Moiré fringe spacing formula ($D=1/|{\mathbf{g}}_1-{\mathbf{g}}_2|$)

Subtract the phases $2\pi {\mathbf{g}}_1\cdot \mathbf{r}$ and $2\pi {\mathbf{g}}_2\cdot \mathbf{r}$ of two projected periodic structures. Their relative phase repeats along the difference vector after distance $D=1/|{\mathbf{g}}_1-{\mathbf{g}}_2|$. For magnitudes $1/d_1$ and $1/d_2$ separated by angle $\theta$, this gives $D=d_1{d}_2/\sqrt{d_1^2+d_2^2-2d_1{d}_2\cos \theta }$. At zero rotation it reduces to $d_1{d}_2/|d_1-d_2|$; equal spacings give $d/[2|\sin (\theta /2)|]$. Finite specimen extent limits how many periods can be observed.