Experimental evidence against extreme paraxial theory

The neglect of the field's component along the propagation direction is a common practice. However, recent experimental data indicates that this extreme approximation is not aplicable to twisted light beam.

Lax [1] showed that electromagnetic fields can be expressed in series of the paraxial parameter or ratio wavelength/waist. He found that, to lowest order, the electric field transversal (longitudinal) to the propagation direction is of zero (first) order in $\lambda/w_0$. This supports the (extreme) paraxial approximation, that neglects the longitudinal component in fields not tightly focused ($\lambda/w_0\ll 1$). However, some authors [2] include the first-order $\lambda/w_0$ longitudinal term in studying twisted light.

Is it really necessary to take into account all component of the field to accurately describe the interaction with matter? We have analyzed recent experimental data on the interaction of twisted light with a single Ca ion, and concluded that indeed we need to consider longitudinal fields, specially in the case of antiparallel fields. 

Figure 1. Quadrupole transitions in a Ca ion.

In the experiment [3] single trapped Ca ions were excited using propagating Laguerre-Gaussian beams with $\ell=0, \pm 1$, producing quadrupole transitions as shown in fig. 1. Transitions 'c' were induced by antiparallel momenta beams with orbital angular momentum $\ell=\pm 1$ and circular polarization or spin angular momentum $\sigma=\mp 1$. In contrast, transitions 'a' and 'e' were induced by parallel momenta beams with $\ell=\pm 1$ and $\sigma=\pm 1$. The Rabi frequency $\Omega_i$ of each transition $i$ was measured.



The Rabi frequency is proportional to the optical matrix element, that can be calculated as shown in earlier posts, for example 1 and 2. Then, we can compare experiment and theory. We actually compare ratios of Rabi frequencies to ratios of matrix elements, for instance $\Omega_c/\Omega_e$, to eliminate irrelevant constants.

In Fig. 2 we compare theory and experiment [4], where all ratios contain a transitions induced by antiparallel beams. Dots are ratios of experimental Rabi frequencies, while bars are ratios of calculated matrix elements. Two types of matrix elements are calculated: wide (narrow) bars correspond to matrix elements of an interaction Hamiltonian having (not having) a longitudinal electric field component. To reinforce the idea: light-orange (narrow) bars represent an extreme paraxial theory that only takes into account transverse components of the electric field. Clearly the predictions from the extreme paraxial theory strongly deviates from the experimental data, while the full theory gives an excellent match.

Figure 2. Comparing theory and experiment. Dots are ratios of experimental Rabi frequencies,
while bars are ratios of calculated matrix elements. Two types of matrix elements are considered:
wide (narrow) bars correspond to matrix elements of an interaction Hamiltonian having
(not having) a longitudinal electric field component.

One can analyse ratios of parallel beams too. This is shown in Fig. 3, where all Rabi frequencies are normalized to the specific transition 'a' -note that the only normalized transition including an antiparallel beam is $\Delta m=0$. One can see that transitions induced by parallel beams and Gaussian beams (transitions 'b' and 'd') are well described by the extreme paraxial theory.

Figure 3.  Comparing theory and experiment. All ratios are to the same specific transition 'a'.

We conclude that the extreme paraxial approximation fails for antiparallel-momenta twisted light beams.

See the related article in PRL:

REFERENCES 

[1] Melvin Lax, From Maxwell to paraxial wave optics, Phys. Rev. A 11, 1365 (1975).
[2] Rodney Loudon, Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics, Phys. Rev. A 68, 013806 (2003).
[3] C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, Transfer of optical orbital angular momentum to a bound electron, Nature communications 7, 12998 (2016).
[4] G. F. Quinteiro, Ferdinand Schmidt-Kaler, Christian T. Schmiegelow, Twisted-light--ion interaction: the role of longitudinal fields, Phys. Rev. Lett. 119, 253203 (2017). Editors' Suggestion