What Is Spatial Coherence?

Spatial coherence is a property of a wave that describes the degree to which the electric field maintains a fixed phase relationship across different points in the transverse plane of a beam at a single instant in time. A wave with perfect spatial coherence has a well-defined, smooth wavefront, meaning any two points on that wavefront oscillate in a fully predictable phase relationship. Spatial coherence is a foundational concept in optics and wave physics, distinct from temporal coherence (which concerns phase stability over time), and governs whether a light source can form stable interference fringes, focus to a tight spot, or illuminate a coherent aperture.

The concept arises from the statistical theory of wave fields. Natural thermal light sources such as incandescent bulbs or sunlight emit radiation from a large number of spatially separated, independently radiating atoms, producing a wavefront with short coherence radius. A single-mode laser, by contrast, enforces a single transverse field distribution through its resonator and emits with high spatial coherence across the full beam aperture. The degree of spatial coherence directly determines beam quality, focusability, and the achievable fringe visibility in any interference measurement.

Mutual Coherence and the Van Cittert-Zernike Theorem

Formally, spatial coherence is quantified by the mutual coherence function, which expresses the correlation between the field amplitudes at two transverse positions. When the mutual coherence function approaches unity between two points, the field at those points is fully correlated; when it approaches zero, the fields are statistically independent. A classical result known as the van Cittert-Zernike theorem connects source geometry to the spatial coherence it produces: the mutual coherence function of the field radiated by an extended incoherent source equals the Fourier transform of the source's intensity distribution. A study published in the Journal of the Optical Society of America formalized this relationship and its implications for determining the minimum spatial coherence required for interferometric measurements.

Measurement via Interferometry

The standard method for measuring spatial coherence is a two-point or double-slit interferometry experiment. The beam is directed through two apertures (or two slits) separated by a variable distance, and the fringe visibility of the resulting interference pattern is recorded as a function of that separation. High fringe visibility at a given separation confirms that the field at those two points is strongly correlated. As the separation exceeds the spatial coherence radius (also called the coherence area or transverse coherence length), fringe visibility drops. The RP Photonics Encyclopedia provides a detailed treatment of spatial coherence measurement approaches including shearing interferometry, in which the beam is interfered with a laterally displaced copy of itself to map the coherence function across the aperture.

Partial Coherence and Practical Sources

Most real-world light sources fall between the extremes of perfect coherence and complete incoherence, a condition known as partial coherence. Partially coherent sources produce a finite coherence area whose radius depends on source size and propagation distance. A small source seen from a large distance (as with starlight) appears spatially coherent across a sizeable aperture, while a large nearby source (as with a white LED at close range) is nearly incoherent. Controlling partial coherence is important in optical lithography, where high-NA illumination systems are deliberately made partially coherent to balance resolution, depth of focus, and speckle suppression. An arXiv preprint on the van Cittert-Zernike theorem discusses how source size can be recovered from coherence measurements using intensity interferometry techniques.

Applications

Spatial coherence has applications in a wide range of fields, including:

• Laser beam focusing and free-space optical communications, where high spatial coherence enables diffraction-limited propagation
• Optical lithography and photomask projection, where coherence controls resolution and depth-of-field trade-offs
• Stellar interferometry and radio astronomy aperture synthesis, using spatial coherence to measure source angular size
• Holography, which requires spatially coherent illumination to record stable interference patterns
• Synchrotron and X-ray optics, where source emittance determines available spatial coherence for nanoscale imaging