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Coma and the Sine Condition |
LA OSC
OSC
c1
LA
Figure 9.4 Typical effect of bending a single thin lens.
lens is such that the equivalent refracting locus is too flat, the marginal focal length will be too long and the OSC will be positive. A thin lens bent to the left meets this condition. Similarly, if the rim of the lens is bent to the right the OSC will be negative. Plotting spherical aberration and OSC against bending for such a lens gives a graph like the one in Figure 9.4.
It should be noted that in any reasonably thin lens, the lens bending for which the spherical aberration reaches an algebraic maximum is almost exactly the same bending as that which makes the OSC zero. For the primary aberrations of a single thin lens, this is easily verified by comparing the value of c1 that makes @LAp0 /@c1 ¼ 0 (Section 6.3.2) with the value of c1 that makes the comap ¼ 0 [Eq. (9-10)]. It will be found that for a variety of refractive indices and a variety of object distances, the c1 for zero coma is always slightly greater than the c1 for maximum spherical aberration.
DESIGNER NOTE
There is, of course, no aperture limit for a nonaplanatic system. A parabolic mirror, for example, has zero spherical aberration for a distant axial object point, but the focal length of each ray is the distance from the mirror surface to the image point, measured along the ray. The focal length continuously increases with increasing incidence height, which means the magnification is changing, as explained in Section 4.3.4. Consequently, the image is afflicted with enormous positive coma. Consider an f/0.25 parabolic mirror as illustrated in Figure 9.5. Notice that the marginal ray heads toward the image orthogonal to the optical axis just as it does for an aplanatic lens (Section 9.2); however, the focal length of the aplanatic lens remains constant as a function of incidence height while the marginal focal length of the parabola is twice that of its axial focal length.
9.3 Offense Against the Sine Condition |
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Figure 9.5 Nonaplanatic f/0.25 parabolic mirror.
Figure 9.6 shows the geometric spot diagram for an f/0.26 parabolic mirror with the paraxial image located only one-third of an Airy disk radius from the optical axis. The small circle in this figure represents the Airy disk. As can be observed, the coma is huge (contains many higher-order terms) compared to the diffraction blur, assuming no aberrations, although the shift in the object is just a fraction of the diffraction disk. This means that any calculations relying on this type of optical system behaving as a linear system will be seriously flawed. In contrast, a well-behaved system will have aberrations that are relatively slow to change as the field angle changes, thereby having regions in the image plane that are spatially stable, such that the shape (aberrations or wavefront) of the point-source image remains constant over an area of at least several Airy disk diameters. Such an image region is called an isoplanatic region or patch. Beware that some optical design programs may blindly compute dif- fraction-based MTF, point spread functions, etc. and produce erroneous results as a consequence of not using a correct modeling construct! Use common sense and test your program with an appropriate example to assure yourself of valid results.3