Luminous efficiency function

File:Fluorescence in beer @ 450nm illumination.jpg in beer. The one watt laser appears much dimmer than the fluorescence it produces, because the camera, like the human eye, is much more sensitive between 500 and 600 nm than at the laser's 450 nm wavelength.]]

There are two luminous efficiency functions in common use. For everyday light levels, the photopic luminosity function best approximates the response of the human eye. For low light levels, the response of the human eye changes, and the scotopic curve applies. The photopic curve is the CIE standard curve used in the CIE 1931 color space.

The luminous flux (or visible power) in a light source is defined by the photopic luminosity function (assuming it is bright enough to activate photopic vision in the eyes). The following equation calculates the total luminous flux in a source of light:

: $\backslash Phi\_\backslash mathrm\{v\}\; =\; 683.002\backslash \; (\backslash mathrm\{lm/W\})\; \backslash cdot\; \backslash int^\backslash infin\_0\; \backslash overline\{y\}(\backslash lambda)\; \backslash Phi\_\{\backslash mathrm\{e\},\backslash lambda\}(\backslash lambda)\backslash ,\; \backslash mathrm\{d\}\backslash lambda,$

where

• Φ_v is the luminous flux, in lumens;
• Φ_e,λ is the spectral radiant flux, in watts per nanometre;
• {{overline|y}}(λ), also known as V(λ), is the luminosity function, dimensionless;
• λ is the wavelength, in nanometres.

Formally, the integral is the inner product of the luminosity function with the spectral power distribution. In practice, the integral is replaced by a sum over discrete wavelengths for which tabulated values of the luminous efficiency function are available. The CIE distributes standard tables with luminosity function values at {{nowrap|5 nm}} intervals from {{nowrap|380 nm}} to {{nowrap|780 nm}}.

The standard luminous efficiency function is normalized to a peak value of unity at {{nowrap|555 nm}} (see luminous coefficient). The value of the constant in front of the integral is usually rounded off to {{val|683|u=lm/W}}. The small excess fractional value comes from the slight mismatch between the definition of the lumen and the peak of the luminosity function. The lumen is defined to be unity for a radiant energy of {{nowrap|1/683 W}} at a frequency of {{nowrap|540 THz}}, which corresponds to a standard air wavelength of {{nowrap|555.016 nm}} rather than {{val|555|u=nm}}, which is the peak of the luminosity curve. The value of {{overline|y}}(λ) is {{val|0.999997}} at {{val|555.016|u=nm}}, so that a value of 683/{{val|0.999997}} = 683.002 is the multiplicative constant.

The number 683 is connected to the modern (1979) definition of the candela, the unit of luminous intensity. This arbitrary number made the new definition give numbers equivalent to those from the old definition of the candela.

The CIE 1924 photopic V(λ) luminosity function, which is included in the CIE 1931 color-matching functions as the {{overline|y}}(λ) function, has long been acknowledged to underestimate the contribution of the blue end of the spectrum to perceived luminance. There have been numerous attempts to improve the standard function, to make it more representative of human vision. Judd in 1951, improved by Vos in 1978, resulted in a function known as CIE V_M(λ). More recently, Sharpe, Stockman, Jagla & Jägle (2005) developed a function consistent with the Stockman & Sharpe cone fundamentals; their curves are plotted in the figure above.

Stockman & Sharpe has subsequently produced an improved function in 2011, taking into account the effects of chromatic adaptation under daylight.{{cite journal |last1=Sharpe |first1=L.T. |last2=Stockman |first2=A. |display-authors=etal|date=February 2011 |title=A Luminous Efficiency Function, V*D65(λ), for Daylight Adaptation: A Correction |journal=COLOR Research and Application |volume=36 |issue=1 |pages=42–46 |doi=10.1002/col.20602 |doi-access=free }} Their work in 2008{{cite journal |last1=Stockman |first1=A |last2=Jägle |first2=H |last3=Pirzer |first3=M |last4=Sharpe |first4=LT |title=The dependence of luminous efficiency on chromatic adaptation. |journal=Journal of Vision |date=15 December 2008 |volume=8 |issue=16 |pages=1.1–26 |doi=10.1167/8.16.1 |pmid=19146268 |doi-access=free}} has revealed that "luminous efficiency or V(l) functions change dramatically with chromatic adaptation".{{cite journal |last1=Stockman |first1=Andrew|date=December 2019 |title=Cone fundamentals and CIE standards |url=http://www.cvrl.org/people/Stockman/pubs/2019%20Cone%20fundamentals%20CIE%20S.pdf |journal=Current Opinion in Behavioral Sciences |volume=30 |issue= |pages=87–93 |doi= 10.1016/j.cobeha.2019.06.005|s2cid=199544026 |access-date=27 October 2023}}

The ISO standard is ISO/CIE FDIS 11664-1. The standard provides an incremental table by nm of each value in the visible range for the CIE 1924 function.{{cite web |url=https://www.iso.org/standard/74164.html |title=Colorimetry -- Part 1: CIE standard colorimetric observers|access-date=December 9, 2018}}{{cite web|url=http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_3.html|title=Kay & Laby;tables of physical & chemical constants;General physics;SubSection: 2.5.3 Photometry|publisher=National Physical Laboratory; UK|access-date=December 9, 2018 |url-status=dead |archive-url=https://web.archive.org/web/20190501164428/http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_3.html |archive-date=May 1, 2019}}

For very low levels of intensity (scotopic vision), the sensitivity of the eye is mediated by rods, not cones, and shifts toward the violet, peaking around {{nowrap|507 nm}} for young eyes; the sensitivity is equivalent to {{val|1699|u=lm/W}} or {{val|1700|u=lm/W}} at this peak. The standard scotopic luminous efficiency function or V{{′}}(λ) was adopted by the CIE in 1951, based on measurements by Wald (1945) and by Crawford (1949).{{Cite web|url=http://www.cvrl.org/database/text/lum/scvl.htm|title=Scotopic luminosity function}}

Luminosity for mesopic vision, a wide transitioning band between scotopic and phototic vision, is more poorly standardized. The consensus is that this luminous efficiency can be written as a weighted average of scotopic and mesopic luminosities, but different organizations provide different weighting factors.[http://www.visual-3d.com/Education/LightingLessons/Documents/PhotopicScotopiclumens_4%20_2_.pdf Photopic and Scotopic lumens - 4: When the photopic lumen fails us]

File:LuminosityCurve3.svg

Color blindness changes the sensitivity of the eye as a function of wavelength. For people with protanopia, the peak of the eye's response is shifted toward the short-wave part of the spectrum (approximately 540 nm), while for people with deuteranopia, there is a slight shift in the peak of the spectrum, to about 560 nm. People with protanopia have essentially no sensitivity to light of wavelengths more than 670 nm.

For older people with normal color vision, the crystalline lens may become slightly yellow due to cataracts, which moves the maximum of sensitivity to the red part of the spectrum and narrows the range of perceived wavelengths.{{cite journal |last1=Artigas |first1=Jose M. |last2=Felipe |first2=Adelina |last3=Navea |first3=Amparo |last4=Fandiño |first4=Adriana |last5=Artigas |first5=Cristina |title=Spectral Transmission of the Human Crystalline Lens in Adult and Elderly Persons: Color and Total Transmission of Visible Light |journal=Investigative Opthalmology & Visual Science |date=25 June 2012 |volume=53 |issue=7 |pages=4076–4084 |doi=10.1167/iovs.12-9471|pmid=22491402 }} A method for estimating the transmittance of the human crystalline lens depending on age is standardized as CIE 203:2012,{{cite web |title=A Computerized Approach to Transmission and Absorption Characteristics of the Human Eye {{!}} CIE |url=https://cie.co.at/publications/computerized-approach-transmission-and-absorption-characteristics-human-eye |website=cie.co.at}} though further improvement has been proposed.{{cite journal |last1=Li |first1=Jiaye |last2=Hanselaer |first2=Peter |last3=Smet |first3=Kevin A. G. |title=Individual Color Matching Functions Estimated from Spectrally Narrow-Band Achromatic Matches Using Physiological Observer Models |journal=LEUKOS |date=20 February 2025 |pages=1–21 |doi=10.1080/15502724.2024.2419640}} For a few more lens transmission functions, see the Lucas (2014) Irradiance Toolbox.{{cite web |title=Irradiance toolbox user guide (The University of Manchester) |url=https://documents.manchester.ac.uk/DocuInfo.aspx?DocID=57020 |website=documents.manchester.ac.uk}}

The wavelength-dependent effect of light is seen not only with vision, but also (in humans) in the circadian rhythm via melanopsin. For reporting the effect of light on the human circadian rhythm, a value called melanopic illuminance is used, defined using a luminous efficiency function specific to the melanopsin. The unit is lux (lx) used in a non-SI-compliant fashion. With CIE S 026:2018, the system has become SI-compliant, with the melanopic equivalent daylight illuminance (M-EDI, unit lx) being derived from melanopic irradiance (unit W/m²). A human being subject to 100 lx of M-EDI of light should have the same alternation to their circadian rhythm as if they are being exposed to 100 lx of daylight.{{cite web |title=Measuring melanopic illuminance and melanopic irradiance |url=https://lucasgroup.lab.manchester.ac.uk/measuringmelanopicilluminance/ |website=Lucas Group}}

Lucas (2014) and the later CIE S 026 also define luminous efficiency function specific to four other human opsins. Lucas uses non-SI-compliant lux while CIE uses SI-compliant EDI lux. Although the CIE standard requires payment, the associated toolbox and its user guide is available for free.{{cite web |url=https://files.cie.co.at/CIE%20S%20026%20alpha-opic%20Toolbox%20User%20Guide.pdf |doi=10.25039/S026.2018.UG |quote=This User Guide relates to the α-opic Toolbox v1.049a, published by CIE Division 6. The Toolbox (DOI 10.25039/S026.2018.TB) and User Guide (DOI 10.25039/S026.2018.UG) are maintained under CIE Division Reportership (DR) 6-45.|title=User Guide to the α-opic Toolbox for implementing CIE S 026|date=2020 |last1=Cie }}

Most non-primate mammals have the a similar luminous efficiency function to people with protanopia. Their insensitivity to long-wavelength red light makes it possible to use such illumination while studying the nocturnal life of animals.{{cite journal|author1=I. S. McLennan|author2=J. Taylor-Jeffs|name-list-style=amp|title=The use of sodium lamps to brightly illuminate mouse houses during their dark phases|journal=Laboratory Animals|year=2004|volume=38|issue=4|pages=384–392|pmid=15479553|doi=10.1258/0023677041958927|s2cid=710605|url=http://la.rsmjournals.com/content/38/4/384.full.pdf}}{{Dead link|date=March 2020 |bot=InternetArchiveBot |fix-attempted=yes }}

Definition of melanopic illuminance and opsin-specific illuminances in the sense of Lucas (2014) are available for rodents. There is a significant difference at short wavelengths (< 420 nm) because the rodent eye filters light differently before the retina compared to the human eye. A 2024 article by Lucas's Group and international researchers calls for better standardization of light levels used in animal experiments using these species-adjusted illuminance measurements, both to improve the reproducibility of light-related experiments and to improve animal welfare. The article includes αopic data for mice, brown rats, macaques, cats, and dogs. It links to two separate toolboxes, one for calculating the species-specific EDI from a spectral power distribution, the other for estimating the species-specific EDI for a given amount of photonic lux and a light source of known spectrum.{{cite journal |last1=Lucas |first1=Robert J. |last2=Allen |first2=Annette E. |last3=Brainard |first3=George C. |last4=Brown |first4=Timothy M. |last5=Dauchy |first5=Robert T. |last6=Didikoglu |first6=Altug |last7=Do |first7=Michael Tri H. |last8=Gaskill |first8=Brianna N. |last9=Hattar |first9=Samer |last10=Hawkins |first10=Penny |last11=Hut |first11=Roelof A. |last12=McDowell |first12=Richard J. |last13=Nelson |first13=Randy J. |last14=Prins |first14=Jan-Bas |last15=Schmidt |first15=Tiffany M. |last16=Takahashi |first16=Joseph S. |last17=Verma |first17=Vandana |last18=Voikar |first18=Vootele |last19=Wells |first19=Sara |last20=Peirson |first20=Stuart N. |title=Recommendations for measuring and standardizing light for laboratory mammals to improve welfare and reproducibility in animal research |journal=PLOS Biology |date=12 March 2024 |volume=22 |issue=3 |pages=e3002535 |doi=10.1371/journal.pbio.3002535|doi-access=free |pmid=38470868 |pmc=10931507 }}

The wavelength-dependent attractive effect on bees and moths have been quantified with a relative arbitrary unit of "attraction". These data have been used to design white LED light sources with lower arthopod attraction at night.{{cite journal |last1=Longcore |first1=Travis |last2=Aldern |first2=Hannah L. |last3=Eggers |first3=John F. |last4=Flores |first4=Steve |last5=Franco |first5=Lesly |last6=Hirshfield-Yamanishi |first6=Eric |last7=Petrinec |first7=Laina N. |last8=Yan |first8=Wilson A. |last9=Barroso |first9=André M. |title=Tuning the white light spectrum of light emitting diode lamps to reduce attraction of nocturnal arthropods |journal=Philosophical Transactions of the Royal Society B: Biological Sciences |date=5 May 2015 |volume=370 |issue=1667 |pages=20140125 |doi=10.1098/rstb.2014.0125|pmid=25780237 |pmc=4375365 }}

• Apparent magnitude
• Color vision
• Quantum efficiency, the image sensor equivalent
• A-weighting and equal-loudness contour, related sound concepts

{{notelist}}

{{Reflist|3|refs=

{{cite book

| title = Digital Video and HDTV: Algorithms and Interfaces

| author = Charles A. Poynton

| publisher = Morgan Kaufmann

| year = 2003

| isbn = 1-55860-792-7

| url = https://books.google.com/books?id=ra1lcAwgvq4C&dq=luminance+dot-product&pg=RA1-PA205

}}

{{cite book

|author1=Wyszecki, Günter |author2=Stiles, W.S.

|name-list-style=amp | title = Color Science - Concepts and Methods, Quantitative Data and Formulae

| edition = 2nd

| publisher = Wiley-Interscience

| year = 2000

| isbn = 0-471-39918-3

}}

{{cite book

|author1=Judd, Deane B. |author2=Wyszecki, Günter

|name-list-style=amp | title = Color in Business, Science and Industry

| edition = 3rd

| publisher = John Wiley

| year = 1975

| isbn = 0-471-45212-2

}}

{{cite journal

| author = Vos, J. J.

| title = Colorimetric and photometric properties of a 2° fundamental observer

| journal = Color Research and Application

| year = 1978

| volume = 3

| issue = 3

| pages = 125–128

| doi = 10.1002/col.5080030309

}}

{{cite journal

| author1 = Stiles, W. S.

| author2 = Burch, J. M.

| title = Interim report to the Commission Internationale de l'Eclairage Zurich 1955, on the National Physical Laboratory's investigation of colour-matching

| journal = Optica Acta

| year = 1955

| volume = 2

| pages = 168–181

| bibcode = 1955AcOpt...2..168S

| doi = 10.1080/713821039

| issue = 4

}}

{{cite journal

| author1 = Sharpe, L. T.

| author2 = Stockman, A.

| author3 = Jagla, W.

| author4 = Jägle, H.

| year = 2005

| title = A luminous efficiency function, V*(λ), for daylight adaptation

| url = http://calendar.arvo.org/5/11/3/Sharpe-2005-jov-5-11-3.pdf

| archiveurl=https://web.archive.org/web/20120426041536/http://calendar.arvo.org/5/11/3/Sharpe-2005-jov-5-11-3.pdf

| archivedate=April 26, 2012

| journal = Journal of Vision

| volume = 5

| number = 11

| pages = 948–968

| doi = 10.1167/5.11.3

| pmid = 16441195

| doi-access= free

}}

{{cite book

| title = Light Pollution Handbook

| author1 = Kohei Narisada

| author2 = Duco Schreuder

| publisher = Springer

| year = 2004

| isbn = 1-4020-2665-X

}}

{{cite book

| title = Handbook of Applied Photometry

| author = Casimer DeCusatis

| publisher = Springer

| year = 1998

| isbn = 1-56396-416-3

}}

}}

{{Reflist|group=cie|refs=

{{cite web

| url = http://www.cie.co.at/index.php/LEFTMENUE/index.php?i_ca_id=298

| title = CIE Selected Colorimetric Tables

| archiveurl = https://web.archive.org/web/20170131100357/http://files.cie.co.at/204.xls

| archivedate = 2017-01-31

}}

16th Conférence générale des poids et mesures Resolution 3, CR, 100 (1979), and Metrologia, 16, 56 (1980).

{{cite book

| author = CIE

| title = Commission internationale de l'Eclairage proceedings, 1924

| publisher = Cambridge University Press, Cambridge

| year = 1926

}}

}}

{{Reflist|group=c|refs=

{{cite web

| title = CIE Scotopic luminosity curve (1951)

| url = http://www.cvrl.org/database/text/lum/scvl.htm

| archiveurl = https://web.archive.org/web/20081228115119/http://www.cvrl.org/database/text/lum/scvl.htm

| archivedate = 2008-12-28

}}

{{cite web

| title = CIE (1931) 2-deg color matching functions

| url = http://www.cvrl.org/database/text/cmfs/ciexyz31.htm

| archiveurl = https://web.archive.org/web/20081228084047/http://www.cvrl.org/database/text/cmfs/ciexyz31.htm

| archivedate = 2008-12-28

}}

{{cite web

| title = Judd–Vos modified CIE 2-deg photopic luminosity curve (1978)

| url = http://www.cvrl.org/database/text/lum/vljv.htm

| archiveurl = https://web.archive.org/web/20081228083025/http://www.cvrl.org/database/text/lum/vljv.htm

| archivedate = 2008-12-28

}}

{{cite web

| title = Sharpe, Stockman, Jagla & Jägle (2005) 2-deg V*(l) luminous efficiency function

| url = http://www.cvrl.org/database/text/lum/ssvl2.htm

| archiveurl = https://web.archive.org/web/20070927222337/http://www.cvrl.org/database/text/lum/ssvl2.htm

| archivedate = 2007-09-27

}}

}}

• [http://www.cvrl.org/lumindex.htm Color and Research Vision Laboratory - luminous efficiency data tables]

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Category:Physical quantities

Category:Photometry