1
29.1 Introduction 2
29.2 Power Series Expansions 8
29.3 Chromatic Aberrations 13
29.4 Primary Aberrations 16
29.4.1 Aperture and Field Dependence 16
29.4.2 Symmetry and Periodicity Properties 18
29.4.3 Presentation of Aberrations and their Impact on Image Quality 20
29.4.4 Calculation of the Seidel Sums 29
29.4.5 Stop Shift Formulae 36
29.4.6 Several Aberration Expressions from the Seidel Sums 38
29.4.7 Thin Lens Aberrations 41
29.5 Pupil Aberrations 45
29.6 High-order Aberrations 50
29.6.1 Fifth-order Aberrations 50
29.6.2 Seventh and Higher-order Aberrations 53
29.7 Zernike Polynomials 55
29.8 Special Aberration Formulae 56
29.8.1 Sine Condition and the Offence against the Sine Condition 57
29.8.2 Herschel Condition 60
29.8.3 Aplanatism and Isoplanatism 61
29.8.4 Aldis Theorem 61
29.8.5 Spherical Aberration, a Surface Contribution Formula 64
29.8.6 Aplanatic Surface and Aplanatic Lens 68
29.9 Literature 70
29
Aberrations
Handbook of Optical Systems: Vol. 3. Aberration Theory and Correction of Optical Systems.
Edited by Herbert Gross
Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-40379-0
29 Aberrations2
29.1
Introduction
In this and in the following section (29.2) we will deal with monochromatic aberra-
tions only. In section 29.3 chromatic aberrations will be introduced and will then be
included in our further discussions. As we have already explained in chapter 11, in
general, a real optical imaging system does not perform ideal imaging. So, rays
emerging from one object point O will not all meet at a single image point O′.An
example with three meridional rays is shown in figure 29-1.
O'
O
optical
system
Figure 29-1: Aberrations of an imaging system.
There are several methods used to describe how the rays miss the image point, see
section 11.2, Description of Aberrations. Very often it is convenient to think in terms of
transverse aberrations. Consider a single object point and a given position of the image
plane and on this plane consider a given reference point, which is the point where the
image should be. Usually the assumed image point will be the Gaussian image point,
but this is in fact not necessary. The position of the image plane as well as the choice of
the image reference point may differ from the Gaussian values if required. For instance,
one reason for choosing the Gaussian image plane but not the Gaussian image height
may be some distortion if distortion is not of interest. In this case the intersection of the
real chief ray with the image plane may be adequate as the image reference point. A very
common reason for choosing image planes, which are different from the Gaussian
image plane, is to study the behavior of the aberrations with a change in the focus.
There are several ways to present transverse aberrations in graphical form. We
will outline some examples for a very simple system, a single biconvex lens as
shown in figure 29-2. Two field points are considered, one field point is on the axis,
while the second field point is off the axis.
Figure 29-2: Biconvex lens, f ′ = 100 mm, F/5, image height 5 mm.
29.1 Introduction
Spot diagrams are a very popular way of presenting transverse aberrations, see
figures 29-3 and 29-4. The through-focus spot diagrams show the characteristics of
a lot of aberrations and suggest the size of the image blur. However, even with this
simple example, it can be seen that the impression of the ray intersection spots
strongly depends on the chosen ray distribution in the pupil. For figure 29-3 a
rectangular pupil grid is used and for figure 29-4 a polar pupil grid is used.
Image height
5 mm
0 mm
Defocus ( mm ) - 0 . 5 - 0.25 0 0.25 0.5
0.4 mm
Used pupil grid :
Rectangular
Figure 29-3: Through-focus spot diagram with rectangular pupil grid.
Image height
5 mm
0 mm
Defocus (mm) -0.5 -0.25 0 0.25 0.5
0.4 mm
Used pupil grid :
Polar
Figure 29-4: Through-focus spot diagram with polar pupil grid.
3
29 Aberrations
In general it is difficult to distinguish typical aberrations from spot diagrams.
However, for this purpose the transverse aberration fans are an adequate means, see
figure 29-5. Here the transverse aberrations of two pupil sections are represented:
For the on-axis image the ray bundle is symmetric with respect to the optical axis, so
all information is contained in the presentation of the meridional section only. The
meridional pupil section is also called the tangential pupil section. For an image
height of 5 mm, in the tangential pupil section, due to the rotational symmetry of
the system, all aberrations are in the image y-direction. From the sagittal pupil sec-
tion the rays have aberration components y′ in image y-direction as well as compo-
nents x′ in the image x-direction and these components exhibit characteristic sym-
metries (point symmetry and mirror symmetry, respectively) as shown in figure
29-5. Usually, for the sagittal section, only the more important x-component of the
aberration is shown. As will be explained later, the transverse aberration fans in
figure 29-5 are clearly dominated by spherical aberration and also by coma.
Image
height 0 mm 5 mm 5 mm 5 mm
A
berration ∆ y' ∆ y' ∆ x' ∆ y'
Pupil section meridional meridional sagittal sagittal
0.2 mm 0.2 mm 0.2 mm 0.2 mm
Figure 29-5: Transverse ray aberrations.
The use of transverse aberrations is a powerful method and all types of aberration,
such as spherical aberration, coma, astigmatism, field curvature and distortion, as
well as axial chromatic aberration and lateral chromatic aberration can be under-
stood and represented as transverse aberrations.
Nevertheless, there are situations where it is desirable to use longitudinal aberra-
tions. For instance, astigmatism and field curvature are easily understood as longitu-
dinal phenomena. But spherical aberration, coma, and axial chromatic aberration
can also be understood as longitudinal aberrations. Figure 29-6 shows the spherical
aberration and the astigmatic field curves of the system considered in figure 29-2 as
longitudinal aberrations. The only aberrations, which cannot be described as long-
itudinal aberrations, are distortion and lateral chromatic aberration.
4
29.1 Introduction
2.0-2.0 0.0
Spherical aberration Tangential & sagittal field
z ∆ ∆
Aperture
-2.0 2.0
0.0
z
Field
T S
Figure 29-6: Longitudinal ray aberration curves.
For optical systems which image to infinity, the transverse as well as the longitu-
dinal aberrations do not make sense, as they both become infinite. In this case, for
the image at infinity, instead of transverse aberrations measured in length units,
angular aberrations measured in angle units will be adequate. So angle aberrations,
which were introduced in section 11.2, Description of Aberrations, correspond to
transverse aberrations and exhibit the respective properties. Also, for longitudinal
aberrations there is an adequate representation when the image is at infinity:
Instead of the longitudinal aberration itself, the reciprocal value of the intersection
lengths is used. As the unit of dimension diopters are normally used. To express an
intersection length S′ in diopters, S′ should be measured in m. Then the corre-
sponding value
~
S′ in diopters is defined as
~
S′ diopter
1
S′ m
. (29-1)
So aberrations measured in diopters correspond to the longitudinal aberrations. For
instance, in vision optics, diopters are the preferred aberration description.
Transverse and longitudinal ray aberrations are easy to understand, and they rep-
resent a complete and powerful method of describing the aberrational behavior of
an optical imaging system. So, for what reason do we need wave aberrations? There
are, in fact, some specific advantages connected with the understanding and the use
of wave aberrations. The most important benefits of a description based on wave
aberrations are as follows.
.
Wavefront aberrations can be measured very easily and very accurately by
means of interferometric methods. This is a big advantage because the mea-
surement of ray aberrations with any comparable completeness and accuracy
is almost impossible.
5