# On-Chip Reconstructive Spectrometer Based on Parallel Cascaded Micro-Ring Resonators
## 3 Simulation Results and Analysis
### Spectral Reconstruction Simulation Results  
Combining the simulation results of the designed on-chip spectrometer with the trained ANN model, the reconstruction results for different types of synthetic spectra selected from the test set are shown in Figure 6a,b. We employ the root mean squared error (\(RMSE\)) and relative error \(\varepsilon\) to evaluate the spectral reconstruction results:  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{m}\left(\Phi-\hat{\Phi}\right)^{2}}{m}} \tag{4}\]  
Figure 5: (**a**) The calculated spectral auto-correlation function and average cross-correlation of the transmission spectra among the 192 sampling channels (64 sampling states). The cross-correlation curve represents the mean value of all the pairwise cross-correlations between the transmission spectra of the 192 channels. (**b**) The HWHM of the auto-correlation function is approximately 0.65 nm. \[\varepsilon=\frac{\left\|\Phi-\delta\right\|_{2}}{\left\|\Phi\right\|_{2}} \tag{5}\]  
where \(m\) is the total number of wavelength points. Across the entire test set, the average RMSE for spectral reconstruction is 0.0051, and the average relative error \(\varepsilon\) is 0.0484. Figure 6a,b illustrates the reconstruction of a spectrum with relatively wide peaks, where the FWHM of each peak is randomly selected from the interval (2 nm, 100 nm) and (2 nm, 10 nm). The reconstructed spectrum closely matches the true spectrum, with an RMSE of 0.0009 and 0.0053 and \(\varepsilon\) of 0.0011 and 0.0175, respectively. In Figure 6c, we demonstrate the reconstruction of a sparse spectrum with several narrow peaks, where the FWHM of each peak is randomly chosen from the interval (1 nm, 2 nm). Despite the presence of sharp and closely spaced peaks, the reconstructed spectrum still accurately captures the key features of the true spectrum, achieving an RMSE of 0.0106 and a relative error \(\varepsilon\) of 0.0837. From the figures, it is evident that the peaks in the synthesized spectra are precisely reconstructed. We also performed reconstruction on the measured spectra of an amplified spontaneous emission (ASE) light source, with the results shown in Figure 6d. The spectral range of the ASE light source is inherently 1500 nm to 1600 nm, originally containing 1001 data points. To facilitate the verification of our chip's ability to reconstruct measured spectra, we expanded it to 2001 data points through interpolation, assuming a wavelength range of 1450 nm to 1650 nm. The RMSE and the relative errors \(\varepsilon\) are 0.0065 and 0.0095, respectively. These results demonstrate that our designed on-chip spectrometer is capable of accurately reconstructing real-world spectra. The low RMSE and relative error values for both the synthesized and measured spectra indicate the robustness and effectiveness of our on-chip spectrometer design and the associated ANN-based reconstruction model. The model's ability to handle diverse spectral features, including  
Figure 6: Simulated reconstruction results of different types of spectra. (**a**–**c**) Randomly selected synthetic spectrum with three different FHWM; (**d**) ASE light source spectra. narrow and closely spaced peaks, highlights its potential for practical applications in various fields where accurate spectral reconstruction is crucial. The ability to resolve closely spaced narrow peaks is crucial for many spectroscopic applications, such as gas sensing, environmental monitoring, and chemical analysis. In these fields, the spectral features of interest often lie in a narrow wavelength range, and the ability to distinguish between adjacent peaks is essential for the accurate identification and quantification of the target analytes. To further demonstrate the spectral resolution of the proposed spectral chip, we simulated the reconstruction of a dual-peak spectrum, where the peaks are separated by 2 nm, and each peak has an FWHM of approximately 1 nm.