Stokes scintillations for vector vortex beams with controllable spatial correlation

The present work reports the theoretical and experimental observation of Stokes scintillations (SS) for focused vector vortex beams (VVBs) with controllable spatial correlation. Initially, a Gaussian-Schell model (GSM) source is utilized to generate the partially coherent VVBs (PC-VVBs). The focusing of PC-VVBs results in the optimization of extremum values of SS parameters across the beam cross section. Additionally, the SS parameters of the focused PC-VVB show spatially structured behavior. It is found that the spatial correlation, the inhomogeneous polarization distribution, and the Poincaré-Hopf index (PHI) of VVBs are important factors for structuring the SS parameters. The SS parameters are observed to exhibit a complementary relationship. Furthermore, the degeneracy associated with the SS parameters is discussed to complete the study. The flower-patterned distribution of specific SS parameters is useful in determining the topological index of the PC-VVBs. Interestingly, the bound condition and complementary nature of SS parameters are useful for scintillation squeezing across the beam cross section in PC-VVBs, where one SS parameter is suppressed at the expense of another. Thus, the source can be finely tuned using the bound condition with correlation and topological features to harness this effect optimally.

Figure 1

The azimuthal variation of SS parameters ${\text{D}}_{nn}^{N\left(s\right)},n=0$ (solid line), 1 (dotted line), and 2 (dashed line) for PC-VVBs with $\left|\eta \right|=1$ (blue), $\left|\eta \right|=2$ (red), and $\left|\eta \right|=3$ (green) at the source plane (SWP plane). Here, ${\text{D}}_{00}^{N\left(s\right)}=1$ and ${\text{D}}_{33}^{N\left(s\right)}=0$ (concentric circles of radius $=1$ and $=0$, respectively, for all indices).

Figure 2

Theoretical plot for the dependency of the SS parameters on correlation width $\delta$ of focused $(f=200\mathrm{mm}$) PC-VVBs $\left[\right|\eta |=1$ (blue), 2 (red), 3 (green)] at (0.5 mm, 0 mm). The selected location aligns with the extremum value (either maximum or minimum) of the SS parameters at the periphery of PC-VVB with a beam waist of 3 mm. (A) The solid lines represent ${\text{D}}_{00}^N$ and dashed lines represent ${\text{D}}_{33}^N$. (B) The solid lines represent ${\text{D}}_{11}^N$ and dashed lines represent ${\text{D}}_{22}^N$. In the insets the SS profiles corresponding to two distinct ${\delta }^{\prime }\mathrm{s}$ (${\delta }_1,{\delta }_2$) are shown.

Figure 3

Schematic of the experimental setup to generate PC-VVBs and investigate the effect of focusing on the Stokes scintillation. MO: microscope objective; PH: pinhole; ${\text{L}}_1,{\text{L}}_2,{\text{L}}_3,{\text{L}}_4$: lens; $M_1,M_2$: mirrors; TS: translational stage; RGGD: rotating ground glass diffuser; P: polarizer; HWP: half-wave plate; SWP: spatially varying HWP; SC: Stokes camera. Translating lens ${\text{L}}_2$ provides a controlled generation of PC-VVBs. The VVB of $\left|\eta \right|=3$ is generated using two SWPs of $\left|\eta \right|=1$ and $\left|\eta \right|=2$ in concatenation with a HWP as shown in the inset (blue-dashed borderline).

Figure 4

Higher-order Poincaré sphere representation of focused PC-VVBs $\left|\eta \right|=1$ (blue), $\left|\eta \right|=2$ (red), and $\left|\eta \right|=3$ (green). The experimentally obtained total intensity $\left(S_0\right), |S_1|$, and $|S_2|$ for ${\delta }_1=1.54\text{mm}$ (cyan), ${\delta }_2=0.85\text{mm}$ (yellow) are shown in the blue, red, and green dashed box for $\left|\eta \right|=1,2,3$, respectively, in row I. The location of two orthogonal pairs (I, III and II, IV) of PC-VVBs are at diametrically opposite coordinate points of HOPS (points 1 and 2). Inner spherical shells can see the correlation-induced depolarization. The theoretical and experimentally obtained SS parameters at two ${\delta }^{\prime }s$ are shown in rows II and IV. The SOP distributions for a particular $\left|\eta \right|$ in row III show SS degeneracy.

Figure 5

Variation of SS parameters for PC-VVBs with $\left|\eta \right|=1$ (blue), $\left|\eta \right|=2$ (red), and $\left|\eta \right|=3$ (green) across the beam cross section for different input spatial correlation values $({\delta }_1=1.54\text{mm},{\delta }_2=0.85\text{mm})$. The solid and dashed lines are theoretical predictions, and various shapes are the experimentally obtained data points.