Question

How can I compute the intensity of a polarized wave going through a Polaroid?

Answer 1

You could use Malus' Law.
Malus' Law tells us that if you have a polarized wave (of intensity `I_0`) passing through a polarizer the emerging intensity ( `I` ) will be proportional to the cosine squared of the angle between the polarizing direction of the incoming wave and the axis of the polarizer.

Or: `I=I_0*cos^2(theta)`

Sounds difficult but look at the picture:

Knowing the angle `theta` and the incoming intensity you´ll be able to evaluate the output intensity.

Special case :
When you have unpolarized light falling upon the polarizer the transmitted intensity will be `1/2` of the incoming intensity, i.e.:
`I=I_0/2`
(to understand this, look at the mathematical explanation that follows and remember that the average value of `cos^2` is `1/2`!!!)

As a mathematical explanation you have:
If `E0` is the amplitude of the electric vector of the incoming wave, then the intensity `I_0` of the wave incident on the polarizer is proportional to `E0^2`.

The electric field vector `E0` of the incoming wave can be resolved into two rectangular components projected upon the polarizer axis:
`E0*cos(theta)` and `E0*sin(theta)`
The analyzer will transmit only the component ( i.e `E0*cos(theta)`) which is parallel to its transmission axis.

But intensity is proportional to the square of the electric vector you have:
`IpropE^2` and:
`Iprop(E0*cos(theta))^2` but:
`I/I_0=(E0*cos(theta))^2/(E0^2)`

and finally:
`I=I_0*cos^2(theta)`