Beams and nanostructures with displaced axes

So far we have only talked about what happens when the light's axis coincides with the symmetry axis of the nanostructure. But what would happen in the more general case when the light is off the nanostructure center?

From an experimentalist point of view, a model describing the "off-center" illumination of nanostructures is probably more realistic and useful. First, it is easier to perform experiments on ensembles of nanostructures, illuminated at normal incidence; here, the majority of the nanostructures will see a beam whose axis is displaced form their centers. Second, for single-structure experiments, an imprecise positioning of the beam's axis might introduce other effects. So, we wonder: how does the electronic excitation change when the beam is displaced?

Figure 1: The TL beam axis is displaced with respect to 

nanoparticle's center, in this case a quantum ring.

Let's imagine a nanostructure centered at the origin of coordinate $\bf x$ and a twisted-light beam centered at the origin of coordinate $\bf y$, both origins displaced by D. For Bessel beams, we can easily find an analytical solution to the optical excitation by using the relation

\begin{equation}
\large J_\ell(q_r y) e^{i \ell \psi} = \sum_s J_{\ell+s}(q_r D) J_s(q_r r) e^{i s\phi}
\end{equation}

A beam that looks from $\bf y$ like a single-$\ell$ beam, is seen from $\bf x$ as a superposition of single-$s$ beams. We want to express everything with respect to the $\bf x$ coordinate.

We have already seen how a single-$\ell$ beam affects a quantum dot or a quantum ring. A displaced beam will produce a superposition of the effects. The mathematical details of the model can be found in Ref. [1]. A pictorial representation of the interaction of an off-centered beam with a quantum ring is shown next.

Figure 2: Valence to conduction band transitions in a quantum ring: (a) on-center $D=0$ irradiation, (b) off-center $D\neq0$ irradiation.

We already described, in previous posts, the transitions shown in Fig. 2(a): it corresponds to a beam of topological charge $\ell=1$ impinging at normal incidence at the center of the quantum ring. When the same beam is displaced a distance $D$, the nanostructure "sees" a superposition of beams having different topological charge $s=0,1,2,..$. The electron in the valence band is promoted to a superposition of states in the conduction band, with relative weights that depend on $D$, the initial and final quantum number $m$ and the original topological charge $\ell$.

It is interesting to mention that, for a particular value of $q_r D$, the original transitions (valence $m=0 \rightarrow m=1$ conduction) does not happen.

Also, a real beam carrying no orbital angular momentum will also exhibit this behavior, due to the finite waist.


Reference

[1] G F Quinteiro, A O Lucero and P I Tamborenea, Electronic transitions in quantum dots and rings induced by inhomogeneous off-centered light beams, J. Phys.: Condens. Matter 22 (2010) 505802 (6pp).