Research note: Calculating spectral irradiance indoors

Transcription

1 Lighting Re. Technol. 217; Vol. 49: Reearch note: Calculating pectral irradiance indoor S Bará PhD a and J Ecofet PhD b a Área de Óptica, Facultade de Óptica e Optometría, Univeridade de Santiago de Compotela, Santiago de Compotela, Galicia, Spain b Departament d Òptica i Optometria, Univeritat Politècnica de Catalunya, Terraa, Catalunya, Spain Received 28 July 216; Revied 11 Augut 216; Accepted 13 Augut 216 The pectral compoition of the light that reache any indoor work plane depend on the characteritic of the light ource and the pectral reflectance of the urrounding urface due to the multiple reflection experienced by the light ray along their path from the ource to the obervation point. We how that in indoor pace, the ource and urface radiance mut obey a definite elf-conitent relationhip derived from the fact that each illuminated urface point act a a econdary ource of light. It i then etablihed that the pectral irradiance on any plane i linearly dependent on the pectral radiance of the light ource. The explicit integral form of thi relationhip provide a theoretical framework for a quantitative decription of the urface effect. Additionally, under very general aumption, we how that the pectral irradiance can be computed from the pectral flux of the ource through a imple multiplication by a wavelength-dependent function. Thi function, with unit of invere urface (1/m 2 ), provide a convenient way for evaluating the effect that arbitrary change in the ource pectrum will produce on the pectral irradiance at the indoor point under tudy. 1. Introduction In indoor etting, the pectral compoition of the light that actually reache the point of interet no matter whether it be on a work plane or at the cornea of the eye depend on the light ource pectrum and on the pectral reflectance of the urrounding urface. 1 Predicting the pectral irradiance at any arbitrarily oriented indoor urface require quantitative model that incorporate the pectral reflectance of the urrounding urface. In thi reearch note, we how that in indoor pace the ource and urface radiance mut obey a elf-conitent relationhip, Addre for correpondence: Salvador Bara, Área de Óptica, Facultade de Óptica e Optometría, Univeridade de Santiago de Compotela, Santiago de Compotela, Galicia, Spain. alva.bara@uc.e from which the linear dependence of the pectral irradiance on the pectral radiance of the ource can be formally etablihed. Furthermore, and under very general aumption, thi linear relationhip allow u to define an effective invere urface function (EISF), with unit m 2, that, once multiplied by the ource pectral radiant flux (Wnm 1 ), directly provide the pectral irradiance (Wm 2 nm 1 ) on the plane of interet. The EISF depend on the poition and orientation of the meaurement plane, on the characteritic of the environment, and on the radiance ditribution of the ource. It doe not depend on the ource pectrum. The EISF allow for a rapid evaluation of the effect that any change of the pectral compoition of the light ource may have on the oberved pectral irradiance, keeping contant the remaining geometrical parameter. ß The Chartered Intitution of Building Service Engineer /

2 Spectral irradiance indoor Light ource in indoor pace The pectral irradiance on any arbitray indoor plane contain contribution from the direct radiance from the light ource and the radiance reflected from the point of the environment acting a econdary ource. Since any point of the environment, in turn, i illuminated not only by the light ource but alo by the light reflected at the remaining point, the reflected radiance are mutually dependent on each other and have to obey the elf-conitent relationhip (ee equation (12) of the Appendix). Another way to conider thi i to ay that light arrive at the defined point after following multiple path. Some ray travel along a traight line connecting the ource to the point without modifiying their pectral compoition, neglecting for all practical purpoe any differential aborption or cattering effect due to the air preent within the room. Other ray, however, reach that plane after having undergone one or more diffue reflection at the urrounding urface. Every time the light i diffuely reflected, it pectral compoition i modified by the local bidirectional pectral reflectance ditribution function. Conequently, the pectral compoition of the light at the plane can be very different from the original pectrum of the ource. The pectral irradiance at any given point from multiple reflection can be mathematically decribed by a erie of recurively neted integral (ee Appendix, equation (14)), each one accounting for the light arriving at that point after a given number of diffue reflection. Thi erie can be traightforwardly deduced from the above quoted elf-conitent relationhip, a hown in the Appendix. A it could be anticipated, the final reult how that the irradiance at the plane of interet, E o ðl; x o, n o Þ, although pectrally different from the radiance ariing from the light ource, L ðl; x, xþ, depend linearly on it. X e X Thi can be formally expreed a E o ðl; x o, n o Þ ¼ l ½L ðl; x, xþš, ð1þ where l tand for linear operator (ee equation (14) and (15) in the Appendix), l i the wavelength, x o i the poition vector of the point on the urface where the irradiance i evaluated, n o i the unit vector perpendicular to the plane at that point, x i the poition vector of the ource, and x i a unit vector along the emiion direction of the ource, a hown in Figure 1. E o ðl; x o, n o Þ i meaured in Wm 2 nm 1 and L ðl; x, xþ ha unit Wm 2 r 1 nm The EISF dw ee w The light emitted by mot practical lamp ha the ame pectral compoition for all emiion direction and all point of their emitting area. The ource radiance L ðl; x, xþ can therefore be factored out a the product of a pectral term ðþand l a geometric one Bðx, xþ a X e dw e n e n o dw o a oe dw oe Figure 1 The pectral irradiance at a point x o of the plane of interet, whoe normal i decribed by the vector n o,i made up from the radiance directly received from the ource (located at x ) and the radiance reflected from the point (x e ) of the urrounding urface X o L ðl; x, xþ ¼ ðþb l ðx, xþ ð2þ

3 124 S Bara and J Ecofet The factor ðþ l and Bðx, xþ can be arbitrarily caled, provided that their product give the correct value for the ource pectral radiance in equation (2). A convenient choice i to identify ðþwith l the pectral radiant flux of the ource, meaured in Wnm 1, and conequently expreing the remaining geometric factor, Bðx, xþ, in unit m 2 r 1. Since the linear operator l act on the ource poition and angular direction variable but not on the wavelength, after ubtituting equation (2) into equation (1), we get where E o ðl; x o, n o Þ ¼ ðlþtðl; x o, n o Þ ð3þ Tðl; x o, n o Þ ¼ l ½Bðx, xþš ð4þ with unit m 2, i the EISF that, when multiplied by the ource pectral radiant flux in Wnm 1, give u the pectral irradiance in Wm 2 nm 1 at the defined point. 4. Dicuion The EISF, Tðl; x o, n o Þ, decribe the way in which the indoor environment modifie the original ource pectrum to generate the pectral irradiance ditribution at the defined point. In a looe ene, Tðl; x o, n o Þact a filter that attenuate or enhance the different wavelength a the light propagate from the ource to the defined point. Note, however, that it i not a filter function in the proper ene of the term, ince it relate two magnitude with different unit, and hence it i not dimenionle. The form of thi function depend on the ditribution of the ource radiance, on the pectral reflectance of the environment, and on the poition and orientation of the meaurement urface, relative to the ource and urface. Different kind of lamp may therefore produce a lightly different functional form of Tðl; x o, n o Þ, depending on their precie radiance ditribution. However, thi function i independent of the ource pectrum. Once computed or meaured for a given ituation, it can be applied to any arbitrary lamp pectrum in order to predict the pectral irradiance at the obervation point. The EISF can be numerically computed uing equation (14) if the radiance pattern of the ource and the reflecting propertie of the urrounding urface are known. Thi calculation involve a recurive erie of integral that, in practice, ha to be truncated to a finite number of term. Thi i a common ituation in any real application with multiple reflection. The number of term that have to be included in the truncated erie increae a the urface reflectance increae. When the pectral reflectance of the urrounding urface are not preciely known, the EISF can be determined experimentally, by meauring the pectral irradiance at the elected point of obervation and dividing it by the pectral radiant flux of the ource a Tðl; x o, n o Þ ¼ E o ðl; x o, n o Þ= ðþ: l ð7þ Continuou pectrum ource hould be ued for thi meaurement, in order to avoid zero or very low value in the denominator that would give rie to trong noie amplification and indeterminate value of T. Declaration of conflicting interet The author declared no potential conflict of interet with repect to the reearch, authorhip, and/or publication of thi article. Funding The author dicloed receipt of the following financial upport for the reearch, authorhip, and/or publication of thi

4 Spectral irradiance indoor 125 paper: Thi work wa developed within the framework of the Spanih Network for Light Pollution Studie (Miniterio de Economı a y Competitividad, grant AYA REDT). Reference 1 Bellia L, Pedace A, Fragliao F. Indoor lighting quality: Effect of different wall colour. Lighting Reearch and Technology Firt publihed on 1 July 215. doi:1.1177/ Appendix Let u denote by x o the poition vector of a defined point located on an arbitrary urface within an indoor pace, and by n o the unit vector perpendicular to the urface at x o (Figure 1). The pectral irradiance at that point, E o ðl; x o, n o Þ, i then E o ðl; x o, n o Þ ¼ L ðl; x, x o Þco o d! o þ L e ðl; x e, x oe Þco oe d! oe e ð8þ where the firt integral repreent the direct contribution of the light ource, being L ðl; x, x o Þ the pectral radiance of the ource point x that arrive at the defined point from the direction of the unit vector x o ¼ ðx x o Þ= kx x o k, with k k tanding for modulu, o ¼ arccoðn o x o Þ i the angle between thi direction and the normal n o,d! o i an infiniteimal olid angle element around x o, and the integral i extended to, the overall olid angle ubtended by the ource a een from x o. The econd integral repreent the radiance reflected toward x o from the point x e of the environment, L e ðl; x e, x oe Þ. The meaning of the magnitude appearing therein (x oe, oe,d! oe and e ) are analogou to the one in the firt integral, but now refer to the point on the urrounding urface intead of to the ource. The olid angle domain e cover all the direction encompaed by the hemiphere above the plane that contain the defined point (2 teradian), excluding the point geometrically obtructed by the ource. Henceforth, the notation will follow the ame convention for the definition of the poition vector, unit vector along the direction joining them, planar angle with repect to the normal to the urface, and olid angle. The pectral radiance L e ðl; x e, x oe Þ directed to the defined point from any generic point x e on the urface i given by the fraction of the light incident on that point that i diffuely reflected toward x o. Note that every urface point i illuminated not only by the primary light ource located at x but alo by the light reflected from the remaining unobtructed urface point x e 6¼ x e. Let u denote by de e ðl; x e, x ee the elementary irradiance incident on x e from an infiniteimal cone of direction d! ee around the unit vector x ee ¼ ðx e x e Þ=jjx e x e jj. The elementary radiance reflected at x e in the direction of the defined point, dl e ðl; x e, x oe Þ, can be related to de e ðl; x e, x ee by mean of the pace-variant bidirectional reflectance ditribution function f ðl; x e, x oe, x ee,defined a the ratio f ðl; x e, x oe, x ee dl e ðl; x e, x oe Þ=dE e ðl; x e, x ee ð9þ But, by definition, de e (l; x e, x ee ¼L e ðl; x e, x ee co ee d! ee, where L e ðl; x e, x ee i the radiance of the urface point x e in the direction of x e, ee ¼ arccoðn e x ee,and n e i the unit vector normal to the urface at x e.

5 126 S Bara and J Ecofet Subtituting thi expreion into equation (9), olving for dl e ðl; x e, x oe Þ, and integrating to the olid angle e ubtended by all urface point een from x e, we get the overall radiance reflected from x e toward x o due to the urrounding urface L e,wall ðl; x e, x oe Þ ¼ f ðl; x e, x oe, x ee L e ðl; x e, x ee e co ee d! ee ð1 On the other hand, the light that arrive at x e directly from the ource contribute to the reflected radiance a L e, ource ðl; x e, x oe Þ ¼ f ðl; x e, x oe, x e Þ L ðl; x, x e Þ co e d! e ð11þ The overall radiance directed from x e toward x o, L e ðl; x e, x oe Þ, which i the um of the contribution of equation (1) and (11), i then given by L e ðl; x e,x oe Þ ¼ fðl; x e,x oe,x e ÞL ðl; x,x e Þco e d! e þ fðl; x e,x oe,x ee L e ðl; x e,x ee e co ee d! ee ð12þ Note that equation (12) hold for any arbitrary urface point x e and any direction x oe. Thi equation expree a elf-conitent condition that hall be obeyed by the radiance of the environment, whereby the radiance reflected at any urface point depend on the radiance reflected at the remaining one, a well a on the radiance received directly from the ource. Subtituting equation (12) into equation (8), we can write the pectral irradiance at the defined point a E o ðl; x o, n o Þ ¼ L ðl; x, x o Þco o d! o þ f ðl; x e, x oe, x e ÞL ðl; x, x e Þ e co e d! e Š co oe d! oe þ f ðl; x e, x oe, x ee e e L e ðl; x e, x ee co ee d! ee Š co oe d! oe ð13þ where the firt term on the right-hand ide correpond to the light that arrive directly from the ource to the defined point x o, the econd correpond to the light from the ource that ha undergone a ingle reflection at the urrounding urface, and the third i the overall contribution of the light that ha undergone more than one reflection at the urface (i.e. the reultant of the reflection at every urface point of the diffue light reflected from the remaining point). The elf-conitent relationhip (equation 12) can be recurively ubtituted for the urface radiance L e ðl; x e, x ee in the integrand of the lat term of equation (13) to rewrite thi equation a a erie of contribution of light cattered at the urface multiple time. Each term of thi erie correpond to the light that underwent a given number of diffue reflection at the wall before reaching x o. The explicit expreion up to the double reflection i E o ðl; x o, n o Þ ¼ L ðl; x, x o Þco o d! o

6 þ f ðl; x e, x oe, x e Þ L ðl; x, x e Þ e i co e d! e co oe d! oe " þ f ðl; x e, x ee, x e Þ e e i L ðl; x, x e Þco e d! e # f ðl; x e, x oe, x ee co ee d! ee co oe d! oe þ ð14þ Note that, a expected, the pectral irradiance at the defined point depend linearly on the pectral radiance of the ource, L ðl; x, xþ and, in a trongly non-linear way, on the pectral reflectance of the urrounding urface. The linear dependence of E o ðl; x o, n o Þ on L ðl; x, xþ through the erie (equation 14) can be formally expreed a E o ðl; x o, n o Spectral irradiance indoor 127 Þ ¼ l ½L ðl; x, xþš, ð15þ where l tand for linear operator. Thi i a wavelength-dependent operator acting on the patial variable (poition and orientation) through the integral in equation (14). It i important to note that although all the equation preented here have been given in term of pectral radiometry they alo apply to pectral photometry.