Ridged mirror

The ridges enhance the quantum reflection from the surface, reducing the effective constant ${\displaystyle ~C~}$ of the van der Waals attraction of atoms to the surface. Such interpretation leads to the estimate of the reflectivity

${\displaystyle \displaystyle r\approx r_{0}\!\left({\frac {\ell }{L}}C,\!~K\sin(\theta )\right)}$,

where ${\displaystyle ~\ell ~}$ is width of the ridges, ${\displaystyle ~L~}$ is distance between ridges, ${\displaystyle \displaystyle ~\theta ~}$ is grazing angle, and ${\displaystyle ~K=mV/\hbar ~}$ is wavenumber and ${\displaystyle ~r_{0}(C,k)~}$ is coefficient of reflection of atoms with wavenumber ${\displaystyle ~k~}$ from a flat surface at the normal incidence. Such estimate predicts the enhancement of the reflectivity at the increase of period ${\displaystyle ~L~}$; this estimate is valid at ${\displaystyle KL\!~\theta ^{2}\ll 1}$. See quantum reflection for the approximation (fit) of the function ${\displaystyle ~r_{0}~}$.

For narrow ridges with large period ${\displaystyle L}$, the ridges just blocks the part of the wavefront. Then, it can be interpreted in terms of the Fresnel diffraction[2][3] of the de Broglie wave, or the Zeno effect;[4] such interpretation leads to the estimate the reflectivity

${\displaystyle ~\displaystyle r\approx \exp \!\left(-{\sqrt {8\!~K\!~L}}~\theta \right)~}$,

where the grazing angle ${\displaystyle \displaystyle ~\theta ~}$ is supposed to be small. This estimate predicts enhancement of the reflectivity at the reduction of period ${\displaystyle ~L~}$. This estimate requires that ${\displaystyle ~\ell /L\ll 1~}$.

For efficient ridged mirrors, both estimates above should predict high reflectivity. This implies reduction of both, width, ${\displaystyle \ell }$ of the ridges and the period, ${\displaystyle L}$. The width of the ridges cannot be smaller than the size of an atom; this sets the limit of performance of the ridged mirrors.[5]