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In
Gaussian optics Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. ...
, the cardinal points consist of three pairs of points located on the
optical axis 300px, Optical axis (coincides with red ray) and rays symmetrical to optical axis (pair of blue and pair of green rays) propagating through different lenses. An optical axis is a line along which there is some degree of rotational symmetry in an opt ...
of a rotationally symmetric, focal, optical system. These are the focal points, the principal points, and the nodal points. For ''ideal'' systems, the basic imaging properties such as image size, location, and orientation are completely determined by the locations of the cardinal points; in fact only four points are necessary: the focal points and either the principal or nodal points. The only ideal system that has been achieved in practice is the
plane mirror A plane mirror is a mirror with a flat (planar) reflective surface. For light rays striking a plane mirror, the angle of reflection equals the angle of incidence. The angle of the incidence is the angle between the incident ray and the surface nor ...

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 a system with many components, allowing the imaging characteristics of the system to be approximately determined with simple calculations.


Explanation

The cardinal points lie on the
optical axis 300px, Optical axis (coincides with red ray) and rays symmetrical to optical axis (pair of blue and pair of green rays) propagating through different lenses. An optical axis is a line along which there is some degree of rotational symmetry in an opt ...
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 In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optical a ...
. The paraxial approximation assumes that rays travel at shallow angles with respect to the optical axis, so that \sin\theta\approx\theta and \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.


Focal 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 infinitely far from the optical system forms an
image An SAR radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island. Lava flows a ...

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. A diaphragm or "stop" 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 300px, Optical axis (coincides with red ray) and rays symmetrical to optical axis (pair of blue and pair of green rays) propagating through different lenses. An optical axis is a line along which there is some degree of rotational symmetry in an opt ...
) 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 camera A digital single-lens reflex camera (digital SLR or DSLR) is a digital camera that combines the optics and the mechanisms of a single-lens reflex camera with a digital imaging sensor. The reflex design scheme is the primary difference between a D ...
s 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 photography and optics, vignetting (; french: vignette) is a reduction of an image's brightness or saturation toward the periphery compared to the image center. The word ''vignette'', from the same root as ''vine'', originally referred to ...
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 the 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, and the linear magnification from one principal plane to the other is +1. 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 Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one, it refers to a reduction in size, ...
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 In optics, the refractive index (also known as refraction index or index of refraction) of a material is a dimensionless number that describes how fast light travels through the material. It is defined as :n = \frac, where ''c'' is the speed of l ...
of 1 (e.g., air or
vacuum A vacuum is space devoid of matter. The word stems from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss id ...
), then the distance from the principal planes to their corresponding focal points is just the
focal length#REDIRECT Focal length#REDIRECT Focal length {{R from other capitalisation ...
{{R from other capitalisation ...

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 A lens may be considered a thin lens if its thickness is much less than the radii of curvature of its surfaces ( and )., 300px In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the 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.


Nodal points

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. (Angular magnification between nodal points is +1.) 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 points, respectively. The nodal points are widely misunderstood in
photography Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It is employed in ...
, where it is commonly asserted that the light rays "intersect" at "the nodal point", that the
iris diaphragm In optics, a diaphragm is a thin opaque structure with an opening (aperture) at its center. The role of the diaphragm is to ''stop'' the passage of light, except for the light passing through the ''aperture''. Thus it is also called a stop (an ap ...

iris diaphragm
of the lens is located there, and that this is the correct pivot point for
panoramic photography Panoramic photography is a technique of photography, using specialized equipment or software, that captures images with horizontally elongated fields of view. It is sometimes known as ''wide format photography''. The term has also been applied to a ...
, so as to avoid
parallax Parallax () is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby object ...
error. These claims 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. (A better choice of the point about which to pivot a camera for panoramic photography can be shown to be the centre of the system's
entrance pupil . The entrance pupil is the image of the physical aperture, as seen through the front (the object side) of the lens. The size and location may differ from those of the physical aperture, due to magnification by the lens. Image:030608 Pupil.jpg, The a ...
. Item #6. On the other hand, swing-lens cameras with fixed film position rotate the lens about the rear nodal point to stabilize the image on the film.Searle, G.F.C. 191
''Revolving Table Method of Measuring Focal Lengths of Optical Systems''
in "Proceedings of the Optical Convention 1912" pp. 168–171.
)


Surface vertices

The surface vertices are the points where each optical 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 cardinal points must be known with respect to the vertices to describe the physical system. In
anatomy Anatomy (Greek ''anatomē'', 'dissection') is the branch of biology concerned with the study of the structure of organisms and their parts. Anatomy is a branch of natural science which deals with the structural organization of living things. It ...
, the surface vertices of the eye's
lens A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), us ...
are called the anterior and posterior ''poles'' of the lens.


Modeling optical systems as mathematical transformations

In
geometrical optics Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances. Th ...
for each ray entering an optical system a single, unique, ray exits. In mathematical terms, the optical system performs a
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Transfo ...
that maps every object ray to an image ray. The object ray and its associated image ray are said to be ''conjugate to'' each other. This term also applies to corresponding pairs of object and image points and planes. The object and image rays and points are considered to be in two distinct optical spaces, ''object space'' and ''image space''; additional intermediate optical spaces may be used as well.


Rotationally symmetric optical systems; Optical axis, axial points, and meridional planes

An optical system is rotationally symmetric if its imaging properties are unchanged by ''any'' rotation about some axis. This (unique) axis of rotational symmetry is the
optical axis 300px, Optical axis (coincides with red ray) and rays symmetrical to optical axis (pair of blue and pair of green rays) propagating through different lenses. An optical axis is a line along which there is some degree of rotational symmetry in an opt ...
of the system. Optical systems can be folded using plane mirrors; the system is still considered to be rotationally symmetric if it possesses rotational symmetry when unfolded. Any point on the optical axis (in any space) is an ''axial point''. Rotational symmetry greatly simplifies the analysis of optical systems, which otherwise must be analyzed in three dimensions. Rotational symmetry allows the system to be analyzed by considering only rays confined to a single transverse plane containing the optical axis. Such a plane is called a ''meridional plane''; it is a cross-section through the system.


Ideal, rotationally symmetric, optical imaging system

An ''ideal'', rotationally symmetric, optical imaging system must meet three criteria: #All rays "originating" from ''any'' object point converge to a single image point (Imaging is ''stigmatic''). #Object planes perpendicular to the optical axis are
conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
to image planes perpendicular to the axis. #The image of an object confined to a plane normal to the axis is geometrically similar to the object. In some optical systems imaging is stigmatic for one or perhaps a few object points, but to be an ideal system imaging must be stigmatic for ''every'' object point. Unlike rays in mathematics, optical rays extend to infinity in both directions. Rays are ''real'' when they are in the part of the optical system to which they apply, and are ''virtual'' elsewhere. For example, object rays are real on the object side of the optical system. In stigmatic imaging an object ray intersecting any specific point in object space must be conjugate to an image ray intersecting the conjugate point in image space. A consequence is that every point on an object ray is conjugate to some point on the conjugate image ray. Geometrical similarity implies the image is a scale model of the object. There is no restriction on the image's orientation. The image may be inverted or otherwise rotated with respect to the object.


Focal and afocal systems, focal points

In afocal systems an object ray parallel to the optical axis is conjugate to an image ray parallel to the optical axis. Such systems have no focal points (hence ''afocal'') and also lack principal and nodal points. The system is focal if an object ray parallel to the axis is conjugate to an image ray that intersects the optical axis. The intersection of the image ray with the optical axis is the focal point F' in image space. Focal systems also have an axial object point F such that any ray through F is conjugate to an image ray parallel to the optical axis. F is the object space focal point of the system.


Transformation

The transformation between object space and image space is completely defined by the cardinal points of the system, and these points can be used to map any point on the object to its conjugate image point.


See also

*
Film plane A film plane is the area inside any camera or image taking device with a lens and film or digital sensor upon which the lens creates the focused image. The film plane varies in distance from the lens focal point in cameras from different manufacture ...
*
Pinhole camera model . The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ''ideal'' pinhole camera, where the camera aperture is described as a poi ...
*
Radius of curvature (optics) Radius of curvature (ROC) has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis. The vertex of the lens surf ...
*
Vergence (optics) Image:Vergence.svg, Vergence of a beam. The vergence is inversely proportional to the distance from the focus in metres. If a (positive) lens is focussing the beam, it has to sit left of the focus, while a negative lens has to sit right of the foc ...


Notes and references

* * Pages 74–76 define the cardinal points.


External links


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Geometrical optics{{Commons category, Geometric optics Geometrical optics deals with rays and their propagation in media. Optics ...
Geometric centers Science of photography de:Brennebene