Wave Optics: Interference and Diffraction notes: ISC Class 12 Physics

1. Wavefronts and Huygens’ principle

Wavefront: the locus of points vibrating in the same phase; it moves at the wave speed. In a uniform medium the rays are perpendicular to the wavefronts.
Shapes: a point source gives spherical wavefronts (amplitude ∝ 1/r, intensity ∝ 1/r²); a line source gives cylindrical wavefronts (amplitude ∝ 1/√r, intensity ∝ 1/r); a very distant source gives plane wavefronts (no spreading, constant intensity).
Huygens’ principle: every point of a wavefront is a source of secondary wavelets travelling forward at the wave speed v; after time t the new wavefront is the forward envelope (common tangent) of the wavelets of radius vt. The backward envelope is ignored (no backward wave is observed).
A spherical wavefront of radius R becomes one of radius R + vt; a plane wavefront moves forward a distance vt and stays plane.

Formulas

Fall of intensity with distance
$I \propto \dfrac{1}{r^2}\ \text{(point)},\qquad I \propto \dfrac{1}{r}\ \text{(line)}$

Key terms

Wavefront: A surface on which all points vibrate in the same phase.
Secondary wavelets: The small spherical waves that, by Huygens’ principle, spread from every point of a wavefront at the wave speed.

2. Reflection and refraction from Huygens’ construction

Reflection: a plane wavefront AB meets the mirror at A; while B travels to C (BC = vt), the wavelet from A grows to AE = vt. The reflected wavefront CE is the tangent from C. Triangles ABC and CEA are right-angled (at B and E), share AC and have BC = AE, so they are congruent: i = r.
Refraction (rarer to denser, v₂ < v₁): BC = v₁t, AE = v₂t, EC tangent from C. sin i = BC/AC, sin r = AE/AC, so $\dfrac{\sin i}{\sin r} = \dfrac{v_1}{v_2} = \dfrac{n_2}{n_1}$ (Snell’s law), with n = c/v.
Denser to rarer: the wavelet is longer than BC and the ray bends away from the normal; when (v₂/v₁) sin i > 1 the wavelet radius exceeds AC, no tangent exists and there is no refracted wave (total internal reflection).
In a medium: frequency unchanged (set by the source); speed v = c/n; wavelength λ_m = λ/n.

Formulas

Snell’s law from Huygens’ construction
$\dfrac{\sin i}{\sin r} = \dfrac{v_1}{v_2} = \dfrac{n_2}{n_1}$
Wavelength in a medium
$\lambda_m = \dfrac{\lambda}{n}$

Key terms

Refractive index: n = c/v, the ratio of the speed of light in vacuum to that in the medium; n₂/n₁ = v₁/v₂.

3. Superposition, coherence and the conditions for interference

Superposition: where waves overlap, the resultant displacement is the sum of the separate displacements; I ∝ (amplitude)².
Phase difference $\varphi = \dfrac{2\pi}{\lambda}\Delta$ for a path difference Δ (sources in phase). Constructive: Δ = nλ, φ = 2nπ, amplitude a₁ + a₂. Destructive: Δ = (2n − 1)λ/2, φ = (2n − 1)π, amplitude |a₁ − a₂|.
$I_{\max} = (\sqrt{I_1} + \sqrt{I_2})^2$ , $I_{\min} = (\sqrt{I_1} - \sqrt{I_2})^2$ .
Coherent sources: same frequency and a constant phase difference. Two independent lamps are incoherent (random, short bursts of emission), so the intensities simply add: I = I₁ + I₂, no fringes.
Conditions for sustained, clear fringes: coherent sources (derived from one source, e.g. by two slits or a laser); monochromatic light; nearly equal amplitudes (good contrast); narrow source slit; slits close together and the screen far enough away for β to be visible.

Formulas

Phase difference from path difference
$\varphi = \dfrac{2\pi}{\lambda}\,\Delta$
Contrast of the fringes
$\dfrac{I_{\max}}{I_{\min}} = \dfrac{(a_1 + a_2)^2}{(a_1 - a_2)^2}$

Key terms

Coherent sources: Sources of the same frequency with a constant phase difference between them.
Path difference: The difference in the distances travelled by two waves from their sources to a point.

4. Young’s double-slit experiment

A narrow slit S lights two slits S₁, S₂ (separation d), which act as coherent sources in phase; the screen is at distance D ≫ d.
Path difference at a point P at height y: S₂P − S₁P ≈ S₂N = d sin θ ≈ yd/D (small angles).
Bright fringes: yd/D = nλ, $y_n = \dfrac{n\lambda D}{d}$ (n = 0, 1, 2, …; the central fringe is bright). Dark fringes: $y_n = \dfrac{(2n - 1)\lambda D}{2d}$ (n = 1, 2, …).
Fringe width $\beta = \dfrac{\lambda D}{d}$ , the same for all fringes; angular fringe width λ/d.
Effects: β ∝ λ (red fringes wider than blue), β ∝ D, β ∝ 1/d; in a medium of index n, β′ = β/n. White light: white central fringe, coloured fringes beyond (violet edge inwards), soon merging. Widening the source slit or making one slit dimmer reduces the contrast; covering one slit removes the fringes.

Worked examples 4 and 6 drawn to the same scale.

Formulas

Path difference
$\Delta = d\sin\theta \approx \dfrac{yd}{D}$
Fringe width
$\beta = \dfrac{\lambda D}{d}$

Key terms

Fringe width: The distance between the centres of two neighbouring bright (or dark) fringes, β = λD/d.

5. Intensity distribution in the double-slit pattern

Result (quoted; the derivation from the wave equations is not in the ISC syllabus): $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\varphi$ . For equal slits, $I = 4I_0\cos^2\dfrac{\varphi}{2}$ .
Sketch: a cos² pattern against y (or θ): maxima 4I₀ at y = nβ, zeros at y = (n − ½)β, all maxima equally bright (narrow slits); the average over many fringes is 2I₀, so energy is conserved and only redistributed.
Unequal slits: the same positions and spacing, but maxima (√I₁ + √I₂)² and non-zero minima (√I₁ − √I₂)²: poorer contrast.
Real slits have a width a, so the fringes are modulated by the single-slit envelope, whose first zeros are at sin θ = ±λ/a.

equal slits, each I₀ (computed)
slits giving I₀ and I₀/4 (computed)

I = I₁ + I₂ + 2√(I₁I₂) cos φ, with φ = 2πy/β.

two slits 0.35 mm apart, each 0.07 mm wide (computed)
single-slit envelope of one slit

Real double-slit fringes: interference fringes (spacing λ/d) inside a diffraction envelope (first zeros at ±λ/a).

Formulas

Equal slits
$I = 4I_0\cos^2\dfrac{\varphi}{2}$
Average intensity
$I_{\text{av}} = 2I_0$

Key terms

Contrast (visibility): How dark the dark fringes are compared with the bright ones; best when the two slits give equal intensities.

6. Single-slit (Fraunhofer) diffraction and the central maximum

Diffraction: the spreading of light into the geometrical shadow when it passes a slit or edge comparable with its wavelength. Fraunhofer case: plane wavefront on the slit, pattern observed far away (or in the focal plane of a lens).
Central maximum: straight ahead, all wavelets from the slit are in phase.
Minima: $a\sin\theta = n\lambda$ (n = ±1, ±2, …). For n = 1, divide the slit into two halves: each wavelet in the upper half has a partner a/2 lower, λ/2 behind, and they cancel in pairs.
Secondary maxima: approximately at $a\sin\theta = \left(n + \tfrac{1}{2}\right)\lambda$ (divide into three, five, … parts: one part is left uncancelled), so they are faint (the first under 1/20 of the central intensity) and fade outwards.
Central maximum: angular width $2\theta_1 = \dfrac{2\lambda}{a}$ , linear width $W = \dfrac{2\lambda D}{a}$ ; twice as wide as each secondary maximum. A narrower slit or longer wavelength widens it; in a liquid λ, and so W, is divided by n.
Compared with double-slit interference: see the table.

slit 0.18 mm, λ = 630 nm (computed)

Minima where a sin θ = nλ; the central maximum is twice as wide as the others and far brighter.

Table comparing double-slit interference with single-slit diffraction: two sources against many, equal fringe widths against a double-width central band, equal brightness against rapidly fading bands, and the conditions for dark fringes
Double-slit interference	Single-slit diffraction
Waves from two coherent slits	Wavelets from every point of one slit
Fringes equally wide ( $\lambda D/d$ )	Central band twice as wide as the others
Bright fringes (nearly) equally bright	Brightness falls off sharply away from the centre
Dark fringes at $d\sin\theta = (n - \tfrac{1}{2})\lambda$	Dark bands at $a\sin\theta = n\lambda$

Formulas

Single-slit minima
$a\sin\theta = n\lambda$
Width of the central maximum
$W = \dfrac{2\lambda D}{a}$

Key terms

Diffraction: The bending and spreading of waves round obstacles or through apertures comparable in size with the wavelength.
Central maximum: The bright band between the first minima on either side of a single-slit pattern, of angular width 2λ/a.

Exam tips

Huygens’ proofs: wavelet from the first point of contact, tangent from the last; equal times for BC and AE.
In a medium: f unchanged, v and λ divided by n; fringe width and central-band width divided by n.
Double slit: bright yd/D = nλ, dark (2n − 1)λ/2, β = λD/d; equal slits give I from 4I₀ to 0.
Single slit: dark at a sin θ = nλ; central width 2λ/a (angle), 2λD/a (screen).

1. Wavefronts and Huygens’ principle

Formulas

Key terms

2. Reflection and refraction from Huygens’ construction

Formulas

Key terms

3. Superposition, coherence and the conditions for interference

Formulas

Key terms

4. Young’s double-slit experiment

Formulas

Key terms

5. Intensity distribution in the double-slit pattern

Formulas

Key terms

6. Single-slit (Fraunhofer) diffraction and the central maximum

Formulas

Key terms

Exam tips

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