View source for Spectral power distribution

{{for|the [[statistics]] and [[signal processing]] concept|Power spectral density}}
[[File:NormSPD with eye.png|thumb|[[Standard illuminant|CIE standard illuminant]] spectral power distribution comparisons referenced to the human visual system [[photopic vision|photopic response]]]]

In [[radiometry]], [[photometry (optics)|photometry]], and [[color science]], a '''spectral power distribution''' ('''SPD''') measurement describes the [[Power (physics)|power]] per unit [[area]] per unit [[wavelength]] of an [[illumination (lighting)|illumination]] ([[radiant exitance]]). More generally, the term ''spectral power distribution'' can refer to the concentration, as a function of wavelength, of any radiometric or photometric quantity (e.g. [[radiant energy]], [[radiant flux]], [[radiant intensity]], [[radiance]], [[irradiance]], [[radiant exitance]], [[Radiosity (heat transfer)|radiosity]], [[luminance]], [[luminous flux]], [[luminous intensity]], [[illuminance]], [[luminous emittance]]).<ref>{{cite book | title = Color Appearance Models | author = Mark D. Fairchild | isbn = 0-470-01216-1 | date = 2005 | publisher = [[John Wiley and Sons]]| url = https://books.google.com/books?id=8_TxzK2B-5MC&pg=PA56&dq=light+source+%22spectral+power+distribution%22&lr=&as_brr=3&ei=wgCJR6bHNpOMtAPpj6HQBQ&sig=dIlnEgPmXu9Sy3YpEVF6wy9m8tU#PPA57,M1 }}</ref><ref name=focal>{{cite book | title = The Focal Encyclopedia of Photography | author = Michael R. Peres | publisher = [[Focal Press]] | date = 2007 | isbn = 978-0-240-80740-9 | url = https://books.google.com/books?id=VYyldcYfq3MC&pg=RA1-PA383&dq=%22spectral+power+distribution%22+555+560+nm&lr=&as_brr=0&as_pt=ALLTYPES&ei=MdVTSaHNEJTckwSOpfQk }}</ref><ref name=intro>{{cite book | title = Introduction to Radiometry and Photometry | author = William Ross McCluney | isbn = 0890066787 | date = 1994 | publisher = Boston: Artech House }}</ref><ref name=meas>{{cite book | title = Optical Radiation Measurements (v. 1) | author = Franc C. Grum | isbn = 0123049016 | date = 1979 | publisher = New York: Academic Press }}</ref>

Knowledge of the SPD is crucial for optical-sensor system applications. [[Optical properties]] such as [[transmittance]], [[reflectivity]], and [[absorbance]] as well as the sensor response are typically dependent on the incident wavelength.<ref name=intro/>

== Physics ==

Mathematically, for the spectral power distribution of a radiant exitance or irradiance one may write:
: <math>M(\lambda)=\frac{\partial^2\Phi}{\partial A\,\partial\lambda}\approx\frac{\Phi}{A\,\Delta\lambda}</math>
where ''M''(''λ'') is the [[spectral irradiance]] (or exitance) of the light ([[SI]] units: [[watt|W]]/m<sup>3</sup> = [[kilogram|kg]]·m<sup>−1</sup>·[[second|s]]<sup>−3</sup>); ''Φ'' is the radiant flux of the source (SI unit: watt, W); ''A'' is the area over which the radiant flux is integrated (SI unit: square meter, m<sup>2</sup>); and ''λ'' is the wavelength (SI unit: meter, m). (Note that it is more convenient to express the wavelength of light in terms of [[nanometer]]s; spectral exitance would then be expressed in units of W·m<sup>−2</sup>·nm<sup>−1</sup><!--= W·m<sup>−9</sup> = W/mm<sup>3</sup> = W/µl-->.) The approximation is valid when the area and wavelength interval are small.<ref name=des>{{cite book | title = Radiometric System Design | author = Clair L. Wyatt | isbn = 0029488001 | date = 1987 | publisher = New York: Macmillan }}</ref>

==Relative SPD==
[[Image:Spectral Power Distributions.png|right|frame|Characteristic spectral power distributions (SPDs) for an [[Incandescent light bulb|incandescent lamp]] (left) and a [[fluorescent lamp]] (right). The horizontal axes are in [[nanometer]]s and the vertical axes show relative intensity in arbitrary units.]]

The ratio of spectral concentration (irradiance or exitance) at a given wavelength to the concentration of a reference wavelength provides the relative SPD.<ref name=meas/> This can be written as:
: <math>M_\mathrm{rel}(\lambda)=\frac{M(\lambda)}{M\left(\lambda_0\right)}</math>
For instance, the [[luminance]] of lighting fixtures and other light sources are handled separately, a spectral power distribution may be normalized in some manner, often to unity at 555 or 560 nanometers, coinciding with the peak of the eye's [[luminosity function]].<ref name=focal/><ref name="WyszeckiStiles">{{cite book | first=Günter | last=Wyszecki |author2=Stiles, Walter Stanley | title=Color Science: Concepts and Methods; Quantitative Data and Formulae | edition=second | publisher=New York: Wiley | date=1982 | isbn=978-0-471-39918-6}}</ref>

==Responsivity==
The SPD can be used to determine the response of a [[sensor]] at a specified wavelength. This compares the output power of the sensor to the input power as a function of wavelength.<ref name=det>{{cite book | title = Radiometry and the Detection of Optical Radiation | author = Robert W. Boyd | isbn = 047186188X | date = 1983 | publisher = New York: Wiley }}</ref> This can be generalized in the following formula:
: <math>R(\lambda)=\frac{S(\lambda)}{M(\lambda)}</math>
Knowing the responsitivity is beneficial for determination of illumination, interactive material components, and optical components to optimize performance of a system's design.

==Source SPD and matter==
[[File:Rayleigh sunlight scattering.svg|thumb|right|250px|Figure showing the greater proportion of blue light scattered by the atmosphere relative to red light.]]The spectral power distribution over the [[visible spectrum]] from a source can have varying concentrations of relative SPDs. The interactions between light and matter affect the absorption and reflectance properties of materials and subsequently produces a color that varies with source illumination.<ref name=mcol>{{cite book | title = The Measurement of Colour | author = William David Wright | date = 1969 | publisher = New York: Van Nostrand Reinhold Co. }}</ref>

For example, the relative spectral power distribution of the sun produces a white appearance if observed directly, but when the sunlight illuminates the Earth's atmosphere the sky appears blue under normal daylight conditions. This stems from the optical phenomenon called [[Rayleigh scattering]] which produces a concentration of shorter wavelengths and hence the blue color appearance.<ref name=intro/>

==Source SPD and color appearance==
[[File:Incand-3500-5500-color-temp-comparison.png|right|thumb|alt=Color temperature comparison of common electric lamps|Color temperature comparison of common electric lamps]]

The human visual response relies on [[trichromacy]] to process color appearance. While the human visual response integrates over all wavelengths, the relative spectral power distribution will provide [[color appearance model]]ing information as the concentration of wavelength band(s) will become the primary contributors to the perceived color.<ref name=mcol/>

This becomes useful in photometry and [[colorimetry]] as the perceived color changes with source illumination and spectral distribution and coincides with [[Metamerism (color)|metamerisms]] where an object's color appearance changes.<ref name=mcol/>

The spectral makeup of the source can also coincide with [[color temperature]] producing differences in color appearance due to the source's temperature.<ref name=meas/>

==See also==
{{cmn|colwidth=30em|
*[[Color]]
*[[Lighting]]
*[[Photometry (optics)]]
*[[Physical quantities]]
*[[Radiometry]]
*[[Spectral density estimation]]
}}

==References==

{{Reflist}}

==External links==
* [https://web.archive.org/web/20120626153615/http://www.gelighting.com/na/business_lighting/education_resources/learn_about_light/distribution_curves.htm Spectral Power Distribution Curves], GE Lighting.

{{Color topics}}

[[Category:Radiometry]]
[[Category:Color]]
[[Category:Lighting]]
[[Category:Physical quantities]]