An image of an off-axis point source: Until this point we only treated light waves which are in phase on a zone plate.  But if parallel light waves incident on the zone plate at a slant the waves are not in phase on the zone plate.  If the angle of incidence is too large, it is supposed that an image of a point source cannot be formed any more.  Calculation of how much inclination  of an incident wave is allowed is described in remark (*3) where   (radian) is the angle between the light wave and the normal to the zone plate.  From this calculation it is found that an image is formed even for an obliquely incident light wave if the condition is satisfied.  Because both the distances to the point source and to the image from the zone plate are same as in the case for the perpendicular incidence, it is assured that not only an image of a point source but also a whole image of a source with a finite size can be built up on the image screen (*3).

and the image points ( and ) and the focal length of the zone plate, and  then derive the location where an image of an off-axis point source is formed.

Formula of an image location: When an image of a point source located at a distance is formed at a distance by a zone plate with a focal length , a relation among , , and is given by the following formula of an image location (*2).

     

From this equation the position of the image is given as a function of the position of the object and the focal length of the zone plate as

     

It should be remarked that the formula of an image location of a zone plate is the same as the well-known Gaussian lens formula.  In the case of a convex lens the Gaussian lens formula is easily proved geometrically by using the two basic features of a light, i.e., “a light proceeding in parallel with the optical axis is deflected at the lens and proceeds to the focal point” and “a light passing through the center of the lens goes straight without being deflected”.  The formula of an image location for a zone plate is also proved easily as shown in remark (*2) on the different ground that light waves from point source at the point go through transparent zones should arrive at the point in phase.

   A light from infinity (green arrow) travels in parallel to the optical axis of a zone plate passing through the zone plate at U arrives at the focal point F in phase with other waves passing through other zones.  A spherical light wave from a point A  passing through the zone plate at U arrives at a point B in phase as above.  In this case the relation among , , and are expressed by the same formula as the Gaussian lens formula.

   ZP: a zone plate, FP: a focal plane, OP: an object plane located away from ZP by a distance , IP: an image plane located away from ZP by a distance .  Green thin lines show light rays from a point source at infinity, which focus at the point F located away from ZP by .  Green bold lines tilting by radian ( ) from the optical axis show oblique light rays from a point source at infinity.  These rays focus at the point on the focal plane as the on-axis rays. 

An example: For example, if we fit a zone plate with a focal length of 55 mm  to a SLR of four-thirds format such as an Olympus E-300 which I use, the angle of view ( the radius of the image circle/the focal length) is given as and square of the view angle is sufficiently smaller than unity.  Therefore, it is assured that a picture of a photogenic object located at a distance from the zone plate is built on an image screen located at a distance from the zone plate.  Actually, by this zone plate pictures are taken normally as shown in Gallery of  the zone plate.