Again, this expression becomes the Gauss equation when Ī and Īʹ are zero. These equations are frequently used to evaluate the astigmatism of optical systems. When using these expressions, it is important to remember that these images are obtained from rays very close to the principal ray (parabasal rays), and thus, both images, sagittal and tangential, are located on this principal ray.
6.4.1.3 General Expression
A general expression for the calculation of the image for rays in a plane between the tangential and sagittal planes can be easily obtained from these two expressions.
The tangential and the sagittal curvatures of an astigmatic refracted wavefront are the minimum and maximum values for a given principal ray path and are called normal curvatures given by
′ = ′ ′ = c ′
s c
s t t
1 and 1. (6.66)
The curvatures for this wavefront in a plane P as shown in Figure 6.12, where the principal ray is perpendicular to this plane, are oriented by an angle α with respect to the tangential plane, according to the Euler formula, as follows:
cα= ′ct′cos2α+ ′cs′sin2α. (6.67) If the angle is measured in the S plane, tangent to the optical surface, and this angle is represented by θ, as illustrated in Figure 6.12, we can write
cos cos
136 Handbook of Optical Design
where Iʹ is the angle of refraction for the principal ray, and
sin cos Now, substituting these functions in Euler’s formula (Equation 6.67):
c c I
After some algebraic manipulations, we obtain the generalized Coddington equa-tion, to be derived by a different method in Chapter 7 (Equation 7.77):
′ ′ − = ′ −
(
′)
This expression is useful in finding the astigmatism in systems of two spherical mirrors with their meridional planes forming any angle between them.
6.4.2 relationsbetween PetzVal curVatureand astiGmatism
With a third-order approximation, if the field is relatively small (semifield smaller than approximately 10°) the sagittal (S), tangential (T), and Petzval (P) surfaces may be represented by spherical surfaces, as shown in Figure 6.13, where these aber-rations are positive. These surfaces touch each other at the principal ray, but the
P S
Principal ray
Meridional plane
Optical surface x' = x
aθ y' y
FIGURE 6.12 Projection of angle α in a plane perpendicular to the principal ray to an angle θ in a plane tangent to the optical surface.
separation between these surfaces at other points follows Equation 6.56. The sagittas for sagittal and tangential surfaces are equal to AstLS + Ptz and AstLt + Ptz, respec-tively. The surface of best definition is between the sagittal and tangential surfaces.
Thus, the sagitta for the surface of best definition is
Best Ptz= + AstTs+AstTt =Ptz+ AstTs
2 2 . (6.72)
If the surface of best definition has to be flat, this sagitta has to be equal to zero;
otherwise, the radius of curvature for this surface is
r h
Ptz AstTs
best= ′
+
2
2( 2 ). (6.73)
We have proved that the Petzval curvature depends only on the total power of the lenses forming the optical system and not on the lens shapes (bending) or on the stop position. Thus, by bending and selecting the stop’s position, we may only change the astigmatism. On the other hand, the tangential and sagittal astigmatisms are always in a 3:1 relation. Hence, these focal surfaces may have different curvatures, as shown in Figure 6.13.
To obtain the best overall image definition, we may position the observation plane (screen, photographic film, or detector) at the places indicated with a dotted line in Figure 6.14. The four figures show the same Petzval surface with increasing amount of the magnitude of astigmatism. If there is no astigmatism, but there is a negative Petzval curvature, as in most cases, as shown in Figure 6.14a, the image is perfect and well defined over the whole Petzval surface. In this case, the observing screen may be curved as in some astronomical instruments, or a field flattener may be used.
P S M T
Ptz AstL AstLS
T X
FIGURE 6.13 Astigmatic surfaces and definitions of sagittal and tangential astigmatism.
138 Handbook of Optical Design
As stated by Conrady (1957), the astigmatism in Figure 6.14b is a better choice for astronomical photography, where the field is not very large. Then, there is some astig-matism, but the optimum focal plane, between the sagittal and tangential surfaces, is flatter. If elongated images are not satisfactory, as in the case of photographic cam-eras, where the field is wide, the large astigmatism in Figure 6.14d is a compromise, where the best-definition surface is flat. The price is a large astigmatism with the size of the image growing toward the edge. The best choice for most practical purposes is to reduce the astigmatism a little bit with respect to that in Figure 6.14d, by choosing a flat tangential field, as in Figure 6.14c.
An important practical conclusion that should always be in mind is that in a sys-tem with a negative Petzval sum, which occurs most of the time, the best overall image is obtained only if positive astigmatism is present.
For semifields larger than approximately 20°, significant amounts of high-order astigmatism may appear, resulting to the sagittal and tangential surfaces deviating strongly from the spherical shape. In this case, high-order aberrations should be used to balance the primary aberrations, as shown in Figure 6.15. The two astigmatic sur-faces should cross near the edge of the field.
P, S, T P S T P S T P S T
(a) (b) (c) (d)
FIGURE 6.14 Astigmatic curves for different amounts of astigmatism with a constant Petzval curvature.
T
P
S h'
FIGURE 6.15 Astigmatic curves with high-order aberrations for a large field.
6.4.3 comaticand astiGmatic imaGes
We have seen that coma and astigmatism are two different aberrations, but they are not independent in a single optical surface. Both are present and closely interrelated through the spherical aberration. Figure 6.16 shows how the aberrations of astig-matism and coma change for different values of the ratio i i/. We may observe that for values of this ratio smaller than one, the coma dominates, but for values greater than one, the situation is reversed. Only in a complex system, with several centered spherical optical surfaces, may a single aberration, coma, or astigmatism be present.
The image structure for each of these aberrations will now be described.
To understand how the coma image appears, let us divide the exit pupil into con-centric rings as in Figure 6.17. The paths for some rays from one of the rings in the pupil are illustrated in this figure. Here, we may appreciate the following facts:
1. Rays symmetrical with respect to the meridional plane, Di and D′i cross each other at a point on the meridional plane, Pi.
2. All the points Pi lie on a straight line, parallel to the principal ray. This line is called the characteristic focal line. A diapoint is defined as the point intersection of an oblique ray and the meridional plane. Thus, the focal characteristic line may also be defined as the locus of diapoints for the rays passing through a ring on the exit pupil.
3. Each circle on the exit pupil generates a characteristic focal line, parallel to the principal ray. The smaller the circle on the exit pupil is, the closer the characteristic focal line gets to the principal ray.
4. The tangential rays from D0 and D4 cross at the tangential focus, on the focal plane.
5. The sagittal rays from D2 and D′2 cross at the sagittal focus, on the focal plane.
0.5 1 1.5
i/i 4
3
2
1
AstTS SphT
ComaS SphT
FIGURE 6.16 Variations in the ratios AstTS/SphT and ComaS/SphT versus the ratio i i/.
140 Handbook of Optical Design
6. Each ring on the exit pupil also becomes a small ring on the focal plane.
However, one turn on the exit pupil becomes two turns on the image.
7. The complete comatic image is formed with all the rings, becoming smaller as they shift along the meridional plane (y axis).
A stereo pair of images showing the ray paths passing through a ring on the entrance pupil in the presence of coma is illustrated in Figure 6.18. The final structure
0
Tangential focus
Sagittal focus Principal ray
D0, D4 D0
P4P3 P2
P1 P0 D1
D2
D3 D4
D2, D'2 D1, D'3
D'1 D'2
D'3
D'1, D3
FIGURE 6.17 (See color insert.) Rays around a ring on the pupil in the presence of coma.
60 mm
FIGURE 6.18 Stereo pair of images showing the ray paths from a ring on the pupil under the presence of coma.
of a comatic image with positive coma is illustrated in Figure 6.19. In this figure, we see graphical definitions of the transverse sagittal and tangential coma.
In an optical system with pure astigmatism (without coma), the rays from a ring on the exit pupil travel as in Figure 6.20. Here, we may appreciate the following:
Exit pupil
Image
Tangential marginal focus
ComaT
ComaS
ComaS
Sagittal marginal focus
Principal ray
FIGURE 6.19 (See color insert.) Formation of the comatic image.
D0 D1
D2
D3 D4 D'3
D'1 D'2
P0
P1 P2P3 P4
Sagittal
plane Tangential
plane
FIGURE 6.20 (See color insert.) Rays forming the image in a system with pure astigma-tism (no coma).
142 Handbook of Optical Design 1. Rays symmetrical with respect to the meridional plane, Di and D′i, cross
each other at a point on the meridional plane, Pi′. The letter i stands for 0, 1, 2, 3, or 4 in Figure 6.20.
2. All the points Pi lie on a straight line, perpendicular to the principal ray.
This line is the characteristic focal line. In this case, this is also the sagittal focus.
3. Each circle on the exit pupil generates a characteristic focal line; all are placed on the corresponding sagittal focus. Thus, in a single optical surface, all characteristic focal lines are parallel to each other and perpendicular to the principal ray. In an optical system without spherical aberration, all characteristic focal lines collapse in a single line.
4. At an intermediate plane between the sagittal and the tangential focus, the image is a small circle.
The astigmatic images at several focal surfaces are illustrated in Figure 6.21. The magnitudes of the sagittal and tangential transverse astigmatisms are shown here.
Now, let us make some general considerations about the characteristic focal line:
1. The focal characteristic line for a single refracting surface is on the auxil-iary axis, as shown in Figure 6.4.
2. In general, there is a focal characteristic line for each concentric ring on the exit pupil, and all lines are parallel to each other.
3. In a system with pure coma, the focal characteristic lines are parallel to the principal ray, and in a system with pure astigmatism, they are perpendicu-lar to the principal ray. Thus, it is clear that in a single surface, we cannot isolate astigmatism and coma, because the auxiliary optical axis is always inclined with respect to the principal ray.
Image
plane Medium
focus Tangential
focus Out of
focus Sagittal
focus Petzval
focus
AstTS
4 AstTS 4 AstTS
AstTT AstTS
AstTS AstTT
PtzT
PtzT
(a) (b) (c) (d) (e) (f)
FIGURE 6.21 Astigmatic images at different focal surfaces.
4. The center of the characteristic focal line is the sagittal focus. The extremes are defined by the marginal tangential rays.
5. In a complete optical system, the characteristic focal line is in general inclined to a certain angle θ with respect to the principal ray. Then, this inclination is given by the relative amounts of these two aberrations as follows:
tanθ = − AstT ′ ComaS u
S k Total
Total
. (6.74)
A little more insight and understanding about the structure of the astigmatic and comatic images may be obtained with a detailed examination of Figure 6.22. This figure plots the locus of the intersections on the focal plane of the light rays passing through a circular ring on the entrance pupil. These plots are taken at different equi-distant focal plane positions, but different for the three cases shown. The large dot represents the principal ray intersection and a line joining them would be the prin-cipal ray. The focal characteristic line (or diapoint locus) is graphically represented in these three cases.