Polarisation

The Half Rule

• When unpolarised light of intensity $I_0$ passes through a single ideal polariser, the transmitted intensity is exactly half of the incident intensity.
• This occurs because the polariser averages all possible planes of oscillation, effectively transmitting only the vertical (or horizontal) components of the randomly oriented waves.

Formula: $I = \frac{I_0}{2}$

Malus's Law

• If polarised light is incident on a second filter (the analyser), the transmitted intensity depends on the angle $\theta$ between the light's plane of polarisation and the analyser's transmission axis.
• The intensity of the light transmitted by the analyser is proportional to the square of the cosine of the angle between the two axes.

Malus's Law: $I = I_{max} \cos^2(\theta)$

• Here, $I_{max}$ is the intensity of the polarised light incident on the analyser, and $\theta$ is the relative angle of rotation.

It is vital to distinguish between the behavior of transverse and longitudinal waves when interacting with filters or grids.

• Polariser vs. Analyser: A polariser is the first filter used to create polarised light from an unpolarised source. An analyser is a second filter used to detect or vary the intensity of that already polarised light.

• Identify the Wave Type: Always check if the wave mentioned is transverse. If a question asks about the polarisation of sound or ultrasound, the answer is usually that it cannot happen.

• The Two-Step Intensity Drop: Remember that if you start with unpolarised light, the first filter reduces intensity by half ($I_0 \rightarrow I_0/2$). Malus's Law then applies to that $I_0/2$ value for any subsequent filters.

• Angle Calculations: Be careful with the angle $\theta$. It is the angle between the transmission axes of the two filters. If a filter is rotated from the vertical to the horizontal, $\theta = 90^\circ$ and the transmitted intensity will be zero.

• Sanity Check: Since $\cos^2(\theta)$ ranges from $0$ to $1$, the transmitted intensity $I$ can never be greater than the incident intensity $I_{max}$.

• Forgetting the Square: A common error is using $I = I_{max} \cos(\theta)$ instead of $\cos^2(\theta)$. The squaring is necessary because intensity is proportional to the square of the amplitude.

• Misinterpreting 90-degree Rotation: Students often think a $90^\circ$ rotation means half the light passes through. In reality, $\cos^2(90^\circ) = 0$, meaning no light is transmitted when axes are perpendicular (crossed polaroids).

• Confusing Polarisation with Reflection: While reflection can cause partial polarisation (glare), they are distinct physical processes. Polarisation refers specifically to the restriction of the oscillation plane.