# Light coherence, an elusive concept

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# Light coherence, an elusive concept

In this article we are going to introduce the concept of coherence, as it plays a major role when selecting a light source for a certain application. It is always mentioned in the articles about light sources in the domain of fiber optics but it isn’t clearly explained most of the times. Although it isn’t an easy concept to grasp, we will try to make it as clear as possible.

Coherence can be defined as the capability of a light source to produce interference. It has to be admitted that the definition isn’t very clear. Furthermore, talking about coherence is a bit ambiguous, as there are two types of coherence: spatial coherence and temporal coherence. In general, we will say a light source is coherent if it is both spatially and temporally coherent. But what does this really mean?

• Spatial coherence: the standard definition says that “a high spatial coherence means a strong correlation (fixed phase relationship) between the electric field values of the light source at different locations of the light beam”.

Keeping it simple: a high spatial coherence means a strong directionality of the light beam. If we consider a holographic display (display that employs light diffraction to create a virtual 3D image of an object), the higher the spatial coherence, the more detailed the reconstructed image will be.

• Temporal coherence: the standard definition says that “a high temporal coherence means a strong correlation (fixed phase relationship) between the electric field values of the light source at different time instants”.

Keeping it simple:the higher the temporal coherence, the narrower the bandwidth of the light source and the other way round (a low temporal coherence implies a wide bandwidth). A high temporal coherence means that the light source can be used in an interferometric set-up with a higher distance between the light source and the target (due to a larger coherence length) and that there will be more problems related to speckle in imaging applications.

• Coherence length: the length the light has to travel so there is no correlation between this wave and the wave that comes from the light source at the same time, thus there is no interference. For a light source with a Gaussian emission spectrum, the coherence length LC is given by:

$L_C=\sqrt{\frac{2\cdot&space;ln2}{\pi&space;\cdot&space;n}}\frac{\lambda^{2}&space;}{\Delta&space;\lambda&space;}$

where n is the refractive index of the medium, λ is the central wavelength and Δλ is the full width half maximum (FWHM). We can easily observe that a high Δλ (wide bandwidth) ⇔ low LC⇔ low temporal coherence, while a low Δλ (narrow bandwidth) ⇔ high LC ⇔ high temporal coherence.

• Speckle: is a type of noise consisting of a random granular pattern that can be seen when a highly coherent light beam is diffusely reflected at a rough surface.The ratio of speckle contrast before and after the spectrum expansion of the light source is given by

$\frac{C}{C_0}=\frac{1}{\sqrt{1+(2\cdot&space;\Delta&space;\lambda&space;\cdot&space;\sigma&space;)^{2}}}$

where C is the new speckle contrast, C0 is the old speckle contrast, Δλ is the spectrum bandwidth and σ is the surface roughness. Combining this equation and the one for the coherence length, we reach the following conclusions: low temporal coherence ⇔ high Δλ⇔ low speckle, while high temporal coherence ⇔ low Δλ⇔ high speckle.

Now that we have made the concept of coherence clear, we include a table with the characteristics of the 3 main light sources used in fiber optics: LEDs, lasers and SLEDs  (see our previous article about SLEDs). Notice that the temporal coherence automatically determines the bandwidth and the speckle, but we list all the three: