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. Phys. 50 455104

Manuscript version: Accepted Manuscript

Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors”

This Accepted Manuscript is © © 2017 IOP Publishing Ltd.

During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 month embargo period.

After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0

Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record.

View the article online for updates and enhancements.

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. of 0.047 cm. Such wavelength corresponds to the absolute index value of 52 53 0.5, which is exactly the same as defined by the retrieval procedure at respective frequency of 1.28 THz 54 (shown below in figure 2(d)). The retrieval procedure operating with S-parameter spectra was the basic 55 one in this work for obtaining spectra of index values. For this procedure, we usually employed S- 56 57 parameter spectra simulated for 5-cell models. However, one-cell models were also checked for 58 59 2 60

Page 3 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2

1 2 3 comparison with the results in [3] and [11]. Index spectra for two types of models did not demonstrate 4 5 significant differences. 6 7 8 9 10 Ex 11 x 12 1 k 1 13 z 14 15 (a)Hy (b) y 16 1 17 0.8 18 0.6 19 S21 0.4 a 0.2 b 20 (c) 0 21 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 22 Frequency (THz) 23 E (V/m) 24 ‐1.0e6 0 1.0e6 25 26 Ex 27 28 (d) 29 Figure 1. (a) One-cell model of adMM, (b) model of 5 stacked cells, (c) simulated S21 spectra for the 30 models presented in (a) - thin curve, and (b) - bold curve, for 2D square-lattice rod array with ε =100, rod 31 r 32 radius 10 μm, and the lattice constant α =100 μm, and (d) snap-shot of E-field intensity at wave 33 propagation through multi-cell model of adMMs with rod and lattice parameters similar to those in (c), 34 sampled at f=1.28 THz. 35 36 As in many other works on MMs, index retrieval was based on employing the equation given in [18]: 37 1 "' 38 nemieln(jk00 nd ) 2  ln( jk nd ) (1)   39 kd0  40 41 where symbols and denote, respectively, real and imaginary parts of the refractive index, and ‘m’ 42 is an integer number, 43 22 44 jk nd S z 1 (1SS ) e 0  21 ,  , and z  11 21 (2) 45 22 1S11 z 1 (1SS ) 46 11 21 47 Correct choice of adjustable coefficient ‘m’ in (1) is challenging, since this coefficient has to be changed, 48 49 for instance, from 0 to 1, at such frequency, at which the value m=0 causes discontinuity in the spectrum 50 of real index component. We usually preferred to avoid such changes of coefficient ‘m’ and, following 51 [19], used m=0 for the entire spectrum. However, in the cases of doubts in the accuracy of retrieved 52 53 results, we controlled obtained index values by employing alternative options for index determination. It 54 is worth noting here, that while verifying the retrieved index values, we followed [20, 21] to account 55 56 properly for the effects of array periodicity on index spectra. 57 58 59 3 60

AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2 Page 4 of 13

1 2 3 Obtained by using retrieval procedure index spectra along with spectra of S-parameters and probe signals 4 5 were analyzed in comparison with dispersion diagrams of infinite rod arrays and with spectra of Mie 6 resonances in single infinitely long dielectric rods. For calculation of dispersion diagrams, we used the 7 8 latest editions of MPB software developed at MIT [22]. This software was also employed for calculating 9 equi-frequency contours (EFCs), which were basically used for distinguishing bands with positive and 10 negative indices. In addition, EFCs were used for verifying the retrieved index values. Spectra of Mie 11 12 resonances were calculated by using the expression for scattering coefficients from [12]: 13 rnJkRJ()(0000 n r kRJkRJ ) n ()( n  r kR ) 14 bn  (3) 15 rnH ()(kRJkRHkRJkR0000 n r ) n ()( n  r ) 16 17 where k0 is wavenumber in free space, R is the radius and εr is relative permittivity of rods, Jn and Hn 18 represent Bessel function of the first kind and Hankel function of the second kind, respectively, while 19 Lagrange’s notation (·)′ is used here to denote first derivatives of these functions. 20 21 3. Surface resonances in energy bandgaps versus Mie resonances 22 23 Figure 2 presents a set of spectra, described in Section II, for an adMM represented by a square lattice of 24 rods with radius R=10 µm and relative permittivity εr=100, at lattice constant 100 µm. Similar adMM in 25 26 [11] demonstrated negativity of real index values in the retrieved index spectra at about 1.1 THz. As seen 27 in figure 2(a), dispersion diagram of this array features as separated by bandgaps transmission branches, 28 so independent on k-vector photonic states at frequencies 1.15 THz and 1.82 THz, which are close to 29 30 characteristic frequencies for coefficients (|b1|) and (|b2|), defined by (3), in the spectrum of Mie 31 resonances of a single rod (figure 2(b)). These coefficients correspond to the magnetic resonance and to 32 33 the higher order resonance, respectively [12], for which characteristic simulated field patterns are shown 34 in inserts above figure 2(b). Similar independent on k-vector photonic states, which could be also related 35 36 to Mie resonances, were observed in dispersion diagrams of 3D arrays composed of dielectric spheres and 37 were called “localized states” [23]. 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 4 60

Page 5 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2

1 2 3 Γ 4 5

6 K‐Vector (a) 7 X 8 9 Ex Hy Ex Hy Ex Hy 10 1 11 |b0| |b1||b2| | 12 n 0.5 13 |b (b) 14 0 15 0 200 16 ‐50 100 0 17 ‐100 S21 (dB)

‐100 Phase (°) 18 (c) ‐150 ‐200 19 4 20 Real 2 21 Imaginary 22 Index 0 23 ‐2 (d) 24 2 1.5 40 25 H 1 26 E 20 0.5 E (MV/m) 27 (e) 0 H (kA/m) 0 28 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 29 Frequency (THz) 30 Figure 2. Spectral characteristics obtained for models of 2D square-lattice rod array with εr=100, R=10 μm, and 31 lattice constant 100 μm: (a) dispersion diagram for infinite array, (b) Mie scattering by a single rod (inserts show 32 field patterns at the maxima of Mie coefficients), (c) S21 magnitude (solid curve) and phase (dashed curve) for the 33 34 model of five cells in a row, (d) retrieved index components for one-cell model, and (e) signals from E- and H-field 35 probes in one-cell model. 36 37 No localized states, which could be associated with electric-type Mie resonance were observed as in our 38 studies, so in [23]. The electric Mie resonance represented by the coefficient |b0| in figure 2(b) could be 39 associated only with the position of fundamental band edge. Figures 3(a) and 3(b) present, respectively, 40 41 E- and H-field patterns in the array cross-section at the frequency of band edge. Field patterns in each cell 42 look corresponding to the patterns observed at the electric Mie resonance in a single rod (first insert above 43 ° 44 figure 2(b)). However, in neighboring cells, they demonstrate 180 phase difference that allows for 45 relating the respective transmission mode to the odd type. In addition, field magnitude is changing from 46 cell to cell along the chain with maximal fields observed in the third cell. These changes can be related to 47 48 the formation of F-P resonance, which are common for PhCs of restricted lengths [18]. As known, F-P 49 resonances cause formation of standing waves, and figures 3(a) and (b) just represent the snapshots of the 50 51 half-wavelength standing wave. Conserved at F-P resonance out-of-phase arrangement of neighboring 52 Mie resonances excludes a possibility of their meaningful effect on effective medium parameters and 53 thus, excludes any contribution of the electric resonance in double negativity of effective parameters. 54 55 56 57 58 59 5 60

AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2 Page 6 of 13

1 2 3 E (V/m) H (A/m) 4 ‐1.0e6 0 1.0e6 ‐1.8e4 0 1.8e4 5 6 Ex Hy 7 8 9 (a) (b) 10 ‐0.3e6 0 0.3e6 ‐0.5e4 0 0.5e4 11 12 Ex Hy 13 14 (c) (d) 15 ‐0.3e6 0.3e6 ‐1.4e4 1.4e4 16 0 0 17 Ex Hy 18 19 20 (e) (f) 21 22 Figure 3. E- and H-field patterns in the cross-section of 2D rod array: (a)-(b) at the edge frequency of st 23 fundamental band (0.42 THz); (c)-(d) at the 1 dip in S21 spectrum (figure 2(c) - 0.8 THz); (e)-(f) at the 24 2nd dip in S21 spectrum (1.05 THz). 25 26 As seen in figure 2(c), S21 spectrum of the respective rod array demonstrates two dips at 0.8 THz and 27 1.05 THz. Similar dips were observed in S-parameter spectra by authors of [3-11], and in some cases 28 29 were assumed to be related to Mie resonances in rods. However, comparison of figures 2(b) and 2(c) does 30 not support such assumption. In addition, it can be seen from comparison of figures 2(a) and 2(c) that 31 32 resonance-type dips in S21 are observed at frequencies corresponding to the bandgap in the dispersion 33 diagram, when waves should not penetrate inside the array. Furthermore, the spectra of signals from E- 34 and H-field probes (figure 2(e)) do not demonstrate typical for resonances enhancement of field 35 36 magnitudes at dip frequencies. Field patterns in figures 3(c)-3(f) confirm that at dip frequencies, 37 evanescent waves do not penetrate further than the first cell, where they cause weak responses barely 38 39 comparable with resonance fields. This means that wave phenomena, which occur at the dips, could only 40 be related to responses in the surface layer and not to resonance phenomena in the array volumes. It is 41 42 worth noting here that observed distortions of surface responses against the patterns in inserts, placed 43 above figure 2(b), could be caused by interaction between resonance fields at two Mie resonances. As 44 seen in figure 2(d), this interaction reveals itself in overlapping of frequency ranges, in which imaginary 45 46 components of index at two “dip resonances” are defined. 47 48 4. Refraction controlled by dispersion of transmission branches 49 After establishing irrelevance of surface resonances to wave phenomena in the volume of 2D rod arrays, 50 51 we had to exclude from further consideration those parts of the retrieved index spectra, which were 52 defined by surface responses due to rod interaction with evanescent waves. To do that, we took into 53 consideration only those areas of the spectra of real index component, in which imaginary component was 54 55 equal to zero, i.e. in which the index had physically meaningful values. In addition, we restricted our 56 analysis by the frequency range around 1.1 THz, since, according to [11], negative refraction for arrays 57 58 under study was expected just in this range. Our data presented in figure 2(d) for arrays with the lattice 59 6 60

Page 7 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2

1 2 3 constant a =100 μm demonstrate a region with negative values of real index component just below 1.1 4 5 THz that agrees with the results in [11]. In order to clarify the reasons for appearance of index negativity, 6 we compared changes in the dispersion diagrams and in the retrieved index spectra at decreasing the array 7 8 lattice constant from 120 µm down to 80 µm. As seen in figure 4, at a =120 µm the values of real index 9 component near 1 THz have positive sign and are defined in relatively wide band from 0.95 THz up to 10 1.07 THz. At a =110 µm, these values remain positive, however, in a narrower band between 1.04 THz 11 12 and 1.1 THz. At a =105 µm, the band with positive indices disappears and a trend of switching to a 13 negative index is observed at 1.08 THz. Further decrease of lattice constant leads to formation of a band, 14 15 in which the values of real index component are negative. While at a =100 µm this band can be found 16 between 1.08 THz and 1.12 THz, then at a =90 µm it extends from 1.08 THz to 1.17 THz and remains 17 almost unchanged at a = 80 µm. 18 Γ 19 Γ a=120μm a=100μm 20 21 K‐Vector 22 X (a) X (g) 23 4 4 24 2 25 2 Index 0 26 (b) (h) 27 0 ‐2 Γ a=110μm Γ a=90μm 28 29 30 K‐Vector X (c) X (i) 31 4 4 32 2 33 2 34 Index 0 (d) (j) 35 0 ‐2 36 Γ a=105μm Γ a=80μm 37

38 K‐Vector (e) 39 X X (k) 4 40 3 41 2

42 Index 0 0 (f) (l) 43 ‐2 ‐3 44 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2 45 Frequency (THz) Frequency (THz) 46 Figure 4. Dispersion diagrams and spectra of real (solid curves) and imaginary (dashed curves) index 47 components obtained for the models of 2D rod arrays (εr=100 and R=10 μm) at various array lattice 48 constants a in the range from 120 μm to 80 μm: (a)-(b) 120 μm, (c)-(d) 110 μm, (e)-(f) 105 μm, (g)-(h) 49 50 100 μm, (i)-(j) 90 μm, (k)-(l) 80μm. 51 52 It is well seen in figure 4 that described above bands with physically meaningful index values exactly 53 coincide with the bands, in which the dispersion diagrams of arrays with respective lattice constants 54 exhibit the 2nd transmission branch. Additionally, it is well seen that at 120 µm > a > 105 µm, the 2nd 55 56 branch has the slope corresponding to positive indices, at a =105 µm it becomes flat, and at 105 µm > a > 57 80 µm it acquires a slope corresponding to negative indices. It should be noted here that the bands in 58 59 7 60

AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2 Page 8 of 13

1 2 3 figure 4, in which physically meaningful indices are defined, have no relation to surface resonances, since 4 5 the latter affect index components in frequency ranges located far from these bands. Index spectra 6 observed at a = 80 µm in [11] did not correspond neither to changes of the dispersion diagram (figure 7 8 4(k)), nor to index spectra presented in figure 4(l). This discrepancy could possibly be caused by 9 employing a step-function for adjustable coefficient ‘m’ in [11] in the expression used for index retrieval. 10 11 Although obtained changes of retrieved index spectra at decreasing the lattice constant of the array 12 exactly corresponded to respective changes of the 2nd branch shape, we decided to additionally verify 13 14 obtained index data by using another well-known approach [23,24], taking into account mentioned above 15 discrepancy with the data in [11]. In particular, we calculated and compared EFCs for adMM with a = 16 nd rd 17 100 µm in 2 and 3 bands located just below and above 1.17 THz (see figure 4(g)) and characterized in 18 retrieved spectrum (figure 4(h)) by negative and positive indices, respectively. We also compared EFCs 19 nd 20 for adMM with a = 80 µm in the fundamental and 2 bands observed below and above the bandgap 21 (figure 4(k)) and also presented in retrieved index spectrum (figure 4(l)) by positive and negative indices, 22 respectively. As seen in figures 5(a) and 5(b) for adMM with a = 100 µm, moving from lower to high kx 23 24 or ky values along the axes of EFCs for the band below 1.17 THz leads to crossing contours for lower 25 frequencies, while for the band above 1.17 THz, such movement, just opposite, leads to crossing 26 27 contours for higher frequencies. This confirms that the band below 1.17 THz can be characterized by 28 negative indices, while the band above 1.17 THZ - by positive indices, in correspondence with retrieved 29 index spectra in figure 4h. EFCs presented in figures 5(c) and 5(d) for adMM with a = 80 µm demonstrate 30 31 opposite difference between two bands chosen for comparison. In the first band, increase of k-values 32 leads to crossing EFCs for higher frequencies, while in the 2nd band we see inverse trend. These data 33 nd 34 confirm that the lower (fundamental) band can be characterized by positive indices, while the 2 band - 35 by negative indices, in exact correspondence with our retrieved index spectra (and in contrary with the 36 data in [11]). 37 2nd band (negative) 3rd band (positive) 1st band (positive) 2nd band (negative) 38 1.12 π 0

1 . 5 1 1 1 9 1 (c) (d) . (a) 1 . (b) . 39 . .6 7 0 1 1

. 1

1 0 1 .45 2 5 5 . 1 40 8 6 . 9 1.5 3π/4 .0 1.124 0 4 1 0 .4 1 . 5 1 1. 0.5 1.1 1.11 41 1 .5 7 0 2 ) .4 1.1 1.4 1. a 6 1 .09 1.1 .11 0 0 1.1 2 42 1 1 1 1 1 . . 1.1 π/2 . .4 4 4 2 1.12 1 1 5 1 .095 1.3 .5 0 1.12 1.1 43 1 1 0.3 .3 ky(1/ 1.1 .7 5 1.1 1.1 0.25 1.13 1.095 1 1 1 1 44 .6 0. 1.1 .3 0 . 1 1 1 2 . 5 0. .3 4 0 π/4 9 1.09 . 2 1 1 .0 1 4 . 3 1 .1 1 .2 0 . 1 1 0.15 4 45 . 1 . 1 1 9 0 . 1 2 5 1 1 0 5 5 . . 1 .08 0 . 1 . 5 1 . 1 . 1 1 1 3 . 2 5 4 1 46 5 0 47 0 π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π 0π/4π/23π/4π0 π/4 π/2 3π/4 π 48 kx(1/a) kx(1/a) kx(1/a) kx(1/a) 49 Figure 5. Equi-frequency contours in various frequency bands: (a) and (b) for adMM with a = 100 µm, (c) 50 and (d) for adMM with a = 80 µm. (a) 2nd band, (b) 3rd band, (c) 1st band, (d) 2nd band. 51 52 5. Origin of the 2nd transmission branch in adMMs and its irrelevance to Lorentz’-type responses 53 The 2nd branch in the dispersion diagram, which affects the sign and the value of refractive index, needs 54 55 to be considered in more details. It could be noticed that near points X of the dispersion diagrams these 56 branches, for all used lattice constants, tend to become parallel to the branch for a =105 µm, which is flat 57 58 and independent on k-vector. The shape of this branch makes possible its association with photonic state 59 8 60

Page 9 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2

1 2 3 comparable to the localized state described in section III (observed at 1.15 THz for a =100 µm) as 4 5 associated with magnetic Mie resonances in arrays. It is worth noting here that the latter state 6 demonstrated full transmission for all arrays, while the former state observed at a =105 µm showed 7 8 transmission at the level less than -50 dB that made phase changes along the rod chain unstable. This 9 created problems for obtaining field patterns, and we were compelled to compare field patterns at two 10 photonic states using arrays with a =100 µm. As seen in figures 6(a) and 6(b), photonic state at 1.15 THz, 11 12 which, following [25], we called a localized state, represented coherent magnetic Mie resonances in all 13 rods of the array. This state could be related to wave propagation with superluminal phase velocity and to 14 nd 15 the even transmission mode [26]. To characterize the second state, although the 2 branch was not flat at 16 a =100 µm, we used this branch near the point X in the dispersion diagram (figure 4(g)), where it still 17 kept the same slope as that at a =105 µm. As seen in figures 6(c) and 6(d), obtained field patterns for the 18 19 second state appeared corresponding to the odd transmission mode, in which incorporated local Mie 20 resonances in neighboring rods were shifted in phase by 180°. 21 22 E (V/m) H (A/m) 23 ‐1.0e6 0 1.0e6 ‐4.6e4 0 4.6e4 24 25 Ex Hy 26 27 (a) (b) 28 29 ‐0.8e6 0 0.8e6 ‐1.4e4 0 1.4e4 30 31 Ex Hy 32 33 (c) (d) 34 35 Figure 6. E- and H-field patterns in the cross-section of 2D rod array with a =100 µm: (a)-(b) at the 36 frequency of localized photonic state (1.15 THz); (c)-(d) at the lower edge (X-point) of the 2nd branch in 37 the dispersion diagram. 38 39 Since both even and odd transmission modes are related to the same type of Mie resonances, the 40 41 formation of two modes could be considered as the result of splitting of the resonance photonic states due 42 to difference in energy of interaction between magnetic dipoles representing resonance responses at even 43 and odd transverse orientations [27]. Observed transition from positive refraction in the 2nd transmission 44 45 band in arrays with a >105 µm to negative refraction in arrays with smaller lattice constants apparently is 46 also related to increased interaction between resonators. 47 48 Based on the revealed rigorous correlation between the transformation of the 2nd branch in the dispersion 49 50 diagrams and the reversal of the index sign occurring at changes of the array lattice constant, it can be 51 concluded that negative refraction in adMMs is defined by the specifics of their dispersive properties and 52 53 not by double negativity of their effective parameters, as it is typically expected at observation of Mie 54 resonances [28]. It should be noted that none of the investigated here array properties, affected by 55 occurrence of Mie resonances, appeared related to the Lorentz-type dispersion model or to the specifics of 56 57 effective material parameters defined by this model. In fact, it is simply impossible to expect that dipoles 58 formed in rod arrays at Mie resonances and incorporated in odd transmission modes could contribute to 59 9 60

AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2 Page 10 of 13

1 2 3 effective medium parameters. In contrary, such contribution could be expected at even modes. However, 4 5 as seen in figure 4, for lattice constants in the range 120 µm > a > 100 µm we observe another 6 transmission band in dispersion diagrams above even localized states, instead of a bandgap, which should 7 8 appear due to negative effective permeability at the Lorentz-type resonance response. 9 10 Considering as established the dispersive origin of negative refraction in adMMs and its irrelevance to the 11 double negativity of effective parameters, we have also confirmed that this phenomenon is related to 12 lefthanded wave propagation. To do this, we simulated spectra of phase changes at wave transmission 13 14 through multi-cell models with different quantity of stacked cells. Such spectra can be used to 15 demonstrate either phase delay in “longer” model compared to phase in “shorter” model, which is 16 17 characteristic for forward wave propagation, or respective phase advance characteristic for backward 18 wave propagation [29]. Figure 7 compares spectra of phase changes for models composed of 9 and 11 19 nd rd 20 cells of adMM with a = 100 µm in the bands located below and above 1.17 THz (2 and 3 bands in 21 figure 4(g)). As seen in figure, the spectrum obtained for the model composed of 11 cells at frequencies 22 corresponding to the 2nd band appears shifted forward along the frequency axis with respect to the 23 24 spectrum obtained for the model composed of 9 cells. This means that at any frequency in this band, 25 waves passing through the longer model attain phase advance with respect to waves passing through the 26 27 shorter model, which is specific for the backward wave propagation. In contrast, at frequencies 28 corresponding to the 3rd band, the phases of waves passing through the model composed of 11 cells 29 appears delayed with respect to that for the model of 9 cells. This behavior is specific for the forward 30 nd 31 wave propagation. Thus, we can conclude that in adMM with a = 100 µm wave transmission in the 2 32 band represents a lefthanded phenomenon. 33 34 200 35 100 36 0 37 ‐100 S21 Phase (a) 38 ‐200 39 1 1.1 1.2 1.3 1.4 1.5 1.6 40 200 41 100 42 43 0

44 S21 Phase ‐100 (b) 45 ‐200 1.07 1.09 1.11 1.25 1.35 1.45 1.55 1.65 46 Frequency (THz) 47 48 Figure 7. Spectra of phase changes at wave transmission through two models of adMM with a = 100 µm: nd 49 (a) at the same frequency scaling in all spectra, (b) at different frequency scaling in 2 and 3d bands to 50 provide better resolution. In (b) the left side is related to the 2nd band, while the right side – to the 3rd band 51 in the dispersion diagram. The model of 11 cells (dashed curves) demonstrates at the wave propagation 52 nd 53 the phase advances in the 2 band and phase delays in the 3rd band against the model of 9 cells (solid 54 curves). 55 56 57 58 59 10 60

Page 11 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2

1 2 3 6. Confirming the dispersive nature of negative refraction in other adMMs. 4 5 To demonstrate that our conclusions are applicable to any other rod array from referenced above 6 publications [3-11], we present below the results of studying the dispersion diagrams and the retrieved 7 8 index spectra for microwave arrays of rods used in [3, 5], taking into account that [3] is the only work, in 9 which the phenomenon of negative refraction in adMMs was confirmed experimentally. In [3, 5], rod 10 arrays had the lattice constant of 3 mm, the relative rod permittivity of 600 and the radius of 0.68 mm. 11 12 The negative refraction was observed at frequencies around 7 GHz. Here, to ensure the possibility of 13 realizing the same, as in figure 4 effects of lattice changes on the array properties, the values of array 14 15 lattice constants were varied from 15 mm down to 3 mm. 16 17 As seen in figure 8, dispersion diagrams of all arrays incorporated, similar to that in figure 4, two 18 localized photonic states, the lower of which slightly changed its position from 6.8 GHz at a = 15 mm up 19 20 to 7.25 GHz at a = 3 mm, while the upper one remained located at the same frequency of 11 GHz. As in 21 the previously considered case, the frequencies of localized states could be associated with the magnetic 22 and the higher order Mie resonances in separate rods. No localized states, which could be related to 23 24 electric-type Mie resonances, were found in the diagrams. As in figure 4, physically meaningful index 25 values of arrays (registered, when imaginary index components were equal to zero) were observed at 26 nd 27 frequencies close to 7 GHz within frequency bands corresponding to the locations of the 2 branches in 28 dispersion diagrams. 29 30 Γ a=15mm Γ a=7mm 31 32 33 K‐Vector (a) X X (e) 34 6 12 35 4 Real 8 Real 36 2 4

Index Imaginary Imaginary 0 0 37 (b) (f) 38 ‐2 ‐4 39 Γ a=13.3mm Γ a=3mm 40

41 K‐Vector (c) 42 X X (g) 6 20 43 Real 44 4 Real 10 2 Imaginary Index Imaginary 0 45 0 (d) (h) 46 ‐2 ‐10 47 024681012024681012 48 Frequency (GHz) Frequency (GHz) 49 Figure 8. Dispersion diagrams and index spectra for real and imaginary components obtained for models 50 representing 2D rod arrays (εr=600 and R=0.68 mm) at various array lattice constants a in the range from 51 52 15 mm to 3 mm: (a)-(b) 15 mm, (c)-(d) 3.3 mm, (e)-(f) 9 mm, (g)-(h) 3 mm. 53 54 At all lattice constants under study, the respective branches looked aligned to the lower of split photonic 55 states associated with the magnetic Mie resonance (although, in this case, at various points of the 56 dispersion diagram - either X or Γ), while decreasing of the lattice constant caused changes of the branch 57 58 slopes with respective reversal of index sign, as well as increase of the branch bandwidth. 59 11 60

AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2 Page 12 of 13

1 2 3 The obtained data additionally confirm that negative refraction in adMMs is defined by their dispersion 4 5 properties, while double negativity of effective medium parameters due to Lorentz-type responses at 6 electric and magnetic Mie resonances is irrelevant to this phenomenon. 7 8 7. Conclusion 9 10 Comprehensive studies of appearance of negative refraction in adMMs at changes of their lattice 11 parameters have shown that there are no signs that Mie resonances cause Lorentz-type response of 12 adMMs’ effective parameters and their switching from positive to negative values. Thus, there are no data 13 14 allowing for relating negative refraction to double negativity of effective parameters. Instead, strict 15 correlation was observed between appearances of bands with negative refraction and respective 16 nd 17 transformations of 2 transmission band in energy diagrams of adMMs at changes of their lattice 18 constants. Obtained data allowed for concluding that the bandwidths of both positive and negative 19 nd 20 refraction in adMMs are unambiguously defined by the shape and the location of 2 branch in dispersion 21 diagrams and thus could be governed by lattice constants. This conclusion provides an important 22 guidance for controlling the property of refraction in low-loss adMMs at various photonic applications. 23 24 Acknowledgments 25 26 This work was supported by the National Science Foundation under Award ECCS-1709991. 27 28 29 30 References: 31 [1]. Corbitt S J, Francoeur M and Raeymaekers B 2015 J. of Quant. Spec. and Rad. Trans. 158 3-16 32 [2]. Jahani S and Jacob Z 2016 Nat Nano 11 23-36 33 [3]. Peng L, Ran L, Chen H, Zhang H, Kong J A and Grzegorczyk T M 2007 Phys. Rev. Lett. 98 157403 34 35 [4]. Schuller J A, Zia R, Taubner T, Brongersma M L 2007 Phys. Rev. Lett. 99 107401 36 [5]. Vynck K, Felbacq D, Centeno E, Căbuz A I, Cassagne D, Guizal B 2009 Phys. Rev. Lett. 102 133901 37 [6]. Lai Y J, Chen C K, Yen T J 2009 Opt. Exp.. 17 12960-70 38 [7]. Peng L, Ran L and Mortensen N A 2010 Appl. Phys. Lett. 96 241108 39 [8]. Wang J, Xu Z, Du B, Xia S, Wang J, Ma H, et al 2012 J. of Appl. Phys. 111 044903 40 [9]. Yahiaoui R, Chung UC, Elissalde C, Maglione M, Vigneras V and Mounaix P 2012 Appl. Phys. Lett. 101 41 042909 42 43 [10]. Yahiaoui R., Mounaix P, Vigneras V, Seu U C C, Elissalde C and Maglione M 2013 Europ. Mic. Conf. 44 (unpublished). 45 [11]. Dominec F, Kadlec C, Němec H, Kužel P and Kadlec F 2014 Opt. Exp. 22 , 30492-503 46 [12]. Van De Hulst H C 1982 Light scattering by small particles by small particles (Dover Publications Inc.: 47 [distributor] Grantham Book Services Ltd). 48 [13]. Lidorikis E, Sigalas M M, Economou E N and Soukoulis C M 1998 Phys. Rev. Lett. 81, 1405-08 49 [14]. Lidorikis E, Sigalas M M, Economou E N and Soukoulis C M 2000 Phys. Rev. B 61 13458-64 50 [15]. Veselago V, Braginsky L, Shklover V and Hafner C 2006 J. of Comp. and Theor. Nano. 3 189-218 51 52 [16]. Valdivia-Valero F J and Nieto-Vesperinas M 2012 Phot. Nano. Fund. Appl. 10 423-34 53 [17]. Semouchkina E, Duan R, Semouchkin G and Pandey R 2015 Sensors 15 9344-59 54 [18]. Chen X, Grzegorczyk T M, Wu B I, Pacheco J and Kong J A 2004 Phys. Rev. E 70 016608 55 [19]. Numan A B and Sharawi M S 2013 IEEE Ant. and Propag. Mag. 55 202-11 56 [20]. Koschny T, Markoš P, Smith D R and Soukoulis C M 2003 Phys. Rev. E 68, 065602 57 58 59 12 60

Page 13 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPhysD-113774.R2

1 2 3 [21]. Koschny T, Markoš P, Economou E N, Smith D R, Vier D C and Soukoulis C M 2005 Phys. Rev. B 71 4 245105 5 6 [22]. http://jdj.mit.edu/wiki/index.php/MIT_Photonic_BandsMPB software 7 [23]. Foteinopoulou S and Soukoulis C M 2003 Phys. Rev. B 67 235107 8 [24]. Parimi P V, Lu W T, Vodo P, Sokoloff J, Derov J S, and Sridhar S 2004 Phys. Rev. Lett. 92 127401 9 [25]. Vandenbem C and Vigneron J P 2005 J. Opt. Soc. Am. A 22 1042-47 10 [26]. Semouchkina E 2012 Metamaterial (Intech) 11 [27]. Liu N and Giessen H 2010 Ang. Chem. Int. Ed. 49 9838-52 12 [28]. Moitra P, Yang Y, Anderson Z, Kravchenko I I, Briggs D P and Valentine J 2013 Nat. Phot. 7 791-5 13 [29]. Chen F, Wang X, Semouchkin G, and Semouchkina E 2014, AIP Advances 4, 107129 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 13 60