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
![Intensity of wave](https://www.sarthaks.com/?qa=blob&qa_blobid=17774033242725582206)
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.