in Figure 3.13 which shows relevant plots for coma.
Figure 3.12 OPD fan for coma.
Figure 3.13 Ray fan for coma.
Since the (vector) transverse aberration for coma is non-symmetric, the blur spot relating to coma has a distinct pattern. The blur spot is produced by filling the entrance pupil with an even distribution of rays and plotting their transverse aberration at the paraxial focus. If we imagine the pupil to be composed of a series of concentric rings from the centre to the periphery, these will produce a series of overlapping rings that are displaced in the y direction.
Figure 3.14 shows the characteristic geometrical point spread function associated with coma, clearly illustrating the overlapping circles corresponding to successive pupil rings. These overlapping rings produce a characteristic comet tail appearance from which the aberration derives its name. The overlapping circles produce two asymptotes, with a characteristic angle of 60°, as shown in Figure 3.14.
Figure 3.14 Geometrical spot for coma.
To see how these overlapping circles are formed, we introduce an additional angle, the ray fan angle, φ, which describes the angle that the plane of the ray fan makes with respect to the y axis. For the tangential ray fan, this angle is zero. For the sagittal ray fan, this angle is 90°. We can now describe the individual components of the pupil function, px and py in terms of the magnitude of the pupil function, p, and the ray fan angle, φ:
(3.26)
From (3.25) we can express the transverse aberration components in terms of p and φ. This gives:
A is a constant
It is clear from Eq. (3.27) that the pattern produced is a series of overlapping circles of radius A√2p2 offset in y by 2Ap2. Coma is not an aberration that can be ameliorated or balanced by defocus. When analysing transverse aberration, the impact of defocus is to produce an odd (anti-symmetrical) additional contribution with respect to pupil function. The transverse aberration produced by coma, is, of course, even with respect to pupil function, as shown in Figure 3.12. Therefore, any deviation from the paraxial focus will only increase the overall aberration.
Another important consideration with coma is the location of the geometrical spot centroid. This represents the mean ray position at the paraxial focus for an evenly illuminated entrance pupil taken with respect to the chief ray intersection. The centroid locations in x and y, Cx, and Cy, may be defined as follows.
(3.28)
By symmetry considerations, the coma centroid is not displaced in x, but it is displaced in y. Integrating over the whole of the pupil function, p (from 0 to 1) and allowing for a weighting proportional to p (the area of each ring), the centroid location in y, Cy may be derived from Eq. (3.27):
(3.29)
(the term cos2φ is ignored as its average is zero)
So, coma produces a spot centroid that is displaced in proportion to the field angle. The constant A is, of course, proportional to the field angle.
3.5.4 Field Curvature
The third Gauss-Seidel term produced is known as field curvature. The OPD associated with field curvature is second order in both field angle and pupil function. Furthermore, there is no dependence upon ray fan angle, as the WFE is circularly symmetric. Unlike in the case for coma, behaviour is identical for the tangential and sagittal ray fans.
From Eq. (3.30), in the case of a single field point, the effect of a quadratic dependence of WFE on pupil function is to produce a uniform defocus. That is to say, a uniform defocus produces a characteristic quadratic pupil dependence in the WFE. The extent of this defocus is proportional to the square of the field angle, producing a curved surface which intersects the paraxial focal plane at zero field angle – the optical axis. If this field curvature were the only aberration, then this curved surface would produce a perfectly sharp image for all these field points. That is to say, with the presence of field curvature, the ideal focal surface is a curved surface or sphere rather than a plane. This is illustrated in Figure 3.15.
Figure 3.15 shows both the tangential and sagittal focal surfaces (S and T), with the optimum focal surface lying between the two. Ideally, for field curvature, the imaging surface should be curved, following the ideal focal surface. If, for instance, only a plane imaging surface is available, then this need not be located at the paraxial focus. This surface can, in principle, be located at an offset, such that the rms WFE is minimised across all fields. In calculating the rms WFE, this would be weighted according to area across all object space, as represented by a circle centred on the optical axis whose radius is the maximum object height.
Figure 3.15 Field curvature.
Figure 3.16 Ray fan plots
