Interference of light

Coherent sources

Two light sources are said to be coherent sources if they emit continuous light wave of same frequency or wavelength, nearly same amplitude and in phase or constant phase different.

The characteristic of the coherent are given below.

Coherent source have same frequency, wavelength and amplitude

Coherent source have same phase difference or constant phase difference.

They can be realized in practice in:

a. Lloyd’s single mirror (real narrow source and virtual image produced by reflection)

b.Fresenlbiprism(two virtual image produced by the same sourceby reflection)

c.Michelson’sinterferometer (dividing amplitude portion of the wave front into two parts by reflection and refraction



Interference of light is process in which the light energy is redistributed in the medium on account of superposition of light wave from two coherent sources. If the wave of two coherent sources traveling in a medium interferes (superpose) at a point in a way that the crest or tough of one wave fall exactly on the crest or trough of other wave then it is called constructive interference. But when crest of one wave falls on the trough of another wave or vice-versa, it is called destructive interference.

1.The conditions for interference of the light are given below:

2.The two sources should be coherent.

3.The two sources must be very closer to each other.

4.The two sources must emit continuous light.

5.The two sources must be travel in same direction

Two independent sources of light cannot produce interference because for producing interference pattern, there is required of two coherent source vibrating in same phase which not possible by two independent sources of light. Two independent sources cannot be coherent because light is emitted by individual atoms. Even a very small source consists of millions of atoms, and emission of light by them takes place independently. An atom emits an unbroken wave of above 10-8 second due to its transition from a higher energy state to a lower state. So, the phase difference between two independent light sources changes after 10-8second. The rapid changes cannot be seen by our eyes and almost a uniform illumination on the screen is observed. So, we cannot use two independent sources to produce interference pattern.

Let us consider the monochromatic light of wavelength λpasses through the two slits P and Q, which act as coherent sources .The light reaching at the point M has the phase difference ϕ.

Let y­1and y2 are the displacement of light eave coming from P and Q respectively.



By superposition principle Y =y1 +y2
=asinwt + asin (wt +ϕ)

= asinwt (1+ cosϕ) + acoswtsinϕ

Put, A cosϕ = a (1+ cosϕ)…………..1

A sinϕ = a sinϕ ………………2



Then,

Y = Acosϕsinwt + Asinϕcoswt

Y = Asin(wt +ϕ)

Squaring 1 and 2
A2 = 4 Ao^2cosϕ^2/2
Since Intensity at a point is given by square of the amplitude.


I=4Iocosϕ2/2 and Io=Ao^2

Constructive interference

Constructive interference occurs whenever waves come together so that they are in phase with each other. This means that their oscillations at a given point are in the same direction, the resulting amplitude at that point being much larger than the amplitude of an individual wave. For two waves of equal amplitude interfering constructively, the resulting amplitude is twice as large as the amplitude of an individual wave. For 100 waves of the same amplitude interfering constructively, the resulting amplitude is 100 times larger than the amplitude of an individual wave. Constructive interference, then, can produce a significant increase in amplitude.

Case. 1

Condition for constructive interference

For the constructive interference I=maximum as if cosϕ2/2=1

i.e. ɸ=2∏nwhere n=0, 1, 2, 3, 4, ……..etc.

Since the phase difference of 2∏ cross ponds to path difference of λ, the constructive path difference is , x=nλ where n=0, 1, 2, 3, 4, ……etc.

Hence the constructive path difference is integral multiple of 2∏ or λ.

On the other hand if the fringth width is equal to the integral multiple of the wavelength of the light i.e. β=(xnd)/D=n λ where n=0, 1, 2, 3, …….n

Destructive interference

Destructive interference occurs when waves come together in such a way that they completely cancel each other out. When two waves interfere destructively, they must have the same amplitude in opposite directions. When there are more than two waves interfering the situation is a little more complicated; the net result, though, is that they all combine in some way to produce zero amplitude. In general, whenever a number of waves come together the interference will not be completely constructive or completely destructive, but somewhere in between. It usually requires just the right conditions to get interference that is completely constructive or completely destructive.

Case. 2

Condition for destructive interference

For destructive, I=minimum

i.e. Cos.2=0
or, ɸ=(2n-1)∏where n= 1,2, 3, 4, ………………….etc.

and hence destructive path difference is the integral multiple of ∏ or λ/2.

On the other hand we also have if the fringe width is equal to the odd multiple of half wavelength of light i.e. β=(xnd)/D={(2n+1)}λ/2 where n=0, 1, 2,3, ……….n.

As we see the intensity of the lightat maximum i.e. 4A2and minimum is 0 so, the variation of interference lies between 0 to 4A2. A/c to the law of conservation of the light energy i.e. the energy neither can be created nor created but is transferred fromplace to another in this case the point of minimum intensity to the point of maximum intensity. Thus the energy which apparently disseapar (dark) at minima has actually been transferred to the maximum (bright) where the intensity is greater than that produce by two beams acting separately. So, interference follows the law of the conservation.