Quasicrystals - Research Log

Quasicrystals

3D Photonic Image-Structure Projection Technology.

This website is interesting: http://unifiedfractalfield.com/

Negative Refraction and Imaging Using 12-fold-Symmetry Quasicrystals

Zhifang Feng, Xiangdong Zhang,  Yiquan Wang, Zhi-Yuan Li, Bingying Cheng, and Dao-Zhong Zhang

Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China Department of Physics, Beijing Normal University, Beijing 100875, China (Received 5 July 2004; published 23 June 2005)

Recently, negative refraction of electromagnetic waves in photonic crystals was demonstrated experi- mentally and subwavelength images were observed. However, these investigations all focused on the periodic structure. Here, we report a new theoretical and experimental finding that negative refraction can appear in some transparent quasicrystalline photonic structures. The photonic quasicrystals (PQCs) exhibit an effective refractive index close to 􏰀1 in a certain frequency window. The index shows small spatial dispersion, consistent with the nearly homogeneous geometry of the quasicrystal. More interest- ingly, a superlens based on the 2D PQCs can form a non-near-field subwavelength image whose position varies with the source distance. These properties make PQCs promising for application in a range of optical devices. 

Looking For Possible Military applications:

Three-dimensional quasicrystalline photonic material with five-fold planar symmetry for visible and infrared wavelengths by holographic assembly of quasicrystalline photonic heterostructures

Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003, USA Current address: Department of Physics & Astronomy, The University of Texas at San Antonio, One UTSA Circle, San Antonio TX 78249, USA
*vkf002@my.utsa.edu

Abstract: 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.


© 2013 Optical Society of America
OCIS codes: (060.3510) Lasers, fiber; (090.1760) Computer holography; (160.5298) Photonic crystals.
References and links

1. S. E. Burkov, T. Timusk, and N. W. Ashcroft, “Optical conductivity of icosahedral quasi-crystals,” J. Phys. Condens. Matter 4(47), 9447–9458 (1992).
2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995).
3. M. Florescu, S. Torquato, and P. J. Steinhardt, “Complete band gaps in two-dimensional photonic quasicrystals,”
   Phys. Rev. B 80(15), 155112 (2009).
   
4. L. Jia, I. Bita, and E. L. Thomas, “Level Set Photonic Quasicrystals with Phase Parameters,” Adv. Funct. Mater.
   22(6), 1150–1157 (2012).
   
5. X. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic crystals,”
   Phys. Rev. B 63(8), 081105 (2001).
   
6. M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, “Optimized Structures for
   Photonic Quasicrystals,” Phys. Rev. Lett. 101(7), 073902 (2008).
   
7. J. Xu, R. Ma, X. Wang, and W. Y. Tam, “Icosahedral quasicrystals for visible wavelengths by optical
   interference holography,” Opt. Express 15(7), 4287–4295 (2007).
   
8. A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and
   G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat.
   Mater. 5(12), 942–945 (2006).
   
9. Y. Roichman and D. G. Grier, “Holographic assembly of quasicrystalline photonic heterostructures,” Opt.
   Express 13(14), 5434–5439 (2005).

10. E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. 69(5), 1974–1977 (1998).
    
11. D.G.Grier,“Arevolutioninopticalmanipulation,” Nature424(6950),810–816(2003).
    
12. E.D.Palik,HandbookofOpticalConstantofSolids (Academic,1985).
    
13. W. Man, Photonic Quasicrystals and Random Ellipsoid Packings: Experimental Geometry in Condensed Matter
    Physics (Princeton University PhD Dissertation, 2005).
    
14. P.M.Chaikin,“PhotonicQuasicrystals, http://www.physics.nyu.edu/~pc86/.
    
15. Version 4.3, “COMSOL Multiphysics Reference Guide,” http://www.comsol.com.
    
16. W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties
    of icosahedral quasicrystals,” 
    
17. T.F.Krauss,R.M.D.L.Rue,andS.Brand,“Two-dimensionalphotonic-bandgapstructuresoperatingatnear
    infrared wavelengths,” Nature 383(6602), 699–702 (1996).
    
18. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” 

Introduction
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 [1]. 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 [7]. Recently, three- dimensional (3D) silicon inverse photonic quasicrystals for infrared wavelengths were designed by direct laser writing and subsequent silicon single-inversion approach [8]. Also, holographic assemblies of two-dimensional (2D) and 3D silica quasicrystalline heterostructures were demonstrated [9].
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.

OPTICAL MATERIALS EXPRESS

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 [13]. 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. This particular domain consists of 190 spheres in 5 layers. Details regarding the design and spacing between layers and vertices.

Quasicrystal constructed from spherical silica particles. (a) Colloidal silica particles trapped in three dimensional configurations with holographic optical traps (Media 1). Brightness of a particle corresponds to different crystalline layer in Z-axis. From (b) to (e): schematic representations of different projections of the quasicrystal. Colors represent different heights of the crystalline planes in Z-axis. Orange, black, red, green and blue correspond to 3.7, 2.9, 0.4, −0.4 and −3.7 μm respectively.
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.

OPTICAL MATERIALS EXPRESS

Fig. 2. The experimental setup consists of a Laser, SLM, Microscope, Charge Coupled Device (CCD) Camera, Optical Fiber and Spectrometer all connected to the Computer. Laser beam passes through a series of optical devices creating an array of a three dimensional traps in microscope’s conventional imaging plane. (a) Real-time spectrum of forward scattered light from quasicrystal is collected by the optical fiber mounted to the second eyepiece port and connected to the Spectrometer. (b) Real-time image of the constructed quasicrystal, obtained from the CCD camera mounted to the first eyepiece port of the microscope.
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).


OPTICAL MATERIALS EXPRESS 1335

Visible and infrared transmission spectra of the quasicrystal sample obtained in far-field measurements along the Z-axis.
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 [14]. 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.
Fig. 4. Transmission spectra calculated for both the TE and TM modes, for the light propagating along the (a) X-axis, (b) Y-axis, (c) Z-axis and (d) XY plane at 0, 45 and 90 degree angles corresponding to X, M and Y directions. The gap at 650 nm shows an evidence of the complete band gap for light propagating in XY plane. Arrows and dashed lines indicate the locations of band gaps.
#192295 - $15.00 USD Received 17 Jun 2013; revised 26 Jul 2013; accepted 30 Jul 2013; published 13 Aug 2013
(C) 2013 OSA 1 September 2013 | Vol. 3, No. 9 | DOI:10.1364/OME.3.001332 | OPTICAL MATERIALS EXPRESS 1336

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 [15].
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 [16].

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. [16] 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.
In general, controlling the propagation of light at visible and infrared wavelengths is an extremely important field of research due to recent advancements in photonic crystal technology [17, 18].
4. Conclusion
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.
Acknowledgments
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.
#192295 - $15.00 USD Received 17 Jun 2013; revised 26 Jul 2013; accepted 30 Jul 2013; published 13 Aug 2013

Wolf Clan Media

I found these papers but cannot access most of them. If anyone with access (students, researchers, etc.) decides to help with the investigation. Please reach out. Thanks.

https://wolfclanentertainment.com/

http://www.sciencedirect.com/science/article/pii/S0921509300013058

$208 USD
https://books.google.com/books?id=6MqMjDPthG8C&lpg=PP2&ots=SLNjF_2VPY&dq=quasicrystals%20military%20applications&lr&pg=PP2#v=onepage&q&f=false

$60

https://books.google.com/books?id=fN0I6f1XyJgC&lpg=PA92&ots=EAsT700qoF&dq=recent%20trends%20in%20thermoelectric%20materials%20research%20SAIC&pg=PA92#v=onepage&q&f=false

Duke University PDF - Thermoelectric Materials

http://webhome.phy.duke.edu/~baranger/articles/thermoelectric/Dresselhausetal_reviewmaterials_03.pdf

Possible applications - Vice news

https://motherboard.vice.com/en_us/article/4x3me3/quasicrystals-are-natures-impossible-matter

NIST on Dan Schetman

https://www.nist.gov/content/nist-and-nobel/person-behind-nobel-prize-dan-shechtman

NIST - Dec. 2001 

https://www.nist.gov/publications/nist-centennial-celebration-crystallographic-highlights

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.90.055501

Abstract

We propose a holographic design of five-beam symmetric umbrella configuration, where there are a central beam and four ambient beams symmetrically scattered around the central one with the same apex angle, for fabrication of three-dimensional photonic crystals with tetragonal or cubic symmetries, and systematically analyzed the band gap properties of resultant photonic crystals when the apex angle is continuously increased. Our calculations reveal that large complete photonic band gaps exist in a wide range of apex angle for a relatively low refractive index contrast. Specifically, the face-centered cubic structure with a relative band gap of 25.1% for ε = 11.9 can be obtained with this recording geometry conveniently where all the beams are incident from the same half-space. These results will provide us with more understanding of this important recording geometry and give guidelines to its use in experiments.

©2006 Optical Society of America

1. Introduction

Photonic crystals (PhCs) are structures in which the dielectric constant is periodically modulated on a length scale comparable to the desired wavelength of operation [1, 2], and the resultant photonic dispersion may exhibit photonic band gaps (PBGs) which are useful in controlling light behavior. In the last decade much attention has been attracted to the fabrication of PhCs with complete PBGs. Various techniques such as the electron-beam lithography, self-assembly, multiphoton polymerization, and holographic lithography (HL) have been proposed and demonstrated with different levels of success [3–7]. Among them the method of HL has some unique features such as inexpensive volume recording.

In HL a desired geometrical structure is formed by multibeam interference with single or multiple exposures [8–11]. We have shown that all 14 three-dimensional (3D) Bravais lattices can be produced this way [12], and different beam designs will result in different PBG properties of the resultant structures [13, 14]. For example, the face-centered-cubic (fcc) lattice can be obtained by the interference of one central beam and three ambient beams symmetrically scattered around the former [15, 16], but the structure made in this geometry has only quite a narrow PBG [16, 17]. An alternative beam design was proposed to fabricate fcc lattice with a large complete PBG, but it requires four beams incident from two opposite surfaces of a sample [18–20], making it difficult to realize in practice. In addition, in real experiments the directions of interference beams may be slightly deviated from their theoretical values for many practical reasons [16]. Therefore a more extensive investigation into the effect of beam angle derivation on the resulting structures and then on the PBGs will be helpful for a clearer understanding of the structure formation with this geometry and for the control of experimental parameters. Recently, a research group fabricated some tetragonal and cubic PhCs with the use of phase masks and analyzed the PBGs of these structures; however, what was employed in their calculations is the woodpile model instead of the real holographic structure [21], and the two cases are usually not the same since in HL the actual structures including the cell’s shape and size and consequently the corresponding PBG are strongly dependent on the specific beam design and light intensity threshold selection.

In this paper we propose a five-beam symmetric umbrella configuration which is more convenient to make 3D PhCs with large complete PBGs. For instance, an fcc lattice with 25.1% relative band gap for a dielectric constant contrast 11.9:1 can be obtained with all the beams incident from the same side of the sample. Furthermore, a systematical study on the structure and band gap evolution for a continuously varying apex angle in this geometry is provided. These analyses and results may serve as guidelines in both theory and practice.

2. Recording geometry and resultant structures

The recording geometry with five-beam symmetric umbrella configuration we proposed here is shown in Fig. 1, where the central beam (C-beam) with wave vector K c is set along the z direction, while the four ambient beams (A-beams) of wave vectors K 1 to K 4 are in the plane yoz or xoz, respectively, with the same apex angle θ with z axis. The five wave vectors can be expressed as functions of the apex angle,

(1)

𝑲1=(2𝜋𝜆)(−sin𝜃,0,cos𝜃),𝑲2=(2𝜋𝜆)(0,sin𝜃,cos𝜃),$${\mathit{K}}_1=\left(\frac{2\pi }{\lambda }\right)\left(-\sin \theta ,0,\cos \theta \right),{\mathit{K}}_2=\left(\frac{2\pi }{\lambda }\right)(0,\sin \theta ,\cos \theta ),$$

𝑲3=(2𝜋𝜆)(sin𝜃,0,cos𝜃),𝑲4=(2𝜋𝜆)(0,−sin𝜃,cos𝜃),$${\mathit{K}}_3=\left(\frac{2\pi }{\lambda }\right)\left(\sin \theta ,0,\cos \theta \right),{\mathit{K}}_4=\left(\frac{2\pi }{\lambda }\right)(0,-\sin \theta ,\cos \theta ),$$

𝑲𝑐=(2𝜋𝜆)(0,0,1).$${\mathit{K}}_c=\left(\frac{2\pi }{\lambda }\right)(0,0,1).$$

This geometry can be realized conveniently with the use of a diffraction beam splitter (DBS) where a zero-order diffracted beam and four symmetric first-order diffracted beams are employed. For polarization, we choose the central beam to be circularly polarized and all the four A-beams linearly polarized. The corresponding unit polarization vectors are

(2)

𝒆1=𝒆3=(0,1,0),𝒆2=𝒆4=(1,0,0),𝒆c=12−−√(1,−𝑖,0).$${\mathit{e}}_1={\mathit{e}}_3=(0,1,0),{\mathit{e}}_2={\mathit{e}}_4=(1,0,0),{\mathit{e}}_{\text{c}}=\frac{1}{\sqrt{2}}(1,-i,0).$$

Fig. 1. Symmetric umbrella recording geometry and the coordinate system used for calculations.

Download Full Size | PPT Slide | PDF

If we use the same amplitude EA for all the A-beams and the amplitude EC for C-beam, and adopt double-exposure approach with the C-beam, beams 1 and 2 open for the first exposure while the C-beam, beams 3 and 4 open for the second exposure, the two exposures have the same exposure time, the total intensity will be

(3)

𝐼(𝑟)=2𝐸2𝐶+4𝐸2𝐴+2−−√𝐸𝐶𝐸𝐴{sin[2𝜋𝜆(𝑥sin𝜃+(1−cos𝜃)𝑧)]$$I\left(r\right)=2E_C^2+4E_A^2+\sqrt{2}{E}_C{E}_A\{\sin \left[\frac{2\pi }{\lambda }\left(x\sin \theta +\left(1-\cos \theta \right)z\right)\right]$$

+cos[2𝜋𝜆(−𝑦sin𝜃+(1−cos𝜃)𝑧)]$$+\cos \left[\frac{2\pi }{\lambda }\left(-y\sin \theta +\left(1-\cos \theta \right)z\right)\right]$$

+sin[2𝜋𝜆(−𝑥sin𝜃+(1−cos𝜃)𝑧)]$$+\sin \left[\frac{2\pi }{\lambda }\left(-x\sin \theta +\left(1-\cos \theta \right)z\right)\right]$$

+cos[2𝜋𝜆(𝑦sin𝜃+(1−cos𝜃)𝑧)]}.$$+\cos \left[\frac{2\pi }{\lambda }\left(y\sin \theta +\left(1-\cos \theta \right)z\right)\right]\}.$$

To get the best contrast we should maximize the ratio of √2EC EA /(2𝐸2𝐶$E_C^2$+4𝐸2𝐴$E_A^2$), and this requirement leads to the optimized beam amplitude ratio EC /EA = √2. For brevity we may write the relative intensity of the spatially varying part in Eq. (3) as

(4)

Δ𝐼(𝒓)=sin[2𝜋𝜆(𝑥sin𝜃+(1−cos𝜃)𝑧)]$$\Delta I\left(\mathit{r}\right)=\sin \left[\frac{2\pi }{\lambda }\left(x\sin \theta +\left(1-\cos \theta \right)z\right)\right]$$

+cos[2𝜋𝜆(−𝑦sin𝜃+(1−cos𝜃)𝑧)]$$+\cos \left[\frac{2\pi }{\lambda }\left(-y\sin \theta +\left(1-\cos \theta \right)z\right)\right]$$

+sin[2𝜋𝜆(−𝑥sin𝜃+(1−cos𝜃)𝑧)]$$+\sin \left[\frac{2\pi }{\lambda }\left(-x\sin \theta +\left(1-\cos \theta \right)z\right)\right]$$

+cos[2𝜋𝜆(𝑦sin𝜃+(1−cos𝜃)𝑧)].$$+\cos \left[\frac{2\pi }{\lambda }\left(y\sin \theta +\left(1-\cos \theta \right)z\right)\right].$$

A common approach in HL is the use of a negative photoresist material such as SU-8. In this case the “underexposed” region can be selectively removed using a developer substance which leaves the “overexposed” region intact. The developed photoresist is then infiltrated with SiO2and burned away, leaving a daughter inverse template. Finally the daughter template is inverted by high temperature infiltration with silicon [20]. In the following analysis we will consider this case. If we denote the light intensity threshold in Eq. (4) as I t and assume that the region of ΔI > I t is filled with a material of high refractive index while the other region is air, we may obtain a periodic microstructure whose concrete shape is determined by the specific apex angle θ and I t.

In general the lattice structures resulting from Eq. (3) or Eq. (4) are tetragonal symmetric structures. The continuous increase of apex angle θ leads to continuous variation of primitive vectors, reciprocal vectors and the irreducible Brillouin zone of the resultant structure. Our calculations show that the Brillouin zone changes from a small tetragonal cake spreading out on the xy plane when θ is very small to a long tetragonal pillar along the z axis when θ is close to 180° (see Fig. 2). When the apex angle θ reaches 70.53° (corresponding to c/a = √2, where a is the period of the interference pattern along x or y direction and c is that in z direction), the structure has fcc symmetry, similar to the diamond structure [22]. If the θ is near 70.53°, a lattice with face-centered-tetragonal (fct) symmetry is obtained. Figure 3 (a) and 3(b) show the real fcc structure and its primitive cell constructed by five-beam symmetric umbrella configuration when θ = 70.53°, which obviously differ from the rhombohedral structure with fcc symmetry and its primitive cell [Fig. 3(c) and 3(d)] formed by four-beam symmetric umbrella configuration [12] when the apex angle is 38.94°. If the apex angle achieves the value of θ = 90° (corresponding to c/a = 1), a body-centered-cubic (bcc) lattice is obtained. When the value of θ is near 90°, the structure has body-centered-tetragonal (bct) symmetry.

Fig. 2. Evolution of irreducible Brillouin zone (Movie, 155K). The irreducible Brillouin zones of tetragonal structures for θ = 80° (a), θ = 90° (b), and θ = 100° (c).

Download Full Size | PPT Slide | PDF

Fig. 3. (a). The real fcc structure formed by five-beam symmetric umbrella configuration when θ = 70.53° and I t = 1.39 in Eq. (4), the corresponding filling ration is 21.7%; (b) the primitive cell of the fcc structure shown in (a); (c) the rhombohedral structure with fcc symmetry fabricated by four-beam symmetric umbrella configuration when the apex angle is 38.94°; and (d) the primitive cell of the structure shown in (c).

Download Full Size | PPT Slide | PDF

3. Band gap calculations

The plane-wave expansion method [23, 24] has been used to study PBG properties of the structures of this kind and search for the corresponding optimum volume filling ratio f yielding maximum relative PBG for each apex angleθ. Calculations reveal that full photonic band gaps exist over a very wide range of apex angle with a relatively low refractive index contrast needed to open them. Figure 4 represents the relative gap sizes of optimized structures for the apex angle range of 50° < θ < 115° with a dielectric constant contrast of 11.9 to 1 corresponding to silicon in air. From Fig. 4 we can see clearly that there are complete PBGs above 10 % in the range of 52° < θ < 112°, and even larger PBGs above 20% for 59° < θ < 92°. The maximum relative gap size of 25.1% appears at θ = 70.53° for the fcc structure, and the relative gap size for the bcc structure formed when θ = 90° is 21.3%. As an example, the band structure of the fcc lattice is given in Fig. 5, obviously a large complete PBG, from 0.330ωa/2πc to 0.425ωa/2πc(where c is the light velocity), exists between the second and third bands. Comparing the band gap result with that of woodpile structure fabricated by woodpile model [21], one can find two results have similar trend that the band gap size is a function of c/a defined in last section or θ, and the fcc structure has the biggest gap size.

Fig. 4. Relative band gap of optimized structures as a function of apex angle for 50°< θ < 115° when ε = 11.9. The solid symbols are the date for fcc and bcc structures when θ = 70.53° and θ = 90°, respectively. When the value of θ is near 70.53° and 90° the fct and bct symmetric structures can be obtained.

Download Full Size | PPT Slide | PDF

Fig. 5. Photonic band structure for the fcc structure with θ = 70.53°. The position of the high symmetry points together with the irreducible Brillouin zone are shown in the inset.

Download Full Size | PPT Slide | PDF

The effect of the dielectric refractive index n of the structure formed by this five-beam symmetric umbrella configuration on the band gap size has also been investigated. As a special case, in Fig. 6, we illustrate the variation of relative band gaps with different filling ratios and different refractive index contrasts for the fcc symmetric structure formed when θ = 70.53°. In this case the minimum refractive index required to open a complete photonic band gap is slightly less than 1.95, lower than 2.05 obtained in one work of band gap calculation for the woodpile model as the lowest requirement to open the gap [25]. When the refractive index is 3.6 and the filling ratio is 21.5%, the relative band gap reaches as high as 27.3%. In addition, it is worth noting that the range of filling ratio yielding a complete band gap for each given refractive index is fairly wide, especially for the high refractive index. For example, when the dielectric refractive index is 3.4 or 3.6, a complete relative band gap larger than 10% can be obtained over a wide range of filling ratio, from 10% to 53%.

Fig. 6. Variation of relative band gaps with different filling ratios for different refractive index contrasts for the fcc symmetric structure with the apex angle θ = 70.53°.

Download Full Size | PPT Slide | PDF

4. Conclusions

In summary, we have designed a five-beam symmetric umbrella configuration for the fabrication of 3D photonic crystals using holographic lithography method. The photonic structures with different symmetries, including face-centered-tetragonal, face-centered-cubic, body-centered-tetragonal and body-centered-cubic, can be produced with this geometry of different apex angles. The theoretical analysis indicates that complete band gaps exist for the structures formed this way over a very wide range of apex angle. Particularly, the fcc and bcc symmetric structures can be obtained when θ = 70.53° and 90°, respectively; and their corresponding relative band gap sizes are as high as 25.1% and 21.3%, respectively, for a dielectric constant contrast 11.9:1. When the value of θ is near 70.53° and 90° the fct and bct symmetric structures can be obtained, respectively. Furthermore, the dielectric constant contrast or equally the refractive index contrast of the structures of this kind required to open complete PBGs is quite low (n ≥ 1.95), and the filling ratio range for a certain structure to assure complete PBGs is fairly large. These discussions give us more understanding of this configuration and its advantages in applications. For example, we can fabricate an fcc lattice with a large complete PBG with all the interference beams arranged in the same half-space of the sample, which is more convenient compared with the geometry where the beams are incident from two opposite sides; the requirement for an exact theoretical apex angle may be relaxed to a certain extent in practical fabrication from the viewpoint of PBG formation; and the filling ratio or equivalently the light threshold selection may also be done more easily. We believe that these results are helpful in both theory and practice of HL.

Acknowledgment

This work is supported by the National Natural Science Foundation (64077005) and the Doctoral Program Foundation of Ministry of Education (20020422047), China.

Holographic assembly of quasicrystalline photonic heterostructures

Yael Roichman

David G. Grier

Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003

Abstract. 

Quasicrystals have a higher degree of rotational and point-reflection symmetry than conventional crystals. As a result, quasicrystalline heterostructures fabricated from dielectric materials with micrometer-scale features exhibit interesting and useful optical properties including large photonic bandgaps in two-dimensional systems. We demonstrate the holographic assembly of two-dimensional and three-dimensional dielectric quasicrystalline heterostructures, including structures with specifically engineered defects. The highly uniform quasiperiodic arrays of optical traps used in this process also provide model aperiodic potential energy landscapes for fundamental studies of transport and phase transitions in soft condensed matter systems.

pacs: (140.7010) Trapping; (090.1760) Computer holography; (120.4610) Optical fabrication

Figure 1. Two-dimensional colloidal quasicrystals organized with holographic optical traps. (a) 5-fold. (b) 7-fold. (c) 8-fold. (d) An octagonal quasicrystal with an embedded structured defect. The scale bar in (a) indicates 5 .

Quasicrystals have long-ranged orientational order even though they lack the translational periodicity of crystals. Not limited by conventional spatial point groups, they can adopt rotational symmetries that are forbidden to crystals. The resulting large number of effective reciprocal lattice vectors endows quasicrystals' effective Brillouin zones with an unusually high degree of rotational and point inversion symmetry (1). These symmetries, in turn, facilitate the the development of photonic band gaps (PBG) (2) for light propagating through quasicrystalline dielectric heterostructures (3); (4); (5), even when the dielectric contrast among the constituent materials is low. Photonic band gaps have been realized in one- (6) and two-dimensional (7) lithographically defined quasiperiodic structures. Here we demonstrate rapid assembly of arbitrary materials into two-dimensional and three-dimensional quasicrystalline heterostructures with features suitable for obtaining interesting and potentially useful diffractive effects in infrared and visible light.

Figure 2. Four views of a rolling colloidal icosahedron. (a) 5-fold axis. (b) 2-fold axis. (c) 5-fold axis. (d) Midplane. Scale bar indicates 5 . The complete assembly and rolling process is shown in the associated movie.

Our approach is based on the holographic optical trapping technique (8); (9); (10) in which computer-generated holograms are projected through a high-numerical-aperture microscope objective lens to create large three-dimensional arrays of optical traps. In our implementation, light at 532 from a frequency-doubled diode-pumped solid state laser (Coherent Verdi) is imprinted with phase-only holograms using a liquid crystal spatial light modulator (SLM) (Hamamatsu X8267 PPM). The modified laser beam is relayed to the input pupil of a  NA 1.4 SPlan Apo oil immersion objective mounted in an inverted optical microscope (Nikon TE-2000U), which focuses it into traps. The same objective lens is used to form images of trapped objects, using the microscope's conventional imaging train (10).

We used this system to organize colloidal silica microspheres 1.53  in diameter (Duke Scientific Lot 5238) dispersed in an aqueous solution of  (wt/wt) acrylamide, -methylenebisacrylamide and diethoxyacetophenone (all Aldrich electrophoresis grade). This solution rapidly photopolymerizes into a transparent polyacrylamide hydrogel under ultraviolet illumination, and is stable otherwise. Fluid dispersions were imbibed into 30  thick slit pores formed by bonding the edges of #1 coverslips to the faces of glass microscope slides. The sealed samples were then mounted on the microscope's stage for processing and analysis.

Figure 3. Holographic assembly of a three-dimensional colloidal quasicrystal. (a) Colloidal particles trapped in a two-dimensional projection of a three-dimensional icosahedral quasicrystalline lattice. (b) Particles displaced into the fully three-dimensional configuration. The shaded region identifies one embedded icosahedron. (c) Reducing the lattice constant creates a compact three-dimensional quasicrystal. (d) Optical diffraction pattern showing ten-fold symmetric peaks. The three-dimensional assembly process is shown in the associated movie.

Silica spheres are roughly twice as dense as water and sediment rapidly into a monolayer above the coverslip. A dilute layer of spheres is readily organized by holographic optical tweezers into arbitrary two-dimensional configurations, including the quasicrystalline examples in Fig. 1. Figures 1(a), (b) and (c) show planar pentagonal, heptagonal and octagonal quasicrystalline domains (11), respectively, each consisting of more than 100 particles. Highlighted spheres emphasize each domain's symmetry. These structures all have been shown to act as two-dimensional PBG materials in microfabricated arrays of posts and holes (3); (12); (13); (14). As a soft fabrication technique, holographic assembly requires substantially less processing than conventional methods such as electron-beam lithography, and can be applied to a wider range of materials. Unlike complementary optical fabrication techniques such as multiple-beam holographic photopolymerization (14); (15); (16); (17), assembly with holographic optical traps lends itself to creating nonuniform architectures with specifically engineered features, such as the channel embedded in the octagonal domain in Fig. 1(d). Similar structures of comparable dimensions created lithographically have been shown to act as narrow-band waveguides and frequency-selective filters for visible light (12); (13); (18); (19).

Holographic trapping's ability to assemble free-form heterostructures extends also to three dimensions. The sequence of images of a rolling icosahedron in Fig. 2 shows how the colloidal spheres' appearance changes with distance from the focal plane. This sequence also recalls earlier reports (20); (21) that holographic traps can successfully organize spheres into vertical stacks along the optical axis, while maintaining one sphere in each trap.

The icosahedron itself is the fundamental building block of a class of three-dimensional quasicrystals, such as the example in Fig. 3. Building upon our earlier work on holographic assembly (22), we construct a three-dimensional colloidal quasicrystal by first gathering the appropriate number of sedimented spheres into a two-dimensional arrangement corresponding to the planar projection of the planned structure, Fig. 3(a). Next, we translate the spheres along the optical axis to their final three-dimensional coordinates in the quasicrystalline domain, as shown in Fig. 3(b). One icosahedral unit is highlighted in Figs. 3(a) and (b) to clarify this process. Finally, the separation between the traps is decreased in Fig. 3(c) to create an optically dense structure. This particular domain consists of 173 spheres in roughly 7 layers, with typical inter-particle separations of 3 .

The completed structure was gelled and its optical diffraction pattern recorded at a wavelength of 632 by illuminating the sample with a collimated beam from a HeNe laser, collecting the diffracted light with the microscope's objective lens and projecting it onto a charge-coupled device (CCD) camera with a Bertrand lens. The well-defined diffraction spots clearly reflect the quasicrystal's five-fold rotational symmetry in the projected plane.

Quasicrystalline heterostructures are likely to be particularly well suited for photonic bandgap applications because their effective Brillouin zones are more nearly spherical than those for photonic crystals and thus optimize the overlap of local bandgaps (23). This property has been exploited in lithographically defined two-dimensional devices with micrometer-scale features. Recently, the photonic band structure of a centimeter-scale three-dimensional icosahedral quasicrystal was measured in the microwave region and shown to feature prominent gaps at the effective Brillouin zone edge (23).

Although photonic band structure calculations generally are lacking for dielectric quasicrystals, the structure in Fig. 3 is similar to the polymeric quasicrystalline lattice in Ref. (23) and has a comparable mismatch in dielectric constant. It therefore should have comparable optical properties at appropriately rescaled wavelengths in the infrared and visible bands. Moreover, we expect bandgaps to be more pronounced in the holographically defined colloidal quasicrystal because of its larger volume fraction. These benefits all could be further enhanced by drying the gel at the triple point of water to further increase both the volume fraction and the dielectric constant mismatch.

Sequential assembly starting from a planar starting configuration ensures that there is one particle at each vertex of the quasicrystal. A similar outcome could be achieved in a system with microfluidic sample control without this initial step, and indeed without imaging. Larger structures then can be fabricated by assembling smaller domains, fixing them through spatially-resolved polymerization (24) or by initiating inter-particle bonds, and then combining the sub-assemblies, in much the same spirit as shoot-and-step lithography commonly used for two-dimensional semiconductor devices.

The use of monodisperse silica spheres for this demonstration was dictated by their commercial availability, low optical absorption, and high density. Holographic assembly lends itself just as easily to other materials, and so can be tailored to particular applications. Deterministic organization of disparate components under holographic control could be used to embed gain media in photonic bandgap cavities, to install materials with nonlinear optical properties within waveguides to form switches, and to create domains with distinct chemical functionalization. Distinctly engineered domains can be combined into larger heterostructures through sequential assembly, with larger-scale structures lending themselves to conventional stamping technologies. In all cases, this adaptable soft fabrication process can be directed toward creating mechanically and environmentally stable materials that can be integrated readily into larger systems.

Beyond the immediate application of holographic trapping to fabricating quasicrystalline materials, the ability to create and continuously optimize such structures provides new opportunities for studying the dynamics (10) and statistical mechanics (25) of colloidal quasicrystals. The optically generated quasiperiodic potential energy landscapes developed for this study also should provide a flexible model system for experimental studies of transport (26) through aperiodically modulated environments.

We are grateful to Paul Steinhardt, Paul Chaikin and Weining Man for illuminating conversations. Support was provided by the National Science Foundation through Grant Number DMR-0451589.

Multiple-beam holography has been widely used for the realization of photonic quasicrystals with

high rotational symmetries not achievable by the conventional periodic crystals. Accurate control of

the properties of the interfering beams is necessary to provide photonic band-gap structures. Here we

show, by FDTD simulations of the transmission spectra of 8-fold quasiperiodic structures, how the

geometric tiling of the structure affects the presence and properties of the photonic band-gap for low

refractive index contrasts. Hence, we show an interesting approach to the fabrication of photonic

quasicrystals based on the use of a programmable Spatial Light Modulator encoding ComputerGenerated

Holograms, that permits an accurate control of the writing pattern with almost no

limitations in the pattern design. Using this single-beam technique we fabricated quasiperiodic

structures with high rotational symmetries and different geometries of the tiling, demonstrating the

great versatility of our technique.

Keywords: computer holography, photonic band-gap materials, microstructure fabrication

________________________________________________________________________________

1. Introduction

Quasicrystals are structures exhibiting long-range

aperiodic order and rotational symmetry [1-3].

Mesoscale quasicrystals may possess photonic

bandgaps (PBGs) [4-6] that are more isotropic than

in conventional photonic and, hence, the PBG

becomes more spherical leading to interesting

properties of light transmission [7], wave guiding

and localization [8], increasing the flexibility of

these materials for many photonic applications, also

due to the possible presence of many non equivalent

defect sites. To construct two-dimensional (2D) or

three-dimensional (3D) quasicrystals is a very

difficult task. Previously used to realize periodic

photonic crystals [9,10], holographic lithography

was recently proposed and used to realize

quasicrystals too at the mesoscale [11-15]. The

holographic lithography is based on the interference

pattern of many coherent light beams, that are

usually obtained by splitting a single laser beam by

suitable grating [11,13], prism [15] or dielectric

beam splitters [16], in a single or multiple exposure

process [12]. Realizing a quasicrystal structure,

exhibiting N-fold symmetry [14], requires to control

the amplitude and phase of N interfering laser

beams, leading to several difficulties. By controlling

the relative phases of the interfering beams,

different geometries in the tiling of the dielectric

medium (typically rods, in a binary pattern,

corresponding to the maxima positions of the light

distribution) are achievable [15]. Moreover,

aperiodic structures cannot be realized even in

principle by multiple-beam interference, as for

instance the Thue-Morse structure, defined by

recursive substitutional sequences [17-20]. Onedimensional

and two-dimensional Thue-Morse

structures are known to exhibit PGB with interesting

omnidirectional reflectance [21-24].

In this communication, we demonstrate the

importance of the tiling geometry in the

arrangement of a quasicrystal structure. We show,

by FDTD (Finite Difference Time Domain)

simulations of the transmission spectra in several 8-

fold quasiperiodic patterns, the influence of different 

2

building tile geometries on the photonic band-gap

for low refractive index contrasts. This demonstrates

the importance of an accurate control of the writing

pattern to produce feasible photonic band-gap

structures with low refractive index materials. The

typical interference patterns obtained changing the

relative phases of eight interfering light beams in a

multiple-beam process (examples are shown in [15])

produce structures with important differences in the

dielectric distribution with respect to the standard

Ammann-Beenker octagonal tiling of space with

“squares” and “rhombuses” of equal side lengths a

[25,26]. We chose the octagonal case because its

building tile is easier to analyze with respect to

other structures like Penrose or dodecagonal. We

analyze and compare both the structures (octagonal

and 8-fold interferential pattern with different unittiles)

to provide a comprehension of the behavior of

the photonic band-gap with respect to the building

tile, the variation of the refractive index difference

Δn and the filling factor (high dielectric constant

area to overall area ratio).

Moreover, in this communication, we report the

fabrication of several 2D quasicrystals in the

mesoscale range, using a single-beam technique

based on the spatial modulation of the optical beam

by means of Spatial Light Modulator (SLM) and

Computer-Generated Hologram (CGH). This

method was discussed in details in a previous work

[27]. Here we show interesting results from the

FDTD simulations about quasiperiodic structures

that are difficult to achieve by multiple-beam

holography, but achievable with the SLM-CGH

technique. That being so, in this work we review the

most important aspects of our holographic technique

and the results already obtained. In fact, this method

permits to control with high accuracy the properties

of the required photonic structure achieving the

desired dielectric distribution. We were able to

produce, with single-beam optical setup, 2D

quasiperiodic patterns of rotational symmetry as

high as 23-fold, and aperiodic patterns that cannot

be realized using N-beam interference whatever

large may be N, as for instance the 2D Thue-Morse

pattern. These results demonstrate well the potential

of our single-beam technique. Our structures were

realized by induced photo-polymerization of liquid

crystal-polymer composites. Holographic Polymer

Dispersed Liquid Crystals (H-PDLCs) are materials

that provide good mechanical and optical properties

and can be switched by applying moderate external

electric fields [16]. Nevertheless, the SLM-CGH

technique might be applied to other kinds of

photosensitive materials.

2. FDTD analysis, results and discussion

The octagonal structure analyzed in this paper was

supposed made of dielectric rods with the AmmannBeenker

tiling of space, in air. The positions of the

cylinders of radius r are coincident with the vertices

of “squares” and “45° rhombuses” with sides of

equal length a. This structure presents a complete

PBG with a very low threshold value for the

refractive index difference (Δn=0.26) between high

and low dielectric materials, whereas the gap width

to midgap ratio becomes close to 5% for Δn=0.45

[28]. Therefore, optoelectronics devices based on

the octagonal photonic quasi-crystal (PhQC)

promise to be realized in silica, a very common

telecommunication optical material, or even in soft

materials like polymer. Permitting to record largearea

photonic quasicrystals in photosensitive

materials and exploiting, typically, such substrates,

holographic lithography, therefore, represents an

important fabrication technique. The writing pattern

of light is usually obtained as multiple-beam

interference. In fact, the interference irradiance

profiles I(r), according to [11,15]

1 1

( ) exp[ ( ) ( )],

N N

mn m n m n

m n

I AA i i ϕ ϕ ∗

= =

r k kr = − ⋅+ − ∑ ∑ (1)

where Am, km, φm, are the amplitudes, the wave

vectors and the initial phases of the interfering

beams, respectively, give quasiperiodic distributions

of the dielectric material in the recording medium.

Depending on the threshold level of the

photosensitive matter and the exposure time, the

filling factor may have different values. Typically,

the maxima positions of the light pattern correspond

to the high dielectric regions, that, usually, can be

approximated with a structure of dielectric rods in

air (or other materials). The number N of the

interfering beams determines the order of the

rotational symmetry of the quasicrystal pattern [29].

The wave vectors of the interfering beams are given

by [15]

22 2 m (sin( )sin ,cos( )sin ,cos ), nm m

N N

ππ π θ θ θ

λ

k = (2)

where the km, m=(1, …, N), are oriented at angle θ

with respect to the longitudinal z-direction, and are

equally distributed along the transverse (x, y)-plane;

n is the average refractive index of the

photosensitive mixture, and λ is the common

wavelength of the beams. By adjusting the

parameters in equation (1), different interference

patterns can be obtained (figure 1). By changing the

relative initial phases of the beams, different spatial

3

Figure 1. (a) Calculated quasiperiodic irradiance profile (IP) with 8-fold rotational symmetry obtained from phase

values φi = 0, for i ={1,…,8}, say, 8-fold(A) pattern; (b) calculated 8-fold IP from phase values φ1 = φ5 =0, φ2 = φ4 = φ6

= φ8 = π/2, φ3 = φ7 = π, say, 8-fold(B) pattern. (c)-(e) Calculated IPs from 9-, 10-, 12-beam interference, respectively.

arrangements of the rods in air may be realized,

depending on the resulting interferential profiles.

We compared the octagonal pattern with the 8-fold

interferential structures obtained for two different

sets of the initial phases. For the first pattern, say, 8-

fold(A), the phases were supposed to be all equal,

that is φ1 = …= φ8 =0, whereas for the second

pattern, say, 8-fold(B), the phases were periodically

shifted of π/2, that is φ1 = φ5 =0, φ2 = φ4 = φ6 = φ8 =

π/2, φ3 = φ7 = π.

Figure 2. (a) dielectric structure of rods from 8-fold(A)

pattern; (b) dielectric structure of rods from 8-fold(B)

pattern; (c) octagonal structure of rods with “squarerhombus”

tile of equal side lengths a.

The intensity profiles calculated for both the 8-

fold(A) and 8-fold(B) structures are shown in figure

1-(a) and 1-(b), respectively. The corresponding

patterns of rods are shown in figure 2-(a) and 2-(b),

respectively. These structures are obtained by

positioning circular dielectric rods of radius r in the

maxima of the interferential intensity pattern IP of

figure 1-(a) and 1-(b), respectively. In figure 2-(c),

the octagonal structure of rods with the AmmannBeenker

tiling of space is depicted. The side length

of the unit-tile of space is a for the octagonal pattern

in 2-(c). Due to the non-geometric procedure of

building the structures depicted in figures 2-(a) and

2-(b), we found more convenient to define as

characteristic length of the patterns a new

parameter, that is the average distance ad between

neighbouring rods along the x-direction.

The 2D finite difference time domain (FDTD)

method with uniaxial perfectly matched layer

(PML) boundary conditions was used in all

simulations. The photonic quasi-crystals examined

were non-periodic in the translational direction and

the supercell approximation, necessary in these

cases, required very long computational time [30].

The FDTD technique, instead, was faster and very

accurate. We employed this approach to obtain

transmission information, through the x-y plane

(figure 2), as a function of propagation direction,

wavelength and polarization. A Gaussian time-pulse

excitation was placed in several points of the

structure (in different simulations). The pulse was

wide enough in frequency domain to cover the range

of frequencies of interest. Several detectors were

placed in particular positions and allowed us to store

field components. Their positions were chosen to

cover the angular range related to the 8-fold

rotational symmetry (45°) and the mirror symmetry

with respect to a line of 22.5° in each 45° sector

[31], with an angular separation from 5° to 15°.

After a sufficiently large time of calculation, the

field was Fourier-transformed to calculate the

transmission spectrum with very high frequency

resolution. The corresponding wavelength range

was (0.1-6.0)μm with a resolution of δ=5.0×10-4

μm.

The transmission coefficient through the system was

calculated for different values of the dielectric

constant of the rods in air, for both polarization TM

(electric field Ez parallel to the rod axis) and TE

(magnetic field Hz parallel to the rod axis) [5], for

each detector, that is for different propagation

directions of the time-pulse excitation. The

discretization grid provided a minimum of 100 grid

points per free space wavelength. The transmission

coefficient, normalized with respect to the incident

power of the source, was calculated as a function of

the filling fraction r/ad, given as the rod radius to

average distance ratio. The parameter r/ad is related

to the filling factor and it was varied in order to

maximize the gap width of the octagonal structure

for fixed refractive index difference Δn=0.65. We

found that interesting values were in the range r/ad ≈

(0.15 – 0.35), with a maximum band width at r/ad ≈

0.24 (r/a ≈ 0.34). The value of r/ad was held fixed in

all simulations analyzed here for comparison.

Figure 3. Transmission spectra (TM polarization) of the

octagonal pattern (Ammann-Beenker tiling) for several

values of the refractive index difference Δn.

The data collected from the detectors placed at

different positions and angular orientations had the

same overall shape for all the transmission spectra

we investigated here, demonstrating the isotropy of

the structures with respect to the propagation

direction of the excitation source and, hence, the

existence of a complete photonic band-gap.

In figure 3, the transmission spectra of the

octagonal pattern of figure 2-(c) is shown for TM

polarization as a function of the refractive index

difference. The spectra are shifted in the vertical

direction to permit comparison. We see that an

index difference of Δn ≈ 0.3 is needed to form the

band-gap at the free space mid-gap wavelength

λm=1.62μm (corresponding to the tile length

a=0.93μm and the average distance ad=1.0μm), with

a gap width to midgap ratio Δλ/λm of 0.3%. The

PBG had Δλ/λm=6.3% at λm=1.74μm for Δn=0.5

and increased up to Δλ/λm=22.6% at λm=2.21μm for

 Δn=1.0 (not shown in figure 3). The mid-gap free

space wavelength λm varied with Δn because the

increasing of the index difference corresponds to

increasing of the average refractive index.

We did not find a clear PBG in the octagonal

pattern of figure 2-(c) for the TE polarization. Our

simulations showed, in fact, only a very narrow

PBG with a relative width <1% for Δn=0.9 around

1.48μm, whereas in other regions of the spectra the

possible presence of a band-gap was completely

hidden by structures of multiple peaks that were

related, probably, to localized modes (not shown

here). Moreover, in the case of the TE polarization,

the data collected from the detectors placed at

different angular orientations demonstrated a

dependence from the propagation direction.

In figure 4-(a) and 4-(b), respectively, the

transmission coefficients in TM polarization of the

8-fold(A) and 8-fold(B) patterns (see figure 2a and

2b) are shown, for a refractive index difference

Δn=1.5. The two structures presented important

differences. The pattern (A) had a PBG of

Δλ/λm=19% around λm=3.75μm, whereas the

pattern (B) had two band-gaps, one with a relative

width Δλ/λm=13% at λm=1.93μm and the other of

12% at 3.30μm. We analyzed the same structures

for several values of the index contrast and TM

polarization. We found that, in comparison with the

octagonal structure, an index contrast larger of about

a factor two was needed to open a PBG of

comparable width to mid-gap ratio. The pattern (A)

had a larger band-gap, but for longer wavelengths

(λm=3.75μm) with respect to the first band-gap of

the pattern (B) and with respect to the PBG of the

octagonal pattern of figure 2-(c), that had λm=2.6μm

for Δn=1.5 (not shown in figure 3). On the other

hand, the 8-fold(B) pattern, even with a smaller

PBG, could work at shorter wavelengths

(λm=1.93μm). By changing the filling fraction the

transmission spectra preserved the overall shape.

Transmission (TM)

Only a shift in the mid-gap wavelength was

observed.

Figure 4. (a) Transmission spectrum (TM polarization)

for the 8-fold(A) pattern with refractive index difference

Δn=1.5; (b) Transmission spectrum (TM polarization) for

the 8-fold(B) pattern with Δn=1.5.

Very interesting are the results related to the

calculation of the transmission spectra for the TE

polarization. In figure 5-(a) and 5-(b), the

transmission information in TE polarization related

to the 8-fold(B) pattern, calculated supposing a

refractive index difference Δn=0.6 and Δn=0.4,

respectively, are shown. Two PBGs, independent

from the propagation direction, were present, one

for Δn=0.4 with a relative width Δλ/λm=32% at the

mid-gap free space wavelength λm=1.25μm and the

other for Δn=0.6 with a relative width

Δλ/λm=31% at λm=1.30μm: very short wavelengths

in both cases compared to the other patterns. The

peaks around 1.3μm were probably due the

existence of localized modes. With the increasing of

the refractive index difference only a shift in the

mid-gap wavelength was observed, up to 2.25μm

for Δn=1.6. The 8-fold(A) pattern, on the other

hand, did not present low index PBG for TE

polarization (not shown here) as is also for the

octagonal structure with the Ammann-Beenker

tiling.

The behaviour of the 8-fold(B) pattern

represents, in our opinion, a surprising result

promising the implementation of reliable low index

contrast PhQC devices in polymeric substrates.

Figure 5. (a) Transmission spectrum (TE polarization)

for the 8-fold(B) pattern with refractive index difference

Δn=0.6; (b) Transmission spectrum (TE polarization) for

the 8-fold(B) pattern with Δn=0.4.

We analyzed also the complementary dielectric

structures of the 8-fold(A) and (B) patterns, that is

circular rods of the low index material (air)

embedded in the high index substrate. Also in this

case no band-gap was observable for the low

refractive index contrasts examined. Nevertheless,

the simulations were performed only for TM

polarization in this case.

3. Holographic patterning and fabrication

As demonstrated in the previous section, accurate

control of the dielectric arrangement is fundamental

to provide the desired low index PBG in

quasicrystal structures. The interferential structure

designated as 8-fold(B) might be a good candidate

for this purpose. For this reason, in this section, we

review the Computer-Generated Holography

technique, a single-beam technique developed in a

previous work [27] that permits to fabricate almost

any kind of structure, interference-based or not, with

high reproducibility and very accurate control of the

light intensity distribution in the writing process.

Computer-generated holography is an attractive

technique that allows to create, with almost any

design, two-dimensional, even three-dimensional,

Transmission (TM)

spatial distributions of the optical beam intensity by

controlling the phase profile of a laser impinging on

a Diffractive Optical Element (DOE) [32,33]. In our

work, we implemented the DOE by means of a

programmable liquid crystal SLM that can encode

the CGH in its LC-display. While in the standard

multi-beam holography is quite difficult to work

with a large number of beams [29], with the CGHSLM

it is possible to encode the desired spatial

distribution in a single beam. Although limited from

the pixel resolution of the SLM, our CGH-SLM

technique alleviates the great difficulty in

maintaining phase coherence of the interfering

beams between subsequent exposures, providing

high reproducibility of the resulting structures.

Moreover, the real-time switching from one pattern

to the other is easily accomplished without changing

the optical alignment. We adopted a liquid crystal

spatial light modulator HoloEye LC-R 3000,

permitting intensity images of 256 grey levels with a

maximum resolution of 1200×1920 pixels. The

CGHs corresponding to the optical phase profiles to

be added to the incident beam have been addressed

to the SLM via computer. By using an iterative

algorithm it is possible to create the desired

irradiance pattern in the Fourier plane of the CGH

[32,34,35]. Alternatively, we encoded the desired

intensity pattern directly into a phase-only CGH

[36], with a “direct imaging” method, discussed in

details in [27].

The spatial light modulator LC-R 3000 (for

visible light at λ=532nm) had a pixel pitch of

9.5µm. We used a mixture of the monomer

dipentaerythrol-hydroxyl-penta-acrylate DPHPA

(60.0% w/w), the liquid crystal BLO38 by Merck

(30.0% w/w), the cross-linking stabilizer monomer

N-vinylpyrrolidinone (9.2% w/w) and a mixture of

the photoinitiator Rose Bengal (0.3% w/w) and the

co-initiator N-phenylglycine (0.5% w/w) [34,37,38].

The polymer had a refractive index np=1.530,

whereas the LC BLO38 had an ordinary refractive

index no=1.527 and an extraordinary refractive

index ne=1.799. The average refractive index was

estimated to be n∼1.57. Although the refractive

index contrast achievable with such a mixture,

usually, is Δn ∼ 0.2 - 0.3, the liquid crystal content

can be removed to increase the index difference up

to ∼ 0.5. Different mixtures or materials can be used

in order to increase the amplitude of the dielectric

modulation.

Here we review the results obtained with our

technique [27]. We realized quasi-periodic patterns

with 8-fold rotational symmetry, in particular the 8-

fold(A) and 8-fold(B) pattern described in the

previous section (see figure 1a-2a and 1b-2b,

respectively). Furthermore, we were able to realize

other two-dimensional quasicrystal structures with

9-, 10-, 12-, 17-, 23-fold rotational symmetry, and

the aperiodic structure based on the Thue-Morse

sequence [17]. Figures from 6-(a) to 6-(h) refer to

these structures [39]. Three insets are shown in each

figure, consisting of: the calculated Irradiance

Profile (IP) sent to our SLM and then imaged at the

sample position; the calculated 2D Fourier

Transform (FT) of this irradiance pattern; and the

experimental Diffraction Pattern (DP) produced by

the written structure. These structures are twodimensional

phase gratings in which the modulation

profile of the average refractive index is not easily

accessible with the scanning electron microscopy.

The similarity between the positions and cone angle

of the spots in the calculated Fourier transform FT

and the observed DP doubtless substantiates the

presence of the index profile and the good quality of

the samples. The DPs show the expected N-fold

symmetry having N points for even symmetry and

2N points for odd symmetry. We estimated the size

d of the “self-similarity cell” (that is the selfrepeating

basic structure observable in the

calculated intensity profiles) from the magnitudes of

the basic reciprocal vectors in the diffraction

patterns, that are related to sensible lengths of the

crystal structures, such as the tile side [16,40].

Although the size d reported in figure 6 (IP insets)

was in the range between 4.8 and 8.6µm, increasing

with the order of the symmetry, the typical

separation between equal dielectric regions in the

fine structure of the self-similarity cell was in the

range of ∼1 - 2µm. In fact, we were able to realize

1D Bragg grating and 2D periodic square lattice

with a pitch of ∼1µm (not shown here). The scale

length of the realized structures depended on the

lateral magnification of the relay lenses [27] (which

also affected the extension of the writing area). The

achievable resolution, lastly, depends on the SLM

pixel size and on the wavelength of the writing light.

Using UV light and state-of-the-art SLM pixel size

[41], the limit in the spatial resolution could be

improved by a factor of 3-4. Anyway, the scalability

of the optical properties permits to use our

quasiperiodic structures to study their optical

behavior. Furthermore, according to the TE

transmission spectra for the 8-fold(B) pattern of

figure 5 presented in the previous section, an

average distance ad∼1.25µm between neighbouring

rods would be sufficient to induce a PBG at

λm=1.55μm. That being so, the SLM-CGH

technique could be employed to produce photonic

quasicrystals for feasible and reliable applications. 

7

Figure 6. (a) Quasiperiodic structure 8-fold(A) with phases φi = 0, for i ={1,…,8}; (b) quasiperiodic structure 8-fold(B)

with phases φ1 = φ5 =0, φ2 = φ4 = φ6 = φ8 = π/2, φ3 = φ7 = π. (c)-(g) Quasiperiodic structure with 9-, 10-, 12-, 17-, 23-

fold rotational symmetry, respectively. (h) Two-dimensional Thue-Morse quasicrystal structure. (a)-(h) Top left inset:

calculated irradiance profile (IP); top right inset: 2D Fourier transform (FT) of the irradiance profile; bottom inset:

observed diffraction pattern (DP); d estimates the self-similarity cell size of the structures we fabricated.

4. Conclusion and discussion

In conclusion, we showed, by FDTD simulations of

the transmission spectra of 8-fold quasicrystals, the

influence of different building tile geometries on the

photonic band-gap, demonstrating the importance of

an accurate control of the writing pattern to produce

feasible photonic band-gap structures with low

refractive index materials. Besides the octagonal

pattern with the Ammann-Beenker geometric tiling,

the 8-fold(B) pattern of figure 2-(b) was found very

interesting among the interferential structures

because it permits to obtain a large PBG for an

index contrast as low as 0.4, as shown in figure 5.

For this purpose we presented a fabrication

technique that alleviates many drawbacks of the

multi-beam holographic lithography that is typically

employed for the PhQCs realization. We used

Computer-Generated Holograms to drive a liquid

crystal Spatial Light Modulator. This permits to

write arbitrary structures with simple and reliable

optical setup. The writing intensity pattern can be

modified by real time change of the CGH addressed

to the programmable SLM. Accurate control of the

pattern designs and of the tiling geometry can be

achieved without mechanical motion and optical

realignment. Together with the 8-fold quasicrystal

patterns discussed in the section 2, we were able to

obtain, with a single-beam technique, quasiperiodic

structures with unprecedented rotational symmetries

up to 17- and 23-fold and two-dimensional ThueMorse

structures too. The possibility of a full

characterization of quite interesting aperiodic

structures not achievable by multiple beam

holography even in principle (like two-dimensional

Thue-Morse), and that have been studied until now

only theoretically [24], is open. Our structures were

written into polymeric liquid crystal films, so to

permit switching by external fields [16]. However,

our holographic technique could be applied, in

principle, to any photosensitive material (e.g. with a

larger index contrast), or to produce patterned masks

and templates for use in lithography of hard

materials [9-16,29]. We expect that the SLM based

CGH technique may have a large impact on the

production of complex photonic structures.

References

[1] Levine D and Steinhartdt P J 1984 Quasicrystals: a

new class of ordered structures Phys. Rev. Lett.

53 2477-80

[2] Steinhartdt P J and Ostlund S 1987 The Physics of

Quasicrystals (River Edge, NJ: Word Scientific)

[3] Stadnik Z M 1999 Physical Properties of

Quasicrystals (New York: Springer)

[4] Zoorob M E, Charlton M D B, Parker G J,

Baumberg J J and Netti M C 2000 Complete

photonic bandgaps in 12-fold symmetric

quasicrystals Nature 404 740-3

[5] Chan Y S, Chan C T and Liu Z Y 1998 Photonic

band gaps in two-dimensional photonic

quasicrystals Phys. Rev. Lett. 80 956-9

[6] Zhang X, Zhang Z Q and Chan C T 2001 Absolute

photonic band gaps in 12-fold symmetric

photonic quasicrystals Phys. Rev. B 63

081105/1-4

[7] Cheng S S M, Li L, Chan C T and Zhang Z Q

1999 Defect and transmission properties of twodimensional

quasiperiodic photonic band-gap

systems Phys. Rev. B 59 4091-99

[8] Jin C, Cheng B, Man B, Li Z, Zhang D, Ban S and

Sun B 1999 Band gap and wave guiding effect

in a quasiperiodic photonic crystal Appl. Phys.

Lett. 75 1848-50

[9] Campbell M, Sharp D N, Harrison M T, Denning

R G and Turberfield A J 2000 Fabrication of

photonic crystals for the visible spectrum by

holographic lithography Nature 404 53-6

[10] Berger V, Gauthier-Lafaye O, Costard E 1997

Fabrication of a 2D photonic band gap by a

holographic method Electr. Lett. 33 425-6

[11] Wang X, Ng C Y, Tam W Y, Chan C T and Sheng

P 2003 Large-area two-dimensional mesoscale

quasicrystals Adv. Mater. 15 1526-28

[12] Gauthier R C Ivanov A 2004 Production of

quasicrystal template patterns using a dual beam

multiple exposure technique Opt. Express 12

990-1003

[13] Wang X, Xu J, Lee J C W, Pang Y K, Tam W Y,

Chan C T and Sheng P 2006 Realization of

optical periodic quasicrystals using holographic

lithography Appl. Phys. Lett. 88 051901/1-3

[14] Gorkhali S P, Qi J and Crawford G P 2005

Electrically switchable mesoscale Penrose

quasicrystal structure Appl. Phys. Lett. 86

011110/1-3

[15] Yang Y, Zhang S and Wang G P 2006 Fabrication

of two-dimensional metallodielectric

quasicrystals by single-beam holography Appl.

Phyis. Lett. 88 251104/1-3

[16] Gorkhali S P, Qi J and Crawford G P 2006

Switchable quasicrystal structures with five-,

seven-, and ninefold symmetries J. Opt. Soc.

Am. B 23 149-158

[17] Macià E 2006 The role of aperiodic order in

science and technology Rep. Prog. Phys. 69 397-

441

[18] Axel F and Terauchi H 1991 High-resolution Xray-diffraction

spectra of Thue-Morse GaAsAlAs

heterostructures: towards a novel

description of disorder Phys. Rev. Lett. 66 2223-

26

[19] Axel F and Terauchi H 1994 Axel and Terauchi

reply Phys. Rev. Lett. 73 1308

[20] Kolar M 1994 High-resolution X-ray-diffraction

spectra of Thue-Morse GaAs-AlAs

heterostructures Phys. Rev. Lett. 73 1307

[21] Dal Negro L, Stolfi M, Yi Y, Michel J, Duan X,

Kimerling L C, LeBlanc J and Haavisto J 2004

Photon band gap properties and omnidirectional

reflectance in Si/SiO2 Thue-Morse quasicrystals

Appl. Phys. Lett. 84 5186-88

[22] Lee H Y and Nam G Y 2006 Realization of

ultrawide omnidirectional photonic band gap in

multiple one-dimensional photonic crystals J.

Appl. Phys. 100 083501/1-5

[23] Lei H, Chen J, Nouet G, Feng S, Gong Q and Jiang

X 2007 Photonic band gap structures in the

Thue-Morse lattice Phys. Rev B 75 205109/1-10

[24] Moretti L and Mocella V 2007 Two-dimensional

photonic aperiodic crystals based on ThueMorse

sequence Opt. Express 15 15314-23

[25] Zhou G and Gu M 2007 Photonic band gaps and

planar cavity of two-dimensional eigthfold

symmetric void-channel photonic quasicrystal

Appl. Phys. Lett. 90 201111/1-3

[26] Socolar J E S 1989 Simple octagonal and

dodecagonal quasicrystals Phys. Rev. B 39

10519-10551

[27] Zito G, Piccirillo B, Santamato E, Marino A,

Tkachenko V and Abbate G 2008 Twodimensional

photonic quasicrystals by single

beam computer-generated holography Opt.

Express 16 5164-70

[28] Romero-Vivas J, Chigrin D N, Lavrinenko A V

and Sotomayor Torres C M 2005 Resonant adddrop

filter based on a photonic quasicrystal Opt.

Express 13 826-35

[29] Mao W, Liang G, Zou H, Zhang R, Wang H and

Zeng Z 2006 Design and fabrication of twodimensional

holographic photonic quasi crystals

with high-order symmetries J. Opt. Soc. Am. B

23 2046-50

[30] Gauthier R C and Mnaymneh K 2005 Photonic

band gap properties of 12-fold quasi-crystal

determined through FDTD analysis Opt. Express

13 1985-98

[31] Hase M, Miyazaki H, Egashira M, Sninya N,

Kojima K M and Uchida S 2002 Isotropic

photonic band gap and anisotropic structures in

transmission spectra of two-dimensional fivefold

and eightfold symmetric quasiperiodic photonic

crystals Phys. Rev. B 66 214205/1-7

[32] Soifer V A 2002 Methods for Computer Design of

Diffractive Optical Elements (New York: John

Wiley & Sons, Inc.)

[33] Lee G, Song S H, Oh C H and Kim P S 2004

Arbitrary structuring of two-dimensional

photonic crystals by use of phase-only Fourier

gratings Opt. Lett. 29 2539-41

[34] Zito G, Piccirillo B, Santamato E, Marino A,

Tkachenko V and Abbate G 2007 Computergenerated

holographic gratings in soft matter

Mol. Cryst. Liq. Cryst. 465 371-8

[35] Goodman J W 1996 Introduction to Fourier Optics

(New York: McGraw-Hill) 

[36] Davis J A, Cottrell D M, Campos J, Yzuel M J and

Moreno I 1999 Encoding amplitude information

onto phase-only filters Appl. Opt. 38 5004-13

[37] Sutherland R L, Natarajan L V, Tondiglia V P and

Bunning T J 1993 Bragg gratings in an acrylate

polymer consisting of periodic polymerdispersed

liquid-crystal planes Chem. Mater. 5

1533-38

[38] Vita F, Marino A, Tkachenko V, Abbate G,

Lucchetta E D, Criante L and Simoni F 2005

Visible and near-infrared characterization and

modeling of nanosized holographic-polymerdispersed

liquid crystal gratings Phys. Rev. E 72

011702/1-8

[39] Figures 6(a)-(h) are taken from figures 1 and 2 of

our previous work [27] with the permission of

the publisher

[40] Kaliteevski M A, Brand S, Abram R A, Krauss T

F, Millar P and De La Rue R M 2001 Diffraction

and transmission of light in low-refractive index

Penrose-tiled photonic quasicrystals J. Phys.:

Condens. Matter 13 10459-70

[41] Apter B, David Y, Baal-Zedaka I and Efron U

2007 Experimental study and computer

simulation of ultra-small-pixel liquid crystal

device 11th Meeting on Optical Engineering and

Science (Israel, Tel Aviv) 

x

x

I believe this material could have been part of the technology used to place "fake airplanes" in the sky on 9/11.

- Christian Hampton - Wolf Clan Media