# Coherent and Incoherent Addition of Waves

Coherence was originally conceived in connection with Thomas Young's double-slit experiment in optics but is now used in any field that involves waves, such as acoustics,electrical engineering, neuroscience, and quantum mechanics. The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and telescope interferometers .

#### Coherent and Incoherent Waves

Two waves produce the interference pattern only if they originate from coherent sources. The process of light emission from ordinary sources such as the sun, a candle, an electric bulb, is such that one to use special techniques to get the coherent sources. In order to obtain the fine interference pattern the path difference between the two waves originating from the sources should be very small. In practice the path difference should not be exceed a few centimeters to observe a good interference pattern.

1)Coherent light is light in which the photons are all in 'step' – other words the change of phase within the beam occurs for all the photons at the same time. There are no abrupt phase changes within the beam. Light produced by lasers is both coherent and monochromatic .

2)Incoherent sources emit light with frequent and random changes of phase between the photons.

Conventional light sources are incoherent sources. The transitions between energy levels in an atom is a completely random process and so we have no control over when an atom is going to lose energy in the form of radiation. The light that comes from a laser, however, is coherent, parallel, monochromatic and in unbroken wave chains .We can make a normal light source more coherent by making it smaller, so reducing the number of atoms that may emit quanta, but if we do this the intensity is reduced.

#### Addition of two waves

The addition of two waves emitting from two sources having intensities I1 and I2 . These two waves interfere each other so that the intensity of the resulting wave will be I as shown in following relation. $$I = I_{\,1} + I_{\,2} + 2(I_{\,1} I_{\,2} \cos {q})^{\frac {1}{2}}$$ Where q is the phase difference between two waves.
For the constructive interference the value of q = 0°, so that the Cos q = 1 and
for the destructive interference the value of q = 90°, so that the Cos q = 0.
In case of the incoherent waves the intensity I = I1 + I2