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It is worthwhile, at this juncture, to examine the four expressions in Eqs. (4.5a)–(4.5d) in some detail and, in particular, those for spherical aberration and coma. Before examining these expressions further, it is worthwhile to cast them in the form outlined in Chapter 3:

(4.6a)

(4.6b)

Figure 4.2 Aplanatic points for refraction at single spherical surface.

There is a clear pattern in these expressions in that both spherical aberration and coma can be reduced to zero for specific values of the object distance, u. Examining Eqs. (4.6a) and (4.6b), it is evident that this condition is met where u = −R. That is to say, where the object is located at the centre of the spherical surface. However, this is a somewhat trivial condition where rays are undeviated by the surface and where the surface would not provide any useful additional refractive power to the system. Most significantly, another condition does exist for u = −(n + 1)R. Here, for this non-trivial case, both third order spherical aberration and coma are absent. This is the so-called aplanatic condition and the corresponding conjugate points are referred to as aplanatic points (Figure 4.2). From Eq. (4.3) we can derive the image distance, v, as (n + 1)R/n. That is to say, the object is virtual and the image is real if R is positive and vice-versa if R is negative.

To be a little more rigorous, we might suppose that refractive index in object space is n₁ and that in image space is n₂. The location of the aplanatic points is then given by:

(4.7)

Fulfilment of the aplanatic condition is an important building block in the design of many optical systems and so is of great practical significance. As pointed out in the introduction, for those systems where the field angles are substantially less than the marginal ray angles, such as microscopes and telescopes, the elimination of spherical aberration and coma is of primary importance. Most significantly, not only does the aplanatic condition eliminate third order spherical aberration, but it also provides theoretically perfect imaging for on axis rays.