Some Basic Ideas in Geometrical Optics

The Concepts of Geometrical Optics

Geometrical optics is used as the basic tool in designing almost any optical system, image-forming or not. We use the intuitive ideas of a ray of light, roughly defined as the path along which light energy travels, together with surfaces that reflect or transmit the light. When light is reflected from a smooth surface, it obeys the well- known law of reflection, which states that the incident and reflected rays make equal angles with the normal to the surface and that both rays and the normal lie in one plane. When light is transmitted, the ray direction is changed according to the law of refraction: Snell’s law. This law states that the sine of the angle between the normal and the incident ray bears a constant ratio to the sine of the angle between the normal and the refracted ray; again, all three directions are coplanar.

A major part of the design and analysis of concentrators involves ray-tracing— that is, following the paths of rays through a system of reflecting and refracting surfaces. This is a well-known process in conventional lens design, but the requirements are somewhat different for concentrators, so it will be convenient to state and develop the methods ab initio. This is because in conventional lens design the reflecting or refracting surfaces involved are almost always portions of spheres, and the centers of the spheres lie on one straight line (axisymmetric optical system) so that special methods that take advantage of the simplicity of the forms of the surfaces and the symmetry can be used. Nonimaging concentrators do not, in general, have spherical surfaces. In fact, sometimes there is no explicitly analytical form for the surfaces, although usually there is an axis or a plane of symmetry. We shall find it most convenient, therefore, to develop ray-tracing schemes based on vector formulations but with the details covered in computer programs on an ad hoc basis for each different shape.

In geometrical optics we represent the power density across a surface by the density of ray intersections with the surface and the total power by the number of rays. This notion, reminiscent of the useful but outmoded “lines of force” in electrostatics, works as follows. We take N rays spaced uniformly over the entrance aperture of a concentrator at an angle of incidence в, as shown in Figure 2.1.

Suppose that after tracing the rays through the system only N' emerge through the exit aperture, the dimensions of the latter being determined by the desired concentration ratio. The remaining N - N' rays are lost by processes that will become clear when we consider some examples. Then the power transmission for the angle 0 is taken as N7N. This can be extended to cover a range of angle в as required. Clearly, N must be taken largely enough to ensure that a thorough exploration of possible ray paths in the concentrator is made.

FIGURE 2.1 Determining the transmission of a concentrator by ray-tracing.

Formulation of the Ray-Tracing

To formulate a ray-tracing procedure suitable for all cases, it is convenient to put the laws of reflection and refraction into vector form. Figure 2.2 shows the geometry with unit vectors r and r" along the incident and reflected rays and a unit vector n along the normal pointing into the reflecting surface. To see this, we know that n bisects -r" and r, so that r" - r is parallel to n, say a n, to find a, we dot both sides by n, and a follows.

Thus, to ray-trace “through” a reflecting surface, first we have to find the point of incidence, a problem of geometry involving the direction of the incoming ray and the known shape of the surface. Then we have to find the normal at the point of incidence—again a problem of geometry. Finally, we have to apply Equation (2.1) to find the direction of the reflected ray. The process is then repeated if another reflection is

FIGURE 2.2 Vector formulation of reflection, r. r" and it are all unit vectors.

FIGURE 2.3 The stages in ray-tracing a reflection, (a) Find the point of incidence P. (b) Find the normal at P. (c) Apply Equation (2.1) to find the reflected ray r<.

to be taken into account. These stages are illustrated in Figure 2.3. Naturally, in the numerical computation the unit vectors are represented by their components—that is, the direction cosines of the ray or normal with respect to some Cartesian coordinate system used to define the shape of the reflecting surface. Ray-tracing through a refracting surface is similar, but first we have to formulate the law of refraction vec- torially. Figure 2.4 shows the relevant unit vectors. It is similar to Figure 2.2 except that r' is a unit vector along the refracted ray.

We denote by «, n’ the refractive indexes of the media on either side of the refracting boundary; the refractive index is a parameter of a transparent medium to the speed of light in the medium. Specifically, if c is the speed of light in a vacuum, the speed in a transparent material medium is c/n, where n is the refractive index. For visible light, values of n range from unity to about 3 for usable materials in the visible spectrum. The law of refraction is usually stated in the form

FIGURE 2.4 Vector formulation of refraction.

where / and /' are the angles of incidence and refraction, as in the figure, and where the coplanarity of the rays and the normal is understood. The vector formulation

contains everything, since the modulus of a vector product of two unit vectors is the sine of the angle between them. This can be put in the form most useful for raytracing by multiplying through vectorially by n to give

which is the preferred form for ray-tracing[1] The complete procedure then parallels that for reflection explained by means of Figure 2.3. We find the point of incidence, then the direction of the normal, and finally the direction of the refracted ray. Details of the application to lens systems are given, for example, by Welford (1974, 1986).

If a ray travels from a medium of refractive index n toward a boundary with another of index n' then as can be seen from Equation (2.2) it would be possible to have sin Г greater than unity. Under this condition it is found that the ray is completely reflected at the boundary. This is called total internal reflection, and we shall find it a useful effect in concentrator design.

Elementary Properties of Image-Forming Optical Systems

In principle, the use of ray-tracing tells us all there is to know about the geometrical optics of a given optical system, image-forming or not. However, ray-tracing alone is

FIGURE 2.5 A thin converging lens bringing parallel rays to a focus. Since the lens is technically “thin,” we do not have to specify the exact plane in the lens from which the focal length/is measured.

of little use for inventing new systems having properties suitable for a given purpose. We need to have ways of describing the properties of optical systems in terms of general performance, such as, for example, the concentration ratio C introduced in Chapter 1. In this section we shall introduce some of these concepts. Consider first a thin converging lens such as one that would be used as a magnifier or in eyeglasses for a farsighted person (see Figure 2.5). By “thin” we mean that its thickness can be neglected for the purposes of our discussion. Elementary experiments show us that if we have rays coming from a point at a great distance to the left, so that they are substantially parallel as in the figure, the rays meet approximately at a point F, the focus. The distance from the lens to F is called the focal length, denoted by / Elementary experiments also show that if the rays come from an object of finite size at a great distance, the rays from each point on the object converge to a separate focal point, and we get an image. This is, of course, what happens when a burning glass forms an image of the sun or when the lens in a camera forms an image on film. This is indicated in Figure 2.6, where the object subtends the (small) angle 2q. It is then found that the size of the image is 2/9. This is easily seen by considering the rays through the center of the lens, since these pass through undeviated. Figure 2.6 contains one of the fundamental concepts we use in concentrator theory, the concept of a beam of light of a certain diameter and angular extent.

The diameter is that of the lens—say, 2a—and the angular extent is given by 20. These two can be combined as a product, usually without the factor 4, giving Oa, a quantity known by various names including extent, etendue, acceptance, and Lagrange invariant. It is, in fact, an invariant through the optical system, provided that there are no obstructions in the light beam and provided we ignore certain losses due to properties of the materials, such as absorption and scattering.

For example, at the plane of the image the etendue becomes the image height Of multiplied by the convergence angle alf of the image-forming rays, giving again Oa. In discussing 3D systems—for example, an ordinary lens such as we have supposed Figure 2.6 to represent—it is convenient to deal with the square of this quantity, a^-O1. This is also sometimes called the etendue, but generally it is clear from the

FIGURE 2.6 An object at infinity has an angular subtense 2(9. A lens of focal length/forms an image of size 2/(9.

FIGURE 2.7 An optical system of acceptance, throughput, or etendue a2q2.

context and from dimensional considerations which form is intended. The 3D form has an interpretation that is fundamental to the theme of this book. Suppose we put an aperture of diameter 2fO at the focus of the lens, as in Figure 2.7. Then this system will only accept rays within the angular range ±0 and inside the diameter 2a. Now suppose a flux of radiation В (in Win"2 sr') is incident on the lens from the left[2] The system will actually accept a total flux (Bp202a2)W thus, the etendue or acceptance 02a2 is a measure of the power flow' that can pass through the system.

The same discussion show's how the concentration ratio C appears in the context of classical optics. The accepted power (Bj)202a2)W must flow out of the aperture to the right of the system, if our preceding assumptions about how' the lens forms an image are correct[3] and if the aperture has the diameter 2fO. Thus, our system is

(2a)2 fa)2

acting as a concentrator with concentration ratio C = - = — for the input

semiangle в. V2/#J l/®,

Let us relate these ideas to practical cases. For solar energy collection we have a source at infinity that subtends a semiangle of approximately 0.005 rad (1/4°) so that this is the given value of q, the collection angle. Clearly, for a given diameter of lens we gain concentration by reducing the focal length as much as possible.

• [1] The method of using Equation (2.4) numerically is not so obvious as it is for Equation (2.2), since thecoefficient of n in Eq (2.4) is actually n'cos/'-«cos/. Thus, it might appear that we have to findr' before we can use the equation. The procedure is to find cos /' via Equation (2.2) first, and thenEquation (2.4) is needed to give the complete three-dimensional picture of the refracted ray.
• [2] In full, В watts per square meter per steradian solid angle.
• [3] As we shall see, these assumptions are only valid for limitingly small apertures and objects.