1.

Introduction

Spectral analysis plays a crucial role in scientific research and industrial production.^1–3 Mainstream spectral analysis is based on spectrometers whose size must be reduced for portable applications and assembling in other systems. Traditional spectrometers face significant challenges in miniaturization and portability due to their complex optical paths and reliance on precision dispersion elements.⁴ In recent years, with the continuous advancement of computer technology and optical manufacturing, computational spectrometers have emerged as a promising solution to overcome these challenges.⁵

The uniqueness of computational spectrometers lies in their utilization of various encoding techniques to acquire spectral data, followed by algorithmic reconstruction, and avoiding the dependence on long optical paths and complex optical components. The performance of computational spectrometers is highly dependent on the specificity of the spectral encoding method and the robustness of the reconstruction algorithm.^6–10 In terms of spectral encoding, a variety of techniques, such as filter arrays,^11–13 detector arrays,^14,15 tunable filters,^16,17 and detectors^18,19 have been proposed. Among these techniques, the quantum dot (QD) filter array is a promising route for computational spectrometers due to the finely tunable absorption spectrum via size and composition and ease of solution processability.^20–22 While most of the study on computational spectrometers has been focused on spectral encoding, there have also been some works dedicated to exploring reconstruction algorithms with higher noise immunity. As a result, a limited number of algorithms, such as least squares regression,²³ Ridge regression,¹⁶ and Lasso regression,²⁴ have been adopted. Furthermore, the robustness of these algorithms needs to be improved. Noise might emerge during the data acquisition process due to variations of transmittance in the filter array, spectral response in the detector array, or repeatability of tunable filters/detectors. The noise affects the accuracy of spectral reconstruction, imposing a higher requirement on the robustness of the algorithms.²⁵ The noise tolerance of the aforementioned algorithms still needs improvement. For instance, ordinary least squares regression has minimal noise tolerance, and Lasso regression and Ridge regression also exhibit sensitivity to noise perturbations.²⁶ In response, new algorithms have been proposed to enhance noise tolerance. For example, an ALM algorithm obtained by combining total variation (TV) regularization with the augmented Lagrange multiplier method improves the noise immunity of the algorithm to some extent, while reconstruction performance decreases under a high noise level.²⁶ Recently, machine learning was also applied to pursue better robustness of algorithms.²⁷ Through processing the detected signals using a pretrained denoising autoencoder network before employing the reconstruction algorithm, the noise immunity of the spectrometer system is substantially improved. However, such an algorithm was designed uniquely for its encoding structure and therefore cannot be directly used quickly and inexpensively for other computational spectrometers, but rather again requires the entire process from database creation to model tuning and training. Therefore, it is highly desirable to develop a robust algorithm with a high level of noise tolerance and good migratory properties.

Against this backdrop, this study aims to enhance the tolerance of computational reconstruction spectrometers to noise, thereby further improving their performance. We fabricated a spectrometer based on a QD filter array and proposed a new algorithm with supreme noise tolerance performance. Benefiting from the robust algorithm, the QD spectrometer is capable of accurately reconstructing spectral features even at signal to noise ratio $(\mathrm{SNR})=-3\mathrm{dB}$, which is remarkably better than the results based on previously reported algorithms. Furthermore, the spectral accuracy and spectral recovery performance can be substantially improved by interpolating the precalibration data using the new algorithm. As a result, with $10\times$ interpolation, the QD spectrometer here achieves an extremely high spectral accuracy as low as 0.1 nm and a spectral resolution of 2 nm in the 400 to 800 nm range, which represents the highest spectral accuracy among QD spectrometers.

2.

Results and Discussion

Figure 1 shows the fabrication process of the QD spectrometer. Overall, we selected four different components of QDs for the spectral encoding, including CdS, ${\mathrm{CdS}}_{\mathrm{x}}{\mathrm{Se}}_{1-\mathrm{x}}$, CdSe, and CdTe QDs. Based on these components, a uniform and complete coverage of the absorption peaks of the synthesized QDs in the range of 400 to 800 nm was achieved by controlling the growth time of the QDs after the hot injection. Subsequently, the QD filter array was fabricated using electrohydrodynamic jet printing on a quartz substrate, which allowed for the small size and uniform distribution of individual QDs in the arrays. Finally, the substrate with the filter array was mounted on top of the CMOS image sensor using a UV-curable adhesive. A robust reconstruction algorithm with supreme noise tolerance performance was proposed to accomplish the spectral reconstruction, which integrates Tikhonov regularization and TV regularization techniques, along with the introduction of the alternating direction method of multipliers (ADMM).

Fig. 1

The strategy for fabricating the QD spectrometer. The absorption spectra of the QDs are tuned via their size and composition, and the QD filter array is fabricated by the electrohydrodynamic jet printing technique. The substrate with the QD filter array is assembled on top of a CMOS image sensor to fabricate the spectrometer. A new reconstruction algorithm is proposed to enhance the noise tolerance performance.

The CdS, CdSe, ${\mathrm{CdS}}_{\mathrm{x}}{\mathrm{Se}}_{1-\mathrm{x}}$, and CdTe QDs were synthesized in a seeded growth manner. For a given composition, small QDs were formed by hot-injection synthesis with a highly excessive Cd precursor; then the chalcogenide precursor was added dropwise to promote continuous growth of the QDs. The synthetic parameters were carefully controlled to maintain the very distinguishable exciton peaks during the whole growth process. Aliquots were withdrawn from the growth solution at different intervals to obtain a series of QDs with increasing sizes. Through the control of composition and size, we managed to produce QDs with absorption peaks covering the visible light range. Figure 2(a) shows some solution samples of the QDs. A series of typical absorption spectra are shown in Fig. 2(b), and all the absorption spectra of the QDs used in this work are shown in Fig. S1 in the Supplemental Material. Typical transmission electron microscope images of the four types of QDs are shown in Fig. S2 in the Supplemental Material. In order to avoid interference from QD photoluminescence (PL), a fluorescence quencher was introduced to quench the PL,²⁸ as shown in Fig. S3 in the Supplemental Material. It is worth noting that the seeded growth is an advisable strategy to synthesize a series of QDs with different sizes. Figure 2(c) shows the absorption peak wavelength as a function of sampling time for the four types of QDs (two starting sizes were used for CdSe QDs), demonstrating the convenience of material preparation for QD spectrometers.

Fig. 2

(a) A picture of some typical QD solutions. (b) Absorption spectra of selected QDs. (c) Plots of QD absorption peak as a function of growth time in the seeded growth process for the QDs with different compositions.

To prepare QD microfilters, the QDs were mixed with organic polymers, and the electrohydrodynamic jet printing technique was used to print them on a quartz substrate. Subsequently, a QD filter array was obtained through solvent evaporation and heat treatment. The native ligands on the QD surface are oleate, making them soluble only in nonpolar solvents, such as n-hexane and n-octane, whereas typical polymers have low solubility in these nonpolar solvents. To address this issue, we replaced the oleate ligand with a thiol ligand, 6-mercaptohexanol (MCH).^29,30 Simultaneously, it was observed that there was a slight change in the absorption peak of the QDs after ligand exchange, as shown in Fig. 3(a). This demonstrates that by controlling the duration and ligand concentration in the ligand exchange process, we can further fine-tune the absorption peak distribution of the QDs, achieving a more uniform spectral encoding effect. In addition to treatment on QDs, the selection of polymers must also meet certain criteria. First, the polymer itself must be highly transparent to visible light to avoid unnecessary light flux loss. Furthermore, the polymer must have good solubility in the chosen solvent and good compatibility with QDs. This helps ensure the uniformity of the printed QD filter, avoiding light intensity loss caused by the optical structure on the filter surface, thus eliminating influence on the final encoding effect.³¹ After comparative analysis (Fig. S4 in the Supplemental Material), we chose a binary solvent system of dimethyl sulfoxide (DMSO) and butanol and used polyvinylpyrrolidone (PVP) as the polymer. Figure 3(b) shows the printed QD filter array, demonstrating the orderly distribution of the microfilters. We tested the height of the printed microfilters using a profilometer, as displayed in Fig. 3(c), confirming that these QD filters have a smooth surface morphology. The QD filter array used for the spectrometer consists of a total of 256 microfilters, with an average size of $250\mu \mathrm{m}$, a space of $70\mu \mathrm{m}$, and an array size of only $0.50\mathrm{c}\mathrm{m}\times 0.50\mathrm{c}\mathrm{m}$. This array size is the smallest one reported to date. After obtaining the QD filter array, we assembled it on top of a CMOS image sensor. Then, precalibration was performed by collecting the data at every 1 nm in the range of 400 to 800 nm, producing transmission spectra of every microfilter; Fig. 3(d) shows some of the transmission spectra.

Fig. 3

(a) Redshift of the absorption peak of QDs with the increase of the MCH ligand. (b) A picture of the QD filter array, in which the scale bar is 0.50 mm. (c) Uniformity and surface profile of the filter units. (d) Transmittance spectra of some filter units.

The CMOS camera was equipped with the QD filter array, which, combined with a spectral reconstruction algorithm, can work as a spectrometer.³² The incident light passes through the spatially distributed 256 QD microfilters and is modified to varying degrees at different wavelengths, and the resulting signals are captured by the camera. The problem of reconstructing the input spectrum can be described as

We initially used the Tikhonov regularization technique, commonly known as Ridge regression, for spectral reconstruction, as employed in the majority of previous works.^14,18,40 Due to the inclusion of the regularization term, the influence of noise on the reconstruction results during the iterative process is effectively suppressed. Using this algorithm, we reconstructed several monochromatic lights and compared the results with those collected by a commercial spectrometer, as shown in Fig. 4(a). For the monochromatic light, Ridge regression demonstrated good reconstruction performance. However, when this algorithm was applied to reconstruct broad spectra, the reconstruction results were not satisfactory, as shown in Fig. 4(b).

Fig. 4

Spectral reconstruction results of the TKVA algorithm for (a) monochromatic light, (b) a white LED spectrum, and (c) the xenon lamp spectrum. For comparison, the results of the Ridge algorithm are presented.

To improve the broad spectrum reconstruction performance of the algorithm, we considered the TV regularization technique commonly used in the field of image processing.⁴¹ The essence of TV regularization is to impose constraints based on the continuity of data, typically expressed as the first-order derivative of the data in mathematical terms. This allows for a more suitable application for spectra with complex features. We named this model $\text{Ridge}+\mathrm{TV}$ and introduced the AdaDelta iterative strategy for parameter control, with detailed calculation procedures available in the Supplemental Material. As shown in Fig. S5 in the Supplemental Material, the introduction of the TV regularization term significantly improved the spectral reconstruction performance of the algorithm for the white LED spectrum.

To further improve the algorithm’s noise tolerance, we believed that the role of the regularization term needed to be better utilized. The ADMM,⁴² evolved from the Douglas–Rachford splitting method and Rockafellar’s method of multipliers, figures out problems by alternately optimizing primal and dual variables, and updates the dual variables at each step using the method of Lagrange multipliers. This process effectively decomposes complex optimization problems into several subproblems for controlled resolution. Therefore, the ADMM was introduced to further enhance control over the regularization term. Moreover, the multistep iteration effect brought about by the method of ADMM allows us to more intuitively observe the impact of parameter changes during the iteration process of spectral reconstruction. We named this algorithm TKVA (algorithm with Tikhonov and TV regularization, and ADMM), and the mathematical model for the reconstruction problem becomes

First, the spectral reconstruction performance of the TKVA algorithm was evaluated. As shown in Figs. 4(a) and 4(b), the TKVA algorithm not only exhibits excellent spectral recovery for simple single peak spectra but also has supreme capability for the broad spectrum compared to the conventional Ridge regression algorithm. It is noteworthy that the TKVA algorithm shows significantly improved spectral recovery for complex broad spectra, such as the xenon lamp spectrum, as shown in Fig. 4(c).

Then, the noise tolerance of the TKVA algorithm was investigated. In fact, in the simulated spectral reconstruction process, the Ridge regression technique also demonstrated excellent reconstruction performance for broad spectra. However, this performance could not be retained in the actual measurement process, evidently due to the influence of noise during the practical detection process. The sources of noise could stem from the measurement process or from measurement errors during the precalibration process. We simulated the influence of noise by adding the noise to the incident spectrum and performed the reconstruction accordingly. By manually adding noise and comparing the simulated reconstruction results with the actual reconstruction results, it was estimated that the SNR in the actual measurement process was $\sim 10\mathrm{dB}$. However, in certain situations, the SNR could decrease to 0 dB or even lower due to tremors, drastic environmental changes, etc.

The noise tolerance performance of the TKVA algorithm was evaluated by introducing varying levels of noise. As shown in Fig. 5(a), at an SNR of 10 dB, the TKVA, the Ridge+TV, and the commonly used Ridge regression and Lasso regression demonstrate good reconstruction results. However, as the SNR gradually decreases, except for TKVA, overfitting becomes increasingly prominent in other algorithms [Fig. 5(b)]. When the SNR drops to $-3\mathrm{dB}$, as shown in Fig. 5(c), only the TKVA is still able to accurately reconstruct the original spectral features. In addition, the TKVA algorithm was also compared to the ALM algorithm²⁶ that was previously proposed to suppress the noise influence. As a result, the TKVA still shows much better noise tolerance performance than the ALM algorithm, as shown in Fig. S6 in the Supplemental Material.

Fig. 5

Spectral reconstruction results of different algorithms at noise levels of (a) 10 dB, (b) 0 dB, and (c) $-3\mathrm{dB}$.

Spectral accuracy is a key figure of merit for spectrometers, and this performance of the QD spectrometer based on TKVA algorithm was evaluated. The computational spectrometer has to be precalibrated before operation, in which we scanned the monochromatic light with an interval of 1 nm. Theoretically, a smaller interval is beneficial for attaining a higher spectral accuracy, thus the 1 nm step might impose a constraint on the maximum accuracy. To overcome this limitation and considering the continuity of spectral distribution, we performed spline interpolation on the results obtained from the precalibration process. We conducted $1\times$, $2\times$, $5\times$, $10\times$, and $20\times$ interpolations and reconstructed a simulated monochromatic light with a minimum full width at a half-maximum of 0.8 nm. The reconstruction results for different interpolation factors are shown in Figs. 6(a) and 6(b) using the TKVA and the Ridge algorithms, respectively. As the interpolation factor increases, the TKVA algorithm shows better reconstruction performance for the narrow monochromatic light. Although data oscillation appears in the flat data regions at higher interpolation factors due to overfitting, it has little impact on distinguishing the peak [Fig. 6(a)]. In contrast, the improvement of the Ridge algorithm is substantially limited [Fig. 6(b)]. The interpolation effect of the ALM algorithm, which shows inferior reconstruction performance than that of the TKVA algorithm at the same level of interpolation, was also studied (Fig. S7 in the Supplemental Material). These results indicate the unique advantage of the TKVA algorithm on the enhancement of spectral accuracy. In practice, we also used data scanned every 3 nm for $15\times$ interpolation for reconstruction and found that the spectral reconstruction performance was not significantly affected (Fig. S8 in the Supplemental Material). Therefore, the newly proposed TKVA algorithm not only offers a convenient and effective way to improve the spectral accuracy but also provides a route to reduce the workload of the precalibration process through interpolation.

Fig. 6

Spectral reconstruction results of the (a) TKVA and (b) Ridge algorithms at different interpolation factors. Reconstruction results of room light (c) without and (d) with $10\times$ interpolation using the TKVA and Ridge algorithms. (e) Reconstruction results of two monochromatic lights with peaks 0.1 nm apart. (f) Spectral resolution as a function of the number of QD filters.

To evaluate the spectral reconstruction performance with the help of interpolation, we reconstructed a complex and broad spectrum (room light) without interpolation and with $10\times$ interpolation, as shown in Figs. 6(c) and 6(d), respectively. After interpolation, the recovery performance of the TKVA algorithm is significantly improved, which not only exhibits more accurate overall reconstruction but also captures small peaks in the spectrum. In contrast, the Ridge algorithm shows little improvement via interpolation and exhibits much poorer reconstruction performance. With a significant improvement in noise tolerance and spectral recovery by interpolation, the TKVA algorithm shows a relatively modest increase in usage of computing resources; for example, under 10-fold interpolation, the TKVA algorithm requires 0.025 s to reconstruct a single-peak spectral, while the $\text{Ridge}+\mathrm{TV}$ algorithm needs 0.008 s.

Finally, we investigated the spectral accuracy of the TKVA-based QD spectrometer with actual monochromatic light input; the results are shown in Fig. 6(e). After calculation, the spectral accuracy of the spectrometer in the 400 to 800 nm range reaches as low as 0.1 nm, representing the highest level for QD spectrometers.^{26,28,29,43,44} In addition, we investigated the influence of the number of QD filters on spectral reconstruction, as shown in Fig. 6(f). As the number increases, the average spectral resolution increases accordingly, which reaches a value of 2 nm, as shown in Fig. S9 in the Supplemental Material. After the number reached 196, further increasing the number of the filter results in a slight improvement in resolution, indicating a limited number of filters are adequate to achieve a high spectral resolution.

3.

Conclusions

In summary, we demonstrate a high spectral accuracy QD spectrometer with excellent noise tolerance by developing a new reconstruction algorithm. Seeded growth is adopted to produce a series of QDs with increasing size, and the absorption peak of all the QDs used is finely tuned by the combination of size and composition in the range of 400 to 800 nm. The electrohydrodynamic jet printing technique is applied to fabricate the QD filter array with a total number of 256, a filter size of $250\mu \mathrm{m}$, and a space of $70\mu \mathrm{m}$. The filter array on a quartz substrate is assembled on top of a CMOS image sensor to form the spectrometer. In order to enhance the noise tolerance performance, a new algorithm, TKVA, was proposed, which integrates Tikhonov regularization and TV regularization techniques, along with the introduction of the ADMM. The TKVA algorithm shows excellent noise tolerance demonstrated by good spectral recovery even at a low SNR of $-3\mathrm{dB}$, outperforming previously reported algorithms. In addition to noise tolerance, the TKVA algorithm offers a unique advantage compared to other algorithms; that is, the spectral accuracy can be improved by interpolation of the data obtained in the precalibration process. In the meantime, the spectral reconstruction performance is also significantly improved especially for complex and broad spectra, which is not observed for other algorithms. Through interpolation, the TKVA can reduce the workload of the precalibration process while maintaining a high spectral recovery performance. Based on the TKVA algorithm, the QD spectrometer shows a spectral accuracy as low as 0.1 nm in the range of 400 to 800 nm, representing the highest level among QD spectrometers. The supreme TKVA algorithm might find broad applications in different types of computational spectrometers.

4.

Appendix: Materials and Methods