Publication 4 E. Ikonen, P. Manninen, and P. Kärhä, Modeling distance dependence of LED illuminance, Light & Engineering Vol. 15, No.

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1 P4 Publication 4. Ikonn, P. Manninn, and P. Kärhä, Modling distanc dpndnc of LD illuminanc, Light & nginring Vol. 15, No. 4, (007). 007 Znack Publishing Hous Rprintd with prmission.

2 Light & nginring Svtotkhnika Vol. 15, No. 4, pp , 007 No. 6, 007, pp MODLING DISTANC DPNDNC OF LD ILLUMINANC* rkki Ikonn 1,, Pasi Manninn 1, and Ptri Krh 1, 1 Mtrology Rsarch Institut, Hlsinki Univrsity of Tchnology (TKK), PO Box 3000, FI-0015 TKK, Finland Cntr for Mtrology and Accrditation (MIKS), PO Box 9, FI-0151 spoo, Finland -mail: rkki.ikonn@tkk.fi A mthod for dtrmining photomtric proprtis of an LD in trms of its axial luminous intnsity, angular intnsity pattrn, and radius and location of th ffctiv LD sourc is prsntd. Th applicability of th mthod was tstd for svntn LD typs with diffrnt packags, angular intnsity distributions, and powr lvls. Whn applying th dvlopd mthod to th masurmnt data, instad of th invrs-squar law of a point sourc, a significant rduction in th distanc dpndnc of apparnt LD luminous intnsity was achivd. 1. INTRODUCTION Th dfinition of masurmnt conditions for th optical proprtis of LD light sourcs continus to b on of th focal points of th Tchnical Committs of th CI. Mthods hav bn prsntd for masuring th position of th LD sourc to dtrmin a rliabl luminous intnsity valu of th LD [1,]. Howvr, ths analyss ncssitat that th xact radius of th mitting ara of th LD is known. Anothr possibility is to trat th LD as just a simpl point sourc. This articl rviws som of th limitations of th convntional point-sourc analysis and prsnts an improvd mthod [3] which can b usd to dtrmin th illuminanc at variabl distancs from an LD. Th mthod is closly rlatd to th arlir analyss of th distanc dpndnc of signal with photomtrs [4] and spctroradiomtrs [5, 6, 7] quippd with diffusrs. * On basis of prsntation on 6 th CI Sssion, July 007, Bijing. To b abl to prdict th LD illuminanc at any distanc, th mthod utilizs a two-aprtur modl with an xtndd sourc and dtctor [3]. It dscribs nrgy transfr btwn paralll circular plats, cntrd on th optical axis of th masurmnt systm. Th mthod can also account for a highly non-lambrtian, dirctional angular intnsity distribution of th LD sourc. Dtails of th drivation of th modl quations ar prsntd for th first tim in this articl. Th bhavior of th LD illuminanc is dscribd by its axial luminous intnsity, angular intnsity pattrn, and radius and location of th ffctiv LD sourc. Th paramtrs of th modl can b dtrmind from illuminanc masurmnts at a fw distancs from th LD. Whn applying th dvlopd mthod to th masurmnt data, instad of a point-sourc modl, th distanc dpndnc of apparnt LD luminous intnsity of up to 50 % rducd to statistical variation of lss than 1 % for all th svntn tstd LD typs [3]. In this articl it is furthr shown that in many cass a simplifid analysis mthod can b usd, which lads to significant rduction of th ndd xprimntal ffort and still yilds rsults of sufficint rliability. Th analysis of th data lads to th satisfactory conclusion that an unambiguous luminous intnsity valu can b usd for calculating th LD illuminanc ovr a broad distanc rang.. LIMITATIONS OF POINT-SOURC ANALYSIS Th luminous intnsitis I v of svntn diffrnt typs of LDs wr dtrmind at various distancs 57

3 Light & nginring Vol. 15, No. 4 Luminous intnsity, cd Luminous intnsity, cd Distanc mm Distanc mm Fig. 1. Luminous intnsitis I v,tip, I v,chip and I v,fr of LD #14 (a) and LD #1 (b) as a function of distanc (s txt for dtails) d. Th masurd illuminanc valus v (d) wr first assumd to oby th invrs-squar law for a point sourc, I v ( d ) v,tip/chip ( d d) (1) rlativ to two rfrnc plans for th LD, i.., th front tip of th LD (distanc offst d = 0) and th sit of th LD chip (d > 0). A clar distanc dpndnc of th obtaind luminous intnsity valus occurrd in th analysis with rspct to both rfrnc plans. Thn th rfrnc plan position d was allowd to vary frly, but still mor than half of th tstd LD typs showd distanc dpndnc of th corrsponding luminous intnsity I v,fr.xampls from masurmnts with two of th LDs ar shown in Fig. 1. Ths wr unsatisfactory rsults du to th nd of larg amount of masurmnts for th full LD illuminanc charactrization. 3. DRIVATION OF MODL QUATIONS Fig. dscribs a circular sourc (radius r 0 ) and a circular dtctor (radiusr 0 ) at a distanc D from ach othr. Th plans of th sourc and dtctor ar paralll and thir cntrs ar locatd on th z axis (optical axis) of a cylindrical coordinat systm. Th radiant flux mittd by a small surfac lmnt da at (r,, 0) to th dtctor surfac lmnt da at ( r,, D) is d = L (r, ; )dad, whr L (r, ; ) is th radianc at position (r,, 0) in th dirction rlativ to th normal of th surfac,da da cos is th projctd ara of da in th dirction, and ddacos R is th solid angl dfind by point (r,, 0) and th surfac lmnt da at th distanc R. A homognous sourc with angular dpndnc n 1 L (r, ; ) =L0 cos of th radianc is assumd, whr L 0 dos not dpnd on r or and th paramtr n indicats th dgr of th dirctivity of th sourc. For a Lambrtian sourc, th dirctivity paramtr has a valu n = 1. With ths dfinitions th spctral radiant intnsity of th sourc bcoms I ()=I n 0 cos, whr I 0 r L 0 0 is th axial radiant intnsity of th sourc. Th total radiant flux mittd by th sourc to th dtctor is obtaind by intgrating d ovr all surfac lmnts da = rddr andda = rd dr. Th distanc dpndnc of spctral irradianc at th dtctor is thus ( D) A A d r 0 I 0 r r 0 0 A A cos n1 dada.() R Sinc cos = D/R and R = D + r + r rr cos( ), q. () can b writtn as r r n1 I D ( D) 0 r r 0 0 rr d d dr dr [ D r r rrcos( )] ( n ) 3. (3) Th intgration of q. (3) can b carrid out analytically whn n = 1 [8]. Nxt w approximat th squar root of th dnominator of q. (3) using th fact that D is much largr than ithr r or r, D r r rrcos( ) 1 D ( n3) 4 58

4 Light & nginring Vol. 15, No. 4 D( n3) n 3 r r rr cos( ) 1 = 4 D D ( n1 ) [ D cos( )], (4) whr = r(n +3) 1/ / and = r (n +3) 1/ /. Trms of ordr ( rd) 4, ( r D) 4, ( rr D), r3r D4, rr 3 D 4, and highr ar nglctd in q. (4). Whn q. (4) is substitutd into q. (3), a rsult is obtaind, rs rd I D ( D) 0 r r S D dddd [ D cos( )], (5) which is of th sam form as q. (3) with n = 1 taking into account that r 0 and r 0 ar rplacd by th ffctiv sourc and dtctor radii r S r 0 r n 3 and rd 0 n 3, (6) rspctivly [3,6]. As mntiond abov, th quation for spctral irradianc can thn b intgratd xactly ovr th sourc and dtctor surfacs [8]. Aftr V()-wightd intgration of th spctral irradianc, th distanc dpndnc of sourc illuminanc is finally obtaind, I v,ff v ( d) gd ( ) ( d d) r r S D, (7) whr I v,ff is th ffctiv luminous intnsity of th sourc on th masurmnt axis, D = d + d is writtn as a sum of th nominal distanc d and distanc offst d, and gd ( ) 1 14r r [( d d) r r ] S D S D (8) is a gomtrical factor, spcific to th circular sourc and dtctor. qs. (6 8) ar applid to rigorous analysis of LD illuminanc data v (d) at diffrnt distancs d from th front tip of th LD. Angular LD intnsity distribution masurmnt first dtrmins th dirctivity paramtr n. Thn q. (7) is fittd to th distanc dpndnc data using I v,ff, d, and r 0 as th fittd paramtrs, whil paramtrs n and r 0 ar fixd to th known valus. Factor g(d) is usually clos to 1, but it is rtaind in th rigorous analysis in its complt form, sinc for highly dirctional LDs, (, r,0) quations (6) amplify th ffct of th sourc and dtctor radii in such a way that th obtaind luminous intnsity may b affctd by a fw prcnts if approximation g(d) 1 is usd. Aftr driving th complt quations for th distanc dpndnc of LD illuminanc, on may ask if th paramtrs n and g(d) ar always ncssary for th analysis. Clarly, if n is clos to 1 and d is much largr than d, r 0, and r 0, a good approximation of q. (7) is givn by r 0 v R I v,ff ( d) ( d d) r r 0 0, (9) which is obtaind by fixing n = g(d)=1inqs.(6)and (7). Th conditions for th validity of q. (9) will b addrssd in Sc. 5 using synthsizd LD illuminanc data basd on th rigorous modl qs. (6 8). 4. XPRIMNTAL D d A commrcial LD photomtr manufacturd by LMT Lichtmsstchnik GmbH was usd in th illuminanc masurmnts. Th photomtr has a circular ntranc aprtur with a radius of r 0 =5.64mm.Aplanardiffusrisusdbhindth aprtur to improv th angular rsponsivity of th photomtr. Th distanc offst of th photomtr rfrnc plan position from th front surfac of th diffusr is zro. Th photomtr was calibratd for th illuminanc rsponsivity against an absolutly charactrizd rfrnc photomtr. Th dvlopd mthod was tstd with svntn diffrnt typs of LDs having a varity of packags, bam gomtris, and powr lvls [3]. Th tstd LDs wr mountd, on by on, on a spcific holdr mad of PVC (polyvinyl chlorid) and aluminum. Constant currnt of 330 ma was usd for th high-powr LDs of 1 W and 0.0 ma for th othr LDs. Th illuminanc masurmnts wr prformdona4.5-mopticalrailusingamagnticlngth masuring dvic with 0.1 mm rsolution. Masur- r 0 ( r,, D) Fig.. Gomtrical paramtrs for drivation of th modl quations 59

5 Light & nginring Vol. 15, No. 4 mnts wr mad at svn distancs in various distanc rangs btwn d =50mmandd = 3000 mm dpnding on th powr lvl of th tstd LD. Th photomtr was mountd on a rail carrir and th tstd LDs wr placd to th othr nd of th optical rail. Th mchanical symmtry axis of th tstd LD was alignd prpndicular to th rciving aprtur of th photomtr and to cross th cntr point of th photomtr aprtur by using a two-bam alignmnt lasr and an alignmnt mirror. To gt th corrct orintations for th tstd LDs, aluminum bodis with bor diamtrs of 6 mm and 1 mm wr usd for th LD alignmnts. For photomtr signal masurmnt, a digital voltmtr was connctd to a masurmnt computr via th I-488 bus. Dark currnt was masurd bfor th signal masurmnts utilizing an lctronic shuttr also controlld through th srial bus. 5. RSULTS AND ANALYSS Th distanc dpndnc of illuminanc was masurd for th diffrnt LDs. Th mthod basd on qs. (6 8) was usd in th analysis of th data. Fig. 3 shows xampls of th rsults for th sam LDs as usd in th masurmnts of Fig. 1. Comparing Figs. 1 and 3, it can b sn that in cas of LD #14 (#1) th distanc dpndnc of LD illuminanc of 50 % (10 %) is rducd to statistical fluctuation with a standard dviation of 0.7 % (0.1 %). Th tstd LD #14 is 10-mm-typ TLOH190P of 4 half-valu angl (full width at half maximum, FWHM) and LD #1 is of typ 1L51-YC1A with = 8. Th fitting rsults of th data of Fig. 3(a) ar I v,ff = 61. cd, d = 73 mm, and r 0 = 10 mm. Th dirctivity paramtr is n = 1915, rsulting in g(d) = 1.15 at th closst usd distanc d = 00 mm. Whn th data analysis was rpatd with a fixd valu g(d) = 1, th valu of fittd luminous intnsity I v,ff incrasd from 61. cd to 64.0 cd and its pak-to-pak variation from.0 % to.4 %. Th fitting rsults of th data of Fig. 3(b) ar I v,ff = 11.7 cd, d = 5.4 mm, and r 0 = 5 mm. Th dirctivity paramtr is n = 590, rsulting in g(d) = at d = 00 mm. Th data analysis by qs. (6 8) gav a valu of r mm for svn LD typs. Ths LDs had n 45 and thy can b rasonabl wll dscribd by a point-sourc modl. For som LDs, th point sourc appars to b locatd at th sit of th LD chip. Howvr, for tn LD typs it was ncssary to allow valus of r 0 btwn 1 and 1 mm in ordr to obtain a luminous intnsity I v,ff which dos not dpnd on th masurmnt distanc. All of ths LDs ar not high-n typs, sinc half of thm had dirctivity paramtr valus lss than 30. Figs. 1 and 3 illustrat th significant improvmnt in th consistncy of th luminous intnsity valus I v,ff as obtaind by th analysis mthod of Sc. 3. It is of intrst to dtrmin such a valu of th dirctivity paramtr n, blow which th luminous intnsity analysis can b mad rliably using th simplifid q. (9). Thus, simulatd illuminanc valus wr calculatd by qs. (6 8) for th distanc rang mm with various valus of th dirctivity paramtr n and typical valus of th distanc offst and sourc radius. Ths synthsizd data wr thn analyzd using th simplifid q. (9) and th fitting rsults wr compard with th original paramtr valus. Fig. 4 shows xampls of th rsults of ths tsts, whr th rlativ dviation of th obtaind luminous intnsity is plottd as a function of th dirctivity paramtr n. Luminous intnsity v,ff, cd Distanc mm Distanc, mm Fig. 3. Luminous intnsity I v,ff as a function of distanc for LD #14 (a) and LD #1 (b). Th rlativ pak-to-pak variations of % (a) and 0.3 % (b) ar smallr than thos in Fig. 1 by a factor of 5 to 0 Luminous intnsity v,ff, cd

6 Light & nginring Vol. 15, No Rlativ dviation of luminous intnsity Width of LD intnsity distribution on (FWHM) =10mm 0 =mm Dirctivity paramtr Fig. 4. Rlativ dviation of th obtaind luminous intnsity using two diffrnt analysis mthods. Th FWHM width of th LD intnsity distribution is indicatd by th uppr scal. Th usd distanc offst was d =10mm It is concludd from ths analyss that data fitting using q. (9) can b mad with good accuracy whn th width (FWHM) of th LD intnsity distribution is largr than 10. Such a width corrsponds to th dirctivity paramtr valu n = CONCLUSIONS Applicability of th dvlopd mthod has bn dmonstratd with svntn diffrnt LD typs, for which also th luminous intnsitis according to th point-sourc approximation wr dtrmind. Whn applying th mthod to th masurmnt data, th distanc dpndnc of apparnt LD luminous intnsity dcrasd dramatically as compard with th analysis rsults basd on th point-sourc approximation. Significant simplification of th dvlopd mthod can b achivd whn analyzing LDs with broad bams. Whn th half-width angl is largr than 10, th luminous intnsity of th LD can b dtrmind sufficintly accuratly without dtaild information on th angular intnsity distribution. Th dvlopd mthod gav notably consistnt rsults for th luminous intnsitis masurd at diffrnt distancs, thir standard dviation bing lss than 0.3 % for most of th tstd LDs. Th LD manufacturrs could masur and spcify luminous intnsity valus for thir LDs using th mthods dscribd in Sc. 3. Th photomtric bhaviour of dsignd light systms could thn b stimatd mor rliably bfor constructing prototyps, which might lad to considrabl savings in dvlopmnt costs. RFRNCS 1. Muray K, Proc. Soc. Photo-opt. 954, 1988, pp Muray K, Appl. Opt. 30, 1991, pp Manninn P, Hovila J, Krh P, and Ikonn, Mas. Sci. Tchnol. 18, 007, pp Hovila J, Mustonn M, Krh P, and Ikonn, Appl. Opt. 44, 005, pp Manninn P, Hovila J, Sppl L, Krh P, Ylianttila L, and Ikonn, Mtrologia 43, S10-S14 (006). 6. Ikonn, Manninn P, Hovila J, and Krh P, in Procdings of Mtrology Symposium 006, Qurtaro, Mxico, 006 (CD, 5 p.). 7. Grbnr J and Blumthalr M, Optics Lttrs 3, 007, pp Walsh J W T, Photomtry (Nw York: Dovr, 1965), pp rkki Ikonn rcivd th M.Sc. and Dr.Sc.(Tch.) dgrs in nginring from th Hlsinki Univrsity of Tchnology, in 198 and 1988, rspctivly. His rsarch for th last 15 yars has bn dirctd toward dvlopmnt of lasr-basd masurmnts, high-accuracy instrumnts for intrfromtry, and improvd mthods for radiomtric and photomtric masurmnts Ptri Karha rcivd his M.Sc. and D.Sc. dgrs in lctrical nginring from th Hlsinki Univrsity of Tchnology in 1993 and 1997 rspctivly Pasi Manninn rcivd his MSc dgr in lctrical nginring from th Hlsinki Univrsity of Tchnology in 004. H is currntly pursuing th doctor of tchnology dgr in TKK 61