[Physics] coherence length


Suppose i have two waves emanating from a point source. The waves start out completely in phase.
Is the coherence length consistently defined as the length at which these two waves achieve a phase difference of 1 radian?
That seems arbitrary to me, you could easily choose a phase difference of $ \frac {\pi}{10}$

I'm kind of lost on this whole topic of wave coherence to be honest…
I'd appreciate someone giving an overview for the example of two standard plane waves (with full mathematical description).

Best Answer

The coherence length is just the coherence time multiplied by the propagation speed.

To understand the coherence time, say you have a wave described, in complex notation, by $$ E(t) = A(t) e^{i \omega t} $$ where $A(t)$ is a slowly varying complex amplitude. You make this wave interfere with a delayed version of itself and collect the intensity $$ |E(t) + E(t-\tau)|^2 = |E(t)|^2 + |E(t-\tau)|^2 + 2\Re\big(E(t)E^*(t-\tau)\big). $$ where $\Re$ means real part and $^*$ means complex conjugate. The interference term is $$ 2\Re\big(E(t)E^*(t-\tau)\big) = 2\Re\big(A(t)A^*(t-\tau)e^{i \omega \tau}\big) $$ If $A(t)$ is constant, or roughly constant within a time interval $\tau$, then this becomes $$ 2|A(t)|^2 \cos(\omega \tau) $$ which is the interference pattern. On the other hand, if $A(t)$ fluctuates sufficiently fast, and $\tau$ is larger than its correlation time, then $A(t)A^*(t-\tau)$ averages to zero and there is no interference. Thus, the coherence time can be simply seen as the correlation time of the complex amplitude $A(t)$.

Now, I'm not sure there is a very quantitative definition of the correlation time. You could define it as the delay where the autocorrelation function drops below some arbitrary threshold. This is equivalent to setting a threshold on the visibility of the interference pattern. The relationship with the shape of the spectral line should also be apparent: the squared modulus of the Fourier transform of $A(t)$ is the shape of the line (the spectrum of the wave shifted by $-\omega$). It is also the Fourier transform of the autocorrelation function of $A(t)$. Thus, when the line is wide, the autocorrelation function is narrow,and the coherence time is short.

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