Cardinal point (optics)
The cardinal points and the associated cardinal planes are a set of special points and planes in an optical system, which help in the analysis of its paraxial properties. The analysis of an optical system using cardinal points is known as Gaussian optics, named after Carl Friedrich Gauss.
The cardinal points and planes of an optical system include:
- The focal points and focal planes
- The principal planes and principal points
- The surface vertices (or vertexes)
- The nodal points
For a lens, there will be two of each of these, identified by "front" and "rear" depending on whether they are on the input or the output side of the lens, respectively.
These points and planes, together with the aperture stop, and the chief and marginal rays of the system, define the locations and sizes of the entrance and exit pupils of the system, as well as its other image-forming properties, such as the focal length and magnification.
More detailed and accurate analysis of an optical system's performance can be achieved by raytracing, either within the paraxial approximation or using "real rays", i.e. rays that refract and reflect according to Snell's law and the law of reflection, without approximation.
The cardinal points lie on the optical axis of the optical system. Each point is defined by the effect the optical system has on rays that pass through that point, in the paraxial approximation. Aperture effects are ignored—rays that do not pass through the aperture stop of the system are ignored in the discussion below.
Focal points and planes
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 an infinite distance away from the optical system forms an image at the rear focal plane. For objects a finite distance away, the image is formed at a different location, but rays that leave the object parallel to one another cross at the rear focal plane.
An aperture at the rear focal plane can be used to filter rays by angle, since:
- It only allows rays to pass that are emitted at an angle (relative to the optical axis) that is sufficiently small. (An infinitely small aperture would only allow rays that are emitted along the optical axis to pass.)
- No matter where on the object the ray comes from, the ray will pass through the aperture as long as the angle at which it is emitted from the object is small enough.
Note that the aperture must be centered on the optical axis for this to work as indicated. Using a sufficiently small aperture in the focal plane will make the lens 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). 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.
Principal planes and points
The two principal planes have the property that a ray emerging from the lens appears to have crossed the rear principal plane at the same distance from the axis that that ray appeared to cross 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. The principal planes are crucial in defining the optical properties of the system, since it is the distance of the object and image from the front and rear principal planes that determines the magnification of the system. The principal points are the points where the principal planes cross the optical axis.
If the medium surrounding the optical system has a refractive index of 1 (e.g., air or vacuum), then the distance from the principal planes to their corresponding focal points 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.
For a thin lens in air, the principal planes both lie at the location of the lens. The point where they cross the optical axis is sometimes misleadingly called the optical centre of the lens. Note, however, that for a real lens the principal planes do not necessarily pass through the centre of the lens, and in general may not lie inside the lens at all.
The surface vertices are the points where each surface crosses the optical axis. They are important primarily because they are the physically measurable parameters for the position of the optical elements, and so the positions of the other cardinal points must be known with respect to the vertices to describe the physical system.
The front and rear nodal points have the property that a ray aimed at one of them will be refracted by the lens such that it appears to have come from the other, and with the same angle with respect to the optical axis. The nodal points therefore do for angles what the principal planes do for transverse distance. If the medium on both sides of the optical system is the same (e.g., air), then the front and rear nodal points coincide with the front and rear principal planes, respectively.
The nodal points are widely misunderstood in photography, where it is commonly asserted that the light rays "intersect" at "the nodal point", that the iris diaphragm of the lens is located there, and that this is the correct pivot point for panoramic photography, so as to avoid parallax error. These claims are all false, and generally arise from confusion about the optics of camera lenses, as well as confusion between the nodal points and the other cardinal points of the system. The correct pivot point for panoramic photography can be shown to be the centre of the system's entrance pupil.
Notes and references
- Kerr, Douglas A. (2005). "The Proper Pivot Point for Panoramic Photography" (PDF). The Pumpkin. Retrieved 2006-03-05.
- van Walree, Paul. "Misconceptions in photographic optics". Retrieved 2007-01-01. Item #6.
- Littlefield, Rik (2006-02-06). "Theory of the “No-Parallax” Point in Panorama Photography" (pdf). ver. 1.0. Retrieved on 2007-01-14.
- Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN 0-8194-5294-7.
- Hecht, Eugene (1987). Optics (2nd ed. ed.). Addison Wesley. ISBN 0-201-11609-X.
- Lambda Research Corporation (2001). OSLO Optics Reference (PDF) (Version 6.1 ed.). Retrieved 2006-03-05. Pages 74–76 define the cardinal points.
- Learn to use TEM
- The Grid — Alain Hamblenne's method for a precise location of the entrance pupil on a DSLR camera