Radiation pattern of two identical emitters driven by a Laguerre-Gaussian beam: An atom nanoantenna

We study the directional properties of a radiation field emitted by a geometrically small system composed of two identical two-level emitters located at short distances and driven by an optical vortex beam, a Laguerre-Gaussian beam which possesses a structured phase and amplitude. We find that the system may operate as a nanoantenna for controlled and tunable directional emission. Polar diagrams of the radiation intensity are presented showing that a constant phase or amplitude difference at the positions of the emitters plays an essential role in the directivity of the emission. We find that the radiation patterns may differ dramatically for different phases and amplitude differences at the positions of the emitters. As a result the system may operate as a two- or one-sided nanoantenna. In particular, a two-sided highly focused directional emission can be achieved when the emitters experience the same amplitude and a constant phase difference of the driving field. We find the general directional property of the emitted field that when the phase differences at the positions of the emitters equal an even multiple of $\pi /4$, the system behaves as a two-sided antenna. When the phase difference equals an odd multiple of $\pi /4$, the system behaves as a one-sided antenna. The case when the emitters experience the same phase but different amplitudes of the driving field is also considered and it is found that the effect of different amplitudes is to cause the system to behave as a unidirectional antenna radiating along the interatomic axis.

Figure 1

A Laguerre-Gaussian laser beam with a doughnut-like intensity profile interacting with two atoms. The beam travels along the $z$ axis but is displaced by a distance $d$ along the $x$ axis. The atoms distance $r_{12}$ from each other are located at positions $(0,r_{12}/2,0)$ and $(0,-r_{12}/2,0)$, respectively.

Figure 2

Intensity pattern (in arbitrary units), up to a radial distance equal to the beam waist, in the case of a Laguerre-Gaussian laser beam with $l=1$ and $p=20$. The beam is assumed to propagate along the $z$ axis. Inset: How the two atoms have to be irradiated by the beam in order to experience different light intensities and thus to ensure different Rabi frequencies.

Figure 3

The two atoms located on the $xy$ plane are separated from each other by $r_{12}$ and are irradiated with an optical Ferris wheel beam propagating along the $z$ direction with helicity equal to $l=15$. Atoms are at points where they experience different intensities of the field. Inset: Intensity pattern at the first quadrant of the $xy$ plane.

Figure 4

Schematic of the system composed of two emitters represented by transition dipole moments ${\vec{\mu }}_1$ and ${\vec{\mu }}_2$ and irradiated with a laser beam propagating in the direction perpendicular to the interatomic axis, ${\vec{k}}_L\perp {\vec{r}}_{12}$. The laser has a cylindrically symmetric (doughnut-shaped) intensity profile. With this variable intensity profile, the two emitters can be excited with arbitrary amplitudes and phases.

Figure 5

Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda /4$ with ${\mathrm{\Delta }}_L=0$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=0.2\mathrm{\Gamma }$, and different values of the phase difference ${\varphi }_d:{\varphi }_d=0$ (dashed-dotted black line), ${\varphi }_d=\pi /4$ [solid (red) line], and ${\varphi }_d=-\pi /4$ [dashed (blue) line]. The positions of atoms with the transition dipole moments polarized perpendicular to the plane of the paper $(\overline{\mu }\perp {\overline{r}}_{12})$ are shown by filled (brown) circles.

Figure 6

Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda /4$ with ${\mathrm{\Delta }}_L=0$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=0.2\mathrm{\Gamma }$, and different phase differences ${\varphi }_d:{\varphi }_d=0$ (dashed-dotted black line) and ${\varphi }_d=\pi /2$ [solid (red) line].

Figure 7

Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda /2$ with ${\mathrm{\Delta }}_L=0$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=0.2\mathrm{\Gamma }$, and different phase differences ${\varphi }_d:{\varphi }_d=0$ (solid black line), ${\varphi }_d=\pi /4$ [dashed-dotted (red) line], and ${\varphi }_d=\pi /2$ [dashed (blue) line].

Figure 8

Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=3\lambda /4$ with ${\mathrm{\Delta }}_L=0$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=0.2\mathrm{\Gamma }$, and different phase differences ${\varphi }_d:{\varphi }_d=0$ (dashed black line), ${\varphi }_d=\pi /4$ [solid (red) line].

Figure 9

Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda$ with ${\mathrm{\Delta }}_L=0$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=0.2\mathrm{\Gamma }$, and different phase differences ${\varphi }_d$. Top: The dashed black line represents ${\varphi }_d=0$; the solid (red) line, ${\varphi }_d=\pi /2$. Bottom: The dashed black line represents ${\varphi }_d=\pi /8$; the solid (red) line, ${\varphi }_d=\pi /4$.

Figure 10

Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda /4$ with ${\mathrm{\Delta }}_L=0$ and ${\varphi }_1={\varphi }_2=0$ and for different Rabi frequencies experienced by the atoms: ${\mathrm{\Omega }}_1=0.4\mathrm{\Gamma },{\mathrm{\Omega }}_2=0$ (dashed black line) and ${\mathrm{\Omega }}_1=0,{\mathrm{\Omega }}_2=0.4\mathrm{\Gamma }$ [solid (red) line].