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Light Production Metrics of Radiation Sources Light production metrics of radiation sources C. Tannous Laboratoire de Magn´etismede Bretagne - CNRS FRE 3117 UBO, 6, Avenue le Gorgeu C.S.93837 - 29238 Brest Cedex 3 - FRANCE∗ (Dated: November 15, 2013) Light production by a radiation source is evaluated and reviewed as an important concept of physics from the Black-Body point of view. The mechanical equivalent of the lumen, the unit of perceived light, is explained and evaluated using radiation physics arguments. The existence of an upper limit of luminous efficacy is illustrated for various sources and implications are highlighted. PACS numbers: 42.66.Si,42.72.-g,85.60.Jb Keywords: Visual perception,light sources,light-emitting diode I. INTRODUCTION 103 White LED Physics students are exposed to Black-Body radia- 102 tion in undergraduate Quantum Physics or in Gradu- 1 ate/Undergraduate Statistical Physics without any clue 10 regarding its significance as to the fundamental role it 100 played in the development of light source calibration, de- −1 lumens 10 velopment and of lighting in general. Perhaps, only students with astrophysics or atmo- 10−2 spheric physics curriculum will be generally more aware 10−3 of radiation physics in connection with the Black-Body 10−4 fundamentals. 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Presently, lighting is undergoing a tremendous evolu- Year tion because of the swift evolution of the light-emission- diode (LED) that is now gaining larger and larger lumi- nous efficacy to a point such that it is now replacing, at a FIG. 1. (Color online) Haitz rule and evolution of white LED (in cyan) after year 2000 when S. Nakamura introduced In very impressive pace, our traditional home lighting, car within GaN1. Before 2000, evolution was driven by red LED headlights, LCD-monitors backlights, street lighting... starting in the 1970's with GaAs, GaAsP, GaP, GaAsP:N, Additionally, LED colors are becoming more versa- GaAlAs and finally GaAlInP. tile and sharper both in the case of traditional inorganic LED's or their organic counterpart, the OLED. The underlying basis of lighting progress is the exis- important not only in Physics and technology but also tence of Haitz rule (see fig. 1) that is similar to Gordon for energy savings and efficiency, renewable energy and Moore rule of Electronics evolution. consequently for sustainable development of the Planet. Light production by a radiation source is described by This report is organized as follows: In section 2, a re- a luminous efficacy ratio ηL given by the product of ηC view of lighting metrology is made, in Section 3 luminous the conversion efficacy ratio of number of photons pro- efficacy of radiation sources is explained and derived and duced (having any wavelength) to input energy (usually in Section 4, we derive the maximum luminous efficacy on electrical but it could also be mechanical, thermal or the basis of the colorimetry standard established by the chemical...) and ηP the light perception efficacy ratio CIE2. This standard is briefly explained and reviewed in arXiv:1311.3504v1 [quant-ph] 14 Nov 2013 or Photometric efficacy ratio (PER). ηP is the ratio of the Appendix that comes after Section 5 carrying discus- number of photons perceived by the human eye (photon sions and conclusions. wavelength in the visible spectrum) to the total number of photons. Some authors call it ηS the spectral efficacy. This work can be taught as an application chapter in a general course of Statistical physics or in Semiconduc- II. METROLOGY OF LIGHTING tor physics at the Undergraduate or Graduate level since physicists can contribute readily in improving light pro- The SI system of units is based on seven entities: the duction level of radiation sources or conversion efficacy meter, the kilogram, the second, the ampere, the kelvin, (through the increase of either ηP or ηC ) once the basis the mole and the candela as the unit of luminous inten- for luminous efficacy is explained and illustrated along sity (Lumen is candela per unit solid angle). some notions of colorimetry and light calibration. Prior to 1979, the SI system of units defines the candela The notions reviewed in this work are primarily con- as follows: cerned with ηP the perception efficacy and are clearly A pure sample of Platinum at its fusion temperature 2 (T=2042 K) emits exactly 60 candelas/cm2 along the It represents the thermal average number of photons per vertical direction to the sample and per unit solid angle. unit volume emitted by a Black-Body at temperature T . The SI system changed the definition during the 16th Since a Black-Body (absorbs and) emits radiation, we General Conference on Weights and Measures in October need a Planck equivalent law for the emitted energy den- 1979 to: sity. By analogy with electrical charge current density J The candela is the luminous intensity, in a given direc- (J = ρv with ρ the charge density and v their velocity), tion, of a source that emits monochromatic radiation of we multiply the average photon density by the velocity frequency 540 ×1012 Hz with a radiant intensity, in that of light c and divide by 4π, the whole space solid angle direction, of 1/683 Watts per unit solid angle. factor. Thus we obtain the Planck distribution of pho- In order to relate these two definitions, despite the ob- ton energy emission at wavelength λ, temperature T and vious equivalence of 555 nm wavelength and 540 ×1012 unit solid angle: Hz frequency, some reminders about Black-Body radia- tion must be made. 2hc2 1 Black-Body radiation was noticed for the first time by fB(λ) = 5 h i (5) 3,4 λ exp( hc ) − 1 Gustav Kirchhoff who used to watch the color change λkB T of cavities present in heated metals worked by black- smiths in his neighborhood. He noticed a systematic Light sensed by the human eye is given by the distri- color change as the metal is being heated along the se- bution of energy average over the eye sensitivity function 2 quence of red, orange, yellow, white and finally blue. called V (λ) by the CIE . Cavity color change did not depend on the nature of This function depicted in fig. 2 peaks at a 555 nm the metal but depended on the size of the cavity. (Yellow-Green) wavelength that is the maximum sen- This means that we have a photon gas in the cavity sitivity of the human eye. The analytical expressions 2 3 λ obeying Planck radiation law with the average density of V (λ) = 1:019 exp(−285 ( 1000 ) − 0:559 ) (sensitiv- of photons having energy ~! of the Bose-Einstein form ity in daylight or photopic sensitivity) with λ expressed (the chemical potential being zero since the number of in nm. The sensitivity in the dark V 0(λ) (scotopic) is photons is not fixed): shifted with respect to V (λ) by 45 nm to shorter wave- lengths and peaks at 510 nm (Purkinje shift). Analyti- 0 λ 2 1 cally V (λ) = 0:992 exp(−321:9 ( 1000 ) − 0:503 ). n(!) = h i (1) exp( ~! ) − 1 kB T 1.2 Photopic The average radiation energy in the [!; ! + d!] is Scotopic 1 5 1 (n(!) + 2 )~!g(!) with g(!) the density of states . The vacuum energy 1 ! should not be included since it is 2 ~ 0.8 2 ) 5 ! λ independent of T . The density of states being π2c3 , we 0.6 get: ), V'( λ V( 0.4 2 ! ~! 0.2 E(!) = 2 3 h i (2) π c exp( ~! ) − 1 kB T 0 400 450 500 550 600 650 700 750 λ In order to get E(λ) the average energy density of pho- (nm) tons having wavelength λ, we use the conservation rule given by: FIG. 2. V (λ) function giving the eye sensitivity versus wave- length in daylight (photopic) and V 0(λ) in the dark (scotopic). They are shifted one with respect to the other by 45 nm, the E(!)d! = E(λ)dλ (3) Purkinje shift. with the transformation from angular frequency ! to The conversion from Radiometry λ 2 [0; 1] to Pho- linear frequency f to wavelength λ: ~! = ~2πf = hc/λ. tometry λ 2 [380nm; 780nm] is carried out by multiply- This yields: ing the total radiant power (in Watts) perceived by eye: 8πhc 1 Z 1 2hc2 1 E(λ) = 5 h i (4) × V (λ)dλ (6) λ exp( hc ) − 1 λ5 h hc i λkB T 0 exp( ) − 1 λkB T This law is derived in many Statistical Physics books by a conversion factor Km called the "Mechanical and is usually enough for standard Physics curriculum. equivalent of the Lumen" such that: 3 Z 1 2hc2 V (λ)dλ light perceived = Km 5 h i (7) 0 λ exp( hc ) − 1 80 λkB T It is important to notice that we are carrying the in- 60 tegration over all positive frequencies and not the visible 40 spectrum given by the wavelength interval [380 nm, 780 PER (lm/Watt) nm] since we are dealing with (unlimited) radiation en- 20 ergy. In order to evaluate Km, we use the old definition of 0 the candela and get: 0 5000 10000 15000 20000 25000 30000 T(K) 60 cd/cm2/sr Km = (8) R 1 2hc2 V (λ)dλ FIG. 3. Photometric Efficacy Ratio ηP versus temperature 5 h i 0 λ exp( hc )−1 of the Black-Body. Notice that the maximum is 95 lm/Watt λkB T and that the temperature is about 7000K. Transforming to SI units, the numerator becomes 6×105 lumens/m2 (since lumen=cd per unit solid angle).
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