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Posteriori

A Posteriori Error Estimator for Photonic Crystals

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This project has been done in collaboration with Prof. Ivan Graham.

Figure 1. A portion of a periodic structure

We derived an a posteriori error estimator based on residuals for eigenvalue problems arising from photonic crystals (PC).
Our goals were: firstly, to compute in an efficient way the gaps in the spectra of periodic media in both the TE and TM modes. Secondly, we used the same error estimator to compute efficiently the trapped mode of photonic crystal fibers with defects.
We proved that our error estimator is reliable and efficient. Moreover, it is able to capture well the interfaces in the media especially in presence of high contrast.

Periodic Media:

Figure 2. A solution with singularities in the gradient around the corners

In Fig. 1 we have an example of periodic structure formed by square inclusions of a different material from the background.
Typical example of solutions for this kind of structure for different value of the contrast between the two materials are illustrated in Fig. 2 and Fig. 3. The solution could be characterise by singularities in the gradient around the corners and sharp edges along the interfaces.

Figure 3. 
Another solution with singularities in the gradient around the corners

Fig. 4 illustrates how the error estimator drives the mesh adaptivity to refine around the corners of the interface of the square inclusion, where the singularities in the gradient of the solutions are situated.
Since the goal is the computation of the spectral gaps of the structure, we conclude with Fig. 5 showing the spectral bands surrounding the gaps. To speed up the computation we use a parallel machine to compute the spectra.

Figure 4. Mesh adaptivity
Figure 5. Spectral gap

Photonic Crystals with Defects:

Figure 6. Structure with defect

A defect in a photonic crystal is a region where the symmetries of the periodic structure are broken. An example is illustrated in Fig.6.
In a well designed defect the light of certain frequencies can be trapped and allowed to travel for long distances, the solutions of the problem corresponding to these trapped frequencies are called trapped modes. An example of trapped mode is illustrated in Fig. 7. Our error estimator can be used to adapt the mesh in way to compute efficiently the trapped modes.
The frequencies of the trapped modes lies in the spectral gaps of the periodic structure. So, comparing the spectrum of the photonic crystal with the defected (Fig. 8) with the spectrum of the same structure without the defect (Fig. 5), it is easy to spot a very narrow band in the spectral gap of the structure. This narrow band is the trapped mode.

Figure 7. Trapped mode in the defect

Further Research:

  • 3D problems
  • use structures with inclusions with smooth interfaces
Figure 8. Spectrum with trapped mode in the gap

References:

S. Giani and I.G. Graham (2012), Adaptive finite element methods for computing band gaps in photonic crystals. – Numerische Mathematik 121(1), 31-64.

S. Giani and I.G. Graham, Adaptive finite element methods for computing band gaps in photonic crystals. – Bath Institute for Complex Systems Preprint number 06/10, (2010)