Ā©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 āinfocusā

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

Lens

Blur disc

Object point

Object distance = s Focal length = f

Image distance = sā

(a) Pinhole (b) Conventional

Film plane

Pinhole Cameras.doc – 2 – Ā©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 )

s

b d s f G ļ£· = ā¢ +

ļ£ø

ļ£¬ ļ£¶

ļ£

= ā¢ ļ£« + ,

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 āfocal lengthā or lens-to-image distance increases. The

diameter of the diffraction blur (bD) depends upon the wavelength of light (Ī»), the aperture

size (d), and the image distance (f), and is given by the equation:

f

d

b 2.44 1 D ā¢ ļ£· ļ£·

ļ£ø

ļ£¶

ļ£¬ ļ£¬

ļ£

ļ£«

= ā¢ Ī» ā¢

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:

( )

2

d

1 M 2.44 f 1ļ£·

ļ£ø

ļ£¶ ļ£¬

ļ£

+ = ā¢ Ī»ā¢ ā¢ļ£« , so

(1 M)

d 2.44 f

+

= ā¢ Ī» ā¢ , where = ļ£·

ļ£ø

ļ£¬ ļ£¶

ļ£

= ļ£«

s

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 (Ī», f, d) must be

in the same units (millimeters or inches, say). Note that the (1+M) term ādisappearsā 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

+

= ā¢ This is the Prober-Wellman equation.

Pinhole Cameras.doc – 3 – Ā©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 ā 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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.01 0.1 1 10 100

Magnification

Pinhole Diameter,

millimeters

Focal Length,

millimeters

250

200

150

125

100

75

50

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[ cameraās focal length in millimeters / (magnifaction+1) ]

When subject at infinity [ magnification equals zero ]

Pinhole size = X * SQRT[ cameraās 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 Ā½ diameter of the

preferred size +2 more stops of light are required.

To find f-stop of camera

f-stop = cameraās focal length in millimeters / pinhole diameter in millimeters

or

f-stop = cameraās focal length in inches / pinhole diameter in inches

To find the distance and view area for PinPLUS cameras

Distance subject to pinhole = (1 / Magnification) * cameraās focal length in inches)

Horizontal view window = (1 / Magnification) * filmās horizontal length in inches)

Vertical view window= (1 / Magnification) * filmās vertical length in inches)