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

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

Interferometers

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

Michelson Morley

Photograph of Michelson Interferometer

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.

Schematic diagram

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.

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.

Haidinger Fringe

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)

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

Fringe system at concordance

2𝑑1=𝑝𝜆1=𝑞 (𝜆1+∆ 𝜆   )

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).

Fringe patterns at discordance

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 )

•Measurement of the coherence length of a spectral line

•Measurement of thickness of thin transparent flakes

•Measurement of refractive index of gases

Newton’s rings

Pair of rays method

Condition for dark rings

2 /n nt r R n

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

A

𝑟𝑛2 ≈𝑈 2𝜃𝑛

2=2𝑛𝜆𝑈 2

𝑑=2𝑛𝜆𝑈 2

( 2𝑈 2

𝑅 )=𝑛𝜆𝑅

⟹𝑟𝑛=√𝑛𝜆𝑅  

Newton’s rings

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