Ray (optics)
Encyclopedia
In optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing
Ray tracing (physics)
In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces,...

. This allows even very complex optical systems to be analyzed mathematically or simulated by computer. Ray tracing uses approximate solutions to Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

. Ray theory does not describe phenomena such as interference and diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

, which require wave theory (involving the phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

of the wave).

## Definition

A light ray is a line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

or curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

that is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the light's wavefront
Wavefront
In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...

s (and is therefore collinear with the wave vector
Wave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...

). Light rays bend at the interface between two dissimilar media and may be curved in a medium in which the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

changes. Geometric optics describes how rays propagate through an optical system.

A slightly more rigorous definition of a light ray follows from Fermat's principle
Fermat's principle
In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light...

, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

## Special rays

There are many special rays that are used in optical modelling to analyze an optical system. These are defined and described below, grouped by the type of system they are used to model.

### Interaction with surfaces

• An incident ray is a ray of light that strikes a surface. The angle between this ray and the perpendicular or normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

to the surface is the angle of incidence
Angle of incidence
Angle of incidence is a measure of deviation of something from "straight on", for example:* in the approach of a ray to a surface, or* the angle at which the wing or horizontal tail of an airplane is installed on the fuselage, measured relative to the axis of the fuselage.-Optics:In geometric...

.
• The reflected ray corresponding to a given incident ray, is the ray that represents the light reflected by the surface. The angle between the surface normal and the reflected ray is known as the angle of reflection. The Law of Reflection says that for a specular
Specular reflection
Specular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction is reflected into a single outgoing direction...

(non-scattering) surface, the angle of reflection always equals the angle of incidence.
• The refracted ray or transmitted ray corresponding to a given incident ray represents the light that is transmitted through the surface. The angle between this ray and the normal is known as the angle of refraction, and it is given by Snell's Law
Snell's law
In optics and physics, Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...

. Conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

requires that the power in the incident ray must equal the sum of the power in the transmitted ray, the power in the reflected ray, and any power absorbed at the surface.
• If the material is birefringent
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

, the refracted ray may split into ordinary and extraordinary rays, which experience different indexes of refraction when passing through the birefringent material.

### Optical systems

• A meridional ray or tangential ray is a ray that is confined to the plane containing the system's optical axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

and the object point from which the ray originated.
• A skew ray is a ray that does not propagate in a plane that contains both the object point and the optical axis. Such rays do not cross the optical axis anywhere, and are not parallel to it.
• The marginal ray (sometimes known as an a ray or a marginal axial ray) in an optical system is the meridional ray that starts at the point where the object crosses the optical axis, and touches the edge of the aperture stop of the system. This ray is useful, because it crosses the optical axis again at the locations where an image will be formed. The distance of the marginal ray from the optical axis at the locations of the entrance pupil
Entrance pupil
In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front of the lens system. The corresponding image of the aperture as seen through the back of the lens system is called the exit pupil...

and exit pupil
Exit pupil
In optics, the exit pupil is a virtual aperture in an optical system. Only rays which pass through this virtual aperture can exit the system. The exit pupil is the image of the aperture stop in the optics that follow it. In a telescope or compound microscope, this image is the image of the...

defines the sizes of each pupil (since the pupils are image
Image
An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...

s of the aperture stop).
• The principal ray or chief ray (sometimes known as the b ray) in an optical system is the meridional ray that starts at the edge of the object, and passes through the center of the aperture stop. This ray crosses the optical axis at the locations of the pupils. As such chief rays are equivalent to the rays in a pinhole camera. The distance between the chief ray and the optical axis at an image location defines the size of the image. The marginal and chief rays together define the Lagrange invariant, which characterizes the throughput or etendue
Etendue
Etendue or étendue is a property of pencils of rays in an optical system, which characterizes how "spread out" light is in area and angle. It may also be seen as a volume in phase space....

of the optical system. Some authors define a "principal ray" for each object point. The principal ray starting at a point on the edge of the object may then be called the marginal principal ray.
• A sagittal ray or transverse ray from an off-axis object point is a ray that propagates in the plane that is perpendicular to the meridional plane and contains the principal ray. Saggital rays intersect the pupil along a line that is perpendicular to the meridional plane for the ray's object point and passes through the optical axis. If the axis direction is defined to be the z axis, and the meridional plane is the y-z plane, saggital rays intersect the pupil at yp=0. The principal ray is both sagittal and meridional. All other sagittal rays are skew rays.
• A paraxial ray is a ray that makes a small angle to the optical axis of the system, and lies close to the axis throughout the system. Such rays can be modeled reasonably well by using the paraxial approximation
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 ....

. When discussing ray tracing this definition is often reversed: a "paraxial ray" is then a ray that is modeled using the paraxial approximation, not necessarily a ray that remains close to the axis.
• A finite ray or real ray is a ray that is traced without making the paraxial approximation.
• A parabasal ray is a ray that propagates close to some defined "base ray" rather than the optical axis. This is more appropriate than the paraxial model in systems that lack symmetry about the optical axis. In computer modeling, parabasal rays are "real rays", that is rays that are treated without making the paraxial approximation. Parabasal rays about the optical axis are sometimes used to calculate first-order properties of optical systems.

### Fiber optics

• A meridional ray is a ray that passes through the axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

of an optical fiber
Optical fiber
An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...

.
• A skew ray is a ray that travels in a non-planar zig-zag path and never crosses the axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

of an optical fiber
Optical fiber
An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...

.
• A guided ray, bound ray, or trapped ray is a ray in a multi-mode optical fiber
Multi-mode optical fiber
Multi-mode optical fiber is a type of optical fiber mostly used for communication over short distances, such as within a building or on a campus...

, which is confined by the core. For step index fiber
Step-index profile
For an optical fiber, a step-index profile is a refractive index profile characterized by a uniform refractive index within the core and a sharp decrease in refractive index at the core-cladding interface so that the cladding is of a lower refractive index. The step-index profile corresponds to a...

, light entering the fiber will be guided if it makes an angle with the fiber axis that is less than the fiber's acceptance angle.
• A leaky ray or tunneling ray is a ray in an optical fiber that geometric optics predicts would totally reflect
Total internal reflection
Total internal reflection is an optical phenomenon that happens when a ray of light strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is...

at the boundary between the core and the cladding
Cladding is one or more layers of material of lower refractive index, in intimate contact with a core material of higher refractive index. The cladding causes light to be confined to the core of the fiber by total internal reflection at the boundary between the two. Light propagation in the...

, but which suffers loss due to the curved core boundary.

• Paraxial approximation
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 ....

• Pencil beam
Pencil beam
In optics, a pencil or pencil of rays is a geometric construct used to describe a beam or portion of a beam of electromagnetic radiation or charged particles, typically in the form of a narrow cone or cylinder....

• Ray transfer matrix analysis
Ray transfer matrix analysis
Ray transfer matrix analysis is a type of ray tracing technique used in the design of some optical systems, particularly lasers...