Optics and radiometric magnitudes: are their connections clear?

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1 Optics a aiometic magitues: ae thei coectios clea? M.A. Illaamei, A. Oleaga, J. Zubia, G. Alabaleteu, G. Duaa, a I. Aambuu Depatameto e Física Aplicaa I, Escuela Técica Supeio e Igeieía e Bilbao - Faculty of Egieeig. Uivesia el País Vasco UPV/EHU. Ala Uquijo s/ E-4803 Bilbao (Spai Depatameto e Electóica y Telecomuicacioes, Escuela Técica Supeio e Igeieía e Bilbao - Faculty of Egieeig. Uivesia el País Vasco UPV/EHU. Ala Uquijo s/ E-4803 Bilbao (Spai ABSTRACT The elatios betwee aiometic magitues a quatities associate to optical popeties of mateials (pocesses of eflectio, tasmissio a emissio of aiat flux by o though mateial meia have bee aalyze. By stuyig some paticula examples, we illustate the epeece of optical popeties of mateials o the aiometic magitue chose a it is show that quatities obtaie fom a aiometic poit of view iffe mathematically a physically fom the coespoig Optics expessios.. INTRODUCTION Raiomety is a system of laguage, mathematics a istumetatio use to escibe a measue the popagatio of electomagetic (EM aiatio, icluig the effects o eflectio, absoptio, tasmissio a scatteig by mateial substaces. May textboos o electomagetism -7 a optical physics 8-7 aalyze these physical pheomea. I most of them the flux of eegy associate to electomagetic aiatio is escibe i tems of the time aveage of the Poytig vecto. This aveage is elate to the squae of the amplitue of the electic fiel a it is calle itesity o powe esity. Liewise, the eflectace a tasmittace at a iteface sepaatig two iffeet meia o the eflectace a tasmittace of a plae paallel plate ae expesse as a fuctio of the iciet, eflecte a tasmitte amplitues of the electic fiel. The basic cocepts of aiomety ae itouce i most uegauate optics textboos 9,4,7. Howeve, these texts o ot explai i aequate etail the elatio betwee the itesity a the aiometic magitue calle iaiace eithe expess the eflectace a tasmittace (eithe at a iteface o of a plate as a fuctio of the aiometic magitues. Othe example whee the li betwee both views (optical a aiometic is fa fom clea is the popagatio of aiatio though a lossy meium. The empiical law which escibes this behavio is the well-ow expoetial ecay of the aiatio with the istace. EM textboos escibe this ecease of aiatio stuyig the ecay of the amplitue of electic fiel with the tavele istace. Howeve, i Optics boos, thee is a geat ispesio i the magitues use to escibe the expoetial law a it seems that expoetial ecay taes always the same fom egaless of the aiometic magitue. The pupose of this aticle is to cotibute to a bette uestaig i the elatio betwee the optical popeties of mateials a aiometic magitues, payig special attetio to the physical cocepts uelyig the equatios a tyig to claify what is somewhat messe. With that pupose i mi, i the ext sectio aiometic magitues ae biefly itouce. I sectio 3, the efiitios of eflectace a tasmittace at a iteface a the popagatio of a elemetal beam of aiatio immese i a lossy meium ae aalyze as a fuctio of aiometic magitues. I sectio 4 we evelop a example which is fou i most textboos: the optical popeties of plae paallel plates. The eflectace a tasmittace by calculatig the powe fluxes at each iteface of the plate have bee obtaie a compae with the coespoig optical expessios fou i optics textboos. Fially, we e with the coclusios.

2 . REMARKS ON RADIOMETRIC MAGNITUDES Let us begi egaig the picipal magitues use i aiomety. They ae isplaye i Table I. The meaigs of most of the quatities ae show by thei efiig equatios. Table I. Raiometic magitues Symbol Defiig equatio Raiometic magitues Name Uit Φ Raiat powe o flux W I Φ I ω ω Φ Raiat itesity W/s M M Φ s s Raiat exitace W/m E E Φ s s Iaiace W/m L L omal Φ Raiace W/sm ay scos ω s ω Usually, the efiitio of the itesity (I as flux pe uit of soli agle, is elate to poit souces. Howeve, the efiitio ca be applie to extee sufaces usig the cocept of aiace (L.The itesity of a ifiitesimal suface s at iectio espect to its omal is efie as: I Lcos s ( whee L is the aiace at s. The efiitio of L, state hee fo a souce, is extee tivially fo a etecto a eve fo a ay, at ay poit alog its path. Fo a souce, aiace may vay fom poit to poit, a fo a fixe poit, it may vay as a fuctio of the iectio. Raiace is the most geeal quatity fo escibig the popagatio of aiatio though space. Its impotace stems maily fom the ivaiace theoem that states that, i ay optical system, the aiace alog the path of a ay is ivaiat 8. Iaiace (E is the most impotat quatity fo escibig aiatio iciet o o leavig a suface whe it is ot essetial to escibe the iectioal istibutio of that aiatio i etail. It oes ot iscimiate, fo example, betwee vey collimate aiatio a aiatio that is impigig fom all agles. I oe to tae ito accout the oietatio of the elemetal sufaces i which the aiatio impiges with espect to the

3 iectio of popagatio of the beam, we popose the use of what we have calle pepeicula iaiace, S, which is the aiat flux which cosses a uit aea pepeicula to the iectio of the flow. The efiitio of this pepeicula iaiace woul be: Φ S ( s which matches the efiitio of the time aveage of the Poytig vecto amplitue of the electic fiel. ε S E o whee E 0 is the μ The pepeicula iaiace is equal to the iaiace whe the suface elemet is pepeicula to the iectio of popagatio of the aiatio. The popagatio of the aiatio is fequetly stuie fo wave plaes, that is, it woul coespo to a paallel beam of aiatio. I this case the suface is usually place pepeicula to the iectio of popagatio (0, so thee is o istictio betwee iaiace a pepeicula iaiace a both magitues ae ietical to the aiatio itesity. I this case, some Optics textboos 3,4,7 call coectly the itesity iaiace. The ecessity of the pepeicula iaiace will be fully eveale whe, i the ext sectios, we pocee to evelop the eflectace a tasmittace coefficiets as fuctios of aiometic magitues as well as i the stuy of the popagatio of aiatio i a lossy meium. 3. PROPAGATION THROUGH AN INTERFACE SEPARATING TWO MEDIA AND A LOSSY MEDIUM We examie the situatio whee a beam of aiatio passes though a smooth suface sepaatig two meia with iffeet efactive iices ( a. The geometic situatio is show i figue. We cosie the extemely thi suface egio of a pefectly smooth homogeeous a isotopic ielectic mateial. This iteface is too thi to absob sigificat quatities of the aiatio iciet o it. The aiatio iciet upo the iteface is split ito two pats: some is eflecte a the est is tasmitte. The agles of iciece a eflectio ( i a ae ietical ue to cosieig a specula eflectio. The cosevatio of eegy at the iteface implies: Φ i Φ + Φ t (3 whee is the elemet of the iciet flux o the aea s, Φ is the elemet of the eflecte flux a Φ t is the elemet of the tasmitte flux. The efiitios of eflectace a tasmittace fo iciet aiatio of a give spectal compositio, polaizatio a geometical istibutio ae the atios of the eflecte o tasmitte flux to iciet aiat flux: Φ a Φ τ t (4 Meium Meium ω i Φ i L i i t L t L Φ t ω t Φ ω Figue. Reflectio a tasmissio of a iciet beam iaiatig a elemetal suface s at the iteface. The agles of iciece a eflectio ae i a ( i a t is the agle of efactio. Let s wite them as fuctio of the iffeet aiometic magitues. By applyig the efiitio of the coespoig aiometic magitue a pefomig simple geometical a mathematical opeatios, we obtai the expessios of eflectace a tasmittace at a iteface as a fuctio of aiometic 3

4 magitues. The obtaie expessios have bee epicte i Table II. Note that eflectace is always the atio of the eflecte quatity to the iciet oe. O the cotay, tasmittace expessio chages with the aiometic magitue. Simila elatios to the expessios obtaie fo pepeicula iaiace have bee fou i some texts 4,7 but they simply call S iaiace istea of pepeicula iaiace, what is a bit misleaig. This is ot the case of the text by Bo 9, whee eflectace a tasmittace ae coectly E efie as the atio of iaiaces E, τ t, the witte i tems of S, a fially, i tems of the Ei Ei electic fiels amplitues. Although i the e, i all cases a τ ae expesse i tems of the amplitues, authos efie them as atios of iffeet magitues. Theefoe, it becomes absolutely ecessay to use coectly the coespoig magitue i oe to escibe accuately the optical popeties of the mateial. Table II. Reflectace a tasmittace at a iteface as a fuctio of aiometic magitues. Depeece with these quatities of the expoetial ecay of aiatio i a lossy meium. ENERGY CONSERVATION REFLECTANCE TRANSMITTANCE EXPONENTIAL LAW Flux Φ i Φ + Φ t Φ τ Φ t Φ Φ cx e L L i L + Lt L Li Lt τ Li L cx Le I Ii I + It cos i cos t I I i τ I t I i cos i cos t I I (s (s e cx E E i E + Et E Ei E τ t Ei (s cx E E e (s (E Si S + St cost cosi S Si S τ t Si cost cosi (s S cx S e (s Suppose ow we have some aiatio leavig the suface elemet s i the iectio a aothe suface s at x istace eceivig this aiatio flux fom iectio. This is illustate o Figue (a. The flux eteig the soli agle ω a leavig s is Φ. The flux eceive by s is Φ. If the two sufaces ae immese i lossy meium whose absoptio of light esults fom liea espose, the flux falls off expoetially with iceasig the istace tavelle i the meium: Φ cx Φe (5 4

5 hee c is the atteuatio coefficiet which we suppose costat. This coefficiet is the absoptio coefficiet whe oly absobig effects ae cosiee. If the sufaces wee withi a lossless meium, the flux woul emai costat. By pefomig the appopiate calculatios we ca expess this equatio as a fuctio of the aiometic magitues. The obtaie equatios have bee iclue i Table II. s ω s x Φ s s Φ x ω Φ (a (b (c Figue. (a A aow beam of aiatio that pass though the elemetal aeas s a s. (b Collimate beam popagatig alog the iectio iicate by the aows.(c Poit souce: Raiatio flux cotaie i the soli agle ω, Φ Iω. If oe has a collimate beam of aiatio, that is, a bule of appoximately paallel ays popagatig i the same iectio with the associate flux cotaie i a small but measuable soli agle (Figue (b, the iaiace is usually cosiee o a plae pepeicula to the ay. I this case, (s s (s s, the iaiace a the pepeicula iaiace, ae ietical a the simple expoetial law is satisfie with all aiometic magitues. Fo the case of a poit souce immese i a lossy meium (Figue 3(c, it ca be emostate that simple expoetial law is satisfie if the magitue chose is the itesity of poit souce. If the magitue use is the iaiace o the pepeicula iaiace the obtaie equatio is the familia ivese-squae law of aiatio fom a poit souce with the exceptio that the aiatio is beig atteuatig by the meium. These esults ae summaize i Table III. Table III. Expoetial law of the aiatio popagatig withi a absobig meium applie to the case of a collimate beam of aiatio a a poit souce. R is the aius of the poit souce (x>>r.i 0 is the itesity of the poit souce a E 0 is the iaiace o the suface of the poit souce. Collimate beam Poit souce I I Iexp( cx I(x I0exp( cx E, S E Eexp( cx R E(x E 0exp( cx x To e with this sectio, we woul lie to emphasize the impotace of the iffeet cases we have stuie hee i oe to elate popely the physical situatio with the appopiate aiometic magitue. The ext sectio illustates a applicatio of the equatios we have just woe out (a isplaye i Table II. 4. CASES STUDIES: PLANE PARALLEL PLATE A mateial boue by two paallel itefaces efies a object that ca eflect, tasmit a absob aiatio iciet o it. (Scatteig pocesses ae cosiee egligible. Let s ow obtai the optical popeties of the plate by sepaatig the powe flux at each iteface ito a outgoig compoet a a icomig compoet. The eflectace, tasmittace a absoptace of this object ae efie espectively 5

6 as the factio of flux iciet upo the object that is eflecte, tasmitte a absobe by the object fo efie iectios of iciece a emegece, polaizatio state a wavelegth. Let s cosie a upolaize collimate beam of aiatio of wavelegth λ at the iectio impigig o a plae paallel plate of a homogeeous a isotopic mateial of ow thicess a efactive iex which is suoue by two meia of iex a. The multiple eflectios a tasmissios of the iciet beam ae show i Figue 3. ( L i S i cx 3ττe 4cx 3ττe cx 3τe cx 3τe 3 τ 3 4cx τe τe 4cx e cx τ 3cx 3τe 3 3 e 3cx τ cx 3τe cx τ3e τ cos e 5cx τ 3cx 3ττ3e cx τ τ3e cx τ τ 3 3e 3 3 cx τ τ 3 3 e 5 5cx ττ3e Figue 3. Multiply eflecte a tasmitte beams i a paallel plate. The value of x is give by x/cos. The values of S i iffeet positios of the beam have bee plotte fo iciet S i. The expessios i paethesis ae the values of aiace (L fo iciet L i. The eflectace a tasmittace of the left iteface ae, τ, a, τ epeig o the iectio of the aiatio (fom meium to mateial o fom the mateial to the meium. I the same way, the eflectace a tasmittace of the ight iteface ae eote by 3, τ 3, 3 a τ 3. The efiitios of 6

7 these iteface magitues as a fuctio of iffeet aiometic magitues have bee isplaye i Table II. They ca be etemie by usig Fesel equatios, which give the atio of eflecte (o tasmitte electic fiel amplitue to the iciet oe. Absoptio is cosiee by taig ito accout that the flux of the ay popagatig acoss the mateial will ecease accoig to the expoetial law. 4. Raiatio view Supposig that the iciet flux is Φ, the total eflectace R of the plate fo this situatio will be the atio of the sum of all the fluxes emegig to the left of the iciet flux: ( Φ R (6 whee the summatoy is extee to the total umbe of iteeflectios i the mateial. Taig ito accout that the iectio of eflecte aiatio is the same fo all the emeget beams a that the agles of eflectio a iciece ae equal, the ext elatios ae satisfie: cos cos ω ω ω s cos ωs cos (collimate beam s s. Applyig these elatios the eflectace ca be expesse as a fuctio of iffeet aiometic magitues as follows: ( Φ ( L ωs cos ( L R (7 Liωs cos Li ( Φ ( E s ( E R (8 Eis Ei ( Φ ( S s cos ( S R (9 Sis cos Si Theefoe, the eflectace of the plate ca be efie as the atio of eflecte to iciet aiace o iaiace o pepeicula iaiace. Let s pefom the same calculatios i oe to obtai the tasmittace of the plate. The total tasmitace T of the plate will be the atio of the sum of all the fluxes emegig to the ight of the plae paallel plate: T ( Φ t Now, the iectios of the tasmitte aiatio ae ietical but they ae iffeet to the iciet iectio. The tasmittace expesse as a fuctio of iffeet aiometic magitues will be: ( Φ t ( L t ωs cos T ( Liωs cos I this case ω s cos ωs cos a we ca wite: ωcos ( L t T ( Liω cos Applyig the Sell laws si si, si si a thei iffeetial equatios the expessio fo the tasmittace is euce to: (0 7

8 ( L t T (3 Li The equatios fo the tasmittace as a fuctio of iaiace a pepeicula iaiace will be: ( Φ t ( Et s ( Et T (4 Eis Ei ( Φ t ( St s cos ( St cos T i Sis cos cos (5 Φ Si It ca be obseve that the expessios o ot epe o the mateial efactive iex ( a the iectio of the iteeflectios ( a that they ae quite simila to the expessios fo a iteface. If the plate is suoue by the same meium (i.e.,, the agles of iciece, eflectio a tasmissio ae equal a the eflectace (tasmittace ca be efie as the atio of eflecte (tasmitte to iciet aiometic magitue, o matte which oe is chose. Let s wite the eflectace a tasmittace of the plate (R a T as a fuctio of the eflectace, tasmittace of the itefaces,, τ,, τ, 3, τ 3, 3 a τ 3. Fo that pupose, afte choosig oe aiometic magitue we must apply its coespoig equatios fom Table II. Figue 3 shows the values obtaie whe usig pepeicula iaiace (S fo the multiple eflectios a tasmissios betwee the two itefaces. If, fo istace, aiace L wee chose, the values isie the mateial a at the ight woul be iffeet fom the S values isplaye o figue 3, while the L values at the left of the figue woul ot chage. The expoetial law fo the eceasig of aiatio has bee applie i the popagatio of the beam isie the mateial. I this case this law oes ot chage with the aiometic magitue a taes its simplifie expessio (fo istace, S S exp( cx ue to (s (s. By pefomig mathematical opeatios the followig expessios fo R a T ae obtaie: cx e R 3ττ τ cx τ e 3 cos + a T (6 cx 3e e cx cos 3 a the law of eegy cosevatio implies that the absoptace A of the plate is give by A-R-T. get I the case that the exteal meium be the same ( a cos cos, usig Fesel equatios we 3, τ τ (, so that equatios (6 ae simplifie to: cx ( e R + a cx e cx ( e T (7 cx e If absoptio pocesses ae eglecte (e -cx, the esult is: R a a + - T a (8 + 8

9 These magitues, that we call aiatio eflectace a tasmittace, oly epe o, that is, o the iciet agle a the efactio iexes. 4. Optical view Let s pay attetio to the coespoig expessios that appea i may Optics textboos fo the last case (i.e., same exteal meium ; these boos 9,0,7 usually povie the total eflecte a tasmitte itesity, fom which the eflectace a tasmittace of the plate ae easily calculate by pefomig the coespoig atios. The eflectio a tasmissio expessios of a plate whe absoptio pocesses ae egligible ae give by: ( cosδ R with 4 (9 + cosδ ( t t T with t 4 t (0 + cosδ 4π whee δ is the phase iffeece of two cosecutive waves δ cos, a a t ae eflectio a λ tasmissio coefficiets; the latte ae efie as the atio of eflecte/tasmitte electic fiel amplitues to ( E0 ( E0 t the iciet electic fiel amplitue,, t, a they ae etemie by applyig the Fesel ( E0 i ( E0 i equatios. Taig ito accout the efiitio of S as a fuctio of the electic fiel amplitue, cosieig omagetic mateials a usig popely the efiitios isplaye i Table II, the coect values fo, τ, τ ae obtaie i this case as cos, t τ, cos τ t. We ca the cos cos tasfom equatios (9-(0 ito: ( cosδ Ropt a + cosδ (- Topt ( + cosδ 4.3 Compaiso Let s compae ow equatios (8 a (. Obviously, they ae ot the same. It ca be otice i equatios ( the epeece o the phase iffeece of two cosecutive waves which implies that these expessios epe o the thicess of the plate. O the cotay, equatios (8 o ot epe o the thicess. As a example, we plot i Figue 4 both tasmittaces as a fuctio of thicess fo a ucoate calcium fluoie wiow at 486m 9 at two agles of iciece. Calcium fluoie has vey low absoptio at this wavelegth, so the equatios (8 a ( ae appopiate fo this case. The cuve lies ae values obtaie fom eq. ( a the staight lie fom eq. (8. As it ca be see, the optical values oscillate aou those calculate fom the aiatio metho a the oscillatios chage with the value of agle of iciece. The maximum ispesio of the R opt a T opt values is ΔR opt ΔT opt 4/(+, which oly epes o. I Figue 5, it ca be see the epeece of both tasmittaces (T a a T opt a the ispesio ΔT opt with the agle of iciece. The ispesio is costat a small at low agles a iceases stogly at highe agles of iciece. As we have otice i the pevious figue, the T opt values oscillate aou the T a values.how ca we explai fom a physical poit of view these esults? 9

10 .05 0 ο.00 T 0.95 ΔT opt ο.00 T 0.95 T a,t opt, ΔT opt ΔT opt ΔT opt Thicess (m Agle of iciece (º Figue 4. Tasmittace values fo a ucoate calcium fluoie wiow at 486 m as a fuctio of thicess at omal iciece (a a at iciece 45º (b. The cuve lies ae the optical values a the blac staight lies ae T a. Figue 5. T a, T opt (0.00m a maximum ispesio of T opt as a fuctio of agle of iciece fo a ucoate calcium fluoie wiow at 486 m. The ashe lies ae the optical values a the blac staight lies ae T a. At fist sight, a ue to the fact that the optical equatios ( have bee obtaie taig ito accout itefeece pocesses, it may seem that pefomig some i of aveagig to the phase iffeece δ 4π ( δ cos i the optical equatios ca lea us to the equatios (8. But, what type of aveage? By λ supposig a homogeeous a isotopic mateial, the et vaiatio of δ ca be expesse as the followig expessio: Δ Δλ Δδ 4π cos + cos + si Δ ( λ λ λ Hece, we coclue that the phase iffeeces pouce by the waves isie the plate ca aopt ay value, if: 0

11 i The plate sufaces ae ough, meaig that the thicess (Δ a the suface omal (Δ vay aomly ii The iciet light is a o-moochomatic aiatio with a spectal bawith (Δλ. By assumig that ay of these effects tae place, the R opt a T opt coul be aveage ove all possible values of δ. If the followig aveages ae pefome: π π ( cosδ < R > Roptδ δ (3 π π cosδ π π ( < T > Toptδ δ (4 π π cosδ we obtai <R>R a a <T>T a, that is, the optical equatios aveage ove δ become ito the aiatio expessios fo eflectace a tasmittace. So, clealy istiguishig these two types of magitues is vey impotat ot oly fom a basic physical poit of view but fom a pactical viewpoit. Accuate eflectace a tasmittace measuemets ae ecessay fo calibatio spectometes o fo etemiatio of the optical popeties of mateials. It is also impotat to istiguish both magitues i optics catalogues 9, whee the tasmittaces of the coloe glass filtes, the eutal esity filtes, the itefeece filtes, the ucoate wiows, etc., ae show. Evietly, the expessios fo tasmittace which escibe the coespoig behavious ae iffeet fo each type of filte; theefoe we must tae ito accout the above cosieatios, such as, the oughess of the plate o the spectal bawith of the aiatio i oe to coectly itepet the give ifomatio. 5. CONCLUSIONS The efiitio give i Electomagetic a Optics textboos of the itesity as the time aveage of the amout of eegy which cosses pe seco a uit aea pepeicula to the iectio of the flow is the aiometic magitue iaiace oly if the iectio of the popagatio is pepeicula to the suface. The eflectace at a iteface ca always be expesse as the atio of the eflecte aiometic magitue to the iciet oe. I cotast, the tasmittace τ expessio chages with the aiometic magitue. Futhemoe, the eceasig expoetial law of the aiatio popagatig withi a absobig meium epes o the aiometic magitue use. The moe covetioal vesio of this law coespos to the case of a collimate beam of aiatio, expesse as a fuctio of iaiace. I this situatio, the simple expoetial law is satisfie with evey aiometic magitue. Fo othe cases, the expessio of the law may chage. So oe has to be vey caeful while itepetig this vey well ow equatio. 3 Equatios fo eflectace R a tasmittace T have bee obtaie fo the case of a plae paallel plate by computig the total powe flux (eflecte a tasmitte at each iteface. These expessios show a epeece o aiometic magitues simila to the oe isplaye by the eflectace a the tasmittace τ at a iteface. Oly if the plate is suoue by the same meium, R a T ca be efie as the atio of the eflecte (tasmitte to the iciet aiometic magitue, o matte which oe is chose. The expessios obtaie fo this simple case have bee compae with the coespoig oes fou i Optics textboos. Not oly the mathematical expessios iffe, but also the ie physical meaig, sice the effect of itefeeces is oly cosiee i the optical view. Both expessios (the aiatio oes a the optical oes coicie if the plate oes ot have smooth a paallel sufaces a/o the aiatio pesets a boa spectal bawith.

12 ACKNOWLEDGMENTS This wo was suppote by the istitutios Miisteio e Eucació y Ciecia, Miisteio e Ciecia e Iovació, Uivesia el País Vasco/Eusal Heio Uibetsitatea, Gobieo Vasco/Euso Jaulaitza, Diputació Foal e Bizaia/Bizaio Fou Aluia, a the Euopea Uio 7th Reseach Famewo Pogamme, ue pojects TEC C03-0, PSS , UE08/6, S-PE08CA0, DIPE08/4, a CE07/-AISHA II, espectively. REFERENCES [] Cheg D K983 Fiel a Wave Electomagetics (Massachusetts: Aiso-Wesley Publishig Compay [] Heal M A a Maio J B 995 Classical Electomagetic Raiatio (Saues Collegue Publihshig [3] Jacso J D 980 Electoiámica clásica ( Mai: Alhamba Uivesia [4] Loai P a Coso DR 979 Campos y oas Electomagéticos ( Mai : Seleccioes cietíficas [5] Reitz J R a Milfo F J 97 Fuametos e la teoía electomagética (Mexico: Uió tipogáfica eitoial hispao-ameicaa [6] Ulaby F T 999 Applie Electomagetics (New Yesey:Petice Hall [7] Wagsess R K 987 Campos Electomagéticos.( Mexico: E. Limusa [8] Aequi R a Boutigy J 976 Optica (Baceloa: E. Revete [9] Bo M a Wolf E 980 Piciples of Optics (Oxfo: Pegamo Pess [0] Casas J 985 Optica (Zaagoza: coopeativa e ates gáficas [] Ditchbu R W 98 Optica (Baceloa: E. Revete [] Guethe R 990 Moe Optics (Joh Wiley&Sos [3] Heaves O S a Ditchbu R W 99 Isigth to Optics ( E. Wiley [4] Hecht E a Zajac A 986 Optica (Wilmigto USA: E: Aiso-Wesley Ibeoameicaa [5] Jeis F A a White H E 98 Fuametals of Optics (E McGaw-Hill [6] Lasbeg G S 984 Optica (Moscu: E Mi [7] Peotti F L a Peotti L S 993 Itouctio to Optics (New Yesey:Petice Hall [8] Nicoemus FE 963 Raiace Am. J. Phys [9] Newpot Resouce Catalog (008