1. Michelson Interferometer
This instrument can produce both types of interference fringes i.e., circular fringes of equal inclination at infinity and localized fringes of equal thickness 1 1 1

2. INTERFEROMETER

3. Albert Abraham Michelson
Michelson Interferometer
Albert Abraham Michelson
( )

4. Michelson-Morley Experiment
In 1878, Michelson thought the detection of motion through the ether might be measurable.
In trying to measure the speed of the Earth through the supposed "ether", you could depend upon one component of that velocity being known - the velocity of the Earth around the sun, about 30 km/s. Using a wavelength of about 600 nm, there should be a shift of about 0.04 fringes as the spectrometer was rotated 360°. Though small, this was well within Michelson's capability.
Michelson, and everyone else, was surprised that there was no shift. Michelson's terse description of the experiment: "The interpretation of these results is that there is no displacement of the interference bands. ... The result of the hypothesis of a stationary ether is thus shown to be incorrect." (A. A. Michelson, Am. J. Sci, 122, 120 (1881))

5. Experimental set up

6. Michelson Interferometer6

7. Michelson
Interferometer 7

8. Circular fringes Effective arrangement of the interferometer
An observer at the detector looking into B will see M1, a reflected image of M2(M2//) and the images S’ and S” of the source provided by M1 and M2. This may be represented by a linear configuration. 8

9. Longitudinal section –Circular fringesP rn N O q S S d D

10. In Young’s double-hole experiment:
For small qm
Radius of nth bright ring

11. Internal reflection implies that the reflection is from an interface to a medium of lesser index of refraction.
External reflection implies that the reflection is from an interface to a medium of higher index of refraction. 11

12. In Michelson interferometer
Order of the fringe:
When the central fringe is dark the order of the fringe is
As d is increased new fringes appear at the centre and the existing fringes move outwards, and finally move out of the field of view.
For any value of d, the central fringe has the largest value of m.

13. In Michelson interferometer
For central dark fringe:
The first dark fringe satisfies:
For small θ

14. Radius of nth dark ring:14

15. Haidinger Fringe

17. 1. Measurement of wavelength of light
Move one of the mirrors to a new position d’ so that the order of the fringe at the centre is changed from mo to m.

18. 2. Measurement of wavelength separation of a doublet (λ1 and λ1+λ)
If the two fringe patterns coincide at the centre: (Concordance)
The fringe pattern is very bright 18

19. Concordance

20. 2. Measurement of wavelength separation of a doublet (λ1 and λ1+λ)
As d is increased p and q increase by different amounts, with
When
the bright fringes of λ1 coincide with the dark fringes of λ1+λ, and vice-versa and the fringe pattern is washed away (Discordance).

22. Discordance
= (q+1/2)

23. 2. Measurement of wavelength separation of a doublet (λ1 and λ1+λ)
- Δ can be measured by increasing d1 to d2 so that the two sets of fringes, initially concordant, become discordant and are finally concordant again.
- If p changes to p+n, and q changes to q+(n-1) we have concordant fringes again. 23

24. Measurement of the coherence length of a spectral line
Measurement of thickness of thin transparent flakes
Measurement of refractive index of gases

25. Displacement Sensitivity: 10-16 cm
LIGO - Laser Interferometer Gravitational Wave Observatory
To detect Gravitational waves, one of the predictions of Einstein’s General Theory of Relativity
When Gravitational waves pass through the interferometer they will displace the mirrors!
Hanford Nuclear Reservation, Washington, Livingston, Louisiana
Arm length: 4 Km
Displacement Sensitivity: cm 25

26. Fabry-Perot Interferometer26 26 26

27. Fabry-Perot Interferometerθ o 30 27

28. Multiple Beam Interference28 28 28

29. Optical Reversibility and Phase Changes on Reflection
G.G. Stokes used the principle of optical reversibility to investigate the reflection of light at an interface between two media.
The reversibility principle states that
If there is no absorption of light, a light ray that is reflected or refracted will retrace its original path if its direction is reversed.

30. r and t are fractional amplitudes reflected and transmitted respectively
According to principle of reversibility, the combined effect of reversing the reflected and transmitted beams should just be the incident beam (in absence of absorption).
© SPK

31. Thin films: multiple beam interference
© SPK

32. Δ= 2μ d cos θ Path difference between rays 2 and 1 = μ (PN)
[(OS + SR)(in film)] – [OM( in air) ]
= [(PS + SR)(in film)] – [OM( in air)]
= [(PR)(in film)] – [OM( in air)]
= μ (PN + NR) – OM
= μ (PN)
= μ (OP Cos θ)
Δ= 2μ d cos θ 32

33. = aor’= - aor = aotr’t’(1+ r’ + r’ + r’ +…) = aotr’t’(1/(1 –r’ ))
CASE - I
If 2μ d cos θm = m λ
then rays 2,3,4, 5, …. are in phase
and 1 out of phase.
Amplitude of ….
= aotr’t’(1+ r’ + r’ + r’ +…) 2 4 6
= aotr’t’(1/(1 –r’ )) 2
= aotr’t’(1/tt’)
= aor’= - aor 33

34. = aor +(- aor) = 0 = ao = aott’(1+ r’ + r’ + r’ +…)
Total reflected Amplitude: 1+(2+3+4+…)
= aor +(- aor) = 0
Amplitude of transmitted beams α, β, γ, δ …
= aott’(1+ r’ + r’ + r’ +…) 2 4 6
= ao 34

35. If 2μ d cos θm = (m+1/2) λ then rays 1,2,4, 6, … are in phase
CASE - II
If 2μ d cos θm = (m+1/2) λ
then rays 1,2,4, 6, … are in phase
and 3,5,… are out of phase.
Rays α, γ, … in phase and rays β , δ, …
are out of phase

36. Optical field in reflected beam
where
: is the incident wave;
 is the phase arising from the extra optical path length.

37. Resultant reflected scalar wave
If the number of terms of the series approaches infinity, the series converges and the resultant becomes
where,

38. Reflected irradiance

39. Optical field in transmitted beam

40. Transmitted irradiance

42. For Transmitted rays
= 2mπ
Path diff.
2μd cos θm = m
= (2m+1)π
Path diff.
2μd cos θm = (2m+1)/2

43. For Reflected rays

44. Interference filter

45. An interference filter is designed for normal incidence of 488 nm light. The refractive index of the spacer is What should be the thickness of the spacer for normal incidence of light.
It will pass different wavelength if the angle of incidence is not 90o.

46. We now introduce
Coefficient of Finesse

48. Airy function A
Airy function represents the transmitted flux-density distribution.
Note:  is related to path difference .
The complementary [1 - A()] represents the reflected flux-density distribution.

49. I0
d or 
Multiple beam interference has resulted in redistribution of energy density in comparison to sinusoidal two-beam patter.

50. IR/I
IT/I
d or 

51. Variation of intensities with phase
d or  51

52. Bright fringes Transmitted rays Dark fringes Reflected rays
Dark fringes Transmitted rays
Bright fringes Reflected rays 52

53. Fabry-Perot Interferometer53 53 53

54. Fabry-Perot Interferometerθ o 30 54

55. The conditions of interference are precisely those discussed earlier
The conditions of interference are precisely those discussed earlier. With =1, the bright fringes in transmission are given by:
2d cosm= m
The radii of the rings are therefore given by the formula obtained in Michelson interferometer i.e.,
Rn ≈ D2m2 = D2n/d
However, there is an essential difference between M.I. and F.P.: One uses a two beam interference while the other uses multiple beam interference. Hence the formula for the intensities and the sharpness of the fringes are quite different.

56. The intensity is given by:
Where F is Coefficient of finesse of the mirror system.
F = (2r/(1-r2))2
and we also know that, for bright fringe : 2d cosm= m
What we can conclude from these equations:
The intensity falls on either side of the maximum.
The fall in intensity is dictated by the value of the Coefficient of finesse F.
The Coefficient of finesse is larger for values of the reflection coefficient r approaching unity. Thus very sharp rings are obtained by increasing the polish of the mirrors.

57. I0 57

58. Transmitted intensity58

59. Full width at half maximum
=IT/Io φm 59

60. wikipedia

61. When two mirrors are held fixed and adjusted for parallelism by screwing some sort of spacer, it is said to be an Etalon.
A quartz plate polished and metal-coated will also serve as an Etalon (with   1).

62. Chromatic resolving power
The ability of the spectroscope or the interferometer to separate the components of multiplets is known as chromatic resolving power (CRP).
In a two beam interferometer, like Michelson interferometer and Young’s double slit set-up, the bright fringes are as broad as the dark fringes. The fringes are not sharp.
For good resolution, the bright fringes must be as sharp as possible.

63. Michelson
fringes
Fabry-Perot
fringes 63

64. Fabry-Perot interferometer
Doublet separation in
Fabry-Perot interferometer 64

65. Resolved wavelengths
s: separation
w: width 65

66. Unresolved wavelengths66

67. Barely resolved 67

68. Chromatic resolving power of Fabry Perot interferometer
- Where, λ is the minimum wavelength interval of a doublet that the instrument is capable of barely resolving.
- The criterion for bare resolution is called the Rayleigh criterion.
- The smaller the value of λ, the higher is the resolving power of the instrument.
Barely resolved
Using: 2d cos θm= mλ ; ( Pabry-Perot - bright fringe in transmission ) 68

69. FWHM: Angular distance at which the intensity falls to half the peak intensity

70. sin(a+b)
=sin a cos b+ cos a sin b ;
&
Using 70

71. Using
F1/2 71

72. λ1= 589.0 nm λ2= 589.6 nm Δλ= 0.6 nm λ/Δλ~1000 CRP<1000
Sodium doublet
λ1= nm
λ2= nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP<1000 72

73. λ1= 589.0 nm λ2= 589.6 nm Δλ= 0.6 nm λ/Δλ~1000 CRP ~ 1000
Sodium doublet
λ1= nm
λ2= nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP ~ 1000 73

74. λ1= 589.0 nm λ2= 589.6 nm Δλ= 0.6 nm λ/Δλ~1000 CRP >1000
Sodium doublet
λ1= nm
λ2= nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP >1000 74

75. λ1= 589.0 nm λ2= 589.6 nm Δλ= 0.6 nm λ/Δλ~1000 CRP >> 1000
Sodium doublet
λ1= nm
λ2= nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP >> 1000 75

76. λ1= 589.0 nm λ2= 589.6 nm Δλ= 0.6 nm λ/Δλ~1000 CRP>>>1000
Sodium doublet
λ1= nm
λ2= nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP>>>1000 76

77. Types of fringes

78. Interference fringes
Real
Virtual
Localized
Non-localized

79. Real fringe Virtual fringe
Can be intercepted on a screen placed anywhere in the vicinity of the interferometer without a condensing lens system.
Virtual fringe
Cannot be projected onto a screen without a condensing focusing system. In this case, rays do not converge.

80. Non-localized fringe
Exists everywhere
Result of point/line source

81. Localized fringe
- Observed over particular surface
- Result of extended source

83. POHL’S INTERFEROMETER
Real
Non-localized
Virtual
Localized

84. Newton’s Ring
U<<R & U>>d