Gitterordnungs-Rechner

List of symbols:

\(AOI\): Angle of incidence
\(\phi\): Angle to the normal of the grating lines of the incident light
\(n_x\): Diffraction order number in x
\(n_y\): Diffraction order number in y
\(k_x\): Wave vector in x
\(k_y\): Wave vector in y
\(k_z\): Wave vector in z (out of the grating plane)
\(\lambda\): Light wavelength
\(\Lambda_x\): Grating period in x
\(\Lambda_y\): Grating period in y (in case of 1D gratings, this becomes infinite)
\(AOD\): Angle of diffraction
\(\psi\): Angle to the normal of the grating lines of the diffracted light


Wavevectors:
The outgoing wavevectors are given by the following equations: \[k_x=2\pi\left(\frac{n_x}{\Lambda_x}+\frac{\sin\left(AOI\right)\cos\left(\phi\right)}{\lambda}\right)\] \[k_y=2\pi\left(\frac{n_y}{\Lambda_y}+\frac{\sin\left(AOI\right)\sin\left(\phi\right)}{\lambda}\right)\] \[k_z=\sqrt{\frac{4\pi^2}{\lambda^2}-k_x^2-k_y^2}\] In case of a 1d grating, the period in y (\(\Lambda_y\)) is set to infinity, and all the y-terms go to 0.


Angles of diffraction:
From these wavevectors, the angle of diffraction (\(AOD\)) and the angle to the x-axis (\(\psi\)) can be found like this: \[AOD=\cos^{-1}\left(\frac{\lambda k_z}{2\pi}\right)\] \[\psi=atan2\left(k_y,k_x\right)\]


Littrow angles:
The Littrow angles in x and y are given by the following equations: \[L_{x}=\sin^{-1}\left(\frac{\lambda}{2\cos\left(\phi\right)\Lambda_x}\right)\] \[L_{y}=\sin^{-1}\left(\frac{\lambda}{2\sin\left(\phi\right)\Lambda_y}\right)\] The Littrow angle is the angle where the -1 reflection order is exaclty parallel to the ingoing beam. In case of a 2D grating, there is one angle where the the [-1,0] reflection order matches the input beam, and another where the [0,-1] reflection order matches the input beam. If \(\phi\) is a multiple of 45°, and the periods in the two directions are the same, the two Littrow angles are the same.


Angle conversions:
Instead of denoting the ingoing beam by an angle to the surface and an angle to the x-axis, the beam can instead be described by an angle in the x-direction and an angle in the y-direction. The conversions back and forth are given in the following way: \[AOD_x=\tan^{-1}\left(\tan\left(AOD\right)\cos\left(\psi\right)\right)\] \[AOD_y=\tan^{-1}\left(\tan\left(AOD\right)\sin\left(\psi\right)\right)\] \[\phi=atan2\left(\tan\left(AOI_y\right),\tan\left(AOI_x\right)\right)\] \[AOI=\tan^{-1}\left(\sqrt{\tan\left(AOI_x\right)^2+\tan\left(AOI_y\right)^2}\right)\]


Dispersion in 1D and \(\phi\)=0:
\[\frac{dAOD}{d\lambda}=\frac{n}{\Lambda\cos\left(AOD\right)}\]