## Abstract

External waves incident on a periodic metamaterial lattice couple to it at frequencies corresponding to the leaky, or second, stop band. The resulting leaky-mode or guided-mode resonance effects are useful in device design and spectral manipulation. Indeed, some of the most important properties of metamaterials are associated with the leaky stopband. Thus motivated, we treat the band dynamics of leaky-mode resonant photonic lattices. In particular, properties of the band gap and conditions for band closure and band flips under multimode conditions are quantified. For a symmetric lattice, the nonleaky band edge hosts a bound state in the continuum whose band transition reverses the modal symmetry of the band edge modes. The leaky edge supports a guided-mode resonant radiative peak that also undergoes band flip upon band closure. We analyze a canonical one-dimensional lattice with exact numerical methods and a semianalytical formulation modified to handle the multimodal case. We show that the band dynamics of the various leaky modes present differ appreciably with, for example, the band associated with the fundamental TE_{0} and the first higher order TE_{1} modes closing at differing values of dielectric-constant modulation. We compare the thin-film lattice with an infinite lattice and find an approximate analytical condition for band closure that we verify with rigorous computations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical bound states in the continuum (BICs) are perfectly localized electromagnetic eigenstates in photonic structures coexisting with continuous radiative modes above the light line [1,2]. Because BICs increase the life time of confined waves and generate sharp spectral responses, their physical properties are among the most interesting scientific issues of the present [3–11]. Different types of BICs have been reported in versatile photonic structures including photonic crystals, metasurfaces, fiber Bragg gratings, and hybrid plasmonic-photonic systems [12–21]. Robust BICs in subwavelength photonic-crystal slab geometry are attractive for practical applications because they are stable. A small variation in the structural parameters simply moves the position of a BIC slightly along the band diagram [22]. State-of-the-art nanofabrication technology enables facile prototyping of resonant leaky-mode lattices even in the UV and visible spectral regions.

In one-dimensional (1D) and 2D subwavelength periodic photonic films, most of the important spectral properties are associated with phase-matched modes at the second stop band. Normally incident waves generate various beneficial zero-order spectral responses via coupling with lateral leaky Bloch modes [23–26]. Very recently, it has been reported that symmetry-protected BICs in 1D photonic lattices transit across the second stop band by interplay between first and second order Bragg processes [27]. With proper lattice parameters, the two Bragg effects due to the first and second Fourier harmonics cancel out, closing the band. Before and after band closure, TE-polarized symmetry-protected BICs are located at the upper and lower band edges, respectively [27]. In the previous study, however, leaky-mode band effects were studied for only the fundamental TE_{0} mode.

In general, multiple resonant quasi-guided modes coexist in periodic dielectric slabs in waveguide geometry. The various modes possess their own leaky bands that differ in spectral location and physical properties; these have hitherto not been quantified. Accordingly, in this paper, we numerically and semi-analytically investigate the leaky-band dynamics of the higher order guided modes contrasting their properties with those of the fundamental TE_{0} mode in representative 1D leaky-mode photonic lattices. An analytical expression, which determines the structural parameters of the lattice when the band closes, is derived from the conventional half-wave stack lattice that is fully transmissive at band gap closure. We show that the leaky stop band closes when the effective optical path lengths of lattice segments with high and low dielectric constants are the same. This conclusion is verified with numerical computations.

## 2. Lattice structure and perspective

Figure 1 illustrates our simple model and the attendant schematic photonic band structure indicating inter-band bound-state transitions. As noted in Fig. 1(a), we investigate a 1D photonic lattice composed of high (ε* _{h}*) and low (ε

*) dielectric constant materials. Thickness of the lattice is*

_{l}*d*and width of high dielectric constant part is ρΛ, where Λ is the period of the grating structure. This simple lattice can support numerous TE-polarized guided waves because its average dielectric constant ε

*= ε*

_{avg}*+ ρ (ε*

_{l}*− ε*

_{h}*) is larger than that of surrounding medium (ε*

_{l}*). In general, we can expect multiple band gaps corresponding to each guided mode provided that fill factor 0 < ρ < 1 and modulation Δε = ε*

_{s}*− ε*

_{h}*> 0 [28]. In this study, we limit our attention to the second band gaps of the fundamental TE*

_{l}_{0}and first higher order TE

_{1}modes because these simple cases suffice to show that band gaps due to different guided modes are closed with different structural parameters. This notion then extends straightforwardly to all higher-order modes. Near the second band gaps, the quasi-guided modes are described by the complex frequency Ω = Ω

_{Re}+

*i*Ω

_{Im}, where the imaginary part Ω

_{Im}represents radiative leakage loss. With the time dependence of electric field as exp(−

*i*Ω

*t*), guided modes lose their electromagnetic energy as time goes on. However, one of the band edge modes suffers no radiation loss and becomes a symmetry-protected BIC with purely real frequency [27,29–31]. As schematically shown in Fig. 1(b), the modal band gaps ΔTE

_{0}and ΔTE

_{1}due to TE

_{0}and TE

_{1}modes, respectively, are opened at

*k*= 0 and symmetry-protected bound states BIC

_{z}_{0}and BIC

_{1}in red circles appear at the upper band edges of the bands when ρ and Δε are small. With proper values of ρ and Δε, each band gap is closed. The spectral locations of BIC

_{0}and BIC

_{1}transit from the upper to lower band edges as the values of ρ and/or Δε increase. In this study, leaky-mode band dynamics are investigated through finite-difference time-domain (FDTD) simulations [32,33] and a semi-analytical approach proposed by Kazarinov and Henry [29]. We show that ΔTE

_{0}and ΔTE

_{1}close with different lattice parameters such as fill factor and index modulation.

## 3. Leaky-band dynamics

Figures 2(a) and 2(b) illustrate the FDTD simulated second stop bands ΔTE_{0} and ΔTE_{1} opened by the fundamental TE_{0} and first higher order TE_{1} modes, respectively, under variation of dielectric-constant modulation Δε. Lattice parameters are set as *d* = 0.7 Λ, ρ = 0.48, ε* _{avg}* = 4.00, and ε

*= 1. As shown in Fig. 2(a), the band gap ΔTE*

_{s}_{0}opens and its size increases as Δε increases from zero. However, the gap size decreases and becomes zero as Δε is further increased. On additional increase in Δε, the band reopens and its size grows again. These leaky-mode band dynamics are associated with bound-state transitions as can be seen clearly by investigating the spatial electric-field distributions of band edge modes represented in the insets of Fig. 2(a). When Δε is 0.30 and 0.50, symmetry-protected BICs are located at the upper band edges. When Δε is 0.59 and 0.70, on the other hand, these non-leaky bound states appear at the lower band edges. Figure 2(b) shows similar leaky-mode band dynamics and bound-state transitions for the band gap ΔTE

_{1}belonging to the TE

_{1}mode. As Δε increases from zero, the size of ΔTE

_{1}increases, decreases, becomes zero, and increases again. BIC

_{1}transits from the upper to lower band edge. However, Figs. 2(a) and 2(b) clearly show that the transition points of ΔTE

_{0}and ΔTE

_{1}are different. The insets show the radiating fields propagating orthogonal to the lattice at the non-BIC edges; these are the externally-observable guided-mode resonances that appear as high-efficiency reflection peaks in the spectrum [34]. Note the field distribution along the

*x*-direction representing the classic TE

_{0}and TE

_{1}mode shapes. The nodes along the

*z*-direction arise on account of counter-propagating modes forming a standing wave along the lattice.

Leaky-mode band dynamics of 1D photonic lattices, including band closure and inter-band bound-state transitions, can be understood by solving the wave equation given by [35]

*E*represents the radiating wave. Near the second stop band, the dispersion relation can be written as

_{rad}_{0}is the Bragg frequency under vanishing index modulation. We note that Eq. (2) is valid for both TE

_{0}and TE

_{1}modes with different coupling coefficients given by

*m*= 0 and

*m*= 1 represent coupling coefficients for TE

_{0}and TE

_{1}modes, respectively, and$G\left(x,{x}^{\text{'}}\right)$denotes the Green’s function for the diffracted field [30,31]. As indicated in Eq. (2), the semi-analytical dispersion model yields two band edges Ω

*= Ω*

^{a,m}_{0,}

*+*

_{m}*h*

_{2,}

*/ (K*

_{m}*h*

_{0,}

*) and Ω*

_{m}*= Ω*

^{s,m}_{0,}

*– (*

_{m}*h*

_{2,}

*+*

_{m}*i*2

*h*

_{1,}

*) / (K*

_{m}*h*

_{0,}

*). One band edge mode with frequency Ω*

_{m}*, which is obtained when the electric field distribution is an asymmetric function (*

^{a,m}*A*= –

*B*), becomes the symmetry-protected BIC because purely real frequency implies no radiation loss. At the other edge with Ω

*, which is obtained when the electric field distribution is a symmetric function (*

^{s,m}*A*=

*B*), there is radiation loss with the imaginary frequency Im (Ω

*) = –2Re (*

^{s,m}*h*

_{1,}

*) / (K*

_{m}*h*

_{0,}

*). The size of the gap is given by Re (|Ω*

_{m}*– Ω*

^{a,m}*|) = 2|*

^{s,m}*h*

_{2,}

*– Im (*

_{m}*h*

_{1,}

*)| / (K*

_{m}*h*

_{0,}

*). We here interpret Im (*

_{m}*h*

_{1,}

*) and*

_{m}*h*

_{2,}

*as the effect of Bragg reflection due to the first and second Fourier harmonics of the periodic dielectric function, respectively. When the two Bragg effects are balanced with Im (*

_{m}*h*

_{1,}

*) =*

_{m}*h*

_{2,}

*, the real parts of the two band edge modes degenerate and the band closes. Before (*

_{m}*h*

_{2,}

*> Im (*

_{m}*h*

_{1,}

*)) and after (*

_{m}*h*

_{2,}

*< Im (*

_{m}*h*

_{1,}

*)) the band closure, relative positions of the two band-edge frequencies with Re (Ω*

_{m}*) and Re (Ω*

^{s,m}*) are reversed.*

^{a,m}For the symmetric lattice studied herein, Im (*h*_{1,}* _{m}*) is positive irrespective of ρ and Δε. However,

*h*

_{2,}

*is positive (negative) when the fill factor is smaller (greater) than 0.5 because the second Fourier coefficient ε*

_{m}_{2}= (Δε / π) sin (2πρ) changes its sign from positive to negative when ρ = 0.5. For the value of fill factors smaller than 0.5, therefore, the condition for band gap closure Im (

*h*

_{1,}

*) =*

_{m}*h*

_{2,}

*can be satisfied with proper lattice parameters such as Δε and ρ. Figures 3(a) and 3(b) show calculated Im (*

_{m}*h*

_{1,}

*) and*

_{m}*h*

_{2,}

*as a function of ρ when Δε = 0.40 and 1.20, respectively. For Δε = 0.40 (1.20), TE*

_{m}_{0}and TE

_{1}modes satisfy the condition for band gap closure when ρ = 0.489 (0.467) and 0.487 (0.461), respectively. Transition points for TE

_{0}and TE

_{1}modes are different because the values of the coupling coefficients for TE

_{0}and TE

_{1}modes are different in general. Moreover, since Im (

*h*

_{1,}

*) and*

_{m}*h*

_{2,}

*are proportional to [ε*

_{m}_{1}]

^{2}= [(2Δε / π) sin (πρ)]

^{2}and ε

_{2}= (Δε / π) sin (2πρ), respectively, the values of fill factors where band gaps close are increasingly pulled away from 0.5 as Δε increases. It can be also inferred from the coupling coefficients that large values of index modulation are required to close the band as the fill factor ρ deviates substantially from 0.5. To check the effect of Δε on the transition point, the size of the band gaps ΔTE

_{0}and ΔTE

_{1}are calculated as a function of Δε and the results are illustrated in Figs. 3(c) and 3(d) for ρ = 0.49 and 0.46, respectively. In both cases, the band for the higher mode TE

_{1}closes first as Δε increases. As the fill factor increasingly deviates from 0.5, a larger modulation is required to close the gaps in both cases as seen in Fig. 3(d).

As seen here, the simple semi-analytical dispersion model explains the formation of the stop band clearly physically. Nevertheless, the band-closure condition Im (*h*_{1,}* _{m}*) =

*h*

_{2,}

*can be obtained only numerically via Green’s functions. It is possible to derive an approximate analytical expression which determines the structural parameters for the band closure and bound-state transition point by association with (nonleaky) thin-film stacks; this is the subject of the following section. This relation holds only at band closure and does not embody the full lattice dynamics inherent in rigorous FDTD computations and the semianalytical Kazarinov-Henry model.*

_{m}## 4. Approximate formula for band closure

To find an analytical formula for the transition point, we first consider the simpler band structure of a half-wave layer stack of infinite lateral extent. As the structure is fully transmissive, there is no reflection and the band is closed. Light with wavelength λ which satisfies the condition (λ / 2) = *t _{h}* =

*t*, where

_{l}*t*and

_{h}*t*denote optical thickness of high (${n}_{h}=\sqrt{{\epsilon}_{h}}$) and low (${n}_{l}=\sqrt{{\epsilon}_{l}}$) index layers, respectively, transmits through each layer without reflection due to consecutive Fabry-Perot resonances. From this condition, we find the relation for the band closure as ρ =

_{l}*n*/ (

_{l}*n*+

_{l}*n*) at which the band gap vanishes. For example, for fixed values of refractive indices

_{h}*n*= 2.60 and

_{h}*n*= 1.40, the band closes when ρ

_{l}*= 0.35. Figures 4(a), 4(b), and 4(c) show the band structure when ρ = 0.25 < ρ*

_{c}*, ρ = 0.35 = ρ*

_{c}*, and ρ = 0.45 > ρ*

_{c}*, respectively. Spatial electric field distributions are also plotted in the insets of Fig. 4. When ρ = 0.25, there is a photonic band gap with the upper (lower) band edge mode with asymmetric (symmetric) field distributions. At ρ = 0.35, the band closes and the band edge modes become degenerate. When ρ = 0.45 the band is open and the field profiles are reversed with the upper (lower) band edge mode being symmetric (asymmetric).*

_{c}We now connect the leaky lattice to the non-leaky film-stack lattice and draw some conclusions. The band closure and band-flip phenomenon in 1D non-leaky photonic lattices can be understood qualitatively from the electromagnetic variational theorem which is analogous to the variational principle of quantum mechanics [37]. According to the theorem, the mode with the smallest frequency will have a field pattern that minimizes the energy functional

*(*

_{c}*t*<

_{h}*t*) as in Fig. 4(a), therefore, we can interpret that symmetric modes tend to locate below asymmetric modes. Since symmetric modes have two nodes in the low dielectric constant part whereas asymmetric modes have one node in each of the low and high dielectric parts, respectively, we can infer reasonably that the electric field is more concentrated in regions of high dielectric constant in the case of ρ < ρ

_{l}*. On the other hand, when ρ > ρ*

_{c}*(*

_{c}*t*>

_{h}*t*), symmetric modes tend to locate above asymmetric modes because symmetric modes have two nodes in the high dielectric constant part whereas asymmetric modes have one node in each of the low and high dielectric parts, respectively. At the intermediate state with ρ = ρ

_{l}*(*

_{c}*t*=

_{h}*t*), the two band edge modes are degenerate and the band closes. Pertinent to this discussion, we note that in the insets in Fig. 4, vertical dotted lines represent the mirror planes (

_{l}*z*= 0) at the center of the high dielectric constant part as in Fig. 2.

For the leaky-mode resonant lattice with finite thickness under study here, we can by analogy infer the condition for the band closure to be

where*N*and

_{h}*N*represent the effective refractive indices of guided modes in a uniform slab waveguide with dielectric constant ε

_{l}*and ε*

_{h}*, respectively, and*

_{l}*L*= ρΛ

_{h}*N*and

_{h}*L*= (1 − ρ)Λ

_{l}*N*are attendant effective optical path lengths. Effective indices

_{l}*N*and

_{h}*N*are functions of frequency due to waveguide dispersion. Before (

_{l}*L*<

_{h}*L*) and after (

_{l}*L*>

_{h}*L*) band closure, the bound states locate at the upper and lower band edges, respectively. To check the validity of our formula, we calculate the values of index modulation, where the band closes, for various values of fill factor and the results are plotted in Fig. 5(a). Red and blue lines represent the values of fill factors ρ

_{l}

_{c}_{,0}and ρ

_{c}_{,1}where the band gaps ΔTE

_{0}and ΔTE

_{1}vanish, respectively, as a function of index modulation. FDTD simulated results, green stars for ΔTE

_{0}and purple circles for ΔTE

_{1}, are also plotted for comparison. Figure 5(a) clearly shows that the calculated ρ

_{c}_{,0}=

*N*

_{l}_{,0}/ (

*N*

_{l}_{,0}+

*N*

_{h}_{,0}) and ρ

_{c}_{,1}=

*N*

_{l}_{,1}/ (

*N*

_{l}_{,1}+

*N*

_{h}_{,1}), where

*N*

_{h}_{,0}(

*N*

_{h}_{,1}) and

*N*

_{l}_{,0}(

*N*

_{l}_{,1}) denote the effective indices of guided modes in the high and low dielectric constant parts for TE

_{0}(TE

_{1}) modes, respectively, via the half-wave stack model agree well with the FDTD results. Hence, we conclude that the formula for the band closure and bound-state transition ρ =

*N*/ (

_{l}*N*+

_{l}*N*) = 1 / [1 + (

_{h}*N*/

_{h}*N*)] is a good approximation. Equation (7) and Fig. 5(a) show that the band closes when the fill factor is smaller than 0.5 always for both ΔTE

_{l}_{0}and ΔTE

_{1}. As the grating modulation strength increases, the transition points ρ

_{c}_{,0}and ρ

_{c}_{,1}are increasingly pulled away from 0.5 because the refractive index ratio

*N*

_{h}_{,0}/

*N*

_{l}_{,0}and

*N*

_{h}_{,1}/

*N*

_{l}_{,1}increases with Δε. As can be seen from Fig. 5(b), moreover, since the index ratio

*N*

_{h}_{,1}/

*N*

_{l}_{,1}is always larger than

*N*

_{h}_{,0}/

*N*

_{l}_{,0}, for a given value of ρ, ΔTE

_{1}closes faster than ΔTE

_{0}as Δε increases. This is consistent with results in Fig. 3.

## 5. Leaky edge Q factors

In this study, we focus on the band closure and band transitions of BICs; these states represent the nonleaky band edges and thus possess infinite Q factors that do not vary on transition across the gap. In contrast, the leaky band edge modes are directly observable via the radiated power associated with the corresponding guided-mode resonance. Under band transition implemented by variation of the spatial modulation strength Δε as in Figs. 2 and 3, there will be a variation in the modal Q factors. Specifically, for the lattice and values of Δε in Fig. 2, the pertinent Q factors are summarized in Table 1. As Δε increases, the Q factors of the leaky modes decrease gradually. Hence, there is no abrupt change in the Q factors before and after the band transition. It is known classically that the linewidth of the resonance increases with Δε [34]. As Q is inversely proportional to the linewidth, the results in Table 1 follow. In this context, we note that Eqs. (2) and (4) show that radiation loss as represented by the real part of *h*_{1,}* _{m}* is strongly dependent on the lattice thickness

*d*. Analysis of the effects of thickness on the leaky-mode Q factor near the band transition points may be of interest in future work detailing the physical properties associated with the band dynamics of related photonic structures.

## 6. Conclusions

In conclusion, we numerically and analytically treat the band dynamics of leaky-mode resonant photonic lattices. We seek understanding of the fundamental physics and attendant band transitions and symmetry properties and hence take the simplest possible 1D symmetric lattice in a uniform host medium. In particular, properties of the band gap and conditions for band closure and band flips are quantified for a lattice supporting multiple lateral Bloch modes. For each mode, the lattice sustains a bound state above the light line at the nonleaky band edge whereas a radiating guided-mode resonance appears at the leaky edge. Our analysis shows that upon both bound and unbound state transitions, that we call band flip, the modal symmetry of the band edge modes reverses before and after band closure. The modal band dynamics including band closure and band flip are executed via variation of structural parameters such as the fill factor and the strength of the refractive-index modulation. Analytical expression for the leaky band closure which connects the fill factor and index modulation is derived from the conventional half-wave stack lattice that is fully transmissive at band closure. This formula shows that the stop band closes when the effective optical path lengths of the high and low dielectric parts within a period are the same. Moreover, we see that the second stop band can be closed only when the fill factor of the high dielectric constant part is smaller than 0.5 at which value ε_{2} = 0. As the grating modulation strength increases, the band closure point is increasingly pulled away from 0.5. For the same value of fill factor, band gaps due to the different guided modes are closed with different values of index modulation because effective indices, or effective optical path lengths, of each guided mode are different. An important conclusion of the study is that the band dynamics of the various leaky modes present differ appreciably with quantitative results presented for the simple lattice treated. For example, the bands associated with the fundamental TE_{0} and the first higher order TE_{1} modes close at differing values of the dielectric-constant modulation. The ideas presented here apply generally to 1D and 2D photonic lattices composed of arbitrary materials appearing in various geometric configurations. Ascertaining their properties with further studies along these lines will support physical understanding and innovative developments in metamaterials and metasurfaces.

## Funding

National Science Foundation (NSF) (ECCS-1809143).

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