1. Interference: Division of wave fronts
Young’s double slit arrangement
Lloyd’s mirror
Fresnel’s mirrors
Fresnel’s bi-prism
Billet’s split lens
Meslin’s split lens

2. Interference: Division of amplitude
Pohl’s patterns
Fringes from a transparent film
Fringes from a wedge
Michelson’s interferometer
Newton’s rings
Fabry-Perot interferometer

3. Interferometers

4. Albert Abraham Michelson (1852-1931)
Michelson measured the velocity of light with amazing delicacy in 1881.
Michelson and Morley showed that the light travels at a constant speed in all inertial frames of reference.
Michelson measured the standard meter in terms of wavelength of cadmium light
( The Nobel Prize: 1907)

5. Michelson
Morley

6. Photograph of Michelson Interferometer

7. 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.

8. Schematic diagram

10. Effective linear arrangement for circular fringe
An observer looking through the telescope will see 𝑀 1 ′ , a reflected image of M1 and the images S’ and S” of the source provided by 𝑀 1 ′ and M2.

11. Order of the fringe, if the central fringe is dark:
Order of fringes
2𝑑 cos 𝜃 𝑚 =𝑚𝜆, 𝑚=0, 1, 2, 3, …, 𝑚 0
Minima:
2𝑑 cos 𝜃 𝑚 =(𝑚+1/2)𝜆, 𝑚=0, 1, 2, 3, …, 𝑚 0
Maxmima:
Order of the fringe, if the central fringe is dark:
𝑚 0 =2𝑑/𝜆
For any value of d, the central fringe has the largest value of m.
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.

13. Haidinger Fringe

15. Measurement of wavelength of light
2𝑑 cos 𝜃= 𝑚𝜆
2𝑑= 𝑚 0 𝜆 (𝜃=0)
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.
2 𝑑 ′ =𝑚𝜆
⟹2 𝑑− 𝑑 ′ = 𝑚 0 −𝑚 𝜆=𝑛𝜆
⟹𝜆=2∆𝑑/(∆𝑚)

16. C0ncordance 2 𝑑 1 =𝑝 𝜆 1 =𝑞 𝜆 1 +∆𝜆 2𝑑= 𝑚 0 𝜆
If fringe patterns due to two wavelengths coincide at the centre,
2 𝑑 1 =𝑝 𝜆 1 =𝑞 𝜆 1 +∆𝜆
The fringe pattern is very bright⟹Concordance
2𝑑= 𝑚 0 𝜆

18. Fringe system at concordance
2 𝑑 1 =𝑝 𝜆 1 =𝑞 𝜆 1 +∆𝜆

19. Measurement of wavelength separation
2𝑑= 𝑚 0 𝜆
2 𝑑 1 =𝑝 𝜆 1 =𝑞 𝜆 1 +∆𝜆
The condition for concordance:
As d is increased 𝑝→𝑝+∆𝑝and 𝑞→𝑞+∆𝑞 with ∆𝑞< ∆𝑝.
When 𝑞=𝑝−1/2, the bright fringe pattern of λ1coincides with the dark fringe pattern of λ1+λ, and vice-versa and the fringe pattern is washed away (Discordance).
The fringe pattern is dark⟹Discroncordance

20. Fringe patterns at discordance

21. 2 𝑑 1 =𝑝 𝜆 1 =𝑞 𝜆 1 +∆𝜆 2 𝑑 2 =(𝑝+𝑛) 𝜆 1 =(𝑞+𝑛−1) 𝜆 1 +∆𝜆
The condition for concordance:
2 𝑑 1 =𝑝 𝜆 1 =𝑞 𝜆 1 +∆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.
2 𝑑 2 =(𝑝+𝑛) 𝜆 1 =(𝑞+𝑛−1) 𝜆 1 +∆𝜆
⟹2( 𝑑 2 −𝑑 1 )=𝑛 𝜆 1 =(𝑛−1) 𝜆 1 +∆𝜆
⟹2 𝑑 2 −𝑑 1 =𝑛 𝜆 1 & 𝑛−1 ∆𝜆= 𝜆 1
⟹2 𝑑 2 −𝑑 1 =𝑛 𝜆 1 ≈ 𝜆 1 2 ∆𝜆 𝑓𝑜𝑟 𝑛≫1
⟹∆𝜆≈ 𝜆 1 2 /2 𝑑 2 −𝑑 1

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

23. Newton’s rings

24. Pair of rays method
Condition for dark rings

25. 1 𝑉 = 1 𝑈 − 2 𝑅 = 𝑅−2𝑈 𝑅𝑈
⟹𝑉= 𝑅𝑈 𝑅−2𝑈
𝑑=𝑉−𝑈= 𝑅𝑈 𝑅−2𝑈 −𝑈≈ 2 𝑈 2 𝑅 , 𝑎𝑠 𝑅≫𝑈
𝑆 1 𝑁 =𝑑 cos 𝜃 𝑚 ≈𝑑 1− 𝜃 𝑚 =𝑚𝜆 𝑓𝑜𝑟 𝑎 𝑑𝑎𝑟𝑘 𝑟𝑖𝑛𝑔
𝐴𝑠 𝑑 𝜆 = 𝑚 0 , 𝑚 0 −𝑚=𝑑 𝜃 𝑚 2 /2𝜆⟹
𝜃 𝑛 2 =2𝑛𝜆/𝑑

26. A𝑠 𝑑≪𝑈,
𝑟 𝑛 2 ≈ 𝑈 2 𝜃 𝑛 2 = 2𝑛𝜆 𝑈 2 𝑑 = 2𝑛𝜆 𝑈 𝑈 2 𝑅 =𝑛𝜆𝑅
⟹𝑟 𝑛 = 𝑛𝜆𝑅

27. Newton’s rings

28. Measurement of R.I. of a liquid
𝑡 𝑛 = 𝑟 𝑛 2 2𝑅 ⟹𝑠𝑎𝑔𝑖𝑡𝑡𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎
For an air film,
2 𝑡 𝑛 =𝑛𝜆⟹ 𝑟 𝑛 2 =2 𝑡 𝑛 𝑅=𝑛𝜆𝑅
For a liquid film of R.I. μ,
2𝜇 𝑡 𝑛 =𝑛𝜆⟹2𝜇 𝑟 𝑛 2 /2𝑅=𝑛𝜆⟹ 𝑟 𝑛 2 =𝑛𝜆𝑅/𝜇
𝑟 𝑛 2 𝑟 𝑛 2 =𝜇