On the numerical evaluation of umbilic diffraction catastrophes

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Riccardo Borghi Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1661

On the numerical evaluation of umbilicdiffraction catastrophes

Riccardo Borghi

Dipartimento di Elettronica Applicata, Università degli Studi “Roma Tre,” Via della Vasca Navale 84,I-00146 Rome, Italy (borghi@uniroma3.it)

Received April 28, 2010; accepted May 17, 2010;posted May 27, 2010 (Doc. ID 127685); published June 21, 2010

A simple computational approach is proposed for the evaluation of umbilic diffraction catastrophes which, to-gether with cuspoids, describe the whole hierarchy of the structurally stable diffraction patterns that can beproduced by optical diffraction. In this paper, after expanding the double integral representations of hyperbolicand 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 re-sults to complex values of the control parameter. Numerical experiments aimed at giving evidence of the ef-fectiveness and implementative ease of the approach are eventually presented. © 2010 Optical Society ofAmerica

OCIS codes: 000.3860, 260.1960, 050.1940.

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. INTRODUCTIONt is well known that focusing of light is a typical source ofright caustics [1,2]. Less known is that such caustics arelassified by Thom’s theorem [3] in a hierarchy which, inhe case of diffraction in three-dimensional space, consistsf five topologically different structurally stable singularonfigurations [1,2]. At the wavelength scale each caustics “decorated” by a characteristic diffraction pattern,

athematically described by standard canonical inte-rals, namely, diffraction catastrophes, which are defineds [1,2]

��C� =1

�2��m/2�¯� dms exp�i��s;C��, �1�

here ��s ;C� denotes a function which is a polynomial ofegree n with respect to the (m-dimensional) state vari-ble s and is linear with respect to the, possibly complex,ontrol variable C that is suitably associated with thehysical coordinates in the observation space [4]. The fiveypes of diffraction catastrophes involved in optical prob-ems are classified as cuspoids and umbilics, dependingn the dimensionality of the corresponding integrals inq. (1), with m=1 for the former and m=2 for the latter

1,5]. A practical problem of pivotal importance in dealingith catastrophe optics, which has stimulated a consider-ble amount of work over a long period [6–17], is repre-ented by the numerical evaluation of the phase integralsf Eq. (1).

In [18,19] a simple computational method has been pro-osed to evaluate the whole class of cuspoid diffraction ca-astrophes. The idea consists of expanding the canonicalntegral in Eq. (1) as a convergent power series with re-pect to the C-parameters and searching for the limit ofuch a series. Such an approach presented two major dif-culties: (i) the evaluation of the single terms of the seriesnd (ii) the slow convergence of the series for “nonsmall”

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alues of the control parameters C. As far as point (i) isoncerned, in [18,19] single series expansions have beenound for all cuspoid diffraction catastrophes, whoseingle terms were expressed in closed form via the use ofeneralized hypergeometric functions [5]. As far as pointii) is concerned, it was shown that the above series can befficiently summed by rearranging the partial sum se-uence via the Weniger transformation (WT) [20], whichurns out to be able to produce strong accelerations of theonvergence for nonsmall values of C. In this way, ithould be possible to reach regions of control space thatre both distant from the origin and on, or close to, theaustic.

The present paper is aimed at completing the paper se-uence of [18,19] by offering a simple computational toolo evaluate the fields associated with the umbilic diffrac-ion catastrophes. These types of catastrophes naturallyccur by focusing light using water droplet “lenses,”hose peripheries are constrained to assume circular [21]

for hyperbolic ones) or triangular [22] (for elliptic ones)hapes. Hyperbolic umbilics also occur in light scatteringrom spheroidal drops [23–26].

Here we shall follow the same path traced in [18,19].ccordingly, the paper is structured into two main parts.

n the first part (Section 2), convergent single power se-ies expansions are derived for umbilic diffraction catas-rophes (both hyperbolic and elliptic) starting from theirouble integral representations, through elementary com-lex integration techniques. Similarly to what was foundn [19], the single terms of such series admit a closed-formxpression given in terms of a finite sum of generalizedypergeometric functions and thus easily evaluable, up torbitrary precision, on the main symbolic computationallatforms like Maple or Mathematica. As a by-product ofhe same section a new, at least up to our knowledge, ana-ytical expression for the on-axis field of umbilics is de-ived, which extends to the complex realm results previ-

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1662 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Riccardo Borghi

usly found by Uzer et al. [27] for the hyperbolic umbilicnd by Berry et al. [22] for the elliptic one. In the secondart of the paper (Section 3), the series expansions previ-usly obtained are summed by using the WT, whose defi-ition is briefly recalled for the reader’s convenience.ome numerical experiments are carried out to illustratehe accuracy and the ease of implementation of the pro-osed approach, as well as to put into evidence its mainomputational limits.

. CONVERGENT SERIES EXPANSIONS FORMBILIC DIFFRACTION CATASTROPHES

n the present section we shall find convergent power se-ies for both hyperbolic and elliptic umbilic diffraction ca-astrophes. Before starting, however, it must be stressedhat several authors used in the past different math-matical definitions of umbilics. In the present paper, wehall refer to the definition given by Nye in [21] for theyperbolic umbilic and that given by Berry et al. in [22]

or the elliptic umbilic.

. Hyperbolic Umbilic Diffraction Catastrophesollowing Nye [21], for the hyperbolic umbilic diffractionatastrophe the integral in Eq. (1) takes on the double in-egral form,

H�x,y,z� =1

2��

−�

+�

d��−�

+�

d�

�exp�i��3 + ��2 − z��2 + �2� − x� − y���. �2�

his definition differs from that, say HB�x ,y ,z�, used else-here [5,22,27],

HB�x,y,z� =1

2��

−�

+�

d��−�

+�

d�

�exp�i��3 + �3 + z�� − x� − y���, �3�

hich in some particular cases gave rise to closed-formxpressions. Note that passing from HB to H (and viceersa) turns out to be straightforward by using the linearransformation,

� →

� +�

�3

21/3 −z

3,

� →

� −�

�3

21/3 −z

3, �4�

nto the integral in Eq. (3), which gives

HB�x,y,z� =21/3

�3

�expi z3

27+

x + y

3z��Hx + y

21/3 ,x − y

21/3�3,

z

22/3� ,

�5�

H�x,y,z� = 2−1/331/2 exp− i4z3

27+

2xz

3 ���HBx + �3y

22/3 ,x − �3y

22/3 ,22/3z� . �6�

quation (6) can now be used to find the exact values ofhe function H across the xy plane and along the z axis. Inact, as far as HB�x ,y ,0� is concerned, in [27] it has beenound that

HB�x,y,0� =2�

32/3Ai−x

31/3�Ai−y

31/3� , �7�

here Ai� � denotes the Airy function [5]. Accordingly,ubstituting Eq. (7) into Eq. (6) gives the well known re-ult [21]

H�x,y,0� = 22/33−1/6� Ai−x + �3y

121/3 �Ai−x − �3y

121/3 � .

�8�

losed-form evaluations of HB�0,0,z� are also availableor real values of z. In particular, in [27] it has beenhown that, for z�0,

HB�0,0,z� = 2�V�z�exp iz3

54� , �9�

here

V�z� =1

36 z

��1/2− i exp i�

3 �H1/6�1� z3

54�+ 31/2 exp−

i�

3 �H1/6�2� z3

54�� , �10�

ith Hn�1� �Hn

�2�� denoting the ingoing (outgoing) Hankelunction of order n [5]. The complex conjugate of the rightand side applies for z0 [27]. However, it is possible toxtend the above results also to complex values of z. This,n particular, is desirable for finding the exact expressionf the on-axis field of the elliptic umbilic, as it will behown in the next section. In Appendix A it is proved that

HB�0,0,z� = 3−1/3G− i,z

31/3� + Gi,−z

31/3�� , �11�

here

G�p,c� =

�� exp c3

18p�9�6p�2/3

6p1/3

5

6�0F1;

5

6;

c6

1296p2�

−61/3c

7

6�0F1;

7

6;

c6

1296p2�� , �12�

ith pFq� � denoting the hypergeometric function [5]. Onubstituting Eqs. (11) and (12) into Eq. (6), after rearrang-ng we obtain

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Riccardo Borghi Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1663

H�0,0,z� = 6−1/331/2 exp− i4z3

27 ��G− i,

22/3z

31/3 � + Gi,−22/3z

31/3 �� . �13�

or arbitrary choices of the control parameters �x ,y ,z�,he evaluation of H must be carried out numerically. Theouble integral in Eq. (2) can be converted into a singlene via the substitution �2=�−z, which leads to [21]

H�x,y,z� =1

��expi

�

4�exp�− ixz�

��CH

d� exp�i�6 + 2z�4 + �z2 − x��2 −y2

4�2�� ,

�14�

here now the complex integration path CH runs from +i�o +�, passing to the upper right of the essential singu-arity at �=0. The integral in Eq. (14) can be further sim-lified by letting �2=s, thus obtaining

H�x,y,z� =1

2��expi

�

4�exp�− ixz�

��−�

+� ds

�sexpis3 + Xs2 + Ys +

Z

s ��=

1

2��expi

�

4��J�X,Y,Z� − iJ��− X,Y,Z��,

�15�

here the asterisk denotes complex conjugation, X=2z,=z2−x, Z=−y2 /4, and the function

J�X,Y,Z� =�0

� ds

�sexpis3 + Xs2 + Ys +

Z

s �� �16�

as been introduced. Later, we will show how the numeri-al evaluation of J�X ,Y ,Z�, which represents the keyroblem for our approach, can be easily implemented.

. Elliptic Umbilic Diffraction Catastrophesor the elliptic umbilic diffraction catastrophe the double

ntegral from Eq. (1) takes on the form [5,22]

E�x,y,z� =1

2��

−�

+�

d��−�

+�

d�

�exp�i��3 − 3��2 − z��2 + �2� − x� − y���.

�17�

s for the hyperbolic umbilic, closed-form expressions ofhe elliptic umbilic are available for particular cases. Forxample, E�x ,y ,0� is expressed through [22]

E�x,y,0� = 2

3�2/3

� Re�Ai− x − iy

121/3 �Bi− x + iy

121/3 �� ,

�18�

here Ai and Bi are the standard linearly independentiry functions [5]. As far as the on-axis expression�0,0,z� is concerned, the following approximate expres-

ions have been derived in [22]:

E�0,0,z� � 2

3�2/3

��3�Ai2�0� − i2

3�2/3

�Ai��0��2z� ,

�19�

E�0,0,z� � −i

2z��3 expi�

2−

4z3

27 �� + 1� , �20�

hich are valid for z�1 and z 1, respectively. Thanks tohe analytical expression of HB�0,0,z� found in the previ-us section, E�0,0,z� can now be evaluated in closed formy using the connection formula between HB and E estab-ished in [22] which, for x=y=0, reads

E�0,0,z� = 2−1/3�exp−i�

6 �HB0,0,22/3 exp i�

3 �z�+ exp i�

6 �HB0,0,22/3 exp−i�

3 �z�� . �21�

quation (21), together with Eqs. (11) and (12), gives thexact on-axis field of the elliptic umbilic. A visual compari-on of the approximate expressions in Eqs. (19) and (20)ith respect to the exact result is shown in Fig. 1, where

he modulus �E�0,0,z�� is plotted as a function of z, asvaluated from Eq. (21) (solid curve), together with thosebtained through Eq. (19) (dashed curve) and through Eq.20) (dotted curve). A quantitatively more significant com-arison is shown in Fig. 2, where now the relative error islotted against z. As for the hyperbolic umbilic, for arbi-rary choices of the control parameters �x ,y ,z�, the ellipticmbilic must be evaluated numerically. In particular, theouble integral in Eq. (17), after the substitution �= ��z /3�1/2, can be expressed through the complex contourne-dimensional integral [28]

ig. 1. Behavior of the modulus �E�0,0,z�� versus z as evaluatedrom Eq. (21) (solid curve), together with the approximations inq. (19) (dashed curve) and in Eq. (20) (dotted curve).

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1664 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Riccardo Borghi

E�x,y,z� =1

�3�exp− i

�

4�expixz

3−

4z3

27 ����

CE

d� exp�i�6 − 2z�4 + �z2 − x��2 +y2

12�2�� ,

�22�

here now the integration path CE runs from −i� to +�,assing to the lower right of the essential singularity at=0. As for the hyperbolic umbilic, the substitution �2=sransforms Eq. (22) into

E�x,y,z� =1

2�3�exp− i

�

4�expixz

3−

4z3

27 ����

−�

+� ds

�sexpis3 − Xs2 + Ys −

Z

3s��=

1

2�3�exp− i

�

4�expixz

3−

4z3

27 ���J− X,Y,−

Z

3� + iJ�X,Y,−Z

3�� , �23�

here the function J, defined in Eq. (16), will be com-uted according to the algorithm described in the nextubsection.

. Evaluation of the Function J„X ,Y ,Z…

he numerical evaluation of the integral J�X ,Y ,Z� can bearried out, as for the cuspoid diffraction catastrophes in18,19], by using elementary tools. To this end, we firstrite Eq. (16) as follows:

J�X,Y,Z� =�0

� ds

�sexpis3 +

Z

s ��exp�i�Xs2 + Ys��,

�24�

hich, on expanding the second exponential as a conver-ent power series, becomes

ig. 2. Relative errors associated with the asymptotic approxi-ations in Eq. (19) (dashed curve) and in Eq. (20) (dotted curve),

gainst the values of z.

J�X,Y,Z� = �k=0

� ik

k!�0

� ds

�sexpis3 +

Z

s ��sk�Xs + Y�k,

�25�

nd, by expressing the term �Xs+Y�k via the Newton for-ula, after rearranging gives

J�X,Y,Z� = �k=0

� ik

k!�j=0

k k

j�XjYk−jIk+j�Z�, �26�

here the function In��� is defined by

In��� =�0

� ds

�ssn expis3 +

�

s �� , �27�

nd, in order for the integral to converge, it is assumedhat Im����0. In Appendix B it is proved that the func-ion In��� can be expressed through the following closed-orm expression:

In��� = i�n+1/2�/3��− i2/3��n+1/2

�− n −1

2�0F3;3 + 2n

6,5 + 2n

6,7 + 2n

6;�3

27�+

1

3�k=0

2 �i2/3��k

k!n − k

3+

1

6��1F41;�1,1 +

k − n − 1/2

3 �,��3,1 + k�;�3

27�� ,

�28�

here the symbol ��q ,a� denotes the sequence

��q,a� = �a

q,a + 1

q, . . . ,

a + q − 1

q � . �29�

hanks to Eq. (29), elliptic and hyperbolic umbilic diffrac-ion catastrophes can be expanded as convergent singleower series whose terms are exactly evaluable for anyndex value. Nowadays the numerical evaluation of gen-ralized hypergeometric functions can be achieved onomputational platforms supporting Mathematica oraple, up to arbitrary precision. Moreover, as already

hown in [18,19], the use of convergent expansions can bextended even to nonsmall values of the control param-ters �x ,y ,z� by using suitable convergence accelerationechniques.

. NUMERICAL RESULTS. Preliminarieshe present section is aimed at illustrating some ex-mples of numerical evaluation of the diffraction patternsssociated with hyperbolic and elliptic umbilics. Beforeroceeding, it is worth giving some preliminary informa-ion about the computational framework within which theubsequent numerical experiments have been carried out.

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Riccardo Borghi Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1665

s already pointed out in [19], the main advantage of theroposed approach resides in its ease of implementation.ll terms of the series expansion in Eq. (26) are given

hrough closed-form expressions, which can be numeri-ally evaluated also for complex values of the control pa-ameters �x ,y ,z�. However, the price to be paid for suchn implementative simplicity is the need to deal with nu-erical precisions greater than, for example, the qua-

ruple precision of FORTRAN. This unavoidably implieshat the above approach must be implemented in a sym-olic language, which will run at a lower speed with re-pect to pure numerical codes. All subsequent numericalxperiments have been carried out by using the symbolicanguage Mathematica running on a commercial personal

ig. 3. This figure should be compared to Fig. 3 of [21]. (a) andection z=2 of the hyperbolic umbilic diffraction catastrophe H�xithmic amplitude in the xy section centered on the point at �26.q. (31). The WT order used in the four figures was set to 70 for (f the X-shape in Fig. 3(d) of [21] has been corrected.

omputer equipped with a Quad central processing unitt 2.4 GHz and having 3 Gbytes random access memory.

. Summary of the Weniger Transformationhe last obstacle to overcome in order to make the abovepproach suitable for an effective computation of theunction J to be achieved is represented by the slow con-ergence of the above series expansions for nonsmall val-es of the control parameters �x ,y ,z�. As successfully em-loyed in [18,19] for cuspoids, such convergence can bereatly accelerated via the use of a powerful nonlinear se-uence transformation, introduced by Weniger [20]. Basi-ally, the WT, when applied to the sequence of the partial

ntours of logarithmic amplitude and phase, respectively, in thes defined in Eq. (2). (c) Amplitude in the section y=0. (d) Loga-2c�, where b and c are the unit cell dimensions defined throughfor (b), 70 for (c), and 80 for (d). Note that the fine-scale breakup

(b) Co,y ,z� a

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1666 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Riccardo Borghi

ums, say �sn�, sn=�j=0n aj �n=0,1, . . .� of a series, converts

t into a new sequence, say ��k� �k=1,2, . . .�, defined by20]

�k =�j=0

k

�− 1�jk

j��1 + j�k−1

sj

aj+1

�j=0

k

�− 1�jk

j��1 + j�k−1

1

aj+1

, �30�

ith � �k denoting the Pochhammer symbol [29]. Despitets unusual aspect, the sequence transformation in Eq.30) is of straightforward implementation and in opticsas recently shown its effectiveness in summing severalypes of divergent and slowly convergent series18,19,30–39]. For a full and rigorous treatment of theheoretical basis of nonlinear transformations, the readers encouraged to consult the available bibliography and,n particular, [20,40–43].

. Hyperbolic Umbilicsn [21], Nye presented a detailed description of the geo-etrical arrangement of the optical vortices on which the

iffraction pattern associated with hyperbolic umbilic ca-astrophes is based. In the present subsection, some of theesults obtained by Nye will be used for testing the effec-iveness and the performances of the proposed method. In

Table 1. Action of the WT When the EllipticUmbilic E„0,0,3… Is Evaluateda

Weniger �n Relative Error

0 −0.513 060 144 7+0.178 725 118 8i 1.60 −0.269 345 263 5+0.116 741 081 8i 0.50 −0.205 468 944 4+0.041 693 795 9i 1.4�10−3

1 −0.205 875 500 8+0.041 802 501 4i 7.4�10−4

2 −0.205 779 099 9+0.041 607 391 4i 3.8�10−4

3 −0.205 736 720 2+0.041 710 907 5i 2.0�10−4

4 −0.205 790 631 5+0.041 692 127 6i 10−5

5 −0.205 765 354 8+0.041 677 963 6i 5.0�10−5

6 −0.205 768 478 5+0.041 692 240 4i 2.5�10−5

7 −0.205 772 785 3+0.041 686 312 3i 1.3�10−5

8 −0.205 769 151 1+0.041 686 676 7i 6.3�10−6

9 −0.205 770 485 7+0.041 687 899 3i 3.1�10−6

0 −0.205 770 495 8+0.041 687 007 5i 1.5�10−6

1 −0.205 770 167 3+0.041 687 295 6i 7.4�10−7

2 −0.205 770 378 7+0.041 687 320 8i 3.6�10−7

3 −0.205 770 319 2+0.041 687 236 5i 1.7�10−7

4 −0.205 770 308 3+0.041 687 285 0i 8.3�10−8

5 −0.205 770 329 0+0.041 687 273 3i 4.8�10−8

6 −0.205 770 318 3+0.041 687 269 7i 1.9�10−8

7 −0.205 770 320 4+0.041 687 274 6i 8.9�10−9

8 −0.205 770 321 5+0.041 687 272 3i 4.2�10−9

9 −0.205 770 320 3+0.041 687 272 7i 1.9�10−9

00 −0.205 770 320 8+0.041 687 273 0i 9.1�10−10

01 −0.205 770 320 8+0.041 687 272 7i 4.2�10−10

02 −0.205 770 320 7+0.041 687 272 8i 1.9�10−10

03 −0.205 770 320 8+0.041 687 272 8i 0xact −0.205 770 320 8+0.041 687 272 8i

aFirst column: sequence index; second column: Weniger transformed partial sumequence; third column: relative error with respect to the exact value provided inubsection 2.B. Apparent convergence is indicated by underline.

articular, some computational artifacts emphasized in21] will also be corrected. Figure 3 is aimed at reproduc-ng Fig. 3 of [21]. In particular, Figs. 3(a) and 3(b) showontour plots of the logarithmic amplitude and phase ofhe hyperbolic umbilic H�x ,y ,z� evaluated at the plane=2, while in Fig. 3(c) the amplitude at the plane y=0 ishown. Figure 3(d) shows the logarithmic amplitude, inhe xy plane, centered in correspondence of a typical hori-ontal dislocation junction of the lattice of vortices pro-uced inside the V-shaped caustic of the hyperbolic um-ilic at the point �26.2,b ,1 / �2c��, where b and c are giveny [21]

b =2�

�x,

c =3�

x. �31�

. Elliptic Umbilicsn the present subsection we will carry out numerical ex-eriments aimed at reproducing some of the results ob-ained in [22]. Before doing this, however, it is worthhowing the way how the WT works on a single pointvaluation when its order is increased. To this aim, con-ider the evaluation of E�0,0,3� which, as evident fromigs. 1 and 2, is located well inside the asymptotic region.his, in turn, is also aimed at showing that the proposedpproach allows a sufficient overlap between convergentower expansions and asymptotic expansions to bechieved. In Table 1, the first column contains the valuesf the sequence index in Eq. (30), the second column con-ains the corresponding values of the elliptic umbilic ob-ained by the WT, and the third column contains the rela-ive error evaluated with respect to the exact valuerovided by theory exposed in Subsection 2.B. Apparentonvergence is indicated by the underline. In Fig. 4 plotsf �E�x ,y ,z��, as a function of x and y, are shown for z=0Fig. 4(a)], z=1 [Fig. 4(b)], z=2 [Fig. 4(c)], z=3.55 [Fig.(d)], z=3.85 [Fig. 4(e)], z=4 [Fig. 4(g)], and z=5 [Fig.(h)]. These pictures are aimed at reproducing Figs.(a)–3(h) of [22], respectively. Finally, to give an idea ofhe fine details of the interference pattern inside the four-ave region, in Fig. 5 the contour plot of the section=4 in Fig. 4(g) is also reported.

. CONCLUSIONSatastrophe optics represents a modern approach totudy diffraction phenomena and describes, followingye, how “to add diffraction to the caustics of ray optics,”

44] in a radically different way from the classical diffrac-ion theory [45]. An issue of pivotal importance for usingatastrophe optics is certainly represented by the numeri-al evaluation of the five structurally stable types of dif-raction catastrophes.

The present paper completes the sequence started in18,19], aimed at giving new computational tools forvaluating the whole hierarchy of stable diffraction catas-rophes, also in regions of control space that are distant

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Riccardo Borghi Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1667

rom the origin and on, or close to, the correspondingaustics. Here a simple approach to evaluate, up to arbi-rary accuracies, the fields associated with hyperbolic andlliptic umbilic diffraction catastrophes has been pro-osed. Similarly as done for cuspoids [18,19], such an ap-roach is based on the numerical summation, operatedia the WT, of converging series expansions of the highlyscillating two-dimensional integrals defining hyperbolicnd elliptic umbilics. Analytical exact expressions for then-axis umbilic field have also been found, which extendo complex values of the control parameter some previ-

ig. 4. Transverse sections of the modulus �E�x ,y ,z��, with re-pect to the z axis, of elliptic umbilic diffraction catastrophes.ictures are evaluated at the planes (a) z=0, (b) z=1, (c) z=2, (d)=3.55, (e) z=3.85, (g) z=4, and (h) z=5. Note that the scales areifferent. This figure should be compared to Figs. 3(a)–3(h) of22].

usly published results. Numerical experiments haveeen carried out to give evidence of the effectiveness ofhe proposed method, as well as of the main computa-ional limits. As in [18,19], the main advantage of our ap-roach is its ease of implementation. Of course, the priceo be paid in any practical implementation is the unavoid-ble resort to symbolic programming languages dealingith numerical evaluations of generalized hypergeomet-

ic functions with arbitrary precision like, for instance,athematica or Maple. We believe that our approach

ould be profitably used also to deal with other types ofptical catastrophes like, for instance, those recentlytudied in [46,47]. Finally, we hope that the availability oflow cost,” in terms of the ease of implementation, effi-ient computational tools like those introduced in this se-uence of papers could be of help in stimulating people topproach this fascinating way to mix diffraction and raysptics.

PPENDIX A: EXACT EVALUATION OFB„0,0,Z…

onsider the function HB�0,0,z�, i.e.,

HB�0,0,z� =1

2��

−�

+�

d��−�

+�

d� exp�i��3 + �3 + z����,

�A1�

hich can be written as

HB�0,0,z� =�−�

+�

d� exp�i�3�1

2��

−�

+�

d� exp�i��3 + z����

= 3−1/3�−�

+�

d� exp�i�3�Ai z

31/3��= 3−1/3G− i,

z

31/3� + Gi,−z

31/3�� , �A2�

ig. 5. Contour plot of �E�x ,y ,z�� as a function of x and y, evalu-ted at the transverse section z=4.

w

hp

a

rB(f

wf

AFC

wigtsl

I

is

w

NtRi

ab

w

1668 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Riccardo Borghi

here the function

G�p,c� =�0

�

d� exp�− p�3�Ai�c�� �A3�

as been introduced. The integral in Eq. (A3) can be ex-ressed through formula 2.11.1.5 of [48], i.e.,

G�p,c� =3−1/6�1/3�

6�p1/3 �k=0

� k +1

3�k!�2/3�k

c3

9p�k

−31/6c�2/3�

6�p2/3 �k=0

� k +2

3�k!�4/3�k

c3

9p�k

, �A4�

nd the two series can be evaluated in closed form as

�k=0

� k +1

3�k!�2/3�k

xk = ��2

3�x1/6I−1/6 x

2�exp x

2� , �A5�

�k=0

� k +2

3�k!�4/3�k

xk = ��4

3�x−1/6I1/6 x

2�exp x

2� , �A6�

espectively, with In� � denoting the nth-order modifiedessel function of the first kind [5]. On substituting Eqs.

A5) and (A6) into Eq. (A4), after rearranging and simpli-ying we eventually obtain

G�p,c� =

�� exp c3

18p�9�6p�2/3

6p1/3

5

6�0F1;

5

6;

c6

1296p2�

−61/3c

7

6�0F1;

7

6;

c6

1296p2�� , �A7�

here pFq� � denotes the generalized hypergeometricunction [5].

PPENDIX B: EVALUATION OF THEUNCTION IN„�…

onsider first the integral

�C

ds

�ssn expis3 +

�

s �� , �B1�

here C denotes the contour depicted in Fig. 6. Since thentegrand has no singularities inside C, the contour inte-ral in Eq. (B1) must be zero. Furthermore, when �→�he contribution associated with C� tends to zero. Con-ider now the contribution associated with C�, which onetting s=� exp�i�� with �� �0,� /6� can be written as

�C�

ds

�ssn expis3 +

�

s ��= − i�n+1/2�

0

�/6

d� expin +1

2���exp�i�3 exp�i3���expi

�

�exp�− i��� . �B2�

t vanishes, for �→0, only if the condition

Re�i� exp�− i��� 0 �B3�

s identically fulfilled for �� �0,� /6�. This condition re-tricts the validity domain of � to the sector

�

6� arg��� � �, �B4�

here the function In��� can then be expressed as

In��� = i�n+1/2�/3�0

�

dttn−1/2 exp− t3 +i2/3�

t � . �B5�

ote, in particular, that for values of � satisfying Eq. (B4)he integral defined by Eq. (B5) converges sincee�i2/3���0. To evaluate the integral we let �=1/ t so that

t becomes

In��� = i�n+1/2�/3�0

�

d� �n−3/2 exp�− �−3 + i2/3���, �B6�

nd, by using formula 2.3.2.14 of [49], after simple alge-ra it is found that

In��� = i�n+1/2�/3��− i2/3��n+1/2

�− n −1

2�0F3;3 + 2n

6,5 + 2n

6,7 + 2n

6;�3

27�+

1

3�k=0

2 �i2/3��k

k!n − k

3+

1

6��1F41;�1,1 +

k − n − 1/2

3 �,��3,1 + k�;�3

27�� ,

�B7�

here the symbol ��q ,a� denotes the sequence

Fig. 6. Contour path for the integral in Eq. (B1).

Asff

Ftw

AIsh

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

3

4

4

Riccardo Borghi Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1669

��q,a� = �a

q,a + 1

q, . . . ,

a + q − 1

q � . �B8�

ctually, since for 0k2 there is always a term of theequence ��3,1+k� which equals 1, the evaluation of theunction in Eq. (B7) involves only 0F3 hypergeometricunctions due to the fact that

1F4�1;b1,b2,b3,1;z� = 0F3�;b1,b2,b3;z�.

inally, by analytical continuation, the representation ofhe function In��� given in Eq. (B8) is extended to thehole domain 0�arg�����.

CKNOWLEDGMENTSam grateful to both reviewers for their comments and

uggestions. I also wish to thank Turi Maria Spinozzi foris help during the preparation of the manuscript.

EFERENCES1. M. V. Berry and C. Upstill, “Catastrophe optics: Morpholo-

gies of caustics and their diffraction patterns,” Prog. Opt.18, 257–346 (1980).

2. J. F. Nye, Natural Focusing and Fine Structure of Light(IOP, 1999).

3. R. Thom, Structural Stability and Morphogenesis (West-view, 1989).

4. J. F. Nye and J. H. Hannay, “The orientations and distor-tions of caustics in geometrical optics,” Opt. Acta 31, 116–130 (1984).

5. National Institute of Standards and Technology (NIST),Digital Library of Mathematical Functions (NIST, 2010);http://dlmf.nist.gov/.

6. T. Pearcey, “The structure of an electromagnetic field in theneighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

7. J. N. L. Connor and D. Farrelly, “Theory of cusped rainbowsin elastic scattering: Uniform semiclassical calculations us-ing Pearcey’s integral,” J. Chem. Phys. 75, 2831–2846(1981).

8. J. N. L. Connor and P. R. Curtis, “A method for the numeri-cal evaluation of the oscillatory integrals associated withthe cuspoid catastrophes: Application to Pearcey’s integraland its derivatives,” J. Phys. A 15, 1179–1190 (1982).

9. J. J. Stamnes and B. Ê. Spjelkavik, “Evaluation of the fieldnear a cusp of a caustic,” J. Mod. Opt. 30, 1331–1358(1983).

0. J. N. L. Connor, P. R. Curtis, and D. Farrelly, “A differentialequation method for the numerical evaluation of the Airy,Pearcey and swallowtail canonical integrals and their de-rivatives,” Mol. Phys. 48, 1305–1330 (1983).

1. J. N. L. Connor, P. R. Curtis, and D. Farrelly, “The uniformasymptotic swallowtail approximation: Practical methodsfor oscillating integrals with four coalescing saddle points,”J. Phys. A 17, 283–310 (1984).

2. J. N. L. Connor and P. R. Curtis, “Differential equations forthe cuspoid canonical integrals,” J. Math. Phys. 25, 2895–2902 (1984).

3. M. V. Berry and C. Howls, “Hyperasymptotics for integralswith saddles,” Proc. R. Soc. London, Ser. A 434, 657–675(1991).

4. D. Kaminski, “Asymptotics of the swallowtail integral nearthe cusp of the caustic,” SIAM J. Math. Anal. 23, 262–285(1992).

5. N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, “An adaptivecontour code for the numerical evaluation of the oscillatorycuspoid canonical integrals and their derivatives,” Comput.Phys. Commun. 132, 142–165 (2000).

6. R. B. Paris and D. Kaminski, “Hyperasymptotic evaluation

of the Pearcey integral via Hadamard expansions,” J. Com-put. Appl. Math. 190, 437–452 (2006).

7. C. A. Hobbs, J. N. L. Connor, and N. P. Kirk, “Theory andnumerical evaluation of oddoids and evenoids: Oscillatorycuspoid integrals with odd and even polynomial phase func-tions,” J. Comput. Appl. Math. 207, 192–213 (2007).

8. R. Borghi, “Evaluation of diffraction catastrophes by usingWeniger transformation,” Opt. Lett. 32, 226–228 (2007).

9. R. Borghi, “On the numerical evaluation of cuspoid diffrac-tion catastrophes,” J. Opt. Soc. Am. A 25, 1682–1690 (2008).

0. E. J. Weniger, “Nonlinear sequence transformations for theacceleration of convergence and the summation of divergentseries,” Comput. Phys. Rep. 10, 189–371 (1989).

1. J. F. Nye, “Dislocation lines in the hyperbolic umbilic dif-fraction catastrophe,” Proc. R. Soc. London, Ser. A 462,2299–2313 (2006).

2. M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilicdiffraction catastrophe,” Proc. R. Soc. London, Ser. A 291,453–484 (1979).

3. P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffrac-tion catastrophe and rainbow scattering from spheroidaldrops,” Nature 312, 529–531 (1984).

4. J. F. Nye, “Rainbow scattering from spheroidal drops: Anexplanation of the hyperbolic umbilic foci,” Nature 312,531–532 (1984).

5. P. L. Marston, “Cusp diffraction catastrophe from sphe-roids: Generalized rainbows and inverse scattering,” Opt.Lett. 10, 588–590 (1985).

6. J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc.R. Soc. London, Ser. A 438, 397–417 (1992).

7. T. Uzer, J. T. Muckerman, and M. S. Child, “Collisions andumbilic catastrophes. The hyperbolic umbilic canonical dif-fraction integral,” Mol. Phys. 50, 1215–1230 (1983).

8. M. V. Berry and C. J. Howls, “Stokes surfaces of diffractioncatastrophes with codimension three,” Nonlinearity 3, 281–291 (1990).

9. I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Se-ries, and Products, 6th ed., A. Jeffrey and D. Zwillinger,eds. (Academic, 2000).

0. R. Borghi and M. Santarsiero, “Summing Lax series fornonparaxial propagation,” Opt. Lett. 28, 774–776 (2003).

1. R. Borghi, “Summing Pauli asymptotic series to solve thewedge problem,” J. Opt. Soc. Am. A 25, 211–218 (2008).

2. R. Borghi, “Joint use of the Weniger transformation and hy-perasymptotics for accurate asymptotic evaluations of aclass of saddle-point integrals,” Phys. Rev. E 78, 026703(2009).

3. R. Borghi, “Joint use of the Weniger transformation and hy-perasymptotics for accurate asymptotic evaluations of aclass of saddle-point integrals. II. Higher-order transforma-tions,” Phys. Rev. E 80, 016704 (2009).

4. R. Borghi and M. A. Alonso, “Free-space asymptotic far-field series,” J. Opt. Soc. Am. A 26, 2410–2417 (2009).

5. R. Borghi, “Asymptotic and factorial expansions of Euler se-ries truncation errors via exponential polynomials,” Appl.Numer. Math. (in press), ISSN 0168-9274, DOI:10.1016/j.apnum.2010.02.002.

6. D. Deng and Q. Guo, “Exact nonparaxial propagation of ahollow Gaussian beam,” J. Opt. Soc. Am. B 26, 2044–2049(2009).

7. J. Li, W. Zang, and J. Tian, “Simulation of Gaussian laserbeams and electron dynamics by Weniger transformationmethod,” Opt. Express 17, 4959–4969 (2009).

8. J.-X. Li, W. Zang, Y.-D. Li, and J. Tian, “Acceleration of elec-trons by a tightly focused intense laser beam,” Opt. Express17, 11850–11859 (2009).

9. J.-X. Li, W.-P. Zang, and J. Tian, “Electron acceleration invacuum induced by a tightly focused chirped laser pulse,”Appl. Phys. Lett. 96, 031103 (2010).

0. J. Cižek, J. Zamastil, and L. Skála, “New summation tech-nique for rapidly divergent perturbation series. Hydrogenatom in magnetic field,” J. Math. Phys. 44, 962–968 (2003).

1. E. J. Weniger, “Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Za-mastil, and Skala. I. Algebraic theory,” J. Math. Phys. 45,1209–1246 (2004).

4

4

4

4

4

4

4

4

1670 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Riccardo Borghi

2. E. J. Weniger, “Asymptotic approximations to truncationerrors of series representations for special functions,” in Al-gorithms for Approximation, A. Iske and J. Levesley, eds.(Springer-Verlag), pp. 331–348 (2007).

3. For an updated review about methods for decoding diverg-ing series, see, for instance, E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura,“From useful algorithms for slowly convergent series tophysical predictions based on divergent perturbative ex-pansions,” Phys. Rep. 446, 1–96 (2007).

4. J. F. Nye, “The relation between the spherical aberration ofa lens and the spun cusp diffraction catastrophe,” J. Opt. A,Pure Appl. Opt. 7, 95–102 (2005).

5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cam-bridge U. Press, 1999).

6. J. F. Nye, “Diffraction in lips and beak-to-beak caustics,” J.Opt. A, Pure Appl. Opt. 11, 065708 (2009).

7. J. F. Nye, “Wave dislocations in the diffraction pattern of ahigher-order optical catastrophe,” J. Opt. A, Pure Appl. Opt.12, 015702 (2010).

8. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Inte-grals and Series (Gordon and Breach, 1986), Vol. III.

9. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Inte-grals and Series (Gordon and Breach, 1986), Vol. I.