Optical Engineering Science. Stephen Rolt
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4.22.
Figure 4.21 Longitudinal chromatic aberration.
Figure 4.22 Transverse chromatic aberration.
The effect illustrated is known as transverse chromatic aberration or lateral colour. Whilst no distinct blurring is produced by this effect, the fact that different wavelengths have different focal lengths inevitably means that system magnification varies with wavelength. As a result, the final image size or height of a common object depends upon the wavelength. This produces distinct coloured fringing around an object and the size of the effect is proportional to the field angle, but independent of aperture size.
Hitherto, we have cast the effects of chromatic aberration in terms of transverse aberration. However, to understand the effect on the same basis as the Gauss-Seidel aberrations, it is useful to express chromatic aberration in terms of the OPD. When applied to a single lens, longitudinal chromatic aberration simply produces defocus that is equal to the focal length divided by the Abbe number. Therefore, the longitudinal chromatic aberration is given by:
f is the focal length of the lens and r the pupil position.
Figure 4.23 Huygens eyepiece.
Similarly, the transverse chromatic aberration can be expressed as an OPD:
Examining Eqs. (4.49a) and (4.49b) reveals that the ratio of transverse to longitudinal aberration is given by the ratio of the field angle to the numerical aperture. In practice, for optical elements, such as microscope and telescope objectives, the field angle is very much smaller than the numerical aperture and thus longitudinal chromatic aberration may be expected to predominate. For eyepieces, the opposite is often the case, so the imperative here is to correct lateral chromatic aberration.
Worked Example 4.5 Lateral Chromatic Aberration and the Huygens Eyepiece
A practical example of the correction of lateral chromatic aberration is in the Huygens eyepiece. This very simple, early, eyepiece uses two plano-convex lenses separated by a distance equivalent to half the sum of their focal lengths. This is illustrated in Figure 4.23.
Since we are determining the impact of lateral chromatic aberration, we are only interested in the effective focal length of the system comprising the two lenses. Using simple matrix analysis as described in Chapter 1, the system focal length is given by:
If we assume that both lenses are made of the same material, then their focal power will change as a function of wavelength by a common proportion, α. In that case, the system focal power at the new wavelength would be given by:
For small values of α, we can ignore terms of second order in α, so the change in system power may be approximated by:
The change in system power should be zero and this condition unambiguously sets the lens separation, d, for no lateral chromatic aberration:
(4.50)
If this condition is fulfilled, then the Huygens eyepiece will have no transverse chromatic aberration. However, it must be emphasised that this condition does not provide immunity from longitudinal chromatic aberration.
Figure 4.24 Abbe diagram.
4.7.3 The Abbe Diagram for Glass Materials
For visible applications, the Abbe number for a glass is of equal practical importance as the refractive index itself. The Abbe diagram is a simple graphic tool that captures the basic refractive properties of a wide range of optical glasses. It comprises a simple 2D map with the horizontal axis corresponding to the Abbe number and the vertical axis to the glass index. A representative diagram is shown in Figure 4.24.
By referring to this diagram, the optical designer can make appropriate choices for specific applications in the visible. In particular, it helps select combinations of glasses leading to a substantially achromatic design. One special and key application is the achromatic doublet. This lens is composed of two elements, one positive and one negative. The positive lens is a high power (short focal length) element with low dispersion and the negative lens is a low power element with high dispersion. Materials are chosen in such a way that the net dispersion of the two elements cancel, but the powers do not. This will be considered in more detail in the next section.
The different zones highlighted in the Abbe diagram replicated in Figure 4.24 refer to the elemental composition of the glass. For example, ‘Ba’ refers to the presence of barium and ‘La’ to the presence of lanthanum. Originally, many of the dense, high index glasses used to contain lead, but these are being phased out due to environmental concerns. The Abbe diagram reveals a distinct geometrical profile with a tendency for high dispersion to correlate strongly with refractive index. In fact, it is the presence of absorption features within the glass (at very much shorter wavelengths) that give rise to the phenomenon of refraction and these features also contribute to dispersion.
4.7.4 The Achromatic Doublet
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.