Pattern formed in the shadow of a solid ball or circular disk
Jearl Walker www.flyingcircusofphysics.com
September 2010 In 1818, Augustin Jean Fresnel submitted a wave model of light to a competition at the French Academy. Simeon D. Poisson, one of the members of the judging committee, argued strongly against the model, attempting to reduce it to absurdity with this thought experiment: If an opaque object with a circular cross section (such as a coin or a ball) is illuminated with a beam of light, Fresnel’s wave model predicted that a bright spot should appear at the center of the shadow cast by the object on a distant viewing screen.
Dominique F. Arago, another committee member, arranged to test the prediction in spite of the absurd conclusion. Surprisingly, he found the bright central spot. Through a quirk of history, the spot is now known as either the Poisson spot or the Arago spot, although neither man initially believed in its existence.
Here is my photograph of the pattern produced by a small opaque disk that was illuminated by the red light of a helium neon laser. The pattern appeared on a sheet of paper, which I photographed at a small angle to the laser beam.
To understand the formation of the spot, first suppose that the source of light is a distant, bright point and that the imaging is done with a solid ball. When the light waves reach the ball, they diffract around its sides, spreading not only radially outward but also into the shadow region of the ball. If a screen is placed well behind the ball, the light forms a small diffraction pattern of bright and dark concentric circles on it. The center of the pattern is a bright point because waves passing on the opposite sides the ball travel the same distance to reach the center.
The waves start out in phase (in step) at the edge of the ball because they are part of the same wave coming from the laser. Because they travel equal distances to reach the center point, they must still in phase when they arrive. Thus, they reinforce each other at the center and undergo constructive interference, giving a bright spot.
The first dark circle surrounding the center is due to destructive interference in which waves arrive out of step and thus cancel each other. Consider the top of the dark circle. Waves passing the bottom of the ball must travel a longer distance to reach that point than waves passing the top of the ball. The extra distance amounts to half a wavelength. Again the waves from top and bottom start out in phase at the edge of the ball. However, because of the difference in the lengths of the path, they arrive at the top of the circle out of step.
The rest of the pattern is due to similar constructive or destructive interference. In some places, waves from opposite sides of the ball differ in the travel distance by an integer number of wavelengths, which results in constructive interference because that puts the waves in phase. In other places the difference in the travel distance amounts to an odd number of half wavelengths, which results in destructive interference because that puts the waves out of phase.
References
Dots · through ··· indicate level of difficulty
Journal reference style: author, journal, volume, pages (date)
Book reference style: author, title, publisher, date, pages
··· Hufford, M. E., "The diffraction ring pattern in the shadow of a circular Object," Physical Review, Series 2, 7, 545-550 (1916)
·· Sommerfeld, A., Optics, Academic Press, 1954, page 216
·· Jenkins, F. A., and H. E. White, Fundamentals of Optics, McGraw-Hill, 1957, pages 359-360
· Pohl, R. W., Optik und Atomphysik, 12th edition, Springer, 1967, pages 92-93
· Rayleigh, Lord, "Shadows" in The Royal Institution Library of Science: Physical Sciences, W. L. Bragg and G. Porter, editors, Elsevier, 1970, vol. 5, pages 54-61
·· Meyer-Arendt, J. R., Introduction to Classical and Modern Optics, Prentice-Hall, 1972, page 200
··· Rinard, P. M., "Large-scale diffraction patterns from circular objects," American Journal of Physics, 44, 70-76 (1976)
· Johnston, J. B., "Projecting Poisson's spot," Physics Teacher, 16, 179 (1978)
·· Hecht, E., and A. Zajac, Optics, Addison-Wesley, 1979, pages 373-375
··· Harvey, J. E., and J. L. Forgham, "The spot of Arago: new relevance for an old phenomenon," American Journal of Physics, 52, 243-247 (1984)
· Walker, J., "A ball bearing aids in the study of light and also serves as a lens" in "The Amateur Scientist," Scientific American, 251, 186-193 (November 1984)
··· English Jr., R. E., and N. George, "Diffraction patterns in the shadows of disks and obstacles," Applied Optics, 27, 1581-1587 (1988)
··· Hovenac, E. A., “Fresnel diffraction by spherical obstacles,” American Journal of Physics, 57, No. 1, 79-84 (January 1989)
··· Sommargren, G. E., and H. J. Weaver, "Diffraction of light by an opaque sphere. 1: Description and properties of the diffraction pattern," Applied Optics, 29, 4646-4657 (1990)
··· Sommargren, G. E., and H. J. Weaver, "Diffraction of light by an opaque sphere. 2: Image formation and resolution consideration," Applied Optics, 31, 1385-1398 (1992)
·· Harrison, M. E., C. T. Marek, and J. D. White, “Rediscovering Poisson’s spot,” Physics Teacher, 35, 18-19 (January 1997)
· Higbie, J., (letter) “More on Poisson’s spot,” Physics Teacher, 35, 197 (April 1997)
· Wong, R. D., (letter) “Still more on Poisson’s spot,” Physics Teacher, 35, 197-198 (April 1997)
··· Wein, G. R., “A video technique for the quantitative analysis of the Poisson spot and other diffraction patterns,” American Journal of Physics, 67, No. 3, 236-240 (March 1999)
·· Kolodziejczyk, A., Z. Jaroszewicz, R. Henao, and O. Quintero, “An experimental apparatus for white light imaging by means of a spherical obstacle,” American Journal of Physics, 70, No. 2, 169-172 (February 2002)
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