A beam of initially unpolarized light passes through a sequence of three ideal polarizers. The angle ϕ 12 between the axes of the first and second polarizers is 23.9 ∘ , and the angle ϕ 23 between the axes of the second and third polarizers is 51.3 ∘ . Three polarizing filters equally spaced along a line colinear with their central axes. The polarization axis of the first filter points straight up. The polarization axis of the second filter is rotated clockwise by an angle phi subscript 1 2 relative to the axis of the first filter. The polarization axis of the third filter is rotated clockwise by an angle phi subscript 2 3 relative to the axis of the second filter. The beam incident on the first filter has intensity I subscript 0. The beam transmitted by the third filter has intensity I subscript 3. What is the ratio of the intensity I 3 of light emerging from the third polarizer to the intensity I 0 of light incident on the first polarizer?

From Malus' law, for the case of (linearly) polarized light falling on a (linear) polarizer, the output intensity Iout from the polarizer to the input intensity Iin, can be related as;

I_out = I_in•cos²(θ)

So we can relate the intensity of the light coming out of the second polarizer, I_2, to the intensity of the light coming into the second polarizer, I_1, by Malus' law:

I_2 = I_1cos²(23.9) = I_o/2(cos²(23.9))

Thus, in the same way, we can relate I_3 to I_2 as;

I_3 = (I_2)cos²(51.3) =I_o/2(cos²(23.9))cos²(51.3)

So, I_3/I_o = (1/2)cos²(23.9)•cos²(51.3)

= 0.5 x 0.8359 x 0.3909 = 0.1634