The longitudinal component of twisted light

The electric- and magnetic-field components in the direction of propagation of a twisted light beam are not only interesting but can help to improve optoelectronics applications.

Complex beams -such as Laguerre-Gaussian, Bessel, and radially-polarized- can present a strong field component in the direction of propagation (here z). In the case of Bessel beams the vector potential is [1]
\begin{eqnarray}
    \mathbf A(\mathbf{r}) &=& A_0 \,\left[ \mathbf{e}_{\sigma} J_{\ell}(q_r r)
    e^{i \ell \varphi}      - i \,\sigma \mathbf{e}_z
    \frac{q_r}{q_z} J_{\ell + \sigma}(q_r r)     e^{i (\ell + \sigma) \varphi} \right]
\,,
\end{eqnarray}

[in the Coulomb gauge $\mathbf E=-\partial_t \mathbf A$] where $\sigma=\pm 1$ indicates circular polarization, and $ \mathbf{e}_\sigma = \mathbf{e}_x + i \sigma \mathbf{e}_y$. For Laguerre-Gaussian is [2]

\begin{eqnarray}
    \mathbf E(\mathbf{r}) &=& E_0 \,\left[
i k (\alpha \mathbf{e}_x + \beta \mathbf{e}_y) u
-(\alpha \partial_x u - \beta \partial_y u)\mathbf{e}_z\right]
\,,
\end{eqnarray}

where $u \propto L_{lp}(r)\exp[-(r/w_0)^2]\exp(i\ell \varphi)$ and the constants $\{\alpha, \beta\}$ give the polarization of the field; in particular the circular polarization is $\sigma = i (\alpha \beta^*-\alpha^* \beta)$. The radially-polarized beam can be easily deduced from the above fields, since it is a superposition of TL with opposite orbital and spin angular momenta [3].

It is interesting that Bessel and LG modes present a strong longitudinal field component for anti-parallel fields - having opposite spin and orbital angular momenta- close to the phase singularity: $q_r r \ll 1$. For Bessel, the field is approximated using that $ J_{m}(x) = (x/2)^m/m! $, which shows that $\mathbf A_z \propto (q_r r)^{\ell+\sigma}$ and is thus stronger than $\mathbf A_\perp$ only when $Sgn(\sigma) \neq Sgn(\ell)$ (antiparallel field). The same happens to the LG beam, that reduces close to the singularity to ($\ell>0$)
\begin{eqnarray}
\label{Eq:Ex}
    E_x (\mathbf r)
&\propto&
    i
   r^\ell
    e^{i\ell \varphi}
\\
\label{Eq:Ey}
    E_y (\mathbf r)
&\propto&
    -\sigma
   r^\ell
    e^{i\ell \varphi}
\\
\label{Eq:Ez}
    E_z (\mathbf r)
&\propto&
    (\sigma-1)
    r^{\ell+\sigma}
    e^{i(\ell+\sigma) \varphi}
\,,
\end{eqnarray}
where for parallel beams ($\sigma=1$)  $E_z=0$. Finally, when a radially-polarized beams is focused it evolves a strong component in the direction of propagation, as Fig. 1 suggests.

Figure 1: By focusing a radially-polarized beam the electric

field is tilted (to "follow the rays"), and develops a strong component in $z$.

The strong $z$-component may provide new ways to control matter. For example, using twisted light at normal incidence, one could excite intersubband transitions in quantum wells or light-hole states in quantum dots, without modifying the sample. In contrast, if one wants to use "normal" light (with no strong longitudinal component), one has to cut the sample, in order to illuminate it from the side and have the electric field pointing parallel to the nanostructure's quantization axis. This is clearly cumbersome in an experiment, but may become a serious problem in applications that for example rely on scalability: one wishes to address many nanoparticles that sit on a single sample by displacing the beam center.



References

[1] G. F. Quinteiro and P. I. Tamborenea, Theory of the optical absorption of light carrying orbital angular momentum by semiconductors, EPL 85, 47001 (2009).

[2] Loudon, Rodney. Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics. Physical Review A 68, no. 1 (2003): 013806.

[3] Quinteiro, G.F., Reiter, D.E. and Kuhn, T., 2017. Formulation of the twisted-light–matter interaction at the phase singularity: Beams with strong magnetic fields. Physical Review A, 95(1), p.012106.