Atelier Bonryu(E)
zone plate photography
Laboratory: Zone Plate Photography
Taking Zone Plate Photographs
- Remark -
These results are rather pessimistic for photographing by the sub-focus. However, from the figures it is expected that the sub-focus may become useful for a larger zone number, and it is tad too early to give up to use the sub-focus. Especially the first order sub-focus of the zone plate with a long focal length looks hopeful. As shown in the main text the actual experience of photographing by using the sub-focus encourages this conclusion.
Remark#7: Sub-Focus
Origin of Sub-Focus: One of features of a zone plate which utilizes the diffraction and interference phenomena of a light is that a zone plate designed for a given light wavelength and a given focal length has a lot of sub-foci (
) beside the main focus (
), where +f means a focal point located at the far side of the light source and -f means a focal point located at the near side of the light source, that is, the former corresponds a focal point of a convex lens and the latter corresponds to that of a concave lens. Other foci than the main focus are called sub-foci and the focal length of a sub-focus is the focal length of the main focus divided by an odd number. By the way only a Fresnel zone plate has the sub-foci but a Gabor zone plate does not have such a sub-focus. And 
represents the focus for a light going straight to infinity. Then what is the origin of the sub-foci? To explain why there are a lot of sub-foci we need to remember in what way the pattern of the Fresnel zone plate is designed. The radius of the boundary of a zone is determined so that the light wave which passes the outer boundary of the zone reaches the focus in retard of the light wave passing the inner boundary of the same zone by an amount of half the wavelength. In this case, if the zones are made transparent and opaque one after the other, a light wave passing through a transparent zone delays from the wave passing through the next inner zone at the focus by an amount of one wavelength and all the light waves passing through the transparent zones reinforce each other. This relation is expressed as 
. As 
in the central zone the radius of the central circle is 
. The radii of the zone boundaries are calculated from the center to the outermost circle and the radius of the n-th boundary zone is calculated as 
.
By the way, what happens if the delay between the light waves passing through the neighboring transparent zones is made as integral multiple of half the wavelength? As a matter of fact a zone plate with even-numbered multiple of half the wavelength does not work because in this case light waves passing through one transparent zone are extinguished by themselves. If the multiplication factor is an odd number the light waves are not extinguished completely and some of them reach together the focal point. For example, for the multiplication factor of 3 an equation corresponding to the above one is given as 
. Then the radius of the central circle is given as 
and the radius of the n-th circle is given as 
. If we put as 
, this equation is expressed as 
. This equation gives the pattern of a zone plate with the main focal length of 
. Consequently, at the position of the main focal length f/3 the lights passing through the transparent zones where the phases of lights from neighboring zones are different by three wavelength.
Intensity of focusing light: Though it is not possible to calculate analytically the ratio of light distributed to the main and sub-foci of an actual zone plate, for the ideal zone plate with infinite zones the ratio is expressed analytically by Fourier series. For the light quantity of unity entering the aperture of a zone plate the ratio of light reaching the foci is calculated as follows. As the areas of the transparent and the opaque zones are same it is easily understandable that a half of the light entering within the area of the aperture is absorbed or reflected at the surface of the opaque zones. Therefore, the half of the entering light reach the foci as 
, where 
is the ratio of light going straight to infinity. From this expression it is easily known that the light intensity becomes small rapidly with increasing the order of the sub-focus N. For example, at the first order sub-focus only 1/9 of the incident light focuses. For other sub-foci the intensity decreases rapidly as 1/25, 1/49,... Therefore, it seems that the higher order sub-focus is useless for photographing. However, the first order sub-focus may be useful in some cases. As the designed focal length is three times as long as the desired focal length when you are to use the first order sub-focus, the advantage to use the sub-focus is that you can make a zone plate with three times larger zone number for a fixed width of the outermost zone. This may be a great advantage when you make a zone plate by using a silver film, because the resolution of a silver film is about 0.005 lines/mm at best. Thereupon a problem to be considered is the background light. If the intensity of the background light is strongin comparison with the the image of the object the photograph becomes of very low contrast. As long as we can understand from the above graph the intensity of the background light is rather low. But it seems a little unnatural. Where does the incident light go which does not focus at the first order sub-focus? To resolve this problem we calculated the light distribution on the wider region on the focal plane of the sub-focus. The following graph shows the calculated result.
The distribution of light passing through the zone plate with the focal length of 100 mm and the zone number of 5. The solid line shows the distribution on the main focal plane (f), the broken line shows the distribution on the sub-focal plane (f/3).
The distribution of light passing through the zone plate with the focal length of 100 mm and the zone number of 15. The solid line shows the distribution on the main focal plane (f), the broken line shows the distribution on the sub-focal plane (f/3).
Above figures show distributions of the light on the main and sub-focal planes when parallel light beam enters a zone plate with the focal length of 100 mm. The left figure shows the result for the case of 5 zones and the right for the 15 zones (same as the figure in the main text). As the figure in the main text is for the region near the optical axis the resolution of the image projected on the sub-focal plane seemed rather good. But at the region far from the optical axis the intensity of the background light becomes extremely large and the resolution is expected to be deteriorated considerably. Especially in the case of 5 zones the intensity of the background light becomes very large and the sub-focus cannot be used for photographing. To see the distribution of light more comprehensively we show the 2-dimensional distribution of light intensities on the plane including the optical axis and the plane perpendicular to the optical axis.
The distribution of the light on the plane including the optical axis (x=0).
The zone plate is located down below and the number (z) is the distance from the zone plate. Both the figures are for the focal length of 100 mm. The left figure is for 5 zones and the right figure for 15 zones. The light intensity is strong at z=100 mm and 33 mm on the optical axis, but in the left figure the light intensity is very strong at z=33 mm even far from the optical axis.
The distribution of light on the focal planes (the focal length of 100 mm and the zone number of 5).
The left figure shows the distribution on the main focal plane (z= 100 mm) and the right for the sub-focal plane (z=33.33 mm).
The distribution of light on the focal planes (the focal length of 100 mm and the zone number of 15).
The left figure shows the distribution on the main focal plane (z= 100 mm) and the right for the sub-focal plane (z=33.33 mm).
