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

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

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Answer 1 · 2014-12-21T12:00:03

You could use Malus' Law.
Malus' Law tells us that if you have a polarized wave (of intensity $I_{0}$ $I_0$ ) passing through a polarizer the emerging intensity ( $I$ $I$ ) 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: $I = I_{0} \cdot {cos}^{2} (θ)$ $I=I_0*cos^2(theta)$

Sounds difficult but look at the picture:
enter image source here

Knowing the angle $θ$ $theta$ 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 $\frac{1}{2}$ $1/2$ of the incoming intensity, i.e.:
$I = \frac{I_{0}}{2}$ $I=I_0/2$
(to understand this, look at the mathematical explanation that follows and remember that the average value of ${cos}^{2}$ $cos^2$ is $\frac{1}{2}$ $1/2$ !!!)

As a mathematical explanation you have:
If $E 0$ $E0$ is the amplitude of the electric vector of the incoming wave, then the intensity $I_{0}$ $I_0$ of the wave incident on the polarizer is proportional to $E 0^{2}$ $E0^2$ .

The electric field vector $E 0$ $E0$ of the incoming wave can be resolved into two rectangular components projected upon the polarizer axis:
$E 0 \cdot cos (θ)$ $E0*cos(theta)$ and $E 0 \cdot sin (θ)$ $E0*sin(theta)$
The analyzer will transmit only the component ( i.e $E 0 \cdot cos (θ)$ $E0*cos(theta)$ ) which is parallel to its transmission axis.
enter image source here
But intensity is proportional to the square of the electric vector you have:
$I \propto E^{2}$ $IpropE^2$ and:
$I \propto {(E 0 \cdot cos (θ))}^{2}$ $Iprop(E0*cos(theta))^2$ but:
$\frac{I}{I_{0}} = \frac{{(E 0 \cdot cos (θ))}^{2}}{E 0^{2}}$ $I/I_0=(E0*cos(theta))^2/(E0^2)$

and finally:
$I = I_{0} \cdot {cos}^{2} (θ)$ $I=I_0*cos^2(theta)$

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

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