Focusing of Laguerre-Gaussian beams

Focused fields exhibit interesting features not present in their paraxial counterpart. We introduce here the topic and one of the main qualitative differences between Laguerre-Gaussian beams and their focused beams.

In older posts we argued that focused beams present interesting features not present in collimated beams, some examples are: $i)$ possible excitation of light-hole states in semiconductor quantum dots to control spin states, $ii)$ possible excitation of intersubband transitions in quantum wells, $iii)$ existence of strong magneto-optical interactions.

Focused beams are created from collimated lasers beams using lenses. The process can be understood in the following way: The incoming light from a laser passes through a lens, and each ray composing the beam is bent towards the center (axis), focusing the entire beam. Most important is that we assume that the electric (and magnetic) field follows each ray, so that it remains always perpendicular to the ray's direction. As a result, a larger $E_z$ component arises. However, this is not the only new feature.

Figure 1: The electric field (blue thin arrows) follows each ray. 

The theory to predict the EM field after the lens is based on papers by Wolf and Richards [1-2]. Several authors [3-5] have studied related problems to the focusing of a Laguerre-Gaussian beam. They agree in that the components of orbital and spin angular momentum, clearly defined for the incoming collimated beam, become mixed. That is, an incoming beam of the type
\begin{eqnarray}
E_x &\propto& r^\ell e^{i \ell \varphi}
\nonumber \\
E_y &\propto& i \sigma r^\ell e^{i \ell \varphi}
\nonumber \\
E_z &=& 0
\end{eqnarray}
is transformed by the lens to
\begin{eqnarray}
E_x &\propto& F_1(r) e^{i \ell \varphi} + F_1(r) e^{i (\ell+2\sigma) \varphi}
\nonumber \\
E_y &\propto& i \sigma [F_1(r) e^{i \ell \varphi} - F_1(r) e^{i (\ell+2\sigma) \varphi}]
\nonumber \\
E_z &\propto& i \sigma F_3(r) e^{i (\ell+\sigma) \varphi}
\end{eqnarray}

with $\sigma$ and $\ell$ the polarization and topological charge, respectively. Note the exponential that shows the mixing of spin and orbital AM. More interesting consequences are explored in Ref. [3-5].

References

[1] E. Wolf, Electromagnetic diffraction in optical systems I. An integral representation of the image field, Proceedings of the Royal Society A Volume 253, issue 1274 (1959).

[2] B. Richards and E. Wolf, Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system, Proceedings of the Royal Society A Volume 253, issue 1274 (1959).

[3] Paula B. Monteiro, Paulo A. Maia Neto, and H. Moysés Nussenzveig, Angular momentum of focused beams: Beyond the paraxial approximation, Phys. Rev. A 79, 033830 (2009).

[4] Konstantin Y. Bliokh, Elena A. Ostrovskaya, Miguel A. Alonso, Oscar G. Rodríguez-Herrera, David Lara, and Chris Dainty, Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems, Optics Express Vol. 19, No. 27 26132 (2011).

[5] Yoshinori Iketaki, Takeshi Watanabe, Nándor Bokor, and Masaaki Fujii, Investigation of the center intensity of first- and second-order Laguerre–Gaussian beams with linear and circular polarization, Optics Express Vol. 32, No. 16 2357 (2007).