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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).

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also... if the only thing that matters is constant relative phase... then they needn't be initially in phase... it's a question of when does the relative phase stop being constant by 1 radian... is that where the critical length is? –  Timtam Jul 27 '11 at 12:59
    
does it matter if they are monochromatic or not? yikes... –  Timtam Jul 27 '11 at 13:03
    
Note the Wiener–Khinchin theorem. It states how the spectrum of the light source is related to the temporal coherence function. There is an analogue realtion for the spatial coherence function. It blew my mind when I learned that you can estimate the diameter of stars by measuring there spatial coherence in a Young interferometer. –  whoplisp Jul 27 '11 at 15:19
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2 Answers

up vote 5 down vote accepted

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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Mandel has defined (a good definition) the coherence time as : $$ \tau_c = \int_{-\infty}^\infty \gamma(\tau)\gamma^*(\tau)d\tau $$ where $\gamma(\tau)$ is the complex degree of coherence. –  daaxix Dec 31 '12 at 7:29
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Is the coherence length consistently defined as the length at which these two waves achieve a phase difference of 1 radian?

No

Coherence length is the maximal difference in way traveled by the to rays without losing the phase relation which allows interference. This lenghts can be some µmeters (eg for white light from a glowing body) or several meters (eg for very narrow lines from discharge lamps) or even more for mode selected/stabilized lasers.

To first order approximation coherence length is inversly to the bandwidth of the light, but divergence of the bundle plays a role too. Coherence length is important in interference, eg why Newtons rings are only a few around the center of that lens for white sunlight, but hundreds when illuminated with a He/Ne laser.

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"Coherence length is the maximal difference in way traveled by the to rays without losing the phase relation which allows interference." i'm sorry but I really need an example w/ full mathematical detail... you seem to know what your talking about... but it's not helping me –  Timtam Jul 27 '11 at 13:08
    
what happens when the they can no longer interfere? –  Timtam Jul 27 '11 at 13:13
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@Timtam: I'm not sure how you can "tell he is a chemist." I'm an optical engineer and this is absolutely a correct answer. He didn't include equations, but everything he said is completely accurate and expressed in the same way that you would hear it from an optical engineer. –  Colin K Jul 27 '11 at 15:47
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@Colin, thanks for the "flowers" :=) –  Georg Jul 27 '11 at 16:11
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Several comments deleted after multiple flags. –  dmckee Jul 28 '11 at 12:32
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