Deriving Bessel beams à la Laguerre-Gauss

Twisted light can be derived in many ways, for example from potentials, a direct application of Maxwells' equations, or a superposition of plane waves. Motivated by the common method to find Laguerre-Gauss beams, we derive through potentials Bessel beams with strong electric fields.

The literature covering twisted light can be roughly divided into works on Laguerre-Gauss beams and works on Bessel beams. The derivation of fields using potentials follows in each case mainly a single method.

A common way to derive Bessel beams through potentials makes use of the Coulomb gauge, and a starting assumption on the transverse (to the propagation direction $z$) components of the vector potential $\mathbf A$. These are chosen so that the wave equation is satisfied, resulting in Bessel modes

\begin{eqnarray} \mathbf A_\perp(\mathbf r,t) = A_0 J_\ell(q_r r) e^{i \ell \varphi } e^{i (q_z z -\omega t)} {\boldsymbol \epsilon}_\sigma ,  ~~~~\text{(1)} \end{eqnarray}

with $ J_\ell(q_r r)$ a Bessel functions, $q_r^{-1}$ the beam waist, ${\boldsymbol \epsilon}_\sigma = (\hat{x} + i \sigma \hat{y})/\sqrt{2}$ with $\sigma=\pm 1$ the polarization vector for

circular polarization, and topological index $\ell$. The Columb gauge $\nabla \cdot \mathbf A = 0$ then determines a null scalar potential $\Phi=0$ and provides the missing longitudinal component  $A_z$. The electric and magnetic fields derived from $\mathbf A$ present an asymmetry: Close to the phase singularity at $r=0$, the magnetic field of antiparallel beams (opposite spin and orbital AM) dominates over the electric field.

In contrast, Laguerre-Gauss (LG) beams are derived from potentials in the Lorenz gauge. A complete transverse vector potential satisfying the paraxial wave equation is chosen, and the Lorenz condition

determines the scalar potential. We stress that here $A_z = 0$ and $\Phi \neq 0$.

But nothing precludes us to use the method just sketched for LG to derive Bessel beams: We postulate a pure transverse vector potential that satisfies the (complete) wave equation, Eq. 1 works. From the Lorenz condition we find a scalar potential. The resulting and complete electric and magnetic fields is given in Eqs. 16a and 16b of Ref 1.

Let us consider the particular case of a TL field with $\{\ell=2, \sigma=-1\}$ close to the phase singularity [compare with beams with strong magnetic fields]. The leading transverse terms are
\begin{eqnarray}
\label{Eq:E_2CA_b}
  {\mathbf E}(\mathbf r)
&=&
  i \frac{E_0}{2} \left(\frac {c q_r}{\omega}\right)^2
  e^{i (q_z z -\omega t)} (\hat{x} + i \hat{y})
\nonumber \\
  {\mathbf B}(\mathbf r)
&=&
  -\frac{B_0}{8 \sqrt{2}} (q_r r)^2 e^{i 2 \varphi}
 e^{i (q_z z -\omega t)}    (\hat{x} - i \hat{y})
\,.
\end{eqnarray}

Excluding the propagation factor $e^{i (q_z z -\omega t)}$ one sees that the electric field is constant while the magnetic field tends to zero as the singularity is approached ($r^2$ dependence). Here the electric field dominates. Also we see that the polarization of each is different.

Then, the symmetry between electric and magnetic fields is restored when we consider both Bessel beams deduced à la Lorenz and à la Coulomb: In the former the electric field dominates while in the latter the magnetic field dominates.

References

[1] G. F. Quinteiro, C. T. Schmiegelow, D. E. Reiter, T. Kuhn, Bessel beams revisited: a generalized scheme to derive optical vortices, arxiv:1808.00390.