# Specular reflection

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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.

A mirror is an object that reflects light in such a way that, for incident light in some range of wavelengths, the reflected light preserves many or most of the detailed physical characteristics of the original light, called specular reflection. This is different from other light-reflecting objects that do not preserve much of the original wave signal other than color and diffuse reflected light, such as flat-white paint.

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 physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.

## Contents

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.

Hero of Alexandria was a mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He is considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition.

The terms anno Domini (AD) and before Christ (BC) are used to label or number years in the Julian and Gregorian calendars. The term anno Domini is Medieval Latin and means "in the year of the Lord", but is often presented using "our Lord" instead of "the Lord", taken from the full original phrase "anno Domini nostri Jesu Christi", which translates to "in the year of our Lord Jesus Christ".

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.

## Background

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 ${\displaystyle \theta _{i}}$.

Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is the ratio of the transmitted to incident electric field.

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.

Paint is any pigmented liquid, liquefiable, or 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 to objects. Paint can be made or purchased in many colors—and in many different types, such as watercolor, synthetic, etc. Paint is typically stored, sold, and applied as a liquid, but most types dry into a solid.

## 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.

In geometric optics, the angle of incidence is the angle between a ray incident on a surface and the line perpendicular to the surface at the point of incidence, called the normal. The ray can be formed by any wave: optical, acoustic, microwave, X-ray and so on. In the figure below, the line representing a ray makes an angle θ with the normal. The angle of incidence at which light is first totally internally reflected is known as the critical angle. The angle of reflection and angle of refraction are other angles related to beams.

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.

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle. In classical physics, the diffraction phenomenon is described as the interference of waves according to the Huygens–Fresnel principle that treats each point in the wave-front as a collection of individual spherical wavelets. These characteristic behaviors are exhibited when a wave encounters an obstacle or a slit that is comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance. Diffraction has an impact on the acoustic space. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays and radio waves.

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength 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 characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

### 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

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.
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.

## Related Research Articles

The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as

In physics refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

In computer graphics, ray tracing is a rendering technique for generating an image by tracing the path of light as pixels in an image plane and simulating the effects of its encounters with virtual objects. The technique is capable of producing a very high degree of visual realism, usually higher than that of typical scanline rendering methods, but at a greater computational cost. This makes ray tracing best suited for applications where taking a relatively long time to render a frame can be tolerated, such as in still images and film and television visual effects, and more poorly suited for real-time applications such as video games where speed is critical. Ray tracing is capable of simulating a wide variety of optical effects, such as reflection and refraction, scattering, and dispersion phenomena.

Total internal reflection is the phenomenon which occurs when a propagated wave 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 greater than the critical angle, the wave cannot pass through and is entirely reflected. The critical angle is the angle of incidence above which the total internal reflection occurs. This is particularly common as an optical phenomenon, where light waves are involved, but it occurs with many types of waves, such as electromagnetic waves in general or sound waves. When a wave reaches a boundary between different materials at an angle of less than 25° with different refractive indices, the wave will in general be partially refracted at the boundary surface, and partially reflected. However, if the angle of incidence is greater than the critical angle – the angle of incidence at which light is refracted such that it travels along the boundary – then the wave will not cross the boundary, but will instead be totally reflected back internally. This can only occur when the wave in a medium with a higher refractive index (n1) reaches a boundary with a medium of lower refractive index (n2). For example, it will occur with light reaching air from glass, but not when reaching glass from air.

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 polarized. This special angle of incidence is named after the Scottish physicist Sir David Brewster (1781–1868).

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 an interface. The reflectance spectrum or spectral reflectance curve is the plot of the reflectance as a function of wavelength.

Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.

A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the half-wave plate, which shifts the polarization direction of linearly polarized light, and the quarter-wave plate, which converts linearly polarized light into circularly polarized light and vice versa. A quarter-wave plate can be used to produce elliptical polarization as well.

The Phong reflection model is an empirical model of the local illumination of points on a surface. In 3D computer graphics, it is sometimes referred to as "Phong shading", in particular if the model is used with the interpolation method of the same name and in the context of pixel shaders or other places where a lighting calculation can be referred to as “shading”.

Specularity is the visual appearance of specular reflections.

Geometrical optics, or ray optics, describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.

Lambertian reflectance is the property that defines an ideal "matte" or diffusely reflecting surface. The apparent brightness of a Lambertian surface to an observer is the same regardless of the observer's angle of view. More technically, the surface's luminance is isotropic, and the luminous intensity obeys Lambert's cosine law. Lambertian reflectance is named after Johann Heinrich Lambert, who introduced the concept of perfect diffusion in his 1760 book Photometria.

The bidirectional reflectance distribution function is a function of four real variables that defines how light is reflected at an opaque surface. It is employed in the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction, , and outgoing direction, , and returns the ratio of reflected radiance exiting along to the irradiance incident on the surface from direction . Each direction is itself parameterized by azimuth angle and zenith angle , therefore the BRDF as a whole is a function of 4 variables. The BRDF has units sr−1, with steradians (sr) being a unit of solid angle.

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.

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.

X-ray optics is the branch of optics that manipulates X-rays instead of visible light. It deals with focusing and other ways of manipulating the X-ray beams for research techniques such as X-ray crystallography, X-ray fluorescence, small-angle X-ray scattering, X-ray microscopy, X-ray phase-contrast imaging, X-ray astronomy etc.

Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light. When the thickness of the film is an odd multiple of one quarter-wavelength of the light on it, the reflected waves from both surfaces interfere to cancel each other. Since the wave cannot be reflected, it is completely transmitted instead. When the thickness is a multiple of a half-wavelength of the light, the two reflected waves reinforce each other, increasing the reflection and reducing the transmission. Thus when white light, which consists of a range of wavelengths, is incident on the film, certain wavelengths (colors) are intensified while others are attenuated. Thin-film interference explains the multiple colors seen in light reflected from soap bubbles and oil films on water. It is also the mechanism behind the action of antireflection coatings used on glasses and camera lenses.