For a one-dimensional (1D) periodic system with inherent mirror symmetry, the value of the geometric “Zak” phase in a bulk band is related to the sign of reflection phase for wavelengths inside the bandgaps sandwiching the bulk band. Here, we designed an interference setup which allows us to measure the reflection phase of 1D phonic crystal fabricated for the optical range; this, in turn, enabled us to determine the Zak phases of the bands. We then found interface states whose existence can be traced to the topological properties of the bandgaps and the geometric phases of the bulk bands.

We present an optical interference model to create chiral microstructures (spirals) and its realization in photoresist using holographic lithography. The model is based on the interference of six equally-spaced circumpolar linear polarized side beams and a circular polarized central beam. The pitch and separation of the spirals can be varied by changing the angle between the side beams and the central beam. The realization of the model is carried out using the 325 nm line of a He-Cd laser and spirals of sub-micron size are fabricated in photoresist.

We have fabricated three-dimensional periodic quasicrystals exhibiting quasiperiodicity of the Penrose structure in the $x\text{-}y$ plane and periodic along the $z$ axis using ten-beam visible light holographic lithography. The quasicrystals show photonic band gaps in the visible range. The band gaps are found to follow a simple relation as a function of periodicity and polymeric volume fraction, in accord with the Bragg’s diffraction relation.

Quasicrystals, realized in metal alloys, are a class of lattices exhibiting symmetries that fall outside the usual classification for periodic crystals. They do not have translational symmetry and yet the lattice points are well ordered. Furthermore, they exhibit higher rotational symmetry than periodic crystals. Because of the higher symmetry (more spherical), they are more optimal than periodic crystals in achieving complete photonic bandgaps in a new class of materials called photonic crystals in which the propagation of light in certain frequency ranges is forbidden. The potential of quasicrystals has been demonstrated in two dimensions for the infrared range and, recently, in three-dimensional icosahedral quasicrystals fabricated using a stereo lithography method for the microwave range and direct laser writing for the IR range. Here, we report the fabrication and optical characterization of icosahedral quasicrystals using a holographic lithography method for the visible range. The icosahedral pattern, generated using a novel 7-beam optical interference holography, is recorded on photoresists and holographic plates. Electron micrographs of the photoresist samples show clearly the symmetry of the icosahedral quasicrystals in the submicron range, while the holographic plate samples exhibit bandgaps in the angular-dependent transmission spectra in the visible range. Calculations of the bandgaps due to reflection planes inside the icosahedral quasicrystal show good agreement with the experimental results.

There is no known simple rule that assures the existence of interface states in photonic crystals (PCs). We show here that one can control the existence or absence of interface states in 1D PCs through engineering the bulk geometrical phase such that interface states can be guaranteed in some or all photonic bandgaps. We verify experimentally the interface state design paradigm in 1D multilayered PCs fabricated by electron beam vapor deposition. We obtain the geometrical phases by measuring the reflection phases at the bandgaps of the PCs and achieve good agreement with the theory. Our approach could provide a platform for generating a PC interface state for various applications.