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 luminous 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 radiation 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 ~ω 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 wavelengths and peaks at 510 nm (Purkinje shift). Analyti- 0 λ 2 1 cally V (λ) = 0.992 exp(−321.9 ( 1000 ) − 0.503 ). n(ω) = h i (1) exp( ~ω ) − 1 kB T 1.2 Photopic The average radiation energy in the [ω, ω + dω] is Scotopic 1 5 1 (n(ω) + 2 )~ωg(ω) with g(ω) the density of states . The vacuum energy 1 ω should not be included since it is 2 ~ 0.8 2 )

5 ω λ independent of T . The density of states being π2c3 , we 0.6 get: ), V'( λ

V( 0.4 2 ω ~ω 0.2 E(ω) = 2 3 h i (2) π c exp( ~ω ) − 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 wavelength 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(ω)dω = E(λ)dλ (3) Purkinje shift.

with the transformation from angular frequency ω to The conversion from Radiometry λ ∈ [0, ∞] to Pho- linear frequency f to wavelength λ: ~ω = ~2πf = hc/λ. tometry λ ∈ [380nm, 780nm] is carried out by multiply- This yields: ing the total radiant power (in Watts) perceived by eye:

8πhc 1 Z ∞ 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 ∞ 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 ∞ 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). The integral of the denominator has Watt/m2 dimensions of 93/683=13.6% and that the ordinary Tungsten light 7 and when it is numerically evaluated we get Km = 679 bulb based on incandescence phenomenom is about 15 lm/Watt (lumens/Watt) which is close to the value of lm/Watt (temperature about 3000K) or 2% only. For 683 lm/Watt adopted by the SI6. a candle considered as a Black-Body at T=1800K, we get 0.6 lm/Watt which corresponds to an efficiency of 0.6/683 or about 0.1%. III. LUMINOUS EFFICACY OF RADIATION The consequence in terms of lighting is that when one SOURCES acquires a 60 Watts bulb of the Tungsten type, the light produced by the bulb is 60 W x 15 lm/Watt= 900 lumens A. Luminous efficacy of Black-Body radiation in total and that has tremendous consequences for the sources quality and cost of the lighting desired.

Having determined Km we are now in the position of determining the luminous efficacy of any radiation B. Luminous efficacy of White radiation sources source. The value of ηP for any radiation source characterized A White source is considered as having a flat power by a power emission spectrum P (λ) is given by: emission spectrum P (λ) over the entire visible interval, nevertheless in practice the interval is limited and one has to define precisely the wavelength interval over which this R P (λ)V (λ)dλ flatness is observed. η = K Dλ (9) P m R P (λ)dλ Two cases are encountered in lighting systems: Dλ 1. White source as a truncated Black-Body source: The wavelength interval Dλ is arbitrary and depends on the radiation source. This is a Black-Body source taken at a temperature In the particular case of a source of the thermal Black- T=5800K with a spectrum limited by definition to Body type, we apply the above formula 9 with P (λ) = λmin=400 nm and λmax=700 nm. The PER is obtained from: fB(λ) and Dλ = [0, ∞]:

λ R ∞ f (λ)V (λ)dλ R max P (λ)V (λ)dλ 0 B λmin ηP = Km ∞ (10) ηP = Km (11) R R λmax 0 fB(λ)dλ P (λ)dλ λmin The result is the temperature dependent PER curve 2hc2 1 Using P (λ) = 5 h i we get a PER of depicted in fig. 3. λ exp( hc )−1 λkB T PER has lm/Watt dimensions, thus we define effi- about 250 lm/Watt. ciency as a dimensionless ratio (in %) yielding the frac- tion with respect to the ideal efficacy of 683 lm/Watt. 2. Equal Energy White Source: It shows that the Sun efficacy is about 93 For instance, an ”Equal Energy White Source” pos- lm/Watt (temperature about 6000K) or an efficiency sesses by definition a flat power emission spectrum 4

over the entire visible spectrum. Mathematically we obtain a PER of 343 lm/Watt which represents P (λ) = W for λ ∈ [380nm, 780nm] i.e. λmin=380 an efficiency of 50%. Usually the conversion efficacy nm, whereas λmax=780 nm. Thus we get: ηC is about 20% which makes the overall efficacy of 68.6 lm/Watt and the total efficiency at 10 %. Such efficiency is quite interesting, however the problem R λmax WV (λ)dλ with fluorescent light is that it suffers from flicker λmin ηP = Km (12) (fluctuating light intensity) due to the ballast and R λmax W dλ λmin the random collision phenomena, besides it relies on Mercury which is a highly polluting source of the This yields about 179 lm/Watt. This leads to the environment. Moreover the ballast circuit might conclusion that flatness is not enough to increase produce annoying noise in some cases. PER. We need a compromise between flatness and wavelength interval length. D. Luminous efficacy of lasers and LED’s

C. Luminous efficacy of fluorescent sources White light, Black-Body and fluorescent radiators are considered as broadband emitters since their radiation spans (at least) the entire visible spectrum. Fluorescent light sources are known as cold sources This is not the case of LED’s and lasers since they are (Eco light is also a fluorescent light source) as op- somehow closer to monochromatic (narrowband) sources. posed to thermal (Black-Body like) or incandescent The new CIE definition of the mechanical equivalent of sources. They need a special circuit called a ballast the lumen is that a monochromatic source that is a power to stabilize current and accelerate electrons in order emission spectrum peaking at λ0=555 nm (at the maxi- to make them collide inelastically with a gas mix- mum sensitivity of the eye) with power of 1 W produces ture of heavy atoms (typically Mercury, Terbium exactly 683 lumens. and Argon) producing radiation. This can be understood readily from the general PER An example power emission spectrum of the three definition: band type is displayed in fig. 4. It shows several peaks over a finite wavelength interval in sharp con- R λmax δ(λ − λ0)V (λ)dλ trast with Black-Body spectrum that is smooth and λmin ηP = Km (13) continuous extending over an infinite wavelength R λmax δ(λ − λ0)dλ interval. λmin

where we used: P (λ) = aδ(λ − λ0) with a a constant and the condition that λ0 ∈ [λmin, λmax]. 1.2 The integral gives the result: PER= KmV (λ0) = Km

1 since V (λ0 = 555nm) = 1. As an application, consider a laser emitting at λ0=570 0.8 nm with a power of 50 mW. It has a PER=K V (λ )

) m 0 λ that produces 30 lumens. 0.6 ), V(

λ Turning to lighting with LED’s, one of the main advan- P( 0.4 tages of LED is that its lifetime is extremely long (on the order of 100,000 hours) because it is a rugged solid state 0.2 device. Moreover it does not rely on a ballast or Mer- 0 cury which makes it safer than Fluorescent lamps whose 350 400 450 500 550 600 650 700 lifetime is about several 1000 hours. λ (nm) The LED power emission spectrum function is usually approximated by a Gaussian or a superposition of several Gaussian functions. In the single Gaussian approxima- FIG. 4. (Color online) Relative power emission spectrum of 2 h (λ−λ0) i a three-band type fluorescent bulb compared to eye sensitiv- tion P (λ) = exp − 2∆λ2 can be used to evaluate the ity curve (in green). Mercury peak is around 450 nm (Data 8 efficacy ηP from eq. 11. adapted from Hoffmann ). The LED is characterized by an average wavelength λ0 and a standard deviation ∆λ. As an example, we Emission by a fluorescent lamp extends over a fi- consider a blue LED with λ0=450 nm and ∆λ=20 nm. nite interval [λmin, λmax] with λmin=380 nm and The PER obtained from eq. 11 is about 39.7 lm/Watt λmax=700 nm for this case. The evaluation of the and therefore an efficiency of 6%. The small efficiency PER is done by spline interpolating the data dis- is due to small overlap between the blue LED spectrum played in fig. 4. Using the general definition eq. 11 and the eye sensibility curve V (λ), moreover that number 5 is further reduced after multiplication by the conversion efficacy ηC which is typically about 20% yielding a total efficiency of 1.2%. PN Piy¯i This is to be contrasted with the present status of x = i=1 c PN White LED who has a very large PER as illustrated by i=1 Pi(¯xi +y ¯i +z ¯i) Haitz law in fig. 1. That might be due to the fact a flat PN i=1 Piy¯i power emission spectrum enhances the PER as previously yc = (18) PN P (¯x +y ¯ +z ¯ ) seen with White sources however the question might be i=1 i i i i asked more generally in specific terms as explained in the next section. This can be transformed into:

IV. MAXIMUM LUMINOUS EFFICACY OF N RADIATION SOURCES X Pi[xc(¯xi +y ¯i +z ¯i) − x¯i] = 0, i=1 An important question can now be asked: For a given N color (chromaticity), is there a maximum PER that can X Pi[yc(¯xi +y ¯i +z ¯i) − y¯i] = 0 (19) be realized with any radiation source? i=1 Mathematically this can be answered with the following set of assumptions: Given a source endowed with a normalized power emission spectrum function P (λ): The problem now is expressed in the standard Simplex form (see Numerical Recipes10, chapter 10): One ought to find a set of N values Pi such that eq. 16 is maximized Z λmax under three constraints given by eq. 17 and eqs. 19. P (λ)dλ = 1 (14) The Simplex method10 results are displayed in fig. 5. λmin The constant PER curves or iso-PER curves represented Is it possible to find the best P (λ) such that the PER in the CIE diagram (see Appendix) get closer to the CIE given by: contour as the PER is increased. Low values of PER are in the blue region of the CIE

Z λmax diagram which explains the result obtained for the blue Y = Km P (λ)¯y(λ)dλ (15) diode, whereas larger PER values occur as we move to- λmin ward the yellow part of the CIE diagram. for some given color represented by chromaticity coor- We find the largest value of 679 lm/Watt, as before, and the corresponding iso-PER curve in the vicinity of dinates xc, yc (see Appendix) is maximized. Note that this stems from the facty ¯(λ) = V (λ) as explained in the the 555 nm region, the area of highest sensibility of the Appendix. eye, confirming the candela standard once again and the 6 Thus we have an optimization problem for an unknown SI metrological data . function P (λ) subject to three constraints: normalization 0.9 eq. 14 and fixed color (xc, yc) constraints eq. A2. 650 lm/W 9 600 lm/W Following Ohta et al. suggestion , we transform the 0.8 550 lm/W 500 lm/W problem into its discrete version by dividing the wave- 0.7 450 lm/W length interval [λ , λ ] into N values λ with a step 555 nm 400 lm/W min max i 0.6 350 lm/W ∆λ such that the objective function to be optimized is: 300 lm/W 0.5 250 lm/W

y 200 lm/W 0.4 150 lm/W Z N 100 lm/W X 0.3 CIE-contour max P (λ)¯y(λ)dλ → max Piy¯i (16) CIE-base i=1 0.2 0.1 Discrete values Pi, x¯i, y¯i, z¯i correspond to 0 P (λ), x¯(λ), y¯(λ), z¯(λ) taken at λ = λi. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 The normalization constraint eq. 14 becomes: x

FIG. 5. (Color online) Constant PER curves in the CIE dia- N X gram. The maximum value of 679 lm/Watt is attained around ∆λ Pi = 1 (17) the 555 nm region (red spot) as adopted by the SI system. i=1 Waviness observed in some curves around borderline are due to slow convergence of the optimization simplex algorithm. whereas the chromaticity constraints (see Appendix) become: 6

1.8 V. DISCUSSION AND CONCLUSION x¯(λ) 1.6 y¯(λ) z¯(λ) Some lighting metrics have been introduced and per- 1.4 spectives for future development regarding the increase 1.2 ) λ of lighting intensity and quality were presented. ( ¯ z

, 1 )

Despite the fact White LED is presently showing tremen- λ ( ¯ y , dous potential in terms of quality and PER increase, the ) 0.8 λ ( ¯ x diagram presented in ﬁg. 5 indicates that a white source 0.6 maximum is between 350 and 400 lm/Watt which is yet 0.4 to be reached by White LED’s. On the other hand, Yellow-Green light sources may 0.2 achieve a large increase since their maximum PER is 0 around the theoretical standard of 683 lm/Watt adopted 350 400 450 500 550 600 650 700 750 800 850 by both CIE and SI organizations. λ (nm) The conclusion is that work must be targeted toward in- creasing rather the conversion eﬃcacy η in the White FIG. 6. (Color online) Color matching functions of the CIE C for eye opening of 2◦ .x ¯ in red,y ¯ in green andz ¯ in blue cover sources case and in particular the White LED case. approximately the corresponding RGB color zones.

Appendix A: CIE Chromaticity Coordinates Z λmax Z = Km P (λ)¯z(λ)dλ, λ In 1931, the CIE undertook a series of historical mea- min (A1) surements called Color Matching experiments in order to calibrate colorimetry and human color perception. A number of ”standard” observers had to make a com- 1.0 parison between a color of a given wavelength λ and a y superposition of three selected wavelengths called RGB 0.9 11 primaries . The weight of each of the three colors to 520 525 0.8 515 530 535 perform the match was recorded. The observations were 510 540 done at a fixed distance of 50 cm with two possibilities 545 0.7 550 for the eye angle opening (defined by the observation di- 505 555 ◦ ◦ 560 ameter value) of 2 and 10 . 0.6 565 570 This led to the existence of color matching functions 500 575 0.5 (CMF)r ¯(λ), g¯(λ), ¯b(λ) corresponding to the red, green, 580 585 blue weight coefficients that matched the color λ. 590 495 0.4 595 D50 2222 2353 2105 2500 2000 600 The study showed that not all matching weights are 2677 2857 3077

D65 3333 605 3636

4000 610 positive and that some values were negative. The alge- D93 4444 0.3 5000 620 490 5714 635 6667 700 braic values of the coefficients meaning that CMF func- 8000 10000 tions took positive and negative values originating from 0.2 485 the overlap versus wavelength between human cone sen- 480 0.1 sitivities. 475 470 This pushed the CIE to perform a linear transfor- 460 mation overr ¯(λ), g¯(λ), ¯b(λ) in order to define three 0.0 380 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x 1.0 strictly positive CMF functionsx ¯(λ), y¯(λ), z¯(λ) displayed in fig. 6. FIG. 7. (Color online) CIE diagram displaying color of points The linear transformation is based on equal area of with coordinates (x, y) and the Black-Body radiation color x¯(λ), y¯(λ), z¯(λ) over the visible spectrum and the choice path as a function of absolute temperature. The various sym- for the middle spectrum functiony ¯(λ) to be taken equal bols DT correspond to Daylight type source (illuminant or to V (λ), the photopic eye sensitivity. synthetic source) at a given temperature T/100. For instance 8 If we have a radiation source characterized by a power D65 is for T=6500K (adapted from Hoffmann ). emission spectrum function P (λ) its tristimulus coordinates are given by: The color of the P (λ) source is given by a point with coordinates (x, y) in the CIE diagram displayed in fig. 7.

Z λmax (x, y) are called chromaticity coordinates with values ex- X = Km P (λ)¯x(λ)dλ, plicitly given by: λmin Z λmax X Y Y = Km P (λ)¯y(λ)dλ, x = , y = (A2) λmin X + Y + Z X + Y + Z 7

The CIE diagram (called tongue or horseshoe diagram) yellow, white and finally blue sequence as tempera- shown in fig. 7 displays several interesting characteristics: ture is increased. This describes heated metals and agrees with Kirchhoff observations3. 1. The contour contains pure colors (completely sat- urated or free of any white content) having wave- 4. Illuminants (artificial daylight sources) indicated lengths indicated on the borderline in nanometers. by D50,D65 and D93 appear at their correspond- The corresponding wavelengths are called domi- ing colour with index (50,65,93) equal to absolute nant since they control the color from pure (on the temperature divided by 100. Black-Body sources border) to White point at the center with coordi- with temperatures of 5000K, 6500K and 9300K 1 1 nates x = 3 , y = 3 . have colours close to the White point. 2. Colors within the horseshoe diagram are unsatu- 5. The CIE contour is closed from below by a straight rated and as we move forward toward the White line (called also the purple line) that does not carry point they become pastel like. This stems from the any dominant wavelength. It means that most pur- increase of white content as we proceed toward the ple colors cannot be obtained by altering the White white point. content of some main (dominant) color as done before. This is another consequence of the cone over- 3. Black-Body color appears on a path as a function lap that resulted in negative CMF weights. of absolute temperature. It follows the red, orange,

∗ [email protected] and a number of polarizations Np = 2. Should their mass 1 J. Ouellette, Phys. Today, December 2007 p. 25. be finite, they would have had Np = 2S + 1 = 3. Thus, the 2 CIE is Commission Internationale de l’Eclairage or Inter- density of states in 3D per unit volume L3 is: national Organization for Lighting based in Vienna (Aus- tria) that sets standards for light and colors and is respon- 3 1 1 4πω2 ω2 sible of the metrology of lighting like the NIST (National g(ω) = Np 2 = 2 3 (A4) Institute of Standards and Technology) in the US. 2π c c π c 3 F. Grum and R. J. Becherer, Optical radiation measure- ments, Volume 1: Radiometry; Volume 2: Color Measure- 6 Report of the 21st meeting (23-24 February 2012) of the ments, Academic Press (New-York, 1979). Consultative Committee for Photometry and Radiome- 4 G. Kirchhoff, On the relation between the radiating and try (CCPR), Bureau International des Poids et Mesures absorbing powers of different bodies for light and heat, Phil. (2012). Mag. Series 4, Volume 20, 1 (1860) (available on Google 7 D. C. Agrawal, H. S. Leff and V. J. Menon, Am. J. Phys. books). 64, 649 (1996). 5 Photons have the dispersion relation ω(k) = c|k| and ac- 8 G. Hoffmann, CIE Color Space, Web document cie- cording to Kittel12 the density of states g(ω) in d dimen- graph17052004.pdf from University of Emden URL: sions for a system of length L is given by: http://www.fho-emden.de/∼hoffmann/howww41a.html 9 N. Ohta and A. R. Robertson, Colorimetry: Fundamentals d Z L dSω and Applications, J. Wiley and Sons, (New-York, 2005). g(ω) = (A3) 10 2π vg W. H. Press, W. T. Vetterling, S. A. Teukolsky and B. P. ω<ω(k)<ω+dω Flannery, ”Numerical Recipes in C: The Art of Scientific where dSω is the differential area element on the constant Computing”, Second Edition, Cambridge University Press energy surface ω(k) = ω and vg the group velocity. Since (New-York, 1992). 11 the dispersion is linear in k, vg = c and the constant energy R.D. Edge and R. Howard, Am. J. Phys. 47, 142 (1979). ω 2 12 C. Kittel, Introduction to Solid State Physics, Wiley, New- surface is a sphere of radius k = c , thus Sω = 4πk = ω 2 York (1975). 4π( c ) . Photons are zero-mass particles with spin S = 1