Читать книгу Optical Engineering Science - Stephen Rolt - Страница 61

The second term, coma, has an WFE that is proportional to the field angle. Its pupil dependence is third order, but it is not symmetrical with respect to the pupil function. The WFE associated with coma is as below:

(3.23)

In the preceding discussions, the transverse aberration has been presented as a scalar quantity. This is not strictly true, as the ray position at the paraxial focus is strictly a vector quantity that can only be described completely by an x component, t_x and a y component t_y. Equation (3.12) should strictly be rendered in the following vectorial form:

(3.24)

The transverse aberration relating to coma may thus be written out as:

(3.25)

From the perspective of both the OPD and ray fans the behaviour of the tangential (y) and sagittal ray fans are entirely different. As an optical designer, the reader should ultimately be familiar with the form of these fans and learn to recognise the characteristic third order aberrations. For a given field angle, the tangential OPD fan (p_x = 0) shows a cubic dependence upon pupil function, whereas, for the sagittal ray fan (p_y = 0), the OPD is zero. The OPD fan for coma is shown below in Figure 3.12.

The picture for the ray fans is a little more complicated. For both the tangential and sagittal ray fans, there is no component of transverse aberration in the x direction. On the other hand, for both ray fans, there is a quadratic dependence with respect to pupil function for the y component of the transverse aberration. The problem, in essence, it that transverse aberration is a vector quantity. However, when ray fans are computed for optical designs they are presented as scalar plots for each (tangential and sagittal) ray fan. The convention, therefore, is to plot only the y (tangential) component of the aberration in a tangential ray fan, and only the x (sagittal) component of the aberration in a sagittal ray fan. With this convention in mind, the tangential ray fan shows a quadratic variation with respect to pupil function, whereas there is no transverse aberration for the sagittal ray fan. Tangential and sagittal ray fan behaviour is shown 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, p_x and p_y 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:

(3.27)

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, C_x, and C_y, 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.