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Interference: Division of wave fronts1. Young’s double slit arrangement 2. Lloyd’s mirror 3. Fresnel’s mirrors4. Fresnel’s bi-prism5. Billet’s split lens6. Meslin’s split lens
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Interference: Division of amplitude
1. Pohl’s patterns2. Fringes from a transparent film3. Fringes from a wedge4. Michelson’s interferometer5. Newton’s rings 6. Fabry-Perot interferometer
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Interferometers
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Albert Abraham Michelson (1852-1931)
( The Nobel Prize: 1907)
• 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
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Michelson Morley
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Photograph of Michelson Interferometer
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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.
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Schematic diagram
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Effective linear arrangement for circular fringe
An observer looking through the telescope will see , a reflected image of M1
and the images S’ and S” of the source provided by and M2.
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Order of fringes 1, 2, 3, …, Minima:
1, 2, 3, …, Maxmima:
Order of the fringe, if the central fringe is dark: 𝑚0=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.
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Haidinger Fringe
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Measurement of wavelength of light
2𝑑 cos𝜃=¿𝑚𝜆¿2𝑑=𝑚0 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.
2𝑑 ′=𝑚 𝜆
⟹2|𝑑−𝑑′|=|𝑚0−𝑚|𝜆=𝑛𝜆
⟹𝜆=2 ∆ 𝑑 /(∆𝑚)
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C0ncordance
2𝑑1=𝑝𝜆1=𝑞 (𝜆1+∆ 𝜆 )
2𝑑=𝑚0 𝜆
If fringe patterns due to two wavelengths coincide at the centre,
The fringe pattern is very bright
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Fringe system at concordance
2𝑑1=𝑝𝜆1=𝑞 (𝜆1+∆ 𝜆 )
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Measurement of wavelength separation2𝑑=𝑚0 𝜆
2𝑑1=𝑝𝜆1=𝑞 (𝜆1+∆ 𝜆 )The condition for concordance:
The fringe pattern is d
As d is increased and with
When 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).
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Fringe patterns at discordance
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The condition for concordance: 2𝑑1=𝑝𝜆1=𝑞 (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.
2𝑑2=(𝑝+𝑛)𝜆1=(𝑞+𝑛−1)(𝜆1+∆ 𝜆 )⟹2(𝑑2−𝑑1)=𝑛𝜆1=(𝑛−1)(𝜆1+∆ 𝜆 )⟹2 (𝑑2 −𝑑1 )=𝑛𝜆1∧(𝑛−1 ) ∆ 𝜆=𝜆1
⟹2 (𝑑2 −𝑑1 )=𝑛𝜆1 ≈𝜆1
2
∆ 𝜆𝑓𝑜𝑟 𝑛≫1
⟹∆ 𝜆≈ 𝜆12/2 (𝑑2 −𝑑1 )
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•Measurement of the coherence length of a spectral line
•Measurement of thickness of thin transparent flakes
•Measurement of refractive index of gases
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Newton’s rings
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Pair of rays method
Condition for dark rings
2 /n nt r R n
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1𝑉
=1𝑈
−2𝑅
=𝑅− 2𝑈𝑅𝑈
⟹𝑉=𝑅𝑈
𝑅−2𝑈
𝑑=𝑉 −𝑈= 𝑅𝑈𝑅−2𝑈
−𝑈 ≈2𝑈 2
𝑅,𝑎𝑠 𝑅≫𝑈
𝑆1𝑁=𝑑cos𝜃𝑚≈ 𝑑(1−𝜃𝑚
2
2 )=𝑚𝜆 𝑓𝑜𝑟 𝑎𝑑𝑎𝑟𝑘𝑟𝑖𝑛𝑔
𝐴𝑠𝑑𝜆
=𝑚0 ,𝑚0 −𝑚=𝑑𝜃𝑚2 /2𝜆⟹𝜃𝑛
2 =2𝑛𝜆/𝑑
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A
𝑟𝑛2 ≈𝑈 2𝜃𝑛
2=2𝑛𝜆𝑈 2
𝑑=2𝑛𝜆𝑈 2
( 2𝑈 2
𝑅 )=𝑛𝜆𝑅
⟹𝑟𝑛=√𝑛𝜆𝑅
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Newton’s rings
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Measurement of R.I. of a liquid
=
𝑡𝑛=𝑟𝑛
2
2𝑅⟹𝑠𝑎𝑔𝑖𝑡𝑡𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎
2 𝑡𝑛=𝑛𝜆⟹𝑟𝑛2 =2 𝑡𝑛𝑅=𝑛𝜆𝑅
For an air film,
For a liquid film of R.I. μ,
𝑟𝑛2
~𝑟𝑛2 =𝜇