# Grating Pair

The grating pair method of pulse compression is used with wavelengths $\leq$ 1.3 um. Why might this be? Why can we not use the grating pair for wavelengths longer than 1.3 um? Well, if we remember earlier discussions about dispersion in fibers, we will recall that 1.3 um is the zero-dispersion wavelength for silica fibers, so $\lambda~=$ 1.3 um marks the transition from positive (normal) dispersion to negative (anomalous) dispersion. From this, we can infer that if our goal is to compress the pulses, and gratings can only act on wavelengths in the normal dispersion regime, then gratings must provide anomalous dispersion. Indeed, this is the case! We shall see exactly how in the coming paragraphs (also covered in the “Ultrafast Optics” portion of this website).

In the case of normal dispersion, the red frequency components lead the blue. Looking at Figure 1 below, it can be seen that the different frequency components are diffracted from the grating at a different angle, leading to a spread of the frequencies in the spatial domain. In particular, higher frequencies are diffracted more, so they have a shorter optical path than the red frequency components.

The angle at which the components are diffracted is given by Equation \eqref{diff_grat}.

$$sin(\theta_r) = \frac{2\pi{c}}{\omega\Lambda}-sin(\theta_i) \label{diff_grat}$$

From these relations and from the figure, we can then see that the total amount of delay would be:

$$t_d(\omega) = \frac{l(\omega)}{c} = \frac{d\phi_c}{d\omega}$$

where, as we might expect from our discussion before, “$l(\omega)$” is the optical path length and “$\phi_c(\omega)$” is the phase shift.

In thinking about the gratings, why would there potentially be drawbacks? We know that the gratings necessarily disperse the frequency components spatially, as this is what provides the differences in optical path lengths. We also know that diffraction of a beam changes the profile, and that the profile change is more pronounced the farther the beam propagates. Because of this, we might infer that the grating could distort the beam, which would be correct! Beam distortion would be problematic for any down-stream applications of the beam, so we would like to compensate this. One main way to compensate the beam distortion is to do a double pass, such that the beam emerging from the second grating is reflected back at a mirror placed on the output. Using this technique, all of the distortions in the beam are compensated in the opposite direction. The double pass is easily done by tilting the mirror so that the backward propagating beam is slightly offset from the forward propagating beam. The distance between the gratings can also be reduced by half, since the pulse would experience twice the amount of dispersion correction once being sent back through the grating. This is actually how I, Optics Girl, do this in my own laboratory!

Another problem with the gratings is the fact that gratings are very lossy. Typically only 60-80% of the original pulse energy remains in the pulse during first order diffraction. One way to get around this is to use a Gires-Tournois interferometer for compression. Another is prism pairs.