14.6.2 The Thin-Lens Predesign of the Bendings

The bendings of our three thin-lens elements are defined by c1, c3, and c5, respectively. Since the stop is assumed to be in contact with lens (b), the astigmatism contribution of that element is independent of its bending. Our procedure, therefore, is to adopt some arbitrary bending of lens (a) and ascertain its AC* by the formulas given in Section 11.7.2. We find the AC of lens (b) by 12 h0y2 fb and then solve for the bending of lens (c) that will make the total astigmatism contribution equal to the specified value of –0.09. Having done this, we go to lens (b) and bend it to give the desired value of the sagittal coma, namely, 0.0025. This will not affect the astigmatism in any way. Finally, knowing the bendings of lenses (b) and (c), we can calculate the spherical contributions of all three lens elements, and plot a point on a graph connecting the spherical aberration with the value of c1. Repeating this process several times with different values of c1 will enable us to complete the graph and pick off the final solution for any desired value of the thin-lens primary spherical aberration.

The contributions of the thin-lens elements to the three aberrations are given by the formulas in Section 11.7.2 involving the G sums for spherical aberration and coma. These contributions are quadratics in terms of the bending parameters c1, c3, and c5 as follows:

Lens (a)

SC* ¼ –23.227833c12 þ 11.968981c1 – 2.011823

CC* ¼ 1.454188c12 – 1.361274c1 þ 0.292417

AC* ¼ –7.374247c12 þ 10.006298c1 – 3.442718

Lens (b)

SC* ¼ 15.229687c32 þ 4.686647c3 þ 1.436793

CC* ¼ 0.667069c3 þ 0.095270

AC* ¼ 2.020947

Lens (c)

SC* ¼ –15.073642c52 þ 5.519113c5 – 0.937665

CC* ¼ –1.089393c52 – 0.130340c5 þ 0.030780

AC* ¼ –6.377286c52 – 3.861014c5 – 0.528703

Collecting these expressions, we find that with a given c1, we first solve for c5 by the quadratic expression

c25 þ 0:6054322c5 þ ð1:15633c21 1:569053c1 þ 0:2917344Þ ¼ 0

Only one of the two solutions is useful; the other represents a freakish lens bent drastically to the left that would exhibit huge zonal residuals; that is, it looks odd and is quite strained (see Endnote 7).

Knowing c1 and c5, we can solve for c3 for coma correction by

c3 ¼ 2:179967c21 þ 2:040679c1 þ 1:633104c25 þ 0:1953917c5 0:6235753

Finally, knowing all three parameters c1, c3, and c5, we can calculate the spherical aberration by

LA0 ¼ 23:227833c21 þ 11:968981c1 þ 15:229687c23

þ 4:686647c3 15:073642c25 þ 5:519113c5 1:512695

Taking a series of values for c1 we find what is shown in Table 14.18. Thus all three lenses are bending to the right together. The spherical sums are plotted on a graph (Figure 14.29), from which we can pick off the desired c1 values for our residual of –0.08. There are obviously two solutions, namely,

c1 ¼ 0:2314 and c1 ¼ 0:3780

Table 14.18

Primary Spherical Aberration versus Bendings of a Triplet Lens

LA′

0.1

0

–0.1

–0.2

–0.3

–0.4

c1

0.15 0.2 0.25 0.3 0.35 0.4

Figure 14.29 Relation between c1 and primary spherical aberration, after correcting field by c5 and coma by c3.

We shall follow up only the left-hand solution since the right-hand solution has more steeply curved surfaces and is likely to exhibit larger zonal residuals. For the left-hand solution, then, we have the following thin-lens curvatures:

14.6.3 Calculation of Real Aberrations

After selecting suitable thicknesses from a scale drawing, scaling the lenses up or down to restore their exact thin-lens powers, and calculating the air spaces to maintain the thin-lens separations between adjacent principal points, we obtain the following thick-lens system:

0.2326236

0.41.62031

0.04031

1.051018

0.2617092

0.251.61644

0.227485

0.986946

0.0152285

0.451.62031

0.2984403

with f 0 ¼ 10.00, l0 ¼ 8.649082, LA0 ( f/4.5) ¼ 0.01267, LZA ( f/6.3) ¼ –0.01051, OSC ( f/4.5) ¼ –0.001302, Ptz ¼ 0.03801 (see Table 14.19).

The lateral color correction is evidently about right. Figure 14.30 shows the plotted spherochromatism graphs, which show that both the spherical and chromatic aberrations are also about right. The astigmatic fields are also plotted in

Table 14.19

Astigmatism, Distortion, and Lateral Color for Final Triplet Lens

Figure 14.30 Aberrations of final triplet lens (f/4.5).

Figure 14.30, where it can be seen that a slight change in the thin-lens sagittal astigmatism in the positive direction might be an improvement.

As far as coma is concerned, we have plotted the graphs of H0 against Q1 for sets of oblique rays at three different obliquities in Figure 14.31. The points VV on each graph represent the limiting vignetted rays, which enter and leave the lens at the initial marginal aperture height of 1.1111, assuming that the front and rear apertures of the lens are limited to that value (Figure 14.32). When the obliquity is increased the vignetting becomes accentuated, and the graphs become shorter at the upper (right-hand) ends. The principal ray along which the astigmatism was calculated is indicated in each case. The slope of the graph at the principal-ray point is, of course, an indication of the Xt0 for that obliquity. To improve the distortion, the design could be repeated with a different value of K, say, 0.9 or 1.1.

On the whole, this seems to be a pretty good design, typical of many triplets using these common types of glass. For a good lens at a higher aperture such as f/2.8, for example, it would be highly desirable to use glasses with much higher refractive indices, such as a lanthanum crown and a dense flint. A search through the patent files will reveal many triplet designs for use at various apertures and angular fields.

14.6.4 Triplet Lens Improvements

As we have mentioned, Dennis Taylor created the Cooke triplet lens over a hundred years ago and you may wonder why we have spent such effort discussing this lens in this chapter.16 The reason is that this lens type continues to be of interest for use in various new systems such as low-cost cameras, printers, copiers, and rifle scopes. As odd as it may seem that such a simple lens

Figure 14.31 Meridional ray plots of triplet lens.

Marginal

f /4.5

Figure 14.32 Final triplet design, showing f/4.5 marginal ray and limiting rays of vignetted 20 beam.

configuration continues to receive interest by lens designers, it has because clever lens designers have been able to find solutions that provide better performance and lower manufacturing costs that satisfy certain product requirements. The cost of a lens system includes at least the following considerations:

. Number of elements

. Diameter of elements

. Volume of elements

. Cemented elements

. Tolerance of radii, thicknesses, decenter, tilt, wedge, and so on

. Surface figure and quality

. Cost of glasses (higher index typically costs more; higher production volume glasses typically cost less)

. Mechanical mounting complexity

. Focusing mechanism complexity

. Coatings

Clearly the lens designer is required to consider far more than just the optical configuration and the performance of the design. It is often said that the optical design of the lens is less than half of the lens designer’s work in completing a project.

In 1962, Hopkins published a systematic study of a region of triplet solutions in the midst of the infinite number of third-order solutions.17 His analysis included both third-order and fifth-order aberrations. Hopkins made a number of observations, but perhaps the most significant was that he found that raising the index of the lenses was productive.

Independently in the mid-1950s, Baur and Otzen improved on the triplet photographic objective lens.18 They pointed out that the known triplet photographic objectives used the highest possible refractive index material for the two positive elements and the lowest possible refractive index material for the negative element in an effort to obtain a low Petzval sum; therefore the field was flattened to achieve a large and useful image area. Their invention was finding structures that provided improved performance where the new lens has greater radii than prior art triplet lenses, requires less glass, and is more economic to manufacture. They found that the refractive index for all three lenses should be in the range of 1.72 to 1.79 for the D spectral line and that the Abbe number should be about 45 for the positive elements and about 28 for the negative element. More particularly, the arithmetic mean of the Abbe numbers of the three elements must obey the relationship,

36 < V1 þ V2 þ V3 < 41: 3

Baur and Otzen teach in their patent the other relationships that are required to achieve the performance they claim. Their example lenses are f/2.8 with field coverage of 26 half-angle.

About a decade later, Kingslake19 disclosed a means to produce a triplet covering a wide field that was remarkably different than that of Baur and Otzen. Ackroyd and Price20 cofiled a patent application also on wide-angle triplets. Interestingly, their patents were sequentially issued. Kingslake’s goals were to markedly increase the useful field-of-view to at least 34 half field-angle with less vignetting than any prior art, to use lower-cost glasses, reduce the size of elements, and minimize the overall length of the lens.

Up to this time, the maximum half field-angle achieved was about 28 and the lens tended toward a reverse telephoto lens, which means that the focal length is shorter than the total length from the front lens vertex to the focal plane. He was able to reduce refractive indices used to about 1.61 and Abbe numbers to about 59 for the positive elements and 38 for the negative element. An f/6.3 lens having good performance was designed with 35 half field-angle and about 0.30 vignetting. The diameter of the front, middle, and rear elements are greater than 30%, 15%, and 18% of the focal length, respectively. Ackroyd and Price, who also worked at Kodak with Kingslake, improved on Kingslake’s design by finding rules for the lens structure that allowed a 34 half field-angle with 0.58 vignetting at f/6.3; thus, the diameter of the front, middle, and rear elements are greater than 24%, 16%, and 23% of the focal length, respectively. However, satisfactory results were claimed if diameters were kept above 20%, 13%, and 20%, respectively.

These lens designers made a dramatic leap forward in developing a smaller, lower-cost, and wide-field objective lens suitable for volume production. In both patents, specific and detailed guidance is provided to teach the design procedure for a lens designer to follow. Figure 14.33 illustrates an example lens from their patents. The ray fans in Figure 14.34 show acceptable spherical aberration and chromatic correction on-axis (0 ) with some negative coma and slight sagittal astigmatism appearing at 12 . By 24 , oblique spherical aberration is dominating in the meridional plane and sagittal astigmatism continues to increase. A small amount of vignetting of the lower rays can be seen. At the edge of the field (34 ), vignetting of both upper and lower rays is evident and necessary to limit the image degradation of the quite strong tangential astigmatism. The sagittal astigmatism has also grown and switched from undercorrected to overcorrected.

The astigmatic field curves in Figure 14.35a show this behavior (see page 432). The tangential astigmatism has been reasonably well controlled to be relatively flat out to about 86% of the field, with the sagittal somewhat less so. At about 29 , the sagittal and tangential curves intersect and then rapidly separate, with the tangential astigmatism becoming more undercorrected and the sagittal astigmatism more overcorrected. This is also another example of using the higher-order astigmatic aberrations to balance against the lower-order terms to achieve a wider field. An estimate of this intersection height is given by Eq. (13-8). The distortion is shown in Figure 14.35b and the 0.25% distortion is quite acceptable for intended application of this lens.

34°

24°

12°

0°

Figure 14.33 Wide-angle triplet lens.

In the early 1990s, Hiroyuki Hirano investigated using the triplet for a lowcost, broad zoom-range lens for a copying system rather than the more typical four-element symmetrical lenses.21 At that time, available copying lenses had a zooming range of about 0.6X to 1.4X with an f-number of 5.6 and total field coverage of about 40 . The new triplet copying lens is shown in Figure 14.36 and is the third of the five examples contained in the patent (see page 433). This lens has a 100-mm focal length, an f-number of 6.7, and total field coverage of 46 . The structure is as follows:

24° 34°

Figure 14.34 Ray fans for wide-angle triplet lens with f ¼ 100 and f/6.3. Ordinate is 0.5 lens units. F light is solid curve, d light is short dashed curve, and C light is long dashed curve.

Figure 14.36 Triplet copying lens with f ¼ 100, f/6.7, and 46 total field coverage.

Table 14.20

Alternative Glass Selections for Triplet Copying Lens

An important innovation Hirano made was to use lower refractive index glasses and the minimum number of elements feasible to solve the design goals. Notice that different crown glasses were used for the two positive elements and also it is seen that the refractive index of each of the three lenses is remarkably different, in contrast to typical triplet lenses. The e spectral line refractive index and the Abbe number for each glass used in the five examples are listed in Table 14.20.

Abbe found that if the condition 0:08 < n3 n1 is met, then additional performance improvement can be realized. In the five examples, n3 n1 is observed to range from 0.09 to 0.25. Note also the significant thickness of the first element, which is utilized to control Petzval and to achieve useful distribution of surface powers for aberration control.

Figure 14.37 presents the longitudinal spherical aberration. It is evident that the lens is adequately corrected for spherical aberration and is achromatic since the e and F curves intersect at the 0.7 zone. The ray fans in Figure 14.38 show the existence of high-order aberrations that are controlled nicely by the lens designer. Some tangential coma can be seen starting at 16 and sagittal oblique spherical aberration is marginally acceptable at 23 . Modest vignetting is employed at 23 to control the strong negative tangential coma.

M

Z

Figure 14.37 Longitudinal spherical aberration for unity magnification of the triplet copying lens. F light is short dashed curve, d light is solid curve, and e light is long dashed curve. The ordinate is in lens units with f ¼ 100.

16° 23°

Figure 14.38 Ray fans at 1X for triplet copying lens with f ¼ 100 and f/6.7. Ordinate is 0.2 lens units. F light is solid curve, d light is short dashed curve, and e light is long dashed curve.