Refractive index of liquid film by interference

Coherent sources

Two sources are coherent when they emit waves of the same frequency (wavelength) with a constant phase difference between them over time. Only coherent sources give a stable, observable interference pattern. In practice, two coherent sources are obtained from one source by division of wavefront (Young's slits, Fresnel biprism) or division of amplitude (thin films, Newton's rings).

Determination of the refractive index of a liquid by Newton's rings

   microscope
       |
   ----+----  glass plate G (45 deg)
       |
   ____|____
   \  lens /   <- plano-convex lens, radius R
  ----------   <- film: air, then liquid
  ==========   <- glass plate

Procedure

1. Place the plano-convex lens on the plate and observe the rings in reflected monochromatic light of wavelength λ with a travelling microscope.
2. With air in the film, measure the diameters of the nth and (n + p)th dark rings. Then D²_(n+p) − D²_n = 4pλR (air, μ = 1).
3. Put a little of the liquid between the lens and the plate (the film is now liquid of index μ). The rings contract. Measure the diameters D′ of the same nth and (n + p)th rings: D′²_(n+p) − D′²_n = 4pλR / μ.
4. Divide the two results:

μ = (D²_(n+p) − D²_n) / (D′²_(n+p) − D′²_n)

Why it works: the optical path in the film is 2μt, with t = r²/2R, so the ring condition becomes D² = 4nλR/μ and the diameters shrink by a factor √μ. R and λ cancel in the ratio, so neither needs to be known.

Precautions: the lens and plate must be clean, monochromatic light must be used, and the microscope should be moved in one direction only to avoid backlash errors.