Coherence is the measure of the relation between phases observed at distinct points on a wave, including both temporal and spatial dimensions. It represents a phenomenon wherein two wave sources are considered coherent when they share identical frequencies and waveforms. Coherence can be described as an ideal property of waves with consistent interference patterns. It broadly explains the association between physical attributes within a single wave or between wave packets.

Temporal coherence measures the phase correlation of a light wave at different locations along its propagation path and helps to understand the degree of monochromaticity exhibited by the source. On the other hand, spatial coherence measures the phase correlation of a light wave at different points perpendicular to its propagation direction which provides the level of uniformity present in the wavefront's phase.

The primary distinction between temporal and spatial coherence lies in their scopes: Temporal coherence describes the correlation among waves observed at separate time instances in the same space. Spatial coherence describes the correlation between waves at distinct points in space, whether transverse or parallel in the same direction.

Temporal Coherence

Temporal coherence refers to the extent to which a wave aligns with itself when subjected to a time shift, delayed by T (time period of oscillation of the wave) at any pair of times. It is used for characterizing the degree of monochromaticity exhibited by a source. Temporal coherence explains how a wave can engage in self-interference during a specific period. The span over which the phase or amplitude of this wave can substantially deviate is termed the coherence time, often represented as “T_c”.

Also, when there's no time delay (T=0), the coherence becomes very strong. But when the delay goes beyond T=T_c, the coherence starts to decrease.

Another important concept is the “coherence length”, often shortened to L_c. This is like the distance a wave can travel during the time of T_c.

Spatial Coherence

Spatial coherence measures the relationship between points on a wave across all moments in time. In certain situations, the wave-like behavior spreads out in one or two directions. This characteristic of spatial coherence helps to explain how two points in space, say X₁ and X₂ (along the wave's reach), can create interference when looking at their average behavior over time.

For instance, when a wave holds the same amplitude value across an infinite distance, it possesses perfect spatial coherence. An essential term in the context of spatial coherence is the “coherence area”, often abbreviated as A_c. This area defines the separation range between two points where significant interference occurs, effectively determining the size of the coherence. The concept of coherence area is particularly relevant for setups like Young's double-slit interferometer. Also, it holds importance in optical imaging systems and various types of astronomical telescopes.

Applications of Spatial and Temporal Coherence

Spatial coherence finds applications in interferometry, enabling precise measurements and holography for 3D image reconstruction. It enhances optical imaging systems like microscopes and telescopes, aids in fiber optics communication, and influences remote sensing technologies like radar and LiDAR systems.

Temporal coherence also plays an important role in interferometry and spectroscopy for light source analysis and material composition studies. It's vital for stable laser emission in medical, industrial, and communication applications. Also, temporal coherence contributes to optical imaging in medical diagnostics and is significant in radar, LiDAR, and other sensing systems for distinguishing signals.