understand that polarisation is a phenomenon associated with transverse waves

Polarisation

1. What is polarisation?

Polarisation is a property of transverse waves in which the direction of oscillation of the wave’s field vector is restricted to a particular orientation.

2. Why only transverse waves can be polarised

• In a longitudinal wave the oscillations are parallel to the direction of propagation, so there is no direction orthogonal to the travel that can be fixed.
• In a transverse wave the oscillations are perpendicular to the direction of travel, providing a plane in which a specific direction can be selected and therefore polarised.

3. Electromagnetic (EM) waves – key properties

• The electric field \(\mathbf{E}\) and the magnetic field \(\mathbf{B}\) are both perpendicular to the direction of propagation \(\mathbf{k}\); EM waves are therefore transverse.
• In free space they travel with the universal speed

  \[

  v = c = 3.00\times10^{8}\ \text{m s}^{-1}.

  \]

• The polarisation of an EM wave is defined by the direction of its \(\mathbf{E}\)-field (the \(\mathbf{B}\)-field is automatically perpendicular to both \(\mathbf{E}\) and \(\mathbf{k}\)).
• This transverse nature is the reason polarisation is a property unique to EM waves (and other transverse waves) and does not occur for longitudinal waves.

4. Types of polarisation

5. Producing polarised light

• Absorbing polarising filters (e.g., Polaroid sheets): molecules are aligned so that one component of the electric field is absorbed, the orthogonal component is transmitted.
• Reflection at Brewster’s angle:

  • When light strikes a dielectric surface at the angle \(\theta_B\) given by

    \[

    \tan\thetaB = \frac{n2}{n_1},

    \]

    the reflected beam is perfectly linearly polarised with its \(\mathbf{E}\)-field perpendicular to the plane of incidence.

• Rayleigh scattering: molecules scatter light preferentially perpendicular to the scattering plane, giving the sky a characteristic polarisation pattern.
• Birefringence (double‑refraction):

  • Nicol prism – a calcite crystal cut so that one polarisation component undergoes total internal reflection and is discarded; the emerging beam is polarised.
  • Wave‑plates (quarter‑ and half‑wave plates) – introduce a controlled phase shift between orthogonal components, converting linear ↔ circular or rotating the plane of linear polarisation.

6. Intensity of polarised light

1. Unpolarised light through a single ideal polariser:

   \[

   I = \frac{1}{2}\,I_0,

   \]

   because only the component of the electric field parallel to the transmission axis is transmitted.

2. Malus’s law (for an ideal analyser and linearly polarised incident light):

   \[

   I = I_0\cos^{2}\theta,

   \]

   where \(\theta\) is the angle between the analyser’s transmission axis and the incident polarisation direction.

7. Worked examples (Cambridge AS & A Level style)

1. Unpolarised light through one polariser

   Given \(I_0 = 200\ \text{W m}^{-2}\), the transmitted intensity is

   \[

   I = \frac{1}{2}I_0 = 100\ \text{W m}^{-2}.

   \]

2. Malus’s law with a single analyser

   A linearly polarised beam of intensity \(I_0 = 100\ \text{W m}^{-2}\) passes through an analyser set at \(\theta = 30^{\circ}\).

   \[

   I = 100\cos^{2}30^{\circ}=100\left(\frac{\sqrt{3}}{2}\right)^{2}=75\ \text{W m}^{-2}.

   \]

3. Two successive polarisers

   Unpolarised light of intensity \(I_0 = 120\ \text{W m}^{-2}\) passes through:

   • Polariser P1 (transmission axis taken as 0°).
   • Analyser P2 whose transmission axis is at \(\theta = 45^{\circ}\) to P1.

   Step 1 – after P1:

   \[

   I1 = \frac{1}{2}I0 = 60\ \text{W m}^{-2}.

   \]

   Step 2 – after P2 (apply Malus’s law):

   \[

   I2 = I1\cos^{2}45^{\circ}=60\left(\frac{1}{\sqrt{2}}\right)^{2}=30\ \text{W m}^{-2}.

   \]

   Thus the final transmitted intensity is one‑quarter of the original unpolarised intensity.

4. Reflection at Brewster’s angle

   Light travelling from air (\(n1=1.00\)) into glass (\(n2=1.50\)).

   \[

   \theta_B = \arctan\!\left(\frac{1.50}{1.00}\right) \approx 56.3^{\circ}.

   \]

   At this incidence the reflected beam is completely linearly polarised with \(\mathbf{E}\) perpendicular to the plane of incidence.

8. Applications of polarisation

• Glare‑reducing sunglasses and camera lenses – absorbing polarisers block horizontally polarised glare.
• Liquid‑crystal displays (LCDs) – a polariser, a liquid‑crystal layer (which rotates the plane of polarisation), and an analyser together control pixel brightness.
• Polariscope for stress analysis – birefringent materials become optically anisotropic under stress, revealing fringe patterns.
• Optical fibre and free‑space communication – polarisation‑division multiplexing allows two independent data streams to be carried on the same carrier frequency.
• 3‑D cinema – left‑ and right‑eye images are projected with orthogonal linear (or circular) polarisations; viewers wear matching polarised glasses.

9. Key points to remember (Cambridge checklist)

• Only transverse waves can exhibit polarisation.
• All electromagnetic waves are transverse and travel at the speed of light \(c\) in free space.
• Linear, circular and elliptical are the three ideal states of polarisation.
• When unpolarised light passes through an ideal polariser, its intensity is reduced to \(\tfrac12 I_0\).
• For ideal polarisers and linearly polarised light, Malus’s law gives \(I = I_0\cos^{2}\theta\).
• Brewster’s angle is \(\displaystyle \tan\thetaB = \frac{n2}{n_1}\); reflected light at this angle is perfectly polarised.
• Polarisation can be produced by absorption (filters), reflection, scattering, and birefringence (Nicol prisms, wave‑plates).
• Polarisation is widely used in everyday technology (sunglasses, LCDs, 3‑D cinema) and in scientific instruments (polariscope, communication systems).