$1.(24$ Marks$)$ Four polarizing sheets are stacked as follows: the first is at $20\mathrm{.}0$ clockwise $($C$)$ with the vertical, the second is at an angle of $30\mathrm{.}0$ counterclock wise $($CCW$)$ with the vertical, the third is at $40\mathrm{.}0$ CW with the vertical, and the fourth is horizontal. The incident light has an

intensity la $=400$ W$/$m$\text{'}.$

Calculate i$,$ the transmitted intensity by the system it a$)$

b$)$

the incident light lo is unpolarized? the incident light lo is vertically polarized?

For the remainder of the problem assume that I, $-400$ Wim' and is vertically polarized.

d$)$

Calculate the transmitted intensity if the third polarizer is removed.

What would be the transmitted intensity if the second and the third polarizer are exchanged?

If the second polarizer is removed, what should be the angle $($with the vertical$)$ of the third polarizer so that maximum intensity can be transmitted through the system? Calculate this maximum intensity.

Light is incident on air $-$ water interface at the Brewster angle $8$g of that interface and reflects toward the original system of $4$ polarizers. Calculate the fraction of the incident light that is transmitted$$by$$the$$system$.$