Figure 9.1 Derivation of the sine theorem: (a) oblique view and (b) plan view.

It is essential to remember that hs0 is the height of the sagittal image for the zone, namely, the intersection of the 90 and 270 rays, and has no relation whatsoever to the height of any other rays from the zone. It is, in particular, not related to the meridional rays in any way.

9.2 THE ABBE SINE CONDITION

Abbe regarded coma as a consequence of a difference in image height from one lens zone to another, and he thus realized that a spherically corrected lens (in his case a microscope objective) would be free from coma near the center of the field if the paraxial and marginal magnifications m ¼ nu/n0u0 and M ¼ n sin U/n0 sin U0 were equal, that is,

This is known as the Abbe sine condition.

For a very distant object, the sine condition takes a different form. As was shown in Section 3.3.4, the Lagrange equation for a distant object can be written as

h0 ¼ ðn=n0Þ f 0 tan Upr

where f 0 is the distance from the principal plane to the focal point measured along the paraxial ray, or f 0 ¼ y1/uk0 . A similar expression can be written for the focal length of a marginal ray, namely,

where F0 is the distance measured along the marginal ray from the equivalent refracting locus to the point where the ray crosses the lens axis. Thus for a spherically corrected lens and a distant object, Abbe’s sine condition reduces to

F 0 ¼ f 0

This relation tells us that in such a lens, called by Abbe an aplanat, the equivalent refracting locus is part of a hemisphere centered about the focal point. The maximum possible aperture of an aplanat is therefore f/0.5, although this aperture is never achieved in practice. The greatest practical aperture is about f/0.65 when the emerging ray slope is about 50 .

There is no equivalent rule for a lens that is aplanatic for a near object, such as a microscope objective. We can, if we wish, assume that in such a case the two principal planes are really parts of spheres centered about the axial conjugate points, but we could just as easily make any other suitable assumption provided the marginal ray moves from one principal “plane” to the other along a line lying parallel to the lens axis, as indicated for paraxial rays in Figure 3.10.

If the refractive index of either the object space or the image space is other than 1.0, we must include the actual refractive index in the f-number:

Thus if the image space were filled with a medium of refractive index 1.5, the highest possible relative aperture would be f/0.33. To realize the benefit of this high aperture, the receiver, film, CCD, or photocell must be actually immersed in the dense medium. Similarly, when a camera is used for underwater photography, the effective aperture of the lens is reduced by a factor of 1.33, which is the refractive index of water.

When the object is not located at infinity, the effective f-number is given by

f -numbereffective ¼ f -number1ð1 mÞ:

NA ¼ n0 sin U 0. If the lens is aplanatic, f -numbereffective ¼ 2NA1 :

9.2.1Coma for the Three Cases of Zero Spherical Aberration

It was shown in Section 6.1.1 that there are three cases in which a spherical surface has zero spherical aberration: (a) when the object is at the surface itself,

(b) when the object is at the center of curvature of the surface, and (c) when the