Simulations of Optical Thin Films

Transcription

1 Dael O Keeffe 7/4/9 Dasy Wllams Smulatos of Optcal Th Flms Th flm terferece, despte eg a terestg physcal pheomea ts ow rght, has umerous applcatos to varous forms of dustry. A few examples are optcal commucato, precso scetfc equpmet ad eve photovoltac cells (solar paels). I fact, t s wdely eleved that the mplemetato of th flm techologes to solar cells could greatly crease ther effectveess ad hece reduce costs. For ths reaso, ad of course just for the sack of dog some awesome physcs, t s mportat to develop a sold uderstadg of the pheomea of th flm terferece. To ths ed, we propose to costruct a mathematcal model of lght travelg, from ar, to a seres of th flm layers of hgh ad low dex of refracto ad try to uderstad the oserved ehavour. I order to test our model, we propose to actually costruct a devce called a Fary-Perot terferece flter (whch s a drect applcato of th flm terferece, these devces are actually a mportat part of producg spectroscopc mages, more aout applcatos wll e dscussed later), ad make a few measuremets whch wll compare to the predctos of our model/smulatos. Such a flter s composed costructed y depostg varyg layers of hgh ad low dex of refracto oto a glass sustrate. The exact detals of how all ths wll e accomplshed wll e dscussed later, ut for ow we must frst uderstad exactly what all ths terferece usess s aout. Frst, we would lke to cosder our model of lght (sce ultmately we are tryg to uderstad how lght s trasmtted through a Fary-Perot flter, we must have a way to uderstad lght!). The model we use s a macroscopc oe, treatg lght as a electromagetc pheomeo, gorg ay Quatum effects (ths s allowale ecause we are terested the macroscopc ehavour or lght, we do t really care aout how each photo s trasmttg through the devce sce all practcal applcatos, we are dealg wth huge esemles of photos). Frst, we must cosder Maxwell s equatos sde of a delectrc medum (a delectrc medum s ascally a sulator, whch s the type of materal our terferece flter wll e costructed out of). See for example, Y.Hamakawa, Th-Flm Solar Cells, Next Geerato Photovoltacs ad ts Applcatos. Sprger 4.

2 D ρ f t D H J f + t Where: D, s the so called electrc dsplacemet feld,, s the magetc feld, s the electrc feld, H, s the so called Auxlary feld, J s the free curret desty (we say free, ecause these f currets are ot produced y ay magetzato of the medum,.e. they are put the medum y some exteral force that we ca cotrol) ad ρ f s the free charge (free for the same reaso as the free curret desty). For our purposes (the costructo of a optcal devce), t suffces to supposed that there are o free currets or free charges our materals. Ths s reasoale smply ecause we are dealg wth a free stadg devce sttg ar, there really s t ay plausle source for ay free currets or charges! Hece, our equatos reduce too: D t D H t We also ote that the dsplacemet ad auxlary felds are related to the electrc ad magetc felds respectvely y: D H

3 3 Where s the electrc permttvty ad s the magetc permealty. s essetally a measure of how well a materal trasmts (or permts ) ad electrc feld. s a lttle more complcated, t s defed as the degree of magetzato of a materal the respods learly to a appled magetc feld. Whe a magetc feld s appled to a medum, so called oud currets ca form sde the materal (essetally, they are result of electros eg pushed aroud medum y the appled feld). The eautful thg aout the way we wrote dow Maxwell s equatos s that we actually do t have to cosder these currets (otce that the equatos oly care aout free currets)! That eg sad, we ca gore ther effect. The ext crtcal assumpto that we make s that our materals are lear (actually, we already made ths assumpto whe we wrote dow the relatoshps etwee D ad ad etwee H ad aove). What exactly does ths mea? It meas that the values of ad do ot vary sde the medum, they are the same from pot to pot, other words they are costats. If we dd t make ths assumpto, the the values of ad would vary from place to place ( fact, they would have to e represeted y tesors, makg the math all that much more complcated). Fally, wth all ths formato, we ca fally wrte: t t If we take the curl of the last two of these equatos (ad usg the frst two whe eeded), we fd the followg: t t

4 4 Whch we recogze as wave equatos havg wave velocty ( fact t s ths very wave ature of lght that allows for terferece, just lke stadg waves o a strg ca terfere wth each other): v c Where c s the speed of lght vacuum ad s a costat called the dex of refracto (whch depeds o the propertes of the materal we are terested ), ad s defed as: o o Where havg umercal values of o ad o are the electrc permttvty ad magetc permealty of vacuum, 8.85 s 4 A / m 3 kg ad.6 6 mkg / s A respectvely (where m s meters, A s Ampere s, s s secods ad kg s klograms). Now we ve got two partal dfferetal equatos that eed to e solved, ths of course meas that we eed some oudary codtos, whch get y cosderg how the electrc ad magetc felds are related o opposte sdes of two meda (say ad ). These are easly suppled y: Here, the symol meas the perpedcular compoet of the felds (.e. perpedcular to the terface etwee the two meda) ad meas the parallel compoets. For the sake of smplcty (ad revty!) we omt the dervato of these oudary codtos, they ca e foud pretty much ay textook o classcal electrodyamcs (see for example Itroducto to

5 5 lectrodyamcs y Davd J. Grffths pages 33 to 333 for all the detals). I order to get to the pot of our model, we oly eed a few more peces of ackgroud formato, so alas, patece! Next, we propose aother assumpto aout our system to help smply ( fact, greatly) the math. We assume that source of lght ( our expermetal measuremets, ths wll e a laser) s far eough away from the flm so that the cdet lght s approxmately a plae wave. Ths assumpto makes sese, f we have a lght source, whch s emttg lght some shape (a good example to keep md s that of a lght ul, or the su, lght s emtted a sphercal fasho), oce we get far eough away from the source, the lght waves wll have spread out to the pot that ay small area o the surface of the wavefrot s approxmately flat (thk of the surface of the arth, from hgh up t looks pretty curved, ut as you get closer t looks suspcously flat). Ths assumpto allows us to propose solutos of the wave equatos that take the followg partcularly smple form: ~ ~ ( r, t) oe ( k r ωt) ~ ~ ( r, t) ( k r t) o e ω Here, we use the tlde to dcate that the quattes (actually they are wave ampltudes) may e complex. It s easy to see that these equatos verfy the wave equatos for ad. We have also used the otato k to represet the wave vector, whch s a vector of magtude π /, where s the wavelegth of the wave. Ths vector pots the drecto of propagato of the wave. The quatty ω s the so called agular frequecy of the wave (measurg how much the wave oscllates, sce the complex expoetals are of course ses ad coses). Seeg how these felds are complex, the actual physcal felds are smply the real parts. relato s: Fally, the last result we eed to cosder s the so called Poytg Vector, S. The S oc ( ) Ths quatty gves the drecto of eergy propagato of a electromagetc wave (aga, the proof s omtted for revty; the referece to Davd J. Grffths ook has all the relevat detals pages 345 to 347). Sce ths vector tells us the drecto whch eergy s travelg, t tells us fact the drecto of propagato of the wave! Notce that ths drecto s

6 6 perpedcular to oth the electrc ad magetc felds. I fact, all we have to do s specfy oe of the feld drectos ad, gve the drecto of propagato, we ca determe the oretato of the remag feld. Tradtoally, t s the drecto of the electrc feld that s specfed (called the polarzato of the wave), ad we wll follow ths tradto what follows. Usg the Poytg vector, we ca express the magetc feld terms of the electrc feld. It wll e coveet to determe the magtude of the feld what follows. It s readly show that: v Now we have all the ackgroud formato that we eed. All we eed to ow (easer sad tha doe) s to apply the oudary codtos that apply to our specfc case of a Fary- Perot terferece flter. The geeral method for attackg prolems lke ths s to setup the magtudes of the electrc ad magetc felds o all sdes of the system ad takg to accout the reflected ad trasmtted parts. Cosder the followg: We assume that our th flm s homogeeous ad sotropc, ad that the flm thckess s of the order of the wavelegth of lght. I our expermet, the th flms are of ZS (Zc Sulphde) ad LF (Lthum Fluorde), oth of whch are homogeeous ad sotropc, ad the thckesses (as wll e later determed), are measured agstroms, whch s approxmately of the order of the laser lght used ths expermet, 63.8 m. Ths s mportat so that the pathlegth dfferece etwee multply reflected ad trasmtted eams remas small compared wth the coherece legth of the moochromatc lght.

7 7 Also, the tagetal compoets of cdet lght o a terface, from Maxwell s equatos, must e cotuous across the terface. Thus, the oudary codtos for the electrc feld at the two terfaces (a) ad () ecomes + + () a r t + () r t Ad the correspodg equatos for the magetc feld are: a θ (3) cosθ r cosθ t cosθt cos θ (4) cosθ t r cosθt t cos t t We, however, kow that the electrc feld ad the magetc feld are related the followg way:, where s the dex of refracto. v c Thus equatos (3) ad (4) ca e expressed terms of electrc feld a cosθ r cosθ t cosθt cos θ t cosθt r cosθt s t cos θ t Or, defg cosθ, cosθ t ad s s cosθ t, a ) ( ) (5) ( r t (6) ( r ) st Notce ow that ad tdffer due to a phase dfferece that develops due to oe trasversal of the flm. We kow that the optcal path legth dfferece Δ due to oe trasversal of a flm s Δ l cosθ t, whle the phase dfferece whch occurs ths case s

8 8 cos t l k θ π Δ, where k s the wave umer assocated wth the correspodg wavelegth. Thus we have t e (7) ad smlarly r e (8). y susttutg equatos (7) ad (8) to equatos () ad (6), we are ale to elmate the felds ad r the oudary codtos at the () terface: t r t e e + (9) ) ( ) ( t s t t e e e e () quatos (9) ad () aove may e solved to get t ad terms of ad : t e + () e () Fally, susttutg equatos () ad () to the expressos () ad (5) for the oudary codtos of (a), we get a e e a e e ut keepg md that uler s Idettes are: e e + cos

9 9 e e s We get: + s cos a (a) cos ) s ( a + () quatos (a) ad () may e rewrtte matrx form as a a cos s s cos (c) Or, for ormal cdece, t θ t θ θ ad thus, ad s s, the also l k Δ π, where s the phase dfferece etwee each succeedg reflecto at each terface. Thus the matrx equato (c), at ormal cdece, ecomes a a l l l l cos s s cos π π π π (d) However, s the wavelegth of lght sde the medum. Usg Sell s Law, we have terms of : ad smlarly, for a secod layer adjacet to the frst, wth wavelegth ', we have:

10 ' ' ' ' ' ' ' ' Also, sce our expermet we are takg the thckess of the flm,l, where the wavelegth of the lght sde the medum, Δ k ecomes l l k π π Δ (x) Thus the matrx equato (d) may e smplfed eve more to e a a l l l l π π π π cos s s cos The x matrx s called the Trasfer matrx. Now, for two dfferet layers of materal of a hgh dex of refracto materal (H) ad a low dex of refracto materal (L) of thckess H H H l 4 4 ad L L L l 4 4 respectvely, where m 63.8, ad where ) H ( ad ) L( are the dces of refracto respectvely (as fuctos of wavelegth), we have the followg trasfer matrces: H H H H H H H H H H l l l l H π π π π ) ( cos ) ( s ) ( s ) ( cos (h)

11 L L L L L L L L L L l l l l L π π π π ) ( cos ) ( s ) ( s ) ( cos (l) A Fary-Perot terferece flter cossts of the followg arragemet of th layers, (HL) m (HH) l (LH) m where H ad L are the trasfer matrces correspodg to each layer, ad m ad l are atural umers. Thus, t s possle to get a resultat matrx, M, after multplyg the correspodg trasfer matrces the order aove, wth certa etres X, Y, Z ad W. W Z Y X LH HH HL M m l m ) ( ) ( ) ( Thus the collecto of the may layers of the Fary-Perot Iterferece flter, matrx otato, ca e treated as oe etty, expressed y the resultat matrx M, ad amely, for the Fary-Perot flter, we have from the matrx expresso (d) a a W Z Y X M However, utlzg expressos (), (), (5) ad (6), we get + ) ( t s t r r W Z Y X Ths gves us the equvalet system of equatos: t s t r Y X + + (3) ) ( t s t r W Z + + (4)

12 Next we dvde oth equatos (3) ad (4) y, ad otg that the reflecto coeffcet, r, ad trasmsso coeffcet, t, are defed as r r ad t t Thus we ota t Y Xt r s + + (5) t W Zt r s ) ( + (6) Solvg the aove system of equatos (5) ad (6) for r ad t, we get the followg expressos W Z Y X t s s W Z Y X W Z Y X r s s s s The reflectace, R, ad trasmttace, T, are defed as * * t t t T r r r R The trasmttace s a measure of how much lght s trasmtted through the medum; t ca take values etwee ad. Smlarly, R s a measure of how much lght s reflected from the medum, t ca also oly take values etwee ad. I fact, R + T. Trasmttace s geerally defed as the rato of the testy of lght that made t through the medum to the testy of lght cdet o the medum (where testy s a measure of the tme average eergy flux. I our preset case, t s the tme average of the eergy eg trasported y the electromagetc felds as they propagate through space).

13 3 We have wrtte a MATLA code that wll allow use to make a few smulatos of what we expect to see for varous values of m ad l (rememer tha the umer m ad l cotrol how may layers). The followg graphs demostrate what the theory predcts for a few values of m ad l (ote that we also eed to kow the dces of refracto of the layers used, see page for all the detals):

14 4

15 5

16 6

17 7

18 8 The last graph show aove s very ecouragg. It shows that our smulatos predct a costat trasmttace through optcal system f we do t have ay layers of hgh ad low dex of refracto. Ths makes perfect sese physcally, sce f there are o layers of materal, the all we have s the glass sustrate upo whch the system s sttg, other words, we just have oe medum for lght to travel through, hece we should t expect aythg other tha a costat trasmttace! We should also ote that the order of the layers s mportat, ad t must follow the coveto set out o page. If we do t use such a coveto, we o loger have a terferece flter ( fact, f oe were to follow a layerg patter lke g(hl) Ha, you would ed up wth a devce called a delectrc mrror. Ulke our terferece flter, that tres to lock out a rage of wavelegths, whle lettg oe pass through, the delectrc mrror attempts to reflected all the lght a certa rage)! If m > l, the terferece flter locks the trasmsso of lght a certa rage, allowg for the trasmsso of lght a arrow rego aroud the peak (ths s called the ad pass). The peak trasmsso occurs for oe wavelegth ths rage, our case ths correspods to 63.8 aometers, sce ths s the wavelegth of lght we are gog to use our expermet, t makes sese that we should costruct our smulatos aroud ths value!

19 9 The phase shft at 63.8m s a teger multple of the wavelegth, whch results a maxmum trasmsso. Phase shfts at all other wavelegths are fractoal values, whch do ot provde maxmum trasmsso, as ca e see from the results of the smulato. Also, as ca e see from the graphs where m>l, the spectral trasmssos of each of those Fary-Perot terferece flter s such that wavelegths the rage of ~53m to ~75m are locked, wth a trasmsso peak cetered at ~63m. Thus, ths expermet, the Fary- Perot flter (desged for a wavelegth of 63.8m) locks wavelegths from most of the vsle spectrum ad a lttle of the frared spectrum (gree lght to frared). Hece depedg o the value for whch the Fary-Perot flter s desged, ot oly s oe ale to choose the wavelegth that wll e trasmtted (trasmsso peak), ut also the rage of wavelegths to e locked. Notce that as m crease (stll the cases where m>l), the trasmsso peak aroud our specal wavelegth of 63.8 m ecomes sharper. Ths meas that, as we crease the value of m, the flter locks out more ad more of the wavelegths the surroudg spectral rego ad we see that the wavelegth that we have costructed the system aroud really ecomes the oly sgfcat source of trasmsso, just as we wat t to! I other words, creasg m decreases the ad pass. As m cotues to crease (such as the cases for whch m 3 ad m ), the trasmsso peak vashes othg at all s trasmtted the rage of aout 55 m to 75 m (.e. the optcal rage), fact for the case l, we fd y expermetg wth varous values of m our smulato code, that complete lockage starts for m 8. Lastly, f l>m, the umer of trasmsso peaks creases. I fact, ths umer of peaks seems to correspod approxmately to the value chose for l, whch are all evely spaced. I case that m 4, l, however, the umer of peaks s smaller, the ad pass for each s arrower, the mmum trasmsso for the wavelegths locked s also lower, ad these regos of locked wavelegths are also more sharply defed (ths ca e attruted to a crease m, as oted aove). Thus, the umer, ad pass, ad sharpess of trasmsso peaks a rego of locked wavelegths may e characterzed y the rato of m to l. Armed wth our smulatos, we would ow lke to see how these compare to a real stuato. We wll costruct two Fary-Perot terfereces flter wth m 3 ad l ad measure ther trasmttace as a fucto of wavelegth. I order to acheve ths we make use of a hgh vacuum deposto techque, whch materal (ZS ad LF) are evaporated ad deposted oto two glass sldes. Frst, we eed to determe how thck we eed our layers to e f we are to use lght of 63.8 m.

20 Usg (that of ar), the materal of hgh dex of refracto s ZS ad the materal of low dex of refracto s LF, where followg way ZS ad LF vary as a fucto of wavelegth the ZS where s measured agstroms ( - m), ad LF (.7376) (3.79) where s measured mcrometers ( -6 m). For ths expermet, 63.8m, therefore the thckesses of the ZS ad the LF layers are ZS lzs (!) 4 4ZS ag. 673ag..673kloag (638ag.).73 7 l LF 4 LF 4 4 LF (.7376) (.638 m) + (.638m) (.7376) m.38kloag 5 m (3.79) (.638 m) + (.638 m) (3.79) See Palk ad Huter ad. Palk ad A Addamao or. Palk. Hadook of Optcal Costats of Solds. Vol. I ad II. Academc Press. 985/989.

21 Ad the dces of refracto at ths specfc wavelegth are ZS ( ) (638ag.) Ad LF ( ) (.7376).9549 (.638 m) + (.638 m) (.7376) (3.79) (.638 m) + (.638 m) (3.79).39 I order to actually evaporate the materal, we place t sde of a crucle wth the vacuum chamer. We wrap wres aroud the crucle ad (oce the vacuum has ee estalshed) pass a curret through the wres. Ths has the effect of heatg up the materal to the pot (evetually) of evaporato. The evaporated materal the rses from ts crucle ad ecouters the glass sldes that are placed aove t. I order to determe the thckess of the layers (whch s supposed to e precse), we use a thckess motor that s placed ear the glass sldes (the motor s also ale to tell us the rate at whch the materal s eg deposted). Oce the desred thckess s acheved, a shutter sde the evaporator s closed; ths shutter s place etwee the glass sldes ad the materal, so that t ca lock the further deposto of materal oto the sldes. We ca the do the exact same thg for the secod materal, ad proceedg ths way, depostg layer after layer, we ca costruct our terferece flter. Of course, lke ay expermet, ths procedure s rddled wth potetal prolems, whch wll e dscussed later. Oce we have our flters, we ca qute try to measure ther trasmsso spectrum to see f they match what our smulatos predct. I order to do ths we she lght from a Mercury lamp (whch has a structured spectrum, ths way we ca test our flter at varous dfferet wavelegths at the same tme) o our flters. The lght s the collected ad fed to a spectrometer. The spectrometer ca sca over a rage of wavelegths ( our expermet ths rage correspods to 38 m to 8 m, whch s from the volet ed of the spectrum to the ultra volet), ad pass whatever lght t gets through at each wavelegth to a photovoltac cell. Ths cell wll the produce a curret that s proportoal to the testy of lght that has ee receved. The cell fuctos ased o the photoelectrc effect whch demostrates that lght o a

22 metallc surface wll produce a curret, ad that ths curret s proportoal to the testy of lght cdet o the surface. Now that we have a curret, we ca measure a voltage (sce there s a resstace as well), ad ths value s recorded y a computer. The spectrometer scas wavelegths at a terval of m/m, takg a measuremet aout every. secods. Wth ths sca rate (ad kowg that we have scaed over a terval of 38 m to 8 m); we ca determe exactly whch wavelegths of lght were tested as follows: 5 ( t) 38m + (m / m) t 38m + ( m / sec) t 3 Wth ths, we ca assocate that each scaed wavelegth wth a voltage measured y the computer. We kow that the voltages are proportoal to the trasmttace, ut we stll eed to determe exactly how. I order to do ths, we do the same expermet expect wth a clear glass slde. We the take the voltages measured for the terferece flters at each wavelegth ad dvde each oe y the correspodg voltage measured at the same wavelegth for the clear slde. Ths rato tells us dfferece the proporto of lght that was trasmtted through the flter compared to whe the flter was ot preset for each wavelegth measured, other words ths rato s exactly the trasmttace value we are lookg for! Whe uldg our flters we amed them wdow ad door. The ames come from the proxmty of the flters to ether a wdow the la or the door to la as they sat sde the evaporator. The reaso for amg them s to e ale to keep track of the results for each dvdual flter ad to see f ther locato sde the evaporator had ay effect o ther overall ehavour (whch t most certaly dd). elow, we show graphs of the results otaed for each flter plotted agast the predcted graph of our smulato for m 3, l, as well as our smulated graph:

23 (Theoretcal graph) 3

24 4 ( Door ) ( Wdow )

25 5 We summarze some of the relevat features the followg tales: Tale : Trasmsso Spectrum Data of the Fary-Perot Iterferece Flter Theoretcal Fary-Perot locked Wth the Spectrum excepto of a (m) Peak cetered at aout (m) Costructed Fary-Perot (Door) locked Wth the Spectrum excepto of a (m) Peak cetered at aout (m) Costructed Fary-Perot (Wdow) locked Wth the Spectrum (m) excepto of a Peak cetered at aout (m) Tale : Vared Parameters for the Theoretcal Smulato of the Fary-Perot Iterferece Flter Smulato for Flter ear DOOR Iputted thckess of Iputted thckess of ZS layer LF layer (kloagstroms) (kloagstroms) Smulato for Flter ear WINDOW Iputted thckess of Iputted thckess ZS layer of LF layer (kloagstroms) (kloagstroms)

26 6 Tale 3: Maxmum Trasmttace Data of the Fary-Perot Iterferece Flter Theoretcal Fary-Perot Maxmum Mmum Trasmttace Trasmttace Costructed Fary-Perot (Door) Maxmum Mmum Trasmttace Trasmttace Costructed Fary-Perot (Wdow) Maxmum Mmum Trasmttace Trasmttace Percet Dfferece of Maxmum Trasmttace from Theoretcal Value (%) The values these tales deserve some explaato, sce they tell us a lot aout what s happeg physcally. Frst of all, f we take a look at the graphs comparg the expermetal data wth the predctos of our smulato, we ca see that they do t exactly match. Ideed the data for the door flter s sgfcatly dfferet from the smulated curve; at least ths s how t seems at frst glace. However, t oe s to look closely at the curves, we ca see that, although the peaks are ot at the same heght or are they cetered at exactly the rght wavelegth, the overall ehavour of the systems are exactly the same. There are may reaso why the curves do t le up exactly though, ad we wll dscuss some of them detal. Frst of all, t should e oted that, the ed of the day, the reaso why are smulatos do t match wth the expermetal values exactly s ecause are smulatos are of a dealzed world, whch all thgs are equal, ufortuately for us the real world s hardly ever so eatly packaged. As s summarzed tale 3, the mmum trasmttaces for the smulated data ad the expermetal data ft pretty closely (. compared wth ), ut the major ssue of cocer s the maxmum trasmttaces. We ca see that the maxmum for the door flter s.4, whereas for the wdow flter t s.53. Whe compared to the smulated value of.63, these do t seem so great; fact they lead to percet errors of:

27 7 theo. value exp. value.63.4 % dfferece % % 6.9% theo. value.63 Ad % dfferece % 5.7%.63 Whle the frst error (for the door flter) s ot so great, the last oe for the wdow flter actually s t that ad cosderg all the possle source of error, of whch there are may. Oe of the prmary reasos s the dffereces thckess that seem to have arse etwee the values that we wated to uld for each layer, compared to the projected thckess that were actually ult. These results are summarzed tale, we kow that (as calculated aove) that the desred thckess values were.673 klo-agstroms for ZS ad.38 kloagstroms for LF, whereas t seems lke the actual thckess were closer to.479 ad.786 klo-agstroms for the door flter ad.533 ad.9 klo-agstroms for the wdow flter. These values were extrapolated from our smulatos y takg the expermetal results for the peak trasmsso wavelegths (summarzed tale ) ad outputtg calculated thckess (these outputs were suppressed the orgal program ecause we already kew what the thckess should e). These values proaly are t exactly the real thckesses values, ut they at least gve us a approxmato to tell us roughly how much we were off our costructo of the flters. I order to determe the actual thckess, we would have to go ack to the la ad measure them usg a laser (actually, measurg the thckesses s ot far off from what we have doe our expermet, the dea s to use lght terferece aga a cleaver way, ths topc s covered pretty much ay stadard textook o optcs). The ew plots, of the smulatos ftted to the expermetal data, are show elow:

28 ( door ) 8

29 9 ( wdow ) The dffereces the thckesses ca help to accout for the shft peak trasmsso wavelegth, ut how exactly dd these dfferece occur? Ths prolem s readly explaed smply y the way whch the flters were costructed. Whe we ult these thgs, we had to e ale to close the shutter quckly oce the thckess motor had dcated that the desred thckess had ee acheved, the shutter was closed y had. Ths leads to a lot of ucertaty ecause we could t actually see the shutter closg, we had to estmate ts posto, ad the oly dcato that deposto had ee successfully locked was whe the thckess motor stopped recordg ay chages. Ths s aother major prolem, ecause the detector sde the apparatus was placed off to the sde of the glass plates. Ths meas that, whle the shutter may have ee completely lockg the detector, t may ot actually have ee lockg oth sldes. Coversely, whe the detector was ucovered ad readg chages thckess, there s o guaratee that the sldes were completely ucovered, so t s extremely dffcult to say whether or ot the layer thckess measured y the motor s exactly what got deposted o the sldes. There are some other prolems that we have to cosder as well. For oe, the smulatos were desged assumg that there was othg the surroudg evromet other tha ar,

30 3 ths s of course a pretty grad approxmato. I fact, ay lttle t of drt of dust o the flters could greatly affect the measuremet of trasmsso. A lttle t of dust would t make much of a dfferece from a optcal pot of vew, ut from the stace of a precso measuremet, a layer of dust ca certaly lock a otceale amout of lght from gettg through the flter to the detector. Ufortuately, we ca t really accout for ths sce we ca t measure how much dust s o the surface. Ths dust/drt also possess aother prolem, f dust s to collect o the surface of the flter ad oe tres to wpe t off, there s actual physcal damage eg doe to the flters. It may ot seem lke t from the pot of vew of someoe who cleag the flter, ut rememer that the layers of deposted materal have thckesses o the order of the wavelegth of lght! Ay small dsturace wll actually physcally damage the surface y dslodgg some of the materal from the top layer, ths has the effect of troducg -homogeetes to the materal whch, f you recall, was oe of our prmary assumptos the costructo of our model! That eg sad, we ca t deftely say that our layers were eve deposted completely evely ether! Sce we have o cotrol over the rate at whch materal s evaporated oto our sldes, t s etrely possle that some regos of the flters receved more materal tha others, whch aga wll cause -homogeetes our flters, ultmately affected our fal results. I order to try ad take ths to accout, we have wrtte a program that attempts to correct for some varatos of the thckess of layers. The program allows the user to to certa toleraces (or errors) o the thckesses, as well as a theoretcal peak wavelegth. The program the geerates some thckesses as follows. Suppose the error gve y the user s Δ l (whch s put as a fracto), the the program computes (lettg l e the thckess): l l Δl l Δl The last le eg the legth of the error terval. Added together, we get: l l Δl + l Δl We ca the geerate thckesses sde ths terval y usg the radom umer geerator MATLA, ths has the effected of smulatg a pseudo-radom thckess for the hgh ad low layers. Whle ths ovously does t deal wth radom (ad -homogeous) alteratg layers of materal, t at least gves us some dea of what happes whe the layer thckesses are shfted

31 3 aroud a t from ther theoretcal deal whch, let s face t, s extremely dffcult to acheve practce wthout very expesve equpmet. The fal expresso we use for the smulato s hece: l l Δl + l Δl rad(,) Where we apply ths to oth the hgh ad low layer (ased o the user put for the For example, our program allows for user put: Δl value). ths produces: We ca choose the theoretcal wavelegth for whch we would lke to costruct our flter,

32 3 We ca the eter the tolerace rage that we are terested (.e. the fracto error that we would lke to us). Oce we have etered these values, we push the plot utto ad we get ack a plot wth the error take to accout.

33 33 The clear utto allows the user to clear the graph ad start aga fresh f they would lke. The ext utto closes the program. Despte the expermetal ssues we have see so far, we ca geeralze our smulatos to take to accout ay theoretcal wavelegth as well as the values for the dces of refracto for the sustrate materals ( our expermet, ad earler smulatos these were ar ad glass) as well as the umer of layers used.

34 34 The program show aove allows the user to alter the theoretcal wavelegth usg a slder, as well as the other values show. very tme a value s chaged, the graph s updated automatcally to the ew oe. The graphs o the rght had sde are plots of the dces of refracto of the hgh ad low layers. These values deped o wavelegth ad ths s take to accout y the program. very tme the wavelegth s chaged y the user, the program automatcally moves the red cursors o these

35 35 graphs to pot to the curret dces of refracto. geerated y varyg the parameters: Here are some examples, whch were (Here, the tal wavelegth has ee chaged, otce how t shfts the trasmsso peak to the ew value, as t should (or else t would t e much of a trasmttace maxmum)). (Here the parameter m has ee chaged; ths s equvalet to varyg the umer of layers used).

36 36 (Ths tme, the parameter l s vared, ths too s equvalet to varyg the layers used (.e. the umer of hgh refractve dex layers the mddle of the system). (Ths oe aove has a dfferet tal refractve dex tha the others; here t s.43 as opposed to the other scree shots, otce how t chages the trasmsso curve).

37 37 (Ths scree shot has a updated value for the fal refractve dex, aga otce how t drastcally chages the trasmsso graph (.e. havg a hgher sustrate dex wll decrease the amout of lght trasmtted through the system!). The ext utto closes the program. Fally, we should cosder aother oe of our assumptos. We desged our terferece flters to have peak trasmsso at oe partcular wavelegth (63.8 m), ths s ot some artrary umer, ad fact t s, coveetly eough, the wavelegth of lght produced y a He-Ne laser. I theory, f we ht our flter wth such a eam we would see exactly the trasmsso predcted (.63) ad othg else. However, realty ths s ot the case, sce eve a laser s ot ths perfect, fact most laser eams are what s called a Gaussa eam. It ca e show that 3 the testy of a Gaussa eam s gve y: ω r I( r, z) I o o exp ω( z) ω ( z) Where I o s the testy at the cetre of the eam, ω(z) s the radus at whch the feld ampltude ad testy drop to e - ad e - of ther axal values, respectvely, ωo s the so called wast sze (ths s the locato alog the eam axs where ω(z) s at ts mmum value) ad r s the dstace 3 See for example, Pedrott 3, Itroducto to Optcs, Pearso ducato c, 7. I partcular, chapter 7 o characterstcs of laser eams.

38 38 from the cetre axs of the eam. The take home message s that the testy s ot uformly dstruted over the spread of the eam! Ideed, the trasmttace that we have ee measurg s smply the rato of trasmtted lght testy (how much got through) to cdet lght testy. Thus, f the cdet lght testy s ot uformly spread alog the eam, the trasmttace measured wll ot ecessarly correspod to what we had predcted theoretcally (sce the cdet testy wll e essetally spread out ). The followg program demostrates the Gaussa eam graphcally.

39 39 We ca choose ay values we lke for the m ad max values o the axes. Movg the slders aroud chages the relevat parameters ad the graph automatcally updates to take these chages to accout. Here are a few plots for some values, otce how the curve ca e really flat a spread out, or ca e very sharp, peaked aout a small rego (ths acheved y varyg the w ad lamda parameters).

40 4 Movg the I slder aroud chages the z axs. Also, we ca chage the m ad max values to whatever we would lke, here s a example:

41 4 Aytme oe of the parameters s vared, the plot updates automatcally to take these chages to accout. The ext utto allows the user to close the program wheever he or she so desres. Oe of the major applcatos of terferece flters (our Fary-Perot style flter fdg applcato here as well), s astroomy. A rach of astroomy called photometry s cocered wth measurg the testy of astrophyscal pheomea. If oe s ale to determe the testy of some radatg pheomea (a classc example s varale stars), ad the lumosty of ths oject s kow the ts dstace may e calculated va a verse square law. Ideed, oe of the est kow examples s that of a type a superova, whch always erupts wth the same, kow, lumosty (the reaso for ths s that a type a superova occurs o whte dwarf stars that reach a certa mass (y suckg materal off of a compao star), oce ths mass has ee acheved there s eough mass for thermouclear fuso to eg aga (a whte dwarf star s for tesve purpose a dead star whch all uclear fuso has ceased), the result s massve uclear reacto that always produces the same amout of eergy ad so the lumosty may e calculated qute accurately). If we measure the testy of ths superova (usg a optcal devce that makes use of our hady terferece flters) we ca determe the dstace of the superova. Ths s oe of the prmary methods for determg dstace the uverse. Ths s just oe of may other applcatos of such flters, other otale applcatos are fer optcs, ad as metoed prevously, solar cells. I cocluso, our project presets a model ad smulatos for the trasmsso of lght through a Fary-Perot terferece flter. We foud that smulatos predcted somewhat dfferet results tha were measured a laoratory expermet o the real thg. We oted

42 4 however, that the overall ehavour of our smulatos matched rather well wth what was actually oserved ad that the prmary reaso for our smulatos to ot predct exactly what would happe the la s several sources of massve expermetal error. Of course, our smulato ad model s lke ay other, smply a approxmato of the physcal world. Physcal pheomea are geerally of a rather complex ature, wth may varales ad so may, seemgly, upredctale possltes, that all we could ever hope for s a halfway decet approxmato. If we were to mprove our expermetal techque, we feel cofdet that our smulatos would etter represet the stuato. For example, f were to costruct ad store our flters a hghly satzed evromet, free of dust ad drt (as customary precso optcs) we could elmate a major source of error. etter costructo techques would also e requred to reduce the errors we ecoutered the thckess of the layers. Nevertheless, our model serves to demostrate ad helps to uderstad a very terestg ad applcale pheomeo.