## Straight Waveguide

In a straight waveguide, the FDE engine solves for modes of the form

$${\bf{E}}(x,y,z)={\bf{f}}(x,y)exp(i\beta z)\\ \beta=n_{eff}k_0=n_{eff}\frac{2\pi}{\lambda_0}$$

Maxwell's equations are solved in a Cartesian coordinate system with the appropriate boundary conditions. There are a discrete set of values of effective index (neff) for which the electromagnetic field is displayed at z=0 where

$${\bf{E}}(x,y,z=0)={\bf{f}}(x,y)$$

## Helical Waveguide

The helical waveguide solver takes a cross section of the waveguide (i.e. the cross section cut by the FDE region) and extrudes that cross section in the propagation direction. In the helical solver option, each point in space is twisted around the origin of the global coordinates (i.e. the helix center) creating a helical structure in the propagation direction. In a helical waveguide, the relationship between the helical coordinate {X,Y,Z} with the Cartesian coordinate {x,y,z} is

$$x=Xcoz(\alpha Z)-Ysin(\alpha Z)\\y=Xsin(\alpha Z)+Ycos(\alpha Z)\\z=Z$$

Where α represents the torsion of the waveguide.

### References

- A. Nicolet, et al, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys., vol. 28, no. 2, Nov. 2004.
- X. Ma, “Understanding and Controlling Angular Momentum Coupled Optical Waves in Chirally-Coupled-Core (CCC) Fibers,” Ph.D. thesis, 2011.
- X. Ma, et al, “Angular-momentum coupled optical waves in chirally-coupled-core fibers,” Optics Express, vol. 19, no. 27, 2011.

### Example modes

**Straight fiber modes (untwisted):**

**Helical fiber modes (twisted):**

The twisted fiber breaks the degeneracy of the the two linearly polarized modes, and created two circularly polarized modes with difference neff. Note that the above mode profiles are taken from an application example - Chirally Coupled Core fiber, please visit this KB example for more information.