An experiment is described in which the effect of instrument aperture size on resolution is easily demonstrated. Using diffraction as a tool rather than a limit to resolution, one can further demonstrate the possibility of detecting the presence of more than one object, even when the instrument cannot resolve the objects.
REFERENCES
1.
For example, see D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics (Wiley, New York, 2001), 6th ed., p. 898. The authors make the comment that “the analysis of such patterns is complex,” but state where the first minimum is located [given by Eq. (1) in this paper].
2.
Young’s experiment is also discussed in most, if not all introductory physics texts. For example, again see Ref. 1, pp. 905–911 for a full discussion.
3.
One hole measured 0.26 mm in diameter and the other measured 0.16 mm, which would suggest that the light from the smaller hole is about 2/3 the intensity from the larger hole.
4.
See Ref. 2.
5.
Alternatively, a section of a clear plastic ruler can be cut to the size of a 35-mm slide mount and used as a slide. The numbers and rulings will be projected and their image easily measured on the screen.
6.
Note the remarkable closeness between the factor of 1.23 in this expression and the correct factor of 1.22 found in the literature. Calculating the factor of 1.22 is a problem of considerable difficulty because it requires a double integration over the surface of the apertures. It was first solved by Airy in 1835 in terms of Bessel functions. For details of this solution, see T. Preston, The Theory of Light (Macmillan, London, 1928), 5th ed., pp. 324–328. Most optics texts simply quote the factor of 1.22 and note the Airy reference. For a more modern reference, see K. D. Möller, Optics (University Science Books, Mill Valley, California, 1988). On p. 347 the author simply states where the first minimum is located, but refers the reader to Problem 7 of Chap. 3. This problem concerns the actual calculation for a circular aperture and the author leads the reader through the problem (pp. 178–181). S. A. Akhmanov and S. Y. Nikitin, Physical Optics (Clarendon, Oxford, 1997), give a full treatment of diffraction by a round aperture on pp. 289–292 and derive the factor of 1.22.
7.
See, for example, A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio Astronomy (Wiley, New York, 2001), 2nd ed. These measurements are discussed in Sec. 1.3 and Chap. 12.
8.
Reference 7 is an excellent general reference on this topic as well as for detailed information.
9.
The light dimmer should be rated for at least 600 W (we use a Lutron D-600RH, available for about $5). Another solution is to put the entire slide projector on a Variac™ (autotransformer). However, the projector fan will slow down as the lamp is dimmed—not a good thing if left on too long. A very long extension cord connects the projector to the Variac, which must be at the front of the lecture hall.
10.
Sensitivity is typically expressed in terms of minimum illumination required to produce a usable picture at a particular f-stop. This definition of sensitivity is not very useful in this application but can be used to compare one camera to another. Sometimes manufacturers express the sensitivity in foot-candles. For reference, lux foot-candle and thus
11.
Orion part # 07127.
12.
Orion part # 5264. For part of the experiment it was not possible to focus the image using these adapter parts. The inexpensive solution worked well—simply put an empty cardboard toilet paper roll between the front of the camera and the telescope eyepiece. Its only purpose is to shield the CCD from light other than that projected onto it by the eyepiece.
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© 2002 American Association of Physics Teachers.
2002
American Association of Physics Teachers
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