Studying Berry Phase In Photonic Crystal

Geometric phase of the wave function is an important concept of a widespread in quantum theory, it can profoundly reflect the holistic nature of quantum system, and its research has been involved in Spectroscopy and optics, atomic and molecular physics, condensed matter physics, quantum field theory and other physical fields. Because geometric phase of the wave function can be researched in different physical background, and the photonic crystals not only have the theoretical value but also have the experimental value, the article chooses the Berry phase in the photonic crystal as a research topic.The first part, I change the Maxwell's equations which are obeyed by the photon into the form of the Schrodinger equation, and prove that the periodic photonic crystal lattice which is similar to the general periodic crystal lattice has the energy band structure, and it make people have a clear physical understanding. Although there are many similarities between photonic crystals and general crystals, the photonic crystal has its own unique physical phenomenon.The second part, we mainly compute the Berry phase in some detail, deriving the Berry vector potential and the Berry curvature concepts, and proving that the Berry curvature is metric invariant.The third part, as we all know, when a Hamiltonian system is cyclical changing with a parameter, its instantaneous eigenfunction after going through a cycle will generate a Berry phase factor. In general, the Hamiltonian system of the photonic crystal is the cyclical changing in momentum space, so the wave function after going through a cycle on the momentum parameter will generate a Berry phase factor. In this section, we study the Berry phase in the linear photonic crystal deeply. In the single-particle model, we obtain the Berry phase factor about k space, and the phase factor expression is similar to Zak phase factor expression. In the two-particle model, we also obtain the expression of the Berry phase factor that is similar to the phase factor expression which is generated by the wave function of the electron rotating in periodic magnetic field after going through a cycle.The fourth part, we study the Berry phase in the nonlinear photonic crystal, the results obtained is different from the Berry phase obtained in the linear photonic crystal.