Abstract
We develop a formalism, based on the mode expansion method, to describe the guided resonances and bound states in the continuum (BICs) in photonic crystal slabs with onedimensional periodicity. This approach provides analytic insights to the formation mechanisms of these states: the guided resonances arise from the transverse Fabry–Pérot condition and the divergence of the resonance lifetimes at the BICs is explained by a destructive interference of radiation from different propagating components inside the slab. We show BICs at the center and on the edge of the Brillouin zone protected by symmetry, BICs at generic wave vectors not protected by symmetry and the annihilation of BICs at lowsymmetry wave vectors.
Introduction
Conventionally, the confinement of waves is achieved by spectrally separating the bound state away from the continuum of radiating waves that can carry energy away—examples include electronic bound states at negative energies and light guided below the light line or inside a photonic bandgap. Bound states in the continuum (BICs) are special states that remain localized and have infinite lifetimes even though they reside inside the continuum^{1}. Historically, BIC was first proposed by von Neumann and Wigner for an electron in an engineered potential^{2}, although such an electron BIC has never been achieved. More recently, optical BICs have been experimentally realized in a range of photonic systems^{3,4,5,6,7,8,9}. In periodic structures, BICs may be found by varying the incident angle without tuning the structure, which makes their realization relatively simple^{3,5,6,10,11,12,13,14,15,16,17,18,19,20,21,22} (Note: Originally, BICs are proposed as localized states integrable in all the three dimensions^{2}. But in periodic structure, BICs are extended in periodical directions while spatially confined in other directions). Photonic crystal (PhC) slabs—dielectric slabs with periodically modulated refractive index^{10,17,23,24}—are particularly attractive given their macroscopic sizes and ease of fabrication. The guided resonances and BICs in PhC slabs have been used for a wide range of applications such as lasers^{25,26,27,28,29}, sensors^{12,30,31} and filters^{32}. When BICs do exist, their robustness can be explained by the zerocrossing of radiating amplitudes^{6} or, more generally, by their topological charges^{18}. However, this does not reveal the physical mechanism that suppresses radiation and leads to the localization. In the presence of multiple resonances, the disappearance of radiation can be explained by the destructive interference of the radiating waves from different resonances^{7,12,13,14,15,33,34}. When only one resonance is present, an explanation based on destructive interference should still be possible, but it is no longer clear which sets of waves are interfering. Furthermore, the theoretical studies of BICs have largely relied on numerical simulations that are time consuming and provide little insight.
Here, we develop a mode expansion method that explains the formation mechanism of guided resonances and BICs in PhC slabs. In this formalism, guided resonances require the roundtrip transverse phase shift for each inslab propagating mode to be an integer multiple of 2π and BICs arise from the destructive interference of radiation from different propagating waves inside the slab. This method is also capable of calculating the resonance frequency, quality factor and field profile efficiently with no approximation other than a truncation of the basis size and we validate these results with finitedifference timedomain (FDTD) simulations. For PhC slabs in a periodic inhomogeneous backgrounds, we find symmetryprotected BICs both at the center and on the edge of the Brillouin zone, in addition to those inside the Brillouin zone where they are not protected by symmetry. Although we only show examples for systems with one dimensional periodicity, our mode expansion method can be extended to systems that are periodic in two dimensions.
Methods
For simplicity, we consider TM modes (H_{x}, H_{y}, E_{z}) in a PhC slab that is periodic in y and uniform in z (Fig. 1a); the generalization to TE modes (E_{x}, E_{y}, H_{z}) and to PhC slabs with twodimensional periodicity is straightforward. We consider structures that are mirrorsymmetric in the normal direction x (this symmetry is necessary for reducing the number of radiation channels^{6}) and where the slab permittivity ε_{1}(y) is uniform in x (common in most fabricated structures of PhC slabs). Here we consider two cases: the permittivity of the surrounding medium ε_{2}(y) is a constant (Fig. 1b) or is periodic with the same period a as the slab (Fig. 1c). Inside (x < 0.5h) and outside (x > 0.5h) the slab, the structure is uniform in x, so the fields can be expanded in the eigenmodes of ε_{1}(y) and ε_{2}(y) with a sinusoidal dependence along x; the expansion coefficients can be determined by continuity at the slab surface x = 0.5h. Specifically, with an outgoing boundary condition in x, an eveninx TM mode with wave vector k_{y} can be written as^{35}
where h is the slab thickness, the cosine inside the slab guarantees the eveninx symmetry and the complex exponential outside the slab guarantees the outgoing boundary condition. TM modes satisfy the wave equation where k_{0} = ω/c, ω is the frequency and c is the vacuum speed of light; inserting Eq. (1) into this wave equation, we find that the eigenfunctions and propagation constants u_{m}(y), ϑ_{m}(y), β_{m}and γ_{m} inside and outside the slab satisfy
where for l = 1, 2 are the Hermitian operators governing the wave equation for the layers inside and outside the slab, subject to periodic boundary condition u_{m}(y + a) = u_{m}(y), ϑ_{m}(y + a) = ϑ_{m}(y). For a given frequency and k_{y}, there will be a finite number of eigenmodes with (or ) that propagate in the x direction, with an infinite number of eigenmodes with (or ) that are evanescent in x. Similar expansion methods were used previously for water waves^{21} and for quantum waveguides^{36}. Oddinx TM modes can be written similarly by replacing the cosine in Eq. (1) with sine and TE modes can be treated by replacing with that for H_{z}. C_{m} and T_{m} are coefficients of the eigenmode expansion and they can be determined via the continuity of E_{z} and ∂E_{z}/∂x at x = 0.5h, which requires
The standing waves inside the slab [the cos (β_{m}x) in Eq. (1)] are superposition of waves propagating in +x and in –x directions, so one can interpret the inslab fields as waves circulating within the slab with reflection and transmission at the two slab surfaces. The transverse phase shift for every propagating component is an integer multiple of 2π after a round trip with two reflections, which is the same resonance condition as the Fabry–Pérot resonances in uniform dielectric slabs; the difference is that here multiple propagating components are coupled due to the periodicity in y.
While the two sets of eigenmodes {ϑ_{m}} and {u_{m}} each form an infinitedimensional basis, the highorder ones correspond to fast oscillating fields that are negligible at low frequencies. Therefore, in our calculations, we truncate down to M terms by expanding the eigenmodes in an Mdimensional basis (details below). In the truncated basis, the transformation between the two bases {ϑ_{m}} and {u_{m}} is given by to an M × M matrix P such that . In this way, the continuity condition Eqs (4, 5) can be written in matrix form as
where T = [T_{1}, …, T_{M−1}, T_{M}]^{T}, C = [C_{1}, …, C_{M−1}, C_{M}]^{T} and γ, B are diagonal matrices γ = Diag(γ_{1}, …, γ_{M}), B = Diag (β_{1 }tan (0.5β_{1}h), …, β_{M} tan (0.5β_{M}h)). Note that the matrix B is purely real since is real for all m. Eq. (4) is a linear equation group for vectors T and C. Substituting Eq. (6) into Eq. (7) yields (iγP + PB)C = 0. Nontrivial solutions exist when
where * denotes the determinant of the matrix. Therefore, solving for Eq. (8) for a given k_{y} yields the dispersion relation ω (k_{y}) for arbitrary resonances and BICs, as well as regular bound states. The vector C corresponding to the zero determinant and the associated vector T from Eq. (6) yield the field profile as given in Eq. (1). At frequencies in the continuum spectrum of the surrounding medium ( for a homogeneous medium), some of the γ_{m}’s are real and f (k_{y}, ω) is generally complexvalued; in such region finding the zeroes of f (k_{y}, ω) requires searching for solutions on the lower half of the complexfrequency plane with ω_{r} = Re(ω) and ω_{i} = Im(ω) being the parameters. The imaginary part of the frequency is the decay rate and the quality factor of the resonance is Q = −ω_{r}/(2ω_{i}).
The transformation matrix P is deduced from Eqs (2, 3). Inspired by ref. 24, we expand both sides of Eqs (2, 3) in Fourier series and truncate to Fourier orders from –N to N (with a total of M = 2N + 1 terms). Then, Eqs (2, 3) can be written as matrix equations
with β = Diag(β_{1}, …, β_{M}) and H_{l} is an M × M matrix whose (m, m′)th element is , where n = m − N − 1, n′ = m′ − N − 1 and ξ_{l} (n) is the nth Fourier coefficient of ε_{l} (y). The mth column of Φ (or Θ) contains the –NtoN Fourier coefficients of u_{m} (or ϑ_{m}). Φ and Θ are transform matrices connecting {u_{m}} and {ϑ_{m}} to the same basis with planewave elements, hence P = Θ^{−1}Φ. Note that when the permittivity is real and mirror symmetric in y, , the matrices H_{l} are real symmetric, so β^{2} and γ^{2} are real; moreover, Φ, Θ and P can be chosen to be purely real.
BICs arise when the decay rate γ = −2ω_{i} of a resonance becomes zero, or equivalently when the amplitudes of the radiating waves vanish: T_{m′} = 0 for all the m′ with . From Eq. (6), T_{m′} is given by
where is the (m′,m)’s element of matrix P, which is the m′th component of the inslab eigenmode u_{m} in the basis of {ϑ_{m′}}. Equation (11) reveals that the radiating wave of port m′ comes from interference of all the contributions from {u_{m}} to the radiating mode ϑ_{m′}. Therefore, T_{m′} = 0 is a result of destructive interference from the inslab eigenmodes to the radiation modes.
Numerically it can be ambiguous to determine whether ω_{i} is very small or identically zero. Therefore, we use a slightly modified scheme to look for exact BICs. Define f ′ as : the propagation constants of the radiating waves γ_{m′} are artificially set to zero; this function f ′ is purely real for a lossless dielectric structure that is symmetric in y (where H_{l} is real symmetric). A BIC not only satisfies f = 0; it also satisfies f ′ = 0 since T_{m′} = 0 for a BIC and from Eq. (5) it can be seen that setting γ_{m′} to zero does not change the solution. Therefore, to search for BICs, we first solve the realvalued equation f ′(k_{y}, ω) = 0 at each k_{y} for a realvalued frequency ω; the solution also provides a mode profile given by T and C. However, such a mode profile will only satisfy the continuity condition, Eqs (4 and 5), if T_{m′} = 0. In this work, we study the frequency range where there is only one leaky channel (only one m′ with ) and we perform a root finding with k_{y} being the free parameter to look for solutions of f ′ = 0 where the amplitude of this radiation channel vanishes, T_{m′} = 0. Once found, such a solution will be a true bound state at a purely real frequency and with no radiation.
Results
Photonic crystal slab in a homogeneous background
In this work we study two systems. The first one is a layered slab in a homogeneous medium (Fig. 1b). It consists of a sequence of dielectric rectangles of size d × h with permittivity ε_{A}, surrounded by a homogeneous material with permittivity ε_{B}. The eigenmodes in the homogeneous medium are simply plane waves with and Θ being an identity matrix. In Fig. 2a, the region with one leaky channel (one real γ_{m′}) is shaded in yellow. In the slab, the Fourier coefficients of the permittivity is ξ_{1}(n) = (d/a)(ε_{A} − ε_{B}) sin c (nd/a) + ε_{B}δ_{n}. For the basis truncation, we take M = 21 Fourier terms and eigenmodes (N = 10), which is enough for the results to converge within the frequency range we consider. As an example, we take ε_{A} = 4.9 (this is a reasonably small dielectric constant and is close to many common optical materials such as silicon nitride, zinc oxide, gallium nitride, indium tin oxide and diamond), ε_{B} = 1, d = 0.5a and h = 1.4a. By solving Eq. (8), we obtain the band structure Re(ω) and the quality factor Q = −ω_{r}/(2ω_{i}) of the resonances, shown as solid curves in Fig. 2a,b. As a validation, we also perform finitedifference timedomain (FDTD) simulations for the same structure. The FDTD simulations are carried out at a spatial resolution of 32^{2} points per area of a^{2} (high enough for the calculated Q to converge) and take significantly longer than our method; the results, shown as circles in Fig. 2a,b, are in perfect agreement with our method.
The quality factor diverges at the BICs. Nonsymmetryprotected BICs occur at k_{y} = 00.3156 (2π/a) for eveninx modes (blue curve) and at k_{y} = 0.1640 (2π/a) for oddinx modes (green curve). Symmetry protected BICs can be found at the Γ point (k_{y} = 0), where radiation vanishes because E_{z} is odd in y for the resonance but even in y for the radiating wave. Such symmetryincompatibility mechanism also holds at the edge of the Brillouin zone (k_{y} = π/a), but at the zone edge of this system modes are either regular bound states (below the yellow shaded area) or states with multiple leaky channels (above the yellow shaded area) for which BICs are harder to come by. We will show zoneedge BICs in the second system. Fig. 2c shows the field profiles of the four BICs.
In Fig. 2d, we plot the coefficients C and T of the eveninx nonsymmetryprotected BIC. Inside the slab (upper panel), the amplitudes C are dominated by two propagating modes (shown in red). Outside the slab (lower panel), there is only one propagating mode and its amplitude vanishes at the BIC. The amplitude of the propagating mode outside the slab is the transmission from the inslab modes, as shown in Eq. (11). Therefore, the disappearance of radiation arises from destructive interference of the transmission from the inslab modes, which, as shown in the upper panel, primarily consist of two propagating modes.
Photonic crystal slab in a periodicallymodulated background
The second system we consider is a PhC slab surrounded by a periodicallymodulated background (Fig. 1c). The outofslab region has the same period as the slab but with a different filling fraction. We consider ε_{A} = 4.9, ε_{B} = 1, d_{1} = 0.5a, d_{2} = 0.2a, h = 1.4a. Figure 3a shows the band structure obtained from Eq. (8), with yellow shading over the region with one leaky channel (, ); the corresponding quality factor is shown in Fig. 3b. Results from FDTD simulations are shown as circles in Fig. 3a,b; again the simulation results quantitatively agree with our method.
In this structure, symmetryprotected BICs can be found both at the Γ point (k_{y} = 0) and at the zone edge (k_{y} = π/a) inside the yellowshaded region. The zoneedge BICs are possible because the periodic modulation in the background breaks the degeneracy of the propagating waves on the zone edge and opens up a finite yellowshaded region where only one leaky channel is present. On the zone edge and within this region, the leaky wave is even under mirror flip around y = 0, but the BICs are odd (as can be seen from the mode profile in Fig. 3c) so they decouple from radiation. Meanwhile, nonsymmetryprotected BICs still exist, as marked by red dashed lines in Fig. 3b and with the mode profiles shown in Fig. 3c.
In this system, we can observe an interesting phenomenon that two BICs annihilate at lowsymmetry k points. On the oddinx band, as we vary the filling fraction in the background from d_{2}/a = 0.23 to d_{2}/a = 0.25, we observe that another nonsymmetryprotected BIC emerges from the Γ point, moves along the k_{y} axis and then annihilates with the other nonsymmetryprotected BIC near k_{y} = 0.14 (2π/a). The annihilation removes both BICs and leaves behind a finite peak in Q, as shown in Fig. 4a. The annihilation can be understood from the radiation coefficient T_{0}, which we plot in Fig. 4b. Each zerocrossing of T_{0} corresponds to a BIC. Since T_{0} is expected to change continuously, two adjacent zerocrossings must have opposite slope and will cancel each other when they meet. This is an example of the topological charge of BICs^{18}; here the two neighboring BICs have opposite charges and can annihilate with each other. Note that near the annihilation of the two BICs, quantitative prediction of the quality factor using FDTD becomes exceedingly hard due to an increased sensitivity on structural variations (which requires an unusually high spatial resolution in FDTD); nonetheless, we can still calculate Q efficiently with our method.
Discussion
In principle, more BICs can lie above the yellow shaded area in Figs 2a and 3a, where there are two or more radiation ports. In this case, three or more independent equations have to be satisfied simultaneously: f ′ = 0, T_{0} = 0, T_{1} =0 … It will require rootfinding with more variables than k_{y} and ω, which means the structure itself has to be finetuned to find BICs; such structuresensitive BICs are less practical as experimental realization will be harder.
Our analysis of PhC slabs with 1D periodicity can be considered an extension of the previous topological vortex work^{18} to the 1D parameter space. As shown in Fig. 4b, here BICs correspond to nodal points where the radiation amplitude crosses zero (instead of vortex centers), which are manifestations of topological charges in 1D.
The preceding examples concern structures where the dielectric is real and symmetric in y, for which the matrices H_{l} are real symmetric and so the function f ′ is realvalued. However, BICs can exist in even more general systems. As long as , H_{l} is real (although not necessarily symmetric and not necessarily Hermitian). In such PTsymmetric systems, if the nonHermiticity is below the PTbreaking threshold, the eigenvalues and eigenvectors of H_{l} can still be real and the function f ′ can still be realvalued. Such systems can also support BICs. However, if the introduction of gain leads to lasing, one will need to account for the nonlinearity resulting from gain saturation^{37,38,39}, which is beyond the linear model considered in this work.
Conclusion
We have presented a mode expansion method that can efficiently and quantitatively describe guided resonances and BICs in PhC slabs and the method also reveals their underlying formation mechanisms. We find symmetryprotected BICs at the Γ point and at the zone edge, as well as BICs not protected by symmetry. The formalism is easily extendable and applicable to a wide range of structures. This is an attractive approach for the study of guided resonances and BICs in periodic structures.
Additional Information
How to cite this article: Gao, X. et al. Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs. Sci. Rep. 6, 31908; doi: 10.1038/srep31908 (2016).
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Acknowledgements
This work was partly supported by the Army Research Office through the Institute for Soldier Nanotechnologies under contract no. W911NF13D0001. B.Z. and M.S were partly supported (analysis and reading of the manuscript) by S3TEC, an Energy Frontier Research Center funded by the US Department of Energy under grant no. desc0001299. B.Z. was partially supported by the United StatesIsrael Binational Science Foundation (BSF) under award no. 2013508. C.W.H. was partly supported by the National Science Foundation through grant no. DMR1307632. H.C., X.G. and X.L. were partly supported by the National Natural Science Foundation of China under Grants No. 61322501, No. 61574127 and No. 61275183, the TopNotch Young Talents Program of China, the Program for New Century Excellent Talents (NCET120489) in University, the Fundamental Research Funds for the Central Universities and the Innovation Joint Research Center for CyberPhysicalSociety System.
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C.W.H., M.S. and H.C. conceived the study. X.G. and C.W.H. developed the theory and performed the computation and data analysis. X.G., C.W.H., B.Z., X.L., J.D.J., M.S. and H.C. discussed and interpreted the results and prepared the manuscript. C.W.H., M.S. and H.C. supervised the project.
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Gao, X., Hsu, C., Zhen, B. et al. Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs. Sci Rep 6, 31908 (2016). https://doi.org/10.1038/srep31908
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