10.6 The Matching Principle

10.6 THE MATCHING PRINCIPLE

If we wish to design an aplanatic lens of such a high aperture that a single doublet is impossible, we resort to the use of two achromats in succession. We now have four degrees of freedom. The subdivision of power between the two components and the air space between them is arbitrary, while the two bendings can be used for the correction of spherical aberration and OSC. Any reasonable types of glass can be used, and by achromatizing each component separately we automatically correct both the chromatic aberration and the lateral color.

The design of lenses of this type has been described in detail by Conrady,5 in particular when used as a microscope objective of medium power. Having decided on suitable values for the two arbitrary quantities, we trace a marginal ray through the system from front to back, solving r3 and r6 by the D – d method, and we then add two paraxial rays, one through the front component from left to right using l1 ¼ L1 and u1 ¼ sin U10 , and the other through the rear component from right to left, taking u60 ¼ sin U60 and l60 ¼ L60 as starting data.

If we can now find such a pair of bendings that the two paraxial rays match in the air space between the lenses, the system will be corrected for both spherical aberration and OSC. This is what is meant by the matching principle.

To make the required trial bendings, we have no problem with the front component, but we must adopt standard entry data for the rear component. We can easily adopt a fixed value for L4 by always choosing a suitable air space between the lenses, but any standard value of U4 that we may adopt will never agree exactly with the emerging slope U30 from the front component. Consequently it becomes necessary to match actual aberrations in the air space rather than trying to match lengths and angles.6 So far as lengths are concerned, we have always L4 ¼ L30 – d, and we require that l4 ¼ l30 – d. Subtracting these tells us that we must select bendings such that

To match the slope angles of the paraxial rays in the air space, we have approximately sin U4 ¼ sin U30 , and we require that u4 should also be equal to u30 . Dividing these gives

Since this kind of OSC does not contain the usual correcting factor for spherical aberration and exit-pupil position (see Sections 4.3.4 and 9.3), we

Table 10.5

Aberrations Versus Bendings for Lister-type Microscope

Figure 10.10 The matching principle.

It is our aim to select such values of c1 and c4 that LA30 ¼ LA4 and simultaneously OSC30 ¼ OSC4. This is done by searching for rectangles that just fit into the four curves, with spherical aberration and coma points, respectively, each being on the same level. In this case, there are four such rectangles to be found, indicating that the curves represent quadratic expressions. The four solutions are shown in Table 10.6.

For many reasons we shall continue the design using solution C. All the other solutions contain stronger surfaces, and moreover both components of solution C contain almost equiconvex crown elements. This starting setup is as follows:

0.081

3.21.52520

0.08394

1.01.62115

0.00685

14.9603 (air)

0.165

2.01.52520

0.14654

0.81.62115

0.03730

with l60 ¼ 7.2095, LA60 ¼ 0.01383, u60 ¼ 0.2361, and OSC60 ¼ 0.00297. For the final OSC60 calculation we assumed that the exit pupil is in such a position that (l0 – l0pr) is about 17.0. This puts the exit pupil about 10 mm inside the rear vertex of the objective.

Although this solution is close, we must improve both aberrations by means of a double graph. Changing c1 by 0.001 and maintaining the D – d solutions, and L4 ¼ 20.0, we find that the aberrations become

LA06 ¼ 0:000829; OSC 06 ¼ 0:002404

Inspection of the graph suggests that we change the original c1 by 0.001 and c4 by –0.01. These changes give

LA06 ¼ 0:000403; OSC 06 ¼ þ0:000474

Unfortunately the numerical aperture of the system is now 0.2381, whereas it should be 0.25. We therefore scale all radii down by 4%, which gives

LA06 ¼ 0:001114; OSC 06 ¼ 0:000221

Further reference to the double graph suggests that we try Dc1 ¼ 0.0005 and Dc4 ¼ 0.002. This change gives the almost perfect solution drawn to scale in Figure 10.9, namely,

0.08578

3.2 1.52520

0.08576

1.0 1.62115

0.009152

13.8043 (air)

0.16320

2.0 1.52520

0.16080

0.8 1.62115

0.02602

with l60 ¼ 6.8925, u60 ¼ 0.2500, LA60 ¼ 0.000004, OSC60 ¼ –0.000095, and LZA60 ¼ –0.00289. In practice, of course, we should apply trifling further bendings to both components to render the crown elements exactly equiconvex. These changes are so slight that they have no significant effect on any of the aberrations.

The zonal aberration tolerance is 6l/sin2 Um0 ¼ 0.053, so that the zonal residual of our objective is about half the Rayleigh limit. To improve it, we would have to go to a flint of somewhat higher index, but the present design would be acceptable as it stands.