Optical spectrometers are ubiquitous tools for spectrum analysis and are widely used in many fields from fundamental scientific research to industrial technology development [1–3]. Conventional spectrometers based on free-space optics are usually bulky, expensive, and power-hungry benchtop instruments. Emerging applications such as airspace detection [4], precision agriculture [5], and lab-on-a-chip systems [6] have set high requirements on the compactness, economic efficiency, and in-situ real-time characterization capability, which promotes the development of chip-scale spectrometers.

To date, most on-chip spectrometers rely on miniaturized dispersive optics [7–9], narrowband filters [10–12], and Fourier transform (FT) interferometers [13–17]. On-chip spectrometers based on dispersive elements and narrowband filters provide a high resolution by spectral channel division. However, the multichannel architecture not only degrades the signal-to-noise ratio (SNR) but also requires a corresponding detector array [1]. On the contrary, FT spectrometers center around the use of an interferometer to modulate the incident light on a single detector over time, thus offering two inherent advantages, known as the multiplexing advantage (Fellgett’s advantage) and high optical throughput (Jacquinot’s advantage) [16,18]. Both advantages tend toward affording a high SNR. Additionally, using a single detector offers a simpler, smaller, and more cost-effective alternative to a detector array. Moreover, the past few years have witnessed advances in employing new computational techniques such as compressive sensing [19], machine learning [20], and forward-backward linear prediction [21] to enhance the spectral resolution and robustness of FT spectrometers. The advancement of computational techniques has also yielded a new paradigm of spectrometers named computational spectrometers, in which computational techniques are used to approximate or reconstruct an incident light spectrum from precalibrated spectral response information encoded spatially within a detector array [22–25] or temporally within a single detector [26–28].

In FT spectrometers, the interferometer needs to be tunable to obtain a temporal interferogram. Compared with conventional thermo-optic [14] and electro-optic [3] tunings that rely on the weak perturbation of the material refractive index, microelectromechanical systems (MEMS) effectively induce modulation by spatially displacing photonic components, consequently improving the tuning efficiency and the spectrometer performance [18]. Among a variety of MEMS actuation mechanisms, electrostatic actuation stands out due to the ease of integration and ultra-low power consumption [29]. Through the local removal of the buried oxide (BOX) layer in the common silicon-on-insulator (SOI) platform, a parallel plate capacitor structure is naturally formed between the silicon (Si) device layer and the Si substrate, enabling out-of-plane electrostatic actuation of the photonic components [30,31]. The directional coupler is a type of interferometer that can be conveniently integrated with and effectively modulated by electrostatic MEMS actuators [32]. Nonetheless, the spectrometer scheme based on an electrostatic out-of-plane modulated directional coupler still suffers from two prominent limitations. First, the spectral resolution is positively (negatively) correlated with the coupling strength (coupling gap) of the directional coupler. Reducing the coupling gap is the most effective way to improve the spectral resolution within a compact device footprint. However, the achievable minimum coupling gap is restricted by the lithography resolution [33]. Second, a large enough modulation depth (and thus tuning range of the coupling gap) is required to ensure the effectiveness of spectrum reconstruction, which further increases with the wavelength [26,34]. However, the out-of-plane electrostatic actuation displacement is limited by the buried oxide (BOX) layer thickness and the pull-in effect [35,36]. Consequently, a non-standard and low-yield flip-chip bonding process is needed to enable the large out-of-plane displacement required for long wavelengths, e.g., in the mid-infrared (MIR) [37].

In this work, we report a new scheme for FT spectrometers using a reconfigurable waveguide coupler actuated by a comb-drive actuator array. The comb-drive actuator can achieve in-plane displacement to break through the lithography resolution limitation of the coupling gap and improve the spectral resolution. At the same time, the proposed scheme inherits the high SNR and single detector advantages of FT spectrometers, as well as the algorithm advantages of computational spectrometers. The demonstrated spectrometer achieves a high resolution of 0.2 nm for dual spectral lines over a large bandwidth of 100 nm (1.5–1.6 μm) within a compact footprint of 75μm×1000μm. Besides, the in-plane actuation circumvents the BOX layer thickness limitation of displacement. Thus, the spectrometer can be easily transplanted to other operation bands by simply scaling the structural parameters. As a proof-of-concept, an MIR spectrometer is further demonstrated with a dual-line reconstruction resolution of 1.5 nm and a bandwidth of 300 nm (4–4.3 μm). Leveraging high-yield standard CMOS fabrication processes, our high-performance and generalized spectrometer shows promising potential to be mass-produced for widespread applications across different spectral bands.

Figure 1(a) illustrates the architecture and working principle of our proposed spectrometer built on a standard SOI platform. Similar to traditional FT spectrometers, our spectrometer centers around time-domain modulation of the optical path difference (OPD) to generate interferograms. In the proposed concept, the OPD modulation is achieved by tuning the coupling strength between the two waveguides of the directional coupler. When the two waveguides are brought into close proximity and couple to each other, two distinct supermodes will be formed, namely, symmetric mode (SM0) and asymmetric mode (SM1). Because of the effective index difference Δn between SM0 and SM1, an OPD Δx is obtained after these two modes propagate through a certain coupling length L, with Δx=LΔn. Consequently, interferograms can be generated at the output port by time-domain tuning of Δn. The power observed at the output port can be described as Poutput(λ,Δn)=B(λ)cos2(πLΔnλ);(1)thus, the input spectrum B(λ) can be retrieved by performing FT of the interferogram Poutput. λ is the incident wavelength. The spectral resolution is given following the Rayleigh criterion [20] δλ=λ2Δxmax=λ2LΔnmax,(2)where Δnmax is the maximum Δn during the whole tuning process, corresponding to the strongest coupling, i.e., the minimum coupling gap between the two waveguides [33]. Therefore, the key to spectral resolution improvement within a certain device footprint is to reduce the minimum coupling gap. For the out-of-plane reconfiguration scheme, the minimum coupling gap is determined at the unactuated state, which is restricted by the lithography resolution. Conversely, in our in-plane reconfiguration scheme, the minimum coupling gap can be decreased to a value significantly below the lithography resolution by the in-plane displacement of the movable waveguide toward the fixed one, as illustrated in Fig. 1(b). The variable Δn can be approximated by a polynomial function: $$\begin{gathered} Δn(λ,g)≈f1(λ)f2(g) \\ =(a1+a2λ)(b1+b2g+b3g2+b4g3+b5g4+b6g5), \end{gathered}$$(3)where g is the coupling gap. The polynomial approximation can be fitted with a 99% R-squared value (see Appendix A). It is clearly seen that Δn increases with decreasing g, and increases drastically at sub-100 nm g beyond the resolution of the deep ultra-violet (DUV) photolithography used in common silicon photonics foundries, leading to potentially much finer spectral resolution than what can be achieved in the out-of-plane reconfiguration scheme.

Appendix A also indicates that the tuning range of g increases with the operation wavelength for the same spanning of the Δn change in order to maintain the effectiveness of spectrum reconstruction. In the out-of-plane reconfiguration scheme, the tuning is enabled by the selective removal of the BOX layer, and the tuning range is limited by its thickness. The accrual of the BOX layer thickness is impeded by the accumulated stress [36]. Moreover, the existence of the pull-in effect further circumscribes the effective range of the out-of-plane modulation. Specifically, given a BOX layer thickness of 3 μm, the effective tuning displacement counting from the plane of the Si device layer can be estimated to be up to 1.38 μm [35]. Differently, in our in-plane reconfiguration scheme, the in-plane tuning displacement achieved by the comb-drive actuator is free from the above-mentioned limitations. As depicted in Fig. 1(a), the comb-drive actuator consists of a set of movable fingers supported by springs as well as a set of fixed comb fingers [38]. The movable waveguide of the directional coupler is connected to the movable comb fingers through a movable shuttle. The fixed and movable comb fingers collectively form a parallel plate capacitor structure [39]. When a bias voltage is applied to the fixed comb fingers, a potential difference arises in the structure, leading to the displacement of the movable comb fingers in the plane driven by the electrostatic force. The comb-drive actuator achieves a stable state when the spring recovery force balances the electrostatic force [40]. Thus, the in-plane displacement is given by $$\begin{gathered} y=g0−g=12kdCdxV2 \\ ≃η2kNε0tdV2, \end{gathered}$$(4)where g0 denotes the un-actuated coupling gap, k is the spring constant, η describes the electro-mechanical coupling factor for the given geometry, N is the number of fingers, ε0 is the permittivity of vacuum, t is the device layer thickness, and d is the transverse distance between the movable and fixed fingers. By tailoring the design parameters of the comb-drive actuator (k, η, N, d) and the bias voltage V, the in-plane displacement can be adjusted to provide the required tuning range for arbitrary wavelength, without the limitations faced by the out-of-plane modulation, as illustrated in Fig. 1(c).

In the operation of our spectrometer, we can apply a time-variant bias voltage on the comb-drive actuator to adjust the in-plane coupling gap between the waveguides, thus realizing the time-domain modulation of Δn. Combining Eqs. (1), (3), and (4), the output interferogram can be given as Poutput(λ,V)=B(λ)cos2[πL·f1(λ)f3(V)λ].(5)

Considering the fact that functions f1 and f3 are difficult to be precisely determined experimentally, we use a regularized regression model for computational spectrum reconstruction instead of the conventional FT, as illustrated in Fig. 1(a).

The proposed spectrometer is fabricated on an SOI wafer with a 220 nm thick device layer and a 3 μm thick BOX layer. The device has a compact footprint of 75 μm in width and 1000 μm in length, the latter depending on the coupling length of the directional coupler. Figure 2(a) illustrates the overall structure of the spectrometer, featuring a directional coupler integrated with a comb-drive actuator array. Figure 2(b) details the structure of a single comb-drive actuator, wherein the comb fingers measure 0.25 μm in width and 4 μm in length. Upon the application of bias voltage between the fixed and movable fingers, the movable waveguide is pushed toward the fixed waveguide, reducing the coupling gap and thereby enhancing the modal coupling strength. Figure 2(c) presents an isometric view of the device. Both waveguides of the directional coupler are designed with a width of 350 nm, which not only ensures single transverse-electric (TE) mode propagation but also achieves a balance between coupling strength and propagation loss. The initial coupling gap is set at 800 nm to ensure that the two waveguides are fully decoupled. Therefore, all the structures can be patterned by DUV photolithography. The movable parts (including comb fingers, shuttle, and waveguide) are defined by locally removing the BOX layer by hydrogen fluoride. During the reconstruction of unknown spectra, the reconstruction performance is improved by increasing the OPD between SM0 and SM1 propagating through the directional coupler. The OPD can be amplified by increasing the Δn between these two supermodes, which, as discussed above, is effectively achieved by reducing the coupling gap.

Figure 3 illustrates the change of the coupling gap under different bias voltages applied to the comb-drive actuator array, which is obtained by finite element method (FEM) simulation using COMSOL Multiphysics. It is seen that the coupling gap is uniformly modulated along the whole coupling length of 1000 μm. The insets of Fig. 3(a) provide zoom-in views of the coupling gap at the second actuator along the waveguide propagation direction. The tiny fluctuations below 0.1 nm induce an ignorable change of Δn and thus the interferogram. This characteristic ensures the stability of spectral reconstruction. Figure 3(b) plots the coupling gap as a function of the applied bias voltage. The coupling gap is reduced with increasing applied voltage and finally reaches 22 nm at an applied voltage of 46 V. The narrow coupling gap of 22 nm in the final state goes far beyond the resolution limitation of the DUV photolithography in common silicon photonics foundries and results in strong coupling between the two waveguides. The mode profiles shown in the insets of Fig. 3(b) confirm the fully decoupled and strongly coupled conditions in the initial and final states, respectively. With such a transition, Δn drastically increases from near 0 to 0.5. Therefore, a sufficient modulation range of OPD is obtained through applying time-variant bias voltage from 0 to 46 V. Due to the small displacement of the actuator at low voltage levels as illustrated in Fig. 3(b), which results in a limited optical response, the voltage interval is not uniformly selected. Instead, a higher voltage interval is chosen at lower bias voltages. The bias voltage is incrementally raised from 0 to 20 V with a 2 V interval, followed by intervals of 1 V between 20 and 36 V, 0.5 V between 36 and 44 V, and finally 0.2 V between 44 and 46 V, which counts 53 steps in total.

The inherent imperfections of the device during the fabrication process can be effectively compensated for through spectral reconstruction using computational algorithms. Simultaneously, this approach simplifies the experimental procedure. The first step for achieving spectral reconstruction is to calibrate the spectrometer. To initiate calibration, a gradual increase in bias voltage is applied to the comb-drive actuator. Concurrently, the incident wavelength varies within the range of 1.5 to 1.6 μm, in a step of 0.1 nm. The output intensity from the through port of the directional coupler is measured at each combination of bias voltage and wavelength, thus forming a matrix M (m×n), where m=53 corresponds to the applied bias voltage steps and n=1001 represents the swept wavelength points. As the laser intensity, coupler efficiency, and detector responsivity vary at different wavelengths, the matrix M needs to be normalized to cancel out these wavelength-dependent testing system features. The swept incident spectrum passing through the reference waveguide on the same chip is measured as a vector P=[p1,p2,…,pn]T with n elements. The calibrated measurement matrix A (m×n) is given as A=M·diag(1p1,…,1pn),(6)where diag represents the diagonal matrix form. This calibration matrix A represents the optical response of the directional coupler structure to different incident spectra under varying applied bias voltages, as illustrated in Fig. 4(a). The high quality of the measured calibration matrix A confirms that the coupling gap gradually reduces to an ultra-small width under applied bias voltage and the two waveguides have not stuck together, while the fluctuation along the waveguide remains tiny. Following the completion of the calibration matrix construction, the incident spectrum can subsequently be reconstructed using the established reconstruction algorithm. For any incident spectrum S, the interferogram I measured by the detector is expressed as A·S=I,(7)where I is a column vector with m elements, and each element corresponds to the interference intensity at a specific bias voltage state. S is a column vector with n elements. For applied bias voltages ranging from V0 to Vm, the m integral equations can be discretized and subsequently decomposed into a matrix equation: (AV1,λ1AV1,λ2⋯AV1,λnAV2,λ1AV2,λ2⋯AV2,λn⋮⋮⋮AVm,λ1AVm,λ2⋯AVm,λn)(Sλ1Sλ2⋮Sλn)=(IV1IV2⋮IVm).(8)

This equation is underdetermined, typically resulting in an infinite number of solutions. To address this issue, a regularized regression method named lasso is employed to acquire the approximate solution and reconstruct the incident spectrum. The lasso method applies a weight α1 for the L1-norm of the spectrum to solve the regularized regression problem minS{||I−AS||22+α1||S||1}.(9)

A grid search scheme is implemented to obtain an appropriate hyperparameter α1 for the spectrometer. A set of 101 interferograms is selected across the whole operation waveband, which is then utilized for reconstruction to obtain the corresponding incident spectrum ST with different α1. The hyperparameter α1 is determined to be 3.039×10−4, corresponding to minimum total mean squared error (MSE) between ST and the reference incident spectrum SR. The determined hyperparameter α1 is consistently applied to the following spectrum reconstruction, which simplifies the reconstruction procedure and achieves satisfying reconstruction performance for different input spectra.

In this study, the scikit-learn is utilized to implement the lasso method in Python. This approach proves effective in preventing overfitting and ensuring stable and non-divergent results. Several measured interferograms at different wavelengths are shown in Fig. 4(b), which are further reconstructed to demonstrate the performance of single-wavelength spectrum reconstruction. As depicted in Fig. 4(c), the input laser wavelengths can be precisely reconstructed with ±0.1nm accuracy over the entire 100 nm bandwidth.

In the assessment of the double-wavelength reconstruction performance of our spectrometer, a double-wavelength input in the near-infrared (NIR) range is achieved by two tunable light sources operating concurrently. By modulating the bias applied to the comb-drive actuator, an interferogram is acquired. The incident spectrum can subsequently be reconstructed from the interferogram using a designated reconstruction algorithm. The wavelength spacing (Δλ) of the incident spectrum is gradually reduced, and the smallest wavelength spacing that can be resolved is regarded as the resolution of the spectrometer. The wavelength spacing decreases from 80 to 0.1 nm, as shown in Fig. 5(a), evaluating both the bandwidth and resolution of the spectrometer. As depicted in Fig. 5(b), the incident spectrum can hardly be reconstructed at a wavelength spacing of 0.1 nm, while in Fig. 5(c), the incident spectrum with a wavelength spacing of 0.2 nm is successfully retrieved. The intrinsic resolution at the Rayleigh criterion is estimated to be 5.9 nm (see Appendix B). The demonstrated resolution of 0.2 nm for double-wavelength reconstruction significantly outperforms the Rayleigh criterion, which is mainly attributed to the use of the computational method and the high sparsity assumption we set for spectrum reconstruction [33]. It is worth noting that although the use of only L1-norm in the reconstruction procedure limits the available features to sparsity, this reconstruction method can be transformed to a more universal one, capable of resolving smooth spectra, by introducing L2-norm [27] or using discrete cosine transform (DCT) [25]. The synergy between L1-norm and L2-norm supports a wider range of solvable features and less stringent sparsity a priori. DCT of a smooth spectrum gives a limited number of low-frequency components, resulting in a convex problem that can be solved by standard numerical methods with L1-norm as a penalty.

A comprehensive comparison of reported NIR on-chip FT spectrometers is given in Fig. 6 [14,15,20,41–48]. For spectral resolution and all the related parameters, we plot values corresponding to both Rayleigh resolution and enhanced resolution for all the compared works. In reported works, resolution has been improved through the utilization of computational methods and/or enhancements in structural designs. The bandwidth-to-resolution ratio (BRR) is defined as the figure of merit to evaluate the overall performance of the spectrometer, considering the pursuit of larger bandwidth and finer resolution. Figure 6(a) shows that our device possesses a commendable but not optimal BRR, if compared with some of the reported works [44,47]. However, when taking into account the power consumption, our device operates at more than three orders of magnitude lower power compared to reported FT spectrometers (see Appendix C). The spectral resolution can be improved by enlarging OPD. A larger OPD can be achieved by increasing Δn or L, which leads to larger power consumption or footprint, respectively. Therefore, it is meaningful to evaluate the performance per unit power and unit footprint by defining the resolution-power product (RPP) and resolution-footprint product (RFP) in Figs. 6(b) and 6(c), respectively. Thanks to the high tuning efficiency of the comb-drive actuator, our device features significantly lower power consumption and smaller footprint, resulting in outstanding RPP and RFP.

The tunable directional coupler based on the integration with the comb-drive in-plane actuator exhibits a sufficiently large modulation range, offering the potential to extend the proposed scheme to the MIR and even longer wavelength bands. As a proof-of-concept, we migrate the scheme to a 500 nm SOI platform to realize an MIR spectrometer design. The structural parameters of the NIR and MIR spectrometers are summarized in Table 1.Table 1.

Structural Parameters of NIR and MIR Spectrometers

Similar to the NIR double-wavelength reconstruction, two tunable lasers with simultaneous incidence are used to generate the double-wavelength incidence spectrum in the MIR. The resolution of the spectrometer is systematically investigated by gradually diminishing the wavelength spacing from 260 to 1 nm, as shown in Fig. 7(a). As can be seen in Figs. 7(b) and 7(c), the incident double-wavelength spectrum with wavelength spacing of 1.5 nm can be reconstructed, which becomes challenging when the wavelength spacing is further narrowed to 1 nm. Our MIR spectrometer demonstrates a double-wavelength reconstruction resolution of 1.5 nm and a bandwidth of 300 nm. Compared to the reported MIR spectrometer utilizing an out-of-plane MEMS actuator [33], our device achieves a twofold better resolution with nearly one-fourth of the waveguide length. The improved performance of the spectrometer is facilitated by the increased OPD at the strong coupling state achieved by the in-plane actuator beyond the limitation of lithography accuracy. We also investigate the relationship between the reconstruction resolution and the tuning range. Calibrated measurement matrices under different in-plane tuning ranges are obtained and employed for the same double-wavelength reconstruction to evaluate the spectral resolution. The results are shown in Fig. 7(d). The spectral resolution is observed to be improved incrementally with the widened tuning range. A tuning range of ∼1.6μm (with the ∼20nm coupling gap in the final state included) is necessary to achieve a spectral resolution of 1.5 nm, which can be easily achieved by our proposed in-plane comb-drive actuator. However, as mentioned above, this tuning range is out of reach for out-of-plane tuning, necessitating a non-standard and low-yield flip-chip bonding process.

This work presents a generalized FT spectrometer implemented by a MEMS-enabled tunable interferometer on the common SOI platform. The in-plane comb-drive actuator is utilized to bring the waveguide into close proximity, thus surpassing the limitations of the photolithography resolution in conventional silicon photonics foundries. By reducing the coupling gap, the OPD is increased, thereby enhancing the performance of the spectrometer. Leveraging this high-precision calibration matrix, two spectral spikes separated by 0.2 nm are successfully resolved in the NIR wavelength range of 1.5–1.6 μm. Meanwhile, the in-plane actuation also provides a substantially large modulation range, enabling convenient migration of the scheme to longer wavelength bands without changing device topology and fabrication process. The generality of the proposed spectrometer is demonstrated, exemplified in the MIR waveband of 4–4.3 μm. Through simply scaling the structural parameters, an MIR spectrometer is constructed, achieving a dual-spike reconstruction resolution of 1.5 nm within the bandwidth of 300 nm. Our proposed spectrometer exhibits significant potential as a universal and powerful on-chip spectrometer scheme applicable in various spectroscopic applications such as portable chemical characterization, on-site medical diagnosis, and personalized health care.