Interference: Concept of Spatial and temporal coherence, coherence length, coherence time

Interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have a greater amplitude or a lower amplitude if the two waves are in phase or out of phase, respectively.

Coherent sources of light

If the phase difference the waves from two sou sources reaching any point in space does not change with time rather it remains constant then these sources of light are known as 'coherent sources of light'.

This property of the waves is known as coherence. 

A wave which appears to be a pure sine wave for an infinitely large period of time or in an infinitely extended space is said to be a perfectly coherent wave.

Thus there are two different criteria of coherence.

(i) Temporal coherence and

(ii) Spatial coherence.

(i) Temporal Coherence

• Definition: Temporal coherence refers to the correlation between the phase of a wave at a given instant of time. 
• It is a measure of how monochromatic a source is.

 The average time interval for which the field remains sinusoidal (i.e. definite phase relationship exists) is known as 'coherence time' or 'temporal coherence' of the light beam.) and is denoted by τ. The distance L for which the field is sinusoidal is

Lc​=c⋅τc​

where:

• $c$ = speed of light in vacuum ($3\times {10}^8$ m/s)
• ${\tau }_c$ = coherence time

Relation to Frequency Bandwidth:

$$$$τc=1Δν

where $\mathrm{\Delta }\nu$ is the frequency bandwidth of the source.

Derivation: Since $\nu =\frac{c}{\lambda }$,

we have $\mathrm{\mid }\mathrm{\Delta }\nu \mathrm{\mid }=\frac{c}{{\lambda }^2}\mathrm{\Delta }\lambda$, 

so ${\tau }_c=\frac{{\lambda }^2}{c\mathrm{\Delta }\lambda }$  

and $L_c=c{\tau }_c=\frac{{\lambda }^2}{\mathrm{\Delta }\lambda }$.

Relation to Spectral Width:

$$$$Lc=λ02Δλ

where:

• ${\lambda }_0$ = central wavelength
• $\mathrm{\Delta }\lambda$ = spectral bandwidth (FWHM)

the Purity fector of spectral line is expressed as,  

Qc​=Δλλ0
​​

thus L=Qλ

(ii) Spatial Coherence

• Definition: Spatial coherence refers to the correlation between the phases of a wave at two different points in space at the same time.
• It is a measure of how uniform the phase is across the wavefront.

$$$$τc=Lcc=5.79×10−43×108