(a) (b)

Figure 11.16 Pincushion distortion. (a) 4%, d ¼ 0.50 mm, r ¼ 676 mm; (b) 10%, d ¼ 1.25 mm, r ¼ 302 mm.

sides of the square due to distortion. The quantity r is the radius of curvature of the sides of the images, which should, of course, be straight. As can be seen, 4% distortion is just noticeable, whereas 10% is definitely objectionable. Consequently, we generally set the distortion tolerance at about 1% since few observers can detect such a small amount. For specialized applications such as aerial surveying and map copying, the slightest trace of distortion is objectionable, and the greatest care must be taken in the design and manufacture of lenses for these purposes to eliminate distortion completely.

11.4.1 Measuring Distortion

Since distortion varies across the field of a lens, it is difficult to determine the ideal Gaussian image height with which the observed image height is to be compared. One method is to photograph the images of a row of distant objects located at known angles from the lens axis and measure the image heights on the film. Since focal length is equal to the ratio of the image height to the tangent of the subtense angle, we can plot focal length against object position and extrapolate to zero object subtense to determine the axial focal length with which all the other focal lengths are to be compared. If the lens is to be used with a near object, we substitute object size for angular subtense and magnification for focal length. The determination can be performed at several object field positions and the coefficients s5; m12; and t20 for Eq. (11-8) can be found.

Note that the two parts of this formula are similar in magnitude, the first being caused by the difference in slope of the principal ray as it enters and leaves the system, the second being derived from the lens surface contributions.

To verify the accuracy of this formula, we take the much-used cemented doublet of Section 2.5 and trace a principal ray entering at 8 through the anterior focal point to form an almost perfectly telecentric system (see Table 11.2).20

The agreement in the results of this calculation between the direct measure of the image height and the sum of the various contributions is excellent. For the distortion itself we first calculate

When this is added to the summation value in Table 11.2 we find the distortion to be 0.0618321, again in excellent agreement. The change in slope of the principal ray has contributed about one-third of the distortion, the remainder coming from the lens surfaces themselves.

Unfortunately, the quantities under the summation sign are not really “contributions” that have merely to be added together to give the distortion. Each lens surface, to be sure, provides an amount to be summed, but it also

Table 11.2

Calculation of Distortion Contributions

Hence H 0 ¼ 1.6701438 0.0454832 ¼ 1.6246606