Zone Plate Equations

Equations for a zone plate

To get constructive interference at the focus, the zones should switch from opaque to transparent at radii where:

{\displaystyle r_{n}={\sqrt {n\lambda f+{\frac {1}{4}}n^{2}\lambda ^{2}}}}

where n is an integer, λ is the wavelength of the light the zone plate is meant to focus and f is the distance from the center of the zone plate to the focus.

When the zone plate is small compared to the focal length, this can be approximated as

{\displaystyle r_{n}\simeq {\sqrt {n\lambda f}}}


For zone plates with many zones, and if you know the radius of the outermost zone, rN, and its width, ΔrN the focus will be exactly:

Or approximated (assuming the zone plate is small compared to the focal length) as

{\displaystyle f={\frac {2r_{N}\Delta r_{N}}{\lambda }}}