CAUSTICS ARE PATTERNS OF BRIGHT points and lines that form when light reaches a surface by refraction or reflection. You can see them on a tablecloth in the light refracted by a glass of white wine. They can also form on the bottom of a swimming pool when sunlight is refracted by the waves at the surface. My favorite example appears when a laser beam is directed through a piece of irregularly rippled plastic and illuminates a screen with a beautiful array of bright lines. Even a slight motion of the plastic across the beam sends the patterns of light dancing into variegated new designs.

The caustic patterns can be simple or complex. Different materials lead to different designs. Is the number of possible patterns endless, or is there some way to classify them in groups of basic designs? Until recently I supposed the number of patterns was infinite. New developments in optics, however, have revealed a tidy way of analyzing the patterns. It turns out that there are only a few basic designs, which are called the elementary catastrophes.

This classification is based on catastrophe theory, a mathematical analysis originated in the 1970's by René Thom of the Institut des Hautes Etudes Scientifiques at Bures-sur-Yvette in France. The application of the theory to optical caustics was the work of Michael V. Berry of the University of Bristol. I have based my study on this work.

To examine caustic patterns I directed the beam from a helium-neon laser through a layer of transparent plastic with a rippled surface. The plastic, which came from the cover over a fixture of fluorescent light bulbs, had what appeared to be a random arrangement of smooth hills and valleys rather than a regular pattern. At some distance from the plastic was a screen on which the caustics appeared.

By moving the plastic through the beam I could rapidly sample many caustic patterns. Some were simple curved lines. Others were quite complex with overlapping bright lines. Interference patterns also appeared with the caustics, indicating that the light waves interfered destructively and constructively at the screen. I concentrated only on the caustics.

The caustic patterns can be separated into several basic units. The commonest type is a smoothly curved line that results from what is called a fold catastrophe. In addition the pattern might have a bright point, a cusp, a swallowtail, a triangle, a butterfly or a corner. These basic units follow from the several elementary catastrophes arising when the beam from the laser refracts through the rippled plastic.

I also obtained caustic patterns from a glass slide coated with an uneven layer of plastic glue. Laser light is advantageous in these experiments because its rays normally spread only slightly. Sunlight, spreading much more, tends to obscure some of the basic caustic designs. Another source of light might serve if it is far enough from the refracting material to appear to be a point source of light. Only then is the spread in the rays from the source sufficiently small so that the refracting material can yield clear caustic designs.

An intriguing caustic pattern can be seen readily in sunlight. Put a drop of water on a glass slide and hold it just above a flat surface. On the surface a pattern appears that almost always has cusp caustics around the perimeter.

To explore catastrophe theory in optics I shall consider the caustics that can be produced by a plane wave of light passing through a transparent material such as a layer of plastic. Before the light reaches the plastic it is traveling in the positive direction of the z axis in the top illustration on the left. The notion of a wave surface aids in tracking the progress of the light wave. This surface is an imaginary one on which all parts of the wave are in phase For example, if light is-considered to be a wave of crests and troughs, at some instant only crests pass through the imaginary surface. A short time afterward only troughs pass through.

The shape of the wave surface is given mathematically by a function f that is. the distance between any point on the wave surface and an underlying x-y plane that serves as a reference plane, When the light is a plane wave, f is simple because the wave surface is flat and parallel to the x-y plane. Thus all points on the wave surface are at the same distance from the x-y plane, and f is merely: a constant.

The notion of a light ray also helps in visualizing the progress of a wave. A ray is a vector pointing in the direction of travel of the light. When a ray is added to a section of the wave surface, it is drawn perpendicular to that section. If the light wave is planar, the light rays are easy to draw because they are all parallel.

When the light wave passes through a layer of rippled plastic, the wave surface and the rays are no longer as easy to draw. I shall describe several possible results since generally the shape of the plastic surface is not known. For simplicity I assume two limitations on the shape. First, the surface must be smoothly rippled with no sharp ridges or intentional patterns. A sharp edge complicates the analysis, and an intentional pattern might dominate the pattern of light cast on the screen. Second, the ripples on the surface should be larger than the wavelength of visible light so that the light pattern is not due merely to wave interference.

When the light wave passes through the plastic layer, it is refracted in many directions. Hence its wave surface is no longer flat and the light rays are no longer parallel to the z axis. The value for the function describing the height of a point on the wave surface is no longer a constant. A point on a hill on the wave surface is far from the x-y plane and a point in a valley is closer.

When rays are added to the picture, they must again be perpendicular to the wave. surface wherever they are drawn. Since that surface is no longer flat, the rays point in many directions. The part of the light wave passing through a given section of the wave surface travels in the direction of the ray assigned to that section. . The travel of the entire light wave is therefore harder to follow once the light passes through the plastic.

Much farther along the z axis lies the screen. That is where you see the results of the distortion of the light wave by the plastic. If the plastic were flat, the screen would be evenly illuminated. With rippled plastic much of the screen may be partially illuminated, but in some places the bright caustics will appear. Each point on a caustic is formed because the plastic layer bunches many rays onto that region of the screen.

My objective was to classify the kinds of pattern that can appear on the screen without specifying any details about the surface structure of the plastic layer. The task is to figure out what kinds of pattern are possible on the screen by studying what kinds of shape a wave surface can have. Catastrophe theory predicts that the wave surface can have only a limited number of distinguishable shapes. Thus only a limited number of basic caustic patterns can appear on the screen. Since any complicated pattern of caustics can be broken down into these basic patterns, one can immediately describe the shape of the wave surface even though the details about the surface of the plastic are not known.

To determine the possible shapes of the caustics one must examine the curvature of the wave surface. Figure 5 shows a cross-sectional slice of a hill on the surface. The x axis of the underlying x-y plane is included. I am interested in how the slice curves with respect to the x axis. Two regions of the slice are curved and one section (the inflection on the side of the hill) is not. Rays from the curved sections are spread over the screen with no bunching. Rays from the straight section are bunched onto one spot on the screen and so contribute to a caustic.

The figure is for only one cross-sectional slice of the wave surface. Next to that slice another one can be depicted. Again part of the underlying x-y plane is to be drawn below the slice. For convenience I call this direction x too even though it probably differs from the direction in the first slice.

Part of the hillside of the second slice lacks curvature. Perhaps this region is a bit higher or lower on the hillside than it was before, but such a detail is not important just yet. The point is that one can always take a slice of any part of the wave surface and examine it for curvature. If part of the slice lacks curvature, it can contribute to a caustic to the screen.

Suppose all the uncurved places on the wave surface are discovered. On the underlying x-y plane imagine a line L that is just below those places. The shape of L determines the shape of the caustics on the screen. Usually the line creates a smoothly curved line of light on the screen, but it can create several other caustic patterns: the other elementary catastrophes.

A graphic analysis greatly aids one's use of catastrophe theory. Reconsider the illustration of a cross-sectional slice through a hill on the wave surface. Rays from the hillside end up at various places on the screen. The axis a shows where they shine. (The illustration is misleading because the screen is necessarily shown close to the wave surface.)

Rays from the bottom of the hill point directly toward the screen. As one ascends the hill the rays began to point more to the left. The extreme is reached in the interesting region that has no curvature. Further ascent of the hill takes one through rays pointing less to the left. Finally, on the hilltop the rays are again pointing directly toward the screen.

Since the screen is distant from the wave surface, the rays both from the flat bottom of the hill and from the hilltop end up along the same region of the screen. The rays that are deflected to the left side of the screen are those from the side of the hill. Points along the wave surface (from the bottom of the hill to the top) are connected by the rays to points along the screen.

The connection is best expressed in terms of the values of x (on the axis below the wave surface) and a (on the axis across the screen). Figure 7 shows the relation of the values. For large values of a two regions on the wave surface contribute a ray. For example, the top and bottom sections of the hill both send rays to approximately the same place. With classical optics one can calculate the intensity of the light at that point on the a axis. Since the rays are not bunched, the intensity is less than it is at a caustic.

Near the fold in the curve the value of a reaches a limit set by the rays from the side of the hill. The curve folds over for the limiting value, which means many rays arrive at that point on a from a range of places on x. Classical optics predicts the intensity to be infinitely large at the foldover point on a. This is a caustic point.

This arrangement is a fold catastrophe. The term implies that the graphic relation between points x (associated with the wave surface) and points a (along the screen) has a fold. The word catastrophe is appropriate too. Points on a away from the caustic have a simple contribution of rays, one ray from a large value of x and another from a small one. As one considers points of a nearer the fold, however, the rays abruptly bunch to produce a bright spot at the fold point on a.

The algebraic relations between values of x and a for the elementary catastrophes are usually given by means of generating functions. The functions are listed in Figure 8. For the fold catastrophe two variables are required. The a is called a control variable, the x a state variable. To gain the relation between x and a for the fold catastrophe its generating function is differentiated with respect to the state variable x and set to zero. The resulting equation is the one plotted in Figure 7.

I have now demonstrated that a section on the wave surface can create a bright point on the screen. Suppose that next to this section on the wave surface other sections also contribute a bright point apiece. The string of bright points on the screen forms a smoothly curved caustic line, the commonest kind of pattern yielded by a rippled layer of plastic inserted in a laser beam. Each point on the caustic line is the result of a fold catastrophe from a section on the wave surface that lacks curvature with respect to a direction on the underlying x-y plane. The line L running below these sections on the wave surface is transformed by means of the rays into the caustic line on the screen.

A cusp catastrophe is somewhat more complex. It involves the same-state variable x for the wave surface but includes a new control variable 6 on the screen in addition to the a already employed. In other words, in a cusp catastrophe the points x for the wave surface-determine the rays in the region of a and h on the screen. The algebraic relation between x, a and is obtained by differentiating the appropriate generating function by the state variable x and then setting the result to zero.

Again a graph aids one's understanding of the algebraic relation. This time the graph is three-dimensional. As is shown in Figure 9 two dimensions of the graph are devoted to the control variables a and b; the vertical dimension is the state variable x.

The bottom plane of the graph is the control space and actually represents the screen. Above it is a folded sheet whose shape is set by the equation obtained from differentiating the generating function. (One should not be misled into believing this sheet is visible or tangible. It is only a mathematical relation between the points labeled x on the wave surface and points a and b on the screen.)

What one sees on the screen is the projection of the folded sheet onto the a-b surface of the graph. The projection is a bright cusp. Hence if a layer of rippled plastic produces the right kind of wave surface, one finds a caustic cusp on the screen instead of the smoothly curved caustic line formed from a series of simple fold catastrophes.

What kind of wave surface produces a cusp? A fold catastrophe results from a region on the wave surface that has no curvature along some direction in the x-y plane. I call that direction x for any slice through the wave surface. The line L runs through the points on the x-y plane that are below all these places on the wave surface without curvature. At each point along L I can construct x and y axes so that x is in the direction of no curvature for the part of the wave surface just above the point.

In general the x axis at each point is in a direction different from that of the line L through the point. If the surface of the plastic is appropriately shaped, however, the wave surface can have a region in which the x axis coincides with the direction of L there. The rays from this region end up forming a bright cusp on the screen.

Figure 10 indicates a candidate for such a region on t-he wave surface. The regions without curvature form a path that runs along the side of a hill, climbs the hill and then again runs along the side. Consider the shape of L just below the path. Also consider the direction of no curvature for points along L. For any point on L away from the climbing region the direction of no curvature is not coincident with L. Those points therefore contribute only simple fold catastrophes. Seen as a composite they are smoothly curved caustic lines on the screen. In the climbing regions the direction of L coincides with the direction of no curvature. Hence rays from that region produce the sharp point on the cusp caustic on the screen.

The graph of the cusp catastrophe illustrates the connection between points of a and b on the screen and points x for a path over a wave surface. As before connection is by means of the rays leaving the hillside for the screen. To show the connection I have numbered e sections on a path over the hillside order to show where rays leaving those sections end up on the screen. Bear mind again that the screen is actually much farther from the wave surface than the illustration suggests. Remember too that the cusp is likely to be much larger than the wave surface.

Rays from sections 1, 3 and 5 end up in the middle of the cusp. As one climbs the hill through section 2 the rays are detected to the left side of the screen. Along section 2 the rays begin to bunch to form the left side of the cusp caustic. Higher on the hill the rays begin to fall inside the cusp again. Section 2 is therefore responsible for the extreme left side of the caustic pattern.

As one begins to descend the hill s through section 4 the rays are sent off to the right side of the screen. The bunching of rays from near section 4 is responsible for the caustic line on the right. Rays from below section 4 on the hill fall inside the cusp.

A similar analysis can be. made on the three-dimensional graph of the cusp caustic. In the illustration I have rotated the cusp so that the fold in the graph can be visualized. Again I have added labels to indicate where the rays originated from the hill on the wave surface. The edge of the cusp labeled 4 lies below the fold on the upper sheet of the graph. The other edge lies below the fold on the lower sheet. Thus the upper fold represents the connection between the right side of the hill and one edge of the cusp, whereas the lower fold is for the left side of the hill and the other edge.

Outside the cusp region the screen is not illuminated. The sections of the folded sheet above these dark regions are extraneous and could be eliminated from the graph. Points at the center of the cusp lie below three sheets. The top sheet represents the connection for the rays leaving the hill on the right side, below section 4. The middle sheet is for the rays from the top of the hill. The bottom sheet is for the rays from the section numbered 1.

Next consider a part of the folded sheet closer to the point where the fold smoothes out. That point is called a singularity. There the graph represents rays that originate in the climbing region and fall on the cusp point on the screen. Farther back on the graph the sheet is not folded. Below this section there is no caustic.

Part of catastrophe theory deals with classifying sections of a catastrophe as either generic or nongeneric. The definitions of these terms can be demonstrated by two lines on the screen that slice through the cusp pattern. Make the first line pass through both edges of the cusp but well away from the cusp point. The line cuts through two fold catastrophes, one catastrophe for each edge of the cusp pattern. Such a slice through the caustic pattern is said to be generic in the sense that a slight change in the line produces no major change in the type or number of catastrophes the path samples. Suppose I move the line slightly toward or away from the cusp point. It still passes over two edges and so still samples two fold catastrophes.

Next consider a line that passes directly through the cusp point. The line samples only one fold catastrophe, the one responsible for the point on the cusp itself. This slice through the catastrophe pattern is said to be nongeneric in the sense that a slight change in the line can have a dramatic effect on what the path samples. If I move the line slightly toward the rear of the graph, it no longer samples any of the catastrophe pattern. If I move the line slightly toward the front of the graph, it samples two fold catastrophes instead of one. Moving from a nongeneric section of a catastrophe pattern to a generic section is said to be unfolding the catastrophe.

The nature of the elementary catastrophes depends on the number of control and state variables involved in the production of a caustic pattern on the screen. To produce a fold catastrophe only one state variable (x) and one control variable (a) are required, but an additional control variable (b) is needed to produce a cusp catastrophe. If light passes through a refracting layer, many more control variables besides the two associated with a position on the screen are possible. For example, if the refracting material is a drop of water, a third control variable might be the pressure inside the drop, because the pressure can alter the shape of the drop.

If a caustic pattern has one state variable (still the line x) and three control variables, the catastrophe is said to be a swallowtail one. The algebraic relation between the variables is obtained by differentiating the generating function with respect to the state variable and setting the result to zero. This relation cannot be graphed because it is four-dimensional.

The projection of the folded sheet onto a three-dimensional space of the control variables can be depicted. The procedure resembles what was done with the cusp catastrophe: the folded sheet was projected onto the bottom of the graph (the plane of the control variables) so that the cusp could be analyzed. The projection of the swallowtail folded sheet onto the control space is shown in Figure 11.

What is seen on the screen with such a catastrophe? It is certainly not the full projection (which is three-dimensional). Rather one sees only a cross-sectional slice through the projection. I have labeled the axes through the projection in order to facilitate making such slices. Two of the dimensions are the control variables a and b associated with places on the screen. The third variable (c) is some other variable one could change in the refraction demonstration.

To visualize the patterns that can appear on the screen imagine a slice through the projection at one value for the third control variable. For example, a slice made near the left side of the illustration resembles a bird, which is the source of the name given to the catastrophe. If the third control variable has the proper value, this pattern can be seen on the screen.

Next imagine cutting slices through the projection for other values of the third control variable. The slices are made progressively toward the rear of the pattern. Eventually a slice has only a simple curve. Slices farther back are similar. Hence a slice from the front of the pattern yields a swallowtail on the screen and a slice from the rear produces a simple curve.

The slice dividing these two possibilities is a nongeneric section. The other slices through the projection are generic. If the third control variable can be controlled, one can vary the caustic pattern from the swallowtail through the smooth curve. Variations of this kind are described as folding or unfolding the catastrophe: For example, changing the slice from the nongeneric section to either the front or the back of the pattern is said to be unfolding the catastrophe.

When there are four control factors and one state factor, the catastrophe is of the butterfly type. The algebraic relation is obtained with the usual differentiating of the generating function. The result is impossible to graph because it represents a five-dimensional folded sheet. Even its projection onto the four-dimensional control space is impossible to draw. At best one can draw three-dimensional sections of the projection. I have not attempted the task but am content to provide the possible patterns that appear on the screen, which is only a two-dimensional slice out of those more complicated configurations. A few of the patterns of this type are shown in Figure 14. The catastrophe derives its name from the butterflylike shape of some of the patterns.

I have been considering the higher dimensions for the control variables while holding the control variable to x. A refracting material has another state variable, y. When there are three control variables and two state variables, the catastrophe is called a hyperbolic umbilic. The algebraic relation between these variables is obtained by differentiating the generating function with respect to x and then to y, with each result set to zero to produce two equations. Although the folded sheet representing these equations cannot be graphed, the projection of it onto the three-dimensional control space can be drawn as in the middle illustration at the right.

What appears on the screen is a slice through this pattern. The position of the slice is set by the value of the third control variable. The nongeneric slice is through the middle of the pattern. On the screen the caustic pattern is a bright corner with a finite angle (in contrast to the angle of zero degrees of a cusp corner). If one could actually-control the third variable, this catastrophe could be unfolded by moving the slice. The pattern appearing on the screen would then change accordingly.

If the slice is moved toward the front or the back of the hyperbolic umbilic, the corner unfolds into a smooth curve lying around a cusp. Suppose such a pattern falls on the screen when a plastic layer is inserted into a laser beam. The pattern is said to be an unfolded hyperbolic umbilic.

A combination of three control variables and two state variables can also yield an elliptic umbilic. A different generating function is assigned and different algebraic equations are derived for the relation between the variables. The non-generic slice provides a point caustic for the screen. Generic slices have triangles with curved sides. If the elliptic-umbilic catastrophe is unfolded, the caustic point turns into triangles.

A further increase in the number of control variables for either one state variable or two is certainly possible, but the resulting higher-order catastrophes become difficult to understand. They are also unlikely. The chances are that only two or three control variables are important in any experiment with a refracting surface in a beam of light. Thus one should expect to find only the patterns from the first six elementary catastrophes on the screen.

You can do experiments with rippled plastic and other transparent materials for these catastrophe patterns. If you come on a pattern from a nongeneric slice through a catastrophe, you might try unfolding the catastrophe. I have had some success by carefully turning a plastic layer in a laser beam. Although I hold the same section of the plastic surface in the beam, the rotation of the plastic on an axis perpendicular to the 11 beam forces the light to pass through the plastic at a new angle. The control variable I am changing is that angle. The result is that the higher-order catastrophe folds and unfolds.

Bibliography

WAVES AND THOM'S THEOREM. M. V. Berry in Advances in Physics, Vol. 25, No. 1, pages 1-26; 1976.

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