# Aberrations in Image-Forming Optical Systems

According to the simplified picture presented in Section 2.3, there is no reason why we could not make a lens system with an indefinitely large concentration ratio by simply decreasing the focal length sufficiently. This is, of course, not so, partly because of aberrations in the optical system and partly because of the fundamental limit on concentration stated in Section 1.2.

We can explain the concept of aberrations by looking again at our example of the thin lens in Figure 2.5. We suggested that the parallel rays all converged after passing through the lens to a single point *F.* In fact, this is only true in the limiting case when the diameter of the lens is taken as indefinitely small. The theory of optical systems under this condition is called paraxial optics or Gaussian optics, and it is a very useful approximation for getting at the main large-scale properties of imageforming systems. If we take a simple lens with a diameter that is a sizable fraction of the focal length—say,//4—we find that the rays from a single point object do not all converge to a single image point. We can show this by ray-tracing. We first set up a proposed lens design, as shown in Figure 2.8. The lens has curvatures (reciprocals of radii) c, and *c _{2},* center thickness

*d,*and refractive index

*n.*If we neglect the central thickness for the moment, then it is shown in specialized treatment (e.g., Welford, 1986) that the focal length/is given in paraxial approximation by

and we can use this to get the system to have roughly the required paraxial properties.

Now we can trace rays through the system as specified, using the method outlined in Section 2.2 (details of ray-tracing methods for ordinary lens systems are given in, for example, Welford, 1974). These will be exact or finite rays, as opposed to paraxial rays, which are implicit in the Gaussian optics approximation. The results for the lens in Figure 2.8 would look like Figure 2.9. This shows rays traced from an object point on the axis at infinity—that is, rays parallel to the axis.

In general, for a convex lens the rays from the outer part of the lens aperture meet the axis closer to the lens than the paraxial rays do. This effect is known as spherical aberration. (The term is misleading, since the aberration can occur in systems witli nonspherical refracting surfaces, but there seems little point in trying to change it at the present advanced state of the subject.)

FIGURE 2.8 Specification of a single lens. The curvature cl is positive as shown, and c2 is negative.

FIGURE 2.9 Rays near the focus of a lens showing spherical aberration.

Spherical aberration is perhaps the simplest of the different aberration types to describe, but it is just one of many. Even if we were to choose the shapes of the lens surfaces so as to eliminate the spherical aberration or were to eliminate it in some other way, we would still find that the rays from object points away from the axis do not form point images—in other words, there w'ould be oblique or off-axis aberrations.

Also, the refractive index of any material medium changes with the wavelength of the light, and this produces chromatic aberrations of various kinds. We do not at this stage need to go into the classification of aberrations very deeply, but this preliminary sketch is necessary to show the relevance of aberrations to the attainable concentration ratio.

# The Effect of Aberrations in an Image-Forming System on the Concentration Ratio

Questions regarding the extent to which it is theoretically possible to eliminate aberrations from an image-forming system have not yet been fully answered. In this book we shall attempt to give answers adequate for our purposes, although they may not be what the classical lens designers w'ant. For the moment, let us accept that it is possible to eliminate spherical aberration completely, but not the off-axis aberrations, and let us suppose that this has been done for the simple collector of Figure 2.7. The effect will be that some rays of the beam at the extreme angle *0* will fall outside the defining aperture of diameter 2*fO.* We can see this more clearly by representing an aberration by means of a spot diagram. This is a diagram in the image plane with points plotted to represent the intersections of the various rays in the incoming beam. Such a spot diagram for the extreme angle *в* might appear as in Figure 2.10. The ray through the center of the lens (the principal ray in lens theory) meets the rim of the collecting aperture by definition, and thus a considerable amount of the flux does not get through. Conversely, it can be seen (in this case at least) that some flux from beams at a larger angle than *в* will be collected.

We display this information on a graph such as in Figure 2.11. This shows the proportion of light collected at different angles up to the theoretical maximum, *6 _{aax}. *An ideal collector would behave according to the full line—that is, it would collect all light flux within

*в*and none outside. At this point it may be objected that all we need to do to achieve the first requirement is to enlarge the collecting aperture slightly, and the second requirement does not matter. However, we recall that our aim is to achieve maximum concentration because of the requirement for high-operating temperature so that the collector aperture must not be enlarged beyond 2

_{тш}*fO*diameter.

Frequently in discussions of aberrations in books on geometrical optics, the impression is given that aberrations are in some sense “small.” This is true in optical systems designed and made to form reasonably good images, such as camera lenses. But these systems do not operate with large enough convergence angles *(a/f* in the notation for Figure 2.6) to approach the maximum theoretical concentration ratio. If we were to try to use a conventional image-forming system under such conditions, we would find that the aberrations would be very large and that they would severely depress the concentration ratio. Roughly, we can say that this is one limitation that

FIGURE 2.10 A spot diagram for rays from the beam at the maximum entry angle for an image-forming concentrator. Some rays miss the edge of the exit aperture due to aberrations, and the concentration is thus less than the theoretical maximum.

FIGURE 2.11 A plot of collection efficiency against angle. The ordinate is the proportion of flux entering the collector aperture at angle q that emerges from the exit aperture.

has led to the development of the new, nonimaging concentrators. Nevertheless, we cannot say that imaging-forming is incompatible with attaining maximum concentration. We will show, later, examples in which both properties are combined.

# The Optical Path Length and Fermat’s Principle

There is another way of looking at geometrical optics and the performance of optical systems, which we also need to outline for the purposes of this book. We noted

FIGURE 2.12 Rays and (in broken line) geometrical wave fronts.

FIGURE 2.13 Fermat’s principle. It is assumed in the diagram that the medium has a continuously varying refractive index. The solid line path has a stationary optical path length from *A* to *В* and is therefore a physically possible ray path.

in Section 2.2 that the speed of light in a medium of refractive index *n* is *c/n,* where *c* is the speed in a vacuum. Thus, light travels a distance s in the medium in time *s/v=ns/c* that is, the time taken to travel a distance 5 in a medium of refractive index *n* is proportional to *ns.* The quantity *ns* is called the optical path length corresponding to the length *s.* Suppose we have a point source *О* emitting light into an optical system, as in Figure 2.12. We can trace any number of rays through the system, as outlined in Section 2.2, and then we can mark off along these rays points that are all at the same optical path length from *О*—say, *P,P2...* We do this by making the sum of the optical path lengths from *О* in each medium the same—that is,

in an obvious notation. These points can be joined to form a surface (we are supposing rays out of the plane of the diagram to be included), which would be a surface of constant phase of the light waves if we were thinking in terms of the wave theory of light^{[1]} We call it a geometrical wave front, or simply a wave front, and we can construct wave fronts at all distances along the bundle of rays from *O.*

We now introduce a principle that is not as intuitive as the laws of reflection and refraction but that leads to results that are indispensable to the development of the theme of this book. It is based on the concept of optical path length, and it is a way of predicting the path of a ray through an optical medium. Suppose we have any optical medium that can have lenses and mirrors and can even have regions of continuously varying refractive index. We want to predict the path of a light ray between two points in this medium—say, *A* and *В* in Figure 2.13. We can propose an infinite number of possible paths, of which three are indicated. But unless *A* and *В* happen to be object and image—and we assume they are not—only one or perhaps a small finite number of paths will be physically possible—in other words, paths that rays of light could take according to the laws of geometrical optics. Fermat’s principle in the form most commonly used states that a physically possible ray path is one for which the optical path length from *A* to *В* is an extremum as compared to neighboring paths. For “extremum” we can often write “minimum,” as in Fermat’s original statement. It is possible to derive all of geometrical optics—that is, the laws of refraction and reflection—from Fermat’s principle. It also leads to the result that the geometrical wave fronts are orthogonal to the rays (the theorem of Malus and Dupin); that is, the rays are normal to the wave fronts. This in turn tells us that if there is no aberration—if all rays meet at one point—then the wave fronts must be portions of spheres. So if there is no aberration, the optical path length from object point to image point is the same along all rays. Thus, we arrive at an alternative way of expressing aberrations: in terms of the departure of wave fronts from the ideal spherical shape. This concept will be useful when we come to discuss the different senses in which an image-forming system can form “perfect” images.

- [1] This construction does not give a surface of constant phase near a focus or near an edge of an opaqueobstacle, but this does not affect the present applications.