A glass lens that has an index of refraction equal to 1.57 is coated with a thin layer of transparent material that has an index
of refraction equal to 2.10. If white light strikes the lens at near-normal incidence, light of wavelengths 495 nm in air and 660 nm in air are absent from the reflected light. What is the thinnest possible layer of material for which this can be accomplished?
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When light moves from a medium with higher refractive index to a medium with lower refractive index, the critical angle is the angle above which there is no refracted light, and all the light is reflected. The value of this angle is given by
where n2 and n1 are the refractive indices of the second and first medium, respectively.
In the first part of the problem, light moves from glass to air (
) and the critical angle is 
. This means that we can find the refractive index of glass by re-arranging the previous formula:
Now the glass is put into water, whose refractive index is
. If light moves from glass to water, the new critical angle will be
where n2 and n1 are the refractive indices of the second and first medium, respectively.
In the first part of the problem, light moves from glass to air (
Now the glass is put into water, whose refractive index is