Wave Optics — NEET Notes

Huygens' principle & the shape of wavefronts

A wavefront is the locus of all points oscillating in phase at a given instant. Drop a stone in still water and the circular ripples you see are wavefronts in two dimensions — every point on a single ring has the same amplitude and the same phase. The wavefront travels outward at the speed of the wave; the direction of energy flow at any point is the line perpendicular to the wavefront, and that perpendicular line is what we call a ray. Wavefronts and rays are two ways of describing the same propagating disturbance.

The shape of the wavefront depends on the source. A point source emitting uniformly in all directions generates concentric spheres — a spherical wavefront. A line source generates concentric cylinders — a cylindrical wavefront. Any source viewed from very far away looks effectively point-like, but the small patch of sphere we intercept appears flat — a plane wavefront. Light reaching Earth from a distant star, or laser light emerging through a collimator, is treated as plane.

Once you know the shape of a wavefront at one instant, Huygens' principle lets you predict its shape at any later instant. The principle has two parts. First, every point on a given wavefront acts as a source of secondary spherical wavelets that spread out in all directions with the speed of the wave. Second, the new wavefront at a later time t is the forward common tangent (or envelope) drawn to all these secondary wavelets, each of radius vt. The original wavefront forgets itself; the envelope of the wavelets becomes the new wavefront.

Each point on a wavefront becomes a source of secondary disturbance; the new wavefront is the forward envelope of those secondary wavelets.

Huygens' principle — NCERT §10.2 / NIOS §22.1

One blemish remains. The secondary wavelets, being spherical, should produce a backward envelope as well — a "backwave" travelling in reverse. No such backwave is observed. Huygens patched this with an ad hoc assumption — the amplitude of secondary wavelets is maximum in the forward direction and zero in the backward — but a proper justification comes only from full wave theory. NEET does not test the backwave problem, but it can ask whether energy travels forward or backward.

Reflection & refraction using Huygens' construction

The real payoff of Huygens' principle is that it derives the laws of reflection and refraction — laws that earlier required Newton's corpuscular model — and it does so with a single geometric construction.

Refraction of a plane wave

Imagine a plane wavefront AB incident at angle i on the interface PP' between two media with wave speeds v₁ (medium 1) and v₂ (medium 2). Let τ be the time taken by the wavefront's right end to traverse the distance BC in medium 1, so that BC = v₁τ. During the very same time τ, the left end of the wavefront — already in medium 2 — emits a secondary wavelet of radius AE = v₂τ. Drawing the tangent CE from C to this sphere produces the refracted wavefront, inclined at angle r to the interface. From the right triangles ABC and AEC:

sin i / sin r  =  v₁ / v₂  =  n₂ / n₁

Snell's law, derived from Huygens — NCERT Eq. 10.6

This is Snell's law, written as n₁ sin i = n₂ sin r. Two consequences follow immediately. First, when light bends toward the normal (r < i), the wave slows down: v₂ < v₁. Newton's corpuscular model predicted the opposite — light should speed up in glass — and Foucault's 1850 measurement of the speed of light in water settled the dispute in Huygens' favour. Second, since the same wavefront crest B reaches C in time τ that crest A reaches E in time τ, the distances BC and AE equal the wavelengths in the two media. So λ₁/λ₂ = v₁/v₂, and the wavelength shortens in the denser medium even as the frequency stays unchanged.

Reflection of a plane wave

The construction for reflection is symmetric. A plane wave AB hits a plane reflector at angle i; the right end advances BC = vt within the same medium, while the left end emits a secondary wavelet of radius AE = vt on the same side. Triangles ABC and EAC are congruent (same hypotenuse, equal legs), forcing angle of reflection = angle of incidence. That is the law of reflection — recovered without any reference to rays.

The same Huygens reasoning also explains lens and prism behaviour. The thicker middle of a convex lens delays the central portion of an incoming plane wavefront more than the edges, producing a converging spherical wavefront. A prism tilts a plane wavefront uniformly because the wave travels through unequal glass thicknesses. The total optical path is equal along every ray — the time from object to image is the same whether the light travels straight through the centre or along an edge.

Principle of superposition & coherence

Wave optics rests on a single algebraic fact. When two or more waves overlap in the same region, the resultant displacement at every point is the vector sum of the individual displacements — this is the principle of superposition. From this one rule emerge interference, diffraction, beats, and standing waves.

Consider two sources S₁ and S₂ producing waves of the same amplitude a and the same frequency at a point P. Let the phase difference at P be φ. The resultant displacement is

y = 2 a cos(φ/2) cos(ωt + φ/2)

Superposition of two equal-amplitude waves — NCERT §10.4

Intensity is proportional to the square of the amplitude, so the resultant intensity at P is I = 4 I₀ cos²(φ/2), where I₀ is the intensity each source alone would produce. When φ = 0, ±2π, ±4π, … the cosine is ±1 and the intensity is 4I₀ — constructive interference. When φ = ±π, ±3π, ±5π, … the cosine is zero and the intensity vanishes — destructive interference. Energy is not created or destroyed; it is redistributed from the dark fringes into the bright ones.

Coherent vs incoherent sources

For a stable, observable interference pattern, the phase difference φ at every point must be constant in time. Two sources that maintain such a constant phase relationship are coherent. If the phase difference jitters randomly — as it does between two independent sodium lamps, where each atomic emission lasts only about 10⁻¹⁰ s — the maxima and minima shift faster than the eye or detector can resolve, and the time-averaged intensity comes out to I = 2 I₀ everywhere. The sources are incoherent and no fringes are seen.

Thomas Young's trick in 1801 was to derive two coherent sources from one. He let sunlight pass through a primary pinhole S, then onto two closely spaced pinholes S₁ and S₂. Because S₁ and S₂ are illuminated by the same wavefront, every phase jiggle in S is mirrored identically in both — locking their relative phase. This is the only practical way to get coherent visible-light sources, and it is the heart of every interference experiment in NCERT.

Young's double slit experiment

Two coherent point-like sources S₁ and S₂, separated by a small distance d, illuminate a screen placed at distance D (where D ≫ d). Consider a point P on the screen at distance x from the central axis. The waves from S₂ travel a slightly longer path than those from S₁; the geometric difference is the path difference Δ = S₂P − S₁P. Under the small-angle approximation, Δ = xd/D.

To convert path difference into phase difference, multiply by 2π/λ: φ = (2π/λ) Δ. Plugging into I = 4 I₀ cos²(φ/2) gives the full intensity pattern on the screen — a cos² function of x that oscillates between 4I₀ (bright) and 0 (dark). The conditions for the two extremes follow at once.

Constructive interference (bright fringe): Δ = nλ, for n = 0, ±1, ±2, …. The position of the n-th bright fringe is x_n = nλD/d.

Destructive interference (dark fringe): Δ = (n + ½)λ, for n = 0, ±1, ±2, …. The position of the n-th dark fringe is x_n = (n + ½)λD/d.

The spacing between successive bright fringes — or between successive dark fringes — is the fringe width:

When YDSE is performed in a medium of refractive index μ, the wavelength becomes λ/μ and the fringe width contracts to β' = λD/(μd). This appeared in NEET 2017 Q.176, where the 8th bright fringe in a medium coincided with the position the 5th dark fringe occupied in air — solving 8(λD/μd) = (9/2)(λD/d) gives μ = 9/(2 × 8) × 2 × … ≈ 1.78. Angular fringe width β/D = λ/d depends only on the wavelength-to-slit-separation ratio; it does not change if you move the screen further away (NEET 2023 Q.22, Statement I).

Diffraction at a single slit

If you remove one of Young's two slits and let monochromatic light pass through a single slit of width a, the bright sharp shadow you might expect does not appear. Instead the screen shows a broad bright central band flanked by progressively fainter dark and bright fringes on both sides — the single-slit diffraction pattern. Diffraction is what happens when a wavefront encounters an aperture or an obstacle whose size is comparable to the wavelength: the wave bends around corners.

To analyse it, treat the slit as a continuous distribution of Huygens secondary sources. A plane wavefront arrives at the slit in phase across its full width. At an observation angle θ, the wavelets from one edge of the slit travel a path longer than those from the other edge by a sin θ. Imagine dividing the slit into two halves: at the angle where a sin θ = λ, the wavelet from the very top of the upper half is exactly λ/2 out of phase with the wavelet from the very top of the lower half — they cancel. Pair every point in the upper half with its partner in the lower half and the whole slit cancels out. This is the first minimum.

Repeat the pairing for four equal strips, six strips, and so on, and you get the full sequence of minima at the angles

a sin θ = n λ,    n = ±1, ±2, ±3, …

Single-slit diffraction minima — NCERT §10.6.1

Between consecutive minima lie secondary maxima, located approximately at a sin θ = (n + ½) λ. These maxima are not equally bright — the first secondary maximum has only about 4.5 % of the central intensity and they fall off rapidly. NEET 2016 Q.154 directly tested the position of the first secondary maximum at a sin θ = 3λ/2, paired with first minimum at 30°.

Two features distinguish diffraction from interference. First, the central maximum is twice as wide as any secondary maximum because its boundaries are the ±first minima, while each secondary peak is bounded by two consecutive minima. Second, the intensity falls off dramatically with order — unlike YDSE where every bright fringe is equally bright. NCERT carries Feynman's wry remark that there is "no specific, important physical difference" between interference and diffraction — only convention separates them. For NEET, however, the operational differences are very testable.

Interference vs diffraction — five exam-critical differences

The double-slit experiment, in fact, contains both phenomena: the observed pattern is the YDSE interference fringes multiplied by the single-slit diffraction envelope of each individual slit. The interference fringes are sharp and equally spaced, but their overall brightness is modulated by a slowly varying diffraction envelope — bright in the centre, dimming toward the edges. NCERT mentions this combined pattern explicitly in §10.6.1.

Resolving power of telescopes & microscopes

Diffraction does not just produce pretty patterns — it sets a hard physical limit on what optical instruments can resolve. Because every aperture diffracts, the image of even a single point source is not a point but a small Airy disc surrounded by faint rings. Two stars (or two cells) can be told apart only if their Airy discs do not overlap too much.

For a telescope with objective aperture of diameter D, the minimum angular separation between two just-resolvable sources is θ_min = 1.22 λ / D. The resolving power is 1/θ_min = D/(1.22 λ). NEET 2020 Q.119 used this directly: starlight of λ = 600 nm through a 2 m objective gives θ_min = 1.22 × 6 × 10⁻⁷ / 2 = 3.66 × 10⁻⁷ rad.

For a microscope imaging through a medium of index n, the minimum resolvable distance between two object points is d_min = 1.22 λ / (2 n sin β), where 2β is the angular cone subtended by the objective at the specimen. The numerical aperture NA = n sin β packages the medium and the cone. Resolving power = 1/d_min = 2 n sin β / (1.22 λ), so it scales as 1/λ. NEET 2017 Q.142 asked exactly this: the ratio of resolving powers at λ₁ = 4000 Å and λ₂ = 6000 Å is 6000/4000 = 3/2.

Polarisation, Brewster's angle & Malus' law

Interference and diffraction are exhibited by every kind of wave — sound, water, light, matter waves. Polarisation is the one phenomenon that is special to transverse waves, and so its existence in light directly proves that light is transverse. In an unpolarised beam, the electric field vector points in every direction perpendicular to propagation, changing randomly in time. A polariser — a Polaroid sheet, for example — selects only the component along its pass-axis and absorbs the rest. The transmitted beam is linearly polarised.

Pass unpolarised light of intensity I_unp through a single Polaroid and exactly half the intensity emerges, polarised along the pass-axis: I_after = I_unp / 2. Now place a second Polaroid (the analyser) at angle θ to the first. The component of the electric field along the analyser's pass-axis is E cos θ, so the transmitted intensity scales as cos² θ. This is Malus' law:

For two crossed Polaroids with a third one sandwiched at angle θ to the first, the final intensity becomes (I₀/2) cos²θ · sin²θ = (I₀/8) sin²(2θ) — maximum at θ = 45°.

Brewster's angle — polarisation by reflection

When unpolarised light strikes a dielectric surface at a particular angle, the reflected ray comes out completely plane-polarised with its electric vector lying in the plane of the surface (perpendicular to the plane of incidence). This special angle is the Brewster angle θ_B. At this angle, the reflected and refracted rays are mutually perpendicular — and using Snell's law with this 90° geometry yields Brewster's law:

tan θ_B  =  n₂ / n₁

Brewster's law — NEET 2018 Q.31, NEET 2020 Q.125

For light going from air (n₁ = 1) into glass (n₂ ≈ 1.5), θ_B = tan⁻¹(1.5) ≈ 56°. For any denser medium n₂ > n₁, tan θ_B > 1, so 45° < θ_B < 90° — the exact result NEET 2020 Q.125 tested. The reflected light is polarised perpendicular to the plane of incidence; the refracted light is partially polarised parallel to it. Polaroid sunglasses exploit this: glare off horizontal surfaces is largely horizontally polarised, and the sunglasses block that component.