Chapter 5. Perspectives on Future Research Directions for 3D Photonics
5.1. Opportunities in PhC Architecture and Topology
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78 5.1.1. Topological Photonics
Topological photonics is another research area that has lately captured a great deal of scientific attention.74–77 Very basically, topology is the branch of mathematics which deals with quantities that do not change when an object is continuously deformed, no matter the extent of deformation. One classic example of a topological property, or topological invariant, is the number of holes within a closed surface. Objects with the same topological invariant are topologically equivalent, and only when a hole is created or removed in the object, by the introduction of a cut, will the topological invariant change, the process being a topological phase transition. In solid state systems, the Brillouin zone is analogous to a surface, and the geometric phase, or Berry phase, looks at changes in crystal momentum, analogous to surface curvature.
In the same way the study of photonic crystals benefited from an understanding of electronic crystals, topological photonics was prompted by the discovery of electronic topological insulators.78,79 Topological insulators are unique due to the existence of protected states occurring at crystal boundaries. Protected states support unidirectional propagation and are immune to a wide class of crystal impurities and defects. Given all the analogs between solid state systems and photonic systems, many quantum topological effects are being translated to the field of photonics. Indeed, quantum topological effects are a consequence of the wave nature of electrons, and light can be characterized as a wave as well. While photons do not experience electric potential as they have no electric charge, their motion is governed by the periodic change in dielectric contrast, which is analogous to electronic potential. For 3D PhC systems, one particular area of study has been the observation of Weyl points in a photonic band structure.80,81 Theoretical and experimental observation of Weyl points is currently being pursued, given the numerous phenomena which can result from this topological feature.82
From a practical standpoint, topological effects will allow for substantially more robust photonic devices. Current PhC fabrication processes are not immune to defects, and topologically insulating structures would relax certain device fabrication constraints. In particular, topological insulators could resolve issues like insertion losses, Fabry–Pérot oscillations, lattice disorder, and unwanted localization.74
79 The fabrication of optically active, topological photonic structures is currently the main challenge of this growing field, and consequently an area for future research.
5.1.2. Photonic Quasicrystals
In this thesis we have discussed at length the properties of 3D periodic structures. Interestingly though, recent research has revealed that quasi-periodic structures, or quasicrystals, also possess unique photonic properties that require further exploration.83–87 Quasicrystals are a class of structures that do not have translational symmetry, but possess local rotational symmetry. Because the structures of photonic quasicrystals have no periodicity, the Bloch wave vector 𝑘⃗⃗ cannot be defined exactly, and it is not possible to draw a conventional photonic band structure such as that for a typical PhC. However, photonic quasicrystals exhibit Bragg scattering of light, and a quasi-Brillouin zone can be defined on the basis of a quasicrystal’s long range order.87 This fact suggests that a pseudo-photonic band structure can be created for photonic quasicrystals, and that some of the band gap and dispersion properties evident in periodic photonic crystals may also be realizable in photonic quasicrystals.
The photonic dispersion relations in a band structure govern the basic properties of light propagation, and in periodic PhCs, a complete photonic bandgap arises when frequency gaps at the Brillouin zone boundaries overlap in all directions. The appearance of pseudo-bandgaps, or directional stop bands, depend on the symmetry of the underlying photonic crystal lattice. In photonic quasicrystals, as the rotational symmetry of the quasicrystal increases, the pseudo-Brillouin zone will become more spherical in 3D, which can result in the formation of a complete bandgap.83,88 A study by Man et. al. discussed how, even for low dielectric contrast structures, the increased isotropy of quasicrystals gives these structures an increased likelihood of possessing a complete bandgap.83 In contrast, for high dielectric contrast, light scattering in an architecture gets stronger, overwhelming the advantage of isotropy and allowing periodic photonic crystals to perform better.83 Nonetheless, these findings imply that photonic quasicrystals can be more advantageous in various photonic bandgap applications relative to periodic PhCs, at threshold material index values.
80 With regards to dispersion properties of quasicrystals, work done by Feng et. al. reported that a 2D photonic quasicrystal with dodecagonal symmetry exhibits negative refraction.86 The photonic quasicrystal under study consisted of a quasiperiodic arrangement of dielectric columns with ε = 8.6, embedded in a styrofoam background and was active in the microwave frequency regime. Refractive properties were investigated by creating wedge samples with different wedge angles, 𝜃0, and measuring the refracted wave intensity versus the angle of refraction of an incident k-vector through the quasicrystal wedge. For 𝜃0= 30°, a peak in intensity was observed at the refraction angle 𝜃 = −32°, corresponding to a negative refractive index of n = -1.06. This result was verified by a numerical simulation based on multiple-scattering theory, and further measurements indicated a refractive index of n ≈ -1 over a substantial range of incident angles. However, opportunities still exist in fabricating 3D photonic quasicrystals capable of AANR, and demonstrating negative refraction in quasicrystal structures at infrared and optical frequencies.
Additionally, a recent report from Kraus et. al. demonstrated that quasicrystals can exhibit topological phases that were previously thought to exist only for systems of higher dimensionality.89 This work posits that the study of topological phases in 3D photonic quasicrystals may lead to the discovery of topological properties that would only have appeared in 6D periodic systems.89 It is therefore clear that the intersection of topological photonics and quasicrystals is a minimally explored frontier that can yield unprecedented insights into higher dimensional photonic properties.
5.2. Future Directions for Index of Refraction Engineering and 3D Photonic Crystal