Reflectance

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Spectral reflectance curves for aluminium (Al), silver (Ag), and gold (Au) metal mirrors at normal incidence.

The 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 the boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light, and is in general a function of the frequency, or wavelength, of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.

Mathematical definitions[edit]

Hemispherical reflectance[edit]

The hemispherical reflectance of a surface, denoted R, is defined as[1]

where Φer is the radiant flux reflected by that surface and Φei is the radiant flux received by that surface.

Spectral hemispherical reflectance[edit]

The spectral hemispherical reflectance in frequency and spectral hemispherical reflectance in wavelength of a surface, denoted Rν and Rλ respectively, are defined as[1]

where

Directional reflectance[edit]

The directional reflectance of a surface, denoted RΩ, is defined as[1]

where

  • Le,Ωr is the radiance reflected by that surface;
  • Le,Ωi is the radiance received by that surface.

This depends on both the reflected direction and the incoming direction. In other words, it has a value for every combination of incoming and outgoing directions. It is related to the bidirectional reflectance distribution function and its upper limit is 1. Another measure of reflectance, depending only on the outgoing direction, is I/F, where I is the radiance reflected in a given direction and F is the incoming radiance averaged over all directions, in other words, the total flux of radiation hitting the surface per unit area, divided by π.[2] This can be greater than 1 for a glossy surface illuminated by a source such as the sun, with the reflectance measured in the direction of maximum radiance (see also Seeliger effect).

Spectral directional reflectance[edit]

The spectral directional reflectance in frequency and spectral directional reflectance in wavelength of a surface, denoted RΩ,ν and RΩ,λ respectively, are defined as[1]

where

Again, one can also define a value of I/F (see above) for a given wavelength.[3]

Reflectivity[edit]

Fresnel reflection coefficients for a boundary surface between air and a variable material in dependence of the complex refractive index and the angle of incidence.

For homogeneous and semi-infinite (see halfspace) materials, reflectivity is the same as reflectance. Reflectivity is the square of the magnitude of the Fresnel reflection coefficient,[4] which is the ratio of the reflected to incident electric field;[5] as such the reflection coefficient can be expressed as a complex number as determined by the Fresnel equations for a single layer, whereas the reflectance is always a positive real number.

For layered and finite media, according to the CIE,[citation needed] reflectivity is distinguished from reflectance by the fact that reflectivity is a value that applies to thick reflecting objects.[6] When reflection occurs from thin layers of material, internal reflection effects can cause the reflectance to vary with surface thickness. Reflectivity is the limit value of reflectance as the sample becomes thick; it is the intrinsic reflectance of the surface, hence irrespective of other parameters such as the reflectance of the rear surface. Another way to interpret this is that the reflectance is the fraction of electromagnetic power reflected from a specific sample, while reflectivity is a property of the material itself, which would be measured on a perfect machine if the material filled half of all space.[7]

Surface type[edit]

Given that reflectance is a directional property, most surfaces can be divided into those that give specular reflection and those that give diffuse reflection.

For specular surfaces, such as glass or polished metal, reflectance is nearly zero at all angles except at the appropriate reflected angle; that is the same angle with respect to the surface normal in the plane of incidence, but on the opposing side. When the radiation is incident normal to the surface, it is reflected back into the same direction.

For diffuse surfaces, such as matte white paint, reflectance is uniform; radiation is reflected in all angles equally or near-equally. Such surfaces are said to be Lambertian.

Most practical objects exhibit a combination of diffuse and specular reflective properties.

Water reflectance[edit]

Reflectance of smooth water at 20 °C (refractive index 1.333).

Reflection occurs when light moves from a medium with one index of refraction into a second medium with a different index of refraction.

Specular reflection from a body of water is calculated by the Fresnel equations.[8] Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection.

A real water surface may be wavy. Reflectance, which assumes a flat surface as given by the Fresnel equations, can be adjusted to account for waviness.

Grating efficiency[edit]

The generalization of reflectance to a diffraction grating, which disperses light by wavelength, is called diffraction efficiency.

Other radiometric coefficients[edit]

Quantity SI units Notes
Name Sym.
Hemispherical emissivity ε Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity εν
 or
ελ
Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity εΩ Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity εΩ,ν
 or
εΩ,λ
Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance Aν
 or
Aλ
Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance AΩ Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance AΩ,ν
 or
AΩ,λ
Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance R Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance Rν
 or
Rλ
Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance RΩ Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance RΩ,ν
 or
RΩ,λ
Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance Tν
 or
Tλ
Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance TΩ Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance TΩ,ν
 or
TΩ,λ
Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient μ m−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient μν
 or
μλ
m−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient μΩ m−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient μΩ,ν
 or
μΩ,λ
m−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

See also[edit]

References[edit]

  1. ^ a b c d "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
  2. ^ Jeffrey Cuzzi, Lindsey Chambers, and Amanda Hendrix (Oct 21, 2016). "Rough Surfaces: is the dark stuff just shadow?". Icarus. 289: 281–294. doi:10.1016/j.icarus.2016.10.018. PMC 6839776. PMID 31708591.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. ^ See for example P.G.J Irwin; et al. (Jan 12, 2022). "Hazy Blue Worlds: A Holistic Aerosol Model for Uranus and Neptune, Including Dark Spots" (PDF). Journal of Geophysical Research: Planets. 127 (6). arXiv:2201.04516. Bibcode:2022JGRE..12707189I. doi:10.1029/2022JE007189. S2CID 245877540.
  4. ^ E. Hecht (2001). Optics (4th ed.). Pearson Education. ISBN 0-8053-8566-5.
  5. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Reflectance". doi:10.1351/goldbook.R05235
  6. ^ "CIE International Lighting Vocabulary". Archived from the original on 2016-06-16. Retrieved 2010-12-04.
  7. ^ Palmer and Grant, The Art of Radiometry
  8. ^ Ottaviani, M. and Stamnes, K. and Koskulics, J. and Eide, H. and Long, S.R. and Su, W. and Wiscombe, W., 2008: 'Light Reflection from Water Waves: Suitable Setup for a Polarimetric Investigation under Controlled Laboratory Conditions. Journal of Atmospheric and Oceanic Technology, 25 (5), 715--728.

External links[edit]