Complex-order Fourier Transforms And Their Optical Implementation

Since Fractional Fourier transforms were firstly introduced in optics in 1993, their application has been exploited in many research domains, such as beam propagation, image processing and signal filtering etc. The extension of real-valued order Fourier transforms to complex regime is natural. This dissertation is devoted to investigate complex-order fractional Fourier transforms and their optical implementation. The main work of the dissertation is summarized as follows.1. Based on the propagation formula of Hermite-Gaussian beams in complex ABCD optical systems, the condition for the implementation of complex-order Fourier transforms is derived. A simple optical setup is proposed to realize general complex-order Fourier transforms.2. It is hard to implement complex-order Fourier transforms for general complex ABCD optical systems, because of the severe conditions. However, we can make a complex-order Fourier transform system out of any complex ABCD optical system, if two lenses or (and) Gaussian apertures with appropriate parameters are appended to the input and output planes.3. By using of the Fresnel diffraction integral, it is demonstrated that the optical fields on two complex-spherical surfaces in a uniform gain medium separated by a distance can be related by complex-order Fourier transforms. The beam propagation between two Gaussian reflectivity mirrors filled with gain medium can be regarded as a complex-order Fourier transform process.4. Complex-order Fourier transforms can not be generally written in terms of the conventional Fourier transform, and so the numerical calculation can not be performed through the Fast Fourier transform (FFT) algorithm. One solution of the problem is to divide the transform into two passages, a real-valued order fractional Fourier transform and a complex-order Fourier transform with the real part of the order being one. Each of the passages can be calculated by single FFT. So numerical calculation results of complex-order Fourier transforms can be obtained after two steps of FFT.