Bessel beam

A Bessel beam referred to in the wave optics, a special, ideal form of electromagnetic waves. One of her most important in the application properties is that they are not bending, ie its extension does not change unlike, say, in Gaussian beams during propagation. Mathematically, they are a set of solutions of the paraxial Helmholtz equation that describes the wave optics in the form of paraxial rays.

Bessel beams were theoretically constructed in 1987 by Jim Durnin and demonstrated by Joseph H. Eberly and Durnin 1987 experimentally.

Mathematical Description

In the paraxial optics light rays are in the general form

Described, place and time t, and the wave vector ( without loss of generality: propagation in the z-direction ) and ω describe the vibrational angular frequency of the electromagnetic wave. The amplitude function must now satisfy the paraxial approximation of the Helmholtz equation:

With the restricted to the x -and y- direction Laplace operator. The solution of this differential equation under the assumption of cylindrical symmetry (ie with only one transverse wave number kT: = kx = ky ) now yields for the Bessel beams:

The Bessel function is first kind of order m.

Usually referred to the special case m = 0 simply as Bessel beam.

Properties of Bessel beams

The lateral amplitude, and thus the lateral intensity distribution

Not depend on the position z in the propagation direction. Therefore, their width changed ( in contrast to Gaussian beams ) not during propagation. One speaks therefore of non-diffracting beams. This also means that the Bessel beams have no focus for the purposes of a point of maximum intensity along the direction of propagation.

Bessel Srahlen also be referred to as " self-healing ", since it may be partially blocked or disturbed at one point of the propagation axis (for example, by a scattering center ), but regain their shape afterwards in the direction of propagation.

Note that the intensity of the Bessel beams significantly lower in the radial direction decreases as of Gaussian beams:

However, for Gaussian beams:

The figure illustrates this 1/ρ-Abfalls Because of an ideal Bessel beam also contains an infinite amount of energy as the integral of the intensity in the radial direction diverges. This is one reason why ideal Bessel beams in reality are not feasible. As can also be seen from the figure, Bessel beams have significant side maxima (also in comparison to an Airy disk ), have the same energy content but a much narrower main peak.

Generation

As already mentioned can be real Bessel beams do not produce because they require an infinite amount of energy, as well as an ideal plane wave. Good approximations can, however, create an axicon by focusing a Gaussian beam using. The resulting Bessel - Gaussian beams, the self-healing properties of an ideal Bessel beam still on over a certain range.

Application

Bessel beams are used because of their self-healing properties of optical tweezers. They also provide for light microscopy disc advantages: The lens is then generated by scanning a Bessel - Gauss beam, leading to a narrower central peak, and therefore better resolution at the same axial is applied energy, such as a Gaussian light slices. In addition, the self-healing allows a greater penetration depth.