Propagation of optical vector vortices of slow light in a coherently prepared tripod configuration

Abstract

We investigate the propagation of optical vector vortices of slow light in a coherently prepared four-level tripod atomic system. The vector vortex consists of superposed pulse pairs with opposite circular polarizations and orbital angular momentum (OAM) charges $\pm l$, weakly interacting with an atomic medium initially prepared in a coherent superposition of two ground states. A third unoccupied state is coupled to a stronger control laser without OAM, creating a phase-dependent configuration. In the linear regime, the vortex OAM is mapped onto the medium, producing symmetrical azimuthally structured absorption patterns, with losses significantly reduced by the control field. For small detunings, complementary spatially dependent amplification and absorption occur for the two circular polarization components. This OAM-structured coherence induces a dynamical anisotropy, affecting both the intensity and polarization of the slow-light vortex. Polarization states evolve periodically between left-circular, linear, and right-circular polarizations during propagation. Once the beam reaches a stationary regime, the ring-shaped intensity transforms into a petal-like structure, and the final polarization states stabilize according to the initial superposition. The rate of polarization transitions is tunable via the control field strength, demonstrating flexible control over slow-light vector vortex dynamics.

Figures (6)

• Figure 1: Schematic representation of the four-level atomic system in a tripod configuration. The excited state $\ket{4}$ is coupled to the ground states $\ket{1}$ and $\ket{2}$ by right- and left-circularly polarized probe fields with Rabi frequencies $\Omega_R$ and $\Omega_L$, respectively, and to the ground state $\ket{3}$ by a strong control field with Rabi frequency $\Omega_c$.
• Figure 2: Absorption (Im[$\chi^{(1)}$]) (a)–(b) and dispersion (Re[$\chi^{(1)}$]) (c)–(f) profiles for the right- and left-handed components of the vector vortex beam with topological charge $|l|=1$ at $z=0$. Panels (a) and (b) show the imaginary part of the susceptibilities evaluated at $r=w$ as functions of azimuthal angle $\phi$ and detuning $\Delta/\Gamma$ for the right- and left-handed components, respectively. Panels (c) and (e) present the corresponding real parts. Panels (d) and (f) display the one-dimensional dispersion profiles, Re[$\chi_R^{(1)}$] and Re[$\chi_L^{(1)}$], plotted versus $\Delta/\Gamma$ for selected azimuthal angles $\phi = 0, \pi/4, \pi/2, 3\pi/4$. In (d) and (f), vertical dashed lines indicate resonance ($\Delta/\Gamma = 0$), while horizontal dashed lines denote Re[$\chi_{R(L)}^{(1)}$] = 0. The parameters used are $\gamma_d = 10^{-3}\Gamma$, $|\Omega_C| = \Gamma$, $\theta = \pi/4$, $\alpha = \pi/4$, and $\psi = 0$. All plots are normalized to $2\frac{c}{\omega}\zeta$.
• Figure 3: Evolution of the transverse absorption profiles of the right- and left-handed components, together with the total intensity distribution, at different normalized propagation distances $\zeta z$ for $|l|=1$. The first row shows the absorption of the right-handed component, the second row shows the absorption of the left-handed component, and the third row shows the combined total intensity of both components. (a) Two-photon resonance $\Delta=0$. (b) Slight detuning from resonance with $\Delta = 0.1\Gamma$. All other parameters are the same as in Fig. \ref{['Absorption at z=0']}.
• Figure 4: Intensity profile and polarization state evolution in the transverse plane of the vector vortex with $|l|=1$ and relative amplitude $\alpha=\pi/8$, such that the initial polarization state at the medium entrance is dominated by the left-handed circular polarization ($|E_L(0)|>|E_R(0)|$). The transverse coordinates ($x,y$) are normalized to the beam waist $w$, with the propagation distance expressed as the dimensionless parameter $\zeta z$. The vector vortex beam is in two-photon resonance with detuning $\Delta=0$, while the other parameters are $\gamma_d=10^{-3}\Gamma$ and $|\Omega_C|=\Gamma$. Darker rings or lobes correspond to higher intensity regions. The red and blue ellipses denote the the left- and right-circular polarization states, respectlively, while yellow lines indicate the linear polarization state. (a) The initial phaseonium state is fixed at $\theta=\pi/4$; the first, second, and third rows correspond to relative phases $\psi=0,\pi/2,\pi$, respectively. (b) The relative phase is fixed at $\psi=0$; the first, second, and third rows correspond to initial phaseonium state $\theta=\pi,\pi/4,3\pi/8$, respectively.
• Figure 5: Intensity profile and polarization state evolution (a), and ellipticity distribution (b) of the vector vortex with $|l|=1$ and relative amplitude $\alpha=\pi/8$, such that the initial polarization state at the medium entrance is dominated by the left-handed circular polarization ($|E_L(0)|>|E_R(0)|$). The propagation distance is expressed as the dimensionless parameter $\zeta z$. The relative phase is fixed at $\psi=0$, while the other parameters are $\gamma_d=10^{-3}\Gamma$ and $|\Omega_C|=\Gamma$. (a) The detuning is fixed at $\Delta=0.1\Gamma$; the first, second, and third rows correspond to initial phaseonium states $\theta=\pi/4,\pi/8,3\pi/8$, respectively. Darker rings or lobes correspond to higher intensity regions. The red and blue ellipses denote the left- and right-handed circular polarization states, respectively, while yellow lines indicate linear polarization states. (b) The initial phaseonium state is fixed at $\theta=\pi/4$; the plot shows the average ellipticity corresponding to linear, left- and right-handed circular polarization states, denoted by white, dark red, and dark blue colors, respectively, at different propagation distances $\zeta z$ and detuning values $\Delta/\Gamma$.