# On-Chip Reconstructive Spectrometer Based on Parallel Cascaded Micro-Ring Resonators
## 3 Simulation Results and Analysis  
### Device Design and Simulation  
The device is designed to be fabricated on a Silicon-on-Insulator (SOI) platform with a top silicon layer thickness of 220 nm and a buried oxide layer thickness of 2 \(\upmu\)m. The waveguide is a single-mode rib waveguide with a height of 220 nm, a width of 450 nm, and slab regions on both sides of 70 nm thickness. Referring to the technology of mainstream silicon photonics SOI platforms, the waveguide's transmission loss is set to 2 dB/cm. The effective refractive index of the TE-polarized fundamental mode at a 1550 nm wavelength obtained using the finite difference eigenmode method is 2.4769, and the group refractive index is 4.0088. The MRs in the filter are racetrack micro-rings with straight waveguide  
Figure 3: The schematic of the spectrum reconstruction algorithm. sections 1.14 um in length and with radii of 5 um, 5.5 um, 6 um, 6.5 um, 7 um, and 7.5 um, respectively. The coupling gap between the micro-ring and the straight waveguide is 100 nm, as is the coupling gap between micro-rings, ensuring that the resonators are in an over-coupled state with coupling coefficients at a 1550 nm wavelength being around 0.5. Using the Ansys Lumerical INTERCONNECT simulation platform, we set two operational states for each of the six phase shifters, introducing phase shifts \(\Delta\varphi\) of 0 and \(\pi\), resulting in a total of 64 sampling states. By considering the 3 output ports of the filters, we ultimately acquire 192 sampling channels from the filters, forming the sampling matrix. Figure 4a shows the heatmap of the sampling matrix of the designed spectrometer based on the programmable filter. Figure 4b displays the transmission spectra of the filter under four different phase-change conditions, which are randomly selected. Each row in the heatmap represents the transmission spectrum of a sampling channel. Autocorrelation and cross-correlation function to characterize the filter's transmission spectra [21; 26]. The half width at half maximum (HWHM) of the auto-correlation function, denoted as \(\delta\lambda\), is a crucial parameter that quantifies the spectral resolution of the reconstructive spectrometer. It represents the minimum wavelength shift required to reduce the correlation between individual sampling channels by 50%. In other words, \(\delta\lambda\) determines the ability of the system to distinguish between neighboring wavelength pixels. The cross-correlation between two distinct sampling channels is a measure of their similarity in the spectral domain. It quantifies the extent to which the spectral responses of the two channels are correlated with each other. In an ideal scenario, the sampling channels should be orthogonal, meaning that they are completely independent. When this condition is satisfied, the cross-correlation between any two distinct sampling channels should be zero at all wavelength points. By assessing the autocorrelation and cross-correlations of the sampling matrix, we can evaluate the effectiveness of our sampling system in providing diverse and complementary spectral information. The calculated autocorrelation and cross-correlation are shown in Figure 5. The narrow autocorrelation with \(\delta\lambda\) = 0.65 nm and low cross-correlation indicates a well-designed sampling matrix contains diverse features with very little cross-correlation signal. The incident light spectrum can be converted into a unique intensity vector after sampling and detection, allowing the subsequent reconstruction algorithm to work effectively. Figure 4: (**a**) Heat map of simulated sampling matrix of the devices with 192 sampling channels; (**b**) Transmission spectra of 4 randomly selected sampling channels. Utilizing the network architecture described in Section 2, we have constructed an ANN-based spectral reconstruction model to achieve the transformation from intensity vectors \(I_{M\times 1}\) to reconstructed spectra \(\hat{\Phi}\).