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If phase-randomizing collisions form a Poisson process of mean interval ${\tau }_c$, field correlation survives only when no such collision occurs during the delay. Thus its magnitude decays as $e^{-|\tau |/{\tau }_c}$. The Wiener-Khinchin theorem gives the normalized Lorentzian line shapeIts full width at half maximum is $2/{\tau }_c$ in angular frequency or $1/(\pi {\tau }_c)$ in ordinary frequency. A population lifetime is a different time constant: lifetime-limited amplitude decay at half the population rate introduces the familiar factor of two. The definition of the supplied lifetime must therefore accompany any numerical linewidth.

Temporal coherence measures correlation between an optical field and a delayed copy. For a stationary scalar complex field with carrier convention $e^{-i\omega t}$,The power spectrum $S(\omega )\ge 0$ and the field autocorrelation form a Fourier transform pair, the Wiener-Khinchin theorem:A Michelson interferometer supplies delay $\tau =\Delta L/c$. With equal arm intensities and matched polarization, its interferometric visibility is $|g^{(1)}(\tau )|$; unequal intensities or mode mismatch lower visibility independently of temporal coherence. The relevant spectrum is the optical field spectrum, not the low-frequency spectrum of detector intensity fluctuations.

For a single Lorentzian optical line of angular-frequency full width at half maximum $\Delta \omega$, use the narrow-line detuning modelSuch a model describes a phase-diffusing monochromatic source. An $e^{-1}$ correlation time is $2/\Delta \omega$, and its coherence length is $2c/\Delta \omega$. The Lorentzian extension over all detunings approximates a narrow optical line with $\Delta \omega \ll {\omega }_0$.

For two equal, mutually incoherent ideal monochromatic spectral lines,Here $\bar{\omega }=({\omega }_1+{\omega }_2)/2$ and $\Delta {\omega }_{12}={\omega }_2-{\omega }_1$. Visibility has zeros and revivals, so the overall spectral span alone does not determine a unique decay time. Giving each line a common finite Lorentzian linewidth multiplies this beating correlation by the corresponding exponential envelope.

These two examples distinguish irreversible envelope decay from cancellation and revival of discrete spectral components. The line profiles are explicit models, not claimed measurements of a particular source.

A non-polarizing beam splitter divides the incident spectrum into a fixed arm and an arm with a translating mirror. Recombining their returned fields gives a detector signal as a function of mirror position $x$. The round-trip optical path difference is $D=2x$ relative to the equal-arm position, so the delay is $\tau =2x/c$ in air approximated as vacuum. Record both detector signal and calibrated position, subtract the constant background, and take a Fourier transform of the interferogram.

For matched arm polarizations and balanced throughput, write the ideal signal aswhere $S(\omega )$ is the incident spectral power density and $C$ is a constant set by the beam splitter and detector. This is the cosine-transform form of Fourier transform spectroscopy, or equivalently the Wiener-Khinchin theorem applied to delayed self-interference.

The delay-range resolution in Fourier transform spectroscopy follows because recording a finite delay range multiplies the correlation by a window. Its transform broadens each spectral line by an instrumental width of order the reciprocal delay range. A mirror travel range $L$ creates a delay range of order $2L/c$, soAt fixed wavelength, absolute wavelength resolution therefore improves inversely with travel range. The numerical coefficient depends on whether $L$ is a one-sided or total travel, on the window or apodization, and on whether resolution means first zero, half width or another criterion. The proportionality is the requested convention-independent result. Position sampling must also be fine enough to sample the fastest fringe variation without aliasing.

A line with lifetime broadening has a Lorentzian line shape. Write its half-width at half-maximum in angular-frequency units as $\kappa >0$. The Wiener-Khinchin theorem, or the Fourier transform of the Lorentzian, givesTherefore in vacuumThe sketch is a cosine fringe pattern about baseline $I_0$, bounded by $I_0[1\pm e^{-\sqrt{2}\kappa |y|/c}]$. Its central value is $2I_0$, its spatial interference fringe spacing is $\lambda /\sqrt{2}$, and its visibility decays on length $c/(\sqrt{2}\kappa )$. For an isolated transition whose population lifetime is $T_1$ and which has no extra dephasing, $\kappa =1/(2T_1)$, so the field coherence time is $2T_1$, not $T_1$. Introducing $\kappa$ avoids confusing population, wave amplitude and spectral-width conventions when no numerical linewidth is given.

A power spectrum resolves the second-moment strength of a stationary process or random field by temporal frequency or spatial wavenumber. For a real, mean-zero stationary process with integrable autocovariance function $C(\tau )=\mathbb{E}[X_{t+\tau }{X}_t]$, use the angular-frequency conventionThis form of the Wiener-Khinchin theorem makes the interpretation concrete: integrating spectral density over all frequencies recovers the variance. A nonzero mean contributes a spectral atom at zero if one transforms the uncentered correlation instead of the autocovariance function. Some stationary processes have discrete spectral lines, so an ordinary integrable spectral density need not exist.

For an Ornstein-Uhlenbeck process with $C(\tau )=v{e}^{-a|\tau |}$, $a,v>0$, the Fourier transform is $S(\omega )=2av/(a^2+{\omega }^2)$. Its area is $2\pi v$ as required. Small $a$ concentrates power near zero and corresponds to slowly decaying fluctuations. Conventions using cyclic frequency rather than angular frequency move the factors of $2\pi$; stating the convention prevents a normalization error when comparing spectra.