Browsing by Subject "Space-bandwidth product"
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Item Unknown Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product(Optical Society of America, 2010-07-30) Oktem, F. S.; Özaktaş, Haldun M.Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space-frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space-bandwidth product. The bicanonical width product, which is a generalization of the space-bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.Item Open Access Fast and accurate algorithm for the computation of complex linear canonical transforms(Optical Society of America, 2010-08-05) Koç A.; Özaktaş, Haldun M.; Hesselink, L.A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs, and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm. The algorithm is based on decomposition of an arbitrary CLCT matrix into real and complex chirp multiplications and Fourier transforms. The samples of the output are obtained from the samples of the input in ∼N log N time, where N is the number of input samples. A space-bandwidth product tracking formalism is developed to ensure that the number of samples is information-theoretically sufficient to reconstruct the continuous transform, but not unnecessarily redundant.Item Open Access Fast and accurate algorithms for quadratic phase integrals in optics and signal processing(SPIE, 2011) Koç, A.; Özaktaş, Haldun M.; Hesselink L.The class of two-dimensional non-separable linear canonical transforms is the most general family of linear canonical transforms, which are important in both signal/image processing and optics. Application areas include noise filtering, image encryption, design and analysis of ABCD systems, etc. To facilitate these applications, one need to obtain a digital computation method and a fast algorithm to calculate the input-output relationships of these transforms. We derive an algorithm of NlogN time, N being the space-bandwidth product. The algorithm controls the space-bandwidth products, to achieve information theoretically sufficient, but not redundant, sampling required for the reconstruction of the underlying continuous functions. © 2011 SPIE.Item Open Access Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals(Optical Society of America, 2010-05-12) Koç A.; Özaktaş, Haldun M.; Hesselink, L.We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in gradedindex media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non- orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case. The algorithm takes ∼ñ log ñ time, where ñ is the two-dimensional space-bandwidth product of the signal. Our method properly tracks and controls the space-bandwidth products in two dimensions, in order to achieve information theoretically sufficient, but not wastefully redundant, sampling required for the reconstruction of the underlying continuous functions at any stage of the algorithm. Additionally, we provide an alternative definition of general 2D-NS-LCTs that shows its kernel explicitly in terms of its ten parameters, and relate these parameters bidirectionally to conventional ABCD matrix parameters.Item Open Access Fundamental structure of Fresnel diffraction: Longitudinal uniformity with respect to fractional Fourier order(Optical Society of America, 2011-12-24) Özaktaş, Haldun M.; Arik, S. O.; Coşkun, T.Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. Transverse samples can be taken on these surfaces with separation that increases with propagation distance. Here, we are concerned with the separation of the spherical reference surfaces along the longitudinal direction. We show that these surfaces should be equally spaced with respect to the fractional Fourier transform order, rather than being equally spaced with respect to the distance of propagation along the optical axis. The spacing should be of the order of the reciprocal of the space-bandwidth product of the signals. The space-dependent longitudinal and transverse spacings define a grid that reflects the structure of Fresnel diffraction.