Optical beam propagation in gain media written in terms of complex-order Fourier transforms

The optical beam propagation in gain media is represented in terms of fractional-order Fourier transforms. It is shown that in a uniform amplifying medium characterized by a linear gain there exists a complex-order Fourier transform between the fields on two ‘complex-spherical’ reference surfaces of the given radii and separation. Analytical expressions for the transform order and scale factors are derived and, as an application example, the resonator with Gaussian reflectivity mirrors and linear gain medium is illustrated.