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Abstract
Silicon nitride (Si3N4) subwavelength medium contrast gratings (MCGs) directly integrated with CMOS photodetectors are a promising option for on-chip label-free biosensing. The narrow spectral features required for sensing are often realized in Si3N4 nanostructures by weakly corrugated gratings which limit design flexibility. We numerically investigate the optical properties of asymmetry-engineered MCG gratings and predict the formation of ultra-sharp spectral features via the excitation of quasi-bound states in continuum (QBIC) resonances. Systematic investigation of the design parameter space shows that sharp spectral features are obtained for a wide range of parameters without requiring ultrathin grating profiles. Transmission-mode refractive index sensing simulations for bulk and surface sensing, considering both wavelength-shift and intensity-shift modalities, indicate performance gains using these structures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Sensors employing optical nanostructures directly integrated with CMOS photodetectors and associated circuitry [1] are highly promising for developing on-chip label-free biosensing platforms [2]. Most plasmonic and all-dielectric materials typically used in photonic nanostructures are unsuitable for CMOS integration owing to process incompatibility [3] and/or absorption in the visible wavelength window (where silicon photodetectors perform well). An additional requirement for portability is that the structures should operate in a normal incidence configuration [4,5] rather than laterally. The spectral changes upon analyte binding can be detected by either calculating the spectral peak shift or by monitoring the transmission intensity shift at a particular excitation wavelength; the latter method is preferable for compact sensors [2]. For sensing, the change in light intensity caused by the resonance wavelength shift should be large, the resonance linewidth must be small and the contrast (the difference in in transmittance at the resonance peak/dip and background transmittance) of the resonance must be large. With narrow resonances, the detection threshold is traded-off against detection dynamic range [6]. Silicon Nitride (Si$_3$N$_4$), a CMOS compatible material, is emerging as a promising material for such integration owing to its moderately high refractive index (RI) and negligible absorption in the visible wavelength window [6]. Some recent reports have explored the use of Si$_3$N$_4$ periodic nanostructures in biosensing applications [1,2,4–8] and in high-sensitivity biological fluorescence assays [7,8].
Periodic subwavelength structures have seen widespread application in photonics [9]. Subwavelength gratings with a near-wavelength pitch, that only support reflected and transmitted zeroth order diffraction for normally incident light, exhibit a rich optical behavior which has been long exploited for spectral filtering applications [9,10]. Early on, the so-called high contrast grating (HCG) (owing to the large refractive index contrast of the grating region) attracted interest as a broadband mirror that was robust in the face of inevitable fabrication tolerances. In contrast, guided mode resonance (GMR) gratings [11,12] attracted interest due to their sharp spectral features. Sharper spectral features are found mainly for weaker gratings (the so-called low contrast grating (LCG) [13–15]). The sharper spectral features in LCGs [13], however, come at the cost of a larger planar device footprint, reduced design flexibility and reduced robustness to fabrication induced variations [16]. Despite its shortcomings, most reports on Si$_3$N$_4 $RI sensors rely on LCG based designs.
Recently, bound state in the continuum (BIC) resonances [17–22] are being explored as a pathway for realizing ultra sharp resonances in periodic structures. BIC resonances are a general wave phenomena involving the spatial localization of a mode’s resonant energy in spite of its resonant frequency being located in the continuous spectra of non-localized radiation modes. The spatial localization leads to extended lifetimes (ideally infinite lifetime) that is atypical of open resonators which have finite lifetimes owing to energy leakage. While true BICs are theoretical constructs occurring in lossless structures of infinite extent, Quasi-BICs (QBICs) [23] with large, but finite, lifetimes are observable in lossy finite structures as well. Depending on the mechanism which prevents the leakage of trapped energy, there are two kinds of BICs [24] that can occur: (1) symmetry protected BICs, where the spatial symmetry of the mode prevents coupling with free space; and, (2) "Accidental" BICs which result from the cancellation of radiation by the destructive interference of two or more radiating modes. While accidental BICs can occur in any structure, they require careful tuning of parameters and often need oblique incidence. Recent works have considered asymmetry engineering in Si based HCG nanostructures as a route for achieving very sharp spectral features [25] resulting from the excitation of symmetry-protected variety of BICs. The quality factors (Q) of these modes degrade gracefully with reduction in device footprint [23] and are robust to fabrication imperfections. This offers an alternate mechanism (the degree of asymmetry in the transverse plane) to grating thickness control for achieving a desired mode lifetime.
Many of the studies of the QBICs are focused on sing Silicon: Si-based HCG gratings [25], Si slotted HCG grating [26], Si based all-dielectric metasurfaces [27,28] or all-dielectric LCGs [15]. Only a few papers have looked at QBIC resonances in Si$_3$N$_4 $ photonic crystal slabs (PhCs) [18,29,30]. The study by Romano and coworkers [29,30] considering Si$_3$N$_4 $2D Photonic Crystal Slabs (PhCS) is the only work that has explored QBIC resonances for RI sensing in the Si$_3$N$_4 $ platform. In this article, we numerically investigate Si$_3$N$_4 $1D MCGs with engineered asymmetry with the aim to create normal-incidence excitable QBIC based ultra sharp spectral response. We consider the most general case of period-doubling (or "dimerizing") asymmetries; specifically, the combined width and displacement perturbations (unlike width or displacement perturbations considered in isolation as in [16]). Such asymmetry-engineered 1D gratings are doubly-resonant for either polarization and can achieve sharp spectral features without relying on ultrathin designs. Due to the low dispersion of Si$_3$N$_4 $, the proposed designs can be easily scaled for operation in the whole visible to near IR range and, being 1D shapes, can be fabricated with better fidelity compared to 2D geometries. We simulate bulk and surface sensing in both spectral-shift and intensity-shift modalities and predict that the performance of the proposed asymmetry-engineered structures to be better than previously reported simple grating based structures and photonic crystal slab geometries. Sensing with dual resonances [5] is an additional advantage offered by the proposed structure.
2. Optical response in asymmetric DMCGs
Symmetry breaking in planar photonic crystals [31] has been reported as a tool for tailoring their optical response. A simple way to break the symmetry of a grating is to break the symmetry of each of the fingers [32] giving rise to ultra sharp resonances but at the disadvantage of requiring non-planar geometries. Preserving the planar geometry, a second grating with twice the period of the original placed in close proximity was considered in an early work by Lemarchand [33] leading to improved angular tolerance of a resonance. Considering a similar two-grating geometry with careful symmetry breaking, Nguyen and coworkers [31] have shown intricate control of the energy-momentum dispersion. Zeng and coworkers reported an alternate approach by placing the two gratings on the same plane [34] called the "diatomic" grating whereby alternate grating fingers are displaced without changing the overall filling fraction. Overvig and coworkers have further generalized the concept of period doubling (or "dimerization" as they term it) to two dimensions [16,21]. The work by Wang and coworkers [26] has considered symmetric and asymmetric slotted gratings.
We consider the general class of fill-fraction preserving period-doubling asymmetries which generalizes on previous reports. The terminology proposed by Overig and coworkers is adopted. Figure 1(a) shows the construction procedure for gap-dimerized MCG (henceforth $g$-DMCG) and the width-dimerized MCG ($w$-DMCG). We consider the case of a dimerized MCG with both width and gap perturbation and term it ($w+g$-DMCG). This case has not been considered by Overig and coworkers. The grating dimensions such as grating period, thickness, rib width and slot size are denoted as $GP$, $t_g$, $w$, $g$ respectively. Perturbations in width and gap are represented as $w_1$ and $g_1$ respectively. The proposed $w+g$-DMCG structure covers all possible period-doubling modifications and is also more general than the slotted grating design considered by Wang and coworkers [26].
Fig. 1. Schematic of the asymmetric 1D gratings and their optical properties. (a) shows the fill-fraction preserving transformation steps for a 1D ${Si_{3}N_{4 }}$ MCG lying on a silica ($SiO_2$) substrate following the dimerization terminology of [16]). (b) even and odd symmetry modes in an unperturbed MCG (i,iv), gap-perturbed DMCG (ii,v) and width-perturbed DMCG (iii,vi). (c) calculated band diagrams of infinitely-extended versions of the 4 kinds of MCGs with $a$ denoting the period of the plain MCG (GP=$a$). The white area is accessible where zeroth order beams are considered. Modes excitable by $p$ ($s$) polarized light are marked black (red). In the case of $w+g$-DMCG, the asymmetry enables $p$ and $s$ polarized light to couple to both modes. The modes at the Gamma point are labelled with letters to help compare with exact numerical simulation results. The geometrical parameters for the $g$-DMCG was $g_1 = 0.06*a$, for the $w$-DMCG was $w_1 = 0.15*a$, and for the $w+g$-DMCA was $w_1=0.15*a, g_1=0.06*a$.
Analytical methods like coupled mode theory [22,35] are not expected to work adequately as the index contrast in these structures is not low. The approach adopted here is to first analyze the optical properties of the structures qualitatively using the concept of mode parity and to then interpret the results of exact numerical calculations using this framework. Electromagnetic (EM) analysis of the above structures has been done using the commercial tool CST Microwave Studio which uses the numerical technique called Finite Element Method (FEM). Periodic boundary conditions are imposed along the $x$ and $y$ axes and absorbing boundary conditions applied along the $z$ axis to simulate a structure of infinite extent in the transverse plane. The computational domain size of the unit cell is $GP$ x 5 nm x 1550 nm (the domain size along the x axis depends on the grating period GP of the device). Tetrahedron shaped meshes with edge lengths varying from 1.5 nm to 130 nm have been used. Dielectric constants of Si$_3$N$_4 $ and quartz are taken as $4$ and $2.1025$ respectively [13]. The output light spectrum was obtained with wavelength resolution of 0.005 nm.
Three structures (see Fig. 1(b)) having the same periodicity and fill-fraction are considered: (a) a plain MCG, (b) a $g$-DMCG and (c) a $w$-DMCG grating. A single unit cell can be considered as a finite length waveguide [23] in the $z$-direction; the waves can bounce at the interfaces forming Fabry-Perot like resonance. The total number of possible modes is determined by the grating thickness and fill-fraction. The mode shapes in the grating plane can be approximated in some cases by a model similar to the tight-binding model used in quantum mechanics in the form of "bonding" and "anti-bonding" modes. Since all three structures considered in Fig. 1(b) possess mirror symmetry along some vertical axis, only modes with strictly even or odd symmetry (with respect to the symmetry axis) are present. For any combination of geometrical parameters, the even modes are only excitable by a $p$-polarized (or TM-polarized) normally incident plane wave and the odd modes only by a $s$-polarized (TE) one. There can be no inter-coupling between the even and the odd modes. The calculated [36] band diagrams for all the four grating variants is shown in Fig. 1(c) considering a 50% duty-cycle and infinite thickness. It is found that upto a thickness of 300 nm, only the first order Fabry-Perot modes occur in the $z$ direction. The band diagrams for all of the DMCGs exhibit only slight variations and to the plain MCG (after accounting for the scaling). Some band flattening effect [31] is also observed in all the dimerized variants at the Gamma point that arises from zone-folding [16]. The labelled points which are Bloch modes correspond to reflection peaks that are observed later in exact numerical simulations.
Only in the case of the $w+g$-DMCG, where both gap and width perturbations are non-zero, mirror symmetry is broken. The modes no longer possess strict even or odd symmetry and they allow finite, albeit low [34], coupling to the radiation continuum of the mismatched polarization. Thus the excitation of QBICs becomes possible. The radiation coupling coefficient can be shown to be proportional to the degree of asymmetry [34]. The exact spatial distribution of the mode with respect to the permittivity distribution determines the mode energy, i.e. its resonance wavelength. Generally, the even mode has a lower energy than the odd mode. The modes of the $w$-DMCG ((iii,vi) in Fig. 1(b)) do not follow this and the $p$-polarization excitable mode has a higher energy as its mode is distributed in air more than the dielectric.
Figure 2 shows the numerically calculated optical response of the different MCG configurations for a set of geometrical parameters. In regards to the spectra, using either the gap or width perturbation alone is seen to lead to one resonance excitable by $p$-polarization and another via $s$-polarization, whereas using the combined gap and width perturbation leads to dual resonances for each polarization. In general, the $p$-polarized resonances are seen to be relatively narrower. Some similarities like resonance wavelength, linewidth and the field distributions are observed among resonances labelled C, F and J which seem to exist in symmetric and asymmetric structures. These are identifiable as GMR resonances involving "antibonding" modes with an odd symmetry (see Figs. 1(b) (i) and (ii) for J and F respectively). These resonances do not appear to achieve much field enhancement at all. B, E and I are also GMR resonances involving bonding modes with relatively reduced free-space coupling coefficient (see Figs. 1(b) (iv) and (v) for I and E respectively). Some amount of field enhancement is seen for these resonances. Arguments from symmetry presented earlier indicate that modes G and H are also GMR modes involving coupling to the even shaped bonding mode and odd-shaped antibonding modes respectively. The dramatic difference between $w$-DMCG and the plain MCG arises from the width differences; a larger width implies tighter mode confinement and a smaller width a looser mode confinement which leads to a reduced coupling constant. Finally, mode A is a QBIC involving coupling of $p$-polarized light with an antibonding mode and $s$-polarized light with a bonding mode. This is evident from the near field plots where the electric and magnetic field patterns are opposite of each other for A and D. The somewhat smaller lifetime for D is seen to lead to reduction in field enhancements which reach high values in the case of A.
Fig. 2. Numerically simulated spectra and near-field plots at resonances for various Si$_3$N$_4 $ on silica-DMCG structures clad in water. (a), (b), (c) show the reflectance spectra for $s$ (red) and $p$ (black) polarized plane-wave illuminations for $w+g$-DMCG, $g$-DMCG and $w$-DMCG respectively. Each figure also shows the reflection response of a symmetric MCG with the same periodicity and fill-fraction in dotted line. (d), (e) show the normalized near field enhancement of the electric and magnetic fields in the $xz$-plane of the gratings at various wavelengths marked A-J. See Visualization 1, Visualization 2, and Visualization 3 for animation of magnetic field ($H_y$) variation with input phase at wavelengths A,B,G shown in (e). See Visualization 4 and Visualization 5 for animation of electric field ($E_y$) variation with input phase at C,D shown in (d). Rib width $w$ is 97.5 nm, gap $g$ is 107.5 nm. The perturbations in width $w_1$ and gap $g_1$ are 40 nm, 97 nm for $w+g$-DMCG. For $g$-DMCG, the width perturbation is zero, for $w$-DMCG, the gap perturbation is zero, the remaining design parameters being same as that for $w+g$-DMCG. For the MCG, the rib width $w$ and gap $g$ are 205 nm each. The gratings are 220 nm thick with grating period of 410 nm.
Geometrical parameter tuning for the $w+g$-DMCG can be done in such a way that the two resonances interfere with each other achieving Electromagnetically Induced Transparency (EIT). On excitation by $p$-polarized light, it was observed that increasing grating period or reducing thickness reduced the spacing between QBIC and GMR. However, the linewidths for the resonances become identical due to which EIT-like feature can not be obtained. EIT can be obtained for the grating under $s$-polarized light illumination as shown in the transmission spectrum in Fig. 3(a). A narrow (FWHM=0.74 nm) and high contrast peak in the transmission (at wavelength marked by $\beta$) can be seen within the low transmission window of the spectrum. This is possible due to considerable difference in linewidths/ Full width Half Maximum (FWHM) of the two resonances. Figures 3(b), (c) show the near field amplitudes of the normalized electric and magnetic fields for the proposed grating. The fields are enhanced at the resonant wavelength ($\beta$), but the enhancement is lower than the QBIC mode excited by $p$-polarized light shown in Fig. 2(a).
Fig. 3. Numerically simulation results of a particular $w+g$-DMCG structure when illuminated by a $s$-polarized light showing Electromagnetically Induced Transparency (EIT) window. (a) shows the transmission spectrum. (b), (c) show the electric and magnetic field enhancements in the near field at wavelengths marked by $\alpha ,\beta ,\gamma$. The rib width $w$ and gap $g$ are 102.5 nm each, width perturbation $w_1$=70 nm, gap perturbation $g_1$=82 nm, grating thickness $t_g$=140 nm, periodicity $GP$=510 nm. See Visualization 6 for animation of electric field ($E_y$) variation with input phase at wavelength $\beta$ shown in (b).
2.1 Influence of geometrical parameters on optical response
We extensively explored the design space of the $w+g$-DMCG structure under both $s$ and $p$-polarizations and found that the dual resonances are seen reliably for a wide range of parameter combinations with the possibility for independently setting the resonance wavelengths and lifetimes of both resonances to some extent. Figure 4 shows the transmission response when grating parameters such as thickness $t_g$, grating period $GP$, width perturbation $w_1$, gap perturbation $g_1$ are varied. In Fig. 4(a), for grating thickness of $t_g$=120 nm, the transmission response is flat showing no resonance. For $t_g$=160 nm, two transmission dips are observed corresponding to GMR (broader resonance) and QBIC (narrower resonance). On increasing the thickness, there is a red shift on both the resonances with the shift being larger for GMR. Also at sufficiently high thicknesses ($t_g$=320 nm), higher order modes are excited. In Fig. 4(b), increase in grating period also causes red shift in the resonances but the shift is larger for the QBIC mode. As a result of this, the wavelength spacing between the two resonances reduces. Besides this, the linewidth is reduced from 3 nm to 0.3 nm in GMR. For no perturbation in rib width in Fig. 4(c), only GMR is observed as expected for a $g$-DMCG resonator. On increasing $w_1$, QBIC mode is excited. As shown in Fig. 4(d), QBIC disappears for $g_1$=0 as seen in a $w$-DMCG resonator. With increase in gap perturbation $g_1$, red shift in GMR is observed whereas there is a blue shift in QBIC, thus increasing the spacing between the resonances. Besides this, linewidth for QBIC is broadened from 0.1 nm to 2.4 nm.
Fig. 4. Effect of geometrical parameters of a width and gap perturbed Medium Contrast Grating ($w+g$-DMCG) on the transmission and reflection spectra. (a) shows variation in grating thickness. Grating Period is varied in (b). (c) shows variation in rib width perturbation $w_1$. Perturbation in gap between the ribs $g_1$ is varied in (d). The grating is illuminated by a $p$ polarized light. The fixed design parameters are same as that used for $w+g$-DMCG in Fig. 2.
Figures 5(a)-(d) show the effect of change in geometrical parameters on EIT in the transmission spectrum. In Fig. 5(a), EIT is observed for thickness of 160 nm and 200 nm, beyond which it splits into QBIC and GMR. A single broad GMR (FWHM$\approx$14 nm) is observed for the grating thickness of 120 nm. In Fig. 5(b), EIT resonance is observed for the grating period of 470 nm and 510 nm. On reducing the periodicity, separated quasi-BIC and GMR are observed. A single GMR is observed on increase of grating period beyond 510 nm. Effect of changing the perturbation in rib width is shown in Fig. 5(c). Narrow linewidth EIT (FWHM$\approx$1 nm) is observed for higher perturbations. EIT is changed to a GMR for unperturbed rib width as expected in a $g$-DMCG grating. In Fig. 5(d), sharp EIT resonance (upto Q$\approx$871) is observed for higher perturbations in the gap between the ribs. The feature disappears for zero perturbation in the gap.
Fig. 5. Effect of change in geometrical parameters on the transmission spectrum Electromagnetically Induced of Transparency (EIT) resonant $w+g$-DMCG structure.(a) shows variation in grating thickness. Grating Period is varied in (b). (c) shows variation in rib width perturbation $w_1$. Perturbation in gap between the ribs $g_1$ is varied in (d). The fixed design parameters are same as that used for $w+g$-DMCG in Fig. 3.
3. Refractive index sensing performance
In this section, the proposed Si$_3$N$_4 $ nanostructure is evaluated for its refractive index (RI) sensing performance. As described in the introduction, the specific interest is in using normal incidence visible wavelength region where silicon photodetectors work best. We consider bulk and surface sensing using spectral-shift and intensity-shift modalities. We consider a structure that is doubly resonant with two cleanly isolated spectral dips for $p$-polarized illumination and another structure whose two resonances have spectral overlap resulting in a narrow EIT transmission peak for $s$-polarized illumination.
RI sensing schemes based on intensity shift and resonant wavelength shift have been numerically and experimentally investigated for CMOS-compatible Si$_3$N$_4 $ based gratings [4–6,11,13,14,29,30]. Table 1 gives a detailed summary of reported sensing performance figures for bulk sensing experiments in Si$_3$N$_4 $ nanostructures operating in the visible wavelength region. In the table, grating structures such as 1D low contrast gratings backed by waveguides (LCG), 1D Medium Contrast Gratings (MCGs) and 2D PhCS having BIC have been included. LCGs as a class exhibit the highest sensitivities and Figure of Merit (FOM), especially, when operating in reflectance mode. Although an ultra high FOM of 2200 has been observed in [13] for reflection mode, it is possible only for ultra thin LCG gratings having thickness below 10 nm. For the case of spectral-shift based modality in transmission-mode, it is seen that GMR based MCGs show experimental bulk sensitivities typically of the order of ${400} {\rm{nm}/\rm{RIU}}$. With a typical linewidth of about 1 nm, this translates to an FOM in the hundreds. For the intensity-shift based sensing scheme in transmission mode, bulk sensitivities of the order of 1700%/RIU [6] have been reported for GMR based MCGs. For 2D PhCS, transmission mode spectral-shift sensitivity and FOM of 103 nm/RIU, 129 have been reported, whereas intensity sensing gives high sensitivity of 4000 %/RIU.
Table 1. Summary of bulk RI sensing performance of recently reported Si$_3$N$_4 $ based nanostructures. $\lambda _0$, $\Lambda$ and FOM specify the operating wavelength, typical linewidth of resonance and sensing Figure of Merit respectively. Exp/Sim indicates if these are reported simulation or experimental results.
From the table, it is clear that sensing studies of BIC resonant gratings is in its nascent stage. Besides this, there is a lack of investigation in surface sensing capabilities of Si$_3$N$_4 $ gratings. The only work in this direction includes 2D PhCs showing resonant wavelength shift of 2 nm for surface binding of 66 nM of p53 proteins [29]. As observed earlier in Fig. 2, QBIC resonances possess narrow linewidth ($\approx$0.05 nm) and high electric field enhancements thus enabling realization of high performance gratings in spectrum shift and intensity shift based modalities while offering design flexibility and overcoming the disadvantages of low contrast nanostructures.
Fig. 6. A comparative study of the sensing performance of $w+g$-DMCG for the spectral-shift modality. (a), (c) and (e) show bulk sensing and (b), (d) and (e) show surface sensing. The transmission spectra progression as a function of analyte RI for a $w+g$-DMCG showing the behavior of the QBIC dip for bulk (a) and surface (b) analytes. In (c) and (d), the shifts of the dip as a function of analyte RI are plotted for four cases: (1) QBIC, (2) GMR resonances of the $w+g$-DMCG, (3) GMR resonance of a 50% duty cycle plain MCG and (4) EIT peak in a second $w+g$-DMCG structure. Differential Sensitivity and FOM performance for quasi-BIC is shown in (c), (f). All the 3 structures are 220 nm thick with grating period of 410 nm. For the first $w+g$-DMCG, rib width and gap are 97.5 nm each. Perturbation in width is 40 nm and gap perturbation is 47.5 nm. The design parameters for the second $w+g$-DMCG with an EIT peak are same as those in Fig. 3.
3.1 Spectrum shift modality
Spectral shift modality involves observing the change in a resonator’s characteristic wavelength in response to changing optical characteristics of the surrounding medium. This change is usually a small perturbation resulting in a shift linearly proportional to the change. The behavior of the QBIC resonance, however, is characteristically different as seen in Fig. 6(a), (b). A non-linear increase in resonant wavelength with increase in analyte RI is observed. To have a detailed view of the trend, resonant wavelength shift corresponding to analytes of different refractive indices is shown in Fig. 6(c). An exponential increase in the resonant wavelength is observed for the quasi-BIC as well as EIT exhibited by $w+g$-DMCG. Such type of exponential behaviour for BIC resonant gratings has also been reported in [30]. As observed in an empirical exponential-growth model in [30], the spectrum shift in BIC increases with decrease in RI contrast between the grating and analyte. Besides this, the E-field is more delocalized for higher RI claddings, as such exponential growth has also been observed in the E-field decay length in the cladding. For the above resonances showing non-linearity in spectrum shift, the differential sensitivity and FOM for every refractive index value of the analyte is calculated as
where $\Delta \lambda =\lambda _2-\lambda _1, \Delta n=n_2-n_1$, $n_1$: analyte refractive index for which sensitivity is to be calculated, $n_2$: consecutive higher refractive index, $\lambda _1$: resonant wavelength corresponding to $n_1$, $\lambda _2$: resonant wavelength corresponding to $n_2$. FWHM($n_1$): FWHM for resonance corresponding to analyte RI of $n_1$.As shown in Fig. 6(e), sensitivities for Quasi-BIC range from 50 nm/RIU-300 nm/RIU for analytes with RI in the range 1.33-1.51. For higher RI analytes (n=1.6) studied in [30], better performance is expected due to such exponential rise in the resonant wavelength shift. GMR in both MCG and $w+g$-DMCG showed linear increase in resonant wavelength having sensitivity of 158.75 nm/RIU and 182.82 nm/RIU respectively. EIT resonant gratings show sensitivities lying in the range 50 nm/RIU-300 nm/RIU, however linewidth broadening in presence of analyte is observed degrading the FOM to the range 70-95 (results not included in the figure). Ultra-narrow linewidth (FWHM$\approx$0.05 nm) in Quasi-BIC gives high FOM ranging from 500 to 2000 for analyte RI varying from 1.33 to 1.51. GMR in $w+g$-DMCG and MCG give low FOM of 234.38, 36.7 respectively. Also for $w$-DMCG gratings, sensitivites lie in the range 58.75 nm/RIU-273 nm/RIU and FOM in the range 422-1350 (results not included in the figure). For surface sensing, the top surface of the grating was covered by a 10 nm thick analyte layer as shown in the inset of Fig. 6(b). In Fig. 6(d), non-linearity in resonance shift has been observed in this scheme as well for QBIC and EIT resonant gratings. Besides this, better performance in terms of FOM has also been observed for QBIC resonator as shown in Fig. 6(f). GMR in MCG and GMR, QBIC in $w+g$-DMCG shows sensitivities of 21.1 nm/RIU, 25.49 nm/RIU, 7-23 nm/RIU respectively and FOM of 4.873, 36.69, 110-170 respectively.
Fig. 7. Comparison of intensity-shift sensing scheme performance for quasi-BIC, GMR, EIT resonances exhibited by $w+g$-DMCG and GMR exhibited by MCG. (a) and (b) show the transmission spectra of $w+g$-DMCG showing EIT and quasi-BIC for surface sensing. Intensity shift for EIT resonance is observed at 740.6 nm, quasi-BIC at 643 nm wavelength. (c) shows intensity shift for all the resonances corresponding to varying refractive indices of analyte. (d) shows sensitivity for quasi-BIC and EIT resonant gratings. The design parameters for each of the gratings are same as that used for spectrum shift modality in Fig. 6.
3.2 Intensity shift modality
Intensity interrogated refractive index sensing involves monitoring the light intensity at a single wavelength instead of the entire spectrum. This enables the use of a simple and low cost sensing system without the need of a spectrometer. Besides this, high throughput analysis is possible with multiplexing using CCD or CMOS cameras [13,37,38]. Sharp features exhibited by QBIC (Q$\approx$11300) and EIT(Q$\approx$1000) are expected to give large intensity shifts at a fixed wavelength when there is a small change in surrounding medium refractive index. Although narrow linewidth limits the dynamic range, tiny variations in medium refractive index can be sensed such as in applications requiring sensing of analytes with very low concentrations.
Surface sensing performance of the above resonances found in $w+g$-DMCG is compared with GMR based MCG as shown in Fig. 7. Tiny variations in the analyte refractive index are done keeping in mind the potential applications. Figures 7(a) and (b) show the transmission spectra of EIT and QBIC resonant $w+g$-DMCG under varying refractive indices of 10 nm thick analyte layer. The wavelengths are fixed at 740.6 nm, 643 nm respectively. Intensity for GMR observed in MCG and $w+g$-DMCG is monitored at the wavelengths of 628.8 nm, 674.5 nm respectively. Figure 7(c) shows intensity shifts for all the resonances corresponding to changing analyte refractive index. Non-linear change in intensity can be observed for EIT and quasi-BIC resonances, with the nature of change being exponential in quasi-BIC. The GMR modes show an almost linear behavior with small deviations distributed evenly around the fitting straight line. The small deviations are speculated to arise from the finite frequency sampling inherent in a FEM simulation. Sensitivities for the approximately linear GMR based structures are 5073%/RIU (w+g-DMCG), 2188%/RIU (MCG). Differential sensitivities for quasi-BIC and EIT resonant gratings are shown in Fig. 7(d). The differential intensity sensitivity is calculated for every value of analyte RI as
where $\Delta I=I_2-I_1, \Delta n=n_2-n_1$, $n_1$: analyte RI for which sensitivity is to be calculated, $n_2$: consecutive higher RI, $I_1$: intensity corresponding to $n_1$, $I_2$: intensity corresponding to $n_2$. An exponential increase in the sensitivity from 1430-32000%/RIU with increasing analyte refractive indices from 1.33-1.35 can be seen for quasi-BIC resonances. The EIT peak can be more sensitive than the QBIC, however, fluctuations in the sensitivity is observed for the case of EIT resonances.4. Conclusion
In summary, we numerically investigated the general class of period-doubling, fill-factor preserving asymmetry operations on 1D gratings, specifically considering Si$_3$N$_4 $ towards applications in transmission-mode refractive index sensing applications. The numerical studies indicate performance gain and design flexibility and is hoped to guide experimental realization.
The proposed structures are highly polarization sensitive and are of interest for polarization selective interrogation of light emission from nanoscale emitters attached to the grating [7]. Future studies are underway to explore the behavior of dipole emitters in the spectral vicinity of the various resonances. The work can be extended by considering gratings operating beyond the subwavelength regime with operations involving trimerization and beyond. Obtaining narrow resonances in a given diffraction order is of interest in metagratings [39–41]. Work is underway to consider extension to two dimensions with both gap and width asymmetries.
Funding
Nano Mission Council, Department of Science and Technology (2016, NM, NS-65, SR).
Disclosures
The authors declare no conflicts of interest.
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