In this paper, we investigate three-dimensional (3D) band gap properties of quasiperiodic structure. We successfully demonstrate the fabrication of a 3D dielectric quasicrystalline heterostructures with five-fold planar symmetry using the holographic optical tweezers technique. Light transmitted through this quasicrystal is collected using the spatially resolved optical spectroscopy technique for both visible and infrared wavelength bandwidths in a far-field region. We investigate and analyze the transmission spectra for the same wavelength bandwidths in a near-field region by using computer simulations. The computational modeling indicates that for both TE and TM modes of propagating light in the XY plane there is a clear transmission band-gap of around 50 nm wide centered at 650 nm. This indicates that there is a rotational symmetry in the constructed quasicrystal along its XY plane. Future directions and applications are discussed.
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One of the major challenges in photonic crystal technology lies in designing materials with specific absorption properties that are able to control and filter light over specific wavelength bandwidths. Such materials are useful for developing new types of photonic band gap filters. Among these materials, dielectric quasicrystalline heterostructures are particularly applicable for creating interesting diffraction effects for visible and infrared light due to their unique crystalline symmetries. Quasicrystals are ordered structures lacking of the translational periodicity of crystals. Interestingly, this deficiency allows quasicrystals to adopt a wide variety of unusual long-ranged rotational symmetries which are absent for crystals. In fact, the large number of effective lattice vectors in the reciprocal space provides quasicrystals with effective Brillouin zones . This leads to the formation of photonic band gaps [2–4] for light traveling through dielectric quasicrystalline heterostructures [5,6], despite the relatively low ratio of dielectric constants among the constituent materials . Recently, three-dimensional (3D) silicon inverse photonic quasicrystals for infrared wavelengths were designed by direct laser writing and subsequent silicon single-inversion approach . Also, holographic assemblies of two-dimensional (2D) and 3D silica quasicrystalline heterostructures were demonstrated .
In this article, we present, a 3D assembly of five-fold symmetry silica quasicrystalline heterostructures using the holographic optical trapping technique [10,11]. The far-field forward scattered light through the constructed quasicrystal along Z-axis was collected by using spatially resolved optical spectroscopy technique. We also analyzed the near-field forward scattered light via computational modeling in order to reveal photonic band gap properties of the constructed quasicrystal. A three-dimensional finite element method (FEM) in commercially available COMSOL multiphysics software was used for the optical modeling of the proposed quasiperiodic structure. A plane electromagnetic wave was assumed to be incident normally along X, M (i.e. direction at a 45 degree angle with respect to X and Y axes), Y and Z-axes on the sample. The energy range of the incident radiation varied from 2.48 to 0.83 eV, which correspond to the wavelength range of ~500-1500 nm. Both the transverse electric (TE) and transverse magnetic (TM) modes were considered. In the notation used in this work, the TE mode is defined as the electric field of light perpendicular to a surface in which the incident light propagates, whereas the TM mode is defined as the magnetic field of light perpendicular to the same surface. Port boundary conditions were used in both the positive and negative Z, X, M and Y-directions and perfect electric and magnetic boundary conditions were used along different axis accordingly. We used a finer built-in free triangular mesh in COMSOL multiphysics. The minimum and the maximum element sizes of the triangular mesh were 4.42 nm and 0.103 μm respectively with the maximum element growth rate of 1.35, and 0.3 resolution of curvature. The relative tolerance was set to 0.001. The dielectric functions used for all the materials in this work can be found elsewhere . The server with a dual Intel i7 Xeon six cores at 3 Ghz, 32 GB RAM memory and 1 TB hard drive memory was used for simulations. Each simulation approximately required 24 hour time period for obtaining transmission spectrum for the energy range of interest.
2. Experimental details
2.1 Quasicrystal fabrication
We prepared polyacrylamide hydrogel from colloidal 1.5 µm diameter silica microspheres (Duke scientific Lot 5238), dispersed in an aqueous solution of Acrylamide, N, N′ - Methylenebisacrylamide and Diethoxyacetophenone (All Aldrich Electrophoresis grade), in the ratio of 180:12:1 by weight. A sample was prepared by dispensing 4 µL of the hydrogel on the microscope slide and covering it with 18x18 cover glass. The cover glass was glued to the microscope slide with 5 minute epoxy. Thus, we obtained the hydrogel sample of 9 µm in thickness which was sealed and ready for use. Next, the sample was mounted on the microscope’s stage in order to experimentally build colloidal quasicrystalline material and analyze its optical properties. All vertices of the quasicrystal were generated by specially designed computer algorithm . Computer generated vertices were then projected as holograms through a high-numerical-aperture microscope lens, creating large 3D arrays of optical traps. Silica particles immersed in the hydrogel were arranged into quasicrystal(consisting of 190 spheres in 5 layers) by these 3D optical traps as shown in Fig. 1(a) (Media 1). This particular domain consists of 190 spheres in 5 layers. Details regarding the design and spacing between layers and vertices are shown in Figs. 1(b)-1(e).
2.2 Experimental setup
The schematic of the experimental setup used for optical study is shown in Fig. 2. A frequency-doubled diode-pumped solid state laser (Coherent Verdi 5W) generated light at 532 nm which was imprinted with phase-only hologram with the help of a liquid crystal spatial light modulator (SLM) (Hamamatsu X7690-16). The imprinted laser beam was translated to the input pupil of a 100x NA 1.4 SPlan Apo oil immersion objective positioned in an inverted optical microscope (Nikon TE-2000U). The objective focused the modified laser beam into traps and revealed images of trapped objects.
After trapping silica particles and arranging them into quasicristalline structure, the sample was illuminated by an ultraviolet light which rapidly photopolymerized the structure. Video demonstrating experimental steps and details during the fabrication process can be seen in Media 1. In order to study optical properties of the constructed quasicrystal, the sample was illuminated by the visible and infrared light from microscope’s condenser. The resulting forward scattered light from the quasicrystal was collected by 500 µm diameter optical fiber connected to USB400 Plug-and-Play Miniature Fiber Optic Spectrometer and NIR512, Near-infrared Spectrometer both by (Ocean Optics Inc.). Note that the fabricated 3D photonic structure may reveal interesting optical properties in different directions due to the presence of 2D (fivefold in XY plane) quasicrystalline symmetries with several plane layers in Z direction.
3. Results and discussion
3.1 Experimental Measurements
The transmission spectra for light propagating along the Z-axis through the quasicrystal measured for both visible and infrared wavelength bandwidths in the far-field region are shown in Fig. 3. We find that the first harmonic in the infrared spectrum is located at 1175 nm while the second harmonic in the visible spectrum is located at 585 nm. The first harmonic corresponds to ~857 nm in the medium which is comparable to the interplanar spacing between the layers of the quasicrystal as shown in Figs. 1(d) and 1(e) (orange-black, and red-green).
The maximum coefficients of the transmission spectra at visible and infrared wavelengths are 0.23 and 0.028, respectively. Large crystals with more layers can increase the interference pattern of the diffracted light and strengthen the intensity of the transmitted light. The construction of a crystal with large numbers of layers is restricted by the experimental setup because the amount of optical traps is limited by SLM. The transmission signal can be also improved by increasing the dielectric contrast of crystal-medium. In our case, the ratio of dielectric constants is ~1.10. We observe that the signal of the first harmonic in the infrared spectrum is weaker than the signal for the second harmonic located in the visible spectrum. This is caused by the optical lenses inside the microscope which are regular (Relay optics) rather than specially designed ones for infrared light.
3.2 Computer Simulations
We investigated light propagation in different directions in the proposed quasicrystalline structure using COMSOL multiphysics software. In order to reveal complete photonic band gaps of the quasicrystal; that is, a range of wavelengths over which light propagation is not permitted for all directions and polarizations, we investigate both TE and TM modes in different directions. Detailed information about differences between TE/TM modes and the band gap theory for one dimensional (1D), 2D and 3D photonic crystals and quasicrystals can be found in . A band gap is defined as the transmission minima with the coefficient nearly equal to zero or less than the simulation tolerance (0.001). Figures 4(a) and 4(b) depict the calculated transmission spectra from the quasicrystal in the near field region for both the TE and TM modes of light propagating along X and Y axes in the XY plane. A clear band gap (around 100 nm wide) centered at 650 nm is observed. This indicates that there is rotational symmetry in XY plane of the crystal for both TE and TM polarizations.
Figure 4(c) shows the transmission spectra for light propagating in the Z direction with ZX(TE) and ZY(TM) polarizations. We observe clear band gaps at 1000 nm and 680 nm, respectively. However, there is no rotational symmetry for both of these directions and as a result gaps are shifted and appear at different wavelengths. Figure 4(d) corresponds to the transmission spectra for the propagation of light in XY-plane at three different angles. We observe a clear rotational symmetry band gap around 50 nm wide centered at 650 nm for both TE and TM polarization which clearly shows evidence of a complete band gap.
The required transmission spectrum is retrieved from calculated S-matrix using the port boundary conditions. The calculation details of the S-parameters can be found at .
We chose monodisperse silica spheres because of their lower optical absorption, higher density and commercial availability. Other materials with specific optical properties can also be used for holographic assembly of quasicrystalline heterostructures. These studies are first trials towards the complete 3D band gap investigations of dielectric quasicrystals for visible and infrared wavelength bandwidths, constructed with the holographic optical trapping technique. Further investigations require construction of larger quasicrystals with icosahedral symmetry and analysis of transmitted light at different angles .
The experimental techniques used in our studies have advantages compared to other experimental methods used for the fabrication of quasicrystals. For instance, W. Man et. al.  experimentally investigated band gap properties of icosahedral quasicrystal constructed by using stereolithography technique. The experimental techniques used in our studies can create better dielectric contrast between the constituent materials of a quasicrystal in different directions compared to stereolithography technique. We are trapping and fixing dielectric spherical particles at specific locations in the medium without connecting them. In the stereolithography fabrication method, vertices of constructed quasicrystal are connected with dielectric rods. This in return may reduce the dielectric contrast between the vertices, and affect the band gap properties of constructed 3D quasicrystal. Thus, using our experimental techniques, we can obtain more accurate band gap picture of 3D quasicrystals.
We demonstrated holographic assembly of 3D dielectric quasicrystalline heterostructures with five-fold planar symmetry. We collected far-field forward scattered light from the quasicrystal using the spatially resolved optical spectroscopy technique. The propagation of light in different directions for both TE and TM modes was studied using finite element method in COMSOL multiphysics software. We observed evidence of the complete band gap of the constructed quasicrystal in the near field region. Results from the experimental measurements and the computational modeling were obtained despite a small dielectric contrast between the constituent materials of the quasicrystal. In conclusion, the techniques used in this work are powerful tools for constructing 3D quasiperiodic structures and investigating their full band gap properties which could lead to the development of new types of 3D photonic band gap filters for industrial applications.
Zaven Ovanesyan greatly appreciates the inspiration and support provided by Prof. David Grier to carry out this research work at the Department of Physics at the New York University. Authors are grateful to Weining Man, Lucas Fernandez Seivane and Zurab Kereselidze for valuable assistance. This work was supported by the Brownian Transport Thru Modulated Potential Energy Landscapes Grant Number DMR-0451589 and FRG: Photonic Quasicrystals and Heterostructures Grant Number DMR-0606415.
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