at this point to quantify the increase in system focal power provided by an aplanatic meniscus lens. Effectively, as illustrated in Figure 4.14, it increases the system numerical aperture in (minus) the ratio of the object and image distance. For the positive meniscus lens in Figure 4.14, the conjugate parameter is negative and equal to −(n + 1)/(n − 1). From Eq. (4.27) the ratio of the object and image distances is given by:
As previously set out, the increase in numerical aperture of an aplanatic meniscus lens is equal to minus the ratio of the object and image distances. Therefore, the aplanatic meniscus lens increases the system power by a factor equal to the refractive index of the lens. This principle is of practical consequence in many system designs. Of course, if we reverse the sense of Figure 4.14 and substitute the image for the object and vice versa, then the numerical aperture is effectively reduced by a factor of n.
Figure 4.14 Aplanatic meniscus lens.
Worked Example 4.4 Microscope Objective – Hyperhemisphere Plus Meniscus Lens
We now wish to add some power to the microscope objective hyperhemisphere set out in Worked Example 4.1. We are to do so with an extra meniscus lens situated at the vertex of the hyperhemisphere with a negligible separation. As with the hyperhemisphere, the meniscus lens is in the aplanatic arrangement. The meniscus lens is made of the same material as the hyperhemisphere, that is with a refractive index of 1.6. All properties of the hyperhemisphere are as set out in Worked Example 4.1.
What are the radii of curvature of the meniscus lens and what is the location of the (virtual) image for the combined system? The system is as illustrated below.
We know from Worked Example 4.1 that the original image distance produced by the hyperhemisphere is −23.4 mm. The object distance for the meniscus lens is thus 23.4 mm. From Eq. (4.39a) we have:
There remains the question of the choice of the sign for the conjugate parameter. If one refers to Figure 4.14, it is clear that the sense of the object and image location is reversed. In this case, therefore, the value of t is equal to +4.33 and the numerical aperture of the system is reduced by a factor of 1.6 (the refractive index). In that case, the image distance must be equal to minus 1.6 times the object distance. That is to say:
We can calculate the focal length of the lens from:
Therefore the focal length of the meniscus lens is 62.4 mm. If the conjugate parameter is +4.33, then the shape factor must be −(2n + 1), or −4.2 (note the sign). It is a simple matter to calculate the radii of the two surfaces from Eq. (4.29):
Finally, this gives R1 as −23.4 mm and R2 as −14.4 mm. The signs should be noted. This follows the convention that positive displacement follows the direction from object to image space.
If the microscope objective is ultimately to provide a collimated output – i.e. with the image at the infinite conjugate, the remainder of the optics must have a focal length of 37.44 mm (i.e. 23.4 × 1.6). This exercise illustrates the utility of relatively simple building blocks in more complex optical designs. This revised system has a focal length of 9 mm. However, the ‘remainder’ optics have a focal length of 37.4 mm, or only a quarter of the overall system power. Spherical aberration increases as the fourth power of the numerical aperture, so the ‘slower’ ‘remainder’ will intrinsically give rise to much less aberration and, as a consequence, much easier to design. The hyperhemisphere and meniscus lens combination confer much greater optical power to the system without any penalty in terms of spherical aberration and coma. Of course, in practice, the picture is complicated by chromatic aberration caused by variations in refractive properties of optical materials with wavelength. Nevertheless, the underlying principles outlined are very useful.
4.5 The Effect of Pupil Position on Element Aberration
In all previous analysis, it is assumed that the stop is located at the optical surface in question. This is a useful starting proposition. However, in practice, this is most usually not the case. With the stop located at a spherical surface, by definition, the chief ray will pass directly through the vertex of that surface. If, however, the surface is at some distance from the stop, then the chief ray will, in general, intersect the surface at some displacement from the surface vertex. This displacement is, in the first approximation, proportional to the field angle of the object in question. The general concept is illustrated in Figure 4.15.
Instead of the stop being located at the surface in question, the stop is displaced by a distance, s, from the surface. The chief ray, passing through the centre of the stop defines the field angle, θ. In addition, the pupil co-ordinates defined at the stop are denoted by rx and ry. However, if the stop were located at the optical surface, then the field angle would be θ′, as opposed to θ. In addition, the pupil co-ordinates would be given by rx′ and ry′. Computing the revised third order aberrations proceeds upon the following lines. All the previous analysis, e.g. as per Eqs. (4.31a)–(4.31d), has enabled us to express all aberrations as an OPD in terms of θ′, rx′, and ry′. It is clear that to calculate the aberrations for the new stop locations, one must do so in terms of the new parameters θ, rx, and ry. This is done by effecting a simple linear transformation between the two sets of parameters. Referring to Figure 4.15, it is easy to see:
(4.40a)
