Optical Engineering Science. Stephen Rolt

Figure 1.5 Generalised optical system and conjugate points.

      1.3.1 Conjugate Points and Perfect Image Formation

We consider an ideal optical system which consists of a point source of light, the object, and an optical system that collects the light and re-directs all rays emanating from this point source or object, such that the rays converge onto a single point, the image point. At this stage, the interior workings of the optical system are undefined; the system behaves as a ‘black box’. The object is said to be located in object space and the image in image space and the pair of points are said to be conjugate points. This is illustrated in Figure 1.5.

      In Figure 1.5, the two points P₁ and P₂ are conjugate. The optical system can be simple, for example a single lens, or it can be complex, containing many optical elements. The description above is entirely generalised. Where the object point lies on the optical axis, its image or conjugate point also lies on the optical axis. In Figure 1.5, the object point has a height of h₁ with respect to the optical axis and its corresponding image point has a height of h₂ with respect to the same axis. The ratio of these two heights gives the system (transverse) magnification, M:

(1.5)

      Points occupying a plane perpendicular to the optical axis are conjugate to points lying on another plane perpendicular to the optical axis. These planes are known as conjugate planes.

      1.3.2 Infinite Conjugate and Focal Points

Where an image or object is located at infinity, all rays emerging from or travelling to these locations will be parallel with respect to each other. In this instance, the point located at infinity is said to be at an infinite conjugate. The corresponding conjugate point to the infinite conjugate is known as a focal point. There are two focal points. The first focal point is located in the object space with the corresponding image located at the infinite conjugate. The second focal point is located in the image space with the object placed at the infinite conjugate. Figure 1.6 depicts the first focal point:

Figure 1.6 Location of first focal point.

As well as focal points, there are two corresponding focal planes. The two focal planes are planes perpendicular to the optical axis that contain the relevant focal point. For all points lying on the relevant focal plane, the conjugate point will lie at the infinite conjugate. In other words, all rays will be parallel with respect to each other. In general, the rays will not be parallel to the optic axis. This would only be the case for a conjugate point lying on the optical axis.

      1.3.3 Principal Points and Planes

All points lying on a particular conjugate plane are associated with a specific transverse magnification, M, which is equal to the ratio of the image and object heights. For an ideal system, there exist two conjugate planes where the magnification is unity. These are known as the principal planes. Thus, there are two principal planes and the points where the optical axis intersects the principal planes are known as principal points. The first principal point (plane) is located in object space and the second principal point (plane) is located in image space. The arrangement is illustrated schematically in Figure 1.7.

Figure 1.7 Principal points and principal planes.

      1.3.4 System Focal Lengths

The reader might be used to ascribing a single focal length to an optical system, such as for a magnifying lens or a camera lens. However, in this general description, the system has two focal lengths. The first focal length, f₁, is the distance from the first focal plane (or point) to the first principal plane (or point) and the second focal length, f₂, is the distance from the second principal plane to the second focal plane. In many cases, f₁ and f₂ are identical. In fact, the ratio f₁/f₂ is equal to n₁/n₂, the ratio of the refractive indices of the media associated with the object and image spaces. However, this need not concern us at this stage, as the treatment presented here is entirely general and independent of the specific attributes of components or media.

In classical geometrical optics, the object location is denoted by the object distance, u, and the image location by the image distance, v. In the context of this general description, the object distance is simply the distance from the object to the first principal plane. Correspondingly, the image distance, v, is the distance from the second principal plane to the image. In addition, the object location can be described by the distance, x₁, separating the object from the corresponding focal plane. Similarly, x₂ represents the distance from the image to the second focal plane. This is illustrated in Figure 1.8.

      1.3.5 Generalised Ray Tracing

      This general description of an optical system is very economical in that the definition of conjugate points, focal planes, and principal planes provides sufficient information to determine the path of a ray in the image space, given the path of the ray in the object space. No assumptions are made about the internal workings of the optical system; it is merely a ‘black box’.

We see how input rays originating in the object space are mapped onto the image space for specific scenarios where the object is located at the input focal plan, the infinite conjugate, or the first principal plane. How can this be extended to determine the output path of any input ray? The general principle is set out in Figure 1.9. First, the input ray is traced from point P₁ as far as its intersection with the (first) principal plane at A₁. We know that this point, A₁, is conjugated with point A₂, lying at the same height at the second principal plane. This follows directly from the definition of principal planes. Second, we draw a dummy ray originating from the first focal point, f₁, but parallel to the input ray and trace it to where it intersects the first principal plane at B₁. We know that B₁ is conjugated with point B₂, lying at the same height on the second principal plane. Since this ray originated from the first focal point, its path must be parallel to the optical axis in image space and thus we can trace it as far as the second