Optical Engineering Science. Stephen Rolt

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.

      4.7.7 Spherochromatism

      In the previous analysis we learned that the basic design of simple doublet lenses allowed for the correction of both chromatic aberration and spherical aberration. Furthermore, this flexibility for correction could be extended to coma for an air spaced lens. However, since the refractive index of the two glasses in a doublet lens varies with wavelength, then inevitably, so does the spherical aberration. As such, spherical aberration can only be corrected at one wavelength, e.g. at the ‘D’ wavelength. This means that there will be some uncorrected spherical aberration at the extremes of the spectrum. This effect is known as spherochromatism. It is generally less significant in magnitude when compared with secondary colour.

4.8 Hierarchy of Aberrations

For some specific applications, such as telescope and microscope objective lenses, the field angles tend to be very much smaller than the angles associated with the system numerical aperture. In these instances, the off-axis aberrations, such as coma, are much less significant than the on-axis aberrations. Therefore, as far as the Gauss-Seidel aberrations are concerned, there exists a hierarchy of aberrations that can be placed in order of their significance or importance:

      1 Spherical Aberration

      2 Coma

      3 Astigmatism and Field Curvature

      4 Distortion

      That is to say, it is of the greatest importance to correct spherical aberration and then coma, followed by astigmatism, field curvature, and distortion. This emphasises the significance and use of aplanatic elements in optical design.

      Of course, for certain optical systems, this logic is not applicable. For instance, in both camera lenses and in eyepieces, the field angles are very substantial and comparable to the angles associated with the numerical aperture. Indeed, in systems of this type, greater emphasis is placed upon the correction of astigmatism, field curvature, and distortion than in other systems.

With these comments in mind, it would be useful to summarise all the aberrations covered in this chapter and to classify them by virtue of their pupil and field angle dependence. Table 4.1 sets out the wavefront error dependence upon pupil and field angle for each of the aberrations.

It would be instructive, at this point, to take the example of the 200 mm doublet and to plot the wavefront aberrations attributable to some of the aberrations listed in Table 4.1 against numerical aperture. Spherochromatism is expressed as the difference in spherical aberration wavefront error between the n_F and n_C wavelengths (486.1 and 656.3 nm). Secondary colour is expressed as the wavefront error attributable to the difference in defocus between the n_F and n_D wavelengths (486.1 and 589.3 nm). A plot is shown in Figure 4.28.

It is clear that for the simple achromat under consideration, at least for modest lens apertures, the impact of secondary colour predominates. If a wavefront error of about 50 nm is consistent with ‘high quality’ imaging, then secondary colour has a significant impact for numerical apertures in excess of 0.05 or f#10. With numerical apertures in excess of 0.2 (f#2.5), higher order spherical aberration starts to make a significant contribution. On the other hand the effect of spherochromatism is more modest throughout. In this context, the impact of spherochromatism would only be a significant issue if secondary colour were first corrected.

Table 4.1 Pupil and field dependence of principal aberrations.

Aberration	Pupil exponent	Field angle exponent
Defocus	2	0
Spherical aberration	4	0
Coma	3	1
Astigmatism	2	2
Field curvature	2	2
Distortion	1	3
Lateral colour	1	1
Longitudinal colour	2	0
Secondary colour	2	0
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