A general description of Bessel beams using spherical coordinates

What is it about?

A plane wave is a solution to the wave equation that is the same at all points in space. It is thus in a sense 'invariant'. However, it has the same magnitude at all points in space as well. It is possible to add a number of plane waves and get a narrow beam of sound that is also invariant, and the Bessel beam is an example. The simplest Bessel beam is formed by a sum of all plane waves propagating at a single angle to an axis (say the vertical z axis). The resulting sound field is a beam that propagates along the z axis and has a radial amplitude described by a cylindrical Bessel function. In practice, the Bessel beam can not be generated with perfect properties and so it does disperse, but it remains invariant over a longer range than other beams. The beam is most easily described in cylindrical coordinates, but there are applications where it is useful to describe the beam in spherical coordinates, such as when a sphere is placed in the axis of the beam. You can get conditions where the beam can act like a tractor beam. My paper reviews the theory of Bessel beams including adding plane waves with a range of different angles from the axis of propagation and shows that beams with other radial amplitudes can be generated, such as a spherical Bessel beam.

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Why is it important?

Bessel beams have a number of applications such as in medical operations and a focused beam that does not disperse is useful for limiting damage done to areas other than the intended area. Other non-dispersive beams could be used which have a narrower width. The spherical Bessel beam is one example, since the spherical Bessel function decays more rapidly than the cylindrical Bessel function.

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