Optical Engineering Science. Stephen Rolt

case, the height of any ray within the collimated beam in image space is directly proportional to the numerical aperture associated with the input ray.

Figure 4.19 illustrates the application of the Abbe sine condition for a specific example. As highlighted previously, the sine condition effectively seeks out the aplanatic condition in an optical system. In this example, a meniscus lens is to be designed to fulfil the aplanatic condition. However, its conjugate parameter is adjusted around the ideal value and the spherical aberration and coma plotted as a function of the conjugate parameter. In addition, the departure from the Abbe sine condition is also plotted in the same way. All data is derived from detailed ray tracing and values thus derived are presented as relative values to fit reasonably into the graphical presentation. It is clear that elimination of spherical aberration and coma corresponds closely to the fulfilment of the Abbe sine condition.

The form of the Abbe sine condition set out in Eq. (4.46) is interesting. It may be compared directly to the Helmholtz equation which has a similar form. However, instead of a relationship based on the sine of the angle, the Helmholtz equation is defined by a relationship based on the tangent of the angle:

      It is quite apparent that the two equations present something of a contradiction. The Helmholtz equation sets the condition for perfect imaging in an ideal system for all pairs of conjugates. However, the Abbe sine condition relates to aberration free imaging for a specific conjugate pair. This presents us with an important conclusion. It is clear that aberration free imaging for a specific conjugate (Abbe) fundamentally denies the possibility for perfect imaging across all conjugates (Helmholtz). Therefore, an optical system can only be designed to deliver aberration free imaging for one specific conjugate pair.

Figure 4.19 Fulfilment of Abbe sine condition for aplanatic meniscus lens.

4.7 Chromatic Aberration

      4.7.1 Chromatic Aberration and Optical Materials

      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.

      4.7.2 Impact of Chromatic Aberration

The most obvious effect of chromatic aberration is that light is broad to a different focus for different wavelengths. This effect is known as longitudinal chromatic aberration and is illustrated in Figure 4.21.

As can be seen from Figure 4.21, light at the shorter, ‘blue’ wavelengths are focused closer to the lens, leading to an axial (longitudinal) shift in the paraxial focus for the different wavelengths. In summary, longitudinal chromatic aberration is associated with a shift in the paraxial focal position as a function of wavelength. Thus the effect of longitudinal chromatic aberration is to produce a blur spot or transverse aberration whose magnitude is directly proportional to the aperture size, but is independent of field angle. However, there are situations where, to all intents and purposes, all wavelengths share the same paraxial focal position, but the principal points are not co-located. That is to say, whilst all wavelengths are focused at a common point, the effective focal length corresponding to each wavelength is not identical. This scenario is illustrated in Figure