ZP_Salon

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Depth of field: The depth of field is defined as a range of the position of light sources corresponding to the position of an image screen so that the blur of the image of a point source is within a permissible limit. In the case of a lens the depth of field is defined on the basis of the diameter of a circle of confusion as in the case of the depth of focus. Similarly in the case of a zone plate it is defined on the basis of the condition that the outermost transparent zone is effective. As the result of a detailed calculation in remark (*4) the depth of field for an object located at the distance

from the camera is from

to

, where

are the number of zones and the focal length. It should be noted that the equation is not applicable for the case of an object at infinity (

). The depth of field for this case is separately calculated as

.

Examples of calculations of the depth of field: In order to get the general feel for the depth of field of a zone plate we calculate some examples of the depth of field for the case of an object at infinity (

). As described above a zone plate focused to an object at infinity is virtually in focus to all the objects from

to infinity. When a zone plate we use is made of a silver film the minimum width of zones (the width of the outermost zone) is determined by the resolution of a film. In other words the maximum number of zones is determined by the resolution of a film. Though the resolution of a silver film depends on kinds of films, development methods, and so on, it is not necessarily appropriate to say generally how large the resolution is. However, as the resolution of about 0.01 mm is attained considerably easily, we determine the minimum width of the resolution

as 0.05 mm on the safe side. By the way the minimum width

is derived as

By substituting the minimum width

, the maximum number of zone is derived as

where we use the wavelength

. Therefore, the depth of field for

is given as

.

From this result the following conclusion is obtained. When we use a “zone plate_body cap” set on a conventional SLR, the focal length is around 50 mm. Therefore, the depth of field is approximately given as

and it is safe to say that this zone plate camera is almost pan-focus. But it should be noticed that with increasing the focal length the depth of field rapidly becomes shallow. For example in the case of a zone plate with

the depth of field for

is

.

Focal length of a zone plate: An important difference between pinhole photography and zone plate photography is whether focusing should be considered explicitly or not. Even in the case of the pinhole photography there is an optimum size of a pinhole for the relevant focal length, but dependence of the pinhole size on the focal length is weak and virtually focusing is not necessary to take a photograph. In the case of a zone plate, on the other hand, as there is a definite relation among variables

,

, and

, focusing is necessary in principle by adjusting the distance from an image plane to a zone plate. In this page we discuss a depth of focus, a depth of field, and effects of focusing.

Focusing of a zone plate: We consider to fit a zone plate to a digital SLR and to take a photograph. Though an image by a zone plate is by far brighter than that by a pinhole, brightness is still low to bring the image into focus by looking at it in a viewfinder of the SLR. Therefore, usually we should calculate a distance from the zone plate to the image screen and adjust the position of the screen.

At first we consider a case when an object is a point source at infinity. In this case it is obvious that an image of the object is formed on the focal plane located at distance

from the zone plate. Therefore, the image blurs if a location of an image screen does not coincide with the focal plane. Thus a “depth of focus” for an object at infinity is defined as the magnitude of error in the actual position of an image screen from the zone plate to keep the blur within a permissible level (*4). The corresponding values for the permissible range of the location of a photogenic object is a “depth of filed” which is the range of field visually in focus. As the magnitude of the permissible error is determined depending on an objective and a device, there are various criteria on the permissible error and various definitions of a depth of focus and a depth of field.

In the case of a usual camera with a glass lens, the criterion of focusing is defined by using a diameter of a circle of confusion. This diameter is expressed as the length of a diagonal of an image sensor (or a film) divided by a coefficient of around 1500. A depth of focus is defined by the range of the focal length where the diameter of the image of a point source at infinity is less than the diameter of the circle of confusion. Similarly the depth of field is defined by the corresponding range of the distance of a photogenic object from the lens. As a diagonal length of an image sensor of a four-thirds format digital camera which I use is 21.63 mm, the diameter of a circle of confusion of this camera is about 0.015 mm. By the way in the case of a zone plate which is used conventionally for taking photographs the diameter of the image of a point source at infinity is already larger than the diameter of the circle of confusion (e.g., for a zone plate with f = 100 mm, 25 zones, the image radius is about 0.06 mm: *5). Thus, in such a zone plate the depth of focus and the depth of field defined for a conventional camera with a lens are meaningless. Therefore, the depth of focus and the depth of field for a zone plate camera are defined as the ranges of the focal distance and the distance to the object so that the light waves from the outermost zone do not become completely out of phase. It means that the light from the outermost zone does not contribute to make the resolution of the image higher when the position of the image screen or the object is out of the permissible range. Though this definition gives a measure of the resolution of the image projected by a zone plate, it should be remarked that these values cannot be compared directly with those of a camera with a glass lens.

The depth of focus for a point source at infinity (a glass lens) is a permissible range of the focal length (

). In the case of a lens, it is determined by comparing the radius of the image of the point source and the radius of the circle of confusion.

The depth of focus for a point source at infinity (a zone plate) is usually determined from the positions where a radius of an image formed by a zone plate with M zones is equal to that by a zone plate with M-1 zones.

Depth of focus: As there is a relation (*1) among the focal length

, the wavelength of the light

, the radii of zones

, the range of the permissible error of the focal length is calculated in accordance with the relation (*4). According to the result of the calculation, in the zone plate with

zones and a focal length of

designed for the light wavelength of

the contribution of diffracted waves from the

-th zone diminishes to zero at the points

and

on the optical axis where

. In other words it can be explained that within the range from

to

the zone plate with

zones is in focus. By the way the increment of the radius of the image of the point source is only about

times the radius of the image at the focal point. As described later concerning the chromatic aberration the permissible range of the wavelength also has a similar functional dependence and for the visible light (400 - 700 nm) the limitation for the wavelength is by far stringent. In comparison with this limitation the condition for the focal length is rather easy to satisfy. Though the above depth of focus is for the object at infinity, a depth of focus is also calculated for an object located at a finite distance from the camera. The range of the location of the image in focus is expressed as