Optis iside aisotropi media Fraçois-Xavier Getit * bstrat DM/DPN. CE alay 99 Gif-sur-Yvette. Frae We aim i those pages at givig the physial foudatios for the program Litrai. he problem is the behaviour of plae waves iside isotropi ad aisotropi media, the trasitio of a plae wave from a ( may be aisotropi) medium towards a other ( may be aisotropi) medium. f the media are isotropi, the alulatios are doe i ay physis boo. We advie Feyma "Letures o physis", olume, hapter. ut if the media are aisotropi, we have ot bee able to fid a text boo otaiig the alulatios doe here. o establish the equatios preseted here, have applied the same methods as the oe preseted i Feyma's boo, but geeralized for the ase where the dieletri tat ε is a tesor. Cotets Chapter : Plae waves iside a aisotropi media (.) - Maxwell equatios i aisotropi media (.) - Number of plae waves with a give diretio of vetor. (.) - Calulatio of the E field whe vetor fixed. (.) - bsorptio legth ad dieletri tat or tesor (.5) - rasitio from the system of optial axis to the world oordiate system (.6) - rasitio from the WC to the C. Chapter : rasitio of light betwee a aisotropi medium ad a isotropi medium (.) - Maxwell equatio () (.) - Maxwell equatios (/) (.) - Maxwell equatios () (.) - Equatios for eah wave (.5) - ystem of equatios for aisotropi to isotropi (.6) - olutios for ad. (.7) - peial ases for equatio (8) (.8) Fial solutio of ()-(7) Chapter : rasitio of light betwee a isotropi ad a aisotropi medium. (.) - Maxwell equatios () (.) - Maxwell equatios (/) (.) - Maxwell equatios () (.) - Equatios for eah trasmitted wave (.5) - ystem of equatios for isotropi -> aisotropi (.6) - olutios for ad. (.7) - peial ases for equatio (78) (.8) Fial solutio of ()-(6) * Correspodig author. el.: () 69 8 8; fax: () 69 8 6 8; e-mail: getit@hep.salay.ea.fr.
Chapter : rasitio of light through a thi slie. - rasitio through a thi slie Chapter 5 : Full alulatio speial ase () 5. - otatio aroud Oz ad vetor parallel to Oz 5. - Goig to the C oordiate system. 5. - olvig equatios () (6) 5. - Full alulatios of all fields 5.5 - Crossig the rystal of legth L 5.6 Exit of the wave from the rystal 5.7 Exit of the wave from the rystal. 5.8 Fial formula for rystal absorptio Chapter : Plae waves iside a aisotropi media. - Maxwell equatios i aisotropi media a aisotropi media, the dieletri tat ε is a symmetri tesor ε. he Maxwell equatios are : () div D ρ () div () rote d () dt rot H j d D dt ε (5) D ε E (6) H (7) P E H ε E a more geeral ase, the mageti permeability would also be a tesor. We do ot go to this level of geerality. P is the Poyitig vetor. We are oly iterested i the ase ρ ad j.
We do ot give here the proof that the dieletri tesor ε has to be symmetri. efer to boos o optis. s the dieletri tesor ε is symmetri, there exists a oordiate system i whih ε is diagoal. efer to text boo i liear algebra. the system i whih the dieletri tesor is diagoal, we write it lie this : ε a ε ε b ε ad we tae the ovetio that ε a < ε b < ε..e. we tae the ovetio to assig to the Ox axis the smallest elemet, to the Oz axis the largest elemet ad to the Oy axis the medium elemet. Notie also the followig defiitios : ε a < ε b < ε birefrigee or total aisotropy differet id of birefrigee ε a < ε b ε egativ birefriget ε a ε b < ε positiv birefriget We will all the system ( Ox Oy, Oz) ε a ε b ε isotrop, i whih the dieletri tesor is diagoal "he system of the optial axis". t is ot to be ofused with what is alled "he optial axis". he optial axis is ot a system of oordiate, it is a isolated axis, whih has a meaig oly i the ase of positiv or egativ birefrigee. t is the axis whih is isolated (for whih the ε value is ot equal to the other equal ε values). o that the optial axis is Ox i ase of egativ birefrigee ad is Oz i ase of positiv birefrigee.. - Number of plae waves with a give diretio of vetor We ited to loo whether we a satisfy the Maxwell equatios with a iput of type plae wave :
i (8) ( ω t x ) i( ω t x) e e E E f we put (8) ito (), we get : (9) ε E Equatio (9) implies that i aisotropi media, E is o more orthogoal to. t is ε or D whih is orthogoal to. Let us ow put equatio (8) ito (). We get : E () E i ω i ω or ( ) E Let us ow tae (8) ad () ito () : (We use the relatio ε ) () ω ε E ( ) E E Or ( E ) E E ε, ε where ω Equatio () is really fudametal. Let us first ider () i the isotropi ase. the isotropi ase, ε is a salar, equatio () ad (5) implies div E i. e. (, ) orthogoal to E ad () beomes E ε E, hee : E. is () ω ε ω relatio betwee ad idex of refratio the isotropi ase, the fat that is orthogoal to E implies that E is aywhere i the plae orthogoal to. We are free to hoose two E vetors : E ' ad E ' ' orthogoal to ad orthogoal to eah other ad assig to eah plae wave E ' ad E ' ' a differet phase, whih is the origi of the ellipti polarizatio. the ase of aisotropy, we will show that thigs are radially differet : E is o more orthogoal to ad we have ot the liberty of seletig the E vetor iside a give plae. O the otrary, if the diretio of the vetor is fixed, i the aisotropi ase, tha the E vetor is fixed ( o plae i whih to rotate it ) but we have the possibility to hoose amog two differet E vetors, eah oe beig fixed.
5 t is beause equatio () has exatly solutios for E. hese solutios have differet values of ( legth of vetor), hee they have differet values of idex. herefore, it is o more possible to spea of ellipti polarizatio, we have to spea of photos, propagatig i the medium at differet speed beause they have a differet idex. esides havig differet idies, the photos also propagate i differet diretios : their P vetor are ot parallel. Let us prove all that. Equatio () is a vetor equatio, so that it defies a system of equatios with uows. he uow are (module of the vetor) ad the diretio (θ,) of the eletri field. Let us itrodue followig defiitio ad the itrodue these defiitios ito equatio () : () κ κ uit vetor legth of () E Eu u uit E vetor E legth of E (5) ( u ( κ, u) κ ) ε u () with () ad () (6) ( u ( κ, u) κ ) u usig (), () ad the defiitio : ε We have defied all it " he ε dieletri tesor". ε dieletri tesor multiplied by mageti permeability. We will Equatio (6) is valid i ay oordiate system, i-e othig prevets us to mae a rotatio of the oordiate system. We hoose to write equatio (6) i the oordiate system i whih the ε dieletri tesor is diagoal. we write a, b, the diagoal elemets of the ε dieletri tesor i this oordiate system. We write (6) ito ompoets : ( - κ - a) u - κ κ u - κ κ u (7) - κ κ u ( - κ - b) u - κ κ u - κ κ u - κ κ u ( - κ - ) u (7) is a liear homogeeous equatio for the vetor u. (6) will oly have a solutio if the determiat of the matrix is zero. We leave to the reader the itermediate alulatios for this determiat, whih leads to : ( a κ b κ κ ) - ( a b ( - κ ) b ( - κ ) a ( - κ )) a b he mirale here is that it is a equatio of the d degree i, ot of the rd degree as we would have thought startig the alulatio of the determiat of (6), where at first a term i
6 6 appears. ut this term aels. o that everythig said above is ofirmed: there are possible values of ( hee of ) whih allow a solutio for equatio (6). Notie that (), the relatio betwee ad, has bee derived i the isotropi ase, but we tae it as a defiitio of the relatio betwee ad ( ad ε ) also i the ase of aisotropy : we are free to defie what we all "idex of refratio" i ase of aisotropy, provided this defiitio is ompatible with the solutio of (6). d if we do that, the ompatibility with the solutio of (6) eessitates that we admit that the suh defied varies with κ, the diretio of. We have foud values of : ad. Puttig these values ito (7) gives differet u vetor, i-e. differet diretios of the E field. his will have as a equee that the Poyitig vetors P will be differet : the possible photos assoiated with the fixed diretio κ of the vetor are goig ito differet diretios.. - Calulatio of the E field whe vetor fixed t is ot always so that we are free to use the most omfortable oordiate system ( the oe i whih the ε dieletri tesor is diagoal ) to solve a problem. For istae, i Litrai, we have to geerate a photo with a give diretio κ of the vetor i the World Coordiate ystem, whih has o reaso at all to be aliged with the oordiate system i whih is diagoal. o, let us try to solve equatio (6) i this more geeral ase. gai, we write (6) ito ompoets ad we use the fat that is symmetri : ( - κ - ) u - ( κ κ ) u - ( κ κ ) u (8) - ( κ κ ) u ( - κ - ) u - ( κ κ ) u - ( κ κ ) u - ( κ κ ) u ( - κ - ) u gai, we have a liear homogeeous system of equatio whih will have oly solutios if the determiat is. Let us alulate the determiat : ( - κ - ) ( - κ - ) ( - κ - ) - > ( - κ - ) ( - κ - ) ( - κ - ) - ( κ κ ) ( κ κ ) ( κ κ ) - ( κ κ ) ( κ κ ) ( κ κ ) - ( κ κ ) ( κ κ ) ( κ κ ) - ( κ κ ) ( κ κ ) ( κ κ ) - ( - κ - ) ( κ κ ) - ( ( - κ ) - ) ( κ κ κ κ ) -
7 ( - κ - ) ( κ κ ) - ( ( - κ ) - ) ( κ κ κ κ ) - ( - κ - ) ( κ κ ) ( ( - κ ) - ) ( κ κ κ κ ) 6 ( - κ ) ( - κ )( - κ ) - (( - κ ) ( - κ ) ( - κ ) ( - κ ) ( - κ ) ( - κ ) )) (( - κ ) ( - κ ) ( - κ ) ) - - 6 (κ κ κ ) - ( κ κ κ κ κ κ κ κ κ ) - (κ κ κ κ κ κ ) - - 6 (κ κ κ ) - ( κ κκ κκκ κ κ κ ) - (κ κ κ κ κ κ ) - - 6 ( - κ ) κ κ ( κ κ - ( - κ ) κκ ) ( κ κ - ( - κ ) ) - 6 ( - κ ) κ κ ( κ κ - ( - κ ) κκ ) ( κ κ - ( - κ ) ) - 6 ( - κ ) κ κ ( κ κ - ( - κ ) ) ( κ κ κκ - ( - κ ) ) Let us group terms i power of : 6 (( - κ ) ( - κ )( - κ ) - κ κ κ - ( - κ ) κ κ - ( - κ ) κ κ - ( - κ ) κ κ ) (6 ) ( - ( - κ ) ( - κ ) - ( - κ ) ( - κ ) - ( - κ ) ( - κ ) - κ κ κ - κ κκ - κκκ ( ) κ κ κ κ κ κ - ( - κ ) κκ - ( - κ ) κ κ - ( - κ ) κκ ) ( ) ( ( - κ ) ( - κ ) ( - κ ) - κ κ - κ κ - κ κ ( ) κ κ - ( - κ ) κ κ - ( - κ ) κ κ - ( - κ ) ) ( ) - - ( ) Now, usig κ κ κ : 6 ( κ κ κ κ κ κ - κ κ κ - κ κ - κ κ - κ κ κ κ κ ) ( 6 )
8 ( - - - κ ( ) κ ( ) κ ( ) - κ κ - κ κ - κ κ ) ( ) ( - κ - κ - κ - κ κ - κ κ - κ κ ( ) κ κ - κ κκ - κ κ κ - κ ) ( ) - - ( ) We see that the term i 6 disappears! We will have oly two solutios for, as aoued. ( ) κ κ κ κκ κκ κ κ (9) ( κ)( ) ( κ )( ) ( κ )( ) ( ) ( ) ( ) κκ κκ κ κ Why have we doe these terribly borig alulatio? We have gaied at least thigs : We have prove that the solutio of equatio (6) leads to a polyomial of the seod order i, ot of the rd order, so that we have, ot, solutios. For a give diretio κ of the vetor of the eletri wave, we have possible values for the idex. We a ow uderstad the ode of the method LitPhoto::FidDieli() whih would otherwise appear as very mysterious. (.) - bsorptio legth ad dieletri tat or tesor Dieletri tat he absorptio of the wave may be desribed by assigig a imagiary part to the dieletri tat or tesor. Let us first loo at the problem i isotropi media. Let us write the dieletri tat with a real ad a imagiary part ε C ε i : We put a mius sig to idiate that if, the we must have a mius sig suh that whe the value is iorporated ito the plae wave, we have a absorptio ad ot a explosio of the wave. Equatio () implies ( i ) ( ε i) part: C ε C. o C aquire also a imagiary
9 () i C ε ε i C ( ) ε ( ) ε ε ( ) ε ε i fat, is very small ompared to ε, so that we have to a very good approximatio: ε ε ordig to (), the vetor a be writte : ( ) κ ω κ ω κ ω κ ω κ i i C he plae wave a ow be writte : ( ) ( ) x x t i E x t i E E κ ω κ ω ω ω exp exp his gives a absorptio fator x κ ω exp for the eletri field. For the itesity of the wave, the absorptio fator is x κ ω exp From there, we see that the absorptio legth L a is L a ω ω L a () L a ω L a ω () ad () idiates how to go from absorptio legth ( whih is more ituitive ) to imagiary part of or to imagiary part of ε. Order of magitude Let us alulate the value of for a typial ase of a wave of wavelegth L w m e -7 m ad for a absorptio legth L a of m :
8 7 π ω Lw Lw π La ad are extremely small as ompared to Dieletri tesor Now, what happes i the ase of aisotropy, with a dieletri tesor ε. s already said, there exists a oordiate system where this tesor is diagoal, with diagoal elemets ε a, ε b, ε. s for the isotropi ase, we a assume that to desribe absorptio, we have to subtrat a imagiary part, but this time for eah of these diagoal elemets : ε a - i a, ε b- i b, ε - i : ε a - i a () ε ε b- i b ε - i a geeral oordiate system, the dieletri tesor will be a symmetri (ot hermitia) omplex tesor. ll alulatios previously doe apply equally, we have just to remid that we are ow worig with omplex umbers. he mai result obtaied before, i.e. that for a give diretio of the vetor, there are possible waves, eah with a fixed diretio of the E field ad eah with a differet value of apply as well. ut ow is omplex, so that obtaiig for a wave of vetor gives at the same time the absorptio for this wave, varyig with. s equatio () justified? Max or, i "Priiples of Optis", page 78, says that i the most geeral ase, we are ot allowed to postulate that the priipal axis of the odutivity tesor are the same as the optial axis, so that i the most geeral ase, () is wrog. However, he says that the priipal axis of the tesors oiide i diretio for rystals of the higher symmetry lasses ( of at least orthorhombi symmetry ) whih are the rystals whih iterest us. partiular, for PbWO, equatio () is true. the followig, we are iterested oly i ases where () is true..5 - rasitio from the system of optial axis to the world oordiate system, be the world oordiate system, ' ' O x, O y, O z ' be the system i whih the dieletri tesor ε is diagoal, with eigevalues ε a, ε b ad ε. We all the ' ' O x, O y, O z ' system "the system of the optial axis". y ovetio, ε a < ε b < ε. y ovetio also, Let ( Ox Oy, Oz)
ε a is assoiated with the O x ' axis, ε b with the O y ' axis ad ε with the O z ' axis. Let M be Ox, Oy, Oz axis to the ' ' O x, O y, O z ' axis : the rotatio matrix whih overts the ( ) O x ' M O x O y ' M O y Ox Oy Oz O z ' M O z other words, M is the matrix whih gives the ' ' O x, O y, O z ' axis i the world oordiate, ad E ' is the same vetor expressed i the system where the dieletri tesor is diagoal, the system of the optial axis ' ' O x O y O z ' ',,, the : E M E ( ad ot E ' M E ). his is the well ow fat that vetors are otravariat if axis vetor are ovariat. Now let us tae the Maxwell equatio i both systems system. f E is a vetor expressed i the the world oordiate system ( Ox Oy, Oz) ( ε diagoal dieletri tesor i system ' ' D O x, O y, O z ' ): ystem ( Ox Oy, Oz), system ' ' O x, O y, O z ' Maxwell equatio rasform Maxwell equatio rewritte ε ' ' D E D ε E ε ' ' ' ' E M E D M D E M E D M D ε ' ' ' ' M D ε M E > D ε M M E ε D ε ε M M < ε M M ε D D ε his implies that owig the dieleti tesor ε i the system where it is diagoal with D diagoal elemets ε a, ε b ad ε, we get the dieletri tesor i the world oordiate system through: ε M ε DM.
.6 - rasitio from the WC to the C We are ot yet fiished with problems of oordiates trasform. fat, the oordiate system i whih we ited to do all the alulatios for the trasitio is ot the world oordiate system ( Ox, Oy, Oz), but the idee Coordiate ystem ( O x O y O z ), oordiate system ( O x, O y, O z ) is defied lie this :,. he C O x is the ormal to the hit fae, poitig forward ( poitig forward meas that its salar produt with the vetor of the wave is positive ). O y is ormal to O x ad is i the plae otaiig the photo. ( O x O y ) O x ad the wave vetor of, build the iidet plae. additio, O y is hose i suh a way that the salar produt of O y with the vetor of the photo is positive. O z is ormal to O x ad O y. ( O x, O y, O z ) build a right orthoormal system. We a proeed with the same type of reasoig as i the preedig page. Let N be the rotatio matrix whih overts the ( Ox, Oy, Oz) axis to the ( O x O y, O z ), axis : O x NOx O y NOy Ox Oy Oz O z NOz N is the matrix whih gives the ( O x O y O z ), a vetor expressed i the the world oordiate system ( Ox Oy, Oz) expressed i the idee Coordiate ystem ( O x O y, O z ), axis i the world oordiate system. f E is, ad E is the same vetor,, the : E N E ( ad ot N E E! ). his is the well ow fat that vetors are otravariat if axis vetor are ovariat. ystem ( Ox, Oy, Oz) system ( O x, O y O z ), Maxwell equatio rasform ε D E D ε E ε E N E D N D E N E D N D ε Maxwell equatio rewritte N D ε ε N E > D ε N N E ε
ε N ε N his implies that owig the dieleti tesor ε i the world oordiate system ( Ox, Oy, Oz), we get the dieletri tesor i C oordiate system ( O x O y, O z ), through : ε N ε N. Why have we got ε N ε N, although we got ε M M i the last page? he trasposed symbol is ot at the same plae??? t is for the followig reaso : the preedig page, the origi oordiate system was the system of the optial axis ' ' O x, O y, O z ' ad the destiatio oordiate system was the world oordiate system ( Ox Oy, Oz),. ad the relatio O x ' M O x ( origi expressed as a futio of destiatio! ). Now the origi oordiate system is the world oordiate system ( Ox Oy, Oz) destiatio oordiate system is the C oordiate system ( O x O y, O z ) ε D, ad the, ad the relatio is O x NOx ( destiatio expressed as a futio of the origi ). his is the reaso of the iversio of!
Chapter : rasitio of light betwee a aisotropi medium ad a isotropi medium.. - Maxwell equatio () ll alulatios for a trasitio betwee media are doe i the "idet Coordiate ystem" C. he iidet oordiate system is defied lie this : Ox is the ormal to the hit fae, poitig outside ( poitig outside meas that its salar produt with the vetor of the wave is positiv ). Oy is ormal to Ox ad is i the plae otaiig Ox ad the wave vetor of the photo. ( Ox, Oy) build the iidet plae. additio, Oy is hose i suh a way that the salar produt of Oy with the vetor of the photo is positive. Oz is ormal to Ox ad Oy. ( Ox Oy, Oz), build a right orthoormal system. We wat to fid the solutio for the trasitio from aisotropi medium towards isotropi medium by expressig iidet, st ad d refleted, ad trasmitted waves by plae waves, lie this : ( ) () iidet wave ( t, x) exp i( ω t x y) E () st refleted wave ( t, x) exp i( ω t x y) E ( ) () d refleted wave ( t, x) exp i( ω t x y) E ( ) () trasmitted wave ( t, x) exp i( ω t x y) E ( ) is a diret equee of the defiitio of the idet Coordiate ystem C. he fat is a equee of ad the Maxwell equatios. his allows us to write the uit vetors κ i the followig way : θ -θ -θ θ (5) κ siθ κ siθ κ siθ κ siθ θ is the agle of iidee
5 θ is the agle of refletio of the st refleted wave θ is the agle of refletio of the d refleted wave θ is the agle of trasmissio of the trasmitted wave he mius sig i frot of for the refleted waves is due to the fat that the agles of refletio are volutarily defied otrary to the trigoometri sese, i order to have θ θ i ase of isotropi medium. () ad (5) implies that we have, for the vetors : (6) ω ω [,si,] ω [,si,] ϑ [,si,] ω [,si,] ϑ ϑ ϑ ϑ ϑ ϑ ϑ is the idex of refratio assoiated with the iidet wave is the idex of refratio assoiated with the st refleted wave is the idex of refratio assoiated with the d refleted wave is the idex of refratio assoiated with the outgoig isotropi medium he reaso why the fator betwee ad κ is ω/ has already bee explaied i (). fat, (6) a be idered as a defiitio for,,. eall that i a aisotropi medium, the idex of refratio is depedig upo the diretio of the vetor, κ, suh that although iidet, st ad d refleted waves are i the same medium, they do ot have the same idex of refratio! Let us also itrodue the followig defiitios : (7), Τ mageti permeability of the iidet (aisotropi) ad trasmitted (isotropi) media. We will ow eter followig defiitios for the E fields. eall that these are oly defiitios about the way to represet E fields by mea of agles. hey are itrodued as a mathematial oveiee, without physial sigifiae : (8) [ siδ, δ, si ] [ siδ, δ, si ] [ siδ, δ, si ] [ si,, si ] δ δ Notie the followig remars oerig this agular splittig: O a mathematial poit of view, suh a splittig is always allowed ad it simplifies the alulatios.
6 he agle δ is equal to ϑ (agle of iidee or refletio or refratio ) i isotropi medium. our ase: δ ϑ. We will prove it below. he hoie (looig strage) of the sigs of the ompoets is suh that δ beomes equal to ϑ i isotropi medium. Let us prove immediatly that δ ϑ. Equatio () implies iside the isotropi medium of the trasmitted wave ( ε is the dieletri tat iside the isotropi medium of the destiatio material ) : div D D ε ε E > div E ( exp ) usig () : div ( i( t x y) ) ω > usig (6) siδ ϑ ad (7) : δ siδ > si δ ϑ δ siδ > δ ϑ Quod erat demostradum ie δ ϑ i isotropi medium, δ is lose to ϑ i aisotropi medium. We itrodue the defiitio of the followig parameters, whih are all lose to : (9) η siϑ siδ ϑ δ η siϑ siδ ϑ δ η siϑ siδ ϑ δ he fat that there are refleted waves is a equee of the aisotropy of the iidet medium. his will be prove by the fat that we will be able to fid a orret solutio, whih would ot have happe if we had started with oly oe refleted wave. t follows from the defiitio of the idet Coordiate ystem C that the trasitio betwee the media ours alog the Ox axis. o it is alog the Ox axis that the derivatives will beome ifiite due to the brutal hage of medium. o it is alog the Ox axis that we have to avoid ifiities iside the Maxwell equatios. he method of Feyma ists i looig oly at what happes alog the Ox axis. Let us write the Maxwell equatio () ito ompoets ad loo oly at the derivatives alog Ox. d d ( ε j E j) ( ε j E j) ( ε j E j) d ( ) j j j dx dy dz d dx ( j E j) j looig oly at d/dx : j ε > () ε j j E tat
7 () implies that the quatity j ε j E j just before x is equal to the same quatity j ε j E j just after x. Just before x, the wave is omposed of the iidet wave ad the refleted waves ad. Just after the itersetio, the wave is omposed of the trasmitted wave. o, taig ()-() at the poit x, we have : ( ε is the dieletri tat iside the isotropi medium of the destiatio material ) j ( exp( i( ω t y) )) ( exp( i( ω t y) )) ε j j j ( exp( i( ω t y) )) exp( i( ω t y) ) ε j j implifyig with exp ( iω t) : j j ε j j ε ( exp( i y) ) ( exp( i y) ) ( exp( i y) ) ε j ε exp j ( i y) j ε j j t is impossible to satisfy this equatio, valid for all possible values of y, without puttig : () i whih ase it get simplified ito : () ε ( j j j) ε j j Let us use (6) i (): () si ϑ siϑ siϑ siϑ geeralizatio of Fresel equatios Notie that usig ompoet of equatio (6), ( u ( κ, u) κ ) u, ad, we have : j ε j j, applied to the waves ompoet () of (6) for ompoet () of (6) for ompoet () of (6) ( ( ) ) ε j j κ κ κ κ j ( si ( si si ) ) ε j j δ ϑ δ ϑ δ ϑ j ε j j siϑ ϑ δ j ( si si ) ϑ δ
8 for ompoet () of (6) for ompoet () of (6) for ε this allows us to rewrite () : ε j j siϑ ϑ δ j ( si si ) ε j j siϑ ϑ δ j ϑ δ ( si si ) ε siϑ ε siϑ siϑ ( siϑ siδ ϑ δ ) siϑ ( siϑ siδ ϑ δ ) siϑ ( siϑ siδ ϑ δ ) ϑ δ simplifyig usig () : usig (7) ad (9) ε ( siϑ siδ ϑ δ ) ( siϑ siδ ϑ δ ) ( siϑ siδ ϑ δ ) ( siϑ siδ ϑ δ ) ( siϑ siδ ϑ δ ) ( siϑ siδ ϑ δ ) () η η η
9 We have obtaied all this usig oly Maxwell equatio ()! eall here our mai result : () si ϑ siϑ siϑ siϑ () η η η Now we have to loo at the other Maxwell equatios!. - Maxwell equatios (/) Let us ow tur to Maxwell equatio () ad otiue to use the Feyma method istig i looig oly at the derivatives alog x. Maxwell equatio () i ompoets gives : d E dy d E d othig dz dt d E dz d E d > d E dx dt dx E left E right () d E dx d E dy d dt d E dx E left E right () First lie gives othig : o derivative alog x. eod lie implies that E just before x is equal to E just after x ad the third lie implies the same for E. Let us ow use the agular deompositio of the E fields itrodued previously : () si si si si () δ δ δ δ Let us ow tur to Maxwell equatio () i ompoet : d () d d d div dx dy dz dx eall equatio () for the field : E. We write it ito ompoets, ot ω forgettig that, by defiitio of the C oordiate system!
( ),, ω ( ),, ω ( ) ( ),,,, ω ω ut reall that we have prove i () that : > (5) ( ),, ω ( ),, ω ( ) ( ),,,, ω ω o that equatio () beomes : () ()! We see that Maxwell equatio () brigs othig ew, we obtai equatio () a seod time!. - Maxwell equatios () We tur ow to Maxwell equatio (): dt dd H rot. (), (6) H ε > dt dd rot ε ompoets : ( We must be areful that ( ) for x< is ot equal to for x> ( ) dt D d dz d dy d ε gives othig, o x derivative ( ) ) (6 dx d dt D d dx d dz d ε
ε d dx d dy d D dt d dx ( ) (5) Let us itrodue the values for the field ompoets foud i (5), the the values for the ompoets of the vetor foud i (6), the the values obtaied for the ompoets of the eletri fields i (8), the the defiitios of the tats i (7), the the defiitios of the η tats i (9), ad fially the fat that δ θ, beause the medium of the trasmitted wave is isotropi. (5) ( ) (5) ( ϑ ϑ ϑ ) ϑ (5) ( ϑ si ϑ si ϑ si ) ϑ si (5) ϑ si ϑ si ϑ si ϑ si (6) ( ) ( ) (6) (6) (6) (6) ( ϑ siϑ ϑ siϑ ϑ siϑ ) ( si ) ϑ ϑ siϑ ϑ siϑ ϑ ϑ siϑ ϑ ( ϑ δ siϑ siδ ) ( ϑ δ siϑ siδ ) ( ϑ δ siϑ siδ ) ( si si ) ϑ δ ϑ δ siϑ ( ϑ δ siϑ si δ ) ( ϑ δ siϑ siδ ) ( si si ) ( si si ) ϑ δ ϑ δ ϑ δ ϑ δ (6) η η η We have foud equatio () a seod time, so that Maxwell equatio () oly brigs equatio (5)!. - Equatios for eah wave We must ot forget that we have established also equatio (8), whih is ot a equatio at the trasitio betwee the media, but a equatio whih must be valid for eah wave iside a aisotropi media. o equatio (8) has to be valid for waves, ad! Let us write
equatio (8) with the defiitios of the ompoets of the uit vetor give i (5), ad with the defiitio of the elemets of the eletri fields give i (8): Equatio (8) for the iidet wave ( - κ - ) u- κ κ ( ) u - ( κκ ) u (8) - ( ) - κκ u ( - κ ) u - ( κκ ) u - ( ) - κ κ κκ u ( ) u ( - κ - ) u pplyig (5) : ( ) ( si ) ϑ u ϑ ϑ u u (8) ( si ) ( ) ϑ ϑ u si ϑ u u ( ) u u u pplyig (8) (7) ( ) si ( si ) si si ϑ δ ϑ ϑ δ (8) ( ) si ( ) si siϑ ϑ δ ϑ δ (9) ( ) si siδ δ Let us do the same for the wave : Equatio (8) for the first refleted wave ( - κ - ) u- κ κ ( ) u - ( κκ ) u (8) - ( ) - κκ u ( - κ ) u - ( κκ ) u - ( κκ ) u - ( κ κ ) u ( - κ - ) u pplyig (5) : (8) ( ) si ( si ) ϑ u ϑ ϑ u u ( si ) ( ) ϑ ϑ u ϑ u u
( ) u u u pplyig (8) () ( ) si ( si ) si si ϑ δ ϑ ϑ δ () ( ) si ( ) si siϑ ϑ δ ϑ δ () ( ) si siδ δ he wave gives idetially : Equatio (8) for the first refleted wave () ( ) si ( si ) si si ϑ δ ϑ ϑ δ () ( ) si ( ) si siϑ ϑ δ ϑ δ (5) ( ) si siδ δ Notie that () -> (5) belog to the system of equatios that we are looig for to solve the problem of the trasitio betwee a aisotropi ad a isotropi media, but ot equatios (7) ->(9)! Equatios (7)->(9) are eessary iput oditios that must be satisfied i order that the problem has a solutio. hese eessary iput oditios (7)-(9) are very welome as a he that the program Litrai wors well. f suddely, after may proesses, a photo has a iidet wave that does ot satisfy (7)-(9), it would be a proof that some alulatio was wrog. his he is welome ad is used i Litrai..5 - ystem of equatios for aisotropi to isotropi t is time ow to group iside a table everythig foud util ow about the equatios to be solved for the trasitio aisotrop to isotrop. We umber all equatios from to 7, givig always i the st olum poiters to the plae where the relevat equatio has bee derived. Equatios for the trasitio from aisotropi to isotropi () (7) ( ) ( )
() (7) ( ) ( ) () (9) η si ϑ siδ ϑ δ () (9) η si ϑ siδ ϑ δ (5) () si ϑ siϑ (6) () si ϑ siϑ (7) () si ϑ siϑ (8) ( ) si ( si ) si () si ϑ δ ϑ ϑ δ (9) () ( ) si ( ) si siϑ ϑ δ ϑ δ () () ( ) si siδ δ () ( ) si ( si ) si () si ϑ δ ϑ ϑ δ () () ( ) si ( ) si siϑ ϑ δ ϑ δ () (5) ( ) si siδ δ () () η η η (5) () si si si si
5 (6) () δ δ δ δ (7) (5) ϑ si ϑ si ϑ si ϑ si t is a o liear system of 7 equatios with 7 uows! d despite of this, it a be solved!!! Notie that equatios (8),(9) ad () a be writte i the form of a system a liear homogeeous equatio with uow : x. he same thig a be said of equatios (),() ad (). his fat is essetial for fidig the solutio of the system of equatios. t is easy ad iterestig to show that equatios ()-(7) give the usual equatios for the trasitio betwee isotropi media if the ε dieletri tesor ε is replaed by the ε dieletri tat ε. We leave this as a exerise for the reader. Let us also mae lear what are the iput oditios ad what are the uow variables to be foud put variables or oditios: defied at (8) defied at (6) ϑ defied at (5) δ defied at (8) defied at (8) defied at (7) η defied at (9) defied at (7) defied at (7) defied at (6) defied at (6) t is required i additio that betwee these iput variables, oditios (7), (8) ad (9) hold. here is a solutio oly if these oditios hold betwee the iput variables. 7 uow variables to be foud usig equatios ()-(7) defied at (8) defied at (6) ϑ defied at (5) δ defied at (8) defied at (8) defied at (7) η defied at (9) defied at (8) defied at (6) ϑ defied at (5) δ defied at (8) defied at (8) defied at (7) η defied at (9)
6 defied at (8) ϑ defied at (5) defied at (8).6 - olutios for ad he system of equatios ()->(7) must be taled by looig first at equatios (8)- >() ad ()->(). Let us remar that exhagig all variables of the wave with the variables of the wave we get the same system of equatios, whih meas that eah solutio that we fid for the wave is also a possible solutio for the wave. Moreover, as already otied, system of equatios (8)->() ad ()-() is of the form of a system a liear homogeeous equatios with uow: x. uh a system has a o trivial solutio oly if matrix has a ull determiat. We will show ow that puttig this determiat to give possible values for ad hee for. t is lear by the first remar that we have to assig the first value to ad the seod to. Let us alulate that determiat : determiat to be put to si si ϑ ϑ ϑ si ϑ ϑ ϑ ( si ϑ )( ϑ )( ) ( siϑ ϑ ) ( ϑ ) ( ϑ ) ( ) si ( si ) ϑ ϑ Usig (5) ad defiig : si ϑ siϑ we a write : ϑ si ϑ ϑ ( )( )( ) ( ) ( ) ( ) ( )( ( ) ) Let us defie :
7 a b ( ) ( ) (7) ( ) ( ( )) d g z ll the parameters a, b,, d, g depeds upo ow quatities. he oly uow is z. he the equatio to be solved is : (8) az d ( bz ) z g z t is the equatio whih is solved iside the lass LitEqdex i Litrai. his equatio is worth a log disussio, beause it is the most diffiult part of the solutio, ad beause may surprisig thigs happe due to this equatio! Let us mae a list of the iterestig questios arisig: bout the square root i the equatio : y the defiitios (6) we have z g ϑ so that the sig of the square root is the sig of ϑ root is egative, it meas that > 9 ϑ f this square. mpossible, at first sight, beause it seems that ϑ > 9 meas that the wave is goig forward ad ot baward as a ie refleted wave! ut this way of reasoig is wrog, beause it is a fat i aisotropi media that the diretio of the wave, the Poyitig vetor P, is ot idetial with the diretio of the vetor! o that it is perfetly possible that ϑ > 9 with the wave goig baward as a ie refleted wave. First olusio, we have to admit that the square root i the equatio may be positive or egative!
8 bout the umber of solutios of equatio (8) : if we isolate the square root ad elevate both members to the square, we will ed with a equatio of the th order with solutios. ut taig ito aout that the square root may be positive or egative, it meas 8 solutios. mog these 8 solutios, have appeared due to the elevatio to the square of equatio (8). o the umber of solutios of equatio (8) is. What a ugly result! We ow that there are oly physial solutios, refleted waves, but the mathematis give us solutios! fat, if we alulate the Poyitig vetors for the solutios, we fid that oly solutios are goig baward as a refleted wave has to, the other solutios are goig forward ad so are meaigless ad have to be rejeted. he bad thig is that we a ow whih solutio is good ad whih is bad oly after havig alulated the Poyitig vetors, whih is a quite aoyig "feature" whih slows dow the program Litrai. t is lear that the majority of the time, good solutios have ϑ > ad bad solutios have ϑ <, but as already stated, there are exeptios! fat, all ombiatios are possible, (,) (,), (,), (,), (,), where the first umber is the umber of solutios with θ > ad the seod umber the umber of solutios with θ <! s the refleted agle equal to the iidee agle? Equatios (7)->(9) for the iidet wave give the same equatio (8) for as the equatios ()->(5) have give for ad s a equee,. is amog the solutios offered by equatio (8)! ut othig a isure you that is amog the good solutios of (8)! Here we have i equee to do a Normad's respose : sometimes we have ( or ), but ot i geeral. f we have, the, beause of (5), we have ϑ ϑ. he orret aswer here is : the refleted agle is i geeral ot equal to the iidet agle, but it may happe. Evaeset waves of the first id. ometimes, the followig very strage behaviour ours : he value uder the square root, (z - g), is egative, meaig that is pure imagiary, but the solutio z is real ad equatio (8) is perfetly satisfied! How is it possible? How is the imagiary part aelled? he imagiary part is aelled beause it appears that the fator i frot of the square root, (b z ), is, givig z -/b ad that this value of z miraulously satisfies the remaiig equatio z a z d! t meas that the oeffiiets of the equatio (8) satisfy i that ase the relatio - a b d b. he fat that ϑ is pure imagiary ad that is real is i itself ot surprisig. fter all, i the ase of total refletio betwee two isotropi media, we also have for the trasmitted wave a ϑ pure imagiary with a idex real. What is ew here is that it appears for a refleted wave, ot for the trasmitted wave, ad also that it requires the strage degeeray - a b d b betwee the oeffiiets of equatio (8). We deide to ame this ase ( ϑ pure imagiary ad is real ) evaeset wave of the first id. Notie that it ever happes that both refleted waves ad are evaeset of the first id. Oly oe of the refleted wave may be evaeset of the first id ad whe it happes we hoose to assig to wave the evaeset solutio. Notie also that whe wave is evaeset of the first id, the alulatio shows that wave arries eergy, exatly as the trasmitted wave, i ase of total refletio betwee two isotropi media arries zero eergy. Evaeset wave of the seod id. We will all evaeset waves of the seod id the ase of a wave for whih ot oly is omplex but also. Does it happe? ϑ ϑ
9 he aswer is yes, but with the followig restritios : beause all oeffiiets of equatio (8), a, b,, d, g are always real by defiitio, it is easy to show that if z is a omplex solutio of (8), the z ( the omplex ojugate of z ) is also a solutio. Here agai, the physis requires that at least oe refleted wave is ot evaeset, so that whe it happes, we will assig the evaesee to the wave as before. Notie also the followig surprise: although the idex is omplex, the alulatio shows that a evaeset wave of the seod id does ot arry ay eergy! t is surprisig i the sese that we ow, i the ase of the refletio from a isotropi medium to a isotropi revetmet with a omplex idex, the ase of a evaeset trasmitted wave arryig eergy! he ase of a wave with a omplex idex but arryig zero eergy is a etirely ew feature of waves i aisotropi media. Positiv irefrigee ε a ε b < ε. eall the defiitio give previously of the ids of birefrigee. We will ivestigate ow the ase of the positive birefrigee, where ε a ε b < ε. that ase, it a be show that if the solutios are all real, the the solutios havig the smallest value of z are equal (degeerate), so that i that ase there are, ot solutios. Negative irefrigee ε a < ε b ε. that ase, it a be show that if the solutios are all real, the the solutios havig the biggest value of z are equal (degeerate), so that i that ase there are, ot solutios. Probably you are ovied ow that equatio (8) is full of surprises! the ext page, we will ivestigate the speial ases of equatio (8) whe by hae the dieletri tesor ε is diagoal i the C oordiate system..7 - peial ases for equatio (8) We mae here a digressio aroud equatio (8) ad we will retur to the solutio of the system of equatio ()-(7) later. Let Oxyz be the iidet oordiate system, Ox y z be the system i whih the dieletri tesor ε is diagoal, with eigevalues ε a, ε b ad ε. We have alled the Ox y z system "the system of the optial axis". y ovetio, ε a < ε b < ε. y ovetio also, ε a is assoiated with the Ox ' axis, ε b with the Oy ' axis ad ε with the Oz ' axis. Let M be the rotatio matrix whih overts the Ox, Oy ad Oz axis to the ' ' ' Ox, Oy ad Oz axis : Ox M Ox Oy M Oy Ox Oy Oz Oz ' M Oz
We have already show that if ε is the dieletri tesor i C ad ε D i the system of the optial axis, the we have : the dieletri tesor ε M ε D M We will examie here the 6 speial ases of equatio (8) whe the dieletri tesor ε appears to be diagoal i the C oordiate system i whih we are doig all the alulatios. Let us remid the followig defiitios iside the C oordiate system : "the trasitio plae" is the plae ( Oz ) Oy, orthogoal to Ox, where the trasitio betwee the media is eoutered. "the iidet plae" is the plae ( Ox, Oy ) otaiig the ormal to the trasitio plae Ox, ad the vetor of the iidet wave. Case ab Ox ' ' ' Ox Oy Oy Oz Oz ε ε a ε b ε M Let us examie the speial ase of equatio (8) where the system of the optial axis is idetial with the C oordiate system i whih we are doig all the alulatios. f the system of the optial axis is idetial with the C oordiate system, it meas that : ε a ε b ε ij for i j Negative birefrigee ε < ε b a ε Positive birefrigee ε a ε b < ε he optial axis is Ox orthogoal to the trasitio plae, iside the iidet plae he optial axis is Oz, iside the trasitio plae, orthogoal to the iidet plae Let us realulate all the oeffiiets of equatio (8) : (7) a ( ) ( ) ε a ε b ε a ε b ε a ε b
d ε ε b ( ) ε a ε a ε aε bε g z Equatio (8) turs ito : (8) z az d his gives, after a legthy but trivial alulatio that we leave to the reader : ase total aisotropy egative birefrigee wave "ordiary wave" ε ε wave "extraordiary wave" ε b ε ( ) ε b ε a ε a ( ) ε ε a ε a positive birefrigee a ε ε We see that i that ase, the wave a be qualified "ordiary" beause it does ot deped upo the vetor (through ). he wave is "extraordiary", beause its idex depeds upo, exept i the ase of positive birefrigee, where aels. t is trivial ow to alulate the 5 other ases. We give just the results : Case ab Ox ' ' ' Ox Oy Oz Oz Oy ε ε a ε ε b M -
ε a ε ε b wave "ordiary wave" wave "extraordiary wave" otal aisotropy ε a < ε b < ε the optial axis has o meaig ε b ε ( ) ε ε a ε a Negative birefrigee ε a < ε b ε the optial axis is Ox orthogoal to the trasitio plae, iside the iidet plae ε ε ( ) ε ε a ε a Positive birefrigee ε a ε b < ε the optial axis is Oy iside the trasitio plae, iside the iidet plae ε a ε ( ) ε ε a ε a Case ba Ox ' ' ' Oy Oy Ox Oz Oz ε ε b ε a ε M - ε b ε a ε wave "ordiary wave" wave "extraordiary wave" otal aisotropy ε a < ε b < ε the optial axis has o meaig ε ε a ( ) ε a ε b ε b Negative birefrigee ε a < ε b ε the optial axis is Oy iside the trasitio plae, iside the iidet plae ε ε a ( ) ε ε a ε
Positive birefrigee ε a ε b < ε the optial axis is Oz iside the trasitio plae, orthogoal to the iidet plae a ε ε Case ba Ox ' ' ' Oy Oy Oz Oz Ox ε ε b ε ε a M ε b ε ε a wave "ordiary wave" wave "extraordiary wave" otal aisotropy ε a < ε b < ε the optial axis has o meaig ε a ε ( ) ε ε b ε b Negative birefrigee ε a < ε b ε the optial axis is Oz iside the trasitio plae, orthogoal to the iidet plae a ε ε Positive birefrigee ε a ε b < ε the optial axis is Oy iside the trasitio plae, iside the iidet plae ε a ε ( ) ε ε a ε a Case 5 ab Ox ' ' ' Oz Oy Ox Oz Oy ε ε ε a ε b M
ε ε a ε b wave "ordiary wave" wave "extraordiary wave" otal aisotropy ε a < ε b < ε the optial axis has o meaig ε b ε a ( ) ε a ε ε Negative birefrigee ε a < ε b ε the optial axis is Oy iside the trasitio plae, iside the iidet plae ε ε a ( ) ε a ε ε Positive birefrigee ε a ε b < ε the optial axis is Ox orthogoal to the trasitio plae, iside the iidet plae ε a ε a ( ) ε a ε ε Case 6 ba Ox ' ' ' Oz Oy Oy Oz Ox ε ε ε b ε a M - ε ε b ε a wave "ordiary wave" wave "extraordiary wave" otal aisotropy ε a < ε b < ε the optial axis has o meaig ε a ε b ( ) ε b ε ε Negative birefrigee ε a < ε b ε the optial axis is Oz iside the trasitio plae, orthogoal to the iidet plae a ε ε
5 Positive birefrigee ε a ε b < ε the optial axis is Ox orthogoal to the trasitio plae, iside the iidet plae ε a ε a ( ) ε a ε ε hese 6 ases are extremely useful whe it omes to measurig the oeffiiets ε a, ε b, ε by a measuremet of the idex of refratios usig equatios (5) ad (6). For istae : for a totally aisotropi rystal, you a obtai ε from ase, ε b from ase ad ε a from ase looig eah time at. for a egative birefriget rystal you a obtai ε from ase ad ε a from ase looig eah time at. for a positive birefriget rystal you a obtai ε from ase ad ε a from ase looig eah time at. We retur ow to the solutios of the system ()-(7)..8 Fial solutio of ()-(7) Notie first that fidig ϑ is a trivial matter usig equatio (7). Now, havig foud ad thas to equatio (8), we get immediately ϑ thas to equatio (5)ad ϑ thas to equatio (6). s already otied, whe talig system of equatios (8)-() ad ()- (), it has the form of a system a liear homogeeous equatios with uows quatities x. he matrix otais oly ad ϑ for the system (8)- (), ad oly ad ϑ for the system ()-(), so that both matries are ow. he vetor x i the system x δ δ, for ()- (). his is exatly the umber of uows whih is expeted, sie matrix has a ull determiat ad is degeerate. Fidig ( δ, ) ad ( δ, ) beomes a trivial problem. ie we have see i the solutio of (8) that ad ϑ may be omplex, it is lear that it is also the ase for ( δ, ). ad a be obtaied from () ad ( sie ad are ow. η ad η a be obtaied from () ad () sie ϑ, δ, ϑ, δ, are ow. Let us repeat the table already preseted idiatig all the iput variables ad all the variables to be foud. otais exatly uows : (, ) for (8)- () ad ( ) iput variables or oditios Ι ϑ δ Ι Ι Ι η Ι (7) 7 uow variables to be foud for 7 equatios ()-(7) ϑ δ η ϑ δ η θ Τ
6 Oly,, ad Τ remai to be foud, whih is easy usig equatios ()-(7). We rewrite these equatios slightly differetly : () η η η (6) δ δ δ ϑ (5) si si si si (7) si ϑ si ϑ si ϑ si ϑ ubtratig (6) from () ad (7) from (5) leave us with a liear system of equatios with uows : ad. o ad are foud. eturig to () ad (5) with these values of ad allow us the to fid ad! he system has bee solved!
7 Chapter : rasitio of light betwee a isotropi ad a aisotropi medium. - Maxwell equatios () Let us ow loo at the symmetri problem of the trasitio betwee a isotropi medium towards a aisotropi medium. s the alulatios will be very similar to the previous ase, we will be shorter i the explaatios. s before, we do all the alulatios i the C [idee Coordiate ystem]. We wat to fid the solutio for the trasitio from isotropi medium towards aisotropi medium by expressig iidet, refleted, st ad d trasmitted waves by plae waves, lie this : (5) iidet wave ( t, x) exp( i( ω t x y) ) E (5) refleted wave ( t, x) exp( i( ω t x y) ) E (5) st trasmitted wave ( t, x) exp( i( ω t x y) ) E (5) d refleted wave ( t, x) exp( i( ω t x y) ) E is a diret equee of the defiitio of the idet Coordiate ystem C. is a equee of ad the Maxwell equatios. his allows us to write the uit vetors κ i the followig way : ϑ ϑ ϑ ϑ (55) κ si ϑ κ si ϑ κ si ϑ κ si ϑ ϑ is the agle of iidee ϑ is the agle of refletio ϑ is the agle of trasmissio of the st trasmitted wave ϑ is the agle of trasmissio of the d trasmitted wave he mius sig i frot of for the refleted waves is due to the fat that the agles of refletio are volutarily defied otrary to the trigoometri sese, i order to have ϑ ϑ i ase of isotropi medium. () ad (5) implies that we have, for the vetors :
8 ϑ ϑ ϑ ϑ (56) ω si ϑ ω si ϑ ω si ϑ ω si ϑ is the idex of refratio i the igoig isotropi medium beause igoig medium isotropi ad () is the idex of refratio assoiated with the st trasmitted wave is the idex of refratio assoiated with the d trasmitted wave Let us also itrodue the followig defiitios : (, mageti permeability of the iidet ad trasmitted media ) (57) We will ow eter followig defiitios for the E fields. eall that these are oly defiitios about the way to represet E fields by mea of agles. hey are itrodued as a mathematial oveiee, without physial sigifiae : (58) [ siδ, δ, si ] [ siδ, δ, si ] [ siδ, δ, si ] [ si,, si ] δ δ Notie the followig remars oerig this agular deompositio : O a mathematial poit of view, suh a deompositio is always allowed ad this deompositio simplify the alulatios. he agle δ is equal to ϑ (agle of iidee or refletio or refratio ) i isotropi medium. our ase: δ ϑ ad δ ϑ. We have already prove it i. he hoie (looig strage) of the sigs of the ompoets is suh that δ beomes equal to ϑ i isotropi medium. ie δ ϑ i isotropi medium, δ is lose to ϑ i aisotropi medium. We itrodue the defiitio of the followig parameters, whih are all lose to : (59) η si ϑ siδ ϑ δ η si ϑ siδ ϑ δ he fat that there are trasmitted waves is a equee of the aisotropy of the outgoig medium. his will be prove by the fat that we will be able to fid a orret solutio, whih
9 would ot have happe if we had started with oly oe trasmitted wave. t follows from the defiitio of the idet Coordiate ystem C that the trasitio betwee the media ours alog the Ox axis. o it is alog the Ox axis that the derivatives will beome ifiite due to the brutal hage of medium. o it is alog the Ox axis that we have to avoid ifiities iside the Maxwell equatios. he method of Feyma ists i looig oly at what happes alog the Ox axis. Let us write the Maxwell equatio () ito ompoets ad loo oly at the derivatives alog Ox. d d d () j ε j E j j ε j E j j ε j E j dx dy dz d looig oly at : dx d dx j ε j E j > (6) ε E j j tat (6) implies that the quatity ε j E j just before x is equal to the same quatity just after x. Just before x, the wave is omposed of the iidet wave ad the refleted waves. Just after the itersetio, the wave is omposed of the trasmitted wave ad. o, taig (5)-(5), at the poit x, we have : ( ε is the dieletri tat iside the isotropi medium of the i material ) ε simplifyig with exp(iωt) : ( i y) exp( i y) exp( i y) ( i y) exp ε ε ε exp j j j j j j t is impossible to satisfy this equatio, valid for all possible values of y, without puttig : (6) i whih ase it get simplified ito : (6) ε ε ε j j ε j j Let us use (56) i (6) to get a geeralizatio of Fresel equatios (6) si ϑ si ϑ si ϑ siϑ j j ie, we have ϑ ϑ
Notie that usig ompoet of equatio (6), ( u ( κ, u) κ ) u ad, we have :, applied to the waves Comp. () of (6) for ε j j j ( ( ) ) κ κ κ κ usig (55) ad (58) Comp. () of (6) for so that (6) beomes usig (6) ad δ ϑ for ad ε j j j ε j j j η siϑ η siϑ ( si si ) δ η siϑ ( ) δ η siϑ η η (6) η η We have obtaied all this usig oly Maxwell equatio ()! eall here our mai results : (6) ϑ ϑ ad si ϑ siϑ siϑ (6) η η Now we have to loo at the other Maxwell equatios!. - Maxwell equatios (/) Let us ow tur to Maxwell equatio () ad otiue to use the Feyma method istig i looig oly at the derivatives alog x. Maxwell equatio () i ompoets gives :
d E dy d E d othig dz dt d E dz d E d > d E dx dt dx E left Eright (6) d E dx d E dy d dt d E dx E left E right (6) First lie gives othig : o derivative alog x. eod lie implies that E just before x is equal to E just after x ad the third lie implies the same for E. Let us ow use the agular deompositio of the E fields itrodued previously : (6) si si si si (6) ϑ ( ) δ δ gai Maxwell equatio () brigs othig ew. We do ot prove it agai.. - Maxwell equatios () We tur ow to Maxwell equatio () : rot H d D dt (), (6) H ε > ε rot d D dt ompoets : ( We must be areful that [ ] for x< is ot equal to [ ] for x>. ε d dy d d D dz dt gives othig, o x derivative ε d dz ε d dx d d D dx dt d d D dy dt d dx d dx ( ) ( ) (66) ( ) ( ) (65)
eall equatio () for the field : ( ) E ω. We write it ito ompoets, ot forgettig that, by defiitio of the C oordiate system! [ ] [ ] [ ] [ ],,,,,,,, ω ω ω ω ut reall that we have prove i (6) that : > [ ] [ ] [ ] [ ],,,,,,,, ω ω ω ω Let us itrodue the values for the field foud here, the the values for the ompoets of the vetor foud i (56), the the values obtaied for the ompoets of the eletri fields i (58), the the defiitios of the tats i (57), the the defiitios of the η tats i (59), ad fially the fat that δ θ δ θ, beause the medium of the refleted wave is isotropi. (65) ( ) ( ) (65) ( ) ( ) ϑ ϑ ϑ ϑ (65) ( ) ( ) ϑ ϑ ϑ ϑ si si si si (65) ( ) ϑ ϑ ϑ si si si si (66) ( ) ( ) (66) ( ) ( ) si si si si ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ (66) δ ϑ δ ϑ δ ϑ δ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ si si si si si si si si
(66) ( ) ( ( si si ) ( si si )) ϑ δ ϑ δ ϑ δ ϑ δ (66) η η (66) η η (66) We have foud equatio (6) a seod time, so that Maxwell equatio () oly brigs equatio (65)!. - Equatios for eah trasmitted wave We must ot forget that we have established also equatio (8), whih is ot a equatio at the trasitio betwee the media, but a equatio whih must be valid for eah wave iside a aisotropi media. o equatio (8) has to be valid for waves ad! Let us write equatio (8) with the defiitios of the ompoets of the uit vetor give i (55), ad with the defiitio of the elemets of the eletri fields give i (58): Equatio (8) for the trasmitted wave ( ) ( ) ( ) κ u κκ u κκ u (8) ( ) ( ) ( ) κ κ u κ u κ κ u ( ) ( ) ( ) κ κ u κ κ u Usig (56) : κ ( ) ( si ) ϑ u ϑ ϑ u u u (8) ( si ) ( ) ϑ ϑ u si ϑ u u ( ) u u u Usig (58) (67) ( ) si ( si ) si si ϑ δ ϑ ϑ δ (68) ( ) si ( ) si siϑ ϑ δ ϑ δ
(69) ( ) si siδ δ t is lear that we get the same for the wave : (7) ( ) si ( si ) si si ϑ δ ϑ ϑ δ (7) ( ) si ( ) si siϑ ϑ δ ϑ δ (7) ( ) si siδ δ.5 - ystem of equatios for isotropi -> aisotropi t is time ow to group iside a table everythig foud util ow about the equatios to be solved for the trasitio isotropi to aisotropi. We umber all equatios from () to (6), givig always i the st olum poiters to the plae where the relevat equatio has bee derived. Equatios for the trasitio from isotrop to aisotrop 57 57 59 η si ϑ siδ ϑ δ 59 η si ϑ siδ ϑ δ 5 6 6 6 si ϑ si ϑ siϑ siϑ 7 67 ( ) si ( si ) si si ϑ δ ϑ ϑ δ
5 8 68 ( ) si ( ) si siϑ ϑ δ ϑ δ 9 69 ( ) si si δ δ 7 ( ) si ( si ) si si ϑ δ ϑ ϑ δ 7 ( ) si ( ) si siϑ ϑ δ ϑ δ 7 ( ) si siδ δ 6 η η 6 si si si si 5 6 ϑ ( ) δ δ si si si 6 65 ϑ ( ) ϑ ϑ si t is a o liear system of 6 equatios with 6 uows! d despite of this, it a be solved!!! Notie that equatios (7),(8) ad (9) a be writte i the form of a system a liear homogeeous equatio with uow : x. he same thig a be said of equatios (),() ad (). t is easy ad iterestig to show that equatios ()-(6) give the usual equatios for the trasitio betwee isotropi media if the ε dieletri tesor ε is replaed by the ε dieletri tat ε. We leave this as a exerise for the reader. Let us also mae lear what are the iput oditios ad what are the uow variables to be foud iput variables ad oditios defied at (58) defied at (56) ϑ defied at (55) δ ϑ see. defied at (58) defied at (57) defied at (57) (7) (9) ε (6) (56) ϑ ϑ (6) 6 uow variables to be foud for 6 equatios ()-(6) defied at (58) defied at (56) ϑ defied at (55) δ defied at (58)