# On-Chip Reconstructive Spectrometer Based on Parallel Cascaded Micro-Ring Resonators
## 1 Introduction  
Spectroscopic analysis techniques are indispensable tools in a multitude of disciplines such as biomedical research, material science, and remote sensing [1; 2; 3]. Despite their significance, traditional benchtop spectrometers suffer from several drawbacks; they are generally bulky, complex, and expensive, rendering them ineffective for emerging applications such as wearable health monitoring and Lab-on-Chip systems [4; 5]. The motivation for our research stems from the urgent need for more accessible and versatile spectroscopic tools that can cater to these innovative applications. Integrated spectrometers, which lack mechanical moving parts, offer an appealing alternative due to their higher reliability, compactness, and field-deployable capabilities. These advantages are particularly relevant for broadening the application scope of spectral analysis. Driven by this vision, the past few decades have seen a concerted effort by both academia and industry to develop miniaturized spectrometers. As a result, over the past few decades, researchers have introduced numerous designs for computationally reconstructed on-chip spectrometers. Thesedesigns leverage advanced technologies such as silicon photonics and nano-fabrication to integrate multiple spectral components onto a single chip, enabling high-performance spectral analysis in a much smaller and more portable form factor. Innovative designs have emerged, replacing traditional diffraction-based elements with novel dispersive ones, such as disordered scattering media [6; 7; 8; 9], metasurfaces [10; 11], photonic crystals [12; 13], quantum dot arrays [14; 15], multimode waveguides [16; 17], or filter arrays [18; 19; 20; 21]. These approaches utilize these novel dispersive elements to create a sampling array, capturing the incident spectrum across multiple channels. The reconstruction of the spectrum is based on the speckle or intensity distribution obtained from these samplings. However, the scalability of the sampling array can limit the resolution and accuracy of the reconstruction. Another design approach employs dynamically tunable elements, such as tunable cavities [22], black phosphorus detectors [23], perovskite detectors [24], and reconfigurable silicon photonic networks [25; 26]. By applying external control signals, the response function of the tunable sampling unit can be dynamically adjusted to achieve variable sampling of the incident spectrum. The challenge for these systems is to achieve accurate and high-resolution spectral reconstruction while avoiding overly complex device architectures or numerous sampling channels. Addressing these challenges, our work presents an on-chip spectrometer based on a silicon photonic platform. The core of our design is a transmission-programmable silicon photonic filter, which serves as a dynamic sampling unit. By manipulating the transmission spectrum of the filter, we can perform diverse samplings of the input spectrum. An ANN model is then employed to reconstruct the incident spectrum from the sampled signals. The programmable silicon photonic filter is ingeniously designed with parallel cascaded MRs. By adjusting the phase variations of light waves within the MRs via thermo-optic phase shifters, the transmission spectrum of the silicon photonic filter can be altered. Our design distinguishes itself from existing filter-based on-chip spectrometers by employing a full MR configuration with mutually coupled cascaded MRs and intentionally designed unequal radii. This unique approach enhances the diversity of sampling responses, resembling random sampling with a limited number of sampling channels, which is essential for computational spectral reconstruction. Furthermore, our chip features three output ports, enabling higher sampling efficiency and improved reconstruction performance.