# Outline: Zone Plate Photography

In the laboratory on the zone plate photography, the principle of the zone plate photography, the history of it, technical details of  photographing method and the subsequent treatment of zone plate photographs, various types of zone plates, applicability of the zone plate photography, and so on are described.  In the first place, the outline of zone plate photography is described to understand roughly about these topics.  It’s possible to understand the basic image of the zone plate photography by reading this part but when you’d like to know more detail, please see pages following the description on the outline.  Technical explanation and numerical formula are avoided as long as possible.  But sometimes it may be better or necessary for understanding to explain by using numerical formula.  Such technical explanations are avoided from the main body of this home page and described in Appendices.  When I studied the zone plate photography I saw many books, papers, and information on internet.  Sources of these data are summarized in the reference page.

### Zone Plate Photography

Zone plate photography is a kind of lensless photography as the pinhole photography.   The principle of the pinhole photography is based on the feature of a light which travels rectilinearly.  In a nutshell, as described in the summary page of the Salon of Pinhole Photography, light rays emitted from points on a surface of a photogenic object passing through a small hole(a pinhole) as a supporting point of a rod (the rectilinearly traveling light ray) draw a similar shape of the object on an image plane.  Therefore, it seems that the size of the pinhole should be small in some measure.  But too small pinhole blurs the projected image as described previously.   Because the area of the entrance (the pinhole) of the light ray is small, the projected image is dark and very long exposure time is necessary for taking a pinhole photograph.  To cope with this difficulty it is contrived to use a zone plate instead of a pinhole.   A zone plate is a plate on which concentric transparent and opaque zones are alternatively placed as a a target of Japanese archery (Fig.1).    As explained minutely in the following pages of the main text and Appendix_1, the diameter of the outermost zone is several millimeters when it is used to take a photograph for a focal length of several centimeters, which is far smaller than a Japanese archery target.

Fig.1  A Pinhole and a Zone plate
A zone plate with the focal length of 50 mm and the zone number of 9 is shown in the right sub-figure. The left sub-figure shows a pinhole with the image distance of 50 mm which is the same as the focal length of the zone plate of the right sub-figure.  The real size of these elements are known by the scale in the center of this figure.  The diameter of the pinhole is made as the same as that of the central zone of the zone plate.  In the case of this figure the diameter of the zone plate (f=50 mm, N=9) is about 1 mm, and the diameter of the corresponding pinhole is about 0.3 mm.  Though there are several formulas to calculate the optimum diameter of a pinhole there is no big difference between the diameters of a pinhole and a central zone of a zone plate.

### Principle of the Zone Plate Photography

Though the area of the central zone of a zone plate is the same as the area of a pinhole with the same focal length (as described previously a pinhole does not have a focal point and the focal length of the pinhole is usually defined as the image distance), a brighter image can be  realized by gathering up lights diffracted  at the outer transparent zones.  Areas of all annular zones (both transparent and opaque zones) are the same as that of the central circular zone.  Other features of a zone plate are described later.

By the way a reason why light rays passing through large circular regions of the zone plate can focalize and build up a picture cannot be explained only by the rectilinear propagation of a light.  This can be explained by diffraction and interference phenomena on the basis of the fact that a light is a wave.  In brief, light rays passing through a transparent zone and diffracting at the zone are intensified at a focal point and weakened in other places due to the interference of light waves.  In order to design the pattern of a zone plate to realize such an optical phenomenon the wavelength of the light ($$\lambda)$$ and the required focal length（$$f)$$ of the zone plate should be given first.  Then we should decide the number of zones ($$N)$$.  Usually the number of zones is defined as the total number of transparent and the opaque zones.   As the area of each circular zone is same, the width of an outer zone is thinner than that of an inner zone.  Therefore, if we make a zone plate without using a special precision instrument, the number of zones is limited by the working accuracy.  It should be also remarked that as the resolution and the chromatic aberration relate with the number of zones as described in the main text, these also limit the number of zones less than a certain number.  Moreover, as a zone pattern is determined as a function of a product of the wavelength of the light and the focal length, the pattern is same each other among zone plates with the same value of this product.  For example, the pattern of a zone plate for the wavelength of 550 nm (1 nm = 0.000001 mm) and the focal length of 300 mm is the same with a zone plate for the wavelength of 660 nm and the focal length of 250 mm.

Fig.2  Principle of a zone plate
We consider light rays perpendicularly incident to a zone plate from left, among which a light ray (A) passes through the central transparent zone and proceeds to the focal point, a (virtual) light ray (B) proceeds from the point in the first opaque zone to the focal point, and a light ray (C) is the ray from the second transparent zone to the focal point.  As the length of the path  becomes longer in the sequence of A, B, and C, the arrival time of the rays delays in this order.  As the (virtual) light ray B delays by a half wavelength at the focal point it weaken the light ray A (destructive interference).  On the other hand, as the ray C delays by one wavelength it enhances the light ray A (constructive interference).  Destructive interference is avoided by cutting light rays such as B  at opaque zones.

### Preparing a Zone Plate

As a pattern of a zone plate is, of course, more complicated than that of a pinhole, it is considerably difficult or impossible to shape a zone plate by drilling a thin metal plate as in the case of a pinhole plate.  A simple way to make a zone plate is as follows: (1) calculate the zone sizes of the necessary zone plate, (2) draw a large pattern of the zone plate such as several 10 to several 100 times larger than the necessary zone plate, (3) take a negative monochromatic photograph of the above pattern by a film camera so that the size of the zone plate image on the film is the same as that of the necessary zone plate, and (4) use this film as a zone plate.

### Characteristics of a Zone Plate Photograph

Though a zone plate is used in various fields of science and engineering, it is not popular as an optical element to take an artistic photograph.  For this reason the features of zone plate photography are not well understood or even misunderstood among photographers.  One of the misunderstandings relates with the fact that a zone plate photograph is very soft in comparison with a pinhole photograph.  Actually a zone plate photograph, generally, looks very soft and it seems that the contrast is very weak.  However, resolution of an image projected through a zone plate is considerably higher than that of a pinhole photograph and it can be theoretically proved   that the resolution is increased by increasing  the number of zones. But, we feel a very soft and vague impression from a photograph taken as it is. This impression is related to several characteristics of a zone plate photograph.

<Background Light>
One of the reasons is a problem concerning the background light of the zone plate photography.  As described previously a main photographic image is projected on a screen by light rays diffracted at the zone plate.  Nevertheless, still more light rays than the diffracted light proceed rectilinearly at a transparent zone, which become a haze at the screen as the background light and make the extremely soft photograph.  Roughly speaking, the projected background light is a pinhole photograph due to an image by a pinhole with an excessively large diameter with the same diameter as that of the zone plate.  As the diameter of the pinhole (diameter of the zone plate) is far larger than the optimum value the image becomes too vague and the image cannot be accepted as the image of the photogenic object.  Consequently it is concluded that a zone plate photograph taken as it is is a photograph composed of a very sharp image due to the first order diffraction light and a very vague image due to a pinhole with an excessively large diameter.  By the way this vague image produces a halo around the image of the object, which is prized as it makes the zone plate photograph very attractive.  When we use a digital camera it is rather easy to remove unnecessary haze from a photograph.  But, as this operation is carried out by removing uniformly the brightness of the background light, it is not effective foe an intense background light.  It should be remarked that the word, “background light” usually means the light from points other than a main object, like an environmental light, but here the light is denoted as the background light, which does not converge to the focus.

<Focal Point>
Next, it should be noted that a zone plate has a focal point and the focal length is defined, though there is not a focus in the case of a pinhole.  Therefore, focusing is necessary when one takes a nearly located object by a zone plate camera.  This means that the macro-photography and the telescopic photography with high magnification factor are executable when focusing is carried out carefully, though except for a zone plate with a long focal length and/or a large zone number almost all zone plates are deep-focus as described in the page of “Focusing and the Depth of Field”.  But as described in the following it is important that the focal length of a zone plate depends strongly on the wavelength of a light.  This means that the focal length of a zone plate used for photographing an object changes when the wavelength of the light from the object changes.  Therefore, it is necessary to adjust the image distance depending on the dominant wavelength of the light from the photogenic object.  Consequently, focusing operation is necessary for the zone plate  photography depending on the wavelength of light though the zone plate is a deep-focus optical element.

<Chromatic Aberration>
Therefore, to know the chromatic aberration of a zone plate is very important.  As described above there exists a very large chromatic aberration by which the focal length of a zone plate is inversely proportional to the wavelength of a light. Moreover, the wavelength range for focusing due to the chromatic aberration is also inversely proportional to the number of zones.  Consequently, if one increase the number of zones for the purpose of higher resolution the wavelength range of the converged light rays becomes narrow abruptly.  Ant it is expected that a colorful image of an object may not be taken.  Even by a zone plate with the zone number of about 20 corresponding to a standard lens of a SLR (f= 50 ~ 100 mm) the wavelength range is narrower than the range of the visible spectrum.  But strangely enough, the color of an image taken by a considerably short focal length does not look unnatural so much.  The reason of this phenomenon is not clearly understood but it may be related to the optical illusion, that a human brain recognizes a color by adjusting a signal spreading in the nerves system.  If this photograph is decomposed into the RGB channels it is usually found that the image of the G channel is clearly taken.  This is because the zone plate is designed for the wavelength of 550 nm and the dominant wavelength of the light from an object is about 550 nm which is the central wavelength of the visible spectrum.  Generally speaking the main light with the wavelength which determines the focal length must be strong relatively among the lights from the photogenic object.

<Black Background and Transparent Color>
The above described features of a zone plate photography are well-known among zone plate photographers from the past.  By taking a lot of zone plate photographs one may find that there are important features of the zone plate photograph.  These features are: a photograph of an object rising up from a black background and a zone plate photograph with vividly transparent colors are often taken.  It is considered that these phenomena happen when the coherence time is short for the case with rather

### Taking a Zone Plate Photograph

Though the zone plate is widely utilized for scientific and engineering applications, it is not very popular in the field of the artistic photography and, consequently, the features of the zone plate photography are not well-understood and are, sometimes, misunderstood in this field.  It is often said that in comparison with a pinhole photograph far a softer photograph is taken by a zone plate.  Actually a photograph taken by a zone plate looks very soft.  However, it is shown theoretically that the resolution of a zone plate photograph is made higher than that of a pinhole photograph for the same focal length and it can be made higher and higher with increasing the number of zones.  Nevertheless, a zone plate photograph untouched as it is taken gives very vague impression.  This is due to various features of the zone plate photography, and among them the most important reason is considered to be an effect of a background light.  As described previously an image projected on the image plane is formed by interference of light waves diffracted at transparent zones, but at the zones a larger quantity of light goes straight in the direction of the incident light than the diffracted light (the zeroth order diffraction).  This straight going light makes a halo around the image.  From the artistic viewpoint the halo around the image of the photogenic object with a rather small size is often considered as giving a favorable impression of a zone plate photograph.  However, the straight going light originating from an ambient light gives haze over the wide area on the image plane.  If the zone plate photograph is taken by a digital camera, it is rather easy to remove the haze by subtracting the nearly uniform “brightness value” corresponding to the background light from that of each pixel.  However, as this process is apt to degrade an image quality and the photograph becomes dark, this method is not effective when the ambient background light is very strong.  But it can be often mitigated by an HDR processing.  These processes will be described in detail in the main text.  By the way, though the term, a “background light” is usually used for a light coming from an object other than the targeting one or an ambient light, in this site we indicate, by this term, the rectilinearly coming light without suffering diffraction at the transparent zones of the zone plate.

In the zone plate photography focusing property and chromatic aberration are no less important than the background light.  At first a zone plate has a definite focal length and focusing operation seems necessary, especially, to take a closely located photogenic object.  Though a zone plate makes a brighter image in comparison with a pinhole, it is still too dark to do focusing through a viewfinder of an SLR and sometimes it is necessary to set the position of the zone plate manually by using a calculated value of distance.  In general, however, from a practical viewpoint focusing is not necessary except for cases of a zone plate with a very long focal length or with a very large zone number.

Next, there is a problem of the chromatic aberration.  As described above a focal length of a zone plate is inversely proportional to the wavelength of the light, which brings a very large chromatic aberration.  Moreover, as the permissible range of a position of the focal point due to the chromatic aberration is inversely proportional to the number of zones, it seems that a colorful object cannot be taken with high resolution by a zone plate with a large zone number.   Though even a usual zone plate camera with a rather short focal length (f=50 – 100 mm) and a small number of zones (10 ~ 20 zones) cannot cover the whole wavelength range of the visible light (400 – 700 nm), color of a photograph by the zone plate camera does not look too unnatural.  However, if one decomposes this photograph into three images of  the RGB channels, one will find that only the image in the G channel (a light of f=550 nm is a green light!!) is in focus and images in the other channels are out of focus.  A reason why a zone plate photograph looks rather natural under such a situation may be explained by the fact that  the shape of the image of the photogenic object is expressed by a sharply focused light and the color of the object is expressed by superposition of the visible lights in all the RGB channels.  To take a zone plate photograph, therefore, the light with a wavelength determining the focal length should be relatively strong in a measure within the viewing frame.

### Applications of Zone Plates

A lens and a mirror reflector are very useful optical components and used extensively for various applications.  On the other hand, lensless optical systems such as a pinhole or a zone plate are not conveniently used as a lens or a mirror reflector because of various constraints associated with them.  However, such lensless systems have two favorable features, (1) no material is indispensable on a pathway of the light for refraction or reflection, and (2) the weight of the system is very light even when it is very large.  By utilizing the features some lensless optical systems have been made or projected to develop.  The first feature is very important to apply a zone plate to an optical system for lights other than visible lights such as infrared, ultraviolet lights, or X-ray, gamma-ray, and various particle beams, and actually the zone plate optical systems are taking active parts in the field of experimental physics.  This is because there is no appropriate optical material as a lens for these lights or particle beams.  For these objects an above-described zone plate made of a film cannot be used, of course, and micro-fabricated metal or ceramic zone plates are used.  The second feature is extremely advantageous for a space telescope placed on a satellite orbit.  As a planning a space telescope with a focal length of several 10 km and diameters of a sensor and a zone plate are more than several meters is considered.  It is almost impossible to construct a telescope of glass lenses or mirrors of the same size because it will become too heavy.  However, it seems still considerably difficult to realize such a large space telescope as it is necessary to control such a large zone plate system within an accuracy of millimeters.

### ゾーンプレートの仲間

Various zone plates

(a) and (b):  Fresnel zone plates consist of white (transparent) zones and black (opaque) zones. The central (first) zone is transparent in the case of (a) and it is opaque in the case of (b).  (c): A Gabor zone plate where the transparency changes sinusoidally in the radial direction.  (d): A photon sieve where transparent zones of a zone plate are replaced by a lot of holes.