Introduction

It has been stressed in previous chapters that perfection at more than one magnification is impossible. Even so, lenses are often used for a variety of object and image distances. Camera lenses as well as enlarger lenses need to form sharp images over a wide range of conjugates. Even high power microscope objectives, notoriously sensitive to variations in object distance, are occasionally pressed into use for three-dimensional imaging. How can we deal with this paradox?

The explanation is that images need not be perfect. All we need is images that are sharp enough to utilize fully the finite resolution of the recording medium. Photographic film is limited by the size of the grain; CCDs are limited by the finite gate size; the retina of the eye is limited by the size of the rods and cones; etc.

For an analysis of incompatible lens requirements it is convenient to describe a lens by one of its eikonal functions. This provides all the information needed to calculate its aberrations at any magnification. The calculations are straightforward, at least in principle: choose a set of rays originating in a specified object point, determine their continuation in the image space, and see where they intersect the image plane.

Unfortunately these calculations can hardly ever be carried out in closed form. The central problem is to calculate x′, y′, L′, and M′ when x, y, L, and M are specified.

Perfect imaging

Aberrations are deviations from perfect image formation. Ideally all the rays that come from any point in the object space should intersect at a single point in the image space. This is, however, not a realistic design goal, as was already intimated in chapter 1, and discussed in more detail in chapter 6. According to Maxwell's theorem, proved in section 6.5, it is incompatible with Fermat's principle for a lens with a finite power to form a perfectly sharp image of more than one object plane. We shall therefore call a lens ‘perfect’ if it forms a perfectly sharp image not of the entire object space, but of a single object surface, plane or curved. The image may be plane or curved as well.

The statement ‘it is incompatible with Fermat's principle that…’ leaves it unclear how the truth or falsity of the statement should be proved. It is preferable to use the assertion ‘no eikonal can be constructed so that …’. As an example we give a second proof of Maxwell's theorem. Take a lens with axial symmetry, and describe it by the angle eikonal W(L, M, L′, M′) from front focal plane to back focal plane. For a magnification β′ the object distance z, measured from the front focal plane, is n/β′A, in which A is the power of the lens. The image distance z′, measured from the back focal plane, is —n′β′/A.

Small fields and large apertures

The theory of third order aberrations is based on the assumption that the aperture and the field of the lens are small enough to neglect terms of degree higher than four in the power series development of the eikonal. In practice this condition is rarely met, and yet the third order theory is quite important in the practice of lens design. The reason is that more often than not small changes in the construction parameters have a much greater effect on the third order aberrations than on the aberrations associated with the higher order terms in the eikonal. As a result it is often a useful design strategy first to make the third order aberrations zero, then to evaluate the magnitude of the higher order aberrations and to reduce them as far as possible, and finally to make minor changes in the construction parameters to introduce small amounts of third order aberration that compensate, as far as possible, for the higher order aberrations that are impervious to all attempts at correction. A numerical recipe to calculate the third order aberrations from the construction parameters is shown in appendix 2; practical routines for the fifth order aberrations may be found in [12]. The total aberrations of a lens are usually calculated by ray tracing, to be discussed in the next chapter. Several commercial computer codes are available to carry out all these calculations.

For certain lens types the series development used so far is wholly inappropriate.

Newton's theory of colours

The modification theory of colours

Why is one object red and another blue? Aristotle believed colours to be a mixture of light and darkness. In his view an object is white when all the light striking it is reflected, without the addition of any darkness, and an object is black because it reflects none of the light falling upon it. The colours of objects derive from the mingling of light and darkness in varying proportions. Darkness may originate in something opaque or, as in the case of the rainbow, in an opaque medium, such as the clouds. Red, the purest colour, is a mixture of light and a small amount of darkness. As the amount of darkness increases, green is observed and eventually violet, the ‘darkest’ colour. The other colours consist of a combination of red, green, and violet, the three primary hues. It is fundamental to this interpretation that colours are a modification of pure and homogeneous white light, resulting from the addition of darkness.

The modification theory of colours, which - like so many of Aristotle's ideas - seemed to fit so well with direct observation, was generally accepted until the second half of the seventeenth century, although with variations. Some writers assumed the existence of two or of four primary colours; others opted for three, but chose different hues than Aristotle's. For example, Athanasius Kircher selected yellow, red, and blue.

Introduction

In the standard historiography of science the eighteenth century is the period in which the emission conception of light was quite generally accepted, certainly after 1740. Euler is usually mentioned as the exception to this rule. Surveys that are more oriented towards Germany add one or another dissident to the list but leave the image unaltered on the whole: The emission tradition ruled the physical optics roost in the eighteenth century. Apparently the picture of the situation in different countries is to a great extent determined by simply declaring the general picture to be valid for every country, without any thorough investigation of the matter. Britain and Ireland are the only countries on which detailed and systematic research has been carried out. G. N. Cantor has provided an exhaustive survey of optical viewpoints in this region. His results do, it is true, lend nuances to the established image, but they introduce no radical change. In Cantor's book only 9 per cent out of a total of sixty-nine optical theorists from the eighteenth century support a medium theory, while the remaining 91 per cent can be located within the emission tradition. In other words, the historical evidence thus far available confirms the strongly dominant position of the emission tradition in the eighteenth century. Nevertheless, it will be argued in this section that a substantially different view of the matter ought to be given for Germany.

Development of the emission tradition

Newton's suggestions

The theoretical tradition in physical optics in which light is regarded as an emission of matter goes back at least as far as Greek Antiquity and has experienced a renaissance in modern times. Newton was a major representative of the emission tradition in seventeenthcentury optics. In the Enlightenment period his ideas dominated this tradition, although he never held the position of an absolute ruler. Despite his unmistakably great importance for the eighteenth-century emission tradition, Newton had a curious status within it. He never unreservedly endorsed the emission hypothesis, in print at any rate. This is connected with his approach to methodology: He attempted to abstain from combining certainties with doubts. Consequently Newton's ideas on the emission of light are mainly encountered in the form of queries forming a supplement to his Opticks. His ideas remained suggestions that were unconnected with one another, provided no solution to some problems, and were occasionally inconsistent. In addition, they changed over time. For this reason we need a detailed study of the different versions of the ‘queries’ in the various editions of the Opticks if we are to obtain a clear and accurate account of Newton's ideas on the nature of light and of the later attempts to systematize his suggestions. The first edition (Opticks, 1704), the first Latin edition (Optice, 1706), and the second English edition of 1717, are the most important for our purposes.

When Leonhard Euler published his treatise ‘Nova theoria lucis et colorum’ (A new theory of light and colours) in 1746, he made a contribution to the medium tradition in physical optics that was without parallel in the eighteenth century. The ‘Nova theoria’ constitutes the most lucid, comprehensive, and systematic medium theory of that century. The significance of Euler's theory can be gauged partly from the fact that it was so widely discussed. No earlier attempts to provide an alternative to the theories developing within the emission tradition had stimulated pens to the same extent as Euler's ‘Nova theoria’. It is remarkable that a relatively short treatise published as part of a collection of articles on a range of subjects created such resonances, and all the more so when we compare this reaction with the limited response to Huygens' Traite - a complete book - and with the way in which Johann II Bernoulli's prize essay was virtually ignored. A partial explanation no doubt lies in the quality of Euler's work, while his authority also made it difficult to ignore his ‘new theory’.

Euler was probably the most important, or at any rate the most fecund, exponent of mathematics and natural philosophy in the Enlightenment. Although his accomplishments in mathematics and mechanics are generally known and acknowledged, Euler's contributions to optics have attracted little scholarly attention up to the present. However, his role in the discovery of achromatic lenses has been the subject of historical study.

In his survey of historical literature on ‘experimental natural philosophy’ in the eighteenth century, J. L. Heilbron rejects attempts to make the rise of ‘Newtonianism’ and its ultimate triumph over ‘Cartesianism’ the guiding historiographical principle. The results reached in the present study on physical optics parallel Heilbron's argument. We have observed that, in the first half of the eighteenth century, the influence of the opposition between ‘Newtonianism’ and ‘Cartesianism’ in the discussion on the nature of light, also perceptible, was certainly not a dominant feature. Furthermore, there is confirmation for Heilbron's remark that the disciplinary borders of eighteenth-century experimental natural philosophy, especially optics, were subject to change and that it is precisely these changes that can provide us with a new historiographical guideline.

If we pursue the latter suggestion further, we meet with a general thesis advanced by T. S. Kuhn in 1975. Among all the available alternatives, this thesis is to my eyes the most suited to making the various developments in the eighteenth century comprehensible and to collecting them within one general viewpoint. However, when I attempted to use Kuhn's point of view for the clarification of eighteenth-century optics, I found that I could not use it as it stood. In this epilogue I shall propose a corrective to Kuhn's ideas, one that will be illustrated by material derived from the previous chapters. Before doing so, I shall present Kuhn's thesis itself, and examine the way in which he and others using it have described the development of optics.

The aim of this work is to make a two-fold contribution to the study of eighteenth-century science. The majority of this book is devoted to a description and analysis of the conceptual development of physical optics in the period, focussing on the origins, contents, and reception of Leonhard Euler's wave theory of light. There will always be a second question in the background of the narrative, which will receive full attention in the last chapter: What does a study of eighteenth-century optics have to teach us about the changing nature of natural philosophy and science in that period?

The title of this study - Optics in the Age of Euler - constitutes a response to the still generally accepted historical image of optics in which the eighteenth century is portrayed as the century of Newton. According to the standard account, ‘Newton's’ particle, or emission, theory of light dominated for more than a century, whereas ‘Huygens’ wave, or medium, theory supposedly did not develop and found few supporters during the same period. This study provides a corrective to this image, with the Swiss mathematician and natural philosopher Leonhard Euler (1707-83) a leading figure in the new historiographical drama. Euler's importance derives from his “Nova theoria lucis et colorum” (A new theory of light and colours), published in 1746. This article was the foremost eighteenth-century contribution to the development of the medium theories of light. Euler's theory of light, rather than Huygens' theory, was the first serious rival to the emission theories.

Resonant energy transfer collisions, those in which one atom or molecule transfers only internal energy, as oppposed to translational energy, to its collision partner require a precise match of the energy intervals in the two collision partners. Because of this energy specificity, resonant collisional energy transfer plays an important role in many laser applications, the He–Ne and CO2 lasers being perhaps the best known examples. It is interesting to imagine an experiment in which we can tune the energy of the excited state of atom B through the energy of the excited state of atom A, as shown in Fig. 14.1. At resonance we would expect the cross section for collisionally transferring the energy from an excited A atom to a ground state B atom to increase sharply as shown in Fig. 14.1. In general, atomic and molecular energy levels are fixed, and the situation of Fig. 14.1 is impossible to realize. Nonetheless systematic studies of resonant energy transfer have been carried out by altering the collision partner, showing the importance of resonance in collisional energy transfer.

The use of atomic Rydberg states, which have series of closely spaced levels, presents a natural opportunity for the study of resonant collisional energy transfer. One of the earliest experiments was the observation of resonant rotational to electronic energy transfer from NH3 to Xe Rydberg atoms by Smith et al.

My intent in writing this book is to present a unified description of the many properties of Rydberg atoms. It is intended for graduate students and research workers interested in the properties of Rydberg states of atoms or molecules. In many ways it is similar to the excellent volume Rydberg States of Atoms and Molecules edited by R. F. Stebbings and F. B. Dunning just over a decade ago. It differs, however, in covering more topics and in being written by one author. I have attempted to focus on the essential physical ideas. Consequently the theoretical developments are not particularly formal, nor is there much emphasis on the experimental details.

The constraints imposed by the size of the book and my energy have forced me to limit the topics covered in this book to those of general interest and those about which I already knew something. Consequently, several important topics which might well have been included by another author are not included in the present volume. Two examples are molecular Rydberg states and cavity quantum electro-dynamics.

Finally, it is a great pleasure to acknowledge the fact that this book would never have been written without the efforts of many people. First I would like to acknowledge the help of my colleagues in the Molecular Physics Laboratory of SRI International (originally Stanford Research Institute).