Editing Specular reflection

{{Short description|Mirror-like wave reflection}}
{{For|the song with identical title|Specular Reflection (song)}}
[[Image:Reflection angles.svg|frame|Coplanar condition of specular reflection, in which <math>\theta _i = \theta _r</math>]]
[[Image:Tso Kiagar Lake Ladakh.jpg|thumb|Reflections on still water are an example of specular reflection.]]

'''Specular reflection''', or '''regular reflection''', is the [[mirror]]-like [[reflection (physics)|reflection]] of [[wave]]s, such as [[light]], from a [[surface]].<ref>{{cite book | chapter = Specularity, Specular Reflectance |title=Computer Vision | last= Tan | first= R.T. | editor-first1= Katsushi | editor-last1= Ikeuchi | publisher = Springer, Boston, MA | year = 2013 | isbn = 978-0-387-31439-6 | doi = 10.1007/978-0-387-31439-6_538 | s2cid= 5058976 }}</ref> 

The '''law of reflection''' states that a reflected [[ray (optics)|ray]] of light emerges from the reflecting surface at the same [[angle]] to the [[surface normal]] as the incident ray, but on the opposing side of the surface normal in the plane formed by the incident and reflected rays. The earliest known description of this behavior was recorded by [[Hero of Alexandria]] ([[Anno Domini|AD]] c. 10–70).<ref>
{{cite book
 |title=A history of Greek mathematics. Volume II: From Aristarchus to Diophantus
 |url={{google books |plainurl=y |id=sl0bUVtZbpEC}}
 |author=Sir Thomas Little Heath
 |isbn=978-0-486-24074-9
 |year=1981
}}</ref> Later, [[Ibn al-Haytham|Alhazen]] gave a complete statement of the law of reflection.<ref>{{cite book |last=Stamnes |first=J. J. |url=https://books.google.com/books?id=dGQ-DwAAQBAJ&dq=alhazen+law+of+reflection&pg=PT15 |title=Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves |date=2017-11-13 |publisher=Routledge |isbn=978-1-351-40468-6 |language=en}}</ref><ref>{{cite book |last=Mach |first=Ernst |url=https://books.google.com/books?id=7dPCAgAAQBAJ&dq=alhazen+incident+ray+reflected+ray+lie+on+same+plane&pg=PA29 |title=The Principles of Physical Optics: An Historical and Philosophical Treatment |date=2013-01-23 |publisher=Courier Corporation |isbn=978-0-486-17347-4 |language=en}}</ref><ref>{{cite book |last=Iizuka |first=Keigo |url=https://books.google.com/books?id=h9n6CAAAQBAJ&dq=alhazen+law+of+reflection&pg=PA7 |title=Engineering Optics |date=2013-11-11 |publisher=Springer Science & Business Media |isbn=978-3-662-07032-1 |language=en}}</ref> He was first to state that the incident ray, the reflected ray, and the normal to the surface all lie in a same plane perpendicular to reflecting plane.<ref>{{cite book |last=Selin |first=Helaine |date=2008 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |p=1817 |url=https://www.google.ca/books/edition/Encyclopaedia_of_the_History_of_Science/czFOvQEACAAJ }}</ref><ref>{{cite book |last=Mach |first=Ernst |url=https://books.google.com/books?id=7dPCAgAAQBAJ&dq=alhazen+first+incident+ray+reflected+ray+lie+on+same+plane&pg=PA29 |title=The Principles of Physical Optics: An Historical and Philosophical Treatment |date=2013-01-23 |publisher=Courier Corporation |isbn=978-0-486-17347-4 |language=en}}</ref>

Specular reflection may be contrasted with [[diffuse reflection]], in which light is scattered away from the surface in a range of directions.

== Law of reflection ==
[[File:Heart of the City water feature Sheffield - geograph.org.uk - 618552.jpg|thumb|Specular reflection from a wet metal sphere]]
[[File:Marble ball - Kongens Have - Copenhagen - DSC07898.JPG|thumb|Diffuse reflection from a marble ball]]
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 (physics)|reflection]] and [[refraction]], are expressed by the difference of the refractive index on both sides of the boundary, whereas [[reflectance]] and [[Absorption (optics)|absorption]] are the real and imaginary parts of the response due to the [[electronic structure]] of the material.<ref>{{cite book|last1=Fox|first1=Mark|title=Optical properties of solids|url={{google books |plainurl=y |id=5WkVDAAAQBAJ}}|date=2010|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-957336-3|page=1|edition=2nd}}</ref>
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]] describe the physics at the optical boundary.

Reflection may occur as specular, or mirror-like, reflection and [[diffuse reflection]]. 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]] [[paint]] and [[Matte (surface)|matte]] 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]] is called the [[angle of incidence (optics)|angle of incidence]] and the plane defined by both directions is the [[plane of incidence]]. 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]].

When the light is incident perpendicularly to the surface, it is reflected straight back in the source direction.

The phenomenon of reflection arises from the [[diffraction]] of a [[plane wave]] on a flat boundary. When the boundary size is much larger than the [[wavelength]], then the electromagnetic fields at the boundary are oscillating exactly in phase only for the specular direction.

=== Vector formulation ===
{{see also|Snell's law#Vector form}}

The law of reflection can also be equivalently expressed using [[linear algebra]]. The direction of a reflected ray is determined by the vector of incidence and the [[surface normal]] vector. Given an incident direction <math>\mathbf{\hat{d}}_\mathrm{i}</math> from the light source to the surface and the surface normal direction <math>\mathbf{\hat{d}}_\mathrm{n},</math> the specularly reflected direction <math>\mathbf{\hat{d}}_\mathrm{s}</math> (all [[unit vector]]s) is:<ref>{{cite book |last=Haines |first=Eric |year=2021 |editor-last1=Marrs |editor-first1=Adam |editor-last2=Shirley |editor-first2=Peter |editor-last3=Wald |editor-first3=Ingo |title=Ray Tracing Gems II |publisher=Apress |pages=105–108 |chapter=Chapter 8: Reflection and Refraction Formulas |doi=10.1007/978-1-4842-7185-8_8 |isbn=978-1-4842-7185-8|s2cid=238899623 }}</ref><ref>{{cite book | last = Comninos | first = Peter | title = Mathematical and computer programming techniques for computer graphics | publisher = Springer | year = 2006 | url = {{google books |plainurl=y |id=Kdb7-YnnOVwC|page=361}}| isbn = 978-1-85233-902-9 | page = 361 | url-status = live | archive-url = https://web.archive.org/web/20180114234526/https://books.google.com/books?id=Kdb7-YnnOVwC&lpg=PR3&dq=isbn%3A9781852339029&pg=PA361 | archive-date = 2018-01-14 }}</ref>
: <math>\mathbf{\hat{d}}_\mathrm{s} = \mathbf{\hat{d}}_\mathrm{i} - 2 \mathbf{\hat{d}}_\mathrm{n} \left(\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}\right),
</math>
where <math>\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}</math> is a scalar obtained with the [[dot product]]. Different authors may define the incident and reflection directions with [[sign convention|different signs]].
Assuming these [[Euclidean vector]]s are represented in [[column vector|column form]], the equation can be equivalently expressed as a matrix-vector multiplication:<ref>{{cite book | last1 = Farin | first1 = Gerald | last2 = Hansford | first2 = Dianne | author2-link = Dianne Hansford | title = Practical linear algebra: a geometry toolbox | url=http://www.farinhansford.com/books/pla/
| publisher = A K Peters | year = 2005 | isbn = 978-1-56881-234-2 | pages = 191–192 | url-status = live | archive-url = https://web.archive.org/web/20100307145056/http://www.farinhansford.com/books/pla/ | archive-date = 2010-03-07 }} {{google books|id=Iq6qVt22RZUC|title=Practical linear algebra: a geometry toolbox}}</ref>
: <math>\mathbf{\hat{d}}_\mathrm{s} = \mathbf{R} \; \mathbf{\hat{d}}_\mathrm{i},</math>
where <math>\mathbf{R}</math> is the so-called [[Householder matrix|Householder transformation matrix]], defined as:
: <math>\mathbf{R} =  \mathbf{I} - 2 \mathbf{\hat{d}}_\mathrm{n} \mathbf{\hat{d}}_\mathrm{n}^\mathrm{T};</math>
in terms of the [[identity matrix]] <math>\mathbf{I}</math> and twice the [[outer product]] of <math>\mathbf{\hat{d}}_\mathrm{n}</math>.

== Reflectivity ==
''[[Reflectivity]]'' 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]].<ref>{{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 |page=100 }}</ref> 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 relations|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]] 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]], some of the light is reflected. If the angle of incidence is greater than the [[critical angle (optics)|critical angle]], [[total internal reflection]] occurs: all of the light is reflected. The critical angle can be shown to be given by
: <math>\theta_\text{crit} = \arcsin\!\left(\frac{n_2}{n_1}\right)\!.</math>

=== Polarization ===
When light strikes an interface between two materials, the reflected light is generally partially [[Polarization (waves)|polarized]]. However, if the light strikes the interface at [[Brewster's angle]], the reflected light is ''completely'' linearly polarized parallel to the interface. Brewster's angle is given by
: <math>\theta_\mathrm{B} = \arctan\!\left(\frac{n_2}{n_1}\right)\!.</math>

=== Reflected images ===
{{See also|Mirror image#In three dimensions}}
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 [[Cartesian coordinate system#In three dimensions|handedness]] of the coordinate system, one axis of the coordinate system appears to be reversed, and the [[Chirality (mathematics)|chirality]] of the image may change. For example, the image of a right shoe will look like a left shoe.

== Examples ==
[[File:Eiffel Tower under cloudy sky.jpg|thumb|Esplanade of the Trocadero in Paris after rain. The layer of water exhibits specular reflection, reflecting an image of the Eiffel Tower and other objects.]]
A classic example of specular reflection is a [[mirror]], which is specifically designed for specular reflection.

In addition to [[visible light]], specular reflection can be observed in the [[ionospheric reflection]] of [[radiowave]]s and the reflection of radio- or [[microwave]] [[radar]] signals by flying objects. The measurement technique of [[x-ray reflectivity]] exploits specular reflectivity to study thin films and interfaces with sub-nanometer resolution, using either modern laboratory sources or [[synchrotron]] x-rays.

Non-electromagnetic waves can also exhibit specular reflection, as in [[acoustic mirror]]s which reflect sound, and [[Atomic mirror (physics)|atomic mirrors]], which reflect neutral [[atom]]s. For the efficient reflection of atoms from a [[Solid-state physics|solid-state]] mirror, very cold atoms and/or [[angle of incidence (optics)|grazing incidence]] are used in order to provide significant [[quantum reflection]]; [[ridged mirror]]s are used to enhance the specular reflection of atoms. [[Neutron reflectometry]] uses specular reflection to study material surfaces and [[thin film]] interfaces in an analogous fashion to x-ray reflectivity.

== See also ==
* [[Geometric optics]]
* [[Hamiltonian optics]]
* [[Reflection coefficient]]
* [[Reflection (mathematics)]]
* [[Specular highlight]]
* [[Specularity]]

== References ==
{{reflist|30em}}

[[Category:Optical phenomena]]

[[de:Reflexionsgesetz]]
[[es:Imagen especular]]
[[fr:Réflexion optique#Les deux formes de la réflexion]]
[[ru:Закон отражения света]]