Malus Law — Definition

Definition

Imagine light as a wave, specifically an electromagnetic wave. Unlike sound waves, light waves can vibrate in many directions perpendicular to their direction of travel. When light vibrates in all possible directions, it's called 'unpolarized light'.

Think of it like a crowd of people dancing randomly. Now, imagine a special filter, called a 'polarizer', which is like a gate that only allows dancers facing a certain direction to pass through. When unpolarized light passes through this polarizer, it becomes 'plane-polarized light' – meaning its vibrations are now confined to a single plane.

The intensity of this light is typically halved compared to the original unpolarized light.

Now, let's introduce a second identical filter, called an 'analyzer'. This analyzer is also a polarizer, but we use it to 'analyze' the polarized light coming from the first polarizer. If we align the analyzer's transmission axis (its 'gate' direction) perfectly with the plane of polarization of the light coming from the first polarizer, all the light that passed through the first polarizer will also pass through the analyzer.

The intensity remains maximum. However, what happens if we start rotating this second filter?

Malus's Law tells us exactly how the intensity of light changes as we rotate the analyzer. It states that the intensity of the light transmitted through the analyzer is proportional to the square of the cosine of the angle between the transmission axis of the analyzer and the plane of polarization of the incident polarized light.

So, if the angle between the two filters' transmission axes is $heta$, the transmitted intensity $I$ will be $I_0 cos^2 \theta$, where $I_0$ is the intensity of the plane-polarized light incident on the analyzer.

Let's break down the formula $I = I_0 cos^2 \theta$:

$I$: This is the intensity of light that successfully passes through the second filter (the analyzer).
$I_0$: This is the intensity of the plane-polarized light that is *incident* on the analyzer. It's important to remember that this $I_0$ is *already* polarized light, not the original unpolarized light. If the original light was unpolarized with intensity $I_{unpol}$, then after passing through the first polarizer, $I_0 = I_{unpol}/2$.
$heta$: This is the angle between the transmission axis of the analyzer and the plane of polarization of the light hitting the analyzer.

Consider the extremes:

If $heta = 0^circ$ (axes are parallel), $cos 0^circ = 1$, so $cos^2 0^circ = 1$. Then $I = I_0 \times 1 = I_0$. Maximum intensity is transmitted.
If $heta = 90^circ$ (axes are perpendicular, or 'crossed'), $cos 90^circ = 0$, so $cos^2 90^circ = 0$. Then $I = I_0 \times 0 = 0$. No light is transmitted. This is why crossed polaroids appear dark.

Malus's Law is a cornerstone in understanding how polarized light interacts with optical components and is fundamental to many applications, from LCD screens to glare-reducing sunglasses.