It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complex-order regime. A complex-order Fourier transform deals with the imaginary part as well as the real part of the exponential function in the integral. As a result, while the optical implementation of fractional-order Fourier transform involves gradient-index media or spherical lenses, the optical interpretation of complex-order Fourier transform is practically based on the convolution operation and Gaussian apertures. The beam width of a Gaussian beam subjected to the complex-order Fourier transform is obtained analytically with the ABCD matrix approach.
© 1995 Optical Society of America
Chun-Ching Shih, "Optical interpretation of a complex-order Fourier transform," Opt. Lett. 20, 1178-1180 (1995)