Spectrometers play a crucial role in various domains, including biomedicine, environmental monitoring, material characterization, and the chemical industry [1–5]. The increasing demand for applications such as wearable health monitoring, portable sensing, and hyperspectral imaging has driven the miniaturization of spectrometers [6–8]. In recent years, several strategies have emerged for miniaturized spectrometers. Dispersive spectrometers commonly use array waveguide grating (AWG) and echelle diffraction grating (EDG) to split the optical path, but combining high resolution with compactness remains challenging [9–14]. Narrowband filters are also developed into spectrometers by segmenting input optical bands, but the limited free spectral range (FSR) constrains the channel capacity [15,16]. Fourier transform spectrometers leverage Fellgett’s advantage and Jacquinot’s advantage to achieve a high signal-to-noise ratio (SNR) [17–19]. However, they also exhibit drawbacks such as high power consumption and centimeter-scale size.

Reconstructive spectrometers (RSs) represent an emerging paradigm designed to reconstruct unknown spectra through the utilization of software algorithms and pre-calibrated spectral information [19–22]. Within this category, random speckle spectrometers (RSSs) stand out for their remarkable performance of high resolution and compact size. RSSs usually leverage multiple scattering or interference of light in disordered systems to generate random speckles [23–29]. Due to the sensitivity of the speckle pattern to the system structure, minor wavelength shifts can induce substantial changes in the pattern, rendering different patterns as exclusive “fingerprints” of distinct spectral information. In this manner, RSSs can achieve a higher resolution while maintaining a smaller size by designing an appropriate structure. However, existing RSSs structures exhibit limitations in the number of sampling channels and spectral decorrelation, thereby constraining resolution improvement to the picometer level.

In this study, we demonstrate an integrated spatial-temporal random speckle spectrometer (STRSS) that utilizes only several Mach–Zehnder interferometers (MZIs) and a planar waveguide with a pixel array as its core structure. Under the effect of random tuning, random interference and random diffraction occur in the MZI array and the tunable diffraction network, respectively. Consequently, a series of spatial-temporal speckles is generated at the output ports. Our spectrometer can generate more sampling channels than the number of physical channels, concurrently achieving high decorrelation for enhanced resolution. In the experiments, we achieve the precise reconstruction of single-peak signals with a linewidth of 50 pm and dual-peak signals with a 100 pm interval. Moreover, the multi-port output allows for the parallel acquisition of optical power information, significantly reducing acquisition time. Our spectrometer features a simple structure, strong portability, and ease of fabrication with ordinary manufacturing technology, making it highly promising for practical applications in portable spectral analysis scenarios.

The principle of the STRSS is outlined as follows. The input light with an unknown spectrum S(λ) passes through a sampling channel with a spectral response A(λ), where λ represents the wavelength and the collected optical power response at the output ports is denoted as O. This process can be mathematically expressed as O=∫λminλmaxA(λ)S(λ)dλ,(1)where λmin and λmax represent the lower and upper limits of the operating wavelength, respectively. Considering N sampling channels with distinct responses, and discretizing the band into M wavelength channels, the equation can be expressed as ON×1=AN×MSM×1.(2)

Each element of O corresponds to the optical power collected by the sampling channel in a different tuning state. The transmission matrix A is obtained through pre-calibration. In the reconstruction process, the unknown spectrum S can be reconstructed by employing a suitable algorithm to solve the inverse problem presented in Eq. (2).

Figure 1(a) illustrates the comparison among spatial, temporal, and spatial-temporal random speckle spectrometers. The primary distinction lies in the construction of the transmission matrix. As the spatial speckle spectrometers are usually not reconfigurable, the transmission matrix is the random speckle generated at the output end after the light passes through the disordered system at one time. Despite their pronounced decorrelation, the number of sampling channels is generally constrained by the number of output ports. Temporal speckle spectrometers collect each row of the transmission matrix at a single output port over different time intervals, and the decoherence performance exceeds the former. However, they require more rounds of acquisition to obtain more sampling channels. In contrast, the spatial-temporal speckle spectrometer collects spatial speckles at multiple output ports and combines different spatial speckles in the time domain. This approach combines the benefits of spatial speckle spectrometers and temporal speckle spectrometers. It allows for the flexible expansion of sampling channels and enhances decorrelation by leveraging various tuning states. The parallel output of multiple output ports reduces the number of acquisition rounds and shortens the acquisition time.

The upper section of Fig. 1(b) displays a schematic of STRSS, comprising an MZI array and a tunable diffraction network. The input light is split into 16 channels and directed into the MZI array. Random voltage is applied to the electrodes on each MZI, resulting in different degrees of phase modulation in the optical path. The tunable diffraction network consists of a planar waveguide and a 16×5 TiN pixel array (a total of 80 pixels). Application of random voltage to the pixels causes localized changes in the effective refractive index near each pixel, leading to random diffraction of the optical path throughout the planar waveguide. Following the random modulation of MZIs and the random diffraction of the diffraction network, the phase and amplitude of distinct optical paths change differently, resulting in the formation of random spatial speckles at output ports. Note that the random voltage set applied to the electrodes and pixels can be changed in the time domain, generating different spatial-temporal speckles. In some previous works, there have also been studies on multiport scanning speckle spectrometers [25,27,30,31]. However, the number of sampling channels for these researches is limited by the number of states of the tuning unit, or by a limited number of paths. In contrast, our spectrometer features continuous tuning states, thereby producing more sampling channels.

The workflow of STRSS comprises three stages: pre-calibration, optical power acquisition, and reconstruction. In the pre-calibration stage, light with a known broadband spectrum is input, and random voltage is applied to both electrodes of the MZI array and pixels in the diffraction network. The transmission spectrum collected at each output port denotes the intensity distribution of different wavelengths. Each transmission spectrum constitutes a row of the transmission matrix. Consequently, each column of the transmission matrix corresponds to the intensity distribution of a specific wavelength under all sampling channels. Therefore, the transmission matrix represents the intensity distribution of all wavelengths in the operating band under all sampling channels, which is the so-called speckle. The transmission spectrum of the decorrelated optical path forms spatial random speckle at the output port. Changing the voltage set results in different spatial random speckle patterns. The lower section of Fig. 1(b) illustrates the process of establishing the transmission matrix, where the Nth output set corresponds to the random voltage set of the Nth round. Thus, the transmission matrix is obtained by sequentially applying different rounds of voltage sets and combining the spatial random speckles in the time domain. Figure 1(c) shows the application process of the random voltage set. The power of each voltage set can be represented as a 16×6 random matrix. The maximum powers of the voltage of electrodes and pixels are denoted as Pπ and Pmax, respectively. Pπ represents the power corresponding to the applied voltage when the phase shift of the MZI reaches FSR while the value of Pmax depends on the maximum voltage that the chip can withstand. The first column of the matrix corresponds to the 16 MZI electrodes, and the second to fifth columns correspond to the 16×5 pixel array in the tunable diffraction network. After the pre-calibration, the unknown spectrum is input. The voltage set corresponding to the pre-calibration stage is successively applied, and the power in each round of the tuning state is successively collected at the output port. In the final reconstruction stage, the unknown spectrum can be reconstructed by using the algorithm described below.

Figure 2(a) shows a micrograph of the proposed spectrometer chip. The chip is fabricated on a silicon-on-insulator (SOI) wafer with 220 nm thick silicon on top and a 2 μm thick buried oxide substrate on the bottom. Figure 2(b) shows the chip packaged on a printed circuit board (PCB) using wire bonding. External light is introduced through the fiber array. The integrated thermoelectric cooler (TEC) is located below the chip and is used to control the temperature precisely at 25°C. During the process of the data collection, the chip is almost unaffected by the ambient temperature. Zoomed-in views of the 1×16 optical splitter, the MZI array, and the tunable diffraction network are presented in Figs. 2(c)–2(e), respectively. The feature size of our device is on the order of waveguide size.

For STRSS, the number of sampling channels has a significant impact on spectral resolution [24,26]. Following the compressive sensing theory, the number of reconstructed wavelength channels M is positively correlated with the number of sampling channels N, that is [32,33]: M∼ClogN,(3)where C represents a constant associated with the structure. In other words, achieving high resolution needs an adequate number of sampling channels. As Fig. 3(a) shows, we compare the results of resolving a dual-peak signal, with the two peaks spaced 100 pm apart, under different sampling channel numbers. When the number of sampling channels is insufficient, distinguishing the dual-peaks becomes challenging. With the increase of the channel number, the resolution is improved, and the shape of the reconstructed spectrum is gradually close to the original spectrum. To value the accuracy of the result of the reconstruction, we introduce the relative error ε defined as [34–39] ε=‖S−S0‖2‖S0‖2,(4)where S represents the reconstruction result, and S0 is the reference spectrum. When ε is less than 0.1, we consider the reconstruction results to exhibit high precision. Figure 3(b) shows that the relative error of the reconstruction of narrowband and broadband spectra varies with the number of voltage sets. The narrowband spectrum and the broadband spectrum are characterized by a single-peak signal with a linewidth of 50 pm and a sinc function curve with a bandwidth of 35 nm, respectively. Since the number of output ports is 16, the number of sampling channels is 16 times the rounds of voltage sets. As the rounds of voltage sets increase, the relative error rapidly decreases and falls below 0.1. Considering both the relative error and the power acquisition time of the output port, we choose to apply 10 rounds of the voltage set to obtain 160 sampling channels.

The decorrelation of the transmission matrix also affects the quality of the reconstructed spectrum [23,38]. The constructed transmission matrix is shown in Fig. 3(c), where different colors represent different transmittance. The period of each row is not identical. To assess the performance of the transmission matrix, we introduce the self-correlation function to estimate the resolution of the proposed spectrometer. The self-correlation function of the transmission matrix can be written as [23] C(Δλ)=⟨⟨I(λ,m)I(λ+Δλ,m)⟩λ⟨I(λ,m)⟩λ⟨I(λ+Δλ,m)⟩λ−1⟩m,(5)where I(λ,m) represents the transmission intensity at sampling channel m for wavelength λ, and ⟨…⟩ denotes the average for wavelength channels or sampling channels. We choose 160 sampling channels generated by 10 rounds of random voltage sets to construct the transmission matrix, then calculate the C(Δλ), and normalize it to 1 at Δλ=0, as shown in Fig. 3(d). The full width at half-maximum (FWHM) of C(Δλ), namely self-correlation width δλ, measures 0.31 nm. This value signifies that a wavelength shift of 0.31 nm is adequate to reduce the correlation of the speckle pattern by half. The value of δλ serves as an indicator of the overall sampling efficiency of a single sampling channel and can be used as an estimate of resolution, but it cannot precisely determine analytical resolution [24,25,37]. Subsequent experiments show that the resolution can exceed δλ. Figure 3(e) shows the cross-correlations among different sampling channels, with the majority falling below 0.1.

Additionally, we use singular value decomposition (SVD) to evaluate the linear independence of the entire vector space in the transmission matrix [40]. The transmission matrix can be written as A=UΣVT,(6)where U and V are two singular spaces, while ∑ denotes a diagonal matrix. The elements in ∑ are the singular values (σi) of the transmission matrix A. If the descent slope of the σi curve is gentle, the transmission matrix can be considered highly orthogonal. We oversample the transmission matrix to a square matrix of 3501×3501. The calculated σi after normalization is shown in Fig. 3(f). The σi curve falls gently, indicating favorable orthogonality of the transmission matrix. Moreover, the horizontal coordinate corresponding to the sudden drop of the singular value curve indicates the upper limit of the channel capacity. Figure 3(f) shows that our spectrometer has 3501 effective sampling channels. The capacity limit can be further increased by breaking the spatial symmetry of the structure. To sum up, the constructed transmission matrix achieves a high degree of decorrelation.

Our experimental process is shown in Fig. 4. In the pre-calibration stage, the amplified spontaneous emission (ASE) light source inputs a reference spectrum with approximately equal intensity at each wavelength into the chip through a single-mode fiber (SMF). The computer controls the voltage source to apply random voltage sets, and the output end is connected to a commercial optical spectrum analyzer (OSA, Yokogawa AQ6370C) to collect the transmission spectrum of each sampling channel. After pre-calibration, the unknown narrowband spectrum and broadband spectrum are output by a tunable laser (TL) and ASE light source, respectively. These spectra are filtered by a commercial programmable optical filter (Finisar Waveshaper 1000s) to generate custom waveform and then pass through the fiber array into the chip, where the corresponding random voltage sets are applied. At the same time, a 1×16 optical power meter (OPM) is connected to the output end to record the optical power response of the 16 output ports.

After collecting the data, it is essential to use a suitable algorithm to accurately reconstruct the unknown spectrum. Generally, we aim to generate S such that the following formula is satisfied [22,41]: MinS‖O−AS‖22.(7)

This is a classic least squares problem. However, the least squares method is not well-suited for an underdetermined system, i.e., when the number of rows in the transmission matrix A is far less than the number of columns. Under such condition, the equations are ill-conditioned, and minor perturbations of experimental noise can lead to large deviations in the reconstructed results from the original spectrum [42]. To address this problem, improvements to Eq. (7) can be made using methods such as the basis pursuit [43] and Tikhonov regularization [44,45]. These methods transform the equation into a convex optimization problem. We can rewrite the equation as [46] MinS(‖O−AS‖22+α1‖S‖1+α2‖ΓS‖22),(8)where α1 and α2 are regularization parameters, and Γ represents the second-order difference operator. The second and the third terms of Eq. (8) serve as the regularization terms for the sparse narrowband spectrum and the smooth broadband spectrum, respectively. The parameters α1 and α2 are chosen through cross-validation analysis [19,47]. The inclusion of two regularization terms at the end of the equation enhances the anti-noise capability of the algorithm.

We utilize the CVX solver, a package for specifying and solving convex programs, to solve Eq. (8) [48]. The reconstruction results are presented in Fig. 5. Figure 5(a) displays the reconstruction results of single-peak spectra with a linewidth of 50 pm at 1534.60 nm, 1546.50 nm, and 1558.74 nm, respectively. The values of ε are all below 0.1, indicating a high level of accuracy. Figure 5(b) shows the reconstruction results of three dual-peaks at different locations, each separated by a distance of 100 pm. These closely spaced dual-peaks are well distinguished. We also reconstruct a triple peak, as illustrated in Fig. 5(c). The spectrum comprises a dual-peak spaced 100 pm apart, and a single peak situated at a more distant location. Figure 5(d) shows the reconstruction results of a dual-peak spaced 200 pm apart, with a lower value of ε. Figure 5(e) shows the broadband spectra, which are generated by the ASE light source and obtained after passing through the programmable optical filter. The left figure is the waveform fitted by the polynomial, while the right figure is the waveform of the sinc function. The results demonstrate that the reconstruction accuracy of broadband spectrum is also remarkably high.

We compare the proposed spectrometer with some prior representative work on relevant indicators, as shown in Table 1. Evidently, many previous studies are confined to resolutions at the nanometer scale, or require more sampling rounds, posing challenges for meeting the requirements of high-precision and fast spectral analysis. During the device fabrication process, complex structures are often designed to attain the desired functions, leading to high manufacturing costs and hindering practical applicability. In contrast, our spectrometer employs a relatively basic combination of optical components and achieves a picometer-level resolution. According to Eq. (3), when the number of wavelength channels reaches a certain threshold, the constant C experiences a sharp increase, resulting in the need for more wavelength channels to slowly improve the resolution. A limitation exists in STRSS in terms of increasing resolution by adding channels, with a certain upper bound. Instead, we can choose to break the spatial symmetry of the structure to obtain higher decorrelation to improve the resolution. The thermal power consumption of the entire chip is approximately 3 W. Additionally, through 10 rounds of parallel data collection, we achieve optical power information acquisition in less than 5 s. It should be noted that the power acquisition time presented is constrained by the response time of the experimental equipment. The use of higher-precision optical switches and detectors could potentially result in even shorter optical power acquisition time. Furthermore, our chip and fiber are coupled by a pair of grating couplers. If the end coupling method is employed, the actual working bandwidth could be expanded significantly. Fabricated on SOI, our devices are complementary metal oxide semiconductor (CMOS) compatible, allowing integration into a diverse array of smart devices.Table 1.

Comparison of Our Spectrometer with Other Works

In this paper, we propose an innovative method for constructing a transmission matrix using the configuration of MZIs and a tunable diffraction network to generate spatial-temporal random speckles. With the help of the reconstruction algorithm, we accurately reconstruct several single-peak spectra with 50 pm linewidth and dual-peak spectra with an interval of 100 pm, showing a high resolution. We also realized accurate reconstruction of broadband spectra covering the entire C-band. Our device structure is simple and independent of advanced manufacturing processes, resulting in a low fabrication cost. These features position the device for practical applications in various convenient spectral analysis scenarios.