The theorem of Malus and Dupin

The vast majority of lenses are used for image formation. Camera lenses, eye glasses, binoculars, and all sorts of other lens systems are useful because they form, each in their own way, an image of an object. To begin a discussion of the image forming process we choose a fixed source point P0 in the object space of a lens that we wish to study, and follow the rays emerging from this point all the way through the lens to the image space. One might hope that the rays emerging into the image space pass through a single point again, but this is rarely the case. Usually each of the emerging rays intersects the nominal image plane at a slightly different point. It is the task of the lens designer to bring, for a specified set of conditions, these intersection points as close together as possible.

Although the emerging rays do not usually pass through one point, the ray patterns in the image space created by a single source point in the object space are not completely arbitrary. Fermat's principle imposes an important restriction. To analyze this restriction we choose in the image space any convenient Cartesian coordinate system (x, y, z) and introduce the path function E(x, y) from the source point P0 to points (x, y, 0) in the z = 0 coordinate plane.

Characterizing a lens

A ray can be specified by one of its points and its direction at that point. In a homogeneous medium we can, for instance, select a ray by specifying the coordinates (x, y) of its point of intersection with the plane z = 0, and the first two components (L, M) of the unit vector in the direction of the ray. The third direction cosine N is not needed because it is the square root of (1 – L2 – M2). The number of parameters needed to specify a ray is clearly four: x, y, L, and M.

Now consider a ray as it approaches a lens. It enters the lens, travels through the lens along a possibly tortuous path, and leaves the lens to enter the image space. Each of the four parameters needed to give a complete description of the ray in the image space is determined by the four parameters that specify the ray in the object space. It seems to follow that four functions of four variables are needed to characterize the lens fully: x′, y′, L′, and M′, each as a function of x, y, L, and M. In this chapter we shall see that this conclusion is incorrect; on account of Fermat's principle one single function of four variables is all that is required. As a result, Fermat's principle puts a severe constraint on the imaging processes a lens can carry out.

Many books on modern optics confine the treatment of lens theory to the paraxial approximation. Aberrations are treated casually as an afterthought, and it usually remains unexplained whether they are due to the laws of physics or due to the limited art of the lens designer. The reader is often left with the notion that lenses must be designed to obey the laws of Gaussian optics as closely as possible. This is regrettable, because it has been known since the eighteenth century that paraxial optics used with finite heights and angles leads to projective geometry, a valuable branch of mathematics which is, however, a poor representation of the behavior of actual lenses.

The development of Fourier optics over the last forty years has brought lens theory and physics much closer together, but again many insights are lost because most authors, in spite of the ubiquity of high aperture lenses in the laboratory, are content with the small angle approximation when dealing with the theory of image formation. Either a clear and convincing demonstration should be given that the small angle approximation can be used with impunity for very large angles, or the theory should be developed honestly, without the small angle approximation. This honest theory exists, but is buried in books and papers providing so much detail that beginners are apt to get lost in the mathematical intricacies.

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).