In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light 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.^[1] This allows even very complex optical systems to be analyzed mathematically or simulated by computer. Ray tracing uses approximate solutions to 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. Ray theory does not describe phenomena such as interference and diffraction, which require wave theory (involving the phase of the wave).

Definition[link]

A light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore collinear with the wave vector). Light rays bend at the interface between two dissimilar media and may be curved in a medium in which the refractive index 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, 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.^[2]

Special rays[link]

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[link]

Diagram of rays at a surface, where Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_\mathrm i

is the angle of incidence, Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_\mathrm r
is the angle of reflection, and Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_\mathrm R

is the angle of refraction.

• An incident ray is a ray of light that strikes a surface. The angle between this ray and the perpendicular or normal to the surface is the angle of incidence.
• 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 (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. Conservation of energy 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, the refracted ray may split into ordinary and extraordinary rays, which experience different indexes of refraction when passing through the birefringent material.

Optical systems[link]

• A meridional ray or tangential ray is a ray that is confined to the plane containing the system's optical axis and the object point from which the ray originated.^[3]
• 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.^[3]
• 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.^[4][5] 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 and exit pupil defines the sizes of each pupil (since the pupils are images 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.^[4][6] 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 of the optical system.^[7] 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.^[5]
• 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.^[3] 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 y_p=0. The principal ray is both sagittal and meridional.^[3] 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.^[8] Such rays can be modeled reasonably well by using the paraxial approximation. 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.^[9][10]
• A finite ray or real ray is a ray that is traced without making the paraxial approximation.^[10][11]
• A parabasal ray is a ray that propagates close to some defined "base ray" rather than the optical axis.^[12] 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.^[13]

Fiber optics[link]

• A meridional ray is a ray that passes through the axis of an optical fiber.
• A skew ray is a ray that travels in a non-planar zig-zag path and never crosses the axis of an optical fiber.
• A guided ray, bound ray, or trapped ray is a ray in a multi-mode optical fiber, which is confined by the core. For step index fiber, 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 at the boundary between the core and the cladding, but which suffers loss due to the curved core boundary.

See also[link]

Wikimedia Commons has media related to: Ray (optics)

• Paraxial approximation
• Pencil beam
• Ray transfer matrix analysis

References[link]

This section should be summarized and a link to Optics provided by using the main template per the guidance in Wikipedia:Summary style.

Geometrical optics, or ray optics, describes light propagation in terms of "rays". The "ray" in geometric optics is an abstraction, or "instrument", which can be used to approximately model how light will propagate. Light rays are defined to propagate in a rectilinear path as far as they travel in a homogeneous medium. Rays bend (and may split in two) at the interface between two dissimilar media, may curve in a medium where the refractive index changes, and may be absorbed and reflected. Geometrical optics provides rules, which may depend on the color (wavelength) of the ray, for propagating these rays through an optical system. This is a significant simplification of optics that fails to account for optical effects such as diffraction and interference. It is an excellent approximation, however, when the wavelength is very small compared with the size of structures with which the light interacts. Geometric optics can be used to describe the geometrical aspects of imaging, including optical aberrations.

Explanation[link]

As light travels through space, it oscillates in amplitude. In this image, each maximum amplitude crest is marked with a plane to illustrate the wavefront. The ray is the arrow perpendicular to these parallel surfaces.

A light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore collinear with the wave vector).

A slightly more rigorous definition of a light ray follows from Fermat's principle, 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.^[1]

Geometrical optics is often simplified by making the paraxial approximation, or "small angle approximation." The mathematical behavior then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.^[2]

Reflection[link]

Diagram of specular reflection

Glossy surfaces such as mirrors reflect light in a simple, predictable way. This allows for production of reflected images that can be associated with an actual (real) or extrapolated (virtual) location in space.

With such surfaces, the direction of the reflected ray is determined by the angle the incident ray makes with the surface normal, a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal.^[3] This is known as the Law of Reflection.

For flat mirrors, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. (The magnification of a flat mirror is equal to one.) The law also implies that mirror images are parity inverted, which is perceived as a left-right inversion.

Mirrors with curved surfaces can be modeled by ray tracing and using the law of reflection at each point on the surface. For mirrors with parabolic surfaces, parallel rays incident on the mirror produce reflected rays that converge at a common focus. Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing the focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration. Curved mirrors can form images with magnification greater than or less than one, and the image can be upright or inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen.^[3]

Refraction[link]

This article should include a summary of or be summarized in another article. See Wikipedia:Summary style for information on how to incorporate it into this article's main text, or the main text of another article.

Illustration of Snell's Law

Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction Failed to parse (Missing texvc executable; please see math/README to configure.): n_1

and another medium with index of refraction Failed to parse (Missing texvc executable; please see math/README to configure.): n_2

. In such situations, Snell's Law describes the resulting deflection of the light ray:

Failed to parse (Missing texvc executable; please see math/README to configure.): n_1\sin\theta_1 = n_2\sin\theta_2\

where Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_2
are the angles between the normal (to the interface) and the incident and refracted waves, respectively. This phenomenon is also associated with a changing speed of light as seen from the definition of index of refraction provided above which implies:

Failed to parse (Missing texvc executable; please see math/README to configure.): v_1\sin\theta_2\ = v_2\sin\theta_1

where Failed to parse (Missing texvc executable; please see math/README to configure.): v_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): v_2
are the wave velocities through the respective media.[3]

Various consequences of Snell's Law include the fact that for light rays traveling from a material with a high index of refraction to a material with a low index of refraction, it is possible for the interaction with the interface to result in zero transmission. This phenomenon is called total internal reflection and allows for fiber optics technology. As light signals travel down a fiber optic cable, it undergoes total internal reflection allowing for essentially no light lost over the length of the cable. It is also possible to produce polarized light rays using a combination of reflection and refraction: When a refracted ray and the reflected ray form a right angle, the reflected ray has the property of "plane polarization". The angle of incidence required for such a scenario is known as Brewster's angle.^[3]

Snell's Law can be used to predict the deflection of light rays as they pass through "linear media" as long as the indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. Additionally, since different frequencies of light have slightly different indexes of refraction in most materials, refraction can be used to produce dispersion spectra that appear as rainbows. The discovery of this phenomenon when passing light through a prism is famously attributed to Isaac Newton.^[3]

Some media have an index of refraction which varies gradually with position and, thus, light rays curve through the medium rather than travel in straight lines. This effect is what is responsible for mirages seen on hot days where the changing index of refraction of the air causes the light rays to bend creating the appearance of specular reflections in the distance (as if on the surface of a pool of water). Material that has a varying index of refraction is called a gradient-index (GRIN) material and has many useful properties used in modern optical scanning technologies including photocopiers and scanners. The phenomenon is studied in the field of gradient-index optics.^[4]

A ray tracing diagram for a simple converging lens.

A device which produces converging or diverging light rays due to refraction is known as a lens. Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation.^[5] In general, two types of lenses exist: convex lenses, which cause parallel light rays to converge, and concave lenses, which cause parallel light rays to diverge. The detailed prediction of how images are produced by these lenses can be made using ray-tracing similar to curved mirrors. Similarly to curved mirrors, thin lenses follow a simple equation that determines the location of the images given a particular focal length (Failed to parse (Missing texvc executable; please see math/README to configure.): f ) and object distance (Failed to parse (Missing texvc executable; please see math/README to configure.): S_1 ):

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}

where Failed to parse (Missing texvc executable; please see math/README to configure.): S_2

is the distance associated with the image and is considered by convention to be negative if on the same side of the lens as the object and positive if on the opposite side of the lens.[5] The focal length f is considered negative for concave lenses.

Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens. Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens. With concave lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on. Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal length, and on the same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens.

Likewise, the magnification of a lens is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): M = - \frac{S_2}{S_1} = \frac{f}{f - S_1}

where the negative sign is given, by convention, to indicate an upright object for positive values and an inverted object for negative values. Similar to mirrors, upright images produced by single lenses are virtual while inverted images are real.^[3]

Lenses suffer from aberrations that distort images and focal points. These are due to both to geometrical imperfections and due to the changing index of refraction for different wavelengths of light (chromatic aberration).^[3]

Underlying mathematics[link]

This article needs attention from an expert on the subject. Please add a reason or a talk parameter to this template to explain the issue with the article. WikiProject Mathematics may be able to help recruit an expert. (June 2009)

As a mathematical study, geometrical optics emerges as a short-wavelength limit for solutions to hyperbolic partial differential equations. In this short-wavelength limit, it is possible to approximate the solution locally by

Failed to parse (Missing texvc executable; please see math/README to configure.): u(t,x) \approx a(t,x)e^{i(k\cdot x - \omega t)}

where Failed to parse (Missing texvc executable; please see math/README to configure.): k, \omega

satisfy a dispersion relation, and the amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): a(t,x)
varies slowly.  More precisely, the leading order solution takes the form

Failed to parse (Missing texvc executable; please see math/README to configure.): a_0(t,x) e^{i\varphi(t,x)/\varepsilon}.

The phase Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi(t,x)/\varepsilon

can be linearized to recover large wavenumber Failed to parse (Missing texvc executable; please see math/README to configure.): k:= \nabla_x \varphi

, and frequency Failed to parse (Missing texvc executable; please see math/README to configure.): \omega := -\partial_t \varphi . The amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): a_0

satisfies a transport equation.  The small parameter Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon\,
enters the scene due to highly oscillatory initial conditions.  Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays.  Assuming coefficients in the differential equation are smooth, the rays will be too.  In other words, refraction does not take place.  The motivation for this technique comes from studying the typical scenario of light propagation where short wavelength light travels along rays that minimize (more or less) its travel time.  Its full application requires tools from microlocal analysis.

A simple example[link]

Starting with the wave equation for Failed to parse (Missing texvc executable; please see math/README to configure.): (t,x) \in \mathbb{R}\times\mathbb{R}^n

Failed to parse (Missing texvc executable; please see math/README to configure.): L(\partial_t, \nabla_x) u := \left( \frac{\partial^2}{\partial t^2} - c(x)^2 \Delta \right)u(t,x) = 0, \;\; u(0,x) = u_0(x),\;\; u_t(0,x) = 0

assume an asymptotic series solution of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): u(t,x) \sim a_\varepsilon(t,x)e^{i\varphi(t,x)/\varepsilon} = \sum_{j=0}^\infty i^j \varepsilon^j a_j(t,x) e^{i\varphi(t,x)/\varepsilon}.

Check that

Failed to parse (Missing texvc executable; please see math/README to configure.): L(\partial_t,\nabla_x)(e^{i\varphi(t,x)/\varepsilon}) a_\varepsilon(t,x) = e^{i\varphi(t,x)/\varepsilon} \left( \left(\frac{i}{\varepsilon} \right)^2 L(\varphi_t, \nabla_x\varphi)a_\varepsilon + \frac{2i}{\varepsilon} V(\partial_t,\nabla_x)a_\varepsilon + \frac{i}{\varepsilon} (a_\varepsilon L(\partial_t,\nabla_x)\varphi) + L(\partial_t,\nabla_x)a_\varepsilon \right)

with

Failed to parse (Missing texvc executable; please see math/README to configure.): V(\partial_t,\nabla_x) := \frac{\partial \varphi}{\partial t} \frac{\partial}{\partial t} - c^2(x)\sum_j \frac{\partial \varphi}{\partial x_j} \frac{\partial}{\partial x_j}

Plugging the series into this equation, and equating powers of Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon , the most singular term Failed to parse (Missing texvc executable; please see math/README to configure.): O(\varepsilon^{-2})

satisfies the eikonal equation (in this case called a dispersion relation),

Failed to parse (Missing texvc executable; please see math/README to configure.): 0 = L(\varphi_t,\nabla_x\varphi) = (\varphi_t)^2 - c(x)^2(\nabla_x \varphi)^2.

To order Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon^{-1} , the leading order amplitude must satisfy a transport equation

Failed to parse (Missing texvc executable; please see math/README to configure.): 2V a_0 + (L\varphi)a_0 = 0

With the definition Failed to parse (Missing texvc executable; please see math/README to configure.): k : = \nabla_x \varphi , Failed to parse (Missing texvc executable; please see math/README to configure.): \omega := -\varphi_t , the eikonal equation is precisely the dispersion relation that results by plugging the plane wave solution Failed to parse (Missing texvc executable; please see math/README to configure.): e^{i(k\cdot x - \omega t)}

into the wave equation.  The value of this more complicated expansion is that plane waves cannot be solutions when the wavespeed Failed to parse (Missing texvc executable; please see math/README to configure.): c
is non-constant.  However, it can be shown that the amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): a_0
and phase Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi
are smooth, so that on a local scale there are plane waves.  

To justify this technique, the remaining terms must be shown to be small in some sense. This can be done using energy estimates, and an assumption of rapidly oscillating initial conditions. It also must be shown that the series converges in some sense.

References[link]

External links[link]