Accurate vectorial finite element mode solver for magneto-optic and anisotropic waveguides

Abstract

In this work, a dielectric waveguide mode solver is presented considering a general nonreciprocal permittivity tensor. The proposed method allows us to investigate important cases of practical interest in the field of integrated optics, such as magneto-optical isolators and anisotropic waveguides. Unlike the earlier developed mode solver, our approach allows for the precise computation of both forward and backward propagating modes in the nonreciprocal case, ensuring high accuracy and computational efficiency. As a result, the nonreciprocal loss/phase shift can be directly computed, avoiding the use of the perturbation method. To compute the electromagnetic modes, the Rayleigh-Ritz functional is derived for the non-self adjoint case, it is discretized using the node-based finite element method and the penalty function is added to remove the spurious solutions. The resulting quadratic eigenvalue problem is linearized and solved in terms of the propagation constant for a given frequency (i.e., γ-formulation). The main benefits of this formulation are that it avoids the time-consuming iterations and preserves the matrix sparsity. Finally, the method is used to study two examples of integrated optical isolators based on nonreciprocal phase shift and nonreciprocal loss effect, respectively. The developed method is then compared with the perturbation approach and its simplified formulation based on semivectorial approximation.

General coupled mode theory in non-Hermitian waveguides

“…Thirdly, the mode set (ψ ψ ψ 2d ) of the adjoint system is selected as the countering propagating modes of the original system (φ φ φ 2d ). As such, the total degree of freedom of unknows is halved [30]. We note that GCMT can also be applied to bianisotropic waveguides [33].…”

“…The particular choice of the mode sets [28][29][30][31] is relevant: (1) it is necessary to get a functional of coupled modes that is independent of z, meaning the terms e ±iβ z in the inner-product between the modes sets φ φ φ and ψ ψ ψ are canceled; (2) the mode pair of counter-propagating modes has a definite relation, hence φ φ φ can be deduced from ψ ψ ψ, and vice versa. It is easy to get the z-independent and source-free wave-equation for the mode profiles φ φ φ 2d = [e e e + , h h h + ] of the original problem as follows .…”

“…(6)), a few remarks may deserve attentions. Firstly, the operator H H H 2d with inner product defined over 2D domain is not self-adjoint [30][31][32], e.g.,H H H a 2d =H H H 2d , for reciprocal medium. Secondly, for self-adjoint system, the adjoint system and the original system can be treated separately.…”

Full vectorial mode solver for design and optimization of magneto-optic devices

“…In cases of practical interest, B ext lies in the x-y plane of the waveguide, so ε xy =0. As a result, it can be shown that [9] ( ) ( ).…”

“…Perfectly matched layer boundary conditions, in both coordinate systems, have been implemented to include radiation loss. Unlike the earlier developed mode solver based on the perturbation method, our approach allows for the precise computation of both forward and backward propagating modes in the nonreciprocal case, ensuring high accuracy and computational efficiency, as shown in [9]. As a result, the nonreciprocal loss/phase shift can be directly computed.…”