Michelson Interferometer

Anatomy of the instrument

Five optical elements, arranged in a cross:

• Source. A coherent laser — single longitudinal mode, narrow linewidth so the coherence length spans the path difference. HeNe (633 nm), Nd:YAG (1064 nm), or stabilised diode lasers are typical.
• Beam splitter (BS). A half-silvered plate or cube at 45° to the input. Reflects half the amplitude, transmits half.
• Mirror A (M_A). Terminates one arm, length L_A from the BS.
• Mirror B (M_B). Terminates the perpendicular arm, length L_B.
• Detector. A photodiode or screen at the recombination output.

Light goes BS → M → BS → detector along each arm. The total path is 2L_A on one side, 2L_B on the other. The two return amplitudes superpose at the detector.

Output intensity

If both arms produce return amplitude E₀/√2 (50/50 splitter, lossless mirrors), the field at the detector is

E_out = (E₀ / 2) · [e^(i 2k L_A) + e^(i 2k L_B)]   with k = 2π/λ

and the intensity

I = I₀ · cos²(k (L_A - L_B)) = I₀ · cos²(π ΔL / λ)

where ΔL = L_A − L_B is the difference in one-way arm length (the factor of 2 from the round trip is absorbed). The detector reads bright when 2ΔL = m·λ and dark when 2ΔL = (m + ½)·λ.

Fringes — circular vs straight

For a perfectly aligned interferometer with a point source, the output is uniform — bright or dark depending on ΔL. With an extended source or expanded beam, rays at different angles take slightly different path lengths, producing visible fringes:

Geometry	Fringe shape	Use
Mirrors exactly orthogonal, finite arm-length mismatch	Circles centred on optical axis	Distance measurement, white-light position locating
Mirrors slightly tilted, ΔL ≈ 0	Straight parallel lines	Alignment, surface profilometry
Mirror curvature or sample insertion	Closed curves / hyperbolas	Lens testing, plasma diagnostics

Worked example — measuring a 5 µm displacement

You stick a HeNe laser (633 nm) on the input and translate mirror M_A by what your stage claims is 5.0 µm. How many fringes pass?

1. Round-trip path change: 2 × 5.0 µm = 10.0 µm = 10 000 nm.
2. Fringes: 10 000 / 633 ≈ 15.80.
3. Count 15 full fringes plus 80% of another by phase-locking to a quadrature detector pair.
4. Stage calibration: 0.80·633 nm / 2 = 253 nm into the next full fringe.

So the actual translation is 15 × λ/2 + 253 nm = 4 747 + 253 = 5 000 nm. The interferometer just measured a micrometre stage to nanometre accuracy.

LIGO — Michelson at planetary scale

LIGO uses every available trick to crank a Michelson's sensitivity by 12 orders of magnitude:

• Arm length. Each arm is 4.0 km of evacuated stainless tube. Longer arm → larger ΔL for a given strain h = ΔL/L.
• Fabry-Perot cavities. Each arm has a partial input mirror; light bounces ≈ 280 times before exiting, multiplying effective arm length to ~1 120 km.
• Power recycling. A mirror upstream of the BS sends unused output power back into the cavity, boosting circulating power from ~20 W input to ~750 kW at the test masses.
• Signal recycling. Another mirror after the BS narrows the detection bandwidth around the kHz range where most gravitational-wave power lives.
• Suspension. Mirrors hang from quadruple pendulums on monolithic silica fibres — passively isolated above 10 Hz.
• Squeezed light. Quantum-squeezed vacuum is injected at the dark port to beat shot noise.

The September 2015 detection of GW150914 corresponded to a strain peak of h ≈ 10⁻²¹, i.e. a 4-km arm change of about 4 × 10⁻¹⁸ m — roughly 1/10 000 the diameter of a proton.

Real-world Michelson applications

System	What's measured
FTIR spectroscopy	Mirror swept; Fourier-transform of fringe vs position yields absorption spectrum
Distance metrology	Sub-nm displacement, alignment of semiconductor lithography stages
Refractive index measurement	Insert sample in one arm; fringe shift gives (n - 1)·L
Vibration analysis	Out-of-plane motion of MEMS, speaker cones, biological membranes
Optical coherence tomography	Low-coherence Michelson — scans interface depths in tissue
Gravitational wave detection	LIGO, Virgo, KAGRA — direct strain measurement of spacetime
Frequency calibration	Compare unknown laser to stabilised reference via beat note
Atom interferometry analogue	Two-arm matter-wave devices share the same architecture

JavaScript — Michelson calculations

// Output intensity given arm-length difference
function michelsonI(deltaL_m, wavelength_m, I0 = 1) {
  return I0 * Math.cos(Math.PI * deltaL_m / wavelength_m) ** 2;
}

// 633 nm HeNe, mirror differential of 100 nm
console.log(michelsonI(100e-9, 633e-9).toFixed(3)); // ≈ 0.418

// Mirror motion that shifts N fringes
function mirrorShiftForFringes(N, wavelength_m) {
  return N * wavelength_m / 2;
}
console.log(`1 fringe @ 633 nm = ${(mirrorShiftForFringes(1, 633e-9) * 1e9).toFixed(1)} nm`); // 316.5

// Number of fringes from a given mirror displacement
function fringesFromMotion(dx_m, wavelength_m) {
  return 2 * dx_m / wavelength_m;
}
console.log(`5 µm @ 633 nm: ${fringesFromMotion(5e-6, 633e-9).toFixed(2)} fringes`); // 15.80

// LIGO effective arm via Fabry-Perot bounces
function effectiveArm(L_m, bounces) { return L_m * bounces; }
const Leff = effectiveArm(4000, 280);
console.log(`LIGO effective arm: ${(Leff/1000).toFixed(0)} km`); // 1120 km

// Strain → ΔL
function strainToDeltaL(h, L_m) { return h * L_m; }
console.log(`h=10⁻²¹ over 4 km: ${(strainToDeltaL(1e-21, 4000) * 1e18).toFixed(2)} aN`); // 4 × 10⁻¹⁸ m

// Shot-noise limit on phase
function shotNoisePhase(P_W, lambda_m, bandwidth_Hz) {
  const h = 6.626e-34;
  const c = 3e8;
  const photonRate = P_W * lambda_m / (h * c);
  // Phase noise = 1/√(N), where N = rate · 1/bw
  return 1 / Math.sqrt(photonRate / bandwidth_Hz);
}
console.log(`Shot phase @ 1 W, 1064 nm, 1 Hz BW: ${shotNoisePhase(1, 1064e-9, 1).toExponential(2)} rad`);
// ~10⁻⁹ rad — equivalent to ~10⁻¹⁶ m mirror displacement

Where Michelson interferometers matter

• Length metrology. Defines the metre indirectly through frequency-stabilised lasers — the most accurate length standard in existence.
• Surface and shape testing. Optical components are routinely tested with phase-shifting Michelsons to λ/100.
• Spectroscopy. Every FTIR molecular spectrometer is a Michelson with a swept mirror.
• Medical imaging. OCT uses a low-coherence Michelson to map retinal layers and arterial walls in vivo.
• Fundamental physics. Tested aether, confirmed special relativity, detected gravitational waves — still the workhorse for tabletop tests of equivalence and Lorentz invariance.
• Sensing. Fibre-based Michelsons embedded in bridges and dams detect strain and seismic activity.
• Quantum optics. Photon-number-resolved Michelsons probe Heisenberg-limited phase estimation.

Common mistakes

• Confusing arm motion with path change. Path is round-trip, so a mirror moves λ/2 to shift one fringe — not λ.
• Ignoring coherence length. If |ΔL| exceeds the laser's coherence length, fringe contrast collapses. A 1 MHz linewidth laser gives ~300 m coherence; a broad-band thermal source, ~10 µm.
• Forgetting the compensator for white light. For broadband fringes you need equal glass in both arms; otherwise dispersion blurs everything.
• Missing the additional π at the splitter. Internal reflection at the high-index side of the splitter adds a 180° phase shift, which is why complementary outputs (transmission and reflection) sum to constant — energy is conserved.
• Counting one direction only. A fringe count gives unsigned displacement. Use quadrature detection (a λ/8 plate) to sense direction of motion.
• Treating LIGO as a bare Michelson. Without arm cavities, power recycling, and quantum squeezing, the basic Michelson can't reach the strains LIGO measures.