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

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## Abstract

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## 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

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**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.

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**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