KAHIBARO
Table of Contents
Polarized light through an analyzer
Malus's law describes how the intensity of polarized light changes when it passes through a polarizing filter. It applies when light that is already linearly polarized enters a second polarizer, often called an analyzer.
If the direction of polarization of the incoming light makes an angle $\theta$ with the transmission axis of the analyzer, then the transmitted intensity is
$$
I = I_0 \cos^2 \theta
$$
Here, $I_0$ is the intensity of the polarized light before it enters the analyzer, and $I$ is the intensity after passing through it.
Malus's law:
$$
I = I_0 \cos^2 \theta
$$
It is valid for linearly polarized light incident on an ideal analyzer.
Meaning of the angle
The angle $\theta$ is the angle between two directions, not between the ray and the filter surface. It is the angle between the electric field direction of the incoming polarized light and the transmission axis of the analyzer.
When the two directions are aligned, $\theta = 0^\circ$, so
$$
I = I_0
$$
and the maximum possible intensity is transmitted.
When the analyzer is turned so that it is perpendicular to the polarization direction, $\theta = 90^\circ$, so
$$
I = I_0 \cos^2 90^\circ = 0
$$
and ideally no light passes through.
Why the cosine squared appears
The analyzer only allows the component of the electric field along its transmission axis to pass. If the incoming electric field has amplitude $E_0$, then the transmitted amplitude is
$$
E = E_0 \cos \theta
$$
Since light intensity is proportional to the square of the electric field amplitude,
$$
I \propto E^2
$$
we get
$$
I \propto (E_0 \cos \theta)^2 = E_0^2 \cos^2 \theta
$$
which leads to
$$
I = I_0 \cos^2 \theta
$$
This is the essential physical idea behind Malus's law.
The analyzer reduces the electric field by a factor of $\cos\theta$, but the intensity is reduced by a factor of $\cos^2\theta$.
Important special cases
Some angles appear often in problems.
| Angle $\theta$ | $\cos^2 \theta$ | Transmitted intensity |
|---|---|---|
| $0^\circ$ | $1$ | $I = I_0$ |
| $30^\circ$ | $\frac{3}{4}$ | $I = 0.75 I_0$ |
| $45^\circ$ | $\frac{1}{2}$ | $I = 0.50 I_0$ |
| $60^\circ$ | $\frac{1}{4}$ | $I = 0.25 I_0$ |
| $90^\circ$ | $0$ | $I = 0$ |
This shows that intensity does not fall linearly with angle. It follows the $\cos^2 \theta$ pattern.
Example calculation
Suppose linearly polarized light with intensity $I_0 = 80 \, \text{W/m}^2$ enters an analyzer at an angle of $60^\circ$.
Using Malus's law,
$$
I = I_0 \cos^2 \theta = 80 \cos^2 60^\circ
$$
Since $\cos 60^\circ = \frac{1}{2}$,
$$
I = 80 \left(\frac{1}{2}\right)^2 = 80 \cdot \frac{1}{4} = 20 \, \text{W/m}^2
$$
So the transmitted intensity is
$$
20 \, \text{W/m}^2
$$
Successive polarizers
Malus's law can be applied more than once if light passes through several polarizers. After each polarizer, the transmitted light becomes polarized along that polarizer's axis. Then the next angle must be measured relative to the new polarization direction.
For example, if polarized light of intensity $I_0$ passes through a second polarizer at angle $\theta$, and then through a third polarizer at angle $\phi$ relative to the second, the final intensity is
$$
I = I_0 \cos^2\theta \cos^2\phi
$$
This step by step use of Malus's law is very common.
For multiple polarizers, always measure each angle relative to the polarization direction of the light entering that polarizer.
Visual picture
The law can be understood geometrically. The electric field direction is projected onto the analyzer axis, and only that projected part survives.
In this sketch, the blue arrow is the incoming electric field, and the red arrow is its component along the allowed direction. The transmitted intensity depends on the square of that component.
Relation to an unpolarized beam
Malus's law itself is for already polarized light. If unpolarized light first passes through an ideal polarizer, the transmitted intensity becomes half of the original,
$$
I_1 = \frac{1}{2} I_{\text{unpol}}
$$
Then, if that light passes through an analyzer at angle $\theta$,
$$
I = \frac{1}{2} I_{\text{unpol}} \cos^2 \theta
$$
This is a very common experimental situation.
Experimental significance
Malus's law provides a simple way to test whether light is polarized and to determine polarization direction. By rotating an analyzer and observing how brightness changes, one can identify the angle of polarization. Maximum brightness occurs when the analyzer axis is parallel to the polarization direction, and minimum brightness occurs when it is perpendicular.
Final reminder
Malus's law is one of the central quantitative results for polarization.
For linearly polarized light passing through an ideal analyzer,
$$
I = I_0 \cos^2 \theta
$$
Maximum transmission at $\theta = 0^\circ$, zero transmission at $\theta = 90^\circ$.