Phase-space analysis for fractional Fourier transform of first-order optical system

The fractional Fourier express of the first-order optical system was derived by decomposing the transfer matrices of first-order optical system into coordinate rotation matrix, scale matrix and chirp matrix in Wigner phase space. The results show that an arbitrary first-order optical system can be expressed as the scaled and chirp modulation fractional Fourier transform. The transfer matrix and diffractive integral formula in frequency domain were acquired by rotating the input and output optical field /2 in the phase space. Accordingly the fractional Fourier transforms of a first-order optical system in frequency domain were also obtained. By comparing the transfer matrices of two first-order optical systems in space and frequency domains respectively, it is found that the two first-order optical systems in different domain can be expressed as two different expressions of one and the same transfer based on different coordinates. At last the condition is derived for an optical system to implement the fractional Fourier transform in space and frequency domains with the same order.