Grating pair dispersion calculator

Input

°

l/mm

Output

ps²

fs³

°

°

°/nm

This calculator computes the second order (GDD) and third order (TOD) dispersion of a double-pass grating pair compressor in Treacy configuration. Additionally, the calculator gives the Littrow angle, which is the incidence angle optimized for highest diffraction efficiency in blazed transmission gratings. The angle of diffraction $\theta_d$ for diffraction order $m=-1$ is calulated using the sign convention for transmission gratings. Finally, the calculator also gives the angular dispersion, i.e. how many degrees the diffracted beam spreads per nanometer of spectral width.

The formulas for second and third order dispersion are

$GDD = -\dfrac{N m^2 \lambda^3 L}{2 \pi c^2 d^2} \left[1 - \left(-m \frac{\lambda}{d} - \sin(\theta_i)\right)^2 \right]^{-3/2}$
$TOD = -\dfrac{3 \lambda}{2 \pi c} \dfrac{1 + \frac{\lambda}{d} \sin\theta_i - \sin^2\theta_i}{1 - \left(\frac{\lambda}{d} - \sin\theta_i\right)^2} \cdot GDD$
where $N$ is the number of passes (in this case 2), $m=-1$ is the diffraction order, $\lambda$ is the center wavelength, $d$ is the grating period (inverse of the line density), $L$ is the physical distance between the two parallel gratings, and $\theta_i$ is the incidence angle.

Littrow angle is calculated as

$\theta_L = \arcsin\left(\dfrac{\lambda}{2d}\right)$

and the angle of diffraction is computed from the grating equation as

$\theta_D = \arcsin\left(\sin\theta_i + m\dfrac{\lambda}{d}\right)$

Finally, the angular dispersion is found by differentiating the grating equation with respect to wavelength

$\dfrac{\partial \theta_D}{\partial \lambda} = \dfrac{m}{d\sqrt{1 - \left(\dfrac{m\lambda}{d} - \sin\theta_i\right)}}$