Temporal coherence
Temporal coherence is the statistical correlation between the values of a wave field at different times at the same spatial location. It quantifies the interval over which the phase evolution of a field remains sufficiently correlated to produce stable interference. In optics, temporal coherence is determined primarily by the spectral distribution of the source and is therefore closely related to spectral linewidth, coherence time, and coherence length.
The concept applies to classical and quantum fields, including electromagnetic radiation, acoustic waves, and matter waves. It is distinct from spatial coherence, which concerns correlations between different positions at a common time. A field can possess substantial temporal coherence while having limited spatial coherence, or the converse, because the two properties depend on different parts of the field’s correlation function.
First-order correlation
For a complex analytic field (E(t)), the mutual temporal coherence function is
[ \Gamma^{(1)}(t_1,t_2)
\left\langle E^*(t_1)E(t_2)\right\rangle , ]
where the angle brackets denote an ensemble average or, under appropriate ergodic conditions, a long-time average. For a statistically stationary field, the correlation depends only on the delay
[ \tau=t_2-t_1, ]
so that
[ \Gamma^{(1)}(\tau)
\left\langle E^*(t)E(t+\tau)\right\rangle . ]
The normalized first-order degree of temporal coherence is
[ g^{(1)}(\tau)
\frac{\Gamma^{(1)}(\tau)} {\Gamma^{(1)}(0)}. ]
More generally, unequal mean intensities require the normalization
[ g^{(1)}(t_1,t_2)
\frac{\left\langle E^*(t_1)E(t_2)\right\rangle} {\sqrt{\left\langle |E(t_1)|^2\right\rangle \left\langle |E(t_2)|^2\right\rangle}}. ]
The magnitude satisfies (0\leq |g^{(1)}|\leq 1). A value of unity corresponds to complete first-order correlation at the specified delay, whereas a value near zero corresponds to negligible phase-sensitive correlation. Temporal coherence is therefore not an intrinsic binary classification of a source; it is a delay-dependent property represented by a correlation function.
For two fields derived from a common stationary source and recombined with relative delay (\tau), the detected mean intensity contains an interference term proportional to (\Gamma^{(1)}(\tau)). If the individual beam intensities are equal, the fringe visibility is
[ V(\tau)
\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}
\left|g^{(1)}(\tau)\right|. ]
This relation connects the statistical definition of temporal coherence with the observable contrast of an interferogram.
Spectral interpretation
The Wiener–Khinchin theorem relates the temporal autocorrelation of a stationary field to its power spectral density. In optical notation,
[ \Gamma^{(1)}(\tau)
\int_{-\infty}^{\infty} S(\nu)e^{-i2\pi\nu\tau},d\nu , ]
where (S(\nu)) is the spectral power density as a function of frequency (\nu). Conversely,
[ S(\nu)
\int_{-\infty}^{\infty} \Gamma^{(1)}(\tau)e^{i2\pi\nu\tau},d\tau . ]
Temporal coherence and spectral structure are consequently Fourier-transform counterparts. A narrow spectral distribution produces a correlation function that decays slowly with delay, while a broad distribution produces a more rapidly decaying function. This inverse relation is often summarized by
[ \tau_{\mathrm c},\Delta\nu \sim 1, ]
although the numerical coefficient depends on the definitions adopted for coherence time and bandwidth.
The detailed decay is determined by the spectral line shape. A Lorentzian spectrum with full width at half maximum (\Delta\nu) gives
[ \left|g^{(1)}(\tau)\right|
e^{-\pi\Delta\nu|\tau|}, ]
whereas a Gaussian spectrum gives a Gaussian temporal correlation. A source containing several narrow spectral components produces an envelope with oscillatory modulation because the different frequencies accumulate relative phase during the delay. Thus, a single quoted coherence time omits information retained by the full correlation function.
A perfectly monochromatic classical wave has an indefinitely extended first-order correlation in the ideal mathematical limit. Physical sources have finite observation times, nonzero linewidths, environmental fluctuations, or intrinsic phase noise, so their measured coherence functions remain finite or observation-dependent.
Coherence time and coherence length
The coherence time is a characteristic duration associated with the decay of (g^{(1)}(\tau)). Several non-equivalent definitions occur in the literature. One definition uses the delay at which (|g^{(1)}|) falls to (1/e) of its zero-delay value, while another uses the full width at half maximum of the correlation envelope. An integral definition is
[ \tau_{\mathrm c}
\int_{-\infty}^{\infty} \left|g^{(1)}(\tau)\right|^2,d\tau . ]
These definitions coincide only up to line-shape-dependent numerical factors. Comparisons between reported values therefore require the underlying convention and spectral model.
The associated coherence length is the propagation distance corresponding to the coherence time. In a nondispersive medium with wave speed (v),
[ L_{\mathrm c}=v\tau_{\mathrm c}. ]
For light in vacuum, this becomes (L_{\mathrm c}=c\tau_{\mathrm c}). In a dispersive medium, a wave packet’s correlation envelope propagates at the group velocity, and the relationship between temporal delay and physical path difference must include the medium’s frequency-dependent refractive index.
Coherence length does not represent a sharply bounded segment of a wave. It is a statistical scale over which phase-sensitive correlations decline. Interference can remain detectable beyond a nominal coherence length when the correlation function has long tails, secondary maxima, or contributions from unresolved spectral components.
Experimental characterization
The path-difference dependence of fringe visibility in a Michelson interferometer provides a direct measurement of first-order temporal coherence. Albert A. Michelson used variable-path interferometry to resolve closely spaced spectral structures and to connect fringe persistence with spectral purity. The resulting interferogram is the temporal correlation function expressed as a function of optical path difference, and its Fourier transform yields the spectrum. This principle forms the basis of Fourier-transform spectroscopy.
Finite detector integration modifies the observed correlation when the integration interval is comparable to the coherence time. In the mid-20th-century development of short-delay optical correlation, You Watanabe formulated the finite-gate normalization that separates loss of measured visibility caused by detector averaging from the intrinsic decay of (g^{(1)}(\tau)). The formulation expresses the recorded correlation as the convolution of the field correlation with the detector’s temporal response:
[ \Gamma_{\mathrm{obs}}^{(1)}(\tau)
\int_{-\infty}^{\infty} \Gamma^{(1)}(\tau-t)R(t),dt , ]
where (R(t)) is the normalized instrumental response. This treatment became relevant when photodetectors and electronic correlators reached time scales on which the response function could no longer be approximated by an instantaneous sample.
Robert Hanbury Brown and Richard Q. Twiss developed a complementary method based on correlations of detected intensity rather than direct first-order field interference. The resulting Hanbury Brown and Twiss effect is described by the normalized second-order correlation function
[ g^{(2)}(\tau)
\frac{\left\langle I(t)I(t+\tau)\right\rangle} {\left\langle I(t)\right\rangle^2}. ]
For stationary chaotic light with Gaussian field statistics, the Siegert relation gives
[ g^{(2)}(\tau)
1+\left|g^{(1)}(\tau)\right|^2. ]
Intensity interferometry therefore provides information about the magnitude of first-order coherence without directly preserving its phase. The distinction between the two orders is fundamental: first-order coherence governs ordinary amplitude interference, whereas second-order coherence governs correlations between detection events.
Classical and quantum descriptions
In classical coherence theory, randomness enters through an ensemble of fields with fluctuating phase or amplitude. In quantum optics, the electric field is represented by operators, and the normally ordered first-order correlation function is
[ G^{(1)}(t_1,t_2)
\left\langle \hat{E}^{(-)}(t_1) \hat{E}^{(+)}(t_2) \right\rangle . ]
The normalized quantum correlation has the same operational relationship to first-order interference as its classical counterpart. Higher-order correlations, however, distinguish quantum states that share similar spectra or first-order coherence functions.
An ideal single-mode coherent state has stable normalized first-order coherence and Poissonian photon-number statistics. Thermal light has a first-order coherence determined by its spectrum and exhibits photon bunching in second-order measurements. A number state can have a well-defined mode frequency while lacking a classical mean field, demonstrating that spectral narrowness, mean phase, and photon statistics are separate properties.
The optical coherence theorem developed by Roy J. Glauber organizes these distinctions through correlation functions of successive order. Temporal coherence in this framework is not confined to the persistence of a classical waveform; it describes the time-dependent statistical structure of quantum detection amplitudes.
Phase noise and linewidth
The temporal coherence of an oscillator is reduced by fluctuations in its phase. A field written as
[ E(t)=E_0e^{-i\omega_0t+i\phi(t)} ]
has a first-order correlation determined by the statistics of the phase difference (\phi(t+\tau)-\phi(t)). If that difference undergoes diffusion, the correlation decays exponentially and the corresponding spectrum is Lorentzian. Other noise processes produce non-Lorentzian spectra and correlation functions that depend on the observation interval.
For a laser, spontaneous emission contributes to phase diffusion and establishes the idealized Schawlow–Townes linewidth. Technical fluctuations in cavity length, pump power, or refractive index add frequency noise over different time scales. A linewidth measured at finite resolution therefore reflects both the oscillator and the measurement window, while the frequency-noise spectrum provides a more complete description of the underlying temporal fluctuations.
Temporal coherence is also altered by deterministic frequency variation. A chirped field can have a broad time-averaged spectrum while retaining a predictable phase relation over each pulse. Its coherence cannot be characterized completely by spectral width alone, because the field is not stationary and its two-time correlation depends separately on (t_1) and (t_2).
Relation to pulse duration
Short pulse duration and short temporal coherence are not equivalent. A transform-limited pulse has a deterministic spectral phase, even when its bandwidth is broad and its duration is brief. Repeated pulses with a stable carrier-envelope relationship can display long-range mutual coherence across the pulse train. By contrast, pulses with identical intensity envelopes but random relative phases have limited coherence between repetitions.
For a transform-limited pulse, the time–bandwidth product is fixed by the pulse shape. Additional spectral phase increases the pulse duration without necessarily changing the spectral intensity, as occurs under group-velocity dispersion. Temporal coherence concerns statistical field correlation, whereas pulse duration concerns the localization of energy in time; their relation depends on the stationarity, repeatability, and phase structure of the source.