Eq. (4.55) into Eq. (4.54) gives the desired quadratic equation:
There are, of course, two sets of solutions to Eq. (4.56), with the following values:
Solution 1: s1 = −0.194; s2 = 1.823
Solution 2: s1 = 3.198; s2 = 2.929
There now remains the question as to which of these two solutions to select. Using Eq. (4.29) to calculate the individual radii of curvature from the lens shapes and focal length we get:
Solution 1: R1 = 121.25 mm; R2 = −81.78 mm; R3−81.29 mm; R4 = −281.88 mm
Solution 2: R1 = 23.26 mm; R2 = 44.43 mm; R3−58.91 mm; R4 = −119.68 mm
The radii R1 and R2 refer to the first and second surfaces of lens 1 and R3 and R4 to the first and second surfaces of lens 2. It is clear that the first solution contains less steeply curved surfaces and is likely to be the better solution, particularly for relatively large apertures. In the case of the second solution, whilst the solution to the third order equations eliminates third order spherical aberration and coma, higher order aberrations are likely to be enhanced.
The first solution to this problem comes under the generic label of the Fraunhofer doublet, whereas the second is referred to as a Gauss doublet. It should be noted that for the Fraunhofer solution, R2 and R3 are almost identical. This means that should we constrain the two surfaces to have the same curvature (in the case of a cemented doublet) and just optimise for spherical aberration, then the solution will be close to that of the ideal aplanatic lens. To do this, we would simply use Eq. 4.29, forcing R2 and R3 to be equal and to replace Eq. 4.55 constraining the total coma, providing an alternative relation between s1 and s2. However, the fact that the cemented doublet is close to fulfilling the zero spherical aberration and coma condition further illustrates the usefulness of this simple component.
The analysis presented applies only strictly in the thin lens approximation. In practice, optimisation of a doublet such as presented in the previous example would be accomplished with the aid of ray tracing software. However, the insights gained by this exercise are particularly important. For instance, in carrying out a computer-based optimisation, it is critically important to understand that two solutions exist. Furthermore, in setting up a computer-based optimisation, an exercise, such as this, provides a useful ‘starting point’.
4.7.6 Secondary Colour
The previous analysis of the achromatic doublet provides a means of ameliorating the impact of glass dispersion and to provide correction at two wavelengths. In the case of the standard visible achromat, correction is provided at the F and C wavelengths, the two hydrogen lines at 486.1 and 656.3 nm. Unfortunately, however, this does not guarantee correction at other, intermediate wavelengths. If one views dispersion of optical materials as a ‘small signal’ problem, and that any difference in refractive index is small across the region of interest, then correction of the chromatic focal shift with a doublet may be regarded as a ‘linear process’. That is to say we might approximate the dispersion of an optical material by some pseudo-linear function of wavelength, ignoring higher order terms. However, by ignoring these higher order terms, some residual chromatic aberration remains. This effect is referred to as secondary colour. The effect is illustrated schematically in Figure 4.26 which shows the shift in focus as a function of wavelength.
Figure 4.26 Secondary colour.
Figure 4.26 clearly shows the effect as a quadratic dependence in focal shift with wavelength, with the ‘red’ and ‘blue’ wavelengths in focus, but the central wavelength with significant defocus. In line with the notion that we are seeking to quantify a quadratic effect, we can define the partial dispersion coefficient, P, as:
(4.57)
If we measure the impact of secondary colour as the difference in focal length, Δf, between the ‘blue’ and ‘red’ and the ‘yellow’ focal lengths for an achromatic doublet corrected in the conventional way we get:
where f is the lens focal length.
The secondary colour is thus proportional to the difference between the two partial dispersions. For simplicity, we have chosen to represent the partial dispersion in terms of the same set of wavelengths as used in the Abbe number. However, whilst the same central (nd) wavelength might be used, some wavelength other than the nF, hydrogen line might be chosen for the partial dispersion. Nevertheless, this does not alter the logic presented in Eq. (4.58). Correcting secondary colour is thus less straightforward when compared to the correction of primary colour. Unfortunately, in practice, there is a tendency for the partial dispersion to follow a linear relationship with the Abbe number, as illustrated in the partial dispersion diagram shown in Figure 4.27, illustrating the performance of a range of glasses.
Thus, in the case of the achromatic doublet, judicious choice of glass pairs can minimise secondary colour, but without eliminating it. In principle, secondary colour can be entirely corrected in a triplet system employing lenses of different materials. More formally, if we describe the three lenses as having focal powers of P1, P2, and P3, with the Abbe numbers represented as V1, V2, and V3 and the partial dispersions as, α1, α2, α3, then the lens powers may be uniquely determined from the following set of equations:
(4.59b)
As
