Atomic form factor for twisted vortex photons interacting with atoms

The relatively new atomic form factor for twisted (vortex) beams, which carry orbital angular momentum (OAM), is considered and compared to the conventional atomic form factor for plane-wave beams that carry only spin angular momentum. Since the vortex symmetry of a twisted photon is more complex that that of a plane wave, evaluation of the atomic form factor is also more complex for twisted photons. On the other hand, the twisted photon has additional parameters, including the OAM quantum number, $\ell$, the nodal radial number, $p$, and the Rayleigh range, $z_R$, which determine the cone angle of the vortex. This Rayleigh range may be used as a variable parameter to control the interaction of twisted photons with matter. Here we address (i) normalization of the vortex atomic form factor, (ii) displacement of target atoms away from the center of the beam vortex, and (iii) formulation of transition probabilities for a variety of photon-atom processes. We attend to features related to experiments that can test the range of validity and accuracy of calculations of these variations of the atomic form factor. Using the absolute square of the form factor for vortex beams, we introduce a vortex factor that can be directly measured.

Figure 1

Envelope of classical light rays associated with a twisted vortex beam at a radial distance $\rho \left(z\right)=w\left(z\right)$ from the beam axis. At $\rho \left(z\right)=w\left(z\right)$ the Gaussian beam intensity has decreased by a factor of $\frac{1}{e^2}$ from the maximum intensity at the beam's center. Each segment shown is a straight line. At $z=0$, all segments shown have the same distance of closest approach, $b$, to the beam center, i.e., $b\equiv \rho \left(0\right)=w\left(0\right)=\sqrt{\lambda z_R/\pi }$. Here $z$ is given in units of the experimentally adjustable Rayleigh range, $z_R$, and $\lambda$ is the wavelength of the light ray.

Figure 2

Two-dimensional side view of envelopes of intensity for a Gaussian beam, which varies as $e^{-2{\rho }^2\left(z\right)/w^2\left(z\right)}$. Each envelope shown corresponds to a surface of constant beam intensity, ranging from maximum intensity along the center line out to an intensity that has decreased exponentially by a factor of $\frac{1}{e^2}\simeq 0.135$. A three-dimensional view of the outermost envelope, shown here in (a), is given in Fig. 1, where $\rho \left(z\right)=w\left(z\right)$. Envelopes of different intensity may not cross, as then the intensities would not be different. Each envelope shown here corresponds to a ring of classical light rays of the same intensity, as shown in Fig. 1, but at a distance $\rho \left(z\right)=b\frac{w\left(z\right)}{w\left(0\right)}$ from the centerline, i.e., a distance changed by a factor of $b/w\left(0\right)$ from that shown in Fig. 1. Thus, as $b$ decreases, the intensity of the rays within this envelope increases. The horizontal scale of $z$ is in units of the Rayleigh range, $z_R$. The vertical scale of $\rho \left(z\right)/\rho \left(0\right)=\sqrt{1+z^2/z_R^2}$ is dimensionless, while $\rho \left(0\right)=b$ is expressed in units of $w\left(0\right)=\sqrt{\lambda z_R/\pi }$. Part (b) shows the paraxial region near the center of the beam where envelopes of the same beam intensity are nearly parallel to the beam axis. Part (c) shows the asymptotic region, far from the beam center, where envelopes of the same beam intensity are well described by cones corresponding to Eq. (8).