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Cardinal Rays and Planes of a Lens

Late in January we began by considering the limitations of lens systems - namely aberrations, but only in a very general way. However, before we start detailed discussions, let me first make it clear that to every optical person in the world, light ALWAYS travels from left to right. It’s just one of those conventions that is universally accepted (and assumed in any optical discussion).

In order to continue our discussion of lenses we need to define several key elements that are common to ALL lenses. These key elements focus on two very critical or “cardinal” rays.

Consider the schematic telescope sketched below. The axial ray is the first cardinal ray to consider. It starts at the axial point of the object and grazes the edge of the aperture stop. The aperture stop is the aperture within the lens that limits the axial bundle of rays coming from the axial point of the object. When I adjust the F/stop of my camera, I am adjusting this aperture. It has a clearly defined place within the lens that cannot be relocated whimsically. We will also soon see that adjusting this aperture stop does not affect the field of view of the lens.

The chief ray is the second cardinal ray and starts at the edge of the field in the object plane and crosses the optical axis at the aperture stop. Notice that, like the axial ray bundle, the chief ray bundle is clipped or “vignetted” symmetrically by the aperture stop. In a narrow field of view system, the chief ray bundle and the axial ray bundle convey very nearly the same amount of energy, assuming that the object is uniformly radiating. This may not be the case as the field of view expands.

While the aperture stop is depicted as a surface in the diagram above, it is in reality, an aperture or “diaphragm”. It is the component that is always adjusted to achieve a different F/# in a lens. Once again, it is the aperture in the lens that limits the axial ray bundle, and therefore determines the irradiance (often mistakenly referred to as the “brightness”) of the image.

Stops, Pupils, and Principle Planes

In addition to the aperture stop of a system, which we have already seen, must be placed at a point that is symmetrically filled by both the axial ray bundle and the chief ray bundle, there are other stops and pupils to a lens. But let me be clear that pupils are simply constructs, not physical apertures. The entrance pupil is defined as the image of the aperture stop formed by the combination of lens elements on the object side of the aperture stop. The exit pupil is similarly defined as the image of the aperture stop formed by the elements on the image side of the aperture stop. These pupils are simply designated planes in space whose locations and diameters are linked to the aperture stop by the intervening lens elements. They are useful because rays that intercept these pupils are assured (in the absence of vignetting) of penetrating the aperture stop as well. It should be noted that since the aperture stop determines the irradiance of the image plane, and the pupils are “proportional” to the aperture stop, the pupil irradiance can also be used to define the irradiance in the image.

Finally the principle planes of a lens are important to understand as it is from these that we determine the true focal length of a lens, and hence the magnification. The rear principle plane is simply a focal length from the image, and in a multi-element lens this is rarely to the rear-most element. (This distance from the image to the rear lens element vertex is generally referred to as the “back focal distance”.) It can lie either somewhere within the lens assembly or may be in the space between the lens assembly and the image plane. As always, the focal length (often referred to as the “effective focal length” or EFL) is determined using an object at “infinity” (defined as many focal lengths away from the lens).

If the lens is flipped end for end, the front principle plane is found in the same manner. It is a focal length away from the “new” image formed by the reversed lens.

Ray tracing may clarify this description. If we trace a ray in object space parallel with the optical axis, then the plane in which it intersects its own final trajectory to the image (after being refracted by all the elements of the lens) is the rear principle plane. The distance from this plane to the image plane is the effective focal length, or EFL.

So let’s finish up the discussion of stops by briefly looking at other types of stops sometimes employed within a lens assembly.

A field stop is a physical diaphragm that can only be used within an assembly that forms an internal image. A rifle telescope is a good example, as are binoculars. (The first diagram on this page shows two places within the assembly where a field stop might be placed.) The field stop limits the field coverage of the instrument by blocking rays farther out in the field of view. Why would one use a field stop? Often the aberrations cannot be economically controlled far out in the field, and the internal field stop is used to block the peripheral part of the image that is highly aberrated, as well as to stop this light from decreasing the contrast in the desired part of the image.

A glare stop is another physical diaphragm that is placed within the assembly to block rays that are reflected by the internal structure of the lens barrel. The best location is generally at one of the pupil locations if they are internal to the assembly. However baffles (also physical apertures) can be used for the same purpose, but have no prescribed location within the assembly. Often these are positioned and sized by trial and error, or by modeling likely scattered ray paths within the assembly using a process referred to as “non-sequential ray tracing”. Baffles are more commonly used since they may be employed in systems with pupils that are external to the lens assembly.

These are the cardinal rays and planes of a more complex lens assembly. A simple single thin lens packs all the stops and pupils essentially at the lens aperture, but generally at the expense of much more severe aberrations.

The next blog will look at the effects of bending a single thin lens on the aberration content, and will also look at the means of combining two lenses of different glass types in order to balance the chromatic aberrations of the combination.

So ‘til next time….

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