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Diffraction

 

Imagine yourself standing on a bridge between two jetties enclosing a calm harbor, looking down at the wave pattern. You will see the parallel waves from the open sea spreading out on the surface of the water inside, roughly as shown in the figure below:

 

 

 

The point to notice here is that the waves continue almost unchanged through the middle of the gap, but spread out at the edges. If the gap were wider, it is clear that the edges would be further apart, and that most of the waveform would be undisturbed. If the gap were narrower, more of the waveform would be affected by the edge effect. This phenomenon is a characteristic of waves in general, including light, and is called diffraction. 

 

The diagram of the harbor entrance above might just as well be the cross-section through a camera lens, with the jetty taking the role of the diaphragm. Just as before, if you close the aperture, the effect of diffraction becomes more important in relation to the total amount of light passing through the lens. As the opening gets very small, the amount of light diffracted (spread out) becomes significant, so that the image of a point source of light is enlarged into a circular blob of light on the film, with a somewhat soft edge. Actually, because of interference effects among the waves (not shown in our harbor diagram), this circle of light is surrounded by a dark ring, then by a dimmer ring of light, and so on successively, as shown in the figure below, but beyond the first dark ring the pattern is so much dimmer than the central disk of light that we can safely ignore it for practical photographic purposes.

 

 

 

 

This central disk of light is known as an Airy disk, after Sir George Airy, the British 19th century scientist and Astronomer Royal, who studied this problem, among many others. The diameter of the Airy disk is customarily taken as the diameter of the darkest point in the first dark ring surrounding the central blob of light, and can be calculated by means of the simple equation:

 

 

 

 

In this equation, λ is the wavelength of the light and x is the aperture number. The wavelength of light is usually measured in nano-meters (billionths of a meter: 1 meter = 109 nano-meters), so the diameter of the Airy disk has the same units. This formula is correct for the image of a point source of light at a great distance, with the lens focused on infinity. For macro work, the calculated D must be increased by the factor:

 

 

 

 

Here, f is the focal length of the lens and e is the lens extension, i.e. the amount by which the lens has been moved forward from its infinity position in order to focus on a nearer object. Both e and f are generally measured in millimeters (mm). 

 

The wavelengths of visible light extend from about 380nm at the violet end of the spectrum, verging on ultra-violet (UV), to about 780nm at the red end, approaching the infra-red (IR) region. If we take an average wavelength of (say) 580nm in the middle of this range, then we can compute the diameter of the Airy disk at this wavelength for any aperture number, as shown in the table below:

 

 

x = f/

D (nm)

1.0

1,415

1.4

2,001

2.0

2,830

2.8

4,003

4.0

5,661

5.6

8,006

8

11,322

11

16,011

16

22,643

22

32,022

32

45,286

45

64,045

64

90,573

90

128,089

 

 

At f/22, for example, the diameter of the Airy disk is about 32000nm, or 32 millionths of a meter, or 0.032 millimeters. This result is very close to the diameter of the circle of confusion used in computing depth-of-field tables (about 0.03mm for the 35mm format - the figure varies slightly from source to source), and this is the reason why 35mm lenses rarely have aperture settings smaller than f/22: because at any such smaller aperture the sharpness only gets worse because of diffraction, and you cannot increase the depth of field any further (unless, of course, you can accept a larger circle of confusion). 

 

Some macro lenses for the 35mm format have apertures up to f/32 or even more, but you should understand that by using such a small aperture you are only gaining extra depth of field at the cost of reduced overall sharpness. In macro photography, this may be a reasonable trade-off, depth-of-field being so tight anyway. However, you will also be limiting the maximum print size that you can get from your image.