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The Spherical Aberration


Center ray from short object reflect off concave mirror. Short Objects & Concave Spherical Mirrors


We being our examination of spherical mirrors with a familiar example. An object stands before a concave mirror. To find its image location we draw a ray to the mirror center. The reflected ray is drawn using the rule that the incident and reflection angles are equal. It is shown in green.

Center ray from short object reflect off concave mirror.

At least two light rays are required to locate an image in an ideal situation. A ray from the top of the object passing through the mirror's center of curvature arrives normal to the mirror surface. That means that its return path follows the incoming path. A ray of this type is drawn in blue in the sketch.

Center ray from short object reflect off concave mirror.
If the image is truly in focus many light rays can be drawn with all focusing at the same place. Our third ray path, shaded in red, initially parallel to the optic axis intersects with the two drawn previously.

Three light rays will be used to demonstrate the "spherical aberration."


Spherical Aberrations

The object is taller here than in the example above. This increase in height means that the curvature of the mirror plays a more important role.

We set about the task of finding the image position using the same routine as that used above. The first two light rays are drawn in the diagrams below.

If the image is in focus a third light ray should confirm the image location as was the case of the short object. (Of course, if that were the case, there would be no need for this section.)

Center ray from short object reflect off concave mirror. Center ray from short object reflect off concave mirror.

Again we choose a light ray initially parallel to the optic axis as our third ray. Here, the third light ray points to a problem. The image position is not at all clear. With three rays we now how three possible locations for our image.

Center ray from short object reflect off concave mirror. Center ray from short object reflect off concave mirror.

One would see the image as distorted or fuzzy. What section of the spherical mirror produces a well-focused image?


Teaspoon Optics

A clear focal point is needed to obtain a crisp image. To find the portion of a spherical mirror that yields a clear image, it is convenient to use a ray diagram.

Consider the left-hand side diagram below. Parallel light rays should cross the optic axis at the focal point for a clean, undistorted image. Some of the light rays appear to intersect at about the same point, thus defining a single focal point. These rays, together with the section of the mirror that they used are shaded in green.

Since this "green" area is a subjective call, an additional section has been colored amber. The three image example above came from this amber section.

Finally, there are areas of the spherical mirror for which there is clearly no well-defined single focal point. These sections and the associated rays are shown in red.

Incoming parallel rays reflect and cross the optic axis. A portion of a spherical mirror

The ray diagrams in section 3 that were used to demonstrate the idea of a focal point were parabolic, rather than spherical in shape. The aqua blue curve in the right-hand side figure above is a parabola. The color-coded spherical mirror outline has been superimposed on top of this. Observe that the green area, for which a clear image is expected, follows the central portion of the parabola.

If you choose to use a spherical mirror to demonstrate basic optics, it is a good idea to restrict your object height so that only a small portion of the mirror is used. A better solution probably can be found in your kitchen. I find that a small teaspoon provides a very nice, inexpensive parabolic mirror for demonstrating focal points, and real and virtual images.

To find a good teaspoon* for your demonstrations or experiments simply start looking through your silverware drawer (or in a store) for a very shiny teaspoon where your reflection is crisp. This is your curved mirror. If you sketch out the shape, you will likely find that it is very nearly parabolic.

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* Tablespoons, servings spoons, and small candy dishes are potential candidates too.


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