Malus Law

Malus' Law is a fundamental concept in optics and wave physics. This law is important to understand the polarization properties of light. It is named after Étienne-Louis Malus, who in 1808 discovered that natural incident light can be polarised when reflected by a glass surface.

Brief Introduction about Polarisation, Polarisers & Analysers

Light is an electromagnetic wave, which is a combination of electric and magnetic fields oscillating at right angles to each other and the direction of propagation. In unpolarized light, these oscillations occur in all directions perpendicular to the direction of travel. But in polarised light, its electric field vibrates in only one direction. This is when light is passed through a device called a polariser.

A polariser is a material or filter that allows only light vibrating in a specific direction to pass through.
An analyser is a second polariser used to study the light after it has passed through the first polariser.

If unpolarized light passes through a polariser, the intensity of the light is reduced by half. This is because only the component of light vibrating in the direction of the polariser’s axis is allowed to pass through. Now, suppose this polarised light passes through a second polariser (the analyser). In that case, the amount of light that passes through depends on the angle between the transmission axes of the two polarisers. This is where Malus' Law is used.

Malus Law

Malus' law states that the intensity of plane-polarised light that passes through an analyser varies as the square of the cosine of the angle between the plane of the polariser and the transmission axes of the analyser.

Formula -

I = I₀ cos²

where,

I - Intensity of transmitted light

I₀ - Intensity of incident polarised light

θ - Angle between the axes of the polariser and analyser

Explanation of Formula

When unpolarised light is incident on a polariser, the intensity of the transmitted light is exactly half that of the incident unpolarised light. The orientation of the polarising axis does not matter.

Note:

An ideal polariser transmits only the component along its axis; for unpolarised light this averages to 50%.
Polariser transmits vibrations in only one direction.
An analyser is used after the polariser to test the degree of polarisation.
If unpolarised light passes through, intensity is I = I₀/2.

Graphical Representation

The graph represents I (Intensity) versus θ. The graph forms a cosine-squared curve. The intensity is maximum when θ = 0 and minimum at θ = 90°.

I is max at θ = 0° (axes aligned)
I = 0 at θ = 90° (axes perpendicular)

Graph Representation

Limitations of Malus’ Law

Malus’ Law applies only to completely polarised light.
Malus’ Law assumes only an ideal polariser. Only the component of the electric field parallel to the analyser’s axis is transmitted.
It also assumes polarisers with no absorption or scattering losses.
The law satisfies only monochromatic light (single wavelength). The results may not be accurate for polychromatic light (multi-wavelength)

Applications

Sunglasses: Malus’ Law is used in Polarised sunglasses to reduce glare from reflective surfaces.
Photography - Photographers use polarising filters to enhance contrast and colours in the images.
Optical Instruments - Instruments like polarimeters it is used to measure the angle
LCD Screens - In LCD screens, liquid crystal molecules are used to polarise light.
Scientific Research - Malus' Law is crucial to understanding experiments involving light polarisation.

Summing Up

Malus’ Law explains how the intensity of polarised light passing through an analyser varies with the square of the cosine of the angle between the polariser and analyser. It is applicable only to completely polarised, monochromatic light and assumes ideal polarisers. It's used in sunglasses, photography, optical instruments, LCD screens, and scientific research involving light polarization.