6.3 Lenses

A lens is merely an optical system which includes two or more refracting surfaces. From the geometry of light rays, it turns out that spherical refracting surfaces are most interesting and practical. We define the first focal point of a lens to be the point such that if a point source of light is placed at , all rays will be parallel after passing through the lens. Similarly, the second focal point is the common point where formerly parallel light rays meet after passing through the lens. In the case of a lens which is relatively thin compared to the distance from one surface to or , the first and second focal lengths are the respective distances from and to the center of the lens. If the refractive index of the materials on either side of the lens is the same (e.g. if the lens is in air), the first and second focal lengths are equal to a common value f.

If, in a thin double convex lens, we take the radius of curvature ( ) to be positive for both surfaces, we can derive the equation

where n is the refractive index of the lens material.

In a lens system the image of an object often appears to be larger than the object itself; this is referred to as magnification. If s is the distance from the object to the lens and the distance from the lens to the image, then the magnification m is given as

Moreover, if d=s-f (i.e. distance from the object to the closest focal point) and , then

In this part of the experiment you will verify these theoretical results and investigate the distortion (called aberration) that spherical lenses introduce into images.

6.3.1 First Focal Length

1. Position the incandescent light source at the left end of the optical bench.
2. Attach a double convex lens to one component carrier and the viewing screen to another.
3. Position the lens and screen at the extreme right end of the bench so that the lens is between the light source and the screen. The further the lens is from the light the more parallel are the rays entering the lens.
4. Adjust the position of the screen until the image is as thin as possible. Then the distance between the middle of the lens and the screen is an approximation of the first focal length. Enumerate all the possible errors. Find the focal length of the other double convex lens. (The focal lengths of the plano-convex lenses are so large compared to the length of the optical bench that the incident light rays are not parallel enough to measure the focal length with any accuracy. We will use another method outlined below.)
5. Attach the plano-concave lens to one side of a component carrier and attach the aperture mask to the other side. Position the assembly about 40 cm from the light source.
6. Place the screen behind the lens and notice that as you move the screen further from the lens, the image expands.
7. Measure the image width and corresponding distance from the lens to the screen for two different screen positions. Using simple geometry, calculate where the rays diverted by the lens would theoretically converge at a point in front of the lens. This is an approximation of the focal length.

6.3.2 Second Focal Length

1. Attach a double convex or plano convex lens to a component carrier.
2. The light source is a line filament, so if we consider only lateral dimensions the source is essentially a point source. The equipment instructions for the source give the distance from the filament to the front surface of the light source to be approximately 21 mm.
3. Adjust the lens position so that the image on the screen is the same width no matter where the screen is placed (i.e. the emergent rays are parallel). The distance from the lens to the filament is the second focal length. Compare your result to the first focal length.
4. You cannot use the above method to measure the focal lengths of the 18 mm F.L. convex lens or the -22 mm F.L. plano-concave lens. These lenses cannot be positioned sufficiently close to the filament. Thus, as an alternative, place the 48 mm F.L. double convex lens between the light and the lens to be measured. The focal point of the first lens now becomes the point source for the second.

6.3.3 Improved Method

1. From the formulae above, if the magnification m= 1.0, then , and f=d. Hence, at such a lens position, .
2. Place a double convex lens or a plano-convex lens between the crossed-arrow target and the screen. (All the components should be on separate carriers.)
3. Adjust the position of the lens and screen until the image is in focus and is the same size as the original object. The target has a millimeter scale on it. At the final position, the distance from target to screen is four times the focal length.
4. With the focal length, compute the radius of curvature for the lenses. Assume for the double convex lenses; the radius of curvature of a plane is infinite. Also verify Gauss' formula:

   

6.3.4 Magnification

1. Position a double convex lens or plano-convex lens between the crossed arrow target and screen.
2. Adjust the position of the lens until the image is focused.
3. Measure the distance from the target to the lens and from the screen to the lens. Calculate the theoretical magnification.
4. Measure the actual magnification by measuring the distance between the scale markings on the image and compare with the theoretical value.

6.3.5 Spherical Aberration

A spherical surface lens does not form a point image of a point object originally positioned on the lens axis. Instead, the image is a line segment collinear with the lens axis. Thus, there is no exact screen position where the image is ``in focus''. Conversely, the point object can be moved slightly along the axis and still appear ``in focus''. The amount of movement which maintains a given image sharpness is called the depth of field.

1. Position the 188 mm F.L. double convex lens in front of the incandescent light source (about 30 cm away). Attach a sharp razor blade to a component carrier and position the assembly to the right of the 18 mm lens. Adjust the assembly position until the razor blade is in the focal plane of the lens. The lens image on the razor blade should be a small dot.
2. Carefully adjust the razor blade on the carrier until the sharp edge cuts across the center of the focussed dot. Place the viewing screen to the right of the razor blade and examine the image. It should resemble one of the patterns shown in Fig. B. Move the razor blade assembly slightly forward or backward and observe the pattern changes. The patterns will only approximate those in Fig. B since razor edges are not usually optically perfect. This technique is the Foucault knife test for lenses. Convince yourself that the patterns are due to spherical aberrations. Try using the 48 mm F.L. lens.

     
   
   Figure B: Spherical aberrations.
   

3. Place the variable diaphragm just in front of the test lens. By varying the opening, notice that the patterns degenerate. A small aperture gives a more unique focal point. If you placed the crossed arrow target between the light source and the variable diaphragm, you can observe that the point where the image is in focus becomes more distinct as the aperture closes. Thus, depth of field varies inversely with aperture size. This phenomenon is familiar to photographers.