Design of Silicon Based Integrated Optical Devices Using the Finite Element Method

Abstract

Among the components needed in photonic integrated circuits, dielectric waveguides and small footprint ring resonators play a key role for many applications and require sophisticated electromagnetic analysis and design. In this work, we present an accurate vectorial mode solver based on the finite element method. Considering a general nonreciprocal permittivity tensor, the proposed method allows us to investigate important cases of practical interest. To compute the electromagnetic modes, the Rayleigh-Ritz functional is derived for the non-self adjoint case, it is discretized using the node elements and the penalty function is added to remove the spurious solutions. Although the use of the penalty function is well known for the waveguide problem, it has been introduced for the first time (to the best of our knowledge) in the ring resonator modal analysis. The resulting quadratic eigenvalue problem is linearized and solved in terms of the propagation constant for a given frequency (i.e., γ-formulation). Unlike the earlier developed mode solvers, our approach allows us to precisely compute both forward and backward propagating modes in the nonreciprocal case. Moreover, it avoids time-consuming iterations and preserves matrix sparsity, ensuring high accuracy and computational efficiency.

Appendix

To compute the matrices in Eq. (12) let us define the following matrices

$$\displaystyle\begin{array}{rcl} R_{ij} =\iint _{S}\phi _{i}\phi _{j}\,dxdy,\quad P_{ij}& =& \iint _{S}\phi _{i}\phi _{j}\,x^{2}dxdy,\quad J_{ ij} =\iint _{S}\phi _{i} \frac{\partial \phi _{j}} {\partial y}\,dxdy, \\ N_{ij} =\iint _{S}\phi _{i} \frac{\partial \phi _{j}} {\partial x}\,dxdy,\quad D_{ij}& =& \iint _{S} \frac{\partial \phi _{i}} {\partial y} \frac{\partial \phi _{j}} {\partial y}\,dxdy,\quad E_{ij} =\iint _{S} \frac{\partial \phi _{i}} {\partial x} \frac{\partial \phi _{j}} {\partial x}\,dxdy, \\ Z_{ij}& =& \iint _{S} \frac{\partial \phi _{i}} {\partial y} \frac{\partial \phi _{j}} {\partial x}\,dxdy, {}\end{array}$$

(13)

where R, P, D and E are symmetric. If $\mathbf{p} =\boldsymbol{\varepsilon }_{ r}^{-1}$, M, L, C and K for the waveguide problem are

$$\displaystyle\begin{array}{rcl} M& =& \left (\begin{array}{*{10}c} -p_{yy}R& p_{yx}R & 0 \\ p_{xy}R &-p_{xx}R& 0 \\ 0 & 0 &\alpha _{p}R \end{array} \right ),\qquad L = \left (\begin{array}{*{10}c} R& 0 & 0\\ 0 &R & 0 \\ 0 & 0 &-R \end{array} \right ), \\ C& =& \left (\begin{array}{*{10}c} p_{zy}J^{t} - p_{yz}J & p_{yz}N - p_{zx}J^{t} &p_{yx}J - p_{yy}N -\alpha _{p}N^{t} \\ p_{xz}J - p_{zy}N^{t} & p_{zx}N^{t} - p_{xz}N & p_{xy}N - p_{xx}J -\alpha _{p}J^{t} \\ p_{xy}J^{t} - p_{yy}N^{t} -\alpha _{p}N &p_{yx}N^{t} - p_{xx}J^{t} -\alpha _{p}J & 0 \end{array} \right ), \\ K& =& \left (\begin{array}{*{10}c} p_{zz}D +\alpha _{p}E &-p_{zz}Z +\alpha _{p}Z^{t}& p_{zy}Z - p_{zx}D \\ -p_{zz}Z^{t} +\alpha _{p}Z & p_{zz}E +\alpha _{p}D & p_{zx}Z^{t} - p_{zy}E \\ p_{xz}D - p_{yz}Z^{t}& p_{yz}E - p_{xz}Z &p_{xy}Z + p_{yx}Z^{t} - p_{xx}D - p_{yy}E \end{array} \right ). {}\end{array}$$

(14)

Note that the matrices for the ring resonator are defined as in Eq. (13) where x ↦ r and y ↦ z. To compact the notation, we also defined the matrices

$$\displaystyle\begin{array}{rcl} S& =& \tfrac{3} {2}R + N,\qquad T = \tfrac{1} {2}R + N,\qquad X = \tfrac{3} {2}J^{t} + Z,\qquad Y = \tfrac{1} {2}J^{t} + Z, \\ U& =& \tfrac{9} {4}R + \tfrac{3} {2}\left (N + N^{t}\right ) + E,\qquad V = \frac{3} {4}R + \tfrac{3} {2}N + \tfrac{1} {2}N^{t} + E, \\ W& =& \tfrac{1} {4}R + \tfrac{1} {2}N + \tfrac{1} {2}N^{t} + E. {}\end{array}$$

(15)

Therefore, M, L, C and K for the ring resonator are

$$\displaystyle\begin{array}{rcl} M& =& \left (\begin{array}{*{10}c} -p_{zz}R& 0 & p_{rz}R \\ 0 &\alpha _{p}R& 0 \\ p_{zr}R & 0 &-p_{rr}R \end{array} \right ),\qquad L = \left (\begin{array}{*{10}c} P & 0 & 0\\ 0 &-P & 0 \\ 0 & 0 &P \end{array} \right ), \\ C& =& \left (\begin{array}{ccc} p_{\theta z}J^{t} - p_{z\theta }J &p_{zr}J - p_{zz}S -\alpha _{p}S^{t}& p_{z\theta }T - p_{\theta r}J^{t} \\ p_{rz}J^{t} - p_{zz}S^{t} -\alpha _{p}S & 0 &p_{zr}S^{t} -\alpha _{p}J - p_{rr}J^{t} \\ - p_{\theta z}T^{t} + p_{r\theta }J^{t} & p_{rz}S -\alpha _{p}J^{t} - p_{rr}J & p_{\theta r}T^{t} -\alpha _{p}T \end{array} \right ), \\ K& =& \left (\begin{array}{ccc} p_{\theta \theta }D +\alpha _{p}U & - p_{\theta r}D + p_{\theta z}X & \alpha _{p}X^{t} - p_{\theta \theta }Y \\ p_{r\theta }D - p_{z\theta }X^{t}&p_{rz}X + p_{zr}X^{t} - p_{rr}D - p_{zz}U &p_{z\theta }V - p_{r\theta }Y \\ \alpha _{p}X - p_{\theta \theta }Y ^{t} & - p_{\theta z}V ^{t} + p_{\theta r}Y ^{t} & p_{\theta \theta }W +\alpha _{p}D \end{array} \right ). {}\end{array}$$

(16)