GEOMETRICAL OPTICS AND THE RADIATIVE TRANSFER EQUATION IN A VARYING REFRACTIVE INDEX MEDIA

Transcription

1 IJRRAS 5 Octobe pf GEOMETRICA OPTICS AND THE RADIATIVE TRANSFER EQUATION IN A VARYING REFRACTIVE INDE MEDIA Owoyo, M. James & Geoge Mocheche Uvesty of Naob, P.O. Bo 397-, Naob MultMea Uvesty of Keya, P.O. Bo 335-, Naob ABSTRACT Ths pape s shows that the gve Raatve tasfe equato pefectly wos the o-absobg / o-scatteg lmt, what was coteste by. Matí-ópez [Opt. Commu. 66, 44 6]. The assumpto that ths equato woul mply a zeo vegece of the ays s eploe. Raatve tasfe a meum wth a spatally vayg efactve e was ecetly at the og of a lot of actvty. Key wos: Raatve Tasfe Equato, spatally vayg efactve e.. INTRODUCTION The followg mofe aatve tasfe equato [3] ca be useful escbg such a poblem, that s:,, l, c t l, a s,, ' ', ', s 4 whee, s the aace, the efactve e, c the spee of lght vacuum, absopto a scatteg coeffcets,,, ' the omalze scatteg fucto,, a whee eotes the gaet opeato wth espect to cooate, whle a a s the the souce tem, s the tasvese gaet opeato wth espect to cooate. Ths epesso was cofme by M. Sheeleva [4], who otes that t was fact fst eve by G.C. Pomag [5]. Futhemoe, G. Bal pove [7] that t coespos to the hgh fequecy lmt of Mawell equatos heteogeeous mea [6]. Ths epesso satsfes the eegy cosevato equato: f we touce the quattes,, a, we ee have the cosevato equato: a E t whee E s a souce tem. We ca pove ths fact the same way as fo the classcal RTE equato whch s wellow to satsfy ths equato [7], by tegatg ove. The emostato s eactly the same please ote that oes ot epe o, ecept fo the tems l that taes to accout the spatal vaatos of the efactve e. The fst oe ca be ealy tegate, a leas to: l, l 4 The seco tem s moe subtle. T. Kha a H. Jag pove [] by og the complete calculato that,, 5 a we popose a moe geeal emostato of ths fact the appe. Oe ca theefoe see that these two cotbutos eactly cacel, whe the othe tems gve the cosevato equato 3. Thee ae theefoe a lot of elemets that attest to the valty of ths mofe RTE. Howeve. Matí-ópez, J. Bouza-Domíguez, R.A. Matíez-Celoo, J.C. Hebe cotest ts valty [4], a ague that t oes ot epla some well-ow esults the o-scatteg / o-absobg lmt. They cosee two tme-epeet poblems ths lmt: 3 3

2 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs a The aato fom a pot-le souce a meum wth a costat efactve e. Oe has to f a vese squae law fo the lght testy. b The evoluto of the testy alog a classcal ay a meum wth a spatally vayg e, whee oe shoul have [8,4] I s s s s s whee s eotes the ac legth alog the classcal ay, a whee s s s the eoal. et us ow eame how equato ca completely solve these poblems a satsfyg mae a let us stat wth the fst oe. I a tme-epeet poblem, wth a pot-le tme-epeet souce tem E P, a wth s a P, the cosevato equato taes the fom: 7 Usg Gauss theoem a the atual symmetes of the poblem, as was also otce [4], that the soluto of ths equato s: P 4 ˆ whee s the stace fom the souce a whee ˆ s the ut vecto. The RTEVI s theefoe ot cotacto wth the vese squae law. Ayway, as ths poblem s a estctve case of the seco oe, let us cose ths othe poblem. No Scatteg Sceeo We ae coseg the o-scatteg o o-absobg lmt, whee, that coespos to staa geometcal optcs, a we wat to follow optcal ays. We theefoe loo at a aace, peae alog a ecto uˆ, so that oe ca cose that the whole eegy popagates alog û. I that case, we ca touce the lght testy I, whch ca be efe as equal to the aveage ffuse testy. We wll the have fo the ffuse flu vecto: uˆ I uˆ 9 whee we have use the fact that, s peae alog uˆ, a theefoe cacels f s sgfcatly ˆ uˆ ffeet fom u. Whle oes ot epe o, epes o, a thee s o cotacto at ths stage. If we set equato 9 the cosevato equato, we have omttg the souce tem: s / a uˆ I I uˆ The quatty uˆ was ee efe as a popety of the aace, 6 8, a shoul ot coespo to geometcal optcs f equato s wog. To show ths pot, let us multply equato by, a tegate ove. The fst tem of the tme-epeet poblem wll be: As, s peae alog uˆ we ca wte: uu ˆ ˆ I uˆ uˆ uˆ I uˆ I uˆ uˆ I uˆ uˆ Usg ths tem theefoe euces to I uˆ uˆ I uu ˆ ˆ l. The th a last tem we have s a appe to show that f s the etty:. Wth the same agumet, the seco tem wll be s moe comple, a we efe the eae to the 33

3 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs 3 I 3ˆ uui ˆ I Theefoe we have: uˆ uˆ l uu ˆ ˆ l what s eactly the ay optcs equato! We ca theefoe see û as the ut vecto taget to the optcal ay, that s: u Puttg 3 to gves usg u ˆ / s : I I I s ˆ 3 o s I I I I s s Ths s equato 6 we wee loog fo. We ca ote that, fo a costat efactve e a a pot-le souce, whee [4], ths equato euces to: I I s that s plaly compatble wth the vese squae law. Eq. theefoe accouts fo geometcal optcs laws the o-scatteg / o-absobg lmt. Aothe pot coces the asseto that ths equato volves the assumpto of a zeo vegece of the ays eveywhee the meum. To aess ths msuestag, let us ecose the whole evato of eq.. The aace,, s efe so that A u 4 s the powe flowg wth the sol agle aou, a though the suface elemet A aou u s a ut vecto othogoal to ths suface eleme. The eegy flow ca be matealze by a vecto fel, whee s a ut vecto paallel at each pot to the geometcal optcal ay passg though ths pot. Note hee that epes o, what s ot the case of. Ths vecto fel theefoe epeset, amog a lot of ays, the pat of the eegy that popagates the ecto. But how ca we eactly efe t, as the paallelsm s ot obvous wth cuve ays? The stuato s summaze o fgues a : Thee ae a lot of ays that coss the suface elemet A, a oe has to select whch wll be cosee as paallel to. The most mmeate choce, we mae [3], s epcte fgue : oe ca smply choose the ays that ae paallel to whe cossg A. But fact, the oly oblgato s to have a ay paallel to at, a oe ca cose a lea evato to ths law whe movg wth A, as fgue. Oe ca ee asset that such a evato wo t mply ay mofcato the flu cossg A at the fst oe A, so that t wo t mofy the efto 4 of. Oe shoul theefoe be able to mae ths choce wthout ay chage, but ow the vegece of oes ot cacel ay moe! et us thus ecose the whole emostato wth ths seco hypothess: the fel evolves a A, whee the mat A we have A ths, the evoluto of s ee govee by the abtay way wth A, that s abtay evato wos oly fo. I the ecto ay optcs equato. 34

4 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs Net, we touce vecto cooates wth Este summato coveto wth space. Fom fome coseatos we ca wte: A a a a Euclea 5 whee s the etty. The ma ffculty, oe to obta equato, s to evaluate the quatty ', ', t ' A',, A whee the pmes sta fo quattes at ' s wth s c t /. We have ow [,3]: A' A sa A whee the vegece y sa A of s a completely abtay umbe. et us ow focus o othogoal to a efe the sol agle y ; as [3], we touce vectos a as: y y et, y, y 6 If y, we ca efe the vecto fel. et us wte the N vaato of fo a ftesmal splacemet alog, s N s s a fucto of that ca be evelope aou : y s s s We ca futhemoe wte, as y : y s s As y s othogoal to, we mmeately have fom 5 that y y A t t so that Ay Ay s a 8, that: y s a, that s: y, a we ca theefoe coclue, by compag 7 y y A 9 s, y, y ' et s, s s, y A s, y y A, s y

5 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs We wll fst otce that / s, whch s othogoal to, s the plae y, up to the fst oe s; a seco ema s that, f othogoal to, we have:, y, y If we ecall that wte: ' et l s s y P a o ot cotbute s the poecto o the plae P, y, y A, we ca wte the same thg wth the opeato A s s P P l sal sa We fally obta, afte some algeba a usg P : s ' sa We theefoe have, up to the fst oe: A' ' s s s s A A. These coseatos allow us to 3 4 whee we ca see that the tems cotag the vegece of eactly cacels, a theefoe o ot ete eq.. Ths vegece s abtay, a t was absolutely legtmate to cacel t by mag the choce of fgue CONCUSION We have clafe some pots coceg the aatve tasfe equato a meum wth a spatally vayg efactve e RTEVI a show that the RTEVI coectly escbes stuatos the geometcal optcs lmt o absopto/ o scatteg, a accouts fo stuatos wth a o-zeo vegece of the optcal ays. Appe Ths appe s evote to the calculus of some tegals volvg the tasvese gaet hee a -mesoal space, whee of a fucto f eas: f f f s a vecto of om alog the ut vecto 36 /. et us cose. The gaet ove what efes the tasvese gaet a -mesoal space. et us ow cose the eteso of the aace as a fucto of. / A As oes ot epe o the om, oe has fom A: et us ow tegate o the sphecal shell of fgue 3: A4 A A3

6 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs 37 whee. The fst tem equato A4 s: The seco tem, the ght of equato A4, ca be evaluate usg Gauss theoem, whee the outwa ut omal fel of the bgge bouay sphee s, whle t s fo the smallest oe see fgue 3: 4 ] [ S S Equato A4 theefoe eas: 4 I meso =, ths coespos to equato 5. The same wo ca be pefome o: A5 We apply the same poceue: the fst tem s whle a so that we have: what gves eactly equato meso thee. REFERENCES []. H.A. Fewea, J. Opt A.: Pue Appl. Opt []. T. Kha, H. Jag, J. Opt A.: Pue Appl. Opt [3]. J.-M. Tualle, E. Tet, Opt. Commu [4]. M. Sheeleva, J. Opt. Soc. Am. A [5].. Matí-ópez, J. Bouza-Domíguez, J.C. Hebe, S.R. Age, R.A. Matíez-Celoo, J. Opt. Soc. Am. A [6].. Matí-ópez, J. Bouza-Domíguez, J.C. Hebe, Opt. Comm [7]. G. Bal, J. Opt. Soc. Am. A [8]. T. Kha, A. Thomas, Opt. Commu [9]. A.M. Zys, E.J. Chaey, S.A. Boppat, Phys. Me. Bol []. AM Zys, D Mas, DY u, SA Boppat, Optcs Epess []. M Sheeleva, JA Molloy, Apple Optcs []. T Kha, A Thomas, Ivese Poblems [3]. C Che, JQ u, K, S Zhao, RS Boc, H Hu, Mecal Physcs

7 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs [4].. Matí-ópez, J. Bouza-Domíguez, R.A. Matíez-Celoo, J.C. Hebe, Opt. Comm [5]. G.C. Pomag, Raato Hyoyamcs Pegamo, Ofo UK, 973. [6].. Ryzh, G. Papacolaou, a J. B. Kelle Wave Moto [7]. Ishmau A. 978 Wave Popagato a Scatteg Raom Mea New Yo: Acaemc. [8]. M. Bo, E. Wolf, Pcples of Optcs, seveth e. epae, Cambge Uvesty Pess, 3., Fgue Rays passg though the suface elemet A. I bol ae the ays mple the efto of,, : ths fgue, these ays ae paallel to whe cossg A. 38

8 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs,, Fgue Same as fgue, but ow the bol ays chose fo the efto of,, evato vesus the splacemet wth A. peset a lea 39

9 IJRRAS 5 Octobe 5 James & Mocheche Geometcal Optcs Fgue 3 The tegato volume cosee s a sphecal shell, wth bouaes a. 4