Non-Existence of 2D Zero-Index Photonic Band Gaps

Parry Y. Chen CUDOS, School of Physics, University of Sydney, Sydney, NSW, Australia Christopher G. Poulton CUDOS, School of Mathematical Sciences, University of Technology Sydney, Sydney, NSW, Australia Ara A. Asatryan CUDOS, School of Mathematical Sciences, University of Technology Sydney, Sydney, NSW, Australia Ross C. McPhedran CUDOS, School of Physics, University of Sydney, Sydney, NSW, Australia

When negative index metamaterials are incorporated into photonic crystals,a band gap arises that is not based on coherent Bragg scattering. Li et al. predicted its existence in a 1D photonic crystal whenever the average refractive index is zero. The cause is negative phase accumulation through negative index media, resulting in zero net phase accumulation. Consequently, properties of zero index and Bragg gaps differ, resulting in insensitivity to lattice constant, angle of incidence, polarization, and aperiodicity, and exhibiting phenomena such as suppression of Anderson localization and formation of omni-directional solitons.

To date, only 1D examples have been reported, and whether the zero phase mechanism is particular to 1D is unknown. In 1D, Fresnel reflections from each interface occur at planes of constant phase, and even for oblique incidence, the phase difference at successive interfaces is uniform. In higher dimensions, the inclusions have more complex boundaries, and this 1D mechanism seems impossible due to the more complex scattering patterns that result.

We derive analytic conditions for the existence of band gaps within 2D photonic crystals consisting of a square array of circular negative index inclusions in a positive index background. The expressions apply in the long wavelength limit, where similarity between the 1D and 2D cases is most likely. The results predict the possibility of a band gap, but zero index is neither a necessary nor sufficient condition. Instead, two conditions are obtained that can be attributed to the scattering of either monopole or dipole fields from the inclusions.

For the transverse magnetic polarization, the monopole condition states that average permittivity must be negative, while the dipole condition resembles a Clausius Mossotti condition satisfied by the permeability. For the opposite polarization, the roles of permittivity and permeability interchange. No zero phase accumulation mechanism exists, and instead the geometry of Fresnel reflection from each interface determines the existence of a band gap.

Results were derived using rapidly converging multipole field expansions and lattice sums, allowing accurate long wavelength results despite truncation to dipole order. Numerical simulations confirm the analytic results.

parche@physics.usyd.edu.au









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