Littrow expansion

The Littrow expansion/compression effect is an anamorphic distortion of single-wavelength image on the focal plane of the instrument, due to a geometric effect surrounding reflection or transmission at the grating. In particular, the angle of incidence ${\displaystyle \theta _{i}}$ and reflection ${\displaystyle \theta _{r}}$ from a flat mirror, measured from the direction normal to the mirror, have the relation

${\displaystyle \theta _{r}=-\theta _{i}\,,}$

which implies

${\displaystyle {\frac {d\theta _{r}}{d\theta _{i}}}=-1}$

so that an image encoded in the angle of collimated light is reversed but not distorted by the reflection.

In a spectrograph, the angle of reflection in the dispersed direction depends in a more complicated way on the angle of incidence:

${\displaystyle \theta _{r}=-\arcsin \left(\sin(\theta _{i})+n\lambda /D\right)\,,}$

where ${\displaystyle n}$ is an integer and represents spectral order, ${\displaystyle \lambda }$ is the wavelength of interest, and ${\displaystyle D}$ is the line spacing of the grating. Because the sine function (and its inverse) are nonlinear, this in general means that

${\displaystyle {\frac {d\theta _{r}}{d\theta _{i}}}\neq -1\,,}$

for most values of ${\displaystyle n}$ and ${\displaystyle \lambda /D}$, yielding anamorphic distortion of the spectral image at each wavelength. When the magnitude is larger, images are expanded in the spectral direction; when the magnitude is smaller, they are compressed.[2]

For the special case where

${\displaystyle n\lambda /D=-2\sin(\theta _{i})\,,}$

the reflected ray exits the grating exactly back along the incident ray, and ${\displaystyle d\theta _{r}/d\theta _{i}=1}$; this is the Littrow configuration,[3] and the specific angle for which this configuration holds is the Littrow angle. This configuration preserves the image aspect ratio in the reflected beam. All other incidence angles yield either Littrow expansion or Littrow compression of the collimated image.