Wave Optics

What this chapter is about

Wave optics treats light as a wave rather than a ray. While ray optics explains reflection and refraction using straight-line paths, wave optics explains phenomena that ray optics cannot: interference, diffraction and polarisation. These effects become noticeable when light interacts with obstacles or apertures whose size is comparable to its wavelength (roughly 400–700 nm for visible light).

You study this chapter now because you already know geometric optics and the basic wave equation from earlier classes. Wave optics connects those ideas and shows that light carries transverse electromagnetic oscillations. Understanding coherence, path difference and fringe formation prepares you for modern topics such as lasers, optical fibres and holography.

After completing the chapter you should be able to explain how two light waves add constructively or destructively, calculate fringe width in Young's experiment, describe single-slit diffraction patterns, and distinguish polarised light from unpolarised light using Malus's law.

Key ideas

• Huygens's principle: Every point on a wavefront acts as a source of secondary spherical wavelets; the new wavefront is the forward envelope of all these wavelets. This explains reflection and refraction without invoking rays.
• Coherent sources: Two sources are coherent if they have a constant phase difference over time. Only coherent sources produce a stable interference pattern; ordinary lamps are incoherent.
• Superposition and interference: When two waves meet, the resultant displacement is the algebraic sum. Constructive interference (bright fringe) occurs when path difference = nλ; destructive interference (dark fringe) occurs when path difference = (n + ½)λ, where n is an integer.
• Young's double-slit experiment: Two narrow slits illuminated by monochromatic light act as coherent sources; alternate bright and dark fringes appear on a distant screen. Fringe width β = λD/d, where D is slit-to-screen distance, d is slit separation, λ is wavelength.
• Diffraction: Bending of light around edges or through narrow apertures. In single-slit diffraction the first minimum occurs at angle θ where a sin θ = λ, with a the slit width.
• Resolving power: The ability of an optical instrument to distinguish two close objects. A smaller aperture or longer wavelength worsens resolution.
• Polarisation: Restriction of the electric-field vibration to a single plane. Polarisation proves light is a transverse wave. Malus's law: I = I₀ cos²θ, where I₀ is intensity after the polariser and θ is the angle between polariser and analyser axes.

Formulas and facts to remember

• Item: Condition for constructive interference · Formula / Statement: Path difference Δ = nλ · Meaning: Waves arrive in phase; bright fringe.
• Item: Condition for destructive interference · Formula / Statement: Path difference Δ = (n + ½)λ · Meaning: Waves arrive out of phase; dark fringe.
• Item: Young's fringe width · Formula / Statement: β = λD / d · Meaning: Larger wavelength or screen distance gives wider fringes; larger slit separation gives narrower fringes.
• Item: Position of n-th bright fringe · Formula / Statement: yₙ = nλD / d · Meaning: Distance from central bright to n-th bright fringe.
• Item: Single-slit first minimum · Formula / Statement: a sin θ = λ · Meaning: Central maximum has angular width 2λ/a.
• Item: Malus's law · Formula / Statement: I = I₀ cos²θ · Meaning: Intensity through analyser depends on angle with polariser.
• Item: Brewster's angle · Formula / Statement: tan θ_B = n₂ / n₁ · Meaning: Reflected light is completely polarised when refracted and reflected rays are perpendicular.
• Item: Speed relation in medium · Formula / Statement: v = c / n · Meaning: Light slows inside a denser medium; wavelength shrinks, frequency unchanged.

Worked examples

Example 1 – Fringe width calculation

Problem: In a Young's double-slit set-up, monochromatic light of wavelength 600 nm illuminates two slits separated by 0.3 mm. A screen is placed 1.5 m from the slits. Find the fringe width.

Solution: Given: λ = 600 nm = 600 × 10⁻⁹ m, d = 0.3 mm = 3 × 10⁻⁴ m, D = 1.5 m.

Fringe width β = λD / d = (600 × 10⁻⁹ × 1.5) / (3 × 10⁻⁴) = (9 × 10⁻⁷) / (3 × 10⁻⁴) = 3 × 10⁻³ m = 3 mm.

Each bright or dark band is separated by 3 mm on the screen.

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Example 2 – Single-slit diffraction angle

Problem: A single slit of width 0.1 mm is illuminated by light of wavelength 500 nm. Calculate the angular position of the first minimum.

Solution: For first minimum: a sin θ = λ, so sin θ = λ / a.

sin θ = (500 × 10⁻⁹) / (0.1 × 10⁻³) = 5 × 10⁻³.

Since the angle is small, θ ≈ sin θ = 5 × 10⁻³ rad = 0.005 rad ≈ 0.29°.

The first dark band appears at about 0.29° from the central axis.

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Example 3 – Malus's law

Problem: Unpolarised light of intensity 80 W/m² passes through a polariser and then through an analyser whose axis makes 60° with the polariser. Find the final intensity.

Solution: After the polariser, intensity becomes I₁ = 80 / 2 = 40 W/m² (half the unpolarised intensity passes).

Through the analyser (Malus's law): I = I₁ cos²θ = 40 × cos²60° = 40 × (0.5)² = 40 × 0.25 = 10 W/m².

The light emerging from the analyser has intensity 10 W/m².

Common mistakes

• Confusing path difference with phase difference → Path difference is in metres; phase difference = (2π/λ) × path difference.
• Using fringe-width formula for diffraction minima → β = λD/d applies only to double-slit interference; single-slit uses a sin θ = mλ.
• Forgetting that unpolarised light loses half its intensity on passing the first polariser → Always halve before applying Malus's law.
• Mixing up conditions for maxima and minima → For double-slit, nλ gives bright; for single-slit diffraction, nλ gives dark (n ≠ 0).
• Assuming wavelength stays the same inside a medium → Wavelength λ_medium = λ_vacuum / n; frequency does not change.

Quick revision

1. Huygens's principle explains wave propagation by treating each wavefront point as a new source.
2. Stable interference needs coherent sources with constant phase difference.
3. Fringe width β = λD/d; larger D or λ widens fringes, larger d narrows them.
4. First single-slit minimum: a sin θ = λ.
5. Polarisation confirms light is a transverse wave; Malus's law: I = I₀ cos²θ.
6. Brewster's angle: tan θ_B = n; reflected ray is fully polarised.