Zone Plate: a quasi scientific explanation

By Guillermo Pe�ate Dec/11/1999

When an opaque object is placed between a source of light and a screen, the object will cast its shadow on the screen. A geometric analysis tell us that there will be a clear separation between the dark shadow and the bright lighted area surrounding it. In practice and contrary to what that pure geometric analysis predicts, the shadow does not have hard and fast edges, instead, this boundary is characterized by an area of penumbra where light and shadow blends together. This mixing of light and darkness is due to a very important physical characteristic of the light and all waves phenomena for that matter, that characteristic is called DIFFRACTION.

Diffraction is an anathema for glass lens photography, we usually hear the phrase: "The diffraction limit of the lens" and that means the performance of a lens is limited by the diffraction the aperture causes to the light. So why are we talking about diffraction if it is something we generally speaking want to minimize?.

Glass lenses effectively focus light by their use of the REFRACTION of light caused when light passes through mediums having different densities (air and glass usually), Zone Plates use DIFFRACTION to achieve the same purpose.

Now, let's pause at this point and show you how a Zone Plate looks like:

The actual size of it would depend on the focal length we want to use it for.

Let's keep going with the explanation:

Diffraction happens when light "skims" the edges of an opaque object. In photography that happens when light skims the edges of the aperture. The smaller the aperture the greater the diffraction will be.

Two types of diffraction are generally considered: Fraunhofer diffraction (or far-field diffraction) and Fresnel diffraction (or near-field diffraction).

The way light waves advance is very similar to the way a water wave advances when we drop a stone in a pool of calm water, like a circle that keep getting bigger and bigger. If we have a source of light and a receiving point (film plane in the case of photography) and an aperture in between and they are FAR from each other, the wave "circle" that originates from the source of light, is very big when it reaches the aperture, in the same manner, the wave "circle" that originates from the aperture is big when it reaches the receiving point, in fact, in both cases is so big with respect to the aperture that the curvature of a small segment of it can for practical purposes be neglected. In these cases, the analysis of the diffraction caused by the aperture is done using FRAUNHOFER diffraction equations.

Fresnel diffraction happens when either the source and aperture or aperture and receiving plane are close or NEAR enough to each other (this is why it is called near-field diffraction) for the curvature of the wave to count. It is Fresnel diffraction that interest us for Zone Plate imaging.

Fresnel diffraction causes FRESNEL ZONES, and in the case of a circular aperture, we can think of this zones as imaginary concentric rings that go from the edge of the aperture to the center of it. These rings meet a very special characteristic: for a given ring, the total distance from the source of light to border of a ring and to the receiving plane is exactly 1/2 wavelength of light longer that the similarly measured distance of the immediate inner or smaller ring and 1/2 wavelength of light shorter than the one of the immediate outer or larger ring.

An ocean wave, has a "peak" and a "valley", when 2 ocean waves meet each other they will form a bigger wave if when they meet they both are at their peak or at their deepest valley, (this two waves are in phase). On the other hand, if one wave is at its peak and the other at its valley, they will cancel each other, the result will be "no wave" (this two waves are not in phase, their phases are 180 degrees apart). Light waves behave in the same way.

We know that the distances source-ring-receiver in fresnel zones increment (or decrease) in half wavelength increment from one ring to the next, that means light diffracted will have to travel 1/2 wavelength more to reach the receiver, it will take to light 1/2 wavelength more time to get to the receiver, therefore, if light diffracted by, let's say, ring #5 is at its peak when it reaches the receiver, light diffracted by ring #4 will, since it takes 1/2 wavelength less time to reach the receiver, reach the receiver at its deepest valley, the result is that the wave of light from ring #5 will cancel the wave of light from ring #4. We can generalize and say that the light diffracted by EVEN ring numbers will cancel out the light diffracted by ODD ring numbers.

If we could eliminate all the light diffracted by EVEN fresnel zone rings, the cancellation of light of EVEN with ODD zone rings would not exist and the total light reaching the receiver would be the sum of light diffracted by ALL the ODD fresnel zones. If we construct a group of concentric rings with radiuses such that comply with the requirement that the distance source to ring to receiver is 1/2 wavelength longer/shorter than the previous/next ring and if we blacken all the EVEN zone ring numbers, effectively eliminating them from the equation, we will cause the effect of all the ODD ring numbers to add up and form a BIGGER WAVE at the receiver (film plane). Such a fresnel zone rings configuration would look similar to a "bull's eye" with alternating black and white rings, this configuration is what it is known as a FRESNEL ZONE PLATE or just a ZONE PLATE.

Zone plates used for photographic purposes are made by drawing a magnified zone plate on paper, for instance a 100 times larger than the actual zone plate we need and then photographing that drawing at a distance such that its size on film will be 100 times smaller, this will give us a zone plate of the right size we need. Use of a very high contrast film is mandatory to make this zone plates. Once the film is developed, we use the zone plate in the same fashion we use a pinhole. Obviously, when drawing the magnified zone plate, it would have to be a "negative" version of the actual zone plate we want (assuming we use negative film to photograph the paper zone plate), that is: if we need a zone plate that starts with a clear inner ring, the inner ring on the paper drawing should be black (if using lith film, it could be painted red also, as lith is not sensitive to wave length of that color). A zone plate is much faster than a pinhole, how much faster depends on the number of clear rings the zone plate has and the density (base+fog) of the clear rings. That fastness is paid by having less "sharpness" than a pinhole, that lack of sharpness is due to a good amount of light the zone plate let through un-diffracted, which causes highlights to have a ghostly effect. This quality (or lack of it) can be used as an advantage for some types of images. Some beautiful samples of zone plates can be seen at the web site of Joao_Ribeiro, an excellent Brazilian professional photographer, check it out: Joao Ribeiro - Zone Plate Photos

The diameter of the rings of a Zone Plate can be found with the following formula:

Diameter = 0.0469 x SQRT( R x F)

Where R is the Ring number (value 1 is used for the inner ring, 2 for the next ring, and so on), F is the Zone Plate focal length and both Diameter and F are given in millimeters.

Scientist calculate the f/stop of a Zone Plate by dividing the distance Zone Plate to film by the diameter of the outer most ring of the Zone Plate. This approach, in my opinion, will not work for when a Zone Plate is used for general photographic purposes. Logic tells me that in order to calculate the f/stop, we should only consider the areas of clear rings. A characteristic of the rings of a Zone Plate is that all of them (the rings) enclose the same area, so in order to find the f/stop of a Zone Plate we should find the area of the center ring (0.7854 x diameter^2), multiply that area for the number of clear rings and find the diameter of the circle that encloses the same area. This found diameter is the one we should use as denominator to find the f/stop of our Zone Plate. If the Zone Plate is made using Tech Pan film, I suggest you consider the f/stop of the Zone Plate as one stop less that the calculated, this will compensate for the base+fog density of the film (about 0.3D in my experience). If you use Kodak Lith (B+F = 0.05D as reported by Joao Riveiro) or Arista Lith film, just approximate the calculated f/stop to the nearest smaller full f/stop.

Pinhole is a very close relative of Zone Plate, in fact, the central ring of a Zone Plate is nothing but a pinhole. We could say that if you have done Pinhole photography, you have done "one ring zone plate photography"!!

For comments, corrections, questions and suggestions, please contact the author.