# Dispersion of Grating Pairs

Grating pairs, as discussed in the “Diffraction” section of the Classical Optics tab as well as in “Nonlinear Compression Techniques” under the Fiber Optics tab, are used to provide anomalous dispersion to compensate for normal dispersion. This means, from our definition of anomalous dispersion, that the blue components are delayed less than red components in a grating pair, which we will indeed verify via derivation in a few paragraphs. Grating pairs can also provide normal dispersion compensation when used with a telescope (also explained in a few paragraphs).

GRATING PAIR DISPERSION

{SALES PITCH} Ever have problems with a lot of positive dispersion that is keeping your pulses long, and wonder how to get rid of it? Well, let me introduce you to the (not-so-new) concept of grating pairs! This hard-to-setup system will provide you with the tunable dispersion amount you need! {END SALES PITCH}

As the above pitch implies, grating pairs provide negative second order dispersion, which can compensate for the positive dispersion accumulated in many gain media such as fibers. We can think of this mathematically as we are adding negative exponential components to positive exponential components in the frequency domain, which means that in the time domain the pulse should be bandwidth-limited. If that last sentence didn’t make any sense to you, ignore it and the full meaning of that sentence should be better illuminated shortly.

We can get a qualitative idea of how gratings work by looking at the image below:

As we can see in the above image, the various wavelengths in the incident light (which should be white for accuracy, but that was hard to make clear in PowerPoint, so the black arrow = broadband white light) are refracted at different angles at the first grating, which introduces a spatial chirp. Then, because the red (i.e. longer) wavelengths will travel a further distance than the blue (i.e. shorter) wavelengths, they will then be delayed in time with respect to the blue wavelengths, which is exactly what we want! We can kind of think of grating pairs as way for the shorter wavelengths to “catch up” with the longer wavelengths when you have positive dispersion.

One issue you may immediately notice is that the light coming out of the second grating is collimated, but is a rather large beam. To fix this, you can either put a mirror at the output of the grating pair and angle it such that the input and output beam do not overlap, or you can put in another two gratings. Since the latter is more expensive both monetarily and in terms of time spent agonizingly aligning, the former is more common (at least in the experience of Optics Girl). These setups would look like the following image:

HIGHER ORDER DISPERSION

POSITIVE GRATING PAIR DISPERSION

Although we said above that grating pairs always provide negative second order dispersion, a smart man by the name of O. A. Martinez figured out how to get tunable dispersion from a grating pair by inserting lenses in between the grating pairs. The basic concept is shown below: