Class 12 · Chapter 10

Wave Optics

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Wave Optics Short Notes

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Condensed revision points for Wave Optics — for quick recall before exams, not a substitute for the full notes.

Newton's corpuscular theory: light = tiny particles. Explained reflection & refraction, but predicted light is faster in denser media — WRONG. Could not explain interference, diffraction, polarisation.
Huygens' wave theory: light travels as a wavefront through ether. Each point on a wavefront acts as a secondary source. Correctly predicted light is slower in denser media.
Wavefront: locus of all points vibrating in the same phase. Normal to wavefront = ray direction.
Spherical wavefront (point source): A ∝ 1/r, I ∝ 1/r².
Cylindrical wavefront (linear source): A ∝ 1/√r, I ∝ 1/r.
Plane wavefront (source at ∞): A = constant, I = constant.
Phase difference between any two points on the same wavefront = 0.

Resultant amplitude: A = √(a₁² + a₂² + 2a₁a₂ cos φ).
Resultant intensity: I = I₁ + I₂ + 2√(I₁I₂) cos φ.
I_max = (√I₁ + √I₂)²; I_min = (√I₁ − √I₂)².
I_av = I₁ + I₂ (average intensity, independent of phase).
Equal amplitudes (I₁ = I₂ = I₀): I_max = 4I₀, I_min = 0, I_av = 2I₀.
Intensity ∝ slit width ∝ (amplitude)²: I₁/I₂ = w₁/w₂ = a₁²/a₂².
Fringe visibility V = (I_max − I_min)/(I_max + I_min). V = 100% when I_min = 0 (equal amplitudes).
Phase ↔ path: φ = (2π/λ)δ = 2π(δ/λ) = 2π(Δt/T).
Constructive: δ = nλ (n = 0,1,2,...). Destructive: δ = (2n−1)λ/2 (n = 1,2,...).
Interference conserves energy — redistributes it from dark to bright regions.

Path difference at point P (height y): δ = yd/D.
nth bright fringe: y_n = nλD/d.
mth dark fringe: y_m = (2m−1)λD/2d.
Fringe width: β = λD/d (same for all fringes).
Angular fringe width: α = λ/d (independent of D).
Central fringe (n=0) is always bright. Closing one slit → no interference pattern.
White light: central fringe = white; nearest fringe on each side = red (longest λ, widest β); outermost visible fringe = blue.
Liquid of refractive index μ: λ′ = λ/μ, so β′ = β/μ. Fringe width decreases.
Increasing D: β increases, intensity decreases, α unchanged.
Fringe coincidence (two λ): n₁λ₁ = n₂λ₂. Fringes in space are hyperbolae.

Division of amplitude: both reflected rays come from the same incident beam (partial reflection at each surface).
Reflection from denser medium → phase change π → extra path λ/2. Reflection from rarer medium → no phase change.
Reflected system — Bright: 2μt cos r = (2n+1)λ/2. Dark: 2μt cos r = nλ.
Transmitted system — Bright: 2μt cos r = nλ. Dark: 2μt cos r = (2n+1)λ/2.
Reflected and transmitted systems are complementary (one bright where the other is dark).
Normal incidence (r = 0): cos r = 1, formulas simplify.
Uses: wavelength measurement, refractive index measurement, holography.

Bending of light around edges of an obstacle or aperture into the geometrical shadow region.
Condition: λ ≈ a (obstacle size ≈ wavelength). If a >> λ, light travels in straight lines.
Discovered by Grimaldi; theoretically explained by Fresnel.
Sound diffracts easily (λ large); light barely does (λ ~ 10⁻⁷ m); X-rays diffract in crystals (λ ~ 10⁻¹⁰ m).
Fresnel distance Z_F = a²/λ: beyond this, diffraction spreading equals slit width — wave optics needed.

Minima: a sin θ_n = nλ → x_n = nλD/a (n = ±1, ±2, ...).
Secondary maxima: a sin θ = (2n+1)λ/2.
Width of central maximum (linear): w_x = 2λD/a.
Angular width of central maximum: w_θ = 2λ/a.
Secondary fringe width (beyond central): β = λD/a (half the central maximum width).
Intensity decreases outward: I₀, I₀/22, I₀/61, ... for central, 1st, 2nd secondary maxima.
Wider slit → narrower central maximum. Narrower slit → wider central maximum.
Central maximum width ∝ λ: red light gives a wider central max than blue light.
Unlike YDSE: fringes are not equally bright or equally spaced.

Rayleigh's criterion: two images just resolved when central max of one falls on first min of the other.
Telescope — Resolving limit: δθ = 1.22λ/a; RP = a/(1.22λ).
Microscope — Resolving limit: RL = 1.22λ/(2μ sin θ); RP = 2μ sin θ/(1.22λ).
μ sin θ = Numerical Aperture (NA). Higher NA → better resolution.
Both RP ∝ 1/λ: shorter wavelength → better resolution.
Resolving power ≠ magnifying power. Magnifying a blur does not resolve it.

Light is transverse — the electric vector E vibrates perpendicular to the direction of propagation.
Unpolarised light: E vibrates symmetrically in all directions perpendicular to propagation.
Plane polarised: E vibrates in one fixed direction only.
Passing unpolarised light through an ideal polariser: intensity halved (I = I₀/2).
Crossed polariser + analyser (θ = 90°): zero intensity.

Brewster's law (reflection): μ = tan θ_p. At polarising angle, reflected beam is 100% polarised; refracted beam is partially polarised. At θ_p: reflected ⊥ refracted (θ_p + θ_r = 90°).
At i = 0° or 90°: reflected beam is unpolarised.
Pile of glass plates (refraction): repeated reflections at Brewster's angle remove the perpendicular component; transmitted beam becomes fully polarised after enough plates.
Dichroism / Polaroids: crystal absorbs one component, transmits the other. Most common practical polariser.
Scattering: light scattered at 90° to the incident direction is completely polarised (explains polarised sky light).
Double refraction (calcite, quartz): splits into O-ray and E-ray, both plane polarised with perpendicular vibrations. Nicol prism isolates E-ray using TIR of O-ray at canada balsam.

I = I₀ cos² θ (I₀ = intensity of polarised light incident on analyser; θ = angle between axes).
θ = 0°: I = I₀. θ = 45°: I = I₀/2. θ = 90°: I = 0.
Unpolarised → polariser → analyser: I = (I_unpol/2) cos² θ.
Malus's law applies only when the input to the analyser is already plane-polarised.

YDSE: nλD/d gives BRIGHT fringe. Single slit: nλD/a gives DARK fringe. Same formula, opposite meaning — don't swap.
Thin film reflected bright: 2μt cos r = (2n+1)λ/2 (odd half-multiples, NOT even). Transmitted bright: 2μt cos r = nλ.
Fringe width in YDSE changes with D and λ, but not with the number of the fringe — all fringes have the same width β.
Polarisation by reflection: at θ_p the reflected beam is 100% polarised; the refracted beam is only partially polarised — NOT 100%.
I₁ = I, I₂ = 4I (common NEET setup): √I₁ = √I, √I₂ = 2√I → I_max = 9I, I_min = I, ratio 9:1.
Intensity ratio of slits given as w₁:w₂ = 1:9 → a₁:a₂ = 1:3 → I_min:I_max = (3−1)²:(3+1)² = 4:16 = 1:4.
Angular fringe width α = λ/d is independent of screen distance D — a favourite trap in MCQs.

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