What Is Coherence?

Coherence is a property of waves that describes the degree to which the phase relationship between wave components at different points in space or at different moments in time remains fixed and predictable. Two wave sources are said to be coherent when they maintain a constant phase difference, enabling stable interference patterns to form wherever the waves overlap. Incoherent sources, by contrast, have randomly fluctuating phase relationships, so any interference pattern averages out over time and becomes unobservable. Coherence arises in optics, acoustics, quantum mechanics, and radio engineering, and its quantification governs the performance of systems that depend on wave interference.

The concept was given rigorous mathematical form through the theory of partial coherence developed by Emil Wolf and Leonard Mandel in the 1950s and 1960s, which introduced mutual coherence functions and the coherence matrix to describe correlations between field values. A detailed treatment of the physical optics framework is maintained by RP Photonics, a reference resource for laser and photonics engineering.

Temporal and Spatial Coherence

Temporal coherence characterizes how well the phase of a wave at a single point in space can be predicted as a function of time. It is inversely related to the spectral bandwidth of the source: a perfectly monochromatic wave has infinite temporal coherence, while a broadband source has very short temporal coherence. The coherence time, commonly denoted as the interval over which the field autocorrelation remains high, is approximately the reciprocal of the bandwidth. Lasers, with their narrow linewidths, exhibit temporal coherence times ranging from nanoseconds for multimode devices to milliseconds or longer for highly stabilized single-mode systems.

Spatial coherence characterizes the correlation between the field at different transverse positions across a wavefront at the same instant. A point source produces high spatial coherence because all field values originate from a single emitter; an extended thermal source produces low spatial coherence because different parts of the source emit independently. High spatial coherence enables the formation of sharp diffraction patterns and is a requirement for holography, telescope aperture synthesis, and optical lithography. The distinction between the two coherence types and their relationship to source geometry is analyzed in Physics LibreTexts.

Quantum Coherence and Decoherence

In quantum mechanics, coherence has a distinct but related meaning. A quantum state is coherent when it exists as a superposition of multiple basis states, with well-defined relative phases between the components. This superposition is what underlies the interference effects observed in quantum systems, from electron diffraction to photon two-slit experiments. Quantum coherence is a fragile property: interactions between a quantum system and its surrounding environment cause decoherence, in which the phase relationships between superposition components are randomized, effectively converting quantum behavior into classical probabilistic behavior. Decoherence is the primary obstacle to maintaining qubits in quantum computing. Research published in Science Advances has examined how the coherence of emitted photons is fundamentally linked to the quantum coherence of the emitting particle, establishing a direct bridge between optical and quantum coherence frameworks.

Applications

Coherence has applications in a wide range of technologies, including:

• Holography, which records and reconstructs three-dimensional wavefronts using temporally and spatially coherent laser light
• Interferometry, used for precision measurement of surface flatness, refractive index, and length standards
• Optical coherence tomography, where deliberately short coherence length provides micrometer-scale depth resolution in medical imaging
• Radio telescope aperture synthesis, where spatial coherence between widely separated antennas enables high-resolution imaging
• Quantum computing, where maintaining qubit coherence is the central engineering challenge for fault-tolerant operation