Loop theory for input-output problems in cavities

The input-output formalism is the basis to study the response of an optical cavity to the external stimulations. The existing theories usually handle cavity systems with only one internal mode. However, there is growing interest in more complex systems, especially the hybrid cavity-matter systems, which contain at least two internal modes, one or more from both the optical cavity and the matter. Here we propose a graphical loop theory to calculate and visualize the reflection and transmission spectrum of such a multimode cavity, resembling the role of Feynman diagrams in the quantum field theory. This loop theory gives a unified picture to interpret the experimental observations of a hybrid magnet-light system and is extremely easy to apply to arbitrary complicated problems without any calculations.