Radiometry and Photometry Units and Conversion Factors [PDF]

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Radiometry and Photometry: Units and Conversion Factors Jurgen R. Meyer-Arendt

Although often colorful and of historical interest, some radiometric and photometric units are redundant and illogical. The essential units are few in number. These, together with their definitions and conversion factors, are presented here.

The following outline is concerned with definitions, symbols, units, and conversion factors as they occur, The and are helpful, in radiometry and photometry. situation here is more complex than in other areas of this is for several reasons. Sometimes, optics; different terms are used for identical quantities. Certain terms such as candlepower are ill-conceived. Others like nox, phot, glim, skot or scot (identical), bril and brill (different), helios, lumerg, pharos, stilb, talbot, and blondel may merely delight the historian.

The impending-we hope-general conversion to the mks system, which should more properly now be referred to as SI, presents us with a unique opportunity for simplifying some of the basic definitions and units, although a solution fully satisfying to all may never be found. The term SI stands for Systbme Interna-

tional-International

System of Units-adopted by the

Eleventh General Conference on Weights and Measures held 11-20 October 1960 in Paris, France, to

which the United States has been a party. In this paper, only SI units are used, but their relationship to other units is also shown. Close cooperation of committees of the International Commission on Illumination, the International Organi-

zation for Standardization, the International Electrotechnical Commission, and the SUN Commission of the International Union of Pure and Applied Physics has resulted in the development of a set of terms and symbols that is receiving acceptance throughout the entire world. The symbols in the U. S. A. Standard Letter Symbols for Illuminating Engineering (USAS

Y10.18-1967) are consistent with the international agreements that have been reached to date.

These

symbols have been used throughout this paper and also are summarized in Table I. The symbols, terms, and notations are also consistent with USAS Z7. 1-1967. Any difference in wording between the definitions in this paper and those in Y10.18-1967 and Z7.1-1967 does not imply any difference in meaning. The outstanding features of the symbols used in Standard Y10.18-1967 are (1) the use of the same set of symbols for radiometric and photometric quantities, with the proviso that when there is need to differen-

tiate these quantities subscripts e and v, respectively, will be used, and (2) the use of a subscript X to designate a spectral concentration and (X)in parentheses to designate a function of wavelength. In some of the following equations, and when otherwise obvious, subscripts e and v have been omitted. Radiant Energy Qe is energy traveling in the form of electromagnetic waves. The term radiation, although widely in use, is deprecated because words that end in -ion should be reserved for processes rather than quan-

tities. The unit of radiant energy is the joule, J. Radiant Energy Density we is radiant energy per unit volume, W = bQ/v.

Unit:

joule/meter3 , J

Radiant Power

ke,

m-3 .

also called radiant flux, is the ra-

diant energy transferred per unit time, i.e., the time The author is with Pacific University, Forest Grove, Oregon 97116. Received 26 October 1967. This paper was written at the request of the Editor after three or

four shorterpapers on units and nomenclature had been submitted for publication.

The author was asked to consolidate these other

papers and try to summarize the preferred units.

This paper does

not constitute an official report of a nomenclature committee, but

neverthelessit was widely circulatedin draft form and represents a reasonableconsensus.

rate of flow of radiant energy: 0 = Qlbt. Radiant power is measured in the same units as power in general. Since power = work/time, and work ( = potential energy) = force X distance, the unit of power is 1 joule_

1soe=

1 second

1 watt,

W = J s-

October 1968 / Vol. 7, No. 10 / APPLIED OPTICS 2081

Table I.

Radiometric and Photometric Symbols and Units

Symbol Radiant

Name of unit

Abbreviation of unit

energy

Qe

joule

J

Luminous energy Radiant energy density Luminous energy density

Q" we w.

talbot joule/meter3 talbot/meter 3

J m-< lm s m-3

Radiant

power (radiant flux)

0e

watt

W

Luminous power (luminous flux) Luminous efficacy

0v K

lumen lumen/watt

lm Im W-

Luminous efficiency

V

Radiant intensity Luminous intensity

I,

Radiant

Ie

exitance

Me

Luminous exitance Radiance Spectral radiance Luminance Irradiance Spectral irradiance

Ml'

Le Lex L, Ee

Ex Ev

Illuminance

watt/steradian candela

W sr-' cd (Im sr-1 )

watt/meter

W m-2

2

lumen/meter 2 watt/meter 2 -steradian watt/meter 2 -steradian-nanometer candela/meter 2 watt/meter 2 watt/meter 2 -nanometer lumen/meter 2 (lux)

Spectral Radiant Power ox is the radiant power per

unit wavelength interval at wavelength , bo/-X, measured in units of watts/nm. The mere specification that certain electromagnetic energy occurs in the visible region of the spectrum says nothing about the visual effectiveness of that energy. A blue lamp, for instance, may emit the same radiant power (in watts) in the visible region as a green lamp,

but the latter will appear brighter because the eye is more sensitive to green light than to blue light.

In the visible region of the spectrum, radiant energy

lm m-2 W m- 2 sr-' W m- 2 sr-' nm cd m-2 W m-2 W m- 2 nm-

im m-2 .

At 555 nm, 1 m is equivalent to 0.00147 ( 1/680) W; or 1 W, at 555 nm, is equal to 680 m. In order to use this conversion anywhere else in the

visible spectrum, the proper luminous efficiencymust be included. For example, the luminous efficiency at 600 nm is 0.63; thus, 1 W of monochromatic light at that wavelength equals (0.63) (680) = 428 lm; or 1 lm is equal to 0.00147/0.63 = 0.00233 W. If the source is not monochromatic, integration is needed.

Outside of the visible part of the spectrum, there can by definition be no light.

Therefore, the terms, units,

is frequently evaluated with respect to its capacity to evoke the sensation of brightness. One should always bear in mind the fundamental difference between radio-

and quantities of photometry do not apply there.

metric and photometric terms. Radiometric terms

ric term luminous energy Q, the product of luminous

apply anywhere in the electromagnetic spectrum. Photometric terms apply to the visible part of the spectrum only, and if we want to convert radiometric values into photometric values, and vice versa, we have

to take into account the relative visibility of the light of the particular wavelength involved. The resulting curve (Fig. 1) is called the luminous efficiency curve and

the ratio of any photometric unit to its radiometric equivalent is called the luminous efficacyK. The peak at 555 nm, the wavelength to which the eye is most sensitive, is the point of maximum visibility or-preferably-the maximum value of luminous efficacy for photopic vision. Luminous Power 0,.

Photometric

Since radiant energy is the product of radiant power

and time, there should also be a unit for the photometpower and time. This unit is the talbot (1 talbot = 1 lumen-second). One lumen, hence, is the luminous power of 1 talbot per second.

Luminous Efficiency V and Luminous Efficacy K. The term efficiency as applied to visible light has a meaning similar to that which it has when it is applied to a machine-it has a maximum value of 1.0 and is dimensionless. The term efficacy as applied to visible 1.0

.

U as 4 U. U.

terms all have

the adjective luminous. Thus, instead of radiant power we now have luminous power (luminous flux). The unit of luminous power is the lumen, m. One lumen is defined as the luminous power emitted (within a unit solid angle) by a point source of luminous intensity (defined below) of one candela. In other words, a point source that radiates uniformly in all directions with a luminous intensity of 1 candela emits a total of 47r lm. 2082 APPLIED OPTICS / Vol. 7, No. 10 / October 1968

M

0.5

Z

400

50

600

700nm

WAVELENGTH

Fig. 1. CIE spectral luminous efficiency curve, showing sensitivityof the human eye to different wavelengths of radiant energy.

light is expressed as the quotient of luminous power (output) to power consumed (input) and has the unit of lumen per watt, lm W- 1.

This may take two forms: radiant exitance Me, with the unit watt/meter2 , W m-2 , and luminous exitance M,, with the unit lumen/meter2 , m m-2.

Radiant Intensity e is defined as the radiant power proceeding from a point source per unit solid angle in the direction considered,

Radiance Le is defined as the radiant power that leaves a surface per unit solid angle and unit projected area of that surface,

I = a/a.

L

Solid angles are measured in units of steradian, sr. A unit solid angle, or 1 sr, is defined as the solid angle subtended at the center of a sphere with a radius R = 1 meter by an area of one square meter on its surface. 2 Therefore, since the surface of a sphere is Ao = 4rR , 4 the total solid angle about a point is w = 7r sr. For an element of surface dA, which subtends a small solid angle at a point at distance D from the element of surface, the solid angle is 2 do = dA cosO/D ,

where a is the angle between the normal to the surface and the direction D.

Unit: watt per steradian, W sr-'. Luminous Intensity I,,. The equivalent psychophysical term for radiant intensity is luminous intensity. Here, as in all luminous quantities, lumen takes the place of watt. The candela, the unit of luminous intensity, is equal to 1 lumen of luminous power per unit solid angle; thus, 1 candela = 1 lumen/steradian, (Luminous intensity has sometimes been lm sr-'. called candlepower, a name that is misleading since intensity is not power.) General Conference on Unit: The Thirteenth Weights and Measures in October 1967 adopted this definition of the unit of luminous intensity, the candela, cd: The candela is the luminous intensity, in the direction of the normal, of a full radiator (blackbody) surface 1/600,000 square meter in area, at the temperature of 20420 K or solidification of platinum-approximately per newtons 101,325 of pressure a 17690C-under square meter. Exitance M is defined as the radiant power per unit area leaving a surface, M = 60/6A.

Table II.

Number of multiplied by equals number

=

2

( 0)/(A

bX coso) =

I/(aA

coso).

(Note: The term intensity is restricted to a point source, while radiance also applies to an extended source. In practice, the finite extent of a source is sometimes neglected if its diameter is less than about 1/20 of the dis-

tance to the irradiated surface.) The term coso, in which 0 is the angle between the normal to 6A and the direction of view transforms area 2 -A to projected area. If the amount of flux 6 k that in all dibw angle solid in bA area of element an leaves rections is proportional to cosO, the surface is said to obey Lambert's law and is often referred to as a Lambertian, or perfectly diffuse radiating (or reflecting or transmitting) surface. Because the projected area is also proportional to coso, the radiance of such a perfectly diffuse surface is independent of the direction of view. Unit: watt per square meter and steradian, W m-2

sr-'. Spectral Radiance Lx is the power emitted per unit projected source area and unit solid angle and unit wavelength band. It can be expressed in units of

W m- 2 sr- nm-.

Luminance L, is the photometric term corresponding to radiance. Its unit is candela per square meter, cd m- 2 (sometimes called nit). There exist numerous other units of luminance, but considering the rigor in using cd m- 2 and the advantages of avoiding some decimals or 7rfactors, most of these are not really essential: 2 2 2 Stilb, candela/cm , cd cm- , equal to 10,000 cd/M , 929.02 2 cd/ft , ir lamberts, or 2.919 X 103 ft-lamberts.

Apostilb (international), equal to 0.1 millilambert, 10-4 lambert,

2 2 2 1 blondel, (1/7r) cd m- , 0.3183 cd m- , 0.00003183 cd/cm , or (1I/7r) 10O-4stilb.

Conversion Factors for Units of Luminance

cd/m a (nit)

cd/cm2 (stilb)

cd/ft 2

cd/in2

apostilb (blondel)

millilambert

footlambert

1 0.0001 0.0929 0.000645 3.1416 0.31416 0.2919

10,000 1 929 6.452 31,416 3141.6 2919

10.764 0.001076 1 0.00694 33.82 3.382 3.1416

1550 0.155 144 1 4869 486.9 452.4

0.3183 0.00003183 0.02957 0.0002054 1 0.1 0.0929

3.183 0.0003183 0.2957 0.002054 10 1 0.929

3.426 0.0003426 0.3183 0.002211 10.764 1.0764 1

2

of

cd/M2 (nit)a cd/cm2 (stilb) cd/ft 2 cd/in2 apostilb (blondel) millilambert foot-lambert

a SI unit. The name nit is not in widespread use. October 1968 / Vol. 7, No. 10 / APPLIED OPTICS 2083

Table IlIl. Conversion Factors for Units of Illuminance

Number of multiplied by -,, equals number

Footcandles

lm/m 2 (lux)a

1

0.0929

lm/m2 (lux)a

10.764

Phot Milliphot

0.00108 1.076

aSI unit.

1

929 10,000

0.0001 0.1

1 1,000

Mr

2

, 0.001076 phot,

phot

The same argument also applies to meter-candle (equal to lux or lm/m2 ), 10-4 phot, 0.092902foot-candles, lm/ft 2 , and to milecandle, lm/mile2.

0.929

Troland, previously called photon [sic], unit of retinal illuminance, produced by luminance of 1 cd m-2 if the apparent cross section of the entrance pupil of the eye, corrected for the StilesCrawford effect, is 1 mm2.

Milli-

Phot

of

Foot-candles

Foot-candle, lm/ft 2 , equal to 10.763910lm

lm/cm 2 ; or 1.076 milliphot.

10

0.001 1

The symbol for lux is lx.

Apostilb (German Hefner candle), equal to 0.09 millilambert. Blondel, equal to (1/7r) [= 0.3183] cd m-' or 0.02957 cd ft- 2 .

Skot, unit of luminance for rod vision, equal to 10-3 apostilb or

The brightness of a surface is not the same as either irradiance or illuminance. Rather, it can be a function of illuminance and reflectivity. Irradiation, and likewise illumination, are processes of exposing an object to electromagnetic energy for a given length of time.

3.2 X 10-4 cd m-2. Lambert, cd cm

2

1000 millilambert, equal to (1/7r) cd cm'2, 0.3183 , (1/7r) 104 cm m'2, 3.18 stilb, or 9.29 X 102 ft-lambert.

Millilambert,

equal to 10 apostilb or 0.0003183 cd cm- 2, roughly

equal to 1 ft-lambert. Microlambert, 0.001 millilambert. Foot-lambert, equal to 1/7r candle/ft2, 1 lm/ft2, 3.4263 cd m- 2 , 3.4263 X 10-4 stilb, or 1.1 X 10-3 lambert. (Table II).

Glim, equal to 10-3 foot-lambert. Candela/ft2 and candela/inch2 are also deprecated.

Irradiance e is defined as radiant power incident upon a surface per unit area,

E= 0/bA. The magnitude of irradiance from a point source follows the inverse square law: E = I/D , where D is the dis-

tance. Unit: watts per square meter, W m-2.

Spectral Irradiance E is the power incident per unit area and per unit wavelength interval, with the unit W m-2 nm-. Illuminance E, is the luminous power per unit area incident on a surface. It is measured in units of lumen/m, Im m-2. The SI name of this unit is lux. (See Table III.) Not rigorous in this context are: W/cm2, Nox, equal to 10-3 lm/m2 , Phot or centimeter-candle, equal to lm/cm 2 , 104 lm/m2, or 929 footcandles. Foot-candle. This is a rather unfortunate term which seems to indicate that the luminous intensity in candles is to be multiplied by a distance, rather than indicating that the unit is defined as the illuminance

of a surface at a unit distance from a point source of

luminous intensity equal to one candela.

2084 APPLIED OPTICS / Vol. 7, No. 10 / October 1968

If radiant energy is incident on an element of surface at an energy flow rate 0, and is reflected by the element of surface at a rate Or,the ratio 0kr/0Ois called the reflectance of the surface. In general, this ratio depends on the wavelength and state of polarization of the incident energy, and the direction of incidence. I would like to acknowledge contributions, suggestions, and criticisms from a great many individuals and organizations. Among these were L. E. Barbrow,

NBS, Washington, D.C.; L. M. Biberman, IDA, Arlington, Va.; \ar. Bodner, Lockheed, Burbank, Calif.; E. Dews, RAND Corp., Santa Monica, Calif.; R. H. Ginsberg, Hughes Aircraft, Culver City, Calif.; J. N. Howard, AFCRL, Bedford, Mass.; D. B. Judd, NBS, Washington, D.C.; W. M. Lyle, University of Waterloo, Ontario, Canada; A. G. McNish, NBS, Washington, D. C.; J. C. Richmond, NBS, Washington, D.C.; C. S. Williams, Texas Instruments, Dallas, Tex.; and

Eastman Kodak Co., Rochester, N.Y. Bibliography

U.S.A. Standard Letter Symbols for Illuminating Engineering USAS Y10.18-1967.

U.S.A. Standard Nomenclature and Definitions for Illuminating Engineering, USAS Z7.1-1967. L. E. Barbrow, Illum. Eng. 62, No. 11 (1967). L. M. Biberman, Appl. Opt. 6, 1127 (1967). D. Deirmendjian, RAND Rep. P-2079, 19 Aug 1960. H. K. Hughes, Anal. Chem. 24, 1349 (1952). D. B. Judd, J. Opt. Soc. Amer. 57, 445 (1967).

H. R. Luxenberg, Inform. Display 2, 39 (May-June 1965). F. E. Nicodemus, Amer. J. Phys. 31, 368 (1963). Optical Society of America, The Scienceof Color (Optical Society of America, Washington, D. C., 1963), pp. 223-233, 254-316. W. Viezee, RAND Rep. RM-2492,

12 July 1960.

J. W. T. Walsh, Photometry (Constable, London, 1958). C. S. Williams, Texas Instrum.

Tech. Rep. No. 08-66-79 (1966).