Browsing by Author "Mendlovic, D."
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Item Open Access About the Wigner distribution of a graded index medium and the fractional fourier transform operation(SPIE, 1993-08) Özaktaş, Haldun M.; Mendlovic, D.; Lohmann, A. W.Upon propagation through quadratic graded index media, the Wigner distribution of the wavefunction of light rotates uniformly. As a consequence, a definition of fractional Fourier transforms based on rotating the functions Wigner distribution, and another based on propagation through graded index media, are equivalent.Item Open Access Anamorphic fractional Fourier transform: optical implementation and applications(Optical Society of America, 1995-11-10) Mendlovic, D.; Bitran, Y.; Dorsch, R. G.; Ferreira, C.; Garcia, J.; Özaktaş, Haldun M.An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics. A different fractional Fourier order along the orthogonal principal directions is performed. A laboratory experimental system shows preliminary results that demonstrate the proposed theory. Applications such as anamorphic fractional correlation and multiplexing in fractional domains are briefly suggested.Item Open Access Applications of the fractional Fourier transform in optics and signal processing-a review(SPIE, 1996) Özaktaş, Haldun M.; Mendlovic, D.The fractional Fourier transform The fractional Fourier transform is a generalization of the common Fourier transform with an order parameter a. Mathematically, the ath order fractional Fourier transform is the ath power of the fractional Fourier transform operator. The a = 1st order fractional transform is the common Fourier transform. The a = 0th transform is the function itself. With the development of the fractional Fourier transform and related concepts, we see that the common frequency domain is merely a special case of a continuum of fractional domains, and arrive at a richer and more general theory of alternate signal representations, all of which are elegantly related to the notion of space-frequency distributions. Every property and application of the common Fourier transform becomes a special case of that for the fractional transform. In every area in which Fourier transforms and frequency domain concepts are used, there exists the potential for generalization and improvement by using the fractional transform.Item Open Access Applications of the fractional Fourier transform to optical pattern recognition(Cambridge University Press, 1998) Mendlovic, D.; Zalevsky, Z.; Özaktaş, Haldun M.; Yu, T. S.; Jutamulia, S.Item Open Access Chirp filtering in the fractional Fourier domain(Optical Society of America, 1994-11-10) Dorsch, R. G.; Lohmann, A. W.; Bitran, Y.; Mendlovic, D.; Özaktaş, Haldun M.In the Wigner domain of a one-dimensional function, a certain chirp term represents a rotated line delta function. On the other hand, a fractional Fourier transform (FRT) can be associated with a rotation of the Wigner-distribution function by an angle connected with the FRT order. Thus with the FRT tool a chirp and a delta function can be transformed one into the other. Taking the chirp as additive noise, the FRT is used for filtering the line delta function in the appropriate fractional Fourier domain. Experimental filtering results for a Gaussian input function, which is modulated by an additive chirp noise, are shown. Excellent agreement between experiments and computer simulations is achieved.Item Open Access Compact optical temporal processors(Optical Society of America, 1995) Mendlovic, D.; Melamed, O.; Özaktaş, Haldun M.Optical signal processing can be done with time-lens devices. A temporal processor based on chirp-z transformers is suggested. This configuration is more compact than a conventional 4-f temporal processor. On the basis of implementation aspects of such a temporal processor, we did a performance analysis. This analysis leads to the conclusion that an ultrafast optical temporal processor can be implemented.Item Open Access Convolution and filtering in fractional fourier domains(Springer-Verlag, 1994) Özaktaş, Haldun M.; Barshan, B.; Mendlovic, D.Fractional Fourier transforms, which are related to chirp and wavelet transforms, lead to the notion of fractional Fourier domains. The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.Item Open Access Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms(1994) Özaktaş, Haldun M.; Barshan, B.; Mendlovic, D.; Onural, L.A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.Item Open Access Design of dynamically adjustable anamorphic fractional Fourier transformer(Elsevier BV * North-Holland, 1997-03-01) Erden, M. F.; Özaktaş, Haldun M.; Sahin, A.; Mendlovic, D.We form optical systems by using only free space portions and cylindrical lenses, and consider these systems as anamorphic fractional Fourier transformers. We dynamically adjust the transform order, scale factor and field curvature of both orthogonal dimensions of anamorphic fractional Fourier transformation by just changing the focal lengths of cylindrical lenses used in the proposed setups. Here, we also consider two approaches for implementing cylindrical lenses with dynamically adjustable focal lengths. There may also be some other methods to obtain cylindrical lenses having adjustable focal lengths which can successfully be used in these proposed setups.Item Unknown Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering(Optical Society of America, 1996-06-10) Özaktaş, Haldun M.; Mendlovic, D.We consider optical systems composed of an arbitrary number of lenses and filters, separated by arbitrary distances, under the standard approximations of Fourier optics. We show that every such system is equivalent to (i) consecutive filtering operations in several fractional Fourier domains and (ii) consecutive filtering operations alternately in the space and the frequency domains.Item Unknown Filtering in fractional Fourier domains and their relation to chirp transforms(IEEE, 1994-04) Özaktaş, Haldun M.; Barshan, Billur; Onural, Levent; Mendlovic, D.Fractional Fourier transforms, which are related to chirp and wavelet transforms, lead to the notion of fractional Fourier domains. The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.Item Unknown Fourier transforms of fractional order and their optical interpretation(Elsevier, 1993) Özaktaş, Haldun M.; Mendlovic, D.Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a=1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Fractional Fourier transforms can extend the range of spatial filtering operations.Item Open Access Fractional correlation(Optical Society of America, 1995) Mendlovic, D.; Özaktaş, Haldun M.; Lohmann, A. W.Recently, optical interpretations of the fractional-Fourier-transform operator have been introduced. On the basis of this operator the fractional correlation operator is defined in two different ways that are both consistent with the definition of conventional correlation. Fractional correlation is not always a shift-invariant operation. This property leads to some new applications for fractional correlation as shift-variant image detection. A bulk-optics implementation of fractional correlation is suggested and demonstrated with computer simulations.Item Open Access Fractional Fourier Optics(Optical Society of America, 1995-04) Özaktaş, Haldun M.; Mendlovic, D.There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.Item Open Access The Fractional Fourier transform and its applications in optics and signal processing(SPIE International Technical Working Group Newsletter, 1999) Mendlovic, D.; Özaktaş, Haldun M.Item Open Access Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators(Optical Society of America, 1994) Özaktaş, Haldun M.; Mendlovic, D.The complex amplitude distributions on two spherical reference surfaces of given curvature and spacing are simply related by a fractional Fourier transform. The order of the fractional Fourier transform is proportional to the Gouy phase shift between the two surfaces. This result provides new insight into wave propagation and spherical mirror resonators as well as the possibility of exploiting the fractional Fourier transform as a mathematical tool in analyzing such systems.Item Open Access Fractional Fourier transform: simulations and experimental results(Optical Society of America, 1995) Bitran, Y.; Mendlovic, D.; Dorsch, R. G.; Lohmann, A. W.; Özaktaş, Haldun M.Recently two optical interpretations of the fractional Fourier transform operator were introduced. We address implementation issues of the fractional-Fourier-transform operation. We show that the original bulk-optics configuration for performing the fractional-Fourier-transform operation 3J. Opt. Soc. Am. A 10, 2181 1199324 provides a scaled output using a fixed lens. For obtaining a non-scaled output, an asymmetrical setup is suggested and tested. For comparison, computer simulations were performed. A good agreement between computer simulations and experimental results was obtained.Item Open Access Fractional Fourier transforms and their optical implementation. II(Optical Society of America, 1993) Özaktaş, Haldun M.; Mendlovic, D.The derivation of a linear transform kernel for fractional Fourier transforms is presented. Discussed in direct relation to fractal Fourier transforms are spatial resolution and the space-bandwidth product for propagation in graded-index media. Results show how fractional Fourier transforms can be made the basis of generalized spatial filtering systems.Item Open Access Fractional Fourier transforms and their optical implementation: I(Optical Society of America, 1993) Mendlovic, D.; Özaktaş, Haldun M.Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Several mathematical properties are derived.Item Open Access Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform(Optical Society of America, 1994) Mendlovic, D.; Özaktaş, Haldun M.; Lohmann, A. W.Two definitions of a fractional Fourier transform have been proposed previously. One is based on the propagation of a wave field through a graded-index medium, and the other is based on rotating a function's Wigner distribution. It is shown that both definitions are equivalent. An important result of this equivalency is that the Wigner distribution of a wave field rotates as the wave field propagates through a quadratic graded-index medium. The relation with ray-optics phase space is discussed.