Reflectance

Mathematical definitions

Hemispherical reflectance

The hemispherical reflectance of a surface, denoted R, is defined as $R = \frac{Φ_{e}^{r}}{Φ_{e}^{i}},$ where Φ_e^r is the radiant flux reflected by that surface and Φ_eⁱ is the radiant flux received by that surface.

Spectral hemispherical reflectance

The spectral hemispherical reflectance in frequency and spectral hemispherical reflectance in wavelength of a surface, denoted R_ν and R_λ respectively, are defined as $R_{ν} = \frac{Φ_{e, ν}^{r}}{Φ_{e, ν}^{i}},$ $R_{λ} = \frac{Φ_{e, λ}^{r}}{Φ_{e, λ}^{i}},$ where

Φ_e,ν^r is the spectral radiant flux in frequency reflected by that surface;
Φ_e,νⁱ is the spectral radiant flux in frequency received by that surface;
Φ_e,λ^r is the spectral radiant flux in wavelength reflected by that surface;
Φ_e,λⁱ is the spectral radiant flux in wavelength received by that surface.

Directional reflectance

The directional reflectance of a surface, denoted R_Ω, is defined as $R_{Ω} = \frac{L_{e, Ω}^{r}}{L_{e, Ω}^{i}},$ where

L_e,Ω^r is the radiance reflected by that surface;
L_e,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. 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

The spectral directional reflectance in frequency and spectral directional reflectance in wavelength of a surface, denoted R_Ω,ν and R_Ω,λ respectively, are defined as $R_{Ω, ν} = \frac{L_{e, Ω, ν}^{r}}{L_{e, Ω, ν}^{i}},$ $R_{Ω, λ} = \frac{L_{e, Ω, λ}^{r}}{L_{e, Ω, λ}^{i}},$ where

L_e,Ω,ν^r is the spectral radiance in frequency reflected by that surface;
L_e,Ω,νⁱ is the spectral radiance received by that surface;
L_e,Ω,λ^r is the spectral radiance in wavelength reflected by that surface;
L_e,Ω,λⁱ is the spectral radiance in wavelength received by that surface.

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

4 more chapters are free with a Lably account.

Reflectivity

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, which is the ratio of the reflected to incident electric field; 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, reflectivity is distinguished from reflectance by the fact that reflectivity is a value that applies to thick reflecting objects. 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.

Surface type

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.

Liquid reflectance

Reflection of light occurs at a boundary at which the index of refraction changes. Specular reflection is calculated by the Fresnel equations. Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection.

A liquid surface may be wavy. Reflectance may be adjusted to account for waviness.

Grating efficiency

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