Coherence length requirement for interference One property of light sources that is usually stated, which is of particular importance when trying to create interference fringes, is the coherence length (or coherence time). The equation for the coherence length is given by $l_c = \frac{c}{n\Delta{f}}$ where $c$ is the speed of light, $n$ is the refractive index of the medium and $\Delta{f}$ is the bandwidth of the source. 
An alternate description that I've seen (but unfortunately have lost a decent reference for) is that given a source emitting with wavelength range $\lambda \pm \Delta{\lambda}$, $l_c$ is the length that light with wavelength $\lambda$ and $\lambda + \Delta{\lambda}$ travels before they go from completely in-phase to completely out of phase. It leads from these definitions that an ideal, monochromatic source has an infinite coherence length and a broadband white light source has a very short coherence length.
This definition I am fairly happy with and can see what $l_c$ is trying to describe. The problem arises when we begin to talk about the limitations on when these non-idea light sources can produce interference. 
Consider a Mach-Zinder interferometer (MZI):

it is always stated that in order to see an interference pattern through the MZI, (or for any other interference experiment) you must match the path-lengths of each arm of the interferometer to within the coherence length of the source.
My question is, quite simply, why? Considering the MZI and the definition of the coherence length above, I cannot seem to form an image in my mind as to why this should be so. I am able to follow the mathematics of the MZI (including the introduction of the degree of first-order temporal coherence: $g^{(1)}(\tau)$ - which is where this question first arose) but creating a physical image in my mind is proving very tricky.
 A: I hope that the image below may clarify the situation.

Laving aside formulas, let me refer to the concept.
Coherence length $l_c$ of a wave-packet is the length of the wave-packet along which its wave-length is stable. The longer $l_c$, the better is for our interference experiments. 
Let me explain. Please see the figure.
What we do in experiments as with the MZ interferometer, is to vary a bit the length of one of the arms and measure the intensity of the beam at one of the outputs of the upper beam-splitter, for instance at $O_{vert}$. 
The coherence length of the packets A and B is the same, since we got the packets from the same "parent" packet at $BS_1$. Now, if $l_c$ is extremely short, and the arms $a$ and $b$ of the interferometer are slightly different, it is like the coherent region (constant $\lambda$) of the packet A, passed through $BS_2$ before he coherent region of B came to $BS_2$. So, no effect.
But if $l_c$ is long, we will see that the intensity at $O_{vert}$ varies with the difference in path-length as cosine square.
Now, coherence time $\tau_c$ in void, is $\tau_c = l_c/c$. 
If $\tau_c$ is long, we will see, for a fixed path-difference, a stable intensity at the examined output for a long time. As I said, varying the path-length difference the intensity for each path-length is stable, but comparing it for different path-lengths we get the cosine square dependence. Now, if $\tau_c$ is short, we will have difficulty in observing reliable results. Moreover, for the path-difference a bit longer, any effect will disappear, practically, we will see only the component of the packet that came later to the beam-splitter.
