An aperture stop is a restriction of the optical beam in a lens system. Lenses have aberrations which are related to the deviation of light rays from the optical axis. An aperture stop eliminates the light rays that deviate more greatly from the optical axis. Thus restricting the rays that enter the lens system will produce a more accurate, more sharply focused image. However when some of the light rays from the object are blocked out the brightness of the image is reduced. In photography this means that the camera shutter will have to be open longer to expose the film adequately.
The time required to properly expose the film is inversely proportional to the intensity of the light reaching the film. . The diagram below shows the relationships involved without an aperture stop and with the light rays from the object being parallel to the optical axis. This presumes the object is at infinity.
In the diagram yo is the radius of the object field and yi is the radius of the image of the aperture. The point F is the focal point of the lens. The focal length is the distance from the midplane of the lens to the focal point. The distance xi is the distance from the focal point F to the image plane. In the diagram it is presumed that the lens is thin enough that the aperture stop can be considered as located in the midplane of the lens.
Let Ao be the area of the object field and Io the intensity of the light per unit area at the lens. This would be the intensity of the light as read by a light meter at the camera. Likewise Ai is the area of the image and Ii is the intensity of the light per unit area at the image.
Let E be the total energy of radiation impinging upon the lens. The energy impinging up on image plane is the same as E. Thus
The required exposure time t is inversely proportional to Ii; i.e.,
where k is a constant.
By the similarity of the triangles involved
This ratio, yi/yo, is called the magnification ratio. From the above equation it is seen that this ratio is determined by the focal length of the lens and the dimensions of the camera.
When an aperture stop is introduced into the system it will reduce the amount of illumination (radiant energy) received by film on the image plane. The ratio of the energy received with the aperture stop in place relative to what would be received without the aperture stop is given by the ratio of the areas. Let D be the diameter of the aperture and hence ya=D/2. Thus
The intensity of the light reaching the film is then
where K=¼Io(xi/yo)². Thus K is a constant determined by the ambient light intensity Io and the geometric characteristics of the camera, yo and xi and does not change as the camera setting is adjusted. The required exposure time t is given by
where the k1 is a constant and a characteristic of the film. Thus k2 is a constant determined by the characteristics of the fim and the
The ratio (f/D) is known as the f-stop or the f-number of the camera setting.
The above equation indicates that the required exposure time t is proportional to the square of the f-stop number. This means that if the f-number is doubled the required exposure time is quadrupled. If the f-number is increased by a factor of √2 then the exposure time is doubled. Cameras typically have settings so that the f-number can be set to 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, or 22. Thus if the camera gives an exposure time of 1/1000 of a second for f/1 then it will give 1/500 seconds for f/1.4 and 1/250 seconds for f/2.
Telescopes as well as cameras have f-numbers. A telescope at the Yerkes Observatory at the University of Chicago has a diameter of 40 inches and a focal lenght of 63 feet. Thus its f-number is 18.90. In contrast the 200 inch telescope at Mount Palomar has a focal length of 666 inches so its f-number is 3.33.
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