# Will these two coherent electromagnetic waves be in phase?

My question is how will I know if two coherent electromagnetic waves are in phase based on their phase difference.

I just solved a problem which stated...

Two coherent sources A & B send electromagnetic waves with wavelength = 2.4 cm. Find the phase difference when A is 2.52m from a point and B is 3.60m away from the same point.

I used the equation $\phi = 2\pi(R_2-R_1)/\lambda$ and I got $90\pi$.

So these two wavelengths have a phase difference of $90\pi$ degrees , are these two wavelengths in phase? How would I know?

Coherent sources- Two sources of light are said to be coherent if the waves emitted from them have the same frequency and are 'phase-linked'; that is, they have a zero or constant phase difference.

The calculation yielded 90.pi ; We know that 2.pi denotes a phase change of zero as the waves will come back to initial phase relation of the coherent sources.The waves advance by one wavelength when the phase change is 2.pi.

Your result to be correct ; its a multiple of 2.pi therefore their phase relations are same as the initial condition that either they are in phase or having the constant phase difference as in the beginning.

• Does the amplitude of the two sources of light affect the coherence? to some extent? – Stefano Feb 2 '18 at 23:10

Coherency has to be over a period of time. Instantaneously any two waves (or signals) at the same freq will have at at one time or point a phase difference. If their freqs stay exactly what they had at one time, they will stay coherent with each other, i.e. The phase differences won't change over the same period of time for the two. Over a period of time the oscillator or clock that determines the freq of a wave, will drift and have random fluctuations. If the drift of two such oscillators or clocks, is zero and their instantaneous fluctuations also zero (impossible unless at zero Kelvin) they stay coherent. Small drifts and small amount of random fluctuations will make them coherent to some extent. So drifts increase over a longer time range, and random fluctuations also affects it - the probability density function of the so called phase noise which causes those random fluctuations (in freq and in phase) is highest at the oscillator center freq, and goes down as you go away, so one must calculate.

Usually coherence can be calculated/analyzed by knowing (somebody measured it and specified it, or you go measure it) the stability of the transmitter (usually set by the oscillator), and is denotes D as eg, 10^(-6) over 100 msec. Means that over 100 msec the difference in freq (and from that you can compute phase differences) is 1 part in 10^6, eg 1 KHz at a freq of 1 GHz. Clearly ok for pulsed Non-coherent detection, but not so good for non-coherent detection (eg, in radar, one would use a signal at that freq and look for Doppler changes, and 1 KHz would be a Doppler you would look for, so this coherency (or stability) would not be good enough in a Doppler radar).

You can estimate the coherency of two signals using standard signal processing techniques. First you must digitize, with a x100 (or dependent on the sigma you can tolerate) time better stability in your oscillator than whatever you want to estimate. Then you do a cross coherency estimate, it is very similar to a cross correlation, both of which done with correlators or Fast Fourier Transforms.