An important question to ask ourselves before delving into the math of any subject is, do I really even know what is physically happening here? What really is dispersion in the first place? First, we should say that the dispersion we are concerned with in this section is chromatic dispersion (as opposed to modal or polarization dispersion). We have touched on this in some other sections, but it is also useful to reiterate the concept here. To explain, let’s look at the following image we constructed when first discussing the idea of a pulse:
Here we can see (very roughly due to the limitations of my drawing abilities) that each of the wavelengths that make up a pulse are coherently combined at the center of the pulse i.e. where they all align makes up the peak of the pulse. Now, with dispersion, the peak of each wavelength shifts a bit, such that our pulse now looks more like the following:
Here we can see (again very roughly…my apologies…) that the different wavelengths have now “shifted” along the pulse, which also has had the effect of lengthening the pulse. This means that now there is a frequency dependence on time. In a slightly inaccurate but roughly correct analogy, this means that if you were to be staring at a pulse coming straight at you, you would see first red (longer) wavelengths, then green, then blue (smaller) wavelengths. This effect is basically the entire reason why we are interested in dispersion!
Material dispersion refers to the amount of extra phase picked up when propagating through a bulk material.