Wave Optics

2.1 Core concepts

• The corpuscular model (Descartes 1637, popularised by Newton in OPTICKS) wrongly predicted higher speed of light in a denser medium; Huygens' wave model (1678) correctly predicted lower speed, confirmed by Foucault's 1850 experiment (NCERT §10.1, p. 255).
• Young's 1801 interference experiment firmly established the wave nature of light; Maxwell later identified light as an electromagnetic wave, explaining how it can travel through vacuum (NCERT §10.1, p. 256).
• A wavefront is a surface of constant phase; energy propagates perpendicular to the wavefront, and the speed of the wavefront equals the wave speed. A point source gives spherical wavefronts; far from the source these become effectively plane (NCERT §10.2, p. 257).
• Huygens' principle: every point on a wavefront acts as a source of secondary spherical wavelets that spread out at the wave speed; the new wavefront at time t is the forward envelope (common tangent) of these wavelets. The backwave is suppressed because the amplitude is maximum forward and zero backward (NCERT §10.2, p. 257-258).
• Refraction by Huygens' construction: if a plane wavefront AB hits the interface and the wavefront travels distance v1τ in medium 1 while a secondary wavelet in medium 2 has radius v2τ, geometry gives sin i / sin r = v1/v2 = n2/n1, i.e., Snell's law n1 sin i = n2 sin r (NCERT §10.3.1, p. 258-259, Eqs. 10.3, 10.6).
• On refraction into a denser medium (v1 > v2) the wavelength and speed decrease but the frequency ν = v/λ is unchanged (NCERT §10.3.1, p. 259-260).
• Refraction at a rarer medium (v2 > v1) bends the wave away from the normal; the critical angle sin ic = n2/n1 defines the onset of total internal reflection for i > ic (NCERT §10.3.2, p. 260, Eq. 10.8).
• Reflection by Huygens' construction gives congruent triangles EAC and BAC, so the angle of incidence equals the angle of reflection (NCERT §10.3.3, p. 260-261).
• The total time taken from object point to image point is the same along every ray — e.g., the central ray through a convex lens traverses less distance but moves slower through thicker glass (NCERT §10.3.3, p. 261).
• Coherent sources maintain a constant phase difference at every point (like two needles oscillating in phase in a trough); when waves of amplitude a superpose in phase, resultant amplitude is 2a and intensity is 4 I0 (NCERT §10.4, p. 262-263).
• For path difference S1P ~ S2P = nλ we get constructive interference (I = 4 I0); for S1P ~ S2P = (n + 1/2) λ we get destructive interference (I = 0) (NCERT §10.4, p. 263-264, Eqs. 10.9, 10.10).
• General intensity formula: I = 4 I0 cos²(φ/2); if two sources are incoherent (rapidly fluctuating phase) intensities just add, giving I = 2 I0 everywhere (NCERT §10.4, p. 264, Eqs. 10.11, 10.12).
• Two independent sodium lamps cannot give interference fringes because each ordinary source undergoes abrupt phase changes on a 10⁻¹⁰ s timescale; Young solved this by deriving S1 and S2 from a single primary pinhole S so the two phases are locked (NCERT §10.5, p. 265).
• Young's double-slit fringe positions: bright fringes at x_n = nλD/d and dark fringes at x_n = (n + 1/2) λD/d; bright and dark fringes are equally spaced (NCERT §10.5, p. 266, Eqs. 10.13, 10.14).
• Single-slit diffraction: for a slit of width a illuminated normally, the central maximum sits at θ = 0; minima occur at θ ≈ nλ/a, n = ±1, ±2, ±3,…; secondary maxima at θ ≈ (n + 1/2) λ/a, growing weaker with n (NCERT §10.6.1, p. 267).
• A double-slit pattern is actually a superposition of single-slit diffraction from each slit on the double-slit interference pattern (NCERT §10.6.1, p. 267).
• Quoting Feynman: there is no sharp physical distinction between interference and diffraction — "interference" is used for a few sources, "diffraction" when there are many (NCERT §10.6.1, p. 267).
• In both interference and diffraction, light energy is only redistributed — no gain or loss — consistent with energy conservation (NCERT §10.6.2, p. 268).
• Polarisation and transverse nature: a wave with displacement at right angles to its propagation direction is transverse; if displacement stays in one plane, the wave is linearly (plane) polarised (NCERT §10.7, p. 269-270, Eq. 10.15).
• Unpolarised wave: plane of vibration changes randomly in very short intervals while remaining perpendicular to the propagation direction; natural light (e.g., sunlight) is unpolarised (NCERT §10.7, p. 270; Summary point 6, p. 272).
• A polaroid has long aligned chain molecules that absorb the electric vector along the molecular axis and transmit the perpendicular component (the "pass-axis"). Unpolarised light through a single polaroid loses half its intensity, independent of orientation (NCERT §10.7, p. 270-271).
• Malus' law: when polarised light of intensity I0 passes through a polaroid whose pass-axis makes angle θ with the polarisation direction, the transmitted intensity is I = I0 cos²θ (NCERT §10.7, p. 270, Eq. 10.18).
• Two crossed polaroids transmit zero intensity; a third polaroid placed between them at angle θ transmits (I0/4) sin²(2θ), maximum at θ = π/4 (NCERT §10.7, Example 10.2, p. 271).
• Polaroid applications: sunglasses, windowpanes, photographic cameras, 3D movie cameras (NCERT §10.7, p. 271).
• Conditions for sustained interference: the two sources must be coherent (constant phase difference), monochromatic (single wavelength), and of nearly equal intensity (else dark fringes are not strictly dark). Path difference must be small compared with the coherence length of the source (NCERT §§10.4–10.5, p. 262–266).
• Fringe width β = λD/d is independent of fringe order; all fringes are equally spaced. β increases if λ is larger, D is larger, or d is smaller; immersing the apparatus in a medium of refractive index n reduces λ to λ/n, so β shrinks by the same factor (NCERT §10.5, p. 266).
• Energy conservation in interference and diffraction: the redistribution of light over the screen produces brighter maxima and darker minima, but the total energy reaching the screen equals what would arrive without interference; light is neither created nor destroyed (NCERT §10.6.2, p. 268).
• Polarisation by scattering, reflection and selective absorption are three distinct ways of producing polarised light from natural unpolarised sunlight; polaroids (selective absorption) are the most common. Scattered sunlight from the sky is partially polarised, which is what polarising sunglasses exploit (NCERT §10.7, p. 270–271).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 10.1 (a),(b) — Spherical wavefront from a point source; plane wavefront at large distance (p. 257).
• Fig. 10.2 — Huygens' construction: spherical wavefront F1F2 → new wavefront G1G2 as envelope of secondary wavelets; backwave D1D2 absent (p. 257).
• Fig. 10.3 — Huygens' construction for a plane wave propagating to the right; lines A1A2, B1B2 are rays (p. 258).
• Fig. 10.4 — Refraction of plane wave AB at interface PP'; CE is refracted wavefront when v2 < v1 (bends toward normal) (p. 258).
• Fig. 10.5 — Refraction at a rarer medium (v2 > v1); wavefront bends away from normal (p. 260).
• Fig. 10.6 — Reflection of plane wave AB by surface MN; CE is reflected wavefront (p. 261).
• Fig. 10.7 (a),(b),(c) — Wavefront refraction by thin prism, by convex lens (focal point F), and reflection by concave mirror (p. 261).
• Fig. 10.8 — Two needles oscillating in phase → coherent sources; nodal (N) and antinodal (A) lines on water surface (p. 262).
• Fig. 10.9 — Constructive interference at point Q (path difference 2λ); destructive interference at point R (path difference 2.5λ) (p. 263).
• Fig. 10.10 — Locus of points where S1P – S2P = 0, ±λ, ±2λ, ±3λ (p. 263).
• Fig. 10.11 — Two sodium lamps illuminating two pinholes give no fringes (incoherent) (p. 265).
• Fig. 10.12 — Young's double-slit arrangement; fringes on screen GG' (p. 265).
• Fig. 10.13 — Computer-generated Young's fringes with d = 0.025 mm, D = 5 cm, λ = 5×10⁻⁵ cm (p. 266).
• Fig. 10.14 — Single-slit geometry: slit LN of width a, midpoint M, path differences for diffraction (p. 267).
• Fig. 10.15 — Intensity distribution + photograph of single-slit fringes; broad central maximum with weaker side maxima (p. 267).
• Fig. 10.16 — Two razor blades held to form a single slit (home experiment) (p. 268).
• Fig. 10.17 — Transverse string wave: y(x,t) = a sin(kx – ωt); linearly/plane polarised (p. 269).
• Fig. 10.18 — Two polaroids P2 and P1: transmitted fraction varies as cos²θ as angle between axes goes from 0° to 90° (p. 271).

2.4 Common confusions / NTA trap points

• Corpuscular vs wave prediction on refraction: corpuscular says light is faster in denser medium, wave theory says slower — Foucault (1850) confirmed wave theory. Distractors often swap this.
• On refraction, frequency stays the same while wavelength and speed change. Students wrongly say wavelength is unchanged.
• Two independent identical sodium lamps are NOT coherent — their phases jump every ~10⁻¹⁰ s, so no fringes are seen. The trick in Young's setup is that S1 and S2 are derived from one primary source S.
• For interference: I_max = 4 I0 (when both sources contribute equal I0) and I_min = 0; with incoherent addition I = 2 I0 everywhere. Many students miss the factor of 4.
• Single-slit minima are at a sin θ = nλ (n ≠ 0). The condition nλ looks identical to the interference maxima condition — but here it's minima, and n = 0 is excluded (it's the central maximum, not a minimum).
• Width of central maximum in single-slit diffraction is 2λ/a (in angular terms) — i.e., from θ = –λ/a to θ = +λ/a — twice the spacing of subsequent minima. NTA loves this trap.
• Malus' law applies when light is already polarised. Unpolarised light through a single polaroid is reduced by exactly half, regardless of the angle of the polaroid.
• Fringe width β = λD/d depends on d (slit separation), not on a (slit width). Confusing the two parameters is a recurring NTA trap.
• Immersing Young's setup in water reduces wavelength by 1/n; fringe width shrinks (β/n), the central fringe stays put because the wavelength reduction is symmetric.
• Diffraction angle is large only when a ≈ λ. For a ≫ λ (e.g. ordinary slit and visible light), the central maximum is very narrow and the pattern resembles geometric shadow.
• Both interference and diffraction conserve total energy — bright maxima borrow intensity from where the minima fall; nothing is "lost" at dark fringes.

2.5 Key formulas table