5.5 PARAXIAL SECONDARY SPECTRUM

We have so far regarded an achromatic lens as one in which the C and F foci are coincident. However, as we have seen, in such a case the d (yellow) focus falls short and the g (blue) focus of the same zone falls long. To determine the magnitude of the paraxial secondary spectrum of a lens in which the paraxial C and F foci coincide, we write the chromatic aberration contribution of a single thin element, for two wavelengths l and F, as

The quantity in parentheses is another intrinsic property of the glass, known as the partial dispersion ratio from l to F. It is generally written PlF. Hence for any succession of thin elements

For the case of a thin achromatic doublet, y is the same for both elements, and Eq. (5-5) shows that fa0Va ¼ –f b0 Vb ¼ F 0(Va – Vb); hence

For any particular pair of wavelengths, say F and g, we can plot the available types of glass on a graph connecting PgF with V, as in Figure 5.7. All the common types of glass lie on a straight line that rises slightly for the very dense flints. Below this line come the “short” glasses, which exhibit an unusually short blue end to the spectrum; these are mostly lanthanum crowns and so-called short flints (KzF and KzFS types). Above the line are a few “long” crowns with an unusually stretched blue spectrum (this region also contains some plastics and crystals such as fluorite). The titanium flints also fall above the line, as can be seen.

If we join the points belonging to the two glasses used to make an achromatic doublet, the slope of the line is given by

tan c ¼ Pa Pb

Va Vb

and clearly the secondary spectrum is given by F 0 tan c. The fact that most of the ordinary glasses lie on a straight line indicates that the secondary spectrum will be about the same for any reasonable selection of glass types. For example,

For a thin system with a very distant object, which is achromatized and also corrected for secondary spectrum, we have the three relationships

X

ðVc DnÞ ¼ F ðpowerÞ

X

ðc DnÞ ¼ 0 ðachromatismÞ

X

ðPc DnÞ ¼ 0 ðsecondary spectrumÞ

For a thin three-lens apochromat, these equations can be extended to

Vaðca; DnaÞ þ Vbðcb DnbÞ þ Vcðcc DncÞ ¼ F

ðca DnaÞ þ ðcb DnbÞ þ ðcc DncÞ ¼ 0

Paðca DnaÞ þ Pbðcb DnbÞ þ Pcðcc DncÞ ¼ 0

These can be solved for the three curvatures as follows:

Note the cyclic order of the terms, and that the coefficient in front of the parentheses is the same in each case.

The meaning of E is as follows: If we plot the three chosen glasses on the P – V graph shown in Figure 5.10 and then join the three points to form a triangle, E is the vertical distance of the middle glass from the line joining the two outer glasses, E being considered negative if the middle glass falls below the line. Algebraically E is computed by

Since E appears in the denominator of all three c expressions, it is clear that the lenses will become infinitely strong if all three glasses fall on a straight line, and conversely, all the elements will become as weak as possible if we select glass types having a large E value. The most usual choice is some kind of crown for

VgC

Figure 5.10 P – V graph of the glasses used in a three-lens apochromat.

lens a, a very dense flint for lens c, and a short flint or lanthanum crown for the intermediate lens b. Once the three curvatures have been calculated, the actual lenses can be assembled in any order.

As an example in the use of these formulas, we will select three glasses forming a wide triangle on the graph of Figure 5.10, namely, Schott’s N-PK51, N-KZFS4, and N-SF15. Since the calculated curvatures depend on the differences between the P numbers, it is necessary to know these to many decimal places, requiring a knowledge of the individual refractive indices to about seven decimals, which is beyond the capability of any measurement procedure. We therefore use the interpolation formula given in the current Schott catalog to calculate refractive indices to the required precision. Failure to do this will result in such scattered points that it is impossible to plot a smooth chromatic spectrum focus curve for the completed design.

To unite the C, e, and g lines at a common focus, we use Table 5.4. In this

case, VgC ¼ ne 1 , analogous to the commonly used Abbe number formula for

ng nC

n n

C, d, and F lines, and Pge ¼ g e . Using these somewhat artificially accurate

ne nC

numbers, we calculate the value of E as –0.009744, and assume that the focal length is F0 ¼ 10 mm, which gives ca ¼ 0.5461855, cb ¼ –0.3830219, and cc ¼ 0.0661602. As illustrated in Figure 5.10, a negative value of E means that the glass having the intermediate dispersion lies below the line connecting the other two glasses.

Depending on the choice of glasses, the chromatic aberration curve for an apochromat can take on other shapes such as shown in Figure 5.11. In this case, both the foci for the d and h lines are slightly back from the e line focus. Note that if the image plane is shifted slightly away from the lens, a common focus for four wavelengths is again obtained and the residual chromatic aberration is reduced.

This system should be called a “superachromat,” since the three glasses satisfy Herzberger’s condition10 for the union of four wavelengths at a common focus. Failure to meet this condition generally ends up with three united wavelengths in the visible spectrum with the fourth wavelength falling far out into the infrared. In Section 7.4, the design of the apochromat will be completed after inserting suitable thicknesses and choosing such a shape that the spherical aberration is also corrected.

DESIGNER NOTE

As has been explained, longitudinal or axial chromatic aberration is a first-order aberration. When using a computer program to aid in the design of a lens, the designer can effectively use the axial color operand to define an optimization defect term that is the axial image distance between two selected wavelengths. For an achromat, the designer might select C and F lines to unite their foci. In the case of an apochromat, defect terms might be formed that measure the axial image distance between, say, g and e, g, and C, and e and C to unite foci for the g, e, and C lines. Always remember the great importance of proper selection of the glasses as the bendings of the lenses have essentially no impact upon the first-order chromatic aberration.

5.7THE SEPARATED THIN-LENS ACHROMAT (DIALYTE)

In the early 1800s, the fabrication of a large achromatic doublet for astronomical objectives was problematic due to the difficulty of obtaining large disks of flint glass. An elegantly simple solution was introduced at that time to mitigate this difficulty, which no longer exists today. This solution was known as “dialyte objectives,” which comprises a positive crown lens with a smaller negative flint lens separated by some modest distance. This is in effect a telephoto lens, since the track length11 of the lens is significantly smaller than the effective focal length. When used as a telescope objective, the dialyte lens has the advantage that both sides of the objective lens are exposed to the atmosphere, thereby allowing quicker and more uniform thermal tracking to maintain sharp imagery

(a)(b)

ya d yb

l ′

f ′ a

Figure 5.12 The dialyte lens with positive lens (a) and negative lens (b).

and spatial stability of the image. Gauss recognized that by choosing the proper separation between the two lenses, it is possible to correct the spherical aberration for two different colors.12

In Figure 5.12 we show the dialyte lens having the two components of a thin achromatic doublet separated by a finite distance d, where we find that the flint element particularly must be considerably strengthened. It is convenient to express d as a fraction k of the focal length of the crown lens, that is, k ¼ d=fa0. Since the chromatic aberration contributions of the two elements in an achromat must add up to zero, we see that

ing these gives

fbVb¼ faVað1 kÞ2

Since the system must have a specified focal length F 0, we have

Combining the last two relationships gives the focal lengths of the two components as