Specular reflection, or regular reflection, is the
mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
-like
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
of
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
s, such as
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
, from a surface.
The law of reflection states that a reflected
ray of light emerges from the reflecting surface at the same
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
to the
surface normal
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
as the incident ray, but on the opposing side of the surface normal in the plane formed by the incident and reflected rays. This behavior was first described by
Hero of Alexandria
Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greece, Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egy ...
(
AD c. 10–70).
Specular reflection may be contrasted with
diffuse reflection
Diffuse reflection is the reflection (physics), reflection of light or other radiation, waves or particles from a surface such that a ray (optics), ray incident on the surface is scattering, scattered at many angles rather than at just one angle ...
, in which light is scattered away from the surface in a range of directions.
Law of reflection
![Marble ball - Kongens Have - Copenhagen - DSC07898](https://upload.wikimedia.org/wikipedia/commons/2/28/Marble_ball_-_Kongens_Have_-_Copenhagen_-_DSC07898.JPG)
When light encounters a boundary of a material, it is affected by the optical and electronic response functions of the material to electromagnetic waves. Optical processes, which comprise
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
and
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
, are expressed by the difference of the refractive index on both sides of the boundary, whereas
reflectance
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
and
absorption
Absorption may refer to:
Chemistry and biology
* Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which ...
are the real and imaginary parts of the response due to the
electronic structure
In quantum chemistry, electronic structure is the state of motion of electrons in an electrostatic field created by stationary nuclei. The term encompasses both the wave functions of the electrons and the energies associated with them. Electro ...
of the material.
The degree of participation of each of these processes in the transmission is a function of the frequency, or wavelength, of the light, its polarization, and its angle of incidence. In general, reflection increases with increasing angle of incidence, and with increasing absorptivity at the boundary. The
Fresnel equations
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
describe the physics at the optical boundary.
Reflection may occur as specular, or mirror-like, reflection and
diffuse reflection
Diffuse reflection is the reflection (physics), reflection of light or other radiation, waves or particles from a surface such that a ray (optics), ray incident on the surface is scattering, scattered at many angles rather than at just one angle ...
. Specular reflection reflects all light which arrives from a given direction at the same angle, whereas diffuse reflection reflects light in a broad range of directions. The distinction may be illustrated with surfaces coated with
glossy
Gloss is an optical property which indicates how well a surface reflects light in a specular (mirror-like) direction. It is one of the important parameters that are used to describe the visual appearance of an object. The factors that affect glo ...
paint
Paint is any pigmented liquid, liquefiable, or solid mastic composition that, after application to a substrate in a thin layer, converts to a solid film. It is most commonly used to protect, color, or provide texture. Paint can be made in many ...
and
matte
Matte may refer to:
Art
* paint with a non-glossy finish. See diffuse reflection.
* a framing element surrounding a painting or watercolor within the outer frame
Film
* Matte (filmmaking), filmmaking and video production technology
* Matte p ...
paint. Matte paints exhibit essentially complete diffuse reflection, while glossy paints show a larger component of specular behavior. A surface built from a non-absorbing powder, such as plaster, can be a nearly perfect diffuser, whereas polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.
Light propagates in space as a wave front of electromagnetic fields. A ray of light is characterized by the direction normal to the wave front (''wave normal''). When a ray encounters a surface, the angle that the wave normal makes with respect to the
surface normal
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
is called the
angle of incidence and the plane defined by both directions is the
plane of incidence
In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave optic ...
. Reflection of the incident ray also occurs in the plane of incidence.
The law of reflection states that the angle of reflection of a ray equals the angle of incidence, and that the incident direction, the surface normal, and the reflected direction are
coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. Howe ...
.
When the light impinges perpendicularly to the surface, it is reflected straight back in the source direction.
The phenomenon of reflection arises from the
diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
of a plane wave on a flat boundary. When the boundary size is much larger than the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
, then the electromagnetic fields at the boundary are oscillating exactly in phase only for the specular direction.
Vector formulation
The law of reflection can also be equivalently expressed using
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
. The direction of a reflected ray is determined by the vector of incidence and the
surface normal
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
vector. Given an incident direction
from the light source to the surface and the surface normal direction
the specularly reflected direction
(all
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
s) is:
:
where
is a scalar obtained with the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. Different authors may define the incident and reflection directions with
different signs.
Assuming these
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
s are represented in
column form, the equation can be equivalently expressed as a matrix-vector multiplication:
[ ]
:
where
is the so-called
Householder transformation matrix, defined as:
:
in terms of the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
and twice the
outer product
In linear algebra, the outer product of two coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...
of
.
Reflectivity
''
Reflectivity
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
'' is the ratio of the power of the reflected wave to that of the incident wave. It is a function of the wavelength of radiation, and is related to the refractive index of the material as expressed by
Fresnel's equations. In regions of the electromagnetic spectrum in which absorption by the material is significant, it is related to the electronic absorption spectrum through the imaginary component of the complex refractive index. The electronic absorption spectrum of an opaque material, which is difficult or impossible to measure directly, may therefore be indirectly determined from the reflection spectrum by a
Kramers-Kronig transform. The polarization of the reflected light depends on the symmetry of the arrangement of the incident probing light with respect to the absorbing transitions dipole moments in the material.
Measurement of specular reflection is performed with normal or varying incidence reflection spectrophotometers (''reflectometer'') using a scanning variable-wavelength light source. Lower quality measurements using a
glossmeter
A glossmeter (also gloss meter) is an instrument which is used to measure specular reflection gloss of a surface. Gloss is determined by projecting a beam of light at a fixed intensity and angle onto a surface and measuring the amount of reflect ...
quantify the glossy appearance of a surface in
gloss units.
Consequences
Internal reflection
When light is propagating in a material and strikes an interface with a material of lower
index of refraction
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
, some of the light is reflected. If the angle of incidence is greater than the
critical angle
Critical angle may refer to:
*Critical angle (optics), the angle of incidence above which total internal reflection occurs
*Critical angle of attack
In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference lin ...
,
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
occurs: all of the light is reflected. The critical angle can be shown to be given by
:
Polarization
When light strikes an interface between two materials, the reflected light is generally partially
polarized. However, if the light strikes the interface at
Brewster's angle
Brewster's angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with ''no reflection''. When ''unpolarized'' light ...
, the reflected light is ''completely'' linearly polarized parallel to the interface. Brewster's angle is given by
:
Reflected images
The image in a flat mirror has these features:
* It is the same distance behind the mirror as the object is in front.
* It is the same size as the object.
* It is the right way up (erect).
* It is reversed.
* It is ''virtual'', meaning that the image appears to be behind the mirror, and cannot be projected onto a screen.
The reversal of images by a plane mirror is perceived differently depending on the circumstances. In many cases, the image in a mirror appears to be reversed from left to right. If a flat mirror is mounted on the ceiling it can appear to reverse ''up'' and ''down'' if a person stands under it and looks up at it. Similarly a car turning ''left'' will still appear to be turning ''left'' in the rear view mirror for the driver of a car in front of it. The reversal of directions, or lack thereof, depends on how the directions are defined. More specifically a mirror changes the
handedness
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more Fine motor skill, dextrous. The other hand, comparatively often the weaker, less dextrous or sim ...
of the coordinate system, one axis of the coordinate system appears to be reversed, and the
chirality
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
of the image may change. For example, the image of a right shoe will look like a left shoe.
Examples
![Eiffel Tower under cloudy sky](https://upload.wikimedia.org/wikipedia/commons/9/94/Eiffel_Tower_under_cloudy_sky.jpg)
A classic example of specular reflection is a
mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
, which is specifically designed for specular reflection.
In addition to
visible light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
, specular reflection can be observed in the
ionospheric reflection
In radio communication, skywave or skip refers to the propagation of radio waves reflected or refracted back toward Earth from the ionosphere, an electrically charged layer of the upper atmosphere. Since it is not limited by the curvature ...
of
radiowaves and the reflection of radio- or
microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
signals by flying objects. The measurement technique of
x-ray reflectivity X-ray reflectivity (sometimes known as X-ray specular reflectivity, X-ray reflectometry, or XRR) is a surface-sensitive analytical technique used in chemistry, physics, and materials science to characterize surfaces, thin films and multilayers.J. ...
exploits specular reflectivity to study thin films and interfaces with sub-nanometer resolution, using either modern laboratory sources or
synchrotron
A synchrotron is a particular type of cyclic particle accelerator, descended from the cyclotron, in which the accelerating particle beam travels around a fixed closed-loop path. The magnetic field which bends the particle beam into its closed p ...
x-rays.
Non-electromagnetic waves can also exhibit specular reflection, as in
acoustic mirror
An acoustic mirror is a passive device used to reflect and focus (concentrate) sound waves. Parabolic acoustic mirrors are widely used in parabolic microphones to pick up sound from great distances, employed in surveillance and reporting of ...
s which reflect sound, and
atomic mirrors, which reflect neutral
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
s. For the efficient reflection of atoms from a
solid-state
Solid state, or solid matter, is one of the four fundamental states of matter.
Solid state may also refer to:
Electronics
* Solid-state electronics, circuits built of solid materials
* Solid state ionics, study of ionic conductors and their use ...
mirror, very cold atoms and/or
grazing incidence The angle of incidence, in geometric optics, is the angle between a ray incident on a surface and the line perpendicular (at 90 degree angle) to the surface at the point of incidence, called the normal. The ray can be formed by any waves, such as o ...
are used in order to provide significant
quantum reflection
Quantum reflection is a uniquely quantum phenomenon in which a compact object, such as a neutron or a small molecule, reflects smoothly and in a wavelike fashion from a much larger surface, such as a pool of mercury. In contrast, a classically beha ...
;
ridged mirror
In atomic physics, a ridged mirror (or ridged atomic mirror, or Fresnel diffraction mirror) is a kind of atomic mirror, designed for the specular reflection of neutral particles (atoms) coming at a grazing incidence angle. In order to reduce the ...
s are used to enhance the specular reflection of atoms.
Neutron reflectometry
Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films, similar to the often complementary techniques of X-ray reflectivity and ellipsometry. The technique provides valuable information over a wide var ...
uses specular reflection to study material surfaces and thin film interfaces in an analogous fashion to x-ray reflectivity.
See also
*
Geometric optics
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
*
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
*
Reflection coefficient
In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected w ...
*
Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of refle ...
*
Specular highlight
A specular highlight is the bright spot of light that appears on shiny objects when illuminated (for example, see image on right). Specular highlights are important in 3D computer graphics, as they provide a strong visual cue for the shape of a ...
*
Specularity
Specularity is the visual appearance of specular reflections.
In computer graphics
In computer graphics, it means the quantity used in three-dimensional (3D) rendering which represents the amount of reflectivity a surface has. It is a key com ...
Notes
References
* {{cite book , first=Eugene, last=Hecht, year=1987, title=Optics, edition=2nd , url={{google books , plainurl=y , id=oiSrQgAACAAJ020111609X, publisher=Addison Wesley, isbn=0-201-11609-X
Geometrical optics
de:Reflexionsgesetz
es:Imagen especular
fr:Réflexion optique#Les deux formes de la réflexion
ru:Закон отражения света