Specular reflection

Specular reflection is the mirror-like reflection of light (or of other kinds of wave) from a surface, in which light from a single incoming direction (a ray) 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, thus the angle of incidence equals the angle of reflection ( $\theta _{i}=\theta _{r}$ 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 (AD c. 10–70).^[1]

Background

When light hits a surface, there are three possible outcomes. Light may be absorbed by the material, light may be transmitted through to the other side, or light may be reflected back. 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 $\theta _{i}$ .

Reflected light can be divided into two sub-types, specular reflection and diffuse reflection. Specular reflection reflects all light at the same angle, whereas diffuse reflection reflects 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 $\mathbf {\hat {d}} _{\mathrm {i} }$ from the surface to the light source and the surface normal direction $\mathbf {\hat {d}} _{\mathrm {n} },$ the specularly reflected direction $\mathbf {\hat {d}} _{\mathrm {s} }$ (all unit vectors) is:^[2]^[3]

\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 $\mathbf {\hat {d}} _{\mathrm {n} }\cdot \mathbf {\hat {d}} _{\mathrm {i} }$ 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:

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

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

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

$\mathrm {T}$ denotes transposition and $\mathbf {I}$ is the identity matrix.

Reflectivity

Reflectivity describes how much incident light gets reflected. Specifically, reflectivity 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 quantify the glossy appearance of a surface in gloss units.

Consequences

Internal reflection

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 may occur if the angle of incidence is greater than a certain critical angle. Specular reflection from a dielectric such as water can affect polarization and at Brewster's angle reflected light is completely linearly polarized parallel to the interface.

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 utilizes specular reflection to study material surfaces and thin film interfaces in an analogous fashion to x-ray reflectivity.

References

^ Sir Thomas Little Heath (1981). A history of Greek mathematics. Volume II: From Aristarchus to Diophantus. ISBN 978-0-486-24074-9.
^ Farin, Gerald; Hansford, Dianne (2005). Practical linear algebra: a geometry toolbox. A K Peters. pp. 191–192. ISBN 978-1-56881-234-2.
^ Comninos, Peter (2006). Mathematical and computer programming techniques for computer graphics. Springer. p. 361. ISBN 978-1-85233-902-9.

[1] Sir Thomas Little Heath (1981). A history of Greek mathematics. Volume II: From Aristarchus to Diophantus. ISBN 978-0-486-24074-9.

[2] Farin, Gerald; Hansford, Dianne (2005). Practical linear algebra: a geometry toolbox. A K Peters. pp. 191–192. ISBN 978-1-56881-234-2.

[3] Comninos, Peter (2006). Mathematical and computer programming techniques for computer graphics. Springer. p. 361. ISBN 978-1-85233-902-9.

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