The figure below shows the structure of a microring resonator. Light propagating in the waveguide couples with the light in the microring resonator through a coupler (red shaded area). The coupler can be modeled as a beam splitter that connects the fields in the waveguide and microring resonator via: where It | + | | = 1. For simplicity, we assume t and x are real positive numbers. The microring resonator's circumference is L, effective refractive index is n, and the roundtrip transmissivity is a. Waveguide Ei Microring resonator (a) Show the relation between E; and E12. (b) Given Eij, derive Er, E; and E, based on the parameters of the microring resonators and wavelength. (c) What are the frequencies that meet the resonance condition of the microring resonator? (d) Derive the free spectral range of the microring resonator. (e) Under the "critical coupling" condition, the transmitted field E,, vanishes when the light is on resonant with the microring resonator. Derive the critical coupling condition