完善修正并且检查下面这句话让它符合sci科技论文结论的书写要求：In conclusion we have derived the resonance lineshape variation of the MRR structure with MRU by the method of the transfer matrix The established model considers the i

In conclusion, we have successfully derived the resonance lineshape variation of the MRR structure with MRU using the transfer matrix method. Our established model comprehensively considers the influence of light transmitted in both directions within the structure and the size of the MRU on the resonance lineshape and intensity. This rigorous analysis addresses the limitations of previous theories.

Furthermore, we have demonstrated the practical application of Fano lineshape shape in single nanoparticle sensing. We have also elucidated the mechanism of lineshape shifting and splitting caused by nanoparticles from the perspective of coupling.

We anticipate that our theoretical work will expand the utilization of microring resonators with MRU structures in the field of sensing. Additionally, we look forward to developing more efficient optic isolators to minimize the impact of potential reflected light on lasers and other on-chip devices