Zero-dispersion wavelength Please help me understanding the zero-dispersion wavelength in fibers.
I found this wiki-article on the topic. Accordingly:

"In a single-mode optical fiber, the zero-dispersion wavelength is the wavelength or wavelengths at which material dispersion and waveguide dispersion cancel one another."

Does the zero-dispersion wavelength only exist in single mode fibers?
Does it depend on the length of the fiber? Will two single mode fibers with identical step-profile, but different length (5o km and 300 km) have the same zero-dispersion wavelength?
 A: Dispersion, $D$ is defined as:
\begin{equation}
D=\frac{\partial^2k}{\partial \omega^2}
\end{equation}
for the whole system, which is length independent (answering second question).
For the first question, then you can see that for highly multimode fibres, waveguinding will not affect propagation much, and the dispersion relation is that of the core's medium. So if it is made of fused silica, then the zero dispersion wavelength will be at approximately $1250\:nm$.
Edit: From what I remember, fibres with big cores just approach the core's propagation, but I might be wrong, after giving it a bit of thought, and multimode fibres might display a stronger dispersion. In any case, the first definition applies, and as you can see its about the dispersion relation of the system.
A: Does the zero-dispersion wavelength only exist in single mode fibers?
In general, the zero-dispersion wavelength does not exist in multimode fibers. This is because the core of multimode fibers is typically much larger than that of single-mode fibers, which causes light to travel through the fiber in many different modes. This results in a wide range of group velocities for different wavelengths of light, which means that there is no single wavelength at which the group velocity dispersion is zero.
Only MMF to have zero-dispersion wavelength that are specially designed to minimise the dispersion might have ZDW like Graded-index multimode fibers (GI-MMF), Large core multimode fibers (LC-MMF), and Dispersion-flattened multimode fibers (DF-MMF)
Does it depend on the length of the fiber?
The zero-dispersion wavelength (ZDW) of a single-mode fiber is determined by the material properties and the refractive index profile of the fiber. It is not dependent on the length of the fiber. Therefore, two single-mode fibers with identical step-profile and material properties, but different lengths (50 km and 300 km), will have the same ZDW.
Dispersion depends on both distance and wavelength. However, zero dispersion wavelength doesn't depends on distance. It's a constant value of any manufactured fiber with a dispersion coefficient D in terms of ps/nm/km @1550 nm
Dispersion will be more in longer fiber.
This zero-dispersion wavelength is shifted in dispersion-shifted fibers and highly-nonlinear fibers. As dispersion directly related to occurance of nonlinear effects in fiber useful/harmful for different applications
