This appendix gives a recipe for calculating the third order aberrations of a lens when its construction data are known. As an example we use the 4 diopter eye glass corrected for astigmatism described in section 30.6. This is a thin lens with a focal length of 250 mm, and the exit pupil 25 mm to the right of the lens.

The first step is to trace two paraxial rays through the system, as shown in table A2.1. The first ray, called the object ray, comes from the axial point of the object and must be chosen such that its direction in the image space is +1. For the eye glass used as an example the object is located at infinity, so the incident ray must be parallel to the axis, and 250 mm below it to obtain the direction +1 in the image space. The first two columns of numbers in table A2.1 show the data for this ray. The unit of length used is the dm (100 mm); this avoids very large and very small numbers. Lines 1, 2, 3, and 6 contain the radii, refractive indices, and surface spacings of the lens. The incident ray is specified by h = –2.5 dm on line 7 and ψ=0 on line and 10. Filling in lines 11 through 15 of the first column yields the direction of the ray after the first surface, and lines 8 and 9 provide the height of the ray at the next surface. These two numbers, 2.0951 for the new direction and –2.5 for the new height, are transferred to the next column, which is then completed to find the height and direction of the ray as it emerges from the second surface. In our case there are only two surfaces; if there are more surfaces the process is repeated till the last surface is reached. Line 16 must be completed as well.

Basic properties

A concentric system is constructed of spherical surfaces, refracting or reflecting, that all have a common center. A solid sphere is a simple example; two other examples are shown in fig. 29.1. A concentric lens is insensitive to rotations around its center; this high degree of symmetry determines the special properties of concentric systems.

For any incident ray the plane containing the ray and the center of the system divides the lens into two symmetric halves. There is no reason for the ray to prefer one of these halves over the other; so the ray will remain in the symmetry plane as it traverses the lens. It follows that every ray travels in a plane containing the center.

As long as the lens is not afocal the angle eikonal W(L, M, L′, M′) can be used. We choose any straight line through the center as the axis, and locate both the (x, y) reference plane in the object space and the (x′, y′) reference plane in the image space right in the center of the system. Then the angle eikonal is the optical distance from the projection P of the center onto the ray in the object space to the projection P′ of the center onto the ray in the image space. On account of the spherical symmetry of the lens a rotation of the entire ray around the center has no effect on the value of the eikonal function.

This appendix describes a ray tracing scheme that can be used to trace a meridional ray by hand. It is a slight modification of one of T. Smith's ray tracing procedures [74]. In this calculation a ray approaching a surface is specified by the sine of the angle ψ with the axis, and the perpendicular distance h from the vertex of the surface to the ray. The calculation yields the sine of the angle ψ′ with the axis after refraction, and the perpendicular distance h′ from the vertex to the refracted ray. An additional calculation determines the change in h′ as the ray travels to the next surface.

Table A3.1 shows a ray traversing a cemented doublet which happens to have a unit focal length. The radii, refractive indices, and thicknesses are found on lines 1, 3, 4, and 5. The ray enters the lens parallel to the axis; so ψ = 0 for the first surface, as shown on line 11. The entrance height is 0.125, corresponding to an F/4 aperture; it is shown on line 6.

The change in direction is calculated in two steps. Lines 12 through 15 yield an approximate value sinθ for sin ψ′. Paraxially this value would be correct, but to obtain exact results a correction must be made. This correction is calculated in lines 18 through 26, followed by lines 16 and 17. The change from h to h′ is generated at the same time (lines 7 and 8).

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.