# On-Chip Reconstructive Spectrometer Based on Parallel Cascaded Micro-Ring Resonators
## 2 Principle and Design  
The on-chip spectrometer we have designed is primarily composed of a programmable silicon photonic filter. The structural schematic of the device is shown in Figure 1. By applying control signals to the phase shifters, we can program the phase shift of each MR, thereby altering the filter's transmission spectrum. Each time the transmission state of the filter is changed, three samplings of the input optical signal can be completed because the filter has three output ports. Through the reasonable setting of control signals, we can ensure that the transmission spectra of sampling channels have unique spectral features with low correlation, leading to diversified sampling of the incident spectrum. The sampled spectral signals are then converted into response vectors by photodetectors at the output end of the filter. The process of spectrum reconstruction is illustrated in Figure 2.  
The designed programmable silicon photonic filter consists of a 6-stage parallel cascaded MR with different radii. Each MR is coupled to adjacent ones to increase the diversity of the optical paths within the device. By employing different radii of the MRs, the Vernier effect is introduced, which further enhances the diversity of the filter's transmission spectrum, thereby improving the sampling efficiency of the filter. By individually controlling the voltage across the six phase shifters, we can leverage the thermo-optic effect to program the phase shift (\(\Delta\phi\)) in each MR, thus enabling flexible adjustment of the filter's transmission spectrum. When the incident light, \(\Phi(\lambda)\), enters the silicon photonic filter from the input port, the transmission spectrum \(T_{i}(\lambda)\) of the filter is modified by adjusting the input voltages of the six phase shifters, which completes the sampling of the incident light spectrum. Then the output power intensity \(I_{i}\) of an unknown incident spectrum \(\Phi(\lambda)\) propagating through a broadband sampling channel can be described as:  
\[I_{i}=\int T_{i}(\lambda)\Phi(\lambda)d\lambda \tag{1}\]  
Equation (1) can be rewritten in a discrete form as:  
\[I_{i}=\sum_{k=1}^{N}T_{i}(\lambda_{k})\Phi(\lambda_{k}) \tag{2}\]  
where \(N\) is the number of spectral pixels in the wavelength domain. By repeatedly changing the sampling state of the filter through external voltages, multiple samplings of the incident light spectrum can be achieved. An intensity vector \(I_{M\times 1}\) and the corresponding \(M\) equations are obtained through \(M\) sampling channels. These equations can be expressed in the following matrix form:  
\[ \begin{bmatrix}I_{1}\\ \vdots\\ I_{M}\end{bmatrix} = \begin{bmatrix}T_{1}(\lambda_{1})&\cdots&T_{1}(\lambda_{N} )\\ \vdots&\ddots&\vdots\\ T_{M}(\lambda_{1})&\cdots&T_{M}(\lambda_{N})\end{bmatrix}\begin{bmatrix}\Phi( \lambda_{1})\\ \vdots\\ \Phi(\lambda_{N})\end{bmatrix}  \tag{3}\]  
where the sampling matrix \(T_{M\times N}\) connects the intensity vector \(I_{M\times 1}\) with the incident light \(\Phi_{N\times 1}\) to be measured. Clearly, there are \(M\) equations and N unknowns \(\Phi(\lambda_{k})\) to solve for in Equation (3). The unknown input spectrum \(\Phi_{N\times 1}\) can be computed by multiplying the  
Figure 1: Schematic diagram of the on-chip computational spectrometer based on programmable silicon photonic filters. The phase shifters are integrated within each ring resonator to individually control the phase shift, enabling the programmability of the filter’s transmission spectrum. Figure 2: Spectral reconstruction procedure. output intensity \(I_{M\times 1}\) by the inverse of the matrix \(T_{M\times N}\). In general, since the inverse matrix is only applicable to square matrices, the number of recoverable wavelength channels is limited by the number of sampling channels. However, with proper design, the number of samples \(M\) required to reconstruct the spectrum of input light can be much smaller than the number of spectral pixels \(N\)[21].