Editing Zone plate (section)

==Design and manufacture==
To get constructive interference at the focus, the zones should switch from opaque to transparent at radii where<ref>{{cite book|chapter-url=http://xdb.lbl.gov/Section4/Sec_4-4.html|publisher=Center for X-ray Optics and Advanced Light Source, Lawrence Berkeley National Laboratory|access-date=13 January 2015|title=X-Ray Data Booklet|chapter=Zone Plates}}</ref>
 
<math display="block">r_n = \sqrt{n\lambda f + \frac{1}{4}n^2 \lambda^2}</math>

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
<math display="block">r_n \simeq \sqrt{n\lambda f}</math>

For plates with many zones, you can calculate the distance to the focus if you only know the radius of the outermost zone, ''r''<sub>''N''</sub>, and its width, Δ''r''<sub>''N''</sub>:
<math display="block">f = \frac{2r_N \Delta r_N}{\lambda}</math>

In the long focal length limit, the area of each zone is equal, because the width of the zones must decrease farther from the center.  The maximum possible [[angular resolution|resolution]] of a zone plate depends on the smallest zone width, 
<math display="block">\frac{\Delta l}{\Delta r_N} \approx 1.22</math>

Because of this, the smallest size object you can image, Δ''l'', is limited by how small you can reliably make your zones.

Zone plates are frequently manufactured using [[photolithography|lithography]].  As lithography technology improves and the size of features that can be manufactured decreases, the possible resolution of zone plates manufactured with this technique can improve.

=== Continuous zone plates ===
Unlike a standard lens, a binary zone plate produces intensity maxima along the axis of the plate at odd fractions (''f''/3, ''f''/5, ''f''/7, etc.). Although these contain less energy (counts of the spot) than the principal focus (because it is wider), they have the same maximum intensity (counts/m{{sup|2}}).

However, if the zone plate is constructed so that the opacity varies in a gradual, sinusoidal manner, the resulting diffraction causes only a single focal point to be formed. This type of zone plate pattern is the equivalent of a [[holography|transmission hologram]] of a converging lens.

For a smooth zone plate, the opacity (or transparency) at a point can be given by:
<math display="block">\frac{1 \pm \cos\left(kr^2\right)}{2}\,</math>

where <math>r</math> is the distance from the plate center, and <math>k</math> determines the plate's scale.<ref>{{cite book|pages=125|author=Joseph W. Goodman|title=Introduction to Fourier Optics|edition=3rd|year=2005|publisher=Roberts and Company Publishers |isbn=0-9747077-2-4}}</ref>

Binary zone plates use almost the same formula, however they depend only on the sign:
<math display="block">\frac{1 \pm \sgn\left(\cos\left(kr^2\right)\right)}{2}\,</math>

===Free parameter===
It does not matter to the constructive interference what the absolute phase is, but only that it is the same from each ring.  So an arbitrary length can be added to all the paths
<math display="block">r_n = \sqrt{(n + \alpha)\lambda f + \frac{1}{4}(n + \alpha)^2 \lambda^2}</math>

This reference phase can be chosen to optimize secondary properties such as side lobes.<ref name="wmm" />