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Wave Optics

Ch. 7Std 12

Easy Overview

Geometric optics treats light as rays that travel in straight lines — and for many purposes, this is good enough. Mirrors, lenses, and the formation of images are all well described by ray optics. But light is a wave, and many phenomena simply cannot be explained by rays alone. Why does light bend around corners? Why do you get a pattern of bright and dark bands when light passes through a narrow slit? Why do some materials become dark when viewed through certain filters? These questions belong to wave optics, which treats light as an electromagnetic wave and explains phenomena like diffraction, interference, and polarisation. The foundation of wave optics is Huygens' principle, proposed by Christiaan Huygens in 1678. It states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront at any later time is the envelope of these secondary wavelets. This principle elegantly explains the laws of reflection and refraction. When a plane wave hits a mirror, the secondary wavelets produce a reflected plane wave obeying the law of reflection. When a wave enters a denser medium where it travels slower, the wavelets are closer together, and the wavefront bends — this is refraction. Diffraction is the bending of waves around obstacles or through apertures. If light were strictly rays, a sharp shadow would be cast by an obstacle. Instead, we observe fringes — alternating bright and dark bands. The single-slit diffraction pattern has a central bright maximum twice as wide as the others. A diffraction grating uses thousands of parallel slits to produce very sharp maxima, allowing precise measurement of wavelengths. Polarisation reveals that light is a transverse wave. Ordinary light vibrates in all transverse directions. Polarised light vibrates in only one direction. Polarisation occurs by reflection (Brewster's angle), by scattering, and by transmission through polaroid sheets. Polarised sunglasses block glare from horizontal surfaces. Liquid crystal displays (LCDs) use polarisation to control pixel brightness. Understanding wave optics is essential for technologies ranging from photography to fibre optics.

Huygens Principle and Wavefronts

Huygens' principle: every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the envelope of these wavelets. A wavefront is a surface of constant phase. For a point source, wavefronts are spheres. Far from the source, they approximate planes. Huygens' principle can derive the laws of reflection and refraction geometrically.

Reflection and Refraction Using Huygens Principle

Using Huygens' principle, we derive the laws of reflection and refraction. For reflection: a plane wave incident on a mirror generates secondary wavelets whose envelope is a reflected plane wave at angle r = i. For refraction: when a wave enters a slower medium, the wavelets travel slower, bending the wavefront toward the normal. Snell's law n₁ sin i = n₂ sin r follows directly.

Coherent and Incoherent Sources

Two sources are coherent if they emit waves with constant phase difference. Ordinary sources (bulbs, candles) are incoherent — phase changes randomly. Laser light is highly coherent. To produce coherent light from ordinary sources: (1) Division of wavefront — use two slits (Young's experiment). (2) Division of amplitude — use partial reflection (thin films, Michelson interferometer).

Interference of Light Waves

For two waves y₁ = A sin(ωt), y₂ = A sin(ωt + δ), the resultant is y = 2A cos(δ/2) sin(ωt + δ/2). Intensity I = 4I₀ cos²(δ/2). Constructive interference: δ = 0, 2π, 4π, ... (I = 4I₀). Destructive: δ = π, 3π, 5π, ... (I = 0). Phase difference δ = (2π/λ) × path difference.

Young's Double-Slit Experiment — Detailed Analysis

Monochromatic light illuminates two slits separated by d. Path difference at angle θ: Δx = d sinθ. Constructive: d sinθ = nλ. Bright fringe position y_n = nλD/d. Fringe width β = λD/d. With white light, coloured fringes appear with white centre. Covering one slit destroys the interference pattern.

Diffraction of Light at a Single Slit

Light passing through a slit of width a diffracts. Minima at a sinθ = nλ (n = ±1, ±2, ...). Central maximum is twice as wide as others. Intensity I = I₀ [sin(α)/α]² where α = (πa sinθ)/λ. If a >> λ, pattern is very narrow — ray optics is recovered.

Diffraction Grating

A diffraction grating has many parallel, equally spaced slits. Principal maxima at (a+b) sinθ = nλ, where (a+b) is the grating element. Intensity ∝ N² (N = number of slits). Angular dispersion dθ/dλ = n/[(a+b) cosθ]. Resolving power R = λ/Δλ = nN. Used in spectrometers to analyse light spectra.

Resolving Power of Optical Instruments

Due to diffraction, even a perfect lens forms an Airy pattern. The angular radius of the central bright spot is θ = 1.22λ/D (circular aperture). Rayleigh criterion: two points are just resolved when the central maximum of one falls on the first minimum of the other. θ_min = 1.22λ/D. For microscopes: d_min = 0.61λ/(NA), where NA = n sinα.

Polarisation of Light

Light is a transverse EM wave. In unpolarised light, the electric field vibrates in all transverse directions. In polarised light, it vibrates in one plane. Polarisation is produced by: (1) Polaroid sheets (dichroic crystals). (2) Reflection at Brewster's angle. (3) Scattering (skylight is polarised). (4) Birefringence (calcite splits light into two polarised beams).

Malus' Law

Malus' law: I = I₀ cos²θ, where θ is the angle between transmission axes of polariser and analyser. θ = 0° → I = I₀ (maximum). θ = 90° → I = 0 (crossed polarisers). For unpolarised light incident on a polariser, transmitted intensity = I₀/2. Used in polarised sunglasses, LCDs, and optical stress analysis.

Polarisation by Reflection — Brewster's Law

Brewster's law: tan θ_B = n₂/n₁. At Brewster's angle, reflected light is completely polarised perpendicular to the plane of incidence. For air to glass (n = 1.5), θ_B ≈ 56.3°. At this angle, reflected and refracted rays are perpendicular (θ_B + r = 90°). Polarised sunglasses block horizontally polarised glare from water and roads.

Polarisation by Scattering and Double Refraction

Sunlight scattered by atmospheric molecules is polarised — maximum polarisation at 90° from the Sun. Bees use this for navigation. Birefringent crystals (calcite) split light into ordinary and extraordinary rays with different speeds and refractive indices. Used in wave plates that convert linear to circular polarisation.

Validity of Ray Optics and the Wave Model

Ray optics is valid when obstacles and apertures are much larger than the wavelength. When sizes are comparable to λ, wave optics is needed. For everyday optics (cameras, eyes), ray optics works well. At resolution limits, wave optics becomes essential. X-ray diffraction (λ ~ 0.1 nm) requires wave optics because wavelengths are comparable to atomic spacings.

Fresnel and Fraunhofer Diffraction

Fraunhofer diffraction (far-field): source and screen at infinite distances, plane wavefronts. Includes single-slit, circular aperture, and diffraction grating patterns. Fresnel diffraction (near-field): source or screen at finite distance, spherical wavefronts. Explains Poisson spot (bright spot at centre of shadow of circular disc) and near-field patterns.

Key Points

  • •Huygens principle: each point on wavefront is source of secondary wavelets.
  • •Laws of reflection and refraction derivable from Huygens principle.
  • •Interference: I = 4I₀ cos²(δ/2), δ = (2π/λ) × path difference.
  • •Young's double-slit: fringe width β = λD/d.
  • •Single-slit diffraction: minima at a sinθ = nλ.
  • •Diffraction grating: (a+b) sinθ = nλ. Resolving power R = nN.
  • •Rayleigh criterion: θ_min = 1.22λ/D.
  • •Malus' law: I = I₀ cos²θ.
  • •Brewster's law: tan θ_B = n₂/n₁.
  • •Polarisation occurs by reflection, scattering, and transmission.
  • •Fraunhofer: far-field. Fresnel: near-field.
  • •Poisson spot: evidence for wave nature of light.

Practice Questions

  • State Huygens principle. Using it, derive the laws of reflection and refraction of light.
  • Explain Young's double-slit experiment and derive fringe width. How does the pattern change when (a) slit separation increases, (b) screen moves farther, (c) white light is used?
  • What is diffraction of light? Derive conditions for minima and secondary maxima in single-slit diffraction.
  • What is resolving power? State Rayleigh criterion. A telescope has objective diameter 10 cm. Find minimum angular separation resolvable for λ = 550 nm.
  • What is polarisation? State and explain Malus' law. Three polaroid sheets at 45° each. Unpolarised light I₀ incident on first. Find intensity from third.
  • State Brewster's law. Show that at Brewster's angle, reflected and refracted rays are perpendicular. Find Brewster's angle for glass (μ = 1.5) in air.
  • Distinguish between Fresnel and Fraunhofer diffraction. Explain formation of Poisson spot.
  • Describe working of a diffraction grating. Derive condition for principal maxima. How does resolving power depend on number of lines?