Introduction

In chapter 9 we derived the laws of paraxial optics by developing the angle eikonal of a lens into a Taylor series and keeping the quadratic terms only. We now investigate the result of taking along the fourth degree terms as well. We base our treatment on T. Smith's celebrated 1921/2 paper ‘The changes in aberrations when the object and stop are moved’ [43]. Other approaches may be found in, for example, Herzberger [7], [8], Buchdahl [11], [12], and Luneburg [27].

A word about the notation. So far it has been shown explicitly how the refractive indices of the object space and the image space enter into the formulas. In this chapter we take a different approach. We assume that all distances in the object space, transverse as well as axial, are multiplied by the object space refractive index, and similarly that all distances in the image space are multiplied by the image space refractive index. In other words: lengths are expressed in units that are a constant multiple of the local wavelength. This is almost always a useful notation; the only exception (and the reason that we have not adhered to it throughout this book) is the paraxial calculations discussed in chapter 10. It is useful to introduce a reduced magnification G as well.

Introduction

In this chapter we begin to forge a connection between the ray theory and the wave theory of light, two topics that so far have been treated as entirely separate and disconnected. Fresnel showed in the early nineteenth century how the idea of straight line propagation can be reconciled with the wave theory by using what are now called Fresnel zones. His reasoning went as follows. A source point S radiates a spherical wavefront towards a circular aperture, as shown in fig. 16.1. To find the amplitude of the light wave at a point P beyond the aperture, each point in the part of the wavefront not stopped by the screen may be considered as a secondary source. The amplitude at P is the sum of the amplitudes contributed by each of these secondary sources. In this summation the relative phase of the contributions plays a crucial role.

To get a handle on the summation, Fresnel divided the wavefront into annular zones. These zones are bounded by circles, chosen such that successive distances SQ1P, SQ2P, SQ3P… differ by half a wavelength. It is not difficult to show that the areas of the zones so constructed are very nearly equal. So the waves arriving at P coming from two adjacent zones have the same amplitude, but a phase difference of 180° on account of the way in which the zones were constructed. The contributions from adjacent zones therefore cancel each other.

Axial symmetry

In the previous chapters (sections 1.3, 6.6, 7.6) we have seen several cases of the image quality becoming progressively worse as the angles between the rays and the axis of the lens are increased. In this chapter we assume that the angles between the rays and the axis are so small that the images formed are essentially perfect. The resulting approximate theory of lenses is called the paraxial approximation, or Gaussian optics. We use in this chapter an abstract method based on eikonal function theory. In the next chapter we use a more down to earth approach, which links the paraxial properties of a lens to its radii, thicknesses, and refractive indices. Later on, when we deal with the problem of wave propagation through lenses, it will become clear why we need both these approaches.

The discussion will be restricted to lenses with axial symmetry around the z-axis. This restricts the possible forms of the eikonal functions, as we now demonstrate for the angle eikonal. As a first step we replace the four variables L, M, L′, and M′ by four angles. In the object space we use the slope angle ψ between the ray and the z-axis, and the azimuth angle φ between the x-axis and the projection of the ray onto the (x, y) plane. In the image space we use similar variables ψ′ and φ′.

Introduction

Geometrical optics is based on the concept that light travels along rays. Rays are lines in space that satisfy Fermat's principle, which states that light travels from one point to another along a path for which the travel time is stationary with respect to small variations in the shape of the path. The theory of geometrical optics can be founded on other ideas as well; for instance, Bruns based his classic paper [5] on the law of Malus and Dupin, which states that a fan of rays jointly perpendicular to a surface in the object space emerges from the lens with again all its rays jointly perpendicular to a surface. We could also simply start with Snell's law. All these starting points are logically equivalent; but broad insights into the properties and limitations of lenses can be obtained most easily by starting with Fermat's principle. One conclusion we can draw immediately: in a homogeneous medium light travels along straight lines.

The speed of light depends on the medium traversed. The ratio of the speed in vacuum and the speed in a medium is called the refractive index, usually denoted by the symbol n. In an inhomogeneous medium the speed, and therefore the refractive index, varies from point to point. In an anisotropic medium the speed depends on the direction of propagation, which makes the specification of the refractive index rather more complicated. Except for propagation in vacuum, the speed of light always depends on the wavelength.

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.