Paraxial sharp-edge diffraction of vortex beams by elliptic apertures.

A semi-analytical computational algorithm to model the wave field generated by paraxial diffraction of a class of Laguerre-Gauss beams by sharp-edge elliptic apertures is here developed. Thanks to such a powerful computational tool, some basic aspects of an intriguing and still unexplored singular optics scenario can be studied, within a geometry as simple as possible, with arbitrarily high accuracies.

"Analytical continuation" of flattened Gaussian beams.

A purely analytical extension of flattened Gaussian beams [Opt. Commun.107, 335 (1994)OPCOB80030-401810.1016/0030-4018(94)90342-5] to any values of beam order is here proposed. Due to it, the paraxial propagation problem of axially symmetric, coherent flat-top beams through arbitrary ABCD optical systems can definitely be solved in closed form via a particular bivariate confluent hypergeometric function.

Nijboer-Zernike's aberration theory: computational achievements via Tchebychev's polynomials approximation theory.

Nijboer-Zernike's diffraction theory of aberration is a nearly abandoned jewel of physical optics. The present paper constitutes an attempt to extend its practical feasibility. It is found that such a task can be achieved by using what is probably the most important property of  Tchebychev's polynomials.

Paraxial sharp-edge diffraction: a general approach.

A general reformulation of classical sharp-edge diffraction theory is proposed within paraxial approximation. The, not so much known, Poincaré vector potential construction is employed directly inside Fresnel's 2D integral in order for it to be converted into a single 1D contour integral over the aperture boundary. Differently from the recently developed paraxial revisitation of BDW's theory, such approach should be applicable, in principle, to arbitrary wavefield distributions impinging onto arbitrarily shaped sharp-edge planar apertures. However, in those cases where such a conversion were not analytically achievable, our approach allows Fresnel's integral to be easily converted, irrespective of the shape and the regularity features of the aperture geometry, into a double integral defined onto a square domain. A couple of interesting examples of application of the proposed method is presented.

Sharp-edge diffraction under Bessel beam illumination: a catastrophe optics perspective.

Boundary diffraction wave theory and catastrophe optics have already proved to be a formidable combination in computational optics. In the present paper, a general paraxial theory aimed at dealing paraxial diffraction of Bessel beams by arbitrarily shaped sharp-edge apertures is developed. A key ingredient of our analysis is the δ-like nature of the angular spectrum of nondiffracting beams. This allows the diffracted wavefield to be effectively represented through two-dimensional integrals defined onto rectangular domains, whose numerical evaluation is easily achievable via standard Monte Carlo techniques. As a byproduct of the present analysis, a simple explanation of a recently observed property of some "heart-like" apertures to flatten the axial intensity of apodized Bessel beams is also provided.

Sharp-edge diffraction under Gaussian illumination: a paraxial revisitation of Miyamoto-Wolf's theory.

A "genuinely" paraxial version of Miyamoto-Wolf's theory aimed at dealing with sharp-edge diffraction under Gaussian beam illumination is presented. The theoretical analysis is carried out in such a way the Young-Maggi-Rubinowicz boundary diffraction wave theory can be extended to deal with Gaussian beams in an apparently straightforward way. The key for achieving such an extension is the introduction of suitable "complex angles" within the integral representations of the geometrical and boundary diffracted wave components of the total diffracted wavefield. Surprisingly enough, such a simple (although not rigorously justified) mathematical generalization seems to work well within the Gaussian realm. The resulting integrals provide meaningful quantities that, once suitably combined, give rise to predictions that are in perfect agreement with results already obtained in the past. An interesting and still open theoretical question about how to evaluate "Gaussian geometrical shadows" for arbitrarily shaped apertures is also discussed.

"Tailoring axial intensity of laser beams with a heart-shaped hole," by Wang et al.: comment.

Some comments about the recently published Optics Letters paper "Tailoring axial intensity of laser beams with a heart-shaped hole," by Wang et al., Opt. Lett.42, 4921 (2017)OPLEDP0146-959210.1364/OL.42.004921, are provided.

Twisting partially coherent light.

Twisted Gaussian Schell-model beams were introduced 25 years ago as a celebrated example of a "genuinely two-dimensional" partially coherent wavefield. Today, a definite answer about the effect that a twist phase should produce on an arbitrary cross-spectral density has not yet been reached. In the present Letter, the necessary and sufficient condition for a typical Schell-model partially coherent CSD endowed with axial symmetry to be successfully mapped onto a bonafide twisted CSD is addressed. In particular, it is proved that any shift-invariant degree of coherence of the form µ(|r1-r2|) is "twistable" if and only if the zeroth-order Hankel transform of the radial function µ(r)exp(ur2/2) (with u being the twist strength) turns out to be a well-defined, non-negative function.

Heart diffraction.

A theoretical study of the plane-wave diffraction by a heart-like sharp-edge aperture corresponding to the involute of a circle is proposed here. Through the recently developed paraxial boundary diffraction wave theory, expressed via the language of catastrophe optics, the presence of pseudo-nondiffracting regions within the three-dimensional spatial intensity distribution of the diffracted wavefield is intuitively explained and quantitatively characterized. The results of some old, beautiful, but nearly forgotten, diffraction experiments have also been reconsidered from such a peculiar and rather unorthodox perspective.

Catastrophe optics of sharp-edge diffraction.

A classical problem of diffraction theory, namely plane wave diffraction by sharp-edge apertures, is here reformulated from the viewpoint of the fairly new subject of catastrophe optics. On using purely geometrical arguments, properly embedded into a wave optics context, uniform analytical estimates of the diffracted wavefield at points close to fold caustics are obtained, within paraxial approximation, in terms of the Airy function and its first derivative. Diffraction from parabolic apertures is proposed to test reliability and accuracy of our theoretical predictions.

Twisted Schell-model beams with axial symmetry.

The problem of when a twist can be impressed on a partially coherent beam is solved for Schell-model 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.

Uniform asymptotics of paraxial boundary diffraction waves.

Starting from the paraxial formulation of the boundary-diffracted-wave 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" hard-edge 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.

Plane-wave Fresnel diffraction by elliptic apertures: a Fourier-based approach.

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.

Uniform approximation of paraxial flat-topped beams.

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.

Numerical computation of diffraction catastrophes with codimension eight.

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.

Optimizing diffraction catastrophe evaluation.

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.

Evaluation of cuspoid and umbilic diffraction catastrophes of codimension four.

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.

Decoding divergent series in nonparaxial optics.

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.

On the numerical evaluation of umbilic diffraction catastrophes.

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.

On a class of electromagnetic diffraction-free beams.

We present a general theory of electromagnetic diffraction-free beams composed of uncorrelated Bessel modes. Our approach is based on the direct application of the nonnegativity constraint to the cross-spectral density tensor describing the electromagnetic field distribution. The field correlation properties are most conveniently derived in the spatial frequency domain, where the angular spectrum takes on the form of an infinitely thin ring. We also present several examples, including a vector generalization of the recently introduced dark and antidark diffraction-free beams.