# On-Chip Reconstructive Spectrometer Based on Parallel Cascaded Micro-Ring Resonators
## 3 Simulation Results and Analysis
### Spectral Reconstruction Simulation Results
The reconstruction results are shown in Figure 7, with an RMSE of 0.0169 and a relative error \(\varepsilon\) of 0.1918, providing strong evidence for the high spectral resolution of our proposed on-chip spectrometer. The two narrow peaks spaced at 2 nm are clearly distinguishable in the reconstructed spectrum, indicating that the spectral reconstruction resolution of our designed spectrometer chip can reach 2 nm under a limited number of sampling channels. The clear resolution of the two peaks demonstrates the capability of our design to resolve fine spectral features, which is a key performance metric for advanced spectroscopic devices. Increasing the number of sampling states of the photonic filter can further reduce the spectral reconstruction error and increase the resolution. We investigate the impact of the filter sampling states on the spectral reconstruction performance by increasing the number of phase shift states of the phase shifters in the programmable silicon photonic filter. We sequentially increase the number of phase shift states of the phase shifters from 2 (0 and \(\pi\)) to 3 (0, \(\pi\)/3, and 2\(\pi\)/3), obtaining sampling matrices with \(M\) = 3 \(\times\) 96, 3 \(\times\) 144, 3 \(\times\) 216, 3 \(\times\) 324, 3 \(\times\) 486, and 3 \(\times\) 729 sampling channels, respectively. Increasing the number of phase shift states of the phase shifters essentially increases the programmable degrees of freedom of the filter, resulting in more diverse sampling channels. When the number of states of the phase shifters increases from 2 to 3, each phase shifter can provide 3 different phase shifts, allowing the transmission spectrum of the filter to exhibit more variations. These variations enable the filter to perform richer and more diverse sampling of the incident spectrum, obtaining more spectral information. This helps to improve the accuracy and resolution of the spectral reconstruction. As shown in Figure 8a, the average relative error \(\varepsilon\) of the reconstructed spectra in the test set steadily decreases as the number of sampling states increases from 64 to 729. Across the entire test set, with 729 sampling states, the average RMSE for spectral reconstruction is 8.912 \(\times\) 10\({}^{-6}\), and the average relative error \(\varepsilon\) is 2.2991 \(\times\) 10\({}^{-5}\). The significant reduction in RMSE and \(\varepsilon\) can be attributed to the fact that when the number of sampling states is 729, the number of sampling channels reaches 3 \(\times\) 729, which exceeds the number of spectral pixels \(N\) = 2001. Consequently, the redundant information captured by the sampling channels helps to reduce the difficulty of spectral reconstruction significantly. Simultaneously, as the number of sampling states increases to 729, the resolution of the on  
Figure 7: Simulated reconstruction results of a narrow double-peak spectrum separated by 2 nm. chip spectrometer for dual-peak spectra improves to 0.1 nm, limited by the resolution of the spectral pixels set in the simulation. This improvement is demonstrated by the dual-peak spectrum reconstruction results in Figure 8b. Figure 8c,d shows the reconstruction results of the proposed on-chip spectrometer for the previously selected narrow spectrum and the measured ASE spectrum, respectively, when the number of sampling states is set to 729. The reconstruction errors (RMSE) are \(1.457\times 10^{-6}\) and \(1.5333\times 10^{-5}\), and the relative errors \(\varepsilon\) are \(1.1512\times 10^{-5}\) and \(2.2297\times 10^{-5}\), respectively. The steady decrease in the relative error \(\varepsilon\) as the number of sampling states increases highlights the direct relationship between sampling diversity and reconstruction performance. With more distinct sampling channels, the filter captures more features in the spectrum, enabling the reconstruction algorithm to better approximate the original spectrum. Furthermore, the increase in sampling states also positively impacts the resolution of the on-chip spectrometer.