Interference

01 / UNDERSTAND & APPLY

1. Wavefronts explain reflection and refraction

A wavefront joins points of equal phase. In an isotropic homogeneous medium, rays are locally normal to wavefronts. Huygens’s construction treats each point on a wavefront as a source of secondary disturbances; the next wavefront is an envelope of their spread after a common time. This gives a geometric way to follow propagation without assuming light is only a collection of rays.

At a reflecting plane, wavefront geometry gives equal incident and reflected angles measured from the normal. At an interface where speed changes from v₁ to v₂, the portions of the front in the second medium advance a different distance during the same time. The resulting geometry gives sin i/sin r = v₁/v₂ = n₂/n₁, or n₁ sin i = n₂ sin r.

To derive the angles, let A be the first point of an incident plane wavefront AB to reach a flat interface. Let the still-incident point B reach C on the interface after time Δt. Because BC is an incident ray and AB is perpendicular to it, triangle ABC is right-angled at B. The angle between AB and the surface AC equals the incident ray angle i to the normal, hence sin i = BC/AC = v₁Δt/AC.

For reflection, the secondary wavelet centred at A has radius AD = v₁Δt in the original medium. The reflected front CD is tangent to this wavelet, so AD is perpendicular to CD. Triangle ADC is right-angled at D and gives sin r = AD/AC. Equal radii BC = AD therefore give sin r = sin i and r = i for angles between zero and 90°. The incident ray, normal and reflected ray lie in the same construction plane.

For refraction, construct instead a wavelet from A in the second medium with radius AD = v₂Δt, and draw the transmitted tangent CD. The transmitted ray AD is normal to CD. Thus sin r = AD/AC = v₂Δt/AC. Dividing the two equal-time triangles gives sin i/sin r = v₁/v₂. With v = c/n, this becomes n₁ sin i = n₂ sin r. The front is steeper relative to the interface when its ray has a larger angle to the normal; confusing a front angle with its complementary ray-to-surface angle reverses this reasoning.

For incidence from higher index to lower index, the required transmitted sine can exceed one. The plane propagating-ray construction then has no real refracted angle: the limiting case r = 90° gives sin ic = n₂/n₁. Above it, total internal reflection occurs in the lossless idealisation, although an evanescent field extends into the second medium. At normal incidence the directions remain normal even though speed and wavelength change.

Frequency is fixed by temporal continuity at a stationary boundary. Speed and wavelength change together: v = fλ and λ₂/λ₁ = v₂/v₁. Entering a higher-index medium reduces speed and wavelength and bends an oblique ray toward the normal. It does not lower photon frequency simply because the light slows in the medium.

The elementary construction predicts wavefront directions but does not by itself provide every reflected/transmitted amplitude or phase relation. Diffraction and interference require amplitudes and relative phases, while polarization needs the transverse vector nature of the electromagnetic field. Each refinement adds information rather than contradicting the ray limit.

02 / UNDERSTAND & APPLY

2. Interference compares amplitudes before intensities

When coherent waves overlap, their electric-field amplitudes add. Intensity depends on the time average of the squared resultant field, producing a cross term that depends on phase. For parallel polarization and mutually coherent monochromatic beams, I = I₁ + I₂ + 2√(I₁I₂)cosδ.

Constructive interference occurs at δ = 2πm; destructive at δ = (2m + 1)π. Maximum intensity is (√I₁ + √I₂)² and minimum is (√I₁ − √I₂)². Complete darkness requires equal amplitudes as well as opposite phase. Unequal source intensities give reduced contrast rather than perfect cancellation.

Coherence means a sufficiently stable phase relationship during observation. Two independent ordinary lamps do not generally produce stationary visible interference fringes because their relative phase fluctuates rapidly. Splitting one source into two paths is a practical way to preserve a relation. Orthogonally polarized beams also lack the same scalar interference cross term unless projected onto a common polarization.

Path difference and phase difference are related through optical path length. In uniform medium, δ = 2πΔ/λ plus any initial or reflection phase difference. In varying media, use optical path ∫n ds and vacuum wavelength. A longer geometric path is not automatically the larger optical path if refractive indices differ.

03 / UNDERSTAND & APPLY

3. Young’s experiment turns path difference into position

For two narrow coherent slits separated by d and a distant screen, the path difference at observation angle θ is approximately d sinθ. Bright fringes satisfy d sinθ = mλ and dark fringes satisfy d sinθ = (m + 1/2)λ when the slits begin in phase. The central bright fringe corresponds to equal path length.

For a screen distance D much larger than d and small observation angle, sinθ ≈ tanθ ≈ y/D. Bright positions are ym = mλD/d and the fringe spacing is β = λD/d. Larger wavelength or screen distance spreads fringes; larger slit separation crowds them. These are small-angle far-field results, not exact formulas for an arbitrarily close screen.

Inserting a thin transparent plate of thickness t and refractive index n in one path replaces air optical length t by nt, adding optical path (n − 1)t at normal incidence. The central fringe shifts toward the slit with the plate because a shorter geometric path through that arm is needed to compensate the added optical length. Magnitude of displacement is D(n − 1)t/d in the same approximation.

If the entire apparatus is immersed in a medium of index n while source frequency is unchanged, wavelength becomes λvac/n and fringe spacing decreases by n. If white light is used, the zero-path-difference central fringe is achromatic, while neighbouring colours separate and eventually overlap. Broad bandwidth reduces the range over which distinct fringes remain visible.

04 / UNDERSTAND & APPLY

4. Reflection phase can reverse a bright/dark condition

A reflected wave undergoes a π phase reversal when reflecting from an optically lower-index medium toward a higher-index medium in the usual nonabsorbing normal-incidence model. Reflection from higher to lower index does not have that reversal. Therefore two reflected rays from a thin film can acquire an extra relative π depending on the surrounding indices.

For a film of index n and thickness t at normal incidence, the extra optical path of the ray traversing the film down and back is 2nt. If exactly one of the two relevant reflections produces a π reversal, reflected-light destructive interference occurs at 2nt = mλvac, while constructive occurs at 2nt = (m + 1/2)λvac. If zero or two reversals occur, these conditions swap.

At oblique incidence the path condition involves 2nt cos r, where r is the angle inside the film. Be explicit about whether the requested pattern is reflected or transmitted. In an ideal lossless system, diminished reflection can coincide with enhanced transmission, but absorption and multiple reflections can modify simple intensity claims.

A wedge-shaped film has thickness changing along the surface, so equal-thickness contours become fringes. Newton’s rings arise from the varying thin air gap between a curved surface and a plane. Their detailed radii require the gap geometry; they are not Young’s equally spaced straight fringes. The common principle is optical path plus reflection phase.

05 / UNDERSTAND & APPLY

5. A finite slit interferes with itself

Diffraction is the spreading and interference associated with a finite aperture or obstacle. In a single slit of width a illuminated coherently, contributions from different points across the slit reach a distant direction with different phases. At certain angles these contributions cancel in pairs, producing minima.

For Fraunhofer single-slit diffraction, minima satisfy a sinθ = mλ for nonzero integers m. The central maximum lies between the first minima at approximately ±λ/a for small angles, so its angular width is 2λ/a and its screen width is 2Dλ/a. The central maximum is twice as wide as the spacing between adjacent minima on either side in the small-angle pattern.

The intensity envelope is I(θ) = I(0)[sinβ/β]² with β = πa sinθ/λ and its limiting central value defined by sinβ/β → 1. Secondary maxima are weaker and do not occur exactly halfway in angle between successive minima. The often-used half-integer rule is an approximation, unlike the exact minima condition within the model.

In a real double-slit experiment each slit has finite width, so rapid two-slit interference fringes are modulated by a single-slit diffraction envelope. Missing orders occur when an interference maximum coincides with an envelope minimum. Keep slit separation d and slit width a distinct; exchanging them exchanges two different spatial scales.

06 / UNDERSTAND & APPLY

6. Aperture size controls angular detail

A diffraction grating combines many equally spaced slits. Principal maxima satisfy d sinθ = mλ, where d is grating spacing, the inverse of the number of lines per unit length. More illuminated slits sharpen the principal maxima without changing their angular positions. A possible order must also satisfy |mλ/d| ≤ 1.

For N illuminated grating lines in order m, the ideal resolving power is λ/Δλ = |m|N under the usual Rayleigh criterion. Resolving two nearby wavelengths is different from merely spreading them apart: sharpness and overlap matter. Higher order and more illuminated lines improve ideal resolution, subject to usable geometry and intensity.

For a circular aperture of diameter Dap, the first dark ring of the Airy pattern occurs near angle 1.22λ/Dap in the small-angle limit. The Rayleigh criterion takes this as an approximate angular resolution threshold for two equal point sources. Larger aperture or shorter wavelength improves diffraction-limited resolution; aberrations, turbulence and detectors may impose stronger practical limits.

Interference and diffraction are not unrelated mechanisms. Both come from superposing wave amplitudes. The labels often distinguish a few separated sources from an extended aperture, but their mathematics shares phase addition. This helps prevent treating a telescope’s finite aperture as a purely geometric opening with unlimited detail.

07 / UNDERSTAND & APPLY

7. Polarization reveals the transverse electric field

In a plane electromagnetic wave, the electric field oscillates transverse to propagation. Linear polarization specifies one fixed oscillation direction. Unpolarized light has fluctuating polarization directions with no preferred transverse orientation over the measurement interval. An ideal linear polarizer transmits the electric-field component along its transmission axis.

For already linearly polarized light meeting an ideal analyser at angle θ between polarization and transmission axes, the transmitted field amplitude is E cosθ and intensity obeys Malus’s law I = I₀ cos²θ. For unpolarized incident light, averaging over transverse orientations gives I = I₀/2 after one ideal polarizer. Apply the half-factor only for unpolarized input, not after every polarizer.

Two crossed ideal polarizers transmit no light from a linearly polarized output of the first. Inserting a third polarizer at an intermediate angle can allow transmission because each stage projects onto a new axis. This is a change in polarization state through filtering, not a violation of the crossed-polarizer result for the original two-stage setup.

At Brewster incidence for nonabsorbing dielectric media, reflected light has no component polarized parallel to the plane of incidence. The condition is tan iB = n₂/n₁, and reflected and refracted rays are perpendicular. This is distinct from the critical angle for total internal reflection, which requires travel from higher to lower index.

08 / UNDERSTAND & APPLY

8. Match the observation to the mechanism

The analytics label “Optical Phenomena” can combine several observations rather than naming a single new law. A useful method is to identify what changes: relative phase suggests interference; finite aperture suggests diffraction; selected transverse orientation suggests polarization; wavelength-dependent refraction suggests dispersion. More than one mechanism may occur in the same experiment.

Coloured soap films primarily involve thin-film interference, while a compact disc’s closely spaced tracks act as a reflective grating. The reduction of glare by a polarizing filter uses polarization-selective transmission of partially polarized reflected light. A prism’s separation of colours arises mainly because refractive index depends on wavelength, not because each colour has a different frequency after entering the prism.

The blue appearance of a clear sky involves stronger scattering of shorter visible wavelengths by small atmospheric molecules, approximately following a λ⁻⁴ dependence in the Rayleigh regime. This is not the same geometry as diffraction by a slit. The observed sky colour also depends on the solar spectrum and human visual response, so the simple inverse-fourth scaling alone does not predict every colour sensation.

For statement or matching questions, insist on a causal chain: geometry/material property → phase, amplitude or polarization effect → observation. Do not use a colour alone as proof of interference, since dispersion, scattering and absorption can also produce colour. This section teaches a cross-phenomenon comparison, not a claim that one formula solves every optical observation.

09 / UNDERSTAND & APPLY

9. Original practice and approximation checks

Use vacuum wavelengths unless the problem explicitly places the entire setup in a medium. Distinguish angle to a normal, angle to a film plane and angle between polarization axes.

• Add amplitudes before computing coherent intensity.
• Young’s fringe spacing needs distant-screen and small-angle approximations.
• Thin films require both optical path and reflection phase.
• Single-slit width and double-slit separation set different scales.
• Polarizers act on field components; intensity follows the square.
• Identify the mechanism before selecting an optical formula.