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To decrease light losses due to reflection from the glass surface the latter is coated with a thin layer of substance whose refractive index n' = sqrt(n), where n is the refractive index of the glass. In this case the amplitudes of electromagnetic oscillations that coating is the glass reflectivity in the direction of the normal equal to zero for light with wavelength lambda? |
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Answer» Solution :When the glass surface is coated with a material of `R.I. n' = sqrt(n) (n = R.I.` of glass) of appropriate thickness, reflection is zero because of interference between various multiply reflected waves. We shown this below. Let a wave of unit amplitide be normal incident from the left. the reflected amplitude is `-r` where `r = (sqrt(n) - 1)/(sqrt(n) + 1)` Its phase is `-ve` so we WRITE the reflected wave as `-r`. The transmitted wave has amplitude `t` `t = (2)/(1 + sqrt(n))` This wave is reflected at the SECOND face and has amplitude -TR (beacuse `(n - sqrt(n))/(n + sqrt(n)) = (sqrt(n) - 1)/(sqrt(n) + 1.))` The emergent wave has amplitude `-tr' r`. We prove below that `-tt' = 1- r^(2)`. there is also a reflected part of emplitude `tr r' = -tr^(2)`, where `r`' is the reflection coefficient for a ray incident from the coating towards air. After reflection from the second face a wave of amlitude `+ tt' r^(3) = + (1 - r^(2))r^(3)` emerges. Let `del` be the pahse of the wave after TRAVERSING the coacting both ways. Then the complete reflected wave is `-r-(1 - r^(2)) re^(i del) + (1 - r^(2)) r^(3) e^(2i del)` `-(1- r^(2)) r^(5) e^(3i del)`........ `= -r- (1- r^(2)) re^(idel) (1)/(1 + r^(2) e^(i del))` `= -r[1 + r^(2) e^(idel) + (1 -r^(2)) e^(i del)](1)/(1 + r^(2) e^(i del))` `= -r(1 + e^(idel))/(1 + r^(2) e^(idel))` This vanishes if `del = (2k + 1) pi`. But `del = (2pi)/(lambda) 2 sqrt(n) d` so `d = (lambda)/(4 sqrt(n)) (2k + 1)` We now deduce `tt' = 1- r^(2)` and `r' = + r`. This follows from the principle of reversibility of light path as shown in the figure below. `t t' + r^(2) = 1` `-r t + r't = 0` `:. tr' = 1 - r^(2)` `r' = + r`. `(-r` is the reflection ratio for the wave entering a denser medium).  
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