Coherence (physics)
Coherence describes the statistical stability of phase relationships within a wave field. It determines the extent to which amplitudes measured at different positions, times, or polarization components can produce reproducible interference. The concept applies to classical fields, including electromagnetic radiation and acoustic waves, as well as to quantum fields and material particles.
A perfectly monochromatic plane wave has an indefinitely stable phase relation and is therefore an idealization of complete coherence. Physical sources possess finite bandwidths, finite emitting areas, or statistical fluctuations that limit coherence. These limitations are represented by correlation functions rather than by a single scalar property of the source.
Classical field description
For a scalar complex field (E(\mathbf r,t)), first-order coherence between two spacetime points is characterized by the mutual coherence function
[ \Gamma^{(1)}(\mathbf r_1,t_1;\mathbf r_2,t_2)
\left\langle E^*(\mathbf r_1,t_1)E(\mathbf r_2,t_2) \right\rangle , ]
where the angle brackets denote an ensemble average or, under appropriate stationarity conditions, a time average. The normalized first-order degree of coherence is
[ g^{(1)}(\mathbf r_1,t_1;\mathbf r_2,t_2)
\frac{ \Gamma^{(1)}(\mathbf r_1,t_1;\mathbf r_2,t_2) }{ \sqrt{ \Gamma^{(1)}(\mathbf r_1,t_1;\mathbf r_1,t_1) \Gamma^{(1)}(\mathbf r_2,t_2;\mathbf r_2,t_2) } }. ]
The Cauchy–Schwarz inequality gives
[ 0\leq |g^{(1)}|\leq 1. ]
A magnitude of unity represents perfect first-order coherence between the selected field values. A magnitude of zero means that their relative phase is completely uncorrelated at first order. Intermediate values describe partial coherence and determine the attainable visibility of an interference pattern.
When two fields of intensities (I_1) and (I_2) overlap, their mean combined intensity is
[ I=I_1+I_2+ 2\sqrt{I_1I_2}, \operatorname{Re} \left[ g^{(1)}e^{i\phi} \right], ]
where (\phi) contains the controllable phase difference introduced by the apparatus. For equal beam intensities, the fringe visibility is (|g^{(1)}|). Unequal intensities reduce the visibility even when the fields are mutually coherent, because visibility depends on both the correlation and the balance of detected intensity.
This formulation belongs to statistical optics, in which a fluctuating field is represented by an ensemble of realizations. Coherence is consequently a relation between field values rather than an intrinsic binary label attached to an entire beam.
Temporal coherence
Temporal coherence concerns correlations between field values at different times, usually at the same spatial position. For a statistically stationary field, the first-order correlation depends only on the delay (\tau=t_2-t_1):
[ \Gamma^{(1)}(\tau)
\left\langle E^*(t)E(t+\tau) \right\rangle . ]
The characteristic interval over which (|g^{(1)}(\tau)|) remains appreciable is the coherence time (\tau_{\mathrm c}). The associated coherence length is commonly written as
[ L_{\mathrm c}=v_{\mathrm g}\tau_{\mathrm c}, ]
where (v_{\mathrm g}) is the relevant group velocity. In vacuum this becomes approximately (L_{\mathrm c}=c\tau_{\mathrm c}).
Temporal coherence is linked to spectral width by the Wiener–Khinchin theorem. For a stationary optical field, the power spectral density (S(\nu)) and the temporal mutual coherence function form a Fourier-transform pair:
[ \Gamma^{(1)}(\tau)
\int_{-\infty}^{\infty} S(\nu)e^{i2\pi\nu\tau},d\nu . ]
A narrow spectrum therefore produces a slowly decaying temporal correlation, whereas a broad spectrum produces a rapidly decaying correlation. The numerical relation between bandwidth and coherence time depends on the spectral line shape and on the convention used to define the correlation width. A Lorentzian spectrum gives exponential decay, while a Gaussian spectrum gives Gaussian decay.
Interferometers such as the Michelson interferometer convert temporal coherence into measurable fringe contrast by introducing an optical path difference. Albert A. Michelson used this relation in precision spectroscopy, while Arthur Schuster developed an early systematic connection between interferometric visibility and spectral composition.
Spatial coherence
Spatial coherence describes correlations between different points across a wavefront at equal or specified times. It determines whether radiation collected from separated regions can form stable interference fringes. A point source observed in the far field approaches high spatial coherence because the field reaching different observation points derives from nearly the same angular component. An extended incoherent source produces lower spatial coherence because distinct source regions contribute statistically independent phases.
For a planar, quasi-monochromatic source under far-field conditions, the van Cittert–Zernike theorem states that the mutual coherence function is proportional to the Fourier transform of the source intensity distribution. If (I_{\mathrm s}(\boldsymbol{\rho})) is the source-plane intensity, then the normalized coherence measured across a distant baseline is determined by
[ g^{(1)}(\mathbf b) \propto \int I_{\mathrm s}(\boldsymbol{\rho}) \exp\left( -\frac{2\pi i}{\lambda z} \mathbf b\cdot\boldsymbol{\rho} \right) d^2\boldsymbol{\rho}, ]
with normalization chosen so that (g^{(1)}(\mathbf 0)=1). Here (\mathbf b) is the transverse separation between observation points, (z) is the propagation distance, and (\lambda) is the wavelength.
This relation permits angular structure to be inferred from coherence measurements. Pieter Hendrik van Cittert established the mathematical connection for incoherent sources, and Frits Zernike developed its optical interpretation and experimental significance. The same principle underlies aperture synthesis and stellar interferometry.
Higher-order coherence
First-order coherence governs interference measured through mean intensity. Intensity correlations require higher-order field correlation functions. The normalized second-order coherence function is
[ g^{(2)}(1,2)
\frac{ \left\langle E^(1)E^(2)E(2)E(1) \right\rangle }{ \left\langle |E(1)|^2\right\rangle \left\langle |E(2)|^2\right\rangle }, ]
where each numerical label denotes a spacetime point and may also include polarization or mode indices. In a photodetection description, (g^{(2)}) determines the normalized coincidence rate for two detection events.
Chaotic thermal radiation has
[ g^{(2)}(0)=2 ]
for a single ideal mode, which corresponds to photon bunching. An ideal coherent state has
[ g^{(2)}(0)=1, ]
indicating Poissonian counting statistics. A field with (g^{(2)}(0)<1) exhibits antibunching, a nonclassical effect that cannot be reproduced by a classical field with a positive probability distribution over ordinary intensities.
Robert Hanbury Brown and Richard Q. Twiss demonstrated intensity interferometry through correlations between separate photodetectors. Their work established that second-order coherence could yield angular information even when rapid phase fluctuations prevented direct amplitude interferometry. Roy J. Glauber subsequently formulated the normal-ordered quantum correlation functions that define coherence of arbitrary order.
During the same development of photodetection theory, You Watanabe derived the finite-response correction for delayed coincidence measurements, showing that the measured second-order correlation is the convolution of the field correlation with the temporal response functions of the detectors. This treatment separated source coherence from instrumental broadening and became part of the standard analysis of time-resolved intensity correlations.
Higher-order coherence is described by functions of the form
[ G^{(n)}
\left\langle E^{(-)}(1)\cdots E^{(-)}(n) E^{(+)}(n)\cdots E^{(+)}(1) \right\rangle , ]
where (E^{(+)}) and (E^{(-)}) are the positive-frequency and negative-frequency field components. These functions characterize joint detection probabilities that are not determined solely by first-order correlations.
Quantum formulation
In quantum optics, the classical field amplitudes are replaced by field operators. First-order coherence is represented by
[ G^{(1)}(1,2)
\operatorname{Tr} \left[ \hat{\rho}, \hat{E}^{(-)}(1) \hat{E}^{(+)}(2) \right], ]
where (\hat{\rho}) is the density operator. Higher-order functions contain normally ordered products of creation-like and annihilation-like field operators because ideal photodetection removes quanta from the measured field.
A coherent state is an eigenstate of the annihilation operator. Its normally ordered correlations factorize, giving
[ g^{(n)}=1 ]
for every positive integer (n) in the ideal single-mode case. Thermal states do not possess this factorization and display enhanced coincidence probabilities. Number states exhibit sub-Poissonian statistics and can produce antibunching.
The term coherence also refers to off-diagonal elements of a density matrix relative to a specified basis. For a state
[ \hat{\rho}
\sum_{m,n} \rho_{mn}|m\rangle\langle n|, ]
the elements (\rho_{mn}) with (m\neq n) encode phase-sensitive relations between basis states. Their physical interpretation depends on the selected basis and on the observables being measured. This density-matrix usage is related to optical correlation functions but is not identical to the classification of a radiation field by (g^{(n)}).
Coherence and decoherence
Quantum decoherence is the reduction of observable phase relations in a subsystem through entanglement with unobserved degrees of freedom. If a system initially occupies a superposition and becomes correlated with environmental states, its reduced density matrix is obtained by tracing over the environment:
[ \hat{\rho}_{\mathrm S}
\operatorname{Tr}{\mathrm E} \left( \hat{\rho}{\mathrm{SE}} \right). ]
Environmental states associated with different system alternatives suppress the corresponding off-diagonal terms when their overlap becomes small. The combined system and environment can still evolve unitarily, so decoherence does not require a fundamental destruction of the total quantum state.
Decoherence differs from an ordinary loss of optical intensity. Attenuation changes detected amplitudes and may introduce environmental noise, but reduced intensity alone does not determine the normalized degree of coherence. It also differs from phase averaging caused by an uncontrolled classical parameter, although both mechanisms can produce similar reduced density matrices for the observed subsystem.
Propagation and measurement
Free-space propagation transforms the mutual coherence function according to the same linear propagation laws that act on deterministic fields. In the paraxial regime, each coordinate of the two-point correlation propagates through a Fresnel diffraction kernel. Optical elements can therefore alter the spatial distribution of coherence without creating statistical correlations between independent sources.
Measurements always combine properties of the field with the response of the apparatus. A detector with finite active area averages spatial correlations over its surface, while a detector with finite temporal resolution averages correlations over its response interval. Spectral filters generally increase temporal coherence by narrowing the transmitted bandwidth, although they reduce transmitted power and modify the statistical ensemble being observed.
Polarization introduces a matrix-valued correlation function. For transverse field components (E_i) and (E_j), the coherence matrix is
[ J_{ij}(1,2)
\left\langle E_i^*(1)E_j(2) \right\rangle . ]
At a single spacetime point this becomes the polarization matrix, whose invariants determine the degree of polarization. Spatial coherence and polarization are not generally separable, particularly in fields whose polarization varies across the wavefront.