Semi-analytic methods for modelling photonic woodpiles
- Publication Type:
- Thesis
- Issue Date:
- 2011
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New semi-analytical methods are presented for modelling the electromagnetic fields of
three-dimensional photonic crystals that are composed of orthogonal layers of cylindrical
rods. Firstly, the multipole method is extended so that cylindrical defects, which act as
optical waveguides, can be introduced into such ‘woodpile’ structures. These waveguides
are important because they offer greater control over the mode dispersion and optical
losses than conventional waveguides do. The multipole method forms the basis of all of
the techniques presented in this thesis, and is employed here because it is considerably
faster than the pre-existing methods for modelling woodpile waveguides.
Two approaches for modelling linear defects are presented. The first approach uses a
grating super-cell to approximate a localised defect, and results are presented for both a
coupled resonator optical waveguide and for a linear waveguide, where each waveguide is
embedded in a finite woodpile cladding. The existence of waveguiding modes is inferred
from the transmission spectra, and is verified by numerically reconstructing the fields.
Furthermore, low loss waveguiding is observed for the linear waveguide.
To complement the super-cell approach, we have generalised the two-dimensional
fictitious source superposition method, whereby the defect modes of a woodpile are computed
directly. The principal advantage of this approach is that it is particularly efficient,
making this approach well-suited to the task of tuning the dispersion relationships of the
defect states; however, the performance gains are achieved by forgoing the ability to deal
with finite structures. The dependence of the dispersion on the refractive index and size
of the defect is investigated, and it is shown that tuning these parameters is an effective
method for optimising the waveguide for operation in the slow-light regime.
Lastly, a comprehensive analysis of the surface modes of photonic woodpiles is performed.
Specifically, the surface modes of both finite and semi-infinite woodpiles are
characterised using transfer matrix and plane wave matrix formulations. In the case of
finite structures, a general mathematical description of the modes that propagate simultaneously
along the top and bottom surfaces is given. It is shown that when the number
of layers is even, such ‘double-interface’ modes only exist for specific directions of the
Brillouin zone. However, when the number of layers is odd, every surface mode is a
double-interface mode and, in this case, the direction of propagation plays an important
role in determining the coupling strength between the two surfaces: for certain directions,
the coupling is negligible even when the number of layers is small. The dispersion curves of two different double-interface modes can anticross or be interwoven, depending on the
symmetry of the modes. A Fabry-P´erot cavity comprising two woodpile barrier regions is
also considered. In particular, the conditions required in order for coupled surface modes
to exist in these ‘compound woodpile’ structures are described.
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