Expression for the Conditions of Constructive and Destructive Interference

Interference of light-

Definition: The phenomenon of redistribution of light energy due to the superposition of two coherent light waves of the same frequency (or wavelength), producing alternate bright and dark fringes, is called interference of light.

What are the coherent sources of light?

Explanation

When two coherent light waves travelling through the same medium superimpose on each other, the resultant intensity at different points changes due to constructive and destructive interference.

Expression for the Conditions of Constructive and Destructive Interference

When two coherent light waves superimpose, the resultant intensity depends on the path difference or phase difference between them.

Let the two wave

Y₁=A₁ sin (ωt )

Y₂=A₂ sin (ωt +Φ)

After superimposition

 Y = Y₁ + Y₂

Y = A₁ sin (ωt )+ A₂ sin (ωt +Φ)

After solving it we get ,

Y = R A₂ sin (ωt +θ)

Where R is the resultant amplitude of the resulting wave

R² = A₁ ² + A₂² +2 A₁A₂ cos (ωt )

But Intensity of light,  I α (amplitude )²

So we can write ,

I = I₁ + I₂ +2√I₁I₂ cos ωt ,

1. Constructive Interference (Bright Fringe)

When two waves meet in the same phase, their amplitudes add up and maximum intensity is obtained.

 For Maximum intensity cos Φ = +1

cos Φ = 2nπ

 Therefore path difference  p = nʎ [ whwre n= 0,1,2,3…],

For maximum intensity I_max = I₁ + I₂ +2√I₁I₂.

Or, I_max = A₁²+ A₂ ² + 2A₁ A₂ = (A₁ +A₂)²

2. Destructive Interference (Dark Fringe)

When two waves meet in opposite phases, their amplitudes cancel each other and minimum intensity is obtained.

Condition: intensity should be minimum i.e. cos Φ = -1

 Phase difference , Φ = (2n-1)π

Path difference p = (2n-1 )ʎ/2

For minimum intensity I_min = I₁ + I₂ – 2√I₁I₂,

OR, I_m = A₁²+ A₂ ² – 2A₁ A₂ = (A₁ -A₂)²

Diffraction of light

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