11.5 Lateral Color

contributes to the slope of the emergent ray in the first term of the distortion expression. The relation just given is therefore mainly of theoretical interest; however, the Buchdahl coefficients for distortion discussed in Chapter 4 do not suffer the above issue.

11.4.3 Distortion When the Image Surface Is Curved

If a lens is designed to form its image on a curved surface, the meaning of distortion must be clearly defined. As always, distortion is the radial distance from the ideal image point to the crossing point of the principal ray; but now the ideal image is represented by the point of intersection of a line drawn through the second nodal point at the same slope as that of a corresponding ray entering through the first nodal point (Figure 11.17b). Then

distortion ¼ ½ðY2 Y1Þ2 þ ðZ2 Z1Þ2&1=2

where subscript 2 refers to the traced principal ray and subscript 1 to the ideal ray through the nodal points.

11.5 LATERAL COLOR

Lateral color is similar to distortion in that it is calculated by finding the height of intercept of principal rays at the image plane, but now we must compare two principal rays in two different wavelengths, typically the C and F lines of hydrogen, although, of course, any other specified lines can be used if desired. Then

lateral color ¼ HF0 HC0

Lateral color can be resolved into a power series, but now there is a first-order term that does not appear in distortion (the first-order term in distortion is the Gaussian image height; see Figure 4.5):

lateral color ¼ aH0 þ bH03 þ cH05 þ . . .

Some people consider that only the first term represents lateral color, all the others being merely the chromatic variation of distortion. No matter how it is regarded, lateral color causes a radial chromatic blurring at image points located away from the lens axis. Of course, both distortion and lateral color vanish at the center of the field.

11.5.1 Primary Lateral Color

The first term of this series, representing the primary lateral color, can be calculated by a method similar to the calculation of the OSC, except that now we

trace paraxial rays in C and F light instead of tracing a marginal and a paraxial ray in brightest light. Thus, writing paraxial data in F in place of the original marginal ray data, and paraxial data in C in place of the original paraxial ray data, our formula Eq. (9-4) becomes

where Df 0 ¼ fF0 – fC0 and Dl0 ¼ lF0 – lC0 . The latter is, of course, the ordinary paraxial longitudinal chromatic aberration. The expression CDM is an abbreviation for chromatic difference of magnification and it is strictly analogous to

OSC.

In a symmetrical lens, or any other lens in which the pupils coincide with the principal planes, l0 – lpr0 ¼ f 0, and Eq. (11-11) becomes

The numerator of this expression is simply the distance between the second principal planes in C and F light. Thus, if these principal planes coincide, there will be no primary lateral color. This is often a convenient computing device for use in the early stages of a design. Later, of course, it is necessary to trace true principal rays in F and C and calculate the difference in the heights of these rays at the focal plane.

The logic of this last relationship can be understood by the diagram in Figure 11.18, which shows the principal rays in C and F, at small obliquity, emerging from their respective principal points and proceeding to the image plane. It is clear that

primary lateral color ¼ z tan U 0pr ¼ zðh0=f 0Þ

Figure 11.18 Primary lateral color depends on z.