Band dynamics of leaky-mode photonic lattices

Sun-Goo Lee; Robert Magnusson

doi:10.1364/OE.27.018180

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 TE₀ 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 TE₀ 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_l) dielectric constant materials. Thickness of the lattice is 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_avg = ε_l + ρ (ε_h − ε_l) is larger than that of surrounding medium (ε_s). In general, we can expect multiple band gaps corresponding to each guided mode provided that fill factor 0 < ρ < 1 and modulation Δε = ε_h − ε_l > 0 [28]. In this study, we limit our attention to the second band gaps of the fundamental TE₀ and first higher order TE₁ 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 Ωt), 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 ΔTE₀ and ΔTE₁ due to TE₀ and TE₁ modes, respectively, are opened at k_z = 0 and symmetry-protected bound states BIC₀ and BIC₁ 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 BIC₀ and BIC₁ 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 ΔTE₀ and ΔTE₁ close with different lattice parameters such as fill factor and index modulation.

Fig. 1 (a) Schematic of a 1D leaky-mode photonic lattice for studying bound-state transitions. TE-polarized leaky guided modes are described by complex frequency Ω = Ω_Re + i Ω_Im. (b) Conceptual illustration of the bound-state transitions. Here, k_z is the wavevector along the z-direction and X = π / Λ. Before and after closing the gaps ΔTE₀ and ΔTE₁, symmetry-protected BIC₀ and BIC₁ in red circles appear at the upper and lower band edges, respectively. The BIC band edges are nonradiative and thus not leaky whereas the opposite edges are leaky and represent the spectral placement of guided-mode resonance peaks.

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3. Leaky-band dynamics

Figures 2(a) and 2(b) illustrate the FDTD simulated second stop bands ΔTE₀ and ΔTE₁ opened by the fundamental TE₀ and first higher order TE₁ 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_s = 1. As shown in Fig. 2(a), the band gap ΔTE₀ 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 ΔTE₁ belonging to the TE₁ mode. As Δε increases from zero, the size of ΔTE₁ increases, decreases, becomes zero, and increases again. BIC₁ transits from the upper to lower band edge. However, Figs. 2(a) and 2(b) clearly show that the transition points of ΔTE₀ and ΔTE₁ 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 TE₀ and TE₁ mode shapes. The nodes along the z-direction arise on account of counter-propagating modes forming a standing wave along the lattice.

Fig. 2 FDTD simulated dispersion relations near the second stop band originating from (a) TE₀ and (b) TE₁ mode. K = 2π / Λ is the magnitude of the grating vector. Insets with blue and red colors illustrate spatial electric field distributions (E_y) of the band edge modes at the y = 0 plane. Vertical dotted lines represent the mirror planes (z = 0), at the center of the high dielectric constant part, in the computational cells. As the index modulation increases, symmetry-protected BICs with asymmetric electric field distributions transit from the upper to lower band edges. In the FDTD simulations, we use the computational cell of size Λ × 8Λ and spatial resolution is set to Δx = Δz = Λ/64.

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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]

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial z^{2}}) E_{y} (x, z) + ε k_{0}^{2} E_{y} (x, z) = 0.

To solve the equation, the periodic dielectric function can be expanded in a Fourier series and the electric field can be expanded as plane waves [36]. We solve the wave equation by retaining only the zeroth, first, and second Fourier harmonics and by approximating the electric field distribution as

E (x, z) = [A \exp (+ i K z) + B \exp (- i K z)] φ (x) + E_{r a d},

where

φ (x)

characterizes the mode profile of the unmodulated waveguide and E_rad represents the radiating wave. Near the second stop band, the dispersion relation can be written as

Ω (k_{z}) = Ω_{0} - (i h_{1} \pm \sqrt{k_{z}^{2} + {(h_{2} + i h_{1})}^{2}}) / (K h_{0}),

where Ω₀ is the Bragg frequency under vanishing index modulation. We note that Eq. (2) is valid for both TE₀ and TE₁ modes with different coupling coefficients given by

h_{0, m} = Ω \int_{- \infty}^{\infty} ε_{0} (x) φ_{m} (x) φ_{m}^{*} (x) d x,

h_{1, m} = i \frac{K_{}^{3} Ω_{}^{4} ε_{1}^{2}}{8} \int_{- d}^{0} \int_{- d}^{0} G (x, x^{'}) φ_{m} (x^{'}) φ_{m}^{*} (x) d x^{'} d x,

h_{2, m} = \frac{K Ω_{}^{2} ε_{2}}{4} \int_{- d}^{0} φ_{m} (x) φ_{m}^{*} (x) d x,

where m = 0 and m = 1 represent coupling coefficients for TE₀ and TE₁ modes, respectively, and

G (x, x^{'})

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^a,m = Ω_0,_m + h_2,_m / (Kh_0,_m) and Ω^s,m = Ω_0,_m – (h_2,_m + i2h_1,_m) / (Kh_0,_m). One band edge mode with frequency Ω^a,m, which is obtained when the electric field distribution is an asymmetric function (A = –B), becomes the symmetry-protected BIC because purely real frequency implies no radiation loss. At the other edge with Ω^s,m, which is obtained when the electric field distribution is a symmetric function (A = B), there is radiation loss with the imaginary frequency Im (Ω^s,m) = –2Re (h_1,_m) / (Kh_0,_m). The size of the gap is given by Re (|Ω^a,m – Ω^s,m|) = 2|h_2,_m – Im (h_1,_m)| / (Kh_0,_m). We here interpret Im (h_1,_m) and h_2,_m 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 (h_1,_m) = h_2,_m, the real parts of the two band edge modes degenerate and the band closes. Before (h_2,_m > Im (h_1,_m)) and after (h_2,_m < Im (h_1,_m)) the band closure, relative positions of the two band-edge frequencies with Re (Ω^s,m) and Re (Ω^a,m) are reversed.

For the symmetric lattice studied herein, Im (h_1,_m) is positive irrespective of ρ and Δε. However, h_2,_m is positive (negative) when the fill factor is smaller (greater) than 0.5 because the second Fourier coefficient ε₂ = (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,_m can be satisfied with proper lattice parameters such as Δε and ρ. Figures 3(a) and 3(b) show calculated Im (h_1,_m) and h_2,_m as a function of ρ when Δε = 0.40 and 1.20, respectively. For Δε = 0.40 (1.20), TE₀ and TE₁ modes satisfy the condition for band gap closure when ρ = 0.489 (0.467) and 0.487 (0.461), respectively. Transition points for TE₀ and TE₁ modes are different because the values of the coupling coefficients for TE₀ and TE₁ modes are different in general. Moreover, since Im (h_1,_m) and h_2,_m are proportional to [ε₁]² = [(2Δε / π) sin (πρ)]² and ε₂ = (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 ΔTE₀ and ΔTE₁ 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 TE₁ 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).

Fig. 3 Calculated coupling coefficients when (a) Δε = 0.40 and (b) Δε = 1.20 as a function of ρ. Size of the band gaps when (c) ρ = 0.49 and (d) ρ = 0.46. As the index modulation increases, the size of the gaps increases, decreases, becomes zero, and grows again. Band gaps ΔTE₀ and ΔTE₁ close at different index modulations.

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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,_m 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.

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_l, where t_h and t_l denote optical thickness of high ( $n_{h} = \sqrt{ε_{h}}$ ) and low ( $n_{l} = \sqrt{ε_{l}}$ ) 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 = n_l / (n_l + n_h) at which the band gap vanishes. For example, for fixed values of refractive indices n_h = 2.60 and n_l = 1.40, the band closes when ρ_c = 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).

Fig. 4 Dispersion relations of the 1D photonic lattice with infinite lateral extent when (a) ρ = 0.25, (b) ρ = 0.35, and (c) ρ = 0.45. Spatial electric field distributions shown in the insets indicate that symmetry properties of band edge modes are reversed before and after the band gap closure. Refractive indices are set as n_h = 2.60 and n_l = 1.40.

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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

U_{f} = \frac{\int {| \nabla \times E (r) |}^{2} d^{3} r}{\int ε (r) {| E (r) |}^{2} d^{3} r} .

The way to minimize the functional is to maximize the denominator by concentrating the electric field in regions of high dielectric constant and to minimize the numerator by decreasing the amount of spatial oscillations while remaining orthogonal to lower frequency modes. At the second stop band, two band edge modes are standing waves with two nodes. When ρ < ρ_c (t_h < t_l) 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_c. On the other hand, when ρ > ρ_c (t_h > t_l), 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_c (t_h = t_l), 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 (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

\frac{λ}{2} = L_{h} = L_{l} or ρ = \frac{N_{l}}{N_{l} + N_{h}}

where N_h and N_l represent the effective refractive indices of guided modes in a uniform slab waveguide with dielectric constant ε_h and ε_l, respectively, and L_h = ρΛN_h and L_l = (1 − ρ)ΛN_l are attendant effective optical path lengths. Effective indices N_h and N_l are functions of frequency due to waveguide dispersion. Before (L_h < L_l) and after (L_h > L_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_c_,0 and ρ_c_,1 where the band gaps ΔTE₀ and ΔTE₁ vanish, respectively, as a function of index modulation. FDTD simulated results, green stars for ΔTE₀ and purple circles for ΔTE₁, 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 TE₀ (TE₁) 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_h) = 1 / [1 + (N_h / N_l)] 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 ΔTE₀ and ΔTE₁. 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, ΔTE₁ closes faster than ΔTE₀ as Δε increases. This is consistent with results in Fig. 3.

Fig. 5 (a) Relation between fill factor ρ and index modulation Δϵ when the modal gaps ΔTE₀ and ΔTE₁ close. (b) Effective-index ratio as a function of dielectric-constant modulation.

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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.

Table 1. Calculated Q factors of leaky band edge modes

View Table

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 ε₂ = 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 TE₀ and the first higher order TE₁ 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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TE₀	Δε	before band transition		after band transition
		0.30	0.50	0.59	0.70
	Q factor	3150	1134	812	568
TE₁	Δε	before band transition		after band transition
		0.25	0.37	0.45	0.55
	Q factor	2054	938	634	424

Band dynamics of leaky-mode photonic lattices

Abstract

1. Introduction

2. Lattice structure and perspective

3. Leaky-band dynamics

4. Approximate formula for band closure

5. Leaky edge Q factors

6. Conclusions

Funding

References

Cited By

Figures (5)

Tables (1)

Equations (7)

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