Optical coherence is the degree of correlation between the values of a light field at different positions or times. In classical optics, it describes how reliably the field maintains relationships of amplitude and phase, determining the visibility of interference patterns. In quantum theory, coherence is characterized through a hierarchy of field-correlation functions that also describe correlations between photon-detection events. Coherence is therefore not simply a synonym for monochromatic light or a property that a source either possesses or lacks: it depends on the points, delays, and statistical measurements being considered. (doi.org)
Field correlations and interference
A classical optical field can be represented by a complex amplitude (E(\mathbf r,t)), whose magnitude and phase describe the oscillating electric field. For a scalar field, or one selected polarization component, the mutual coherence function is
[ \Gamma(1,2)=\left\langle E^*(\mathbf r_1,t_1) E(\mathbf r_2,t_2)\right\rangle. ]
Here (1) and (2) denote two space–time points, the asterisk denotes complex conjugation, and the brackets denote a statistical average. An experimental time average can represent this ensemble average when the required ergodicity conditions hold. The normalized complex degree of coherence is
[ g^{(1)}(1,2)= \frac{\Gamma(1,2)} {\sqrt{\langle|E(1)|^2\rangle\langle|E(2)|^2\rangle}}. ]
For nonzero intensities, the Cauchy–Schwarz inequality implies (0\leq |g^{(1)}|\leq1). Unit magnitude means complete first-order coherence between the selected points; zero means no first-order correlation; intermediate values indicate partial coherence. (sites.science.oregonstate.edu)
When two overlapping beams with matched polarization are combined, their mean intensity is
[ I=I_1+I_2+ 2\sqrt{I_1I_2}, \operatorname{Re}!\left[g^{(1)}e^{i\phi}\right], ]
where (\phi) is an adjustable relative phase. Scanning this phase gives the fringe visibility
[ V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}} =\frac{2\sqrt{I_1I_2}}{I_1+I_2}|g^{(1)}|. ]
Thus (V=|g^{(1)}|) only for equal beam intensities and otherwise ideal overlap. Unequal intensities, polarization mismatch, or unresolved spatial variations can reduce measured contrast without demonstrating intrinsic incoherence of the incident field. (web.mit.edu)
Temporal coherence
Temporal coherence compares a field at one position with a delayed version of itself. For a stationary process, the correlation depends on the delay (\tau=t_2-t_1), rather than on the two times separately:
[ g^{(1)}(\tau)= \frac{\langle E^*(t)E(t+\tau)\rangle} {\langle|E(t)|^2\rangle}. ]
A variable-delay interferometer measures this correlation through the changing visibility of its fringes. A broadband source can produce interference near equal optical path lengths, even when its visibility falls rapidly at larger path differences. (ocw.mit.edu)
The Wiener–Khinchin theorem relates the temporal correlation of a stationary field to its power spectrum. With a suitable frequency-sign convention,
[ g^{(1)}(\tau)= \frac{\int S(\nu)e^{i2\pi\nu\tau},d\nu} {\int S(\nu),d\nu}. ]
Consequently, spectral shape matters as well as spectral width: the correlation is a normalized Fourier transform of the spectrum, not merely a number inferred from bandwidth. (atomoptics.uoregon.edu)
A coherence time (\tau_c) characterizes the width or decay of this correlation. Its numerical value depends on whether one uses an amplitude-decay threshold, a full width at half maximum, or an integral definition. Generally,
[ \tau_c\sim\frac{1}{\Delta\nu}. ]
For a Lorentzian spectrum of full width at half maximum (\Delta\nu),
[ |g^{(1)}(\tau)|=e^{-\pi\Delta\nu|\tau|}, ]
so its amplitude (1/e) decay time is (1/(\pi\Delta\nu)). The corresponding vacuum coherence length is (L_c=c\tau_c), where (c) is the speed of light. This is a path-difference scale, not the distance light can travel before it disappears. (courses.ece.ucsb.edu)
These bandwidth relations require attention to the averaging procedure and stationarity assumptions. A reproducible broadband pulse can have definite phase relationships among its spectral components. Broad bandwidth therefore does not, by itself, establish incoherence under every definition; Glauber’s theory explicitly allows coherent fields with arbitrary spectra. (doi.org)
Spatial coherence
Spatial coherence compares optical fields at different positions, commonly across a transverse plane at equal time or with compensated propagation delays. Young’s double-slit experiment samples this correlation: the visibility obtained from two apertures measures their mutual coherence after intensity and polarization effects are accounted for. (ocw.mit.edu)
An extended source with mutually uncorrelated emitting regions generally has limited spatial coherence. Different regions contribute different relative phases at the observation apertures, reducing the averaged interference contrast. Under appropriate far-field and quasi-monochromatic conditions, the van Cittert–Zernike theorem relates the spatial coherence to the Fourier transform of the source’s angular brightness distribution. A smaller apparent source generally produces a larger transverse coherence scale, approximately
[ \ell_{\mathrm{sp}}\sim\frac{\lambda}{\theta}, ]
where (\lambda) is wavelength and (\theta) is angular source size; numerical factors depend on source shape and the chosen criterion. (pas.rochester.edu)
Spatial and temporal coherence are distinct. A broadband source viewed through a sufficiently small aperture can have high spatial coherence but short temporal coherence. Conversely, narrowband illumination from an extended source can have relatively long temporal coherence while remaining poorly correlated across widely separated transverse positions. (ocw.mit.edu)
Higher-order and quantum coherence
First-order coherence determines ordinary interference, but does not completely describe the statistics of light. In quantum mechanics, the field is represented by operators, and photodetection correlations involve normally ordered products of those operators. For a single optical mode, the zero-delay second-order correlation is
[ g^{(2)}(0)= \frac{\langle \hat a^\dagger\hat a^\dagger\hat a\hat a\rangle} {\langle\hat a^\dagger\hat a\rangle^2}
\frac{\langle n(n-1)\rangle}{\langle n\rangle^2}, ]
where (\hat a) annihilates a photon and (n) is photon number. This quantity measures pair-detection statistics rather than the phase correlation measured by an ordinary interferometer. (atomoptics-nas.uoregon.edu)
Important idealized cases include:
- A coherent state has (g^{(2)}(0)=1) and Poisson photon-number statistics.
- Single-mode thermal light has (g^{(2)}(0)=2), exhibiting photon bunching.
- An ideal one-photon number state has (g^{(2)}(0)=0), because it cannot supply two photons for simultaneous detection.
Multimode collection and finite detector resolution can change measured values, particularly for thermal light. Zero-delay (g^{(2)}(0)<1) cannot be explained by a classical fluctuating-intensity model; photon antibunching is more specifically identified through the comparison of zero-delay and delayed correlations. (atomoptics-nas.uoregon.edu)
Coherence at one order does not establish coherence at every order. Fields can share first-order interference properties while having different higher-order detection statistics. Glauber’s hierarchy formalized this distinction between traditional optical coherence and more complete statistical descriptions of the electromagnetic field. (doi.org)
Measurement and applications
Interferometry provides a direct measurement of first-order coherence. A Michelson interferometer varies the delay between two copies of a beam to examine temporal coherence; separated apertures or displaced wavefront copies probe spatial coherence. Intensity-correlation experiments instead measure second- and higher-order properties. (web.mit.edu)
Different applications require different coherence properties:
- Astronomical imaging: correlations measured across separated telescopes constrain the angular brightness distribution of a source through the van Cittert–Zernike relation. (pas.rochester.edu)
- Holography and interference patterning: mutual coherence allows an object or signal field to interfere reproducibly with a reference field. (courses.ece.ucsb.edu)
- Optical coherence tomography: broadband illumination provides a narrow coherence gate for depth discrimination. Short temporal coherence is useful here rather than a defect. Fourier-domain methods recover depth information from spectrally resolved interference; swept-source implementations obtain analogous information by scanning wavelength. (biophotonics.illinois.edu)
- Image formation: spatially coherent imaging propagates and combines field amplitudes, whereas spatially incoherent imaging combines intensities from uncorrelated object points. The corresponding transfer functions differ, so image behavior cannot be predicted from intensity alone without specifying coherence. (ocw.mit.edu)
Historical development
Traditional coherence concepts developed around wave interference and the conditions needed for stable fringes. Statistical descriptions extended this treatment to fluctuating fields and partially coherent illumination. The emergence of the laser and photon-correlation experiments made distinctions between field coherence and photon statistics especially important. (nobelprize.org)
In 1963, Roy J. Glauber published The Quantum Theory of Optical Coherence, introducing a quantum description based on field-correlation functions and successive orders of coherence. He received half of the 2005 Nobel Prize in Physics for this contribution. (doi.org)
References
- The Quantum Theory of Optical Coherencedoi.org
- One Hundred Years of Light Quanta (Nobel Lecture)sites.science.oregonstate.edu
- Optical Interferometryweb.mit.edu
- Lecture 22: Coherent and incoherent imagingocw.mit.edu
- MITOCW: Lecture 23, MIT 2.71 Optics, Spring 2009ocw.mit.edu
- Gaussian Beams and Coherencecourses.ece.ucsb.edu
- Astronomy 203/403: Lecture 16pas.rochester.edu
- Classical and Modern Opticsatomoptics.uoregon.edu
- Quantum and Atom Opticsatomoptics-nas.uoregon.edu
- MITOCW: Quantum Description of Lightocw.mit.edu
- Optical Coherence Imagingbiophotonics.illinois.edu
- The Nobel Prize in Physics 2005: Supplementary Informationnobelprize.org