The Michelson Interferometer & Optical Metrology

Young's slits divide a wavefront, sampling it at two points, so the source must be small or the two samples carry unrelated phases. The Michelson divides the amplitude instead: a half-silvered plate sends part of every wavefront down arm 1 and part down arm 2, and the mirrors return both parts to recombine. Each point of the lamp interferes only with its own light, so a broad source gives bright fringes instead of washing them out. Seen from the output, mirror M₂ appears as a virtual image M₂′ parallel to M₁ and a distance d behind it: the instrument is a virtual air film whose thickness d you drive with a screw. A ray leaving at angle θ to the axis returns with path difference 2d cos θ, so bright rings sit where 2d cos θ = mλ. The centre carries the highest order, m₀ = 2d/λ = 3663 for d = 1.00 mm and λ = 546 nm, and the p-th ring out from a bright centre sits near θₚ ≈ √(pλ/d), 1.34° for p = 1, because cos θ ≈ 1 − θ²/2 turns 2d(1 − θ²/2) = (m₀ − p)λ into dθ² = pλ.

Let the incident intensity be I₀ and the splitter 50:50. Each arm carries I₀/2 out and back, and each returning beam sends half of itself, I₀/4, to the output port. Two equal beams with phase difference δ superpose to I = (I₀/4)(1 + 1 + 2 cos δ) = (I₀/2)(1 + cos δ) = I₀ cos²(δ/2), with δ = (2π/λ)(2d) plus a constant fixed by the reflections. The output swings between I₀ and zero: the interferometer does not destroy energy, it steers it, and at a dark fringe the complementary I₀ sin²(δ/2) leaves through the input port toward the source. Two checks: 4πd/λ is a length over a length, so δ is a pure number as a phase must be; and at d = 0 every wavelength shares the same δ, which is the seed of the white-light fringe. Real beams are never exactly equal, so write I = (I₀/2)[1 + V cos δ] with the visibility V = (Iₘₐₓ − Iₘᵢₙ)/(Iₘₐₓ + Iₘᵢₙ). V = 1 is perfect contrast and V = 0 is no fringes; everything this instrument measures is δ changing by 2π while V stays high enough to see it.

Moving the mirror by Δd changes the path difference by 2Δd, so δ changes by 2π(2Δd)/λ and N = 2Δd/λ fringes cross a reference mark. Each fringe is λ/2 of mirror travel: 316.4 nm for He-Ne light at 632.8 nm. Drive the mirror 0.100 mm and 316 fringes pass; judge the last one to a quarter and you have measured 100 μm to ±80 nm, about a part in a thousand, with no ruler finer than a lamp. The identity runs backwards too: a calibrated screw and a counted N give λ = 2Δd/N, which is how wavelengths were first tied to the metre. Put a gas cell of length L in one arm and nothing moves; admitting gas raises the index from 1 to n and adds optical path 2L(n − 1), so N = 2L(n − 1)/λ. A 50.0 mm cell filled with room air under sodium light at 589.3 nm passes 2 × 0.0500 m × 2.72 × 10⁻⁴ ÷ 589.3 nm ≈ 46 fringes, and reading N to ±1 fixes n − 1 to about 2%.

The splitter is a glass plate of thickness t coated on one face. Follow the two beams: the one reflected at the coating first crosses the glass three times in all, the other once, so one arm carries extra optical path 2n(λ)t that the other does not. For a laser this is harmless: absorb the constant into δ and move the mirror. For white light it is fatal, because n depends on λ. In 6.0 mm of crown glass, n(486 nm) − n(656 nm) = 1.5224 − 1.5143 = 0.0081, so the extra path differs between blue and red by 2 × 6.0 mm × 0.0081 = 97 μm; no single mirror position is zero-path-difference for both colours, and coloured fringes never form. The compensator is a second plate of the same glass and thickness, set parallel to the splitter in the other arm, so both beams cross equal glass and the dispersion cancels for every λ at once. Only then does d = 0 mean zero path difference for the whole spectrum: one achromatic central fringe, unmistakable, with a few coloured fringes on either side.

A real source spans a band of width Δλ about λ. Each wavelength makes fringes with its own spacing, and they all agree only at zero path difference. Track the order m = 2d/λ across the band: differentiating, |Δm| = 2dΔλ/λ². While Δm ≪ 1 the colours are in step and the pattern is sharp; when Δm reaches 1 the two edges of the band are a whole fringe apart, every colour between fills the gap, and the fringes vanish. The path difference where that happens is the coherence length l_c = λ²/Δλ, and the number of fringes you can count before reaching it is l_c/λ = λ/Δλ. White light, λ ≈ 550 nm and Δλ ≈ 300 nm, gives l_c ≈ 1 μm and about two fringes each side of zero, which is exactly why white-light fringes locate d = 0 so well. An LED at 630 nm with Δλ = 20 nm allows 20 μm of path; a multimode He-Ne with Δλ ≈ 2 × 10⁻³ nm allows about 20 cm. The sharp cut-off is the flat-band idealisation: a Gaussian line fades smoothly with no revival, and a doublet fades and returns with mirror period λ²/(2Δλ), 0.29 mm for the sodium D lines.

Write the uncertainty budget for a displacement and the wavelength comes last. Counting: ±1 fringe is ±λ/2 = ±0.32 μm, and a photodiode with a counter cuts that to a hundredth of a fringe. Calibration: a lever-and-micrometer drive quoted at 1% turns 0.250 mm of travel into ±2.5 μm, eight times the count error. Temperature: an aluminium arm of 120 mm (α = 23 × 10⁻⁶ K⁻¹) warming by 0.5 K lengthens by 1.4 μm, and the double pass makes that 4.7 spurious fringes at 589 nm, drift of the same order as the effect being measured. Vibration: a footfall that moves a mirror by λ/4 = 160 nm shifts the pattern by half a fringe, and building vibration of a few micrometres at a few hertz smears the fringes into grey unless the table is isolated and the count is fast. Alignment: a tilt ε between M₁ and M₂′ lays 2εD/λ straight fringes across an aperture D, so fewer than one fringe across 20 mm at 633 nm needs ε < λ/(2D) = 1.6 × 10⁻⁵ rad, about three arcseconds. The wavelength enters only as a multiplier, and a laser's part-in-10⁶ uncertainty is invisible beside any of these.