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Abstract
An on-chip reconstruction spectrometer provides a powerful approach for miniaturized and portable applications. Owing to the cross-correlation and line shape of broadband responses, the reconstruction spectrometer exhibits limited resolution and a fixed bandwidth. Here we demonstrate a cascaded nanobeam spectrometer with assistance of a reconstruction algorithm. The transmissions of tunable Fano-enhanced nanobeams (FENs) are of good orthogonality and have improved quality factors. We retrieved a narrowband signal of 0.16 nm linewidth and a dual peak at 0.32 nm distance for resolution characterization. This result breaks the limit of the full width at half maximum for narrowband filter spectrometers. An expanded bandwidth is realized by a three-channel cascaded device, and signal spectra in the 16 nm range are reconstructed successfully. Each FEN unit has an ultra-compact footprint of ${18} \times {18}\;\unicode{x00B5}{\rm m}^2$. In view of the improved performance and flexibility, our concept is an ideal candidate for precise and miniaturized applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
Optical spectrum analysis has been reckoned as a notable method in science studies. To identify the intensity of different frequency components in signal light, spectrometers are of great importance in various areas such as medical treatment, chemical industry, and environmental monitoring [1–5]. Bulky systems utilized for conventional spectrum analyzers give access to high resolution and broad bandwidth, but they may hit a bottleneck in view of their physical dimensions, power consumption, and expensive components. Miniaturized spectrometers required in critical and integrated applications have attracted more and more attention [6,7]. Silicon photonics provides compact and low-cost devices for lab-on-a-chip systems, and the silicon-on-insulator (SOI) platform has emerged as one of the most promising technologies in light of its compatibility with complementary metal oxide semiconductor (CMOS) manufacturing technology [8–12]. Therefore, this technological process is expected to be a practical option for the design and fabrication of microspectrometers.
Recent years have witnessed significant demonstrations of spectrometer shrink-down, which thus provides an approach to miniaturized application scenarios [7,13–15]. Reported dispersive-optics spectrometers have used a variety of schemes for the separation of spectral components [16–19]. Similar to conventional grating-dispersion-detection solutions, microspectrometers of that category may bring about a reduction in performance when miniaturized. The main reason is that the resolution is inversely proportional to the light path passing through the dispersive element. Fourier transform (FT) planar spectrometers, which are generally based on Mach–Zehnder interferometers (MZIs), convert interferograms to a spectrum via FT [20–23]. But due to the inverse proportion between the maximum optical path length difference (OPD) and the resolution, high performance FT spectrometers generally require movable components, extra footprints, or high power consumption.
Nowadays, reconstructive spectrometers have received great attention due to the compatibility of diverse spectral responses and the improvement of retrieved spectra [24–29]. Detectors with distinct and well-tailored transmissions sample the incident signal, and the solution of a linear equation system involving pre-calibrated information is optimized to be the reconstructed spectrum. But the diversity, or rather the orthogonality, of all detection units, is a challenging issue that substantially determines the performance of the spectrometer. Broadband filters or detectors can improve the signal-to-noise ratio (SNR) to a certain extent, but it tends to be more difficult to construct spectral responses with sharp features and high contrast at each wavelength point. It has been pointed out that the similarity and smoothness in an individual filter’s spectrum are the main restrictions of spectral resolution [27]. Increasing the number of detectors is not always a guarantee of an obviously better performance. In addition, the pre-read responses are usually inseparable, which thus blocks the scalability of bandwidth. In other words, one may encounter a dilemma in simply reducing the number of filters for a smaller bandwidth and more compact size, or increasing it for a larger bandwidth, with the constancy of resolution. However, the inherent characteristics of narrowband filters can provide a solution to the abovementioned problems. The transmission spectra, especially for tunable resonators, are mutually orthogonal, and all units can operate either individually or collectively to satisfy the requirement for proper bandwidth. To date, the full width at half maximum (FWHM) has been considered as the limit of spectrometers based on narrowband filters [7]. But the applicability of good orthogonality and scalability to reconstruction methods gives us an approach to break this restriction.
Fig. 1. Overview of the cascaded nanobeam spectrometer. (a) Schematic and (b) zoom-in of the device. (c) Scanning electron microscope (SEM) image of a nanobeam unit. (d) Computational reconstruction process. GCV, generalized cross-validation.
In this paper, we propose and demonstrate a cascaded scalable nanobeam spectrometer assisted by computational reconstruction. The device consists of three tunable Fano-enhanced photonic crystal nanobeam cavities (PCNCs), and each unit can work individually as a single spectrometer with a smaller bandwidth but consistent resolution. A Fano-enhanced nanobeam (FEN) module has a compact size of ${18} \times {18}\;\unicode{x00B5}{\rm m}^2$, consisting of a nanobeam of ${18} \times {0.7}\;\unicode{x00B5}{\rm m}^2$ and a partially transmitting element (PTE) of ${0.7} \times {9}\;\unicode{x00B5}{\rm m}^2$. By combining the advantages of both narrowband filters and reconstructive spectrometers, each point on the resonant line shape contributes to the reconstruction of the incident signal, and the device has broken through the restriction of linewidth. Our spectrometer obtains a high resolution of 0.32 nm, and is able to retrieve a narrowband peak of 0.16 nm. The three-channel device reconstructs a signal with 16 nm spectral range, and the bandwidth can be further extended owing to the scalability of PCNC design.
2. RESULTS
A. Reconstruction Principle and Overall Design
The reconstruction method is generally according to the following principles. For the signal to be measured $I(\lambda)$ (where $\lambda$ is wavelength), the detected power of the output ports ${P_m}$ ($m$ refers to different thermal-tuning states) can be reckoned as an integral with respect to the intrinsic response ${F_m}(\lambda)$ and mathematically written as
Figure 1(a) shows the schematic of the cascaded three-channel FEN spectrometer, whose basic unit is illustrated in Fig. 1(b). The input light is guided and coupled into the PCNC structure and repeatedly reflected by the one-dimensional etched holes [shown in Fig. 1(c)] in both ends. The nanobeam cavity concentrates most energy in the center area of the photonic crystal and produces a resonant whispering-gallery mode. With the assistance of PTE located in one port of the drop waveguide [in Fig. 1(b)], the incident light component that satisfies resonant conditions is downloaded to the other port [31]. The finely designed parameters of each PCNC unit ensure unique resonant wavelengths individually, which are 1547.44 nm, 1552.80 nm, and 1558.21 nm. On the other hand, non-resonant light is regarded as the through-put and transmitted to the next unit, which makes the cascade series possible. To guarantee the coverage of operation bandwidth, active tuning is introduced and thermal electrodes [purple bars in Fig. 1(a)] are configured on the cladding for each FEN unit. In Fig. 1(d), the transmission spectra of the three ports in different tuning states are detected and converted to the elements of matrix ${\boldsymbol X}$ in Eq. (2) in advance for pre-calibration. These spectra have decent orthogonality, since most energy is focused in the resonant wavelength and the cross-correlation is effectively suppressed. When the signal is input and downloaded to the output ports, detectors will measure the power, and the vector ${\boldsymbol P}$ can be constructed. By obtaining a good estimate of the coefficient vector $\alpha$ and substituting it into Eq. (S3) in Supplement 1, the unknown incident signal can be reconstructed. Note that the three channels can work both individually for a miniaturized application and cooperatively for a broadband scenario. The cascaded structures are realized by coupling adjacent waveguides, and each FEN operates as an independent unit, which makes the scalability much simpler.
B. Fano-Enhanced Nanobeam Characterization
The proposed cascaded FEN spectrometer consists of three units. In Fig. 2(a), one can see the schematic of a single FEN, which involves 40 mirror-symmetrically etched ellipse holes. We define the number of holes in the right half from one to 20, and each of them is given the main parameters of horizontal axis diameter ${h_x}$, longitudinal axis diameter ${h_y}$, and period $a$. The first seven holes have a tapered structure (for all three parameters) to efficiently confine the light and reduce scattering loss. The photonic bandgap information is shown in Fig. 2(b), where the dashed line referring to the resonant wavelength intersects the energy band of center holes but goes into the bandgap of the holes in higher order. Namely, the light with a resonant wavelength has a Bloch mode allowed by the center holes and will be reflected by the edge holes. The last 13 holes (from eight to 20) share the same parameters, which aims at increasing reflectivity of the photonic crystal cavity and thus improving the quality factor (${\rm Q}$). Unlike microring resonators (MRRs), the nanobeam cavity has a quite large free spectral range (FSR), and there is merely one single peak that can be detected in the required band, which provides convenience for signal detection.
Fig. 2. Fano-enhanced nanobeam (FEN) characterization. (a) Schematic of a FEN unit. (b) Photonic bandgap of tapered etching holes. (c) Transmission with and without Fano enhancement. (d) SEM image of etched holes of three nanobeams in a FEN spectrometer device.
As a four-port waveguide–cavity system, extra coupling loss will be introduced if the output light is not effectively controlled, hence deteriorating the ${\rm Q}$ factor and extinction ratio (ER). Therefore, a PTE structure [shown in Fig. 1(b)] is designed so as to upgrade the performance. Detailed analysis can be found in Supplement 1. The PTE with high reflectivity results in a Fano enhancement of resonance, and we can see from Fig. 2(c) that the Lorentzian line shape is transferred to a Fano line shape with higher ${\rm Q}$ (from 2766 to 4319), higher ER (from ${\sim}{14}$ to ${\gt}{26}\;{\rm dB}$), and lower insertion loss (from 7.2 to 4.7 dB). This improvement contributes to a better performance of the resolution and SNR of the spectrometer.
To expand the bandwidth of the device, FEN units with different resonant wavelengths are arranged. As mentioned above, the property of the cavity depends on the parameters of the etched holes and the way they change. The photonic band information [in Fig. 2(b), for example] enables us to control the parameters in a reliable method. We have revised these parameters and guaranteed all three FENs working in the required wavelengths. The detailed parameters can be found in Supplement 1. The scanning electron microscope (SEM) image of the etched holes in the FENs is shown in Fig. 2(d). It is necessary to point out that the manufacturing deviation is inevitable and may lead to a shift of resonant wavelength. But the calibration process and thermal tuning will handle this issue and increase fabrication tolerance.
C. Thermal Tuning
Due to the decent thermo-optical (TO) effect of silicon, the fabricated FENs are configured with heaters [shown in Fig. 3(a)] to control the resonant states and shift the transmission spectra. The resonant wavelength ${\lambda _r}$ of the nanobeam can be approximately expressed as ${\lambda _r} = {2}{n_{{\rm eff}}}{\Lambda _{{\rm eff}}}$, where ${\Lambda _{{\rm eff}}}$ denotes the equivalent period of the photonic crystal, and the effective refractive index ${n_{{\rm eff}}}$ depends on the material property and structure parameters. Assuming that the temperature variation hardly changes the mode field distribution, we can consider that ${n_{{\rm eff}}}$ changes linearly with the heating power and so does ${\lambda _r}$. Note that the PTE adopted in one of the drop ports requires equal heating power to keep the Fano enhancement constant. As a result, we configured two identical heaters in parallel connection for the nanobeam and PTE [shown in the partial view of Fig. 3(a)].
Fig. 3. Schematic of thermal-tuning method. (a) Optical micrography of the device. (b) Relation between resonance wavelength and heating power. The error bars denote S.D. (c) Normalized spectral transmission of FEN spectrometer with different heating powers ${P_h}$.
We tested the fabricated device, and the relation between resonant wavelength and heating power is plotted in Fig. 3(b) (for one of the units). This proportional relationship indicates that a lager bandwidth can be achieved simply by increasing the heating power. Moreover, for a fixed bandwidth requirement, this approach may reduce the number of units involved, and thus significantly reduce the device footprint. But with higher power applied, the line shape of the transmission spectrum begins to deteriorate seriously, which requires another unit to cover a higher band. For a 20% tolerance of ${\rm Q}$ factor, the maximum wavelength tuning range is commonly ${\sim}{9}\;{\rm nm}$ in our chips. The calibration process begins as a set of voltages are imposed, and the corresponding transmission spectra [shown in Fig. 3(c)] are pre-stored as well as the heating powers ${P_h}$. The dotted lines, solid lines, and dashed lines correspond to nanobeams with numbers 1, 2, and 3 in Fig. 3(a), respectively. As the signal is incident from the bus waveguide, the same set of ${P_h}$ is imposed, and the corresponding received powers ${P_m}$ are measured for the following calculation.
D. Reconstruction Results
To characterize the performance of the FEN spectrometer, various kinds of incident signals are input and reconstructed by the approach mentioned above. A broadband amplified spontaneous emission (ASE) source is utilized to supply incident light. After being exported by a single mode fiber (SMF), broadband light is sent into a programmable optical filter (Finisar Waveshaper 1000s), which is capable of generating the required signal. Fiber-to-chip coupling is realized by a pair of grating couplers, and the output light is finally measured by an optical spectrum analyzer (OSA, Yokogawa AQ6370C) for calibration or detected by an optical power meter for reconstruction. The filter is set in a full-pass state when calibrating. A figure of the experimental setup can be found in Supplement 1. As a reference, the output of the filter is directly measured by the commercial OSA as well. All of the results are illustrated in Fig. 4.
Fig. 4. Reconstruction results. (a) 0.16 nm FWHM single peak. (b) Duel peak with 0.32 nm distance. (c) Two different peaks. (d) Multiple peaks. (e), (f) Broadband incident light with a bandwidth of 16 nm.
Table 1. Comparison of Our Work with a Range of Microspectrometers
For a single peak containing narrowband components in Fig. 4(a), our demonstration realizes the reconstruction of 0.16 nm linewidth. To characterize the resolution, a dual peak in Fig. 4(b) with a distance of 0.32 nm is input and retrieved. We also reconstructed an asymmetric dual peak, multi-peaks, and two broadband signals, as shown in Figs. 4(c)–4(f). The FEN spectrometer exhibits great performance and good scalability in these cases. Actually, this demonstration is beyond the FWHM limit of transmission responses. An experiment (in Supplement 1) is conducted to prove this point, in which the device is utilized as a conventional narrowband filter spectrometer.
One may notice that some noisy sidebands occur especially in narrowband reconstructions. In fact, if we attempt to construct signals much narrower than that in Fig. 4(a), these sidebands will become unacceptably larger and overwhelm the signal, which is the main restriction of the proposal. Another issue is the reconstruction errors in the broadband results, because the mechanical instability of fiber–chip coupling leads to a mismatch of detected power among the three units. A well-packaged chip with fiber arrays will improve the deviation to some extent.
3. DISCUSSION
As a kind of structure-engineering design, PCNC has the advantages of being ultra-compact, scalable, and transplantable. It has been proved that by simply scaling up or down the size of the prototype without extra design [32], the resonant wavelength of a PCNC can be increased or decreased in equal multiples. Therefore, our spectrometer proposal has the potential to be generally transplanted to other operating bands or fabrication platforms such as ${{\rm Si}_3}{{\rm N}_4}$ [33]. Although we have adopted the Gaussian function to reduce the size of the matrix in the reconstruction algorithm, the calculation will still become complicated when multiple FEN units are involved. A solution is to segment the wave band into small sub-bands corresponding to one or a few nanobeam units.
We compare our current device with respect to other spectrometers in Table 1. The proposed system is an attractive solution for spectral measurements with its high resolution, scalable bandwidth, and compact size.
In conclusion, we propose and experimentally demonstrate a high performance and scalable cascaded nanobeam spectrometer. The single-peak response of narrowband filters has decent orthogonality and is quite suitable for reconstruction spectrometers. With the assistance of Fano enhancement, our device obtains a high resolution of 0.32 nm and can retrieve a narrow peak with 0.16 nm linewidth, breaking the FWHM limit of narrowband filters. The FEN unit has an ultra-compact size of ${18} \times {18}\;\unicode{x00B5}{\rm m}^2$ and can work individually or cooperatively. The scalable multiplexing is simple to realize, and we have demonstrated a three-channel device that is capable of reconstructing a signal of 16 nm bandwidth. Our proposal has potential for miniaturized spectrometer applications in integrated and portable systems.
Funding
Innovation Project of Optics Valley Laboratory (OVL2021BG001); National Natural Science Foundation of China (61805090, 62075075).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
REFERENCES
1. F. Martinelli, R. Scalenghe, S. Davino, S. Panno, G. Scuderi, P. Ruisi, P. Villa, D. Stroppiana, M. Boschetti, L. R. Goulart, C. E. Davis, and A. M. Dandekar, “Advanced methods of plant disease detection. A review,” Agron. Sustain. Dev. 35, 1–25 (2015). [CrossRef]
2. Z. K. Li, X. Y. Wei, H. L. Yan, and Z. M. Zong, “Insight into the structural features of Zhaotong lignite using multiple techniques,” Fuel 153,176–182 (2015). [CrossRef]
3. P. R. Mahaffy, C. R. Webster, M. Cabane, et al., “The sample analysis at mars investigation and instrument suite,” Space Sci. Rev. 170, 401–478 (2012). [CrossRef]
4. J. Kim, U. Jeong, M. H. Ahn, et al., “New era of air quality monitoring from space: geostationary environment monitoring spectrometer (GEMS),” Bull. Am. Meteorol. Soc. 101(1), E1–E22 (2020). [CrossRef]
5. Y. W. Chen and J. L. Wang, “Preparation and characterization of magnetic chitosan nanoparticles and its application for Cu(II) removal,” Chem. Eng. J. 168, 286–292 (2011). [CrossRef]
6. H. J. Butler, L. Ashton, B. Bird, G. Cinque, K. Curtis, J. Dorney, K. Esmonde-White, N. J. Fullwood, B. Gardner, P. L. Martin-Hirsch, M. J. Walsh, M. R. McAinsh, N. Stone, and F. L. Martin, “Using Raman spectroscopy to characterize biological materials,” Nat. Protoc. 11, 664–687 (2016). [CrossRef]
7. Z. Y. Yang, T. Albrow-Owen, W. W. Cai, and T. Hasan, “Miniaturization of optical spectrometers,” Science 371, eabe0722 (2021). [CrossRef]
8. X. T. He, E. T. Liang, J. J. Yuan, H. Y. Qiu, X. D. Chen, F. L. Zhao, and J. W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10, 872 (2019). [CrossRef]
9. D. X. Xu, J. H. Schmid, G. T. Reed, G. Z. Mashanovich, D. J. Thomson, M. Nedeljkovic, X. Chen, D. Van Thourhout, S. Keyvaninia, and S. K. Selvaraja, “Silicon photonic integration platform—have we found the sweet spot?” IEEE J. Sel. Top. Quantum Electron. 20, 9700501 (2014). [CrossRef]
10. H. T. Lin, Z. Q. Luo, T. Gu, L. C. Kimerling, K. Wada, A. Agarwal, and J. J. Hu, “Mid-infrared integrated photonics on silicon: a perspective,” Nanophotonics 7, 393–420 (2018). [CrossRef]
11. A. Rahim, E. Ryckeboer, A. Z. Subramanian, et al., “Expanding the silicon photonics portfolio with silicon nitride photonic integrated circuits,” J. Lightwave Technol. 35, 639–649 (2017). [CrossRef]
12. T. Hu, B. W. Dong, X. S. Luo, T. Y. Liow, J. F. Song, C. Lee, and G. Q. Lo, “Silicon photonic platforms for mid-infrared applications [Invited],” Photon. Res. 5, 417–430 (2017). [CrossRef]
13. S. Yuan, D. Naveh, K. Watanabe, T. Taniguchi, and F. Xia, “A wavelength-scale black phosphorus spectrometer,” Nat. Photonics 15, 601–607 (2021). [CrossRef]
14. Z. Zhang, Y. Li, Y. Wang, Z. Yu, X. Sun, and H. K. Tsang, “Compact high resolution speckle spectrometer by using linear coherent integrated network on silicon nitride platform at 776 nm,” Laser Photon. Rev. 15, 2100039 (2021). [CrossRef]
15. H. Li, L. Bian, K. Gu, H. Fu, G. Yang, H. Zhong, and J. Zhang, “A near-infrared miniature quantum dot spectrometer,” Adv. Opt. Mater. 9, 2100376 (2021). [CrossRef]
16. G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light Sci. Appl. 3, e203 (2014). [CrossRef]
17. K. Ma, K. Chen, N. Zhu, L. Liu, and S. He, “High-resolution compact on-chip spectrometer based on an echelle grating with densely packed waveguide array,” IEEE Photon. J. 11, 4900107 (2019). [CrossRef]
18. E. Ryckeboer, A. Gassenq, M. Muneeb, N. Hattasan, S. Pathak, L. Cerutti, J. B. Rodriguez, E. Tournié, W. Bogaerts, R. Baets, and G. Roelkens, “Silicon-on-insulator spectrometers with integrated GaInAsSb photodiodes for wide-band spectroscopy from 1510 to 2300 nm,” Opt. Express 21, 6101–6108 (2013). [CrossRef]
19. P. Cheben, J. H. Schmid, A. Delâge, A. Densmore, S. Janz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D. X. Xu, “A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with sub-micrometer aperture waveguides,” Opt. Express 15, 2299–2306 (2007). [CrossRef]
20. S. N. Zheng, J. Zou, H. Cai, J. F. Song, L. K. Chin, P. Y. Liu, Z. P. Lin, D. L. Kwong, and A. Q. Liu, “Microring resonator-assisted Fourier transform spectrometer with enhanced resolution and large bandwidth in single chip solution,” Nat. Commun. 10, 2349 (2019). [CrossRef]
21. L. Zhang, J. Chen, C. Ma, W. Li, Z. Qi, and N. Xue, “Research progress on on-chip Fourier transform spectrometer,” Laser Photon. Rev. 15, 2100016 (2021). [CrossRef]
22. D. M. Kita, B. Miranda, D. Favela, D. Bono, J. Michon, H. Lin, T. Gu, and J. Hu, “High-performance and scalable on-chip digital Fourier transform spectroscopy,” Nat. Commun. 9, 4405 (2018). [CrossRef]
23. M. C. M. M. Souza, A. Grieco, N. C. Frateschi, and Y. Fainman, “Fourier transform spectrometer on silicon with thermo-optic non-linearity and dispersion correction,” Nat. Commun. 9, 665 (2018). [CrossRef]
24. Z. Yang, T. Albrow-Owen, H. Cui, J. Alexander-Webber, F. Gu, X. Wang, T.-C. Wu, M. Zhuge, C. Williams, P. Wang, V. Zayats Anatoly, W. Cai, L. Dai, S. Hofmann, M. Overend, L. Tong, Q. Yang, Z. Sun, and T. Hasan, “Single-nanowire spectrometers,” Science 365, 1017–1020 (2019). [CrossRef]
25. Z. Wang, S. Yi, A. Chen, M. Zhou, T. S. Luk, A. James, J. Nogan, W. Ross, G. Joe, A. Shahsafi, K. X. Wang, M. A. Kats, and Z. Yu, “Single-shot on-chip spectral sensors based on photonic crystal slabs,” Nat. Commun. 10, 1020 (2019). [CrossRef]
26. B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7, 746–751 (2013). [CrossRef]
27. A. Li and Y. Fainman, “On-chip spectrometers using stratified waveguide filters,” Nat. Commun. 12, 2704 (2021). [CrossRef]
28. W. Hadibrata, H. Noh, H. Wei, S. Krishnaswamy, and K. Aydin, “Compact, high-resolution inverse-designed on-chip spectrometer based on tailored disorder modes,” Laser Photon. Rev. 15, 2000556 (2021). [CrossRef]
29. J. Bao and M. G. Bawendi, “A colloidal quantum dot spectrometer,” Nature 523, 67–70 (2015). [CrossRef]
30. G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21,215–223 (1979). [CrossRef]
31. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]
32. Z. Cheng, Y. Zhao, J. Zhang, H. Zhou, D. Gao, J. Dong, and X. Zhang, “Generalized modular spectrometers combining a compact nanobeam microcavity and computational reconstruction,” ACS Photon. 9, 74–81 (2021). [CrossRef]
33. M. Khan, T. Babinec, M. W. McCutcheon, P. Deotare, and M. Loncar, “Fabrication and characterization of high-quality-factor silicon nitride nanobeam cavities,” Opt. Lett. 36, 421–423 (2011). [CrossRef]