Interference of Light

Interference of Light - Principle of Superposition of Waves

1. Principle of Superposition

   The principle of superposition states that when two or more waves meet at a point in space, the resultant displacement at that point is the algebraic sum of the displacements of the individual waves.

   Mathematically, if \(A_1\) and \(A_2\) are the amplitudes of two waves, and \(\phi_1\) and \(\phi_2\) are their respective phases, then the resultant displacement \(D\) at any point is given by:

   \[ D = A_1 \cos(\omega t + \phi_1) + A_2 \cos(\omega t + \phi_2) \]

2. Interference of Light

   Light is an electromagnetic wave, and it exhibits wave-like properties, including interference.

   When two light waves meet, they interfere with each other, leading to either constructive interference or destructive interference.

   Constructive interference occurs when the peaks of one wave align with the peaks of another wave, resulting in a wave with an amplitude equal to the sum of the individual amplitudes.

   Destructive interference occurs when the peak of one wave aligns with the trough of another wave, resulting in a wave with reduced or zero amplitude.

3. Mathematical Representation of Interference

   Consider two light waves with amplitudes \(A_1\) and \(A_2\), and phases \(\phi_1\) and \(\phi_2\). The resultant wave can be represented as:

   \[ D = A_1 \cos(\omega t + \phi_1) + A_2 \cos(\omega t + \phi_2) \]

   Depending on the relative phases of the two waves, interference can be constructive (\(\phi_2 - \phi_1 = 2\pi n\), where \(n\) is an integer) or destructive (\(\phi_2 - \phi_1 = (2n + 1) \pi\)).

4. Double-Slit Interference

   A classic example of interference in optics is the double-slit experiment. When light passes through two closely spaced slits, it creates an interference pattern on a screen.

   The interference pattern consists of alternating bright and dark fringes. Bright fringes result from constructive interference, and dark fringes result from destructive interference.