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I read in a book about optical fibers that the different spectral components of a light pulse transmitted in the fiber propagate with different velocities due to a wavelength dependent refractive index. Can someone explain that? Why is that silica refractive index depends on the wavelength/frequency of the wave?

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The fundamental reason for the wavelength dependance of refractive index ($n$), in fact the fundamental description of refraction itself, is the domain of quantum field theory and is beyond my understanding. Hopefully somebody else can provide an answer on that subject.

However, I can state that it isn't just silica that has a wavelength dependent $n$. In fact, every material has some wavelength dependence, and this property is called dispersion. In optical materials, the dispersion curve is very well approximated by the Sellmeier Equation: $$ n^2(\lambda) = 1 + \sum_k \frac{B_k \lambda^2}{\lambda^2 - C_k} $$

usually taken to $k=3$, where $B_k$ and $C_k$ are measured experimentally. As far as I know this equation is not derived from theory; it is completely empirical.

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You can derive the Sellmeier Equation by assuming a set of independent driven harmonic oscillators, yielding the Lorentzian curve (and then use $\omega=2\pi c/\lambda$ and remember that polarization and $n^2$ are linearly connected for non-magnetic materials) –  Tobias Kienzler Jan 12 '12 at 15:24

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