Nonlinear continuous orbital-angular-momentum modulation of linearly polarized Bessel beams

We numerically and experimentally demonstrate the nonlinear continuous modulation of orbital angular momentum (OAM) of a linearly polarized high-order Bessel beam in a biased photorefractive (PR) crystal. According to the simulation results, we identify three nonlinear dynamic regimes depending on the strength of the applied electric field, and we successfully stabilize the OAM via the modulation of a background illumination. As a result, we predict a parabolic trend for the maximal OAM modulation range in the stable regime. In addition, with the comparative analysis between vortex and differently truncated Bessel beams, we report the existence of exploitable plateaus, enabling a more extensive modulation range in the unstable regime. The experimental results are consistent with the simulations, confirming the practical realization of the flexible control of the OAM in both stationary and dynamic regimes. These results contribute to new reconfigurable components in OAM-based communications.

Figure 1

(a) Principle scheme of a high-order Bessel beam illuminating a SBN crystal. (${\mathrm{b}}_1$ and ${\mathrm{b}}_2$) Two-dimensional (2D) intensity and phase distributions at the output face of the crystal for a fourth-order Bessel beam ($l=4, k_t=0.2\mu {\mathrm{m}}^{-1}, {\omega }_0=20r_0, I_{\mathrm{in}}/I_d=1$) linearly propagating in the $2\text{−}\mathrm{cm}$ SBN crystal ($E_0=0\mathrm{V}/\mathrm{cm}$).

Figure 2

(a) Variation tendency of the available OAM values versus electric field $E_0$ for the high-order Bessel beam ($l=1,2,3,4$) with $k_t=0.2\mu {\mathrm{m}}^{-1}, {\omega }_0=20r_0$, and $I_{\mathrm{in}}/I_d=1$ propagating in the 2-cm SBN crystal. (${\mathrm{b}}_1$–${\mathrm{d}}_2$) 2D output intensity and phase distributions of the fourth-order Bessel beam propagating under the condition of (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$) $E_0=1\mathrm{kV}/\mathrm{cm}$, (${\mathrm{c}}_1$) and (${\mathrm{c}}_2$) $E_0=1.5\mathrm{kV}/\mathrm{cm}$, and (${\mathrm{d}}_1$) and (${\mathrm{d}}_2$) $E_0=2\mathrm{kV}/\mathrm{cm}$. (e) The OAM spectrum on the normalized intensity for the cases of $E_0=1$ kV/cm, $E_0=1.5\mathrm{kV}/\mathrm{cm}, E_0=2\mathrm{kV}/\mathrm{cm}$.

Figure 3

(a) The variation of the output OAM versus the electric field $E_0$ for the second-order Bessel beam ($l=2, k_t=0.2\mu {\mathrm{m}}^{-1}, {\omega }_0=20r_0, I_{\mathrm{in}}/I_d=1$) propagating in the $2\text{−}\mathrm{cm}$ SBN crystal. (b–d) Temporal behavior of the output OAM at (b) $E_0=1.6\mathrm{kV}/\mathrm{cm}$, (c) $E_0=3\mathrm{kV}/\mathrm{cm}$, and (d) $E_0=4\mathrm{kV}/\mathrm{cm}$. (${\mathrm{c}}_1$–${\mathrm{c}}_3$) The intensity distribution at the moments (${\mathrm{c}}_1$) $t=1.25T_d$, (${\mathrm{c}}_2$) $t=1.625T_d$, and (${\mathrm{c}}_3$) $t=2.25T_d$. (${\mathrm{c}}_4$) The OAM variation along the propagation length at different moments. (e) The achievable minimum stable OAM varying with the electric field and the intensity ratio.

Figure 4

(a) Variation tendencies of the output OAM versus electric field $E_0$ for the fourth-order Bessel beam with $k_t=0.2\mu {\mathrm{m}}^{-1}$ and the fourth-order vortex beam with $r_0=10\mu \mathrm{m}$ propagating in the $2\text{−}\mathrm{cm}$ SBN crystal. (${\mathrm{b}}_1$ and ${\mathrm{b}}_2$) 2D output intensity and phase distributions of the fourth-order vortex beam at $E_0=5\mathrm{kV}/\mathrm{cm}$ at the moment $t=4.75T_d$. (${\mathrm{c}}_1$ and ${\mathrm{c}}_2$) 2D output intensity and phase distributions of the fourth-order Bessel beam at $E_0=5\mathrm{kV}/\mathrm{cm}$ at the moment $t=3.625T_d$.

Figure 5

(a) 1D intensity profiles of fourth-order Bessel beams truncated by the Gaussian term with $a_t=1/7r_0, a_t=1/10r_0$, and $a_t=1/25r_0$ (normalized by their individual maximum intensities). (b) Temporal behaviors of the fourth-order vortex beam ($r_0=10\mu \mathrm{m}$) and three fourth-order Bessel beams with the truncation parameter of $a_t=1/7r_0, a_t=1/10r_0$, and $a_t=1/25r_0$ ($l=4, k_t=0.2\mu {\mathrm{m}}^{-1}$) propagating in the $2\text{−}\mathrm{cm}$ SBN crystal under the electric field of $E_0=5\mathrm{kV}/\mathrm{cm}$.

Figure 6

(a) Temporal behaviors of the OAM measured in the experiments of the fourth-order BB propagates ($k_t=0.2\mu {\mathrm{m}}^{-1}, l=4$) in the $1\text{−}\mathrm{cm}$ SBN crystal under the nonlinear conditions with different strength of the electric field. (${\mathrm{b}}_1$–${\mathrm{d}}_2$) Intensity and phase distributions: (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$) under $E_0=1\mathrm{kV}/\mathrm{cm}$ at $t=120\mathrm{ms}$, (${\mathrm{c}}_1$) and (${\mathrm{c}}_2$) under $E_0=2\mathrm{kV}/\mathrm{cm}$ at $t=150\mathrm{ms}$, and (${\mathrm{d}}_1$) and (${\mathrm{d}}_2$) under $E_0=5\mathrm{kV}/\mathrm{cm}$ at $t=220\mathrm{ms}$, corresponding to the blue crosses marked in (a). (e and f) Intensity distributions: (e) under $E_0=2\mathrm{kV}/\mathrm{cm}$ at $t=400\mathrm{ms}$ and (f) under $E_0=5\mathrm{kV}/\mathrm{cm}$ at $t=500\mathrm{ms}$, corresponding to the red dots marked in (a).