Certain feedback loops can be used in photorefractive optical schemes to implement periodic states and create spatial gratings characterized by extremely high or low diffraction efficiencies. This highly nonlinear phenomenon is studied both experimentally and numerically. An analytical method is developed for analyzing periodic states with the use of symmetries of time-dependent diffraction equations and fast feedback response. The method is applied to describe the properties of periodic states, including their spatial structure, diffraction-efficiency oscillation period and amplitude, and characteristics of feedback-controlled strong phase modulation.