## Abstract

It is pointed out that a theoretical relation exists between the distribution of energy in the image of an edge (edge trace) and the distribution in the image of a line (line spread-function). Experimental data are presented in support of this relation and the method of determining these distributions experimentally is outlined. It is also pointed out that, since the spot diagram of a lens represents the point spread-function, the line spread-function, needed for the foregoing procedure, can be found by mechanical summation of this diagram. An example is given to show that the edge trace of the finished lens can be thus predicted from the spot diagram.

© 1958 Optical Society of America

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### Figures (6)

Fig. 1

Method of forming and scanning a point spread-function (left) and a line spread-function (right).

Fig. 2

Geometrical illustration of the convolution of a semi-infinite plane with the line spread-function of a lens. Each linear element of the illuminated area (of which four are shown) forms its own spread function A(x), and the sum of these is the image of the entire edge. The summation is indicated for x0.

Fig. 3

Comparison between line spread-function of a certain lens as computed from the edge trace in Fig. 4 (circles) and as measured (curve).

Fig. 4

Comparison between edge trace as computed from the spread function of Fig. 3 (circles) and as measured (curve).

Fig. 5

Comparison between photomicrographs of images of a point object at selected places in the field of a certain lens (lower row) with the corresponding spot diagrams computed from the design data (upper row). Image (A) is on the axis, (B) is 5° away, and (C) is 24° away.

Fig. 6

Comparison between edge trace formed by a certain lens as computed from the spot diagram (circles) and as measured (curve).

### Equations (3)

$A ( x ) = ∫ - ∞ ∞ a ( x , y ) d y .$
$E ( x 0 ) = ∫ ∞ x o A ( x ) d x .$
$d E / d x = A ( x ) .$