1. INTRODUCTION

Catastrophe optics [1, 2, 3] (CO henceforth) was developed at the end of the 1970s as a modern and rather unorthodox (at least with respect to classical diffraction theories [4]) way to describe the wave fields produced when light is focused by irregular surfaces, typically involved in natural lensing phenomena, or is propagating through nonhomogeneous media [5]. Classical examples are given by the bright patterns that can be seen at the bottom of a swimming pool, or in a coffee cup, on a sunny day, the scintillation of stars, or the focusing of light by liquid drops [3]. Rather than attempting to solve directly the wave equation, the CO solution of a typical diffraction problem is built up starting from its geometrical optics (GO henceforth) limit, which is characterized by a bright light “skeleton” made of caustics, each of them, when seen at the wavelength scale, surrounded by a characteristic diffraction pattern. In a sense, and by quoting Nye [3], CO is aimed at “adding” wave optics to GO, in such a way that the asymptotic approximations of the Helmholtz equation so built do not diverge at caustics, i.e., following Berry and Upstill, “…at the most important places, where the light is brightest” [2]. Diffraction patterns produced by natural focusing of light in the three-dimensional (3D) space typically display a limited number of topologically different caustic distributions, classified according to a celebrated theorem by Thom [6], which forms the basis of the catastrophe theory [7]. Such caustics are called structurally stable, in the sense that they are resistant to small perturbations of the external physical parameters that produce the field, and, as far as optical diffraction in three-dimensional space is concerned, form a hierarchy consisting of five configurations [2, 3]. The simplest stable catastrophe is the fold, denoted (in the catastrophe theory language) by $A_2$, which occurs, for example, in the well-known phenomenon of rainbows [3]. Cusp-shaped caustics are commonly observed, for instance, inside a cup of tea (or coffee) illuminated by a parallel bundle of rays. The structurally stable configurations, denoted by $A_3$, are called cusps [3]. On growing the dimensionality of the physical observation space, the so-called codimension, the complexity of the caustic topologies increases, too. While the fold displays its essential features in a one-dimensional (1D) space and the cusp in a two-dimensional (2D) space, catastrophes with codimension three need all 3D space variables for their complete unfolding. The corresponding stable configurations, which are in number of three, are classified as the swallowtail, denoted by $A_4$, and the elliptic and the hyperbolic umbilics, denoted by $D_4^{+}$ and $D_4^{-}$, respectively. All of them can be experimentally observed, for example, in the space filled by the light reflected by metalized plastic films [2, 3], or in the radio wave propagation in nonhomogeneous plasmas, like the ionosphere [5]. Moreover, such types of catastrophes naturally occur by focusing light using water droplet “lenses,” whose peripheries are constrained to assume circular [8] (for hyperbolic ones) or triangular [9] (for elliptic ones) shapes. Hyperbolic umbilics also occur in light scattering from spheroidal drops [10, 11, 12, 13].

In CO, an adequate mathematical description of the optical field, at the wavelength scale, at those points in the observation space that are on, or close to, the caustics, is given in terms of certain canonical integrals, each of them being associated to a particular stable caustic topology, aimed at “clothing” the caustic skeleton by superimposing to them a characteristic diffraction pattern [2, 3]. Such integrals, called diffraction catastrophes, are defined as follows [2]:

In the present paper, we shall deal with the two catastrophes of codimension four, which are of interest not only in optics [2, 3, 35, 36], but also, for example, in seismology [37, 38], as well as in plasma physics [5]. They are the so-called butterfly (of cuspoidal type), denoted by $A_5$, and the parabolic umbilic, denoted by $D_5$, which, as its name suggests, is intermediate between the elliptic and the hyperbolic umbilics [3]. As said above, $A_5$ and $D_5$ live in a four-dimensional (4D) space, so that they cannot appear in front of an initial wavefront, unless the latter is variable under the control of a parameter, for instance, the time [3]. Experimental optical realizations of diffraction catastrophes associated to $A_5$ and $D_5$ are shown, for example, in Figs. 7.1 and 7.6 of [3], respectively, where butterflies and parabolic umbilics were produced by light reflection by metalized Mylar films (for the former) and by light diffraction through water drops placed in a V-shaped hole (for the latter), with the gravity acting as the variable (by suitably tilting the drops) parameter. Moreover, a theoretical treatment of light diffraction through liquid droplet lenses has been developed in the past, by Nye, and used to interpret the results of several experimental investigations (see again [3] and references therein). However, it should be noted that, differently from the five structurally stable diffraction catastrophes, detailed numerical evaluations of those of codimension four appear to be quite rare in the available literature. In particular, we were able to find just a single example, concerning the butterfly, reported in the recent review by Kryukovskii et al. [27], whereas numerical evaluations of diffraction catastrophes associated to $D_5$ are, at least to our knowledge, missing. In the present paper, on following, almost slavishly, the schedule outlined in [30, 31, 32], convergent series representations of diffraction catastrophes associated to both $A_5$ and $D_5$, will be established. As for the diffraction catastrophes examined in [30, 31, 32], all single terms of the series are evaluable, up to arbitrary precisions, on the main symbolic computational platforms (like Maple or Mathematica), with series being summed via the WT to generate 2D maps of the associated intensity patterns. The convergence of the above series turns out to be extremely slow, especially for “nonsmall” values of the control C, but can be greatly improved by rearranging the sequence of the partial sums via a nonlinear sequence transformation, the so-called δ or WT, introduced in [33]. The paper is structured in two main sections. In the first (Section 2), the convergent series are derived for both diffraction catastrophes on using the approach used in [30, 31, 32]. Only the main technical results have been reported and detailed there, whereas the underlying mathematics is confined within Appendixes A, B, C, and D, aimed at providing sufficient support for the interested reader. The second part of the paper (Section 3) is entirely devoted to presenting several examples of evaluation of the diffraction patterns associated to the optical catastrophes $A_5$ and $D_5$. Finally, in Section 4, some conclusive remarks are given.

2. SERIES EXPANSIONS OF DIFFRACTION CATASTROPHES OF CODIMENSION FOUR

2A. Butterfly $A_{\mathrm{5}}$

The butterfly diffraction catastrophe is defined through the integral in Eq. (1) where $m=1$ (it is a cuspoid) and with the generating function Φ given by

From Eq. (1), the butterfly diffraction catastrophe, say ${\mathrm{\Psi }}_{A_5}(\mathit{C})$, will be written in the form

2B. Parabolic Umbilic $D_{\mathrm{5}}$

The parabolic umbilic diffraction catastrophe is defined via the integral in Eq. (1), where $m=2$ and with the generating function Φ being given by

2C. Discussion

Equations (7, 8), together with Eqs. (18, 19), provide the desired convergent series representations for the two diffraction catastrophes of codimension four. Similarly to what happened in [30, 31, 32], as far as diffraction catastrophes with smaller codimension were concerned, the numerical evaluation of the single terms of the series can be easily achieved on any computational platform supporting Mathematica or Maple, up to arbitrary accuracies. The use of the WT [33] to accelerate the series convergence will be illustrated in Section 3 when the diffraction patterns associated to the caustic skeletons shown in Figs. 1, 2 will be computed. Before doing this, however, it is worth spending some remarks aimed at pointing out, together with some undisputable advantages, an unavoidable computational limit, with respect to the algorithms developed for the case of diffraction catastrophes of codimension three, and, precisely, the fact that the single terms of each series are expressed through double (although finite) sums. Nevertheless, although some symmetry, as well as recurrence, properties might be exploited to improve the algorithm performance (in terms of convergence speed), the above formulas are already able to provide a reasonably fast generation of the 2D maps of the diffraction patterns associated to Figs. 1, 2. For this reason, in the next section we shall use the formulas as clearly as possible, leaving the problem of their optimization, which is out of the scope of the present paper, open.

3. NUMERICAL RESULTS

All single terms of the series expansions in Eqs. (7, 8) can be numerically evaluated (also for complex values of the control C), up to arbitrary accuracies, on using Maple or Mathematica. And, as pointed out in Section 1, the use of Taylor-like series expansions to evaluate diffraction catastrophes turns out to be insensitive at all to the closeness to the caustics. As we did in [30, 31, 32], as far as the computation of diffraction catastrophes of codimensions $\le 3$ was concerned, all subsequent numerical experiments have been carried out by using the symbolic language Mathematica running on a commercial PC equipped with a Quad CPU at $2.4\mathrm{GHz}$ and having $3\mathrm{Gbytes}$ RAM. The convergence of the series in Eqs. (7, 8) will be accelerated through the use of the WT [33], whose recent employment in optics revealed to be extremely profitable [39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. Since the definition and the implementation details of the WT have already been recalled in [30, 31, 32], they will be not repeated in the present paper, and the reader interested in a full and rigorous treatment of the theoretical basis of nonlinear transformations is encouraged to consult the available bibliography and, in particular, [33, 49, 50, 51, 52]. Figure 3 provides the “clothing,” promised at the beginning of the present paper, of the caustic skeletons of $A_5$ shown in Fig. 1. As already pointed out, the numerical codes built up to produce such a figure was not optimized. This means, for instance, that a fixed value of the WT transformation has been used for all the $(C_1,C_2)$ plane points in each of the four plots, whereas it could be observed that the desired convergence (once the number of digits has been preliminarily fixed) of the series is achieved for values of the transformation order depending on the point under investigation. For our unoptimized codes, the WT orders have been set to 25 for Fig. 3a, 50 for Fig. 3b, and 40 for Figs. 3c, 3d, with computational times of the order of a few hours to generate each grid of $200\times 200$ points.

The results of the same type of numerical experiments, carried out on the caustic distributions depicted in Fig. 2, are shown in Fig. 4, where the diffraction patterns associated to the various caustic topologies have been generated and then superimposed to the formers. Similarly as we did for the butterfly, the WT orders has been kept fixed for each plot, and precisely to 120 for Fig. 4a, 135 for Fig. 4b, 115 for Fig. 4c, 80 for Fig. 4d, 130 for Fig. 4e, and 60 for Fig. 4f. The WT orders have been chosen in order to reach a numerical accuracy on the diffraction catastrophe fields at the second digit, which proved adequate for visualization purposes. The result of a further numerical experiment concerning the parabolic umbilic is finally shown in Fig. 5, where the 2D map of the modulus is plotted, in the $(C_2,C_1)$ plane, for $C_3=-7$ and $C_4=-1/2$, together with the corresponding caustic (thick curve). In the present case, a naive optimization about the choice of the WT order has been carried out by dividing the whole interval of interest into several rectangular sectors, in each of which such orders were suitably set in order to reach a two-digit accuracy.

4. CONCLUSIONS

In the present paper, numerical evaluations of the two diffraction catastrophes of codimension four occurring in focusing and diffraction of light, i.e., the butterfly and the parabolic umbilic, have been carried out. The computational approach, successfully used in [30, 31, 32] to evaluate the wave field of all five structurally stable diffraction catastrophes of codimension $\le 3$, has here been applied to evaluate those associated to $A_5$ and $D_5$. In particular, on using basically the same mathematics, converging single series expansions of the related wildly oscillating phase integrals have been derived in terms of generalized hypergeometric functions, and used, jointly to the WT, to produce 2D maps of the diffraction catastrophe wave fields. Several numerical experiments have been carried out to give evidence of the effectiveness of the proposed method, as well as of the main computational limits. In particular, our feeling is that such an approach could be successfully employed also in the numerical evaluation of several other classes of diffraction catastrophes, for instance, those associated to edge catastrophes ($B_N$, $C_N$, $F_N$, and so on) [27, 53], which are of great interest in electromagnetic theory, as well as those recently studied in [54, 55]. Moreover, we believe that the present approach could also be used to develop alternative strategies for the computation of diffraction catastrophes associated to $A_5$ and $D_5$, for instance, similar to those recently proposed in [41, 42] for the evaluation of all cuspoids of codimension $\le 3$, or based on the numerical solution of canonical partial differential systems [19, 27, 29]. In this perspective, the availability of reference values is of pivotal importance to assess the performances of new computational methods and, due to the aforementioned extreme rarity (if not complete absence) of butterfly and parabolic umbilic field numerical evaluations, the results obtained in the present paper could even assume a sort of “paradigmatic” character, also for helping the dissemination of the CO way of thinking to the widest scientific audience.

APPENDIX A: CAUSTIC DISTRIBUTIONS OF THE BUTTERFLY

Since the internal state s is 1D, caustics are obtained on solving the following system:

(A1)$$\{\begin{array}{c}{\partial }_s\mathrm{\Phi }=0,\\ {\partial }_s^2\mathrm{\Phi }=0,\end{array}$$

which, on taking Eq. (2) into account, leads to

(A2)$$\{\begin{array}{c}6s^5+4C_4{s}^3+3C_3{s}^2+2C_{2}s+C_1=0,\\ 15s^4+6C_4{s}^2+3C_{3}s+C_2=0,\hfill \end{array}$$

which can be rearranged as follows:

(A3)$$\{\begin{array}{c}{C}_1=s^2(8C_{4}s+3C_3+24s^3),\\ C_2=-3s(2C_{4}s+C_3+5s^3),\end{array}$$

to give an s-parametric representation of the caustics across the plane $(C_1,C_2)$, for fixed values of $(C_3,C_4)$.

APPENDIX B: PROOF OF EQS. (7) AND (8)

To convert the phase integral in Eq. (6) into a convergent integral, it is sufficient, as was done in [31] for the swallowtail diffraction catastrophe, to let $\mathrm{i}{s}^6=t^6$ and to use Jordan’s lemma. After simple algebra, Eq. (6) takes on the form

(B1)$$(-\mathrm{i}{)}^{1/6}{B}_{1/2}(x,y,z,w)={\int }_0^{\infty }\mathrm{d}t\exp (-t^6+\xi t^4+\eta t^3+\zeta t^2+\omega t),$$

where $\xi =x{\mathrm{i}}^{5/3}$, $\eta =y{\mathrm{i}}^{3/2}$, $\zeta =z{\mathrm{i}}^{4/3}$, and $\omega =w{\mathrm{i}}^{7/6}$. To obtain the convergent series expansion for the integral in Eq. (B1), we first rewrite it in the following form:

(B2)$$\begin{gathered} (-\mathrm{i}{)}^{1/6}{B}_{1/2}(x,y,z,w) \\ =\sum _{k=0}^{\infty }\frac{1}{k!}{\int }_0^{\infty }\mathrm{d}t\exp [(-t^6+\xi t^4)(\eta t^3+\zeta t^2+\omega t{)}^k, \end{gathered}$$

and, by using the Newton formula, after simple algebra, obtain

(B3)$$\begin{gathered} (-\mathrm{i}{)}^{1/6}{B}_{1/2}(x,y,z,w)=\sum _{k=0}^{\infty }\frac{{\mathrm{i}}^{7k/6}}{k!} \\ \times \sum _{j=0}^k\sum _{\ell =0}^j{\mathrm{i}}^{\frac{\ell +j}{6}}(\begin{array}{c}k\\ j\end{array}\left)\right(\begin{array}{c}j\\ \ell \end{array})y^{\ell }{z}^{j-\ell }{w}^{k-j}{\mathcal{J}}_{k+j+\ell }(x{\mathrm{i}}^{5/3}), \end{gathered}$$

which coincides with Eq. (7) and with the function ${\mathcal{J}}_n(\mu )$ being defined through

(B4)$${\mathcal{J}}_n(\mu )={\int }_0^{\infty }{t}^n\exp (-t^6+\mu t^4).$$

The integral can be evaluated in closed form on first letting $\tau =t^4$, thus obtaining

(B5)$${\mathcal{J}}_n(\mu )=\frac{1}{4}{\int }_0^{\infty }\mathrm{d}\tau {\tau }^{\frac{n-3}{4}}\exp (-{\tau }^{\frac{3}{2}}+\mu \tau ),$$

and then on using formula 2.3.2.13 of [56], which leads, after some algebra, to Eq. (8).

APPENDIX C: CAUSTIC DISTRIBUTIONS OF THE PARABOLIC UMBILIC

Caustics are obtained by solving the system [2]

(C1)$$\{\begin{array}{c}{\partial }_1\mathrm{\Phi }=0,\hfill \\ {\partial }_2\mathrm{\Phi }=0,\hfill \\ {\partial }_{11}^2\mathrm{\Phi }{\partial }_{22}^2\mathrm{\Phi }=({\partial }_{12}^2\mathrm{\Phi }{)}^2,\end{array}$$

where ${\partial }_{\mathrm{i}}$ denotes partial derivative with respect to the variable $s_{\mathrm{i}}$ ($\mathrm{i}=1,2$). On substituting from Eq. (10) into Eq. (C1), after rearranging, we obtain

(C2)$$\{\begin{array}{c}{C}_1=-2C_3{s}_1-4s_1^3-s_2^2,\hfill \\ C_2=-2s_2(C_4+s_1),\hfill \\ s_2^2=(C_4+s_1)(C_3+6s_1^2).\hfill \end{array}$$

On first eliminating $s_2$ and then on replacing $s_1$ by s, we have

(C3)$$\{\begin{array}{c}{C}_1=-[4s^3+2s{C}_3+(6s^2+C_3)(s+C_4)],\\ C_2=\pm 2(6s^2+C_3{)}^{1/2}(s+C_4{)}^{3/2},\hfill \end{array}$$

which gives an s-parametrization of the caustics across the plane $(C_2,C_1)$, for given values of $(C_3,C_4)$.

APPENDIX D: PROOF OF EQS. (18) AND (19)

We first write Eq. (17) as follows:

(D1)$$\begin{gathered} P_{1/2}(X,Y,Z,W)={\int }_0^{\infty }\frac{\mathrm{d}s}{\sqrt{s}}\exp \left[\mathrm{i}\right(s^4+\frac{W}{s}\left)\right] \\ \exp [\mathrm{i}(X{s}^3+Y{s}^2+Zs)], \end{gathered}$$

which, on expanding the second exponential as a convergent power series, becomes

(D2)$$\begin{gathered} P_{1/2}(X,Y,Z,W) \\ =\sum _{k=0}^{\infty }\frac{{\mathrm{i}}^k}{k!}{\int }_0^{\infty }\frac{\mathrm{d}s}{\sqrt{s}}\exp \left[\mathrm{i}\right(s^4+\frac{W}{s}\left)\right](X{s}^3+Y{s}^2+Zs{)}^k\mathrm{.} \end{gathered}$$

On expanding the factor $(X{s}^3+Y{s}^2+Zs{)}^k$ via the Newton’s formula, i.e.,

(D3)$$(X{s}^3+Y{s}^2+Zs{)}^k=\sum _{j=0}^k\sum _{\ell =0}^j(\begin{array}{c}k\\ j\end{array}\left)\right(\begin{array}{c}j\\ \ell \end{array})X^{\ell }{Y}^{j-\ell }{Z}^{k-j}{s}^{k+j+\ell },$$

after substituting into Eq. (D2) and rearranging, Eq. (18) follows, with the function ${\mathcal{I}}_n(\mu )$ being defined by

(D4)$${\mathcal{I}}_n(\mu ){\int }_0^{\infty }\frac{\mathrm{d}s}{\sqrt{s}}{s}^n\exp \left[\mathrm{i}\right(s^4+\frac{\mu }{s}\left)\right],$$

and, in order for the integral to converge, with $\mathrm{Im}\{\mu \}>0$. The integral in Eq. (D4) has a similar structure to that evaluated in Appendix B of [32] for the elliptic and hyperbolic umbilic diffraction catastrophes, so that the involved mathematics is practically the same. However, since the final expression of ${\mathcal{I}}_n(\mu )$ cannot be trivially deduced from that obtained in [32], we decided to detail in the present appendix its derivation. We thus start from the complex integral

(D5)$${\oint }_{\mathcal{C}}\frac{\mathrm{d}s}{\sqrt{s}}{s}^n\exp \left[\mathrm{i}\right(s^4+\frac{\mu }{s}\left)\right],$$

where $\mathcal{C}$ denotes the contour depicted in Fig. 6. Since the integrand has no singularities inside $\mathcal{C}$, the contour integral in Eq. (D5) must be zero. Furthermore, when $\rho \to \infty$, the contribution associated with $C_{\rho }$ tends to zero. Consider now the contribution associated with $C_{ϵ}$, which, on letting $s=ϵ\exp (\mathrm{i}\phi )$ with $\phi \in [0,\pi /8]$, can be written as

(D6)$$\begin{gathered} {\int }_{C_{ϵ}}\frac{\mathrm{d}s}{\sqrt{s}}{s}^n\exp \left[\mathrm{i}\right(s^4+\frac{\mu }{s}\left)\right]=-\mathrm{i}{ϵ}^{n+1/2}{\int }_0^{\pi /8}\mathrm{d}\phi \exp \left[\mathrm{i}\right(n+\frac{1}{2}\left)\right] \\ \times \exp [\mathrm{i}{ϵ}^3\exp (\mathrm{i}4\phi )] \\ \exp [\mathrm{i}\frac{\mu }{ϵ}\exp (-\mathrm{i}\phi )]. \end{gathered}$$

It vanishes, for $ϵ\to 0$, only if the condition

(D7)$$\mathrm{Re}\{\mathrm{i}\mu \exp (-\mathrm{i}\phi )\}\le 0$$

is identically fulfilled for $\phi \in [0,\pi /8]$. This condition restricts the validity domain of μ to the sector

(D8)$$\frac{\pi }{8}<\arg \{\mu \}<\pi ,$$

where the function ${\mathcal{I}}_n(\mu )$ can then be expressed as

(D9)$${\mathcal{I}}_n(\mu )={\mathrm{i}}^{\frac{n+1/2}{4}}{\int }_0^{\infty }\mathrm{d}t{t}^{n-\frac{1}{2}}\exp ({-t}^4+\frac{{\mathrm{i}}^{3/4}\mu }{t}).$$

To evaluate the integral, we let $\xi =1/t$, so that it becomes

(D10)$${\mathcal{I}}_n(\mu )={\mathrm{i}}^{\frac{n+1/2}{4}}{\int }_0^{\infty }\mathrm{d}\xi {\xi }^{-n-\frac{3}{2}}\exp (-{\xi }^{-4}+{\mathrm{i}}^{3/4}\mu \xi ),$$

and, by using formula 2.3.2.14 of [56], after simple algebra, Eq. (19) follows and which, by analytical continuation, is naturally extended to the whole domain $0<\arg \{\mu \}<\pi$.

ACKNOWLEDGMENTS

I thank Turi Maria Spinozzi for his help during the preparation of the manuscript.

 

Fig. 1 Representation, in the plane $(C_1,C_2)$, of the caustic distributions associated to $A_5$, for (a) $(C_3,C_4)=(0,3)$, (b) $(3,0)$, (c) $(0,-3)$, and (d) $(1,-3)$. Note that the scales are different.

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Fig. 2 Representation, in the plane $(C_2,C_1)$, of the caustic distributions associated to $D_5$, for (a) $(C_3,C_4)=(-4/7,-1)$, (b) $(0,5/4)$, (c) $(-169/100,-106/100)$, (d) $(-4,-64/100)$, (e) $(-3,-1)$, and (f) $(-5,-1/10)$. Note that the scales are different.

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Fig. 3 2D maps of the modulus, in the plane $(C_1,C_2)$, of the diffraction catastrophes associated to the caustic distributions depicted in Fig. 1. (a) $(C_3,C_4)=(0,3)$, (b) $(3,0)$, (c) $(0,-3)$, and (d) $(1,-3)$. The WT order has been kept fixed for each plot, and, precisely, it was set to 25 for (a), 50 for (b), and 40 for (c) and (d).

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Fig. 4 2D maps of the modulus, in the plane $(C_2,C_1)$, of the diffraction patterns associated to the caustic distributions depicted in Fig. 2. (a) $(C_3,C_4)=(-4/7,-1)$, (b) $(0,5/4)$, (c) $(-169/100,-106/100)$, (d) $(-4,-64/100)$, (e) $(-3,-1)$, and (f) $(-5,-1/10)$.

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Fig. 5 2D contour map of the modulus distribution of a parabolic umbilic diffraction catastrophe with $(C_3,C_4)=(-7,-1/2)$. The choice of the WT order has been chosen in order to reduce the computational time. The accuracy has been chosen up to the second digit.

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Fig. 6 Contour path for the integral in Eq. (D5).

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