We have almost always chosen to work directly with twisted light, because Laguerre-Gaussian and Bessel beams form a good basis to construct any possible beam. However, the emphasis placed on plane waves is overwhelming, and it's interesting to decompose twisted light (our complex beam) into plane-waves and consider the interaction of each component with the bulk semiconductor.
A modal decomposition, that is writing the field in terms of a sum/integral of elementary plane-waves, is a standard procedure to analyze complex beams. To name only one (more familiar to me) there is the problem of reflection/refraction, that leads to the shift of the beam's centroid. The phenomena are known as Goos-Hänchen and Imbert-Fedorov effects, and they are analyzed in terms of reflections of single plane-waves using Fresnel coefficientes.
A collimated Bessel beam $ {\bf{E}}({\bf{r}},t)= E_0 (\hat{x} + i \sigma \hat{y}) e^{-i \omega t} J_\ell(q_r r) e^{i\ell \varphi} e^{i q_z z} $ is decomposed as
\begin{eqnarray} {\bf{E}}({\bf{r}},t)=\frac{(-i)^\ell}{2\pi} E_0 (\hat{x} + i \sigma \hat{y}) e^{-i \omega t}\int_{-2\pi}^0 e^{i\ell \eta} e^{i{\bf{q}}(\eta) \cdot {\bf{r}}} d\eta
\end{eqnarray}
(for more details see post) The beam is written in terms of plane-waves $ e^{i{\bf{q}}(\eta) \cdot {\bf{r}}}$ with wave-vector $\mathbf q(\eta)=q_r \cos(\eta)\hat{x}+q_r \sin(\eta)\hat{y}+q_z\hat{z}$ lying on a cone surface in momentum q-space.
Note that we only need plane-waves parametrized by a single number ($\eta$). This is in contrast to a modal decomposition of a Laguerre-Gaussian beam, that would require plane-waves parametrized by two parameters. This is another advantage of Bessel beams, see post.
Also, each plane-wave has a particular phase $e^{i\ell \eta}$ given by the orbital angular momentum index $\ell$.
In the interaction with matter we can now think on each separate plane-wave, all contributing to the total optical transition. The modal decomposition is particularly good when treating the bulk case, since here we can use Bloch states to represent electronic states. Before we had found [1] that a twisted-light beam produces a cone-like transition from a single valence-band state to a superposition state in the conduction band.
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| Figure 1: An optical transition induced by twisted light in a bulk semiconductor. The semiconductor is represented by Bloch states, and the transition is a one-to-many (superposition) states. |
From the point of view of a modal decomposition, we understand the result as follows. An electron in a single valence-band state, as in Fig. 1, will experience all plane-waves in the decomposition. Each plane-wave, carrying a linear momentum $\mathbf q(\eta)$ would induce a transition that transfers energy and momentum to the electron, as it is clear from the photon-drag effect. For example, one particular plane-wave produces the transition represented by the arrow in Fig. 1. What we observe in the conduction band is the total effect, that translates into an electron's superposition state. Even more, in our early study [1] we also found the relative phase (carried by the light) among electron states. So, we see a good agreement between the two viewpoints: using the Bessel/Laguerre-Gaussian mode and the modal decomposition.
[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).
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