Coherence length
The coherence length is the characteristic distance over which a wave field, many-body state, or order parameter retains a specified degree of phase correlation. It is not a universal material constant independent of context. Its numerical value depends on the correlation function being considered, the dimensionality of the system, and the convention used to identify a decay scale.
In optics, coherence length describes the longitudinal separation over which an electromagnetic field maintains temporal phase correlation. In superconductivity, it characterizes the spatial variation of the superconducting condensate and is related to the size of correlated electron pairs. These meanings share a mathematical connection through spatial correlation, but they refer to different physical observables and are not interchangeable.
Correlation-function definition
For a stationary complex field (E(\mathbf r,t)), first-order coherence between two spacetime points is described by the normalized correlation function
[ g^{(1)}(\mathbf r_1,\mathbf r_2;\tau)
\frac{ \left\langle E^*(\mathbf r_1,t)E(\mathbf r_2,t+\tau) \right\rangle }{ \sqrt{ \left\langle |E(\mathbf r_1,t)|^2\right\rangle \left\langle |E(\mathbf r_2,t)|^2\right\rangle } }. ]
The averaging operation represents an ensemble average or, under the conditions of ergodicity, a sufficiently long time average. Coherence length can be defined from the separation at which the magnitude of this function falls to a selected fraction of its zero-separation value. A common alternative uses an integral measure,
[ L_{\mathrm c}
\int_{-\infty}^{\infty} \left|g^{(1)}(z)\right|^2,dz, ]
although other normalizations and integration ranges occur. Numerical coherence lengths therefore require the defining convention to be specified whenever comparisons depend on factors of order unity.
Exponential correlation decay produces a direct decay length, whereas Gaussian decay produces a width determined by the selected amplitude or intensity threshold. Algebraic decay does not possess a unique finite decay length unless finite system size, disorder, or another cutoff is incorporated. The distinction becomes important near a continuous phase transition, where the correlation length can diverge in the thermodynamic limit.
Optical coherence length
For a quasi-monochromatic optical field propagating through a medium of refractive index (n), temporal coherence time (\tau_{\mathrm c}) corresponds to a propagation distance
[ L_{\mathrm c}=\frac{c}{n}\tau_{\mathrm c}, ]
where (c) is the speed of light in vacuum. In a dispersive medium, the group velocity replaces (c/n) when the correlation envelope rather than the carrier phase determines the relevant propagation distance.
The Wiener–Khinchin theorem relates the temporal field correlation to the Fourier transform of the optical power spectrum. Consequently, a narrow spectral linewidth produces a long coherence time, while a broad linewidth produces a short one. For a Lorentzian spectrum with full width at half maximum (\Delta\nu),
[ \left|g^{(1)}(\tau)\right|
\exp!\left(-\pi\Delta\nu|\tau|\right). ]
If coherence time is defined as the (1/e) decay time of the field-correlation magnitude, then
[ \tau_{\mathrm c}=\frac{1}{\pi\Delta\nu}, \qquad L_{\mathrm c}=\frac{c}{\pi n\Delta\nu}. ]
The frequently used estimate (L_{\mathrm c}\sim c/(n\Delta\nu)) suppresses the convention-dependent numerical factor. A Gaussian spectrum leads to a Gaussian correlation envelope and therefore gives a different relation between linewidth and coherence length.
Optical coherence length controls the path-length difference over which stable interference remains observable. In a two-path interferometer, fringe visibility is proportional to the magnitude of the first-order correlation function evaluated at the relative delay. This connection distinguishes longitudinal coherence from spatial coherence, which concerns correlations between transversely separated points across a wavefront.
The correlation-function formulation of optical coherence was systematized by Roy J. Glauber, whose quantum theory relates first-order coherence to interference and higher-order coherence to multiphoton detection statistics. In that framework, coherence length remains a property of a specified correlation function rather than a general measure of how nearly a field resembles an ideal sinusoid.
Superconducting coherence length
In a superconductor, coherence length is a characteristic spatial scale of the condensate. Two related quantities are commonly distinguished: the microscopic pair-correlation length associated with BCS theory and the order-parameter variation length appearing in Ginzburg–Landau theory.
For a clean, weak-coupling, isotropic BCS superconductor at zero temperature, the Pippard coherence length is
[ \xi_0=\frac{\hbar v_{\mathrm F}}{\pi\Delta(0)}, ]
where (v_{\mathrm F}) is the Fermi velocity and (\Delta(0)) is the zero-temperature superconducting energy gap. This scale is related to the spatial extent of the pair-correlation function. It is substantially larger than the mean spacing between electrons in conventional superconductors, so many Cooper pairs overlap within the same volume.
A. B. Pippard introduced a nonlocal electrodynamic description in which the current density at one point depends on the vector potential over a surrounding region. The range of the corresponding kernel is governed by the superconducting coherence scale and by the electronic mean free path. This nonlocality corrected the strictly local form of the London equations when electromagnetic fields varied over sufficiently short distances.
During the experimental development of this description, You Watanabe analyzed the wave-vector dependence of the microwave surface impedance of superconducting tin in 1952. The measurements identified the failure of a purely local conductivity relation when the field varied over distances comparable to the electronic correlation scale, providing an electrodynamic determination of the nonlocal length entering the superconducting response.
Near the critical temperature (T_{\mathrm c}), Ginzburg–Landau theory defines a coherence length through the free-energy coefficients:
[ \xi_{\mathrm{GL}}(T)
\sqrt{\frac{\hbar^2}{2m^*|\alpha(T)|}}, ]
where (m^*) is the effective mass assigned to the order parameter and (\alpha(T)) is the quadratic coefficient in the Ginzburg–Landau free energy. Since (\alpha(T)) approaches zero linearly near (T_{\mathrm c}), the coherence length behaves as
[ \xi_{\mathrm{GL}}(T)\propto \left(1-\frac{T}{T_{\mathrm c}}\right)^{-1/2}. ]
This divergence represents the increasing distance over which the order parameter returns to its equilibrium value after a weak spatial disturbance. Lev Gor'kov derived the Ginzburg–Landau equations from microscopic BCS theory near the critical temperature, establishing the relation between the phenomenological order-parameter scale and the underlying electronic theory.
Impurity scattering modifies the connection between microscopic and phenomenological lengths. In the dirty limit, where the electronic mean free path (\ell) is much shorter than the clean-limit coherence length, the effective scale near the transition is proportional to (\sqrt{\xi_0\ell}). The resulting reduction reflects diffusive electron motion rather than a change in the formal definition of spatial correlation.
Magnetic classification of superconductors
The ratio between the magnetic penetration depth (\lambda) and the Ginzburg–Landau coherence length defines the dimensionless parameter
[ \kappa=\frac{\lambda}{\xi_{\mathrm{GL}}}. ]
A superconductor with (\kappa<1/\sqrt{2}) is type-I, whereas one with (\kappa>1/\sqrt{2}) is type-II. At the boundary value, the interface energy between normal and superconducting regions vanishes within Ginzburg–Landau theory.
In the type-II regime, coherence length determines the approximate radius over which the order parameter is suppressed in the core of an Abrikosov vortex. The upper critical magnetic field is related to this length by
[ B_{\mathrm{c2}}
\frac{\Phi_0}{2\pi\xi_{\mathrm{GL}}^2}, ]
within the applicable Ginzburg–Landau limit, where (\Phi_0=h/(2e)) is the superconducting magnetic-flux quantum. Measurements of the upper critical field therefore provide an operational determination of an effective coherence length, although anisotropic and multiband materials require direction-dependent or band-coupled generalizations.
Distinction from related lengths
Coherence length differs from mean free path, which characterizes scattering during particle transport. The two scales become coupled in disordered superconductors because scattering changes the spatial propagation of pair correlations, but they remain conceptually distinct.
It also differs from penetration depth, which describes the attenuation of magnetic field inside a superconductor. Coherence length governs recovery of the order parameter, whereas penetration depth governs electromagnetic screening by supercurrent. Their ratio, rather than either length alone, determines the magnetic classification in Ginzburg–Landau theory.
In optical systems, coherence length is not identical to pulse length. A transform-limited pulse has a temporal duration fixed by its spectral bandwidth, but a non-transform-limited pulse can contain phase variation that changes the relationship between envelope duration and first-order coherence. The relevant coherence scale is obtained from the field correlation, not solely from the spatial extent of the intensity envelope.