When high energy lasers with the power to do serious damage to animals, plants, and minerals were being built, it was necessary to measure how much power was being deposited in a small region in the focal plane. The description generally grew to the point that one wanted to determine how much power could be deposited in a bucket of a certain size placed at the target. Thus rose the name power in the bucket.
In some cases the bucket was a physical entity, like a high power calorimeter. In the laboratory, it can be as simple as a specific size pinhole backed by an integrating detector. Without worrying too much about the details, it is just a well-known bit of trivia among optical engineers that "the power in a bucket (where the bucket is the size of the first Airy ring in the diffraction pattern of an unaberrated beam in a circular aperture) contains 84% of the total power." So it is written.
Another less known bit of trivia is that Zernike polynomials provide a mathematically convenient way to describe the phase of an optical beam. We can represent phase in a number of ways. One way is just to say the beam has so much focus, so much astigmatism, so much coma, and so forth. Being scientists, and therefore creatures of mathematics, we decided that a series of functions with terms that represent those aberrations would be a useful thing to have around.
We can lay out the phase of an electric field (the beam of light) as a three-dimensional surface, the height of which is the phase advance (or retardation) of the beam. If the beam has just focus, we find that it has a different spherical shape than that which would propagate smoothly to a focus in the focal plane; the different, out of focus shape looks like a bowl.
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