Collective Lattice Resonances in All-Dielectric Nanostructures under Oblique Incidence

: Collective lattice resonances (CLRs) emerging under oblique incidence in 2D ﬁnite-size arrays of Si nanospheres have been studied with the coupled dipole model. We show that hybridization between the Mie resonances localized on a single nanoparticle and angle-dependent grating Wood–Rayleigh anomalies allows for the efﬁcient tuning of CLRs across the visible spectrum. Complex nature of CLRs in arrays of dielectric particles with both electric dipole (ED) and magnetic dipole (MD) resonances paves a way for a selective and ﬂexible tuning of either ED or MD CLR by an appropriate variation of the angle of incidence. The importance of the ﬁnite-size effects, which are especially pronounced for CLRs emerging for high diffraction orders under an oblique incidence has been also discussed.

Collective lattice resonances (CLRs) arising in arrays of NPs originate from the strong interaction between NPs composing the lattice, which usually occurs under the illumination with a wavelength close to Wood-Rayleigh anomalies (WRAs) [42,43] of the array. In this case, a majority of NPs are excited with the same phase, which results in ultra-narrow high-Q spectral features. CLRs have been well studied for nanostructures from plasmonic NPs for a long time [44][45][46][47][48][49][50][51][52][53][54][55][56], while the all-dielectric analogues have gained attention only a decade ago [57]. In contrast to plasmonic NPs (in most of the cases characterized by weak magnetic and strong electric responses), all-dielectric NPs with pronounced electric and magnetic optical resonances [58] give rise to a rich variety of tunable CLRs that emerge even in regular rectangular-shaped arrays [32]. Moreover, 2D structures from all-dielectric NPs with two distinct electric dipole (ED) and magnetic dipole (MD) resonances exhibit inherently more sophisticated and intriguing behavior compared to the respective situations in purely ED-responsive plasmonic arrays, for example, in disordered [36,59] and finite-size [37,38,[60][61][62] lattices.
Most of the numerical and theoretical studies of CLRs deal with infinitely large arrays of NPs under a normal incidence; however, it can be easily anticipated that, under oblique incidence, all-dielectric arrays may exhibit a plethora of properties overlooked in the literature. Our expectations are well justified by the reported results for plasmonic arrays [63,64] (with only ED response), which imply that, for all-dielectric NPs with ED and MD resonances, one may expect to observe even more effects. Thus, in this work, we address this problem and study electromagnetic properties of 2D arrays of all-dielectric NPs under oblique illumination. Moreover, we focus on finite-size arrays and reveal a role of the array size (in terms of a total number of NPs composing the lattice) on CLRs emerged under such conditions, which is more relevant to the experimental setups than infinite-array approximation.

Coupled Dipole Approximation
Consider an array from N tot spherical NPs embedded in a vacuum and illuminated by a plane wave, which, at any location r, reads as where E 0 = E 0x , E 0y , E 0z and H 0 = H 0x , H 0y , H 0z are amplitudes of the electric and magnetic fields, and k is a wave vector. The time dependence exp(−iωt) is assumed and suppressed throughout a paper. In the framework of point-dipole approximation, electric, d i , and magnetic, m i , dipole moments induced on a given i-th NP under such an incidence are coupled to the respective dipoles on other j = i NPs and to the external field as [57,65,66] (unlike these works, we use Gauss units) where r i is the position of the i-th NP center, α e = 3ia 1 2k 3 and α m = 3ib 1 2k 3 are electric and magnetic dipole polarizabilities, where a 1 and b 1 are scattering coefficients [67], k = |k| = 2π/λ, and λ is a wavelength. Tensors G ij and C ij describe the interaction between dipoles induced on i-th and j-th NPs: where I is a 3 × 3 unit tensor, ⊗ denotes a tensor product, r ij = |r ij | = |r i − r j | is center-to-center distance between i-th and j-th NPs, and ε αγβ is Levi-Civita symbol with α, β, γ denoting Cartesian components of the tensors.
For an array with a given geometry and composition of NPs, the solution of the linear system of Equations (1) yields d i and m i induced on each i-th NP; thus, the electromagnetic response of the array to the incident excitation can be explicitly found. Particularly, in this work, we consider the total amount of the electromagnetic energy scattered and absorbed by the array normalized to the sum of the cross sectional area of all NPs, i.e., the extinction efficiency [57,66]: where the asterisk denotes a complex conjugate, R is the radius of the NP, and takes the imaginary part.

Wood-Rayleigh Anomalies
CLRs emerge at wavelengths close (slightly red-shifted) to WRAs, which for a general case of a regular 2D lattice (with pitches h x and h y along x and y axes, as shown in Figure 1a) takes place if where K x = (2π/h x )x and K y = (2π/h y )ŷ are reciprocal lattice vectors, k = (k x , k y ) is wave vector of a wave propagating in the lattice plane, k σ is projection of the incident wave vector on the lattice plane, [p, q] is a pair of integers which denotes the order of the anomaly, and symbolˆdenotes a unit vector.
Explicitly, x and y components in Equation (3) read as where θ x and θ y are angles between the z-axis and projections of k to XOZ and YOZ planes (see Figures 1a and 2a, respectively). In a homogeneous environment, the wave vector of a wave propagating in the lattice plane is |k | 2 = k 2 x + k 2 y = (2π/λ) 2 , which along with Equation (4) provide the quadratic equation in λ: where, for a given combination of integers [p, q], one can get a corresponding spectral position λ p,q of WRA of [p, q] order. We emphasize that the hybridization between localized Mie resonances and [±1, 0] or [0, ±1] WRAs is usually considered in a solid body of the literature [32,36,37,57]. For a special case of normal incidence (θ x = θ y = 0), these WRAs are simply λ ±1,0 = h x and λ 0,±1 = h y . However, Equation (5) immediately implies that the broad variation of θ x and/or θ y may result in CLRs emerging from the hybridization with WRAs of higher order (i.e., |p|, |q| > 1), which are studied below.

Results
We consider regular arrays from Si NPs with R = 65 nm, arranged in a 2D rectangular lattice with h y = 480 nm and h x = 580 nm. A direct comparison with full-field simulations ( [31] Figure 1) ([38] Figure 3) has confirmed a reliability of the coupled dipole approximation (1) for arrays with similar pitches and the same R. Under a normal incidence with E 0 = (E 0x , 0, 0) and H 0 = 0, H 0y , 0 , arrays with these geometrical parameters exhibit ED and MD CLRs ( [37] Figure 2b Since the efficient tuning of ED and MD CLRs occurs if h y,x are changed in a direction perpendicular with respect to the polarization of E 0 or H 0 ( [32] Figures 2 and 4), ( [36] Figure 2), it is insightful to consider an incidence with only one θ x,y varied keeping the other θ y,x = 0. Following this approach, it is possible to study separately ED and MD CLRs, while, for any other oblique incidence with θ x = 0 and θ y = 0, one can expect the optical response to be a superposition of the studied examples. Moreover, even for θ y = 0, variation of θ x implies gradual blue-shift of [0, ±1] WRA, which allows for fine-tuning of ED CLRs for θ x < 25 • . Such angle-dependent hybridization between Mie resonances on single NP and WRAs paves a way for the efficient tuning of ED CLRs in the 450-540 nm range. It is noteworthy that MD CLR vanishes quite rapidly with a slight change of θ x , since λ ±1,0 strongly depends on θ x (cf. Equation (5)); thus, for θ x > 5 • , only an MD resonance of a single NP is observed. As it might be expected from Ref. [37], the extinction efficiency at the CLR regime grows with N tot ; thus, the CLRs that have emerged from the interaction with high-order WRAs are more pronounced for larger arrays, which can be clearly seen by the following Figure 1b   On the contrary, by changing θ y and keeping θ x = 0 constant, one can control the spectral position of MD CLRs, as clearly shown in Figure 2. In this case, however, ED CLR does not vanish so rapidly for 0 < θ y < 5 • (as it does MD CLR from Figure 1 for the opposite case of 0 • < θ x < 5 • ). ED resonance on a single NP efficiently couples to [0, −1] WRA and thus corresponding ED CLR can be tuned all the way up to ≈ 570 nm, and, finally, overlap with [±1, 0] MD CLR around 580 nm under θ y ≈ 12 • incidence. MD CLR, however, can be efficiently tuned only for [±1, 0] WRA, while, wavelengths of high-order WRAs appear to be quite far away from MD resonance, and only [±1, −1] efficiently interacts with MD resonance, but for quite large angles of incidence.  3 further elaborates on discussed effects and shows several extinction spectra for the clearer presentation. Indeed, under oblique incidence, one can observe efficient tuning of the optical properties of the 2D lattice from Si NPs. The "gradual" quadratic λ p,q (θ 2 ) and "rapid" linear λ p,q (θ) dependence with one of the θ x , y being zero (see Equation (5)) allows for a flexible control of ED and MD CLRs. For a strong coupling of single-particle resonance with WRAs (i.e., for spectral regions where they almost overlap), the finite size effects are of particular importance, while, for a weakly coupled case (i.e., for spectral regions where they are sufficiently far from each other), these effects have a minor impact. For example, from Figure 3a, one can see that Q ext rapidly grows with increasing N × N for ED CLR strongly coupled to [0, ±1] around λ ≈ 490 nm, while MD CLR for [1, 0] WRA becomes almost independent on N tot with increasing θ x .
Finally, Figure 4 demonstrates angle-dependent Q-factors of ED CLRs from Figure 1, for arrays with different N × N. As it might be expected, the Q-factor is generally larger for arrays with larger N × N. Interestingly, for 0 • ≤ θ x ≤ 24 • incidence, with increasing θ x , i.e., weakening coupling between single-particle resonance and [0, ±1] WRA, Q-factor gradually converges to ≈ 30 value for any array size at θ x ≈ 24 • . It is noteworthy that for CLRs that have emerged from the hybridization with high-order [−2, 0] WRA, Q-factor is about two times larger than that of commonly considered CLRs that have emerged from the interaction with [0, ±1] WRA.

Conclusions
To conclude, we have considered the features of collective lattice resonances emerging in regular 2D arrays of all-dielectric nanoparticles under an oblique incidence. For a particular case of Si constituents with fixed pitches h x,y , we have shown that high-order Wood-Rayleigh anomalies appear to be within a visible range and close to the optical resonances of a single Si nanoparticle. Under such conditions, an efficient hybridization between either electric dipole or magnetic dipole resonance of a single nanoparticle with, for instance, [−1, ±1], [−2, 0] or [±1, −1] Wood-Rayleigh anomalies leads to the appearance of collective lattice resonances, which can only be observed under an oblique incidence. Moreover, by adjusting the angle of illumination, one can efficiently tune the spectral position of such collective lattice resonances across the whole visible spectrum. We emphasize that all the results presented in this work correspond to a single lattice (with given N × N). It means that the optical response of a considered nanostructure can be tuned to a variety of scenarios by simply inclining the array with the respect to the incident illumination, which, in some cases, might be more preferable compared to other strategies used to tune the wavelength of the collective lattice resonances [69,70]. Finally, we show that the total number of nanoparticles composing arrays may play a crucial role for collective lattice resonances under an oblique incidence, depending on the coupling strength between Wood-Rayleigh anomalies and single-particle resonance. Thus, results reported in this manuscript might be used in the design of photonic devices where the tuning of the resonant response can be achieved without complex technologies.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: CLR collective lattice resonance ED electric dipole MD magnetic dipole NP nanoparticle WRA Wood-Rayleigh anomaly