05. Diffraction
Description
This article is from the Photographic Lenses Tutorial, by David Jacobson with contributions by others.
05. Diffraction
When a beam of parallel light passes through a circular aperture it spreads out a little, a phenomenon known as diffraction. The smaller the aperture, the more the spreading. The normalized field strength (of the electric or magnetic field) at angle phi from the axis is given by
2 J1(x)/x, where x = 2 phi Pi R/lambda
and where R is the radius of the aperture, lambda is the wavelength of the light, and J1 is the first order Bessel function. The normalization is relative to the field strength at the center. The power (intensity) is proportional to the square of this function.
The field strength function forms a bell-shaped curve, but unlike the classic E^(-x^2) one, it eventually oscillates about zero. Its first zero is at 1.21967 lambda/(2 R). There are actually an infinite number of lobes after this, but about 86% of the power is in the circle bounded by the first zero.
Relative field strength
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Angle from axis (relative to lambda/diameter_of_aperture)
Approximating the aperture-to-film distance as f and making use of the fact that the aperture has diameter f/N, it follows directly that the diameter of the first zero of the diffraction pattern is 2.43934*N*lambda. Applying this in a normal photographic situation is difficult, since the light contains a whole spectrum of colors. We really need to integrate over the visible spectrum. The eye has maximum sensitivity around 555 nm, in the yellow green. If, for simplicity, we take 555 nm as the wavelength, the diameter of the first zero, in mm, comes out to be 0.00135383 N.
As was mentioned above, the normally accepted circle of confusion for depth of field is .03 mm, but .03/0.00135383 = 22.1594, so we can see that at f/22 the diameter of the first zero of the diffraction pattern is as large is the acceptable circle of confusion.
A common way of rating the resolution of a lens is in line pairs per mm. It is hard to say when lines are resolvable, but suppose that we use a criterion that the center of a dark band receive no more than 80% of the light power striking the center of a light band. Then the resolution is 0.823 /(lambda*N) lpmm. If we again assume 555 nm, this comes out to 1482/N lpmm, which is in close agreement with the widely used rule of thumb that the resolution is diffraction limited to 1500/N lpmm. However, note that the MTF, discussed below, provides another view of this subject.
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