12.5 Achromatic Double Lenses

Of course, the design should be finalized using these specific glasses since the ne and V values are a bit different.

It is perhaps not obvious which of these two designs would be the better. For a narrow field such as 22 , the lens in Figure 12.10 is to be preferred, while for a wider field such as 33 the lens in Figure 12.11 would obviously be better. It is interesting to see how the small changes in the design have made such a large difference to the tangential field at the wider field angles.

The large spherical aberration is a definite disadvantage of the new-achromat form. This was corrected by Paul Rudolph in his Protar design which will be discussed in Section 14.4.

12.5 ACHROMATIC DOUBLE LENSES

12.5.1 The Rapid Rectilinear

The Rapid Rectilinear, or aplanat lens is one of the most popular photographic lenses ever made. The lens is symmetrical, and the rear half is spherically corrected and has a flat field. In order to keep the lens compact, a large amount of positive coma is required in the rear component. This implies that a graph of spherical aberration against bending should rise high above the zero line, much higher than is usual for telescope objectives. To achieve this, the V difference between the old-type crown and flint glasses should be small, but a large index difference is helpful. The exact V difference depends on the aperture and field required. For a normal lens of f/6 or f/8 aperture, a V difference of about 7.0 is satisfactory. A smaller V difference can be used for a wide-angle lens of f/16 aperture, while a larger V difference leads to a longer lens of higher aperture, suitable for portraiture applications.

All three of these variations have been used by different manufacturers. At first, two flint glasses were utilized, but after about 1890 it was common to find an ordinary crown in combination with a light barium flint (see “A NewAchromat Combination” in Section 11.2.2).

LA′

0.10

0.05

0

–0.05 c1

–0.6 –0.5 –0.4

Figure 12.12 Bending curve for the rear component of a Rapid Rectilinear.

To initiate the design procedure, we will select the following glasses:

(a)Light Flint: ne ¼ 1.57628, Dn ¼ nF – nC ¼ 0.01343, V ¼ 42.91

(b)Flint: ne ¼ 1.63003, Dn ¼ nF – nC ¼ 0.01756, V ¼ 35.87

The Abbe number difference is Va – Vb ¼ 7.04. In designing the rear component, the procedure already described for telescope doublets is followed, except that because of the strongly meniscus shape of the lenses, the preliminary G-sum analysis is not very helpful and will be omitted.

Using these glasses for a focal length of 10, the (ca, cb) formulas give, ca ¼ 1:0577; cb¼ 0:8089

Assuming that c1 will be about one-half ca with negative sign, we make a drawing of the lens at a diameter of about one-tenth the focal length, enabling us to set the thicknesses at 0.3 for the crown and 0.1 for the flint.

Taking a few bendings and solving each for perfect achromatism by the D – d method on a traced f/16 ray, we can plot the graph in Figure 12.12. Recalling Figures 7.2 and 9.4, it should be evident that we want to select a value for c1 in the neighborhood of the left-hand solution, where the coma is positive; the stop position will be in front of the rear component. The right-hand solution with negative coma is useless since it would require the stop to be behind the lens to flatten the field. Since this is a photographic lens, we desire a small amount of spherical overcorrection to offset the zonal undercorrection shown in Figure 12.13a, which suggests that we try c1 ¼ –0.5 for further study.

This lens has a focal length of 10.806, LAm0 ¼ þ0.026, and LZA ¼ –0.0178. To find the stop position for a flat tangential field, we plot the H0 – L graph at 20 for a succession of L values as illustrated in Figure 12.13b. Remember that such plots are easily generated using an optical design program by filling the lens aperture (assuming the lens is the temporary stop) with meridional rays and then viewing the tangential ray fan plot for that obliquity. As already stated, the abscissa is reversed between the two plots. We now observe that the minimum point falls at L ¼ –0.2, which is the distance from the stop to the front (concave) surface.

We now assemble two of these lenses together about a central stop, as illustrated in Figure 12.14a, and find that the focal length is 5.6676. It is best to

scale this immediately to a focal length of 10.0, yielding the prescription in the following table.

Figure 12.13 Aberrations of the rear component of a Rapid Rectilinear: (a) spherical aberration; (b) the H 0 – L curve at 20 obliquity.

Figure 12.14 The final Rapid Rectilinear design: (a) layout; (b) longitudinal spherical aberration; (c) astigmatic field curves where the sagittal field is the solid curve and the tangential field is the dashed curve.

In the table at the top of 341, we have f 0 ¼ 10.00, l0 ¼ 9.0658, lens diameter ¼ 1.8, and Petzval sum ¼ 0.0630. The f/8 axial ray from infinity gives LA0 ¼ 0.0350, and it also tells us that the f/8 stop diameter must be 1.110. An f/11.3 zonal ray gives LZA ¼ –0.0108, enabling us to plot the spherical aberration graph in Figure 12.14b.

To plot the fields, we now add two other principal rays having slope angles in the stop space of 28 and 12 , respectively. The principal-ray slope angles in the stop space between the lenses are generally somewhat different than the entering or outside slope angle (see Section 12.5.2). The sagittal and tangential fields traced along these principal rays are shown in Table 12.7.

The fields are plotted in Figure 12.14c and closely resemble those of the rear half-system. Both the spherical aberration and the astigmatism are thus very stable in this type of lens for changes in the object distance, which was one of the reasons for its great popularity.

Table 12.7

Astigmatism and Distortion for Lens Shown in Figure 12.14a

12.5.2.A Flint-in-Front Symmetrical Achromatic Doublet

There is, of course, a companion system to the Rapid Rectilinear in which the rear component is a flint-in-front spherically corrected achromat. To design such a lens we may use the same glasses as for the Rapid Rectilinear, and we plot a graph of spherical aberration at f/16 against bending, of course in the region of the left-hand solution where the coma is positive (Figure 12.15). For each plotted point the last radius is solved for strict achromatism by the D – d method, and the curvatures are scaled to a focal length of 10, keeping the thicknesses at 0.1 and 0.3 as before.

We recall that when we were designing a telescope objective, we found that the left-hand solution for a flint-in-front doublet has a much smaller zonal residual than the left-hand crown-in-front doublet (Section 7.2). Consequently we shall plan the present design to be a “portrait” lens with an aperture of f/4.5 and covering a somewhat narrower field than the Rapid Rectilinear.