2D Gauss Diffraction Gratings

2D diffraction gratings based on Gauss lattices are a class of nonperiodic lattice in which the sites are located at $R_{j_1,j_2}=j_1^{n}d\hat{x}+j_2^{n}d\hat{y}$ with $j_1,j_2\in \{\text{…},-1,0,1,\text{…}\}$, $n\in \{2,3,4,\text{…}\}$, $d\hat{x}$ and $d\hat{y}$ orthogonal primitive vectors in the plane, and $d$ the lattice constant. Gauss lattices are treated for various orders $n$, and discuss applications for gratings separable in the $x$ and $y$ directions. These gratings, while geometrically very simple, produce complex pseudorandom diffraction patterns, though they exhibit rotational invariance and strong correlations along the $x$ and $y$ directions. Then how to generalize the approach is discussed to attain nonseparable gratings where such features are suppressed. The result is an intensity distribution like that of diffuse light, the effect originating in the breaking of the hidden translational invariance of the Gauss lattice.