II.7. Basic photometric quantities

One of the central problems of optical measurements is the quantification of light sources and lightning conditions in numbers directly related to the perception of the human eye. This discipline is called “Photometry”, and its significance leads to the use of separate physical quantities, which differ from the respective radiometric quantities only in one respect: Whereas radiometric quantities simply represent a total sum of radiation power at various wavelengths and do not account for the fact that the human eye’s sensitivity to optical radiation depends on wavelength, the photometric quantities represent a weighted sum with the weighting factor being defined by either the photopic or the scotopic spectral luminous efficiency function (see § II.6). Thus, the numerical value of photometric quantities directly relates to the impression of “brightness”. Photometric quantities are distinguished from radiometric quantities by the index “v” for “visual”, and photometric quantities relating to scotopic vision are denoted by an additional prime, for example Φv’. The following explanations are given for the case of photopic vision, which describes the eye’s sensitivity under daylight conditions, thus being relevant for the vast majority of lighting situations (photopic vision takes place when the eye is adapted to luminance levels of at least several candelas per square meters, scotopic vision takes place when the eye is adapted to luminance levels below some hundredths of a candela per square meter. For mesopic vision, which takes place in the range in between, no spectral luminous efficiency function has been defined yet). However, the respective relations for scotopic vision can be easily derived when replacing V(λ) by V’(λ) and replacing Km ( = 683 lm / W ) by K’m (= 1700 lm / W) (see definition in § VII.2).

As the definition of photometric quantities closely follows the corresponding definitions of radiometric quantities, the corresponding equations hold true – the index “e” just has to be replaced by the index “v”. Thus, not all relations are repeated. Instead, a more general formulation of all relevant relations is given in the Appendix.

II.7.a.Luminous flux Φv
Luminous flux Φv is the basic photometric quantity and describes the total amount of electromagnetic radiation emitted by a source, spectrally weighted with the human eye’s spectral luminous efficiency function V(λ). Luminous flux is the photometric counterpart to radiant power. The unit of luminous flux is lumen (lm), and at 555 nm, where the human eye has its maximum sensitivity, a radiant power of 1 W corresponds to a luminous flux of 683 lm. In other words, a monochromatic source emitting 1 W at 555 nm has a luminous flux of exactly 683 lm. The value of 683 lm / W is abbreviated by the symbol Km (the value of Km = 683 lm/W is given for photopic vision. For scotopic vision, Km’=1700 lm/W has to be used). However, a monochromatic light source emitting the same radiant power at 650 nm, where the human eye is far less sensitive and V(λ) = 0.107, has a luminous flux of 0.107*683 lm = 73.1 lm. For a more detailed explanation of the conversion of radiometric to photometric quantities, see paragraph II.7.f.

II.7.b. Luminous intensity Iv
Luminous intensity Iv quantifies the luminous flux emitted by a source in a certain direction. It is therefore the photometric counterpart of the radiometric quantity “radiant intensity” Ie. In detail, the source’s (differential) luminous flux dΦv emitted in the direction of the (differential) solid angle element dΩ is given by

v = Iv · dΩ

and thus

The unit of luminous intensity is lumen per steradian (lm / sr), which is abbreviated with the expression “candela” (cd):

1 cd = 1 lm / sr

II.7.c. Luminance Lv
Luminance Lv describes the measurable photometric brightness of a certain location on a reflecting or emitting surface when viewed from a certain direction. It describes the luminous flux emitted or reflected from a certain location on an emitting or reflecting surface in a particular direction (the CIE definition of luminance is more general. Within the frame of this tutorial, the most relevant application of luminance describing the spatial emission characteristics of a source is discussed).

In detail, the (differential) luminous flux dΦv emitted by a (differential) surface element dA in the direction of the (differential) solid angle element dΩ is given by

v = Lv cos(θ) · dA · dΩ

with J denoting the angle between the direction of the solid angle element dΩ and the normal of the emitting or reflecting surface element dA.

The unit of luminance is

1 lm m-2 sr-1 = 1 cd m-2

II.7.d. Illuminance Ev
Illuminance Ev describes the luminous flux per area impinging upon a certain location of an irradiated surface. In detail, the (differential) luminous flux dΦv upon the (differential) surface element dA is given by

v = Ev · dA

Generally, the surface element can be oriented at any angle towards the direction of the beam. Similar to the respective relation for irradiance, illuminance Ev upon a surface with arbitrary orientation is related to illuminance Ev,normal upon a surface perpendicular to the beam by

Ev = Ev,normal cos(J)

with J denoting the angle between the beam and the surface’s normal.

The unit of illuminance is lux (lx), and

1 lx = 1 lm m-2

II.7.e. Luminous exitance Mv
uminous exitance Mv quantifies the luminous flux per area, emitted or reflected from a certain location on a surface. In detail, the (differential) luminous flux dΦv emitted or reflected by the surface element dA is given by

v = Mv · dA

The unit of luminous exitance is 1 lm m-2, which is the same as the unit for illuminance. However, the abbreviation lux is not used for luminous exitance.

II.7.f. Conversion between radiometric and photometric quantities
Monochromatic radiation: In the case of monochromatic radiation at a certain wavelength λ, a radiometric quantity Xe is simply transformed to its photometric counterpart Xv by multiplication with the respective spectral luminous efficiency V(λ) and by the factor Km = 683 lm / W. Thus,

Xv = Xe * V(λ) * 683 lm / W
with X denoting one of the quantities Φ, I, L, or E.

Example: An LED (light emitting diode) emits nearly monochromatic radiation at λ = 670 nm, where V(λ) = 0.032. Its radiant power amounts to 5 mW. Thus, its luminous flux equals

Φv = Φe * V(λ) * 683 lm / W = 0.109 lm = 109 mlm

As V(λ) changes very rapidly in this spectral region (by a factor of 2 within a wavelength interval of 10 nm), for accurate results the LED’s light output should not be considered monochromatic. However, using the relations for monochromatic sources still results in an approximate value for the LED’s luminous flux which might be sufficient in many cases.

Polychromatic radiation: If a source emits polychromatic light described by the spectral radiant power Φλ(λ), its luminous flux can be calculated by spectral weighting of Φλ(λ) with the human eye’s spectral luminous efficiency function V(λ), integration over wavelength and multiplication with with Km = 683 lm / W, so

In general, a photometric quantity Xv is calculated from its spectral radiometric counterpart Xλ(λ) by the relation

with X denoting one of the quantities Φ, I, L, or E.