Cardinal point (optics)

In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the focal points, the principal points, and the nodal points; there are two of each. For ideal systems, the basic imaging properties such as image size, location, and orientation are completely determined by the locations of the cardinal points. For simple cases where the medium on both sides of an optical system is air or vacuum four cardinal points are sufficient: the two focal points and either the principal points or the nodal points. The only ideal system that has been achieved in practice is a plane mirror, however the cardinal points are widely used to approximate the behavior of real optical systems. Cardinal points provide a way to analytically simplify an optical system with many components, allowing the imaging characteristics of the system to be approximately determined with simple calculations.

The cardinal points lie on the optical axis of an optical system. Each point is defined by the effect the optical system has on rays that pass through that point, in the paraxial approximation. The paraxial approximation assumes that rays travel at shallow angles with respect to the optical axis, so that ${\textstyle \sin \theta \approx \theta }$, ${\textstyle \tan \theta \approx \theta }$, and ${\textstyle \cos \theta \approx 1}$. Aperture effects are ignored: rays that do not pass through the aperture stop of the system are not considered in the discussion below.

The front focal point of an optical system, by definition, has the property that any ray that passes through it will emerge from the system parallel to the optical axis. The rear (or back) focal point of the system has the reverse property: rays that enter the system parallel to the optical axis are focused such that they pass through the rear focal point.

The front and rear (or back) focal planes are defined as the planes, perpendicular to the optic axis, which pass through the front and rear focal points. An object infinitely far from the optical system forms an image at the rear focal plane. For an object at a finite distance, the image is formed at a different location, but rays that leave the object parallel to one another cross at the rear focal plane.

A diaphragm or "stop" at the rear focal plane of a lens can be used to filter rays by angle, since an aperture centred on the optical axis there will only pass rays that were emitted from the object at a sufficiently small angle from the optical axis. Using a sufficiently small aperture in the rear focal plane will make the lens object-space telecentric.

Similarly, the allowed range of angles on the output side of the lens can be filtered by putting an aperture at the front focal plane of the lens (or a lens group within the overall lens), and a sufficiently small aperture will make the lens image-space telecentric. This is important for DSLR cameras having CCD sensors. The pixels in these sensors are more sensitive to rays that hit them straight on than to those that strike at an angle. A lens that does not control the angle of incidence at the detector will produce pixel vignetting in the images.

The two principal planes of a lens have the property that a ray emerging from the lens appears to have crossed the rear principal plane at the same distance from the optical axis that the ray appeared to have crossed the front principal plane, as viewed from the front of the lens. This means that the lens can be treated as if all of the refraction happened at the principal planes, and rays travel parallel to the optical axis between the planes. (Linear magnification between the principal planes is +1.) The principal planes are crucial in defining the properties of an optical system, since the magnification of the system is determined by the distance from an object to the front principal plane and the distance from the rear principal plane to the object's image. The principal points are the points where the principal planes cross the optical axis.

If the medium surrounding an optical system has a refractive index of 1 (e.g., air or vacuum), then the distance from each principal plane to the corresponding focal point is just the focal length of the system. In the more general case, the distance to the foci is the focal length multiplied by the index of refraction of the medium.