# Lines of Quasi-BICs and Butterworth Line Shape in Stacked Resonant Gratings: Analytical Description

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. ω—k_{x} Lorentzian Line Shape in a Single Resonant Grating

#### 2.1. Scattering Matrix

#### 2.2. $\omega -{k}_{x}$ Lorentzian Line Shape in a Symmetric Structure

#### 2.3. $\omega -{k}_{x}$ Lorentzian Line Shape in a Structure without a Horizontal Symmetry Plane

#### 2.4. Numerical Example

#### 3. ω—k_{x} Resonant Approximation for Stacked Resonant Gratings

## 4. Butterworth Filters Based on Stacked Resonant Gratings

#### 4.1. Second-Order Butterworth Filter for Temporal Signals

#### 4.2. Fourth-Order Quasi-Butterworth Filter for Spatial Signals

## 5. BICs and Lines of Quasi-BICs in Stacked Resonant Gratings

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Quaranta, G.; Basset, G.; Martin, O.J.; Gallinet, B. Recent advances in resonant waveguide gratings. Laser Photonics Rev.
**2018**, 12, 1800017. [Google Scholar] [CrossRef] - Suh, W.; Fan, S. Mechanically switchable photonic crystal filter with either all-pass transmission or flat-top reflection characteristics. Opt. Lett.
**2003**, 28, 1763–1765. [Google Scholar] [CrossRef] - Jacob, D.K.; Dunn, S.C.; Moharam, M.G. Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters. Appl. Opt.
**2002**, 41, 1241–1245. [Google Scholar] [CrossRef] [PubMed] - Ko, Y.H.; Magnusson, R. Flat-top bandpass filters enabled by cascaded resonant gratings. Opt. Lett.
**2016**, 41, 4704–4707. [Google Scholar] [CrossRef] - Doskolovich, L.L.; Bezus, E.A.; Bykov, D.A.; Golovastikov, N.V.; Soifer, V.A. Resonant properties of composite structures consisting of several resonant diffraction gratings. Opt. Express
**2019**, 27, 25814–25828. [Google Scholar] [CrossRef] [PubMed] - Song, H.Y.; Kim, S.; Magnusson, R. Tunable guided-mode resonances in coupled gratings. Opt. Express
**2009**, 17, 23544–23555. [Google Scholar] [CrossRef] - Gippius, N.A.; Weiss, T.; Tikhodeev, S.G.; Giessen, H. Resonant mode coupling of optical resonances in stacked nanostructures. Opt. Express
**2010**, 18, 7569–7574. [Google Scholar] [CrossRef] - Weiss, T.; Gippius, N.A.; Granet, G.; Tikhodeev, S.G.; Taubert, R.; Fu, L.; Schweizer, H.; Giessen, H. Strong resonant mode coupling of Fabry–Perot and grating resonances in stacked two-layer systems. Photonics Nanostructures Fundam. Appl.
**2011**, 9, 390–397. [Google Scholar] [CrossRef] - Letartre, X.; Mazauric, S.; Cueff, S.; Benyattou, T.; Nguyen, H.S.; Viktorovitch, P. Analytical non-Hermitian description of photonic crystals with arbitrary lateral and transverse symmetry. Phys. Rev. A
**2022**, 106, 033510. [Google Scholar] [CrossRef] - Gromyko, D.A.; Dyakov, S.A.; Tikhodeev, S.G.; Gippius, N.A. Resonant mode coupling approximation for calculation of optical spectra of stacked photonic crystal slabs Part I. Photonics Nanostructures Fundam. Appl.
**2023**, 53, 101109. [Google Scholar] [CrossRef] - Gromyko, D.A.; Dyakov, S.A.; Tikhodeev, S.G.; Gippius, N.A. Resonant mode coupling approximation for calculation of optical spectra of stacked photonic crystal slabs Part II. Photonics Nanostructures Fundam. Appl.
**2023**, 53, 101110. [Google Scholar] [CrossRef] - Butterworth, S. On the theory of filter amplifiers. Wirel. Eng.
**1930**, 7, 536–541. [Google Scholar] - Hsu, C.W.; Zhen, B.; Stone, A.D.; Joannopoulos, J.D.; Soljačić, M. Bound states in the continuum. Nat. Rev. Mater.
**2016**, 1, 16048. [Google Scholar] [CrossRef] [Green Version] - Marinica, D.C.; Borisov, A.G.; Shabanov, S.V. Bound states in the continuum in photonics. Phys. Rev. Lett.
**2008**, 100, 183902. [Google Scholar] [CrossRef] [PubMed] - Bykov, D.A.; Doskolovich, L.L.; Golovastikov, N.V.; Soifer, V.A. Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating. J. Opt.
**2013**, 15, 105703. [Google Scholar] [CrossRef] - Bykov, D.A.; Doskolovich, L.L. ω−k
_{x}Fano line shape in photonic crystal slabs. Phys. Rev. A**2015**, 92, 013845. [Google Scholar] [CrossRef] [Green Version] - Bykov, D.A.; Bezus, E.A.; Doskolovich, L.L. Coupled-wave formalism for bound states in the continuum in guided-mode resonant gratings. Phys. Rev. A
**2019**, 99, 063805. [Google Scholar] [CrossRef] [Green Version] - Sun, K.; Jiang, H.; Bykov, D.A.; Van, V.; Levy, U.; Cai, Y.; Han, Z. 1D quasi-bound states in the continuum with large operation bandwidth in the ω∼k space for nonlinear optical applications. Photonics Res.
**2022**, 10, 1575–1581. [Google Scholar] [CrossRef] - Gippius, N.A.; Tikhodeev, S.G.; Ishihara, T. Optical properties of photonic crystal slabs with an asymmetrical unit cell. Phys. Rev. B
**2005**, 72, 045138. [Google Scholar] [CrossRef] [Green Version] - Liu, X.; Chen, S.; Zang, W.; Tian, J. Triple-layer guided-mode resonance Brewster filter consisting of a homogenous layer and coupled gratings with equal refractive index. Opt. Express
**2011**, 19, 8233–8241. [Google Scholar] [CrossRef] - Sang, T.; Wang, Y.; Li, J.; Zhou, J.; Jiang, W.; Wang, J.; Chen, G. Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence. Opt. Commun.
**2017**, 382, 138–143. [Google Scholar] [CrossRef] - Moharam, M.G.; Grann, E.B.; Pommet, D.A.; Gaylord, T.K. Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings. J. Opt. Soc. Am. A
**1995**, 12, 1068–1076. [Google Scholar] [CrossRef] - Li, L. Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings. J. Opt. Soc. Am. A
**1996**, 13, 1024–1035. [Google Scholar] [CrossRef] [Green Version] - Bykov, D.A.; Doskolovich, L.L. Numerical methods for calculating poles of the scattering matrix with applications in grating theory. J. Light. Technol.
**2012**, 31, 793–801. [Google Scholar] [CrossRef] [Green Version] - Bykov, D.A.; Bezus, E.A.; Morozov, A.A.; Podlipnov, V.V.; Doskolovich, L.L. Optical properties of guided-mode resonant gratings with linearly varying period. Phys. Rev. A
**2022**, 106, 053524. [Google Scholar] [CrossRef] - Bykov, D.A.; Doskolovich, L.L.; Soifer, V.A. Temporal differentiation of optical signals using resonant gratings. Opt. Lett.
**2011**, 36, 3509–3511. [Google Scholar] [CrossRef] - Golovastikov, N.V.; Bykov, D.A.; Doskolovich, L.L. Resonant diffraction gratings for spatial differentiation of optical beams. Quantum Electron.
**2014**, 44, 984–988. [Google Scholar] [CrossRef] - Blanchard, C.; Hugonin, J.P.; Sauvan, C. Fano resonances in photonic crystal slabs near optical bound states in the continuum. Phys. Rev. B
**2016**, 94, 155303. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**(

**a**) Geometry of the considered guided-mode resonant grating: period $\mathsf{\Lambda}=700\mathrm{nm}$, grating height ${h}_{\mathrm{gr}}=70\mathrm{nm}$, waveguide layer thickness ${h}_{\mathrm{wg}}=290\mathrm{nm}$, grating ridge width $w=40\mathrm{nm}$, refractive indices ${n}_{1}=1.99$ (${\mathrm{Si}}_{3}{\mathrm{N}}_{4}$), ${n}_{2}=1.45$ (${\mathrm{SiO}}_{2}$), and ${n}_{\mathrm{env}}=1$. (

**b**) Reflectance ${\left|r\left(\omega \right)\right|}^{2}={\left|{r}_{\mathrm{u},\mathrm{d}}\left(\omega \right)\right|}^{2}$ and transmittance ${\left|t\left(\omega \right)\right|}^{2}$ of the grating for the case of a TE-polarized normally incident wave. Dashed lines show the approximations calculated using Equation (8); solid lines show the rigorously calculated spectra. (

**c**) Reflection coefficient $\left|r\left({k}_{x},\omega \right)\right|$ of the grating calculated using RCWA (left half, ${k}_{x}<0$) and using the resonant approximation (8) (right half, ${k}_{x}>0$). Approximation parameters: ${\omega}_{\mathrm{p}1}=\left(2147.11-0.80\mathrm{i}\right)\cdot {10}^{12}{\mathrm{s}}^{-1}$, ${\omega}_{\mathrm{p}2}=2.1640\hspace{0.17em}\cdot {10}^{15}\hspace{0.17em}{\mathrm{s}}^{-1}$, ${v}_{\mathrm{g}}=0.695\hspace{0.17em}\mathrm{c}$, $\phi =-2.72$, $\xi =-0.32$.

**Figure 3.**Rigorously calculated reflectance of the stacked structure satisfying the condition (17) vs. angular frequency at ${k}_{x}=0$ (

**a**) and vs. tangential wavevector component at $\omega =\mathrm{Re}{\omega}_{p1}$ (

**b**) (solid lines); squared absolute values of the “model” reflectance ${\left|{r}_{2}\left({k}_{x}=0,\omega \right)\right|}^{2}$ of Equation (20) (

**a**) and ${\left|{r}_{2}\left({k}_{x},\omega =\mathrm{Re}{\omega}_{\mathrm{p}1}\right)\right|}^{2}$ of Equation (22) (

**b**) of the corresponding Butterworth filters (dashed red lines). Dotted lines show the reflectance of the single resonant grating calculated using RCWA.

**Figure 4.**Magnitude of the reflection coefficient $\left|{r}_{2}\left({k}_{x},\omega \right)\right|$ of the stacked structure calculated using RCWA (left half, ${k}_{x}<0$) and using the resonant approximation (14) (right half, ${k}_{x}>0$) (TE polarization) with the intermediate layer thickness $l=6.522\mathsf{\mu}\mathrm{m}$ The insets show the magnified fragments of the left part of the figure.

**Figure 5.**Rigorously calculated quality factors (solid black lines) of the eigenmodes as functions of ${k}_{x}$ (

**a**) for the single grating near the BIC at ${k}_{x}=0$, $\omega ={\omega}_{\mathrm{p}2}$, (

**b**) for the stacked structure with $l=6.522\mathsf{\mu}\mathrm{m}$ near the BIC at ${k}_{x}=0$, $\omega =\mathrm{Re}{\omega}_{\mathrm{p}1}$, and (

**c**) for the stacked structure with $l=6.469\mathsf{\mu}\mathrm{m}$ near the BIC at ${k}_{x}=0$, $\omega ={\omega}_{\mathrm{p}2}$. Dotted, dash-dotted, and dashed red lines show the ${k}_{x}^{-2}$, ${k}_{x}^{-4}$, and ${k}_{x}^{-6}$ decay laws, respectively.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Golovastikov, N.V.; Bykov, D.A.; Bezus, E.A.; Doskolovich, L.L.
Lines of Quasi-BICs and Butterworth Line Shape in Stacked Resonant Gratings: Analytical Description. *Photonics* **2023**, *10*, 363.
https://doi.org/10.3390/photonics10040363

**AMA Style**

Golovastikov NV, Bykov DA, Bezus EA, Doskolovich LL.
Lines of Quasi-BICs and Butterworth Line Shape in Stacked Resonant Gratings: Analytical Description. *Photonics*. 2023; 10(4):363.
https://doi.org/10.3390/photonics10040363

**Chicago/Turabian Style**

Golovastikov, Nikita V., Dmitry A. Bykov, Evgeni A. Bezus, and Leonid L. Doskolovich.
2023. "Lines of Quasi-BICs and Butterworth Line Shape in Stacked Resonant Gratings: Analytical Description" *Photonics* 10, no. 4: 363.
https://doi.org/10.3390/photonics10040363