Coherence, phase & superposition of light

Where two beams overlap the field is E = E₁ cos(ωt) + E₂ cos(ωt + δ); nothing in Maxwell's equations lets the intensities add instead. A detector integrates over millions of optical cycles, so what it records is ⟨E²⟩, and with ⟨cos² ωt⟩ = ½ and ⟨cos ωt cos(ωt + δ)⟩ = ½ cos δ the average is ½E₁² + ½E₂² + E₁E₂ cos δ. Since I ∝ ⟨E²⟩, that is I = I₁ + I₂ + 2√(I₁I₂) cos δ. The phasor picture says the same thing: two arrows of length E₁ and E₂ set δ apart, and the cosine rule gives the resultant. For equal beams of intensity I₀ the total swings from 4I₀ at δ = 0 to zero at δ = π, averaging 2I₀ across the fringes — energy is not created, only moved. The cross term is the entire content of interference, and every later result in this section is a statement about what happens to cos δ.

A wave of vacuum wavelength λ₀ travelling a distance s through a medium of index n has wavelength λ₀/n there, so it accumulates phase 2πns/λ₀. The product ns is the optical path length, and the phase difference between two routes is δ = (2π/λ₀) × (difference in optical path). Inserting a 1.00 mm slab of glass (n = 1.52) into one route replaces 1.00 mm of air by 1.52 mm of optical path, an excess of 0.52 mm — at 550 nm that is 945.45 wavelengths, a phase of 5940 rad. What the detector sees depends only on the fractional part, here 0.45 of a wavelength, so the slab shifts the fringes by 0.45 of a fringe; but thinning the slab by just 1 μm changes the excess by 0.52 μm and moves them by another 0.95 of a fringe. This is the double edge of interferometry: phase is exquisitely sensitive to optical path, so fringes measure thickness or index to nanometres, and predicting a fringe position from three-figure data is hopeless. Reflection adds its own bookkeeping — a shift of π when the wave reflects off a higher-index medium — which the thin-film lesson takes up.

No source is monochromatic. An atom radiates for a limited time before a collision or re-excitation resets its phase, so light arrives as a sequence of wave trains whose phases are unrelated, and the Fourier transform of a train of duration τ_c has bandwidth Δν ≈ 1/τ_c. Read the other way, that is the useful statement: a source of bandwidth Δν has coherence time τ_c ≈ 1/Δν and coherence length ℓ_c = cτ_c ≈ c/Δν. Because ν = c/λ, a small spread in ν maps to Δλ = λ₀²Δν/c, giving the form spectroscopists use, ℓ_c ≈ λ₀²/Δλ. The range is enormous. White light spanning 400 to 700 nm has ℓ_c ≈ 550²/300 nm ≈ 1 μm, about two wavelengths. A red LED with Δλ = 20 nm at 630 nm has ℓ_c ≈ 20 μm. A multimode He–Ne laser with Δν ≈ 1.5 GHz has ℓ_c ≈ 0.20 m; a frequency-stabilised one at 1 MHz reaches 300 m. Coherence length is not a property of laser light as against lamp light but a number every source has, and each interference lesson that follows quietly assumes its path difference sits well inside it.

Split light of bandwidth Δν between two paths that differ by Δ. Each frequency ν in the spectrum makes its own fringe term cos(2πνΔ/c), and terms at different frequencies do not interfere with one another — their cross terms oscillate at the difference frequency and average away — so the total is the intensity-weighted sum I(Δ) ∝ ∫ S(ν)[1 + cos(2πνΔ/c)] dν. For a flat-topped spectrum of width Δν centred on ν₀ the integral is elementary and gives 1 + sinc(πΔνΔ/c) cos(2πν₀Δ/c): the fast cosine is the fringe pattern and the sinc is its envelope. Writing ℓ_c = c/Δν, the visibility is |γ(Δ)| = |sin(πΔ/ℓ_c)/(πΔ/ℓ_c)|. It is 1 at Δ = 0, 0.64 at Δ = ℓ_c/2, and zero at Δ = ℓ_c, where the fringes of the reddest and bluest components are a whole fringe out of step, so every component in the band is cancelled by the one half a bandwidth away. The weak revivals beyond that belong to the sharp-edged model; a Gaussian line gives a Gaussian envelope with none. In general the envelope is the Fourier transform of the spectrum, which is the principle behind Fourier-transform spectroscopy and the Michelson lesson later in this section.

Temporal coherence compares a beam with a delayed copy of itself. Spatial coherence asks instead whether two points across a wavefront share a phase. A point source answers yes for any pair, because both points lie on the same spherical wave. Widen the source to an angular size θₛ and every source point sends its own wave, reaching two apertures a distance d apart with its own path difference. Points at opposite edges of the source differ in that path by about dθₛ, so their fringe patterns are displaced; once dθₛ reaches λ₀ the shifts span a whole fringe and the sum is a uniform wash. So the transverse coherence width is ℓₜ ≈ λ₀/θₛ, and for a uniform strip source the visibility falls as |sinc(πdθₛ/λ₀)|, the same shape as the temporal case with dθₛ playing the role of Δ; a uniform disc gives an Airy factor with its first zero at 1.22λ₀/θₛ. The Sun subtends 9.3 mrad, so at 550 nm sunlight is coherent across only about 59 μm. That is why Young first passed sunlight through a pinhole: a 0.5 mm hole seen from 2 m subtends 0.25 mrad and widens ℓₜ to 2.2 mm, well beyond his slit spacing.

Three approximations sit under every fringe formula, and each fails its own way. Full coherence fails gradually rather than suddenly: the cross term is really 2√(I₁I₂)|γ| cos δ, with |γ| the degree of coherence, and V simply tracks |γ| — partial coherence is the general case, and visibility is how you measure it. Monochromaticity fails when Δ becomes comparable to ℓ_c, as the last two moves showed. The scalar treatment fails with polarisation: the cross term is 2⟨E₁ ⋅ E₂⟩, a dot product, so two beams polarised at right angles superpose perfectly and show no fringes at all — the Fresnel–Arago result the polarisation lesson uses. Two independent sources are not a failure of superposition but of the average: their fields add at every instant, and a detector fast enough to follow the phase sees beats — two lasers a few megahertz apart on a fast photodiode show exactly that — but over any ordinary integration time the phase difference of two lamps wanders through many cycles and ⟨cos δ⟩ is zero. Coherence is a comparison between how fast δ wanders and how long the detector looks.