Chapter 7. Interference

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1 Chape 7 Inefeence

2 Pa I Geneal Consideaions

3 Pinciple of Supeposiion

4 Pinciple of Supeposiion When wo o moe opical waves mee in he same locaion, hey follow supeposiion pinciple Mos opical sensos deec opical inensiy I.

5 Supeposiion of Two Plane Waves Conside wo poin souces, and, emiing monochomaic waves of he same fequency in a homogeneous medium, and conside only linealy polaized waves 1(, ) 01 cos( k1 1), ) cos( k ) ( 0

6 Supeposiion of Two Plane Waves 1 The esulan wave is given by ) ( ) ( Le s se Taking he ime aveage of boh sides, one obains 1 1 I I I I ,, I and I I I Then, I 1 I I Theefoe,

7 Supeposiion of Two Plane Waves The inefeence em I cos( k1 1 k ). 1 O can be e-wien as I cos k k ( 1 1 is he phase diffeence aising fom a combined pah-lengh and iniial phase-angle diffeence. ) If If, hen, hen I 0 I. 1 I 1 I I cos.

8 Supeposiion of Two Plane Waves The inefeence em can be wien as I 1 I1I cos Thus I I 1 I I1I cos Thus, he esulan inensiy can be geae, less han, o equal o, depending on he value of. A maximum I is obained when I max I1 I I1I 0,, 4, Toal consucive inefeence When he esul is known as consucive inefeence I 1 I I Imax

9 Supeposiion of Two Plane Waves A minimum I is obained when I min I1 I I1I, 3, 5, Toal desucive inefeence When he condiion is called desucive inefeence I 1 I I I min If I 1 I I 0, I Unde his condiion I 0 (1 cos ) 4I 0 cos I 0 I I. min max 4 0

10 Supeposiion of Two Spheical Waves Fo he spheical waves emied by and, hey can be expessed as 1( 1, ) 01( 1 ) cos[( k1 1)], ) ( ) cos[( k )] ( 0 and ae he adii of he spheical wavefons ovelapping a P. Suppose he wo waves have simila souce senghs, and 01( 1 ) // 0( ). A fa field, I 1 I I 0.

11 Supeposiion of Two Spheical Waves The phase is k ( 1 ) ( 1 ) Then, I I cos [ k ( ) ( )] The maximum inensiy occus when ( 1 1 ) [m ( )] k m= m 0, 1,, P 1 and minimum when ( 1 1 ) [ m ( )] k S 1 S a m = M m 1, 3, 5

12 Supeposiion of Two Plane Waves Inefeence paens: 1) Viewing sceen pependicula o he S 1 -S axis: Concenic Rings. ) Viewing sceen paallel o he axis: Paallel finges (Young s expeimen). 3) Maximum ode of ings: -M< m <M, whee M =a/l. (-a < 1 - <a )

13 Condiions fo Inefeence Fo he wo ligh wave souces 1. Polaizaion: same polaizaion. Fom a sable phase diffeence: Fequency and wavelengh should be he same o vey close; No iniial phase diffeence vaiaion The conas of inefeence paen When = 1, one obain he mos clea inefeence paen. A his condiion, I 1 = I.

14 Coheen Lengh Coheence and Monochomaic No coheence beween wo ligh bulbs Coheence - wo o moe waves ha mainain a consan phase elaion. Some lae ime o disance Coheence ime Coheence lengh monochomaic - a wave ha is composed of a single fequency.

15 Pa II Wavefon Spliing Inefeomees

16 Young s Double-Sli Inefeence In 1801, Thomas Young expeimenally poved ha ligh is a wave. He did so by demonsaing ha ligh undegoes inefeence, as do wae waves, sound waves, and waves of all ohe ypes.

17 Young s Double-Sli Inefeence Consucive inefeence d sin = ml, m = 0,1,,3 m = 0: zeoh ode, m =1: fis ode, ec. Desucive inefeence d sin = (m+1/)l, m = 0,1,,3

18 Bigh spo: O, Young s Double-Sli Inefeence d sin m l m = 0, 1,, 3 d an ml y d m L ll y m d ml m Disance beween finges: y Ll d y m Inensiy disibuion of he finges: I k( 1 ) 4I0 cos 4I0 cos 4I0 cos dy Ll

19 Young s Double-Sli Inefeence

20 ffecs of Finie Coheen Lengh Ligh fom each sli has a coheen lengh l c. Fo sunligh l c 3l. The waves fom wo slis can only inefee if 1 < l c. The conas of he finges degades when he amoun of he ovelap beween uncoelaed wave pockes inceases. B A P l c B A

21 Fesnel s Double Mio Slis S 1 and S ac as viual coheen souces. They ae images of sli S in he wo mios. Space beween finges: y Ll d

22 Fesnel s Double Pism Inefeence beween ligh efaced fom he uppe and he lowe pisms. The pisms poduce wo viual coheen souce S 1 and S. Quesion: Whee ae S 1 and S? L

23 Lloyd s Mio Inefeence beween ligh fom souce S and image S in he mio. Glancing incidence causes a phase shif of, heefoe he finges ae complemenay o hose of Young s. I 4I0 sin dy ll L y l d d L

24 Inefeence Lihogaphy wih a Lloyd s Mio

25 Pa III Ampliude Spliing Inefeomees

26 Thin Film Inefeence Oil Soap bubble

27 Dielecic Films Double-Beam Inefeence Finges of equal inclinaion Conside he fis wo eflecions (ohe eflecions ae weak). Opical pah lengh diffeence: n d ( AB nd n cos nd n cos cos BC ) n AC sin d nd nd sin cos cos n 1 1 i AD an sin 1 i d S n 1 n 3 n i A D B C P 7

28 Reflecion and Tansmission Coefficiens 1.0 Coefficiens Bewses angle 56.3 o Inciden Angle i

29 Reflecion and Tansmission Coefficiens 1.0 Coefficiens Bewses angle 33.7 o 41.8 o Ciical angle Inciden Angle i

30 Dielecic Films Double-Beam Inefeence Phase diffeence: Assume n > n 1, n > n 3. Thee is a phase shif of beween exenal and inenal eflecions (when he inciden angle is no lage). 4n k d cos l 4 d cos l Bigh spo: Dak spo: l d cos (m 1) 4 l d cos m 4 cos nd sin l cos

31 Dielecic Films Double-Beam Inefeence xended souce: All ays inclined a he same angle aive a he same poin. Finges of equal inclinaion: Acs ceneed on he pependicula fom he eye o he film. Haidinge finges: The finges of equal inclinaion viewed a nealy nomal incidence. Concenic cicula bands. 31

32 Dielecic Films Double-Beam Inefeence Finges of equal hickness Fizeau finges: Conous fom a non-unifom film when viewed a nealy nomal incidence. Thickness: d = x Inefeence maximum: n d m = (m + ½)l d m l l ( m 1), xm (m 1). 4 4 Disance beween finges: x = l / n 1 x d n n 3

33 Dielecic Films Double-Beam Inefeence Finges of equal hickness

34 Dielecic Films Double-Beam Inefeence Inefeence paen fom an ai film beween wo glass sufaces. Newon s ings

35 x d Thickness: Bigh ings: n d m = (m + ½)l, R x d d R d x ) ( Dielecic Films Double-Beam Inefeence Newon s ings R. m x m l Dak ings:. 1 ) 0,1, (, 4 1) ( R m x m m d m m l l

36 Pa IV Mioed Inefeomees

37 Michelson Inefeomee 1. Compensaion plae: Negaes dispesion fom he beam splie. Collimaed souce: Finges of equal hickness 3. Poin souce: Inefeence of spheical waves 4. xended souce: Finges of equal inclinaion Movable mio Beam splie P S S 1 S S M 1 M

38 Michelson Inefeomee S S 1 S P Movable mio S M 1 M Beam splie Opical pah lengh diffeence: d cos Phase diffeence: Dak finges: d cos 4 d cos l m ml Applicaion: Accuae lengh measuemen. fom he splie. ( WIU OpoLab) 38

39 Inefeence beween muliple eflecion and muliple efacions: d S i n 0 1 Muliple Beam Inefeence,... ] 1) ( [ 3) ( 0 ) ( ) ( N i N N i i i e e e e d n k d n l cos 4 cos...]} ) (... ) ( ) ( [1 {...}... { ) ( 0 1) ( 3] [ N i i i i i N i N i i i N e e e e e e e e e 1 i e ] 1 [ 0 i i i e e e

40 Muliple Beam Inefeence In he case of zeo absopion, he Sock s elaions hold: and,hen i i (1 e ) 0e [ ] i 1 e R (1 e (1 e ) (1 e ) (1 e (1 (1 cos ) ) cos i i / 0 i i 4 ) ) Using cos 1 sin ( / ), we have R (1 4 4 ) sin ( / ) [1 sin ( / )] (1 () ) sin () ( / ) sin ( / ) F sin ( / ) 1 F sin ( / ) wih F being he coefficien of F ( 1 ) finess, definded as

41 Similaly fo ansmied beams:... Thus 1 3 N T ( e 0 i 4 Muliple Beam Inefeence e e i( ) i( ) ( N 1) i 0e ) 1 (1 () ) e e i[ ( n1) ] i sin ( / ) 1 d S 0 1 F sin n i ( / )

42 Muliple Beam Inefeence Phase diffeence: 4n d cos l Tansmission: T 1 F 1 sin ( / ) Reflecion: R 1 F sin F sin ( / ) ( / ) Coefficien of finesse: Aiy funcion: F 1 1 A ( ) 1 F sin ( / ) F

43 Muliple Beam Inefeence Finge conas T R max min 0 1 T R min max 1/(1 F /(1 F) F) T R F F 1 T F = 0. ( = 0.046) F = 1 ( = 0.17) F = 00 ( = 0.87) /

44 Muliple Beam Inefeence Boh and ae funcions of l Al 50 nm SiO on Reflecance R Ai Si Wavelengh l(nm) hp:// a8b7a84fb9b&gclid=cp7dw-xnq7ocfuvp7aodj4afw

45 Only is a funcions of d Muliple Beam Inefeence 0.8 Diffeen hickness SiO on Si 0.7 Reflecance R nm 300 nm nm Wavelengh l(nm) hp:// a8b7a84fb9b&gclid=cp7dw-xnq7ocfuvp7aodj4afw

46 1) High esolving powe ) Pooype of lase caviy A( ) 1 1 F sin 1/ ( / ) acsin Half-widh of ansmission: Finesse: Faby-Peo Inefeomee F F 1 F 1 F 1 / o ealon (Fo lage F) 4 F 4n F 1 A() d cos l S P d 46

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