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Interferometers and Coherence

Build a Michelson or Mach–Zehnder interferometer, move a mirror by fractions of a wavelength, and see how the source spectrum, source size, polarization and beam-splitter ratio set the fringe visibility. Visibility versus path difference is compared with the Fourier transform of the spectrum (Wiener–Khinchin).

Every slider has a number box: type a value and press Enter • Click the fringe plot to move the mirror • Settings are kept in the URL
Δ — Optical path difference
P₁ — Port 1 power
P₂ — Port 2 power
V — Visibility (port 1)
lc — Coherence length (V = ½)

Interferometer and path phasors Michelson

Beam widths grow with the power in each segment; arm lengths are drawn to scale with each other, while the displacement d is exaggerated. The phasor panels add the field that each arm delivers to a port at λ₀ (monochromatic, on axis), rotated so that arm 1 points right. Only the part of arm 2 parallel to arm 1 can interfere; the orthogonal part (ψ ≠ 0) adds power without fringes.

Output powers vs displacement P / Pin

Port powers as fractions of the input power while the mirror moves; the vertical line is the current d. Click or drag to set d.

Port 1 detector image circular

Port-1 intensity per direction, relative to the input (colour bar 0–1).

Source spectrum s(ν) peak-normalised

Normalised power spectral density against vacuum wavelength. Dots are the spectral samples that are summed incoherently.

Fringe visibility scan

Dots: visibility measured from the simulated port-1 signal by phase stepping (fringe amplitude divided by the summed single-arm powers). Solid: V₀|γ(τ)|, where γ is the Fourier transform of the sampled, normalised spectrum (Wiener–Khinchin). Dashed: closed form. On the other scans, the dots and dashed lines are port 1 (cyan) and port 2 (gold). Click the plot to set the scanned variable.

OPD Δ (on axis)
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Fringe order Δ/λ₀
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Phase kΔ + φ (mod 360°)
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P₁ / P₂
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P₁ + P₂ (absorbed)
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V measured, port 1 / 2
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V₀·|γ|·Fsrc (theory)
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lc at V = ½ (OPD)
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Mandel lc = c∫|γ|²dτ
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λ₀²/Δλ
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Doublet beat (OPD / mirror)
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Source-size factor Fsrc
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Image fringe scale
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Spectral samples N (bin δν)
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💡 How to use

Learn with this tool

Learning objectives

  • Trace complex amplitudes through a unitary beam splitter and predict which port is bright, including reflection phases and the effect of splitter ratio and loss on contrast.
  • Relate mirror displacement to optical path difference (Δ = 2d) and use fringe counting to measure a wavelength or a displacement.
  • Connect fringe visibility versus delay to the source spectrum through the Wiener–Khinchin theorem, and estimate coherence length, linewidth and doublet separation from a measurement.

Prerequisites

  • Two-beam interference: I = I₁ + I₂ + 2√(I₁I₂)|γ| cos Δφ
  • Polarization: Jones vectors and their overlap
  • Complex numbers, phasors and the Fourier transform

Model

A field of unit input power, linearly polarised along x̂, enters beam splitter BS1. Each splitter is lossless and symmetric, with the declared convention

S = [[t, i r], [i r, t]],  t = √(1 − R),  r = √R,  S S† = 1

so reflection adds +π/2 relative to transmission. A path of optical length Λ multiplies the amplitude by eikΛ (time convention e−iωt). Each mirror reflects with −1, and since every arm has exactly one mirror this common phase cancels. Arm j transmits power ηj. Arm 2 also has an achromatic phase shifter eiφ and a polarization rotator by ψ. For port p, the field is Ep = cp1eikΛ₁ + cp2eikΛ₂ and the power is Pp = |Ep|². Different frequencies do not interfere, so the detected power is the sum of spectral intensities:

Pp(Δ) = Σk wk Pp(νk, Δ) = Ap + 2 Re[Xp γ(Δ/c)],   γ(τ) = ∫ s(ν) ei2πντ dν

Here Ap = |cp1|² + |cp2|², Xp = cp1†cp2, and s(ν) is the normalised power spectrum (∫s dν = 1). The visibility is V = V₀|γ|, where V₀ = 2|Xp|/Ap. This is the Wiener–Khinchin connection: the fringe envelope is the magnitude of the Fourier transform of the spectrum. The Michelson detector port has V₀ = 1 for any lossless R because both arms reflect once and transmit once. The return port has V₀ = 2RT/(R² + T²).

Λ₁, Λ₂
arm optical paths. Michelson: 2L₁ and 2(L₂ + d). Mach–Zehnder: L₁ and L₂ + 2d (delay stage).
Δ = Λ₂ − Λ₁
optical path difference (OPD); τ = Δ/c is the delay
Δν = cΔλ/λ₀²
spectral width: FWHM of the intensity spectrum for the Gaussian and Lorentzian, full width for the rectangle
wk
declared spectral weights: samples of s(ν) on a uniform grid of bin width δν, normalised to Σw = 1. Each bin's fringe is integrated across the bin, which gives a factor sinc(πδν τ). The Lorentzian is truncated at ±100Δν and renormalised, which removes about 0.3 % of its power.
θₛ
angular radius of an extended, uniform source (Michelson spatial coherence)
ψ
net rotation of the arm-2 polarization; the interfering fraction of the field is cos ψ

Closed forms used for comparison (FWHM Δν of the intensity spectrum): Gaussian |γ| = exp[−(πΔντ)²/(4 ln 2)], so V = ½ at Δ = (2 ln 2/π)·c/Δν ≈ 0.441 λ₀²/Δλ. Lorentzian |γ| = exp(−πΔν|τ|). Rectangle |γ| = |sinc(πΔντ)|, with zeros at Δ = m λ₀²/Δλ. Doublet |γ| = |w₂ + w₁ei2πΔν₁₂τ|, which beats with OPD period λ₁λ₂/Δλ₁₂ between 1 and |w₂ − w₁|.

Derivation: ports, conservation and the Michelson source-size factor

Michelson. The input goes into port a. The arms receive t (to M₁) and ir (to M₂). On the way back the same S acts on the returning fields e₁ = −t√η₁eikΛ₁ and e₂ = −ir√η₂eiφeikΛ₂. The detector port gets ir·e₁ + t·e₂ = −irt(√η₁eikΛ₁ + √η₂eiφeikΛ₂). Both terms carry the same factor rt, so the fringe is bright at Δ = 0 and has V₀ = 1 when η₁ = η₂. The return port gets t·e₁ + ir·e₂ = −t²√η₁eikΛ₁ + r²√η₂eiφeikΛ₂. Adding the two port powers, the cross terms cancel (2r²t² − 2r²t² = 0), which leaves P₁ + P₂ = Tη₁ + Rη₂. For lossless arms that equals 1 at every phase: S is unitary.

Mach–Zehnder. BS2 turns arm fields a₁ = −t₁√η₁eikΛ₁ and a₂ = −ir₁√η₂eiφeikΛ₂ into D₁ = t₂a₁ + ir₂a₂ and D₂ = ir₂a₁ + t₂a₂. When balanced, D₁ = −½(eikΛ₁ − eiφeikΛ₂), so port 1 is dark at Δ = 0, φ = 0.

Spectrum. For each frequency, P = A + 2Re[X ei2πνΔ/c]. Weighting by s(ν) and integrating gives P = A + 2Re[X γ(Δ/c)]. Because A does not depend on the spectrum, the mean power is fixed and only the fringe term fades.

Source size. The Michelson acts like an air plate of thickness Δ/2, so a ray at angle θ sees OPD Δ cos θ (these are fringes of equal inclination, which appear as rings at a lens focus). A source of uniform radiance filling a cone θ ≤ θₛ spreads μ = cos θ uniformly over [cos θₛ, 1]. Averaging eikΔμ over μ gives eikΔ(1+cos θₛ)/2 sinc[kΔ(1 − cos θₛ)/2]. The on-axis fringe vanishes when Δ(1 − cos θₛ) = λ, that is Δθₛ² ≈ 2λ. This is the Jacquinot étendue limit of Fourier-transform spectrometers. An aligned Mach–Zehnder has no angle-dependent OPD, so its visibility does not depend on source size.

Exercise 1 — Counting fringes

  1. A HeNe laser (λ₀ = 632.8 nm) lights a Michelson. How far must M₂ move for P₁ to go from one maximum to the next? How far for maximum to minimum?
  2. Load λ/2 step. Type d = 0 into the number box, then d = 0.3164 µm, then d = 0.1582 µm.
  3. Record P₁ at each value and count the fringes that cross the cursor in the fringe plot while you press Step 16 times.
  4. Explain why the period is λ/2 of mirror travel and not λ.
Show answer

P₁ = 1, then 1, then 0. The light crosses the arm twice, so Δ = 2d and one fringe (Δ → Δ + λ) needs d = λ/2 = 316.4 nm. Maximum to minimum needs λ/4 = 158.2 nm. Sixteen steps of λ/16 add up to λ, which is two fringes. Counting N fringes over a travel L measures λ = 2L/N; this is how the metre was once compared with the cadmium red line.

Exercise 2 — Coherence length of a Gaussian source

  1. A source has a Gaussian spectrum with λ₀ = 560 nm and FWHM Δλ = 120 nm. At what OPD does the visibility fall to ½? And to 1/e?
  2. Load White light and keep the scan variable at "Path difference".
  3. Read lc and find where the measured dots cross 0.5 and 0.368. Check that the dots lie on both the FT curve and the dashed closed form.
  4. Why is the fringe envelope the Fourier transform of the spectrum rather than of the field?
Show answer

Δν = cΔλ/λ₀² = 114.7 THz and c/Δν = λ₀²/Δλ = 2.613 µm. V = ½ at Δ = (2 ln 2/π)(c/Δν) = 1.153 µm. V = 1/e at Δ = (2√(ln 2)/π)(c/Δν) = 1.385 µm. Only a handful of fringes are visible, and this is how white-light interferometry finds zero OPD. The detector averages over the random phases of different frequencies, so only intensities add: the cross term is ∫s(ν)ei2πνΔ/cdν. The spectral phase never enters.

Exercise 3 — Limiting cases (challenge)

  1. Predict the visibility for (a) an equal-weight doublet at Δ = λ₁λ₂/(2Δλ), (b) a Mach–Zehnder with R₁ → 100 %, and (c) any spectrum as Δλ → 0.
  2. (a) Load Equal doublet. (b) Load Balanced MZ and choose the "BS1 reflectance" scan. (c) Choose a Gaussian spectrum and drag Δλ to the left.
  3. Read V and P₁ + P₂ in each case.
  4. Which limits remove the fringes because the arms become distinguishable, and which because the fringes of different frequencies cancel?
Show answer

(a) V = |w₂ − w₁| = 0 at Δ = 290.8 µm (mirror travel 145.4 µm): the two lines' fringe patterns are exactly out of step and cancel, while P₁ + P₂ stays 1. (b) V₁ = 2√(T₁T₂R₁R₂)/(T₁T₂ + R₁R₂) → 0 as R₁ → 1, because all the light takes arm 2 and there is nothing to interfere with. This is "which-path" information, not decoherence. (c) γ → 1 for every Δ, so V = V₀ at any OPD: the monochromatic limit.

Exercise 4 — Source size and circular fringes

  1. A Michelson has Δ = 2.5 mm (L₂ − L₁ = 1.25 mm) and uses a 632.8 nm extended source. What source angular radius θₛ makes the on-axis fringe vanish? How many rings then fill the field?
  2. Load Source size and choose the "Source size θₛ" scan.
  3. Find the first zero of the measured visibility and count the bright rings in the detector image.
  4. Why does a Mach–Zehnder with the same source keep V = 1?
Show answer

The zero is at Δ(1 − cos θₛ) = λ, so θₛ = arccos(1 − λ/Δ) = 22.5 mrad (Δθₛ² ≈ 2λ). Further zeros follow at Δ(1 − cos θₛ) = mλ (31.8 and 39.0 mrad). At the first zero the fringe order runs from Δ/λ = 3950.7 at the centre to 3949.7 at the edge. Exactly one ring fills the cone, so the on-axis detector averages a full fringe to zero. With θₛ = 40 mrad the image shows about 3.2 rings. In an aligned Mach–Zehnder, tilting a ray lengthens both arms equally, so the OPD does not depend on angle.

Worked example — measuring the sodium D splitting

A student lights a Michelson with a sodium lamp and records the mirror positions where the fringes almost disappear: d = 0.1454 mm, 0.4362 mm and 0.7270 mm. Consecutive minima are Δd = 0.2908 mm apart, so the OPD beat period is 2Δd = 0.5816 mm. Minima occur when the two lines' fringes are in antiphase, Δ(1/λ₁ − 1/λ₂) = m + ½, so the beat period is Λbeat = λ₁λ₂/Δλ. With λ̄ = 589.46 nm (vacuum):

Δλ = λ̄²/Λbeat = (589.46 nm)² / 0.5816 mm = 0.5974 nm

That agrees with the accepted 0.597 nm. The minimum visibility gives the line-strength ratio: Vmin = (w₂ − w₁)/(w₂ + w₁), so Vmin = 1/3 means D₂:D₁ = 2:1. The slow decay of the beat maxima gives the linewidth. For Gaussian (Doppler) lines of FWHM 0.002 nm, the envelope reaches ½ at 2 ln 2 λ²/(πΔλ) ≈ 76.7 mm of OPD. Load Na doublet and choose the "Path difference" scan to check all three numbers. The dotted markers show the beat period.

When the model fails

  • Dispersion: the arms are dispersion-free, and so is the phase shifter. A real beam-splitter substrate without a compensator plate adds a wavelength-dependent OPD. White-light fringes then become asymmetric and chirped, and the envelope is no longer |FT s|.
  • Real coatings: R, T and the reflection phase depend on polarization, angle and wavelength, and the plate adds ghost reflections. Here the splitter is ideal, symmetric and achromatic.
  • Beams and alignment: fields are plane waves (or a lens-focused angular spectrum). Finite beam size, wavefront curvature, shear between the returning beams and fringe localisation with a tilted mirror and an extended source are not modelled. The source-size factor applies only to the on-axis detector and to the circular-fringe image.
  • Sources: the source must be stationary. The spectrum is sampled with bin width δν. Past the listed revival OPD c/δν the sampled comb would repeat; the bin integration suppresses that repeat but does not remove it entirely. Real lines have Voigt profiles, hyperfine structure and self-absorption.
  • Detection: detectors are ideal and noiseless and average over many optical periods. Vibration, air turbulence and shot noise, which limit real displacement measurements, are absent.

References

  • E. Hecht, Optics, 5th ed., §9.4 (Michelson and Mach–Zehnder interferometers) and §12.1–12.2 (coherence, Fourier-transform spectroscopy).
  • M. Born and E. Wolf, Principles of Optics, 7th ed., §7.5 (fringes of equal inclination, Michelson) and §10.3–10.4 (mutual coherence and visibility).
  • B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., §2.5 (interferometers) and §12.1–12.2 (temporal coherence, Wiener–Khinchin, coherence time).
  • L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (1995), §4.3 (coherence time defined as ∫|γ|²dτ).
  • R. Loudon, The Quantum Theory of Light, 3rd ed., §3.2 (symmetric lossless beam splitter and its unitarity relations).
  • P. Jacquinot, "The luminosity of spectrometers with prisms, gratings, or Fabry–Perot etalons", J. Opt. Soc. Am. 44, 761 (1954) (étendue limit of interferometric spectrometers).