Читать книгу Optical Engineering Science - Stephen Rolt - Страница 91

As introduced previously, the achromatic doublet is an extremely important building block in a transmissive (non-mirror) optical design. The function of an achromatic doublet is illustrated in Figure 4.25.

Figure 4.25 The achromatic doublet.

The first element, often (on account of its shape) referred to as the ‘crown element’, is a high power positive lens with low dispersion. The second element is a low power negative lens with high dispersion. The focal lengths of the two elements are f₁ and f₂ respectively and their Abbe numbers V₁ and V₂. Since the intention is that the dispersions of the two elements should entirely cancel, this condition constrains the relative power of the two elements. Individually, the dispersion as measured by the difference in optical power between the red and blue wavelengths is proportional to the reciprocal of the focal power and the Abbe number for each element. Therefore:

(4.51)

From Eq. (4.51), it is clear that the ratio of the two focal lengths should be minus the inverse of the ratio of their respective Abbe numbers. In other words, the ratio of their powers should be minus the ratio of their Abbe numbers. The power of the system comprising the two lenses is, in the thin lens approximation, simply equal to the sum of their individual powers. Therefore, it is possible to calculate these individual focal lengths, f₁ and f₂, in terms of the desired system focal length of f:

Thus, the two focal lengths are simply given by:

(4.52)

In the thin lens approximation, therefore, light will be focused at the same point for the red and blue wavelengths. Consequentially, in this approximation, this system will be free from both longitudinal and transverse chromatic aberration. The simplicity of this approach may be illustrated in a straightforward worked example.