Phase evolution of inverse Doppler effect in two-dimensional photonic crystal

The inverse Doppler effect in two dimensional photonic crystal with negative index was proved. In order to analysis the phase evolution of light propagated through this effect, the negative refraction was simulated in static photonic crystal by the Finite Difference Time Domain (FDTD) method. Subsequently, the Bloch wave along the propagating direction was analyzed based on the Fast Fourier Transform (FFT) method. Then, the components were retrieved by iFFT method from the filtered spectrum. The phase evolution of each component was analyzed for the extraction of the backward wave component related to the negative refraction. Furthermore, the phase evolution of signal beam and reference beam on the detecting surface was deduced by dividing the continuous movement of the platform into a series of static moment, where the variation of phase difference with time is the beat frequency. These values obtained with this method are closed to the theoretical values, with an error of 20%, which can explain the phase evolution of light in the inverse Doppler effect. The result reveals the phase evolution of the backward wave in photonic crystal, and provides a new way in dealing with the case of light passing through moving objects.