When a number of waves travelling in a medium simultaneously, with each wave being independent of the other, their effects get added together. The resultant wave is obtained by the superposition principle of waves which states that when a number of waves in a medium superpose each other, resultant displacement at any given point at a given instant is equal to vector sum of the displacements of individual waves i.e., y= y1 + y2 + y3 + ….+ yn,

When the two waves have their crest falling on the crest of another wave, i.e., they are in phase, their displacements are added. It is called constructive interference. However, if the waves are in opposite phases, crest of one falling on trough of other, we get destructive interference where their displacements are subtracted.

Conditions for constructive and destructive interference

Let the displacement of two waves be given by the following relations:

y1 = a1 sin ωt    and y2 = a2 sin (ωt+Φ)

Where, a1  and a2 are the wave amplitudes and Φ is the constant phase difference.

Resultant displacement is given by:

y= y1 + y2 = a1 sin ωt +a2 sin (ωt+Φ) = (a1+ a2cos Φ)sin ωt  +a2 sinΦcosωt

Put a1+ a2cos Φ = Acosθ ……………..(1) 

and a2 sinΦ = Asinθ…………….(2)

We get, y=  Acosθ sin ωt + Asinθcosωt = Asin( ωt+θ)

Squaring and adding (1) and (2), A2cos2θ + A2sin2θ = (a1+ a2cos Φ)2+a2 2sin2Φ

 = a12 + a22 +2a1a2cosΦ…………..(3)

Since, intensity of a wave ∝ (amplitude)2

Therefore, I = I1 + I2 +2√I1I2, cos Φ…………(4)

For constructive interference, intensity will be maximum when cos Φ=1 or Φ = 0, 2π, 4π,…

Since, 2π means a path difference of λ, if p is the path difference between two waves, then, 

2πp/λ = 0, 2π, 4π……….

Or, p= 0, λ, 2λ,3λ,……= n λ

For destructive interference, intensity will be minimum when cos Φ=-1 or Φ = π, 3π….

2πp/λ = π, 3π, 5π,………

Or, p=  λ/2, 3λ/2, 5λ/2,…. = (2n-1)λ/2

Coherent and incoherent sources

2 sources of light emitting light waves of the same frequency having zero or constant phase difference between them are called coherent sources. 

Incoherent sources are those which do not emit light waves having constant phase difference between them.

Observable interference or sustained interference

It is necessary that the maxima and minima positions do not fluctuate rapidly to observe interference pattern. The interference pattern, in which maxima and minima positions of intensity do not keep on changing with time is called a sustained or observable interference. 

The conditions necessary to obtain a sustained interference are:

1. Two sources should emit coherent waves continuously.
2. Amplitudes of two waves should be equal for better contrast between maxima and minima.
3. Two sources should be narrow.
4. Interfering waves must travel approximately along the same direction.
5. Two sources should be monochromatic.
6. Interfering waves should have the same polarisation state.
7. Distance between two coherent sources should be small while distance between source and screen should be large.

Application of coherence 

• Holography: it is a class of methods that are used for recording and reconstructing 3-D images based on interference phenomenon. The used laser source in this technique needs to have long coherence lengths that should be larger than the maximum path length differences in the recording apparatus.

• Non- optical wavefields: In quantum optics, the coherence is usually used for the state of light-emitting atoms/ions. There, coherence means a phase relationship between the complex amplitudes corresponding to electronic states. This finds importance in lasing without inversion.

1. 1. A low degree of temporal coherence can also be useful in laser projection displays, imaging, pointer applications,etc. as it lowers the likelihood for laser speckle or any similar interference effects.

• Modal analysis: Coherence is employed to probe the quality of the transfer functions (FRFs) that have to be measured. Low coherence can be a result of  poor signal to noise ratio or insufficient resolution of frequency.

• Optical coherence tomography: It is a high resolution optical imaging technique applied to transparent or translucent substances having limited depth such as biological tissues. It is based on interference effects for which low coherence is needed.

Conclusion

When a number of waves are travelling in a medium independent of each other, their effects may add up. According to the superposition principle, when the crest of one wave falls on the crest of another wave, their resultant displacement adds up and the condition is called constructive interference. However, if the waves are out of phase i.e., crest of one wave falls on the trough of another wave, their resultant displacement gets subtracted and such condition is then referred to as destructive interference. 

There are basically two types of light sources: coherent sources and incoherent sources. The sources from which waves having the same frequency and constant phase difference are obtained are called coherent sources whereas sources emitting waves that are out of phase are called incoherent sources.

For observable interference or sustained interference, maxima and minima should not keep changing rapidly with time. For that coherent sources are required so that the waves are in phase and monochromatic.

Coherence finds a lot of uses. Some of the applications of coherence include holography that requires larger coherence lengths; in non optical wavefields for lasing without inversion; in laser projection displays; in modal analysis as well as in optical coherence tomography for which low coherence is required.