Optimum Pinhole Camera Design

©2002 Paul Prober & Bill Wellman

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Optimum Pinhole Camera Design
A pinhole camera forms images on film by using a very small aperture in place of a
photographic lens. Its extremely small aperture and simple geometry give it extraordinary
depth of field. A pinhole lens gives sharper average focus….
over extreme changes in
object distance, although an ordinary lens gives a sharper image for objects within its
more limited focus range.
An conventional photographic lens collects rays through a relatively large aperture, and
converges them to a point of focus on the film. Depth of focus is limited by the fact that
object points at widely different distances are cannot be brought to the same focal
distance. Focus is maintained only with modest variations in object distance, because
lens-to-image distance increases as object distance decreases, particularly at short
distances. A pinhole lens has a greater depth of field, because it creates focus simply by
limiting the diameter of the aperture, and not by converging rays of a broad ray bundle.
Figure 1 illustrates these effects, for a camera focussed at infinity and viewing a point on
a nearby object.
Figure 1. Camera Image Formation
For a conventional camera, the point of best focus is determined by setting an equal blur
for near and far objects at the nearest and furthest distances in the scene. This range of
focus typically is marked on the lens focusing ring, and depends on f-number. The “infocus”
range of the lens is maximized by setting the lens to the hyperfocal distance,
which is the nearest focus for which objects at infinity have an acceptable blur.
For a pinhole camera, virtually all points are in the same focus, which is determined by
aperture size. Best focus is achieved by choosing an optimum aperture diameter, which
depends upon the object distance and focal length. If the aperture is too small, blurring
increases due to diffraction effects. If the aperture is too large, blurring increases due to
geometric effects. However, the optimum size can be determined by a simple formula,
which now will be derived.
Geometric blur of a pinhole lens was shown in Figure 1a. Light rays emanating from a
point on the object are limited by the small aperture to a very narrow cone, which gives
rise to a uniform blur circle on the film. This blur is made smaller by making the aperture
Pinhole aperture
(diameter = d)
Film plane
Blur disc
Object point diameter = b
Object distance = s Focal length = f
Blur disc
Object point
Object distance = s Focal length = f
Image distance = s’
(a) Pinhole (b) Conventional
Film plane
Pinhole Cameras.doc – 2 – ©2002 Paul Prober & Bill Wellman
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smaller, which is the reason why it becomes a pinhole. The exact diameter of this
geometric blur (bG) depends upon aperture diameter (d), image distance (f), and object
distance (s), and is given by the equation:
d (1 M )
b d s f G  = • +

 

= •  + ,
where M is the magnification, equal to (f/s)
Diffraction blur in a pinhole camera is caused by a slight bending of light as it passes
through the aperture, which spreads a perfect point image into a Fraunhofer diffraction
ring pattern. Such diffraction rings are familiar to telescope users where great angular
magnification makes even slight diffraction noticeable. Most of the energy is in this
pattern lies in the central bright disc, which can be considered the blur diameter. In a
pinhole camera, diffraction is noticeable because diffraction bending increases as the
aperture becomes smaller. In addition, this bending is an angular effect, so blur also
increases as the camera “focal length” or lens-to-image distance increases. The
diameter of the diffraction blur (bD) depends upon the wavelength of light (λ), the aperture
size (d), and the image distance (f), and is given by the equation:
b 2.44 1 D •  

 

= • λ •
The total blur is given by the sum of these two components, bG+bD. Through calculus, the
minimum of this sum is derived, as follows:
( )
1 M 2.44 f 1

 

+ = • λ• • , so
(1 M)
d 2.44 f
= • λ • , where = 

 

= 
M f magnification
This equation defines the optimum pinhole aperture diameter for close-up work, as well
as for more distant work. In using this equation, all distance measures (λ, f, d) must be
in the same units (millimeters or inches, say). Note that the (1+M) term “disappears” at
larger distances, so this equation simplifies to what is more often cited in the literature.
For f in millimeters, and for visible light, f =0.0006 millimeters, the equation is simplified
and made easy to use.
(1 M)
d 0.038 f
= • This is the Prober-Wellman equation.
Pinhole Cameras.doc – 3 – ©2002 Paul Prober & Bill Wellman
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The figure below plots this equation for a typical range of f and M values for distant and
close-up work. Separate curves for each focal length are color-coded as shown by the
legend. (For those viewing a black-and-white copy, the sequence of the curves is the
same in the graph as the focal lengths in the legend, with the longest focal length being
at the top of the sequence.)
Aperture diameter is rather constant for small magnifications, when the object distance
(s) is much larger than the image distance (f). However, as the object moves closer, the
aperture size must be decreased to realize the best focus.
With an optimum aperture, a pinhole camera can realize one of its strongest points – the
ability to take extreme close-up photos of small objects, with an unusually large depth of
focus. The image remains consistently clear over a full range of depth of the object, and
even into a distant background. An ordinary lens gives sharper focus at one distance,
but becomes extremely blurred for close-up objects that have some depth.
A well-designed pinhole camera, at any magnification or focal length, also will give
pictures that are virtually distortionless, which is particularly useful for wide-angle and
close-up work.
Optimum Pinhole Camera Aperture
0.01 0.1 1 10 100
Pinhole Diameter,
Focal Length,
Values for different wavelengths of light
The Prober-Wellman formula gives largest acceptable pinhole size, and the lowest
f-stop pinhole for close-up and macro pictures. Table below has values for different
color temperatures for different light sources, corrected values for removing round off
errors, and values for direct values of pinholes in inches.
Color Wavelength Value for X Value for X
In millimeters Pinhole in millimeters Pinhole in inches
Infrared 0.00075 0.04278 0.001684
Red 0.00065 0.03982 0.001568
Daylight 0.00056 0.03696 0.001455
Green 0.00055 0.03663 0.001442
Blue 0.00045 0.03314 0.001305
Prober-Wellman Formula with expanded for color temperature and pinhole
diameters in metric or imperial.
To find the pinhole for close-ups and macro pictures
For close-ups and macro picture pinhole sizes.
Pinhole size = X * SQRT[ camera’s focal length in millimeters / (magnifaction+1) ]
When subject at infinity [ magnification equals zero ]
Pinhole size = X * SQRT[ camera’s focal length in millimeters ]
Note! Smaller pinhole than formula size is an acceptable pinhole size for the picture,
but the higher f/stop may be a handicap in taking the picture. For ½ diameter of the
preferred size +2 more stops of light are required.
To find f-stop of camera
f-stop = camera’s focal length in millimeters / pinhole diameter in millimeters
f-stop = camera’s focal length in inches / pinhole diameter in inches
To find the distance and view area for PinPLUS cameras
Distance subject to pinhole = (1 / Magnification) * camera’s focal length in inches)
Horizontal view window = (1 / Magnification) * film’s horizontal length in inches)
Vertical view window= (1 / Magnification) * film’s vertical length in inches)

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