Atelier Bonryu(E)
pinhole photography
Laboratory: Pinhole Photography
Taking Pinhole Photographs - Remark
Remark#5: Resolving Power of Pinhole Photography
In the Remark#4 we derived the optimum diameter of a pinhole for the given focal length and the given wavelength of the light. Then how much resolving power is attained by such an optimized “pinhole lens”? In other words, how small sunspot can be recognized when one observes sunspots by using the optimized “pinhole lens”?
Limit of the resolving power: It can be obtained by calculating the distribution of the light intensity of the image of a point source at infinity. Though the image of a point source should be a point without area by considering geometrically, the diffracted light forms a circle with a radius 
on the focal plane (Figs. 1 and 2, 
are the wavelength of the light and the focal length).
<=Fig.1
An image of a point source is a circle with a radius b on the focal plane. Concentric zones of weak light intensity are seen around the circle.
Fig.2 =>
A 3 dimensional schematic figure of the light intensity distribution.
If images of two point sources keep away by d and the distance d is smaller than the above-mentioned radius b (d<b), it is impossible to distinguish these two images (Fig.3). Contrary for d>b two images are distinguishable (Fig.4). Therefore, it is appropriate to consider the distance d as the limit of the resolving power. In a case of a telescope, b/f is the definition of the resolving power.
<=Fig.3
Images of two point sources placed aloof by d(<b) are not distinguishable.
Fig.4=>
Images of two point sources placed aloof by d (>b) are distinguishable.
Relative Resolving Power: When one would take a photograph of sunspots so that the image of the sun fills up the whole picture plane, relation between the size S of the picture plane (a sensor) and the resolving power b becomes important and we define the relative resolving power G(=S/b). The relative resolving power is a function of a wavelength of the light 
, the focal length 
, and the size of the sensor S, as 
. Though in the case of a photograph or a computer display the resolving power is expressed as “a number of pixels per unit length”, the relative resolving power defined here is “a number of pixels per a side of a square sensor”. As astronomical objects such as the sun are located at infinity a size of an object is expressed not by a length but by an angle viewing the object. As the relation between the viewing angle 
and the size of the object is expressed as 
, the relative resolving power is derived as 
. This means that the relative resolving power is increased by increasing the focal length and the magnification factor. If the apparent diameter of the sun, about 32’(=0.00931 radian), as adopted as the viewing angle, the relative resolving power for the sun is 
, where the unit of 
is millimeter. It should be noted that the equation is derived for the case of constant viewing angle 
. Therefore, the size of the picture plane becomes larger with increasing focal length. When one takes a photograph by loading the “pinhole lens” to a ready-made camera, such as a SLR,the formula of the relative resolving power should be a function of S, 
,
and 
as 
.
Summary: For reference we show some useful graphs: a pinhole diameter as a function of the focal length, 
, the relative resolving power for observation of the sun, 
, and the resolving power as a function of the size of the picture plane S, 
, where the wavelength of the incident light is 550 nm.
Fig.5: Pinhole diameter versus focal length
Fig.6: Pinhole diameter versus focal length
Fig.7: Relative resolving power versus focal length for a fixed viewing angle (the Sun).
Fig.8: Resolving power versus focal length for a fixed size of a picture screen.
