Wave optics: interference, diffraction and polarisation

In short

Wave optics treats light as an electromagnetic wave and explains the effects geometrical optics cannot: interference (two or more coherent waves adding), diffraction (bending and spreading at apertures and edges) and polarisation (the orientation of the electric-field vibration). These three effects are the physical basis of anti-reflection coatings, interferometric metrology, spectrometers, LCD displays, polarising filters, photoelastic stress analysis and sugar-industry polarimetry. In Indian exams it is a heavy-weight block in B.Sc Optics papers and in GATE (PH), JAM, JEST and CSIR-UGC NET, where numerical questions on fringe width, grating orders, resolving power, Brewster angle and wave-plate thickness recur almost every year. Employers who test it include ISRO centres (SAC Ahmedabad, LEOS Bengaluru), CSIR labs (CGCRI, NPL, CSIO Chandigarh), Bharat Electronics, Tonbo Imaging, Holmarc, Optics & Allied Engineering, display and lens manufacturers, and sugar mills and testing laboratories that run saccharimeters.

Explained

Wave nature, superposition and coherence

• Light is a transverse electromagnetic wave; the optical field usually means the electric vector E. Intensity I ∝ ⟨E²⟩, so detectors see intensity, never phase directly.
• Superposition: resultant field is the vector sum. For two waves of amplitudes a₁, a₂ with phase difference δ, I = I₁ + I₂ + 2√(I₁I₂)·cos δ. The last term is the interference term.
• Path difference to phase difference: δ = (2π/λ)·Δ for a path difference Δ in vacuum; in a medium use the optical path μ·geometric path.
• Coherence is the condition for a steady fringe pattern: constant (or slowly varying) phase relation.
  • Temporal (longitudinal) coherence: coherence length L_c = λ²/Δλ, coherence time τ_c = L_c/c. A white-light source has L_c of a few micrometres; a laser, metres to kilometres.
  • Spatial (transverse) coherence: depends on source size; a wide source smears fringes, hence the slit S in Young's experiment.
• Two independent sources (two bulbs, two lamps) never interfere observably because their relative phase changes randomly in ~10⁻⁸ s. Both beams must come from one source — this is why interference set-ups use division of wavefront or division of amplitude.
• Conditions for good fringes: coherent sources, comparable amplitudes (best contrast at I₁ = I₂), small separation of sources, monochromatic or narrow-band light, same state of polarisation (Fresnel–Arago laws).
• Fringe visibility V = (I_max − I_min)/(I_max + I_min); V = 1 for equal amplitudes, 0 for incoherent addition.

Interference by division of wavefront

• Young's double slit: two slits separated by d, screen at distance D (D ≫ d). Path difference ≈ y·d/D.
  • Bright fringes: y = nλD/d; dark: y = (2n+1)λD/2d; fringe width β = λD/d, equally spaced.
  • Immersing the whole apparatus in a liquid of index μ reduces β by μ (λ shortens).
  • White light gives a white central fringe with coloured fringes either side; violet nearest the centre.
  • Fringe shift: a transparent sheet (index μ, thickness t) over one slit adds optical path (μ − 1)t; the pattern shifts by N = (μ − 1)t/λ fringes towards that slit.
• Fresnel's biprism: two thin prisms base-to-base produce two virtual coherent images separated by d = 2a(μ − 1)α (a = source-to-prism distance, α = refracting angle). Used to measure λ and the thickness/index of thin sheets; proves the fringes come from two coherent virtual sources, not from diffraction at slit edges.
• Other wavefront-division arrangements: Fresnel's two mirrors, Lloyd's single mirror (here one beam suffers a π phase change on reflection, so the central fringe is dark — the standard distinguishing question).
• Stokes' treatment / phase change on reflection: reflection at a denser medium (rare → dense) adds a phase change of π, i.e. an extra path of λ/2; reflection at a rarer medium adds none.

Interference by division of amplitude: films, Newton's rings, interferometers

• Thin parallel film of index μ, thickness t, refraction angle r:
  • Reflected system: maxima 2μt cos r = (2n − 1)λ/2; minima 2μt cos r = nλ. (The λ/2 accounts for the π change at the top surface.)
  • Transmitted system: conditions are exactly reversed, so reflected and transmitted patterns are complementary (energy conservation).
  • Very thin film (t → 0) appears dark in reflection — explains why soap films thin to blackness before breaking.
• Wedge-shaped film of small angle θ: straight fringes parallel to the edge of the wedge, spacing β = λ/(2μθ). Used for testing flatness of surfaces and measuring thin wire or foil thickness ("air wedge").
• Newton's rings: plano-convex lens of radius R on a flat plate; air film thickness t = r²/2R gives circular fringes, localised, centre dark in reflection.
  • Dark rings: r_n² = nλR. Bright rings: r_n² = (2n − 1)λR/2. Radii ∝ √n, so rings crowd outward.
  • λ from two rings: λ = (D_{n+m}² − D_n²)/(4mR); refractive index of a liquid between lens and plate: μ = (D_{n+m}² − D_n²)_air /(D_{n+m}² − D_n²)_liquid.
• Antireflection (AR) coating: quarter-wave film with μ_film = √(μ_substrate) gives zero reflection at the design λ; thickness t = λ/(4μ_film). High-reflectance stacks use alternating λ/4 high/low index layers (dielectric mirrors, laser cavity mirrors).
• Michelson interferometer: beam splitter + two mirrors + compensating plate. Moving one mirror by d changes the count by N = 2d/λ, so λ = 2d/N. Gives circular, straight, curved or white-light fringes depending on mirror tilt and path difference; used for λ measurement, wavelength-difference (sodium doublet), coherence length and index of gases. Its modern descendants include LIGO and LIGO-India (approved 2023, under construction in Hingoli, Maharashtra).
• Fabry–Perot interferometer / etalon: multiple-beam interference between two highly reflecting parallel surfaces; sharp fringes, high resolving power, used in laser cavities and spectroscopy. Sharpness grows with reflectivity (finesse).

Diffraction: Fresnel and Fraunhofer classes

• Diffraction is the bending of waves around obstacles/apertures, explained by Huygens–Fresnel: every point on a wavefront is a secondary source, and the secondary waves interfere.
• Fresnel diffraction: source and/or screen at finite distance; wavefront curvature matters; no lenses. Treated by half-period zones (Fresnel zones), amplitude of successive zones alternating in sign; a zone plate blocks alternate zones and acts like a lens, r_n = √(n f λ). Examples: straight edge, circular aperture, Cornu spiral treatment.
• Fraunhofer diffraction: source and screen effectively at infinity (parallel light, lenses used); mathematically simpler and the examined case for slits and gratings.
• Single slit (width a):
  • Minima: a sin θ = nλ (n = 1, 2, …); maxima approximately a sin θ = (2n + 1)λ/2; central maximum is the brightest and twice as wide as the others.
  • Intensity: I = I₀ (sin β/β)², β = (πa sin θ)/λ.
  • Linear width of central maximum on a screen at distance D: 2λD/a. Narrower slit → wider pattern.
• Double slit (slits a, separation d): I = I₀ (sin β/β)² cos²γ, γ = (πd sin θ)/λ — a diffraction envelope modulating interference fringes. Missing orders occur where an interference maximum falls on a diffraction minimum: m = (d/a)·n.
• Plane transmission grating: N slits, grating element d = (a + b) = 1/(lines per unit length).
  • Principal maxima: d sin θ = mλ (grating equation); m = 0, ±1, …; maximum usable order m_max ≤ d/λ.
  • Between them, (N − 1) minima and (N − 2) secondary maxima; large N makes the principal maxima very sharp.
  • Angular dispersion dθ/dλ = m/(d cos θ) — spreads a spectrum; overlapping orders and missing orders are standard exam points.
  • Grating spectra versus prism spectra: grating is normal dispersion (nearly linear in λ) and based on diffraction; prism spectrum is non-linear and based on material dispersion; gratings give higher resolution, prisms give more light in one spectrum.
• Circular aperture: Airy pattern; first dark ring at sin θ = 1.22 λ/D — the origin of the Rayleigh criterion.

Resolving power

• Rayleigh criterion: two point objects are just resolved when the central maximum of one falls on the first minimum of the other.
• Telescope / circular aperture: θ_min = 1.22 λ/D (radian). Larger aperture → finer resolution; this drives the aperture of the 3.6 m Devasthal Optical Telescope (ARIES, Nainital) and the 2 m Himalayan Chandra Telescope (IIA, Hanle).
• Microscope: resolvable separation = 1.22 λ/(2 n sin α) = 0.61λ/NA, where NA = n sin α is the numerical aperture; oil immersion raises n and hence resolution.
• Grating: R = λ/Δλ = m·N, with N the total number of illuminated lines — so resolution depends on the illuminated width, not the line density alone.
• Prism: R = t·(dμ/dλ), t = base length.
• Fabry–Perot has the highest R of the three (10⁵–10⁶) but a very small free spectral range.

Polarisation: production, states and analysis

• Only transverse waves can be polarised; polarisation is direct evidence that light is transverse.
• States: unpolarised (random, rapidly changing vibration direction), plane/linearly polarised, circularly polarised (two perpendicular components of equal amplitude, π/2 out of phase), elliptically polarised (general case), partially polarised (degree of polarisation P = (I_max − I_min)/(I_max + I_min)).
• Methods of production:
  1. Reflection at the Brewster angle: tan θ_B = μ₂/μ₁; the reflected ray is completely plane-polarised perpendicular to the plane of incidence, and reflected and refracted rays are at 90°. Used for Brewster windows in lasers.
  2. Refraction / pile of plates — partial polarisation of the transmitted beam.
  3. Double refraction (birefringence) in calcite, quartz: an ordinary (O) ray obeying Snell's law and an extraordinary (E) ray that does not; both plane-polarised, vibrations mutually perpendicular. Negative crystal: n_o > n_e (calcite); positive: n_e > n_o (quartz). The optic axis is the direction along which both travel alike.
  4. Selective absorption (dichroism) — Polaroid sheets (aligned polyvinyl alcohol–iodine); the basis of sunglasses, camera filters and LCD panels.
  5. Scattering — Rayleigh scattering (∝ 1/λ⁴) polarises skylight at 90° to the Sun and explains the blue sky and red sunsets.
• Nicol prism: calcite rhomb cut and cemented with Canada balsam; the O-ray is totally internally reflected at the balsam layer and absorbed, the E-ray passes — gives plane-polarised light. Works as polariser or analyser.
• Malus' law: I = I₀ cos²θ between polariser and analyser; crossed Nicols (θ = 90°) give zero transmission. An ideal polariser passes half the intensity of unpolarised light.
• Retarders (wave plates): a birefringent plate of thickness t introduces a phase difference δ = (2π/λ)(n_o − n_e)t.
  • Quarter-wave plate: t = λ/[4(n_o − n_e)] — converts linear (at 45° to the axis) to circular, and vice versa; used to analyse elliptical light.
  • Half-wave plate: t = λ/[2(n_o − n_e)] — rotates the plane of linear polarisation by 2θ.
• Analysis of a beam: rotate an analyser — uniform intensity means unpolarised or circular; complete extinction twice per revolution means linear; a non-zero minimum means partial or elliptical. Insert a quarter-wave plate before the analyser to distinguish circular (becomes linear, extinguishable) from unpolarised (stays uniform).

Optical activity, electro-optic effects and applications

• Optical rotation: quartz, sugar solutions and turpentine rotate the plane of polarisation — dextrorotatory (right, +) or laevorotatory (left, −). Fresnel's explanation: plane light splits into two oppositely circular components travelling at different speeds.
• Specific rotation S = θ/(l·c), θ in degrees, l path length in dm, c concentration in g/mL; sucrose ≈ +66.5° dm⁻¹ g⁻¹ mL at 20 °C for sodium D light. Measured with a Laurent half-shade or biquartz polarimeter; saccharimeters in sugar mills report °Z on the ICUMSA scale, and such instruments come under Legal Metrology verification in India.
• Photoelasticity: stress makes a transparent material temporarily birefringent; between crossed polaroids a loaded model shows isoclinics and isochromatics, giving stress distribution — used in mechanical design validation and in toughened-glass inspection.
• Faraday effect (magnetic field rotates the plane, used in optical isolators), Kerr effect (electric field induces birefringence, fast shutters), Pockels effect (linear electro-optic, laser Q-switching), liquid-crystal electrically switched retardance (LCD/LCoS displays).
• Interference applications: AR and dielectric coatings, surface flatness testing, Twyman–Green testing of lenses, thin-film thickness measurement, laser interferometric length standards used by NPL New Delhi for traceability of the metre, and gravitational-wave detection.
• Diffraction applications: grating spectrometers for chemical analysis, X-ray diffraction for crystallography, acousto-optic modulators, CD/DVD track readout, optical fibre Bragg-grating sensors.

Key points

• Interference - redistribution of energy by superposition of coherent waves; total energy is conserved.
• Coherence - fixed phase relation; coherence length L_c = λ²/Δλ limits the usable path difference.
• Fringe width - β = λD/d; independent of fringe order, so fringes are equally spaced.
• Phase change on reflection - π (extra λ/2) when reflecting at a denser medium; the reason a thin film is dark in reflection and Lloyd's mirror gives a dark central fringe.
• Thin-film maxima (reflection) - 2μt cos r = (2n − 1)λ/2; transmitted pattern is complementary.
• Newton's rings - r_n² = nλR for dark rings in reflection; radii ∝ √n; centre dark.
• Michelson - λ = 2d/N; one fringe per λ/2 of mirror travel.
• Fresnel vs Fraunhofer - finite source/screen distance with curved wavefronts versus effectively infinite distance with plane wavefronts.
• Single-slit minima - a sin θ = nλ; central maximum width on screen = 2λD/a; narrower slit spreads light more.
• Double-slit envelope - interference fringes inside a single-slit envelope; missing orders at m = (d/a)n.
• Grating equation - d sin θ = mλ; m_max ≤ d/λ; sharpness grows with the number of lines.
• Grating resolving power - R = mN (N = illuminated lines), far better than a prism of similar size.
• Rayleigh criterion - θ_min = 1.22λ/D for a circular aperture; 0.61λ/NA for a microscope.
• Polarisation proves transversality - longitudinal waves (sound) cannot be polarised.
• Brewster's law - tan θ_B = μ; reflected and refracted rays then differ by 90°.
• Malus' law - I = I₀cos²θ; an ideal polariser transmits half of unpolarised intensity.
• Double refraction - O-ray obeys Snell's law, E-ray does not; both plane-polarised at right angles.
• Wave plates - quarter-wave converts linear ↔ circular; half-wave rotates linear polarisation by 2θ.
• Specific rotation - S = θ/(l·c), l in dm, c in g/mL; basis of sugar-content measurement.

Formulas and examples

Example 1 — Young's fringes and a mica sheet. Sodium light λ = 589 nm, d = 0.50 mm, D = 1.00 m.

• β = λD/d = (589 × 10⁻⁹ × 1.00)/(0.50 × 10⁻³) = 1.178 × 10⁻³ m = 1.18 mm.
• A mica sheet μ = 1.58, t = 10 µm over one slit: N = (μ − 1)t/λ = (0.58 × 10 × 10⁻⁶)/(589 × 10⁻⁹) = 9.8 fringes, shifted towards the covered slit, i.e. 9.8 × 1.178 mm ≈ 11.6 mm.
• If the whole set-up is immersed in water (μ = 1.33), β becomes 1.178/1.33 = 0.886 mm.

Example 2 — Newton's rings. Plano-convex lens R = 1.00 m, λ = 589.3 nm, air film.

• 10th dark ring: r₁₀ = √(nλR) = √(10 × 589.3 × 10⁻⁹ × 1.00) = √(5.893 × 10⁻⁶) = 2.428 × 10⁻³ m, diameter 4.86 mm.
• 20th dark ring diameter: r₂₀ = √(1.1786 × 10⁻⁵) = 3.433 mm → 6.87 mm. Check by formula: λ = (D₂₀² − D₁₀²)/(4 × 10 × R) = (47.14 − 23.57) × 10⁻⁶/40 = 5.89 × 10⁻⁷ m ✓.

Example 3 — Grating orders and resolving power. Grating with 5,000 lines/cm, illuminated width 2.0 cm, λ = 589.3 nm.

• d = 1/(5,000 cm⁻¹) = 2.0 × 10⁻⁶ m.
• First order: sin θ = λ/d = 589.3 × 10⁻⁹/2.0 × 10⁻⁶ = 0.2947 → θ₁ = 17.1°. Second order: sin θ = 0.5893 → θ₂ = 36.1°.
• Highest order: m ≤ d/λ = 3.39 → 3 orders visible.
• N = 5,000 × 2.0 = 10,000 lines; in the second order R = mN = 20,000, so Δλ = 589.3/20,000 = 0.029 nm — far finer than the sodium doublet gap (589.0 and 589.6 nm, 0.6 nm), which is therefore easily resolved.

Example 4 — Single slit and telescope resolution. λ = 600 nm, slit a = 0.20 mm, screen D = 2.0 m.

• First minimum: sin θ = λ/a = 3.0 × 10⁻³ → y₁ = Dλ/a = 2.0 × 3.0 × 10⁻³ = 6.0 mm; central maximum width = 12 mm.
• A telescope of aperture 0.30 m at λ = 550 nm: θ_min = 1.22 × 550 × 10⁻⁹/0.30 = 2.24 × 10⁻⁶ rad = 2.24 × 10⁻⁶ × 206,265 ≈ 0.46 arcsecond.

Example 5 — Polarisation numbers.

• Brewster angle for crown glass μ = 1.52: θ_B = tan⁻¹(1.52) = 56.7°; the refracted ray then makes 33.3° with the normal.
• Two polaroids with axes 30° apart in unpolarised light: I = (I₀/2)cos²30° = 0.5 × 0.75 I₀ = 0.375 I₀.
• Quarter-wave plate from calcite (n_o = 1.658, n_e = 1.486) for λ = 589.3 nm: t = λ/[4(n_o − n_e)] = 589.3 × 10⁻⁹/(4 × 0.172) = 0.857 µm.
• Sugar solution, 20 g sucrose in 100 mL (c = 0.20 g/mL), tube 20 cm (l = 2.0 dm), observed rotation 26.6°: S = 26.6/(2.0 × 0.20) = +66.5° dm⁻¹ g⁻¹ mL — the standard value for sucrose.

Where it is used

• Optical coatings and lens manufacture: AR coatings on spectacles, camera and telescope optics, and high-reflectance laser mirrors — made by Indian firms and by CSIR-CGCRI (Kolkata) and CSIO (Chandigarh); coating thickness is monitored by reflectance interference in situ.
• Metrology and inspection: interferometric flatness and surface-form testing, Michelson/Twyman–Green test benches, fibre-Bragg-grating strain and temperature sensors; NPL New Delhi maintains length traceability by laser interferometry.
• Spectroscopy and instrumentation: grating monochromators and spectrometers in analytical labs, ISRO's SAC Ahmedabad imaging spectrometers for remote sensing, and astronomical spectrographs at ARIES Devasthal and IIA Hanle.
• Displays, imaging and consumer optics: LCD/LCoS panels (crossed polarisers plus liquid-crystal retardance), polarising sunglasses and camera filters, 3D cinema using circular polarisation, anti-glare screens — BIS has standards for sunglasses and spectacle lenses; verify the current IS number and edition before citing it.
• Process and quality control: polarimeters and saccharimeters for sucrose in sugar mills and FSSAI-notified food laboratories; polarimetry in pharmaceutical chirality testing; photoelastic stress analysis of plastic and glass components and of toughened glass.
• Job roles: optical design and thin-film engineer, metrology/calibration engineer, photonics test engineer, instrumentation scientist in DRDO/ISRO/CSIR labs, QC chemist operating polarimeters, and academic research in optics and photonics.

Interview questions

• Q: Why do two separate bulbs never produce interference fringes? A: Their phases change randomly in about 10⁻⁸ s, so the interference term averages to zero; both beams must originate from one source to stay coherent.
• Q: In Young's experiment, what happens to the fringes if the whole apparatus is immersed in water? A: The wavelength falls by the factor μ, so the fringe width β = λD/d shrinks by μ (about 1.33 times narrower), with the pattern unchanged in form.
• Q: Why is the centre of Newton's rings dark in reflected light? A: At the contact point the air-film thickness is zero, so the only path difference is the λ/2 introduced by reflection at the denser glass plate — a condition for destructive interference.
• Q: Why is a very thin soap film dark in reflection but the same film bright in transmission? A: The π phase change at the first surface makes the reflected beams cancel as t → 0; energy conservation then requires the transmitted system to be complementary, hence bright.
• Q: How does energy conservation hold in interference? A: Energy is not destroyed but redistributed — what is missing at the minima appears at the maxima, and the average intensity equals the sum of the individual intensities.
• Q: Distinguish Fresnel and Fraunhofer diffraction. A: In Fresnel diffraction the source or screen is at a finite distance and wavefronts are curved; in Fraunhofer diffraction both are effectively at infinity (parallel beams, usually lenses), which makes the analysis and the patterns simpler.
• Q: What are missing orders in double-slit diffraction? A: Interference maxima that coincide with single-slit diffraction minima and so vanish; they occur at orders m = (d/a)·n where d is slit separation and a the slit width.
• Q: Does the resolving power of a grating depend on the number of lines per cm or on the total number of lines? A: On the total number of illuminated lines, since R = mN; widening the beam on the same grating improves resolution.
• Q: Why is a bigger telescope aperture better even on a perfect night? A: Diffraction limits resolution to θ_min = 1.22λ/D, so larger D gives finer angular resolution as well as more light.
• Q: What does polarisation tell us about light? A: That light is a transverse wave — longitudinal waves such as sound cannot be polarised.
• Q: At the Brewster angle, what is special about the reflected and refracted rays? A: The reflected light is completely plane-polarised perpendicular to the plane of incidence, and the reflected and refracted rays are exactly 90° apart.
• Q: How would you tell circularly polarised light from unpolarised light? A: Both give constant intensity through a rotating analyser; insert a quarter-wave plate first — circular light becomes linear and can be extinguished, unpolarised light stays constant.
• Q: Why does the sky look blue and why is skylight polarised? A: Rayleigh scattering by molecules varies as 1/λ⁴, favouring blue; scattering is strongest for vibrations perpendicular to the viewing direction, so light scattered at 90° to the Sun is strongly polarised.

Common mistakes

• Forgetting the λ/2 (π) phase change on reflection at a denser medium — this flips every thin-film and Newton's-rings condition, and is the single most common error.
• Mixing up the reflected and transmitted thin-film conditions, or quoting Newton's-rings bright-ring formula for dark rings.
• Treating Newton's rings as equally spaced; radii go as √n, so spacing decreases outward.
• Using the interference formula d sin θ = nλ for the minima of a single slit: for one slit a sin θ = nλ gives minima, not maxima.
• Taking grating resolving power as proportional to lines per cm only, and ignoring the illuminated width N.
• Unit slips: measuring ring diameters but substituting them as radii, leaving d in cm while λ is in nm, or using degrees where radians are required in θ_min = 1.22λ/D.
• In Malus' law, forgetting the first polariser halves unpolarised intensity, and measuring θ from the wrong reference axis.
• Confusing a quarter-wave plate with a half-wave plate, or using the wrong (n_o − n_e) sign/magnitude for the crystal.
• In polarimetry, substituting path length in cm instead of dm, or concentration in g/L instead of g/mL.

Related topics

Geometrical optics · Lasers and fibre optics · Electromagnetic waves and Maxwell's equations · Fourier optics and holography · Spectroscopy and X-ray diffraction · Crystal optics and liquid crystals