alt="equation"/>
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.
Worked Example 4.1 Microscope Objective
The ‘front end’ of many high power microscope objectives exploits the principle of single surface aplanatic points through the use of a hyperhemisphere co-located with the object. The hyperhemisphere consists of a sphere that has been truncated at one of the aplanatic points which also coincides with the object location, as illustrated in Figure 4.3.
Using the hyperhemisphere, we wish to create a ×20 microscope objective for a standard optical tube length of 200 mm. In this example, it is assumed that two thirds of the optical power resides in the hyperhemisphere itself; other components collimate the beam. In other words:
Figure 4.3 Hyperhemisphere objective.
The refractive index of the hyperhemisphere is 1.6. What is the radius, R, of the hyperhemisphere and what is its thickness?
For a tube length of 200 mm, a ×20 magnification corresponds to an objective focal length of 10 mm. As two thirds of the power resides in the hyperhemisphere, then the focal length of the hyperhemisphere must be 15 mm. Inspecting Figure 4.2, it is clear that the thickness of the hyperhemisphere is −R × (n + 1)/n, or −1.625 × R. To calculate the value of R, we set up a matrix for the system. The first matrix corresponds to refraction at the planar air/glass boundary, the second to translation to the spherical surface and the final matrix to the refraction at that surface. On this occasion, translation to the original reference is not included.
From the above matrix, the focal length is −R/0.6 and hence R = −9.0 mm. The thickness, t, we know is −1.625 × R and is 14.625. In this sign convention, R is negative, as the sense of its sag is opposite to the direction of travel from object to image space.
The (virtual) image is at (n + 1) × R from the sphere vertex or 2.6 × 9 = 23.4 mm.
In summary:
4.2.2 Astigmatism and Field Curvature
Unlike spherical aberration and coma, there is less scope for correction of astigmatism and field curvature. In Eqs. (4.5c) and (4.5d), astigmatism is corrected at the aplanatic point and field curvature at the radial points. However, the convention used in Eq. (4.5c) to describe astigmatic correction corresponds to zero sagittal ray defocus. On the other hand, using the alternative convention set out in Chapter 3 we have:
From Eq. (4.8a), it is evident that at the aplanatic condition where u = −(n + 1)R, the astigmatism vanishes, as does the spherical aberration and coma. It is interesting to see what might happen to the field curvature where this condition is fulfilled:
This is related to the Petzval field curvature, which, by definition, is the field curvature that arises when the astigmatism in the system is zero. Relating this to Eq. 4.8b, then the field curvature may be expressed as:
(4.10)
