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For vacuum wavelength endpoints ${\lambda }_{\min }<{\lambda }_{\max }$, the corresponding frequency span is $\Delta \nu =c(1/{\lambda }_{\min }-1/{\lambda }_{\max })$. The reciprocal-bandwidth estimate of optical coherence time is ${\tau }_c\sim 1/\Delta \nu$, giving ${\ell }_c\sim c/\Delta \nu$. This endpoint conversion is valid even for a broad band, whereas ${\ell }_c\sim {\lambda }_0^2/\Delta \lambda$ requires a relatively narrow optical spectrum.

For constant spectral density per unit frequency in the band, the normalized first-order optical coherence is $g^{(1)}(\tau )=e^{-i2\pi {\nu }_0\tau }\sin (\pi \Delta \nu \tau )/(\pi \Delta \nu \tau )$. Its first zero occurs at $\tau =1/\Delta \nu$. A different spectral shape or a different contrast criterion changes the numerical optical coherence length; endpoints alone do not define an exact universal value.

An exact teaching model makes the peak suppression explicit. Assume the narrowband field correlation separates as $g_{12}^{(1)}(B,u)=\gamma (B)g_t^{(1)}(u)$ after delay compensation. Average each channel's light intensity over the same rectangular gate of duration $T$. The two time integrals have triangular relative-delay weightIntegrating the Siegert relation against this weight givesThe factor $p$ is one for a selected polarization of waves and $1/2$ for two independent equally populated polarizations. The signal fraction ${\rho }_j=s_j/(s_j+b_j)$ uses mean stellar signal $s_j$ and independent additive background $b_j$ in each detector. The photon correlation loss and background model then supplies ${\rho }_1{\rho }_2$. This assumes no correlated sky background or electronic coupling.

For the original model $|g_t^{(1)}(u){|}^2=e^{-|u|/{\tau }_0}$, elementary integration givesThe area-based optical coherence time of this intensity envelope is ${\tau }_c=\int |g_t^{(1)}(u){|}^{2}du=2{\tau }_0$. For $T\gg {\tau }_0$, $\beta \approx {\tau }_c/T$; for $T\ll {\tau }_0$, $\beta \to 1$. With the chosen ${\tau }_0=1\mathrm{ps}$, $T=50\mathrm{ps}$, $p=1$, and ${\rho }_1={\rho }_2=0.8$, $\beta \approx 0.0392$ and $g_{\mathrm{obs}}^{(2)}(0,0)=1.025088$. At $B=10\mathrm{m}$ the stipulated disk gives approximately $1.013617$. A nearly unit measured $g^{(2)}$ can therefore conceal a substantial $V^2$.

This rectangular-gate calculation specifies a counting model. An analog instrument's amplifier bandwidth and impulse response instead weight the fluctuations it actually passes; the Sirius amplifiers were bandpass devices. Both require measuring the response and calibrating the zero-baseline excess. When $p$, ${\rho }_j$, and the temporal response are unchanged, the ratio of excesses recovers $V^2$:A sufficiently unresolved calibration source can supply the denominator; dividing the full $g^{(2)}$ values would give the wrong result. Narrower optical bandwidth lengthens optical coherence time, whereas wider usable electronic bandwidth can resolve faster fluctuations; these are distinct bandwidths. Background changes, unequal spectral responses, drift, or averaging over changing interferometric baselines can invalidate a common calibration. Resolving a disk means establishing a reproducible decline of calibrated correlation, supported by those controls.

Broadening shared by otherwise identical emitters, often described by a common optical coherence time $T_2$. Exponential field-coherence decay produces a Lorentzian line shape with full angular-frequency width $2/T_2$. Population decay, dephasing and collisions can all contribute, with their time constants distinguished.

In the absence of additional dephasing, a transition whose excited-state population has lifetime $T_1$ has a field-coherence decay time $2T_1$. Its Lorentzian line shape has angular-frequency half-width $1/(2T_1)$ and full width $1/T_1$. Population decay time, amplitude decay time and linewidth conventions must therefore be distinguished. Extra dephasing shortens the optical coherence time without necessarily changing the population lifetime.

In this temporal sense, the optical coherence length is $L_c=c{\tau }_c$, where $c$ is the vacuum speed of light and ${\tau }_c$ is an explicitly chosen optical coherence time. It measures a delay as a vacuum-equivalent path difference; it does not measure transverse spatial optical coherence. ^[ref].

For the stipulated $1\mathrm{GHz}$ profiles, the $1/e$ convention gives $L_c\approx 9.54269\mathrm{cm}$ for the Lorentzian and $15.8896\mathrm{cm}$ for the Gaussian. In a vacuum Michelson interferometer, changing one mirror position by $x$ changes the round-trip delay by $2x/c$, so the mirror travel to either of these delays is $L_c/2$. In a medium, convert physical travel using the actual group delay, which can depend on frequency. For a narrow line centered at vacuum wavelength ${\lambda }_0$, differentiating $\nu =c/\lambda$ gives $\Delta \nu \approx c\Delta \lambda /{\lambda }_0^2$; this small-width approximation must use consistently defined spectral widths.

An optical coherence time summarizes the delay scale of temporal optical coherence, but requires a convention. Two useful choices are the first positive delay ${\tau }_{1/e}$ where $|g^{(1)}|=e^{-1}$, for a monotonically decaying envelope, and the integral definitionThe latter counts correlation over both signs of delay and may diverge for ideal discrete lines. ^[ref].

For the stipulated Lorentzian and Gaussian profiles in temporal optical coherence, direct substitution and a Gaussian integral giveThe respective dimensionless coefficients are $0.318310$ for both Lorentzian conventions, and $0.530021$ and $0.664282$ for the Gaussian conventions. The familiar scale $1/\Delta \nu$ is consequently an estimate, not a convention-independent equality. A source with narrow discrete lines can have repeated revivals instead of one monotonically decaying envelope, so recording the full $|g^{(1)}(\tau )|$ can be more informative than one optical coherence time.

For the right-angle geometry of temporal coherence from crossed beams, a physical arm-path delay $\Delta \ell$ gives $I(y)=I_0+\mathrm{Re}\Gamma (\sqrt{2}y/c+\Delta \ell /c)$, with the sign set by the delayed arm. A quarter-cycle optical phase delay requires $\Delta \ell =\lambda /4$; it shifts the fringe pattern by a quarter of its period, $\lambda /(4\sqrt{2})$. A path delay equal to that latter length is a different operation. Replacing the shifted correlation by a phase multiplier additionally requires the delay to be small compared with the envelope optical coherence time.

An optical path length measures how much propagation wave phase a path accumulates, expressed as the distance that would accumulate the same wave phase in vacuum. A centimetre in a uniform medium with refractive index $3/2$ has optical path length $1.5\mathrm{cm}$, although its geometric length remains $1\mathrm{cm}$. This makes optical path length useful whenever two contributions must arrive with matching wave phases, from an optical lens to a Michelson interferometer. ^[ref]

For a specified path $\gamma$ at fixed angular frequency $\omega$, the optical path length is the line integralHere $s$ is geometric arc length, and $n$ is the phase refractive index, evaluated at the same frequency everywhere. The propagation interpretation assumes a stationary linear medium with isotropy, a real positive scalar refractive index and negligible loss, on a valid geometrical optics branch. The path can curve; at sharp interfaces apply the appropriate matching conditions to each segment. A prescribed curve has a mathematical line integral, but it predicts a propagation wave phase only when it represents the physical branch. ^[ref] Optical fibers can require a mode wavenumber rather than the bulk refractive index along a guessed light ray. ^[ref]

Fix the vacuum wavelength ${\lambda }_0=c/f=2\pi c/\omega$, with frequency $f=\omega /(2\pi )$ and vacuum wavenumber $k_0=2\pi /{\lambda }_0$. In a locally uniform medium the wavelength is $\lambda ={\lambda }_0/n$, while the frequency stays unchanged at a stationary interface. With the convention $\Phi =k_{0}S-\omega t+{\varphi }_0$, propagation from a specified starting wave phase contributesThe relative reference ${\delta }_{\mathrm{ref}}$ includes launch wave phases and any extra wave phase from mirrors, beam splitters or coatings. Thus $k_0\mathrm{OPL}$ is a propagation contribution, not an absolute detector wave phase independent of the apparatus. If the reference contributions stay fixed, a change in optical path difference $h$ changes the phase difference by $2\pi h/{\lambda }_0$. Using the material wavelength again in this formula would count the refractive index twice. ^[ref]

A motivating measurement turns invisible optical oscillations into a slowly changing electrical signal. MIT's teaching experiment uses a laser, a beam splitter, two returning mirrors, an amplified photodiode, an oscilloscope and a calibrated mirror actuator driven by a function generator. The returning beams overlap on the photodiode; one channel records detector voltage and another the actuator drive. The measured observable is the changing light intensity, including its maxima and minima, rather than direct timing of an optical cycle. A calibrated displacement per drive voltage and the spacing of the extrema relate geometric motion to the phase difference. The guide infers the laser wavelength from known actuator calibration; conversely, with independently known vacuum wavelength and ambient refractive index, the same relation calibrates mirror motion. These two unknowns cannot both be inferred from fringe spacing alone. ^[ref]

For a proposed change-of-medium measurement, keep the mirrors fixed and put a transparent cell of internal length $L$ in one arm. Compare its contents at refractive indices $n_a$ and $n_b$, leaving its windows in place. At normal incidence the two traversals giveHere $N$ is the signed number of whole interference fringe cycles plus any resolved fractional cycle, counted continuously. A mirror displacement $x$ instead gives $\Delta \mathrm{OPL}=2n_{\mathrm{amb}}x$; distinguish this double-pass factor from a single pass in a Mach-Zehnder interferometer. OpenStax's gas-cell worked example provides the same double-pass reasoning; it is a teaching calculation rather than a published measurement dataset.^[ref]

An exact toy calculation makes the sensitivity visible. Stipulate ${\lambda }_0=600\mathrm{nm}$, $L=10\mathrm{mm}$ and $n_b-n_a=3/100000=0.00003$. Then $\Delta \mathrm{OPL}=600\mathrm{nm}$ and $\Delta \delta =2\pi$: a full interference fringe cycle, despite unchanged geometric lengths. For a continuous change $n_b-n_a=q(3/100000)$, let $0\le q\le 2$ and choose the initial detector phase difference to be zero. Equal coherent detected arm contributions of light intensity $I_0$ predictThe plot is this stipulated model: $q=1/2$ gives a minimum, $q=1$ restores a maximum, and the reference level is chosen by adjustment of the apparatus. It contains no measured values or assigned properties of a real gas. The equivalent mirror travel in vacuum is $300\mathrm{nm}$ per cycle. A single endpoint light intensity cannot distinguish $q=0$ from $q=1$; the continuous fringe count carries the missing information.

For an interpretable cell measurement, establish the initial phase difference with a small calibrated mirror scan, then track slow changes through successive interference fringes. Measure the dark signal and each arm separately, keep photodiode gain within its linear range, align both beams on the same region and match their linear polarizations. Monitor interferometric visibility and keep arm delays within the useful optical coherence time. Record cell length, contents, pressure and temperature; return to the starting condition to check drift. Changes in window deformation, cell length, external air or mirror position also change optical path length, so a reference cell or an independently monitored mechanical path is needed to isolate the contents' refractive index. These controls are deductions from the model, not a claim that the illustrated apparatus was operated. Loss of fringe count leaves a whole-cycle ambiguity; at an intensity extremum the first derivative with respect to phase difference vanishes, so a scan around phase quadrature is more sensitive to small changes. Broadband illumination, unstable optical coherence, vibration and strong loss can invalidate a simple one-frequency inference.

Rabi oscillations require a coherent drive with a stable phase over several population cycles and an isolated, appreciably coupled pair of quantum states. For large-amplitude oscillations, the Rabi detuning must be comparable to or smaller than the Rabi angular frequency. The population lifetime, optical coherence time, transit time and drive coherence time should all exceed $2\pi /{\Omega }_R$ by enough to resolve the cycles. An ensemble also needs sufficiently uniform Rabi frequencies and Rabi detunings that phase averaging does not wash the oscillations out. The usual sinusoidal formula additionally assumes the rotating-wave approximation.

The Einstein coefficients describe incoherent transition rates, rather than the evolving optical probability amplitudes. Their simplest use is appropriate for broadband, randomly phased radiation, with field correlation time short relative to population evolution, or for rapidly dephased chemical atoms whose optical coherences can be eliminated. The radiation energy density per unit frequency must be smooth over the relevant spectral line. Stimulated upward and downward transitions and spontaneous emission then enter rate equations, without predicting coherent population return. A short observation time can give a small excitation probability even for a coherent field; that condition alone does not justify an Einstein coefficients rate description. Coherent laser irradiation can also admit steady rate equations after optical coherence has relaxed, with the appropriate saturation response.

A spectral line has finite width when the emitted or absorbed field loses coherence, or when different emitters have different transition frequencies. It is useful to distinguish homogeneous spectral broadening of each emitter from inhomogeneous spectral broadening across the ensemble.

Finite lifetime and dephasing. For an upper-state population lifetime $\tau$ and a stable lower state, the radiating amplitude decays as $e^{-t/(2\tau )}$. Its Fourier spectrum is Lorentzian, with full width at half maximumMore generally, an optical coherence time $T_2$ gives $\Delta {\omega }_{\mathrm{hom}}=2/T_2$. Both level lifetimes and pure dephasing can contribute to $1/T_2$. Distinguishing population lifetime from amplitude coherence time prevents the usual factor-of-two error.

Thermal motion. To first order in velocity, $\delta \nu ={\nu }_0{v}_{\parallel }/c$. A Maxwell-Boltzmann velocity distribution therefore produces a Gaussian profile, withThis is inhomogeneous doppler broadening: different velocity classes have different resonance frequencies. A distribution of bulk velocities can add further broadening, while a common bulk velocity shifts the line.

Interactions with the surroundings. Collisions can interrupt the dipole wave phase and perturb the energy levels, producing pressure broadening and shifts. In the impact approximation, approximately exponential wave phase relaxation produces a Lorentzian contribution; its rate increases with perturber density times an appropriate velocity-averaged collision cross section. Slowly varying electric microfields can instead produce non-Lorentzian Stark profiles. Neutral perturbers, charged perturbers and resonance interactions need not produce the same line shape. These mechanisms and the Gaussian thermal limit are distinguished in NIST's atomic-spectroscopy compendium.^[ref]

Finite observation time limits coherence; transit-time broadening also occurs for particles crossing a narrow optical beam. Strong resonant excitation causes power broadening: for an ideal lifetime-limited two-level transition, $\Delta \omega =\Gamma \sqrt{1+s}$ with population decay rate $\Gamma$ and saturation parameter $s$. Unresolved fine structure, hyperfine structure, isotope components, or spatially varying Zeeman and Stark shifts may look like a single broadened feature even though several frequencies are present.

The observed spectrum is additionally convolved with the instrument response. A Gaussian Doppler contribution convolved with Lorentzian homogeneous broadening gives a Voigt profile; their full widths do not generally add in quadrature. An informative experiment varies temperature, perturber pressure and excitation intensity separately, calibrates the spectrometer response with a narrower reference, and checks optical thickness or self-absorption. Extrapolation to weak excitation and low pressure helps isolate the lifetime contribution, while a Doppler-free technique is needed if velocity spread still dominates.

The coherence length from a finite spectral band is most safely estimated by converting its endpoints to frequency, because the quoted band is broad:The reciprocal-bandwidth optical coherence time and corresponding longitudinal optical coherence length areThe given temperature is consistent with the spectral peak: Wien's displacement law gives ${\lambda }_{\mathrm{peak}}=2.898\times 10^{-3}\mathrm{m}\mathrm{K}/5778\mathrm{K}\simeq 502\mathrm{nm}$. Using the common narrow-band estimate ${\lambda }_{\mathrm{peak}}^2/\Delta \lambda$, with $\Delta \lambda =600\mathrm{nm}$, gives about $0.42\mu \mathrm{m}$. Its closeness here does not justify the narrow-band approximation for arbitrary broad spectra.

The $0.40\mu \mathrm{m}$ result is a bandwidth-scale estimate. A flat spectral density per unit frequency would make it the first-zero optical coherence length; a truncated Planck law spectrum is not flat and would have a different correlation function. Neither the stated endpoints nor the word “coherence” specifies an exact visibility threshold. Propagation to Earth sets the angular size and transverse scale, but does not by itself narrow the solar optical spectrum or lengthen this longitudinal scale.

Resolve the weak broadband radiation into narrow frequency intervals whose relative phases are uncorrelated. The transition probabilities, rather than the coherent amplitudes, can then be added after ensemble or time averaging. For the x-polarized field, the energy density in one interval is $\rho (\omega )d\omega ={ϵ}_0{\mathcal{E}}_{\omega }^2/2$, so its squared coupling is $|d_{21}{|}^2{\mathcal{E}}_{\omega }^2/{\hslash }^2$. Inserting this into the weak rotating-wave result givesFor times long compared with the optical coherence time of the field, the narrow resonant kernel samples a nearly constant $\rho ({\omega }_0)$. Provided the resonance is well away from the zero-frequency boundary, extend the Rabi detuning integral to the full real line. Its value is $2\pi t$, yieldingThis is the stimulated absorption rate for the specified polarization and spectral radiation energy density per angular-frequency interval. The averaging window should also satisfy $Wt\ll 1$ while the lower state is treated as undepleted. A deterministic phase-locked broadband pulse would not in general permit adding these probabilities without its interference terms.

For isotropic unpolarized radiation with total spectral radiation energy density ${\rho }_{\mathrm{tot}}$, orientational averaging instead gives $W=\pi |{\mathbf{d}}_{21}{|}^2{\rho }_{\mathrm{tot}}({\omega }_0)/(3{ϵ}_0{\hslash }^2)$. The factor one third is not inserted into the original x-polarized formula. If ordinary frequency is used, ${\rho }_{\nu }=2\pi {\rho }_{\omega }$ and the isotropic coefficient becomes $|{\mathbf{d}}_{21}{|}^2/(6{ϵ}_0{\hslash }^2)$ multiplying ${\rho }_{\nu }({\nu }_0)$. The spectral convention and polarization must be stated together to avoid spurious factors of three or $2\pi$.