Wave Optics— Notes, Formulas & Revision

Huygens' Principle: every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront at a later instant is the forward envelope of these wavelets.

A wavefront is a surface of constant phase. Light travels perpendicular to the wavefront, in the direction of the ray.

The principle explains rectilinear propagation, reflection, and refraction without invoking corpuscles — and is consistent with Snell's law.

It works for any wave (sound, water, light) and is the wave-optics foundation of every phenomenon you'll meet in this chapter — interference, diffraction, polarization.

Newton's corpuscular theory could not explain interference and diffraction; Huygens' wave model can.

The 'obliquity factor' (1 + cos θ)/2 in the modern Fresnel-Kirchhoff form means wavelets are stronger in the forward direction — there is no backward wavefront, resolving an old objection to Huygens.

Coherent monochromatic light incident on two narrow parallel slits S₁ and S₂ produces alternating bright and dark fringes on a distant screen — Young's classic 1801 experiment.

Path difference at a point P on the screen: Δ = d sin θ ≈ dy/D, where d = slit separation, D = slit-to-screen distance, y = distance from central axis.

Bright fringes (constructive): Δ = nλ. Dark fringes (destructive): Δ = (n + ½)λ.

Fringe width β = λD/d. β depends only on the wavelength, geometry, and is the same for all orders.

Intensity on the screen: I(θ) = 4I₀ cos²(πd sin θ / λ). All bright fringes have equal intensity in this idealization (ignoring diffraction envelope).

Central bright fringe is at the screen point equidistant from both slits — i.e., the perpendicular bisector of the two slits hits the screen.

Inserting a thin glass slab in front of one slit shifts the fringe pattern by a distance (μ − 1)t·D/d toward that slit, where μ is the slab's refractive index.

When monochromatic light passes through a single slit of width a, it spreads out and produces a diffraction pattern on a distant screen.

The pattern consists of a bright central maximum flanked by alternating dark minima and weaker secondary maxima.

Minima occur where a sin θ = nλ (n = ±1, ±2, ...). NOT n = 0 — that's the central max.

The central maximum is twice as wide as any secondary maximum. Its angular half-width is λ/a.

Intensity envelope: I(θ) = I₀(sin α / α)², where α = πa sin θ / λ. Secondary maxima approximately at α = ±1.43π, ±2.46π, ...

First secondary max ≈ 4.5% of central max. Diffraction is more pronounced when a is comparable to λ.

Fraunhofer diffraction = far-field (source and screen at infinity). Fresnel diffraction = near-field. JEE assumes Fraunhofer.

Coherent sources have a constant phase difference and the same frequency — necessary for stable, observable interference.

Two independent ordinary light sources (e.g., two bulbs) are NOT coherent — their phase relationship fluctuates randomly on ~ns timescales, averaging the interference pattern to zero.

In practice, coherent sources are obtained from a single source by either: (a) division of wavefront (e.g., Young's slits, Fresnel's biprism) or (b) division of amplitude (e.g., thin films, Michelson interferometer).

Coherence length L_c = c·τ_c, where τ_c is the coherence time. For a laser, L_c can be metres; for sunlight ~µm.

When coherent sources superpose: I = I₁ + I₂ + 2√(I₁I₂) cos δ. For incoherent sources, the cosine term averages to zero: I = I₁ + I₂.

Lasers are highly coherent (both temporal and spatial). Sunlight has low temporal coherence — that's why thin-film colors require very thin films (within coherence length).

Malus' Law: When linearly polarized light of intensity I₀ passes through an analyzer whose transmission axis makes angle θ with the polarizer's axis, the transmitted intensity is I = I₀ cos²θ.

Unpolarized light through a single polarizer becomes linearly polarized with half the intensity: I = I₀/2.

Crossed polarizers (θ = 90°) block all light: I = 0. Parallel polarizers (θ = 0°): full transmission.

A third polarizer placed between two crossed polarizers, at angle θ to the first, transmits a non-zero intensity — a surprising result of Malus' Law.

Polarized sunglasses use Malus' Law (and Brewster reflection) to block horizontally polarized glare from water/road surfaces.

Liquid Crystal Displays (LCDs) use crossed polarizers with a tunable liquid-crystal layer to control pixel brightness.

Sky polarization is at 90° from the sun (Rayleigh scattering) — useful for navigation by polarized-light-sensitive insects (e.g., bees).

Brewster's Law: When light strikes a transparent medium at the Brewster angle θ_B, the reflected ray is completely polarized with E perpendicular to the plane of incidence (s-polarization).

Brewster's law: tan θ_B = n₂/n₁. For light going from air (n=1) to glass (n=1.5): θ_B ≈ 56.3°.

At θ_B, the reflected and refracted rays are perpendicular to each other: θ_B + θ_t = 90°.

Only the s-component reflects; the p-component (E in the plane of incidence) is transmitted 100% into the denser medium.

Brewster's law is used to make polarizers (Brewster pile-of-plates) and in laser-tube end-windows to minimize reflection loss for one polarization.

Polarized sunglasses block horizontally polarized glare from horizontal surfaces (water, roads) reflected near the Brewster angle.

Reflected intensity at general angles is given by Fresnel equations; the p-component goes to zero exactly at θ_B.

A diffraction grating is a large number N of equally-spaced parallel slits of width a and spacing d.

Principal maxima at d sin θ = mλ — same condition as YDSE bright fringes — but sharpened by the multi-slit interference.

Peak intensity of a principal max scales as N²·I₀, but the angular width shrinks as 1/N — so total power scales as N (energy is conserved).

Between two principal maxima there are (N − 2) secondary maxima — all very faint compared to principal.

Resolving power R = λ/Δλ = m·N. Larger N or higher order m → finer wavelength discrimination.

Single-slit diffraction envelope (sin α/α)² modulates the entire grating pattern. Missing orders when this envelope hits a zero.

Used in spectrometers to disperse light by wavelength — different m's spread different λ's at different angles.

Rayleigh criterion: two point sources are just resolved when the central maximum of one falls on the first minimum of the other.

For a circular aperture (lens, telescope, eye): θ_min = 1.22 λ/D, where D = aperture diameter.

The 1.22 factor comes from the first zero of the Bessel function J₁ — different from a rectangular slit's λ/a.

Resolving power R = 1/θ_min. Larger D and shorter λ → better resolution.

Telescope resolving power: R_telescope = D/(1.22 λ). For Hubble (D = 2.4 m, λ = 550 nm): R ≈ 3.6 × 10⁶ rad⁻¹.

Microscope (Abbe limit): d_min = 1.22 λ/(2 NA), where NA = n sinα is the numerical aperture. Shorter λ and higher NA improve resolution.

Diffraction-limited optical microscopes can resolve ~200 nm with visible light. Electron microscopes use λ_e ~ 10⁻¹² m for atomic-scale resolution.

Light reflects from both the top and bottom surfaces of a thin film, producing interference between the two reflected beams.

Path difference between the two reflected rays: 2nt cos θ_t, where n = film index, t = film thickness, θ_t = angle inside the film.

When reflection occurs at a denser medium, the reflected wave undergoes a π phase shift (equivalent to extra λ/2 path).

Soap film in air (n_film > n_air, single denser-medium reflection at top): bright when 2nt cos θ_t = (m + ½)λ.

Anti-reflection coating on glass: film of index √n_glass, thickness λ/(4n_film) — destructive interference suppresses reflection at one wavelength.

The vivid colours of soap bubbles, oil slicks, and butterfly wings come from selective wavelength interference — thickness varies, so different colours appear at different places.

Newton's rings: a special thin-film geometry (plano-convex lens on flat glass) — concentric dark/bright rings centered on contact point.