Grating Order Calculator

Period/Resolution:

Period [nm]:Line density [l/mm]:

Period y [nm]:Line density y [l/mm]:

Grating type

AOI [°]:AOI_x [°]:

Phi [°]:AOI_y [°]:

AOI type:

Start wavelength [nm]:Start wavenumber [mm

^{-1}]:End wavelength [nm]:End wavenumber [mm

^{-1}]:
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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: \[k_x=2\pi\left(\frac{n_x}{\Lambda_x}+\frac{\sin\left(AOI\right)\cos\left(\phi\right)}{\lambda}\right)\] \[k_y=2\pi\left(\frac{n_y}{\Lambda_y}+\frac{\sin\left(AOI\right)\sin\left(\phi\right)}{\lambda}\right)\] \[k_z=\frac{2\pi}{\lambda}\sqrt{1-\left(\frac{\lambda n_x}{\Lambda_x}+\sin\left(AOI\right)\cos\left(\phi\right)\right)^2-\left(\frac{\lambda n_y}{\Lambda_y}+\sin\left(AOI\right)\sin\left(\phi\right)\right)^2}\] \[k_z=\sqrt{\frac{4\pi^2}{\lambda^2}-k_x^2-k_y^2}\] 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: \[AOD=\cos^{-1}\left(\frac{k_z}{\sqrt{k_x^2+k_y^2+k_z^2}}\right)\] \[AOD=\cos^{-1}\left(\frac{\lambda k_z}{2\pi}\right)\] \[\psi=atan2\left(k_y,k_x\right)\]

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:

\[\frac{dAOD}{d\lambda}=\frac{n}{\Lambda\cos\left(AOD\right)}\]

## Want to know more?

For further information see below.