Exact half-period zone radii ($r_n^2=nf\lambda +n^2{\lambda }^2/4$)

For a plane wave incident on a flat zone plate, the path difference from radius $r_n$ to axial focal distance $f$ is $\sqrt{f^2+r_n^2}-f$. Setting it equal to $n\lambda /2$ gives the title relation. The paraxial limit is $r_n\simeq \sqrt{nf\lambda }$, and the fine outer-zone width is approximately $f\lambda /(2r_n)$. Alternately removing half-period annuli makes the remaining contributions reinforce at the principal focus. Choosing the number of zones sets the aperture and fabrication scale, not the principal focal distance independently of these radii.