This article is really an excuse to talk about the geometrical optics of spheres, but the heiligenschein is an interesting phenomenon and an application of the optics of a sphere. The word Heiligenschein is German for the halo around the head of a saint ("holies'-shine"), which so far has only appeared in art, not in nature. In German, we must distinguish between its use for the optical phenomenon under discussion, for the phenomenon called the glory in English, and for the halos that are the effect of refraction and reflection in ice crystals. In English, we can use the term heiligenschein, uncapitalized, and it means only the subject of this article.
The heiligenschein is usually seen when one steps outside early in a cool morning when dew is glistening on the grass, and stands so that the sun is behind you, and you gaze at your shadow on the grassy lawn. Your head is surrounded by a lustrous halo of slightly green light to an angular distance of a degree or so from the centre of the shadow of your head. If a companion stands beside you, the head of her shadow is not likewise ennobled, just your own. If you photograph the heiligenschein with a double-lens reflex held at your waist, the glorious halo will be centered on your belly.
This is only the most impressive of many similar arrangements when light coming from your position is strongly reflected back in the same direction in which it came, often with the aid of transparent spheres. These may be the water spheres of the heiligenschein, the glass spheres of a reflectorized sign, or the living spheres of a cat's eyes. When you take a flash photograph, your subject's eyes show a red light of the same kind coming from the pupil.
A related phenomenon, the glory (q.v.) is also seen around the shadow of your head, but a shadow in a cloud or fog, where the drops are much smaller, say 20 μm in diameter, than in the dewy grass, where the drops are millimeter sized. The glory is not a luminous patch, but colored rings surrounding an aureole, and is much more delicate than the heiligenschein. In the glory, the light is reversed by the effects of surface waves, perhaps aided by refraction and reflection at the back of the drop. The heiligenschein is often seen, the glory very rarely.
The water drops that make the heiligenschein also can make a rainbow, and often a dew-bow is seen at the same time. It is much larger, 42° from the antisolar point, and besides is brightly colored. When you are observing the heiligenschein, drops glistening with color are often seen.
Ray-tracing in a sphere is delightfully simple. The ray lies in a plane defined by the incident ray and the center of the sphere, so any ray is a principal ray. A ray path through a sphere is shown in the figure. The angle of exit is the same as the angle of entrance, and the ray inside the sphere follows a symmetrical path. Angle r is found from Snell's Law, and the height h of the ray from trigonometry. Now, anything you want to know can be calculated. The axis is a line parallel to the incident ray through the center C. The figure can be rotated about this axis. For a paraxial ray, the sines are replaced by the angles (in radians) and we have i = nr and h = ai. This makes the ray tracing even easier, and the paraxial properties of the sphere are easily found. The inclination of the exit ray is s = 2r - i. Paraxially, s = i(2 - n)/n = [(2 - n)/na]h. The focal length f is defined as h = sf, so f = na/(2 - n).
The paraxial ray construction for the sphere is shown at the left. The parallel incident rays have been drawn at an angle α with the axis. They come to a focus at the tip of the arrow. Because of the symmetry of the sphere, it is clear that y = -αf, which also defines the focal length. The principal plane for axial rays is HH, and that for the rays at an angle α is H'H'. The incident rays are drawn up to this plane, and then are continued through the focal point F at a distance f from C. The focal plane of the sphere is curved, with a radius equal to f. By drawing rays either parallel to the axis or through the centre, the image location for any object location can be found in the familiar way.
The focal length for a water sphere, with n = 4/3, is 2a, the diameter of the drop. A glass sphere with n = 3/2 has f = 3a/2, so the focal point is closer to the vertex. The focusing properties of a sphere can be found from the thick-lens formulas, using r1 = a, r2 = -a and d = 2a. The focal length and the location of the principal plane is also easily found from paraxial ray tracing. The principal plane is normal to the axis at the point of intersection of the incident ray and the exit ray, produced backwards.
Glass spheres may be obtained as marbles, but it is hard to find a clear and perfect one. Water spheres may be small Florence flasks filled with water. I have found a 125 ml flask satisfactory. These can be used as burning glasses in the sun, a use that has been common since antiquity. It is inconvenient to use a sphere for a magnifying glass, but it does work. A glass marble 12 mm in diameter has a focal length of 9 mm, and makes a 28X magnifier. A 2 mm sphere makes an 83X magnifier, but is exceedingly hard to use. Nevertheless, van Leeuwenhoek and Hooke made microscopic observations with such tiny spheres. The 125 ml flask has a diameter of 60 mm, so it makes about a 4X magnifier, and is easy to use. Place your eye next to the flask, and bring the object to be viewed up from the other side. The magnifying effect of transparent spheres must surely have been known in antiquity, but there is no written record of this, though the use as burning glasses was known.
With respect to paraxial optics, the sphere is as simple as the thin lens. The Gaussian ray construction is shown in the figure at the right, and is seen to be identical to that for a thin lens. The principal points, nodal points and optical centre of the sphere all fall on the centre of the sphere. The principal plane H divides the sphere into hemispheres. F is the primary focal point and F' the secondary focal point, in the usual terminology. Three rays that are easy to locate are drawn. One passes through the optical centre, while the other two pass through the focal points. Only two of these rays are necessary to locate the image when an object is known. The object P is imaged at the image Q. In the paraxial approximation, these are stigmatic images. In reality, of course, the imaging is not quite stigmatic. It is easy to find the expressions for the linear magnification M from similar triangles, and eliminating y/y' between two of them we find the Gaussian lens formula 1/s - 1/s' = 1/f. The reader should sketch ray diagrams for an object to the left of F, and for an object in contact with the sphere, as practice. Object and image heights are positive upward, object and image distances s and s' are positive as shown.
The Coddington magnifier, apparently first made by Sir David Brewster, is simply a sphere used as a magnifier. A cross-section is shown at the left. A deep groove is cut around the equator, which serves as an aperture stop, limiting the rays to those close to the axis, which minimizes spherical aberration.
We should recognize the general principle of the reversibility of light rays. If a ray is a possible path in one direction, it is also a possible path in the reverse direction. This is, in general, the key to the heiligenschein. The droplets of dew are spherical and of various diameters, though 1 mm is typical. This is enough larger than the wavelength of light that diffraction effects will be small. The droplets do not touch the grass stalks on which they have formed, but are supported on fine hairs some distance from the surface of the grass. If the droplets are disturbed, say by treading on them, they are no longer delicately supported droplets, and they no longer show the heiligenschein.
The light traverses the droplet, and is more or less focused on the leaf, where it produces a bright spot. Light emitted from this spot then retraces the path of the incident light, and exits as collimated light returned in the direction that it approached. Although the light emitted from the illuminated spot is diffusely scattered, the f-number is high enough that a good proportion is in the cone of acceptance of the droplet. The light is not deeply colored by the green of the leaf, because it does not penetrate to that level. Experiments also confirm this assertion. Actually, some coloration is observed in the light of the heiligenschein. This explanation was first given by Lommel, and is the one favored by Tricker, and, of course, myself.
Pernter ascribed the heiligenschein to reflection from the front surface of the droplet, but this scatters light evenly in all directions, and does not provide sufficient intensity. Others have suggested that the brightness seen along the path of the incident ray is due to the lack of shading by other leaves in that direction, which is present when looking at an angle to the incident light. This effect certainly does exist, and is seen in grain fields and grass, but is not strong enough to explain the heiligenschein. Besides, Lommel's theory is supported by all the artificial retroreflectors that are widely used, which consist of transparent spheres and a reflecting background.
A very similar effect occurs in eyes. The bright image formed on the retina becomes a secondary source, and emits light in the same direction as it arrived. Looking for a cat at night with a flashlight is good evidence of this. Cats, and some other animals, have a reflecting layer behind the retina so that it gets two passes of the light. The strong reflection is not an evolutionary disadvantage, since few predators are self-luminous, and if they stand in front of the light they are easily seen anyway. Even if there is no specific reflecting layer behind the retina, the bright image is always a secondary source, and some light exits the pupil in any case. Indeed, this light is employed in the ophthalmoscope to view the interior of the eye. Humans and some other animals have a dark absorbing layer behind the retina that reduces glare, the choroid. The deep red color of this blood-rich layer colors the light that is reflected from it, as is seen in the "red-eye" phenomenon of flash photography.
Reflectorized surfaces consist of transparent spheres embedded in a reflecting backing. Even if a bright image is formed in contact with the rear surface of a sphere, a good proportion of its diffusely reflected light is returned. The "cat's eyes" used in road centre and side lines, and in older road signs, are larger spheres with a reflector designed to maximize the reflected light. It should be noted that the light is returned to its source at any angle of incidence within reason, not just in one particular direction, and that the image-forming properties of the spheres are essential for creating a bright collimated beam. Also, colored filters can modify the color of the returned light, taking white incident light and returning it as red, yellow or green.
Heiligenschein experiments can easily be carried out with a 125-ml Florence flask filled with water (my favorite refracting sphere), or glass or plastic spheres, a flashlight, and various backing materials, such as white or colored cardboard. Observe the effect of different backing materials at different distances from the back vertex. Move to various positions to check that the bright reflection follows you. Note that the illumination fills the exit pupil of the sphere.
Reflective materials can also be examined to find out the diameters of the spheres, and the nature of the backing. Does a sphere silvered on the back make a good retroreflector? How about a plane mirror behind a sphere?
R. A. R. Tricker, Introduction to Meteorological Optics (London: Mills and Boon, 1970). Chapter. II
M. Minnaert, The Nature of Light and Colour in the Open Air (New York: Dover, 1954). pp. 230-234.
F. A. Jenkins and H. E. White, Fundamentals of Optics, 2nd ed. (New York: McGraw-Hill, 1950). The thick-lens formulas are on p. 69 in this edition.
Composed by J. B. Calvert
Created 4 August 2003
Last revised 13 August 2003