Blazed grating

Littrow configuration

The Littrow configuration is a special geometry in which the blaze angle is chosen such that diffraction angle and incidence angle are identical. For a reflection grating, this means that the diffracted beam is back-reflected into the direction of the incident beam (blue beam in picture). The beams are perpendicular to the step and therefore parallel to the step normal. Hence it holds in Littrow configuration ${\displaystyle \alpha =\beta ={\theta }_B}$. All other geometries yield anamorphic Littrow expansion or compression of the beam.

Diffraction angles at the grating are not influenced by the step structure. They are determined by the line spacing and can be calculated according to the in-plane version of the grating equation:

${\displaystyle d\left(\sin \alpha +\sin \beta \right)=m\lambda }$

where:

${\displaystyle d}$ = line spacing,

${\displaystyle \alpha }$ = incidence angle,

${\displaystyle \beta }$ = diffraction angle (angle taken in the same direction as ${\displaystyle \alpha }$, meaning the red ${\displaystyle \beta }$ would have a negative sign in the picture above if ${\displaystyle \alpha }$ is positive),

${\displaystyle m}$ = diffraction order (taken to be positive for blazed orders),

${\displaystyle \lambda }$ = wavelength of incident light.

For the Littrow configuration, this becomes ${\displaystyle 2d\sin {\theta }_B=m\lambda }$. By solving for ${\displaystyle {\theta }_B}$ the blaze angle can be calculated for arbitrary combinations of diffraction order, wavelength and line spacing:

${\displaystyle {\theta }_B=\arcsin \frac{m\lambda }{2d}.}$

In a near-Littrow configuration, or another geometry that illuminates grating grooves with a facet angle ${\displaystyle {\theta }_B}$, the Fraunhofer diffraction equation yields ${\displaystyle \beta =2{\theta }_B-\alpha }$, similar to specular reflection from a tilted surface.

Blazed transmission grating

Blazed gratings can also be realized as transmission gratings. In this case the blaze angle is chosen such that the angle of the desired diffraction order coincides with the angle of the beam refracted at the grating material.

Echelle grating

A special form of a blazed grating is the echelle grating. It is characterized by particularly large blaze angle (>45°). Therefore, the light hits the short legs of the triangular grating lines instead of the long legs. Echelle gratings are mostly manufactured with larger line spacing but are optimized for higher diffraction orders.

Echelle gratings are useful in planet-finding astronomy, and are used on the successful HARPS and PARAS (PRL Advanced Radial-velocity All-sky Search) spectrograph.