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      Finally this gives the solution for s as:

(4.39a)

      Accordingly the solution for t is

      (4.39b)

      Of course, since the equation for spherical aberration gives quadratic terms in s and t, it is not surprising that two solutions exist. Furthermore, it is important to recognise that the sign of t is the opposite to that of s. Referring to Figure 4.10, it is clear that the form of the lens is that of a meniscus. The two solutions for s correspond to a meniscus lens that has been inverted. Of course, the same applies to the conjugate parameter, so, in effect, the two solutions are identical, except the whole system has been inverted, swapping the object for image and vice-versa.

An aplanatic meniscus lens is an important building block in an optical design, in that it confers additional focusing power without incurring further spherical aberration or coma. This principle is illustrated in Figure 4.14 which shows a meniscus lens with positive focal power.

It is instructive, 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.

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