1. Introduction

In the field of near-infrared spectroscopy, ultra-narrow bandpass filters with wide out-of-band suppression capability have always been a focal point in optical research. These filters are widely used in a variety of optical devices and systems, such as radar, optical communication, imaging, nonlinear interactions, and biological research, becoming a core component of modern optical devices [1–10]. Particularly, the grating-based guided-mode resonance filter (GMRF) has attracted significant attention due to its high diffraction efficiency and narrow bandwidth characteristics.

Compared to traditional thin-film bandpass filters, GMRF couples the incident light into the leaky Bloch mode through the grating waveguide layer, thereby exciting the GMR perpendicular to the grating slits, achieving ultra-narrowband pass filtering [11–14]. Currently, there are three main methods to achieve grating waveguide transmission filtering. The first one utilizes waveguide gratings combined with multiple layers of dielectric stacks [15–17]. Multilayer dielectric stacks provide a low transmission background based on thin-film interference effects, and the periodic gratings excite the transmission GMR, achieving narrowband transmission filtering. However, multilayer dielectric stacks support multiple resonance modes, introducing multiple hard-to-adjust resonance peaks and unable to achieve broadband suppression single-channel filtering. Moreover, multilayer dielectric stacking also leads to excessive device volume, not conducive to integration. The second method uses strong modulation gratings to provide high-efficiency transmission resonance peaks at the resonant wavelength [18–21]. However, this method limits the choice of high-refractive-index materials and requires changing multiple parameters such as grating period, thickness, duty cycle, and material refractive index to design the appropriate resonant wavelength, increasing the complexity of design and fabrication. The third method combines phase-gradient metasurfaces with grating GMR [22]. By deflecting the transmitted light beam to the first-order diffraction through the phase-gradient metasurface, it suppresses the transmission light in the wave-vector direction, then excites specific wavelengths of light in the wave-vector direction through the grating GMR, achieving narrowband filtering. However, this scheme's transmission peak efficiency is very low, and the level of out-of-band attenuation is difficult to control below 1%. Each of these methods has its advantages, but there are also some limitations, such as the inability to simultaneously achieve wide working bandwidth, high Q-values, and high filtering efficiency. Therefore, the aim of this study is to enhance the out-of-band suppression capability of the filter while maintaining high Q-values and high filtering efficiency, expand its operational wavelength range, and simplify the tuning of the operational wavelength position.

In this study, we explore a single-layer grating GMR narrowband bandpass filter, which can achieve single-channel filtering with a working bandwidth of up to 400 nm under near-infrared TM light incidence through periodic groove perturbations and the interference between groove radiations. We designed the suitable grating through the Particle Swarm Optimization (PSO) algorithm [23]. Inspired by previous studies on high-Q phase gradient metasurfaces, we incorporated periodic groove perturbations in the y-direction of each grating [24]. These perturbations specifically excite high-Q GMRs propagating parallel to the grating, providing high-Q filter responses. Additionally, the PSO-optimized grating inherently supports GMRs propagating perpendicular to the grating, resulting in broad out-of-band suppression. Next, by introducing constructive and destructive interference between groove radiations, we eliminated redundant GMRs, retaining only one, ultimately achieving a single-channel filter with an ultra-wide working bandwidth. By further adjusting the groove period, we were able to precisely tune the filter's central wavelength within a 1350 nm to 1750nm operational spectral range. Our designed structure can achieve a maximum Q-factor of over 100,000 while maintaining an attenuation level below 1% outside the passband. Our design also demonstrates high robustness against fabrication deviations, with resonance peak position shifts less than 0.5 nm and stable resonance peak transmission rates and Q-values within a reasonable deviation range. Finally, we conducted an in-depth analysis of the obtained simulation results and discussed the underlying physical mechanisms.

2. Methods

In order to simplify the simulation calculation, our model sets the following conditions: the grating is infinitely large in the longitudinal direction, the periodic array is infinite, and the materials used are lossless and non-dispersive. First, we utilize the PSO algorithm to design a single-layer sub-wavelength GMR grating, aiming to achieve wideband low-transmission functionality. This lays an important foundation for achieving a wide transmission stopband range and deep stopband suppression. In the PSO algorithm, we consider the grating's period, thickness, and filling factor as optimization parameters, and we adopt the root mean square fitness function to ensure the numerical quality of the solution.

To calculate the linear spectrum of the designed structure, we employed the RCWA method. The RCWA method can accurately simulate the interaction between electromagnetic waves and periodic grating structures, strictly satisfying the electromagnetic boundary conditions for each polarization, and can quickly calculate the related reflection and transmission spectra. Additionally, we used COMSOL Multiphysics finite element software to simulate the field distribution and band structure of the structure. On the four side boundaries of the computational domain, we applied periodic boundary conditions, while the top and bottom were set as perfectly matched layers to absorb reflected and transmitted light. We used the MUMPS solver when performing calculations with COMSOL. Additionally, to improve simulation accuracy, we adopted an extremely fine mesh density in critical areas (such as silicon gratings and grooves), ensuring the maximum mesh element size is less than one-fifth of the structural size. For example, in the case of a groove width of 50 nm, the maximum mesh size in the corresponding area is set to be less than 10 nm. The quality factor Q was calculated as the ratio of the resonant wavelength (λ) to its full width at half maximum (FWHM, Δλ’), i.e., Q = λ / Δλ’.

3. Results and discussion

In this paper, all structures are based on periodic silicon nanogratings on SiO₂ substrates. Our simulation results demonstrate how to achieve excellent filtering performance with a wide working bandwidth, high peak transmittance, and narrow linewidth on this simple platform by integrating three unique design strategies.

3.1 Single-channel filter

As shown in Fig. 1(a), we introduced a silicon grating structure designed using the PSO algorithm. Our goal is for the grating structure to have a transmittance T₀ = 0 under TM polarized light incidence within the wavelength range of 1350-1750nm. To achieve this target transmittance, the parameters needing optimization include the grating period (p_x), thickness (h), and fill factor (F), it’s noteworthy that the w_g is the product of the p_x and F. Figure 1(b) displays the reflectance and transmittance of the designed element, where p_x = 703 nm, h = 461 nm, F = 0.7473, with the refractive indices of silicon, silicon dioxide, and air being 3.48, 1.48, and 1, respectively. The spectral bandwidth Δλ = 400 nm with T₀ < 1%. The grating structure based on these parameters is referred to as TMG.

 

Fig. 1. Concept and numerical design of silicon photonic grating metasurfaces for transmission suppression and single-channel filtering. (a) The schematic diagram shows the optimized Si grating array, which reflects all incident light. Parameters: p_x = 703 nm, F = 0.7473, h = 461 nm, w_g = p_x * F. (b) The transmission and reflection spectra of the grating, where the transmittance T₀ < 1% in the 1350-1750nm band. (c) The mode dispersion curves show several low-order modes of the grating waveguide. The vertical axis is the free-space wavelength (λ), and the horizontal axis is the guided-mode wavelength (2π/k). Mode 1-4 are represented by black, red, blue, green solid lines, respectively, and their degenerate states are shown with corresponding dotted lines. TMx and TEx modes propagating along the x-direction are shown with purple solid and dotted lines. The right side of the figure reveals the electric field distribution of mode 1-4, where |E|/|E₀| represents the normalized electric field strength. (d) In the Si grating metasurface, a groove with a period of p_y is introduced, with groove dimensions: l = 100 nm, w = 100 nm. (e) Transmission spectra for different groove periods (p_y = 690 nm, 720 nm, 750 nm, 780 nm, and 810 nm). (f) The left side of the figure presents the transmission spectrum for the groove period p_y = 744 nm. The right side shows the electric field distribution for the 1550 nm resonance peak, where the electric field enhancement factor |E|/|E₀| > 100.

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The TMG structure's capacity for broad bandwidth low transmission is fundamentally due to the notably higher refractive index of its periodic silicon layer (Si) relative to adjacent materials (air, SiO₂), thereby establishing a unique grating waveguide. The incident TM waves, upon phase-matched coupling with the periodic grating, initiate multiple interacting GMRs along the x-axis, evidenced by three dips in transmission in Supplement 1 Fig. S1(a). Supplement 1 Fig. S1(b) showcases the standing waves formed at the grating's reflective end by these GMRs [25]. The spectral response of this device is significantly influenced by both the periodic coupling within the grating and its structural profile, leading to complex resonant interactions and the generation of a wide-bandwidth, low-transmission spectrum. This design approach provides vital theoretical foundations for the exploration of filters operating over wide bandwidths [13,26,27].

We introduced a virtual periodic boundary condition in the y-direction, in addition to the aforementioned low-Q GMR, this TMG structure can also support guided modes (GMs) with infinite Q factors [28]. We primarily focus on the GM at the Γ point in the first Brillouin zone in the momentum space of TMG, using the commercial software COMSOL Multiphysics to virtually vary the periods in the y-direction of the grating, rapidly calculating the band characteristic frequencies of these GMs, resulting in the mode dispersion curves shown in Fig. 1(c). In the wavelength range we discussed, there are four types of GMs propagating along the y-direction, corresponding to the black, red, blue, and green groups of dispersion curves in the figure, with the normalized electric field distributions of these four modes on the right side of Fig. 1(c). Additionally, two x-direction GMs exist, represented by the purple solid lines and scatter plots in Fig. 1(c), unaffected by changes in the virtual period p_y.

Further analysis reveals that each type of GM propagating along the y-direction is accompanied by a set of degenerate states. These degenerate states correspond to the black, red, blue, and green solid lines and scatter plots in Fig. 1(c). The electric field diagrams of these GMs along the propagation direction are shown in Supplement 1 Fig. S2. We named the mode with three antinodes as the symmetric mode, and the mode with two antinodes as the antisymmetric mode. These modes predominantly localize within the silicon grating waveguide layer and are characterized by a momentum greater than that of free-space plane waves, categorizing them as “dark modes” [24].

The second key design strategy adopted in this study is the introduction of periodic grooves in the y-dimension of the silicon grating, as shown in Fig. 1(d). These periodic grooves provide a reciprocal lattice vector of 2π / p_y to compensate for the momentum mismatch between free-space waves and GMs, that is, 2π / p_y = n_eff 2π / λ₀. Here, p_y is the groove period, n_eff is the effective refractive index of the mode, and λ₀ is the wavelength of the mode. When appropriate period grooves are introduced to impart additional momentum, GMs will match the momentum of light in free space and be excited as bright modes, resulting in the formation of GMR peaks in the linear spectrum. Under TM-polarized incident light, only the four modes propagating along the y-direction and TMx shown in the solid lines of Fig. 1(c) can be excited, while the other four degenerate states and TEx modes shown in the scatter plots require TE-polarized light to be excited, which we will discuss further in Fig. 2(a).

 

Fig. 2. Transmission spectrum and electric field distribution of single-groove grating excited by TM incident light. (a) The upper and lower images on the left are the xy plane electric field distribution maps of a set of degenerate modes of mode 1. The black dots represent the hypothetical position of the groove, located at the antinode and node of the mode, respectively. The two images on the right correspond to the electric field distribution and electric field lines on the xz plane at the antinode and node, respectively. The electric field lines at the antinode are mainly the E_x component, and those at the node are mainly the E_y component. (b) Under the excitation of TM incident light, the transmission spectrum of the single-groove grating, where the groove period p_y = 640 nm. The inset is a 3D schematic diagram of the structure. (c) The electric field distribution along the propagation direction of the modes corresponding to the five GMR peaks in Fig. 2(b), with the groove position indicated by a black dashed line frame.

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We first selected mode 3 of the TMG structure for study, as it has a significant gap with adjacent modes in its dispersion curve. In this way, under TM-polarized light incidence, we can excite a GMR over a broader wavelength range individually. For instance, as shown in Fig. 1(c), at the intersection of the two blue shaded areas, mode 3 enables us to achieve single-channel filtering within the range of 1490 nm to 1610 nm. In this region, regardless of how the size of p_y is adjusted, only one GMR of mode 3 is excited.

Figure 1(e) shows that the variation of the groove period p_y from 690-810 nm correspondingly changes the wavelength position of the transmission peak. Within this range, the resonance peak shifts from 1495 nm to 1605 nm, covering a single wavelength transmission filtering of 120 nm bandwidth. And the peak transmission values at different groove periods are all close to 1.

Figure 1(f) displays the transmission spectrum of the TMG with grooves of period p_y = 744 nm. The groove length l = 100 nm, width w = 100 nm. The center wavelength of the GMR peak is 1550 nm, with a full width at half maximum (FWHM) of about 0.11 nm, and a Q value of approximately 14000. The high Q value can be attributed to several factors: In the near-infrared region, silicon's absorption loss is negligible. Thus, the resonance life is mainly controlled by radiative losses. The smaller groove structure means the area of interaction between light and matter is limited. This localized interaction leads to reduced absorption and scattering losses caused by structural defects or inhomogeneities. Additionally, the compact groove structure may alter the radiation characteristics of the mode, resulting in less energy being radiated outwards. Therefore, a longer dissipation time narrows the half-width and increases the Q value [29].

The center wavelength of the GMR peak in Fig. 1(f) is 1550 nm, however, according to the blue curve in the energy band Fig. 1(c), when the groove period p_y = 744 nm is introduced, the intersection with mode 3 is about 1577 nm, which shows a 27 nm blue shift compared to the actual calculated center wavelength of the GMR peak. This blue shift originates from the introduction of grooves reducing the effective refractive index of the grating waveguide, causing an upward shift in the dispersion curve, thus causing a blue shift in the intersection of the groove period p_y with the dispersion curve. In addition, the introduction of grooves also causes Bragg scattering, leading to band splitting in the GM dispersion, which also causes the resonance peak position to shift [24].

The right side of Fig. 1(f) shows the electric field distribution of the groove grating at the 1550 nm resonance wavelength, which matches very well with the electric field distribution of mode 3 in Fig. 1(c). In the near field, the extended resonance life increases the electric field strength enhancement factor |E|/|E₀| (where |E₀| represents the electric field amplitude of the incident light), and in this groove grating, |E|/|E₀| exceeds 100. The significant field strength enhancement in the groove area is the key radiation channel of the structure.

We further explored the impact of groove size variation on the GMR peak position and FWHM. As shown in Supplement 1 Fig. S3, we conducted a detailed study on the groove width w and length l. In the simulated transmission spectra of Supplement 1 Fig. S3(a), we set the groove length to l = 100 nm, while varying the groove width w between 85-115 nm. The calculations reveal that for every 5 nm increase in groove width w, the center wavelength of the GMR shifts by approximately 2 nm. Furthermore, Supplement 1 Fig. S3(b) reveals a close relationship between the variation in groove width and the center wavelength position and Q value of the GMR. Hence, we can clearly observe that as the groove width w decreases, the Q value tends to diverge. This phenomenon can be attributed to the fact that the mode we introduced is a bound state dark mode, which would be completely confined within the grating waveguide in the absence of groove perturbations. However, as the groove size increases, the radiative losses within each groove period also increase, leading to a sharp decline in Q value. In Supplement 1 Fig. S3(c) and Fig. S3(d), we fixed the groove width at w = 100 nm and adjusted the groove length within the range of l = 40-160 nm. Similarly, we observed the phenomenon of the GMR center wavelength changing linearly with groove length, and the Q value diverging as the groove length l decreases. Overall, we can achieve precise tuning of the filter's wavelength and FWHM through the fine design of groove dimensions.

The above simulation results verify the key and universal nature of our second design strategy. That is, in sub-wavelength one-dimensional gratings with GMs, we can achieve momentum matching between dark mode GMs and light in free space by appropriately introducing periodic grooves, thus converting dark mode GMs into bright mode GMRs. This provides theoretical support for achieving narrow-band filtering.

Following the validation of embedding high Q-value resonances in the transmission spectrum of low-transmission metasurface gratings, as discussed earlier, we next explored other opportunities brought by this design principle. First, we demonstrated how to achieve multi-channel filtering across a broader bandwidth.

3.2 Muti-channel filter

In the previous section, we mentioned the four GM modes propagating along the y-direction, which exhibited a set of degenerate states: symmetric electric fields with three antinodes and antisymmetric electric fields with two antinodes. The left side of Fig. 2(a) displays the degenerate electric field distribution of mode 1. After introducing virtual p_y-period grooves (black dots in the figure), the symmetric and antisymmetric electric field modes’ grooves are respectively located at the antinodes and nodes. As shown on the right side of Fig. 2(a), we extracted the electric fields and their field lines at the antinodes and nodes, where the electric field at the antinodes is the E_x component, while that at the nodes is the E_y component. Therefore, we predict that under TM incident light, only the symmetric modes with grooves at the antinodes will be excited, while under TE incident light, only the antisymmetric modes with grooves at the nodes will be excited.

Figure 2(b) shows the transmission spectrum under TM light excitation in the range of 1350 nm to 1750nm with a fixed p_y of 640 nm, groove length l = 100 nm, and width w = 100 nm. In this 400 nm bandwidth, the linear spectrum displays five high Q-value GMR peaks. Figure 2(c) presents the electric field distribution of these five GMR peaks, which reveals the excitation of symmetric modes of mode 1-4 and TMx, as expected. Supplement 1 Fig. S4 shows the transmission spectrum under TE incident light and its electric field diagram, revealing the excitation of the antisymmetric modes of mode1-4 and TEx under TE incident light. This design provides an innovative approach to multi-channel filtering.

3.3 Single-channel filter implemented by destructive interference

In seeking solutions suitable for wide operational bandwidth and high-Q value filters, our study demonstrates that for a specifically optimized low-transmission grating TMG, it has a working bandwidth of 400 nm, and our single-groove design generates multiple GMR resonances within this bandwidth. Although this approach offers a new perspective for multi-channel filtering, its effective operational range for single-channel filtering applications is limited to 120 nm. This shows certain limitations for single-channel filtering applications that require a broader working bandwidth.

Therefore, in Fig. 3, we introduce a third ingenious design strategy: introducing a pair of grooves of the same size within the py period of the grating, each with a length of l = 100 nm and a width of w = 50 nm. These two grooves are positioned at a y-directional offset parameter d = p_y / 2, and we name this grating structure as TMG-d. These two grooves act as radiation sources, and their initial phase differences result in constructive or destructive interference at the far-field radiation end. We hope this design strategy will suppress specific modes in the linear spectrum, thereby expanding the operational bandwidth of single-channel filtering.

 

Fig. 3. Transmission spectrum and electric field distribution of the TMG-d structure. (a) The intrinsic mode electric field distribution of mode 1-4 antisymmetric modes and TMx mode in the xy plane. The black dots indicate the positions of the grooves on both sides of the grating, which are at the antinode where E_x dominates. The distance between the two grooves is d = p_y/2. (b) Displays the transmission spectra of the double groove TMG-d grating structure with different groove periods (p_y = 460 nm, 505 nm, 550 nm, 595 nm, and 640 nm). The dimensions of the grooves are: l = 100 nm, w = 50 nm. The left inset is a 3D schematic diagram of the structure, and the right inset shows the electric field distribution corresponding to the GMR resonance peak at p_y = 550 nm, where the white dashed frame indicates the position and size of the grooves.

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In Fig. 2(a), we explored how TM incident light excites modes predominantly composed of E_x components. For ease of understanding, Fig. 3(a) and Supplement 1 Fig. S5 display the E_x electric field distributions of the antisymmetric and symmetric modes of mode 1-4, as well as the E_x electric fields of the TMx and TEx modes. In these figures, black dots virtually represent the grooves on both sides of the grating within one py period. As shown, in the antisymmetric modes of mode 1-4, the two grooves are located at the antinodes primarily composed of E_x, hence requiring TM incident light for excitation. Conversely, in the symmetric modes, the two grooves are at the nodes primarily composed of E_y, necessitating TE incident light for excitation. In the TMx and TEx modes, the main electric field components are E_x and E_y, respectively, thus they are respectively excited by TM and TE incident light.

We focused on the case of TM incident light, so unlike the excitation of symmetric modes by a single groove, our TMG-d structure can only excite the antisymmetric modes of mode1-4 and the TMx mode. As shown in Fig. 3(a), the E_x electric field of mode 2 is not only antisymmetric relative to the x-axis but also to the y-axis. Therefore, the two grooves with offset parameter d have the same initial phase, resulting in constructive interference at the far-field radiation end. However, mode 1, mode 3, and mode 4 are antisymmetric relative to the x-axis but symmetric to the y-axis, leading to a π initial phase difference between the two grooves, with opposing E_x electric field line components, thus producing destructive interference at the far-field radiation end. Similarly, the two grooves in the TMx mode also have a π initial phase difference. We expect that this design will suppress other modes, leaving only the GMR peak of mode 2 in the spectrum, achieving a single-channel high-Q filter with a wide operational bandwidth.

Figure 3(b) presents the simulated transmission spectrum of our TMG-d structure, achieving high-Q single-channel transmission filtering throughout the entire 400 nm working bandwidth. The inset shows the E_x electric field distribution of the GMR corresponding to mode 2 when p_y = 550 nm, which is highly consistent with the intrinsic mode electric field diagram of mode 2 in Fig. 3(a). In Fig. 3(b), by simply adjusting the groove period py of the TMG-d structure from 460 nm to 640 nm, we effectively shifted the resonance peak wavelength from 1350 nm to 1750nm. This design strategy achieves efficient single-channel filtering coverage over a 400 nm wavelength range, with the resonance peak's central wavelength transmission rate approaching 1.

Although it is reasonable to ignore the absorption of silicon in the near-infrared band, the silicon in actual devices may have a certain degree of absorption. Therefore, we evaluated the impact of smaller absorption on the device Q value and transmittance in Supplement 1 Fig. S6. Specifically, we introduced absorption into the silicon TMG-d structure shown in Fig. 3 (p_y = 550 nm) and calculated the corresponding transmission spectrum, setting the k-value range from 10⁻³ to 10⁻⁶. The results show that when the k-value is less than 10⁻⁵, its impact on the device can almost be ignored. However, when the k-value is greater than 10⁻⁴, due to the high Q value of the device design, the field enhancement effect at the resonance position leads to higher absorption, thereby reducing the peak efficiency of transmission and Q value. According to literature reports [30], within the discussed wavelength range (1350-1750nm), silicon's absorption is extremely low, with k-values typically far less than 10⁻⁶, thus even considering the absorption of silicon in actual devices, it will not affect our conclusions.

3.4 Analysis of fabrication deviations

In the fabrication process of micro-nano gratings, particularly when processing smaller grooves, fabrication deviations are a common and inevitable issue. These deviations can cause slight changes in our designed structures, thus affecting the performance of the devices. Therefore, examining the impact of these fabrication deviations on device functionality is a crucial and indispensable step. The structure we designed can be fabricated through standard lithographic procedures. In fact, existing studies have successfully fabricated structures with dimensions similar to our design using electron-beam lithography (EBL) technology and achieved high-Q phase gradient metasurfaces [24]. The mean absolute percentage deviation (MAPD) and root mean square deviation (RMSD) of EBL are 7% and 2.7 nm respectively [31]. Hence, in the fabrication deviation discussion in Fig. 4, we set the structural size deviation range of the grooves to ±10%, and the relative positional deviation range to ±10 nm. Such design parameters significantly exceed the fabrication deviation values of EBL, thus ensuring the feasibility of the plan.

 

Fig. 4. The impact of four typical groove fabrication deviations on the performance of narrowband filters. (a) Deviation I: Processing deviation in the groove length l. Three sets of data are randomly generated, each containing six values corresponding to the lengths l_1-6 of six grooves over three periods p_y, and their transmission spectra are calculated. The left inset shows the corresponding structural schematic, where the white dashed line frame indicates the position and size of the groove without processing deviation. The right inset is an enlarged view of the resonance peak. (b) Deviation II: Fabrication deviation in the groove width w. (c) Deviation III: Fabrication deviation in the position of the groove. (d) Deviation IV: There are deviations in the length, width, and relative position of the groove.

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We first analyzed the impact of fabrication deviation on the broadband low transmission background of the TMG grating. As shown in Supplement 1 Fig. S7, we adjusted the duty cycle (i.e., width), period, and thickness of the grating, respectively, and observed that within the deviation range of ±5 nm, these deviations had almost no effect on the transmission background of the TMG grating, and the transmittance within the entire 400 nm working wavelength range still remained at less than 1%.

Next, We simulated and evaluated the lengths, widths, and relative positional parameters of six grooves in three periodic units to analyze the impact of processing deviations on the performance of narrow-band filters. The processing deviations are categorized into four types: Deviation I, deviation in groove length; Deviation II, deviation in groove width; Deviation III, deviation in relative position of grooves; Deviation IV, combined deviation in length, width, and relative position of grooves.

For type-I deviation, we used Matlab to randomly generate six sets of different Δl_1-6 values, ranging from ±(l*10%). These values were assigned to six grooves, forming new groove lengths l_1-6 = l+Δl_1-6. We randomly generated Δl_1-6 values three times and calculated the transmission GMR peaks under these deviations, with the transmission spectra shown in Fig. 4(a). Compared to the grooves without processing deviations, structures with length deviations showed slight shifts in the GMR peak, with a shift range of less than 0.2 nm. Additionally, our design utilizes destructive interference to suppress redundant resonant peaks. However, fabrication deviation can give rise to imperfect destructive interference, thereby engendering a series of weak, yet high-Q, disordered resonant peaks. These peaks exhibit a significant disparity in both intensity and Q-value compared to the filtering peaks under discussion. Given their magnitude difference, these redundant resonant peaks are essentially inconsequential in experimental scenarios and can be disregarded.

We used the same method to randomly generate different Δw_1-6 and Δd_1-6 values and conducted calculations for Deviation II-IV. The range of variation for Δw_1-6 was defined as ±(w*10%), while that for Δd_1-6 was ±10 nm. The transmission spectra are shown in Fig. 4(b-d), with the relevant structural parameters and results presented in Table 1. In all discussed processing deviation types, compared to the grooves without processing deviations, the GMR peaks exhibited slight shifts, with the maximum shift not exceeding 0.5 nm. Furthermore, there were no significant changes in the Q value and maximum transmission rate of all resonance peaks. The above discussion confirms the significant robustness of our designed structure against fabrication deviations. Therefore, even with potential fabrication deviations in actual processing, the performance of the filter can still maintain high standards.

Table 1. Groove Fabrication Deviation Related Parameters and Transmission Peak Information

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4. Conclusion

This study details a design method for a narrowband filter based on sub-wavelength gratings, successfully achieving efficient narrowband filtering in simulations. During the design process, we initially employed the PSO algorithm to develop a single-layer sub-wavelength grating. This grating effectively suppressed the transmission output through multiple GMRs propagating perpendicular to the grating slits, attaining a transmittance of less than 1% across a wide 400 nm band. Building upon this, we further introduced periodic grooves to excite GMR propagating parallel to the grating slits, thereby creating narrowband GMR transmission peaks with a maximum Q-factor exceeding 100,000. Ultimately, by introducing destructive interference between groove radiations, we suppressed redundant modes, thereby expanding the operational wavelength range of the single-channel filter. We achieved precise tuning of the central wavelength of the passband within the 1350 nm to 1750nm wavelength range, simply by adjusting the groove period. Moreover, the designed structure demonstrated high robustness against fabrication deviations.

Overall, our proposed filter design strategy not only provides significant insights for the design of photonic filters but also reveals its immense potential in efficient spectral control and communication photonics applications. This novel and efficient approach represents a flexible strategy, readily adaptable to different spectral domains including ultraviolet, visible light, infrared, and even microwave frequencies, by altering structural parameters and materials. We anticipate this methodology to make a significant and lasting impact in the field of photonics.