# Specular reflection

Specular reflection, also known as regular reflection, is the mirror-like reflection of waves, such as light, from a surface. In this process, each incident ray is reflected 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 incident and reflected rays. The result is that an image reflected by the surface is reproduced in mirror-like (specular) fashion.

The law of reflection states that for each incident ray the angle of incidence equals the angle of reflection, and the incident, normal, and reflected directions are coplanar. This behavior was first described by Hero of Alexandria (AD c. 10–70).[1] It may be contrasted with diffuse reflection, in which light is scattered away from the surface in a range of directions rather than just one.

Coplanar condition of specular reflection, in which ${\displaystyle \theta _{i}=\theta _{r}}$.
Reflections on still water are an example of specular reflection.

## Background

Specular reflection from metal spheres
Diffuse reflection from a marble ball

When light hits a surface, there are three possible outcomes.[2] Light may be absorbed by the material, light may be transmitted through the surface, or light may be reflected. Materials often show some mix of these behaviors, with the proportion of light that goes to each depending on the properties of the material, the wavelength of the light, and the angle of incidence. For most interfaces between materials, the fraction of the light that is reflected increases with increasing angle of incidence ${\displaystyle \theta _{i}}$.

Reflected light can be divided into two sub-types, specular reflection and diffuse reflection. Specular reflection reflects all light which arrives from a given direction at the same angle, whereas diffuse reflection reflects that light in a broad range of directions. An example of the distinction between specular and diffuse reflection would be glossy and matte paints. 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, whereas polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.

## Law of reflection

The law of reflection describes the angle of reflected light: the angle of incident light is the same as the angle of the reflected light.

The law of reflection arises from diffraction of a plane wave with small wavelength 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.

### Vector formulation

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 ${\displaystyle \mathbf {\hat {d}} _{\mathrm {i} }}$ from the surface to the light source and the surface normal direction ${\displaystyle \mathbf {\hat {d}} _{\mathrm {n} },}$ the specularly reflected direction ${\displaystyle \mathbf {\hat {d}} _{\mathrm {s} }}$ (all unit vectors) is:[3][4]

${\displaystyle \mathbf {\hat {d}} _{\mathrm {s} }=2\left(\mathbf {\hat {d}} _{\mathrm {n} }\cdot \mathbf {\hat {d}} _{\mathrm {i} }\right)\mathbf {\hat {d}} _{\mathrm {n} }-\mathbf {\hat {d}} _{\mathrm {i} },}$

where ${\displaystyle \mathbf {\hat {d}} _{\mathrm {n} }\cdot \mathbf {\hat {d}} _{\mathrm {i} }}$ is a scalar obtained with the dot product. Different authors may define the incident and reflection directions with different signs. Assuming these Euclidean vectors are represented in column form, the equation can be equivalently expressed as a matrix-vector multiplication:

${\displaystyle \mathbf {\hat {d}} _{\mathrm {s} }=\mathbf {R} \;\mathbf {\hat {d}} _{\mathrm {i} },}$

where ${\displaystyle \mathbf {R} }$ is the so-called Householder transformation matrix, defined as:

${\displaystyle \mathbf {R} =\mathbf {I} -2\mathbf {\hat {d}} _{\mathrm {n} }\mathbf {\hat {d}} _{\mathrm {n} }^{\mathrm {T} };}$

in terms of the identity matrix ${\displaystyle \mathbf {I} }$ and twice the outer product of ${\displaystyle \mathbf {\hat {d}} }$.

## 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.[5] 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 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, total internal reflection occurs: all of the light is reflected. The critical angle can be shown to be given by

${\displaystyle \theta _{\text{crit}}=\arcsin \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}$

### 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, the reflected light is completely linearly polarized parallel to the interface. Brewster's angle is given by

${\displaystyle \theta _{\mathrm {B} }=\arctan \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}$

### 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 of the coordinate system, one axis of the coordinate system appears to be reversed, and the chirality of the image may change. For example, the image of a right shoe will look like a left shoe.

## Examples

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 radiowaves 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 mirrors which reflect sound, and atomic mirrors, which reflect neutral atoms. For the efficient reflection of atoms from a solid-state mirror, very cold atoms and/or grazing incidence are used in order to provide significant quantum reflection; ridged mirrors 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.

## Notes

1. ^ Sir Thomas Little Heath (1981). A history of Greek mathematics. Volume II: From Aristarchus to Diophantus. ISBN 978-0-486-24074-9.
2. ^ Fox, Mark (2010). Optical properties of solids (2nd ed.). Oxford: Oxford University Press. p. 1. ISBN 978-0-19-957336-3.
3. ^ Farin, Gerald; Hansford, Dianne (2005). Practical linear algebra: a geometry toolbox. A K Peters. pp. 191–192. ISBN 978-1-56881-234-2. Archived from the original on 2010-03-07. Practical linear algebra: a geometry toolbox at Google Books
4. ^ Comninos, Peter (2006). Mathematical and computer programming techniques for computer graphics. Springer. p. 361. ISBN 978-1-85233-902-9. Archived from the original on 2018-01-14.
5. ^ Hecht 1987, p. 100.

## References

• Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. ISBN 0-201-11609-X.
Atomic mirror

In physics, an atomic mirror is a device which reflects neutral atoms in the similar way as a 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 attracting tails of the van der Waals attraction (see quantum reflection). Such reflection is efficient when the normal component of the wavenumber of the atoms is small or comparable to the effective depth of the attraction potential (roughly, the distance at which the potential becomes comparable to the kinetic energy of the atom). To reduce the normal component, most atomic mirrors are blazed at the grazing incidence.

At grazing incidence, the efficiency of the quantum reflection can be enhanced by a surface covered with ridges (ridged mirror).

The set of narrow ridges reduces the van der Waals attraction of atoms to the surfaces and enhances the reflection. Each ridge blocks part of the wavefront, causing Fresnel diffraction.

Such a mirror can be interpreted in terms of the Zeno effect. We may assume that the atom is "absorbed" or "measured" at the ridges. Frequent measuring (narrowly spaced ridges) suppresses the transition of the particle to the half-space with absorbers, causing specular reflection. At large separation ${\displaystyle ~L~}$ between thin ridges, the reflectivity of the ridged mirror is determined by dimensionless momentum ${\displaystyle ~p={\sqrt {KL~}}~\theta ~}$, and does not depend on the origin of the wave; therefore, it is suitable for reflection of atoms.

Backscatter

In physics, backscatter (or backscattering) is the reflection of waves, particles, or signals back to the direction from which they came. It is a diffuse reflection due to scattering, as opposed to specular reflection as from a mirror. Backscattering has important applications in astronomy, photography, and medical ultrasonography. The opposite effect is forward scatter, e.g. when a translucent material like a cloud diffuses sunlight, giving soft light.

Bragg plane

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, ${\displaystyle \scriptstyle \mathbf {K} }$, at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.

Considering the adjacent diagram, the arriving x-ray plane wave is defined by:

${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\cos {(\mathbf {k} \cdot \mathbf {r} )}+i\sin {(\mathbf {k} \cdot \mathbf {r} )}}$

Where ${\displaystyle \scriptstyle \mathbf {k} }$ is the incident wave vector given by:

${\displaystyle \mathbf {k} ={\frac {2\pi }{\lambda }}{\hat {n}}}$

where ${\displaystyle \scriptstyle \lambda }$ is the wavelength of the incident photon. While the Bragg formulation assumes a unique choice of direct lattice planes and specular reflection of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the Huygens principle. Each scattered wave contributes to a new plane wave given by:

${\displaystyle \mathbf {k^{\prime }} ={\frac {2\pi }{\lambda }}{\hat {n}}^{\prime }}$

The condition for constructive interference in the ${\displaystyle \scriptstyle {\hat {n}}^{\prime }}$ direction is that the path difference between the photons is an integer multiple (m) of their wavelength. We know then that for constructive interference we have:

${\displaystyle |\mathbf {d} |\cos {\theta }+|\mathbf {d} |\cos {\theta ^{\prime }}=\mathbf {d} \cdot ({\hat {n}}-{\hat {n}}^{\prime })=m\lambda }$

where ${\displaystyle \scriptstyle m~\in ~\mathbb {Z} }$. Multiplying the above by ${\displaystyle \scriptstyle {\frac {2\pi }{\lambda }}}$ we formulate the condition in terms of the wave vectors, ${\displaystyle \scriptstyle \mathbf {k} }$ and ${\displaystyle \scriptstyle \mathbf {k^{\prime }} }$:

${\displaystyle \mathbf {d} \cdot (\mathbf {k} -\mathbf {k^{\prime }} )=2\pi m}$

Now consider that a crystal is an array of scattering centres, each at a point in the Bravais lattice. We can set one of the scattering centres as the origin of an array. Since the lattice points are displaced by the Bravais lattice vectors, ${\displaystyle \scriptstyle \mathbf {R} }$, scattered waves interfere constructively when the above condition holds simultaneously for all values of ${\displaystyle \scriptstyle \mathbf {R} }$ which are Bravais lattice vectors, the condition then becomes:

${\displaystyle \mathbf {R} \cdot \left(\mathbf {k} -\mathbf {k^{\prime }} \right)=2\pi m}$

An equivalent statement (see mathematical description of the reciprocal lattice) is to say that:

${\displaystyle e^{i(\mathbf {k} -\mathbf {k^{\prime }} )\cdot \mathbf {R} }=1}$

By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if ${\displaystyle \scriptstyle \mathbf {K} ~=~\mathbf {k} \,-\,\mathbf {k^{\prime }} }$ is a vector of the reciprocal lattice. We notice that ${\displaystyle \scriptstyle \mathbf {k} }$ and ${\displaystyle \scriptstyle \mathbf {k^{\prime }} }$ have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, ${\displaystyle \scriptstyle \mathbf {k} }$, must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, ${\displaystyle \scriptstyle \mathbf {K} }$. This reciprocal space plane is the Bragg plane.

Copy stand

In photography, a copy stand is a device used to copy images and/or text with a camera. The stand consists of a board onto which the media is placed and a tripod-mount parallel to it, usually with an adjustable height. Light is provided by bright lamps mounted on either side of the media at 45° angles. This provides uniform lighting and reduces specular reflection, keeping glare low.

In film cameras, copy stands are traditionally used with slide film. The fine resolution of slide film allows the images to be reproduced with high fidelity when they are projected.

Diffuse reflection

Diffuse reflection is the reflection of light or other waves or particles from a surface such that a ray incident on the surface is scattered at many angles rather than at just one angle as in the case of specular reflection. An ideal diffuse reflecting surface is said to exhibit Lambertian reflection, meaning that there is equal luminance when viewed from all directions lying in the half-space adjacent to the surface.

A surface built from a non-absorbing powder such as plaster, or from fibers such as paper, or from a polycrystalline material such as white marble, reflects light diffusely with great efficiency. Many common materials exhibit a mixture of specular and diffuse reflection.

The visibility of objects, excluding light-emitting ones, is primarily caused by diffuse reflection of light: it is diffusely-scattered light that forms the image of the object in the observer's eye.

Gloss (optics)

Gloss is an optical property which indicates how well a surface reflects light in a specular (mirror-like) direction. It is one of important parameters that are used to describe the visual appearance of an object. The factors that affect gloss are the refractive index of the material, the angle of incident light and the surface topography.

Apparent gloss depends on the amount of specular reflection – light reflected from the surface in an equal amount and the symmetrical angle to the one of incoming light – in comparison with diffuse reflection – the amount of light scattered into other directions.

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 reflected light at an equal but opposite angle.

There are a number of different geometries available for gloss measurement, each being dependent on the type of surface to be measured. For non-metals such as coatings and plastics the amount of reflected light increases with a greater angle of illumination, as some of the light penetrates the surface material and is absorbed into it or diffusely scattered from it depending on its colour. Metals have a much higher reflection and are therefore less angularly dependent.

Many international technical standards are available that define the method of use and specifications for different types of glossmeter used on various types of materials including paint, ceramics, paper, metals and plastics. Many industries use glossmeters in their quality control to measure the gloss of products to ensure consistency in their manufacturing processes. The automotive industry is a major user of the glossmeter, with applications extending from the factory floor to the repair shop.

Jingpo Lacus

Jingpo Lacus is a lake in the north polar region of Titan, the planet Saturn's largest moon. It and similarly sized Ontario Lacus are the largest known bodies of liquid on Titan after the three maria (Kraken Mare, Ligeia Mare and Punga Mare). It is composed of liquid hydrocarbons (mainly methane and ethane). It is west of Kraken Mare at 73° N, 336° W, roughly 240 km (150 mi) long, similar to the length of Lake Onega on Earth. Its namesake is Jingpo Lake, a scenic lake in China.

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 variety of scientific and technological applications including chemical aggregation, polymer and surfactant adsorption, structure of thin film magnetic systems, biological membranes, etc.

Neutron reflector

A neutron reflector is any material that reflects neutrons. This refers to elastic scattering rather than to a specular reflection. The material may be graphite, beryllium, steel, tungsten carbide, or other materials. A neutron reflector can make an otherwise subcritical mass of fissile material critical, or increase the amount of nuclear fission that a critical or supercritical mass will undergo. Such an effect was exhibited twice in accidents involving the Demon Core, a subcritical plutonium pit that went critical in two separate fatal incidents when the pit's surface was momentarily surrounded by too much neutron reflective material.

Polishing

Polishing is the process of creating a smooth and shiny surface by rubbing it or using a chemical action, leaving a surface with a significant specular reflection (still limited by the index of refraction of the material according to the Fresnel equations.) In some materials (such as metals, glasses, black or transparent stones), polishing is also able to reduce diffuse reflection to minimal values. When an unpolished surface is magnified thousands of times, it usually looks like mountains and valleys. By repeated abrasion, those "mountains" are worn down until they are flat or just small "hills." The process of polishing with abrasives starts with coarse ones and graduates to fine ones.

Reflection (physics)

Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The law of reflection says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. Mirrors exhibit specular reflection.

In acoustics, reflection causes echoes and is used in sonar. In geology, it is important in the study of seismic waves. Reflection is observed with surface waves in bodies of water. Reflection is observed with many types of electromagnetic wave, besides visible light. Reflection of VHF and higher frequencies is important for radio transmission and for radar. Even hard X-rays and gamma rays can be reflected at shallow angles with special "grazing" mirrors.

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 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 surface has narrow ridges.

Schlick's approximation

In 3D computer graphics, Schlick's approximation, named after Christophe Schlick, is a formula for approximating the contribution of the Fresnel factor in the specular reflection of light from a non-conducting interface (surface) between two media.

According to Schlick's model, the specular reflection coefficient R can be approximated by:

{\displaystyle {\begin{aligned}R(\theta )&=R_{0}+(1-R_{0})(1-\cos \theta )^{5}\\R_{0}&=\left({\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right)^{2}\end{aligned}}}

where ${\displaystyle \theta }$ is the angle between the direction from which the incident light is coming and the normal of the interface between the two media, hence ${\displaystyle \cos \theta =(N\cdot V)}$. And ${\displaystyle n_{1},\,n_{2}}$ are the indices of refraction of the two media at the interface and ${\displaystyle R_{0}}$ is the reflection coefficient for light incoming parallel to the normal (i.e., the value of the Fresnel term when ${\displaystyle \theta =0}$ or minimal reflection). In computer graphics, one of the interfaces is usually air, meaning that ${\displaystyle n_{1}}$ very well can be approximated as 1.

In microfacet models it is assumed that there is always a perfect reflection, but the normal changes according to a certain distribution, resulting in a non-perfect overall reflection. When using Schlicks's approximation, the normal in the above computation is replaced by the halfway vector. Either the viewing or light direction can be used as the second vector.

Slit lamp

A slit lamp is an instrument consisting of a high-intensity light source that can be focused to shine a thin sheet of light into the eye. It is used in conjunction with a biomicroscope. The lamp facilitates an examination of the anterior segment and posterior segment of the human eye, which includes the eyelid, sclera, conjunctiva, iris, natural crystalline lens, and cornea. The binocular slit-lamp examination provides a stereoscopic magnified view of the eye structures in detail, enabling anatomical diagnoses to be made for a variety of eye conditions. A second, hand-held lens is used to examine the retina.

Specularity

Specularity is the visual appearance of specular reflections.

Sun glitter

Sun glitter is a bright, sparkling light formed when sunlight reflects from water waves. The waves may be caused by natural movement of the water, or by the movement of birds or animals in the water. Even a ripple from a thrown rock will create a momentary glitter.

Light reflects from smooth surfaces by specular reflection. A rippled but locally smooth surface such as water with waves will reflect the sun at different angles at each point on the surface of the waves. As a result, a viewer in the right position will see many small images of the sun, formed by portions of waves that are oriented correctly to reflect the sun's light to the viewer's eyes. The exact pattern seen depends on the viewer's precise location. The color and the length of the glitter depend on the altitude of the Sun. The lower the sun, the longer and more reddish the glitter is. When the sun is really low above the horizon, the glitter breaks because of the waves, which could sometimes obstruct the sun and cast a shadow on the glitter.Sun glitter can be bright enough to damage one's eyes. Caution should be exercised while observing the glitter.

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