The rhinestones in costume jewelry are glass with index of refraction 1.50. To make them more reflective, they are often coated with a layer of silicon monoxide of index of refraction 2.00. What is the minimum coating thickness needed to ensure that light of wavelength 527 nm and of perpendicular incidence will be reflected from the two surfaces of the coating with fully constructive interference

 t = 6.6 10⁻⁸ m

Explanation:

In this case we have that the reflection in the film is constructive, we take into account that

* When the light is reflected from the air (n = 1) to the film (n=2) it has a phase change of 180º

* within the film the wavelength is modulated by the index of refact

          λₙ = λ₀ / n

taking into account these two aspects the expression for constructive interference is

         2n t = (m + ½)  λ₀

let's find the minimum thickness (t) of the film

         t = (m + ½)  λ₀ / 2n

for m = 0

        t = ½   527 10⁻⁹ /(2 2.00)

        t = 6.6 10⁻⁸ m

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