Question

White light is shone on a very thin layer of mica \((n=1.57)\), and above the mica layer, interference maxima for light of two wavelengths (and no other in between) are seen: \(516.9 \mathrm{nm}\) and \(610.9 \mathrm{nm}\). What is the thickness of the mica layer?

Short answer

Answer: The thickness of the mica layer is approximately 329.05 nm.

Step-by-step solution

Understanding interference in thin films

When white light is shone on a thin layer/film, some of the light gets reflected from the top surface of the film while the remaining light enters the film and gets reflected from the bottom surface. The light reflected from the top and the bottom surfaces can either interfere destructively or constructively depending on the phase difference between them. For constructive interference (interference maxima) to occur, the optical path difference between the two reflected rays should be an integral multiple of their wavelength. The optical path difference is affected by the thickness of the thin film and the refractive index of the material.

Formula for thin film interference

Considering the interference condition for a thin film Optical path difference = 2 * thickness * refractive index For constructive interference (maxima), Optical path difference = m * lambda, where m is the order of interference and 'lambda' is the wavelength of the light. So, we can rewrite the equation as 2 * thickness * refractive index = m * lambda We are given the refractive index (n) of mica = 1.57, and wavelengths of light for which maxima are seen (516.9 nm and 610.9 nm). We will now use this equation to find the thickness of the mica layer.

Using known values to find the thickness of the mica layer

For 516.9 nm light, let the order of interference be m1, so: 2 * thickness * 1.57 = m1 * 516.9 For 610.9 nm light, let the order of interference be m2, designate it as m2+1 since it's the next order of interference: 2 * thickness * 1.57 = (m2+1) * 610.9 Now let's divide both equations: (m2+1)*610.9 / (m1*516.9) = 1 In order to find integers for m1 and m2 we close in on m1 = 2 and m2 = 3: 3*610.9 / (2*516.9) = 1.0116 which is close to 1 Now we can use either m1 or m2 to calculate the film thickness. We'll use m1=2: 2 * thickness * 1.57 = 2 * 516.9 thickness = 516.9 / 1.57 = 329.05 nm The thickness of the mica layer is approximately 329.05 nm.