A 60° prism is made of a non-magnetic dielectric of refractive index n. From air

Question

A 60° prism is made of a non-magnetic dielectric of refractive index n. From air, a beam of power Pi1 mW enters the prism from one side and exits from another as shown. It is founed that when the entrance angle ?| is smaller than 300, the exit power P4 is zero (i) What is the refractive index n? (ii) At what angle ?! is the reflected power P2-0? And at what angle ?| is the reflected power P3-0? What is the polarization of the incident beam when that happens? (???) When P2-0, what is the exit power P? If the polarization is then rotated 90°, what is P4? air P 600 KXXKXXKXXK2XKKZKKZAKRKAXK3XEREEKHKK XEK2XK22XKXEKXEREXKXXK23XXXK8XX22K2XX3

Explanation / Answer

For the given prism

i. phi = 60 deg

refractive index = n

P1 = 1 mW

for theta1 < 30 deg, P4 = 0

hence

from snell's law

sin(theta1)/sin(r) = n

sin(r) = sin(30)/n = 1/2n

also, from the triangle

60 + (90 - r) + w = 180

w = 30 + r

and

sin(90 - w)/sin(90) = 1/n

cos(w) = 1/n = cos(30 + r) = cos(30)cos(r) - sin(30)sin(r) = sqrt(3)cos(r)/2 - sin(r)/2

now, sin(r) = 1/2n

cos(r) = sqrt(1 - 1/4n^2)

hence

1/n = sqrt(3 - 3/4n^2)/2 - 1/4n

25 = 12n^2 - 3

12n^2 = 28

n = 1.527525231651

ii. when the reflected and incident rays are perpendicular and the P1 is perpendicular to the surface of the interface

then P2 = 0

hence tan(theta1) = n

theta1 = 56.78908 deg

iii. when P2 = 0

exit angle = r'

sin(56.78908)/sin(r) = n

r = 33.2109 deg

and

sin(90 - 30 - r)/sin(r') = 1/n

r' = 43.508 deg

hence

P2 = 0

P3 and P4 have divided the intensities of the P1 among them

From fresnel's equations

now the incident light is not parallel to the surface

hence s-polarised light

hence

I4 = 1 - Rs*P1

Rs = |(ncos(60 - 33.2109) - sqrt(1 - (n*sin(60 - 33.2109))^2))/(ncos(60 - 33.2109) + sqrt(1 - (n*sin(60 - 33.2109))^2))|^2

Rs = 0.24292861

hence

P4 = 0.75707138395 mW

now, polarisation is rotated 90 deg

then

P2 is non zero as all the polarisation of P2 comes from P1 ( the component parallel to the surface)

hence

P2 = P1

and P4 = 0

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