Superprism effect in all-glass volumetric photonic crystals

Transcription

1 OPTO ELECTRONICS REVIEW 20(3), DOI: /s Superprism effect in all-glass volumetric photonic crystals A. FILIPKOWSKI 1,2, R. BUCZYŃSKI *1,3, A.J. WADDIE 2, I. KUJAWA 1, D. PYSZ 1, M.R. TAGHIZADEH 2, and R. STĘPIEŃ 1 1 Institute of Electronic Materials Technology, 133 Wólczyńska Str., Warsaw, Poland 2 School of Engineering and Physical Sciences, Heriot Watt University, Edinburgh EH14 4AS, Scotland, UK 3 Faculty of Physics, University of Warsaw, 7 Pasteura Str., Warsaw, Poland This paper focuses on the superprism effect which can be obtained in low contrast photonic crystals. The modelling is related to the newly developed method for all dielectric photonic crystals. This places material constraints on the simulated crystals which limit the refractive index difference to 0.1 for all glass photonic crystals and 0.6 for air glass structures and forces us to focus on hexagonal lattices. The simulations show the existence of superprism effect in both types of structure for realistic glasses. In both cases various linear filling factors are studied in order to maximize the frequency range of the superprism ef fect. For the air F2 glass structure it reaches normalized frequencies and for the air NC21 glass structure it reaches 0.99 normalized frequencies for TM polarization. For the double glass structures, the largest range is for the F2/NC21 photonic crystal and spans normalized frequencies. In the F2/NC21 crystal the frequency range reaches for TE polarization. Keywords: photonic crystal, superprism effect, photonic band gap. 1. Introduction A photonic crystal (PC) is a periodic structure built from dielectric materials. Usually, it takes form of localized in clusions in a host substrate, with a difference in permittivity and conductivity between materials [1,2]. The first reports concerning PCs were published in the late 1980s [3,4] and since then a significant body of research has been developed covering many different geometries and applications. Mo dern research focuses mostly on 2D crystals, due primarily to their simplified production methods and geometrical sim plifications that can be applied in their computational mod elling. However, there are several works related to 3D PCs [5]. The optical properties of PCs are defined by the lattice type (hexagonal, rectangular), the lattice constant, the diameter of the inclusions and the material properties of the constituent materials. Modification of these parameters changes the spatial properties of the electromagnetic wave propagating through crystal. The best known effect in PCs is the existence of the photonic band gaps (PBG) bands of frequencies that cannot propagate inside the PC [6]. The effect that this article focuses on is the large dispersion observed at the edge of a PBG. Small changes of angle (or wavelength of incident light) result in a large change in the direction of propagation of the wave inside the PC. This effect is known as the superprism effect and was first shown by Russel and Zengerle [7,8]. The superprism effect has * e mail: rbuczyns@igf.fuw.edu.pl been considered for use in various applications such as beam steering [9], subwavelength imaging [10], compact frequency demultiplexers [11] and in nonlinear optics [12]. In this paper we show that the modelled results of the superprism effect are strongly correlated with technological research devoted to development of all dielectric PC. We are aware that the refractive index contrast is lower than in case of air semiconductor structures. However, the exis tence of a full photonic band gap is not a necessary condi tion for the superprism effect [13]. Moreover, the consi dered stack and draw technology allows high volume fabri cation at a very low cost comparable to the costs of photonic crystal fibre fabrication. 2. Novel technological approach to development of all-dielectric photonic crystals PCs are developed using lithographic, holographic or self assembly methods [2]. In this paper we consider the use of the stack and draw method widely applied to optical pho tonic crystal fibre development (Fig. 1) [14]. We have recently shown that this method can be successfully applied to the development of PC [15], as well as nanostructured components [16,17]. While using the stack and draw method the PC structure is assembled with individual glass rods or tubes. We use two types of soft glass rods with a high difference between refractive indices. Use of thermally and mechanically matched soft glasses allows us to obtain a relatively high Opto Electron. Rev., 20, no. 3, 2012 A. Filipkowski 267

2 Superprism effect in all glass volumetric photonic crystals Fig. 1. Stack and draw method. contrast when compared to silica glasses [18]. By varying the rod or tube diameter, or by mixing rods and tubes, a wide range of structures can be constructed. Structures constructed in this manner are processed in a fibre drawing tower. The prepared preform is lowered into the furnace inside the drawing tower (Fig. 2) and the drawing process is controlled by varying the furnace temperature and the speed with which a fibre is pulled out of the furnace. This method allows the relatively simple production of large volume crystals (with the diameter in the order of few millimetres and height in the order of metres), and allows for the easy manipulation of the desired crystal structure. This stems from the fact that all changes to the lattice are done during construction of a preform and they are performed on a macroscopic structure, not a microscopic one. The main difference between PC and PCF developed with the same method is related to number of microrods elements assembled in the structure. In the case of PCFs a photonic cladding is created with a few tens or hundreds of elements, which result is a photonic cladding of a diameter of 10 to 80 microns. A PC structure is created with few hun dred thousand elements (3 orders of magnitude more than in case of PCFs). As a result, regular structures with a diameter of the order of a few millimetres can be obtained. For the technological verification of PC development, two pairs of glasses were used F2/NC21 and SK222/Zr3. Since both glasses from each pair are heated simultane ously, matched softening temperatures and thermal expan sion coefficients are important. The expansion coefficients for the first pair of glasses are similar (F C = K 1 ; NC C = K 1 ) ensuring a stable structure during processing. Viscosity graphs for both of these glasses show that at temperatures around 700 C, simultaneous treatment is possible (Fig. 3). However, the large difference between the glass softening (70 C) and Fig. 3. Viscosity of NC 21 and F2 glasses. melting temperatures (230 C) can lead to diffusion between the two glasses. Initial fabrication runs with the F2/NC21 glasses showed that diffusion does indeed occur for this pair and it can be observed as a blurred area at the border between two glasses in scanning electron micrographs (Fig. 4). It is important to note that this method allows only for an estimate of the diffusion to be made and, furthermore, the standard methods for measurement of phase contrast cannot be applied due to the small feature sizes in the PC [19]. F2 is a commercially available lead silicate glass form Schott with n d = 1.619, while NC21 is a silicate glass syn thesized in house at the Institute of Electronic Materials Technology (ITME) in Warsaw, Poland (n d = 1.533). The weight composition of the NC21 glass is: SiO 2 = 55.0%, Al 2 O 3 = 1.0%, B 2 O 3 = 26.0%, Li 2 O = 3.0%, Na 2 O = 9.5 %, K 2 O = 5.5%, As 2 O 3 = 0.8%. Because of the mismatch between the softening and melting temperatures of a F2/NC21 pair, the second pair of glasses was considered. Zr3/XV is barium zirconia boro silicate glass with n d = while SK222 is a soda lime silicate glass with a refractive index of n d = Both glasses are synthesized in house at ITME. The chemical Fig. 2. Schematics of stack and draw fibre production method. Fig. 4. Structure of photonic crystal made from F2 and NC21 glasses. Blurred borders between glasses identify diffusion between glasses. 268 Opto Electron. Rev., 20, no. 3, SEP, Warsaw

3 Fig. 5. Viscosity of Zr3 and SK222 glasses. composition of the Zr3/XV and SK222 glasses are (in weight %): 40.5 SiO 2, 17.5 ZrO 2,10B 2 O 3, 12 BaO, 5 CaO, 12 Na 2 O,3K 2 O and 68.4 SiO 2, 2.4 Al 2 O 3,2B 2 O 3, 12.3 Na 2 O, 0.7 K 2 O, 7.1 CaO, 7.1 ZnO, respectively. Thermal expansion coefficients for both glasses are similar (Zr3/XV C = K 1 ; SK C = K 1 ) and their softening temperatures differ by only 20 C. Viscosity graphs for this pair of glasses show that a simulta neous treatment is possible at a temperature of around 650 C (Fig. 5). The second pair of glasses (Zr3/XV and SK222) is better matched thermally and the diffusion prob lem of the first glass system does not appear (Fig. 6). Since both all glass structures have relatively low re fractive index differences, an air glass structure was also considered for simulation. Air glass structures are more dif ficult to manufacture than double glass ones as even small irregularities used in the glass can cause ruptures in the capi llaries and the subsequent escape of air. This results in an irrevocable closing of the capillary and final irregularities in the produced structure (Fig. 7). Fig. 7. Samples of structure irregularities in air glass photonic crystals. In order to verify the possibility of using the developed material as a PC, we have measured the transmission of a broadband spectrum through the PC samples. For the pur poses of these measurements we used a PC made of NC21 and F2 glasses with the lattice constant = 0.8 μm and the filling factor f = The samples were 4 mm thick with polished surfaces perpendicular to the crystal structure (off plane). As an input source we used a broadband superconti nuum source coupled into a single mode fibre and collima ted with a microscope objective [20]. The output signal behind the test sample was collected using another micro scope objective, coupled into a multimode fibre and detec ted using a spectrometer. Since the intensity of the super continuum source varies with wavelength, the relative trans mission is defined as the normalized difference between spectrum of the supercontinuum source and spectrum trans mitted through the measured PC sample. The normalized transmission is presented in Fig. 8. The transmission results show clearly the existence of photonic band gaps at 572 and 850 nm. Fig. 6. Structure of photonic crystal made from Zr3 and SK222 glasses. Fig. 8. Transmission characteristics of photonic crystal made of F2 and NC21 glasses with the lattice constant = 0.8 μm and the filling factor of Opto Electron. Rev., 20, no. 3, 2012 A. Filipkowski 269

4 Superprism effect in all glass volumetric photonic crystals 3. Superprism effect in photonic crystal The superprism effect is governed by the band structure of a given photonic crystal. A photonic band structure is defined by the dispersion relation between the frequency and the wave vector k. Its geometrical representation is an iso frequency surface for each photonic band. For isotropic materials, these surfaces are spherical, but in a periodic medium their shape is complex. This is caused by the fact that iso frequency surfaces must cross the borders of primi tive cells in reciprocal space at right angles [21]. The energy flow of an electromagnetic wave is perpendicular to the wave front and since the group velocity is the gradient of the dispersion relation, it must be perpendicular to the iso fre quency surface directed towards higher frequencies. It can be shown that the wave and group velocity vectors point to the same direction, e. g., the direction of wave propagation inside the crystal [21]. The second law that governs the superprism effect is the conservation of the wave vector component parallel to the PC border which can be derived from the translational symmetry of the PC. These two facts allow the calculation of the direction of propagation for an electromagnetic wave inside the photonic crystal. The refractive index of the medium changes with the wavelength of electromagnetic wave propagating through the PC. Changes in refractive index mean changes in the iso frequency curves for different wavelengths, which, in turn, mean different propagation directions inside the crys tal [22]. This method is used to obtain the superprism effect based on frequency changes of the propagating wave. This method is useful in telecommunication for wave demultiplexing. The path defined by the conservation of the parallel component of the wave vector is known as the construction line. This generally crosses the iso frequency curve at more than one point but only those that define propagation vector away from the crystal border are viable. For hexagonal lat tices, the construction line crosses the iso frequency curves at different points in subsequent Brillouin zones. From the Bloch function it can be deduced that those cross points can be reduced into the first Brillouin zone [22], increasing the number of viable cross points, and, in turn, increasing the number of directions in which the wave can propagate inside the crystal, which is not a desirable effect. we focus on frequencies around photonic band gaps where only one band for the given frequency can be found. After determining the frequency and corresponding mode, MPB is used to calculate iso frequency surfaces for given para meters. For the first band there is also a frequency range where no other bands are observed, but iso frequency sur faces for the first band are too regular and the superprism effect cannot be obtained. All the simulations are performed with normalized units, however, material parameters of the glass for 850 nm are assumed. The simulations reported below are performed based on the thermally matched glasses with good rheological pro perties, used for PC development as shown in Sect. 2. The Zr3/SK222 PC composed of glasses with refractive indices n 850 = and n 850 = , respectively, shows the re latively low contrast below 0.1. The lattice constant of the fabricated PC is = 1.1 μm and the inclusion diameter is equal to d = 0.7 μm. Simulations showed no in plane PBGs for this structure. The second developed PC was constructed from NC21 and F2 glasses. They have refractive indices n 850 = and n 850 = , respectively, giving a rela tively high contrast above 0.1. For this PC the lattice con stant equals = 0.8 um with linear filling factor f = 0.5. Based on the simulations for the in plane propagation, we conclude that the existing photonic band gaps are not in the vicinity of 850 nm. Based on those results, the first set of simulations were performed for air glass PCs built from F2 glass, as those structures have refractive index contrast of which is the highest possible with the given materials. As expected, the highest range of frequencies for which we can have a viable superprism effect can be seen for the hexagonal structure (Fig. 9). For TM polarization, the frequency range is above 0.1 normalized frequency defined as /, where is defined as the lattice constant for the given PC structure with the linear filling factor between and 0.425, with a maximum of for the filling factor of 0.40 [(Figs. 10(a), (b) and (c)]. For TE polarization the maximum frequency range (0.006) is observed for filling factors between 0.4 and 0.45 (Fig. 11). A similar hexagonal air glass PC made from NC21 glass was also considered. The index contrast for such a PC is equal to The changes introduced by this to the 4. Simulation results The MIT photonic bands (MPB) package is an iterative eigensolver used to compute the definite frequency eigen states of Maxwell s equations in photonic crystals using a plane wave basis [23]. Using the frequency domain method is favourable because we are interested in finding the superprism effect for the lowest possible band. This method allows us to obtain the frequencies and eigenstates at the same time, which helps to quickly differentiate modes and create band diagrams. Since each additional band for a given frequency means additional propagation directions, Fig. 9. Frequency range for superprism effect in F2 glass air hexa gonal structure depending on linear filling factor. 270 Opto Electron. Rev., 20, no. 3, SEP, Warsaw

5 Fig. 10. (a) Iso frequency curves for linear filling factor of and polarization TM. Curves show normalised frequency range from to In plane incident angle is calculated from conservation of wave vector component parallel to the PC border. (b) Iso frequency curves for linear filling factor of 0.40 and polarization TM. Curves show normalised frequency range from to (c) Iso frequency curves for linear filling factor of and polarization TM. Curves show normalised frequency range from to Fig. 12. Frequency range for superprism effect in NC21 glass air hexagonal structure depending on linear filling factor. Fig. 11. Iso frequency curves for linear filling factor of 0.40 and po larization TE. Curves show normalised frequency range from to superprism effect frequency range are shown in Fig. 12. As expected, the frequency range for TM polarisation is smaller in the NC21 based crystal than in the F2 based one. It reaches the maximum for a filling factor between and 0.425, with the maximum (0.099 normalized frequency) for 0.4 [(Figs. 13(a), (b) and (c)]. For TE polarization, the freqency range is constant over a filling factor range of 0.35 to and equals normalized frequency [(Fig. 14(a) and (b)]. Fig. 13. (a) Iso frequency curves for linear filling factor of and polarization TM. Curves show normalised frequency range from to In plane incident angle is calculated from conservation of wave vector component parallel to PC border. (b) Iso frequency curves for linear filling factor of and polarization TM. Curves show normalised frequency range from to (c) Iso frequency curves for linear filling factor of and polarization TM. Curves show normalised frequency range from to Opto Electron. Rev., 20, no. 3, 2012 A. Filipkowski 271

6 Superprism effect in all glass volumetric photonic crystals Fig.15. Frequency range for superprism effect in F2/NC21 based crystal depending on linear filling factor. Fig. 14. (a) Iso frequency curves for linear filling factor of and polarization TE. Curves show normalised frequency range from to In plane incident angle is calculated from conserva tion of wave vector component parallel to PC border. (b) Iso fre quency curves for linear filling factor of and polarization TE. Curves show normalised frequency range from to Simulations were performed for structures based on the NC21/F2 crystal with different lattice constants. Due to the low refractive index contrast and exploiting the fact that hexagonal structures have a higher chance of generating a PBG, only hexagonal lattices were considered. The linear filling factor was changed from 0.2 to 0.45 (Fig. 15). For TM polarization, the superprism effect can be observed for a maximum wavelength span equal to normalized fre quencies for all linear filling factors. For TE polarization the superprism effect reaches normalized frequencies for a linear filling factor between and [(Figs. 16(a) and (b)]. Fig. 16. Iso frequency curves for linear filling factor of and polarization TE. Curves show normalised frequency range from to (b) Iso frequency curves for linear filling factor of and polarization TE. Curves show normalised frequency range from to Opto Electron. Rev., 20, no. 3, SEP, Warsaw

7 Fig.17. Frequency range for superprism effect in NC21/F2 based crystal depending on linear filling factor. The last set of simulations was performed for the double glass hexagonal structures with inverted areas of high and low refractive index (Fig. 17). Due to this inversion, the TM polarization covers a larger frequency range than the TE polarization, similarly as for the air glass structures. For TE polarization, the superprism effect can be observed for the maximum wavelength span of normalized frequen cies for all linear filling factors. For TM polarization the superprism effect reaches normalized frequencies for a linear filling factor between and [(Figs. 18(a) and (b)]. 5. Conclusions The superprism effect has been modelled in low contrast all dielectric photonic crystals. We have shown that such volumetric PCs can be developed by using a modified stack and draw technology, which allows for high volume, low cost fabrication of micro optical systems for telecommuni cations and sensing applications. Simulation results show that the highest frequency range for the superprism effect is obtained for those crystals with the highest refractive index contrast. For the air F2 glass structure with contrast, it reaches normalized frequencies. Assuming that the lattice constant of the PC is 1000 nm that corresponds to 368 nm wavelength range in which superprism effect can be observed. For the air NC21 glass structure (contrast ) it reaches 0.99 normalized frequencies, in both instances for TM polarization. The TE polarization superprism effect never exceeds norma lized frequencies. However, practical realization of such a large volume PCs with the proposed stack and draw tech nique is very difficult from technological point of view. For all glass structures, the frequency range for the TM polarization superprism effect is reduced by an order of magnitude. The largest range is for the NC21/F2 crystal and spans of normalized frequencies. Assuming that the lattice constant of the PC is 1000 nm that corresponds to 65 nm wavelength range in which superprism effect can be observed. In the F2/NC21 crystal, because of the inversion of high and low refractive index areas, the TM polarization superprism effect never exceeds normalized frequen Fig. 18. (a) Iso frequency curves for linear filling factor of and polarization TM. Curves show frequency range from to (b) Iso frequency curves for linear filling factor of 0.35 and polar ization TM. Curves show frequency range from to cies, while for TE polarization the frequency range reaches 0.005, which would correspond to 11 nm and 27 nm wave length ranges for which superprism effect can be observed. Based on the obtained results, we conclude that the super prism effect over a limited frequency range can be obtained in the two glass low contrast structures. The wavelength range over which superprism effect can be observed would make those structures suitable for the use in telecommunica tions as wave multiplexers/demultiplexers. The major advantage of the all glass structures is that they can be rela tively easily and at low cost realized with the proposed stack and draw approach. Successful fabrication of all glass PCs has been performed. Moreover, the simula tions show that by changing the materials used, we cannot only change the frequency range of the superprism effect, but also the polarization in which the effect occurs. Opto Electron. Rev., 20, no. 3, 2012 A. Filipkowski 273

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