[Show abstract][Hide abstract] ABSTRACT: The difference between two Gaussian Schell-model cross-spectral densities can give a new genuine correlation function if suitable conditions are met. Generally speaking, the structure of such cross-spectral density changes in a complicated way upon propagation. We consider here the notable exception of shape-invariant beams, and we investigate their intensity and coherence properties. The modal analysis of this class of cross-spectral densities is exploited to devise a synthesis scheme for this type of beam.
Journal of the Optical Society of America A 05/2015; 32(5). DOI:10.1364/JOSAA.32.000790 · 1.56 Impact Factor
[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. Two-dimensional 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.
Journal of the Optical Society of America A 10/2014; 31(10). DOI:10.1364/JOSAA.31.002120 · 1.56 Impact Factor
[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.
(8.4-4)] which is a powerful nonlinear Levin-type 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.
[Show abstract][Hide abstract] 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 high-school students, is proposed. Minimal knowledge of elementary calculus and three-dimensional Euclidean geometry are required.
European Journal of Physics 02/2014; 35(2). DOI:10.1088/0143-0807/35/2/028003 · 0.63 Impact Factor
[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 first-year 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.
European Journal of Physics 08/2013; 34(5):1287. DOI:10.1088/0143-0807/34/5/1287 · 0.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A uniform asymptotic theory of the free-space 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.
Journal of the Optical Society of America A 06/2013; 30(6):1099-106. DOI:10.1364/JOSAA.30.001099 · 1.56 Impact Factor
[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 cross-correlation of the various data-streams. The signals received at each antenna are digitized after low noise amplification and frequency down-conversion, in order to allow subsequent digital post-processing. 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 cross-correlated 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.
Digital Signal Processing 03/2013; 23(2):522–529. DOI:10.1016/j.dsp.2012.10.007 · 1.26 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The Fourier-based 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 so-called "besinc"
function, well known in physical optics.
[Show abstract][Hide abstract] 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 hard-limiter, 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 cross-correlation coefficient between the signal and its reconstructed version. As a result of our analysis, the best achievable cross-correlation 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.
Digital Signal Processing 12/2012; 22(6):1044–1055. DOI:10.1016/j.dsp.2012.07.006 · 1.26 Impact Factor
[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 first-year undergraduates. A central role in our
analysis is played by the so-called "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.
European Journal of Physics 11/2012; 34(2). DOI:10.1088/0143-0807/34/2/359 · 0.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A didactical revisitation of the so-called 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 time-based 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.
European Journal of Physics 09/2012; 33(5):1407-1420. DOI:10.1088/0143-0807/33/5/1407 · 0.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that the cross-spectral density in the far zone of a homogeneous spherical source can be described as a low-pass 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 ξ has a functional structure of the form √(1-ξ2), for 0≤ξ≤1. The cases of spatially incoherent and Lambertian sources are treated as significant examples.
[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 so-called 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 Schell-model 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 propagation-invariant partially coherent beams.
Journal of optics 03/2012; 14(3). DOI:10.1088/2040-8978/14/3/035701 · 2.06 Impact Factor
[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 by-product 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 three-dimensional 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.
Journal of the Optical Society of America A 05/2011; 28(5):887-96. DOI:10.1364/JOSAA.28.000887 · 1.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A theoretical analysis aimed at investigating the divergent character of perturbative series involved in the study of free-space nonparaxial propagation of vectorial optical beams is proposed. Our analysis predicts a factorial divergence for such series and provides a theoretical framework within which the results of recently published numerical experiments concerning nonparaxial propagation of vectorial Gaussian beams find a meaningful interpretation in terms of the decoding operated on such series by the Weniger transformation.
[Show abstract][Hide abstract] ABSTRACT: A simple computational approach is proposed for the evaluation of umbilic diffraction catastrophes which, together with cuspoids, describe the whole hierarchy of the structurally stable diffraction patterns that can be produced by optical diffraction. In this paper, after expanding the double integral representations of hyperbolic and elliptic umbilics as convergent power series, the action of the Weniger transformation on them is studied. Exact expressions for the "on-axis" umbilic field have also been found, which extend previously published results to complex values of the control parameter. Numerical experiments aimed at giving evidence of the effectiveness and implementative ease of the approach are eventually presented.
Journal of the Optical Society of America A 07/2010; 27(7):1661-70. DOI:10.1364/JOSAA.27.001661 · 1.56 Impact Factor