_{3} + e ε_{2}} + \frac{n ε_{4}}{w ε_{4} + e ε_{1}}),

a_{y y}^{P} = \frac{{\tilde{r}}_{P}^{2}}{R^{2}} [\frac{s (d_{1} + d_{2})}{(n + s) (w ε_{3} + e ε_{2})} + \frac{n (d_{3} + d_{4})}{(n + s) (w ε_{4} + e ε_{1})}],

with

d_{1} = w ε_{3} (k_{0}^{2} ε_{2} - \frac{2}{w^{2}} - \frac{2}{s^{2}} + \frac{1}{w {\tilde{r}}_{P}}),

d_{2} = e ε_{2} (k_{0}^{2} ε_{3} - \frac{2}{e^{2}} - \frac{2}{s^{2}} - \frac{1}{e {\tilde{r}}_{P}}),

d_{3} = w ε_{4} (k_{0}^{2} ε_{1} - \frac{2}{w^{2}} - \frac{2}{n^{2}} + \frac{1}{w {\tilde{r}}_{P}}),

d_{4} = e ε_{1} (k_{0}^{2} ε_{4} - \frac{2}{e^{2}} - \frac{2}{n^{2}} - \frac{1}{e {\tilde{r}}_{P}}),

for [A_yy].

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant 60978005.

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R(μm)	n_r		log₁₀(n_i)
R(μm)	Present method	FMM	present method	FMM
1	2.40762	2.40791	−3.68456	−3.63721
2	2.39421	2.39433	−6.41594	−6.48982
3	2.39204	2.39224	−9.37884	−9.30488
4	2.39128	2.39142	−9.98148	−9.97048
5	2.39091	2.39093	−10.39617	−10.36615

Vector analysis of bending waveguides by using a modified finite-difference method in a local cylindrical coordinate system

Abstract

1. Introduction

2. Mathematical formulation

3. Numerical results

4. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (53)

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