Polarisation & Optical Activity – Comprehensive Study Notes

Introduction to Polarisation

• Light previously proved wave-like via interference & diffraction, but those phenomena do not reveal whether waves are longitudinal or transverse.

• Maxwell’s electromagnetic theory ➜ light = transverse electromagnetic (EM) wave; electric ((\vec E)) & magnetic ((\vec B)) fields oscillate mutually perpendicular & both ⟂ to propagation.

  • Usually describe light with (\vec E) only because:

  • (|\vec E| = c |\vec B|) so magnitude larger.

  • Human eye responds mainly to electric field component.

• Polarisation arises because EM waves are transverse.

• Practical relevance: wave-guides, optical fibres, LCDs, etc.

Unpolarised vs Polarised Light

• Unpolarised light: superposition of many waves emitted by independent atoms; (\vec E) vectors have random orientations in the plane ⟂ propagation (e.g., X–Y plane if travelling along Z).

  • Equivalent to two perpendicular, equal-intensity, coherent plane-polarised components.

  • If one component exceeds the other ➜ partially polarised.

• Plane (linearly) polarised light: (\vec E) oscillates in a single fixed direction ⟂ propagation.

• Plane of polarisation: plane containing propagation direction & (\vec E) vibration.

• Eye cannot distinguish polarised from unpolarised directly; need crystals/polaroids as analysers.

Polarisation as Proof of Transversality

• Transverse waves can be polarised; longitudinal cannot.

• Light (transverse) ➜ shows polarisation; sound (longitudinal) ➜ cannot.

• Interference of polarised beams occurs only when their planes of polarisation coincide.

Polarisers & Analysers

• Calcite, quartz, tourmaline crystals, Nicol prisms, Polaroid sheets serve as polarisers or analysers depending on use.

• Single polariser halves intensity of unpolarised light (average of (\cos^2\theta) = 1/2).

• Adding analyser: transmitted intensity varies with relative angle (Malus’ law).

Malus’ Law

• For plane-polarised light incident on analyser at angle (\theta):
  $I = I_0 \cos^2\theta$

• For unpolarised input: first polariser gives (I_0/2); subsequent analyser still follows (\cos^2\theta).

• Examples:

  • Crossed polarisers ((90^{\circ})) ➜ zero intensity.

  • Rotate analyser so that (25\%) transmission ⇒ (\theta = 30^{\circ}).

Methods to Obtain Plane Polarised Light

(a) Polarisation by Reflection (Brewster’s Law)

• At incidence angle (i_p) (polarising/Brewster angle) on interface air–medium: reflected light completely plane-polarised, vibrations ⟂ plane of incidence.

• Brewster’s law: $\mu = \tan i_p$

  • Glass: (\mu \approx 1.5 \Rightarrow i_p \approx 56.3^{\circ})

  • Water: (\mu \approx 1.33 \Rightarrow i_p \approx 53.1^{\circ})

• At (i_p), reflected & refracted rays are perpendicular: $r + r' = 90^{\circ}$

(b) Polarisation by Refraction (Pile of Plates)

• Stack 15–20 microscope slides; incidence at (i_p) (≈56°).

• Each reflection removes ⟂-to-plane vibrations (~15 % per surface).

• After many plates, transmitted beam is almost completely plane-polarised with (\vec E) parallel to plane of incidence.

Optically Isotropic vs Anisotropic Media

• Isotropic: refractive index same in all directions (glass, water, NaCl) → single refraction.

• Anisotropic: refractive index varies with crystallographic direction → double refraction (birefringence).

  • Two categories:

  • Uniaxial (one optic axis) → two indices (\muo,\mue). Examples & values: see Table (ice, quartz, calcite, etc.).

  • Biaxial (two optic axes) → three indices (\mu\alpha, \mu\beta, \mu_\gamma). Examples: epsom salt, borax, mica, etc.

Double Refraction (Birefringence)

• When unpolarised light enters anisotropic crystal (e.g., calcite): splits into

  • Ordinary ray (o-ray): obeys Snell; index (\mu_o); spherical wavefront; constant speed.

  • Extraordinary ray (e-ray): violates Snell; index (\mu_e); ellipsoidal wavefront; speed depends on direction.

• Two rays are plane-polarised in perpendicular planes.

• Huygens construction: superposed spherical & ellipsoidal wave surfaces. If ellipsoid outside sphere ⇒ positive crystal ((\mue > \muo)). If inside ⇒ negative ((\mue < \muo)).

• Optic axis: direction with no double refraction; both rays coincide.

Distinguishing o- & e-Rays

• Obey/violate Snell, refractive index constant/variable, speed same/different, polarisation ⟂, etc.

Nicol Prism (Double Refraction for Polarisation)

• Calcite cut & cemented with Canada balsam.

• o-ray undergoes total internal reflection & absorbed; only e-ray emerges → plane-polarised output.

Wave Plates (Retarders)

• Thin birefringent plates introducing specific phase retardation (\delta) between o & e.

  • Phase: $\delta = \frac{2\pi}{\lambda}(\mu<em>e - \mu</em>o)t$

• Quarter-wave plate (QWP): optical path difference (\frac{\lambda}{4}) ⇒ $t<em>{\lambda/4} = \frac{\lambda}{4(\mu</em>e-\mu_o)}$ produces (\delta = \frac{\pi}{2}).

• Half-wave plate (HWP): path difference (\frac{\lambda}{2}) ⇒ $t<em>{\lambda/2} = \frac{\lambda}{2(\mu</em>e-\mu_o)}$ produces (\delta = \pi).

• General solutions include multiples: QWP if ((\mue-\muo)t = (2n+1)\frac{\lambda}{4}); HWP if ((\mue-\muo)t = (2n+1)\frac{\lambda}{2}).

• Example calculations (from transcript):

  • Quartz QWP at (\lambda=6500\,\text{Å}) with (\mue=1.553,\muo=1.544) ⇒ $t \approx 1.8\times10^{-4}\,\text{m}$.

  • Ice HWP at (\lambda=590\,\text{nm}) ((\mue=1.313,\muo=1.309)) ⇒ $t \approx 7.38\times10^{-4}\,\text{m}$.

Phase-Retarding Plates

• General term for QWP, HWP, etc.; fabricated from calcite, quartz, mica; introduce designed (\delta).

Generating Special Polarisations

Circular Polarisation

1. Unpolarised → polariser → plane-polarised.

2. Incidence on QWP at (45^{\circ}) to optic axis.

3. o & e emerge with equal amplitudes & (\delta=\frac{\pi}{2}). Superposition ⇒ constant |(\vec E)| rotating helix.

Elliptical Polarisation

• Same setup but incident (\vec E) makes arbitrary angle (\neq45^{\circ}) with optic axis ⇒ unequal amplitudes + (\delta=\frac{\pi}{2}) ⇒ ellipse.

Using HWP

• Incident linearly polarised at (45^{\circ}) → HWP rotates plane by (90^{\circ}) (handedness reversal for circular/elliptical).

Detecting the State of Polarisation

1. Pass unknown light through analyser (rotatable).

   • Intensity extinguishes twice per rotation ⇒ plane-polarised.

   • Intensity constant ⇒ either unpolarised or circular.

   • Varies but never zero ⇒ partially or elliptically.

2. Insert QWP before analyser:

   • Extinction after QWP ⇒ originally circular or elliptical (converted to linear).

   • Still no extinction ⇒ partially polarised or unpolarised (distinguish by constancy).

Optical Activity & Specific Rotation

• Optical rotation: rotation of polarisation plane as light traverses optically active material (crystals: quartz, cinnabar; solutions: sugar, tartaric acid, turpentine, quinine, etc.).

  • Dextro-rotatory (+): rotates right (clockwise looking toward source).

  • Laevo-rotatory (−): rotates left (counter-clockwise).

• Specific rotation (for wavelength (\lambda) at temp (T)): $[\alpha]_{\lambda}^{T} = \frac{100\,\theta}{l c}$ where

  • (\theta) = observed rotation (°)

  • (l) = path length (cm; often convert to dm in definition)

  • (c) = concentration (g cm⁻³) or density for pure liquids.

• Example: Sugar solution (0.2 g cm⁻³, 18 cm) rotates 23.4° ⇒ $[\alpha] = 65^{\circ}$.

Polarimeter (Saccharimeter)

• Components: monochromatic source + lens → polariser Nicol (N₁) → half-shade plate (or biquartz) → sample tube → analyser Nicol (N₂) + eyepiece.

• Half-shade plate (Laurent): semicircular quartz HWP + glass; creates brightness contrast aiding null detection.

• Biquartz: two semicircles of dextrorotatory & laevorotatory quartz; colour contrast (yellow-tone equality) improves sensitivity with white light.

• Procedure: set N₁ & N₂ crossed (equal darkness), insert sample, rotate analyser to regain equality; angle difference = (\theta).

• Plot (\theta) vs (c) ➜ straight line; slope gives specific rotation.

Applications of Optical Rotation

• Sugar industry (syrup concentration).

• Medical diagnostics (blood glucose).

• Chemical characterisation & purity testing.

• Mineralogy (thin-section identification).

• Polarisation control in optics.

Negative vs Positive Crystals & Birefringence

• Birefringence magnitude: $\Delta \mu = \mu<em>e - \mu</em>o$

  • Positive: (\Delta \mu > 0) ((\mue > \muo)) → e-ray slower.

  • Examples: quartz, ice, MgF₂, calomel.

  • Negative: (\Delta \mu < 0) ((\mue < \muo)) → o-ray slower.

  • Examples: calcite, beryl, tourmaline, ruby, sapphire.

Summary of Light States

• Unpolarised: random orientations; treat as incoherent sum.

• Linearly polarised: single vibration direction.

• Partially polarised: mixture of linear & unpolarised.

• Elliptically polarised: two coherent perpendicular components, unequal amplitudes, (\delta=\frac{\pi}{2}).

• Circularly polarised: special ellipse with equal amplitudes; |(\vec E)| constant; components in quadrature.

Key Formulae Recap

• Brewster: $\mu = \tan i_p$

• Malus: $I = I_0 \cos^2\theta$

• Phase retardation: $\delta = \frac{2\pi}{\lambda}(\mu<em>e - \mu</em>o)t$

• QWP thickness: $t<em>{\lambda/4} = \frac{\lambda}{4(\mu</em>e-\mu_o)}$

• HWP thickness: $t<em>{\lambda/2} = \frac{\lambda}{2(\mu</em>e-\mu_o)}$

• Specific rotation: $[\alpha] = \frac{100\,\theta}{lc}$

These bullet-point notes cover every major and minor concept, equations, examples, and practical implications discussed in the provided transcript, giving a self-contained study guide on Polarisation & Optical Activity.