field curvature.
Clearly, the OPD fan for field curvature is a series of parabolic curves whose height is proportional to the square of the field angle. There is no distinction between the sagittal and tangential fans. Similarly, the ray fans show a series of linear plots whose magnitude is also proportional to the square of the field angle. A series of ray fan plots for field curvature is shown in Figure 3.16.
In view of the symmetry associated with field curvature, the geometrical spot consists of a uniform blur spot whose size increases in proportion to the square of the field angle. In addition, this spot is centred on the chief ray; unlike in the case for coma, there is no centroid shift with respect to the chief ray.
3.5.5 Astigmatism
The fourth Gauss-Seidel term produced is known as astigmatism, literally meaning ‘without a spot’. Like field curvature, the WFE associated with astigmatism is second order in both field angle and pupil function. It differs from field curvature in that the WFE is non-symmetric and depends upon the ray fan angle as well as the magnitude of the pupil function. That is to say, the behaviour of the tangential and sagittal ray fans is markedly different.
(3.31)
In some respects, the OPD behaviour is similar to field curvature, in that, for a given ray fan, the quadratic dependence upon pupil function implies a uniform defocus. However, the degree of defocus is proportional to cos2φ. Thus, the defocus for the tangential ray fan (cos2φ = 1) and the sagittal ray fan (cos2φ = −1) are equal and opposite. Clearly, the tangential and sagittal foci are separate and displaced and this displacement is proportional to the square of the field angle. The displacement of the ray fan focus is set out in Eq. (3.32):
A is a constant
As suggested previously, for a given field angle, the OPD fan would be represented by a series of quadratic curves whose magnitude varies with the ray fan angle. Similarly, the ray fan itself is represented by a series of linear plots whose magnitude is dependent upon the ray fan angle. This is shown in Figure 3.17, which shows the ray fan for a given field angle for both the tangential and sagittal ray fans.
For a general ray, it is possible to calculate the two components of the transverse aberration as a function of the pupil co-ordinates.
Figure 3.17 Ray fan for astigmatism showing tangential and sagittal fans.
Figure 3.18 Geometric spot vs. defocus for astigmatism.
According to Eq. (3.33), the blur spot produced by astigmatism (at the paraxial focus) is simply a uniform circular disc. Each point in the uniform pupil function simply maps onto a similar point on the blur spot, but with its x value reversed. However, when a uniform defocus is added, similar linear terms (in p) will be added to both tx and ty, having both the same magnitude and sign. As a consequence, the relative magnitude of tx and ty will change producing a uniform elliptical pattern. Indeed, as mentioned earlier, there are distinct and separate tangential and sagittal foci. At these points, the blur spot is effectively transformed into a line, with the focus along one axis being perfect and the other axis in defocus. This is shown in Figure 3.18.
Due to the even (second order) dependence of OPD upon pupil function, there is no centroid shift evident for astigmatism. For Gauss-Seidel astigmatism, its magnitude is proportional to the square of the field angle. Thus, for an on-axis ray bundle (zero field angle) there can be no astigmatism. This Gauss-Seidel analysis, however, assumes all optical surfaces are circularly symmetric with respect to the optical axis. In the important case of the human eye, the validity of this assumption is broken by the fact that the shape of the human eye, and in particular the cornea, is not circularly symmetrical. The slight cylindrical asymmetry present in all real human eyes produces a small amount of astigmatism, even at zero field angle. That is to say, even for on-axis ray bundles, the tangential and sagittal foci are different for the human eye. For this reason, spectacle lenses for vision correction are generally required to compensate for astigmatism as well as defocus (i.e. short-sightedness or long-sightedness).
3.5.6 Distortion
The fifth and final Gauss-Seidel aberration term is distortion. The WFE associated with this aberration is third order in field angle, but linear in pupil function. In fact, a linear variation of WFE with pupil function implies a flat, but tilted wavefront surface. Therefore, distortion merely produces a tilted wavefront but without any apparent blurring of the spot. The WFE variation is set out in Eq. (3.34).
Thus, the only effect produced by distortion is a shift (in the y direction) in the geometric spot centroid; this shift is proportional to the cube of the field angle. However, this shift is global across the entire pupil, so the image remains entirely sharp. The shift is radial in direction, in the sense that the centroid shift is in the same plane (tangential) as the field offset. So, the OPD fan for the tangential fan is linear in pupil function and zero for the sagittal fan. The ray fan is zero for both tangential and sagittal fans, emphasising the lack of blurring.
Taken together with the linear (paraxial) magnification produced by a perfect Gaussian imaging system, distortion introduces another cubic term. That is to say, the relationship between the transverse image and object locations is no longer a linear one; magnification varies with field angle. If the height of the object is hob and that of the image is him, then the two quantities are related as follows:
Figure 3.19 (a) Pincushion (positive) distortion. (b) Barrel (negative) distortion.
M0
