Michigan Technological University Digital Commons @ Michigan Tech Michigan Tech Publications 10-20-2017 All-dielectric metamaterials: irrelevance of negative refraction to overlapped Mie resonances Navid Gandji Michigan Technological University George Semouchkin Michigan Technological University, [email protected] Elena Semouchkina Michigan Technological University, [email protected] Follow this and additional works at: https://digitalcommons.mtu.edu/michigantech-p Part of the Civil and Environmental Engineering Commons, and the Computer Sciences Commons Recommended Citation Gandji, N., Semouchkin, G., & Semouchkina, E. (2017). All-dielectric metamaterials: irrelevance of negative refraction to overlapped Mie resonances. Journal of Physics D: Applied Physics, 50(45). http://dx.doi.org/ 10.1088/1361-6463/aa89d3 Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/655 Follow this and additional works at: https://digitalcommons.mtu.edu/michigantech-p Part of the Civil and Environmental Engineering Commons, and the Computer Sciences Commons Journal of Physics D: Applied Physics PAPER All-dielectric metamaterials: irrelevance of negative refraction to overlapped Mie resonances To cite this article: N P Gandji et al 2017 J. Phys. D: Appl. 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This content was downloaded from IP address 141.219.44.85 on 22/10/2019 at 20:54 Page 1 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2 1 2 3 4 All-dielectric metamaterials: irrelevance of negative refraction 5 6 to overlapped Mie resonances 7 8 9 N P Gandji, G B Semouchkin and E Semouchkina 10 Electrical and Computer Engineering Department, Michigan Technological University, 11 12 Houghton, Michigan, USA 13 14 E‐mail: [email protected] 15 16 Abstract. All-dielectric metamaterials comprised of identical resonators draw a lot of attention as 17 low-loss media providing for negative refraction, which is commonly attributed to double 18 negativity of effective material parameters caused by overlapping of Mie resonances. We study 19 dispersion diagrams of such metamaterials composed of dielectric rod arrays and show that 20 bandwidths of positive and negative refraction and its type are irrelevant to negativity of effective 21 parameters; instead, they are unambiguously defined by the shape and the location of the 2nd 22 23 transmission branch in dispersion diagrams and thus can be controlled by the lattice constants. 24 25 26 1. Introduction 27 Rapid progress in the field of photonics causes increased interest in all-dielectric metamaterials (adMMs), 28 29 which can be practically lossless at optical frequencies [1, 2]. In difference from conventional microwave 30 MMs composed of complementary metallic resonators, adMMs usually consist of identical “atoms” and 31 32 thus are expected to demonstrate common features with ordinary photonic crystals (PhCs). Such specifics 33 should be accounted for at the analysis of the most intriguing phenomenon of negative refraction 34 observed in adMMs, which are composed of Mie-type resonators [3-11]. Originally, Mie theory [12] 35 36 described wave scattering by single dielectric particles (spheres or infinite rods), and its extension to 37 resonances in arrays implied negligible interaction between particles. 38 39 Before emergence of MM concepts, it was thought that Mie resonances in PhCs could create their own 40 41 photonic states with localization lengths comparable to lattice constants [13]. These states were expected 42 to contribute to transmission due to wave transfer/hopping between neighboring resonators. Further 43 44 studies [14] conveyed that respective states could define transmission branches in PhC’s dispersion 45 diagrams and even control bandgaps. 46 47 After MMs’ implementation, PhCs with Mie resonances became typically viewed as MMs, complying 48 49 with the effective medium theory and the Lorentz’s dispersion model. Although Veselago et al. [15] 50 expressed doubts in such views, which neglected the role of periodicity in PhCs, it became common 51 practice to analyze adMMs’ properties by using spectra of effective parameters, retrieved from scattering 52 53 data for one cell [3-11]. Justification of this practice was based on the assumption that resonators’ 54 dimensions were relatively small compared to wavelengths of radiation. As the result, negative refraction 55 56 in adMMs [3-11] was attributed to double negativity of effective material parameters, which could be 57 expected at overlapping of the “tails” of electric- and magnetic-type Mie resonances, although no proofs 58 of the “double negativity” effect and no data about the formation of hybrid modes, which had to combine 59 60 electric and magnetic resonance modes in particles of one type, were presented. Serious problems with AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2 Page 2 of 13 1 2 3 application of the effective medium concepts to the description of wave propagation in adMMs were 4 5 revealed in [16], where it was proposed to relate the observed phenomenon of negative refraction to the 6 Bragg diffraction, although no investigation of the band diagrams of respective adMMs was performed. 7 8 In this work, we investigate the origin of negative refraction in adMMs by analyzing their dispersive 9 10 properties defined by the structure periodicity, along with Mie resonances. The objects of our studies 11 were represented by 2D arrays of infinitely long round dielectric rods. The diameters of rods and the 12 properties of rod dielectrics were taken similar to those used in either [3] or [11] to represent the range of 13 14 parameters covered by the set of works [3-11]. In particular, relative dielectric permittivity of rods εr was 15 taken equal to either 100 or to 600. As in all works of this set, TM wave incidence, with E-field directed 16 17 along the axes of rods and wave propagation vector (k-vector) normal to these axes, was employed. 18 19 2. Methodology of conducted studies 20 To simulate electromagnetic responses of adMMs, we employed various models of rod arrays and used 21 various software. In particular, we worked as with the single cell models typically used for characterizing 22 23 homogenized MMs (figure 1(a)), so with models better suitable for PhCs representation, such as a row of 24 cells stacked in the k-vector direction (figure 1(b)). The row was usually composed of 5 cells as, 25 26 according to [17], it was sufficient for representing PhC’s response. However, models involving 9 and 27 even 11 cells were also employed. Periodicity in the directions normal to k-vector was provided by proper 28 29 boundary conditions at cell faces. Two full-wave software packages (COMSOL Multiphysics and CST 30 Microwave Studio) were used for the studies of S-parameter spectra, wave propagation patterns and 31 distributions of field intensities in arrays at the resonances. CST Microwave Studio was found preferable 32 33 for monitoring field distributions in cross-sections of the basic models and for obtaining spectra of S- 34 parameters and spectra of signals from field probes used to control resonance fields in the rods. Figure 1c 35 36 allows for comparing S21 spectra obtained for the single-cell model and for the model composed of 5 37 cells. As seen in the figure, the spectrum obtained for the latter model clearly demonstrates a set of 38 transmission bands divided by bandgaps and thus, provides the data comparable with those resulting from 39 40 dispersion diagrams, although calculation of the latter suggests infinite array samples. In difference from 41 dispersion diagrams, S21 spectra demonstrate transmission fringes caused by Fabry-Perot (F-P) 42 43 resonances characteristic for PhCs of finite size [17]. COMSOL Multiphysics software was mostly used 44 for studying multi-cell chains representing wave processes in adMMs with close-to-zero index values 45 (snap-shot exemplifying wave pattern in such chain is presented in figure 1(d)). Snap-shots similar to that 46 47 in figure 1(d) were used to estimate wavelengths and, then, absolute index values at frequencies of 48 c 49 interest: n 0 . Considering figure 1(d) it is easy to ensure that the depicted snap-shot represents the 50 f 51 wavelength of 470 microns, i.e.