A caustic surface is the envelope of a family of light rays. It is essentially a concept of geometrical optics, but the region of a caustic displays interesting wave effects. The word "caustic" is from Latin, and implies burning, because the intensity of the light is increased near a caustic. However, the intensity is not increased enough actually to make burning a fact. A caustic is a boundary between regions in which the light intensity is nonzero and zero. A wave cannot be sharply cut off, as the phenomena near the boundary of the shadow of an obstacle show. On one side, we have an intensity decreasing rapidly to zero, and on the other a system of interference fringes typical of two coherent superimposed beams. This is clearly seen in the Fresnel diffraction pattern of a straight edge. However, in the case of a caustic there is no edge, but a natural termination of the light field. Inside the caustic, there are two wavefronts that interfere to produce the observed fringes.
It is easy to see a caustic using a flashlight and a cylindrical coffee mug with a flat bottom. Shine the light obliquely into the mug, and note the reflection on the bottom of the mug. The cusp at the focal point, about halfway between the centre and the edge, and the cardioid-like lines curving around on both sides to right angles with the direction of the light, should be evident. Caustics can be produced by reflection, as here, or by refraction, in a variety of ways, and are evidenced on a large scale by bright lines. On a small scale, fringes can often be seen. They are much more common than would be believed, in things like the starlight bands produced by atmospheric refraction, or the active bright lines on the bottom of a swimming pool, or in the appearance of a point source seen through a bathroom window with a pebbled surface. Minnaert mentions the caustics produced by drops of water on eyeglasses, and the resulting fringes. One of the most familiar sights produced by a caustic is the rainbow. The bright colors occur at the caustic surface between scattered light and where light is not scattered. In fact, the dark area is the Alexandrine dark space, bounded by rainbows on both edges.
The caustic produced by reflection at a spherical surface can easily be demonstrated by ray-tracing. Draw a semicircle, then trace the rays reflected at the circle by constructing equal angles. To do this, draw an arc and lay off equal distances by dividers, which is much more accurate than using a protractor. What we are doing is drawing a cross-section. The caustic surface is obtained by rotating this cross-section about the axis. In this case, the reflected rays are superimposed upon the incident rays. Note that rays from a considerable area are concentrated near the cusp, so the intensity can be expected to be higher there. Indeed, the cusp is the focal point of the reflecting surface, halfway between the vertex and the centre of curvature.
It is a little more work to trace rays through a lens, but a similar result is obtained. There is a cusp at the focal point, and the caustic surface opens out like a horn toward the lens. Now the refracted and incident light is not superimposed. The intensity in a transverse plane near the focal point is the Airy pattern, a disc surrounded by rings. These fringes are produced by the lens aperture. Even with a perfect lens, the distribution of intensity on the caustic surface and near the focal point is complex.
The above two caustics are both real caustics, in which the light rays actually are in the caustic surface. As in the case of a virtual image, a virtual caustic can be formed by rays projected back beyond the refracting or reflecting surface. Consider a luminous point O a depth h below the surface of a liquid of index of refraction n. A ray from O leaving the surface at a distance a from the axis seems to come from a virtual source O' at a depth h', where h' = a tan r, where r is the angle of refraction. Now, h = a tan i, and n sin i = sin r. When a is small, a = hi = h'r and ni = r, so h' = h/n. If we project the rays back until they intersect the axis, we form a virtual caustic with a cusp at the point O', the apparent location of the source.
A caustic surface is a cusp-locus of the wavefronts. That is, within the caustic there are two wavefronts making an angle with each other that touch at the caustic, where their normals lie in the caustic. These wavefronts are from the same source, and so are coherent and interfere, producing the characteristic fringes.
R. A. R. Tricker, Introduction to Meteorological Optics (London: Mills and Boon, 1970). Airy's analysis of the intensity near a caustic is presented in Chapter VI, pp. 169-190, and applied to the rainbow.
R. W. Wood, Physical Optics, 3rd ed. (New York: Dover, 1967). p. 187f, pp. 53-59.
M. Minnaert, The Nature of Light and Colour in the Open Air (New York: Dover, 1954). p. 168f.
Composed by J. B. Calvert
Created 12 October 2004
Last revised