## Abstract

The three-dimensional pattern of the hyperbolic umbilic diffraction catastrophe is computed from an integral representation. A detailed description is given of the geometrical arrangement of the wave dislocation lines (optical vortices) on which the diffraction pattern is based. From a crossed grid of nodal lines in the focal plane, two bundles of dislocation lines spring out symmetrically into the regions of 4-wave interference. Each dislocation line then follows a chain of curved segments which approximate successive steps along lattice vectors in the space group *Fmmm*. The result is a bundle of helices of non-circular cross-section that gradually straighten out until, far from the focal plane, they become the dislocations of the Pearcey diffraction pattern for the cusp catastrophe. A new phenomenon is the multiple puncturing of the caustic surface by a series of helical dislocations.

## 1. Introduction

The generic caustics of geometrical optics form a hierarchy, starting with the fold, the cusp, the swallowtail and the elliptic and hyperbolic umbilics. Each is associated with a diffraction pattern, called a diffraction catastrophe, that is described by a certain diffraction integral. These particular patterns are worthy of detailed study precisely because they are generic, that is, they occur naturally without special preparation, and they are structurally stable (Berry 1981; Nye 1999). The pattern for the fold caustic, of codimension one, is described by the Airy function, and that for the cusp caustic, of codimension two, was computed by Pearcey (1946). Among the diffraction catastrophes of codimension three the swallowtail has been computed (Wright 1977; Connor *et al*. 1983, 1984; Connor 1990); the elliptic umbilic has been thoroughly studied (Berry *et al*. 1979) and the hyperbolic umbilic has been computed by F. J. Wright (1978, unpublished work) and by Berry & Howls (2006). To describe the geometry of such three-dimensional patterns, it is useful to concentrate on the network of wave dislocation lines (singularities of phase, wave vortices, zeros of amplitude) that they contain, because these are structurally stable features of the field; the dislocation network encapsulates the diffraction pattern. The connectivity of the dislocation network for the elliptic umbilic is well understood, but this is not yet the case for the hyperbolic umbilic and the swallowtail. The purpose of the present paper is to describe the details of the three-dimensional dislocation network for the hyperbolic umbilic.

A physical realization of the hyperbolic umbilic diffraction catastrophe is provided by placing a small drop of water on a vertical glass slide, constraining its periphery to be circular, and using it as a lens to focus a distant point source of monochromatic light (Nye 1999, §4.6 and fig. 6.15). The caustic at the focus survives severe changes in the shape of the periphery, as also does its associated diffraction pattern.

## 2. The caustic and the diffraction integral

The caustic consists of two intersecting fold surfaces (figure 1), the distinctive feature being a cusp line (ribline) CC that touches a fold at the focus O. In the focal plane *z*=0, the section is just two semi-infinite straight lines meeting at an angle at the origin.

The diffraction integral we will be concerned with represents the optical pattern in the neighbourhood of the focus in the limit of vanishing wavelength. Out of the various alternative and equivalent forms for the hyperbolic umbilic we choose (Poston & Stewart 1978)(2.1)where *x*, *y*, *z* are dimensionless Cartesian coordinates with origin at the focus (control parameters), and are dimensionless Cartesian coordinates on the initial wavefront (state space). Just as with the elliptic umbilic, the complex scalar diffraction field is given, apart from a constant complex factor, by(2.2)where *K* is a dimensionless phase gradient that depends on the specific problem. The real physical diffraction pattern of course depends on the wavelength, the focal length and the aberration (that is, the departure of the initial physical wavefront from a sphere centred on the focus). The scalings used to reduce the physical problem to the dimensionless form (2.1) are given explicitly in Nye (2003).

The justification for having the phase factor in (2.2) varying only in the *z* direction is provided by the analysis of the orientation of the hyperbolic umbilic in Nye & Hannay (1984). But, actually, this phase factor is not essential to our purpose, because it has no effect on the amplitude of the field or on the positions and senses of its dislocations, and it will therefore be ignored. (In further explanation, a dislocation line, curved in general, has not only a position but also a sense, which is conserved along its length, the sense being simply the sense of the circulation of phase around it. The sense is not affected by the phase factor because close enough to a dislocation the rate of change of phase from the dislocation in approaches infinity and so overwhelms any change from the phase factor.)

In the pattern of complex amplitude described by equation (2.1), *y*=0 is a mirror plane, and *z*=0 reflects to give the complex conjugate, thus and . This implies that in the plane *z*=0, *H* is pure real, the phase is 0 or *π*, and the dislocations become nodal lines (figure 2*a*). In fact, the whole pattern simplifies in the plane *z*=0. First make a linear change of state variables in equation (2.1), , . This converts the term to and yields the product of two Airy functions:(2.3)A shaded contour plot of is shown in figure 2*b*.

Figure 3*a*,*b* shows the amplitude and phase in an *xy* section through the diffraction pattern with . A system very like Airy fringes is seen running parallel to the fold and there is some resemblance to the Pearcey pattern within the cusp caustic. Figure 3*c* maps the amplitude in the plane of symmetry .

Geometrical optics shows that there are four contributing rays at any point within the cusp caustic (figure 3*a*,*b*), two rays at points in the region between the cusp and the fold, and no rays outside the fold. Correspondingly, the integrand in equation (2.1) has 4, 2 and 0 points of stationary phase in these regions. To a certain approximation, therefore, we can expect to see a 4-wave interference pattern within the cusp caustic and a 2-wave pattern between the two caustics. The 2-wave pattern is simply the Airy pattern of fringes running parallel to the fold caustic. In an isolated Airy pattern, the wave function is pure real and there are exact zeros of amplitude, corresponding to nodal surfaces, running along the dark fringes. However, in the case in hand, where there is perturbation from the cusp caustic, these exact zero surfaces disappear, without leaving dislocations.

We can now describe the whole three-dimensional pattern of dislocations. The symmetry plane *z*=0 itself contains, within the V-shaped caustic, a crossed grid of straight dislocations (figure 2*a*) given by equation (2.3) and making a pattern of small parallelograms. Note that the dislocations in *z*=0 extend into the region of very small amplitude, where there are no real rays. The senses are indicated by arrows (one way of indicating the sense of a dislocation is to associate its sense of circulation with a direction along the dislocation line by a right-hand rule; the arrow represents an axial rather than a polar vector.) At each crossing, two arrows point inwards and two outwards, as they must to conserve the dislocation strength. The zeros of the Airy functions are not uniformly spaced, and as a result all the rows of lattice points, except those parallel to the arms of the caustic, are slightly curved. From near the centre of each parallelogram edge, two dislocations spring out of the plane, one up and one down, as shown by small circles. These junctions alternate in character, but, like those entirely in the plane *z*=0, each has two arrows inwards and two outwards.

We now trace what then happens to these out-of-plane dislocations; owing to the symmetry it will usually only be necessary to refer to the half-space *z*>0. If we follow the one springing upwards from the typical point P in *z*=0 close to the *x*-axis (figure 2*a*), its path is as shown in figure 4*a*, which represents it (and other dislocation lines) as projections on to the three coordinate planes. Thus, its projection on to the *z*=0 plane is a zigzag of nearly straight lines. Projected on to the *y*=0 plane, the dislocation first rises vertically but then turns abruptly to the horizontal, goes back to vertical at the point a and so on. The same dislocation projected on to the *x*=0 plane appears as a helix with nearly vertical and horizontal segments. The figure also shows the dislocation starting from the neighbouring node L (with the same *x* value but a larger *y*). On the scale of the figure, it seems to contact the track from P at the place marked *, but close inspection shows that in fact the two tracks do not come together. This is crucial for the description of the dislocation pattern, because topologically it would have been possible for the opposite hyperbolic connections to have been made by reconnection (Nye 2004), thus forming an arch from P to L, with disconnected loops above it. After this first near-contact, the dislocations from P and L tend to diverge slightly from each other, thus removing any threat of contact at larger *x* and *z* values. All these dislocations, which in three dimensions are helical, thus remain separate (and this includes those starting from points nearer the origin than P). Note that the projections of the dislocations from P and L on to the *y*=0 plane are indistinguishable. The same figure also shows tracks from the nodes N and Q at (approximately) the same *x* value; their behaviour is the same, and in fact is typical for dislocations throughout the pattern. The general effect of the successive steps is to increase *x* and *z* while keeping *y* constant, on average.

Figure 4*b*,*c* shows dislocations from the first two junctions in *z*=0 closest to the origin (B+ and B−) and from the next two (C+ and C−). It can be seen that they are helical and the projections on to *y*=0, which is the plane in which the ribline lies, show that their axes run parallel to this ribline. Ultimately, far from the origin and the perturbing effect of the fold caustic, the dislocation pattern becomes the Pearcey pattern in which the helical dislocations straighten out completely and run parallel to the ribline. In figure 4*a–c*, we are seeing the beginning of this approach to the Pearcey limit. Just the same approach was found in the corresponding pattern for the elliptic umbilic (Berry *et al*. 1979).

## 3. Dislocations on the cusp caustic

Whereas all these dislocations start inside the V-shaped caustic in *z*=0, the Pearcey pattern has rows of dislocations just outside the cusped caustic. It should therefore be expected that the dislocations from the outermost rows of marked circles in figure 2*a* would cross the caustic at some values of *z*, thereby creating exact zeros of amplitude at points where geometrical optics predicts brightness—punctures in the caustic as it were. This is 4-wave interference, with two of the rays coinciding and exactly reinforcing each other, but being cancelled out by the other two. Moreover, since the dislocations are helical, it is clearly possible that a single one will cut the caustic more than once, and in fact this is just what happens. In figure 4*b*,*c*, parts of the dislocations that lie inside the cusp caustic are indicated by thickened lines; the lines first exit the 4-wave region at fairly small values of *z* but return several times as successive coils cut the caustic, and eventually stay outside. Far from being repelled by the caustic the dislocation lines seem to be oblivious to it. Table 1 lists the coordinates of the lowest piercings on the dislocations indicated. To apply the stationary phase method at points like this would require a special treatment.

## 4. The idealized lattice and its symmetry

It is instructive to regard the tracks like those in figure 4*a* as based on an ideal lattice; the departures from an ideal lattice will be detailed in §6. The method of stationary phase would regard the pattern within the caustic as arising from interference between four rays. So, as a preliminary, let us consider the general problem of the interference of four uniform plane waves. This model will be most accurate in the 4-wave region near the *x*-axis and far from O. We shall find quite generally that four uniform plane waves produce a repeating amplitude pattern with a lattice. Owing to the phase prefactor in equation (2.2), the phase pattern of the diffraction field differs from problem to problem, unlike the amplitude distribution, and therefore, as a whole, the phase pattern is not of primary significance. Nevertheless, the positions of the singularities and the circulations of phase around the dislocations, their senses, are two features of the phase pattern that *are* significant precisely because they are not affected by the phase prefactor.

To see why a lattice is produced, consider the interference of four quite general uniform plane waves in three dimensions. First, temporarily shift the origin to the point of interest and construct the wave functionwhere (real and non-zero) are amplitudes and are (local) phase gradients (the physical wave vector was already scaled out in the derivation of the diffraction integral). It might perhaps be thought, since the vectors are in general unrelated, that there could be no lattice. In fact, there is always a lattice for amplitude, as already stated, but not usually one for phase. This may be seen as follows. Take out a factor and write as , whereandNow define the three vectors reciprocal to in the crystallographic sense, thus(4.1)and similarly for . If **r** is replaced by , it may be verified that is unchanged, and thus has the spatial periodicity . In the same way, also has the periodicities and . define a space lattice. It is important to note that although the field is periodic, itself is not. Nevertheless, its amplitude is periodic, because . (The selection of the factor was arbitrary; a different choice would have led to a different unit cell but the same space lattice.) Although is triply periodic, the same does not apply to . In fact, and, if is one of the three lattice vectors , it follows that . In other words, on shift of by , changes by an amount that is independent of position. This means that the equiphase surfaces of repeat in their positions but change their phase values (labels). The phase pattern (without labels) is periodic, but since will not generally be a rational multiple of the phase itself will not generally repeat on any scale. Figure 5 shows a two-dimensional example of this.

The actual 4-wave interference pattern of the hyperbolic umbilic far from the caustic has an imperfect lattice for amplitude, as we have seen. The full stationary phase approximation, valid everywhere in the 4-wave region except very close to the caustic, has been deduced by M. V. Berry (2005, personal communication). However, the lattice appears most clearly from the further stationary phase approximation (also due to M. V. Berry) that is valid in the 4-wave region near the *x*-axis not close to O. Explicitly, this is(4.2)where

We now find the lattice. Ignoring the overall phase factor , are found by taking the gradients of , and to find the unit cell on the *x*-axis we set . The difference vectors are then found as:

The repeat vectors, from equations like (4.1), are found to beOther lattice points can be constructed by combining in integer combinations. In particular, we can find lattice vectors parallel to the *x*, *y*, *z* axes, thus:Figure 6*a* shows the unit cell thus defined. The *x* dependences emphasize the curvature of the lattice ‘planes’. The further combinations,show that this cell also has lattice points at the centres of its faces, so making four lattice points per cell.

Figure 7 shows an *xz* section of the pattern of in a region near the *x*-axis distant from the origin, where the lattice is seen without much distortion. The value of *y* has been chosen so that the junctions of the nodal lines in *z*=0 are not intersected; this means that the nodal lines are intersected in pairs and are seen in the figure as separate true dislocations, rather than as merged and degenerate ones. Inspection of the amplitude pattern shows that it has mirror planes perpendicular to the coordinate axes, so that the point-group symmetry is orthorhombic *mmm*. Out of the 230 possible space groups (International Tables 1965, p. 159) only two with this symmetry are face-centred: *Fmmm* and *Fddd*. Inspection shows that, in fact, we have *Fmmm*. (The mirror planes imply that there are also centres of symmetry, glide planes perpendicular to the coordinate axes, twofold rotation axes and twofold screw axes parallel to the *x*, *y*, *z* axes.) An alternative unit cell is outlined in figure 6*a*, which has an angle of 60° and is body-centred, containing two lattice points.

Although the angle at the base of the alternative cell in figure 6*a* is 60°, this is not demanded by any symmetry element. In crystallographic terms, the cell shape is pseudo-trigonal. If, instead of starting with the diffraction integral (2.1), we had used the alternative form with the term in place of , as mentioned just before equation (2.3), the corresponding unit cell would have had a 90° angle at the base instead of 60°, but again with no symmetry element to enforce it; the unit cell would have been pseudo-tetragonal. The angle seems to have little significance.

The alternative unit cell of figure 6*a* is shown again in figure 6*b*, together with the grid of dislocation lines it contains. (The dislocation pattern can repeat without the phase itself repeating.) The lines have the senses indicated by the arrows, which denote axial vectors (using a right-hand rule). In discussing the presence of planes of symmetry in this structure, it is necessary to bear in mind that the arrows represent not directions as such but circulations of phase. An ordinary mirror plane *m* preserves the sense of a dislocation crossing it perpendicularly (figure 8*a*), but reverses the sense of a dislocation lying parallel to it (figure 8*b*), and so does not allow a dislocation to lie in the plane. But there is a different kind of mirror plane *m*′, an ‘anti-mirror plane’, that has just the opposite effects on the senses (figure 8*c*,*d*). The 230 ordinary space groups do not allow for such a symmetry element; for this the 1651 black–white or magnetic groups must be invoked. With these definitions, there is a set of *m*′ planes in figure 6*b* normal to the *z*-axis (containing dislocations) and sets of *m* planes normal to the *x* and *y* axes. Bradley & Cracknell (1972) list five black–white space groups within the ordinary space group *Fmmm*, of which the appropriate one is *Fm*′*mm* (No. 523, p. 591), or *Fmmm*′ with our axes. Without the arrows, the symmetry of the dislocation grid is just that of the amplitude. There are two types of junctions in figure 6*b*, those lying in vertical planes and those in horizontal planes, like the ones at the body-centre and at the corners of the cell. They all have two arrows pointing inwards and two pointing outwards. Those in horizontal planes are responsible for the dark X-shapes seen in *xy* sections of the amplitude pattern (figure 3*a*,*d*).

## 5. The symmetry of the idealized phase pattern

In figure 7, the repeat of the amplitude is clear, as is also the repeat of the pattern of phase lines. However, the labelled phase lines do have a property not found in a general field of four plane waves, but present here because of the symmetry of the four constituent waves, namely that the labels repeat. In fact, with labels included, the repeat in the *z* direction is 2*c*, but the *x* and *y* repeats remain as *a* and *b*. There are *m*′ planes at *z*=0, and and also intermediate glide planes. The absolute value of the phase is 0 or *π* on *z*=0, as expected.

Some further comment is required on the symmetry of the phase pattern. We have mentioned the effect of an *m*′ plane on the senses of the dislocations. Its effect on a general field of phase should be stated in a way that is independent of an overall additive constant phase; what matters is rather the gradient of phase at each point. Formally, if is an *m*′ plane, and and denote the normal and transverse components of the phase gradient, and . Likewise, the conditions for an *m* plane are and . In the present problem, the symmetry of the unlabelled phase lines is exactly the same as that of the amplitude pattern, but with gradients included the symmetry is *Fmmm*′ (conventionally called *Fm*′*mm*) with the same unit cell, which now has *m*′ planes at , , etc.

## 6. The true pattern of dislocations

The grid in figure 6*b* is an idealized one, appropriate to the interference of four infinite and uniform plane waves, whereas, in fact, the interfering waves of the stationary phase approximation are neither infinite nor uniform nor plane. As a result, the actual grid of dislocations is not ideal in several ways: of course it is not infinite, if only because the four real waves do not exist outside the cusp caustic, and it is curved, but most importantly, all the junctions in figure 6*b* where dislocations cross in the ideal grid, except those in the plane *z*=0, disconnect into hyperbolas with two branches. The total effect of all these departures from regularity is to produce the pattern of stepped helical dislocations illustrated in figure 4*a*. A typical dislocation follows the series of segments indicated in figures 4*a* and 6*b* by the letters abcdef; *x* and *z* progressively increase while *y* oscillates, as previously noted.

## 7. Summary and conclusion

The pattern of dislocations that underlies and supports the hyperbolic umbilic diffraction catastrophe is essentially simple. It is based on a non-uniform two-dimensional crossed grid in the plane of focus, filling the space between the arms of the V-shaped caustic. From this two-dimensional grid, helical dislocation lines spring out symmetrically on both sides of the focal plane into the 4-wave regions. The cross-sections of the helices are far from circular; each begins by approximately following selected segments of a three-dimensional slightly curved grid. The amplitude of the field pattern corresponding to a perfect grid would have the symmetry of the space group *Fmmm*. To describe the corresponding symmetry of the dislocations themselves, and to respect their senses, the magnetic space groups are required, the group here being *Fm*′*mm*. The real dislocation lines do not exactly follow the ideal grid; what would be dislocation crossings break up into hyperbolic pairs. The result is a sheaf of separate helices each of which eventually straightens out completely to become a dislocation of the Pearcey pattern for the cusp. The outer rows of dislocations cross to the outside of the cusp caustic to finish as the outer dislocations of the Pearcey pattern.

## Acknowledgements

I would like to thank John Hannay for much valuable discussion, and Michael Berry for advice on the contour integration and information on the stationary phase approximation.

## Footnotes

- Received November 1, 2005.
- Accepted January 20, 2006.

- © 2006 The Royal Society