Definition

Aberrations are deviations from the perfect geometrical imaging case. An understanding of the influence and correction of aberrations obviously requires that somewhat more detail be developed.

Ideal image formation requires that the relation between object and image would follow paraxial rules, and all rays from each object point would pass through its paraxial conjugate image point, with all rays having the same optical path length from object to image. The quantitative measure of the aberration at any field location is a spread of ray intercepts on the image plane or an associated optical path difference error evaluated on the exit pupil. The paraxial point is usually taken as the image reference point, and the image errors with real rays measured with respect to this point. Perfect imagery also can be defined as zero wavefront error in which the exiting wavefront coincides with the exit pupil reference sphere. (Additional considerations relate to the distribution of rays in the pupil, as pupil aberrations may indicate a change of aberration with field position, even though the optical path errors are zero at some specified field location.)

The aberration at a given field point produces a distribution of rays about the image reference point, the symmetry of which is determined by the magnitude and combination of aberrations that are present in the lens. In the case of low-order aberrations, such as lateral displacement or a focus position error, the choice of a new reference position can negate the aberration.

We will now deal with the application of the concepts discussed in the previous chapters by applying these methods to the design of a number of lenses. These examples have been selected to provide a view of the techniques used in design at several greater levels of complexity. The majority of this chapter will deal with the detailed design of somewhat traditional, or basic, design problems, as they actually provide the basis for most of the optics done in this world. Some of the newer opportunities for design will be discussed in the latter portion of the chapter.

The goals set for the designs are somewhat arbitrary, but realistic, versions of those encountered in real life. Expanding beyond these goals will serve as a good learning experience for any individual who wishes to increase his or her knowledge of the methods of lens design.

Several different design programs will be used in these examples. A difficulty with this is that it may lead to some confusion as to the meaning of various program-dependent features that are necessary in working with a computer. The great advantage is that the essential nature of the optical problems will emerge. The design programs used, CODE V, OSLO, and ZEMAX, happen to be those used by the author for many years in teaching a course in lens design at the Optical Sciences Center.

Design and analysis of lens systems uses numerical calculations based upon geometrical optics. The calculation of the form of the image requires interpretation of these geometrical results by the use of physical optics. Closedform mathematical solutions are available only in a few relatively simple cases so that all realistic lens design is based upon manipulation of computed evaluations of the imagery produced by a lens.

The geometrical optical model is sufficient to define the properties of image formation by a lens and to relate them to the construction parameters of the lens. The geometrical ray-based model permits determination of image location and aberrations, and enables calculation of the pupil function describing the wavefront. Evaluation of the image quality requires the introduction of concepts of physical optics and wave propagation. The intensity distribution in the image is calculated by applying a diffraction integral to the pupil function.

The basic optics of image location and size is established by the application of paraxial or first-order optics. Ray tracing is the basic tool used in optical design. Aberrations are defined as the deviation of the ray path for real rays from the paraxial basis coordinates. Physical optics and beam propagation extend the geometrical model to include diffraction and interference. Investigation of these models permits determination of the limits on image formation.

Optical glasses are the most commonly used materials, although many other materials have become important in modern optical design.

In the preceding chapters the material necessary for studying photoionization processes in atoms using synchrotron radiation and electron spectrometry was presented. The discussion will now be completed with some examples of current research activities. These include:

1. photon-induced electron emission around the 4d ionization threshold in xenon from which a complete mapping of these spectra can be obtained and many features characteristic of inner- and outer-shell photoprocesses are well visualized;

2. a complete experiment for 2p photoionization in magnesium which also provides a detailed illustration of the role that many-electron effects have on main photolines;

3. an investigation of discrete satellite lines in the outer-shell photoelectron spectrum of argon which demonstrates for a simple case the origin of satellite processes in electron correlations, and also the importance that instrumental resolution has on the determination of satellite structures;

4. a complete experiment for 5p3/2 photoionization in xenon which includes a measurement of the photoelectron's spin polarization;

5. a quantitative study of postcollision interaction (PCI) between 4d5/2 photoelectrons and N5–O2, 3O2, 31S0 Auger electrons in xenon which also serves as an example of energy calibration in accurate experiments;

6. the determination of coincidences between 4d5/2 photoelectrons and N5–O2, 3O2, 31S0 Auger electrons in xenon which allows a spatial view of the angular correlation pattern for this two-electron emission process;

7. a near-threshold study of state-dependent double photoionization in the 3p shell of argon in which the cross section approaches zero and two electrons of extremely low kinetic energy have to be measured in coincidence.

Description of the K–LL Auger spectrum

Inner-shell ionization is accompanied by subsequent radiative and non-radiative decay. In the context of electron spectrometry, the non-radiative or Auger decay is of special interest, because the emitted Auger electron can be detected. After some remarks on the general description and classification of Auger transitions following 1s ionization in neon, the calculation of K–LL Auger transition rates and the formulation of intermediate coupling in the final ionic state of the K–LL Auger transition will be addressed. This information then provides the basis for a detailed analysis of the experimental K–LL Auger spectrum of neon which is organized similarly to the previous discussion of photoelectrons: namely, with respect to line positions, linewidths, line intensities, and angular distributions.

General aspects

In addition to the photoelectron lines, other discrete structures appear in the electron spectrum of neon if the photon energy is higher than the threshold for 1s ionization. These lines are due to radiationless transitions called Auger transitions [Aug25]; the 1s-hole created by photoionization is filled by a subsequent two-electron transition induced by the Coulomb interaction between the electrons. This interaction causes one outer-shell electron to jump down, filling the 1s-hole, simultaneously ejecting another outer-shell electron, the Auger electron, into the continuum. This process has been sketched schematically in Figs. 1.3 and 2.5.