LED array: where does far-field begin?

Transcription

1 LE array: where oe far-fel begn? Ivan Moreno 1, Chng-Cherng Sun 2 1. Una Acaemca e Fca, Unvera Autonoma e Zacateca, 98060, Zacateca, Mexco. 2. epartment of Optc an Photonc, Natonal Central Unverty, Chung-L 320, Tawan. ABSTRACT Any cluter of lght-emttng oe (LE) can be moele a a rectonal pont ource f the far-zone conton met. A general conton erve for the tance beyon whch the far-zone approxmaton can be ue n meaurng or moelng propagaton of lght from an LE array. A mple equaton gve the far-fel conton n functon of parameter of nfluence, uch a LE raaton pattern, array geometry, an number of LE. We calculate the nearzone extenon of cluter wth LE raaton pattern of practcal nteret; for example Lambertan-type, batwng, an e emttng. The far-fel conton hown to be conerable horter for hgh packagng enty LE array. Moreover, the far-fel ramatcally change n functon of the beam vergence of the LE raaton pattern. For example, the near-zone of a quare LE array wth hghly rectonal LE (mall half-ntenty vewng angle) can exten to more than 70 tme the cluter ze. Th value far from the clacal rule of thumb (5 tme the ource ze). Key wor: lght-emttng oe, LE array, far-fel regon. 1. INTROUCTION LE array are everywhere, n many hape an ze (Fg. 1), an cover a we range of applcaton. However, th varety alo create pecal ffculte n meaurng ther lght output, 1 whch can lea to ncontence between fferent meaurement ytem. For th reaon, the TC2-50 CIE Techncal Commttee preparng a techncal report to gve recommenaton for meaurement metho an conton of the optcal properte of LE cluter. 2 In partcular, an mportant meaurement conton the tance between the LE cluter an the etector urface for meaurng raant (W/r) or lumnou (lm/r) ntenty. Intenty efnton only apple to pont ource. However, n theory, any LE array can be moele a a rectonal pont ource f a far-fel conton met. Geometrcally, a ol angle mut have a pont a t apex; therefore, the trct efnton of ntenty apple only to a pont ource. In practce, however, the raaton emanatng from a ource whoe menon are neglgble n comparon wth the tance from whch t etecte may be conere a comng from a pont. Fg. 1. Some LE array example. 8th Internatonal Conference on Sol State Lghtng, Ian T. Ferguon, Tunemaa Taguch, Ian E. Ahown, Seong-Ju Park, Etor, Proceeng of SPIE Vol (2008) 2008 SPIE ISSN: X (prnt) $ 15.00

2 epenng on the workng tance, both the optcal moelng an the expermental characterzaton of a lght ource mut be performe n a fferent way. Bacally, there are two workng conton: near-fel an far-fel. In near fel a ource moele a an extene area, an t uually aume that the tance to the llumnate target horter than 5 tme the maxmum ource menon. 3-5 A far-fel approach aume that the target farther from the ource than th nomnal eparaton, an the ource can be moele or meaure a an emttng pont. However, th rule of thumb fal for LE array becaue of: the crete nature of the ource, an the we varety of both array geometre an ntenty trbuton (LE are avalable n many fferent beam pattern). Then, a general efnton of the far-fel conton neceary to correctly elmtate the near zone from the far zone. 2. FAR-FIEL CONITION In the far zone, an extene lght ource can be ealy mulate or meaure a a pont ource wth pecfc angular ntenty trbuton. A a rule of thumb, t uual to aume the far-zone begn at a tance of fve tme the larget menon of the lght ource. In that regon, the meaure raant ntenty or lumnou ntenty practcally nepenent upon tance from the ource. Hence, an anwer to the queton where oe far-fel begn? nvolve computng the ntenty of an arbtrary extene ource, whch mut be compare wth an equvalent pont ource. Raant (or lumnou) ntenty I the raant (or lumnou) flux per ol angle n a gven recton from the ource,.e. I=Φ/Ω. In practce, ue to the fnte ze of any etector the meaurement not exactly the ntenty, t alway an average ntenty. The meaure ntenty the rato of the flux Φ collecte by the etector aperture over the ol angle Ω ubtene by the etector on the ource center,.e. I=Φ /Ω. Typcally, the angular ntenty trbuton meaure by holng the ource tatonary an movng a etector on a hemphere centere n the ource. At each tance r the ol angle Ω, reman contant an each meaurement ecrbe only by t polar coornate (r,θ,φ) an the flux Φ collecte at that pont. Then n what follow, we focu n the calculaton of flux Φ. We begn the calculaton of flux Φ by conerng an extene flat ource lyng n the plane z=0 of a utable reference frame an emttng lght wth a raance (W/m 2 r) or lumnance (lm/m 2 r), L. Let coner both ource an etector wth arbtrary hape, an enote ther area by A an A (ee Fg. 2). The total amount of raaton tranferre from the ource to the etector gven by the ntegral over both area: 6 Φ L coα coα = 2 r r A A, (1) where r a poton vector for every emttng pont n the ource, an r a poton vector for every recevng pont n the etector. Fg. 1 llutrate the geometry ue n Eq. (1). Here t aume the etector behave a f t were a mple aperture that repon equally to lght at any pont acro t urface an from any recton,.e. a cone correcte etector. The angular ntenty trbuton n polar coornate I(r,θ,φ)=Φ (r,θ,φ)/ω (r), where Ω.A /r 2. Vrtually, for any ource the ntenty I become nepenent of tance r f the etector locate far enough away,.e.. I f(θ,φ), o the extene ource become a pont ource by comparon. It the far-fel conton. In aton, the nvere-quare law can only be ue n the far zone. Th law tate that the rraance or llumnance (W/m 2 or lm/m 2 ) at any pont on the etector urface (pontng through the ource) vare rectly wth the ntenty I(θ,φ), an nverely a the quare of the tance r from the ource,.e. E=I /r 2.

3 Fg. 2. Geometry for efnton of flux tranfer from a flat ource to etector. Here n an n are untary vector that are normal to the ource an etector urface, repectvely. The fve tme rule of thumb val for a ource wth crcular hape an Lambertan emon (L=contant). 3-5 Th rule tate that for a tance 5 tme the ource ameter, the error from ung the nvere-quare law 1 %. 3-5 The et up for th conton coner a etector locate on the optcal ax,.e. θ= Therefore, th conton equvalent to get an error of 1% when ung I(θ=0,φ=0) ntea of I(r,θ=0,φ=0). A mplfcaton n Eq. (1) acheve when θ=0 an the etector mall enough. Fg. 3 llutrate the geometry nvolve n th mplfcaton. In practce, the etector aperture can be choen mall enough to coner α an L nepenent of r, an then the ntegral over A can be carre on rectly. Furthermore, becaue θ=0 the angular epenence can be mplfe by α =α =α. After thee aumpton, Eq. (1) now: an the ntenty on ax gven by I(r)=Φ (r)r 2 /A. A 4 Φ ( r ) = L( r, r)co α ( r, r) A 2, (2) r (a) (b) Fg. 3. Geometry ue n the efnton of the far fel conton. (a) Show the geometry for a contnuou ource, an (b) llutrate the geometry for an LE cluter. The ntenty of reference I can be obtane by totally neglectng the ource ze. The apparent ource ze zero when the tance r. Therefore, the ntenty of reference I =Φ (r) r 2 /A can be obtane by makng coα(r, r )=1 an by takng the raance only along the z-recton L(r ) =L(r,r ). The raance ecrbe both the patal (gven by r ) an angular charactertc of the ource. The angular charactertc are gven by polar angle (θ,φ ) that ecrbe the

4 recton of optcal raaton emanatng at any pont r. Therefore, L(r ) =L(r,θ =0,φ =0). After thee aumpton the flux of reference vare wth the nvere-quare law: Φ = A L ( ) r 2 A r. (3) A cluter of LE can be terme a a lght ource wth contnuou raance. Becaue n mot cae the far-fel conton of an array longer than that of a ngle LE, 7 each LE of a cluter can be vewe a a pont lght ource wth certan angular ntenty trbuton. In uch cae, the ntegral n Eq. (2) mut be change by a um,.e. A Σ. Raance L alo mut be change by the angular ntenty trbuton I of the -th LE,.e. each LE conere a rectonal pont ource (I=LAcoα=I 0 coα). The calculaton become eay becaue mot manufacturer report the angular raaton pattern of ther LE, whch can be analytcally mulate n a mple way. 8 Hence the flux collecte by a mall etector on ax at tance r from an LE array wth N LE : N A 3 Φ ( r ) = I( α ) co α, (4) 2 r an agan, the ntenty on ax gven by I(r)=Φ (r)r 2 /A. The collecte flux of reference, equvalent to Eq. (3) for an LE array : an alo the ntenty of reference on ax I =Φ (r) r 2 /A. N ( 0) A I Φ = 2 r, (5) A tate before, the far-fel conton atfe when the fference between I(r) an I a gven percentage ΔI (n the 5 tme rule of thumb ΔI=1%). Takng nto account that I(r)> I, the far-fel conton r=r mn obtane by olvng the equaton: I( r) = (1 ΔI) I. (6) An extene flat ource wth raance L an area A poe a far-fel conton gven by: 4 L( r, r)co α( r, r) L( r ) A A = ( 1 ΔI ). (7) On nertng Eq. (4) an (5) nto Eq. (6), we fn the far-fel conton for a cluter e N LE by olvng where N I ( α ) N I co ( 0) 3 α = ( 1 ΔI ), (8)

5 2 2 x + y α = arctan. (9) r Here x an y are the Cartean coornate of the -th LE place to poton (x,y ) over the plane of the LE cluter. I the angular ntenty trbuton of the -th LE. Computaton of the far-zone conton r=r mn reuce to olve the Eq. (7) or (8). A ymbolc oluton for thee general equaton prove ffcult. However, the oluton can be ealy obtane wth any mathematcal program. In what follow we coner ΔI=1%, then Eq. (7) an (8) gve a far-fel conton r mn n whch the ntenty accurate wthn one percent (a n the 5 tme rule of thumb). Table 1 preent the far-fel conton for everal Lambertan ource (L=contant), whch normalze to the ource ze,.e. r mn /. Table 1 gve the conton value for everal ource hape, nclung the clacal 5 for a unform crcular hape. Even though the raance a contant, the rato r mn / change conerable n functon of hape. From Eq. (8), t oberve that the far-fel conton affecte by the trbuton of LE an the raaton pattern of every LE. The trbuton of LE epen on two factor: the packagng enty an the hape of cluter. In next ecton, we apply Eq. (8) to how how the far-fel conton epen on the packagng enty an the LE raaton pattern. Table 1. Far-fel conton of a flat Lambertan ource calculate wth Eq. (7) for everal ource hape. Source Geometry Far-fel conton, r mn Lne, Crcle, tme rule of thumb Square, Crcular rng, 3. EPENENCE ON PACKAGING ENSITY LE cluter are avalable n many fferent packagng ente. The ze of an LE array epen on everal factor, for example: cot, avalable pace, thermal problem, an aethetc egn. The followng Fgure are meant for llutratng the far-fel conton of LE array wth everal packagng ente. Increang the packagng enty equvalent to ncreae the number of LE wthn a cluter f the ze of the array kept contant. Therefore, n the next fgure we ncreae the number of LE n the cluter keepng contant.

6 Fg. 4 how the far-fel conton n functon of the number of LE for a quare array wth Lambertan raaton pattern LE. The plot begn wth a low packagng enty 2x2 array, an contnue up to a large array of 10x10 LE. The ot lne ncate the value of for a unform quare Lambertan ource (alo hown n Table 1). The far-fel conton of the array lowly converge to th value, e.g. r mn /=5.786 for a hypothetcal 100x100 LE array. Fg. 5 how the far-fel conton of a cluter of LE wth e emon raaton pattern (LUXEON Se Emtter from Lumle Phlp, for whte, green, cyan, blue an royal-blue color). Th fgure how r mn / n functon of the number of LE n a quare array. A reference value, for a very large array of 100x100 LE the far-fel conton r mn /= Fg. 4. Far-fel conton n functon of the number of LE n a quare array wth Lambertan LE. It hown from a low packagng enty 2x2 array to one 10x10 LE array. For a 100x100 LE array r mn /= The blue ot lne ncate the value of for a unform quare Lambertan ource. Fg. 5. Far-fel conton n functon of the number of LE n a quare array wth e emttng LE. From a low packagng enty 2x2 array to one 10x10 LE array. For a 100x100 LE array r mn /=2.552.

7 The hexagonal array, alo calle trangular, a popular LE ource owng to t packagng capablte. Fg. 6 how the far-fel conton n functon of the number of LE for a hexagonal array wth Lambertan raaton pattern LE. The plot begn wth a mall 3 LE array, an contnue up to a large array of 115 LE. For reference, the ot lne ncate the value of (by ung Eq. (7) for a contnuou ource x 3/2) for a unform rectangular Lambertan ource. The far-fel conton of th array lowly converge to 5.372, e.g. r mn /=5.411 for a hypothetcal very large array wth 9751 LE. Fg. 7 how the far-fel conton of a hexagonal cluter of LE wth e emttng pattern (LUXEON Se Emtter from Lumle Phlp, for whte, green, cyan, blue an royal-blue color) n functon of the number of LE. From all thee fgure, t event that an LE array wth hgher packagng enty ha a horter near-fel. Th behavor become event becaue n a lute cluter the contrbuton of LE locate at the corner become more mportant. In aton, comparng Fg. (4) an (6) or Fg. (5) an (7) eemngly ncate that the hexagonal array ha a horter far-fel conton. Th behavor may be becaue LE can be more effcently package n a hexagonal cluter than n a quare array. Fg. 6. Far-fel conton n functon of the number of LE n a hexagonal array wth Lambertan LE. It hown from a mall 3 LE array to one wth 115 LE. The blue ot lne ncate the value of for a unform rectangular Lambertan ource. Fg. 7. Far-fel conton n functon of the number of LE n a hexagonal array wth e-emtter LE.

8 4. EPENENCE ON LE RAIATION PATTERN LE are avalable n many fferent beam pattern, o that there not a ngle far-fel conton acro the multple LE type. We ue a mple analytc repreentaton to mulate the raaton pattern of the lght emtte from an LE. 8 The angular ntenty trbuton I moele by ang 2 or 3 Gauan term. Fg. 8 how the far-fel conton of a 2 2 quare LE array n functon of the LE beam charactertc. Th fgure how how the far-fel behave for everal LE raaton pattern: (a) LUXEON Se Emtter from Lumle Phlp (whte, green, cyan, blue an royalblue color); (b) LUXEON Batwng from Lumle Phlp (green, cyan, blue an royal-blue), upper boun; (c) LUXEON K2 (green, cyan, blue an royal-blue) from Lumle Phlp, upper boun; () Perfect Lambertan (β 1/2 =60 ); (e) XLamp XR-E LE from Cree; (f) LUXEON Rebel (green, cyan, blue an royal blue) from Lumle Phlp, upper boun; an (g) Lambertan rectonal (mperfect Lambertan), wth β 1/2 =30. Fg. 9 how the far-fel conton of a quare LE array for the ame LE raaton pattern. Comparng Fg. (8) an (9) t event the hortenng of the far-fel conton n functon of the packagng enty. One can ee n Fg. (8) an (9) that the far-fel conton horter f the LE raaton pattern wer, an longer f the LE angular trbuton more rectonal. Th behavor clearly llutrate n Fg. 10, whch how the varaton of r mn / n functon of the egree of LE beam rectonalty. For mplcty, t plotte for an mperfect Lambertan LE wth half-ntenty vewng angle (half wth half maxmum angle) rangng from 5 to 60. Th trong epenence of the far-fel conton on the LE rectvty mportant n many applcaton. For example, t mut be conere when the vewng angle of a LE veo creen characterze. 9 Fg. 8. Far-fel conton for everal LE raaton pattern: 2 2 quare LE array. Fg. 9. Far-fel conton for everal raaton pattern: quare LE array.

9 Fg. 10. Far-fel conton for everal half-angle β 1/2. 5. CONCLUSION A mple equaton for the farfel conton of an LE array wa erve. The far-zone conton an explct functon of the man parameter of nfluence, uch a LE raaton pattern, array geometry, an number of LE. The reultng equaton wa apple to cluter wth LE of practcal nteret, nclung: Lambertan-type, batwng, an e emttng angular ntenty trbuton. We calculate the far-fel for everal cae, an foun that t conerably longer for low package LE array. Moreover, the far-fel ramatcally change n functon of the beam vergence of the LE raaton pattern. For example, a hown n Fg. 10, the near-zone of a quare LE array wth hgh rectonal raaton pattern LE can exten up to 70 tme the cluter ze. Th near-zone far away of the clacal 5 tme the ource ze rule of thumb. REFERENCES 1. R. Young, Meaurng lght emon from LE, SPIE Proc. Vol. 6355, paper 63550H (2006). 2. CIE Techncal Commttee TC2-50: Meaurement of the optcal properte of LE cluter an array (from G. Sauter, Germany to J. Schuette, Germany). 3. IESNA [1985]. Photometrc Tetng of Inoor Fluorecent Lumnare, IES LM , Illumnatng Engneerng Socety of North Amerca, N.Y., NY. 4. IEEE 100 The Authortatve ctonary of IEEE Stanar Term, Seventh Eton, IEEE Pre (2000). 5. A. Ryer, Lght Meaurement Hanbook, (1998) E. F. Zalewk, Raometry an Photometry, n Hanbook of Optc (Vol. II, 2n e.), M. Ba, E. W. Van Strylan,. R. Wllam, W. L. Wolfe, E (McGraw-Hll, 1995). 7. P. Mannnen, J. Hovalta, P. Karha, E. Ikonen, Metho for analyzng lumnou ntenty of lght-emttng oe, Mea. Sc. Technol. 18, (2007). 8. I. Moreno, C.C. Sun, Moelng the raaton pattern of LE, Optc Expre, vol. 16 (3), (2008). 9. L. Svlan, LE rectvty meaurement n tu," Meaurement, (n pre 2007).