# Luminance

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 from, or is reflected from a particular area, and falls within a given solid angle.

Brightness is the term for the subjective impression of the objective luminance measurement standard (see Objectivity (science) § Objectivity in measurement for the importance of this contrast).

The SI unit for luminance is candela per square metre (cd/m2), as defined by the International System of Units (SI is from the French Système international d'unités) standard for the modern metric system. A non-SI term for the same unit is the nit. The unit in the Centimetre–gram–second system of units (CGS) (which predated the SI system) is the stilb, which is equal to one candela per square centimetre or 10 kcd/m2.

## Description

Luminance is often used to characterize emission or reflection from flat, diffuse surfaces. Luminance levels indicate how much luminous power could be detected by the human eye looking at a particular 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 a luminance of about 1.6×109 cd/m2 at noon.[1]

Luminance is invariant in geometric optics.[2] 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.

## Health effects

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

## Luminance meter

A luminance meter is a device used in photometry that can measure the luminance in a particular direction and with a particular solid angle. The simplest devices measure the luminance in a single direction while imaging luminance meters measure luminance in a way similar to the way a digital camera records color images.[3]

## Mathematical definition

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

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

where

• Lv is the luminance (cd/m2),
• d2Φv is the luminous flux (lm) leaving the area dΣ in any direction contained inside the solid angle dΩΣ,
• dΣ is an infinitesimal area (m2) of the source containing the specified point,
• dΩΣ is an infinitesimal solid angle (sr) containing the specified direction,
• θΣ is the angle between the normal nΣ to the surface dΣ and the specified direction.[4]

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

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

where

• dS is the infinitesimal area of S seen from the source inside the solid angle dΩΣ,
• dΩS is the infinitesimal solid angle subtended by dΣ as seen from dS,
• θS is the angle between the normal nS to dS and the direction of the light.

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

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

where

• dG is the etendue of an infinitesimally narrow beam containing the specified ray,
• dΦv 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:

{\displaystyle {\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 ΩΣ, and

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

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

## Units

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

One candela per square metre is equal to:

### List

SI photometry quantities
Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol[nb 2]
Luminous energy Qv[nb 3] lumen second lm⋅s TJ The lumen second is sometimes called the talbot.
Luminous flux, luminous power Φv[nb 3] lumen (= candela steradians) 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 L−2J 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−2J Luminous flux incident on a surface
Luminous exitance, luminous emittance Mv lumen per square metre lm/m2 L−2J Luminous flux emitted from a surface
Luminous exposure Hv lux second lx⋅s L−2TJ Time-integrated illuminance
Luminous energy density ωv lumen second per cubic metre lm⋅s/m3 L−3TJ
Luminous efficacy (of radiation) K lumen per watt lm/W M−1L−2T3J Ratio of luminous flux to radiant flux
Luminous efficacy (of a source) η[nb 3] lumen per watt lm/W M−1L−2T3J Ratio of luminous flux to power consumption
Luminous efficiency, luminous coefficient V 1 Luminous efficacy normalized by the maximum possible efficacy