Coherence

Coherence is a central concept in optics, describing how much different values of the complex optical field (amplitude and phase) are correlated in space and/or time. A light field is called coherent when there is a stable, predictable phase relationship — statistically, a high first-order field correlation — between the electric field at different locations or at different times.

Partial coherence means the correlation is less than perfect. Several standard metrics quantify this, as outlined below.

It is also common to call certain processes or techniques coherent or incoherent. In that usage, coherent essentially means phase-sensitive. For example, coherent beam combining relies on mutual coherence between beams, whereas spectral (incoherent) beam combining does not.

Note that coherence can have a different meaning in quantum optics, as explained at the end of this article.

There are two complementary aspects of coherence:

• Spatial coherence is the degree of correlation (a fixed phase relationship) between electric fields at different transverse positions across a beam. Within the cross-section of a diffraction-limited laser beam (e.g. a Gaussian beam), the fields at different points oscillate in a highly correlated way — even if the temporal waveform contains many frequency components. High spatial coherence underlies the strong directionality and focusability of laser beams, and is thus the basis for high beam quality.
• Temporal coherence is the degree of correlation between the electric field at one point in space but at different times. For example, a single-frequency laser exhibits very high temporal coherence: The field evolves with an almost perfectly sinusoidal oscillation over long time intervals.

Figures 2–4 illustrate the difference between spatial and temporal coherence. For reference, Figure 2 shows a monochromatic Gaussian beam, exhibiting perfect spatial and temporal coherence.

Figure 3 shows a beam with high spatial coherence, but poor temporal coherence. The wavefronts are formed as above, and the beam quality is still very high, but the amplitude and phase of the beam varies along the propagation direction. Note that both the local amplitude and the spacing of the wavefronts vary to some extent. Such a beam can be generated e.g. from the output of a supercontinuum source.

Figure 4 shows a beam with reduced spatial coherence but high temporal coherence. The wavefronts are distorted, leading to higher divergence and degraded beam quality, while the beam remains monochromatic so the average wavefront spacing is constant. This can occur when a single-frequency beam passes through optically inhomogeneous material that varies across the aperture.

There are different ways to quantify the degree of coherence:

• Mutual coherence function / complex degree of coherence. First-order field correlations are quantified by the mutual coherence function; its normalized form is the complex degree of coherence ($\gamma^{(1)}(\tau)$). Its magnitude directly relates to fringe visibility in two-beam interference.
• Coherence time. This is the characteristic delay over which first-order temporal coherence decays.
• Coherence length is the product of coherence time and the vacuum velocity of light; it characterizes temporal (not spatial) coherence as a path-length scale over which high fringe visibility is maintained.
• Linewidth: For a single-frequency laser, a narrower linewidth (higher monochromaticity) implies higher temporal coherence.
• Second-order coherence: The normalized intensity correlation ($g^{(2)}(\tau)$) describes intensity fluctuations on the photon level (e.g. photon bunching or anti-bunching). Typical values: ($g^{(2)}(0) = 2$) for thermal light, ($g^{(2)}(\tau) = 1$) for coherent states, and ($g^{(2)}(0) \ll 1$) for single-photon sources.

Quantitative measures may come from statistical theory or from measurements. Most measurements use interferometry — e.g., recording fringe visibility versus path-length difference to obtain ($\gamma^{(1)}(\tau)$) — or Hanbury Brown–Twiss–type setups for ($g^{(2)}(\tau)$).

While the general concept for measuring coherence relies on interferometry, specific techniques have become standard for different regimes of coherence:

• Self-delayed heterodyne detection: This is the standard method for measuring the linewidth (and thus temporal coherence) of narrow-linewidth lasers (e.g., single-frequency fiber lasers or diode lasers). The beam is split; one arm is delayed by a fiber much longer than the coherence length, and the other is frequency-shifted by an acousto-optic modulator. The beat note reveals the linewidth.
• Michelson interferometers with variable delay: For sources with shorter coherence lengths (e.g., broadband sources or superluminescent diodes), a scanning interferometer measures the visibility contrast as a function of path difference directly yielding the coherence function.
• Shearing interferometry: This is often used for spatial coherence and wavefront measurements. By interfering a wavefront with a shifted (sheared) copy of itself, one can deduce the spatial phase correlations.
• Hanbury Brown and Twiss (HBT) setup: Used specifically for second-order coherence ($g^{(2)}$) measurements in quantum optics to distinguish between thermal, coherent, and single-photon light sources.

Some applications require very high spatial and temporal coherence: for example, many forms of interferometry, holography, precision optical sensors and coherent beam combining.

Other applications benefit from low temporal coherence. A prime example is optical coherence tomography (OCT), where high axial (depth) resolution relies on a short coherence length (broad bandwidth), typically combined with high spatial coherence for good focusing. Suitable sources include amplified spontaneous emission (ASE) in an optical amplifier (→ superluminescent sources) or broadband supercontinuum generation.

A reduced degree of temporal coherence also helps suppress unwanted laser speckle and interference artifacts in imaging, laser projection displays and some pointer applications.

In quantum optics, coherence often refers to phase relations between quantum states (off-diagonal elements of the density matrix), not simply classical optical phase. To stress this meaning, one often says quantum coherence.

Examples of quantum coherence include coherent superpositions in multi-level atoms driven by optical pumping, leading to phenomena such as Rabi oscillations, electromagnetically induced transparency (EIT), coherent population trapping, and lasing without inversion.

Note also the term coherent states of the light field, which is a different, specific meaning of coherent. These are special quantum states similar to classical light fields.