Photonic crystal bragg fiber

The FDE solver can be used to accurately calculate the modes of arbitrarily complicated structures, including photonic crystal Bragg fibers. In this example, we calculate the modes of the PC Bragg fiber described by Vienne et al and further analyzed by Uranus et al.

Simulation setup

The simulation file bragg_PCfiber.lms contains a parameterized group object that can draw the structure provided by Uranus et al. Initially, the simulation is set up using Anti-Symmetric boundary conditions at x-min and y-min, with Metal boundary conditions at x-max and y-max. The Anti-Symmetric boundary conditions are used to save time, by allowing us to simulate only 1/4 of the structure. However, we must be careful not to miss important modes which may require Symmetric conditions, or a combination of Symmetric and Anti-Symmetric conditions.

Results

We can switch to Analysis Mode by pressing the run button and clicking Calculate Modes in the Analysis window. We see one of the guided modes with an effective index of approximately 0.998. Below is a figure of Hr in cylindrical coordinates.

To study the loss of this type of structure, it is necessary to switch the boundary conditions to perfectly matched layer (PML) absorbing boundaries at x-max and y-max, as shown below. We did not do this initially because it increases the computation time and can make it harder to find the effective index of the guided modes.

When we recalculate the modes, we can look near the index of 0.998 and identify different modes

Mode TE₀₁

Hr

Hphi

Mode HE^b₁₁

Hr

Hphi

Almost 20 modes will be calculated. Scroll down the list, using the field profiles to identify the modes of interest.  Mode 7 is TE₀₁. Mode 8  is HE^b₂₁. The figures above show the radial and angular components of the magnetic field, which can be compared to the results from Uranus et al. We can compare the effective index and loss values with the results from Uranus et al.

The MODE effective index results are very close to that of Uranus et al. Measuring loss is more difficult for this type of structure which is very sensitive to small changes in the numerical mesh (as well as real manufacturing imperfections) and some convergence testing is necessary to find a more accurate result.

Convergence testing

We start by copying the two modes of interest to the global DECK, and renaming them TE and HE, as shown below.

We can now test the convergence by running the sweep contained in the optimization and sweeps window.  This sweep calculates the modes several times using an increasing number of mesh cells. At each step, it calculates the modes, then identifies the TE₀₁ and HE^b₂₁ modes as the ones that gives the best overlap with the modes we have already stored in the DECK. It then records the effective index and loss for these modes as a function of the number of mesh cells used.

The final results, shown below, can be plotted in the Visualizer.

Effective index vs mesh cells

Loss vs mesh cells

We see that the effective index is beginning to converge by the time we reach 500x500 mesh cells but more mesh cells would be necessary to have more accuracy. Depending on the amount of memory on your computer, it may be possible to increase the maximum number of cells tested to 600x600 or more. The loss shows a reasonable amount of variation with the number of mesh cells but is also beginning to converge by 500x500 cells. Again, it may be necessary to increase the number of steps and increase the maximum number of mesh cells to get a more accurate final result. The results at 500x500 mesh cells are

The agreement in the effective index is excellent, and the loss is converging towards the result Uranus et al.

Related publications

1. G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, T.P. Hansen, B.H. Larsen et al., "First demonstration of air-silica Bragg fiber," Post Deadline Paper PDP25, Optical Fiber Conference 2004, Los Angeles, 22-27 Feb. 2004.
2. H. Uranus and H. Hoekstra, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004)

http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-12-2795

See also

Bragg fiber