A sheet of glass is coated with a 500-nm-thick layer of oil (n = 1.42).

a. For what visible wavelengths of light do the reflected waves interfere constructively?

b. For what visible wavelengths of light do the reflected waves interfere destructively?

c. What is the color of reflected light? What is the color of transmitted light?

a) 473.33 nm

b) 568 nm and 406 nm

c) bluish green and blue

Explanation:

a) As the light traverses the layer of oil it first reflects at the front surface of the oil. Here the index of refraction increases from that of air to that of the oil, so a phase change occurs. The light then reflects from the rear surface of oil. The index of refraction increases from that of the oil to that of the glass, so again a phase change occurs. Thus two phase changes occur.

In thin-film interference with 0 or 2 phase changes, condition for constructive interference is:

2t=mλ/n

So:

λ = 2tn/m

For m=1

λ=1420 nm

For m=2

λ=710 nm

For m=3

λ=473.33 nm

For m=4

λ=355 nm

Thus the only wavelength in the visible spectrum (400 - 700 nm) that will give constructive interference is 473.33 nm

b)

In thin-film interference with 0 or 2 phase changes, condition for destructive interference is:

2t = (m+1/2) λ/n = (2m+1) * λ/2n

so;

λ=4tn / (2m+1)

For m=1

λ=946.667 nm

For m=2

λ=568 nm

For m=3

λ=405.33 nm

For m=4

λ=315.56 nm

Thus the wavelengths in the visible spectrum (400 to 700 nm) that will give destructive interference are 568 nm and 406 nm

c) The color of reflected light is bluish green since the wavelength is 473.3 nm. We know that the colors of reflected and transmitted light are complimentary to each other. Thus the color of transmitted light is blue (due to the combination of wavelengths 568 nm (green) and 406 nm (deep violet).