Читать книгу Optical Engineering Science - Stephen Rolt - Страница 87

Hitherto, we have only considered the classical monochromatic aberrations. At this point, we must introduce the phenomenon of chromatic aberration where imperfections in the imaging of an optical system are produced by significant variation in optical properties with wavelength. All optical materials are dispersive to some degree. That is to say, their refractive indices vary with wavelength. As a consequence, all first order properties of an optical system, such as the location of the cardinal points, vary with wavelength. Most particularly, the paraxial focal position of an optical system with dispersive components will vary with wavelength, as will its effective focal length. Therefore, for a given axial position in image space, only one wavelength can be in focus at any one time.

Dispersion is a property of transmissive optical materials, i.e. glasses. On the other hand, mirrors show no chromatic variation and their incorporation is favoured in systems where chromatic variation is particularly unwelcome. Such a system, where the optical properties do not vary with wavelength, is said to be achromatic. As argued previously, a mirror behaves as an optical material with a refractive index of minus one, a value that is, of course, independent of wavelength. In general, the tendency in most optical materials is for the refractive index to decrease with increasing wavelength. This behaviour is known as normal dispersion. In certain very specific situations, for certain materials at particular wavelengths, the refractive index actually decreases with wavelength; this phenomenon is known as anomalous dispersion.

Although dispersion is an issue of concern covering all wavelengths of interest from the ultraviolet to the infrared, for obvious reasons, historically, there has been particular focus on this issue within the visible portion of the spectrum. Across the visible spectrum, for typical glass materials, the refractive index variation might amount to 0.7–2.5%. This variation in the dispersive properties of different materials is significant, as it affords a means to reduce the impact of chromatic aberration as will be seen shortly. Figure 4.20 shows a typical dispersive plot, for the glass material, SCHOTT BK7®.

Figure 4.20 Refractive index variation with wavelength for SCHOTT BK7 glass material.

Because of the historical importance of the visible spectrum, glass materials are typically characterised by their refractive properties across this portion of the spectrum. More specifically, glasses are catalogued in terms of their refractive indices at three wavelengths, nominally ‘blue’, ‘yellow’, and ‘red’. In practice, there are a number of different conventions for choosing these reference wavelengths, but the most commonly applied uses two hydrogen spectral lines – the ‘Balmer-beta’ line at 486.1 nm and the ‘Balmer-alpha’ line at 656.3, plus the sodium ‘D’ line at 589.3 nm. The refractive indices at these three standard wavelengths are symbolised as n_F, n_C, and n_D respectively. At this point, we introduce the Abbe number, V_D, which expresses a glass's dispersion by the ratio of its optical power to its dispersion:

(4.48)

The numerator in Eq. (4.48) represents the effective optical or focusing power at the ‘yellow’ wavelength, whereas the denominator describes the dispersion of the glass as the difference between the ‘blue’ and the ‘red’ indices. It is important to recognise that the higher the Abbe number, then the less dispersive the glass, and vice versa. Abbe numbers vary, typically between about 20 and 80. Broadly speaking, these numbers express the ratio of the glass's focusing power to its dispersion. Hence, for a material with an Abbe number of 20, the focal length of a lens made from this material will differ by approximately 5% (1/20) between 486.1 and 656.3 nm.