# Geometrical Optics

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## Geometrical optics

The geometry of light rays and their images, through optical systems. In the modern view of the wave nature of light, geometrical optics as a model is simply wrong. In spite of this geometrical optics is remarkably robust, remaining as a most practical tool in the solution of optical problems where at first glance it would seem to be totally inappropriate. The principal application of geometrical optics remains in the field of optical design.

Light is a form of energy which flows from a source to a receiver. It consists of particles (corpuscles) called photons. The speed with which the particles travel depends on the medium. In a vacuum this speed is 3 × 10^{8} m · s^{-1}(1.86 × 10^{5} mi · s^{-1}) for all colors. In a material medium, whether gas, liquid, or solid, light travels more slowly. Moreover, different colors travel at different rates. The ratio of the speed in a vacuum to the speed in the medium is called the refractive index of the medium. The variation in refractive index with color is called dispersion. *See* Color, Dispersion (radiation), Refraction of waves

The paths that particles take in going from the source to the receiver are called rays. The product of the refractive index and the path length is called the optical path length along the ray. The optical path length is equal to the distance that the particle would have traveled in a vacuum in the same time interval.

A point source is an infinitesimal region of space which emits photons. An extended source is a dense array of point sources. Each point source emits photons along a family of rays associated with it. For each such family of rays there is also a family of surfaces, each of which is a surface of constant transit time from the source for all the particles, or alternatively, a surface of constant optical path length from the source. These surfaces are called geometrical wavefronts, because they are often good approximations to the wavefronts predicted by a wave theory.

The ray path which any particle takes as it propagates is determined by Fermat's principle, which states that the ray path between any two points in space is that path along which the optical path length is stationary (usually a minimum) among all neighboring paths. In a homogeneous medium (one with a constant refractive index) the ray paths are straight lines.

In a system that consists of a sequence of separately homogeneous media with different refractive indices and with smooth boundaries between them (such as a lens system), the ray paths are straight lines in each medium, but the directions of the ray paths will change in passing through a boundary surface. This change in direction is called refraction, and is governed by Snell's law, which states that the product of the refractive index and the sine of the angle between the normal to the surface and the ray is the same on both sides of a surface separating two media. The normal in question is at the point where the ray intersects the surface.

The primary area of application of geometrical optics is in the analysis and design of image-forming systems. An optical image-forming system consists of one or more optical elements (lenses or mirrors) which when directed at a luminous (light-emitting) object will produce a spatial distribution of the light emerging from it which more or less resembles the object. The latter is called an image. *See* Optical image

In order to judge the performance of the system, it is first necessary to have a clear idea of what constitutes ideal behavior. Departures from this ideal behavior are called aberrations, and the purpose of optical design is to produce a system in which the aberrations are small enough to be tolerable. *See* Aberration (optics)

In an ideal optical system the rays from every point in the object space pass through the system so that they converge to or diverge from a corresponding point in the image space. This corresponding point is the image of the object point, and the two are said to be conjugate to each other (object and image functions are interchangeable).

The geometry of the object and image spaces must be connected by some mapping transformation. The one generally used to represent ideal behavior is the collinear transformation. If three object points lie on the same straight line, they are said to be collinear. If the corresponding three image points are also collinear, and if this relationship is true for all sets of three conjugate pairs of points, then the two spaces are connected by a collinear transformation. In this case, not only are points conjugate to points, but straight lines and planes are conjugate to corresponding straight lines and planes.

Another feature usually incorporated in the ideal behavior is the assumption that all refracting or reflecting surfaces in the system are figures of revolution about a common axis, and this axis of symmetry applies to the object-image mapping as well. Every object plane containing the axis, called a meridional plane, has a conjugate which is a meridional plane coinciding with the object plane. In addition, every object plane perpendicular to the axis must have a conjugate image plane which is also perpendicular to the axis, because of axial symmetry. In the discussion below, the terms object plane and image plane refer to planes perpendicular to the axis.

An object line parallel to the axis will have a conjugate line which either intersects the axis in image space or is parallel to it. The first case is called a focal system, and the second an afocal system.

The point of intersection of the image-space conjugate line of a focal system with the axis is called the rear focal point (see illustration). It is conjugate to an object point on axis at infinity. The image plane passing through the rear focal point is the rear focal plane, and it is conjugate to an object plane at infinity. Every other object plane has a conjugate located at a finite distance from the rear focal point, except for one which will have its image at infinity. This object plane is the front focal plane, and its intersection with the axis is the front focal point.

Now take an arbitrary object plane and its conjugate image plane. Select a point off axis in the object plane and construct a line parallel to the axis passing through the off-axis point. The conjugate line in image space will intersect the axis at the rear focal point and the image plane at some off-axis point. The distance of the object point from the axis is called the object height, and the corresponding distance for the image point is the image height. The ratio of the image height to the object height is called the transverse magnification, and is positive or negative according to whether the image point is on the same or the opposite side of the axis relative to the object point.

Every pair of conjugate points has associated with it a unique transverse magnification. The conjugate pair which have a transverse magnification of +1 are called the (front and rear) principal planes. The intersections of the principal plane with the axis are called the principal points. The distance from the rear principal point to the rear focal point is called the rear focal length, and likewise for the front focal length. *See* Focal length

The focal points and principal points are four of the six gaussian cardinal points. The remaining two are a conjugate pair, also on axis, called the nodal points. They are distinguished by the fact that any conjugate pair of lines passing through them make equal angles with the axis. The function of the cardinal points and their associated planes is to simplify the mapping of the object space into the image space.

In addition to the transverse magnification, the concept of longitudinal magnification is useful. If two planes are separated axially, their conjugate planes are also separated axially. The longitudinal magnification is defined as the ratio of the image plane separation to the object plane separation.

In the case of afocal systems, any line parallel to the axis in object space has a conjugate which is also parallel to the axis. The transverse and longitudinal magnifications are constant for the system. Cardinal points do not exist for afocal systems.

The most common use of an afocal system is as a telescope, where both the object and the image are at infinity. The angular magnification is defined as the ratio of the transverse magnification to the longitudinal magnification. The power of a telescope or a pair of binoculars is the magnitude of the angular magnification. *See* Magnification, Telescope

Real optical systems cannot obey the laws of the collinear transformation, and departures from this ideal behavior are identified as aberrations. However, if the system is examined in a region restricted to the neighborhood of the axis, the so-called paraxial region, where angles and their sines are indistinguishable from their tangents, a behavior is found which is exactly congruent with the collinear transformation. Paraxial ray tracing can therefore be used to determine the ideal collinear properties of the system.

The above discussion does not take into account the fact that the sizes of the elements of any optical system are finite, and the light that can get through the system to form the image is limited. A circular aperture which limits the cone of rays from an axial object point that gets through the system and participates in image formation is called the aperture stop of the system. An observer who looks into the front of the system from the axial object point sees not the aperture stop itself (unless it is in front of the system), but the image of it formed by the elements preceding it. This image of the aperture stop is called the entrance pupil, and is situated in the object space on the system. The image of the aperture stop formed by the rear elements is the exit pupil, and is situated in the image space of the system.

## Geometrical Optics

a branch of optics in which the laws of propagation of light are studied on the basis of concepts of rays of light; a ray of light is taken to be the line of propagation of a flow of light energy. This concept is consistent with reality only to the extent that light diffraction on optical inhomogeneities can be neglected, which is permissible only when the wavelength is much smaller than the dimensions of the inhomogeneities. The laws of geometrical optics make it possible to create a simplified but, in most cases, sufficiently accurate theory of optical systems. For the most part it explains the formation of optical images and makes possible the calculation of aberrations in optical systems, the development of methods for correcting them, and the derivation of the energy relationships for light beams passing through optical systems. At the same time, not all wave phenomena (including diffraction phenomena) that affect the quality of images and determine the resolving power of optical instruments are treated in geometrical optics.

The concept of light rays originated in the science of antiquity. Euclid, by summarizing the achievements of his predecessors, formulated the principle of rectilinear light propagation and the principle of mirror reflection of light. Geometrical optics developed vigorously during the 17th century in connection with the invention of a variety of optical instruments (the terrestrial telescope, the magnifying glass, the telescope, and the microscope) and the beginning of their general use. A major role in this development was played by J. Kepler, R. Descartes, and W. Snell (who discovered Snell’s law of the refraction of light). The construction of theoretical principles until the mid-17th century was completed by the establishment of Fermat’s principle, which asserts that a ray of light passing from one point through several mediums with arbitrary boundaries and varying indexes of refraction arrives at another point in minimum (more accurately, extremal) time. For a homogeneous medium, Fermat’s principle reduces merely to the law of rectilinear propagation of light. The laws of refraction and reflection, which historically were discovered earlier, are also results of this principle, which has played an important role in the development of other branches of physical theory. After 1700 the methods of geometrical optics for calculating optical systems were improved, and they developed as an applied science. After the creation of classical electrodynamics it was shown that the formulas of geometrical optics could be derived from Maxwell’s equations as a limiting case where the wavelength is vanishingly short.

Geometrical optics is an example of a theory that has provided many important practical results from a small number of fundamental concepts and laws (the assumption of light rays and the laws of reflection and refraction). It has retained to the present time great significance in the theory of optical equipment.

### REFERENCE

Landsberg, G. S.*Optika*, 4th ed. Moscow, 1957. (

*Obshchii kurs fiziki*, vol. 3.)