A piece of glass with index of refraction 1.50 is coated with a coating of thickness 138 nm and index of refraction 1.22. Light traveling perpendicular to the surface of the glass reflects at the interface between air and the coating, and at the interface between the coating and the glass. What is the largest vacuum wavelength for which the interference between the two reflected rays is completely destructive

Explanation:

At both interface light is travelling from a medium of lower refractive index to medium of higher refractive index. So the phase difference of 180⁰ occurring  at both interface will cancel out .

Net path difference = 2 μ t where μ is refractive index of thin layer and t is its thickness

For destructive interference

2 μ t  = (2n + 1 )λ / 2 where λ is wavelength of light , n is any number

λ = 4  μ t / ( 2 n + 1 )

For largest wavelength , n = 0

λ = 4  μ t

= 4 x 1.22 x 138 nm

= 673.44 nm .

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