1. Introduction

The miniaturization of optical spectrometers is crucial for widespread field deployment of optical sensing technologies. Spatially arrayed narrowband filters (with differing center wavelengths) is an approach that has several advantages over alternatives [1]; it can be closely mated with image sensor arrays for very compact systems [2–7]. In order to achieve performance constraints, filter arrays must be suited to economical manufacturing with individual filters exhibiting narrow linewidths, small device footprints, and good transmission efficiency. The filter array should ideally also possess adequate spectral coverage, spectral uniformity, and reduced inter-pixel crosstalk [2,3]. Conventional filters using absorbing pigments, dyes are not suitable for such applications due to low coupling efficiency and broader absorption spectrum [8]. The realizations of filter arrays can be broadly categorized into: (1) Fabry-Pérot etalon array filters, (2) plasmonic metasurfaces, (3) Mie resonant metasurfaces and (4) resonant waveguide gratings. Each realization has its own set of advantages and disadvantages. Fabry-Pérot cavity filters use conventional fabrication techniques with multiple deposition steps (they typically vary cavity length to alter center wavelength) and can achieve narrow linewidth and small pixel pitches in the near IR region. However, in the visible wavelengths, low efficiency is a concern and the achievable linewidths are of the order of 16 nm [7]. A wide variety of structural color filters operating in reflection and transmission mode have been reported in plasmonic and Mie-resonant all-dielectric metasurfaces [9–11]. Although plasmonic nanoresonators using nanorod or nanohole arrays possess high light confinement capabilities, their inherent metallic losses lead to broader linewidth and lower coupling efficiencies [12]. On the other hand, high-index dielectric metasurfaces exhibit somewhat reduced optical absorption, but the high radiative damping leads to broader spectral linewidths. Another disadvantage relates to the fabrication of metasurface based spectral filters which often rely on low-throughput nanofabrication techniques. Guided mode resonance (GMR) is a resonant optical phenomenon observed in 1D or 2D sub-wavelength patterned periodic structures [13–15]. Under virtue of its optical properties such as narrow linewidth, high efficiency, sensitivity to surrounding medium refractive index variations, GMR gratings have been widely investigated for applications such as filtering [1–6,16–22] and optical sensing [23–29].

GMR is associated with a sharp Fano-shaped reflection peak arising out of interference between laterally propagating Bloch modes and vertically excited Fabry Perot modes [30]. However, the attractive spectral features are only possible in large device footprints. It is well known that miniaturization of GMR gratings leads to linewidth broadening [16–19] leading to finite-size related performance degradation in spectral filter and sensing applications. This degradation becomes more severe with reduction in refractive index contrast between grating and cladding materials (in the so-called low contrast gratings (LCGs) and medium contrast gratings (MCGs)) as the GMR modes possess large propagation lengths as a result of weak Bragg scattering and edge scattering losses. To emulate the sharp spectral features of GMR resonant gratings under small footprints, the addition of in-plane mirrors (Bragg [29,31] or metallic reflectors [3]) at the grating ends have been proposed in the past. This leads to complete reflection at the ends, reducing the lateral leakage from the grating ends. However, mirroring has the following demerits: increased device fabrication complexity, unavoidable increase in the absorption losses (due to ohmic losses in metals), increase in device footprint (due to large number of periods required to realizing highly reflective Bragg mirrors).

The spectral response for high contrast gratings (HCGs) is marginally affected by footprint reduction since they contain GMR modes with small propagation lengths (caused by strong in-plane Bragg scattering effects). HCGs possess broadband spectral features useful for applications requiring broadband mirrors [32]. Recently, narrow linewidths were obtained in HCGs by realizing quasi bound states in the continuum (QBIC) modes in the NIR wavelength regime [30,33]. QBICs have been reported in sub-wavelength dielectric gratings, metasurfaces and photonic crystals (PhCs) and are of interest in sensing, spectral filtering and non-linear optics [34–39]. Two kinds of BIC resonances exist: accidental radiation cancellation type and symmetry-protection type. The symmetry-protected QBIC mode can be excited by breaking the mirror symmetry of unit cells and is particularly relevant to compact optical systems which typically involve normal incidence illumination. In the case of high refractive index contrast nanostructures exhibiting QBICs, Nanfang Yu et al. [30] have predicted sharp spectral features whose linewidths broaden gracefully upon size reduction. Radiation cancellation type QBICs exhibited by few unit cells of silicon PhCs have demonstrated high Q [40–42]. Most of the studies related to QBIC in miniaturized structures have been carried out in the NIR and mid-IR wavelength regions where silicon is used as a grating material. The mirroring strategies mentioned earlier have also been reported mostly in the near IR region. Due to silicon being lossy in the visible regime, the mentioned solutions can not be useful for high-performance sensing, imaging applications in the visible wavelength region which also includes the important therapeutic window. Recent studies have explored MCGs using transparent $Si_3N_4$ in visible wavelengths which can be effectively mated with high-responsivity silicon based photodetector arrays. Silicon nitride has very reduced optical absorption compared to silicon making $Si_3N_4$ nanostructures highly attractive. Recently, we have reported on symmetry-broken $Si_3N_4$ MCGs predicting promising sensing capabilities [43].

As most of the studies of LCG and MCG structures exhibiting QBIC resonances are often performed for periodic (infinite extent) configurations, it remains unclear whether such structures can perform adequately for spectral filtering and refractive index sensing applications in the face of finite-size related performance degradation. We extend our previous work [43] and numerically investigate the effect of miniaturization in LCGs and MCGs with broken mirror symmetry in the context of surface refractive index sensing and multispectral imaging filters in the visible wavelength bands. Following this, the rest of the paper is organized as follows: the geometry and simulation methodology, optical response, and systematic parametric analysis are found in section 2; the device filtering performance has been evaluated in section 3.1 and the surface refractometric sensing performance for the spectral shift based modality have been evaluated and compared with unperturbed MCGs in section 3.2.

2. Geometry, optical response, and parametric analysis

2.1 Structure and simulation methodology

Figure 1(A) shows the construction procedure for MCG dimerization. Dimerization strategies involving only width perturbation (width perturbed dimerized medium contrast grating, w-DMCG) or gap perturbation (gap perturbed dimerized medium contrast grating, g-DMCG) were reported by Nanfang Yu et al. [30]. It is noted that both the w-DMCG and the g-DMCG structures continue to possess mirror inversion planes. By combining both width and gap perturbations, the resulting structure (width and gap perturbed dimerized medium contrast grating, w+g-DMCG) lacks mirror symmetry. Further details of dimerization strategies and the optical response of structures lacking mirror symmetry can be found in our previous work [43]. The grating dimensions such as thickness, periodicity, rib width, gap between the ribs are denoted as $t_g$, $GP$, $w$, $g$ respectively. The grating material is silicon nitride ($Si_3N_4$), with silica/water cladding (depending on filtering/sensing application) resting on a silica substrate. Perturbations in width and gap are denoted as $w_1$ and $g_1$ respectively. In this paper, w+g-DMCG and MCG geometries with miniaturized (finite-sized) and infinitely large lateral extent are studied. The miniaturized geometries are finite along the direction of refractive index modulation (’x’ axis in Fig. 1(A)) and infinite along the remaining direction (’y’ axis). The infinite sized gratings have been considered infinite along ’x’ and ’y’ axes. The incident wave is oriented along ’z’ axis. Figure 1(B) shows the schematic of the pixel array, each pixel element consisting of a finite-sized grating (gratings 1 to 4) integrated with a photodiode. The grating geometry consists of a finite-sized w+g-DMCG (consisting of 4 unit cells in the above case) functioning as spectral filters, associated with a resonant wavelength ($\lambda _1$ to $\lambda _4$). When vertically illuminated, the grating supports laterally propagating Bloch modes at their resonant wavelengths, which couple back into the reflected light. However, miniaturization of gratings can lead to scattering losses at the finite ends (as shown in the computational model in Fig. 1(C), unlike for infinite-sized gratings, where only $0^{th}$ order transmission and reflection take place.

 

Fig. 1. A - Schematic showing dimerization strategies for one dimensional medium contrast grating (MCG). The MCG can be dimerized by perturbing rib width (w-DMCG), gap (g-DMCG), or both (w+g-DMCG). Only the w+g-DMCG (proposed geometry) lacks a mirror-inversion symmetry. B Schematic of arrayed grating band-reject filters mated with a photodiode array and illuminated by plane wave under normal incidence. Each filter has a w+g-DMCG geometry with a varying geometrical parameter. C Computational model of the finite sized grating employed in this paper.

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All the numerical simulations reported in this paper were obtained with the commercial electromagnetic solver, CST Microwave studio. Transmission of infinite-sized grating is computed in frequency domain using Finite Element Method (FEM) where tetrahedral meshing and periodic boundary conditions are applied along x and y axes and absorbing boundary conditions are applied along the z axis. The optical properties for the miniaturized gratings can be characterized by the spectrally dependent scattering cross section (SCS) and by the spectral dependence of the transmittance recorded by a finite-sized photodetector situated beneath the grating. These are computed in CST Microwave studio using the finite integration technique (FIT). For the FIT simulations, the computational domain is divided into hexahedral mesh cells, periodic boundary conditions are applied along y axes, perfectly matched layer (PML) are applied along the remaining axes. The SCS can be computed as: $\frac {P_{sca}}{I_{inc}}$, where $I_{inc}$ is the intensity of the incident plane wave and $P_{sca}$ is the scattered power calculated by integrating the Poynting vector in the far-field zone. The far field quantities (electric and magnetic fields) required for the calculation of $P_{sca}$ are obtained by applying near-to-far-field transformation (NTFF) in the region close to the grating (near field region) just before the PML boundaries.

The photodiodes considered for the study are assumed to be reflection-less, since the corresponding back reflections can be suppressed by using anti-reflecting coatings as discussed in [44]. Thus, instead of modeling a photodiode layer, power monitors are added at its location in the computational model shown in Fig. 1(C). The transmittance to the photodetector underneath the grating filter is computed using a collection plane represented area $A_{out}$ (as shown in Fig. 1(C)) at a suitable distance beneath the grating as: $\displaystyle \frac {I_{tr(avg)}}{I_{inc}}$, where $I_{tr(avg)}$ is the average transmitted light intensity computed in the collection plane, $I_{inc}$ is the time-averaged intensity of incident plane wave (excitation plane is represented by area $A_{in}$, where $A_{in}$=$A_{out}$). The incoming light intensity is computed as $I_{inc}=0.5E^2/2\eta$, $\eta$ is the background medium (outer cladding of the grating), $E$ is the electric field amplitude of the wave. The materials $Si_3N_4$, glass (or silica), air and water are modeled by refractive index values of 2, 1.45, 1, 1.33 respectively.

2.2 Optical response

The optical response for a simple MCG and a mirror-symmetry broken w+g-DMCG under s and p-polarized normal incidence are compared in Fig. 2. A detailed analysis of the optical response of various infinite-extent dimerized gratings was reported previously in an earlier work [43]. First, some general observations are made about the spectra seen in Figs. 2(A–D). For the finite structures, the ECS and transmittance to the photodetector are seen to correlate well. Both exhibit an asymmetrical lineshape. The transmittance of the finite structures is seen to correlate well with those of the corresponding infinite case, albeit, showing both linewidth broadening and contrast reduction effects. For w+g-DMCG, the transmission and ECS (normalized w.r.t peak value) spectra predict dual QBIC modes for different grating sizes under s-polarized plane wave illumination. In contrast, the MCG (s-polarized plane wave illumination) exhibits single GMR mode with Q factors considerably lowered (Q=12.9 for 10 unit cells, Q=15.5 for 20 unit cells, Q=16.8 for infinite sized grating). Under p-polarized plane wave illumination, sharper spectral features are predicted for the modes in w+g-DMCG, and MCG. In Fig. 2(A), as the grating size is increased from 10 (N=10) to 20 unit cells (N=20), the spectral positions for the resonant modes 1 and 2 are predicted to converge towards that of the infinite sized grating modes. Consider mode 1 in the transmission spectrum for all the grating sizes. The Q factor is predicted to improve typically from 47 to 73 on increasing the grating size from N=10 to 20. The mode has Q factor comparable to the mode excited for infinite-sized grating (Q=150). A similar trend can be observed for the remaining cases in Figs. 2(B-D).

 

Fig. 2. Comparison of the optical response of mirror-symmetry broken grating (w+g-DMCG) (A,C) and a simple grating (MCG) (B,D). One infinite and two finite gratings (10 and 20 unit cells) are considered in each case for normally incident illumination of s (A,B) and p (C,D) polarized light. While the infinite structures are characterized by transmission, the finite structures are characterized by normalized extinction cross section (ECS) and transmission to an underlying photodetector (see Fig. 1(C)). The near-field electric field amplitude responses for infinite-sized MCG, w+g-DMCG modes are shown in E-G. The corresponding responses for finite-sized grating (N=20) are shown in H-J respectively. Magnetic field responses for infinite MCGs are shown in N-P, the corresponding responses for finite sized grating (N=20) are shown in K-M respectively. The near field plots shown in the figure correspond to s-polarized light incidence. i-iii, iv-vi show zoomed-in views of one of the unit cells in H-J, K-M respectively. The gratings are 220 nm thick, $w$=102.5 nm, $g$=205 nm, $w_1$=45 nm, $g_1$=26 nm and are surrounded by water.

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The field distributions for the modes in w+g-DMCG, MCG of size 20 unit cells for the case of s-polarized illumination are shown in Fig. 2(H-M). Finiteness leads to lateral cavity formation in both the gratings. Breaking the mirror symmetry in miniaturized MCGs results in modes with asymmetric field distribution (refer Fig. 2(I,J)) compared to symmetric MCG modes (refer Fig. 2(H)). Due to asymmetric field distribution, the mode coupling to the radiation continuum is low in the dimerized MCGs as compared to the unperturbed MCGs with symmetric modes. Thus, they possess a higher Q factor than the simple MCGs. On comparing the infinite-sized grating shown in Fig. 2(E-G,N-P) with the grating having 20 unit cells (figure H-J,K-M), the E and H field mode distributions remain qualitatively unchanged. In Fig. 2(H-M), miniaturized gratings show edge diffraction effects due to finite ends thus affecting the Q factor. Comparing the mode in Fig. 2(E) of w+g-DMCG with the MCG Fig. 2(F) having infinite sizes, E-field is enhanced by a factor of 2.5. Reducing the grating size to 20 unit cells results in a comparable field enhancement of 2 as shown in Fig. 2(H,I). On comparing the magnetic field distribution of the same modes, an enhancement of 2.7 (Fig. 2(N,O)) is predicted for the infinite sized gratings, compared to twice the improvement predicted (Fig. 2(K,L)) for the miniaturized gratings.

2.3 Influence of geometrical and material parameter variations

This section includes the effect of variations in the geometrical parameters as well as the cladding material on the spectral response of the miniaturized grating. A detailed computational analysis of geometrical parameter variations for the infinite-sized grating was previously reported in [43]. As shown in Fig. 3(A, B), changing the perturbation in width or gap reduces linewidth for one mode while it is increased for the other mode. In Fig. 3(C), with increasing grating thickness $t_g$, the grating linewidth degrades due to increasing refractive index contrast between core and cladding. Unlike in GMR resonant gratings, linewidth reduces with an increase in duty cycle $DC$ as shown in Fig. 3(D). Although narrowband modes can be obtained by tuning $t_g$ or $DC$, the resulting coupling efficiency is lowered as a result of reduced coupling to the radiation continuum. Thus, a moderate thickness and duty cycle are essential to achieve a trade-off between Q and coupling efficiency.

 

Fig. 3. Effect of geometrical and material parameter variations on the transmission response of a miniaturized (20 unit cells) width and gap perturbed medium contrast grating (w+g-DMCG). A-D - variation in grating geometrical parameters, width perturbation ($w_1$), gap perturbation ($g_1$), thickness ($t_g$), duty cycle ($DC$) respectively. G, H show the miniaturized grating response in air and water cladding respectively. The effect of adding glass substrate, laterally adding Aluminium mirrors at the ends is shown. E, F show the effect of adding substrate for the infinite sized grating in air and water cladding respectively for comparison. Other geometrical parameters are the same as that in Fig. 2.

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The cladding materials such as air, water, or silica affect the refractive index contrast and thus the Q factor of the grating. As shown in Figs. 3(E-H), the grating with higher index cladding (water) supports dual modes as compared to a single mode observed in cladding with lower index (air). Considering the case of infinite-sized w+g-DMCG in air (Fig. 3(E)) and water background (Fig. 3(F)), there is marginal improvement in the grating linewidth on the addition of silica substrate. A similar trend has been predicted for the miniaturized gratings in the air (Fig. 3(G)) and water (Fig. 3(H)) claddings. As done for HCG in [3], the effect of laterally adding mirror at the grating is also studied. The mirror distance is suitably decided ($d_m=0.5GP$) so that the reflected waves interfere constructively with grating Bloch modes. The length of the mirror ($l_m$ greater than mirror material skin depth) is chosen to be sufficiently large enough to obtain complete reflection. The addition of mirror only marginally improves linewidth in the finite-sized grating regardless of the cladding.

3. Applications

3.1 Multi-spectral filtering

In this section, we have studied the filter array design using finite-extent w+g-DMCG including the mating with photodiodes to obtain pixel array as shown in Fig. 1(B). Specifically, while optimizing the geometrical parameters of the structure, it is seen that trade-off between various performance aspects can be achieved. Figure 4(A) shows a typical finite-sized w+g-DMCG filter transmission response and the definition of various performance metrics like spectral full width at half maximum (FWHM) and contrast in transmission dip $\Delta T$ (also termed as efficiency in spectral filter literature) for the main band and sideband efficiency for the undesired spectral dips. The lateral footprint $L_x$ of filter is also an important geometrical parameter.

 

Fig. 4. Transmission response of a miniaturized width and gap perturbed dimerized medium contrast grating (w+g-DMCG) filter array for multispectral imaging. A Definition of filter performance parameters. B The grating periodicity $GP$ is varied in order to achieve tunability in resonant wavelengths in different upper cladding material (air, silica). C An illustration of the pixel array output intensity when excited by input sources 1,2 is also included. D shows the effect of crosstalk between adjacent pixel filters 4 and 5 on their transmission spectra. The spectra are computed for different vertical distances from the grating. The grating is surrounded by silica based upper and lower claddings. The geometrical parameters for the grating are same as that in Fig. 2.

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Figure 4(B) represents the result of array design considering air as well as silica claddings. The proposed filter array typically spans 525 to 800 nm wavelength range with comparable main band efficiencies. However, smaller FWHM can be obtained by adding silica cladding. An individual filter element with a particular grating periodicity exhibits dual modes in silica cladding compared to one mode in air. Filtering using dual resonant modes has the advantage of adding temporal stability to the output. The intensity fluctuations and background noise of the input source can be canceled out by using different fiber outputs between the 2 modes [45]. Besides this, using silica cladding suppresses the sidebands considerably. Quantitative details of the study carried out for a set of 9 filters are shown in Table 1. The grating modes possess efficiencies in the range of 43-60% and FWHM of 3.5 nm-12.6 nm. The sideband modes are suppressed in the spectral range except for small $GP$ (consider the case of $GP$=360 nm, $GP$=340 nm).

Table 1. Numerical prediction of performance parameters for an array of 9 spectral filters each consisting of w+g-DMCG geometry with a particular grating period. FWHM specifies full width half maximum.

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Figure 4(C) illustrates the identification of an unknown narrowband source spectrum (source linewidth $\leq$ filter FWHM) using a filter array of 7 pixels. The photodetector (PD) array transmittance values (measured as currents at individual PDs) can be considered as a spectral "fingerprint" signal. Consider the input light source ’1’ being incident on the pixel array. The input spectrum overlaps with one of the resonant modes of the filters 4, 7. It causes low intensity at the pixels 4 and 7, while keeping the remaining pixels bright. Similarly, source ’2’ overlaps with the other resonant mode of pixels 4 along with pixel 1, reducing their intensities except others. Thus, for any source, a unique combination of pixels shows low intensity enabling a unique "fingerprint".

Miniaturization in gratings leads to scattering losses which could lead to radiation leakage into into the adjacent pixel of interest. This can cause crosstalk between the adjacent pixels, thereby degrading its filtering performance. The effect of crosstalk in two adjacent filters corresponding to pixels 4,5 is shown in Fig. 4(D). As shown in the schematic, separate power monitors have been applied to calculate transmission at distance $d_v$ beneath each of the pixels 4,5 (pixel 4 monitor is represented by area $A_4$, pixel 5 monitor is represented by area $A_5$ ). The incoming light is assumed to be incident on both the pixels (the plane of excitation is represented by area $A_{in}$). Transmission is calculated using $\displaystyle \frac {I_{tr(avg)}}{I_{inc}}$, where $I_{tr(avg)}$ is the average transmitted light intensity for a pixel in the collection plane (represented by $A_4$ or $A_5$), $I_{inc}$ is the time-averaged intensity of incident plane wave calculated in plane $A_{in}$ ($I_{tr(avg)}=I_{tr(avg)4}$ for pixel 4 calculated in plane $A_4$, $I_{tr(avg)}=I_{tr(avg)5}$ for pixel 5 calculated in plane $A_5$). Transmission for isolated pixels have been calculated by the method shown in Fig. 1(c). As shown in the transmission response for pixel 5 in Fig. 4(D), an increase in transmission value at 620 nm and the corresponding decrease at 680 nm is observed. These spectral locations match with the resonant modes possessed by the adjacent pixel 4. Thus the observed crosstalk effects are due to coupling of leaky Bloch modes of pixel 4 into or away from pixel 5 resulting in increased or reduced transmission. Similar crosstalk effects can be observed for pixel 4 at 695 nm (near resonant mode of grating in pixel 5). Besides this, with increase in power monitor distance, the spectra are unaffected which shows that the edge diffraction effects from the nearby pixel are not prominent.

Table 2 gives a detailed comparison of the proposed geometry with other filter array realizations reported in the literature such as Fabry Perot etalons, plasmonic nanohole array, GMR based metal /dielectric grating waveguides.The proposed filter array has an individual pixel footprint typically lying in the range of other filters. The spectral coverage for the array is moderate which can be optimized and extended further. Due to low optical losses in $Si_3N_4$, the grating modes can be optimized to operate across the visible-NIR wavelength regime. Besides this, they have moderate efficiency along with linewidths narrower than other filters in the visible wavelength regime. These structures can be fabricated by one step of e-beam lithography (similar to metal grating waveguides [4]) compared to two steps required in silicon grating waveguides [3]. The proposed w+g-DMCG gives narrow FWHM, optimum efficiency under small footprints.

Table 2. Performance parameter comparison of the proposed geometry with recently proposed optical filters for multispectral filtering application. FWHM specifies full width half maximum. Sim/exp indicates reported simulation or experimental results.

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3.2 Refractive index sensing

An arrayed grating exhibiting multiple modes can help in extracting diverse and complete information about the biochemical environment [13] like the biofilm thickness, its refractive index, and changes in solution refractive index. A highly multiplexed sensing system can be realized by using arrayed grating configuration integrated with compact photodetector readouts as shown in Fig. 1(B). A study is carried out comparing simple gratings (MCG) and the proposed mirror-symmetry broken gratings (w+g-DMCG) in the infinite and finite cases in terms of sensing performance parameters such as sensitivity (S) and figure of merit (FOM). Additionally, finite-sized simple MCGs with lateral mirrors have been considered. The objective of the study is to investigate the sensing performance in finite-sized gratings by breaking their mirror symmetry in order to achieve a. improvement over finite sized simple MCGs b. mitigation of performance degradation on reducing the grating size from infinitely large to finite.

As shown in Fig. 5(A), the bioreceptor-analyte molecule binding process has been numerically emulated by a homogeneous layer of thickness $t_a$=20 nm with refractive index $n$. The device sensing performance has been evaluated using 2 methods: calculating spectral shift on (a) varying $n$ from 1.33 to 1.45, for a constant $t_a$=20 nm, (b) varying $t_a$ from 3 nm to 21 nm, for a constant $n$=1.45. Variation in $n$ or $t_a$ represent different instances of the grating surface coverage by analyte molecules [47]. In Fig. 5(D), for infinitely large gratings, creating asymmetry improves sensitivity by 3 times and FOM by 10 times, where FOM is calculated as $\displaystyle \frac {S}{FWHM}$. On reducing the device footprint ($L_x$=8.2 µm) in Fig. 5(E), marginal reduction in sensitivity is predicted for a given grating type. As a result of linewidth broadening in miniaturized gratings, FOM for mode 2 of w+g-DMCG is reduced from 19 to 8.2. However, it is still 4 times larger than that predicted for the modes in simple miniaturized MCGs (with/ without mirror). Unlike in HCGs [3], lateral addition of Aluminium mirrors marginally improves the device Q for simple MCGs. Mode 2 in finite sized w+g-DMCG has a Q factor of 107, compared to a Q of 67 observed in mirrored MCG. Based on analyte thickness variation studies carried out for the gratings, creating asymmetry improves the device sensitivity by 3 times for the case of both infinite (Fig. 5(F)) and small footprint gratings (Fig. 5(G)).

 

Fig. 5. Comparison of surface refractive index sensing performance for a mirror symmetry broken medium contrast grating (w+g-DMCG) with a simple MCG under infinite and finite extent (20 unit cells). A Analyte bound to the grating surface modeled as a homogeneous layer. The modulation in transmission spectra of infinite, finite sized w+g-DMCG due to varying analyte refractive index $n$ (keeping analyte thickness $t_a$ constant) are shown in B, C respectively. D-G show mode shift in the gratings with respect to varying analyte refractive index (D:infinite, E:finite with/without mirror), thickness (F:infinite, G:finite with/without mirror). The grating rests on a silica substrate surrounded by water. The grating material and geometrical parameters are same as that in Fig. 2.

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4. Conclusion

In conclusion, we have numerically investigated the optical response of a miniaturized, mirror symmetry broken MCG under normal plane wave illumination. It is found that mirror symmetry breaking excites QBIC modes which mitigate the finite-size related performance degradation in low to medium refractive index contrast arrayed gratings-based applications. Specifically, arrayed spectral filters with footprints of 8 µm without mirrors at its ends can be realized with a typical FWHM of 7 nm over a wide spectral span in the visible wavelength region and marginal spectral crosstalk effects. For the application of surface refractive index sensing, the proposed device gives approximately four times improvement in FOM as compared to simple gratings of similar footprint. Further work can consider the extension of the spectral span into the near IR region where $Si_3N_4$ continues to exhibit negligible optical losses. This work considers the effect of finiteness along x direction, which can be further extended to consider the effect of finiteness in both the in-plane directions (x, y axes) for its functioning as a 3D microcavity. Higher Q can be obtained by using heterostructuring at the gratings ends similar to that done for silicon PhCs supporting radiation cancellation type QBICs [40]. Additionally, breaking mirror symmetry in 2D low/medium contrast gratings for the application of polarization-insensitive spectral filters/ cavities can be explored further, similar to that done for 2D dimerized high contrast gratings [30]. The algorithmic aspects of spectral reconstruction and aspects related to spectrum reconstruction in the presence of noises are also avenues for further study.