A light beam is incident from a medium with n_1 through a thin layer of a substa

Question

A light beam is incident from a medium with n_1 through a thin layer of a substance with n_2 into a medium with n_3 (layer darkness in the figure does not indicate the relative magnitudes of the indices of refraction). If n_1 = 1.2, n_2 = 2.2, and n_3 =1.7, and the thickness of the middle layer is 120 nm, for which (visible) wavelength does the reflected beam undergo destructive interference? If instead n_2 = 1.7 and n_3 = 2.2, which one of the following would be true for the same wavelength? Interference would still be destructive. Interference would now be constructive. Neither constructive nor destructive.

Explanation / Answer

1)Interference occurs in this case because the reflection causes path difference of one half wavelength.The path difference for destructive interference ,

a= 2d + (0.5)(wavelength in front surface)+ (0.5)(wavelength in rear surface)

Wavelength in any medium= wavelength (air)/ n

The wavelength entering the second surface would be the original wavelength/n1

In this case the light is reflected by the front and back surfaces of medium of refractive index n2.

Hence a = (m+1/2)(wavelength /(n1*n2)) = 2d + ( wavelength/(n1*n2))

(m +(1/2)-1)(wavelength) = 2(n2*n1)d

( m -(1/2))( wavelength) = 2*2.2*1.2*120nm

  . = 6.336e-7 m

m=1 wavelength= 1268nm

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