Interferometers: Michelson and Mach-Zehnder

Why it matters

Interferometers turn a change in optical path — far smaller than a wavelength — into a measurable change in intensity. That makes them the most sensitive length and phase measuring tools we have: laser displacement interferometers calibrate machine tools and coordinate measuring machines, Fourier-transform spectrometers use a scanning Michelson, fibre Mach–Zehnder interferometers form acoustic, temperature and strain sensors, and lithium-niobate Mach–Zehnder modulators encode data in long-haul optical links.

Key ideas

Two-beam interference. When two coherent beams of intensities I₁ and I₂ meet with phase difference Δφ, the detected intensity is I = I₁ + I₂ + 2√(I₁I₂)·cos Δφ. With equal beams, I swings between 0 and 4I₁, and the fringe visibility V = (I_max − I_min)/(I_max + I_min) is 1. The phase difference comes from the optical path difference (OPD): Δφ = (2π/λ)·OPD, where OPD = Σ(n·L) along one path minus the other.

Coherence. Fringes appear only if the OPD is shorter than the source's coherence length L_c ≈ λ²/Δλ. A broadband LED (Δλ of tens of nm) gives fringes only near zero OPD — the basis of white-light interferometry and optical coherence tomography — whereas a stabilised laser gives fringes over metres.

Michelson interferometer. A beam splitter divides the input; one beam goes to a fixed mirror and one to a movable mirror; both return through the same splitter and recombine at the detector. Because light travels to the mirror and back, moving the mirror by d changes the OPD by 2d. Each change of OPD by λ moves the pattern by one fringe, so the number of fringes counted is N = 2d/λ, and each fringe represents a mirror motion of λ/2 (316.4 nm for a He–Ne laser at 632.8 nm). Placing a cell of length L in one arm and changing its refractive index by Δn gives N = 2L·Δn/λ. Commercial displacement interferometers use two-frequency (heterodyne) lasers and quadrature detection to obtain direction and sub-nanometre resolution; air temperature, pressure and humidity must be measured to correct the wavelength in air.

Mach–Zehnder interferometer (MZI). Two beam splitters (or fibre couplers) are used: the first splits the light into two separate arms, the second recombines them. Light passes each arm only once, so the phase difference is Δφ = (2π/λ)·Δ(nL), without the factor 2 of the Michelson. With ideal 50:50 splitters the two output ports are complementary: I₁ = I₀·cos²(Δφ/2) and I₂ = I₀·sin²(Δφ/2); energy is conserved and simply shifts between them. Separate arms make the MZI convenient for inserting a sample, gas cell or sensing fibre, and its two outputs allow balanced (differential) detection.

MZI as a modulator or switch. In an integrated LiNbO₃ MZI, a voltage changes the index of one arm through the electro-optic effect. The voltage giving Δφ = π is the half-wave voltage V_π; switching between 0 and V_π moves the light from one output to the other. Biased at quadrature (Δφ = π/2) the response is most linear and most sensitive, which is also the preferred operating point for MZI sensors.

Fibre-optic interferometric sensors. A long sensing fibre in one arm and a reference fibre in the other make a highly sensitive temperature, strain, pressure or acoustic sensor: the phase changes as Δφ = (2π/λ)·[d(nL)/dX]·ΔX, where X is the measurand. The same sensitivity makes interferometers vulnerable to unwanted temperature drift, vibration and polarisation fading, so practical systems use path matching, polarisation control and active or passive phase demodulation.

Formulas

I = I₁ + I₂ + 2√(I₁I₂)·cos Δφ I₁, I₂ beam intensities (W/m²), Δφ phase difference (rad).

Δφ = (2π/λ)·OPD λ free-space wavelength (m), OPD optical path difference (m).

V = (I_max − I_min)/(I_max + I_min) V fringe visibility (0 to 1).

N = 2d/λ (Michelson, mirror displacement d); N = 2L·Δn/λ (cell of length L) N number of fringes counted.

I₁ = I₀·cos²(Δφ/2), I₂ = I₀·sin²(Δφ/2) (ideal MZI outputs) I₀ input intensity.

L_c ≈ λ²/Δλ L_c coherence length (m), Δλ source spectral width (m).

Worked examples

Example 1 (standard). In a Michelson interferometer lit by a He–Ne laser (λ = 632.8 nm), 1000 fringes pass the detector as the mirror moves. How far did it move?

1. N = 2d/λ gives d = Nλ/2.
2. d = 1000 × 632.8 × 10⁻⁹/2 = 3.164 × 10⁻⁴ m. Answer: d ≈ 316.4 µm.

Example 2 (GATE level). (a) A 5.0 cm gas cell in one arm of a Michelson interferometer (λ = 632.8 nm) is filled from vacuum to atmospheric pressure and 40 fringes are counted. Find the refractive index of the gas. (b) A fibre MZI temperature sensor at 1550 nm has a 1 m sensing arm with (1/L)·d(nL)/dT = 1.08 × 10⁻⁵ /K. Find the phase sensitivity and the phase change for 0.01 K.

1. (a) N = 2LΔn/λ gives Δn = Nλ/(2L) = 40 × 632.8 × 10⁻⁹/(2 × 0.05) = 2.53 × 10⁻⁴, so n = 1.000253.
2. (b) dφ/dT = (2π/λ)·L·(1/L)d(nL)/dT = (2π/1.55 × 10⁻⁶) × 1 × 1.08 × 10⁻⁵ = 43.8 rad/K (about 7 fringes per kelvin).
3. For 0.01 K: Δφ = 43.8 × 0.01 = 0.438 rad. Answer: (a) n ≈ 1.000253; (b) 43.8 rad/K, 0.44 rad for 0.01 K.

Example 3 (coherence). Compare the coherence lengths of an 850 nm LED with Δλ = 40 nm and a 1550 nm laser with Δλ = 0.1 nm.

1. LED: L_c = (850 × 10⁻⁹)²/(40 × 10⁻⁹) = 18.1 µm.
2. Laser: L_c = (1550 × 10⁻⁹)²/(0.1 × 10⁻⁹) = 24.0 mm. Answer: about 18 µm versus 24 mm — an LED gives fringes only when the arms are matched to within micrometres.

Common mistakes

• Forgetting the factor 2 in the Michelson (light goes to the mirror and back), or wrongly adding it in the Mach–Zehnder.
• Using geometric length instead of optical path n·L.
• Assuming one MZI output going dark means light is lost; it has moved to the other port.
• Expecting fringes from a broadband source at large path difference.
• Ignoring the refractive index of air in precision displacement measurement.

For GATE IN

• Fringe counting: displacement, wavelength or refractive index from N.
• Phase difference from path difference, and MZI output intensities for a given phase.
• Coherence length and fringe visibility.
• Conceptual comparison of Michelson and Mach–Zehnder layouts, and their use in modulators and sensors.

Quick check

1. A Michelson mirror moves 0.5 µm. What is the change in optical path difference?
2. An MZI has Δφ = π/3. What fraction of the light leaves the cos² port?
3. How many fringes are counted at 632.8 nm for a mirror displacement of 158.2 µm?
4. Which interferometer has two separate arms traversed once? Answers: 1. 1.0 µm. 2. 0.75. 3. 500. 4. Mach–Zehnder.