VS203B Lecture Notes Spring, Topic: Thin Film Interference

Transcription

1 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene Thin Film Inerferene When ligh his a surfae, i an be absorbed, ransmied or refleed. Ligh an refle from any inerfae where here is a hange in refraive index. When ligh his a maerial ha has muliple layers, eah layer an refle ligh. The ligh refleed from any one of hese layers migh inerfere wih he ligh ha has refleed off any of he oher layers. Eah layer reaes is own image of a poin soure. These soures will be muually oheren beause hey all arise from a single original soure. The basi idea is illusraed in he figure below, whih shows a glass surfae wih a hin oaing. There are wo main soures of refleion from he surfae (ignoring muliple inernal refleions) labeled and. These wo refleions wil linerfere. How hey inerfere depends on phase differene beween hem. The phase differene depended on he index of refraion of he oaing, he hikness of he oaing and wha happens o he phase upon refleion. air n oaing (film) n glass n 3 n <n <n 3 So, wha are he rules for how waves refle from surfaes - wha happens o he phase? The rule for refleion is ha when he index of refraion afer he inerfae is higher hen he index of refraion before he inerfae hen he ligh, upon refleion, undergoes a 80 degree phase shif. When he index of refraion afer he inerfae is less han he index before he inerfae, hen here is no phase shif in he refleed ligh. Iniden wave Refleed wave (80 phase shif) n <n n n Iniden wave Refleed wave (no phase shif) n >n n n 34

2 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene Beause eah inerfae is a a ransiion from a lower o a higher refraive index, here is 80 deg phase shif a eah refleion (for and ). So hey undergo he same phase shif and we an herefore ignore i in his example. All we need o do is alulae he phase differene beween rays and due o he pah lengh differene. If here is desruive inerferene, hen here will be a minimum in he amoun of refleed ligh (his is he design of an anirefleion oaing). If ligh onsruively inerferes, here will be an inrease in he amoun of refleion (his is he design of highly refleive mirror). Sine ligh has o go somewhere, he ligh ha is no refleed will have o be ransmied, so boh ransmission and refleion are affeed by hin film oaings. The differene in aual pah lengh beween and is (assuming ha he angles are small): pah, The differene in he number of waves beween and is n waves, n For desruive inerferene o our, we require ha rays and be 80 degrees ou of phase, or ha he number of waves of differene be / n des 4 Similar inerferene an our when he differene in he number of waves is.5, ,, 5, 7 m... des, m0,,... 4n 4n 4 n 4 n n Bu reall he requiremen ha inerferene an only our when he differene in pah lengh of wo muually oheren soures of ligh be wihin he oherene lengh. So as he hikness inreases, many soures will no longer be muually oheren. n The ondiion for inerferene is ha he differene in disane beween he soures of ligh (from he refleions) be wihin he oherene lengh of he ligh. Diversion: Coherene Lengh. Defined as he lengh in spae over whih he ligh has a prediable phase. A laser, for example, has a long oherene lengh. Alhough here are random fluuaions in phase over ime hey our afer he waves have raveled some meers in disane. A inandesen bulb on he oher hand, has remendously fas variaions so he phase is prediable over only a very shor period of spae. Consider he double sli experimen: The inerferene arises from he differene in pah lenghs beween he wo muually oherene soures of ligh. For high angles, he pah differene an be ome very large (muliple wavelenghs). For a laser, whih has a long oherene lengh, he wo beams will sill reain a fixed phase differene beween hem. For a normal ligh soure, here will be so muh differene in disane beween he wo waves ha here will have been ime for random phase jumps o our. So he wo soures are essenially muually inoheren. The pah differene mus be wihin he oherene lengh for inerferene o our. For a laser, he oherene lengh is on he order of meers. For an inandesen ligh bulb, he oherene lengh is only a few miromeers. For he sun, he oherene lengh is on he order of millimeers. For some lasers, he oherene lengh an be many kilomeers. 35

3 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene The same holds for onsruive inerferene. Consruive inerferene will our when he differene in waves beween rays and is some ineger muliple of wavelenghs.,,,..., m0,,... n 3 m ons ons n n n n n So, imagine ha you ould vary he hikness of a oaing on a piee of glass, he amoun of refleion would vary as he hikness requiremens for onsruive and desruive inerferene were me. Manipulaing he oaings on surfaes an be used for many appliaions, whih I will alk abou in he nex ouple of leures. Ani Refleion Coaings ARCs To make an anirefleion oaing, or ARC, you pu a oaing on he surfae of your glasses wih he appropriae hikness ha minimizes he refleion. By minimizing he refleion, you auomaially inrease he ransmission. Lens wihou anirefleion oaings Lens wih ARC 4% off he firs surfae 0.5% off firs surfae 4% off he seond surfae 0.5 % off seond surfae 9% ransmiane. 99% ransmiane air n air oaing (film) n n air <n <n g glass n g Bu here is a seond ondiion ha needs o be saisfied in order o maximize he amoun of desruive inerferene. For omplee desruive inerferene, we also require ha he ampliudes of ray and be he same. To mee his requiremen we need o have he ondiion ha he refleiviy of he air oaing inerfae and he oaing glass inerfae be he same. Reall he refleane equaion for ampliude: nn nn r for inensiy: Rr nn nn For ray : n r n n n air air For ray : ng n r n n g For ligh ha is ou of phase, he resulan ampliude is he differene beween he wo ampliudes so oal desruive inerferene ours when r r 0 n n nair g n For non-ideal refraive n nair ng n indies, he inensiy is he square of he resulan nonzero ampliude. yields he resul: n n g 36

4 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene To summarize, an effeive ARC mus mee he following wo ondiions: Pah ondiion (desruive inerferene): des 4 n Ampliude ondiion: n ng So, wha are he problems wih ARCs? They an make oaings of exa hiknesses, bu he onsrain is on available maerials ha mee he ampliude ondiion requiremen. Problem : If we sar wih glass of index.5, we are fored o find a suiable oaing maerial ha has an index.. Suh maerials are no readily available so manufaurers simply ry o ge as lose as possible. They ypially use a oaing made of MgFl, whih has an index of.38. The ideal glass for ha oaing would have an index of.9, whih is no available. So he higher he index of he glass, he beer he ARC, when you are using MgFl as a oaing. Problem : Wavelengh dependeny. The pah ondiion an only be exaly me for one wavelengh. Ideally, we hoose o opimize he oaing for wavelenghs in he middle of he sperum. So while 550 does no refle, he lower and higher wavelenghs do, whih gives he refleion from ARC glasses a purplish hue. no ARC 4 refleane (%) 3 average A ypial ARC has an average of abou % refleane Problem #3: Only works as designed on axis. When you look a he refleion off axis, i hanges olor beause he pah lenghs hrough he oaing are inreased. I beomes a muh more ompliaed problem 37

5 VS03B Leure Noes Spring, 03 0 Topi: Example: Wha is he refleane of a glass (n=.5) surfae wih a MgFl oaing (n=.38) opimized for 550 nm ligh for 550 nm ligh? 400 nm ligh? Soluion: Sep : Wha is he hikness of he oaing? des nm 4n 4.38 Sep : Wha is he ampliude of refleane a he surfaes? r n nair n n.38 air r ng n n n.5.38 g Sep 3: For 550 nm ligh. oheren os I E E A A AA p p p p 80 sine hey are ou of phase I A A AA ( ) oheren % Sep 4: For 400 nm ligh, wha is he phase differene? Sep 5: For 400 nm ligh oheren oheren waves waves phase radians os I E E A A AA p p I os % 400 nm ligh is more refleive han 550 nm ligh, as expeed. Refleane was no zero for 550 nm ligh beause he ampliude ondiion was no me. 38

6 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene Ani-Refleion Coaing 80 deg phase hange 80 deg phase hange n n <n n n <n 3 n 3 Soap bubble film 80 deg phase hange No phase hange air: n n <n waer: n n >n 3 air: n 3 So he ondiion is opposie o ha of an ani-refleion oaing, beause here is already a 80 deg phase differene beween he wo waves beause one of he phases is no shifed. So, if he oal pah hrough he film is one half of a wavelengh, hen he ligh onsruively inerferes, no desruively as was he ase for he ARC. Beause he hikness of a soapy film onsanly hanges, you ll see a sperum of differen olors aross he soap bubble. Wha happens righ before he bubble burss? The layer beomes very very hin so ha he hikness is nearly zero. A his poin here is no signifian pah differene. The only hing hanging he phase beween he wo rays is he 80 phase hange from he refleion off he firs surfae. Sine here is a 80 phase hange, here is a minimum in refleiviy. This holds for all wavelenghs, and so he bubble has no refleion a he momen before i breaks. Oil film (like a soap bubble) The ypial index of refraion of oil is abou.5, whih is similar o glass. 80 deg phase hange no phase hange air: n n <n oil: n n >n 3 waer: n 3 The hikness of he oil layer varies as well as your viewing angle, so you see a sperum of olors. Like he soap bubble, he oil has low refleane when he layer is very hin. Tha is why he refleion from an oily surfae on a puddle has a blakish appearane. 39

7 VS03B Leure Noes Spring, 03 0 Topi: If he phase hanges are ommon o boh surfaes (eg ARC), hen des m, m0,,... n ons m, m0,,... n If he phase hanges are no ommon o boh surfaes (eg soap bubble, or oil), hen des m, m0,,... n ons m, m0,,... n 40

8 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene Fringes of equal hikness ligh band ransparen band ligh band 3 ransparen no phase hange n g =.5 ransparen 80 deg phase hange n air = ransparen n g =.5 When = /4 you ge onsruive inerferene, when = / you ge desruive inerferene. In his ase, he film is air so he wavelengh need no be adjused. pah ondiion for onsruive inerferene n 3,,... bu n m, m0,,... m or ons, m0,,... pah ondiion for desruive inerferene n0,,,... bu n m, m0,,... m or des, m0,,... 4

9 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene Speial ase: Newon s rings Same as mirosope slide inerferene exep ha he hange in hikness is is no linear and follows a very prediable shape. 4

10 VS03B Leure Noes Spring, 03 0 Topi: Thin Film Inerferene Example ) Two fla mirosope slides, 0 m long are ouhing on one side and are separaed by 3 mirons on he oher. How many dark inerferene bands will appear on he slide if you look a he refleion for 450 nm ligh? no phase hange 80 deg phase hange pah ondiion for desruive inerferene m, m0,,... m0 0 m 0.5 m m3... a dark band ours whenever he hikness hanges by 0.5 mirons , so he las band ours when m here are 4 dark bands (here is one for m 0) 43