When the light passing through a lens is limited by a circular aperture of radius ρa, the variation of exposure follows the cosine-fourth law for a small Lambertian source at an infinite distance. For an object plane at a distance z from the lens, where z≫ρa, the fractional deviation from the cosine-fourth law is 3(ρa/z)2 sin2ψc where ψc is the angle between the source and the optical axis measured at the center of the aperture. Thus, if the object plane is more than 15ρa from the aperture, the cosine-fourth law deviates less than 1% from the exact expression when within 60° of the optical axis. The cosine-fourth law always gives pessimistic results, so that in the absence of vignetting the illumination for a wide-aperture lens used for macrophotography may be much more uniform than indicated by this approximation. To reduce the exposure of a near object plane, the diaphragm opening can be reduced, which increases the angular variation of exposure; or the center may be blocked so that an annular aperture is employed, thereby improving the uniformity of exposure.
If the front lens element is the limiting aperture, the analysis applies without modification provided that the outer surface is plane or concave. Otherwise, the chord and radius of curvature of this surface determine a right circular cone with axis along the optical axis and vertex in front of the lens within which the exposure follows the same relations. For other locations of the object, or when the diaphragm is within the lens, distortion of the physical boundary of the aperture by lens aberrations prevents a general analysis. The results for the outer lens as the limiting element are pertinent, however, as other apertures can serve only to decrease the received energy further.
The limitations on a general treatment of exposure apply even more severely to an analysis of vignetting. The gross characteristics of vignetting can be described to advantage, however, by means of a simple model consisting of circular apertures spaced along the optical axis. Thus it is shown that: (i) If only one additional aperture limits the light passing through the entrance pupil, then the smaller the diaphragm opening, the greater the angle from the optical axis before the onset of vignetting and the smaller the angle at which all light flux is blocked. (ii) If there are limiting apertures either side of the diaphragm, then the angle at which complete blocking of the light flux occurs is independent of the diaphragm radius, provided that the radius exceeds a certain minimum value.
© 1960 Optical Society of AmericaFull Article | PDF Article
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