Phase diagram for the transition from photonic crystals to dielectric metamaterials

Photonic crystals and dielectric metamaterials represent two different classes of artificial media but are often composed of similar structural elements. The question is how to distinguish these two types of periodic structures when their parameters, such as permittivity and lattice constant, vary continuously. Here we discuss transition between photonic crystals and dielectric metamaterials and introduce the concept of a phase diagram, based on the physics of Mie and Bragg resonances. We show that a periodic photonic structure transforms into a metamaterial when the Mie gap opens up below the lowest Bragg bandgap where the homogenization approach can be justified and the effective permeability becomes negative. Our theoretical approach is confirmed by microwave experiments for a metacrystal composed of tubes filled with heated water. This analysis yields deep insight into the properties of periodic structures, and provides a useful tool for designing different classes of electromagnetic materials with variable parameters.

T he concepts of phase transitions and phase diagrams are among the most fundamental concepts in physics. Here we reveal and analyse a rather unusual type of phase transitions between two classes of artificial electromagnetic structures: photonic crystals (PhCs) 1,2 and metamaterials (MMs) 3,4 . Rapidly growing attention to the study of PhCs and MMs is due to their unique electromagnetic properties and the exciting new physics these structures may offer for novel applications in photonics [5][6][7][8][9] . The resonant properties of PhCs are driven by the familiar Bragg scattering from a periodic structure when the lattice constant becomes comparable with the wavelength of light. MMs are regarded as effective media with averaged parameters which in turn can be engineered by changing the properties of their subwavelength elements modifying the propagation of electromagnetic waves with the wavelengths much larger than the lattice constant.
Many properties of conventional MMs are usually defined by the optically induced magnetic response of metallic wires or splitring resonators, which has been demonstrated experimentally in many studies for both microwave and near-infrared spectral regions 10,11 . However, the scaling of such MMs to higher frequencies encounters many problems, including strong absorption of electromagnetic radiation in metals. Recently suggested electromagnetic periodic structures composed of high-permittivity dielectric elements [12][13][14] can avoid these problems, while still delivering an effective magnetic response. The main physics underlying the properties of such all-dielectric MMs is based on the resonant modes of the Mie scattering 15 supported by dielectric particles 16,17 . Mie resonances provide a mechanism for the generation of an optically induced resonant magnetic response based on the displacement current, and they offer a simpler and more versatile route towards the fabrication of isotropic MMs and metasurfaces operating at optical frequencies.
PhCs and all-dielectric MMs can be composed of the similar structural elements arranged in the similar geometries. Such structures offer a unique opportunity to study a transition from PhCs to MMs when their parameters, such as dielectric permittivity and lattice period, vary continuously. The transition we describe here can be mapped onto a phase diagram; however, it is quite different from phase transitions well known in other fields such as thermodynamics. The thermodynamical phase transitions appear as transformations of a system from one state or 'phase' to another with changing external variables such as pressure, temperature, chemical potential and so on. Here we reveal and analyse a rather unusual type of phase transitions between two regimes of light propagation, that is, the regime where the propagation of light is mainly influenced by the interference between multiple Bragg scattered (PhC 'phase'), and the regime where the propagation of light is mainly determined by the properties of each single element of the materials (MM 'phase').
First, we identify two different scenarios for the transition of a two-dimensional (2D) square lattice of dielectric rods from PhC to MM (Fig. 1). The first scenario that has never been discussed previously occurs when the lattice constant a decreases in comparison with the fixed radius of rods r and dielectric permittivity e. This transformation leads to an increase of the filling ratio r/a, and it makes possible the homogenization 18 of the periodic dielectric structure with the negative effective permeability (mo0) at higher values of e (ref. [19][20][21][22][23].
The second scenario occurs when the dielectric permittivity of rods e varies in a fixed square lattice. This transformation leads to an increase of the Mie scattering wavelength, and it can also provide the conditions for the homogenization approach with negative m. While photonic structures with varying e were studied earlier theoretically 24,25 , the transition between PhCs and MMs has never been analysed previously to our best knowledge, especially in terms of the phase diagram, as discussed below.
We notice that the aim of both scenarios is the same: to tune the wavelength of the Mie mode l Mie higher than the Bragg wavelength l Br that can be achieved by either decreasing the Bragg wavelength or by increasing the Mie wavelength. Decreasing the Bragg wavelength means decreasing the lattice constant (building block size) a, which will be smaller than the operating wavelength for the MM behaviour, l Mie .
Here we integrate both scenarios and introduce a novel concept of a phase transition between all-dielectric PhC and MM and suggest the corresponding phase diagram. The phase diagram PhC-MM is represented on the plane 'filling ratio versus dielectric permittivity' for an example of 2D structure composed of dielectric circular rods. Our theory is confirmed by very careful experimental design and analysis.

Results
An interplay between Bragg and Mie resonances. To analyse the PhC-MM phase transition, we investigate theoretically electromagnetic properties of the dielectric periodic medium depending on the dielectric contrast, filling ratio and incident wave polarization. As an illustrative example, we consider here 2D square lattices of parallel dielectric circular rods infinitely long in the z direction with the radius r and real frequency independent permittivity e. The rods are embedded in air (e air ¼ 1) and the arrays can be characterized by the filling ratio r/a. We have derived three key sets of spectroscopic data, namely: (i) spectra of the Mie scattering by an isolated rod; (ii) photonic band structure of an infinite 2D square lattice composed of rods; and (iii) transmittance of a 2D square lattice of 10 layers in length. All data sets have been calculated for a wide range of the rod permittivity 1 e 100 with the step of De ¼ 1 (e-scenario) and in the range of filling ratio 0 r=a 0:5 with the step of 0.01 (r/a-scenario). In the r/a-scenario cylinder radius is supposed to be as a fixed parameter. Note that at r/a ¼ 0.5 all rods touch each other and at r/a40.5 the rods penetrate into each other and the structure appears as inverted one of air holes in a dielectric matrix.
First, we consider the Mie scattering by an isolated rod. The far-field scattering can be expanded into orthogonal electromagnetic dipolar and multipolar modes, described by circular Lorenz-Mie resonant coefficients a n (transverse-electric (TE) polarization) and b n (transverse-magnetic (TM) polarization) corresponding to electric and magnetic moments, respectively 15 . Figure 2 presents the numerically calculated spectra of the Mie scattering efficiency Q sca;n ¼ 2 x a n j j 2 (ref. 26) for the dipole TE 0 and multipole TE n n ! 1 ð Þmodes in the range of dimensionless frequencies a/l from 0 to 0.8. Each mode consists of an infinite set of quasi-equidistant modes TE nk (ref. 27). The Mie modes of the rod are denoted as TE nk , where n is integer related to the symmetry of the mode, and k ! 1 identifies resonance number for the modes with fixed symmetry n. The calculations reveal a strong decrease of the Mie eigenfrequencies and narrowing of the resonant Mie bands with e increasing, as can be clearly seen for the lowest TE 01 magnetic Mie resonance in Fig. 2.
The photonic band structures for the 2D square lattice of circular rods were computed numerically using the plane wave expansion method 28 . The square 2D lattice has a square Brillouin zone, with the three special points G, X and M corresponding to We restrict here our discussion to the case for which the wave vector k of the eigenmodes is oriented in the G-X direction of the first Brillouin zone displaying the lowest Bragg frequencies in comparison with the G-M direction (Fig. 2). The photonic band structures were obtained with 16,384 (128 Â 128) plane waves.
The transmission spectra were calculated using the CST Microwave Studio software for the wave vector of the incident beam parallel to the G-X direction. We considered the 2D square lattice with different thicknesses along x axis that leads to creation of the photonic bandgaps, including thicknesses of 10 lattice layers as shown in Fig. 2.
The calculated data allow us to analyse precisely the evolution of the low-frequency region of the photonic band structure formed by a complicated mixture of the dispersion curves originating from Mie and Bragg resonances and propagating modes. Since our goal is to discuss the PhC-MM transition, we concentrate on the TE polarization where the lowest TE 01 magnetic dipole resonance of the rods results in a negative magnetic permeability in the Mie gap spectral region 19 .
The physics of the PhC-MM transition is displayed by the gap map (Fig. 3) that relates the Bragg and Mie gaps positions to the rod permittivity e for given r/a ¼ 0.25. These illustrate essential features such as strong decreasing of Mie resonance frequencies with e increase followed by intersection of the Mie and Bragg bands. The Mie resonances of individual rods form both non-dispersive flat bands in the photonic structure and Mie photonic gaps in the energy spectrum (Figs 2 and 3  permittivity e, the Mie resonances of single rods and correspondingly Mie bands of the structure demonstrate a shift from the higher to low frequencies, crossing the strongly dispersive Bragg bands. The creation of the lowest Mie gap driving the artificial magnetism is clearly seen both in the band structure and in the transmission spectra (Figs 2 and 3).
Two crossover scenarios for a metacrystal. Armed with the results of our calculations, we verify our concept experimentally by engineering a metacrystal composed of plastic circular tubes filled with water and forming a 2D square lattice with variable lattice constant a (Fig. 4). The structure of the metacrystal enables us to perform two types of experiments according to the r/a-and e-scenarios. The r/a-scenario was realized using a special construction of the metacrystal that makes it possible to change the distance between tubes synchronously in x and y directions keeping the 2D square symmetry of the lattice (Fig. 4). For the e-scenario, we employ the advantages of strong temperature dependence of the dielectric permittivity of water that is characterized by e ¼ 80 at 20°C and e ¼ 50 at 90°C in the microwave frequencies range from 1 to 6 GHz (refs 29,30). On the basis of the calculated results and the parameters of the metacrystal, we chose intervals of most interest for each scenario, namely 60 e 80 at 0:7 r=a 0:8 for the e-scenario and 0:65 r=a 1:9 at e ¼ 60 for the r/a-scenario. The transmission spectra of the metacrystal installed in an anechoic chamber were measured in the 1-3-GHz range. A rectangular horn antenna (TRIM 0.75-18 GHz; DR) connected to a transmitting port of the vector network analyser Agilent E8362C was used to approximate plane wave excitation. A similar horn antenna was employed as a receiver. For the electromagnetic wave incident perpendicular to the tube axis z, the two transverse polarizations appear decoupled and the vector wave problem can be reduced to two independent scalar equations. As a result, the modes of 2D PhCs can be classified as either TE modes (E x , E y and H z ) with the electric field confined to the x À y plane and the magnetic field polarized along the axes of the rods (Fig. 4) or TM modes (H x , H y and E z ) for which the magnetic field is oriented perpendicular to the rod axis z. For the TM polarization, such structures can not show any magnetic activity 31 , so we only consider TE polarization. Figure 5 demonstrates excellent agreement between the measured and calculated transmission spectra both for r/a and e dependencies. For the r/a-scenario, we started our experiment from the PhC phase (r/a ¼ 0.08) and compressed the structure by decreasing the lattice constant a (Fig. 5a). The lowest Bragg gap demonstrates remarkable shift towards high-frequency region, then nearly almost disappears at r/aE0.09 when it approaches the unchanging TE 01 Mie gap from the low-frequency side, and finally appears again at r/aE0.095 on the high-frequency side of the Mie gap. As a result, the lowest Bragg band and the TE 01 Mie band change their positions in the energy scale inaugurating the transition into MM phase. In contrast, for the e-scenario, the Bragg gap position changes very slowly as opposed to the strong shift of the TE 01 Mie band that finally leads to the same general result (Fig. 5d). Figure 6a shows our main result: the PhC-MM phase diagram for the 2D square lattice of circular rods. The axes represent the dielectric permittivity e and filling ratio r/a. The phase diagram was obtained based on 50 gap maps (such as in Fig. 3a) calculated in the permittivity region 1 e 100 with the step r/a ¼ 0.01 for TE polarization. The diagram is divided into two areas, which represent the PhC and MM phases. The transition from PhC phase to MM phase can be identified by critical points, which are defined by a special combination of permittivity e and filling ratio r/a. The curvature of the second dispersion curve around the G point decreases and finally becomes perfectly flat along the GX path (at e ¼ 19.5, r/a ¼ 0.25, Fig. 2k). With future increase in e or change in r/a, the TE 01 Mie gap completely splits from complicated coupled Mie-Bragg band becoming the lowest gap in the spectrum and the MM phase appears. Within this Mie gap, the magnetic permeability is negative and all waves in the medium are evanescent. Since all the Bragg gaps are located at higher frequencies, any diffraction losses in the vicinity of the Mie resonance are absent. The transition from the PhC phase to the MM phase is completed and the medium is classified as the MM with mo0. A phase transition is usually characterized by an order parameter that basically arises from symmetry breaking but can also be defined for non-symmetry-breaking transitions 32 . Here  TE 01   TE 11   TE 12  TE 02  TE 21   TE 0k  TE 1k  TE 2k  TE  for the unusual phase transition PhC-MM, we define the order parameter Z as a measure of splitting in the energy scale between the TE 01 Mie and Bragg gaps. It is zero in the PhC phase and nonzero in the MM phase. Figure 6b presents the order parameter depending on both external variables e and r/a. Now, we review some features of the PhC-MM phase diagram. At low filling ratio r/ao0.05, the concentration of the completely isolated rods in the lattice is too small for achieving negative average permeability at reasonable values of e. In contrast, when distance between rods become very small (r/a40.45), the inter-rod interaction become strong and the hopping of the Mie photons from one rod to another dominates. This means that Mie resonances are not localized anymore and they finally exhibit the character of propagating Bloch waves. All requirements for the appearance of the MM phase are broken and the PhC phase extends its region of existence to very high e. We notice that for the metacrystals with a small lattice constant a, the frequency of the Mie resonance is increasing. Therefore, there is no transition to the MM phase in this limit.

Discussion
Identification of candidates for dielectric low-loss materials is critical for designing a practical MM for applications in different ranges of the electromagnetic spectrum. As an example, in Fig. 6a, we mark the regions of the MM phase for water in the microwave range (region 55 e 87) 31 , the chalcogenide glass Ge 2 Sb 2 Te 5 (eE30) 33 and tellurium (eE25) 34 in infrared. Thus, the well-studied 2D square lattice of germanium (eE16) and silicon (eE13) 34 rods has only PhC phase.

Conclusion
We have studied a metacrystal composed of dielectric circular rods and introduced the concept of a phase diagram in the permittivity-filling ratio plane. The boundaries of the MM phase have been obtained theoretically and confirmed experimentally. On the basis of the proposed approach, one can obtain different PhC-MM phase diagrams by altering the dimension, symmetry, composition, size and geometry of the structural elements within a unit cell. The underlying mechanism of the PhC-MM transition provides unique criteria for goal-oriented search of novel low-loss r /a-scenario ( =62) -scenario (r/a = 0.08)  MM with a wide range of applications at infrared and visible frequencies.