Fast Light and Focusing in 2D Photonic Quasicrystals

Transcription

1 PIERS ONLINE, VOL. 5, NO. 5, Fast Light and Focusing in 2D Photonic Quasicrystals Y. Neve-Oz 1, T. Pollok 2, Sven Burger 2, Michael Golosovsky 1, and Dan Davidov 1 1 The Racah Institute of Physics, The Hebrew University of Jerusalem Jerusalem 91904, Israel 2 Zuse Institute, Takustrasse 7, Berlin 14195, Germany Abstract We present numerical studies of the electromagnetic wave propagation in a metamaterial built from dielectric rods. The rods were arranged in a quasicrystalline lattice based on 10-fold Penrose tiling. We find wide isotropic band gaps and localized states, as expected in a quasicrystalline lattice. Our most important finding is the presence of the band gap edge states that are characterized by a very small refractive index (fast light). We use this phenomenon to design a focusing device a plano-concave lens. 1. INTRODUCTION Photonic crystals and metamaterials attract much interest due to their ability to manipulate light in unique ways that are impossible to achieve using uniform materials. For periodic crystals, the highest level of rotational symmetry does not exceed six, therefore it is almost impossible to avoid significant anisotropy in their electromagnetic properties. This anisotropy precludes wide omnidirectional band gaps. In contrast to periodic photonic crystals, quasicrystals have higher local rotational symmetry, hence their optical properties are more isotropic and they can exhibit wide omnidirectional band gaps [1 6]. Previous studies of photonic quasicrystals focused mostly on localized states inside the band gap since this feature is crucial for lasing [7 10]. In this work, we study electromagnetic wave propagation through planar photonic quasicrystals and focus on the band gap edge states. In distinction to previous works [11 14] that focused on the states with a low group velocity, we find propagating states with a very high phase velocity fast light. We demonstrate how this phenomenon can be used for focusing with a plano-concave lens. So far, focusing with such device was thought to be associated with negative refraction [15 17]. 2. METHODOLOGY Our seed structure is the 10-fold Penrose tiling shown in Fig. 1. The dielectric rods are mounted in the vertices of this tiling. Fig. 1(c) shows that this arrangement indeed has 10-fold symmetry. To simulate electromagnetic wave propagation through this array we used the software JCMsuite [18] that is based on a time-harmonic, adaptive, higher-order finite-element method. Our numerical model uses 520 dielectric rods arranged in a quasicrystalline lattice as shown in Fig. 1. The rod diameter is d = 2.5 mm, the distance between the rods is a = 5.4 mm, in such a way that the filling factor is 0.2. The dielectric constant of the rods is ɛ r = 12 (this corresponds to Si). The size of the array is 160 mm 80 mm and the plane wave impinges on the wide side of the array, whereas the electric field is parallel to the rod s axes. The model assumes transparent boundary conditions at all boundaries of the computational domain. 3. RESULTS Figure 2 shows electromagnetic wave transmission through our array. The incident T M wave propagates from top to bottom. We considered transmission at different incident angles. To this end the direction of propagation was kept fixed and the crystalline structure was rotated around the vertical axis located in the lower left corner of the model shown in Fig. 1. The rectangular shape of the array was preserved. Transmission plot demonstrates two band gaps: at GHz and at GHz. The band gap positions do not depend on the angle of incidence, as expected for quasicrystals. The band structure found in our work should be compared to that found in Ref. [6] who considered a quasicrystalline array of dielectric rods with almost the same sizes, symmetry and filling factor. The two works used different numerical methods: JCM software based on finite-element method (this work) as compared to direct solution of Maxwell equations using a plane wave method [6]. It is rewarding to observe that both works found a very similar band structure.

2 PIERS ONLINE, VOL. 5, NO. 5, (a) (b) (c) Figure 1: (a) A 10-fold Penrose tiling. The circles denote the vertices. (b) Photonic quasicrystal based on the dielectric rods mounted at the vertices of the tiling shown in (a). (c) Fourier transform of the configuration shown in (b). Note a 10-fold symmetry. Figure 2: Millimeter-wave transmission through the photonic quasicrystal shown in Fig. 1 at different angles of incidence. Transmission spectrum is nearly independent of the incident angle. The first band gap appears at GHz and the second band gap appears at GHz. The inset shows expanded view of the frequency range where fast light appears. We performed field mapping for several chosen frequencies in order to determine the shape of the wave front and to calculate the phase velocity. At the high frequency edge of the second band gap (Fig. 2, inset) the electromagnetic wave propagating inside the array is characterized by especially long wave length. The long wavelength corresponds to a very high phase velocity (fast light). In what follows we studied this phenomenon in detail.

3 PIERS ONLINE, VOL. 5, NO. 5, Figure 3 shows spatial distribution of the magnitude and phase of the electric field for one of the frequencies exhibiting fast light. The phase distribution in the array is nearly uniform (Fig. 3(b)), as if it were a standing wave. Fig. 3(c) indicates that the field magnitude exhibits a strong maximum inside the array suggesting that this is a localized resonant state. In what follows we study transmission through the array in which there is one localized state. Fig. 4 shows that the transmission has a sharp, resonant dependence on frequency. The transmission at resonance is close to 3.5 dbm and the Q-factor is about Fig. 4 shows that the maximum field magnitude inside the array closely follows the transmission curve and this is an additional indication of the resonant transmission. Our results suggest that the fast light is associated with the resonant transmission through one or several resonance states. Similar states were found in Refs. [6, 14, 20]. In particular, the Ref. [14] showed that the propagating states at the band gap edge evolve in abrupt way when the sample size is changed a clear indication that these states are associated with transmission through localized states. This is very similar to resonant tunnelling through tunnel barriers in electronic transport in solids [21]. The fast light phenomena can be used for device fabrication. To achieve focusing we constructed a plano-concave lens with the width 300 mm and the curvature R = 180 mm (Fig. 5). The plane wave impinges on the plane side of the lens. Because of the very long wavelength inside the lens, the phase of the wave at the concave exit face remains the same (Fig. 5(b)). Hence, this lens transforms the plane wave front of the incident wave to the wave front determined by the exit face of the lens. (a) (b) (c) Figure 3: Plane mm-wave (32.6 GHz) propagation through the photonic quasicrystal shown in Fig. 1. (a) Numerical model. (b) Phase distribution. The wave fronts of the incident wave are clearly seen as horizontal blue lines. The wavelength is equal to the distance between subsequent horizontal lines of the same color. The color in the interior of the photonic crystal slowly changes from the dark-blue at the top to the blue at the bottom. This indicates a very small phase gradient along the sample, i.e., a very long wavelength (fast light). (c) Intensity distribution. Note increased intensity in the central part of the crystal. The intensity of the incident wave is equal to 1.

4 PIERS ONLINE, VOL. 5, NO. 5, Figure 4: Transmission as a function of frequency for the photonic quasicrystal shown in Fig. 3 (red line). The arrow shows the frequency for which the field distribution in Fig. 3 has been obtained. A narrow transmission peak indicates resonant transmission. The blue circles denote the maximum field intensity inside the sample. Note the correlation between the transmission and the maximum field intensity inside the array. (a) (b) (c) Figure 5: Focusing by a plano-concave lens made from the photonic quasicrystal shown in Fig. 1. (a) Design. The number of rods is The blue cross indicates the focus. (b) Phase distribution. Note the plane wave front in the incident wave, almost constant phase inside the lens, and a converging cylindrical wave front in the outgoing wave. (c) Intensity distribution. Note bright red spot in the outgoing wave that indicates focusing. The intensity at the focus is ten times higher than the intensity of the incident wave. This face is curved hence the outgoing radiation is focused to a spot (Fig. 5(c)). The intensity in the focus exceeds the intensity in the incident wave by a factor of ten. To estimate focal length we use the lens-makers equation for a plano-concave lens: 1 f = n 1 (1) R Since n 0, f R (see Fig. 5(c)). 4. DISCUSSION We observe fast light at the high frequency edge of the second band gap, but not at the edges of the first band gap. It was shown earlier [3, 6, 19] that the large omnidirectional band gap in quasicrystals is associated with the short-range electromagnetic interactions, while higher band gaps are associated with long-range interactions. Hence, it is not a surprise that the fast light (very long-range phase coherence) is observed in the second band gap and not in the first. So far,

5 PIERS ONLINE, VOL. 5, NO. 5, narrow beam focusing using quasicrystals was thought to be associated with negative refraction [13] although the later work [12] challenged this view and attributed focusing to short range interactions. In this work, we clearly demonstrate plane wave focusing by a quasicrystalline lens where focusing arises from the long-range interactions. REFERENCES 1. Chan, Y. S., C. T. Chan, and Z. Y. Liu, Photonic band gaps in two dimensional photonic quasicrystals, Physical Review Letters, Vol. 80, , Bayindir, M., E. Cubukcu, I. Bulu, and E. Ozbay, Photonic band-gap effct, localization, and waveguiding in the two-dimensional Penrose lattice, Physical Review B, Vol. 63, , Della Villa, A., S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, Band gap formation and multiple scattering in photonic quasicrystals with a penrose-type lattice, Physical Review Letters, Vol. 94, , Kaliteevski, M. A., S. Brand, and R. A. Abram, Directionality of light transmission and reflection in 2D Penrose tiled photonic quasicrystals, J. Phys. Condens. Matter, Vol. 16, 1269, David, S., A. Chelnokov, and J. M. Lourtioz, Isotropic photonic structures: Archimedean-like tilings and quasi-crystals, IEEE Journal of Quantum Electronics, Vol. 37, , Wang, K., Structural effcts on light wave behavior in quasiperiodic regular and decagonal Penrose-tiling dielectric media: A comparative study, Physical Review B, Vol. 76, , Man, W., M. Megens, P. J. Steinhardt, and P. M. Chaikin, Experimental measurement of the photonic properties of icosahedral quasicrystals, Nature, Vol. 436, , Notomi, M., H. Suzuki, T. Tamamura, and K. Edagawa, Lasing action due to the twodimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice, Physical Review Letters, Vol. 92, , Freedman, B., G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, Wave and defect dynamics in nonlinear photonic quasicrystals, Nature, Vol. 440, 1166, Matsui, T., A. Agrawal, A. Nahata, and Z. Valy Vardeny, Transmission resonances through aperiodic arrays of subwavelength apertures, Nature, Vol. 446, , Zhang, X., Z. Li, B. Cheng, and D. Z. Zhang, Non-near-field focus and imaging of an unpolarized electromagnetic wave through high-symmetry quasicrystals, Optics Express, Vol. 15, 1292, Gennaro, E. D., C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Evidence of local effcts in anomalous refraction and focusing properties of dodecagonal photonic quasicrystals, Physical Review B, Vol. 77, , Feng, Z., X. Zhang, Y. Wang, Z. Y. Li, B. Cheng, and D. Z. Zhang, Negative refraction and imaging using 12-fold-symmetry quasicrystals, Physical Review Letters, Vol. 94, , Lai, Y., Z. Q. Zhang, C. H. Chan, and L. Tsang, Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals, Physical Review B, Vol. 76, , Parazzoli, C. G., R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, Experimental verification and simulation of negative index of refraction using Snell s law, Physical Review Letters, Vol. 90, , Vodo, P., P. V. Parimi, W. T. Lu, and S. Sridhar, Focusing by planoconcave lens using negative refraction, Applied Physics Letters, Vol. 86, , Saado, Y., Y. Neve-Oz, M. Golosovsky, D. Davidov, and A. Frenkel, Photonic crystal superlattice as a left-handed planoconcave lens, Journal of Applied Physics, Vol. 104, , JCMwave GmbH, Haarer Str. 14a, Putzbrunn, Germany. 19. Mnaymneh, K. and R. C. Gauthier, Mode localization and band-gap formation in defect-free photonic quasicrystals, Optics Express, Vol. 15, , Della-Villa, A., S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V.Galdi, Localized modes in photonic quasicrystals with Penrose-type lattice, Optics Express, Vol. 14, , Bending, S. J. and M.R. Beasley, Transport properties via localized states in thin a-si tunnel barriers, Physical Review Letters, Vol. 55, , 1985.