Temporal Coherence and Coherence Length

Split a beam of light, send the two halves down paths of unequal length, and bring them back together — the heart of every Michelson interferometer. The returning wave does not meet an independent partner; it meets a delayed copy of itself, shifted in time by $\tau = \Delta/c$ , where $\Delta$ is the path difference. Whether fringes appear comes down to a single question: does the field still remember the phase it had a time $\tau$ ago? Temporal coherence is the precise measure of that memory — the correlation between $E(t)$ and $E(t+\tau)$ — and the delay beyond which the memory fails is the coherence time $\tau_c$ . The distance light travels in that time, $\ell_c = c\,\tau_c$ , is the coherence length: the largest path difference over which the interferometer still shows contrast. The reason the memory is finite is that no real source is perfectly monochromatic. Every source radiates over some spectral width $\Delta\nu$ — set by the finite lifetime of excited atoms, by collisions, or by thermal Doppler motion — and different frequency components drift out of step as the delay grows, scrambling an initially sharp phase relationship. The bookkeeping is exact and beautiful: the visibility of the fringes, scanned against delay, is the Fourier transform of the source's spectrum (the Wiener–Khinchin theorem). A narrow line coheres over a long delay; a broad continuum dies almost at once. This is why white light shows only a handful of coloured fringes near equal path, while a stabilized single-mode laser interferes across kilometres — and why the same interferometer that reveals coherence can be turned around to measure a spectrum.

Key takeaways

Temporal coherence measures how well a light field stays correlated with a time-delayed copy of itself; the delay over which the correlation survives is the coherence time $\tau_c$ .
The coherence length $\ell_c = c\,\tau_c$ is the maximum optical path difference over which interference fringes keep appreciable contrast.
Coherence is the Fourier partner of the spectrum (Wiener–Khinchin): $\tau_c \sim 1/\Delta\nu$ , so $\ell_c \approx c/\Delta\nu = \lambda^2/\Delta\lambda$ — a narrow line means long coherence.
The number of usable fringes is $N \approx \ell_c/\lambda = \lambda/\Delta\lambda = \nu/\Delta\nu$ , the same figure as the spectral resolving power.
White light coheres over ~1 µm (a couple of wavelengths); a multimode HeNe over ~20 cm; a single-mode stabilized laser over hundreds of metres.
Key idea: the visibility $V(\tau)=|\gamma(\tau)|$ is the modulus of the Fourier transform of the normalized power spectrum; its width sets the coherence time $\tau_c$ , and $\ell_c=c\tau_c\approx\lambda^2/\Delta\lambda$ is the path difference over which interference persists.

The core insight

A light field stays in step with a delayed copy of itself only for a limited time $\tau_c$ , fixed by its spectral bandwidth through $\tau_c\sim 1/\Delta\nu$ . Over a path difference up to $\ell_c=c\tau_c\approx\lambda^2/\Delta\lambda$ the two beams still interfere with visible contrast; beyond it the many frequency components have dephased and the fringes wash out to uniform grey.

Where this lives today

The idea that an interferogram is a spectrum runs a whole class of instruments. In Fourier-transform infrared (FTIR) spectrometers, a Michelson mirror scans the delay while a detector records $I(\tau)$ ; inverse-transforming the interferogram yields the full infrared spectrum at once, with a throughput advantage over slit spectrometers and a resolution set purely by how far the mirror travels, $\delta\nu\approx 1/(2\,\Delta_{\max})$ . Optical coherence tomography (OCT) inverts the priority: it deliberately uses low-coherence superluminescent diodes so that fringes appear only when the sample and reference arms match to within a micron, and that short $\ell_c$ becomes the axial resolution that images the layers of a living retina. At the opposite extreme, the longest coherence lengths ever built serve gravitational-wave detectors: LIGO's kilometre arms only hold their fringe contrast because the laser's linewidth is sub-hertz, giving a coherence length far larger than any arm imbalance. And the optical frequency comb — a laser whose coherence is preserved across hundreds of thousands of evenly spaced modes — turns temporal coherence into a ruler for frequency itself, underpinning today's optical atomic clocks. Every one of these instruments is, at bottom, a machine for exploiting or measuring $|\gamma(\tau)|$ .

The physical picture

Picture the emitted wave not as an endless perfect sinusoid but as a long ribbon whose phase drifts randomly on a timescale $\tau_c$ . Overlap the ribbon with a copy of itself shifted by less than $\tau_c$ and crest still meets crest in a fixed way — the fringes are crisp. Shift by more than $\tau_c$ and the two copies are sampling uncorrelated stretches of the ribbon; the relative phase is random, and the fringes blur into flat grey. A perfectly monochromatic wave would be an infinitely long ribbon with perfect memory; real light is a bundle of many frequencies whose phasors, initially aligned, fan out as $\tau$ grows. Their shrinking vector sum is the coherence, and the wider the frequency spread, the faster it collapses.

Figure

A finite coherence ribbon of length ℓc = c·τc, and the enormous spread of coherence lengths across sources: white light ≈ 1 µm, multimode HeNe ≈ 20 cm, single-mode laser ≈ 300 m

Think of temporal coherence as spectroscopy in disguise. Scanning the delay in a Michelson interferometer and recording the fringe visibility is mathematically identical to measuring the Fourier transform of the light's spectrum. A sharp spectral line gives a slowly decaying interferogram — long memory, many fringes. A broad continuum gives one that dies within a wavelength or two. The most vivid demonstration is a source with two close lines: the sodium doublet. Its two fringe systems drift in and out of step as the path grows, so the visibility does not simply decay — it beats, collapsing to zero and reviving periodically. Fizeau read the doublet spacing straight off that beat pattern in 1862, decades before anyone could resolve the two lines with a prism.

Figure

Sodium-doublet visibility beating in a Michelson: V(Δ) = |cos(πΔ/0.578 mm)| collapses to zero at Δ = 0.289 mm and revives at 0.578 mm — the D-line splitting read straight off the path difference

The core equations

Fix a point in the beam and compare the field with a copy of itself delayed by $\tau$ . The first-order (field) correlation, normalized so that a monochromatic wave has unit modulus, is the complex degree of temporal coherence. Everything measurable follows from its modulus.

\gamma(\tau)=\frac{\langle E^*(t)\,E(t+\tau)\rangle}{\langle |E(t)|^2\rangle},\qquad 0\le |\gamma(\tau)|\le 1 \qquad (1)

where: $E(t)$ — complex analytic signal of the field at the point; $\tau$ — time delay between the two copies; $\langle\cdot\rangle$ — time (ensemble) average over the stationary field; $|\gamma|=1$ fully coherent, $|\gamma|=0$ incoherent. The numerator is the self-coherence $\Gamma(\tau)=\langle E^*(t)E(t+\tau)\rangle$ ; the denominator is the mean intensity $\Gamma(0)=I$ . Recombine the field with its delayed copy in a balanced (equal-intensity) interferometer and average. The oscillating fringe term carries exactly $|\gamma(\tau)|$ , so the fringe contrast reads the coherence off directly — a calibration-free measurement.

V(\tau)=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}=|\gamma(\tau)| \qquad (2)

where: $I_{\max},I_{\min}$ — local fringe maxima and minima as the fast phase $\omega_0\tau$ is scanned; $V$ — Michelson fringe visibility. For unequal beams this becomes $V=\tfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma|$ ; the balanced case isolates $|\gamma|$ . The link between coherence and spectrum is not a heuristic but a theorem: for a stationary field, the self-coherence function and the power spectral density are a Fourier pair. Measure the interferogram and you have measured the spectrum.

S(\nu)=\int_{-\infty}^{\infty}\Gamma(\tau)\,e^{-i2\pi\nu\tau}\,\mathrm{d}\tau,\qquad \gamma(\tau)=\frac{\int S(\nu)\,e^{i2\pi\nu\tau}\,\mathrm{d}\nu}{\int S(\nu)\,\mathrm{d}\nu} \qquad (3)

where: $S(\nu)$ — power spectral density; $\nu$ — optical frequency. The normalized coherence is the Fourier transform of the normalized spectrum — a narrow $S(\nu)$ gives a broad $\gamma(\tau)$ and vice versa, which is the whole content of the reciprocity below. The width of $|\gamma(\tau)|$ is the coherence time; the path it corresponds to is the coherence length. Converting the frequency width to a wavelength width gives the compact form used at the bench.

\tau_c\sim\frac{1}{\Delta\nu},\qquad \ell_c=c\,\tau_c\approx\frac{c}{\Delta\nu}=\frac{\lambda^2}{\Delta\lambda} \qquad (4)

where: $\tau_c$ — coherence time; $\ell_c$ — coherence length (max useful path difference); $\Delta\nu,\Delta\lambda$ — spectral bandwidth in frequency and wavelength; $\lambda$ — mean wavelength. The order-unity prefactor in $\tau_c\Delta\nu\sim 1$ depends on the line shape ( $1/\pi$ for a Lorentzian, $\approx 0.66$ for a Gaussian) and should never be treated as exact.

Derivation ACoherence length in wavelength form and the fringe-count rule

Our aim is to turn the reciprocity $\tau_c\sim 1/\Delta\nu$ into the two numbers an experimenter actually uses: the coherence length written in terms of the wavelength spread, $\ell_c=\lambda^2/\Delta\lambda$ , and the maximum number of high-contrast fringes, $N\approx\lambda/\Delta\lambda$ . Both come from a single differentiation of $\nu=c/\lambda$ , and the fringe count turns out to equal the spectral resolving power — the same figure of merit a prism or grating would quote. This is the article's owned result.

Step 1. Start from the order-of-magnitude reciprocity $\ell_c\approx c/\Delta\nu$ , itself the statement that a spectral width $\Delta\nu$ dephases the components after a delay $\sim 1/\Delta\nu$ . Why: $\ell_c$ is the path over which the fastest and slowest frequency components in the band slip by about one radian relative to each other — that slip is $2\pi\Delta\nu\,\tau$ , of order one when $\tau\sim 1/\Delta\nu$ .

Step 2. Differentiate $\nu=c/\lambda$ to convert the frequency width into a wavelength width: $|\Delta\nu|=(c/\lambda^2)\,\Delta\lambda$ . Why: the bench measures wavelengths, not frequencies, so we need the local conversion factor between the two widths at the mean wavelength.

Step 3. Substitute into $\ell_c=c/\Delta\nu$ : the mean $c$ cancels and $\ell_c=c\lambda^2/(c\,\Delta\lambda)=\lambda^2/\Delta\lambda$ . Why: this is the working form — coherence length grows as the square of the wavelength and inversely with the wavelength spread, so purer colour buys longer reach quadratically.

Step 4. Count the fringes: the number that fit within the coherence length is $N\approx \ell_c/\lambda=\lambda/\Delta\lambda=\nu/\Delta\nu$ . Why: this is exactly the resolving power $\mathcal R=\lambda/\Delta\lambda$ of a spectrometer, so "how many fringes can I see" and "how finely can I split colours" are the same question — the interferometer is a spectrometer whose resolution is its fringe count. *Verified: $\ell_c=\lambda^2/\Delta\lambda$ and $N=\lambda/\Delta\lambda$ checked symbolically and numerically for white light, HeNe, and the sodium doublet in verify_temporal-coherence-length.py.*

Go deeper · Derivation B · 1 minExponential visibility from a Lorentzian line

Derivation BExponential visibility from a Lorentzian line

Here we compute the visibility law for a collision- or natural-broadened source, whose spectrum is a Lorentzian of full width $\Delta\nu$ . The Wiener–Khinchin theorem hands us the coherence as the Fourier transform of that line, and a Lorentzian transforms to a two-sided decaying exponential — so the visibility falls off exponentially with delay and defines a clean coherence time $\tau_c=1/(\pi\Delta\nu)$ . This is the case that governs gas-discharge lamps and quiet single-mode lasers.

Step 1. Write the normalized Lorentzian spectrum, $S(\nu)=\dfrac{\Delta\nu/2\pi}{(\nu-\nu_0)^2+(\Delta\nu/2)^2}$ , centred at $\nu_0$ with FWHM $\Delta\nu$ . Why: collisions interrupt the emitted wave train at random, and the exponential distribution of interruption times Fourier-transforms into precisely this Lorentzian line shape.

Step 2. Apply Wiener–Khinchin, $\gamma(\tau)=\int S(\nu)e^{i2\pi\nu\tau}\,\mathrm d\nu$ . The integral is a standard contour transform of a Lorentzian and gives $\gamma(\tau)=e^{i2\pi\nu_0\tau}\,e^{-\pi\Delta\nu|\tau|}$ . Why: the pole of the Lorentzian at complex frequency $\nu_0\pm i\Delta\nu/2$ sets the exponential decay rate; the real centre frequency $\nu_0$ becomes the fast carrier phase.

Step 3. Take the modulus: $|\gamma(\tau)|=e^{-\pi\Delta\nu|\tau|}\equiv e^{-|\tau|/\tau_c}$ with $\tau_c=1/(\pi\Delta\nu)$ . Why: the visibility is $|\gamma|$ , so contrast decays exponentially and drops to $1/e$ exactly at $\tau=\tau_c$ — a sharp, measurable definition of the coherence time.

Step 4. Convert to a length, $\ell_c=c\tau_c=c/(\pi\Delta\nu)$ . Why: for a Lorentzian the naive estimate $c/\Delta\nu$ overstates the $1/e$ length by the factor $\pi$ ; quoting the prefactor is the difference between 30 cm and 10 cm for a real lamp. *Verified: the Lorentzian↔exponential pair, $\tau_c=1/(\pi\Delta\nu)$ , and $|\gamma(\tau_c)|=1/e$ are checked (including a numerical FFT of the line shape) in verify_temporal-coherence-length.py.*

Go deeper · Derivation C · 1 minTwo-line beating and the sodium doublet

Derivation CTwo-line beating and the sodium doublet

Finally we explain the beating visibility in the punchline figure. A source with two equal, close lines separated by $\Delta\nu_D$ produces two fringe systems that drift in and out of register as the delay grows, so the visibility oscillates as $|\cos(\pi\Delta\nu_D\,\tau)|$ rather than decaying monotonically. Reading the first collapse and revival off the path difference measures the line spacing directly — Fizeau's 1862 trick, and the reason a sodium lamp is a poor white-light-fringe source.

Step 1. Superpose two equal-amplitude components, $E(t)=E_0\big(e^{-i2\pi\nu_1 t}+e^{-i2\pi\nu_2 t}\big)$ with $\nu_{1,2}=\nu_0\mp\Delta\nu_D/2$ . Why: the sodium D-lines are mutually incoherent in phase but each individually narrow, so over the delays of interest the pair behaves as two stable tones whose fringe patterns superpose.

Step 2. Form the coherence, $\gamma(\tau)=\tfrac12(e^{i2\pi\nu_1\tau}+e^{i2\pi\nu_2\tau})=e^{i2\pi\nu_0\tau}\cos(\pi\Delta\nu_D\,\tau)$ . Why: the average of the two phasors factors into a fast carrier at the mean frequency times a slow cosine at the half-difference — the mathematical signature of a beat.

Step 3. Read the visibility, $V(\tau)=|\gamma(\tau)|=|\cos(\pi\Delta\nu_D\,\tau)|$ , zero first at $\tau=1/(2\Delta\nu_D)$ and revived at $\tau=1/\Delta\nu_D$ . Why: when the two fringe systems are exactly a half-fringe out of step the bright fringes of one fill the dark fringes of the other, and the contrast vanishes even though each system is perfectly sharp.

Step 4. Put in numbers for sodium, $\Delta\lambda_D=0.60$ nm at $\lambda=589$ nm so $\Delta\nu_D=c\Delta\lambda_D/\lambda^2\approx 519$ GHz: the first null sits at $\Delta=c/(2\Delta\nu_D)=0.289$ mm and the revival at $0.578$ mm. Why: those are the marked features in the punchline figure, and they let you infer $\Delta\lambda_D=0.60$ nm from a ruler reading of the mirror position — spectroscopy without a spectrometer. *Verified: $V=|\cos(\pi\Delta\nu_D\tau)|$ , $\Delta\nu_D=519$ GHz, null at 0.289 mm, revival at 0.578 mm in verify_temporal-coherence-length.py.*

Limiting cases

Case	Result	Meaning
Monochromatic ( $\Delta\nu\to 0$ )	$\|\gamma(\tau)\|=1$ for all $\tau$	infinite memory; fringes at any path difference — the idealization real light never quite reaches
Lorentzian line (collision/natural)	$\|\gamma(\tau)\|=e^{-\|\tau\|/\tau_c}$ , $\tau_c=1/(\pi\Delta\nu)$	exponential contrast decay; the generic lamp/quiet-laser envelope
Gaussian line (Doppler)	$\|\gamma(\tau)\|=e^{-(\pi\Delta\nu\tau)^2/4\ln 2}$ , $\tau_c\approx 0.66/\Delta\nu$	faster-than-exponential decay at large delay; the hot-vapour envelope
Two close lines (doublet)	$\|\gamma(\tau)\|=\|\cos(\pi\Delta\nu_D\tau)\|$	visibility beats — collapses and revives; reads the line spacing
White light (broadband)	$\ell_c=\lambda^2/\Delta\lambda\approx 1\,\mu\text{m}$	only a few coloured fringes near equal path; the zero-order fringe is the alignment landmark
Single-mode stabilized laser	$\Delta\nu\sim$ MHz $\Rightarrow\ \ell_c\sim$ 100 m–km	contrast survives across the lab and beyond; long-baseline interferometry becomes possible

Common trap“The coherence length is the physical length of the emitted wave train (or of a photon)”

It is tempting to picture $\ell_c$ as a little rod of light — the length of one atom's emitted pulse, or the "size of a photon". It is neither. Coherence length is a statistical correlation width of the field: the path difference over which the delayed copy still shares a definite phase with the original, averaged over the light's fluctuations. A continuous-wave single-mode laser has a coherence length of hundreds of metres while emitting a beam that is, in any ordinary sense, unbroken — there is no metres-long "wave packet" flying through the room. Two further traps live here: $\ell_c=c\tau_c$ uses the vacuum $c$ (or the medium's group velocity in matter, not the phase velocity), and the prefactor tying $\tau_c$ to $1/\Delta\nu$ is line-shape-dependent — treating $\tau_c\Delta\nu=1$ as exact will mis-estimate a Lorentzian's reach by a factor of $\pi$ .

Worked examples

1 · Coherence length of a helium–neon laser, multimode vs single-mode (confidence builder)

A red HeNe laser ( $\lambda=633$ nm) oscillates on several longitudinal modes spanning $\Delta\nu=1.5$ GHz. A frequency-stabilized version is locked to a single mode with linewidth $\Delta\nu=1$ MHz. Find each coherence length.

Step 1. Multimode, order estimate: $\ell_c\approx c/\Delta\nu = (3.00\times10^{8}\,\text{m/s})/(1.5\times10^{9}\,\text{Hz}) = \mathbf{0.20\ m}$ . This is why a garden-variety HeNe interferes comfortably across a benchtop but starts to lose contrast if one arm is a metre longer than the other.

Step 2. Single-mode: $\ell_c\approx c/\Delta\nu = (3.00\times10^{8})/(1\times10^{6}) = \mathbf{300\ m}$ . Locking to one mode and narrowing the linewidth by a factor of $1500$ lengthens the coherence by the same factor — the fringes now survive an arm imbalance longer than a football pitch.

Step 3. Fringe count as a cross-check: $N\approx\ell_c/\lambda = 0.20/633\times10^{-9}\approx 3.2\times10^{5}$ fringes for the multimode laser. Equivalently $N=\nu/\Delta\nu=(4.74\times10^{14})/(1.5\times10^{9})\approx 3.2\times10^{5}$ — the two routes agree, as they must.

Step 4. The lesson: coherence length is set by linewidth, not by output power or wavelength. A 1 mW stabilized diode outreaches a 10 W multimode lamp by orders of magnitude. Multimode ℓc ≈ 0.20 m; single-mode ℓc ≈ 300 m.

2 · White light and the sodium doublet (exam level)

Estimate the coherence length of white light ( $\lambda\approx 550$ nm across $\Delta\lambda\approx 300$ nm) and of a sodium lamp ( $\lambda=589$ nm, doublet spacing $\Delta\lambda_D=0.60$ nm). At what path difference do the sodium fringes first vanish?

Step 1. White light: $\ell_c=\lambda^2/\Delta\lambda = (550\times10^{-9})^2/(300\times10^{-9}) = 1.01\times10^{-6}\ \text{m}=\mathbf{1.0\ \mu m}$ — about two wavelengths, i.e. only a handful of fringes straddling zero path. This is exactly why white-light fringes are the landmark for finding equal-path in an interferometer.

Step 2. Sodium (each line narrow): treat the doublet spacing as the effective width, $\ell_c=\lambda^2/\Delta\lambda_D=(589\times10^{-9})^2/(0.60\times10^{-9})=5.78\times10^{-4}\ \text{m}=\mathbf{0.578\ mm}$ — the revival period of the beat, three orders of magnitude longer than white light because each D-line is sharp.

Step 3. First fringe collapse: convert the spacing to a frequency, $\Delta\nu_D=c\,\Delta\lambda_D/\lambda^2=(3.00\times10^{8})(0.60\times10^{-9})/(589\times10^{-9})^2=5.19\times10^{11}\ \text{Hz}$ . The visibility $|\cos(\pi\Delta\nu_D\tau)|$ first hits zero at $\Delta=c\tau=c/(2\Delta\nu_D)=\mathbf{0.289\ mm}$ , exactly the red marker in the punchline figure.

Step 4. Reading it backwards: measuring the first collapse at $0.289$ mm and the revival at $0.578$ mm on the mirror micrometer returns $\Delta\lambda_D=0.60$ nm — Fizeau resolved the sodium doublet this way in 1862, long before gratings could. White light ℓc ≈ 1.0 µm; sodium ℓc ≈ 0.58 mm, first collapse at 0.289 mm.

Physics of the portrait

Temporal coherence is not a person but an idea — the memory a light field keeps of its own phase — and its history is a chain of experimenters who learned to read a spectrum off a moving mirror.

Origins · 1 mintemporal coherence and how this was discovered

temporal coherence

the field's memory of its own phase · Fizeau to Fourier · from sodium fringes to the frequency comb

The concept was born at the interferometer long before it had a name. In 1862 Hippolyte Fizeau noticed that fringes from a sodium flame did not fade smoothly as he lengthened one path — they pulsed, vanishing and reappearing with a period he could measure, and from that period he inferred that the yellow sodium light was really two close lines. Albert Michelson turned the idea into metrology: through the 1890s he pushed his interferometer to ever larger path differences, found that the red line of cadmium kept its fringes out to roughly 20–30 cm, and used that unusually long-coherence line to express the metre in wavelengths of light. The mathematics caught up in 1930–1934, when Norbert Wiener and Aleksandr Khinchin proved for stationary random signals the theorem that now bears their names: the autocorrelation of a fluctuating field and its power spectrum are a Fourier-transform pair. That single result explains why Fizeau's beating fringes and Michelson's decaying ones are Fourier images of the source spectrum — and it made Fourier-transform spectroscopy inevitable. The modern heirs are the femtosecond frequency comb (Hänsch and Hall, Nobel 2005), whose coherence links optical frequencies to a clock, and the kilometre-scale interferometers of gravitational-wave astronomy, which demand the longest coherence lengths ever engineered.

History

1862

— Fizeau observes periodic fringe collapse and revival with a sodium source, inferring the doublet from the beat period

1893

— Michelson measures the coherence length of the cadmium red line (~20–30 cm) and proposes light wavelengths as a length standard

1930

— Wiener formulates the autocorrelation–spectrum relation for stationary random signals

1934

— Khinchin gives the general theorem now called Wiener–Khinchin

1960

— the first lasers deliver coherence lengths from metres to kilometres, transforming interferometry

Going further

Texts. Hecht, Optics, 5th ed., ch. 7 (Coherence) and ch. 9 (Interference) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 3 (first-order coherence) · Saleh & Teich, Fundamentals of Photonics, ch. 11 (Statistical Optics) · Foot, Atomic Physics, ch. 8 (line shapes and broadening) · Demtröder, Laser Spectroscopy, Vol. 1, ch. 3 (spectral line profiles) · Mandel & Wolf, Optical Coherence and Quantum Optics, ch. 4.

Provenance. The definitions of $\Gamma(\tau)$ , $\gamma(\tau)$ , and $V=|\gamma|$ follow Loudon ch. 3, Saleh & Teich ch. 11, and Mandel & Wolf ch. 4; the Wiener–Khinchin theorem and the Lorentzian↔exponential / Gaussian↔Gaussian transform pairs are standard (Hecht ch. 7). Line-shape prefactors ( $\tau_c=1/\pi\Delta\nu$ Lorentzian; $\approx 0.66/\Delta\nu$ Gaussian) match Demtröder Vol. 1 ch. 3 and Foot ch. 8. All quoted numbers — white light $\ell_c\approx 1\,\mu$ m, multimode HeNe $\approx 20$ cm, single-mode $\approx 300$ m, sodium doublet revival $0.578$ mm and first collapse $0.289$ mm — are reproduced in verify_temporal-coherence-length.py from the stated $\lambda$ , $\Delta\lambda$ , $\Delta\nu$ .

Prerequisites. Superposition Principle of Electromagnetic Waves · Two-Wave Interference and the Interference Term · Optical Path Difference and Phase Retardation · Fringe Visibility and Contrast. Leads to. Mutual Coherence Function and Degree of Coherence · Spatial Coherence and the Van Cittert–Zernike Theorem · Wavefront-Splitting and Amplitude-Splitting Interferometers · Conditions for Observable Interference.

Is coherence time the same as the lifetime of an excited atom?

They are closely related but not identical. The natural (radiative) linewidth is set by the excited-state lifetime through $\Delta\nu\sim 1/(2\pi\tau_{\text{life}})$ , and for a source limited only by that lifetime the coherence time is of the same order. But most real sources are broadened by other mechanisms — collisions and Doppler motion usually dominate — so their coherence time is much shorter than the radiative lifetime would allow. Coherence time reflects the total spectral width from all broadening mechanisms combined, not just spontaneous decay. The clean statement is $\tau_c\sim 1/\Delta\nu_{\text{total}}$ .

If I put the interferometer arms exactly equal, is coherence length irrelevant?

At exactly zero path difference every frequency component is in step regardless of bandwidth, so even white light gives a bright fringe — the coherence length only limits how far you can imbalance the arms. In practice this is why the equal-path (zero-order) fringe is so useful: it is the one fringe that survives for any source, so you hunt for it to establish the balance point, then work outward. The coherence length tells you how far out you can go before the fringes fade, which sets the usable scanning range and hence, in a Fourier-transform spectrometer, the spectral resolution.

Does a Lorentzian or a Gaussian line lose coherence faster?

It depends where you look. For equal FWHM, near zero delay the Lorentzian falls faster because its exponential envelope $e^{-\pi\Delta\nu|\tau|}$ has a sharp cusp at $\tau=0$ , whereas the Gaussian $e^{-(\pi\Delta\nu\tau)^2/4\ln2}$ starts out flat. At large delay the roles reverse: the Gaussian's quadratic exponent eventually overwhelms the Lorentzian's linear one, so the Gaussian envelope plunges below the Lorentzian's long exponential tail. Collision- and lifetime-broadened lamps are Lorentzian; Doppler-broadened hot vapours are Gaussian; many real lines are a Voigt convolution of both.

How can visibility "revive" after dropping to zero — isn't the coherence gone?

For a true broadband source the visibility decays monotonically and does not revive. Revival is special to a structured spectrum, most simply two close lines. There the two fringe systems each stay perfectly sharp; what oscillates is their relative registration. At the first collapse the bright fringes of one line sit on the dark fringes of the other and the contrast cancels — but push the delay further and they come back into step, so the combined pattern regains full contrast. The information was never lost; it was hidden in the phase between two still-coherent systems. Any periodic modulation of $|\gamma(\tau)|$ is a fingerprint of periodic structure in the spectrum.

Why must I use the group velocity, not the phase velocity, for ℓc in glass?

Coherence length tracks the envelope of a wave packet — the region over which the delayed copy still overlaps coherently — and it is the envelope, not the individual crests, that travels at the group velocity $v_g=d\omega/dk$ . In a dispersive medium the phase velocity $v_p=\omega/k$ can differ substantially from $v_g$ , and using $v_p$ would misplace where the fringes fade. So the correct conversion inside matter is $\ell_c=v_g\,\tau_c$ . Worse, dispersion also stretches the coherence envelope as it propagates (group-velocity dispersion), which is why deep interferometry through glass or fibre must compensate the dispersion to keep fringes sharp.

Is temporal coherence a purely classical idea, or does it survive in quantum optics?

It survives essentially unchanged. The classical degree of coherence $\gamma(\tau)$ maps directly onto the quantum first-order correlation $g^{(1)}(\tau)$ , and fringe visibility measures $|g^{(1)}|$ in both pictures — a laser and a filtered thermal lamp with the same spectrum give the same fringes. What the field-correlation $g^{(1)}$ cannot see is photon statistics: bunching and antibunching live in the second-order correlation $g^{(2)}$ , measured by coincidence counting rather than interference. So temporal coherence is the quantity that classical and quantum optics agree on, and the deeper quantum signatures require a different experiment.