Abstract

We theoretically and numerically investigate resonant optical properties of composite structures consisting of several subwavelength resonant diffraction gratings separated by homogeneous layers. Using the scattering matrix formalism, we demonstrate that the composite structure comprising N gratings has a multiple transmittance zero of the order N. We show that at the distance between the gratings satisfying the Fabry–Pérot resonance condition, an (N – 1)-degenerate bound state in the continuum (BIC) is formed. The results of rigorous numerical simulations fully confirm the theoretically predicted formation of multiple zeros and BICs in the composite structures. Near the BICs, an effect very similar to the electromagnetically induced transparency is observed. We show that by making the proper choice of the thicknesses of the layers separating the gratings, nearly rectangular reflectance or transmittance peaks with steep slopes and virtually no sidelobes can be obtained. In particular, one of the presented examples demonstrates the possibility of obtaining an approximately rectangular transmittance peak with a significantly subnanometer width. The presented results may find application in the design of optical filters, sensors and devices for optical differentiation and transformation of optical signals.

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

1. Introduction

The effect of optical resonance is utilized in a wide class of photonic devices possessing unique optical properties [1]. Diffraction gratings (DGs) constitute one of the most widespread classes of resonant photonic structures. Despite a long history (resonances in DGs were first observed by R. Wood in 1902 [2]), resonant DGs remain the subject of intensive research due to numerous extraordinary optical effects arising in resonant conditions [3,4].

In recent years, significant research attention was dedicated to the investigation of the so-called bound states in the continuum (BICs) supported by various photonic structures and, in particular, by diffraction gratings [510]. Under BICs, a class of the eigenmodes of the structure is understood, which, although coexisting with a continuous spectrum of radiating waves, remain perfectly confined and, therefore, have an infinite quality factor and a real frequency. The leakage of the mode energy to the open scattering channels can be eliminated, for example, by properly choosing the parameters of the structure so that the amplitudes of the outgoing waves vanish due to destructive interference. A small detuning from the BIC condition enables obtaining resonances with extremely high quality factors. This leads to many potential applications of photonic structures supporting BICs, including narrowband filtering and sensoring.

In the present work, we investigate composite dielectric diffractive structures consisting of several identical resonant DGs with subwavelength period separated by homogeneous dielectric layers. A distinctive feature of resonant subwavelength dielectric gratings is the presence of zeros in the transmittance spectrum [1113]. At the corresponding frequencies and angles of incidence, the radiation incident on the grating is totally reflected. Existence of the transmittance zeros makes it possible to use resonant DGs as optical filters [4,1416], sensors [4,17,18], and even devices for analog optical differentiation of optical signals [12,19]. In this work, using the scattering matrix formalism, we show that a composite structure comprising N identical DGs having a transmittance zero enables obtaining a multiple transmittance zero of the order N. It is demonstrated that at the distances between the DGs satisfying the Fabry–Pérot resonance condition, $N - 1$ bound states in the continuum, which are degenerate, are formed in the composite structure. In the vicinity of the BICs, an effect very similar to the electromagnetically induced transparency is observed. The presence of several resonances (and BICs) in the composite structure provides additional opportunities for controlling the shape of the spectra by choosing the thicknesses of the dielectric layers separating the DGs. In particular, one of the presented examples demonstrates the possibility of obtaining a flat-top resonant transmittance peak having a substantially subnanometer width.

2. Scattering matrix formalism

The optical properties of a diffraction grating can be described by a scattering matrix. The scattering matrix S relates the complex amplitudes of the plane waves incident on the grating with the amplitudes of the transmitted and reflected diffraction orders [11,13,2023]:

$$\left( {\begin{array}{{c}} {\textbf T}\\ {\textbf R} \end{array}} \right) = {\textbf S}\left( {\begin{array}{{c}} {{{\textbf I}_u}}\\ {{{\textbf I}_d}} \end{array}} \right),$$
where ${\textbf R}$ and ${\textbf T}$ are the vectors of complex amplitudes of the reflected and transmitted diffraction orders, respectively, and ${{\textbf I}_u}$ and ${{\textbf I}_d}$ are the vectors of complex amplitudes of the plane waves impinging on the DG from the superstrate and the substrate regions, respectively. In what follows, we assume that the elements of the scattering matrix of Eq. (1) are functions of the angular frequency $\omega$ of the incident light, whereas the incidence angle is fixed. In the case of the so-called planar diffraction [24] and fixed polarization (transverse electric or transverse magnetic), the optical properties of a subwavelength DG, in which only the 0th reflected and transmitted diffraction orders propagate (are non-evanescent), can be described by a $2 \times 2$ scattering matrix
$${{\textbf S}_1}(\omega )= \left( {\begin{array}{{cc}} {{t_1}(\omega )}&{{r_{d,1}}(\omega )}\\ {{r_{u,1}}(\omega )}&{{t_1}(\omega )} \end{array}} \right).$$
Here, ${t_1}(\omega )$ is the complex transmission coefficient of the DG (complex amplitude of the 0th transmitted diffraction order) for a unit-amplitude wave incident from the superstrate or the substrate region, and ${r_{u,1}}(\omega )$ and ${r_{d,1}}(\omega )$ are the complex reflection coefficients (complex amplitudes of the 0th reflected diffraction orders) for unit-amplitude waves impinging on the DG from the superstrate and the substrate regions, respectively. It is worth noting that the scattering matrix of Eq. (2) does not describe the near-field effects, which are associated with the evanescent diffraction orders of the DG. In addition, let us mention that the matrix of Eq. (2) also allows one to describe the optical properties of multilayer diffractive structures containing homogeneous layers and subwavelength diffraction gratings.

Let us consider a composite structure (composite DG) consisting of two subwavelength DGs described by scattering matrices of Eq. (2) and separated by a homogeneous dielectric layer with the thickness l and the refractive index n (Fig. 1). In this case, assuming that the layer is thick enough so that the near-field interaction between the gratings can be neglected and that $n_{env} = n$, where $n_{env}$ is the refractive index of the surrounding medium, we can express the scattering matrix of the composite DG through the matrix ${{\textbf S}_1}(\omega )$ in the form [20,22,25]

$${{\textbf S}_{\,\textrm{2}}}(\omega )= {{\textbf S}_1}(\omega )\ast {\textbf L}(\omega )\ast {{\textbf S}_1}(\omega ),$$
where the symbol $\ast $ denotes the Redheffer star product [20] defined as
$$\begin{array}{l} \left( {\begin{array}{{cc}} {{a_{1,1}}}&{{a_{1,2}}}\\ {{a_{2,1}}}&{{a_{2,2}}} \end{array}} \right) \ast \left( {\begin{array}{{cc}} {{b_{1,1}}}&{{b_{1,2}}}\\ {{b_{2,1}}}&{{b_{2,2}}} \end{array}} \right) = \\ = \frac{1}{{1 - {a_{1,2}}{b_{2,1}}}}\left( {\begin{array}{{cc}} {{b_{1,1}}{a_{1,1}}}&{{b_{1,2}} - {a_{1,2}}({{b_{1,2}}{b_{2,1}} - {b_{1,1}}{b_{2,2}}} )}\\ {{a_{2,1}} - {b_{2,1}}({{a_{1,2}}{a_{2,1}} - {a_{1,1}}{a_{2,2}}} )}&{{a_{2,2}}{b_{2,2}}} \end{array}} \right), \end{array}$$
and ${\textbf L}(\omega )$ is the scattering matrix of the homogeneous dielectric layer. Upon the propagation through this layer, the plane waves corresponding to the 0th diffraction orders (which are assumed to be the only propagating ones) acquire only the phase shift
$$\psi (\omega )= l\sqrt {{{({n\;{\omega \mathord{\left/ {\vphantom {\omega \textrm{c}}} \right.} \textrm{c}}} )}^2} - k_x^2(\omega )} ,$$
where ${k_x}(\omega )= ({{\omega \mathord{\left/ {\vphantom {\omega \textrm{c}}} \right.} \textrm{c}}} ){n_{\textrm{env}}}\sin \theta$ is the tangential component of the wavevectors of the waves impinging on the composite structure from the superstrate and the substrate regions, c is the speed of light in vacuum, $\theta$ is the angle of incidence, and ${n_{env}}=n$ is the refractive index of the surrounding medium. Thus, the matrix ${\textbf L}(\omega )$ reads as
$${\textbf L}(\omega )= \exp \{{\textrm{i}\psi (\omega )} \}{\textbf E},$$
where ${\textbf E}$ is the $2 \times 2$ identity matrix. By substituting Eqs. (2) and (6) into Eq. (3), we obtain the scattering matrix of the composite DG in the form
$$\begin{array}{l} {{\textbf S}_{\,\textrm{2}}}(\omega )= \left( {\begin{array}{{cc}} {{t_2}(\omega )}&{{r_{d,2}}(\omega )}\\ {{r_{u,2}}(\omega )}&{{t_2}(\omega )} \end{array}} \right) = \frac{1}{{1 - \exp \{{2\textrm{i}\psi } \}{r_{u,1}}{r_{d,1}}}} \cdot \\ \cdot \left( {\begin{array}{{cc}} {\exp \{{\textrm{i}\psi } \}t_1^2}&{{r_{d,1}}[{1 - \exp \{{2\textrm{i}\psi } \}({{r_{u,1}}{r_{d,1}} - t_1^2} )} ]}\\ {{r_{u,1}}[{1 - \exp \{{2\textrm{i}\psi } \}({{r_{u,1}}{r_{d,1}} - t_1^2} )} ]}&{\exp \{{\textrm{i}\psi } \}t_1^2} \end{array}} \right). \end{array}$$
The obtained Eqs. (3)–(7) are convenient for the analysis of the optical properties of composite structures carried out in the rest of the present work.

 

Fig. 1. Geometry of a composite structure containing two identical diffraction gratings with period $\Lambda $, height h, and refractive indices ${n_1}$ and ${n_2}$ separated by a dielectric layer with the thickness l and refractive index n. The structure is located in a symmetric dielectric environment with refractive index ${n_{env}}$. The arrows denote the incident, reflected and transmitted waves.

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3. High-order zeros of the transmission coefficient of composite structures

Let us prove that if the initial DG has a transmission coefficient zero at a certain frequency $\omega = {\omega _0}$ (i.e. ${t_1}({{\omega_0}} )= 0$, $|{{r_{({u,d} ),1}}({{\omega_0}} )} |= 1$), then the composite structure described by Eq. (3) will have a transmission coefficient zero of the second order. We start by expanding the elements of the scattering matrix of Eq. (2) into Taylor series in the vicinity of the frequency $\omega = {\omega _0}$ up to linear terms:

$$\begin{array}{l} {t_1}(\omega )= \tau ({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2},\,\,\\ {r_{u,1}}(\omega )= \exp \{{\textrm{i}{\varphi_u}} \}+ {\rho _u}({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2},\\ {r_{d,1}}(\omega )= \exp \{{\textrm{i}{\varphi_d}} \}+ {\rho _d}({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2}, \end{array}$$
where $\tau$, ${\rho _u}$ and ${\rho _d}$ are the complex-valued coefficients corresponding to the linear terms in the Taylor series expansions, whereas ${\varphi _u}$ and ${\varphi _d}$ are the phases of the reflection coefficients at $\omega = {\omega _0}$.

Let us also expand the function $\exp \{{\textrm{i}\psi (\omega )} \}$ in Eq. (6) up to the linear term:

$$\exp \{{\textrm{i}\psi (\omega )} \}= \exp \{{\textrm{i}{\psi_0}} \}+ \gamma ({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2},$$
where ${\psi _0} = \psi ({{\omega_0}} )$, and $\gamma$ is the complex-valued expansion coefficient. Substituting the expansions (8) and (9) into Eq. (7), after some simple transformations we obtain the elements of the “composite” scattering matrix ${{\textbf S}_{\,\textrm{2}}}(\omega )$ in the form
$$\begin{array}{l} {t_2}(\omega )= \frac{{\exp \{{\textrm{i}\psi } \}{\tau ^2}}}{{1 - \exp \{{2\textrm{i}({{\psi_0} + \varphi } )} \}}}{({\omega - {\omega_0}} )^2} + \textrm{O}{({\omega - {\omega_0}} )^3},\\ {r_{(u,d),2}}(\omega )= \exp \{{\textrm{i}{\varphi_{(u,d)}}} \}+ {\rho _{(u,d)}}({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2}, \end{array}$$
where $\varphi = {{({{\varphi_u} + {\varphi_d}} )} \mathord{\left/ {\vphantom {{({{\varphi_u} + {\varphi_d}} )} 2}} \right.} 2}$. Thus, we obtained that the transmission coefficient of the composite structure has a zero of the second order, whereas the expansions of the reflection coefficients ${r_{({u,d} ),2}}$ still contain both the zero- and first-order terms. Let us note that the representations given by Eq. (10) cannot be used if the condition
$${\psi _0} + \varphi = \pi m,\,\,m \in {\mathbb N}$$
holds. The condition (11) describes the formation of the Fabry–Pérot resonance between the two DGs. In this case, the denominator of the fraction in the expression for the coefficient ${t_2}(\omega )$ in Eq. (10) vanishes. By assuming in Eq. (9) that $\exp \{{\textrm{i}{\psi_0}} \}= \exp \{{\textrm{i}({\pi m - \varphi } )} \}$ and carrying out a derivation similar to the derivation of Eq. (10), one can easily show that under the condition of Eq. (11), the transmission coefficient of the composite structure has only a first-order zero:
$${t_2}(\omega )= - \frac{{\exp \{{\textrm{i}\varphi } \}{\tau ^2}}}{{2\exp \{{3\textrm{i}\varphi } \}\gamma + {{({ - 1} )}^m}({\exp \{{\textrm{i}{\varphi_d}} \}{\rho_u} + \exp \{{\textrm{i}{\varphi_u}} \}{\rho_d}} )}}({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2}.$$
Let us note that expressions similar to Eqs. (10)–(12) can also be obtained in a more general case when the second diffraction grating is laterally shifted with respect to the first one along the periodicity direction. In this case, the last scattering matrix in the Redheffer star product in Eq. (3) is multiplied by an exponential “phase” term describing this shift. However, due to the complex form of the Redheffer star product, this complicates Eqs. (10)–(12). Therefore, for the clarity of the further analysis, in the present work we restrict our consideration to the case when the diffraction gratings constituting the composite structure are not shifted with respect to each other. At the same time, the lateral shifts between the gratings in the composite structure can be considered as additional degrees of freedom; the possibility of tailoring the spectra of the composite structures using these shifts will be investigated elsewhere.

In what follows, let us show that a composite structure consisting of N DGs described by the scattering matrix of Eq. (2) has a transmission zero of the order N. This can be easily proven by induction. Indeed, at $N = 2$, this statement has already been proven. Let us now show that if a composite structure described by the scattering matrix ${{\textbf S}_{\,N}}(\omega )$ has a zero of the order N, then the composite structure described by the matrix

$${{\textbf S}_{\,N + 1}}(\omega )= {{\textbf S}_N}(\omega )\ast {\textbf L}(\omega )\ast {{\textbf S}_1}(\omega )$$
will have a zero of the order $N + 1$. Let the elements of the scattering matrix ${{\textbf S}_{\,N}}(\omega )$ have the form
$$\begin{array}{l} {t_N}(\omega )= \frac{{\exp \{{\textrm{i}({N - 1} )\psi } \}{\tau ^N}}}{{{{[{1 - \exp \{{2\textrm{i}({{\psi_0} + \varphi } )} \}} ]}^{N - 1}}}}{({\omega - {\omega_0}} )^N} + \textrm{O}{({\omega - {\omega_0}} )^{N + 1}},\,\,\\ {r_{(u,d),N}}(\omega )= \exp \{{\textrm{i}{\varphi_{(u,d)}}} \}+ {\rho _{(u,d)}}({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2}. \end{array}$$
Let us note that at $N = 2$, Eq. (14) turns into Eq. (10). By substituting Eqs. (8), (9) and (14) into Eq. (13), we obtain the elements of the scattering matrix ${{\textbf S}_{\,N + 1}}(\omega )$:
$$\begin{array}{l} {t_{N + 1}}(\omega )= \frac{{\exp \{{\textrm{i}N\psi } \}{\tau ^{N + 1}}}}{{{{[{1 - \exp \{{2\textrm{i}({{\psi_0} + \varphi } )} \}} ]}^N}}}{({\omega - {\omega_0}} )^{N + 1}} + \textrm{O}{({\omega - {\omega_0}} )^{N + 2}},\\ {r_{(u,d),N + 1}}(\omega )= \exp \{{\textrm{i}{\varphi_{(u,d)}}} \}+ {\rho _{(u,d)}}({\omega - {\omega_0}} )+ \textrm{O}{({\omega - {\omega_0}} )^2}. \end{array}$$
The obtained expressions (15) show that the composite structure described by the scattering matrix ${{\textbf S}_{N + 1}}(\omega )$ of Eq. (13) has a transmission zero of the order $N + 1$. It is worth noting that the first two terms in the Taylor series expansions of the coefficients ${r_{(u,d),N + 1}}(\omega )$ do not depend on N, and, in particular, are equal to the corresponding terms in the expansions for ${r_{(u,d),1}}(\omega )$ in Eq. (8). As in the case of $N = 2$, the expression for ${t_{N + 1}}(\omega )$ in Eq. (15) cannot be used if the Fabry–Pérot condition of Eq. (11) holds. By induction, one can show that in this case the transmission coefficient ${t_{N + 1}}(\omega )$ has only a first-order zero.

4. Bound states in the continuum in composite structures

As a rule, the zeros in the transmittance and reflectance spectra of a diffraction grating are associated with the quasiguided eigenmodes supported by the structure. The frequencies of the eigenmodes correspond to the poles of the scattering matrix (the poles of the reflection and transmission coefficients) considered as a function of the complex frequency $\omega \in {\mathbb C}$ [13,21,26,27]. In the previous section, we demonstrated that a composite structure comprising N DGs has a multiple transmission zero of the order N, which turns into a simple (first-order) zero in the case when the Fabry–Pérot condition of Eq. (11) holds. To gain a deeper understanding of this effect, it is worth investigating how it is connected with the behavior of the eigenmodes supported by the composite structure.

In order to simplify the further analysis, let us assume that the transmittance zero of the initial DG is associated with a resonance having a Lorentzian line shape [23,28,29] in reflection. This line shape is often observed in subwavelength dielectric (lossless) DGs [14,28]. In the case of a Lorentzian resonance, the elements of the scattering matrix of Eq. (2) can be approximated by the following expressions [28]:

$$\begin{array}{c} {r_{u,1}}(\omega )= \exp \{{\textrm{i}{\varphi_u}} \}\frac{{ - \textrm{i}\,{\mathop{\rm Im}\nolimits} {\omega _p}}}{{\omega - {\omega _p}}},\,\,{r_{d,1}}(\omega )= \exp \{{\textrm{i}{\varphi_d}} \}\frac{{ - \textrm{i}\,{\mathop{\rm Im}\nolimits} {\omega _p}}}{{\omega - {\omega _p}}},\\ {t_1}(\omega )= \exp \{{\textrm{i}\varphi } \}\frac{{\omega - {\mathop{\rm Re}\nolimits} {\omega _p}}}{{\omega - {\omega _p}}}, \end{array}$$
where ${\omega _p}$ is the complex frequency of the eigenmode supported by the DG, and $\varphi = {{({{\varphi_u} + {\varphi_d}} )} \mathord{\left/ {\vphantom {{({{\varphi_u} + {\varphi_d}} )} 2}} \right.} 2}$. Let us note that the phases ${\varphi _u}$ and ${\varphi _d}$ have the same meaning as in Eq. (8), and the resonant representations of Eq. (16) are fully consistent with the series expansions of Eq. (8) at $\omega = {\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. The form of the reflection and transmission coefficients used in the resonant representations (16) ensures the unitarity of the scattering matrix ${{\textbf S}_1}(\omega )$. According to Eq. (16), the transmission coefficient $t(\omega )$ vanishes at ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. The width of the resonance is determined by the imaginary part of the pole ${\omega _p}$ and can be characterized by the quality factor $Q = {{{\mathop{\rm Re}\nolimits} {\omega _p}} \mathord{\left/ {\vphantom {{{\mathop{\rm Re}\nolimits} {\omega_p}} {2|{{\mathop{\rm Im}\nolimits} {\omega_p}} |}}} \right.} {2|{{\mathop{\rm Im}\nolimits} {\omega_p}} |}}$. From Eq. (16), it is easy to obtain that the width of the resonant reflectance peak $R(\omega )= {|{{r_{(u,d),1}}(\omega )} |^2}$ (resonant transmittance dip $\,T(\omega )= {|{{t_1}(\omega )} |^2}$) amounts to $\Delta = 2|{{\mathop{\rm Im}\nolimits} {\omega_p}} |$ at the 0.5 level.

Next, we consider composite structures comprising DGs with Lorentzian spectra described by Eq. (16). For the simplicity of the analysis, let us assume that the phase shift $\psi (\omega )$ acquired upon propagation through the layers separating the gratings [see Eq. (5)] can be approximated by the constant ${\psi _0} = \psi ({{\omega_0}} )$. The made assumption is valid when the layers separating the gratings are not too thick and the considered spectral range is narrow enough (i.e. the quality factor of the resonance of the DG is high enough: ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p} > > {\mathop{\rm Im}\nolimits} {\omega _p}$). In this case, the function $\exp \{{\textrm{i}\psi (\omega )} \}$ in Eqs. (5)–(7) can be approximated as $\exp \{{\textrm{i}{\psi_0}} \}$, and the scattering matrix of the layer separating the gratings can be considered frequency-independent:

$$\,{\textbf L} = \exp \{{\textrm{i}{\psi_0}} \}{\textbf E} = \exp \left\{ {\textrm{i}\frac{{{\omega_0}}}{c}l\sqrt {{n^2} - {{({{n_{\textrm{env}}}\sin \theta } )}^2}} } \right\}{\textbf E}.$$
From Eqs. (7) and (15)–(17), it follows that in the general case, a composite structure containing N DGs has a transmission zero of the order N and N poles. However, under the condition of Eq. (11), only one zero and one pole remain. Let us discuss this effect in more detail for the composite structures consisting of two and three resonant DGs. Substituting the resonant representations (16) into Eq. (7), after some simple transformations we obtain the transmission coefficient of a composite structure consisting of two DGs in the form
$${t_2}(\omega )= \exp \{{\textrm{i}({{\psi_0} + 2\varphi } )} \}\frac{{{{({\omega - {\mathop{\rm Re}\nolimits} {\omega_p}} )}^2}}}{{({\omega - {\omega_{p,1}}} )({\omega - {\omega_{p,2}}} )}},$$
where
$$\begin{array}{l} {\omega _{p,1}} = {\mathop{\rm Re}\nolimits} {\omega _p} + \textrm{i}[{1 - \exp \{{\textrm{i}({{\psi_0} + \varphi } )} \}} ]{\mathop{\rm Im}\nolimits} {\omega _p},\\ {\omega _{p,2}} = {\mathop{\rm Re}\nolimits} {\omega _p} + \textrm{i}[{1 + \exp \{{\textrm{i}({{\psi_0} + \varphi } )} \}} ]{\mathop{\rm Im}\nolimits} {\omega _p}. \end{array}$$
In accordance with the proof of the multiplicity of the transmission zero given above, the considered composite grating has a second-order transmission zero at ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$ and two poles defined by Eq. (19). Under the condition of Eq. (11), one of the poles becomes real and coincides with the real-valued transmission zero. For the sake of argument, let ${\mathop{\rm Im}\nolimits} {\omega _{p,1}} = 0$. In this case, the integer m in Eq. (11) is even, and ${\omega _{p,2}} = {\mathop{\rm Re}\nolimits} {\omega _p} + 2\textrm{i}{\mathop{\rm Im}\nolimits} {\omega _p}$. The eigenmode of the composite structure having the frequency ${\omega _{p,1}} = {\mathop{\rm Re}\nolimits} {\omega _p}$ is a bound state in the continuum (BIC) [510]. BIC is a mode that coexists with a continuum of radiating waves but has an infinite quality factor (a real frequency). The considered structure consisting of two DGs has only two scattering channels corresponding to the 0th reflected and transmitted diffraction orders. The leakage to these channels is canceled if the condition of Eq. (11) holds, which describes the Fabry–Pérot resonance formed between the diffraction gratings constituting the composite structure.

A similar behavior is observed in the composite structure consisting of three DGs. Calculating the scattering matrix ${{\textbf S}_{\,3}}(\omega )= {{\textbf S}_2}(\omega )\ast {\textbf L} \ast {{\textbf S}_1}(\omega )$, we obtain the transmission coefficient of this composite structure in the form

$${t_3}(\omega )= \exp \{{\textrm{i}({2{\psi_0} + 3\varphi } )} \}\frac{{{{({\omega - {\mathop{\rm Re}\nolimits} {\omega_p}} )}^3}}}{{({\omega - {\omega_{p,1}}} )({\omega - {\omega_{p,2}}} )({\omega - {\omega_{p,3}}} )}},$$
where
$$\begin{array}{l} {\omega _{p,1}} = {\mathop{\rm Re}\nolimits} {\omega _p} + \textrm{i}({1 - \sigma } ){\mathop{\rm Im}\nolimits} {\omega _p},\\ {\omega _{p,2}} = {\omega _p} + \textrm{i}\frac{{\sigma - \sqrt {\sigma ({8 + \sigma } )} }}{2}{\mathop{\rm Im}\nolimits} {\omega _p},\\ {\omega _{p,3}} = {\omega _p} + \textrm{i}\frac{{\sigma + \sqrt {\sigma ({8 + \sigma } )} }}{2}{\mathop{\rm Im}\nolimits} {\omega _p}, \end{array}$$
where $\sigma = \exp \{{2\textrm{i}({{\psi_0} + \varphi } )} \}$. According to Eq. (20), the transmission coefficient of the composite structure comprising three DGs has a third-order zero and three poles. Under the condition of Eq. (11), the poles defined by Eq. (21) take the form
$${\omega _{p,1}} = {\omega _{p,2}} = {\mathop{\rm Re}\nolimits} {\omega _p},\quad {\omega _{p,3}} = {\mathop{\rm Re}\nolimits} {\omega _p} + 3\textrm{i}{\mathop{\rm Im}\nolimits} {\omega _p}.$$
Thus, if the condition of Eq. (11) holds, the poles ${\omega _{p,1}}$ and ${\omega _{p,2}}$ become identical and real and coincide with the real-valued transmission zeros. This means that in the composite structure with three DGs, two bound states in the continuum exist. Since the BIC frequencies coincide, their arbitrary linear combination is also a bound state; therefore, the BICs supported by the structure described by Eqs. (11) and (20)–(22) turn out to be double-degenerate.

Finally, let us discuss the behavior of the composite structure comprising N DGs. As it was shown above, the composite structure consisting of N DGs has a transmittance zero of the order N and N poles. Therefore, the transmission coefficient of such a composite structure can be written as

$${t_N}(\omega )= \exp \{{\textrm{i}\xi } \}\frac{{{{({\omega - {\mathop{\rm Re}\nolimits} {\omega_p}} )}^N}}}{{\prod\limits_{m = 1}^N {({\omega - {\omega_{p,m}}} )} }},$$
where $\xi$ is a certain constant, and ${\omega _{p,m}},\,\,m = 1,\ldots ,N$ are the poles of the transmission coefficient. Under the Fabry–Pérot resonance condition of Eq. (11), only a first-order transmission zero and a single pole remain in the composite structure. This means that $N - 1$ poles of the composite DG become real and coincide with the real-valued transmittance zero ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. Thus, if the condition of Eq. (11) holds, (N – 1)-degenerate BICs are formed in the composite structure.

Let us note that the formation of a Fabry–Pérot BIC in the composite structure consisting of two resonant structures is known. In particular, Fabry–Pérot BICs formed in a structure consisting of two DGs were considered in [68]. At the same time, to the best of our knowledge, the effect of the formation of multiply-degenerate BICs in the composite structure consisting of N DGs has not been considered yet. Let us note that the Lorentzian shape of the resonance supported by the single DG is not necessary for the formation of a set of BICs. The representations given by Eq. (16) were chosen solely for the convenience of the analysis of the structures composed of two and three DGs. In addition, for Lorentzian resonances, one can easily prove by induction that if the condition of Eq. (11) holds, the transmission coefficient of the composite structure takes the form

$${t_N}(\omega )= {({ - 1} )^{N - 1}}\exp \{{\textrm{i}\varphi } \}\frac{{\omega - {\mathop{\rm Re}\nolimits} {\omega _p}}}{{\omega - {\mathop{\rm Re}\nolimits} {\omega _p} + \textrm{i}\,N{\mathop{\rm Im}\nolimits} {\omega _p}}},$$
i.e. the only remaining pole ${\omega _{p,N}} = {\mathop{\rm Re}\nolimits} {\omega _p} + \textrm{i}\,N{\mathop{\rm Im}\nolimits} {\omega _p}$ has an imaginary part N times greater than that of the pole ${\omega _p}$ of the transmission coefficient of the initial DG.

5. Numerical simulation results

Let us numerically investigate the resonant properties of the considered composite structures. For this, let us first discuss the resonant behavior of the diffraction grating used as a building block of the composite structures.

5.1. Resonant diffraction grating

As the initial DG, let us use a single-layer (binary) resonant grating with the parameters taken from [14]. The reflectance and transmittance spectra of the grating are shown in Fig. 2, and the parameters of the DG are given in the figure caption. The inset to Fig. 2 shows the geometry of the grating. The spectra were calculated using an in-house implementation of the Fourier modal method (also known as rigorous coupled-wave analysis) [20] for the case of normal incidence of a TE-polarized plane wave. Due to the existence of a horizontal symmetry plane of the DG, the coefficients ${r_{u,1}}(\omega )$ and ${r_{d,1}}(\omega )$ are equal: ${r_{u,1}}(\omega )= {r_{d,1}}(\omega )= {r_1}(\omega )$. From Fig. 2, it is evident that the spectra have an approximately Lorentzian shape. Rigorous calculation shows that the scattering matrix of the considered DG has a pole with the complex frequency ${\omega _p} = 3.5863 \cdot {10^{15}} - 6.0108 \cdot {10^{12}}\textrm{i}\,\,{\textrm{s}^{ - 1}}$ [26]. The transmission coefficient vanishes at $\omega = {\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$ (at the wavelength ${\lambda _0} = 525.2\,\textrm{nm}$). At this frequency, $r({{\omega_0}} )= \exp \{{ - \,0.0284\textrm{i}} \}$, i.e. ${\varphi _{u,1}} = {\varphi _{d,1}} = - \,0.0284$.

 

Fig. 2. Spectra $R(\omega )= {|{r(\omega )} |^2}$, $T(\omega )= {|{t(\omega )} |^2}$ of the subwavelength resonant diffraction grating with the following parameters: period $\Lambda = 300\,\textrm{nm}$, grating height $h = 130\,\textrm{nm}$, refractive indices of the grating materials ${n_1} = 2.1$ and ${n_2} = 1.9$, width of the region with the refractive index ${n_1}$ $w = {\Lambda \mathord{\left/ {\vphantom {\Lambda 2}} \right.} 2}$, refractive index of the surrounding medium ${n_{env}} = 1.52$. The normally incident plane wave is TE-polarized. The inset shows the geometry of the grating.

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5.2. Bound states in the continuum

Figures 3(a) and 4(a) show the rigorously calculated dependencies of the reflectance ${R_N}({\omega ,l} )= {|{{r_N}({\omega ,l} )} |^2}$ and transmittance ${T_N}({\omega ,l} )= {|{{t_N}({\omega ,l} )} |^2}$ of the composite structures at $N = 3$ and $N = 4$ on the varying angular frequency of the incident light $\omega$ and the distance l between the DGs. Horizontal dashed lines depict the distances, at which the condition of Eq. (11) is fulfilled, i.e. the Fabry–Pérot resonances are formed. Vertical dashed lines show the “central” frequency ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. In the vicinity of intersections of these lines, resonant features manifested in the transmittance maxima and reflectance minima are clearly visible. When approaching the intersections, the width of the resonances decreases (the quality factor increases). Note that the sharp transmittance peaks arise against the background of a relatively smooth dip, the latter being caused by the existence of a multiple zero of the transmission coefficient. The resulting transmittance spectrum shape is very similar to the spectral shape associated with the electromagnetically induced transparency effect. At the dashed lines, the resonances vanish, which confirms the formation of BICs. The vanishing of the resonant features is clearly demonstrated in the magnified fragments of the transmittance ${T_N}({\omega ,l} )$ shown in Figs. 3(b) and 4(b). It is evident from these magnified fragments that the distance l, at which the resonances vanish, is exactly predicted by the condition of Eq. (11) [see the horizontal dashed lines in Figs. 3(b) and 4(b)]. Let us note that the number of the vanishing resonances in the vicinity of each intersection of the dashed lines coincides with the number of the formed BICs (two BICs at $N = 3$ and three BICs at $N = 4$). For the sake of illustration, Figs. 3(c) and 4(c) show the “near-BIC” transmittance spectra of the composite structures at $l = 1043\,\textrm{nm}$ [depicted with the horizontal dotted lines in Figs. 3(b) and 4(b)], which slightly differs from the value $l = 1038\,\textrm{nm}$ satisfying the Fabry–Pérot resonance condition of Eq. (11) and depicted with the horizontal dashed lines in Figs. 3(b) and 4(b). Two and three resonant peaks are clearly visible in Figs. 3(c) and 4(c), respectively. The vanishing of the resonances under the Fabry–Pérot resonance condition is also confirmed by Figs. 3(d) and 4(d), which show the transmittance spectra of the composite structures at $l = 1038\,\textrm{nm}$. In accordance with the theoretical results of Section 4, only one resonance can be observed in the rigorously calculated Figs. 3(d) and 4(d), which, according to Eq. (24), has a significantly lower Q-factor.

 

Fig. 3. (a) Reflectance ${R_3}({\omega ,l} )$ and transmittance ${T_3}({\omega ,l} )$ of the composite structure at $N = 3$ vs. the angular frequency of the incident wave and the distance l between the adjacent DGs. Horizontal black and white dashed lines show the distances corresponding to the Fabry–Pérot resonances. Red dashed lines indicate the middle distances between adjacent Fabry–Pérot resonances. Vertical dashed lines show the frequency ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. (b) Magnified fragment of the transmittance in the region shown with a dashed rectangle in (a). (c), (d) Transmittance spectra of the composite structure near ($l = 1043\,\textrm{nm}$, horizontal dotted line in (b)) and under ($l = 1038\,\textrm{nm}$, horizontal dashed line in (b)) the Fabry–Pérot resonance condition, respectively (red solid lines). Dashed blue line in (d) shows the transmittance spectrum of the initial diffraction grating.

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Fig. 4. (a) Reflectance ${R_4}({\omega ,l} )$ and transmittance ${T_4}({\omega ,l} )$ of the composite structure at $N = 4$ vs. the angular frequency of the incident wave and the distance l between the adjacent DGs. Horizontal black and white dashed lines show the distances corresponding to the Fabry–Pérot resonances. Red dashed lines indicate the middle distances between adjacent Fabry–Pérot resonances. Vertical dashed lines show the frequency ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. (b) Magnified fragment of the transmittance in the region shown with a dashed rectangle in (a). (c), (d) Transmittance spectra of the composite structure near ($l = 1043\,\textrm{nm}$, horizontal dotted line in (b)) and under ($l = 1038\,\textrm{nm}$, horizontal dashed line in (b)) the Fabry–Pérot resonance condition, respectively (red solid lines). Dashed blue line in (d) shows the transmittance spectrum of the initial diffraction grating.

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5.3. Investigation and optimization of the spectra of composite structures

Generally speaking, the utilization of composite (cascaded) structures supporting several resonances gives additional capabilities for controlling the shape of the reflectance and transmittance spectra. In particular, such structures enable obtaining nearly flat-top spectral shapes [15,16]. This phenomenon can also be observed in the considered composite structures. Indeed, in addition to the formation of a set of BICs, the spectra in Figs. 3 and 4 have another important feature, which is caused by the existence of a multiple zero of the transmission coefficient at $\omega = {\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. The existence of a multiple zero makes the shape of the spectra very different from that of the initial DG, and enables obtaining bands with near-zero transmittance and near-unity reflectance centered at the frequency ${\omega _0}$. These bands have the most “regular” shape in the middle between two Fabry–Pérot resonances, i.e. at

$${\psi _0} + \varphi = \pi ({m - {1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}} ),\,\,m \in {\mathbb N}.$$
The distances l satisfying the condition of Eq. (25) are shown in Figs. 3 and 4 with red dashed lines. As an example, Fig. 5 shows the reflectance and transmittance spectra of the composite structures at $l = 948\,\textrm{nm}$ [$m = 3$ in Eq. (25)]. Let us remind that throughout this work, the chosen distances l are large enough so that the near-field interaction between the gratings constituting the composite structure can be neglected. For example, at the distance $l = 948\,\textrm{nm}$, the amplitudes of the evanescent diffraction orders of the gratings decay by at least $5 \cdot {10^{ - 5}}$ times. Figure 5 shows that under the condition of Eq. (25), the transmittance dip becomes closer to a rectangle with an increase in N. At the same time, sidelobes appear near the main transmittance dip (reflectance peak), which are especially noticeable at $N = 4$ [Fig. 5(b)]. These sidelobes are caused by the appearance of additional poles of the transmission and reflection coefficients. For comparison, dashed lines in Fig. 5 show the spectra of the initial DG.

 

Fig. 5. Transmittance (solid black lines) and reflectance (solid red lines) of the composite structures with $N = 3$ (a) and $N = 4$ (b) at $l = 948\,\textrm{nm}$. Dashed lines show the transmittance (black) and reflectance (red) of the initial DG.

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The existence of a multiple zero and N poles in a composite structure containing N diffraction gratings makes it possible to control the shape of the resonant peak (or dip) by changing the thicknesses of the dielectric layers separating the DGs. Indeed, in the general case, the poles ${\omega _{p,m}},\,\,m = 1,\ldots ,N$ in the expression for the transmission coefficient given by Eq. (23) are the functions of the layer thicknesses ${l_j},\,\,j = 1,\ldots ,N - 1$. Above, we restricted our consideration to the case when ${l_j} = l,\,\,j = 1,\ldots ,N - 1$. However, by considering the ${l_j}$ values as optimization parameters, one can obtain a transmittance dip with a desired shape. Let us define the required transmittance profile ${T_N}(\omega )= {|{{t_N}(\omega )} |^2}$ by the function

$${D_N}(\omega )= \frac{{{{[{{{({\omega - {\omega_0}} )} \mathord{\left/ {\vphantom {{({\omega - {\omega_0}} )} \sigma }} \right.} \sigma }} ]}^{2N}}}}{{1 + {{[{{{({\omega - {\omega_0}} )} \mathord{\left/ {\vphantom {{({\omega - {\omega_0}} )} \sigma }} \right.} \sigma }} ]}^{2N}}}},$$
which describes a smooth nearly rectangular dip with the width $2\sigma$ and no sidelobes. The representation (26) is natural for the considered composite structures, since it generalizes the Lorentzian line shape and takes into account the fact that the transmission coefficient ${t_N}(\omega )$ has a zero of the order N. It is interesting to note that the function
$${P_N}(\omega )= 1 - {D_N}(\omega )= \frac{1}{{1 + {{[{{{({\omega - {\omega_0}} )} \mathord{\left/ {\vphantom {{({\omega - {\omega_0}} )} \sigma }} \right.} \sigma }} ]}^{2N}}}},$$
which describes the required reflectance peak, coincides with the squared modulus of the transfer function of the Butterworth filter of the order N [30].

Let us demonstrate the possibility to control the shape of the resonant transmittance dip (reflectance peak) for the composite structures consisting of four ($N = 4$) and six ($N = 6$) diffraction gratings. The distance between the sidelobe maxima in the transmittance spectrum of Fig. 5(b) ($N = 4$) amounts to $2\Delta \omega \approx 2.2 \cdot {10^{13}}{\textrm{s}^{ - 1}}$. The spectra in Fig. 5(b) were calculated at ${l_j} = l = 948\,\textrm{nm,}\,\,j = 1,2,3$ [$m = 3$ in Eq. (25)]. Using these values as starting points, the distances ${l_j}$ were optimized from the condition of obtaining a transmittance spectrum described by the function ${D_4}(\omega )$ at $\sigma = \Delta \omega \approx 1.1 \cdot {10^{13}}{\textrm{s}^{ - 1}}$. As a result of the optimization, the following values were obtained: ${l_1} = {l_3} = 952\,\textrm{nm}$ and ${l_2} = 1037\,\textrm{nm}$. The spectra of the composite structure with $N = 4$ and the optimized distances ${l_j}$ are shown in Fig. 6(a). Similarly, Fig. 6(b) shows the spectra of the composite structure with $N = 6$ obtained by optimization with respect to the values ${l_j}\textrm{,}\,\,j = 1,\ldots ,5$ from the condition of achieving the transmittance spectrum described by the function ${D_6}(\omega )$. From Fig. 6, it is evident that the obtained flat-top spectra are in good agreement with the functions ${D_N}(\omega )$ and ${P_N}(\omega )$ both at $N = 4$ [Fig. 6(a)] and at $N = 6$ [Fig. 6(b)].

 

Fig. 6. Transmittance (solid black lines) and reflectance (solid red lines) of the composite structures at $N = 4$ and ${l_1} = {l_3} = 952\,\textrm{nm}$, ${l_2} = 1037\,\textrm{nm}$ (a) and at $N = 6$ and ${l_1} = {l_5} = 1126\,\textrm{nm}$, ${l_2} = {l_4} = 1050\,\textrm{nm}$, ${l_3} = 943\,\textrm{nm}$ (b). Dashed lines show the spectra of the initial DG. Dash-dot lines show the functions ${D_N}(\omega )$ and ${P_N}(\omega )$ at $N = 4$ (a) and $N = 6$ (b).

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The width of the transmittance dips (reflectance peaks) in Fig. 6 amounts to $\Delta \omega \approx 2.3 \cdot {10^{13}}{\textrm{s}^{ - 1}}$ or $\Delta \lambda \approx 3.4\,\textrm{nm}$. Obtaining resonant peaks or dips with an essentially subnanometer size is also of great interest. Resonant features with an arbitrarily small width can be obtained if the thicknesses of the separating layers ${l_j},\,\,j = 1,\ldots ,N$ are close to the thickness l satisfying the Fabry–Pérot resonance condition of Eq. (11). As an example, Fig. 7(а) shows the transmittance of the composite structure with $N = 4$ at ${l_j} = 1033\,\textrm{nm},\,\,j = 1,2,3$. This value is close to the distance $l = 1038\,\textrm{nm}$ satisfying the Fabry–Pérot resonance condition and shown with the upper horizontal dashed lines in Fig. 4(a). In Fig. 7(a), one can clearly see three sharp transmittance peaks against the background of a smooth minimum associated with the multiple zero of the transmission coefficient. The full widths at half maximum of the resonant peaks in Fig. 7(a) with respect to the wavelength amount to $\Delta {\lambda _1} \approx 3.8 \cdot {10^{ - 4}}\,\textrm{nm}$ (left peak), $\Delta {\lambda _2} \approx 3.5 \cdot {10^{ - 3}}\,\textrm{nm}$ (central peak), and $\Delta {\lambda _1} \approx 5.7 \cdot {10^{ - 2}}\,\textrm{nm}$ (right peak). This effect is very similar to the electromagnetically induced transparency effect (EIT) and, in the considered case, is associated with the zeros of the reflection coefficient. Let us note that the reflection coefficient of the composite structure containing four DGs ($N = 4$) has four poles (which are the same as the poles of the transmission coefficient) and three zeros, which lie in the vicinity of the multiple zero of the transmission coefficient. It is interesting to mention that in the case when the condition of Eq. (11) holds, these three zeros become real, coalesce with the multiple zero of the transmission coefficient, and are cross-canceled with the real poles corresponding to the three BICs supported by the structure. This leads to the formation of a “simple” (single-pole) resonance having a Lorentzian line shape [see Fig. 4(d)].

 

Fig. 7. Transmittance spectra (solid blue lines) of the composite structure with $N = 4$ at ${l_1} = {l_2} = {l_3} = 1033\,\textrm{nm}$ (a) and of the optimized structure with ${l_1} = {l_3} = 1027\,\textrm{nm}$ and ${l_2} = 950\,\textrm{nm}$ (b). Red dash-dot lines show the desired peak shape ${P_4}(\omega )$. Black dashed lines show the transmittance spectrum of the initial DG. The inset shows the transmittance spectrum of the optimized structure in a wider frequency range.

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The positions and widths of the transmittance peaks in Fig. 7(a) depend on the thicknesses ${l_j},\,\,j = 1,2,3$. Therefore, by adjusting these parameters, one can try to make the peaks coalesce, forming a single peak with a subnanometer width and a required shape. Let us demonstrate this possibility. As the desired transmittance peak shape, we chose the function ${P_4}(\omega )$ defined by Eq. (27) with $\sigma = 8.75 \cdot {10^{10}}{\textrm{s}^{ - 1}}$, which is shown with dash-dot lines in Fig. 7. The peak width with respect to the wavelength amounts to $\Delta {\lambda _1} \approx 2.6 \cdot {10^{ - 2}}\,\textrm{nm}$. As a result of the optimization, the following thickness values were obtained: ${l_1} = {l_3} = 1027\,\textrm{nm}$ and ${l_2} = 950\,\textrm{nm}$. The transmittance spectrum of the corresponding optimized structure is shown in Fig. 7(b). The spectrum of the composite structure in Fig. 7(b) contains a single transmittance peak on an almost zero background. The shape of the obtained peak is quite close to the function ${P_4}(\omega )$. The inset in Fig. 7 shows the transmittance spectrum of the optimized composite structure in a wider frequency range and demonstrates the EIT-like behavior.

6. Conclusion

In this work, using the scattering matrix formalism, we investigated resonant optical properties of composite structures consisting of several subwavelength resonant diffraction gratings separated by homogeneous layers. We demonstrated that the composite structure comprising N gratings has a multiple transmission zero of the order N. At the distance between the DGs satisfying the Fabry–Pérot resonance condition, (N – 1)-degenerate bound states in the continuum are formed in the composite structure. The presented theoretical results were confirmed by full-wave numerical simulations. It was shown that in the vicinity of the BICs, an effect very similar to the electromagnetically induced transparency is observed.

We also demonstrated the possibility to control the shape of the resonant transmittance dip (reflectance peak) by choosing the thicknesses of the layers separating the DGs in the composite structure. In the case when the DGs constituting the structure support a Lorentzian-shape resonance, one can obtain resonant transmittance dips (reflectance peaks) of approximately rectangular shape. The shape of the reflectance peak is well described by the transfer function of the Butterworth filter, whereas the width of the formed peak is very close to the width of the initial Lorentzian resonance supported by a single DG.

The presence of $N - 1$ BICs in the composite structure provides additional degrees of freedom for controlling the shape of the spectra in the vicinity of these BICs. In particular, the presented example demonstrates that by choosing the thicknesses of the dielectric layers separating the DGs, one can obtain an approximately rectangular transmittance peak having a significantly subnanometer width.

The obtained results may find application in the design of optical filters, sensors and devices for analog optical computing and transformation of optical signals. In particular, since a composite structure comprising N diffraction gratings has a multiple transmission zero of the order N, it can optically implement the computation of the N-th derivative of the envelope of the incident optical pulse [19].

Funding

Russian Science Foundation (19-19-00514); Russian Foundation for Basic Research (18-37-20038); Russian Federation Ministry of Science and Higher Education (State contract with the "Crystallography and Photonics" Research Center of the RAS under agreement 007-GZ/Ch3363/26).

Acknowledgments

The investigation of the bound states in the continuum in the composite structures (Sections 4, 5.2) was supported by Russian Science Foundation; the studies regarding the design and investigation of the composite structures providing the required spectral peak shape (Sections 3, 5.3) were supported by Russian Foundation for Basic Research; the implementation of the simulation software and the investigation of the initial resonant grating (Section 5.1) were supported by the Russian Federation Ministry of Science and Higher Education.

References

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2. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4(21), 396–402 (1902). [CrossRef]  

3. P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018). [CrossRef]  

4. G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018). [CrossRef]  

5. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]  

6. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008). [CrossRef]  

7. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009). [CrossRef]  

8. R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010). [CrossRef]  

9. Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017). [CrossRef]  

10. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013). [CrossRef]  

11. N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005). [CrossRef]  

12. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Single-resonance diffraction gratings for time-domain pulse transformations: integration of optical signals,” J. Opt. Soc. Am. A 29(8), 1734–1740 (2012). [CrossRef]  

13. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986). [CrossRef]  

14. S. S. Wang and R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. 34(14), 2414–2420 (1995). [CrossRef]  

15. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters,” Appl. Opt. 41(7), 1241–1245 (2002). [CrossRef]  

16. K. Yamada, K. J. Lee, Y. H. Ko, J. Inoue, K. Kintaka, S. Ura, and R. Magnusson, “Flat-top narrowband filters enabled by guided-mode resonance in two-level waveguides,” Opt. Lett. 42(20), 4127–4130 (2017). [CrossRef]  

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18. P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014). [CrossRef]  

19. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Temporal differentiation of optical signals using resonant gratings,” Opt. Lett. 36(17), 3509–3511 (2011). [CrossRef]  

20. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13(5), 1024–1035 (1996). [CrossRef]  

21. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002). [CrossRef]  

22. N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18(7), 7569–7574 (2010). [CrossRef]  

23. M. Nevière, E. Popov, and R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12(3), 513–523 (1995). [CrossRef]  

24. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]  

25. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017). [CrossRef]  

26. D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31(5), 793–801 (2013). [CrossRef]  

27. A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002). [CrossRef]  

28. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]  

29. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]  

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References

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  1. M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
    [Crossref]
  2. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4(21), 396–402 (1902).
    [Crossref]
  3. P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
    [Crossref]
  4. G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
    [Crossref]
  5. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
    [Crossref]
  6. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
    [Crossref]
  7. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009).
    [Crossref]
  8. R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
    [Crossref]
  9. Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
    [Crossref]
  10. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
    [Crossref]
  11. N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
    [Crossref]
  12. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Single-resonance diffraction gratings for time-domain pulse transformations: integration of optical signals,” J. Opt. Soc. Am. A 29(8), 1734–1740 (2012).
    [Crossref]
  13. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986).
    [Crossref]
  14. S. S. Wang and R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. 34(14), 2414–2420 (1995).
    [Crossref]
  15. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters,” Appl. Opt. 41(7), 1241–1245 (2002).
    [Crossref]
  16. K. Yamada, K. J. Lee, Y. H. Ko, J. Inoue, K. Kintaka, S. Ura, and R. Magnusson, “Flat-top narrowband filters enabled by guided-mode resonance in two-level waveguides,” Opt. Lett. 42(20), 4127–4130 (2017).
    [Crossref]
  17. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
    [Crossref]
  18. P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
    [Crossref]
  19. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Temporal differentiation of optical signals using resonant gratings,” Opt. Lett. 36(17), 3509–3511 (2011).
    [Crossref]
  20. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13(5), 1024–1035 (1996).
    [Crossref]
  21. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
    [Crossref]
  22. N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18(7), 7569–7574 (2010).
    [Crossref]
  23. M. Nevière, E. Popov, and R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12(3), 513–523 (1995).
    [Crossref]
  24. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995).
    [Crossref]
  25. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017).
    [Crossref]
  26. D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31(5), 793–801 (2013).
    [Crossref]
  27. A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002).
    [Crossref]
  28. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
    [Crossref]
  29. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).
    [Crossref]
  30. H.-C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011).
    [Crossref]

2018 (2)

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

2017 (4)

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

K. Yamada, K. J. Lee, Y. H. Ko, J. Inoue, K. Kintaka, S. Ura, and R. Magnusson, “Flat-top narrowband filters enabled by guided-mode resonance in two-level waveguides,” Opt. Lett. 42(20), 4127–4130 (2017).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017).
[Crossref]

2016 (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

2014 (1)

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

2013 (2)

D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31(5), 793–801 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (2)

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18(7), 7569–7574 (2010).
[Crossref]

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

2009 (1)

2008 (1)

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

2005 (1)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

2003 (1)

2002 (4)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters,” Appl. Opt. 41(7), 1241–1245 (2002).
[Crossref]

1996 (1)

1995 (3)

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986).
[Crossref]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4(21), 396–402 (1902).
[Crossref]

Basset, G.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Bogdanov, A. A.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Borisov, A. G.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

Bykov, D. A.

Chang-Hasnain, C. J.

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

Chua, S. L.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Doskolovich, L. L.

Dunn, S. C.

Fan, S.

Fehrembach, A.-L.

Gallinet, B.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Gaylord, T. K.

Giessen, H.

Gippius, N. A.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18(7), 7569–7574 (2010).
[Crossref]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Golovastikov, N. V.

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017).
[Crossref]

Grann, E. B.

Hermannsson, P. G.

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Hsu, C. W.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Inoue, J.

Iorsh, I. V.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Ishihara, T.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Jacob, D. K.

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

Johnson, S. G.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Kintaka, K.

Kivshar, Y. S.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Ko, Y. H.

Koshelev, K. L.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Kristensen, A.

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Lavrinenko, A. V.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Lee, K. J.

Li, L.

Limonov, M. F.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Liu, H.-C.

Liu, V.

Magnusson, R.

Malureanu, R.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Marinica, D. C.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

Martin, O. J. F.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986).
[Crossref]

Maystre, D.

A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002).
[Crossref]

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33(5), 607–619 (1986).
[Crossref]

Moharam, M. G.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Ndangali, R. F.

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

Nevière, M.

Poddubny, A. N.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Pommet, D. A.

Popov, E.

Povinelli, M.

Qiao, P.

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

Quaranta, G.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Reinisch, R.

Rybin, M. V.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Sadrieva, Z. F.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Samusev, A.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Sentenac, A.

Shabanov, S. V.

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

Sinev, I. S.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Smith, C. L. C.

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Soifer, V. A.

Soljacic, M.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

Stone, A. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Suh, W.

Takayama, O.

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Tikhodeev, S. G.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18(7), 7569–7574 (2010).
[Crossref]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Ura, S.

Vannahme, C.

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Wang, S. S.

S. S. Wang and R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. 34(14), 2414–2420 (1995).
[Crossref]

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

Weiss, T.

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4(21), 396–402 (1902).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Yamada, K.

Yang, W.

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

Yariv, A.

Zhen, B.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref]

ACS Photonics (1)

Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017).
[Crossref]

Adv. Opt. Photonics (1)

P. Qiao, W. Yang, and C. J. Chang-Hasnain, “Recent advances in high-contrast metastructures, metasurfaces, and photonic crystals,” Adv. Opt. Photonics 10(1), 180–245 (2018).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

P. G. Hermannsson, C. Vannahme, C. L. C. Smith, and A. Kristensen, “Absolute analytical prediction of photonic crystal guided mode resonance wavelengths,” Appl. Phys. Lett. 105(7), 071103 (2014).
[Crossref]

Comput. Opt. (1)

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Temporal differentiation and integration of 3D optical pulses using phase-shifted Bragg gratings,” Comput. Opt. 41(1), 13–21 (2017).
[Crossref]

J. Lightwave Technol. (1)

J. Math. Phys. (1)

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

J. Opt. Soc. Am. A (6)

Laser Photonics Rev. (1)

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12(9), 1800017 (2018).
[Crossref]

Nat. Photonics (1)

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Nat. Rev. Mater. (1)

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Figures (7)

Fig. 1.
Fig. 1. Geometry of a composite structure containing two identical diffraction gratings with period $\Lambda $, height h, and refractive indices ${n_1}$ and ${n_2}$ separated by a dielectric layer with the thickness l and refractive index n. The structure is located in a symmetric dielectric environment with refractive index ${n_{env}}$. The arrows denote the incident, reflected and transmitted waves.
Fig. 2.
Fig. 2. Spectra $R(\omega )= {|{r(\omega )} |^2}$, $T(\omega )= {|{t(\omega )} |^2}$ of the subwavelength resonant diffraction grating with the following parameters: period $\Lambda = 300\,\textrm{nm}$, grating height $h = 130\,\textrm{nm}$, refractive indices of the grating materials ${n_1} = 2.1$ and ${n_2} = 1.9$, width of the region with the refractive index ${n_1}$ $w = {\Lambda \mathord{\left/ {\vphantom {\Lambda 2}} \right.} 2}$, refractive index of the surrounding medium ${n_{env}} = 1.52$. The normally incident plane wave is TE-polarized. The inset shows the geometry of the grating.
Fig. 3.
Fig. 3. (a) Reflectance ${R_3}({\omega ,l} )$ and transmittance ${T_3}({\omega ,l} )$ of the composite structure at $N = 3$ vs. the angular frequency of the incident wave and the distance l between the adjacent DGs. Horizontal black and white dashed lines show the distances corresponding to the Fabry–Pérot resonances. Red dashed lines indicate the middle distances between adjacent Fabry–Pérot resonances. Vertical dashed lines show the frequency ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. (b) Magnified fragment of the transmittance in the region shown with a dashed rectangle in (a). (c), (d) Transmittance spectra of the composite structure near ($l = 1043\,\textrm{nm}$, horizontal dotted line in (b)) and under ($l = 1038\,\textrm{nm}$, horizontal dashed line in (b)) the Fabry–Pérot resonance condition, respectively (red solid lines). Dashed blue line in (d) shows the transmittance spectrum of the initial diffraction grating.
Fig. 4.
Fig. 4. (a) Reflectance ${R_4}({\omega ,l} )$ and transmittance ${T_4}({\omega ,l} )$ of the composite structure at $N = 4$ vs. the angular frequency of the incident wave and the distance l between the adjacent DGs. Horizontal black and white dashed lines show the distances corresponding to the Fabry–Pérot resonances. Red dashed lines indicate the middle distances between adjacent Fabry–Pérot resonances. Vertical dashed lines show the frequency ${\omega _0} = {\mathop{\rm Re}\nolimits} {\omega _p}$. (b) Magnified fragment of the transmittance in the region shown with a dashed rectangle in (a). (c), (d) Transmittance spectra of the composite structure near ($l = 1043\,\textrm{nm}$, horizontal dotted line in (b)) and under ($l = 1038\,\textrm{nm}$, horizontal dashed line in (b)) the Fabry–Pérot resonance condition, respectively (red solid lines). Dashed blue line in (d) shows the transmittance spectrum of the initial diffraction grating.
Fig. 5.
Fig. 5. Transmittance (solid black lines) and reflectance (solid red lines) of the composite structures with $N = 3$ (a) and $N = 4$ (b) at $l = 948\,\textrm{nm}$. Dashed lines show the transmittance (black) and reflectance (red) of the initial DG.
Fig. 6.
Fig. 6. Transmittance (solid black lines) and reflectance (solid red lines) of the composite structures at $N = 4$ and ${l_1} = {l_3} = 952\,\textrm{nm}$, ${l_2} = 1037\,\textrm{nm}$ (a) and at $N = 6$ and ${l_1} = {l_5} = 1126\,\textrm{nm}$, ${l_2} = {l_4} = 1050\,\textrm{nm}$, ${l_3} = 943\,\textrm{nm}$ (b). Dashed lines show the spectra of the initial DG. Dash-dot lines show the functions ${D_N}(\omega )$ and ${P_N}(\omega )$ at $N = 4$ (a) and $N = 6$ (b).
Fig. 7.
Fig. 7. Transmittance spectra (solid blue lines) of the composite structure with $N = 4$ at ${l_1} = {l_2} = {l_3} = 1033\,\textrm{nm}$ (a) and of the optimized structure with ${l_1} = {l_3} = 1027\,\textrm{nm}$ and ${l_2} = 950\,\textrm{nm}$ (b). Red dash-dot lines show the desired peak shape ${P_4}(\omega )$. Black dashed lines show the transmittance spectrum of the initial DG. The inset shows the transmittance spectrum of the optimized structure in a wider frequency range.

Equations (27)

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( T R ) = S ( I u I d ) ,
S 1 ( ω ) = ( t 1 ( ω ) r d , 1 ( ω ) r u , 1 ( ω ) t 1 ( ω ) ) .
S 2 ( ω ) = S 1 ( ω ) L ( ω ) S 1 ( ω ) ,
( a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 ) ( b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ) = = 1 1 a 1 , 2 b 2 , 1 ( b 1 , 1 a 1 , 1 b 1 , 2 a 1 , 2 ( b 1 , 2 b 2 , 1 b 1 , 1 b 2 , 2 ) a 2 , 1 b 2 , 1 ( a 1 , 2 a 2 , 1 a 1 , 1 a 2 , 2 ) a 2 , 2 b 2 , 2 ) ,
ψ ( ω ) = l ( n ω / ω c c ) 2 k x 2 ( ω ) ,
L ( ω ) = exp { i ψ ( ω ) } E ,
S 2 ( ω ) = ( t 2 ( ω ) r d , 2 ( ω ) r u , 2 ( ω ) t 2 ( ω ) ) = 1 1 exp { 2 i ψ } r u , 1 r d , 1 ( exp { i ψ } t 1 2 r d , 1 [ 1 exp { 2 i ψ } ( r u , 1 r d , 1 t 1 2 ) ] r u , 1 [ 1 exp { 2 i ψ } ( r u , 1 r d , 1 t 1 2 ) ] exp { i ψ } t 1 2 ) .
t 1 ( ω ) = τ ( ω ω 0 ) + O ( ω ω 0 ) 2 , r u , 1 ( ω ) = exp { i φ u } + ρ u ( ω ω 0 ) + O ( ω ω 0 ) 2 , r d , 1 ( ω ) = exp { i φ d } + ρ d ( ω ω 0 ) + O ( ω ω 0 ) 2 ,
exp { i ψ ( ω ) } = exp { i ψ 0 } + γ ( ω ω 0 ) + O ( ω ω 0 ) 2 ,
t 2 ( ω ) = exp { i ψ } τ 2 1 exp { 2 i ( ψ 0 + φ ) } ( ω ω 0 ) 2 + O ( ω ω 0 ) 3 , r ( u , d ) , 2 ( ω ) = exp { i φ ( u , d ) } + ρ ( u , d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 ,
ψ 0 + φ = π m , m N
t 2 ( ω ) = exp { i φ } τ 2 2 exp { 3 i φ } γ + ( 1 ) m ( exp { i φ d } ρ u + exp { i φ u } ρ d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 .
S N + 1 ( ω ) = S N ( ω ) L ( ω ) S 1 ( ω )
t N ( ω ) = exp { i ( N 1 ) ψ } τ N [ 1 exp { 2 i ( ψ 0 + φ ) } ] N 1 ( ω ω 0 ) N + O ( ω ω 0 ) N + 1 , r ( u , d ) , N ( ω ) = exp { i φ ( u , d ) } + ρ ( u , d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 .
t N + 1 ( ω ) = exp { i N ψ } τ N + 1 [ 1 exp { 2 i ( ψ 0 + φ ) } ] N ( ω ω 0 ) N + 1 + O ( ω ω 0 ) N + 2 , r ( u , d ) , N + 1 ( ω ) = exp { i φ ( u , d ) } + ρ ( u , d ) ( ω ω 0 ) + O ( ω ω 0 ) 2 .
r u , 1 ( ω ) = exp { i φ u } i Im ω p ω ω p , r d , 1 ( ω ) = exp { i φ d } i Im ω p ω ω p , t 1 ( ω ) = exp { i φ } ω Re ω p ω ω p ,
L = exp { i ψ 0 } E = exp { i ω 0 c l n 2 ( n env sin θ ) 2 } E .
t 2 ( ω ) = exp { i ( ψ 0 + 2 φ ) } ( ω Re ω p ) 2 ( ω ω p , 1 ) ( ω ω p , 2 ) ,
ω p , 1 = Re ω p + i [ 1 exp { i ( ψ 0 + φ ) } ] Im ω p , ω p , 2 = Re ω p + i [ 1 + exp { i ( ψ 0 + φ ) } ] Im ω p .
t 3 ( ω ) = exp { i ( 2 ψ 0 + 3 φ ) } ( ω Re ω p ) 3 ( ω ω p , 1 ) ( ω ω p , 2 ) ( ω ω p , 3 ) ,
ω p , 1 = Re ω p + i ( 1 σ ) Im ω p , ω p , 2 = ω p + i σ σ ( 8 + σ ) 2 Im ω p , ω p , 3 = ω p + i σ + σ ( 8 + σ ) 2 Im ω p ,
ω p , 1 = ω p , 2 = Re ω p , ω p , 3 = Re ω p + 3 i Im ω p .
t N ( ω ) = exp { i ξ } ( ω Re ω p ) N m = 1 N ( ω ω p , m ) ,
t N ( ω ) = ( 1 ) N 1 exp { i φ } ω Re ω p ω Re ω p + i N Im ω p ,
ψ 0 + φ = π ( m 1 / 1 2 2 ) , m N .
D N ( ω ) = [ ( ω ω 0 ) / ( ω ω 0 ) σ σ ] 2 N 1 + [ ( ω ω 0 ) / ( ω ω 0 ) σ σ ] 2 N ,
P N ( ω ) = 1 D N ( ω ) = 1 1 + [ ( ω ω 0 ) / ( ω ω 0 ) σ σ ] 2 N ,

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