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**Coherence Length** and **Coherence Time** are concepts often encountered in physics, particularly in the study of waves and quantum mechanics. They both refer to the temporal and spatial properties of waveforms and are crucial in understanding the behavior of systems involving interference and coherence.

**Coherence** refers to the extent to which electromagnetic radiation retains a closely consistent phase connection, covering both temporal and spatial dimensions. The time for which coherence remains intact is referred to as **coherence time**. And the length that a signal could travel through a vacuum within this timeframe is denoted as the **coherence length**.

For example, in the case of light-emitting diodes, the coherence time typically falls within half a picosecond, while the associated coherence length is approximately 15 microns. Conversely, a basic laser might exhibit a coherence time of roughly half a nanosecond, accompanied by a coherence length of around 15 centimeters.

In the case of a high-quality laser with a narrow linewidth, the coherence time could extend to about a microsecond, while the coherence length could potentially stretch up to 200 meters.

The range of signal variation (referred to as linewidth) and the coherence length, as well as coherence time, are inversely proportional.

Where Δλ is the linewidth and λ is the centre wavelength.

**Coherence Length**

**Coherence length** is a measure of the distance over which a wave maintains a consistent phase relationship. In other words, it is the distance over which a wave retains its characteristic waveform before significant phase changes or wave interference occurs. In optics, coherence length is frequently used to describe the quality of light sources, such as lasers. A longer coherence length indicates that the light waves from the source are more synchronized and maintain their phase relationship over a larger distance.

The coherence length (L_{c}) is related to the wavelength (λ) and the degree of spectral bandwidth (Δλ) of the wave by the formula:

where a smaller spectral bandwidth or a longer wavelength results in a longer coherence length. Coherence length is important in various applications, including interferometry, holography, and optical communication systems.

**Coherence Time**

**Coherence time** is a measure of the time duration over which a wave maintains its coherence or phase relationship. In quantum mechanics, coherence time is often associated with the duration in which a quantum system can remain in a superposition of states before decoherence sets in and the quantum properties are lost due to interactions with the environment. Coherence time is a critical factor in quantum computing and quantum information processing, where maintaining the delicate quantum states is essential for performing complex computations.

The coherence time (T_{c}) can be related to the spectral linewidth (Δν) of the wave by the formula:

Similar to coherence length, a smaller spectral linewidth leads to a longer coherence time. Coherence time is a crucial parameter in fields such as quantum optics, nuclear magnetic resonance (NMR) spectroscopy, and quantum cryptography.