Bessel Beams: why studying them?

By far the most common beam carrying orbital angular momentum is the Laguerre-Gauss. However, there are other possibilities, in particular Bessel Beams, why should one choose to work with them?

Twisted light or optical vortices are characterized by their phase singularity encoded in the extra complex phase $\large \exp(i \ell \varphi)$, where $\large \ell$ is an integer known as the topological charge, and $\large \varphi$ is the angle of the cylindrical coordinate system $\large \{r, \varphi, z\}$. For instance, the electric field reads
\begin{eqnarray}
\large
\mathbf E(\mathbf r,t) \propto e^{i (k z - \omega t)} e^{i \ell \varphi} + c.c.
\end{eqnarray}
However, we haven't said anything about the radial profile of the beam. It is here where we have to decide whether we use Laguerre-Gauss (LG) or Bessel beams, for example.

Laguerre-Gaussian fields are specially interesting because they are paraxial beams that can be produce using the normal output of laser beams. They have electric fields with strong components perpendicular to the direction of propagation, and a small longitudinal ($\large z$) component.

On the other hand, one could work with Bessel beams. I advocate for the use of these fields, for several reasons:

• non-diffracting: The shape of the beam remains the same during propagation [1],
• solution of the full wave equation (Helmholtz): focusing the beam doesn't change its form; this means that the same mathematical expressions can be used to describe paraxial and tight focusing phenomena [for more see link],
• mathematical simplicity: only Bessel functions of order $\ell$ and $\ell\pm 1$ are needed, and the mathematical expressions look simpler than those for LG beams [for more see link],
• simple plane-wave decomposition: a superposition of plane waves with wavenumbers on the surface of a cone suffices to represent the Bessel beam [for more see link].

Figure 1: Axicon. The light rays pass through different points of the axis, and form a ring.

To produce Bessel beams one can resort to an axicon [2-3], a conical optical element that transforms the beam into a ring by mapping each source point into the optical axis as depicted in Fig. 1. Experiments have demonstrated that Bessel beams with phase singularities can be produced, see Ref. [4-5].


References

[1] J. Durnin, J. J. Miceli, Jr., and J. H. Eberly "Diffraction-free beams" Phys. Rev. Lett. 58, 1499.

[2] Zbigniew Jaroszewicz, Anna Burvall, and Ari T. Friberg, "Axicon - the Most Important Optical Element," Optics & Photonics News 16(4), 34-39 (2005).

[3] Axicon by Edmund
 
[4] Martin Bock, Jürgen Jahns, and Ruediger Grunwald, "Few-cycle high-contrast vortex pulses," Opt. Lett. 37, 3804-3806 (2012).

[5] Kazak, Nikolai Stanislavovich, Nikolai Anatolevich Khilo, and Anatolii Anatolevich Ryzhevich. "Generation of Bessel light beams under the conditions of internal conical refraction." Quantum Electronics 29, no. 11 (1999): 1020-1024.