Publications (140)211.28 Total impact
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ABSTRACT: The problem of when a twist can be impressed on a partially coherent beam is solved for Schellmodel fields endowed with axial symmetry. A modal analysis can be performed for any such beam, thus permitting evaluation of whether it will withstand the twisting process. Beyond exemplifying some twistable beams, it is shown that, for certain correlation functions, the beam cannot be twisted, no matter how the numerical parameters are chosen. (C) 2015 Optical Society of America  [Show abstract] [Hide abstract]
ABSTRACT: The difference between two Gaussian Schellmodel crossspectral densities can give a new genuine correlation function if suitable conditions are met. Generally speaking, the structure of such crossspectral density changes in a complicated way upon propagation. We consider here the notable exception of shapeinvariant beams, and we investigate their intensity and coherence properties. The modal analysis of this class of crossspectral densities is exploited to devise a synthesis scheme for this type of beam.  [Show abstract] [Hide abstract]
ABSTRACT: Starting from the paraxial formulation of the boundarydiffractedwave theory proposed by Hannay [J. Mod. Opt. 47, 121–124 (2000)] and exploiting its intrinsic geometrical character, we rediscover some classical results of Fresnel diffraction theory, valid for “large” hardedge apertures, within a somewhat unorthodox perspective. In this way, a geometrical interpretation of the Schwarzchild uniform asymptotics of the paraxially diffracted wavefield by circular apertures [K. Schwarzschild, Sitzb. München Akad. Wiss. Math.Phys. Kl. 28, 271–294 (1898)] is given and later generalized to deal with arbitrarily shaped apertures with smooth boundaries. A quantitative exploration is then carried out, with the language of catastrophe optics, about the diffraction patterns produced within the geometrical shadow by opaque elliptic disks under plane wave illumination. In particular, the role of the ellipse’s evolute as a geometrical caustic of the diffraction pattern is emphasized through an intuitive interpretation of the underlying saddle coalescing mechanism, obtained by suitably visualizing the saddle topology changes induced by letting the observation point move along the ellipse’s major axis.  [Show abstract] [Hide abstract]
ABSTRACT: A simple theoretical approach to evaluate the scalar wavefield, produced, within paraxial approximation, by the diffraction of monochromatic plane waves impinging on elliptic apertures or obstacles is presented. We find that the diffracted field can be mathematically described in terms of a Fourier series with respect to an angular variable suitably related to the elliptic parametrization of the observation plane. The convergence features of such Fourier series are analyzed, and a priori truncation criterion is also proposed. Twodimensional maps of the optical intensity diffraction patterns are then numerically generated and compared, at a visual level, with several experimental pictures produced in the past. The last part of this work is devoted to carrying out an analytical investigation of the diffracted field along the ellipse axis. A uniform approximation is derived on applying a method originally developed by Schwarzschild, and an asymptotic estimate, valid in the limit of small eccentricities, is also obtained via the Maggi–Rubinowicz boundary wave theory.  [Show abstract] [Hide abstract]
ABSTRACT: Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series $\mathcal{E}(z) \sim \sum_{n=0}^{\infty} (1)^n n! z^n$ is a very important model for the ubiquitous factorially divergent perturbation expansions in physics. In this article, we analyze the summation of the Euler series by Pad\'e approximants and the delta transformation [E. J. Weniger, Comput. Phys. Rep. Vol.10, 189 (1989), Eq. (8.44)] which is a powerful nonlinear Levintype transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a new factorial series representation of the truncation error of the Euler series [R. Borghi, Appl. Num. Math. Vol.60, 1242 (2010)]. We derive explicit expressions for the transformation errors of Pad\'e approximants and of the delta transformation. A subsequent asymptotic analysis proves \emph{rigorously} the convergence of both Pad\'e and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Pad\'e. This is in agreement with previous numerical results. 

Article: On Newton’s shell theorem
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ABSTRACT: In the present letter, Newton’s theorem for the gravitational field outside a uniform spherical shell is considered. In particular, a purely geometric proof of proposition LXXI/theorem XXXI of Newton’s Principia, which is suitable for undergraduates and even skilled highschool students, is proposed. Minimal knowledge of elementary calculus and threedimensional Euclidean geometry are required.  [Show abstract] [Hide abstract]
ABSTRACT: The full text of this article is available in the PDF provided.  [Show abstract] [Hide abstract]
ABSTRACT: An elementary introduction to the adiabatic invariants of the Kepler problem is proposed. Unlike the other didactical expositions already present in the literature, which are based on the Hamilton–Jacobi theory of mechanics, our derivation is suitable to be grasped even by firstyear undergraduates. A central role in the present analysis is played by an elementary proof of the virial theorem for the Kepler problem which is based on the chain rule for derivatives. As a byproduct of our analysis, an interpretation of Keplerian orbit eccentricities in terms of the time average of the position vector direction is also provided.  [Show abstract] [Hide abstract]
ABSTRACT: A uniform asymptotic theory of the freespace paraxial propagation of coherent flattened Gaussian beams is proposed in the limit of nonsmall Fresnel numbers. The pivotal role played by the error function in the mathematical description of the related wavefield is stressed.  [Show abstract] [Hide abstract]
ABSTRACT: In Very Long Baseline Interferometry, signals from far radio sources are simultaneously recorded at different antennas, with the purpose of investigating their physical properties. The recorded signals are generally modeled as realizations of Gaussian processes, whose power is dominated by the system noise at the receiving antennas. The actual signal coming from the radio source can be detected only after crosscorrelation of the various datastreams. The signals received at each antenna are digitized after low noise amplification and frequency downconversion, in order to allow subsequent digital postprocessing. The applied quantization is coarse, 1 or 2 bits being generally associated to the signal amplitude. In modern applications the sampling is typically performed at a high rate, and subchannels are then generated by filtering, followed by decimation and requantization of the signal streams. The redigitized streams are then crosscorrelated to extract the physical observables. While the classical effect of quantization has widely been studied in the past, the decorrelation induced by the filtering and requantization process is still characterized experimentally, mainly due to its inherent mathematical complexity. In the present work we analyze the above problem, and provide algorithms and analytical formulas aimed at predicting the induced decorrelation for a wide class of quantization schemes, with the unique assumption of weakly correlated signals, typically fulfilled in VLBI and radio astronomy applications.  [Show abstract] [Hide abstract]
ABSTRACT: The Fourierbased analysis customarily employed to analyze the dynamics of a simple pendulum is here revisited to propose an elementary iterative scheme aimed at generating a sequence of analytical approximants of the exact law of motion. Each approximant is expressed by a Fourier sum whose coefficients are given by suitable linear combinations of Bessel functions, which are expected to be more accessible, especially at an undergraduate level, with respect to Jacobian elliptic functions. The first three approximants are explicitely obtained and compared with the exact solution for typical initial angular positions of the pendulum. In particular, it is shown that, at the lowest approximation level, the law of motion of the pendulum turns out to be adequately described, up to oscillation amplitudes of $\pi/2$, by a sinusoidal temporal behaviour with a frequency proportional to the square root of the socalled "besinc" function, well known in physical optics. 
Article: Efficient reconstruction of sampled 1bit quantized Gaussian signals from sine wave crossings
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ABSTRACT: We study the reconstruction of a Gaussian random signal, subject to extreme clipping. The reconstruction is achieved by adding a high frequency sinusoidal reference signal prior to the hardlimiter, and by low pass filtering the output. Such a scheme belongs to the area of signal reconstruction from Sine Wave Crossings (SWC). In the present paper we study in detail the effect of sampling in time domain on the reconstruction algorithm, and we carry out an analysis, valid for high sampling rates, leading to approximate analytical expressions of the crosscorrelation coefficient between the signal and its reconstructed version. As a result of our analysis, the best achievable crosscorrelation coefficient, together with the corresponding setting of the configuration parameters, i.e., the frequency and power of the reference signal, is obtained as a function of the sampling rate. Asymptotic closed form formulas are derived in the limit of very large sampling rates.  [Show abstract] [Hide abstract]
ABSTRACT: A didactical exposition of the classical problem of the trajectory determination of a body, subject to the gravity in a resistant medium, is proposed. Our revisitation is aimed at showing a derivation of the problem solution which should be as simple as possible from a technical point of view, in order to be grasped even by firstyear undergraduates. A central role in our analysis is played by the socalled "chain rule" for derivatives, which is systematically used to remove the temporal variable from Newton's law to derive the differential equation of the Cartesian representation of the trajectory, with a considerable reduction of the overall mathematical complexity. In particular, for a resistant medium exerting a force quadratic with respect to the velocity our approach leads, in an elementary way, to the differential equation of the trajectory, which is subsequently solved by series expansion. A comparison of the polynomial approximants obtained by truncating such series with the solution recently proposed through a homotopy analysis is also presented. 
Article: On the tumbling toast problem
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ABSTRACT: A didactical revisitation of the socalled tumbling toast problem is presented here. The numerical solution of the related Newton's equations has been found in the space domain, without resorting to the complete timebased law of motion, with a considerable reduction of the mathematical complexity of the problem. This could allow the effect of the different physical mechanisms ruling the overall dynamics to be appreciated in a more transparent way, even by undergraduates. Moreover, the availability from the literature of experimental investigations carried out on tumbling toast allows us to propose different theoretical models of growing complexity in order to show the corresponding improvement of the agreement between theory and observation.  [Show abstract] [Hide abstract]
ABSTRACT: We show that the crossspectral density in the far zone of a homogeneous spherical source can be described as a lowpass filtered version of that existing across the source surface. We prove that, to an excellent approximation, the corresponding filter with respect to a (normalized) spatial frequency. has a functional structure of the form root 1  xi(2), for 0 <= xi <= 1. The cases of spatially incoherent and Lambertian sources are treated as significant examples. (C) 2012 Optical Society of America  [Show abstract] [Hide abstract]
ABSTRACT: A computational strategy, aimed at evaluating diffraction catastrophes belonging to the X(9) family is presented. The approach proposed is based on the use of power series expansions, suitably derived for giving meaningful representation of the whole (0)X(9) subfamily, jointly with a powerful sequence transformation algorithm, the socalled Weniger transformation. The convergence features of the above series expansions are investigated, and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the Weniger transformation, as well as the ease of implementation of the whole approach.  [Show abstract] [Hide abstract]
ABSTRACT: A general procedure is presented for the evaluation of the modes of a thin annular scalar source, whose angular mutual intensity is of the Schellmodel type. Starting from the knowledge of the modes, the coherence properties of the field propagated from the source in paraxial conditions can be evaluated. When the propagated field is collimated by a suitable converging lens, presented results apply to the synthesis of propagationinvariant partially coherent beams.  [Show abstract] [Hide abstract]
ABSTRACT: A theoretical analysis is proposed, aimed at investigating the character of those power series expansions recently considered for the evaluation of several types of diffraction catastrophes. A hyperlinear convergence is found to be the signature for such expansions, so that the results of the numerical experiments recently carried out find a meaningful interpretation in terms of the accelerating action operated by the Weniger transformation. As an important byproduct of our analysis, simple criteria, aimed at numerically optimizing the diffraction catastrophe evaluations, are provided through analytical expressions.  [Show abstract] [Hide abstract]
ABSTRACT: The evaluation of the two diffraction catastrophes of codimension four, namely, the butterfly and the parabolic umbilic, is here proposed by means of a simple computational approach developed in the past to characterize the whole hierarchy of the structurally stable diffraction patterns produced by optical diffraction in threedimensional space. In particular, after expanding the phase integral representations of butterfly and parabolic umbilic in terms of (slowly) convergent power series, the retrieving action of the Weniger transformation on them is investigated through several numerical experiments. We believe that the methodology and the results presented here could also be of help for the dissemination of catastrophe optics to the widest scientific audience.
Publication Stats
3k  Citations  
211.28  Total Impact Points  
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Institutions

19972015

Università Degli Studi Roma Tre
 • Department of Electronic Engineering
 • Department of Applied Electronics
 • Department of Mathematics and Physics
Roma, Latium, Italy


1999

University of Rome Tor Vergata
Roma, Latium, Italy


19951998

Sapienza University of Rome
 Department of Physics
Roma, Latium, Italy
