Grating pair dispersion calculator










About this calculator

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 θd\theta_d for diffraction order m=1m=-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=Nm2λ3L2πc2d2[1(mλdsin(θi))2]3/2GDD = -\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=3λ2πc1+λdsinθisin2θi1(λdsinθi)2GDDTOD = -\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 NN is the number of passes (in this case 2), m=1m=-1 is the diffraction order, λ\lambda is the center wavelength, dd is the grating period (inverse of the line density), LL is the physical distance between the two parallel gratings, and θi\theta_i is the incidence angle.

Littrow angle is calculated as

θL=arcsin(λ2d)\theta_L = \arcsin\left(\dfrac{\lambda}{2d}\right)

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

θD=arcsin(sinθi+mλd)\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

θDλ=md1(mλdsinθi)\dfrac{\partial \theta_D}{\partial \lambda} = \dfrac{m}{d\sqrt{1 - \left(\dfrac{m\lambda}{d} - \sin\theta_i\right)}}