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Abstract
Guided-mode resonance (GMR) bandpass filters have many important applications. The tolerance of fabrication errors that easily cause the transmission wavelength to shift has been well studied for one-dimensional (1D) anisotropic GMR gratings. However, the tolerance of two-dimensional (2D) GMR gratings, especially for different design architectures, has rarely been explored, which prevents the achievement of a high-tolerance unpolarized design. Here, GMR filters with common 2D zero-contrast gratings (ZCGs) were first investigated to reveal their differences from 1D gratings in fabrication tolerance. We demonstrated that 2D ZCGs are highly sensitive to errors in the grating linewidth against the case of 1D gratings, and the linewidth orthogonal to a certain polarization direction has much more influence than that parallel to the polarization. By analyzing the electromagnetic fields, we found that there was an obvious field enhancement inside the gratings, which could have a strong effect on the modes in the waveguide layer through the field overlap. Therefore, we proposed the introduction of an etch-stop (ES) layer between the gratings and the waveguide-layer, which can effectively suppress the interaction between the gratings and modal evanescent fields, resulting in 4-fold increased tolerance to the errors in the grating linewidth. Finally, the proposed etch-stop ZCGs (ES-ZCGs) GMR filters were experimentally fabricated to verify the error robustness.
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1. Introduction
All-dielectric guided-mode resonance (GMR) bandpass filters have been proposed and applied in the frequency and mode selection of vertical-cavity surface-emitting lasers [1,2], sensitivity enhancement for infrared photodetectors [3,4], spectrometer systems [5,6], biosensors [7–9], and so forth [10–12]. They have the advantages of high transmission efficiency, good spectral fineness, thin thickness, and easy integration compared with traditional multilayer film devices and other approaches [13–16]. The theoretical design and explanation of the underlying physics have been well studied in previous related works. They demonstrated that the bandpass transmission of GMR comes from the integration of multiple leaky-mode resonances and coupling [17]. Moreover, feasible architectures with single-layer gratings or combined with a few thin-film layers have been successfully created using optimization algorithms [18,19]. However, due to the high-Q feature of the GMR spectrum, their fabrication processes are usually difficult with current nanofabrication technology, and the fabrication errors easily cause the operating wavelength to shift and create a mismatch with the design value [20]. For example, Foley et al. [21] had fabricated a transmission resonance filter in the mid-IR domain but with a ∼100 nm shift of the peak position. Therefore, a practical GMR design with high tolerance to fabrication errors is very important for applications.
Previous studies on fabrication tolerance of GMR filters have mostly focused on the 1D gratings. Single-layer GMR gratings are highly sensitive to the grating geometric parameters of the period, height, and linewidth, as the gratings act as both the diffraction-layer and the waveguide-layer [22]. The tolerances of grating period and waveguide-layer thickness are difficult to improve due to the natural limit of the GMR excitation principle [1], but they are usually easier to control in the lithography and material deposition process [23]. Zero-contrast gratings (ZCGs) that include a high-index partially-etched grating-layer have shown minimal sensitivity to the grating linewidth error because the waveguide-layer had translated into the residual thin-film layer. In addition, ZCGs GMR filters have experimentally demonstrated good spectral consistency [24]. To further decrease the difficulties in the depth control of partial-etching, an etch-stop layer was added between grating-layer and waveguide-layer to form an etch-stop ZCGs (ES-ZCGs) filter while maintaining a high tolerance of the grating linewidth [25]. However, to the best of our knowledge, the fabrication tolerance of 2D GMR gratings is rarely discussed.
From 1D to 2D gratings, the more complex physical mechanisms from the additional dimension needed to be taken into account and have prevented us from easily designing an efficient filter with the desired fabrication tolerance. Ko et al. [26] and Macé et al. [27] reported the theoretical design of unpolarized GMR filters created using 2D mesh ZCGs, but did not study the fabrication tolerances or perform experimentation. Through our calculations, we were surprised to find that the given 2D GMR designs in their works would be seriously sensitive to errors in the grating linewidth, which was quite different from the 1D case. A successful experimental demonstration of a high-performance 2D GMR filter is impractical due to the challenge of eliminating linewidth errors during fabrication. Key issues in understanding the poor tolerance of the grating linewidth for 2D ZCGs GMR and determining if there is another grating structure that is insensitive to the errors of the grating linewidth remain unclear.
In this work, the fabrication tolerance of 2D ZCGs GMR filters was first investigated. The underlying reason for the poor tolerance of the grating linewidth had been comprehensively analyzed and recognized in electromagnetic fields coupling between the grating-layer and the waveguide-layer. Then, the influence of an etch-stop layer as the buffer layer on the grating linewidth tolerance was explored via a comparative study. We demonstrated that the fabrication tolerance of the grating linewidth can increase by a factor of 4 through the introduction of an etch-stop layer to suppress the overlap between grating and modal evanescent fields. Finally, we verified the robustness of the 2D ES-ZCGs GMR filters through experimentation.
2. Tolerance analysis of the 2D ZCGs GMR filters
As demonstrated in Ref. [26], the schematic structure of the so-called 2D ZCGs GMR filters is composed of a partially etched 2D grid grating-layer of hydrogenated silicon (nSi:H) and a residual Si:H film as the waveguide-layer on the fused silica substrate (nSub), as shown in Fig. 1(a). The grating period and grating linewidth are respectively defined as P and W, while Dg and Dw are the grating height and thickness of the waveguide layer. To achieve depolarization under normal incidence, the period and linewidth of the gratings in the X and Y directions are such that Px = Py and Wx = Wy. The optical communication wavelength of 1550 nm was selected as the example under the assumption that all materials exhibited no absorption or dispersion. The structural parameters were as follows: P = 1020 nm, W = 204 nm, Dg = 390 nm, Dw = 284 nm, nSub = 1.45, and nSi:H = 3.5. The incident light source was a plane wave, the incident direction was normal incidence, and the incident medium was air. The influence of temperature, humidity, pressure and other environmental parameters on the spectral performance were not the focus of this paper, so they were not taken into consideration.
Fig. 1. Design and tolerance analysis of 2D ZCGs GMR filters. (a) Schematic structure and (b) theoretical spectrum of the design. Parametric dependences of the spectrum on the (c) grating linewidth, (d) height, (e) period, and (f) waveguide layer thickness.
Rigorous coupled wave analysis [28–30] is a common electromagnetic calculation method that can be used to calculate the diffraction efficiency and electromagnetic field distribution of the sub-wavelength structures. The corresponding spectrum and electromagnetic calculation results presented in this article are all based on this method. From the transmission spectrum results shown in Fig. 1(b), it can be seen that the designed GMR filter using ZCGs can excite a narrowband transmission peak with a transmittance close to 100% at 1550 nm, and the full width at half-maximum (FWHM) of the transmission channel is only about 2 nm. In the spectral range of 1530–1570 nm, the filter has a sideband with a transmittance close to 0%, and the shape of that transmission channel is very symmetrical and close to a perfect Lorentz line.
Nevertheless, in practical nanofabrication processes, inevitable fabrication errors prevent the achievement of the desired performance. For the ZCGs, although the sum of the height of the Si:H gratings and the thickness of Si:H waveguide-layer can be controlled precisely by thin-film deposition technology, the individual errors resulting from the partial etching process cannot be fully eliminated [25]. In addition, the errors in the linewidth and period of the gratings could originate from the lithography process. Therefore, a tolerance analysis concerning these four major errors is necessary to understand their influences on the final spectrum. The parametric dependences of the GMR spectrum regarding W, Dg, P, and Dw were thus swept, as shown in Figs. 1(c–f). We mainly compared the sensitivity of the transmission peak shift to the fabrication errors to evaluate the tolerance, as the peak transmittance and FWHM were slightly influenced by fabrication errors. The equation of the tolerance is defined as δ=ΔR (T>=90%, @1550 nm), where δ is the value of fabrication tolerance for each parameter, ΔR refers to the corresponding parameter’s variation range when the transmittance at the working wavelength (1550 nm) is greater than 90% and the transmission spectrum needs to substantially keep the Lorentz line. Sensitivity is the rate of variation of the working wavelength with the parameter, and we define it as S= Δλ/ΔR. It is worth noting that for some parameters such as the grating height Dg, when the value changes, the operating wavelength is almost unchanged, so the sensitivity approaches zero. The fabrication tolerance and sensitivity of each parameter of 2D ZCGs GMR filters are shown in Table 1.
Table 1. Fabrication tolerance and sensitivity of 2D ZCGs GMR filters
For the linewidth error, the transmittance at 1550 nm drops to 90% when W changes by ±3 nm. Thus, its fabrication tolerance is only 6 nm (∼1.35%). In the same analysis, we derived the fabrication tolerances of the grating period P and the waveguide-layer thickness Dw as only 0.5 nm and 0.7 nm, respectively. Such low tolerances indicated that P and Dw were limited by the natural mechanism of GMR according to the phase-matching condition [1]. The least sensitive error was the height of the gratings Dg which had a large fabrication tolerance of about 150 nm, as it has almost no influence on the waveguide mode or the phase-matching conditions. Considering the current lithography technologies used for the preparation of the gratings, e.g., electron beam lithography or laser interference lithography, have achieved sufficient control in periodicity, we believe that linewidth and waveguide-layer errors are more serious problems [25,31]. Increasing the fabrication tolerance of the grating linewidth and improving the control of the waveguide-layer thickness in the nanofabrication process are two key issues to be resolved.
Understanding why the transmission channel is so sensitive to the grating linewidth for 2D ZCGs GMR is a priority. Through comparison with 1D ZCGs GMR, whose fabrication tolerances had been investigated in previous studies and also demonstrated in our supplementary materials, the difference was found to lie in the tolerance of the linewidth. It was found that 1D ZCGs GMR is not sensitive to the grating linewidth (see in Fig. (S1)) in contrast with the high sensitivity of the 2D case. 2D GMR is an extension of 1D GMR with symmetric design in the other dimension to realize depolarization, which can be seen from the almost same parameters in the 1D and 2D GMR designs. However, it is the additional dimension that results in the deterioration of the linewidth tolerance for 2D GMR. To illustrate this, the linewidth tolerances of Wx and Wy were respectively explored at certain TE polarization, which presented completely different phenomena, as shown in Fig. 2. The spectral shift is very small when varying Wy, in correspondence with the 1D GMR results (see in Fig. (S2)), while undergoing a significant spectral shift when varying Wx. Symmetrically, a serious spectral sensitivity to Wy rather than Wx would be achieved at TM polarization. Therefore, 2D ZCGs GMR filters would exhibit the poor tolerances for both Wx and Wy under unpolarized incidence.
Fig. 2. Spectral sensitivity in linewidth Wx and Wy at TE polarization of 2D ZCGs GMR filters. (a) The robust spectrum and (b-d) $|E |$ field distributions for different Wy. (e) The susceptible spectrum and (f-h) $|E |$ field distributions for different Wx. All the $|E |$ field distributions plotted here are on the YZ plane located at the cross center of the grid. The solid line region is the waveguide-layer and dotted line region is the grating-layer. To quantitatively describe the variations of the field distribution with the linewidth, the correlation coefficient R which is calculated using the corr2 function in MATLAB that describes their similarity has been listed here. The field distributions (c, g) with a linewidth of 204 nm were used as the calibration respectively, and the closer R is to 1, the greater the similarity between the field distributions is. Through our calculation, the correlation coefficient R of (b, d, f, h) are 0.9999, 0.9999, 0.8997 and 0.7624, respectively.
The above rule can also be reflected by exploring the mode behavior of GMR. We observed the electric field distributions on the YZ plane at the designed resonance wavelength of 1550 nm and TE polarization. It can be seen that their electric fields present a uniform standing wave distribution phenomenon locally concentrated in the waveguide layer, which is the typical field distribution of GMR and can be understood as follows: Along the Y direction, the propagating high-order evanescent wave couples with the TE waveguide mode, which excites the GMR of the TE mode. For different Wy shown in Figs. 2(b–d), we found that the GMR is very insensitive to the grating linewidth parallel to the polarization direction. For different Wx shown in Figs. 2(f–h), in contrast, the electric field distribution of GMR showed clear variation as the linewidth orthogonal to the polarization direction.
To interpret the underlying mechanism of the linewidth tolerance, a physical picture depicting the electromagnetic field overlap was adopted to describe the coupling effect between the grating-layer and waveguide-layer, as shown in Fig. 3(a). The fields of the waveguide modes would be mainly bounded in the high-index waveguide-layer but partly leak outside in the form of the evanescent field. Then, the modal evanescent field can overlap with the local field in the grating-layer and be interpreted as they interact. In a separate analysis of the electromagnetic field distribution of the gratings, a strong local field enhancement was found in the grating-layer, as shown in Figs. 3(b-g), which could have a strong impact on the waveguide modes and the corresponding GMR excitation. Moreover, this effect is sensitive to the grating linewidth because the field distribution in the grating-layer showed clear variation as the linewidth changed. In Figs. 3(b-d), we can see that the field distributions on the YZ plane in the grating-layer are almost unchanged with different linewidth Wy at the TE polarization. However, in Figs. 3(e-g), there are clear differences between the three field distributions with different linewidth Wx, and field changes mainly appear on the grid line in the X direction. This can also explain why the 2D-ZCGs GMR has different spectral sensitivities towards the grating linewidths Wx and Wy. This observation can be attributed to how the electromagnetic wave would be more influenced by structural changes in the direction orthogonal to the polarization than parallel to the polarization.
Fig. 3. Physical picture and mechanism analysis of the linewidth tolerance. (a) The model describing the electromagnetic field overlap between grating-layer and waveguide-layer for illustrating the coupling effect on the waveguide modes. (b–d) $|E |$ field distributions in separate grating-layer on the YZ plane located at the cross center of the grid at TE polarization for different linewidth Wy; (e–g) The identical $|E |$ field distributions for different linewidth Wx. The dotted line region refers to the grating-layer. The correlation coefficient R of (b, d, e, g) with respect to the calibration (c, f) are 0.9887, 0.9965, 0.9858 and 0.9656, respectively.
Additionally, the interactions between the grating-layer and the waveguide mode are sensitive to the grating linewidth, which is the core reason for the poor fabrication tolerance toward the grating linewidth of the 2D-ZCGs GMR filter. If the coupling effect between the waveguide mode and the gratings can be completely eliminated, the fabrication tolerance toward the grating linewidth can be greatly increased. It is worth considering whether the above purpose can be realized when we introduce a low-index etch-stop layer as the buffer layer between the grating-layer and waveguide-layer as a 2D ES-ZCGs GMR architecture to enable the modal evanescent field to decay in the buffer layer and block the coupling between the gratings and the waveguide modes.
3. Effect of the etch-stop layer on the tolerance for 2D ES-ZCGs GMR filters
We next investigated whether the fabrication tolerance of grating linewidth could be improved using the 2D ES-ZCGs GMR filters. As shown in Fig. 4(a), the 2D ES-ZCGs architecture consists of the top Si:H grid gratings, a SiO2 buffer layer, and a Si:H waveguide layer on the fused silica substrate. The design was achieved using the global optimization algorithm PSO (particle swarm optimization) [32,33]. To facilitate the comparison with 2D ZCGs GMR filters, we set the aimed spectrum in the same 1530–1570 nm range as the optimization target with a transmission channel at 1550 nm, a peak transmittance of 100%, a FWHM of 2 nm, and a sideband transmittance of 0%. The grating period P, linewidth W, height Dg, and the thicknesses of the etch-stop layer De and the waveguide layer Dw were the optimization variables. The fitness function is the root mean square error function expressed as:
Fig. 4. Design and tolerance analysis of the 2D ES-ZCGs GMR filters. (a) Schematic structure and (b) theoretical spectrum of the 2D ES-ZCGs GMR filter. Parametric dependences of the spectrum on the (c) grating linewidth, (d) height, (e) period, and (f) waveguide layer thickness.
As shown in Fig. 4(b), we found that the achieved 2D ES-ZCGs GMR filters can achieve nearly the same performance as the ZCGs GMR filters. Then, we further analyzed the fabrication tolerance of this 2D ES-ZCGs design, whose dependences of the spectral properties with W, Dg, P, and Dw are given in Figs. 4(c–f), respectively. It was found that the fabrication tolerance of the grating linewidth had greatly improved from 6 nm to 20 nm (∼5%), increasing by nearly 4-fold, and the transmission channel was relatively unchanged when the linewidth was changed, which first validated the feasibility and validity of our idea. The tolerance of the period, height, and waveguide layer thickness are close to those of ES-ZCGs GMR design because they were determined by the principle of GMR. The fabrication tolerance and sensitivity of each parameter are shown in Table 2. This buffer layer also acts as an etch-stop layer [25] during fabrication, taking advantage of the high selectivity ratio of Si:H and SiO2 in the etching process. Therefore, the grating height and the thicknesses of the buffer and waveguide layers can be precisely controlled by thin-film deposition, in contrast to the great difficulties of partial-etching for the ZCGs structures.
Table 2. Fabrication tolerance and sensitivity of 2D ES-ZCGs GMR filters
Moreover, the GMR spectrum and electromagnetic field distribution of the 2D ES-ZCGs design with the changes of linewidth Wx and Wy were examined again. The transmission channel showed no shift when varying both Wy and Wx (see in Figs. 5(a) and 5(e)). The electric field distributions of the TE mode resonance still present a uniform standing wave distribution phenomenon of GMR in the waveguide layer. Notably, this moment in the GMR of the TE mode corresponds to a higher guided-mode order excited by different diffraction orders, causing the field distribution to differ from that of the ZCGs GMR structure. The GMR field distribution also exhibits strong robustness regardless of Wy shown in Figs. 5(b–d) or Wx shown in Figs. 5(f–h). Thus, it is reasonable to assume that the introduced etch-stop layer had effectively blocked the coupling effect between the waveguide mode and the local field in the top grating-layer. Such progress can make the waveguide mode and the final spectral property of 2D ES-ZCGs GMR filters no longer sensitive to the grating linewidth, effectively increasing the fabrication tolerance of grating linewidth. The above results well demonstrated our hypothesis that the proposed 2D ES-ZCGs GMR filters can produce an unpolarized GMR filter with a better fabrication tolerance than the 2D ZCGs structures reported in previous literatures.
Fig. 5. Spectral sensitivity in linewidth Wx and Wy at TE polarization of 2D ZCGs GMR filters. (a) The robust spectrum and (b-d) $|E |$ field distributions for different Wy. (e) The susceptible spectrum and (f-h) $|E |$ field distributions for different Wx. All the $|E |$ field distributions plotted here are on the YZ plane located at the cross center of the grid. The solid line region is the waveguide-layer and dotted line region is the grating-layer. The correlation coefficient R of (b, d, f, h) with respect to the calibration (c, g) are 1.0, 1.0, 0.9999 and 0.9999, respectively.
4. Fabrication and characterization of the 2D ES-ZCGs GMR filters
To experimentally confirm the claimed performance of the proposed 2D ES-ZCGs GMR filters, a standard lithography process including the thin-film deposition, exposure mask, and masked etching were used, in turn, to prepare the sample. Since Si exhibits little absorption in the considered wavelength range, we selected Si:H as the high-index material with almost no absorption. The Si:H/SiO2 multilayer films were coated using an NSC-15 magnetron sputtering deposition system from Optorun. In this process, Si was used as the starting material. By adding H2 to Ar sputtering plasma during the sputtering deposition process, the structural defects and voids in the amorphous Si film that causes absorption at low photon energies can be reduced by orders of magnitude. For a quick proof-of-concept demonstration with a sample size of 400 × 400 µm, electron beam lithography (EBL) was used to expose the gratings. It should be noted that EBL equipment is already available for wafer-level fabrication [34]. A positive electron beam resist (ZEP520A) was initially spin-coated (4000 rpm, 2 min) on top of the Si:H layer and baked at 180 °C on a hotplate for 5 min. The final thickness of the resist was about 120 nm. The grating pattern was exposed on the resist by EBL using a Raith EBPG-5200 and then developed in standard developing liquid. Afterward, reactive ion etching (RIE) was performed for the Si:H masked etching using CHF3 and SF6 at flow rates of 50 and 15 sccm, respectively, and at a pressure of 15 mT. Inductively coupled plasma power etching (1200 W) and RIE (25 W) were performed. Finally, we used O2 plasma to remove the residual photoresist mask.
Using scanning electron microscopy (SEM), we characterized the geometrical parameters of the fabricated sample. The top-view and 45° cross-sectional SEM images are shown in Fig. 6(a) and (b). It was found that the uniformity and steepness of the side walls of our sample were quite good. With the help of EBL and deposition technology, we accurately determined the grating period and height, but the measured grating linewidth (115 nm) deviated slightly from the desired value (104 nm) with an error of ∼10 nm. Although the linewidth precision can be improved by experimental iteration and optimization, this is unnecessary for the proposed 2D ES-ZCGs GMR filters with high fabrication tolerance. Fourier-transform-based angle-resolved spectroscopy was employed to verify the spectral property of the sample. The measurements realized angular resolution by focusing the diffractive beam in the Fourier plane of the lens. It should be mentioned that the source of our spectral measurement system was not strictly collimated light, but is conical light with a ∼2$^\circ $ cone angle. In Fig. 6(c), we calculated the simulated spectrum curves of the 2D ES-ZCGs GMR filters at different cone angles by integrating the spectrum at different incident and azimuth angles as a gaussian distribution weight. It can be seen that this issue can seriously affect the practical measurement results. Based on our literature survey, this seems to be a common problem in the spectral characterization of GMR, especially for microscopic spectral measurement, as described in Ref. [35–37]. The measured and theoretical spectrum of our sample are displayed in Fig. 6(d), and the simulated curve of the 2° cone angle is also labeled in the figure with a red dashed line. We can see that the actual measured spectrum is in good agreement with the result considering the cone angle. However, it should be noted that the position of the transmission channel in the measured spectrum is only offset by less than 1 nm compared with the theoretical design. This was achieved despite fabrication errors (∼10 nm, 10%) existing in the grating linewidth, where the background and system noise also significantly influenced the results. Overall, these experimental results demonstrate the robustness of the 2D ES-ZCGs GMR filters against fabrication errors, which will play an important role in the practical applications of GMR.
Fig. 6. Experimental results of the 2D ES-ZCGs GMR filters. (a, b) SEM images of the fabrication sample. (c) Spectrum curves simulated by considering the influence of conical light with different cone angles. (d) Measured spectrum curve compared with the theoretical design and simulated spectrum with a 2° cone angle.
5. Conclusion
In summary, the fabrication tolerance of the 2D GMR bandpass filters was investigated and the effect of the etch-stop layer on the fabrication tolerance was clarified. We demonstrated that the 2D ZCG GMR design is highly sensitive to the errors in the grating linewidth, and the waveguide mode will be deeply affected by the local field in the top grating-layer through field overlapping. This is the main reason for the poor tolerance of 2D ZCGs GMR filters toward the grating linewidth. By introducing a SiO2 buffer layer between the top Si:H gratings and beneath the waveguide layer, the influence of grating resonance on the waveguide mode can be effectively suppressed, improving the tolerance of the 2D ES-ZCGs GMR filters toward the grating linewidth by 4-fold. In addition, the buffer layer can also serve as the etch-stop layer to improve the control of the grating height and waveguide layer thickness. The fabricated sample with 10% grating linewidth errors still exhibited good accuracy in the transmission spectrum, proving its robustness and high fabrication tolerance. Our findings provide a deeper understanding of the fabrication tolerance of 2D GMR bandpass filters and offer insights into improving their performance.
Funding
the Shanghai Municipal Science and Technology Major Project and the Fundamental Research Funds for the Central Universities(2021SHZDZX0100); China Postdoctoral Science Foundation (2020TQ0227, 2021M702471); the “Shu Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education (17SG22); Science and Technology Commission of Shanghai Municipality (17JC1400800, 20JC1414600, 21JC1406100); National Natural Science Foundation of China (61621001, 61925504, 6201101335, 62020106009, 62192770, 62192772).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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