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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.

An extension of the ray-pulse matrix formalism is presented that includes self-phase-modulation, self-focusing, bandwidth limitations, and Gaussian apertures. With this more complete set of matrices it is shown how self-mode-locked lasers can be modeled and how the different pulse-shaping mechanisms interact in order to provide a stable femtosecond output.