Zone plate

A concentric diffractive focusing element with successive zone boundaries chosen to differ by half a wavelength of optical path to a focus. Alternating transmitting and absorbing zones suppress cancellation between neighboring half-period zones. For an incident plane wave, the paraxial boundaries satisfy $r_j^2=j\lambda f$. The outermost zone width controls the finest transverse scale, and the focal length varies with wavelength.

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.

For uniform coherent plane-wave illumination, an ideal binary amplitude zone plate transmits alternate Fresnel half-period zones. If $N_o$ complete odd zones are open, every one contributes approximately twice the unobstructed axial complex amplitude, in the same phase, and $I_f/I_0\simeq 4N_o^2$. This is a local light intensity gain relative to removing the plate and outer stop. It differs from comparison with a finite clear aperture: an aperture containing an odd integer number of half-period zones has axial intensity $4I_0$, giving plate-to-aperture ratio $N_o^2$. Neither gain is a total power efficiency.