VLT-TRE-KMO

Transcription

1 Document Ttle Document Numbe Monte Calo modellng o the adance non-unomty o the calbaton unt ntegatng phee VLT-TRE-KMO Iue 1 Date 30th Mach 006 Sgnatue Date Pepaed by: Alexande Pokhoov Val Inc. Revewed by: Appoved by: Paul Clak Releaed by: Paul Cate

2 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page o 7 CHANGE RECORD Iue Date Secton aected Change Decpton 1 30/3/006 Ft eleae

3 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 3 o 7 TABLE OF CONTENTS 1. INTRODUCTION Scope 5 1. Abbevaton 5. DOCUMENTS 6.1 Applcable Document 6. Reeence Document 6 3. OBJECTIVE 7 4. GEOMETRICAL MODEL OF THE INTEGRATING SPHERE 8 5. REFLECTION MODELS BRDF denton and popete Lambetan BRDF MRPV BRDF RAY TRACING ALGORITHM Backwad ay tacng Ray-phee nteecton Shadow ay method Method o dependng tal RESULTS OF NUMERICAL EXPERIMENTS Intal geometcal data Fou BRDF model Cae o Lambetan BRDF. Evaluaton o computatonal accuacy Angula dtbuton o adance o MRPV BRDF 7.5 Radant luxe deence 5 8. CONCLUSIONS 6

4 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 4 o 7 9. REFERENCES 6

5 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 5 o 7 1. INTRODUCTION 1.1 Scope Th document contan a epot pepaed by Alexande Pokhoov o Val Inc. whch contan the eult o Monte Calo modellng o the adance non-unomty at the ext pot o the KMOS calbaton unt ntegatng phee. The epot conclude that non-unomty ove each pot could be wthn the ange 0.05% to 0.1% meetng the goal o patal (non-)unomty peced n the pck-o module equement [AD ecton ]. Howeve the epot alo ecommend that thee eult hould be condeed pelmnay and need to be conmed ate expemental meauement o the BRDF o nteed PTFE. Th aleady planned a documented n RD1 ecton Abbevaton BRDF FOV MRPV PTFE B-dectonal Relectance Dtbuton Functon Feld O Vew Matonchk-Rahman-Pnty-Vetaete PolyTetaFluooEthylene

6 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 6 o 7. DOCUMENTS The latet veon and ue date o each document ae contaned n the KMOS Conguaton Item Data Lt [AD1]..1 Applcable Document Re. Document Ttle Document Numbe AD1 KMOS Conguaton Item Data Lt VLT-LIS-KMO AD KMOS Pck-O Sub-Sytem Requement Speccaton VLT-SPE-KMO Reeence Document Re. Document Ttle Document Numbe RD1 KMOS Pck-O Sub-Sytem Degn and Analy Document VLT-TRE-KMO In cae o any conlct between thee document and the Applcable Document the nomaton contaned n the Applcable Document hall take pecedence.

7 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 7 o 7 {The ognal epot epoduced hee wth mno change to the wodng and omattng only} {Note: the coodnate ytem ued n th epot (ee Fg &3) doe not match that o the ntument tel} Monte Calo modellng o adance non-unomty at the ext pot o the KMOS Calbatng Unt Integatng Sphee 3. OBJECTIVE Th modelng wok ha thee man objectve: 1. To develop an algothm and compute code to allow the calculaton wth the accuacy o eveal hundedth o a pecent o the adant lux and angula dtbuton o adance wthn the eld o vew o the detecto obevng the ntegatng phee wall though the output pot.. To evaluate the deence between adant lux and adance dtbuton o detecto vewng the phee wall though the uppe and lowe output pot. 3. To povde the ecommendaton o ntegatng phee calbato degn mpovement (optonally computed chaactetc tun out to be unatactoy) The pncpal dculty wth the compute modellng cont o the ollowng. The ntenal uace o the ntegatng phee (a om o nteed PTFE) ha vey hgh electance n the vual and nea-ir pectal ange. To obtan the convegence o computatonal poce wth the gven accuacy a lage numbe o conecutve electon o adaton nde the phee hould be modelled. The elatvely mall ze o the aea adated by the ncdent beam and the necety to egte the adance n the gven decton o adant lux n the gven mall old angle deepen the complexty o the poblem. Moeove up to date uch computaton o ntegatng phee wee peomed o due (Lambetan) [1-3] o pecula-due [4-6] electon model o the phee ntenal wall. Snteed PTFE and elated mateal have electon vey cloe to Lambetan only n cetan ange o ncdence and electon angle theeoe mple Lambetan appoxmaton could be nucent and the angula electon popete o eal mateal mut be condeed. To date the ay-tacng baed Monte Calo method the only technque that accommodate the dect mulaton o non-lambetan uace. The oundaton o th technque the pobabltc teatment o adaton-matte nteacton. Th appoach allow contucton o a tochatc model o the ytem unde condeaton and an evaluaton o t paamete wth a lage numbe o ay tacng mplementaton. The numbe o ealzaton o a tochatc poce detemne the accuacy o the oluton.

8 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 8 o 7 4. GEOMETRICAL MODEL OF THE INTEGRATING SPHERE Fo mplcty we condeed the detecto a a pont object elmnated the oldng mo and tanomed the detecto' FOV nto ght-ccula cone (ee Fg. 1). Fg. 1. Elmnaton o the oldng mo. The global Catean coodnate ytem top vew and two ecton o a geometcal model o the ntegatng phee ae depcted n Fg. -4. We aumed that the phee ha zeo thckne (the ncome o electon om cylndcal duct o pot neglgble). I thee ae α N u uppe output pot cente and N l lowe the angula coodnate ae Δα l ( 1 ); αu k = Δαu ( k 1) + ; = 1... Nl; k 1... Nu l = Δαl = (1) whee Δ α = π / N ; Δα = π / N. l l u u Due to the qua-ymmety o the modelled ytem about t z ax t ucent to nvetgate one abtay pot om the uppe ow and one om the lowe ow. We wll conde the pot coepondng to = 1 and k = 1 n Eq. 1.

9 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 9 o 7 y Detecto α u α l B A A x B R Uppe output pot Lowe output pot Fg.. Schematc o the KMOS Calbatng Unt Integatng Sphee (top vew). A-A z z d B-B z z d z p z pl β u z a ψ l x δ ψ u δ l x d z al β l Fg. 3. Schematc o the KMOS Calbatng Unt Integatng Sphee (ecton A-A). Fg. 4. Schematc o the KMOS Calbatng Unt Integatng Sphee (ecton B-B). In ode to compute the angula dtbuton o adance wthn the concal old angle ubtended by the uppe o lowe output pot and havng the vetex at the detecto' poton the ollowng mappng wa peomed.

10 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 10 o 7 z Tangent plane Detecto z p ζ η Ψ x p x Fg. 5. Scheme o adance mappng. We placed a tangent plane on the phee at the cente o the pot unde condeaton (ee Fg. 5) and aanged the local Catean coodnate ytem ( η ζ ) o that thee local coodnate and global coodnate (x y z) ae connected by elatonhp: whee ( x y z ) p p p x = x y = y z = z p p p + ζ nψ = x + η + ζ coψ = z p p ζ z + R p ζ x + R ae the global coodnate o an output pot. To elmnate the addtonal coodnate tanomaton we algned the x ax wth the poton o an output pot (uppe o lowe) unde condeaton and e-computed the poton o all uppe and lowe output pot. The unom gd wa upempoed on a tangent plane; the adance value o ay pang though the detecto' pont and evey node o a meh computed. p ()

11 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 11 o 7 5. REFLECTION MODELS 5.1 BRDF denton and popete The angula adatve popete o an opaque uace can be exhautvely decbed by the pectal b-dectonal electance dtbuton uncton (BRDF) [7]: ( λ ) dl ( λ ) ( λ ) dlλ ( λ ) ( λ ) cod λ = = (3) deλ Lλ ω L L whee λ wavelength λ and λ ae the pectal adance o ncdent and elected adaton epectvely E λ pectal adance due to ncdent adaton ( ) and ( ) ae the phecal coodnate o the decton o ncdence and electon epectvely; dω the element o old angle aound the decton o ncdence (ee Fg. 6). BRDF ha the dmenon o -1. Dectonal-hemphecal electance can be expeed a an ntegal ove a hemphecal old angle: ρ ( λ ) = ( λ ) co d ω. (4) π ω z dω ω dω y x Fg. 6. Denton o BRDF n a phecal coodnate ytem. Evey BRDF mut obey the enegy conevaton law: ( λ ) co dω ( λ ) (5) π

12 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 1 o 7 a well a Helmholtz ecpocty pncple: ( λ ) ( λ ) =. (6) Heenate o mplcty we wll ue the monochomatc quantte and omt the dependence om wavelength λ. We wll alo uppoe that the adaton unpolazed and neglect the polazaton eect at the electon. 5. Lambetan BRDF I the adance o adaton elected om a uace doe not depend on ncdent no vewng angle uch a uace obey Lambet' law (cone law o adant ntenty); t BRDF then equal to: ρ 0 Lambet = (7) π whee ρ 0 the contant nomal (o dectonal-hemphecal) electance. Lambetan BRDF an dealzed model; due to t mplcty t wdely ued at the oluton o adatve-tane poblem and could be condeed a a zeo-ode appoxmaton model o th tudy. Dependng on the computatonal tak we ued not only the conventonal method [8] to geneate andom decton n accodance wth a Lambetan BRDF but alo a method decbed n [17]. The conventonal method nvolve computaton o the coodnate and o the local phecal coodnate ytem ate mple tanomaton o a pa o peudo-andom numbe η and η unomly dtbuted on (0 1] egment accodng to = acn η = (8) πη wth the ubequent tanomaton to the local Catean coodnate ytem and then to the global one. An altenatve method baed on the ollowng act: evey phee whch tangent to the electng uace o a Lambetan electo n the pont o electon a uace o unom adance. Theeoe n ode to model the equence o due electon pont on the ntenal uace o a phee t ucent to geneate the equence o pont unomly dtbuted ove th uace. The algothm o G. Maagla [9] ued to obtan the pont unomly dtbuted on the phecal uace x + y + z = 1. The next pa o peudo-andom numbe ηx and η y undegoe the lnea tanomaton u x η 1; u = η 1. (9) = x y y The pont wth coodnate y = ae unomly dtbuted wthn the x. I 1 a pont outde the ccle o unt adu the pa o quae ( 1 < y < 1) x = ux and u y = u x + u y >

13 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 13 o 7 peudo-andom numbe η x and η y ejected and new pa geneated. Othewe the coodnate o the pont on the uace o unt phee ae x = u 1 ; y = u 1 ; z = 1. (10) x Fo a phee o adu the coodnate o the pont n Eq. (10) mut be multpled by the value o. The eult o tet modelng ung both method ae peented n Fg. 7. y Fg. 7. Lambetan BRDF n phecal coodnate: let analytc expeon; cente conventonal method n Monte Calo modellng 10 9 ay ued; ght Maagla' method 10 9 ay ued. 5.3 MRPV BRDF The em-empcal BRDF model developed by Rahman Pnty and Vetaete [10 11] o emote enng applcaton wa moded by Matonchk [1] and then employed o the modellng o Spectalon MISR due [13 14]. The MRPV (Matonchk- Rahman-Pnty- Vetaete) BRDF model ha ou adjutable paamete and able to mmc the electon popete o mateal wth pedomnately volumetc catteng uch a nteed PTFE Spectalon Halon etc.: k 1 bg ρ MRPV ( A k b ρ ) = A[ co co ( co + co )] e 1+ (11) 1+ G whee A the calng acto A > 0; k Mnnaet exponent k 0; b aymmety paamete < b < ; ρ the aveage electance; ( ) g = co co + n n co ; (1) [ tan + tan tan tan co( )] 1 G =. (13) A a pecal cae at A = ρ k = 1 b = 0 ρ 0 MRPV BRDF nclude the Lambetan BRDF. 0 = To geneate ay decton wth the pobablty denty coepondng to MRPV BRDF the acceptance-ejecton method [15] wa employed. Fo evey ncdence angle the hgh bound value k 1 ρ B h = A [ co( 1+ co ) ] (14) π

14 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 14 o 7 computed; then the andom decton ( ) accodng to Lambetan BRDF geneated. I o next andom numbe η the ollowng nequalty ( A k ρ ) η B (15) h < MRPV b ullled the decton o electon ( ) accepted othewe new value o ( ) ae geneated. and η 6. RAY TRACING ALGORITHM 6.1 Backwad ay tacng Becaue owad ay tacng (om adaton ouce to detecto) cannot be appled to an etmaton o adance n a gven decton o lux n a gven pont the backwad ay tacng baed on the ecpocty pncple wa ued [16]. To model the adance angula dtbuton ay wee conecutvely taced om the detecto though the node o the unom gd coveng the output pot. Radant lux Φ allng onto detecto wthn concal old angle Ω equal to whee ( ) ( ) Φ = L co dω (16) Ω L the (pectal) adance n the decton that om the angle wth the nomal to the uace o a detecto. To compute Φ we geneated the decton unomly dtbuted wthn a old angle Ω and etmated the ntegal n Eq Ray-phee nteecton Ate detemnng the global coodnate ( ω ω ω ) x o the unt vecto o the decton o the due electon ω the paametc equaton o the elected ay (pmed coodnate belongng to the ntal pont o a ay) y z x = x + ω t; y = y + ω t; z = z + ω t (17) x hould be olved togethe wth the equaton o a phee x + y + z = R to detemne the x y z le on the phee value o the paametet. I ntal pont ( ) y z

15 Iue: 1 Date: 30/3/006 Monte Calo modellng o the adance non-unomty o the calbaton unt ntegatng phee Page 15 o 7 ( ) z y x z y x t ω ω ω + + =. (18) then the coodnate ( ) z y x can be ound. Ate each electon the adance o a ay multpled by the value o electance at the angle o ncdence. Ray tacng end when a ay ecape the phee though one o openng o when the enegy caed by the ay become le than the pecbed thehold value. 6.3 Shadow ay method To acceleate the convegence o computatonal poce we ued the o-called "hadow ay" method that wa decbed n detal n [17 18] o Lambetan phee wall. We have expanded th method o non-lambetan uace. The hadow ay method mple computaton o ncome due to ouce dect adaton nto adance n each pont o electon. In ou cae the collmated beam wth unom dtbuton o optcal enegy ove the beam ecton adate the phee aea that condeed a a econday ouce. The adance o adaton elected om th aea connected wth the adance due to the ntal collmated beam by the elatonhp (ubcpt "" o ouce): ( ) ( ) ( ) de dl = (19) ( ) ( ) ( ) S Cont de L = = (0) The ncome o the ouce nto adance elected by a pont on the phee: ( ) ( ) ( ) co co co d Cont d L L ω = = Ω (1) Snce all ay o a collmated beam cay the ame enegy we can aume that Cont = 1. The adance o a ay ncdent on the detecto (ee Fg. 8) can be expeed a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = k j j n k k d L L L L d d d ' co co... ' ' ' '... co co ' ' ' co co ' ' co co ' ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ()

16 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 16 o ' 1 1 L c c 1 c Fg. 8. The chematc o hadow ay method. 6.4 Method o dependng tal The method o dependng tal (o by anothe name the method o coelated amplng) [19 0] allow eult to be obtaned o eveal (pectal) electance o phee wall multaneouly ung the ame et o ay tajectoe. It aumed that the hape o the BRDF doe not change. Fo MRPV BRDF th mean that the calng acto A can cont o A A... 1 A m. eveal component ( )

17 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 17 o 7 7. RESULTS OF NUMERICAL EXPERIMENTS 7.1 Intal geometcal data All geometcal paamete ued o modellng ae hown n Table 1. Table 1. Intal geometcal paamete (lnea dmenon ae n mm angula n deg.) Paamete Value Comment Numbe o Uppe Output Pot N u 1 Numbe o Lowe Output Pot N l 1 Sphee Intenal Radu R 90 Input Pot Radu R p 5 Collmated Beam Radu R cb 4.9 Vaable om 0.1 mm to 4.9 mm Equato-to-FOV Vetex Dtance H d Z Ax-to-FOV Vetex Dtance L d Output Pot Radu R op 9 Uppe FOV Ax-to-Sphee Cente Dtance δ u Lowe FOV Ax-to-Sphee Cente Dtance δ l Uppe FOV Ax-to-Z Ax Angle β u Lowe FOV Ax-to-Z Ax Angle β l 3.7 Uppe FOV Vetex Angle Ω u 5.7 /10 cae Lowe FOV Vetex Angle Ω l 5.7 /10 cae Each pot omed a a ecton o the phee by the ght ccula cylnde whoe ax pae though the phee cente 7. Fou BRDF model Ate analyzng the avalable publhed data on nteed PTFE BRDF [ ] we elected ou BRDF model: Model 0. Lambetan BRDF; method o dependng tal allow the multaneou modellng o the phee wall wth ρ0 = and Th the zeo appoxmaton model. Model 1. MRPV BRDF wth paamete A = ; k = 1.01; b = 0.01; ρ = 1. The component o A wee choen o that nomal electance = ρ( 0)= and Th the "optmtc" model. ρ n Model. MRPV BRDF wth paamete A = ; k = 1.0; b = 0.0; ρ = 1. The component o A wee alo choen o that nomal electance = ρ( 0)= and Th the "pemtc" model. ρ n

18 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 18 o 7 Model 3. MPRV BRDF wth paamete A = ; k = 1.; b = 0.; ρ = 1. The component o A wee alo choen o that nomal electance = ρ( 0)= and Th model a a-om-lambetan ρ n one and wa ued only to demontate the ecacy o the algothm and code. 3-Dmenonal epeentaton o the model BRDF n a phecal coodnate ytem o thee ncdence angle ae depcted n Fg. 9. The ecton by the plane o ncdence ae hown n Fg. 10. Fg D mage o BRDF model n a phecal coodnate ytem o thee value o ncdence angle

19 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 19 o 7 Fg. 10. BRDF Model 1 and 3 o = and 80 n plane o ncdence The dectonal-hemphecal electance o the BRDF Model 1 and 3 wee computed by numecal ntegaton o the appopate BRDF and plotted agant ncdence angle (ee Fg. 11).

20 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 0 o 7 Fg. 11. Dectonal-hemphecal electance o BRDF Model 1 and 3 plotted agant ncdence angle 7.3 Lambetan BRDF. Evaluaton o computatonal accuacy Colou map o the elatve dtbuton o adance ove the uppe (let-hand map) and lowe (ght-hand map) pot ae peented n Fgue 1 3. The tandad devaton om the mean value hown at the top o each map computed o a vey coae gd havng 9 9 node. Fo each node 10 6 ay wee taced. Fg. 1. Uppe pot; ρ 0 = Fg. 13. Lowe pot; ρ 0 = 0.95.

21 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 1 o 7 Fg. 14. Uppe pot; ρ 0 = Fg. 15. Lowe pot; ρ 0 = Fg. 16. Uppe pot; ρ 0 = Fg. 18. Lowe pot; ρ 0 = Fg. 19. Uppe pot; ρ 0 = Fg. 0. Lowe pot; ρ 0 = 0.98.

22 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page o 7 Fg. 1. Uppe pot; ρ 0 = Fg.. Lowe pot; ρ 0 = The adance dtbuton depcted n the Fgue 1 exhbt no egula tuctue. We peomed addtonal numecal expement o N = 10 7 and 10 8 ay taced. It wa ound that 1 the value o tandad devaton deceae a N wth the N nceaed. Th mean that the tandad devaton chaacteze the andom dgtal noe and the adance dtbuton unom (wthn the amewok o Lambetan appoxmaton). We obtaned the tval eult that could be obtaned wthout numecal modellng. Indeed an ntegatng phee wth Lambetan wall doe not nclude bale o othe auxlay element one can pedct the unomty o adance dtbuton ove the whole phee uace except o an ntally adated aea. Openng o abtay numbe and hape can deceae the mean level o adance but cannot volate the unomty o dtbuton. Becaue the conguaton acto between two abtay pont on the ntenal uace o Lambetan phee depend only on phee adu the ncdent lux the ame o the ente phee uace a well a adaton loe though the openng beng the ame. Th good example o a cae when mple Lambetan model o electon nucent. Howeve the numecal expement peomed allow the evaluaton o the tandad devaton o andom uncetanty o computaton: about 0.08% o 10 6 ay taced and about 0.0% o 10 7 ay. 7.4 Angula dtbuton o adance o MRPV BRDF Compute modellng wth MRPV BRDF eveal tme lowe than that o Lambetan BRDF. To ave tme we abandoned the modellng o D adance mappng even on a coae gd. Due to the qua-ymmety o ull ytem aound the z ax one can pedct that the adance gadent n a vetcal decton wll be geate than that n a hozontal decton. Accodngly we modelled only one-dmenonal dtbuton n hozontal and vetcal decton aco the cente o uppe and lowe pot. The eult o the modellng ae peented n Table 7 and n Fg. 3. All eult wee obtaned wth 10 6 taced ay.

23 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 3 o 7 Table. MRPV BRDF Model 1 vetcal can. y (mm) Uppe Pot o Nomal Relectance Lowe Pot o Nomal Relectance E E-05.96E E E E E-05.96E E E E E-05.93E E E E E-05.93E E E E E-05.94E E E E E-05.94E E E E E-05.97E E E E E-05.9E E E E E-05.94E E E E E-05.93E E E E E-05.96E E E E E-05.94E E E E E-05.9E E E E E-05.97E E E E E-05.93E E E E E-05.93E E E E E-05.94E E E E E-05.96E E E-05 Std. Dev % 0.053% 0.05% 0.050% 0.047% 0.051% 0.055% 0.060% 0.065% 0.071% Table 3. MRPV BRDF Model 1 hozontal can. x (mm) Uppe Pot o Nomal Relectance Lowe Pot o Nomal Relectance E E-05.93E E E E E-05.9E E E E E-05.95E E E E E-05.97E E E E E-05.90E E E E E-05.94E E E E E-05.9E E E E E-05.94E E E E E-05.9E E E E E-05.93E E E E E-05.90E E E E E-05.96E E E E E E E E E E-05.93E E E E E-05.93E E E E E-05.94E E E E E-05.90E E E E E-05.9E E E-05 Std. Dev % 0.065% 0.070% 0.076% 0.083% 0.04% 0.047% 0.05% 0.059% 0.067% Table 4. MRPV BRDF Model vetcal can. y (mm) Uppe Pot o Nomal Relectance Lowe Pot o Nomal Relectance E E E E E-05.18E E E E E E-05.48E E E E-05.17E E E E E E E E E E-05.19E E E E E E E E E E-05.19E E E E E E E E E E-05.18E E E E E-05.14E E E E E-05.17E E E E E E E E E E-05.0E-05.48E-05.79E E E E E E E E-05.E E E E E E E E E E-05.0E E E E E-05 Std. Dev % 0.10% 0.104% 0.107% 0.110% 0.066% 0.069% 0.07% 0.076% 0.08%

24 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 4 o 7 Table 5. MRPV BRDF Model hozontal can. x (mm) Uppe Pot o Nomal Relectance Lowe Pot o Nomal Relectance E E E E E-05.19E E E E E E E E E E-05.19E E E E E E E E E E-05.19E E E E E E E E E E-05.16E E E E E E E E E E-05.19E E E E E-05.17E E E E E-05.17E E E E E E E E E E-05.17E E E E E E E E E E-05.17E E E E E E E E E E-05.19E E E E E-05 Std. Dev % 0.063% 0.067% 0.07% 0.077% 0.056% 0.061% 0.066% 0.073% 0.080% Table 6. MRPV BRDF Model 3 vetcal can. y (mm) Uppe Pot o Nomal Relectance Lowe Pot o Nomal Relectance E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05.00E E E E E E E E E E E E E E E E E E E E E-05 Std. Dev % 0.848% 0.88% 0.809% 0.790% 0.506% 0.506% 0.505% 0.505% 0.505% Table 7. MRPV BRDF Model 3 hozontal can. x (mm) Uppe Pot o Nomal Relectance Lowe Pot o Nomal Relectance E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05 Std. Dev % 0.109% 0.11% 0.116% 0.10% 0.049% 0.050% 0.05% 0.053% 0.054%

25 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 5 o 7 Model3 Vetcal can Model 3 Hozontal can.0e-05.0e E E-05 Radance (el. unt) 1.8E E-05 Radance (el. unt) 1.8E E E E E E E Poton on tangent plane (mm) 0.95U 0.96U 0.97U 0.98U 0.99U 0.95L 0.96L 0.97L 0.98L 0.99L 1.4E Poton on tangent plane (mm) 0.95U 0.96U 0.97U 0.98U 0.99U 0.95L 0.96L 0.97L 0.98L 0.99L Fg. 3. MRPV BRDF Model 3; vetcal and hozontal can o adance dtbuton; "U" and "L" ate nomal electance value n the legend ndcate uppe and lowe output pot epectvely. 7.5 Radant luxe deence We computed the adant luxe allng onto a detecto wthn a concal old angle wth the angle at a vetex o 5.7º (/10 cae). The elatve deence between the eult o uppe and lowe pot wa computed a ( Φu Φl ) ( Φ + Φ )/ 100% γ = (3) whee ubcpt "u" and "l" denote uppe and lowe pot epectvely. u It wa ound that the convegence o the computatonal poce o adant lux 5 to 10 tme ate than that o adance. The eult o modellng ae peented n Table 8. l

26 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 6 o 7 Table 8. Relatve deence γ between adant luxe om uppe and lowe pot. Sphee wall γ (%) nomal electance Model 1 Model Model % -0.17% -1.37% % -0.11% % % % % % % -1.03% % -0.10% % The numecal expement how a weak dependence o adant lux deence upon the adu o the ncdent collmated beam that cannot have pactcal mpotance. 8. CONCLUSIONS 1. The Lambetan model o electon too cude an appoxmaton o olvng the poblem tated.. On the ba o a MRPV BRDF model and hadow ay method the veatle algothm o Monte Calo ay tacng developed; t allow the computng o the adance dtbuton and adant luxe wth a andom uncetanty o eveal hundedth o a pecent. 3. Fo eveal phycally plauble BRDF model o nteed PTFE the non-unomte o adance dtbuton ove the uppe and lowe output pot wee computed. It hown that dependng on the phee wall BRDF non-unomty ove each pot could be wthn the ange om 0.05% to 0.1%; the elatve deence o the mean adance between uppe and lowe pot about 0.%. The elatve deence o appopate adant luxe not geate than 0.13%. 4. All above-mentoned eult mut be condeed a a pelmnay aement only. They hould be ened ate accuate meauement o the BRDF o the choen phee mateal have been made. 9. REFERENCES 1. M. W. Fnkel Integatng phee theoy. Opt. Commun. 5-8 (1970).. R. L. Bown A numecal oluton o the ntegal equaton decbng a photometc ntegatng phee Joun. Re. Nat. Bueau o Standad A. Phyc and Chemty 77A (1973). 3. Y. Ohno Integatng phee mulaton: applcaton to total lux cale ealzaton Appl. Opt (1994). 4. A. Pokhoov S. Mekhontev L.Hanen. Evaluaton o peomance o ntegatng phee o ndect emttance meauement. Poc. o 8th Intenatonal Sympoum on Tempeatue and Themal Meauement n Induty and Scence. Vol June 001 Beln Gemany pp L. M. Hanen Eect o non-lambetan uace on ntegatng phee meauement Appl. Opt (1996).

27 Iue: 1 Monte Calo modellng o the adance non-unomty o the calbaton unt Date: 30/3/006 ntegatng phee Page 7 o 7 6. A. Zegle H. He H. Schmpl Rechnemulaton von Ulbchtkugeln Optk (1996). 7. F. E. Ncodemu et al. Geometcal condeaton and Nomenclatue o Relectance NBS Monogaph 160 US Depatment o Commece Natonal Bueau o Standad (1977). 8. R. Segel J. R. Howell Themal Radaton Heat Tane. 3 d Ed. Taylo & Fanc Wahngton DC (199). 9. G. Maagla Choong a pont om the uace o a phee The Annal o Math. Stattc (197). 10. Rahman H. Vetaete M.M. Pnty B. Coupled Suace-Atmophee Relectance (CSAR) Model. 1. Model decpton and nveon o ynthetc data. Jounal o Geophycal Reeach D Rahman H. Pnty B. Vetaete M.M. Coupled Suace-Atmophee Relectance (CSAR) Model.. Semempcal uace model uable wth NOAA Advanced Vey Hgh Reoluton Radomete Data. Jounal o Geophycal Reeach D Engelen O. B. Pnty M. M. Vetaete and J. V. Matonchk. Paametc Bdectonal Relectance Facto Model: Evaluaton Impovement and Applcaton EC Jont Reeach Cente Techncal Repot No. EUR 1646 EN 114 p S. P. Flae M. M. Vetaete B. Pnty C. J. Buegge. Modelng Spectalon' bdectonal electance o n-lght calbaton o Eath-obtng eno. Poc. SPIE vol (1993) 14. C. J. Buegge N. Chen D. Hane. A Spectalon BRF data bae o MISR calbaton applcaton. Remote Sen. Envon. v (001) 15. Dagpuna J. Pncple o Random Vaate Geneaton. Claendon Pe Oxod P. Shley Realtc Ray Tacng A K Pete Natck MA (000). 17. A. V. Pokhoov S. N. Mekhontev and L. M. Hanen. Monte Calo modelng o an ntegatng phee electomete. Appled Optc v. 4 No (003) 18. A. V. Pokhoov L. M. Hanen Numecal Modelng o an Integatng Sphee Radaton Souce Poc. SPIE 4775 Modelng and Chaactezaton o Lght Souce C. Benjamn Wooley Ed (00). 19. I. Manno. Intoducton to the Monte-Calo Method. Akadema Kado Budapet (1999). 0. J. Spane and E. M. Gelbad Monte Calo Pncple and Neuton Tanpot Poblem Addon-Weley Readng MA (1969). 1. P. Y. Bane E. A. Ealy. Due electance o nteed and peed polytetaluooethylene (PTFE). Poc. SPIE vol (1998).. B. T. McGuckn D. A. Hane R. T. Menze. Multangle Imagng Spectoadomete: optcal chaactezaton o the calbaton panel. Appled Optc vol. 36 No (1997). Lat Page