Interference of Light

Light is said to have a dual nature – it is either a particle or a wave. Interference of light is one proof that light is a wave

When two or more waves overlap in space, they result to interference. According to the principle of superposition, when two or more waves overlap, the resultant displacement at any point and at any instant is found by adding the instantaneous displacements that would be produced at the point by the individual waves if each were present alone.

Interference of light may be constructive or destructive.

Constructive Interference

When waves from two or more sources arrive at a point in phase, they reinforce each other. The amplitude of the resultant wave is the sum of the amplitudes of the individual waves and we call this as constructive interference. Waves could produce constructive interference if their path lengths differ by an integral number of wavelengths:

$$r_2 – r_1 = m\lambda \;\text{where}\; m = 0 , \pm 1, \pm 2, \pm 3, … \;\;\;\;\;\; (1)$$

Destructive Interference

Waves interfere destructively if their path lengths differ by a half-integral number of wavelengths:

$$r_2 – r_1 = (m + \frac 12)\lambda \; \text{where} \; m = 0 , \pm 1, \pm 2, \pm 3, … \;\;\;\;\; (2)$$

Suppose waves from two sources arrive at the same point exactly a half-cycle out of phase. A crest of one wave arrives at the same time as a crest in the opposite direction of the other wave. The amplitude of the resultant wave is the difference between the amplitudes of the two individual waves. If the two waves are equal, then it follows that the amplitude of the resultant wave is zero. This cancellation or partial cancellation of each wave is called as the destructive interference.

Consider the figure below:

The figure shows a single source $$S_1$$ of sinusoidal waves and some of the wave fronts produced by this source in which the red curves show all the positions in which the constructive interference occurs. These curves are called antinodal curves. The curves that show all the positions to which destructive interference occurs are called the nodal curves. A nodal curve lies between each two adjacent antinodal curves.

Equations (1) and (2) will hold only if the two sources have the same wavelength and are always in phase. Because of the way light is emitted, there is no such practical way for light waves to achieve a constant phase relationship or coherence with two independent sources. For ordinary sources of light, atoms gain excess energy through thermal agitation or by colliding with accelerated electrons. They become excited atoms which begin to emit energy and continue until they lose all the energy they can. Such number of atoms in a source ordinarily emit in an unsynchronized and random phase relationship, and so the light emitted from two different sources has no definite phase relationship.

However, the light emitted from a single source can be split and so parts of it may occur in two or more regions of space, and form two or more secondary sources. Any random phase change in the source equally affects these secondary sources and cause no change in their relative phase.

Phase Difference

The wavelength of light depends on the index of refraction of the medium to which they travel. Suppose two lights are travelling through different media, the wavelength of one light differ from the wavelength of the other light, but as soon as they leave the different medium, they could have the same wavelength which is the wavelength of light in air.  Since they have different wavelength at two different media, the two waves may no longer be in phase. Such phenomenon is called as the phase difference. The phase difference between two light waves can change if the waves travel through different materials having different index of refraction.