53. Influence of Diaphragm Aperture on the Different Aberrations. The employment of dia phragms with smaller apertures always (at any rate up to a certain limit) improves the definition given by a lens which is incompletely corrected, but the degree of improvement is different for the different aberrations.
Chromatic aberration varies almost directly as the diameter of the stop ; spherical aberration on the axis varies as a rule almost as the " cube " of the diameter (the product of the diameter multiplied by itself twice). Astigmatism - and curvature of the field are approximately pro portional to the diameter and the square (pro duct when multiplied by itself) of the slope of the secondary axis to the optical axis ; coma is pro portional to this slope and to the square of the diameter, approximately.
As a general rule, it can be stated that no lens is completely corrected, but fortunately the blur produced instead of the ideal point image is not of uniform intensity of illumination over the area derived from geometrical considerations.
There is generally a maximum intensity over a small fraction of the area of the blur, so that the photographic image is always better defined when the exposure is a minimum than when a prolonged exposure is given, which allows the poorly illuminated parts of the image patch to be recorded on the plate. These variations of sharpness with exposure are much more apparent with a badly corrected lens.
As has been indi cated, there is a limit beyond which the diaphragm aperture cannot be diminished. with the hope of reduc ing aberrations. Till now we have con sidered the rays of light obeying the laws of geometrical optics, which is a perfectly legitimate convention, since the results agree with experiment when pencils of light of ffi sucient angular aper- ture are considered, but when the aperture is reduced to less than o•o4 in., or less than about one-seventieth of the distance from the image, geometrical optics fails. By reason of the propagation of light in concentric waves, the image of a point formed by an optical instrument is always a patch, even if the instrument is perfectly stigmatic and aplanatic. The distribution of light in such an area is shown diagrammatically on an enlarged scale in Fig. 34 for the image of a point on the optical axis (bright disc surrounded by concen- tric rings alternately black and faintly bright).
The diameter of this diffraction disc is greater the narrower the pencil of light, and the photo- graphic definition would be spoilt if the limits mentioned above were passed. An exceedingly minute diaphragm would produce an image hardly better than that given by a pinhole.
Astronomers, and especially microscopists, know that to get an image as sharp and detailed as possible it is necessary to use lenses of large aperture, the limit of resolution (minimum dis tance between two parallel lines that can be reproduced separately) is smaller the larger the aperture used.' 54. Distribution of Light in the Field. No objective can give a uniformly bright image of a uniformly illu minated surface, even if this is of small extent.
This can be explained by comparing the effects of a beam directed along the axis with one in an oblique direction, forming the images F and respectively (Fig. 35). Viewed from the point P the lens has the appearance of a uniformly illuminated circle, whilst viewed from P' the appearance is an ellipse, the area of which is smaller than that of the circle by an amount which increases with the obliquity. Further more, F' is further from 0 than F, and, as is well-known, the illumination diminishes when the source of light is farther away. Finally, the oblique beam illuminates the screen or sensitive surface in the plane perpendicular to the optical axis, less than it would a screen placed at pp, perpendicular to its mean direction.
Combining the effects of these different causes it is possible to calculate the maximum illumina tion at different angles.' The numerical values ination at the edges of the field than is accounted for above, because the oblique pencils are partly intercepted by the mount. To explain this without excessive complication let us see what would happen in the case of a lens mount without glasses. For a certain aperture of the diaphragm DD (Fig. 37) all those light-beams more oblique than AA would be partially inter cepted by the lens cell and the tube. If the diaphragm is replaced by a smaller one D'D' the limit of obliquity for which there is no cutting off is increased, since in these circumstances the beam BB passes freely.