White light is sent downward onto a horizontal thin film that is sandwiched betw

Question

White light is sent downward onto a horizontal thin film that is sandwiched between two materials. The indexes of refraction are 1.86 for the top material, 1.71 for the thin film, and 1.53 for the bottom material. The film thickness is 5.22 times 10^-7 m. Of the visible wavelengths (400 to 700 nm) that result in fully constructive interference at an observer above the film, which is the (a) longer and (b) shorter wavelength? The materials and film are then heated so that the film thickness increases. (c) Does the light resulting in fully constructive interference shift toward longer or shorter wavelengths?

Explanation / Answer

The thickness of the film, t = 5.22×10^-7 m
The index of refraction for thin film, n = 1.71

From interference in a thin film, the condition for the constructive interference is
2×n×t = m×lambda
The wavelength of the light is
Lambda = 2×n×t/m

for m = 1,
Lambda_1 = 2nt/1
= 2×1.71×5.22×10^-7
= (1785.24×10^-9 m)(1 nm/10-9 m)
= 1785.24 nm

for m = 2,
Lambda _2 = 2nt/2
= 1.71×5.22×10^-7
= 892.62 nm

for m = 3,
Lambda_3 = 2nt/3
= 2×1.71×5.22×10^-7/3
= 595.08 nm

for m = 4,
Lambda_4 = 2nt/4
= 2×1.71×5.22×10^-7/4
= 446.31 nm
  
  
(a)
From above results, the longer wavelength in region
between 400 nm to 700 nm is
= 595.08 nm

(b)
The shorter wavelength is
= 446.31 nm

(c) there is a shift towards longer wavelengths.

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