Coherent sources and Newton's ring law

Coherent sources

Two sources are coherent if they emit light waves of the same frequency with a constant phase difference between them. Two independent ordinary sources are not coherent, because their atoms emit light in random, short bursts lasting only about 10⁻⁸ s, so the phase difference keeps changing.

How to obtain them

Both coherent sources are derived from one source:

1. Division of wavefront: Young's double slit, Fresnel's biprism, Lloyd's mirror. The wavefront from a single slit is divided into two parts, which act as two (real or virtual) coherent sources.
2. Division of amplitude: the beam is partly reflected and partly transmitted by a thin film, wedge or beam splitter (Newton's rings, Michelson interferometer).

D² ∝ n in Newton's rings

              lens (radius R)
          _______|_______
          \  r  |  t   /
   ________\____|_____/_____ glass plate
             O  film

1. A plano-convex lens of radius of curvature R rests on a plane glass plate. At distance r from the contact point, the air film has thickness t.
2. Geometry: R² = (R − t)² + r² → r² = 2Rt − t² ≈ 2Rt, so t = r²/(2R).
3. In reflected light, ray 2 (reflected at the plate) gets an extra λ/2, so the path difference is 2t + λ/2 (normal incidence, air).
4. Dark ring: 2t + λ/2 = (2n+1)λ/2 → 2t = nλ.
5. Substitute t: r²/R = nλ → r² = nλR. With D = 2r:

D_n² = 4 n λ R

Since λ and R are constants, D_n² ∝ n: the squares of the diameters of the dark rings are in the ratio 1 : 2 : 3 ... of the natural numbers, so the rings crowd closer together as n increases. The centre (n = 0) is dark.