for trim diameter ¼ 2.2, f 0 ¼ 10.3663, l0 ¼ 10.1227, LA0 ¼ 0.00010, LZA ¼0.00176, and OSC ¼ 0.00060. This would make an excellent objective. The undercorrected zonal residual is very small, being only 40% of that found for the common achromat in Section 7.2.1, Table 7.3.

DESIGNER NOTE

The comparative smallness of this zone is due to the use of higher-index glasses. It is found that with a given flint, the zonal residual is large for low-index crowns, drops to a minimum for some medium-index crowns, and then rises again for high-index crowns. There are clearly two opposing tendencies. Raising the crown index weakens the front radius, but it also lowers the index difference across the cemented interface thereby requiring a stronger curvature at the interface.

Somewhat in defiance of this well-known behavior, Ditteon and Feng developed an analytical method for designing a cemented aplanatic doublet where they discovered a pair of glasses, namely FK-54 and BaSF-52, that corrected both coma and secondary spectrum at the same time.1 The FK-54 (437907) is a very-low index crown while the BaSF-52 (702410) is a medium index flint. The approach was to lower the spherical aberration parabola (see Figure 7.2) by having a large difference in Abbe values to have a single zero spherical aberration solution rather than two. Coma is essentially zero as we learned in Chapter 7. The secondary spectrum is corrected by having the partial dispersions of the glasses be essentially equal.

10.4 A TRIPLE CEMENTED APLANAT

Another way to obtain the additional degree of freedom necessary for OSC correction is to divide the flint component of a cemented doublet into two and place one part in front of and the other part behind the crown component to make a cemented triplet. Of course, alternatively the crown component could be divided in this way, but it is generally better to divide the flint.

278 Design of Aplanatic Objectives

Table 10.2

Thin Lens Formulas for Triple Cemented Aplanat

Conrady2 has given a very complete study of this system on the basis of the spherical and coma G sums. To apply such an analysis, for each element we need net curvature c, bending parameter c1, and reciprocal object distance v. In this triplet lens we have two absolute degrees of freedom, which we may call x and y: the amount of flint power in the front element and a bending of the whole lens. We therefore define x ¼ c1 – c2 ¼ ca, and y ¼ c2.

The total powers of crown and flint are found by the ordinary (ca, cb) formulas (Eq. 5-4); they will be referred to here as Cr and Fl. Hence for the three thinlens elements we have what is shown in Table 10.2. Here na ¼ nc is the flint index and nb the crown index. To draw a section of the lens to determine suitable thicknesses, we note that the thin-lens value of c4 is x þ y – (Cr þ Fl).

In performing the G sum analysis, we find that the spherical aberration expression is quadratic, while the coma expression is linear. Hence there will be two solutions to the problem. To reduce the zonal residual and to have as many lenses as possible on a block, we choose that solution in which the strongest surface has the longer radius. (A “block” refers to the tool to which the lens blanks are affixed for grinding and polishing. The working diameter of a short radius tool is less than that for a longer radius tool, which means that, for a given lens diameter, more elements can be mounted on the longer radius block.3)

As an example, we will design a low-power cemented triplet microscope objective, with magnification 5 and tube length 160 mm. This represents a focal length of 26.67 mm. The numerical aperture, sin U40 , is to be 0.125; therefore the entering ray slope is sinU1 ¼ 0.025. We will use the following common glass types:

(a)Flint: F 3, ne ¼ 1.61685, Dn ¼ 0.01659, Ve ¼ 37.18

(b)Crown: BaK 2, ne ¼ 1.54211, Dn ¼ 0.00905, Ve ¼ 59.90

with Vb – Va ¼ 22.72. The (ca, cb) formulas give for the total crown and flint powers

Cr ¼ 0:1824; Fl ¼ 0:0995

This graph indicates the required very small changes in x and y from the starting setup to make both aberrations zero, namely, Dx ¼ 0.0017 and Dy ¼ þ0.0014. These changes give the following solution to the problem:

with L1 ¼ –160 mm, sin U1 ¼ 0.025, LA0 ¼ 0.00042 mm, LZA ¼ –0.04688 mm, OSC ¼ –0.00002, l0 ¼ 30.145 mm, 1/m ¼ –4.927, and NA ¼ 0.123.

Since the numerical aperture is slightly below our desired value of 0.125, we may shorten the focal length in the ratio 0.123/0.125 ¼ 0.984 by strengthening all the radii in this proportion. The small zonal residual is far less than the Rayleigh tolerance of 0.21 mm and is negligible.

10.5AN APLANAT WITH A BURIED ACHROMATIZING SURFACE

The idea of a “buried” achromatizing surface was suggested by Paul Rudolph in the late 1890s.4 Such a surface has glass of the same refractive index on both sides, but because the dispersive powers are different, it can be used to control the chromatic aberration of the lens. Thus achromatism can be left to the end and the available degrees of freedom can be used for the correction of other aberrations.

Some possible matched pairs of glasses from the 2009 Schott catalog are shown in Table 10.3.

Table 10.3

Matched Glass Pairs Suitable for Buried Achromatizing Surface

DESIGNER NOTE

An exact index match is unnecessary; particularly since the actual index of standard delivery fine annealed glass may depart from the catalog values by as much as0.0005% and the Abbe number by 0.8%. Within a given lot of fine annealed glass, the refractive index variation is 0.0001 and about twice that within a lot of pressings. The lens designer should be attentive to the impact such variations may have on the lens performance. Also, for critical applications it is wise to use the actual melt data for the glass purchased and make final adjustments to the curvatures and thicknesses of the lens before making the lens elements. (Glass is typically delivered with a test report according to ISO 10474.)

The measurements are performed with an accuracy of 3 10 5 for refractive index and 2 10 5 for dispersion. Data are provided to five decimal places. The reported values are the median value of the samples taken from the lot. Consequently, the actual value of a part made from the lot may vary by the aforementioned refractive index tolerance. Most lens design programs include tolerancing analysis and some include tolerance sensitivity mitigation during lens optimization.

As an example of the use of a buried surface, we will design a triple aplanat in which the third surface will be buried. The remaining three radii will be used for spherical aberration and coma, the last radius being in all cases solved for the required focal length. We will maintain a focal length of 10.0 and an aperture of Y ¼ 1.0 (f/5), allowing sufficient thicknesses for a trim diameter of 2.2 for the crown and for the insertion of the buried surface in the flint. For the crown lens we will use K 5 glass (nD ¼ 1.5224, Dn ¼ 0.00876, and VD ¼ 59.63). For the flint we will use the glasses in selection (3) in Table 10.3, performing the ray tracing with the average index of 1.6222.

A convenient starting system is

0.16

0.42 1.5224

0.26

0.35 1.6222

(solve for f 0)

The last curvature comes out to be 0.069605, giving LA0 ¼ 0.01013 and OSC ¼ 0. We now make a trial change in c1 by 0.01, giving LA0 ¼ 0.02059 and OSC ¼ 0.00096. Returning to the original setup and changing c2 by 0.01 gives LA0 ¼ 0.00241 and OSC ¼ 0.00016. These values are plotted on the double graph of Figure 10.8, and we conclude that we should make a further