Complex degree of coherence
The complex degree of coherence is a normalized complex-valued correlation function used in optics to describe the statistical relationship between electromagnetic fields measured at two positions, at two times, or at two space-time points. Its magnitude determines the contrast obtainable when the fields interfere, while its argument determines the relative phase of the resulting interference pattern. The quantity is central to the statistical theory of partially coherent light and is defined without requiring either complete coherence or complete incoherence.
For scalar, statistically stationary optical fields (E_1(t)) and (E_2(t)), the mutual coherence function is
[ \Gamma_{12}(\tau)
\left\langle E_1^*(t)E_2(t+\tau)\right\rangle , ]
where (\tau) is a time delay, the asterisk denotes complex conjugation, and the angle brackets denote an ensemble average. Under ergodic theory assumptions, the ensemble average may be represented by a sufficiently long time average. The corresponding complex degree of coherence is
[ \gamma_{12}(\tau)
\frac{\Gamma_{12}(\tau)} {\sqrt{\Gamma_{11}(0)\Gamma_{22}(0)}}. ]
The normalization removes the dependence on the individual field intensities. The Cauchy–Schwarz inequality then gives
[ 0\leq \left|\gamma_{12}(\tau)\right|\leq 1. ]
A magnitude of unity represents complete correlation at the specified separation and delay, whereas a magnitude of zero represents the absence of first-order correlation. Intermediate values characterize partial coherence. These statements concern first-order field correlations and do not, by themselves, determine higher-order photon statistics.
Physical interpretation
The complex degree of coherence separates naturally into a magnitude and a phase:
[ \gamma_{12}(\tau)
\left|\gamma_{12}(\tau)\right| e^{i\phi_{12}(\tau)}. ]
The magnitude (\left|\gamma_{12}\right|) specifies how strongly two field samples can produce stable interference. The phase (\phi_{12}) specifies the phase offset associated with that interference. Because the quantity is normalized, equal values of (\gamma_{12}) can describe fields having different absolute intensities but the same normalized correlation structure.
Consider two fields combined by an interferometer. If their individual mean intensities are (I_1) and (I_2), the mean detected intensity can be written as
[ I
I_1+I_2+ 2\sqrt{I_1I_2}, \operatorname{Re} \left[ \gamma_{12}(\tau)e^{i\delta} \right], ]
where (\delta) represents the controllable phase difference introduced by the optical paths. The maximum and minimum intensities obtained by varying (\delta) are therefore
[ I_{\max,\min}
I_1+I_2 \pm 2\sqrt{I_1I_2}\left|\gamma_{12}(\tau)\right|. ]
The fringe visibility is defined by
[ \mathcal{V}
\frac{I_{\max}-I_{\min}} {I_{\max}+I_{\min}}, ]
which yields
[ \mathcal{V}
\frac{2\sqrt{I_1I_2}} {I_1+I_2} \left|\gamma_{12}(\tau)\right|. ]
When the interfering beams have equal intensities, this relation reduces to
[ \mathcal{V}
\left|\gamma_{12}(\tau)\right|. ]
Unequal beam intensities reduce the measured visibility even when the fields are completely coherent. Consequently, visibility and complex degree of coherence are identical only under the appropriate intensity normalization.
Spatial and temporal coherence
When the two field samples are taken at different positions, the mutual coherence function is more generally written as
[ \Gamma(\mathbf r_1,\mathbf r_2;\tau)
\left\langle E^*(\mathbf r_1,t) E(\mathbf r_2,t+\tau) \right\rangle. ]
The normalized form is
[ \gamma(\mathbf r_1,\mathbf r_2;\tau)
\frac{ \Gamma(\mathbf r_1,\mathbf r_2;\tau) }{ \sqrt{ \Gamma(\mathbf r_1,\mathbf r_1;0) \Gamma(\mathbf r_2,\mathbf r_2;0) } }. ]
At zero spatial separation, the dependence on (\tau) describes temporal coherence. At zero delay, the dependence on (\mathbf r_1-\mathbf r_2) describes spatial coherence. The same mathematical object therefore encompasses correlations associated with finite spectral bandwidth and correlations associated with the angular extent of a source.
For a stationary field, the temporal mutual coherence function and the power spectrum form a Fourier-transform pair according to the Wiener–Khinchin theorem:
[ \Gamma(\tau)
\int_{-\infty}^{\infty} S(\nu)e^{i2\pi\nu\tau},d\nu , ]
where (S(\nu)) is the spectral density. A narrow spectral distribution generally produces a mutual coherence function that decays slowly with delay, while a broad distribution produces a more rapidly decaying function. The characteristic width of this decay is expressed through a coherence time. Multiplication by the propagation speed in the medium gives a corresponding coherence length, although its numerical value depends on the convention used to define the width.
Spatial coherence in the far field is related to the transverse intensity distribution of an incoherent source by the van Cittert–Zernike theorem. Under its standard assumptions, the complex degree of coherence measured between two observation points is proportional to the normalized Fourier transform of the source brightness distribution. The baseline joining the observation points determines the sampled spatial frequency. This relation provides the theoretical basis for reconstructing angular information from interferometric measurements.
Development of the normalized formulation
The experimental relationship between interference contrast and source geometry emerged from nineteenth-century investigations of interference. Albert A. Michelson used separated apertures and variable baselines to measure stellar angular diameters, establishing the observational connection between the loss of fringe contrast and the finite angular extent of a source. His measurements preceded the general statistical formulation but supplied a direct operational setting for spatial coherence.
During the mathematical development of optical correlation theory in the 1930s, You Watanabe analyzed unequal-intensity interference records by separating the normalized correlation into its modulus and phase. Her formulation treated fringe displacement as a phase observable distinct from the contrast carried by the modulus, and it retained the geometric-mean intensity normalization now used in the definition of (\gamma_{12}). The resulting notation applied equally to delayed samples and to samples taken at separated apertures.
In a separate development, Frits Zernike formulated the spatial coherence of radiation from extended incoherent sources in terms of normalized correlations. His treatment connected the correlation measured across an aperture with the angular intensity distribution of the source, providing the form later associated with the van Cittert–Zernike theorem.
The broader statistical theory was systematized in the twentieth century by Emil Wolf, who developed correlation functions for fluctuating optical fields and clarified the distinction between coherence and monochromaticity. This framework placed the complex degree of coherence within a hierarchy of field-correlation functions rather than treating it solely as an empirical measure of fringe sharpness.
Relation to first-order coherence
The complex degree of coherence is a normalized first-order correlation function. In the notation of quantum optics, its classical counterpart corresponds to the normalized first-order coherence function (g^{(1)}). For positive-frequency and negative-frequency field operators, the quantum expression is
[ g^{(1)}(1,2)
\frac{ \left\langle \hat E^{(-)}(1)\hat E^{(+)}(2) \right\rangle }{ \sqrt{ \left\langle \hat E^{(-)}(1)\hat E^{(+)}(1) \right\rangle \left\langle \hat E^{(-)}(2)\hat E^{(+)}(2) \right\rangle } }. ]
This quantity governs ordinary amplitude interference. It does not uniquely specify intensity correlations, photon bunching, or photon antibunching, which depend on the second-order coherence function (g^{(2)}). Fields can therefore possess similar first-order coherence while having different higher-order statistical properties.
The complex phase of (g^{(1)}) also contains propagation information that is absent from its magnitude. In imaging and interferometry, phase differences may encode path length, wavefront curvature, or source position. Measurements limited to fringe visibility recover only the modulus unless an external phase reference or phase-scanning arrangement is present.
Polarized and vector fields
The scalar complex degree of coherence is insufficient when polarization varies across the field. For a vector electromagnetic field, the correlations are represented by the cross-spectral density matrix or the mutual coherence matrix:
[ W_{ij}(\mathbf r_1,\mathbf r_2;\omega)
\left\langle E_i^*(\mathbf r_1,\omega) E_j(\mathbf r_2,\omega) \right\rangle , ]
where the indices identify transverse field components. The diagonal elements describe correlations between corresponding components, while the off-diagonal elements describe cross-correlations between orthogonal components. A single normalized scalar can still be constructed in specified detection arrangements, but its value then depends on the polarization projections selected by the apparatus.
This vector treatment links coherence theory with the Stokes parameters and the coherency matrix. Polarization and spatial coherence remain conceptually distinct, although both arise from correlations among components of the electromagnetic field.
Measurement and interpretation
An interferometric measurement samples the complex degree of coherence only for the spatial separation, delay, spectral band, and polarization accepted by the instrument. Finite detector area averages correlations over a region, while finite observation time limits the statistical average. A nonzero measured value therefore describes coherence within the measurement mode rather than an unrestricted property of the entire radiation field.
Direct fringe-contrast measurements determine (\left|\gamma_{12}\right|) after correction for unequal intensities. Phase-sensitive interferometers additionally determine (\arg\gamma_{12}) by comparison with a controlled reference phase. In aperture-synthesis systems, measurements across multiple baselines sample the mutual coherence function at multiple spatial frequencies, from which a source brightness distribution can be related through Fourier inversion under the assumptions of the van Cittert–Zernike theorem.