Spatial coherence

Spatial coherence describes how strongly the optical field at one point of a wavefront is correlated with the field at another point, observed at the same time. If the correlation is high, light taken from the two points and brought together forms interference fringes of high contrast; if it is low, the fringes wash out. The transverse distance over which the correlation stays high is the coherence width. For direct sunlight at 550 nm it is about 70 µm, because the Sun's disk subtends 0.53°; for the light of a single star it can be metres; across the output of a single-transverse-mode laser it spans the entire beam.

Spatial coherence is independent of temporal coherence, which concerns the correlation of the field with itself at a later time, at one point. A narrowband source can be spatially incoherent (a sodium lamp), and a broadband source can be spatially coherent (a supercontinuum from a single-mode fiber). The general comparison between the two kinds of light is in coherent vs incoherent light.

Complex degree of coherence and fringe visibility

The quantity measured is the complex degree of coherence $\mu_{12}$, the normalized correlation of the fields $E_1$ and $E_2$ at points 1 and 2. Its magnitude runs from 0 (incoherent) to 1 (fully coherent). In a Young's two-pinhole arrangement with equal intensities through both pinholes, the fringe visibility equals $|\mu_{12}|$ directly:

$$V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = |\mu_{12}|.$$

With unequal intensities $V$ is reduced further by the factor $2\sqrt{I_1 I_2}/(I_1 + I_2)$, so a measurement must correct for it. Measuring $V$ as the pinhole separation is varied maps $|\mu_{12}|$ as a function of separation; this is the standard laboratory measurement, performed with the double-slit experiment or with a lateral-shearing interferometer that overlaps a beam with a displaced copy of itself.

The van Cittert-Zernike theorem

A source whose points radiate independently, such as a lamp filament, an LED or a star, is spatially incoherent at its surface, yet its light becomes partly coherent after propagating. The van Cittert-Zernike theorem states that the degree of coherence in a distant plane is the normalized Fourier transform of the source's intensity distribution. For a uniform circular source that subtends a full angle $\theta$ at the observer,

$$|\mu_{12}(d)| = \left|\frac{2 J_1(v)}{v}\right|, \quad v = \frac{\pi \theta d}{\lambda},$$

where $d$ is the separation of the two points and $J_1$ is the first-order Bessel function. The correlation first falls to zero at

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

For the Sun, $\theta = 0.53° = 9.25$ mrad, and at 550 nm $d_0 = 72.5$ µm. Pinholes closer than a few tens of micrometres give sunlight fringes; pinholes a millimetre apart do not. A 1 mm diameter source viewed from 1 m ($\theta = 1$ mrad) gives $d_0 = 0.67$ mm at the same wavelength. Since $\theta$ shrinks as the observer moves away, the coherence width of a fixed source grows in proportion to distance, which is why starlight is spatially coherent over large baselines.

Stellar interferometry

Michelson and Pease used this relation in reverse in 1920 to measure the angular diameter of Betelgeuse, mounting two outrigger mirrors on the 100-inch Hooker telescope and increasing their separation until the fringes disappeared at a baseline of about 3 m. At 575 nm that baseline corresponds to an angular diameter of 0.047 arcseconds, a value no single telescope of the time could resolve. Modern stellar interferometers measure $\mu_{12}$ over many baselines and invert this Fourier relation to form images.

Lasers and beam quality

A laser oscillating in a single transverse mode (TEM₀₀) emits a beam whose field is fully correlated across its cross section, so $|\mu_{12}| \approx 1$ for any two points in the beam. This is what allows such a beam to be focused to a diffraction-limited spot and collimated over long distances. A laser running on many transverse modes, such as a broad-area diode or a multimode fiber laser, has reduced spatial coherence and a correspondingly larger beam parameter product, quantified by beam quality M². Passing a partially coherent beam through a pinhole at a focus, a spatial filter, raises its spatial coherence at the cost of power.

Where it matters in practice

High spatial coherence produces speckle when laser light scatters from a rough surface, and illumination systems for projection and microscopy often reduce it deliberately with rotating diffusers or mode scramblers. In microscopy the spatial coherence of the illumination, set by the condenser aperture, changes image contrast and resolution: coherent illumination has a cutoff at $\mathrm{NA}/\lambda$, fully incoherent illumination at $2\,\mathrm{NA}/\lambda$.

A common pitfall is attributing low fringe contrast to the source linewidth when the cause is source size, or the reverse. Changing the pinhole separation at fixed path difference tests spatial coherence; changing the path difference at fixed geometry tests temporal coherence.

Common questions

What is the difference between spatial and temporal coherence?

Spatial coherence compares the field at two different points at the same time and is set by the size and angular extent of the source. Temporal coherence compares the field at one point at two times and is set by the spectral width, through the coherence length.

Is sunlight spatially coherent?

Partly. The correlation is high across a few tens of micrometres at 550 nm, enough for sunlight to produce Young's fringes through closely spaced pinholes.

How is spatial coherence increased?

By reducing the angular size of the source seen from the point of use: moving farther away, imaging the source onto a small pinhole, or coupling the light into a single-mode fiber, which transmits only one spatial mode.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 10; L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), Ch. 4; J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015); A. A. Michelson, F. G. Pease, Astrophys. J. 53, 249 (1921).