azimuthally and radially polarized beams

A simple superposition of twisted light beams yields fields with either azimuthal or radial polarization. They have received a lot of attention for their potential in applications, but are also interesting from a theoretical perspective.

More precisely, radially and azimuthally polarized beams can be built as a superposition of two antiparallel twisted light beams having $\large \{\ell=1,\sigma=-1\}$ and $\large \{\ell=-1,\sigma=1\}$. In their head-on or on-center interaction with nanoparticles we apply the approximation of small $\large q_r r \ll 1$.

Fig. 1: Electric field of azimuthally (left) and radially (right) polarized beams.

Azimuthally polarized fields are given by the sum of those TL beams:
\begin{eqnarray}
\large E^{\mathrm{(az)}}_\varphi (\mathbf r,t) &=&  \large  E_0  (q_r r) e^{i(q_z z - \omega t)} +\textrm{c.c.}
\nonumber \\
&--& \nonumber \\ \large B^{\mathrm{(az)}}_r (\mathbf r,t) &=&  \large  -  B_0 (q_r r) e^{i(q_z z - \omega t)} +\textrm{c.c.}
\nonumber \\
\large B^{\mathrm{(az)}}_z  (\mathbf r,t)  &=&  \large  - 2  i B_0 \frac{q_r}{q_z} e^{i(q_z z - \omega t)} +\textrm{c.c.} \,.
\,,
\end{eqnarray}
with all other components equal to zero. On the other hand the radially polarized fields given by the difference of the two antiparallel beams are
\begin{eqnarray}
\large E^{\mathrm{(rad)}}_r  (\mathbf r,t) &=& \large  i E_0  (q_r r) e^{i(q_z z - \omega t)} +\textrm{c.c.}
\nonumber \\
\large E^{\mathrm{(rad)}}_z  (\mathbf r,t)  &=& \large - 2 \frac{q_r}{q_z} E_0  e^{i(q_z z - \omega t)} +\textrm{c.c.}
\nonumber \\
&--& \nonumber \\ \large B^{\mathrm{(rad)}}_\varphi  (\mathbf r,t)  &=& \large i B_0 \left[ 1 +  \left(\frac{q_r}{q_z}\right)^2 \right] (q_r r) e^{i(q_z z - \omega t)} +\textrm{c.c.} \,.
\,,
\end{eqnarray}
with all other components equal to zero.

For both types the in-plane components vanish at the origin. In contrast, at $\large r=0$ the azimuthally polarized beam is characterized by a non-vanishing $z$-component of the magnetic field while the radially polarized beam exhibits a non-vanishing $z$-component of the electric field. Thus, close to the beam center both fields are dominated by their longitudinal contributions.

Due to their strong longitudinal-field component with high intensity and degree of focusing, these fields are useful in micro-Raman spectroscopy [1], material processing [2, 3], and as optical tweezers for metallic particles [4]. We also proposed the use of the strong longitudinal component to excite intersubband transitions in quantum wells [5] and light-hole states in quantum dots [6]. These particular transitions in QWs and QDs are technologically challenging to address, since conventional fields can only excite them if the beam propagates perpendicular to the growth direction of the sample, which typically requires cleaving the structure. From a theoretical perspective it has been also demonstrated that these fields can be classically entangled in a way similar to what we find in quantum mechanical systems [7].


References

[1] Y. Saito, M. Kobayashi, D. Hiraga, K. Fujita, S. Kawano, N. I. Smith, Y. Inouye, and S. Kawata, J. Raman Spectroscopy 39, 1643 (2008).
[2] H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[3] M. Meier, V. Romano, and T. Feurer, Appl. Phys. A 86, 329 (2007).
[4] Q. Zhan, Optics Express 12, 3377 (2004).
[5] B. Sbierski, G. Quinteiro, and P. Tamborenea, J. Phys. Cond. Matter 25, 385301 (2013).
[6] G. F. Quinteiro and T. Kuhn, Phys. Rev. B 90, 115401 (2014).
[7] C. Gabriel, A. Aiello, W. Zhong, T. Euser, N. Joly, P. Banzer, M. F¨ortsch, D. Elser, U. L. Andersen, C. Marquardt, et al., Phys. Rev. Lett. 106, 060502 (2011).