9.3 OFFENSE AGAINST THE SINE CONDITION

We may express the magnitude of the sagittal coma by the dimensionless ratio QS/QM in the marginal image plane, and we call this ratio the “offense against the sine condition,” or OSC (see also Section 4.3.4 and Eq. (10-3)). Thus

OSC ¼ QS ¼ SM QM ¼ SM 1

QM QM QM

The length SM is the h0s given by the sine theorem; the length QM is obtainable from the paraxial image height h0 by

For a near object we can insert the values of h0 and h0s by the Lagrange and sine theorems, respectively, giving

where M and m are, respectively, the image magnification for the finite and paraxial rays.

The bracketed quantity, which contains data relating both to the spherical aberration of the lens and the position of the exit pupil, can be readily modified to

!

1

LA0

L0 l0pr

and for a very distant object, M/m can be replaced by F 0/f 0. Hence for a distant object, Eq. (9-4) becomes

Conrady1 states that the maximum permissible tolerance for OSC is 0.0025 for telescope and microscope objectives. This large tolerance is because in those instruments the object of principal interest can always be moved into the center of the field for detailed study. A very much smaller tolerance applies to photographic objectives.

9.3.1 Solution for Stop Position for a Given OSC

Since the exit-pupil position (l0pr) appears in the formulas for OSC, it is clear that as we shift the stop along the axis the OSC will change provided there is some spherical aberration in the lens. If the spherical aberration has been corrected, then shifting the stop will have no effect on the coma. We can thus solve for the value of l0pr to give any desired OSC by inverting Eqs. (9-4) and (9-5).

For a near object,

These formulas find use in the design of simple eyepieces and landscape lenses for low-cost cameras.

9.3.2 Surface Contribution to the OSC

By a process similar to that used for determining the surface contribution to spherical aberration (Section 6.1), we can develop a formula giving the surface contribution to the OSC. For this derivation, we trace a marginal ray and the paraxial principal ray. The development given in Section 6.1 indicates that in our present case we have

We can see from the diagram in Figure 9.3 that S0 ¼ (L0 – l0pr) sin U0, and similarly for the incident ray. Hence, dividing Eq. (9-6) by the Lagrange invariant and substituting for S and S0 we get

Now h0/u0pr ¼ (l0 – l0pr), and h/upr ¼ (l – lpr). Also, by the Lagrange and sine theorems we have

It should be noted that any spherical aberration in the object leads to a contribution to the OSC. Also, the factor outside the summation, u1/sinU1, becomes y1/Q1 for a distant object.

Example

As an example of the use of this contribution formula, we will take our old familiar telescope doublet (Section 2.5) and trace a paraxial principal ray through the front vertex at an entering angle of, say, –5 (tan( 5 ) ¼0.0874887), with the results shown in Table 9.1. Hence,

For the OSC contribution formula, we pick up the data of the marginal ray from Table 6.1, giving the tabulation shown in Table 9.2.

Table 9.1

Trace of Paraxial Principal Ray

Table 9.2

OSC Tabluation for Example Doublet

For this formula, the Lagrange invariant has the value (h0n0u0) ¼ 0.1749774. The excellent agreement between the direct and contribution calculations is evident. Also see Section 4.3.4 for an alternative OSC formula using the y-coordinate ray intercept data from a sagittal ray and a principal ray:

Yðr; 900; H; xÞ Yð0; 00; H; xÞ :

Yð0; 00; H; xÞ