A Posteriori
Error Estimator for Application with Photonic Crystals 

We
derived an a posteriori error estimator for hpadaptive
continuous Galerkin methods for photonic crystal (PC)
applications.
Our goal was to compute efficiently the behaviour of light inside
PC devices.
The error estimator that we propose is based on the residual of the discrete
problem and we show that it leads to very fast convergence in all considered
examples when used with hpadaptive refinement techniques.
We proved that our error estimator is reliable and efficient.
Line Defect:
Figure
1. Photonic crystal with a line defect

In
Fig. 1 we have an example of a photonic crystal with a line defect.
This type of configurations can be used to study PC waveguide.
In Fig. 2 we present a TE eigenfunction trapped in the the gap of the structure and
in Fig. 3 the corresponding hpadapted mesh.
As can be seen the method heavily refine in h around the corners of the inclusions where
the eigenfunction has singularities in the gradient.

Figure
2.
TE Eigenfunction trapped in the waveguide


Figure
3.
hpadapted mesh for the trapped mode


Vbend Crystal:
Figure
4. Photonic crystal with a Vbend

The trapped modes in the bend are important in practice
because exciting these modes it is possible to make an electromagnetic wave
to go around a bend.
In
Fig. 4 we have an example of a photonic crystal with a Vbend.
This type of configurations can be used to study PC waveguide with bends.
In Fig. 5 we present a TE eigenfunction trapped in the the bend of the structure and
in Fig. 6 the corresponding hpadapted mesh.
As can be seen the method heavily refine in h around the corners of the inclusions where
the eigenfunction has singularities in the gradient.

Figure
5.
TE Eigenfunction trapped in the bend


Figure
6.
hpadapted mesh for the trapped mode


Surface of a PC:
Figure
7. Surface of a photonic crystal

In this section we approximate a mode localized on the surface of a semiinfinite
PC.
In
Fig. 7 we have an example of a the surface of a photonic crystal.
This type of configurations can be used to study PC waveguide.
In Fig. 8 we present a TE eigenfunction trapped on the surface of the crystal and
in Fig. 9 the corresponding hpadapted mesh.
As can be seen the method heavily refine in h around the corners of the inclusions where
the eigenfunction has singularities in the gradient.

Figure
8.
TE Eigenfunction trapped on the surface


Figure
9.
hpadapted mesh for the trapped mode


References:
S. Giani (2013), An a posteriori error estimator for hpadaptive continuous Galerkin methods for photonic crystal applications.
 Computing 95(5), 395414.
