Nonlinear geometric phase in optics: Fundamentals and applications

Pancharatnam–Berry phase, the manifestation of geometric phase in optics, which originates from the adiabatic evolution of light in the anisotropic medium, has been widely explored in the past decades. With the rapid development of nonlinear metamaterials and metasurfaces, the concept of geometric phase has been extended to the regime of nonlinear optics and attracts growing interest due to its capability of manipulating the nonlinear optical fields. On the one hand, nonlinear geometric phase offers a platform to investigate the fundamental principles in light–matter interactions during the nonlinear optical processes. On the other hand, it can be applied to design nonlinear optical elements with versatile functionality, such as wavefront engineering, optical imaging, and optical holography. To give a comprehensive review of nonlinear geometric phase, here, we overview the fundamental mechanisms, including the symmetry selection rules, the origin of the geometric phase in harmonic generations, and the characteristics of nonlinear geometric phase. We summarize the related applications based on the concepts of nonlinear geometric phase and symmetry selection rules, mainly focusing on the recent progresses from nonlinear optical sources to nonlinear optical field manipulation and beyond. In addition to reviewing the achievements in a wide variety of applications based on the nonlinear geometric phase, we also give prospects on their future development. With distinctive features, nonlinear geometric phase may find its importance in not only scientific research but also industrial applications.