Light beams carrying out topological singularities in the form of phase vortices continue to be among the most investigated “optical objects,” since the pioneering work by Allen et al. [1–3]. The use of sharp-edge diffraction to unveil the hidden structure of phase singularities carried out by such vortex beams has inspired a great deal of work, both theoretical and experimental [4–6]. Most of the above papers explored a similar scenario, in which the Laguerre–Gauss (LG) vortex beams were diffracted by hard-edge half planes. Sharp-edge diffraction of the LG vortex beams has also been investigated with polygonal as well as circular apertures [7–16], although most of such investigations have been limited to far-field (Fraunhofer) analyses.

A different scenario, which seems to be still largely unexplored so far, has to do with sharp-edge diffraction of vortex beams by noncircular smooth apertures. To point out its relevance, a few words by Michael Berry [17] deserve to be quoted: “for general smooth boundaries, focusing of edge waves occurs on caustic curves (envelope of normals to the boundary).” Berry’s words must be contextualized within the framework of the so-called catastrophe optics [18,19]. Caustics generated from plane wave diffraction by smooth sharp-edge apertures are approximately arranged along the geometric evolute of the aperture rim [20,21]. The possibility of using elliptic apertures to explore paraxial vortex sharp-edge diffraction represents the main idea of the present Letter.

Ellipses have smooth, curved rims whose shapes are controlled by a single real parameter (the eccentricity or, equivalently, the aspect ratio). Ellipse’s geometrical evolute is a symmetric closed curve called astroid, containing as many as four cusps. Ellipses are the simplest smooth closed curves not endowed with radial symmetry (except for a null eccentricity). They are natural candidates, together with LG beams, for building up a scenario mathematically as simple as possible, in which the interaction between singularities of different nature (optical vortices and caustics) can be explored.

In the first part of the Letter, a semi-analytical algorithm will be developed to evaluate, up to arbitrarily high (in principle) accuracies, the optical field generated by the paraxial diffraction of a monochromatic (wavelength $\lambda$) vortex LG beams carrying out a given topological charge $m\in \mathbb {Z}$ by a sharp-edge elliptic aperture $\cal{A}$ of a given aspect ratio $\chi \in (0,1)$, placed in a planar, opaque screen and delimited by the boundary $\Gamma =\partial \cal{A}$. The screen coincides with the plane $z=0$ of a suitable cylindrical reference frame $(\boldsymbol{r} ;z)$, whose $z$ axis coincides with the ellipse symmetry axis, as well as with the mean propagation axis of the impinging LG beam. On denoting $\psi _0(\boldsymbol{r} )$, the disturbance distribution of the latter at $z=0^-$, the field distribution, say $\psi$, at the observation point $P\equiv (\boldsymbol{r} ;z>0)$, is given, within paraxial approximation and apart from an overall phase factor $\exp (\textrm{i} k z)$, by the following dimensionless version of Fresnel’s integral:

As we shall see in a moment, the numerical evaluation of the last integrals does not present any problems, even for nonsmall Fresnel’s numbers and/or nonsmall values of $X$. Moreover, as far as the convergence of the Fourier series into Eq. (5) is concerned, a truncation criterion similar to that proposed in Ref. [23] will be adopted, according to which the index $n$ will run within the range $[-N,N]$, where

In the second part of the Letter, the scenario above described will be initially explored with a LG beam with topological charge $m=3$. Such a value guarantees a reasonably high ratio between the values of the impinging optical intensity $|\psi _0|^2$ at the aperture boundary and those in the central region. In this way, the main contribution to the diffracted beam $\psi$ is expected to come from the rim $\Gamma$ (with the three singularities being initially all located at the beams’ axis). For simplicity, the spot size of the incident LG beams will be assumed to be much greater than the aperture sizes, a choice which also represents a sort of “computational worst case,” as far as the evaluation of the integrals into Eq. (5) is concerned. In formulas, $\gamma \simeq 0$ and thus $V\simeq U$.

Our exploration starts from the far-field limit $U\to 0$, where the diffracted field into Eq. (1) takes on, apart from unessential overall phase and amplitude factors, the following form:

In what follows, elliptic frames (in place of classical rectangular frames) will be employed to visualize amplitude and phase diffraction patterns. Such a technical expedient turns out to be useful to optimize the pattern visualization. In all figures, the elliptic frames will be chosen in order for all involved singularities, i.e., vortices and evolutes, to be included.

To give a single example of far-field evaluation, in Fig. 1, 2D maps of the intensity (first row) and of the phase (second row) distributions of the wave field produced by the diffraction of an ideal plane wave carrying out a topological charge $m=3$ by an elliptic aperture with $\chi =9/10$ are shown for “low” values of Fresnel’s numbers, i.e., for $U <10$. In all figures, the smooth white curve represents the elliptic aperture, while the cusped white curve is the corresponding evolute, defined through the parametric equation $\left \{(1-\chi ^2) \cos ^3t, \frac {1-\chi ^2}\chi \sin ^3t\right \}$, with $t\in [0,2\pi ]$. Moreover, the two white points correspond to the above defined “far-field estimates” of the outer singularities, where in the present case, $\bar p_x\,\simeq \,{2.074901}$. From the figure, it can be appreciated how the far-field estimate of the outer singularities works reasonably well also for Fresnel’s number values out of the far-field range, as for example $U=7$ certainly is. In other words, it is expected that the rectilinear trajectories penetrate inside the near-field zone, in agreement with some of the experimental results provided in [8] for a rhomb-shaped aperture. On further increasing $U$, not only the outer singularities continue to get closer and closer, but it is also possible to appreciate a counterclockwise (for $m>0$) rotation, with respect to the $x$ axis, of the segment joining them. In other words, the outer singularity trajectories in the observation space $(\boldsymbol {r};U)$ deviate from the far-field rectilinear path, up to intersect the aperture evolute. To give an example, in Fig. 2, the phase distributions of the diffracted field are shown for $\chi =7/10$ and $U=19/2$ (a), as well as for $\chi =925/1000$ and $U=89/2$ (b). Note that, in this figure, the white points now represent the “actual positions” of all three singularities. Moreover, the Fresnel numbers have been chosen in order for the outer singularities to “touch” the astroid-shaped caustics. On further increasing $U$, all three singularities will be located inside the diamond-shaped region. To appreciate, at least qualitatively, such a new (topologically speaking) scenario, Fig. 3 shows 2D maps of the intensity (first row) and phase (second row) distributions for $\chi =9/10$ and for four increasing different Fresnel’s numbers.

 

Fig. 1. 2D maps of the intensity (first row) and of the phase (second row) distributions of the wave field produced by the diffraction of an ideal plane wave carrying out a topological charge $m=3$ by an elliptic aperture with $\chi =9/10$. Fresnel’s numbers are $U=1$ (first column), $U=3$ (second column), $U=5$ (third column), and $U=7$ (fourth column).

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Fig. 2. 2D maps of the sole phase distributions for $\chi =7/10$ and $U=19/2$ (a) as well as for $\chi =925/1000$ and $U=89/2$ (b). The white points denote the “actual positions” of the three singularities.

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Fig. 3. 2D maps of the intensity (first row) and of the phase (second row) distributions of the wave field produced by the diffraction of an ideal plane wave carrying out a topological charge $m=3$ by the elliptic aperture ($\chi =9/10$) at different transverse observation planes. $U=19$ (first column), $U=41$ (second column), $U=59$ (third column), and $U=79$ (fourth column).

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Several other numerical experiments, not shown here, have been carried out for different values of $\chi$ as well as of $m$. All of them seem to confirm the key role the aperture evolute should play in discriminating the main topological features of the diffraction patterns. A partial, qualitative confirm of such conjecture can be found in [24], where the propagation of LG vortex beams apertured by smooth amplitude and phase Gaussian filters was studied. In particular, the integration domain of the resulting diffraction integrals coincides with the whole $\mathbb {R}^2$ plane, while it is just the finiteness as well as the compactness of the integration domain (the elliptic aperture) that makes our intensity and phase patterns structurally different from those obtained in [24]. Such a structural difference becomes more and more evident when Fresnel’s number takes on large values. In Fig. 4 the 2D maps of the intensity and phase are shown for $U=101$ and for three different aperture shapes, namely, $\chi =5/10$, $\chi =7/10$, and $\chi =9/10$. In can be appreciated how the diffracted intensity rests onto the geometrical optics caustics (the cusped curve), which is “dressed,” to use a catastrophe optics terminology, by the characteristic Pearcey-based diffraction catastrophe [21]. Our last numerical experiment, showed in Fig. 5, is aimed at showing a visual comparison of diffraction patterns generated by different topological charges, namely, $m \in \{1, 2, 3, 4\}$. For simplicity, the chosen aperture corresponds to $\chi =9/10$, while the observation plane is located at $U=307$. It appears indubitably true that, when $U\gg 1$, the intensity and phase patterns inside the astroid display spectacularly regular lattices. Thanks to the high resolution of the pictures (each of them contains $300\times 300$ points), it can be appreciated, for instance, a dependence of some features of the astroid-lattice topology on the parity of $m$ rather than its absolute value. For instance, due to symmetry reasons, odd values of $m$ must necessarily display one of the singularities exactly at the center of the diffraction pattern, whereas even values of $m$ do not. This, in turn, implies that the corresponding intensity patterns would share some topological features, as it can be checked simply by expanding the pictures. Moreover, for the lowest value of $m=1$, the contribution coming from the central region of the illuminated aperture turns out to be not negligible with respect to that coming from the aperture rim. On the contrary, for instance, when $m=4$, the diffracted wave field distribution could be ascribed to the sole edge waves, similarly to what happened in the plane wave diffraction by opaque elliptic obstacles (see, i.e., Fig. 10 of [23]).

 

Fig. 4. 2D maps of the intensity (first row) and of the phase (second row) distributions of the wave field produced by the diffraction of an ideal plane wave carrying out a topological charge $m=3$ at $U=101$, by elliptic apertures of different shapes. $\chi =5/10$ (first column), $\chi =7/10$ (second column), and $\chi =9/10$ (third column). White cusped curves represent the corresponding aperture evolutes.

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Fig. 5. 2D maps of the intensity (first row) and of the phase (second row) distributions of the wave field produced by the diffraction of an ideal plane wave carrying out a topological charge $m=1$ (first column), $m=2$ (second column), $m=3$ (third column), and $m=4$ (fourth column), at $U=307$. The aperture shape is given by $\chi =9/10$. White cusped curve represents the aperture evolute.

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In the present Letter, a semi-analytical algorithm for studying paraxial diffraction of vortex LG beams by centered elliptic apertures has been developed. The main conjecture of our paper is that the interaction between the singularities (phase vortices) carried out by the incident (plane wave) field and the singularities generated by the edge waves (the cusp-shaped evolute) is mainly responsible for the resulting topological complexity of the diffraction patterns. In particular, our preliminary results seem to confirm the key role played by aperture’s evolute in determining distinct topological scenarios, as far as intensity and phase distributions are concerned, inside and outside of it, over a wide range of Fresnel’s numbers.

The nature of the present Letter is merely computational. Our algorithm, based on rapidly convergent Fourier series, presents only one bottleneck: the numerical evaluation of the integrals into Eq. (5). However, it has been shown how high-resolution maps of the diffracted wave field with, in principle, arbitrarily high accuracies can be (and have been) achieved even for Fresnel numbers of the order of some hundreds. The availability of such an accurate and powerful computational tool should be welcomed by the optical community for several reasons. In primis, the results presented here, thanks to their high accuracy, could become a benchmark for testing new numerical methods able to deal with more general scenarios.

From an experimental perspective, we hope what is contained here could stimulate future activities. In particular, due to the intrinsic structural stability of cusped singularities, we do hope experimental setups similar to other conceived in the past for similar purposes [7,25,26] could be conceived for experimentally exploring the vortex/caustics interaction described here.

Theoretically speaking, several open problems would deserve deeper investigations, the most intriguing being the study of the complex topology of the diffraction patterns inside the astroid. To this end, the joint use of our simulations with more or less sophisticated algorithms of singularity tracking could greatly improve our understanding of the high-Fresnel number dynamics. That the edge waves generated at the aperture rim still focalize along the aperture evolute seems to be valid also for the class of vortex beams here considered and not only under plane wave illuminations [21]. This is something which could have not been taken for granted a priori. Accordingly, our hope is that the beautiful interference patterns produced, in the limit of $U\gg 1$, inside the astroid could be decoded in the future by using the powerful language of catastrophe optics. To this end, a rigorous extension of the paraxial BDW theory [21] to plane waves carrying out optical vortices would certainly represent an important achievement to be reached.