A. Two Planes Waves, Same Frequency Visible light

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Ἀντιόπη Γκόφας
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1 Interference 1

2 A. Two Planes Waves, Same Frequency EE 1 rr, tt = EE 0,1 cccccc αα 1 ωω tt αα 1 kk 1. rr + εε 1 EE 2 rr, tt = EE 0,2 cccccc αα 2 ωω tt αα 2 kk 2. rr + εε 2 ωω = HHHH Visible light EE rr, tt = EE 1 rr, tt + EE 2 rr, tt = EE 0,1 cccccc αα 1 ωω tt + EE 0,2 cccccc αα 2 ωω tt

3 SS = nn εε oo cc EE 2 EE 2 = EE. EE = EE 1 + EE 2. EE 1 + EE 2 = EE 1. EE 1 + EE 2. EE EE 1. EE 2 3

4 EE 2 = EE 1. EE 1 + EE 2. EE EE 1. EE 2 = EE 1. EE 1 + EE 2. EE EE 1. EE 2 = EE 0,1 cccccc αα 1 ωω tt. EE 0,1 cccccc αα 1 ωω tt + EE 0,2 cccccc αα 2 ωω tt. EE 0,2 cccccc αα 2 ωω tt + 2 EE 0,1 cccccc αα 1 ωω tt. EE 0,2 cccccc αα 2 ωω tt 4

5 EE 2 = EE 0,1 2 cccccc αα1 ωω tt cccccc αα 1 ωω tt + EE 0,2 2 cccccc αα2 ωω tt cccccc αα 2 ωω tt + 2 EE 0,1. EE 0,2 cccccc αα 1 ωω tt cccccc αα 2 ωω tt = EE 0, cccccc 2αα 1 2ωω tt + EE 0, cccccc 2αα 2 2ωω tt + 2 EE 0,1. EE 0, cccccc AA cccccc BB = 1 2 cccccc AA BB + 1 cccccc AA + BB 2 2 cccccc αα 1 αα cccccc αα 1 + αα 2 2ωω tt 5 0

6 EE 2 = 1 2 EE 0, EE 0,2 2 + EE 0,1. EE 0,2 cccccc αα 1 αα 2 II EE 2 II 1 EE 1 2 = 1 2 EE 0,1 2 II 2 EE 2 2 = 1 2 EE 0,2 2 cccccc γγ EE 0,1. EE 0,2 δδ αα 1 αα 2 II = II 1 + II II 1 II 2 cccccc γγ cccccc δδ 6

7 kk 1 kk 1 kk 1 θθ θθ θθ kk 2 kk 2 γγ = θθ kk 2 cccccc γγ = 0 = 1 cccccc γγ = θθ 1 cccccc γγ = 90 = 0 Ideal to observe interference, regardless of θθ Will attenuate interference term, attenuation will depend on θθ No interference term, regardless of θθ 7

8 Assume from now on: cccccc γγ = 1 II = II 1 + II II 1 II 2 cccccc δδ δδ = kk 1. rr + εε 1 kk 2. rr εε 2 cccccc δδ = cccccc kk 1 kk 2. rr + εε 1 εε 2 rr = 0 cccccc δδ = cccccc εε 1 εε 2 8

9 Two waves from two independent light sources TT 1 νν tt TT tt averaging time = tt > TT δδ = RRRRRRRRRRRR 0, 2ππ cccccc δδ = 0 II = II 1 + II 2 +2 II 1 II 2 cccccc δδ 9

10 Two waves from one single light source LL 1 LL 2 TT tt averaging time = tt > TT tt δδ = RRRRRRRRRRRR 0, 2ππ cccccc δδ = 0 II = II 1 + II 2 +2 II 1 II 2 cccccc δδ 10

11 εε 1 εε 2 = εε εε/ωω < TT εε = 2 ππ λλ oo 2 nn LL 1 2 nn LL 2 LL 1 LL 2 tt TT averaging time = tt > TT tt εε cccccc δδ = cccccc εε ωω < TT { II = II 1 + II II 1 II 2 cccccc δδ cccccc εε 11

12 Required condition to observe interference εε < ωω TT εε = 2 ππ λλ oo 2 nn LL 1 2 nn LL 2 = 2 ππ λλ oo OOOOOO 1 OOOOOO 2 ωω TT = ωω cc cc TT = 2 ππ λλ oo ll OOOOOO 1 OOOOOO 2 = OOOOOO < ll II = II 1 + II II 1 II 2 cccccc εε 2 ππ OOOOOO λλ 1 OOOOOO 2 oo 12

13 From now on: OOOOOO 1 OOOOOO 2 = OOOOOO < ll cccccc δδ = cccccc δδ = cccccc kk 1 kk 2. rr + εε 1 εε 2 13

14 II = II 1 + II II 1 II 2 cccccc δδ Constructive interference cccccc δδ = 1 δδ = 0, ±2ππ, ±4ππ, 2 II mmmmmm = II 1 + II II 1 II 2 = II 1 + II 2 Destructive interference cccccc δδ = 1 δδ = ±ππ, ±3ππ, ±5ππ, II mmmmmm = II 1 + II 2 2 II 1 II 2 = II 1 II

15 Interference of two plane waves yy yy kk 1 θθ θθ kk 1 kk 2 xx kk 2 δδ = kk 1 kk 2. rr + εε 1 εε 2 = 2 kk ssssss θθ y + εε 1 εε 2 II = II 1 + II II 1 II 2 cccccc 2 kk ssssss θθ y + εε 1 εε 2 15

16 II = II 1 + II II 1 II 2 cccccc 2 kk ssssss θθ y + εε 1 εε 2 Constructive interference δδ = 2 kk ssssss θθ y + εε 1 εε 2 = 0, ±2ππ, ±4ππ, II mmmmmm = II 1 + II II 1 II 2 = II 1 + II 2 2 Destructive interference δδ = 2 kk ssssss θθ y + εε 1 εε 2 = ±ππ, ±3ππ, ± 5ππ, II mmiiii = II 1 + II 2 2 II 1 II 2 = II 1 II

17 Separations between two constructive (destructive) interference fringes δδ = 2 ππ = 2 kk ssssss θθ y yy = 2 2 ππ λλ ssssss θθ y Λ y Λ = λλ 2 ssssss θθ 17

18 Visibility: contrast between bright and dark fringes VVVVVVVVVVVVVVVVty II mmmmmm II mmmmmm II mmmmmm + II mmmmmm = 2 II 1 II 2 II 1 + II 2 18

19 Mathematica 19

20 Sub-Micron Surface-Relief Grating 1) Holographic Exposure 2) Photoresist Development He-Cd 442 nm Loyd s mirror He-Ne λ = nm developer tank photoresist detector 3) Ion Beam Etching 4) Surface Relief Grating 20

21 B. Interference of two spherical waves 21

22 rr 1 rr 2 δδ αα 1 αα 2 = k rr 1 k rr 2 + εε 1 εε 2 = k rr 1 rr 2 + εε 1 εε 2 22

23 II = II 1 + II II 1 II 2 cccccc kk rr 1 rr 2 + εε 1 εε 2 Constructive interference δδ = kk rr 1 rr 2 + εε 1 εε 2 = 0, ±2ππ, ±4ππ, hyperbola II mmmmmm = II 1 + II II 1 II 2 = II 1 + II 2 2 Destructive interference δδ = kk rr 1 rr 2 + εε 1 εε 2 = ±ππ, ±3ππ, ± 5ππ, hyperbola II mmiiii = II 1 + II 2 2 II 1 II 2 = II 1 II

24 How to create coherent waves? Take a single light source and apply one the following approaches: a) Wavefront Division Interferometer b) Amplitude Division Interferometer LL 1 LL 2 24 OOOOOO < ccccccccccccccccc lllllllllll

25 Wavefront Division Interferometer 25

26 Young s interferometer 26

27 Mathematica 27

28 Phase difference in Young s interferometer yy aa 2 rr 2 θθ RR ss RR ss 2 + yy 2 aa 2 rr rr 1 = ss 2 + yy + aa 2 rr 2 = ss 2 + yy aa 2 aa RR 11 rr 1 rr 2 aa yy RR = aa ssssss θθ II = II 1 + II II 1 II 2 cccccc kk aa yy RR + εε 1 εε 2 28

29 Geometrical visualization of far-field approximation aa RR 11 rr 1 rr 2 aa yy RR = aa ssssss θθ rr 2 yy aa θθ θθ rr 1 rr 2 aa ssssss θθ RR rr 1 ss 29

30 Constructive interference: bright fringes δδ = kk aa yy bb RR + εε 1 εε 2 = 0, ±2ππ, ±4ππ, = mm ee ππ mm ee = 0, ±2, ±4, yy bb = mm ee λλ RR 2 aa + εε 2 εε 1 λλ RR 2 ππ aa = mm ee λλ RR 2 aa + yy 0 yy 0 εε 2 εε 1 λλ RR 2 ππ aa 30

31 II = II 1 + II II 1 II 2 cccccc kk aa yy RR + εε 1 εε 2 Constructive interference δδ = kk aa yy bb RR + εε 1 εε 2 yy bb = mm ee λλ RR 2 aa + yy 0 = 0, ±2ππ, ±4ππ, = mm ee ππ mm ee = 0, ±2, ±4, Destructive interference δδ = kk aa yy dd RR + εε 1 εε 2 yy dd = mm oo λλ RR 2 aa + yy 0 = ±ππ, ±3ππ, ± 5ππ, mm oo = ±1, ±3, 31

32 Separation between two constructive (destructive) interference fringes δδ = 2 ππ = kk aa y RR Λ y Λ = λλ RR aa 32

33 i) Same phase at input screen εε 2 = εε 1 yy 0 εε 2 εε 1 λλ RR 2 ππ aa = 0 yy rr 2 yy 0 rr 1 II = II 1 + II II 1 II 2 cccccc kk aa yy RR = II 1 II II1 II 2 cccccc 2 kk aa yy 2 33 RR

34 ii) Phase delay in one input port εε 2 εε 1 = 2 ππ λλ oo nn gg nn dd gg yy 0 εε 2 εε nn 1 gg dd gg εε 2 εε 1 = nn gg rr 2 nn 1 nn gg nn dd gg ll λλ RR 2 ππ aa RR dd gg aa yy yy 0 nn rr 1 II = II 1 + II II 1 II 2 cccccc kk aa yy yy 0 RR 2 = II 1 II II1 II 2 cccccc 2 kk aa yy yy 0 2 RR34

35 iii) Inclination of incident wave yy εε 2 εε 1 = 2 ππ λλ aa ssssss θθ ii yy 0 εε 2 εε 1 θθ ii rr 2 aa θθ ii RR yy 0 εε 2 εε 1 = RR ssssss θθ ii λλ RR 2 ππ aa θθ ii rr 1 II = II 1 + II II 1 II 2 cccccc kk aa yy yy 0 RR 2 = II 1 II II1 II 2 cccccc 2 kk aa yy yy 0 2 RR35

36 Extended light source yy dd,1 = λλ RR 2 aa yy 0 = RR ssssss θθ ii yy aa θθ ii yy dd,1 = yy 0 ssssss θθ ii < λλ 2 aa θθ ii ssssss θθ ii = λλ 2 aa Interference pattern will be observed. ssssss θθ ii > λλ 2 aa Interference pattern will be washed out. 36

37 Spatial coherence for extended light sources θθ ii aa aa λλ 2 ssssss θθ ii spatially coherent λλ 2 ssssss θθ ii = 0.55 μμμμ 2 ssssss μμμμ 37

38 Michelson Stellar Interferometer aa VVVVVVVVVVVVVVVVVVVV aa = II mmmmmm II mmmmmm II mmmmmm + II mmmmmm IIIIIIIIIIIIIIIIII II mmmmmm VVVVVVVVVVVVVViitttt aaa = 0 Angular width of star: II mmmmmm aaa aa 2 ssssss θθ ii = λλ aa 38

39 Other examples of wavefront division interferometers Fresnel double prism Fresnel double mirror Loyd s mirror 39

40 Amplitude Division Interferometer LL 1 LL

41 Pathlength for wave (a) (aa) (bb) d nn 1 nn ff θθ A D θθ ff θθ C (aa) nn 2 AAAA = ssssss θθ 2 ππ λλ oo OOOOOO aa = 2 ππ λλ oo AAAA B nn 1 AAAA = 4 ππ λλ oo AAAA 2 dd = tttttt θθ ff nn ff dd ssssss2 θθ ff cccccc θθ ff 41

42 Pathlength for wave (b) (aa) (bb) d nn 1 nn ff θθ A D θθ ff θθ C (bb) nn 2 AAAA = BBCC B AAAA = 2 ππ λλ oo OOOOOO bb = 2 ππ λλ oo nn ff AAAA + BBBB = 4 ππ λλ oo dd cccccc θθ ff nn ff dd 1 cccccc θθ ff 42

43 Pathlength difference (aa) (bb) nn 1 θθ d nn ff θθ ff nn 2 2 ππ λλ oo OOOOOO bb OOOOOO aa = 4 ππ λλ oo nn ff dd cccccc θθ ff 43

44 Total phase difference (aa) (bb) nn 1 θθ d nn ff θθ ff nn 2 δδ = 4 ππ λλ oo nn ff dd cccccc θθ ff + φφ rr,bb φφ rr,aa pathlength reflection 44

45 Examples 45

46 1. Soap-water film (aa) (bb) nn 1 θθ nn 1 d nn ff θθ ff nn ff nn 2 nn 2 δδ = OOOOOO bb OOOOOO aa + φφ rr,bb φφ rr,aa = 4 ππ λλ oo nn ff dd cccccc θθ ff + 0 π = mm eeeeeeee π mm oooooo π constructive interference destructive interference 46

47 nn ff dd cccccc θθ ff = mm oooooo λλ oo 4 mm eeeeeeee λλ oo 4 constructive interference destructive interference 47 dd 47

48 2. Anti-reflection coating (aa) (bb) nn 1 θθ nn 1 d nn ff θθ ff nn ff nn 2 nn 2 δδ = OOOOOO bb OOOOOO aa + φφ rr,bb φφ rr,aa = 4 ππ λλ oo nn ff dd cccccc θθ ff + π π = mm eeeeeeee π mm oooooo π constructive interference destructive interference 48

49 nn ff dd cccccc θθ ff = mm eeeeeeee λλ oo 4 mm oooooo λλ oo 4 constructive interference destructive interference 49

50 3. Air gap on wedges cccccc θθ ff 1 nn gggg nn aaaaaa xx nn gggg nn ff dd = mm oooooo λλ oo 4 constructive interference xx mm eeeeeeee λλ oo 4 destructive interference dd HH LL xx 50

51 4. Newton s rings nn ff dd = dd xx2 2 RR mm oooooo λλ oo 4 mm eeeeeeee λλ oo 4 constructive interference destructive interference xx xx xx 51

52 5. Surface flatness 52

53 6. Michelson interferometer LL 1 LL

54 Laser Interferometer Gravitational-Wave Observatory (LIGO) 54

55 55

56 Requirement: L m

57 Michelson interferometer 57

58 Mirror 1 δδ = 4 ππ λλ oo OOOOOO 2 OOOOLL 1 cccccc θθ ff OOOOOO 1 Mirror 1 OOOOOO 2 Mirror 2 II = II 1 + II II 1 II 2 cccccc δδ 58

59 cccccc θθ ff 1 nn, ll δδ = 4 ππ λλ oo OOOOOO 2 OOOOOO 1 = 4 ππ λλ oo nn ll = 2 ππ mm nn ll = λλ oo 2 mm 59

60 7. Mach Zehnder interferometer 60

61 Application: integrated optical modulator 61

62 7. Sagnac interferometer 62

63 Application: Fiber optic gyroscope 63

64 What happen if: 100 mmmm 100 mmmm 90 mmmm 10 mmmm??? mirror mirror two identical 10 mmmm mirrors 90 mmmm??? 100 mmmm 64

65 1 ρρ 1 ρρ2 ρρ 3 ρρ 1 = rr ρρ 4 ρρ 2 = tt ee iiδδ rr ee iiδδ ttt ρρ 3 = tt ee iiδδ rr ee iiδδ rr ee iiδδ rrree iiδδ ttt nn 1 ρρ 4 = tt ee iiδδ rr ee iiδδ rr ee iiδδ rrree iiδδ 2 ttt nn 2 nn 1 ττ 1 = tt ee iiδδ tt ττ 2 = tt ee iiδδ rr ee iiδδ rrree iiδδ tt ττ 1 ττ 2 ττ 3 ττ 3 = tt ee iiδδ rr ee iiδδ rrree iiδδ 2 tt ττ 4 ττ 4 = tt ee iiδδ rr ee iiδδ rrree iiδδ 3 tt 65

66 ρρ ττ ii = rr + tt ee iiδδ rr ee iiδδ ttt 1 + rr ee iiδδ rrree iiδδ ii=1 + rr ee iiδδ rrree iiδδ 2 + rr ee iiδδ rrree iiδδ 3 + = rr + tt eeiiδδ rr ee iiδδ ttt 1 rr ee iiδδ rrree iiδδ ττ ττ ii = tt ee iiδδ tt 1 + rr ee iiδδ rrree iiδδ ii=1 + rr ee iiδδ rrree iiδδ 2 + rr ee iiδδ rrree iiδδ 3 + tt ee iiδδ tt = 1 rr ee iiδδ rrree iiδδ 66

67 ττ = tt ee iiδδ tt 1 rr ee iiδδ rrree iiδδ rr rr ee iiδδ rr = tt ee iiδδ tt 1 rr 2 ee 2iiδδ rr ee 2iiδδ ττ 2 = TT RR 2 2RR cccccc 2δδ rr + 2δδ cccccc 2δδ rr + 2δδ = 1 2ssssss 2 δδ rr + δδ = TT 2 1 RR 2 + 4RR ssssss 2 δδ rr + δδ FF 4RR 1 RR 2 = 2 TT 1 RR 1 + FF ssssss 2 δδ rr + δδ 67

4RR FF 1 RR 2 R F 1.00E-01 4.94E-01 5.00E-01 8.00E+00 8.00E-01 8.00E+01 9.00E-01 3.60E+02 9.50E-01 1.

68 4RR FF 1 RR 2 R F 1.00E E E E E E E E E E E E E E+09 ττ 2 = 2 TT 1 RR 1 + FF ssssss 2 δδ rr + δδ ττ 2 TT 1 RR 2 δδ rr + δδ 68

69 Peaks δδ = mm ππ δδ rr ππ δδ = 2 ππ λλ oo nn 2 dd cccccc θθ ff = 2 ππ νν cc nn 2 dd cccccc θθ ff 2 ππ vv mm cc nn 2 dd cccccc θθ ff = mm ππ δδ rr δδ = 2 ππ νν cc nn 2 dd cccccc θθ ff = ππ νν FFFFFF = cc 2 nn 2 dd cccccc θθ ff vv mm = mm δδ rr ππ νν FFFFFF 69

70 Full Width at Half Maximum (FWHM) γγ FF ssssss 2 mm ππ + γγ 2 = 1 ssssss2 γγ 2 = 1 FF γγ 2 FF 2 ππ νν cc nn 2 dd cccccc θθ ff = 2 FF νν FFFFFFFF = 2 νν FFFFFF ππ FF 70

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