Paul J. Dolan, Jr.*, 5709 Bennett Hall, Dept. of Physics and Astronomy, University of Maine, Orono, ME 04469-5709, liquidhelium@hotmail.com

Abstract

Chromatic aberration in lenses is an important concept in practice, but one that is not frequently studied in introductory courses, particularly in introductory labs. An addition to the typical lenses lab is presented, in which chromatic aberration can be easily measured using the standard equipment used by students in studying image formation. The qualitative and quantitative results should be well within the abilities of most students in these courses.
 
 
 
 

Typically, in the second semester of the introductory physics sequence, one or more lab exercises are done in optics, including finding the focal length of a lens. A standard apparatus uses an incandescent light source with a ground glass diffusing screen, fronted by an ‘object’, such as a star or non-symmetric letter (like an ‘F’), cut into a metal sheet. The students are asked to measure the focal length of several lenses, by adjusting source-to-lens (‘object’) and lens-to-screen (‘image’) distances, to which they apply the thin-lens formula. Students are asked to judge visually when the image is ‘in focus’. In performing this experiment it is assumed that the index of refraction (n) is independent of wavelength, which is not strictly true. The goal of this work is to show that the ‘chromatic aberration’ can in fact be measured as part of such as exercise, primarily using the equipment that would normally be used for this exercise.
 
 

Experiment

The equipment used is a 2-meter optical rail, to which are mounted an incandescent light source, a lens holder, and a white screen (the image plane). The ‘object’ is a star cut into a metal sheet, which fits into a slot in front of the ground glass screen. The points of the star do not all have the same shape, which allows the student to see if the final image is erect or inverted. Standard glass lenses are used, though the best results on chromatic aberration are obtained for relatively large focal length lenses (30 cm or greater). The new wrinkle is that color filter sheets are inserted between the ground glass and the object.

The filters used were obtained from Arbor Scientific¹, set 33-0190. The three filters of this set used were the red, blue, and green. None of these pass only a single color (wavelength), which is good, since in that case the intensity of the image would be too weak to be observed. The red is a ‘high pass’ filter, allowing wavelengths above about 636 nm to pass. The green and blue are ‘band-pass’ filters, the green being centered at about 521 nm, and the blue being centered at about 428 nm, as per the transmittance spectra that are included with the filter set. (The yellow, cyan, and magenta filters of the set each pass a broader range of wavelengths.) Measurements of the focal length were made using each of these filters, as well with white light (no filter), using a nominal 30 cm focal length lens, and at several image-to-object distances. The results are shown in Table I.

The measurements were made by adjusting the position of the lens to obtain a sharply focused image. To avoid a bias in the measurements, the image was ‘defocused’ when each new color filter was inserted. However, on inserting a different color, one could qualitatively observe that the previously focused image was no longer sharp.

While the calculated change in the focal length is relatively small (on the order of 0.5 cm), the distance that the lens is moved to achieve focus can be several centimeters. This precision should be well within the capabilities of the careful student. Lenses with shorter focal lengths were also tried, but in these cases the change in position of the lens, as the color is changed, is within the uncertainty of the measurement (0.1 – 0.2 cm); also, the ‘range’ of focus of even white light is so narrow that convincing results can not be obtained with standard introductory lab equipment, ‘by eye’.
 
 

For the More Advanced Students

The experiment described above should be quite sufficient to demonstrate that chromatic aberration is indeed worth considering. However, even with this equipment, additional information can be obtained. In particular, one can determine the index of refraction at each chosen wavelength, and compare this variation with theoretical predications.

The lens used in this experiment was a plano-convex lens, with a measured radius of curvature (R) of 16.7 cm. Using the nominal focal length of 30 cm, and the lens maker’s formula:

(1) 1/f = (n-1)(1/R)

this would indicate that n is 1.558, a reasonable value for glass. Using the measured focal lengths, one can use equation (1) to obtain a value of n(l ) for each wavelength; these are shown in Table I. By inspection, one can see that, as expected, n does decrease as the wavelength increases.

The simplest model of the variation of the index of refraction with wavelength indicates that:

(2) 1/(n²-1) = C/?² + C/?_o²

where C is a constant that can be derived from the interaction (on an atomic scale) of light with matter, and ?_{o is the resonant wavelength of the material². A plot of this function is shown in Figure 1. While admitting that we do have only three measured data points, a reasonably good linear fit is observed. The experimental values determined from the graph are C = -6.6x 10^-9} and ?_o = 72 nm. This value of ?_{o is quite reasonable, for glass, being well into in the ultraviolet.}

Conclusion and Further Suggestions

To obtain a different range of ‘aberration’, a plastic lens or other imaging devices such as a Fresnel Zone Plate, may be used. One might also be able to enhance the aberration by using other schemes for finding the focus, such as the autocollimation method, in which the light passes twice through the lens, which should enhance the aberration. These possibilities are under investigation.

This experiment should serve as a useful addition to an introductory optics experiment, and will give the students an opportunity to investigate the ‘real’ properties of lenses.

* Permanent address: Dept. of Physics, Northeastern Illinois University, 5500 N. St. Louis Ave.

Chicago, IL 60625

1.)    Arbor Scientific Color Filters, #33-0190, P.O. Box 2750, Ann Arbor MI 48106-2750

2.)    See for example E. Hecht, Optics, 4^th ed. (Addison-Wesley-Longman, 2001).
 

Table I: Measured values of focal length and n(l ) for various color filters.
 

Color of Filter	Wavelength of Maximum Transmittance (nm)	Average measured focal length, f (cm)	Index of Refraction  n(l )
Blue	428	29.40	1.569
Green	521	29.74	1.562
Red	636	30.07	1.556

 

Chromatic Aberration: (n ² – 1) ^–1 vs. l ^-2
 
 

Figure 1. Measured values of n(l ) vs l , showing the expected linear dependence of 1/(n²-1) vs 1/l².