Superposition Principle and Coherent Sources

Superposition Principle

When two or more waves meet, they interfere and produce a resultant wave whose properties can be calculated by using the principle of a preposition. According to this principle, when two or more wave motions travelling through a medium superimpose one another, a new wave is formed in which resultant displacement \((\vec y)\) at any instant is equal to the vector sum of the displacements due to individual vectors \((\vec y_1 ,\vec y_2, \dots )\) at that instant,

$$ \text {i.e.} \: \vec y = \vec y_1 + \vec y_2 + \vec y_3 \dots $$

If there are only two waves, then the resultant displacement is given by:

$$ \vec y = \vec y_1 + \vec y_2 $$

Here, \(\vec y, \vec y_1 \: \text {and} \: \vec y_2 \) are the functions of time and space.

For example, when the crest of one wave falls on a crest of other, the amplitude of the resultant wave is a sum of the amplitude of two waves, as shown in the figure. When the crest of one wave falls on the trough of the other, the amplitude of the resultant wave, as shown in the figure.

The superposition principles hole good for mechanical waves as well as for electromagnetic or light waves.

Coherent Sources

Two sources of light, which continuously emit light waves of same wavelength or frequency and are always in phase differences are called coherent sources.

Coherent sources can be obtained from a single sources S as shown in the figure. Light waves which are in phase at S reach in phase on slits S₁ and S₂ as both S₁ and S₂ lie on the same wave front and are at equal distance from s. since they are derived from a single wavefront, the waves from S₁ and S₂ have same frequency or wavelength and so they are coherent sources of light.

The coherent sources obtained as above are from the division of a wavefront. Such coherent sources are also obtained from the division of amplitude as in Lloyd’s single mirror method in which a monochromatic light source and its image act as two coherent sources. Similarly, two virtual images obtained from the refraction through Fresnel’s biprism are also two coherent sources.

Two independent sources of light cannot be coherent sources. This is because light emitted by individual atoms of the light sources is the random and so light source is random and so the light from these atoms are not in phase. Thus, two independent sourced of light are incoherent sources.

Interference of Light

When light waves from two coherent sources P and Q as shown in figure superpose, they produce a non-uniform distribution of energy in different directions. At some points in this region where the crest of one wave falls on the crest of the other, resultant amplitude becomes maximum. Hence, the intensity of light is maximum at these points. At certain other points, a crest of one wave falls on the trough of the other and the resultant amplitude becomes zero. So, the intensity of the light is zero. This non-uniform distribution of energy is called interference of light. Thus, interference of light is the phenomenon of non-uniform distribution of light energy in a medium due to a superposition of light waves from two coherent sources.

There are two types of interference i.e. constructive and destructive interference.

Constructive Interference

When light waves from two coherent sources P and Q reach at a point Y in phase as shown in the figure, the amplitude of a resultant wave is equal to the sum of the amplitude of the two waves. The intensity of light at such point then is maximum that is constructive interference. In the constructive interference, a crest of one wave falls on the crest of the other and the trough of one fall on the trough of the other. If PX is greater than QX by a whole number of wavelength λ, the wave from P will be in phase with the wave form Q. So, for constructive interference, the path difference between the waves from two sources is

$$ \text {path difference,} \: QX – PX = n\lambda $$

where n is the integer number.

Destructive Interference

If waves from P and Q reaching Y are out of phase or differ in phase by 180^o as shown in a figure, the resultant amplitude is the difference in amplitudes of two waves which becomes zero. So, no light energy is obtained at Y and this is called destructive interference. In destructive interference, the crest of one wave falls on the through of the other wave and vice versa. If the distance PY is greater than QY by a half wavelength, then waves at Y will be in out of phase, and so for destructive interference, we have

$$ \text {path difference,} \: QY – PY = (n + \frac 12) \lambda $$

where n is the integer number.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa,et alPrinciple of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.