AN ORDINARY MIRROR WILL REFLECT a beam of light to its source only if the mirror is perpendicular to the beam. A retroreflector, on the other hand, returns a beam to the source regardless of the angle at which the beam strikes the device. With an array of small retroreflectors one can do a novel experiment: if the beam is distorted by, say, a ruffled sheet of plastic before it reaches the array, the array removes the distortion by reversing the beam back through the plastic. David M. Pepper of the Hughes Research Laboratory has written a manuscript, on which my discussion is based, describing this experiment and others that demonstrate the phenomenon. The experiments were based primarily on research by Harrison H. Barrett and Stephen F. Jacobs of the University of Arizona.

Retroreflecting arrays are available at sporting-goods stores and automobile-accessory shops in the form of flexible plastic sheets sold as safety reflectors. Sewn onto clothing, fastened to a bicycle, glued to a highway stripe or attached to a variety of other objects, the reflectors make the objects more visible at night. For example, when the headlights of an automobile strike an array, the light comes strongly back as a warning to the driver. The success of the application is actually due to an imperfection in the arrays If the retroreflectors returned the beam of light precisely instead of spreading it slightly, the entire beam would return to the headlights rather than to the driver.

To do Pepper's experiments you should buy a retroreflecting flexible plastic sheet that is at least three centimeters square. If you can find only narrow strips, cut them up to form the square. Tape it onto a sturdy box. Set up a slide projector so that it shines on the array. If possible, remove the projector's front lens. Mount on the front of the projector a thick piece of cardboard through which you have made a hole two or three millimeters in diameter. The pinhole serves as a point source of light. If you do not have a projector, set up a bright flashlight or even an ordinary incandescent bulb to shine through the pinhole.

In order to see the shape of the returned beam, you must split off part of it and direct it to a viewing screen. You can do this by placing a glass pane in the beam at an angle of about 45 degrees. The glass should be clean and clear, but it does not have to be of excellent optical quality. Glass from a picture frame will do. Hold it in place with modeling clay or with the frame from which the cardboard backing has been removed.

When the beam from the pinhole reaches the glass, it is split into two beams. One beam continues on to the array and the other is reflected out into the room. The beam reflected from the array comes back to the glass, which makes another split. One beam passes through the glass, returning to the pinhole. The other beam is reflected from the glass to a sheet of paper that serves as a viewing screen. Since light is reflected from both the front and the back surface of the glass, the screen displays two slightly displaced images. You will get less displacement and a clearer image with a thin piece of glass serving as the beam splitter.

The beam splitter should be equidistant from the screen and the pinhole so that light returned by the array travels the same distance to each. When everything is arranged, you will see a small, bright circle on the screen. It is an image of the pinhole produced by light that would have returned to the pinhole except for the action of the beam splitter. The image is about the same size as the pinhole but has a fuzzy edge because of the double reflection by the glass and the imperfection of the retroreflecting array.

If you look at the screen through the beam splitter, you will see a bright spot that seems to be on the screen. You might well mistake it for the image returned by the array. Instead it is an image of the pinhole formed by your eye from the light reflected toward you by the beam splitter. Such images, called virtual images, are commonly produced by mirrors. In the experiment the beam splitter acts as a weak or semitransparent mirror. If you take the proper angle of view, you can align the virtual image with the real image on the screen. To avoid the complication of the virtual image do not look through the beam splitter.

If you change the distance between the retroreflecting array and the beam splitter, the image of the pinhole on the screen does not change. Neither does it change if you slide the array across the beam. If you rotate the array with respect to the beam, the image does not change until the angle is large. In short, the array sends a beam of light back toward the light source even when the plane of the array is not perpendicular to the beam.

Replace the array with a flat mirror. The entire screen is bathed with dim illumination, and you will not see an image of the pinhole. Try a concave mirror. It produces a clear image of the pinhole on the screen only when its focus is on the pinhole. If you move the mirror along the beam or rotate it with respect to the beam, the image broadens, becomes distorted and finally disappears.

The action of these mirrors and of the retroreflecting array can be understood with the help of Figure4. Part a depicts the reflection by a flat mirror. The pinhole is represented by a point source of light. Waves of light flow from the point source. Each one displays a wavefront that is part of a sphere. One such wavefront is shown in cross section as part of a circle centered on the point source. As the wavefront travels away from the point source, the radius of the circle increases. When the wavefront reaches the mirror, sections of it are reflected to form a new wavefront traveling in the opposite direction. The reflected wavefront is part of a circle centered on an imaginary point source of light to the right of the mirror. The imaginary source is as distant from the mirror as the true source is. The reflected wavefront cannot form an image of the pinhole because it continuously grows larger, never attaining a focus. When the beam splitter directs part of the light to the screen, the entire screen is illuminated by the expanding wavefront.

Part b of the illustration demonstrates the action of the concave mirror. The mirror reflects and focuses the incident wavefront, but the light returns to the source only if the pinhole lies on the focal point of the mirror. Since the screen is as far from the beam splitter as the pinhole is, a clear image of the pinhole is formed on the screen. For other positions of a concave mirror the light spreads more broadly over the screen. If the mirror is not oriented symmetrically on the beam, the focusing of the light is skewed. The result is either a distorted image or no image.

Part c of the illustration represents the action of a highly specialized mirror system that is related to the commercial retroreflecting array. It is called a phase-conjugate mirror [see "Optical Phase Conjugation," by Vladimir V. Shkunov and Boris Ya Zel'dovich, Scientific American, December, 1985, and "Applications of Optical Phase Conjugation," by David M. Pepper, Scientific American, January]. This ideal retroreflector exactly reverses the light rays incident on it. Moreover, the reflected wavefront preserves the shape and orientation of the incident wavefront. With such an ideal mirror all the returned light that passes through the beam splitter travels through the pinhole. The fraction of the light that travels to the screen forms an exact image of the pinhole.

The retroreflecting sheets Pepper employs are less than ideal. Each retroreflector in a sheet returns a section of the wavefront, preserving its shape and orientation as is shown in part d of the illustration. Because the retroreflectors lie in a plane, however, the sections do not fit together to re-form the initial wavefront. Instead they are spread along a curved line that is identical with the wavefront reflected by the flat mirror. For this reason the sheets are said to be pseudoconjugators. Each section of the wavefront returns to the pinhole or to the image of the pinhole on the screen, but together they lack the coordination that would be seen with a true phase-conjugate mirror. Nevertheless, a fuzzy image of the pinhole forms on the screen as piece after piece of the wavefront arrives.

The retroreflectors in plastic sheets come in two types. One type consists of prisms embedded in the plastic. An arriving light ray is reflected from one of the internal faces of the prism and then from a second and possibly a third internal face. It travels out of the plastic parallel to the incident ray. The retroreflector is not perfect because the returned ray is displaced from the incident ray and because the section of the wavefront is inverted by the prism. (The retroreflector is so small that the section of wavefront can be regarded as straight.) The Reflexite Corporation manufactures this type of retroreflecting array. The prisms are clustered in hexagonal cells, each cell .15 millimeter across. Areas of the array that are between the cells do not take part in retroreflection.

Scotchlite, made by the 3M Corporation, consists of glass beads embedded in a flexible backing. An arriving ray of light is refracted to the back of the bead, from which it is reflected back to the front. When it leaves the bead, it is parallel to the incident ray. This retroreflector is imperfect in that it displaces the final ray and inverts the final wavefront.

One novel application of retroreflectors involves removing the distortion of a beam of light that passes through a nonuniform material. Place a concave mirror in the path of the light from the pinhole, positioning the mirror to get the best image of the pinhole on the screen. Insert a distorting element in front of the mirror. The element could be one of those ruffled plastic sheets that normally cover fluorescent bulbs, a glass slide covered with airplane glue, the plastic "bubble" material used in packing or almost any other nonuniform, transparent material. The distorting element destroys the image of the pinhole, scattering the light into a complex pattern that fills the screen. If you shift the distorting element across the opening of the mirror, the pattern on the screen changes chaotically.

Now replace the mirror with a retroreflecting array. The exact position and orientation of the array are not important, but the distorting element should be directly in front of it and both the array and the distorting element should be illuminated. Although the distorting element jumbles the light and seemingly ruins any information about the pinhole's shape, an image of the pinhole appears on the screen. It is not perfect, being dim and fuzzy, but nonetheless it is there.

The image appears because the array almost exactly reverses the light rays emerging from the distorting element. The rays then travel back through almost the same distorting features they encountered on their first passage. The second passage reverses the distortion of each ray, the beam is reconstituted and the image of the pinhole is formed on the screen. Pepper suggests you can consider that the rays have been time-reversed by the array. If you move the distorting element across the array, the pinhole image changes little or not at all. The reason is that the time the light takes to travel from an irregular feature in the element to the array and back is so short that the feature is essentially stationary during the trip.

Note that the distortion is removed because the light passes through the distorting element twice. (If you place the element in front of the pinhole or the screen, the light passes through it only once.) The light rays must make their second pass through the element along approximately the same paths they followed on the first pass. Remember the retroreflectors return light rays that are slightly displaced from the incident rays. If the element and the array are separated too much, the returned rays encounter different distorting features on their second passage through the element and the distortion is not removed. A large separation may also mean that less light is intercepted and returned by the array. A dimmer image then results.

You can do similar demonstrations if you replace the projector and pinhole with a laser. The image is marred, however, by extra interference in the light. The interference gives rise to a complex pattern of bright and dark lines. Some of the interference is due to the diffraction of light by dust motes on the beam splitter or to nonuniformities in the glass. Some of the light reaching the screen has been distorted only once, and so it retains the distortion. The periodicity of the beads or hexagonal clusters on the retroreflector sheet also introduces interference in the light reaching the screen.

When you do this demonstration, you may be confused by an extra spot of light on the screen. Because of the imperfect construction of the retroreflecting array? part of the light returning from it is reflected in the same way as light is reflected from a flat mirror. If the plane of the array is perpendicular to the beam of light, this mirrorlike reflection reaches the screen and can be confused with the pinhole image formed by retroreflection. The extra spot of light is easy to identify because it shifts over the screen if you tilt the array with respect to the beam. You can remove the spot by erecting a cardboard folder between the array and the screen so that the laser beam just clears the edge of the folder. Then rotate the array until the mirrorlike reflection appears on the folder and is thus blocked from the screen.

Many experiments can be done with the laser, the retroreflecting array and common mirrors and lenses. I particularly enjoyed blowing smoke or throwing fine powder into the path of the laser beam in order to follow its travel. In one arrangement the beam passed through the beam splitter, was reflected from a tilted flat mirror and then was returned from a retroreflecting array. The trip demonstrates two distinct types of reflection: the familiar reflection from the mirror and the effect of retroreflection, which folds the beam back on itself.

I also enjoyed inserting diffraction gratings and other distorting elements in front of the array. In each case retroreflection removes the distortion imposed on the light in its first passage through the element. Some of the laser light, however, is reflected from the front surface of the distorting element and therefore never makes a second passage through it. This distortion is not removed. To block the mirrorlike reflection rotate the distorting element along with the array so that the reflection appears on the cardboard folder. What then remains on the screen is the light that was retroreflected from the array and made a second passage through the distorting element.

More than one pinhole can be imaged using the retroreflecting array. Mount on the front of the projector a mask bearing a series of pinholes arranged to form one or two small letters. All the experiments can be repeated with this design. For example, when the distorting element and the retroreflecting array are in position, a fuzzy image of the letters appears on the screen. Each pinhole creates its own image; the composite forms the letters on the screen.

To improve the image install a convex lens between the beam splitter and the retroreflecting array at a distance of two focal lengths from the pinholes. Place the array two focal lengths on the other side of the lens. Pepper says that the focal length of the lens is not critical but that the experiment works well when the length is between 10 and 30 centimeters. A simple magnifying glass will do.

To tune the alignment move the lens and the array until a sharp, inverted image appears on a sheet of paper held just in front of the array. Remove the paper and move the viewing screen until the image is as sharp as you can make it. A distorting object placed directly in front of the array makes the image dimmer and fuzzier, but it is still visible. A lens is needed in this demonstration to compensate for the slight spreading of the beam produced by the retroreflectors. With this arrangement you might try replacing the cardboard mask on the front of the projector. Remove the front lens and put an ordinary slide inside the projector.

Pepper has also investigated how a retroreflecting array can undo the distortion introduced by vibration. He glued a flat, lightweight mirror to a wood tongue depresser, which he attached to a speaker cone with doublesided adhesive tape. Then he illuminated the mirror with the light passing through the beam splitter. (The source of the light can be either a laser or a projector with a pinhole mask.) The light was reflected from the mirror onto a retroreflecting array. Next Pepper drove the speaker with the amplified signal from an audio oscillator. As the mirror vibrated, the angle at which the beam was reflected to the array changed continuously. Nevertheless, the screen still displayed a stationary image of the laser's exit aperture (or of the pinhole). Even when the mirror vibrated rapidly, it moved considerably slower than the speed of light. Hence it was effectively stationary while the light moved from the mirror to the array and back. When Pepper replaced the array with a flat mirror, the screen showed a smeared, kinetic pattern.

Pepper did a final demonstration that involved removing the distortion imposed by a turbulence. He set up the basic arrangement of equipment with a mask of pinholes on the projector and a retroreflecting array at one end of the optical path. He also placed a convex lens between the beam splitter and the mirror and installed an unlighted gas burner just below the optical path between the lens and the mirror. (It could have been between the lens and the beam splitter.) The pinhole, the viewing screen and the array were each two focal lengths from the lens. Pepper tuned the alignment of the system and lighted the burner.

One would expect the image to dance and flicker as the beam passed through the turbulent convection rising from the flame. It would do so with a flat mirror, but in this case it remained frozen in place. Although the distortion of the beam varied constantly in the turbulence, the array returned a clear image because the light gets through the disrupting features twice before they have time to change.

Pepper then placed a beaker of water on the burner and in the path of the light. As the water heated, the convection in it distorted the light beam, and again the retroreflecting array removed the distortion. When the water began to boil, the bubbles prevented some of the light from reaching the array. As a result the image on the screen became somewhat dimmer.

Here is a puzzle for you to solve. If the arrays are mirrors, why do you not see a retroreflection of your face when you look at an array in normal room illumination? (You will see your face if the room is brightly lighted and the array is oriented just right, but that reflection is due to imperfect construction of the array.)

You will probably find plenty of other demonstrations to do with the retroreflecting arrays. Pepper and I should like to hear about your experiments.

Last December I described a kaleidoscope with polarizing filters and a curved reflecting sheet inside the tube This Karascope, which was commissioned by New York's Museum of Modern Art, was invented and patented by Judith Karelitz of New York City. It is for sale from the museum for $20.75 postpaid; the address of the museum is 11 West 53 Street, New York, N.Y. 10019. Please be sure to mark correspondence to the attention of the Mail Order Department.

Bibliography

RETROREFLECTIVE ARRAYS AS APPROXIMATE PHASE CONJUGATORS. H. H. Barrett and S. F. Jacobs in Optics Letters, Vol. 4, No. 6, pages 190-192; 1979.

EXPERIMENTS WITH RETRODIRECTIVE ARRAYS. Stephen F. Jacobs in Optical Engineering, Vol. 21, No. 2, pages i 281-283; March/April, 1982.

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