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Abstract
The symmetry breaking in a typical dielectric GMR-grating structure allows the coupling of the incident wave with the so-called Symmetry-Protected Modes (SPM). In this present work, the excitation conditions of such particular modes are investigated. A parametric study including the grating dimensions is carried out to exploit them for a blood refractive index sensing with higher Sensitivity (S) and Figure Of Merit (FOM). To our knowledge, the performances obtained by FDTD calculations (Q = 2.1 × 104, S = 657 nm/RIU and FOM ≃ ~9 112 RIU−1) and FMM calculations (Q = 3 × 106, S = 656 nm/RIU and FOM ≃ ~1.64 × 106 RIU−1) are the highest level reached.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the first theoretical papers of Magnusson and Wang [1,2], the Guided-Mode Resonances (GMR) and their several applications have raised an ever-increasing interest in the last three decades [3–9]. In dielectric structures combining a periodic subwavelength grating and a planar waveguide layer, the diffraction modes induced by the interaction of light with the grating result in propagating guided modes in the planar waveguide. A part of the latter are leaky along the grating and then coupled with the external electromagnetic field giving rise to the so-called GMR or leaky modes [1,2]. In [10], and for a periodic grating without the waveguide layer, a non-leaky mode has been highlighted in the band structure in addition to the leaky ones. This non-leaky mode is associated with a Bound state In the Continuum (BIC) related to the high degree of symmetry of the studied structure. This mode, also known as Symmetry-Protected Mode (SPM), is a localized state with an infinite lifetime that can be turned into a resonant one with a finite lifetime and a high quality factor (${{Q=\displaystyle \frac {\lambda }{\Delta \lambda }}}$ where $\Delta \lambda$ is the Full Width at Half Maximum (FWHM) of this resonant mode at the wavelength $\lambda$) making it of great scientific interest. The SPM are set to be dark because they can be excited only under certain illumination conditions. In fact, they can be perturbed to become bright and then radiate as Fano resonances by an intrinsic symmetry breaking such as modifying the geometry of the structure [11–14]. Another possibility to get this resonance is by an extrinsic symmetry breaking through the direction of incidence for instance [15,16].
Thanks to the tunability of their optical properties, 1D or 2D-periodic photonic devices working under GMR are widely developed in a broad range of optical applications such as waveguide reflectors [4,8,17], filters [2,7] and sensors [6,9,18]. For the latter, many recent researches were especially related to free-label biosensing based on GMR resonances [9,19–21] and BICs or Quasi-BICs resonances [22–24]. Among them, the refractive index biosensors are characterized by a high sensitivity of the resonance spectral position ($\lambda$) to the refractive index (RI) variations. Improving the sensor sensitivity ($S$), defined as the change in $\lambda$ position per refractive index unit (RIU) (${{S=\displaystyle \frac {\Delta \lambda }{\Delta n}}}$ in nm/RIU), was the main focus of attention in the literature. Several theoretical and experimental researches suggest various and more or less complex structures with optimized optical and structural parameters in order to enhance the electromagnetic field confinement into the sensing area and then improve the sensing ability [7,25,26]. Another important indicator for the sensing performance is the Figure Of Merit ($\textrm {FOM}$), defined as the ratio of the sensitivity ($S$) to the FWHM of the resonance ($\textrm{FOM} = \frac{S}{{\textrm{FWHM}}}$ in RIU$^{-1}$) [6,27]. Improving the $\textrm{FOM}$ of the sensor involves reducing the detection limits. The most significant results for theses two sensing markers for GMR-grating structures are summarized in Table 1. In the blood sensing domain, the major challenge to achieve high sensing performances still relevant today because of the stability lack and complex fabrication processes for the designed structures. We can mention the photonic crystal nanocavity and wave-guide based biosensor providing a sensitivity of ${272.43}$ nm/RIU for glycated haemoglobin detection [20]. We can also cite the high sensitivity of ${13\,936}$ nm/RIU reached in case of Mach-Zehnder interferometric optical fiber for IgG/anti-IgG immunosensing [28]. However on the one hand, for the most GMR sensors, the studies have focused on optimizing one of the two sensing performances ($S$ or $\textrm {FOM}$) but rarely the both together. As generally reported in the literature, searching for a narrower resonant linewidth (higher $\textrm {FOM}$), results in lower sensitivity because of the high confinement of the electromagnetic field in the dielectric medium rather than in the sensing area. On the other hand, the quest for a higher sensitivity results in a broader resonant linewidth (lower $\textrm {FOM}$), which limits the measurement of small resonant wavelength shifts.
Table 1. RI sensor performances for GMR-grating structures (with $S$ in nm/RIU and FOM in RIU$^{-1}$). The underlined values are calculated by us and added for better readability. Exp/Theo indicates whether the results are experimental or theoretical.
In this paper, we present a typical two-dimensional basic GMR grating structure used for biosensing of blood for the refractive index range $1.345-1.380$. The proposed structure presents the advantage of simple geometry in addition to operate in the telecoms near infrared spectral range for which numerous optical components and light sources are widely available at low cost. The excitation of symmetry-protected guided mode resonances in such a structure is investigated and optimized to achieve high $Q$-factor allowing reducing the limits of detection and then enabling the measurement of small wavelength shifts according to small variations in the refractive index. Both Fourier Modal Method (FMM) and Finite-Difference Time-Domain (FDTD) method are used to simulate and compare transmission spectra in order to determine the illumination conditions to excite such dark modes. Finally, a structural parametric study is done to optimize the structure dimensions and improve the biosensing performances.
2. Designed structure and used methods
The schematic of our GMR dielectric structure, illustrated in Fig. 1, is basically chosen identical to what is proposed in [6,11,33]. It consists of a subwavelength grating periodic in the $X$ direction (with a period $\Lambda$ smaller than the excitation wavelength in order to get the only zero-order transmission), invariant in the $Y$ direction and finite in the $Z$ one. The grating, made of a dielectric material with high refractive index ($n_{g}=2.4$, TiO$_{2}$ for instance), has a rectangular profile of height $H$ and width $W$. The dielectric grating, bounded up by blood with refractive index $n_{c}$, is deposited on a glass substrate ($n_{s}=1.45$) in order to meet the experimental conditions of design and characterization. The mediun of incidence is blood while the transmission medium is glass. For a TM-polarized incident plane wave (magnetic field along the $Y$-direction and electric field in the $XZ$ plane), we will demonstrate the existence of symmetry-protected GMR modes, with very high quality factor. For these particular modes, the grating dimensions, which are period ($\Lambda$), height ($H$) and width ($W$), in addition to the angle of incidence ($\phi$), will be varied in order to examine the sensing performances of the blood refractive index variation due to the presence of impurities or genes.
In order to determine and optimize the sensing performances of our structure, two homemade codes based on two different methods are used to calculate the zero-order transmission spectra and normalized field maps. The first method is the Finite-Difference Time-Domain (FDTD) that allows to simulate the electromagnetic wave propagation in the general case [34]. The spatio-temporal partial derivatives that appear in Maxwell’s equations are approximated by centered finite differences according to the Yee scheme [35]. Within the framework of this method, the oblique incidence case is processed by the Split-Field Method (SFM-FDTD) [36]. Operating with the FDTD algorithm offers the advantage to provide a broadband description. In fact, a short temporal pulse excitation is sufficient to determine the optical response of the system over a wide spectral range via a simple Fourier transform. Nevertheless, this method have the drawback of requiring a hardly CPU-time and memory resulting from the mesh refinement. In Our 2D-SFM-FDTD code, the periodicity along the $X$ direction is described by the Floquet-Bloch’s boundaries conditions. In addition, Berenger’s Perfectly Matched Layer (PML) have been chosen to simulate the free space along the $Z$ direction [37,38]. The second method is the Fourier Modal Method (FMM) [39], also known as Rigorous Coupled Wave Analysis (RCWA). The latter is one of the most reliable, efficient and versatile methods used in diffractive optics and specifically adapted for our 1D-periodic grating structure [40]. Unlike the FDTD based on the direct definition of the structure in the real space, the FMM consists of developing all the electromagnetic variables in Fourier series. Their truncation for numerical implementation purpose is done using the principle of energy conservation on which the convergence of the method is based. In homogeneous medium, the diffracted field is given by the Rayleigh development. In the grating, the integration of Maxwell’s equations is reduced to solve an eigenvalue problem. The implementation of the continuity conditions for the electromagnetic variables on the interfaces constituting the different layers allows to calculate the electromagnetic field everywhere and then the diffraction efficiencies. For our configuration, the FMM method converges practically for a truncation order M = 15 (i.e. 31 Fourier harmonics). However, for more precision, we have fixed M = 35 (i.e. 71 Fourier harmonics).
Fig. 1. Schematic of the studied GMR grating structure, with its different dimensions including the period ($\Lambda$), height ($H$) and width ($W$), illuminated at angle of incidence $\phi$ with TM-polarized light. $n_{c}$, $n_{g}$ and $n_{s}$ are the refractive index of the superstrate, dielectric grating and substrate respectively.
3. Simulation results
3.1 Investigating SPMs
The zero-order transmission calculation, for both FDTD and FMM simulations, is carried out on our designed dielectric grating. Despite the dispersion nature of the blood, its refractive index is chosen to be constant within our wavelength range and set to the indirectly measured value by [41] around $1550$ nm ($n_{c}=1.3456$). After multiple simulations, we have chosen the following geometrical parameters: $\Lambda =960$ nm, ${H}=900$ nm and $W=300$ nm, allowing obtaining a resonant mode around $\lambda =1550$ nm. Let us note that, for all the FDTD simulations performed in the study, the spatial discretization cell was fixed to $\Delta =\Delta x=\Delta z=20$ nm.
At normal incidence in TM-polarization, the transmission spectra, illustrated in Figs. 2(a) and (b), reveal a GMR resonance around the same wavelength ($\lambda _{1}=1476$ nm) for both FDTD and FMM calculations but an additional sharp resonance ($\lambda _{2}=1540.5$ nm) with very high quality factor ($Q_{2}=2.40\times 10^{4}$) only appears in the FDTD result. This is the signature of an SPM excitation, resulting from the intrinsic symmetry break numerically induced by the spatial discretization in the FDTD algorithm [42]. Let us recall that this algorithm is useful to take into account the possible geometrical imperfections induced by the fabrication process [16]. Based on the FDTD calculations, Figs. 2(c) and (d) display the distribution of the normalized electric field intensity ($\left |E\right |^{2}$) and the direction and magnitude of the Poynting vector (see cyan arrows) at the two resonance wavelengths. It can be noticed a strong confinement of the electric field energy between the dielectric grating lines for the SPM ($\lambda _{2}$) with a maximum of the normalized electric field intensity exceeding $2\times 10^{4}$, $300$ times higher than the leaky-one ($\lambda _{1}$). For the latter, the electric field energy is mostly located at the edges between the sensing and dielectric medium as previously reported for a photonic crystal slab with periodic slits at normal incidence [17]. From the spatial distribution of the Poynting vector, the non-leaky character for the second resonance is confirmed by the electromagnetic power loop (flowing up and streaming down). This leads to longer photon lifetime between the grating lines and thus to an enhanced interaction with the sensing area (blood). The calculated electric field vectors, not shown to avoid cluttering Figs. 2(c) and (d), reveal similar behavior to that observed in [16] with rotating field, a symmetry that is not consistent with the linear polarization of the incident plane wave. This SPM excitation, is corroborated by FDTD calculations for different discretization cell size as in [16].
Fig. 2. Zero-order transmission spectra for normal incidence with TM polarization, for the designed grating structure ($\Lambda =960$ nm, $H=900$ nm and $W=300$ nm), calculated by (a) FDTD with $\Delta =20$ nm and (b) FMM. Distributions of the normalized electric field intensity and Poynting vector (cyan arrows) in $XZ$ plane corresponding to the two resonances (c) $\lambda _{1}$ and (d) $\lambda _{2}$ obtained by FDTD simulations for $\Delta =20$ nm.
The evolution of the resonance spectral position and the quality factor for the two modes as a function of the mesh cell size $\Delta$ is depicted in Fig. 3. If the behavior of the spectral positions is almost the same for the two resonances with a slight redshift when the mesh cell size increases (Fig. 3(a)), the $Q$-factor values and behavior are completely different (Fig. 3(b)). While the first resonance ($\lambda _{1}$) has a lower $Q$-factor value independent from $\Delta$, the second one ($\lambda _{2}$) exhibits a very high value (compared to the first one) that decreases with increasing $\Delta$. This can be explained by the fact that the more the symmetry of the structure is broken (important) the less the mode is protected and the more the width of the resonance increases decreasing, consequently, the quality factor. All this confirms the SPM nature of the additional resonance observed in the FDTD spectra [42]. However, this specific mode can appear in the FMM transmission spectrum (see Figs. 4(a) and (b)) as a result of an extrinsic symmetry breaking such as a slight oblique incidence that can easily occur during experimental characterizations. As can be seen from Figs. 4(c) and (d), this mode is very sensitive to the variation of the angle of incidence. Indeed, the more we break the symmetry ($\phi$ increases) the more the spectral position of the second resonance ($\lambda _{2}$) redshifts and more it widdens. Thus its $Q$-factor strongly decreases confirming its SP nature [16,43]. As for the intrinsic symmetry breaking case, the first resonance ($\lambda _{1}$) does not have an SPM behavior because its spectral position slightly blueshifts with increasing $\phi$ while its $Q$-factor is almost constant. Despite the good agreement between the two simulation methods (FMM and FDTD) for the SPM spectral positions, let us note that a gap of more than $50$ pm is observed between these two methods in the FWHM calculation as $\phi \longrightarrow 0^{\circ }$ (see Fig. 4(e)). This difference, which gradually disappears as the angle of incidence increases (and thus the $Q$-factor decreases), can be explained by the double symmetry break (intrinsically induced by the mesh and extrinsically induced by the oblique incidence), which leads to a wider resonance in the case of the FDTD simulations. This can explain the large difference observed for the ${\textrm {FOM}}$ and $Q$ values calculated by these two methods in the literature and even in our present work as presented further on. It is worth noting, from the Poynting vector distribution at $\lambda _{2}$ presented in Fig. 4(c), that the electromagnetic power loop turns into a surface propagation along the grating direction with respect to grating symmetry (see Fig. 5). We also note that the orientation of this propagation can be controlled according to the sign of the angle of incidence due to the experimental excitation conditions as shown in Figs. 5(a) and (b).
Fig. 3. Evolution of (a) the spectral position and (b) the quality factor in logarithmic scale for the two resonances with respect to the mesh cell $\Delta$.
Fig. 4. Zero-order transmission spectra, around $\lambda _{2}$, for different angles of incidence in $\textrm {TM}$ polarization, for the grating structure ($\Lambda =960$ nm, $H=900$ nm and $W=300$ nm), calculated by (a) FDTD and (b) FMM. Evolution of (c) the spectral position and of (d) the quality factor in logarithmic scale for the two resonances with respect to the angle of incidence $\phi$. (e) Comparison between the FDTD and the FMM results of the SPM’s FWHM as function of $\phi$
Fig. 5. Distributions of the normalized electric field intensity and the Poynting vector (cyan arrows) corresponding to the SPM resonance ($\lambda _{2}$) obtained by FDTD simulation for $\Delta =20$ nm at oblique incidence with (a) $\phi =-0.1^{\circ }$, (b) $\phi =0.1^{\circ }$ and (c) $\phi =1^{\circ }$ .
3.2 Parametric study and sensing performances
The second part of our calculations is devoted to an FDTD parametric study involving the grating dimensions in case of a slight off-normal incidence ($\phi =0,1^{\circ }$) with a TM polarization in order to optimize the spectral position in the desired optical range and the $Q$-factor of the SPM resonance for biosensing applications.
Figure 6 illustrates the effect of the height, period and width on the spectral position and the FWHM of the SPM resonance respectively. As shown in Fig. 6(a), increasing the grating height, for a fixed filling factor ${\displaystyle f=\frac {W}{\Lambda }}$, makes it possible to shift the resonance towards higher wavelength values [33]. The bandwidth of the SPM resonance is also affected by slightly decreasing with increasing the grating height as generally predicted by the waveguide theory [5] and the simulation results of [31] (see Fig. 6(b)). For a fixed height, the filling factor is another important parameter affecting its spectral position and its FWHM. Fig. 6(c) shows a linear dependence of the spectral position when the period increases, for fixed width. This linear variation satisfies the effective medium approximation for grating given by [2]:
where $n_{\textrm {eff}}$ is the effective waveguide index of refraction and $m$ is the integer labeling the mode supported by the waveguide gratting.Fig. 6. Spectral position and FWHM of the SPM resonance at oblique incidence ($\phi =0.1^{\circ }$) as a function of (a,b) height $H$ with $W=220$ nm and $\Lambda =960$ nm, (c,d) period $\Lambda$ with $W=220$ nm and $H=940$ nm and (e,f) width $W$ with $\Lambda =960$ nm and $H=940$ nm.
The bandwidth variation depicted in Fig. 6(d) shows that, contrary to the grating height, the resonance peak becomes wider (and then $Q$ decreases) when the period increases [33]. The second parameter of the filling factor is the grating width $W$ for which it is noting a non linear dependence of the wavelength shift (see Fig. 6(e)) and a similar behavior as for the period with a slight increase of the resonance bandwidth (see Fig. 6(f)). More importantly, all the results of Fig. 6 confirm that the SPM’s FWHM is not significantly affected by the grating dimension. This denotes the robustness of the resonant mode properties which can be considered as a real advantage against manufacturing imperfections.
Based on the results of our parametric study, the sensing performances are calculated for our GMR-based optical sensor working into the desired optical range (from $1530$ nm to $1550$ nm). Under TM-polarized and oblique incident plane wave ($\mathrm {\phi }=0.1^{\circ }$), FDTD and FMM transmission spectra are calculated for a blood refractive index variation in the range $1.3456-1.38$. An example of these spectra for a set of the grating dimensions ($\Lambda =960$ nm, ${W}=220$ nm and ${H}=940$ nm) is depicted in Figs. 7(a) and (b). By plotting the variation of the SPM spectral position with the refractive index variation, the same sensitivity (${S}=657$ nm/RIU) is obtained by the two methods (see Figs. 7(c) and (d)). However, due to differences in the accuracy of each method for calculating the FWHM, two different values are obtained for the $\textrm {FOM}$ ($9\,112$ RIU$^{-1}$ by FDTD and $1.64\times 10^{6}$ RIU$^{-1}$ by FMM). The same parametric study as in Fig. 6 is carried out for these two sensing properties ($S$ and $\textrm {FOM}$). It can be seen that the sensitivity increases with $H$, $\Lambda$ and $W$ but decreases when the angle of incidence becomes higher (see Figs. 8(a)-(d)). Fig. 8(e) shows that the $\textrm {FOM}$, for its part, increases with the height (due to increasing sensitivity and decreasing $\textrm {FWHM}$). Despite the significant increase in sensitivity with $\Lambda$ and $W$, the FOM decays under the influence of the resonance widening as depicted in Figs. 8(f)-(g). As expected from the effect of the angle of incidence on the $Q$-factor, the FOM undergoes a sharp decrease as soon as the symmetry is broken by deviating from the normal incidence (see Fig. 8(h)). Among the many configurations studied in our parametric study, detailed results (spectral position, $\textrm {FWHM}$, $Q$-factor, $S$ and $\textrm {FOM}$) for three of them are summarized in Table 2. The highest sensitivity is achieved within the second configuration for the highest grating height. This can be explained by the increase of the interaction volume between the blood and the SPM between the grating lines. However, a highest $\textrm {FOM}$ is obtained for the third configuration by reducing the grating width, and then the FWHM, as predicted by the parametric study. Hence, we can conclude that high sensing performances are reached for these configurations with the commonly 2D-grating geometries proposed in the literature [6,23,33]. In addition to the huge values reached by the FMM calculation, the FDTD calculations gave rise to high quality factor of order $2\times 10^{4}$ (almost $2$ times the value of [31]), high $\textrm {FOM}$ value of the order of $10^{4}$ ($2$ times the value of [9]) and a very important sensitivity value of $650$ nm/RIU higher than the values obtained in [32,33].
Fig. 7. Zero order transmission spectra under TM-polarized oblique incidence ($\phi =0.1^{\circ })$ for different refractive index values of the blood calculated by (a) FDTD and (b) FMM. Variation of the resonance spectral position as function of the blood refractive index by (c) FDTD and (d) FMM. The grating dimensions are $\Lambda =960$ nm, ${W}=220$ nm and ${H}=940$ nm.
Fig. 8. Sensitivity (a-d) and FOM (e-h) of the SPM resonance at oblique incidence ($\phi =0.1^{\circ }$) as a function of (a,e) height $H$ with $W=220$ nm and $\Lambda =960$ nm, (b,f) period $\Lambda$ with $W=220$ nm and $H=940$ nm, (c,g) width $W$ with $\Lambda =960$ nm and $H=940$ nm and (d,h) angle of incidence with $\Lambda =960$ nm, $H=940$ nm and $W=220$ nm.
Table 2. Resonance spectral positions (at $n_c=1.3456$), FWHM, $Q$-factor, $S$ and $\textrm {FOM}$ values calculated for different configurations using the FDTD and FMM methods.
4. Conclusion
In this work we have demonstrated that our basic GMR-grating structure exhibits resonance peaks of SPM type for asymmetrical environment. These SPM are induced by a symmetry break that can be caused by the fabrication imperfections and revealed by the FDTD method in case of normal incidence. Moreover, these particular modes can also be excited in case of FMM calculations by a small deviation from the normal incidence. Thus, the choice of the calculation method and the illumination conditions is important to excite and tune the spectral position and the quality factor of these modes. Nevertheless, the FDTD method remains more suitable because it allows to take into account the manufacturing errors by adapting the spatial discretization to the precision of the fabrication process . Finally, based on a parametric study involving the grating dimensions, high sensing performances are reached for blood detection by FDTD calculations $(Q=2.1\times 10^{4}$, ${S}=657$ nm/RIU and $\textrm {FOM}\simeq 9\,112$ RIU$^{-1})$ and FMM calculations $(Q=3\times 10^{6}$, ${S}=656$ nm/RIU and $\textrm {FOM}\simeq 1.64\times 10^{6}$ RIU$^{-1})$, openning the way to the development of a new generation of miniaturazed biological detectors.
Acknowledgments
This work has been partially supported by the EIPHI Graduate School (contract ANR-17-EURE-0002) by the Bourgogne-Franche-Comté Region, and by the Algerian "Direction Générale de la Recherche Scientifique et du Développement Technologique" (DGRSDT).
Disclosures
The authors declare no competing interests
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7(8), 1470–1474 (1990). [CrossRef]
2. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32(14), 2606–2613 (1993). [CrossRef]
3. W. Yu, D. Wu, X. Duan, and Y. Yi, “Subwavelength grating wideband reflectors with tapered sidewall profile,” MRS Adv. 1(23), 1683–1691 (2016). [CrossRef]
4. S. Zhang, Y. H. Ko, and R. Magnusson, “Broadband guided-mode resonant reflectors with quasi-equilateral triangle grating profiles,” Opt. Express 25(23), 28451–28458 (2017). [CrossRef]
5. G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photonics Rev. 12, 1800017 (2018). [CrossRef]
6. Y. Zhou, B. Wang, Z. Guo, and X. Wu, “Guided mode resonance sensors with optimized figure of merit,” Nanomaterials 9(6), 837 (2019). [CrossRef]
7. T. Sang, Z. Wang, J. Zhu, L. Wang, Y. Wu, and L. Chen, “Linewidth properties of double-layer surfacerelief resonant brewster filters with equal refractive index,” Opt. Express 15(15), 9659–9665 (2007). [CrossRef]
8. W. Yu, M. Ye, and Y. S. Yi, “Impacts of tapered sidewall profile on subwavelength grating wideband reflectors,” J. Nanophotonics 9(1), 093058 (2015). [CrossRef]
9. Y. Zhou, X. Li, S. Li, Z. Guo, P. Zeng, J. He, D. Wang, R. Zhang, M. Lu, S. Zhang, and X. Wu, “Symmetric guided-mode resonance sensors in aqueous media with ultrahigh figure of merit,” Opt. Express 27(24), 34788–34802 (2019). [CrossRef]
10. R. Magnusson, H. Hemmati, D. J. Carney, K. J. Lee, Y. H. Ko, and S. Lee, “Fundamentals and applications of resonant leaky-mode photonic lattices,” in 2019 IEEE Aerospace Conference, (2019), pp. 1–8.
11. J. M. Foley and J. D. Phillips, “Normal incidence narrowband transmission filtering capabilities using symmetry-protected modes of a subwavelength, dielectric grating,” Opt. Lett. 40(11), 2637–2640 (2015). [CrossRef]
12. S. Campione, S. Liu, L. I. Basilio, L. K. Warne, W. L. Langston, T. S. Luk, J. R. Wendt, J. L. Reno, G. A. Keeler, I. Brener, and M. B. Sinclair, “Broken symmetry dielectric resonators for high quality factor Fano metasurfaces,” ACS Photonics 3(12), 2362–2367 (2016). [CrossRef]
13. 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]
14. X. Yin, T. Sang, H. Qi, G. Li, X. Wang, J. Wang, and Y. Wang, “Symmetry-broken square silicon patches for ultra-narrowband light absorption,” Sci. Rep. 9(1), 17477 (2019). [CrossRef]
15. J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301 (2015). [CrossRef]
16. A. Hoblos, M. Suarez, N. Courjal, M.-P. Bernal, and F. I. Baida, “Excitation of symmetry protected modes in a lithium niobate membrane photonic crystal for sensing applications,” OSA Continuum 3(11), 3008–3018 (2020). [CrossRef]
17. X. Cui, H. Tian, Y. Du, G. Shi, and Z. Zhou, “Normal incidence filters using symmetry-protected modes in dielectric subwavelength gratings,” Sci. Rep. 6(1), 1–6 (2016). [CrossRef]
18. J. Hu, X. Liu, J. Zhao, and J. Zou, “Investigation of Fano resonance in compound resonant waveguide gratings for optical sensing,” Chin. Opt. Lett. 15, 030502 (2017). [CrossRef]
19. B. T. Cunningham, M. Zhang, Y. Zhuo, L. Kwon, and C. Race, “Recent advances in biosensing with photonic crystal surfaces: A review,” IEEE Sensors J. 16(10), 3349–3366 (2016). [CrossRef]
20. S. Olyaee, M. Seifouri, and H. Mohsenirad, “Label-free detection of glycated haemoglobin in human blood using silicon-based photonic crystal nanocavity biosensor,” J. Modern Opt. 63(13), 1274–1279 (2016). [CrossRef]
21. A. Shakoor, M. Grande, J. Grant, and D. R. S. Cumming, “One-dimensional silicon nitride grating refractive index sensor suitable for integration with CMOS detectors,” IEEE Photonics J. 9, 1 (2017). [CrossRef]
22. S. Romano, A. Lamberti, M. Masullo, E. Penzo, S. Cabrini, I. Rendina, and V. Mocella, “Optical biosensors based on photonic crystals supporting bound states in the continuum,” Materials 11(4), 526 (2018). [CrossRef]
23. D. N. Maksimov, V. S. Gerasimov, S. Romano, and S. P. Polyutov, “Refractive index sensing with optical bound states in the continuum,” Opt. Express 28(26), 38907–38916 (2020). [CrossRef]
24. H. Vyas and R. S. Hegde, “Improved refractive-index sensing performance in medium contrast gratings by asymmetry engineering,” Opt. Mater. Express 10(7), 1616–1629 (2020). [CrossRef]
25. Z. Wang, T. Sang, J. Zhu, L. Wang, Y. Wu, and L. Chen, “Double-layer resonant brewster filters consisting of a homogeneous layer and a grating with equal refractive index,” Appl. Phys. Lett. 89(24), 241119 (2006). [CrossRef]
26. J. Ju, Y.-A. Han, and S.-M. Kim, “Design optimization of structural parameters for highly sensitive photonic crystal label-free biosensors,” Sensors 13(3), 3232–3241 (2013). [CrossRef]
27. H. Lu, M. Huang, X. Kang, W. Liu, C. Dong, J. Zhang, S. Xia, and X. Zhang, “Improving the sensitivity of compound waveguide grating biosensor via modulated wavevector,” Appl. Phys. Express 11(8), 082202 (2018). [CrossRef]
28. B.-T. Wang and Q. Wang, “An interferometric optical fiber biosensor with high sensitivity for IgG/anti-IgG immunosensing,” Opt. Commun. 426, 388–394 (2018). [CrossRef]
29. Y.-K. Tu, M.-Z. Tsai, I.-C. Lee, H.-Y. Hsu, and C.-S. Huang, “Integration of a guided-mode resonance filter with microposts for in-cell protein detection,” Analyst 141(13), 4189–4195 (2016). [CrossRef]
30. Y. Zhou, Z. Guo, W. Zhou, S. Li, Z. Liu, X. Zhao, and X. Wu, “High-Q guided mode resonance sensors based on shallow sub-wavelength grating structures,” Nanotechnology 31(32), 325501 (2020). [CrossRef]
31. G. Lan, S. Zhang, H. Zhang, Y. Zhu, L. Qing, D. Li, J. Nong, W. Wang, L. Chen, and W. Wei, “High-performance refractive index sensor based on guided-mode resonance in all-dielectric nano-silt array,” Phys. Lett. A 383(13), 1478–1482 (2019). [CrossRef]
32. X. Lu, G. Zheng, and P. Zhou, “High performance refractive index sensor with stacked two-layer resonant waveguide gratings,” Results Phys. 12, 759–765 (2019). [CrossRef]
33. S. Isaacs, A. Hajoj, M. Abutoama, A. Kozlovsky, E. Golan, and I. Abdulhalim, “Resonant grating without a planar waveguide layer as a refractive index sensor,” Sensors 19(13), 3003 (2019). [CrossRef]
34. A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time–Domain Method (Artech House, Norwood, MA, 2005), 2nd ed.
35. Yee Kane, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307 (1966). [CrossRef]
36. A. Belkhir and F. I. Baida, “Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: Application to the study of a radar dome,” Phys. Rev. E 77(5), 056701 (2008). [CrossRef]
37. J.-P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Computat. Phys. 114(2), 185–200 (1994). [CrossRef]
38. D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guid. Wave Lett. 4(8), 268–270 (1994). [CrossRef]
39. I. M. Fradkin, S. A. Dyakov, and N. A. Gippius, “Fourier modal method for the description of nanoparticle lattices in the dipole approximation,” Phys. Rev. B 99(7), 075310 (2019). [CrossRef]
40. H. Yala, B. Guizal, and D. Felbacq, “Fourier modal method with spatial adaptive resolution for structures comprising homogeneous layers,” J. Opt. Soc. Am. A 26(12), 2567–2570 (2009). [CrossRef]
41. E. Lazareva and V. Tuchin, “Blood refractive index modelling in the visible and near infrared spectral regions,” J. Biomed. Photonics Eng. 4(1), 010503 (2018). [CrossRef]
42. A. Hoblos, M. Suarez, B. Guichardaz, N. Courjal, M.-P. Bernal, and F. I. Baida, “Revealing photonic symmetry-protected modes by the finite-difference-time-domain method,” Opt. Lett. 45(7), 2103–2106 (2020). [CrossRef]
43. S. Han, M. V. Rybin, P. Pitchappa, Y. K. Srivastava, Y. S. Kivshar, and R. Singh, “Guided-mode resonances in all-dielectric terahertz metasurfaces,” Adv. Opt. Mater. 8, 1900959 (2020). [CrossRef]
