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Interference Patterns Made By Motes on Dusty Mirrors

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by Jearl Walker
August, 1981

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TWO BEAUTIFUL EXAMPLES OF OPTICAL interference can be observed if you look into a mirror covered with dust or mist as you hold a small light in front of your face. A burning match works well. What you will see are colored rings centered around the reflected image of the light. Another set of interference rings and colors overlaps the first but may not be centered at the same place. This additional pattern is more difficult to see because your eye and the source of the light must be nearly on the same line perpendicular to the mirror. With careful adjustment of the position of the source and your eye the second set of rings can be made to coincide with the first. Under some circumstances the colors in these patterns are stunning.


Figure 1: A Fraunhofer pattern (top) and a Whewell-Quételet pattern (bottom) in laser light

Both patterns require a mirror with the reflecting surface on the back, which is the case with ordinary glass mirrors. The pattern that is always centered on the reflected image of the light source is usually called the Fraunhofer pattern. It has been thoroughly studied, not only because it affords an excellent demonstration of optical diffraction but also because the interference is responsible for the colored ring occasionally seen close to the moon or the sun.

The other interference pattern that can be seen in a dusty or misty mirror was studied intensely in the 1 9th century but has been virtually forgotten since then. I shall begin my story with this less familiar interference. Then I shall return to the Fraunhofer pattern and the lunar or solar ring.

Isaac Newton was apparently the first to study colored interference patterns in a dusty mirror. He made a shaft of sunlight fall on a pinhole in a screen in front of a hemispherical concave mirror. Light passing through the pinhole was reflected from the mirror and returned to the pinhole because it was positioned at the center of curvature of the mirror. The interference pattern appeared on the screen around the pinhole.

The irony of much of Newton's work in optics was that he discounted the possibility that light consisted of waves. Holding fast to the notion that light consisted of particles, he failed to explain any of the demonstrations of interference he investigated, including the one with the dusty mirror. The proper understanding did not emerge until Thomas Young convincingly demonstrated the wave nature of light. He explained briefly the colors that could be seen in Newton's dusty mirror. Later work on the pattern was done with flat mirrors by William Whewell in England and by Lambert Adolphe Jacques Quetelet in Belgium; as a result the pattern is now known as the Whewell-Quetelet interference pattern.

Almost any small particles-room dust, chalk dust, lycopodium powder-on a mirror can generate the Whewell-Quetelet pattern. The moisture that condenses when you breathe on the mirror will do. You can also create a thin layer of dried milk globules. Mix one part of milk with three parts of water and coat the front surface of a mirror. Tilt the mirror so that the excess liquid runs off. Allow the remainder to dry. The milk globules left on the mirror serve in the same way as the dust particles. The dusty material does not have to be directly on the mirror. Sir John Herschel demonstrated the interference pattern by tossing wig powder into the air in front of a mirror.

The interference pattern depends on the scattering of light by the particles on the mirror or just in front of it. For the sake


Figure 2: A Fraunhofer pattern from lycopodium powder in white light

of simplicity I shall assume that the observer and the light source are on the same perpendicular line from the mirror and that the light from the source consists of only one wavelength. Later I shall discuss interference arising from white light and explain how the colors are dispersed.

Suppose two rays travel to the mirror from the source as part of one wave front. They are coherent (the light waves of the two rays are in phase) and nearly parallel. Although the light source is incoherent (unless it is a laser), it does emit coherent waves in short bursts over short distances.

One ray scatters from a particle toward the back of the mirror, is reflected and travels to the observer. The other ray enters the mirror without scattering from the particle, is reflected, then scatters from the same particle and travels to the observer. Although the rays were initially in phase, the phase relation now depends on the lengths of the paths they have taken. If the paths are exactly the same length, the rays are once more in phase and they interfere constructively The observer sees relatively bright light from the position of the particle on the mirror. When the observer and the light source are on the same perpendicular line from the mirror, this situation results with a particle directly in front of the observer, that is, at the foot of the perpendicular line.

A particle slightly off the line sends the observer rays with a different phase relation. If the difference in path lengths results in a phase difference of half a wavelength, the rays interfere destructively. The observer sees a dark spot. Moreover, because of a circle of dust particles around the perpendicular line where the geometry is correct for destructive interference the observer also sees a dark ring around the perpendicular line. It is the centermost of the dark rings in the interference pattern.

Particles a bit farther from the center of the pattern scatter pairs of rays that end up a full wavelength out of phase when they reach the observer. The result is constructive interference and a bright ring. More bright and dark rings are seen somewhat farther from the line. They are caused by progressively larger differences in the path lengths of pairs of light rays.

Interference patterns are normally labeled according to the phase difference between the participating rays. The bright place at which the phase difference is zero is labeled n = 0 and is called the zero-order fringe of constructive interference. In the geometry of observer and light source I have described, this fringe is at the foot of the perpendicular line. The next bright fringe, where the phase difference is a full wavelength, is labeled n = 1 and is called the first-order fringe of constructive interference. It is the central bright ring in the interference pattern.


Figure 3: How light is scattered from a dust particle

In nearly all standard examples of interference the numerical labels for the bright fringes increase with distance from the center of the pattern. The interference pattern from the dusty mirror is an exception, at least in principle. If the bright rings still farther from the center of the pattern could be seen, their order would begin to decrease. The reason is that when two rays return to the observer from a point relatively far from the perpendicular line, they have traveled almost equal distances. The phase difference between them is therefore smaller than the difference between the rays of another pair scattering from a particle somewhat closer to the line. Unfortunately the rings where the numerical sequence reverses cannot be observed in an ordinary mirror.

The width of the interference rings also varies with distance from the center of the pattern, narrowing at first and then (too far from the center to be observed normally) widening. The width of a ring for any particular order of interference depends on several factors. The ring is larger for a longer wavelength, for a larger index of refraction of the glass in the mirror and for a thinner mirror. The width also depends on the relation between the distance of the light source from the mirror and the distance of the observer from the mirror.

Why do a pair of rays participating in the creation of an interference pattern scatter from the same particle? A ray from one particle could interfere with a ray from another, but the interference would not contribute to a composite pattern. The scattering of light from a particle causes a shift in the phase of the light that depends on the size and shape of the particle. If two rays scatter from the same particle, the phase shift is the same. The phase difference then depends only on the geometric relations of the light source, the observer and the particle.


Figure 4: The mode of formation of a Whewell-Quételet pattern

The type of scattering from a particle on the mirror depends on the size of the particle. Particles larger than about 100 micrometers scatter light through either reflection or diffraction. Smaller particles, such as those of lycopodium powder, scatter light through a more complex process that involves surface waves. The Mie theory of scattering provides the basis for a detailed understanding of this scattering. If the particles are quite close to one another on the mirror, the interference between rays that are diffracting through the space between particles becomes important. I shall not consider such complications here.

In white light the rings are colored. Each interference order has a sequence of colors from blue to red, ranging outward from the center of the pattern. The separation of colors arises because the phase difference between a pair of rays participating in the interference depends on the wavelength.

Suppose a particle slightly off the center of the pattern causes constructive interference of blue light in the innermost bright ring (n = 1). Two rays contributing to this ring travel along a path difference equivalent to one full wavelength of blue light. For the same phase difference in red light (a longer wavelength) a pair of red rays must scatter from a particle a bit farther from the center, so that their path difference is equivalent to a full wavelength of red light. Hence the ring is blue on the inside and red on the outside. Intermediate colors lie between the blue and the red.

I have described the general interference pattern that is seen when the observer and the light source are on the same perpendicular line. If the observer moves, the center of the pattern moves away from the directly reflected image of the light source and is either bright or dark, depending on the displacement. As the observer moves, new rings seem to emerge from the center. The narrower rings of higher order can be seen near the reflected image of the source. Eventually the center of the pattern is so displaced that it disappears. If the observer moves far enough to the side, the entire pattern becomes obscure and disappears.

The structure of the higher-order fringes has been studied by employing a thin layer of mica as a mirror. Such a layer, split from a thicker sheet, is coated on one side so that it reflects. Then the interference pattern can be seen even if both the light source and the observer are at highly oblique angles.

The interference pattern can also be seen with mirrors that are coated for reflection from the front surface. Lightly dust an uncoated sheet of glass and put it in front of the mirror with the dusted side facing the reflecting surface. Insert thin strips of mica or paper between the mirror and the glass as separators. (Without separation a pair of rays normally responsible for the interference would travel the same distance regardless of the location of the dust particle on the glass, and there would be no interference pattern.)


Figure 5: A setup for viewing the patterns

To investigate the interference pattern I held a small flame between one eye and a mirror I misted by breathing on it. The reflective surface was-on the back of the mirror. With the room lights off I could see both the Fraunhofer pattern and the Whewell-Quetelet pattern. As I moved the flame closer to my eye the rings of the Whewell-Quetelet pattern became less curved. With the flame close to my eye they were almost straight lines. When a friend held the flame behind me (still positioned so that my head did not block all the light), the rings again curved.

I then sprinkled a light coating of lycopodium powder over the mirror, gently shaking the mirror to achieve as even a distribution as I could. (If the layer is too thick, no pattern will be seen.) The mirror was illuminated with an incandescent lamp or a sodium lamp placed 10 meters away so that it approximated a point source. (I could have inserted a pinhole between a closer source and the mirror.) The Fraunhofer rings were immediately evident, but the Whewell-Quetelet fringes were harder to find. I had to put my head almost in the path of the beam of light before I could see them. A helium-neon laser gave similar results.

To more easily observe the Whewell-Quetelet pattern I placed an uncoated sheet of glass in the path of the light falling on the mirror, angling it to cast a reflection of the mirror perpendicular to the beam. When I looked into the glass, I could see both interference patterns on the mirror. By carefully adjusting the angles of the mirror and the uncoated sheet I got the same view of the mirror that I would have if I had been in line with the light source.

Such an arrangement was first employed by Eugen Lommel in Germany 100 years ago to project an image of the interference pattern onto a screen by means of a convex lens. The arrangement will also serve in making a photograph of the interference pattern. The camera lens is adjusted for a large focal length because in principle the rays reflected from the mirror converge to form a real image of the Whewell-Quetelet pattern only at a considerable distance from the mirror. With the lens in place the image is brought into focus on the film.

If you use a laser as the light source, be extremely circumspect about the light reflected from the mirror and from the sheet of glass inserted into the beam. Both reflections can be bright enough to hurt your eye.


Figure 6: The development of a diffraction pattern from dust on glass

The other major principle of interference-the Fraunhofer pattern-is produced by the diffraction of light by the dust particles on the surface of a mirror coated for reflection from the back surface. To explain what is seen I shall first describe a simpler demonstration of diffraction by small particles. If you look at a small light source through a misty window, the image of the source is surrounded by concentric interference rings. They are Fraunhofer rings resulting from the diffraction of light rays by the water droplets on the window.

When a beam of light passes a small particle such as a dust particle or a water droplet, the light diffracts around the sides of the particle. The diffracted rays create an interference pattern when they are intercepted by a screen after passing the particle. If the particle presents to the light an approximately circular cross section, the pattern consists of concentric bright rings surrounding a relatively bright central spot and interspersed with relatively dark rings.

The bright rings are created when rays of light diffracting around the particle interfere constructively. The dark rings are caused by destructive interference. The kind of interference depends on the geometry of the rays. The difference in their path lengths causes phase differences between them.

At the center of the pattern all the rays arrive in phase because they all travel the same distance. Rays arriving slightly off the center have path differences resulting in a phase difference of half a wavelength. These rays interfere destructively. Since the same condition develops in a circle around the bright center of the pattern, a dark band appears around the center. The next ring is bright because the rays interfere constructively.

You see a similar pattern when you look through a dusty or misty sheet of glass, provided the light source is small enough or far enough away to occupy a small angle in your field of view. You see not the full diffraction pattern from each particle on the glass but a composite pattern encompassing parts of the pattern cast by each particle. The particles on the axis between you and the light source send the bright centers of their patterns; the particles slightly off the axis send part of their innermost dark ring. The result from all the particles is again a pattern of alternating bright and dark rings. The particles on the glass not only diffract the light into the pattern of Fraunhofer rings but also disperse the white light into its component colors so that the rings are colored.

The Fraunhofer ring around the moon or the sun is called a corona. (The term should not be confused with the solar corona seen during a total eclipse of the sun.) The particles responsible for the ring are water droplets or ice crystals in the thin clouds between the observer and the moon or the sun.

I can see a similar corona when condensation mists my eyeglasses or car windows. Distant streetlights and the headlights of cars are surrounded by colored rings. Sometimes I see Fraunhofer rings when distant headlights illuminate the rearview mirror in my car because my mirror is normally some what dusty! (Another kind of corona that is seen surrounding distant light sources has nothing to do with the diffraction I am discussing. This faint "entoptic corona" is created in the observer's eye.) I once saw a remarkable example of the diffraction pattern when I was about a meter away from a misty mirror that was illuminated by bright light from a window some five meters behind me. Superimposed on each of my eyes in the mirror was a Fraunhofer pattern.


Figure 7: The separation of color in a diffraction pattern

The corona caused by diffraction is seen in a dusty mirror because the pattern diffracted forward by the dust particles is reflected by the back of the mirror. Regardless of the observer's angle of view, the pattern of rings and colors always surrounds the reflected image of the light source. This consistency is the primary clue for distinguishing the Fraunhofer rings from the Whewell-Quetelet rings, which surround the reflected image only if the observer and the light source are on the same perpendicular line from the mirror.

The clarity of the Fraunhofer rings is strongly correlated with the size of the particles. The smaller the particle, the larger the angular spread of the diffraction pattern. A large diversity of particle sizes gives rise to an indistinct Fraunhofer pattern because the resulting diffractions overlap.

I get distinct Fraunhofer rings when I illuminate an uncoated sheet of glass or a mirror that is lightly covered with lycopodium powder. The particles in this powder are fairly uniform in size and therefore generate diffraction patterns of about the same size. I get less distinct (sometimes barely visible) Fraunhofer rings when I illuminate a sheet of glass or a mirror covered with condensation from my breath or from a hot shower. The droplets involved in the interference are apparently more diverse in size. They may be more closely spaced than dust particles usually are, so that additional interference effects may obscure the simple diffraction pattern I have described.

The photograph of the interference patterns in Figure 1 was made with light from a helium-neon laser. The laser beam was reflected by a mirror dusted with lycopodium powder. The primary reflection traveled to a screen behind the laser. Surrounding the brightest area of the display is the Fraunhofer pattern. Toward the bottom of the photograph, almost in the shadow of the laser, is the center of the Whewell-Quetelet rings.

Is the pattern I have called the Fraunhofer pattern truly reflected by the rear surface of a mirror as I have described? To find out I scraped off part of the reflective coating on the back of an inexpensive mirror. After lightly dusting the front surface with Lycopodium powder I illuminated the mirror. The Fraunhofer rings appeared as strongly as usual everywhere except over the nonreflecting area. There they were comparatively weak. A faint amount of pattern remained because the back surface of the nonreflecting area still reflected about 2 percent of the light.

To eliminate virtually all reflection from the scraped area I sprayed the back of the mirror with a flat black paint. After the paint had dried I again dusted the front surface of the mirror. No interference pattern was visible from the scraped area.

I wondered why the source of light for the two interference patterns seen in a dusty mirror had to be either small or distant from the mirror. When I held a large diffuse source of light, such as a light bulb, near a dusty mirror, I could see no pattern. The explanation lies in two features of the light: the angular spread of the rays and the coherence of the light falling on the dust particles. The pair of light rays giving rise to an interference pattern must be coherent and approximately parallel at a particle. If they are incoherent, their phase relation shifts from instant to instant. The amount of interference between them when they reach the observer also shifts, and he does not see a consistent interference. If the rays are not parallel, the interference pattern is obscured by the overlapping of many patterns sent out at slightly different angles from a particle.

An astute observer will note that the Fraunhofer patterns created by water droplets differ in two respects from those created by Lycopodium powder. The center spot is surrounded by a dark area in the first instance and by a bright area in the second one. Moreover, the patterns differ in form when the observer's view is not perpendicular to the surface of the mirror or of the uncoated sheet of glass. A careful observer will also note that the patterns have an intriguing granular structure, particularly near the brightest region. These and other puzzles must wait for another time.

Bibliography

ON THE COLOURS OF THICK PLATES. Sir George Gabriel Stokes in Mathematical and Physical Papers: Volume 3. Johnson Reprint Corporation, 1966.

INTERFERENCE IN SCATTERED LIGHT. A. J. de Witte in American Journal of Physics, Vol. 35, No. 4, pages 301-313. April, 1967.

THE CORONA. R. A. R. Tricker in Introduction to Meteorological Optics. American Elsevier Publishing Company, Inc., 1970.

THE CORONA. Robert Greenler in Rainbows, Halos, and Glories. Cambridge University Press, 1980.

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