Wave Optics and Interference

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Huygens’ Principle — Wavefront Construction:

Every point on a wavefront acts as a secondary spherical source of wavelets that propagate forward at the same speed as the original wavefront. The new wavefront is the envelope (tangent) to all these secondary wavelets. This principle explains reflection, refraction, and diffraction of light as wave phenomena.

Young’s Double Slit Experiment — Proof of Light’s Wave Nature:

Thomas Young’s 1801 experiment demonstrated interference of light through two narrow slits. When monochromatic light (wavelength λ) passes through two slits separated by distance d, bright (constructive) and dark (destructive) fringes form on a screen at distance D (D >> d).

Fringe positions:

• Bright fringe: d sinθ = mλ (m = 0, ±1, ±2, …)
• Dark fringe: d sinθ = (m + ½)λ (m = 0, ±1, ±2, …)

Fringe width β (distance between adjacent bright fringes): β = λD/d. This is directly proportional to λ — red light (longer λ) gives wider fringes than blue light.

⚡ ECAT Tip: In the double slit experiment, the central bright fringe (m = 0) is at θ = 0. For small angles (θ in radians, sinθ ≈ θ ≈ tanθ ≈ y/D where y is fringe position on screen): y_m = mλD/d for bright fringes, and y_(dark) = (m + ½)λD/d.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Diffraction — Single Slit:

When light passes through a single slit of width a, it produces a diffraction pattern with a central bright maximum that is twice as wide as the other bright fringes. Minima occur when: a sinθ = mλ (m = ±1, ±2, ±3, …), where m = 0 is the central maximum (not a minimum). The angular width of the central maximum = 2λ/a.

Key difference from double slit: double slit gives sharp bright lines with dark regions between; single slit gives a broad central maximum with many narrower side maxima.

Thin Film Interference — Colours in Soap Bubbles:

When light reflects off the top and bottom surfaces of a thin film (like oil or soap film), the two reflected beams interfere. A phase change of π (half wavelength) occurs when light reflects from a medium of higher refractive index. No phase change occurs when reflecting from a lower refractive index medium.

Condition for constructive interference (bright film) in reflected light:

• For air-film-air (higher n on bottom): 2nt = (m + ½)λ (destructive for transmitted)
• For film thickness t, refractive index n: optical path difference = 2nt cosθ_r + possible π shift

Newton’s rings: When a plano-convex lens is placed on a flat glass plate, concentric interference rings form in the air film between them. In reflected light (from above): dark rings at r_m² = mλR, bright rings at r_m² = (m + ½)λR. Newton’s rings are used to test the quality of lens surfaces and to determine the wavelength of light.

⚡ ECAT Tip: Polarisation proves that light is a transverse wave. Unpolarised light has electric field vectors vibrating in all directions perpendicular to propagation. Polarised light has electric fields vibrating in only one plane (the plane of polarisation). The intensity of polarised light through an analyser: I = I_max cos²θ, where θ is the angle between the polariser and analyser axes (Malus’ law: I = I₀ cos²θ).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Diffraction Grating — High-Resolution Spectroscopy:

A diffraction grating has many equally spaced parallel slits (N slits per unit length). The condition for principal maxima: d sinθ = mλ (same as double slit), but the peaks are much sharper because the many slits cause more destructive interference at angles slightly away from the maxima. The angular width of each principal maximum is approximately λ/(Nd) in radians.

Resolving power R = λ/Δλ = mN, where N is the total number of illuminated slits and m is the order number. A grating with N = 10,000 slits illuminated in the first order (m = 1) has resolving power R = 10,000 — it can distinguish wavelengths that differ by 1 part in 10,000. This is why diffraction gratings are used in spectrometers for precise wavelength measurement.

Fresnel and Fraunhofer Diffraction:

• Fraunhofer diffraction: far-field approximation — incoming waves are planar (parallel rays). Achieved experimentally by placing the screen at the focal plane of a converging lens. Double slit and single slit experiments are Fraunhofer.
• Fresnel diffraction: near-field, wavefronts are spherical. The diffraction pattern of a straight edge, a wire, or a circular aperture in the near field shows characteristic Fresnel half-period zones.

Brewster’s Law — Polarisation by Reflection:

When unpolarised light reflects from a dielectric surface (e.g., glass, water), the reflected light is partially polarised. At one specific angle of incidence (the Brewster angle θ_B), the reflected light is completely polarised. Brewster’s law: tan θ_B = n₂/n₁, where n₁ is the refractive index of the first medium and n₂ is that of the second.

At Brewster’s angle, the reflected ray and refracted ray are perpendicular to each other. No reflection occurs for the component of light polarised in the plane of incidence — only the component perpendicular to the plane of incidence reflects. This is why sunglasses with polarised lenses reduce glare from road surfaces and water — they block horizontally polarised reflected light.

⚡ ECAT Pattern: ECAT frequently tests: (1) double slit fringe spacing calculations using β = λD/d; (2) diffraction grating equation for spectral analysis; (3) Brewster’s angle for glass or water surfaces; (4) Malus’ law I = I₀ cos²θ for two polaroids; and (5) thin film interference conditions for constructive and destructive fringes. A typical ECAT question: “A diffraction grating has 5000 lines/cm. Calculate the angle for the second order maximum of light with wavelength 600 nm.” d = 1/(5000 cm⁻¹) = 2 × 10⁻⁶ m = 2000 nm. For m = 2: sinθ = mλ/d = 2 × 600/2000 = 0.6 → θ = sin⁻¹(0.6) = 36.9°.