5.9 Chromatic Aberration at Finite Aperture

Figure 5.17 The emerging wavefronts from an achromat.

together and cross the axis at the same point. Under these circumstances

XX

ðDnÞC ¼ ðDnÞF

along the marginal ray. However, since all points on a wavefront have the same

P

value of (Dn), it is clear that in an achromat

X X X

ðD dÞnC ¼ ðD dÞnF or ðD dÞðnF nC Þ¼ 0 (5-11)

This is Conrady’s condition for the best possible state of achromatism in a lens that suffers from other residuals of aberration. The presence of spherochromatism, for example, causes the two emerging wavefronts in C and F light to separate between the axis and margin of the aperture, while the presence of spherical aberration causes the wavefronts to assume a noncircular shape.

In stating this condition, we are tacitly assuming that the values of D within all the lens elements are equal for C and F light. This is certainly not true, but we shall make only a very small error if we trace the marginal ray in brightest light, which is usually d or e for C – F achromatism, and calculate the distances D along that ray. The argument breaks down if there is a long air space between unachromatized or only partially achromatized separated components, but in most cases it is surprisingly accurate.

The D – d relation would be impossibly difficult to use if we had to calculate every D value from the original object right up to the emerging wavefront, but the method is saved by the fact that the dispersion Dn ¼ nF – nC of air is zero. For this reason we must calculate D – d only for those sections of the marginal ray that lie in glass. The length D is found by the usual relation

D ¼ ðd þ Z2 X1Þ=cosU10

where Z ¼ r[1 – cos(I U)] as explained in Section 2.3. The choice of dispersion values depends on the region of the spectrum in which achromatism is desired. For ordinary visual achromatism, we trace the ray in d or e light and use Dn ¼ nF – nC; for photographic achromatism, we may prefer to trace the marginal ray in F light and use Dn ¼ ng – nD for the dispersion. The process for interpolating dispersions suggested in Chapter 1 is of value here; however, all

modern lens design programs include data tables for optical materials and the appropriate interpolating dispersion equation for each.

5.9.2Achromatization by Adjusting the Last Radius of the Lens

To achromatize a lens then, we must make the sum P (D – d) Dn equal to zero by some means or other. Commonly we calculate that value of the last radius of the lens that will accomplish this. Alternatively we may design the lens using any suitable refractive indices, and then at the end search the glass catalog for glass types with dispersion values that will make the (D – d) Dn sum zero. To use the first method, suppose that the value of the D – d sum for all the lens ele-

P

ments prior to the last element is 0; then for the last element we must have

XX

ðD dÞDn ¼

0

We now calculate the Z and Y at the next-to-last surface, and knowing the desired value of D in the last element to achieve achromatism, we calculate

Z2 ¼ D cos U 01 þ Z1 d and Y2 ¼ Y1 D sin U 01

(Here the indices 1 and 2 refer to the first and second surfaces of the last element.) The radius of curvature of the last surface is given by

r ¼ ðZ2 þ Y2Þ

2Z

and the problem is solved. As a check on our work, we may wish to trace zonal rays in F and C light through the whole lens; if everything is correct, these rays should cross the axis at the same point in the image space.

5.9.3 Tolerance for the D – d Sum

Conrady17 suggests that in a visual system the tolerance for the D – d sum is about half a wavelength. However, there is no point in achieving perfect achromatism for the 0.7 zonal rays, which the D – d method does, if there is considerable spherochromatism in the lens since this will swamp the excellent color correction. Therefore we have found that a more reasonable tolerance is about 1% of the contribution of either the crown or the flint element in the lens. If these contributions are small, it indicates that the spherochromatism will be small and a tight tolerance for the D – d sum is sensible.

Example

As an example in the use of the D – d method, we return to the cemented doublet lens used as a ray-tracing example in Section 2.5, and compute the (D – d) Dn sum along the traced marginal ray (Table 5.7). It will be seen that there is a small residual of the sum, amounting to –0.0000578, which is about 1% of the separate contributions of the crown and flint elements. We must therefore regard this lens as noticeably undercorrected for chromatic aberration. That is the reason why the C and F curves in Figure 5.3a (see page 141) cross somewhat above the 0.7 zone of the aperture.

If we wish to achromatize this lens perfectly, we can solve for the last radius by the method described in Section 5.9.1. This tells us that a last radius of –16.6527 would make the D – d sum exactly zero. As this radius is decidedly different from the given radius of –16.2225, we see once again that it is necessary to change a lens drastically if we wish to affect the chromatic correction.

As an alternative method for achromatizing, we could calculate what value of Dn for either the crown or the flint glass would be required to eliminate the D – d sum. The numbers shown in Table 5.6 (see page 162) tell us that we could achromatize with the given crown if we had a flint with Dn ¼ 0.01941; this represents a V number of 33.43 instead of the given 33.80. Or we could retain the given flint and seek a crown with Dn ¼ 0.00792; this represents a V number of 65.26 instead of the given 64.54. In both cases the required change in V number is only slightly larger than the normal factory variation in successive glass melts, indicating that the small residual of chromatic aberration in this lens is really almost insignificant.

Table 5.7

Calculation of the (D – d) Dn Sum