Coherent vs incoherent light

Light is coherent when the phase of its field at one time or place predicts the phase at another, and incoherent when the phase changes randomly, faster than a detector can follow and differently from point to point. The distinction matters because waves add as fields, so two overlapping beams give an intensity $I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi$. The cross term, the interference, survives only if the relative phase $\varphi$ stays fixed during the measurement; for incoherent light it averages to zero and the intensities simply add. Coherence is a matter of degree, and real sources are partially coherent. It has two independent parts: how long the phase stays predictable (temporal coherence) and over what distance across the beam (spatial coherence).

A laser is coherent because stimulated emission adds light in phase with the field already in the cavity, so the output builds up in one or a few cavity modes. A lamp or an LED emits by spontaneous emission from many independent atoms or carriers, each starting a wave of random phase.

Temporal coherence

Temporal coherence is set by the spectrum. A source of spectral width $\mathrm{\Delta }\nu$ has a coherence time ${\tau }_c\approx 1\mathrm{/}\mathrm{\Delta }\nu$ and a coherence length ${\mathrm{\ell }}_c=c{\tau }_c={\lambda }^2\mathrm{/}\mathrm{\Delta }\lambda$ in vacuum, the largest path difference over which its beams still give clear fringes:

The table uses the common convention ${\tau }_c=1\mathrm{/}\mathrm{\Delta }\nu$. Mandel's definition gives $1\mathrm{/}\pi$ of these values for the Lorentzian line of a single-frequency laser, 95 m for the DFB laser; the coherence length entry and the coherence length calculator give both. A laser running on several longitudinal modes behaves differently: its modes beat, so the fringe visibility falls within the coherence length of the envelope and then recovers at path differences of $2n_{g}L$, one optical round trip of the cavity.

Spatial coherence

Spatial coherence is set by the angular size of the source seen from the point of observation. By the van Cittert-Zernike theorem, an incoherent circular source of uniform brightness that subtends an angle $\theta$ produces a field whose correlation first vanishes between points

$$d_0=\frac{1.22\lambda }{\theta }$$

apart. The Sun subtends 9.25 mrad, so sunlight at 550 nm is coherent over about 70 µm; a 1 mm source seen from 1 m gives 0.67 mm. A laser running in a single transverse mode, a Gaussian beam, is coherent across its whole width, which is why it can be focused to a diffraction-limited spot or expanded into a beam that stays parallel. An extended source can be made spatially coherent only by selecting a small part of it with a pinhole or a single-mode fiber, at the cost of most of its power: passive optics cannot reduce étendue or raise radiance. A 1 mW helium-neon laser has over a hundred times the radiance of the Sun's surface.

What coherence changes

In a coherent imaging system the point-spread function acts on the field, so light from neighboring object points interferes in the image; in an incoherent system the intensity point-spread functions simply add. The incoherent system therefore passes spatial frequencies up to 2NA/$\lambda$, twice the coherent cutoff of NA/$\lambda$; this is the limit the modulation transfer function entry uses, $1\mathrm{/}(\lambda N)$ at f-number $N$. Coherent illumination adds ringing at edges and speckle from dust and roughness, and microscopes use partially coherent light, set by the condenser aperture, as a compromise between contrast and resolution.

Coherence in optical communication

A coherent receiver mixes the signal with a local oscillator, and its digital carrier recovery tracks the phase over a block of symbols. What matters there is the combined linewidth of transmitter and local oscillator multiplied by the symbol duration; the length of the link does not enter. For 16QAM with feedforward carrier recovery, a product of about ${10}^{-4}$ costs about 1 dB (Seimetz): a combined linewidth of 3.2 MHz at 32 GBd and 6.4 MHz at 64 GBd. External-cavity tunable lasers of about 100 kHz leave a wide margin for 16QAM and room for denser constellations.

Intensity-modulated links meet coherence in two other ways. Double reflections make a delayed copy of the signal interfere with it, and the beat converts the laser's phase noise into intensity noise, spread over a band of the order of the linewidth once the delay exceeds the coherence time. A laser running on several longitudinal modes adds mode partition noise once fiber dispersion separates its modes in time.

Low coherence as a tool

Short coherence locates a reflection in depth. Optical coherence tomography and white-light interferometry see fringes only where the two arms match within the coherence length, so the source bandwidth sets the depth resolution, $(2\ln 2\mathrm{/}\pi ){\lambda }_0^2\mathrm{/}\mathrm{\Delta }\lambda$ in air for a Gaussian spectrum. A source 50 nm wide at 840 nm gives 6.2 µm; micrometer resolution needs a few hundred nanometers of bandwidth, 1.1 µm for 260 nm at 800 nm. Broadband sources also keep speckle low in illumination and projection.

Partial coherence

The complex degree of coherence is the normalized correlation of the field at two points and times,

$${\gamma }_{12}(\tau )=\frac{⟨E_1^{\ast }(t)E_2(t+\tau )⟩}{\sqrt{⟨\mathrm{\mid }{E}_1{\mathrm{\mid }}^2⟩⟨\mathrm{\mid }{E}_2{\mathrm{\mid }}^2⟩}},$$

whose magnitude runs from 1 for fully coherent light to 0 for incoherent light. Two beams of intensities $I_1$ and $I_2$ give fringes of visibility $V=\mathrm{\mid }{\gamma }_{12}\mathrm{\mid }2\sqrt{I_1{I}_2}\mathrm{/}(I_1+I_2)$, so with equal intensities the visibility measures $\mathrm{\mid }{\gamma }_{12}\mathrm{\mid }$ directly. By the Wiener-Khinchin theorem, $\gamma (\tau )$ at a single point is the Fourier transform of the normalized power spectrum, which is how the spectrum fixes the coherence time and why a Fourier-transform spectrometer can recover the spectrum from the interferogram recorded against delay.

Reducing speckle

Coherent light scattered by a rough surface forms speckle of contrast 1, and averaging $N$ independent speckle patterns lowers the contrast to $1\mathrm{/}\sqrt{N}$. The independent patterns come from lowering the spatial coherence with a moving diffuser or a vibrating multimode fiber, from a broader spectrum, or from two orthogonal polarizations, which on their own give $1\mathrm{/}\sqrt{2}$, or 0.71.

Classical and quantum coherence

Everything above concerns first-order coherence, the correlation of the field. Second-order coherence, the correlation of the intensity, is measured by $g^{(2)}(0)$ and describes the photon statistics: 1 for a laser, 2 for thermal light and 0 for a single-photon source. The two are independent. A narrow filter gives thermal light a long coherence time while its photon statistics stay thermal, and attenuating a laser to a fraction of a photon per pulse leaves both its first-order coherence and its Poisson statistics unchanged. Nonclassical light, such as single photons and squeezed light, is told apart by its intensity correlations and noise; its first-order coherence can be high or low.

Common questions

Is laser light always coherent?

Its spatial coherence is high when it runs in one transverse mode, but its temporal coherence depends on the linewidth: about 300 m of path difference for a DFB laser of 1 MHz linewidth, and under a millimeter for a Fabry-Perot laser diode whose modes spread over 3 nm, with revivals at multiples of the cavity round trip.

Is sunlight coherent?

Partially. Its broad spectrum limits the temporal coherence to about 1 µm of path difference, and the Sun's angular size limits the spatial coherence to about 70 µm across the beam at 550 nm, so Young's fringes in sunlight need pinholes closer together than a few tens of micrometers.

Why does laser light look grainy on a wall when LED light does not?

The wall's roughness scatters each source into many waves with random path differences. Laser light, coherent in time and space, interferes into fully developed speckle with a contrast of 1. An LED's light is spatially incoherent and tens of nanometers wide, so its many independent speckle patterns average to a nearly uniform intensity.

References: L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 12; M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 10; M. Seimetz, "Laser linewidth limitations for optical systems with high-order modulation employing feed forward digital carrier phase estimation," Optical Fiber Communication Conference, OTuM2 (2008).