In previous posts we have described the interaction with solids using the minimal-coupling Hamiltonian
\begin{eqnarray}\label{Eq:minimal_coupling}
H
&=&
\frac{1}{2m}
\left[
\mathbf p -q \mathbf A(\mathbf r, t)
\right]^2 ~~~~~~ (1)
\,,
\end{eqnarray}
with the vector potential $A(\mathbf r, t)$ and canonical momentum of the charged particle $\mathbf p$. The scalar potential $U(\mathbf r)$ is zero in this case. This form is complete in the sense that it contains information about the interaction with external magnetic and electric fields. To change the mathematical form of the interaction one performs gauge transformations. In other gauges, the interaction looks different and may be partially or totally written in terms of electric and magnetic fields, instead of the vector potential.
In actual calculations -that require approximations- one form of the interaction may be more accurate than others. And in spite of its fundamental role the minimal coupling is sometimes not the best choice. Also, mathematical results expressed in terms of electric and magnetic fields are easier to interpret and to connect to experimental results.
A very common and well-known expression is the dipole-moment $-q \mathbf r \cdot \mathbf E(0, t)$, for the interaction with a particle centered at $r=0$. Can we use it to study problems with twisted-light? The quick answer is a clear no: The field of TL has zero intensity at the origin, so $\mathbf E(0, t)=0$ and there is no interaction at all!
What about using the improved version $-q \mathbf r \cdot \mathbf E(\mathbf r, t)$, typical in solid-state physics? It makes more sense, since at least it is not zero for a twisted light beam. But we have showed [1] that a more precise expression can be found for fields with phase singularities.
Note that the two expressions above describe only the interaction with the electric field. But in our post on parallel and antiparallel beams we showed that the antiparallel TL fields have strong magnetic fields, and the magnetic interaction may dominate. Then, an only-electric Hamiltonian clearly fails to capture the most important (magnetic) effects. On the other hand, for the parallel beams we are safe and the interaction is dominated by the electric field.
Let us study now paraxial parallel beams (OAM $\ell>0$ and polarization $\sigma=1$) and show how to get, in an intuitive way, the TL gauge transformation (for a precise derivation see Ref. [1]).
We see in Fig. 1 (from parallel and antiparallel beams) the in-plane electric fields of parallel and antiparallel beams. In the Coulomb gauge (Eq. 1) $\mathbf A(\mathbf r, t) \propto \mathbf E(\mathbf r, t)$, so it is clear that the curl of the vector potential $\nabla \times \mathbf A(\mathbf r, t)$ is larger in the case of antiparallel beams (left side of Fig. 1). In fact, one can show that $\nabla \times \mathbf A(\mathbf r, t) \simeq 0$ close to $r=0$ for the parallel beam. Since the curl of the gradient is always zero, one should be able to find a function $\chi$ that satisfies $\nabla \chi = -\mathbf A(\mathbf r, t)$, so that the new (primed) vector potential vanishes, according to the general formula $\mathbf A'(\mathbf r, t)=\mathbf A(\mathbf r, t) + \nabla \chi(\mathbf r, t)
$. The differential equation $\nabla \chi = -\mathbf A(\mathbf r, t)$ solves to
\begin{eqnarray}\label{Eq:gaugefunction}
\chi(\mathbf r, t)
&=&
-\frac{1}{\ell+1}
\mathbf r \cdot \mathbf A(\mathbf r, t) ~~~~~~ (2)
\,.
\end{eqnarray}
$\chi$ is our gauge transformation function! Using $U'(\mathbf r, t)=U(\mathbf r, t) - \partial_t \chi(\mathbf r, t)$ for the new scalar potential we obtain:
$$\mathbf A'(\mathbf r, t) \simeq 0 \, , ~~~~~~~ U'(\mathbf r, t) = -\frac{1}{\ell+1}\mathbf r \cdot \mathbf E(\mathbf r, t) \,.$$
Replacing in Eq. 1 we get an interaction Hamiltonian written solely in terms of electric fields:
\begin{eqnarray}\label{Eq:ham}
H
&=&
\frac{\mathbf p^2}{2m}-\frac{q}{\ell+1}
\mathbf r \cdot \mathbf E(\mathbf r, t) \,. \end{eqnarray}
In summary, our expression is similar to the improved version of the dipole-moment approximation. It leads to the same selection rules, but its magnitude is modified by the value of the topological charge $\ell$. Moreover, our expression contains no approximation in the angular coordinate and is the lowest order "pole" in the radial coordinate (depending on the value of $\ell$ it is a quadrupole, an octopole, etc).
References
[1] G. F. Quinteiro, D. E. Reiter, and T. Kuhn, Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge, Phys. Rev. A 91, 033808 (2014).
H
&=&
\frac{1}{2m}
\left[
\mathbf p -q \mathbf A(\mathbf r, t)
\right]^2 ~~~~~~ (1)
\,,
\end{eqnarray}
with the vector potential $A(\mathbf r, t)$ and canonical momentum of the charged particle $\mathbf p$. The scalar potential $U(\mathbf r)$ is zero in this case. This form is complete in the sense that it contains information about the interaction with external magnetic and electric fields. To change the mathematical form of the interaction one performs gauge transformations. In other gauges, the interaction looks different and may be partially or totally written in terms of electric and magnetic fields, instead of the vector potential.
In actual calculations -that require approximations- one form of the interaction may be more accurate than others. And in spite of its fundamental role the minimal coupling is sometimes not the best choice. Also, mathematical results expressed in terms of electric and magnetic fields are easier to interpret and to connect to experimental results.
A very common and well-known expression is the dipole-moment $-q \mathbf r \cdot \mathbf E(0, t)$, for the interaction with a particle centered at $r=0$. Can we use it to study problems with twisted-light? The quick answer is a clear no: The field of TL has zero intensity at the origin, so $\mathbf E(0, t)=0$ and there is no interaction at all!
What about using the improved version $-q \mathbf r \cdot \mathbf E(\mathbf r, t)$, typical in solid-state physics? It makes more sense, since at least it is not zero for a twisted light beam. But we have showed [1] that a more precise expression can be found for fields with phase singularities.
Note that the two expressions above describe only the interaction with the electric field. But in our post on parallel and antiparallel beams we showed that the antiparallel TL fields have strong magnetic fields, and the magnetic interaction may dominate. Then, an only-electric Hamiltonian clearly fails to capture the most important (magnetic) effects. On the other hand, for the parallel beams we are safe and the interaction is dominated by the electric field.
Let us study now paraxial parallel beams (OAM $\ell>0$ and polarization $\sigma=1$) and show how to get, in an intuitive way, the TL gauge transformation (for a precise derivation see Ref. [1]).
 |
| Figure 1: In-plane components of anti-parallel and parallel beams of twisted light. We are studying here the parallel fields (right side). |
$. The differential equation $\nabla \chi = -\mathbf A(\mathbf r, t)$ solves to
\begin{eqnarray}\label{Eq:gaugefunction}
\chi(\mathbf r, t)
&=&
-\frac{1}{\ell+1}
\mathbf r \cdot \mathbf A(\mathbf r, t) ~~~~~~ (2)
\,.
\end{eqnarray}
$$\mathbf A'(\mathbf r, t) \simeq 0 \, , ~~~~~~~ U'(\mathbf r, t) = -\frac{1}{\ell+1}\mathbf r \cdot \mathbf E(\mathbf r, t) \,.$$
Replacing in Eq. 1 we get an interaction Hamiltonian written solely in terms of electric fields:
\begin{eqnarray}\label{Eq:ham}
H
&=&
\frac{\mathbf p^2}{2m}-\frac{q}{\ell+1}
\mathbf r \cdot \mathbf E(\mathbf r, t) \,. \end{eqnarray}
In summary, our expression is similar to the improved version of the dipole-moment approximation. It leads to the same selection rules, but its magnitude is modified by the value of the topological charge $\ell$. Moreover, our expression contains no approximation in the angular coordinate and is the lowest order "pole" in the radial coordinate (depending on the value of $\ell$ it is a quadrupole, an octopole, etc).
References
[1] G. F. Quinteiro, D. E. Reiter, and T. Kuhn, Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge, Phys. Rev. A 91, 033808 (2014).
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