Bent optical waveguide finite element analysis with a 3D envelope Maxwell model

Abstract

With the goal of accurately extracting the optical field losses in a three-dimensional (3D), circularly coiled waveguide (e.g., bent optical fiber), this effort presents the numerical methodologies that are implemented for an envelope Maxwell model that propagates electromagnetic fields as an entirely boundary value problem. Our unique modeling approach includes an ultraweak variational formulation of the envelope Maxwell model in the curved geometry of the bending, which is discretized by the discontinuous Petrov-Galerkin (DPG) method, which permits residual-driven mesh and polynomial-order adaptivity. This also, then, requires a unique approach for constructing perfectly matched layers (PMLs) as absorbing boundary conditions in both the direction of optical field propagation and in the tangential directions, where unguided energy escapes the waveguide. Our coiled waveguide modeling technology extracts the mode confinement losses from the propagation of the coherent optical field through the bent waveguide. We verify our simulations against the semi-analytical results from the analogous bent slab waveguide problem, and we successfully demonstrate stable convergence to loss values for the 3D coiled optical fiber problem, which has never been done previously for our specific modeling approach.

Figures (17)

• Figure 1: This diagram depicts a straight optical fiber (lower segment) fused (spliced) seamlessly with a circularly bent optical fiber (upper segment). This allows one to launch the optical field into the perfectly guiding (i.e. lossless) straight segment so that it can propagate into the coiled segment, where it is no longer perfectly guiding and thus experiences confinement loss. For realistic portrayals of bent waveguides, the fiber's radius is much smaller than the bend radius: $\rho_{\text{fiber}} \ll r_{0}$.
• Figure 2: Cross-section of a straight, step-index optical fiber waveguide: (left) schematic of geometrical sizes and (right) its corresponding radially dependent refractive index profile.
• Figure 3: Circularly coiled fiber schematic that illustrates the PML at the end of the waveguide used to absorb the optical field exiting the fiber. In this case, perfect electric conductor (PEC) boundary conditions are implemented in the transverse directions to this direction of propagation; i.e., the circumferential direction.
• Figure 4: Geometry setup and boundary conditions for numerical experiment 1: we apply PEC b.c. at the curved boundaries, a PML in $\theta$, and prescribe the mode on the bottom face (waveguide input facet). The parameters defining the PML length in $\theta$ are illustrated in Figure \ref{['fig:pml_theta_diagram']}.
• Figure 5: Real component of the mode profiles for the first and second mode, with corresponding propagation constant $\beta$, of the vacuum waveguide with PEC boundary, bend radius $r_0 = 1300$, and width $2a=1$.

Theorems & Definitions (1)

• Remark 1