Comparing EME and FDTD: Bends

EME and FDTD are the industry’s go-to waveguide simulation methods; both as rigorous solutions to Maxwell’s equations. When both methods are applicable and configured correctly there is no reason their result should differ. 

So how to pick which method to use?
This page shows pros and cons of each method for creating ring resonators as used in WDM, extended cavity lasers, and modulators.

Gradual adiabatic bends can take up a large bounding volume despite very little of that volume contributing to the simulation. 

• FDTD faces a great challenge as its simulation time scales with this volume and the duration of the simulation; doubling the dimensions of a bend leads to an simulation taking 8 times longer. 
• EME does not scale with the volume but the number of unique cross sections in the simulation. Doubling the size has no effect on this, making EME scale invariant.

Both EME and FDTD find advantages with the simple case of simulating constant curvature bends; take for instance a 180 bend in SOI.

• With FDTD a simulation of just part of the bend can be used to predict the full result as bend losses compound multiplicatively.
   
• With EME only a single waveguide cross section is needed! Along the full 180 degree bend there is just one cross section so EME can describe propagation with just one mode list, a basis set of the waveguide’s bend modes.

A common candidate for lossless bends is the Euler bend which increases and decreases in curvature linearly along its length; starting/ ending with zero curvature makes for a zero insertion loss with straight waveguides. The bend’s transmission can be improved by introducing a constant curvature region at the maximum curvature of the bend.

You may compare this to how you would turn a corner in a car: starting with your wheel straight, moving to a maximum turning curvature, holding the wheel at this position for some time, then gradually turning the wheel back to straight.

FIMMPROP EME describes a non-constant bend with a series of cross sections (similarly to a taper), each with a different curvature. Once those cross sections are defined, they can be reused between similar simulations; this can be used when optimising Euler bends to find what fraction of the bend should be held at constant maximum curvature. In FIMMPROP EME, the scan to find the plot below is near instant once the initial simulation is performed.

Illustration: partial euler bends with constant curvature fraction increasing left to right, 0.1, 0.5, 0.9. White lines show cross sections that FIMMPROP would calculate unchanged between calculations.

Materials such as x-cut lithium niobate change in refractive index depending on the orientation in the plane of the PIC. Conventionally this would make the simulation impossible in EME however recent development of FIMMPROP’s EME does permit this. Reach out to learn more on how and see validation as compared to FDTD.

Other relevant materials that show anisotropy include lithium tantalate and barium titanate, all three used for modulation. Since all modulators consist of a splitter and two separate arms, simulating bends that connect these parts is inevitable for the modulator designer. 

FDTD being a time domain method will produce transient results and a spectral response. This makes FDTD a solution for non-linear materials which EME cannot simulate the time evolving effects of.

Further than this however, the typical goals of simulating a common lossless bends see little benefit to the offerings.

In FIMMPROP EME, absorbing boundary conditions like PMLs are applied to absorb any radiation emitted from a lossy bend. FDTD uses the same principle but on boundary conditions that don't conform to the waveguide bend). This allows for light to radiate from a bend and couple back into other parts of the device (though this uncommonly has a first order effect in common designs).

In general, the correct implementation of PMLs has much more impact in EME PMLs with an excessive width can affect the basis set of modes.