Icosahedral quasicrystals for visible wavelengths by optical interference holography

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. 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 quasicrytals 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.

Quasicrystalline structures (quasicrystals), discovered in metal alloys in the early eighties, have a higher point group symmetry than ordinary periodic crystals. 3,4 They exhibit long-range aperiodic order and rotational symmetries that fall outside the traditional crystallographic classification schemes. 1,2 It was suggested 5,6 and realized in 2D 11-14 that mesoscale quasicrystals may possess photonic bandgaps (in which electromagnetic wave propagation is forbidden), a character of a new class of materials called photonic crystals 7,8 . Furthermore, because of their higher rotational symmetry, the bandgaps of quasicrystals are more isotropic and thus are more favourable in achieving complete bandgaps than conventional photonic crystals. [7][8][9][10] Photonic crystals have been fabricated by techniques such as the self-assembly of colloidal microspheres 17,18 or micro-fabrication [19][20] , and, recently, holographic lithography [21][22][23][24] and multi-photon direct laser writing 25 . However, it is difficult to fabricate quasicrystals by the self-assembly and micro-fabrication techniques, and is very time consuming to use the direct laser writing technique. Icosahedral quasicrystals exhibiting sizeable bandgaps in the microwave range have been fabricated using a stereo lithography method 15 . However, it is still a challenge to fabricate quasicrystals in the visible range.
Recently, it has been demonstrated that holographic lithography can be used to fabricate 2D and quasi-3D quasicrystals 26,27 in photoresists in the submicron range. Furthermore, one of us has shown that it is possible to obtain interference patterns with icosahedral symmetry using a novel 7-beam configuration 28 . Here we report the realization and optical characterization of icosahedral quasicrystals for the visible range using the optical interference holography technique suggested in ref. 28.
Quasicrystals can be classified either as physical 3D projections of higherdimensional periodic structures or in terms of wave vectors in the reciprocal space corresponding to the diffraction patterns of the quasicrystals [1][2]29 . It is the reciprocal space approach that provides the basis for fabricating quasicrystals using optical interference holography 29 . Figure 1a shows the novel 7-beam interference configuration, with five evenly spaced side beams surrounding two opposite central beams with incidence angle ϕ, for the icosahedral lattice shown in Fig. 1b. The six lattice basis vectors for the icosahedral quasicrystal are given by Eq. (1) in Table 1.
The corresponding face-centered reciprocal basis vectors Table 1 29 . One of us has shown that the six reciprocal basis vectors can be generated by Table 1) as shown in Fig. 1a with ϕ = 63.4 o using Eq. (4) in Table 1 28 . (Note that 0 k r is along the 5-fold axis OF in Fig. 1b.) The interference pattern of the seven coherent beams can be written as where l E r and l δ are the electric field, with amplitude taken as unity, and its phase for lattice spacings ~ 200nm, are slightly over exposed, presumably due to the finite resolution of the optical interference system, for attempts with lower exposures were not successful in producing "less-connected" samples. Furthermore, the microstructures of the samples are not sufficiently uniform (due to deformations during developing processes and shrinkage of the photoresist) for optical measurements, even though spectral reflections can be observed at some angles.
In view of the non-uniformity obtained using the SU8 photoresist, we have also used the high-resolution dichromate gelatin (DCG) holographic plate PFG-04 (Slavich, Russia) as the recording medium. However, due to the dielectric mismatch between the gelatin (~24 µm thick and n DCG ~ 1.57 before exposure at 488 nm), the glass substrate During the developing processes, the well-exposed regions inside the DCG would generate submicron air-voids, confirmed by SEM images 30 , while the under-exposed bandgaps at 560 nm, 500 nm, and 400 nm for normal incidence as shown in Fig. 4a.
The lower-left photo insets of Fig. 4a show the appearance for the normal reflection (green) and transmission (light purple, due to overall high transmittance) using diffused white light, consistent with the optical spectra.
To obtain the angular dependent transmission spectrum, the DCG sample was are asymmetric for the perpendicular case (Fig. 4c). Furthermore, the bandgaps are separated into four groups, with three groups corresponding to the three normal transmission bandgaps shown in Fig. 4a, and the fourth one arising only at large incidence angles. It turns out that these bandgaps can be explained by the reflections of planes inside the icosahedral quasicrystal. For every reciprocal vector given by for i,j = 0 to 6, planes with spacing given by and normal in the direction of can be identified from the icosahedral structure inside the DCG.
For example, for the reciprocal vector