Question

Explain coherent sources.

Answer 1

Coherent sources: Two sources are said to be coherent if they emit waves of the same frequency and with constant phase difference or no phase difference.

Consider two sources producing electric fields \(\overrightarrow E_1 \) and \(\overrightarrow E_2\) respectively at a given point.

The total electric field at a point, due to superposition of field is

\(\overrightarrow E = \overrightarrow E_1 + \overrightarrow E_2\)

∴ Intensity of wave at that point is I ∝ E²

I = kE²

The term 2k  \((\overrightarrow E_1 . \overrightarrow E_2)\) is called interference term. All practical measurements give the intensity of light averaged over many oscillations of the wave. Therefore, the average value of the interference term is to be considered.

Special case

(i) When \((\overrightarrow E_1 \perp \overrightarrow E_2)\) i.e., θ = 90°,

∴ 2k \((\overrightarrow E_1 . \overrightarrow E_2)\) = 2E₁, E cos 90° = 0

∴ From Eq. (1), I = I₁ + I₂

i.e., Intensities of two sources just get added up.

(ii) When \((\overrightarrow E_1 \parallel \overrightarrow E_2)\) i.e., θ = 0

∴ 2 \((\overrightarrow E_1 . \overrightarrow E_2)\) = 2E₁E₂ ............(2)

Let a₁ = amplitude of wave for one source

ω₁ = angular frquency of source

Φ₁ = initial phase

a₂ = amplitude of wave for 2nd source

ω₂ = angular frequency of source

Φ₂ = initial phase for 2nd source

∴ From Eq. (2),

2kE₁ E₂ = 2ka₁ cos (ω₁t + Φ₁) x a₂ cos (ω₂t + Φ₂)

= 2ka₁a₂ cos (ω₁t + Φ₁) cos (ω₂t + Φ₂)

= ka₁a₂ [cos {(ω₁ + ω₂)t + Φ₁ + Φ₂} + cos {(ω₁ - ω₂) t + Φ₁ + Φ₂}] ...........(3)

[∵ 2 cos A cos B = cos (A + B) + cos (A - B)]

(a) It is clear that if two frequencies ω₁ and ω₂ are different, then both the terms in eq. (3) are of the form cos θ and θ varies with time.

The average value of cos θ is zero if angle θ is allowed to vary between 0° and 360°.

∴ For two different frequencies, the interference term has a zero average.

(b) If ω₁ = ω₂

∴ From Eq. (3),

2E₁E₂ = a₁a₂ cos (Φ₁ - Φ₂)

∴ Interference term depends upon the phase difference (Φ₁ - Φ₂)

If this phase difference varies from 0° to 360°,

∴ the average value will be zero.

But if two sources have the same frequency and either possess phase difference or constant phase difference then this term will survive and will give rise to interference i.e. rise or fall in resultant intensity. Such sources are called Coherent Sources.

Sources.

Conditions for two sources to be coherent are:

(i) They must have the same frequency.

(ii) The phase difference should be stable or there should be no phase difference between the waves.

Lloyd single mirror, Fresnel double mirror, Fresnel biprism, Young's double slit etc. are the methods of obtaining coherent sources.