# Diffraction Gratings

Is it possible to use the sawtooth shape diffraction grating instead of rectangular grating if yes then which parameters will affect at the output ? Secondly, what is special in blaze grating comparison with rectangular shaped diffraction grating ?

Yes indeed. For gratings, the use of different shapes of the rills (triangular, square-wave and so forth) changes the character of the line splitting achieved by the grating. The grating's output is essentially the Fourier transform of the incoming field that is phase or amplitude modulated by the grating in the sense that if $\psi(x, y)$ is the incoming field (scalar optical theory here - so think of $\psi$ as a transverse electromagnetic field component) and the grating modulates this by a phase or amplitude function:

$g(x,y) = \left(\sum_k \left(\alpha_k \exp(i K_g x) + \alpha_k^* \exp(-i K_g x)\right) \right)e^{i \sum_k \left(\beta_k \exp(i K_g x) + \beta_k^* \exp(-i K_g x)\right)}$

then the Fraunhofer diffracted output is:

$\Psi(x,y) = \frac{1}{\sqrt{2\,\pi}} \int_{\mathbb{R}^2} \exp(i \frac{k}{R}(x u + y v)) \psi(u, v) g(u, v)\,dx \,dy$

Here the grating's spatial period is $\frac{2\pi}{K}$ and the coefficients $\alpha_k$ are the Fourier components of the amplitude modulation of the grating and the $\beta_k$ are the Fourier components of the phase and frequency modulation of the grating. The different harmonics $\alpha_k$ and $\beta_k$ in the Fourier transform correspond to copies of the spectrum of the input light projected to the output (called diffraction orders). So fiddling with the Fourier components can change the relative strength of the different orders of the grating - often called the "efficiencies" of the different grating output "diffraction orders". This is wholly analogous to frequency up-converting followed by frequency or amplitude modulation for a time varying signal.

In particular, blazed gratings are often designed to maximize the efficiency of coupling to one diffraction order at the expense of all others for light at a particular incoming angle and wavelength region.

In fibre optics, gratings can realize reflectors. Chirped gratings, where the grating is no longer periodic, can reflect different frequency components of the light with different delays (if the light first encounters high spatial frequencies, the blue components are reflected sooner than the red ones). So this kind of device can be used to cancel dispersion. This is of great use in communications, and also in femtosecond experimental physics, where a chirped grating can shape dispersed laser pulses so that pulses only a few femtoseconds long can be shaped. There is quite a developed inverse scattering theory for designing such gratings for specific transfer functions. See the works of Leon Paladian in the mid to late 1990s on this topic.

Lastly, as you get more and more elaborate with gratings, the distinction between gratings and holograms becomes blurred. Gratings tend to process light spectrums, whereas the word hologram tends more to be kept for monochromatic, coherent light processing. Something monochromatic that is more "grating" than "hologram" is the computer generated hologram, which is designed to impose a very specific optical aberration on an optical wavefront to cancel aberration effects. CGHs are used to stretch the dynamic range of interferometry (interferometers tend to find it hard to image highly aberrated fields) and thus find application in optical testing as well as astronomy.