Exciton-like electromagnetic excitations in non-ideal microcavity supercrystals

We study localized photonic excitations in a quasi-two-dimensional non-ideal binary microcavity lattice with use of the virtual crystal approximation. The effect of point defects (vacancies) on the excitation spectrum is investigated by numerical modelling. We obtain the dispersion and the energy gap of the electromagnetic excitations which may be considered as Frenkel exciton-like quasiparticles and analyze the dependence of their density of states on the defect concentrations in a microcavity supercrystal.

fields of the eigenmodes of neighboring microcavities is taken into account, so that photons are allowed to move along the surface of the microcavity array. For the sake of generality, we assume that each cell of the photonic supercrystal lattice may contain an arbitrary number of elements.
Hamiltonian H of the model system we consider (for more details see also Ref. 3) writes: Subscripts n and m are two-dimensional integer lattice vectors, a and b numerate sublattices, whose total number is s. E na ;"v na , where v na is the frequency of photonic mode localized in the na-th site (cavity). Quantity A namb defines the overlap of optical fields of the na-th and mb-th cavities and the transfer of the corresponding excitation, Y z na , Y na are bosonic creation and annihilation operators describing the photonic mode. Hamiltonian (1) is formally identical to the tightbinding excitonic Hamiltonian in a semiconductor crystal 16,17 , for which reason the studied electromagnetic excitations can naturally be referred to as exciton-like. It is worth stressing that we discuss photonic supercrystal excitations and no electronic transitions are involved. Nevertheless, it will be seen below that the dispersion relations of purely electromagnetic crystal excitations in the studied system are quite similar to the Frenkel exciton bands in molecular crystals 16,20 .
Let us consider a topologically ordered non-ideal lattice of microcavities with point-like defects, namely vacancies and non-typical microcavities. In such a system, Hamiltonian (1) is no more translation invariant, hence the quantities v na and A namb are configurationally dependent. A convenient tool to study the quasiparticle excitation spectrum in a system with randomly distributed defects consists in configurational averaging of the resolvent of the corresponding Hamiltonian 18 . An averaged resolvent is translation invariant, hence the corresponding elementary excitation spectrum can be characterized by a wave vector k. This type of calculation can only be carried out if adopting a certain approximation specific to the considered system. A widespread method of computation of quasiparticle states in disordered media is the virtual crystal approximation (VCA) 18,19 . It proves sufficient to elucidate the transformations of elementary excitation spectra under varying defect concentrations.
In what follows we rely on this method to compute and analyze the spectrum of electromagnetic excitations as well as the corresponding optical characteristics of the considered non-ideal supercrystal.
Since the VCA consists in replacement of configurationally dependent Hamiltonian parameters with their averaged values, Hamiltonian of a ''virtual'' crystal H ph in our case reads as follows: Here angular brackets denote configurational averaging. In an imperfect lattice of coupled cavities quantities E na and A namb are configurationally dependent and can be written in terms of the random variables where g n,m na,mb~1 if the na (mb)-th supercrystal cell is occupied by a n(a)-th or m(b)-th type of cavity (the total number of types is s(a) and r(b) correspondingly) and g n,m na,mb~0 otherwise. Configurational averaging of Eqs. (3) carried out in accordance with the VCA (similarly to the quasiparticle approach 15,22 ) yields where C n a  (2) are found via its diagonalization by means of the Bogolyubov-Tyablikov transformation 16,17 , and are ultimately determined by the system of algebraic equations of the order s: u l (k) are eigenfunctions of the s3s matrixL whose elements are expressed through the corresponding characteristics of the Hamiltonian (2): r na being the radius-vector of a resonator belonging to the a-th sublattice of the n-th elementary cell. The solvability condition of the system (5) yields the dispersion law v l (k) of electromagnetic excitations in the considered photonic supercrystal.

Results and Discussion
Consider localized electromagnetic excitations in a two-sublattice (s52) system of cavities. The left-hand side of Eq. (7) is then a second-order determinant, which when equated to zero gives the following dispersion of photonic excitations: Here 12 (k) and L 21 (k)52A 21 (k) are the matrix elements of operatorL.
To be more specific, let us consider a spectrum of electromagnetic excitations in a binary system where each sublattice contains only two types of cavities. In such a case, the quantities E na h i and A namb are given by Being applied to the supercrystal lattice of microcavities where the only defects are vacancies, these expressions take the form is vacancy concentration in the 1st and/or 2nd sublattices. Concentrations must obviously satisfy the relations C In (9)

:
A 21 characterize the overlap of optical fields of cavities pertaining to the same sublattice but different cells.
The energy spectrum of exciton-like electromagnetic excitations is defined by the type of the considered sublattices and the quantities E na h i and A namb . Below we carry out a nearest-neighbor calculation for the case of a square Bravais lattice of period d 3 . Location of cavities is defined by the radius-vector r na 5r n 1r a , hence their location in the zero elementary cell (r n 50) is defined by vectors r 01 50 and r 02~d 2 e 1 ze 2 ð Þrespectively, where e 1 and e 2 are the basis vectors of the rectangular coordinate system (Fig. 1). In the adopted approximation the matrix elements A ab (k) can with reasonable accuracy be written as: and thus the corresponding matrix elements of operatorL take the form In (10) the overlap characteristic of optical fields A 11 (22) (d) defines the transfer probability of electromagnetic excitation between the nearest neighbors in the first (second) sublattice, and A 12(21) (0) is the excitation transfer probability between cavities in the first (second) and second (first) sublattices in an arbitrary cell. Substitution of expressions (10) for L ab (k) into Eq. (8) gives the dispersion law v 6 (k) for electromagnetic excitations (Fig. 2a,b,c). We performed calculation for modeling frequencies of resonance photonic modes in the cavities of the first and second sublattices v 1 : Eñ 1 h i= h6 : 10 15 Hz and v 2 : Eñ 2 h i= h~8 : 10 14 Hz respectively and for the overlap parameters of resonator optical fields A 11 /2 53?10 14 Hz, A 22 / 2 55?10 13 Hz and A 12 /2 <A 21 /2 55?10 13 Hz. The lattice period was set equal to d53?10 27 m.   Fig. 3. The surface Dv C V 1 ,C V 2 À Á proves non-monotonic and turns to zero in a certain range of In other words in a certain region of ð Þ electromagnetic excitations pass unhindered through the binary two-sublattice microcavity system. Surfaces in Fig. 2a, plotted for C V 1~0 :55, C V 2~0 :1 and in Figs 2b,c plotted for C V 1~0 :84, C V 2~0 :2 and C V 1~0 :9468, C V 2~0 :7 exemplify the cases respectively. The presence of two dispersion branches v 6 (k) (see Eq. (8)) reflects a two-sublattice structure of the resonator system. For molecular crystals with two molecules in a cell an analogous occurrence of two branches in the dispersion law is referred to as the Davydov splitting of exciton zone 20 .
It is important to know how the specificities of the spectrum of the studied quasiparticles are manifested in their density of states . For a non-ideal two-dimensional system with a square lattice the function g v,C V 1 ,C V 2 À Á is given by an integral (see Ref. 21): where integration is carried out along an isofrequency contour v n (k)5v in the (k x ,k y )-plane (Figs. 4a,b,c,d,e,f). Dispersion of quasiparticles in the considered system (Figs. 2a,b,c) has nine critical points in the k-space (where = k v n (k)50), which is indicative of a possible occurrence of singularities in the density of states g v,C V 1 ,C V 2 À Á . These do in fact always arise as demonstrated in Figs. 5a,b,c,d,e,f. Points k x ,k y À Á~0 ,0 ð Þ, +p=d,+p=d ð Þ , +p=d, ð +p=dÞ fail to yield a singularity because there the tending to zero gradient = k v n (k) is offset by a shrinking integration contour, which ultimately gives a finite result of integration in (11) (Figs. 4a,b,c,d,e,f).
for the low-frequency dispersion branch are due to the critical points (k x ,k y )5(0,6p/d),(6p/d,0) (indicated by black diamonds in Figs. 4d,e,f). The high-frequency branch in its turn can behave in a more peculiar way. In the case Dv?0 the singularities of the high-frequency density of states g 1 are due to the already mentioned points(k x ,k y )5(0,6p/d),(6p/d,0) (black diamonds in Fig. 4a). For Dv50 however the corresponding points may fall inside the Brillouin zone on either the centerlines of the (k x ,k y )-square (Fig. 4b) or on its diagonals (Fig. 4c). In Figs. 5a,d   Fig. 2a; b), e) upper and lower surfaces in Fig. 2b; c), f) upper and lower surfaces in It turns out that in the region Dv C V 1 ,C V 2 À Á =0 the density of states g 1 is all but independent of C V 2 , while g 2 is almost unaltered by variations in C V 1 . This is explained by the smallness of the term L 12 (k)L 21 (k) as compared to [L 11 (k)2L 22 (k)] 2 in Eq. (8). Figs. 5b,c and 5e,f give examples of the typical g 1 and g 2 curves for concentration values corresponding to Dv50 (here we took C V 1~0 :84, C V 2~0 :2 and C V 1~0 :9468, C V 2~0 :7). Their evident non-monotonic and discontinuous character is similar to the analogous dependence g(v) obtained in Ref. 21 for phonon excitations.

Conclusion
A number of recent experimental works indicate that microcavity supercrystals may have interesting applications, in particular for creating the optical clockworks of unprecedented accuracy [22][23][24] . We have used the virtual crystal approximation to model the effect of lattice point defects (vacancies) on the spectrum of exciton-like electromagnetic excitations in a quasi-2D binary microcavity lattice. The energy spectrum of electromagnetic excitations affects the density of states of electromagnetic excitations and alters propagation of normal electromagnetic waves. The obtained dispersions of electromagnetic excitations are noticeably more complex than those of primitive lattices. This complexity is due to the non-ideality of the structure and to the presence of two sublattices. The latter entails multiple manifestations in experimentally observable integral characteristics of optical processes. Evaluation of excitation spectra in more complex photonic systems requires the use of more sophisticated computational methods. Depending on particular cases such can be the one-or multiple-node coherent potential method 18 and the averaged T-matrix method 25 along with their various modifications. Our study contributes to the modeling of novel functional materials with controllable propagation of electromagnetic excitations.  Fig. 3). Solid lines correspond to Fig. 2a. Curves a) are valid for any value of C V 2 in the range (0…1). Curves d) are valid for any value of C V 1 in the range (0…0.8). b) and e) show the densities of states for the upper and lower surfaces, respectively, in Fig. 2b. c) and f) show the densities of states for surfaces in Fig. 2c. www.nature.com/scientificreports SCIENTIFIC REPORTS | 4 : 6945 | DOI: 10.1038/srep06945