Luminance

Not to be confused with Luma (video) or Luminescence.

For other uses, see Luminance (disambiguation).

Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted or reflected from a particular area, and falls within a given solid angle. The SI unit for luminance is candela per square metre (cd/m²). A non-SI term for the same unit is the "nit". The CGS unit of luminance is the stilb, which is equal to one candela per square centimetre or 10 kcd/m².

Explanation

Luminance is often used to characterize emission or reflection from flat, diffuse surfaces. The luminance indicates how much luminous power will be detected by an eye looking at the surface from a particular angle of view. Luminance is thus an indicator of how bright the surface will appear. In this case, the solid angle of interest is the solid angle subtended by the eye's pupil. Luminance is used in the video industry to characterize the brightness of displays. A typical computer display emits between 50 and 300 cd/m². The sun has luminance of about 1.6×10⁹ cd/m² at noon.

Luminance is invariant in geometric optics. This means that for an ideal optical system, the luminance at the output is the same as the input luminance. For real, passive, optical systems, the output luminance is at most equal to the input. As an example, if one uses a lens to form an image that is smaller than the source object, the luminous power is concentrated into a smaller area, meaning that the illuminance is higher at the image. The light at the image plane, however, fills a larger solid angle so the luminance comes out to be the same assuming there is no loss at the lens. The image can never be "brighter" than the source.

Definition

Parameters for defining the luminance

The luminance of a specified point of a light source, in a specified direction, is defined by the derivative

$L_{\mathrm {v} }={\frac {\mathrm {d} ^{2}\Phi _{\mathrm {v} }}{\mathrm {d} \Sigma \,\mathrm {d} \Omega _{\Sigma }\cos \theta _{\Sigma }}}$

where

• L_v is the luminance (cd/m²),
• d²Φ_v is the luminous flux (lm) leaving the area dΣ in any direction contained inside the solid angle dΩ_Σ,
• dΣ is an infinitesimal area (m²) of the source containing the specified point,
• dΩ_Σ is an infinitesimal solid angle (sr) containing the specified direction,
• θ_Σ is the angle between the normal n_Σ to the surface dΣ and the specified direction.

If light travels through a lossless medium, the luminance does not change along a given light ray. As the ray crosses an arbitrary surface S, the luminance is given by

$L_{\mathrm {v} }={\frac {\mathrm {d} ^{2}\Phi _{\mathrm {v} }}{\mathrm {d} S\,\mathrm {d} \Omega _{S}\cos \theta _{S}}}$

where

• dS is the infinitesimal area of S seen from the source inside the solid angle dΩ_Σ,
• dΩ_S is the infinitesimal solid angle subtended by dΣ as seen from dS,
• θ_S is the angle between the normal n_S to dS and the direction of the light.

More generally, the luminance along a light ray can be defined as

$L_{\mathrm {v} }=n^{2}{\frac {\mathrm {d} \Phi _{\mathrm {v} }}{\mathrm {d} G}}$

where

• dG is the etendue of an infinitesimally narrow beam containing the specified ray,
• dΦ_v is the luminous flux carried by this beam,
• n is the index of refraction of the medium.

Relation to Illuminance

The luminance of a reflecting surface is related to the illuminance it receives:

${\begin{aligned}\int _{\Omega _{\Sigma }}L_{\mathrm {v} }\mathrm {d} \Omega _{\Sigma }\cos \theta _{\Sigma }&=M_{\mathrm {v} }\\&=E_{\mathrm {v} }R\end{aligned}}$

where the integral covers all the directions of emission Ω_Σ, and

• M_v is the surface's luminous exitance
• E_v is the received illuminance, and
• R is the reflectance.

In the case of a perfectly diffuse reflector (also called a Lambertian reflector), the luminance is isotropic, per Lambert's cosine law. Then the relationship is simply

$L_{\mathrm {v} }=E_{\mathrm {v} }R/\pi$

Units

A variety of units have been used for luminance, besides the candela per square metre.

One candela per square metre is equal to:

• 10⁻⁴ stilbs (the CGS unit of luminance)
• π apostilbs
• π×10⁻⁴ lamberts
• 0.292 foot-lamberts

Health effects

See also: Laser safety

Retinal damage can occur when the eye is exposed to high luminance. Damage can occur due to local heating of the retina. Photochemical effects can also cause damage, especially at short wavelengths.

See also

• Orders of magnitude (luminance)
• Diffuse reflection
• Etendue
• Exposure value
• Lambertian reflectance
• Lightness, property of a color
• Luma, the representation of luminance in a video monitor
• Lumen (unit)
• Radiance, radiometric quantity analogous to luminance
• Brightness, the subjective impression of luminance
• Glare
• Luminance meter

SI photometry quantities

Quantity		Unit		Dimension	Notes
Name	Symbol	Name	Symbol
Luminous energy	Qv	lumen second	lm⋅s	T⋅J	The lumen second is sometimes called the talbot.
Luminous flux, luminous power	Φv	lumen (= candela steradian)	lm (= cd⋅sr)	J	Luminous energy per unit time
Luminous intensity	Iv	candela (= lumen per steradian)	cd (= lm/sr)	J	Luminous flux per unit solid angle
Luminance	Lv	candela per square metre	cd/m2 (= lm/(sr⋅m2))	L−2⋅J	Luminous flux per unit solid angle per unit projected source area. The candela per square metre is sometimes called the nit.
Illuminance	Ev	lux (= lumen per square metre)	lx (= lm/m2)	L−2⋅J	Luminous flux incident on a surface
Luminous exitance, luminous emittance	Mv	lumen per square metre	lm/m2	L−2⋅J	Luminous flux emitted from a surface
Luminous exposure	Hv	lux second	lx⋅s	L−2⋅T⋅J	Time-integrated illuminance
Luminous energy density	ωv	lumen second per cubic metre	lm⋅s/m3	L−3⋅T⋅J
Luminous efficacy (of radiation)	K	lumen per watt	lm/W	M−1⋅L−2⋅T3⋅J	Ratio of luminous flux to radiant flux
Luminous efficacy (of a source)	η	lumen per watt	lm/W	M−1⋅L−2⋅T3⋅J	Ratio of luminous flux to power consumption
Luminous efficiency, luminous coefficient	V			1	Luminous efficacy normalized by the maximum possible efficacy
See also: SI Photometry Radiometry

External links

• A Kodak guide to Estimating Luminance and Illuminance using a camera's exposure meter. Also available in PDF form and Google Docs online version
• Autodesk Design Academy Measuring Light Levels