14.6 The Cooke Triplet Lens

The triplet objective is tricky to design because a change in any surface affects every aberration, and the design would be impossibly difficult without a preliminary thin-lens predesign using Seidel aberrations. We assign definite required residuals for each primary aberration, and then by ray tracing determine the actual aberrations of the completed thick-lens system. If any aberration is excessive, we adopt a different value for that primary aberration and repeat the entire predesign. The thin-lens residuals used in the following example are the result of experience with prior designs that result in the final thick system being satisfactory. Of course, in making a design differing from this in any important respect such as aperture, field, or glass selection, we would require a different set of Seidel aberration residuals, which would have to be found by trial.

14.6.1The Thin-Lens Predesign of the Powers and Separations

If we place the stop at the negative thin element inside the system, we can solve for the powers and separations of the three elements to yield specified values of the overall focal length and primary chromatic aberration, primary lateral color, Petzval sum, and one other condition that will eventually be used for distortion control. This last requirement might be the ratio of the two separations, the ratio of the powers of the outside elements, the ratio of the power of the combination of elements a and b to the power of the system, or some other similar criterion. We thus have five variables (three powers and two separations) with which to solve five conditions, after which we shall have three bendings to correct for the three remaining aberrations: spherical, coma, and astigmatism. Without this convenient division of the aberrations into two groups, those depending only on powers and separations and those depending also on bendings, the entire design process would be hopelessly complicated and almost impossible to accomplish.

The first part of the thin-lens predesign can be performed in several ways, the one employed here having been introduced by K. Schwarzschild around 1904. It uses the formulas for the contributions of a thin element to power, chromatic aberration, and Petzval sum, given in Section 11.7.2. These contributions may be written for each aberration in turn, as follows:

These three equations are linear in the three lens powers, and they can be easily solved for the powers once we know the three axial-ray heights ya, yb, and yc. The first of these, ya, is known when the focal length and f-number

are known, but yb and yc must be found by trial to satisfy the remaining two conditions, namely, the correction of lateral color and the ratio of the two separations S1/S2 ¼ K. Reasonable starting values of the other ray heights are yb ¼ 0.8ya and yc ¼ 0.9ya .

As an example, we will proceed to design an objective of focal length 10.0 and aperture f/4.5 covering a field of 20 . We shall assume that K ¼ 1, and use the following types of glass:

ða; cÞ SK-16; nD ¼ 1:62031; nF nC ¼ 0:01029; V ¼ 60:28

ðbÞ F-4; nD ¼ 1:61644; nF nC ¼ 0:01684; V ¼ 36:61

In our predesign we shall aim at the following set of thin-lens residuals, hoping that these will give a well-corrected system after suitable thicknesses have been inserted:

with ya ¼ 1.111111, yb ¼ 0.888888, and yc ¼ 0.999999. Solving the three Schwarzschild equations for the three powers gives

fa ¼ 0:192227; fb ¼ 0:291104; fc ¼ 0:156285

The paraxial ray and the paraxial principal ray passing through the middle of the negative lens have the values shown in Table 14.17. Inspection of this table shows that, for the paraxial ray,