Statistical properties of partially coherent polarization singular vector beams

We investigate the statistical properties of partially coherent polarization singular beams embedded with a V-point polarization singularity. An analytical formula for the cross-spectral density matrix is derived for the family of partially coherent polarization singular vector beams (PSVBs) propagating through a paraxial ABCD optical system. It is observed that the far-field intensity profiles and the coherence-induced depolarization effect in partially coherent PSVBs depend on both the input spatial coherence length and the Poincaré-Hopf index (PHI) of the beam. Interestingly, it is found that in this process of coherence degradation, the polarization (Stokes, $S_{12}$) vortices are preserved. The depolarization is due to an enhanced unpolarized light field that in turn modulates the beam profile, the transverse distribution of the degree of polarization (DOP) and the degree of coherence (DOC). Furthermore, the Gaussian distribution of the DOC evolves into a non-Gaussian profile in the far-field with the number of ring dislocations equal to the magnitude of PHI of the beam. The degeneracy associated with the intensity profile, the Stokes intensity distribution, the DOP, and DOC profiles of these partially coherent PSVBs carrying opposite polarity of PHI are also discussed to complete this study. Subsequently, all of these findings are experimentally verified by generating a family of partially coherent PSVBs with controllable spatial coherence. The modulation of the spatial coherence length in the source plane leads to efficient control of its intensity, the DOC and DOP profiles on propagation, which are of importance in particle trapping, material thermal processing, free-space optical communications, and detection of a phase object.

Figure 1

Evolution of the intensity distribution of various index partially coherent PSVBs ($\left|\eta \right|=1$, 2, and 3) on propagation. The far-field beam profiles are computed at propagation distances $z=0$, 260, 280, and $300\text{mm}$ for $\delta =1.2\text{mm}$. The intensity distribution of unpolarized (red-borderline) and polarized parts (green-borderline) are also shown with their respective color-maps.

Figure 2

Dependency of the polarized and unpolarized components' power of partially coherent PSVBs of $\left|\eta \right|=1$ (blue curve), 2 (red curve), and 3 (black curve) on (a) spatial coherence length, $\delta$ and (b) propagation distance, $z$. The semidepolarization point is marked with a filled circle in each case.

Figure 3

Dependency of the DOP profile of various index partially coherent PSVBs ($\left|\eta \right|=1$, 2, and 3) on propagation distance. The evolution of the DOP is investigated considering two $\delta$ values $2.5\text{mm}$ (red curve) and $1.5\text{mm}$ (blue curve) at $z=200$, 250, and 300 mm. The DOP-distribution is shown as an inset in respective line-scans. It can be readily observed that the coherence-induced DOP degradation is more prominent in a higher-index partially coherent PSVBs.

Figure 4

Far-field DOC characteristics of partially coherent PSVBs having $\left|\eta \right|=1$, 2, and 3 respectively. The evolution of the DOC on propagation is shown as the line-scan at $z=0$, 100, 200, and 300 mm. The input spatial coherence length is considered to be $\delta =2.3\text{mm}$ (red curve) and $\delta =1.2\text{mm}$ (black curve). The focused DOC contains ring-dislocations ($=\left|\eta \right|$), which are shown as insets.

Figure 5

Schematic describing the inherent degeneracies present in the partially coherent PSVBs. Beams with $\left|\eta \right|=1$, 2, and 3 are considered to carry out this study. The SOP distributions and the Stokes phases of the polarized component of partially coherent PSVBs for all four types of V-point polarization singularities are shown. The SOP distribution is preserved in the polarized part of the beam even on reducing the spatial coherence length ($\delta =1.2\text{mm}$) of the beam.

Figure 6

Schematic of the experimental setup used to synthesize and investigate the far-field propagation characteristics of partially coherent PSVBs. Abbreviations are as follows: $L_1,L_2,L_3,L_4,L_5$: lens; MO: microscope objective; PH: pinhole; P: polarizer; SWP: spatially varying wave plate; HWP: half-wave plate; RGGD: rotating ground glass diffuser; CCD: charged coupled device; and SC: Stokes camera. Translating lens $L_2$ provides the control over the spatial coherence length of the generated partially coherent PSVBs. The SOP distribution of the output beam is governed by the orientation of the polarizer and the HWP. The PSVB of $\left|\eta \right|=3$ is generated using two SWPs of $\left|\eta \right|=1$ and 2 in concatenation with a HWP as shown in the inset (yellow background). The SC plane is imaged onto the CCD camera using the configuration shown in the green-background inset. RGGD is replaced with a beam profiler to measure the focal spot size (pink background inset).

Figure 7

Experimental plots of focused intensity of partially coherent PSVBs of PHI $\left|\eta \right|=1$ (cyan background), $\left|\eta \right|=2$ (yellow background), and $\left|\eta \right|=1$ (green background) for different values of $\delta$. The insets encircled with red and yellow are the polarized and unpolarized parts of the total intensity. The beam-shaping aspect with reducing $\delta$ can be readily observed. The transformation from a doughnut to a Gaussian beam profile occurs slowly for higher $\eta$ partially coherent PSVBs.

Figure 8

Changes observed in the intensity profile in far-field propagation of partially coherent PSVBs ($\left|\eta \right|=1$, 2, and 3) are presented as the line-scan of intensity distribution for $z=260$, 280, and 300 mm. The input spatial coherence length was fixed at $\delta =2.3\text{mm}$. Here the solid line is the predicted theoretical profile, and the filled spheres are the experimentally obtained data points. The profile transition rate from a doughnut to a Gaussian is slower for a higher-index partially coherent PSVB.

Figure 9

Changes observed in the DOP in far-field propagation of partially coherent PSVBs ($\left|\eta \right|=1$, 2, and 3) are presented as the line profile of a DOP for $z=0$, 270, and 300 mm. These DOP profiles are extracted for the input $\delta =2.3\text{mm}$. Here the solid line is the predicted theoretical profile, and filled spheres are the experimentally obtained data points. It can be observed that the depolarization of higher-index partially coherent PSVBs is more rapid with a wider zero-dip profile.

Figure 10

Distribution of the degree of coherence of different index partially coherent PSVBs ($\left|\eta \right|=1$, 2, and 3) having $\delta =2.5\text{mm}$ under far-field propagation.

Figure 11

Experimental results: Degenerate distributions of partially coherent PSVBs of $\left|\eta \right|=1$, 2, and 3, respectively. The SOP distributions of all four types of V-point polarization singularity for a particular $\left|\eta \right|$ are also presented corresponding to ${\delta }_1=3.3\text{mm}$ and ${\delta }_2=2.3\text{mm}$. The invariance of SOP-distribution on reducing field-correlation ($\delta$) can be observed.