Fractional Fourier transform(FRFT) is an extension of the conventional Foureir transform. The FRFT has an order that determines the signal domain such as space and spatial-frequency domains. For example, the FRFT with the order of one is identical with the conventional Fourier transform which deals with the information in the spatial-frequency domain. In optics, like the conventional Fourier transform, the FRFT can be implemented very easily, and its added degree of freedom afforded by the order improves performances in a variety of applications. It requires no additional equipment or alignment to implement the FRFT instead of the conventional Fourier transform. In this thesis, the FRFT is applied to the holographic memory(chapter 3 and 4), the joint transform correlator(chapter 5), and the Vander Lugt correlator(chapter 6). The holographic memory with the FRFT gives a spectral distribution with no high peak and lower reconstruction error, and the cross-talk noise of the volume holographic memory with the FRFT is not worse than that of the conventional Fourier plane hologram. The fractional correlation implemented by combining the FRFT and the nonconventional joint transform correlator provides a spectral distribution with no high peak and sharper output distribution, however, it does not show the shift-invariance any more. Generalized Vander Lugt correlator(GVLC), which has been proposed as an optical neural network, has the same optical configuration with the Vander Lugt correlator except for employing the FRFTs instead of the conventional Fourier transforms. The GVLC is applied to pattern classification and the optimal learning rate is developed in order to improve the learning convergence and the classification performance.
