Читать книгу Optical Engineering Science - Stephen Rolt - Страница 95

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:

(4.58)

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 (n_d) wavelength might be used, some wavelength other than the n_F, 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:

(4.59a)

(4.59b)

(4.59c)

As indicated previously, Figure 4.27 exemplifies the close link between primary and secondary dispersion, with a linear trend observed linking the partial dispersion and the Abbe number for most glasses. It is easy to demonstrate by presenting Eqs. (4.59a)–(4.59c) in matrix form that, if a wholly linear relationship exists between partial dispersion and Abbe number, then the matrix determinant will be zero. In this instance, a triplet solution is therefore impossible. Furthermore, the same analysis suggests that for a set of glasses lying close to a straight line on the partial dispersion plot will necessitate the deployment of lenses with very high countervailing powers. It is clear, therefore, that an optimum triplet design is afforded by selection of glasses that depart as far as possible from a straight-line plot on the partial dispersion diagram. In this context, the isolated group of glasses that appear in Figure 4.27, the fluorite glasses, are especially useful in correcting for secondary colour. These glasses lie particularly far from the general trend line for the ‘main series’ of glasses. Lenses which are corrected for both primary and secondary colour are referred to as apochromatic lenses. These lenses invariably incorporate fluorite glasses.

Figure 4.27 Plot of partial dispersion against Abbe number.