Specular reflection
Encyclopedia
Specular reflection is the mirror
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...

-like reflection
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

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...

(or of other kinds of wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

) from a surface, in which light from a single incoming direction (a ray
Ray (optics)
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. This allows even very...

) is reflected into a single outgoing direction. Such behavior is described by the law of reflection, which states that the direction of incoming light (the incident ray), and the direction of outgoing light reflected (the reflected ray) make the same angle with respect to the surface 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...

, thus the angle of incidence equals the angle of reflection ( in the figure), and that the incident, normal, and reflected directions are coplanar. This behavior was first discovered through careful observation and measurement by Hero of Alexandria
Hero of Alexandria
Hero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...

Anno Domini
and Before Christ are designations used to label or number years used with the Julian and Gregorian calendars....

c. 10
10
Year 10 was a common year starting on Wednesday of the Julian calendar. At the time, throughout Roman Empire, it was known as the year of the consulship of Dolabella and Silanus...

70
70
Year 70 was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Augustus and Vespasianus...

).

## Explanation

Specular reflection is distinct from diffuse reflection
Diffuse reflection
Diffuse reflection is the reflection of light from a surface such that an incident ray is reflected at many angles rather than at just one angle as in the case of specular reflection...

, where incoming light is reflected in a broad range of directions. An example of the distinction between specular and diffuse reflection would be glossy and matte paint
Paint
Paint is any liquid, liquefiable, or mastic composition which after application to a substrate in a thin layer is converted to an opaque solid film. One may also consider the digital mimicry thereof...

s. Matte paints have almost exclusively diffuse reflection, while glossy paints have both specular and diffuse reflection. A surface built from a non-absorbing powder, such as plaster, can be a nearly perfect diffuser. On the opposite side, polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.

Even when a surface exhibits only specular reflection with no diffuse reflection, not all of the light is necessarily reflected. Some of the light may be absorbed by the materials. Additionally, depending on the type of material behind the surface, some of the light may be transmitted through the surface. For most interfaces between materials, the fraction of the light that is reflected increases with increasing angle of incidence . If the light is propagating in a material with a higher index of refraction than the material whose surface it strikes, then total internal reflection
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...

may occur if the angle of incidence is greater than a certain critical angle. Specular reflection from a dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

such as water can affect polarization and at Brewster's angle
Brewster's angle
Brewster's 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 is incident at this angle, the light that is reflected from the surface is therefore perfectly...

reflected light is completely linearly polarized parallel to the interface.

The law of reflection arises from 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...

of a plane wave with small 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...

on a flat boundary: when the boundary size is much larger than the wavelength then electrons of the boundary are seen oscillating exactly in phase only from one direction – the specular direction. If a mirror becomes very small compared to the wavelength, the law of reflection no longer holds and the behavior of light is more complicated.

Waves other than visible light can also exhibit specular reflection. This includes other electromagnetic waves, as well as non-electromagnetic waves. Examples include ionospheric reflection
Ionospheric reflection
Ionospheric reflection is a bending, through a complex process involving reflection and refraction, of electromagnetic waves propagating in the ionosphere back toward the Earth....

Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...

Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

signals by flying objects, acoustic mirror
Acoustic mirror
An acoustic mirror is a passive device used to reflect and perhaps to focus sound waves.- Overview :Prior to World War II and the invention of radar, acoustic mirrors were built as early warning devices around the coasts of Great Britain, with the aim of detecting airborne invasions...

s, which reflect sound, and atomic mirror
Atomic mirror (physics)
In physics, an atomic mirror is a device which reflects neutral atoms in the similar way as the conventional mirror reflects visible light. Atomic mirrors can be made of electric fields or magnetic fields, electromagnetic waves or just silicon wafer; in the last case, atoms are reflected by the...

s, which reflect neutral atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s. For the efficient reflection of atoms from a solid-state
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...

mirror, very cold atoms and/or grazing incidence are used in order to provide significant quantum reflection
Quantum reflection
Quantum reflection is a physical phenomenon involving the reflection of a matter wave from an attractive potential. In classical physics, such a phenomenon is not possible; for instance when one magnet is pulled toward another, you do not expect one of the magnets to suddenly Quantum reflection is...

; ridged mirror
Ridged mirror
In atomic physics, a ridged mirror is a kind of atomic mirror, designed for the specular reflection of neutral particles coming at the grazing incidence angle, characterised in the following: in order to reduce the mean attraction of particles to the surface and increase the reflectivity, this...

s are used to enhance the specular reflection of atoms.

The reflectivity of a surface is the ratio of reflected power to incident power. The reflectivity is a material characteristic, depends on the wavelength, and is related to the refractive index of the material through Fresnel's equations. In absorbing materials, like metals, it is related to the electronic absorption spectrum through the imaginary component of the complex refractive index. Measurements of specular reflection are performed with normal or varying incidence reflectometers using a scanning variable-wavelength light source. Lower quality measurements using a glossmeter
Glossmeter
A glossmeter measures specular reflection. The light intensity is registered over a small range of the reflection angle. The intensity is dependent on the material and the angle of illumination. In case of non-metals the amount of reflected light increases with the increase of the illumination angle...

quantify the glossy appearance of a surface in gloss units.

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 appears to be laterally inverted, in other words left and right reversed.
• It is virtual, meaning that the image appears to be behind the mirror, and cannot be projected onto a screen.

## Direction of reflection

The direction of a reflected ray is determined by the vector of incidence and the surface 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...

vector. Given an incident direction from the surface to the light source and the surface normal direction the specularly reflected direction (all unit vectors) is:

where is a scalar obtained with the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

. Different authors may define the incident and reflection directions with different signs
Sign convention
In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...

.
Assuming these Euclidean vectors 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:

denotes transposition
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

and is the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

.

• Mirror
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...

, a device designed for specular reflection
• Atomic mirror
Atomic mirror (physics)
In physics, an atomic mirror is a device which reflects neutral atoms in the similar way as the conventional mirror reflects visible light. Atomic mirrors can be made of electric fields or magnetic fields, electromagnetic waves or just silicon wafer; in the last case, atoms are reflected by the...

, a device for specular reflection of atoms
• Reflection coefficient
Reflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...

• Fresnel equations
Fresnel equations
The Fresnel equations , deduced by Augustin-Jean Fresnel , describe the behaviour of light when moving between media of differing refractive indices...

which describe the behavior of light when moving between media of differing refractive indices.
• Geometric optics
• Specular highlight
Specular highlight
A specular highlight is the bright spot of light that appears on shiny objects when illuminated . Specular highlights are important in 3D computer graphics, as they provide a strong visual cue for the shape of an object and its location with respect to light sources in the scene.-Microfacets:The...

• Hamiltonian optics
Hamiltonian optics
Lagrangian optics and Hamiltonian optics are two formulations of geometrical optics which share much of the mathematical formalism with Lagrangian mechanics and Hamiltonian mechanics.-Hamilton's principle:...