Newton's Rings

Consider a flat surface in contact with a spherical surface.

Figure 1

x 2 + l 2 = R 2 x 2 + l 2 = R 2

d = R − l d = R − l

d 2 = R 2 + l 2 − 2 R l = R 2 + R 2 − x 2 − 2 R l = 2 R 2 − x 2 − 2 R ( R − d ) = 2 R 2 − x 2 − 2 R 2 + 2 R d d 2 = R 2 + l 2 − 2 R l = R 2 + R 2 − x 2 − 2 R l = 2 R 2 − x 2 − 2 R ( R − d ) = 2 R 2 − x 2 − 2 R 2 + 2 R d or x 2 = 2 R d − d 2 x 2 = 2 R d − d 2 For small d d we have d 2 → 0 d 2 → 0 so x 2 ≈ 2 R d x 2 ≈ 2 R d

In this case there is no phase change at the spherical surface but there is at the flat surface. So again there is an offset in the phase and the condition of constructive interference is 2 d = n λ / 2    n = 1 , 3 , 5 , … 2 d = n λ / 2    n = 1 , 3 , 5 , … or 2 d = x 2 / R = n λ / 2    n = 1 , 3 , 5 , … 2 d = x 2 / R = n λ / 2    n = 1 , 3 , 5 , …

For a spherical interface one observes rings of light and dark. This in fact is used to check the sphericity of the lenses.