Light enhancement by quasi-bound states in the continuum in dielectric arrays

The article reports on light enhancement by structural resonances in linear periodic arrays of identical dielectric elements. As the basic elements both spheres and rods with circular cross section have been considered. In either case, it has been demonstrated that high-$Q$ structural resonant modes originated from bound states in the continuum enable near-field amplitude enhancement by factor of $10$--$25$ in the red-to-near infrared range in lossy silicon. The asymptotic behavior of the $Q$-factor with the number of elements in the array is explained theoretically by analyzing quasi-bound states propagation bands.


I. INTRODUCTION
Light trapping by periodic structures is a mainstream idea in thin film photovoltaics [1,2]. The idea dates back to the paper by Sheng, Bloch, and Steplemann who fist proposed to employ a periodic grating substrate in amorphous silicon solar cells [3]. Since then the periodic structures that diffract light and increase the optical length within the absorbing material have been a subject of numerous publications [4][5][6][7][8]. Most remarkably the dielectric gratings have allowed for large enhancement factors [9] which exceed the fundamental limit for a Lambertian cell [10] that holds well only if there is a high density of optical modes in the plane of the structure. At the same time as one approaches the nanophotonic regime the wave nature of light comes into play to call for theoretical approaches taking into account optical resonances [11].
In this article we consider light enhancement by high-Q structural resonances in linear one-dimensional periodic arrays of identical dielectric elements. In what follows the term structural resonance will be applied to a resonance whose position and width are dictated by the spatial distribution of dielectric elements. Thus, the structural resonances can be contrasted to both natural material resonances and Mie resonances of individual dielectric elements. As the basic elements both spheres and rods with circular cross section will be considered. To achieve high-Q resonances and, consequently, high enhancement factors we propose to tune the parameters of the arrays to bound states in the continuum (BSCs) which are localized states with the eigenfrequency embedded into continuum of propagating solutions [45]. With the state of the art in the BSC research thoroughly reviewed in the above reference here we simply mention the key experiments on optical BSCs in periodic dielectric structures [46][47][48][49][50][51].
The BSCs in linear periodic arrays of dielectric cylinders were predicted theoretically in the earlier paper [52] where the role of the BSCs in wave scattering was also highlighted.
More recently, the problem of BSCs in arrays of dielectric rods [53] was revisited with the existence domains in the plane of radius and dielectric constant of the cylinders determined through extensive numerical sumulations. Unlike the guided solution below the line of light [54,55] the qusi-bound modes in the parametric vicinity of a true BSC are resonantly coupled with the freely propagating plane waves to emerge in the scattering cross-section as sharp Fano resonances. In fact the collapse of a Fano resonance is a generic feature inherent to BSCs [56][57][58][59] including quasi-BSCs in plasmonics [60]. In the field of optics the above feature has allowed for design of narrow band normal incidence filters [61][62][63]. The BSCs in linear arrays of dielectric spheres were first reported in [64]. As before for arrays of cylinders we contrast the BSCs in arrays of spheres to the guided solutions below the line of light [32,33,39,40,65,66]. The effect of BSCs on plane wave scattering was considered [67,68] with the trapping of light with orbital angular momentum theoretically predicted. Light enhancement by symmetry protected BSCs in photonic crystal slab was demonstrated in [69][70][71]. Here, along the same line we propose to employ arrays of finite number of dielectric elements. Although formally BSCs do not exist in finite structures [72] their traces could be observed in form of high-Q structural resonances [41] similarly to the traces of photonic band structure emerging in scattering on small clusters of spherical particles [73].
The article is organized as follows. In Section II we briefly review the BSCs in infinite  Table I. arrays in dielectric rods. Then we demonstrate structural resonances in the parametric vicinity of BSCs in finite arrays. A simple theory predicting the positions of the structural resonances is proposed. In Section III we show that the same theory applies for the arrays of dielectric spheres. In Section IV we elaborate asymptotic behavior of the Q-factor and the enhancement factor for various types of the structural resonances with the increasing number of elements in the array. Finally, we conclude in Section V.

II. QUASI-BSCS IN ARRAYS OF DIELECTRIC RODS
We consider an array of identical dielectric rods of radius R arranged along the x-axis with period a. The axes of the rods are collinear and aligned with the z-axis. The crosssection of the array in x0y-plane is shown in Fig. 1 (left panel). The numerical procedure for finding electro-magnetic field in arrays of rods is well-known in literature [74] and was adapted for BSCs in [52]. Suppose that the array is illuminated by a stationary TM plane wave with its electric vector pointing along the axes of the rods, then according to the above references we have the scattered field inside the rods in the following form where j enumerates the rods, θ j , ρ j are the local polar coordinates for the j rod, J m (k 0 √ ǫρ j ) -the Bessel function, ǫ -the dielectric permittivity, c m (j) -the expansion coefficient, and k 0 -the frequency. Outside the rods we have a similar expression for the scattering function where H (1) m (k 0 ρ j ) is the outgoing Hankel function. The summation over m in Eqs. (1) and (2) runs to infinity. The only approximation in the numerical method is truncation to a finite number of summands in Eqs. (1) and (2). This approximation makes possible to produce a finite interaction matrix of the scattering system and subsequently a finite number of equations for c m (j), c m (j) as described in [52]. The method is known to converge rapidly with the number of multipoles [74]. In our computations we used m = 0, 1, ..., 9 which results in the relative truncation error 10 −5 as defined in [41]. We refer the reader to [52] for the lengthy equations for amplitudes c m (j), c m (j).

A. BSCs in infinite arrays
In case of infinite arrays the summation over index j in Eqs. (1) and (2) is run from minus to plus infinity. The translation invariance allows to apply the Bloch theorem in the form [74] where k x is the x-axis component of the wave vector (Bloch vector). Although the solution (2) is a superposition of outgoing functions it still can be totally decoupled from the far field outgoing waves due to destructive interference between the waves emanating from different rods [52]. In that situation a BSC exists in the system even without the array being illuminated from the far zone. In dependence on k x the BSCs are exceptional points in the quasi-BSCs propagation bands where the imaginary part of the resonant frequency k 0 turns to zero [41]. In general the asymptotic behavior of the imaginary and real parts of the resonant frequency in the vicinity of a BSCs is described by the following formulas and where k BSC 0 is the BSC frequency, k BSC x -the BSC wave vector along the array axis, and ν, µ -the leading coefficient of polynomial expansion [41]. The physical meaning of Eq. (4) is simple; once the system is detuned in k x -space from a true BSC point k BSC  Table I we collect the parameters of six BSCs found for infinite arrays with ǫ = 15 (Silicon). The values of a µ and a ν in Table I were extracted from the data by a polynomial fit. For a typical picture of the band structure of Bloch BSCs the reader is addressed to [41].
There are two generic types of BSCs in arrays of dielectric rods [52], symmetry protected, and unprotected by symmetry. Field patterns of symmetry protected BSCs 2, 3 are shown in the left panel in Fig. 1 where one can see that those BCSs are symmetrically mismatched with a plane wave normally incident to the array. The asymptotic behavior of the resonance live-times in the vicinity of a BSC was recently considered by Yuan and Lu [76] who argued that for symmetry protected BSCs the leading term should be ν = 2. There were no, however, reliable estimate for a ν=2 . In our numerical simulations we found that both ν = 2 and ν = 4 are possible. In Table I we presented the data on four symmetry protected BCSs a couple for each ν = 2 and ν = 4. The asymptotic behavior of unprotected BSCs was also considered in [76] with a mathematically rigorous result that for the the standing wave BSCs unprotected by symmetry the leading term is ν = 4 which is in accordance with our findings for BSC 5. For Bloch BSC 6 we found ν = 2.

B. Quasi-BSCs in finite arrays
Next we discuss the the light scattering by various types of quasi-BSCs in finite arrays to highlight the features related to the asymptotic behavior described by Eqs. (4) and (5).
The array now consists of N elements and summation over j in Eqs. (1) and (2)  Notice that only positive k x were considered for the system is symmetric with respect to the inversion of the x-axis. For simplicity the response was recorded as the mean of the absolute value of the leading coefficient in the expansion Eq. (1) where m 0 is the index of the coefficients c m 0 (j) in Eq. (1) with the largest absolute value (see top panel in Fig. 2 where m 0 = 1).
In both cases one can see "bright" spots following the asymptotic expression for the real part of the resonant frequency (5). These spots correspond to the structural resonances with the widths of the spots along k 0 -axis proportional to the inverse Q-factors which will be considered in Section IV. The nature of the structural resonances could be easily understood by close examination of the upper panel in Fig. 2 where we plotted the expansion coefficients c m (j) for the first resonant spot. One can see from the upper panel in Fig. 2 that the solution is nothing but a sinusoidal standing wave locked between the edges of the array. Thus, to recover the spectrum of the structural resonances one can write down the following condition for the k (p) x of the p structural resonance.
Then by virtue of Eq. (5) we have with p corresponding to the number of half-wavelengths between the edges and the field patterns where f (x, y) = f (x + a, y) is the periodic functions corresponding to the BSCs profiles in Fig. 1. Remarkably, despite the BCSs are standing waves the position of the first structural resonance p = 1 is shifted from k x = 0 according to Eqs. (7) and (8). The field pattern of the first resonance is shown on top of the upper panel in Fig. 2 to visualize Eq. (9). Notice

III. QUASI-BSCS IN ARRAYS OF DIELECTRIC SPHERES
The system under scrutiny is schematically depicted in Fig. 4. For arrays dielectric spheres we will follow the recipes from [64] where the method by Linton, Zalipaev, and Thompson [66] was adapted for finding BSCs. The parameters of two BSCs, one symmetry protected, and one unprotected by symmetry are given in Table II. In the above referenced method the magnetic and electric vectors could be found in terms of Mie coefficients a m n (j), b m n (j). For instance, outside the spheres one has for the the scattered EM field electric vector E(r) where j the number of the sphere in the array, r j -the coordinates of the j sphere, m -azimuthal number, m * = max(1, m), and N m n (r), M m n (r) are spherical vector harmonics [77]. Again in the case of infinite arrays we have according to the Bloch theorem a m n (j) = a m n (0)e ikxaj , b m n (j) = b m n (0)e ikxaj .
In this paper we restrict ourself we with m = 0, though BSCs with higher orbital angular momentum are possible [67,68]. The field patterns of BSCs from Table II are shown in the left panel in Fig. 4. Notice that BSC 7 in Table II is a wave of the TM-type which means H x =0, and b 0 n (j) = 0 [64]. In contrast BSC 8 is a TE-wave with E x =0, and b 0 n (j) = 0. Let us now consider the resonant response of finite array with N spheres. Despite the similarity of Eq. (10) to Eqs. (1) and (2) the numerical method is now computationally much more expensive due to the necessity to evaluate lattice sums with involved special functions [66]. Though limited to the power of a desktop computer we can still circumvent the computational difficulties by recollecting the resonant picture from the previous section.  Table II under [78,79]. It should be noted, however, that the imaginary part in the same range can vary by orders of magnitude. Thus, one may expect that the effect of light enhancement will be limited by the material losses in silicon.

A. Q-factors
Here we use the standard definition of the Q-factor where ω is the resonance frequency, and γ is the resonance width, both quantities extracted from the scattering data described in the previous sections Figs. 2  the Q-factor for resonances with ν = 2 scales as Q ∼ N 2 , however, for ν = 4 we observed Q ∼ N 3 which, seemingly, contradicts our theoretical predictions. The contradiction can be resolved by noticing that Eq. (4) only describes one specific mechanism of the radiation losses equally applicable for both finite and infinite arrays, namely, the radiation from the sides of the arrays due to detuning from the true BSC point in the parametric space. In the case of finite arrays there is another radiation mechanism due to the losses at the edges as the wave bounces between them to form the sinusoidal resonant mode shape Eq. (9). with k 0 a = 2.052, k x a = π, R/a = 0.40. The black dash line shows the best result found in [33].
The horizontal lines show the limits due to material losses in silicon.
Obviously, for the guided modes below the line of light that would be the only possible radiation mechanism because the radiative losses at the sides are forbidden by total internal reflection. In fact, the scaling law for the edge radiation is already obtained in [33] where the authors demonstrated that the Q-factor scales as Q ∼ N 3 . Thus, one can conclude that in case of ν = 4 the radiative losses at the sides are suppressed by the radiative losses at the edges resulting in the scaling law for resonances formed by guided modes below the line of light [33].
In accordance with our findings we can now define side-coupled resonances with Q ∼ N 2 for BSCs with ν = 2, and a wider class of edge-coupled resonances with Q ∼ N 3 which encompasses both BSCs with ν = 4 as well as the below continuum resonances (BCRs).
Now it is instructive to compare our results with the Q-factors for the BCRs. To do that in Fig. 6 we plot the Q-factors for two BCR one for rods and spheres each. The parameters of the BCRs are given in the caption to Fig. 6. One can see that both ν = 2 BSCs and BCR demonstrate a similar dependance against N.
Finally, let us access the role of the material losses. The Q-factor limits were estimated based on the data on the imaginary part of refractive index at λ = 500, 700nm by [78], and at λ = 1000nm by [79]. One can easily see from Fig. 6 that Q ≈ 10 3 -10 4 resonances can be observed with silicon in the red-to-near-infrared range for arrays of N ≈ 10-15 elements.

B. Enhancement factors
We define the enhancement factor F as the average of the square root of the field intensity I(r) within the volume V = πR 2 Na (or area 2RNa) containing the whole of the finite array divided by the square root of the field intensity carried by the incoming wave I 0 .
Although according to Eq. (9) the EM-field varies significantly in the vicinity of the array the enhancement factor F defined through Eq. (13) is able to quantify to what extent the field amplitude is enhanced by resonant scattering. In Fig. 7 we plot the enhancement factors for the first (p = 1 in Eq. (8)) structural resonance of BSCs 2, 3, 7, and 8 as well as the BCRs used in Fig. 6. The arrays were illuminated with a plane wave with frequency equal to the frequency of the resonance. The k x -component of the incident wave each time was tuned to the maximum response which in case of BSC were given by Eq. (7). For BCR the maximum response was found at the strict normal incidence. One can see that only for the side-coupled resonances BSC 2,7 the enhancement factor is nicely fit by the expected dependence F ∼ √ Q ∼ N [71]. Notice that in the above reference the enhancement was defined through intensity rather that amplitude. For the edge-coupled resonances we were unable to produce a simple polynomial fit which is probably due to the residual side-coupling emerging through the specific excitation of the resonance by a plane wave. More interesting, though, is that the side-coupled resonances BSC 2,7 produce significantly larger enhancement factors than the edge-coupled ones, and hence should be opted for as the operating resonances for light enhancement. In lossy silicon the limits for the enhancement factor of the side-coupled resonances will be placed at the same arrays lengths as were found from Fig. 6 for the Q-factors. Thus, comparing Fig.7 against Fig.6 one can conclude that the amplitude enhancement factors F ≈ 10-25 can be achieved with arrays of 10-15 silicon subwavelength nanospheres in the red-to-near-infrared range with the side-coupled resonances. It should be pointed out that in terms of both the Q-factor asymptotics and compactness our set-up is no match to ultra-compact Fabry-Perot cavities [80] where the Q-factor scales exponentially with the number of periods in Bragg mirrors. Those asymptotics, however, are hard to achieve in silicon nonophotonics as the subwavelength spheres are scaled down in size to the visible wavelength where the material losses become significant. Our simulations show that in the red-to-near-infrared range the Q-factors of 10 3 -10 4 and the amplitude enhancement factors F ≈ 10-25 can be achieved with arrays of 10-15 silicon subwavelength nanospheres.
In contrast to familiar Fano resonances [56][57][58][59] the structural resonances in finite systems are shown to generate a complicated picture in the resonant response to illumination by a plane wave. With the frequency of the impinging wave tuned to the structural resonance we observed the oscillating behavior of the enhanced EM-field amplitude under variation of the wave vector component directed along the array. In real systems the effect of finiteness will be always eventually obscured by material losses. One may expect that with the increase of the length of the array the many pronounced resonances will merge into a Fano feature.
We speculate that this picture could be explained by a properly set coupled mode theory [81] based on analytical estimates for resonant frequencies and mode shapes obtained in this paper. Along with asymptotic scaling law for edge-coupled resonances enhancement factor this constitutes a goal for future studies.
The bound states in the continuum were recently employed for engineering high-Q resonators for compact nanophotonic lasers [82] providing access to new coherent sources with intriguing topological properties for optical trapping, biological imaging, and quantum communication. To provide useful guidelines for practical implementations of structures supporting the bound states the effects of both structural fluctuations [83] and substrate coupling [84] have recently been considered. As it was mentioned earlier in the introduction the BSCs are only allowed in infinite systems [72] so that in design of BSC-based devices the researchers will always be bound with residual radiation losses. We believe that our model of the BSC related structural resonances sheds some light onto important aspects of emerging BSC nanophotonics [45,85].