# Full Stokes Polarization Imaging Based on Broadband Liquid Crystal Polarization Gratings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle

_{3}is the normalized Stokes parameter corresponding to the ellipticity of the incident light. It needs to be emphasized that factor Q is firstly determined by the azimuthal pattern of the aligned LC directors. When the surface alignment satisfies the continuous distribution of the cycloidal pattern, it forms a birefringent grating with LC directors following the surface alignment, which is called the single-wavelength PG. In this case, factor Q equals 1 at a certain wavelength, which is called the half-wave condition (∆nd = λ/2). In addition, when a multi-layer, twisted structure (shown in Figure 2a) is introduced by adding chiral doping, the high-efficient range is broadened from a single wavelength to a wide spectral band. Our results in Figure 2b show high diffraction efficiency (>93%) in the most visible wavelengths (e.g., 450–800 nm). As for the polarization of diffractive beams, the ellipticity |S

_{3}| of ±1 diffraction orders shows that the diffractive beams are perfectly circular-polarized even when the efficiency is away from the peak value.

^{2}.

_{PG}of the LCPG is the sum of 0th and ±1st orders, which are written with absorption and scattering neglected as follows:

_{retarder}is the Muller matrix of phase retarder. Here, let us assume that the incident wavelength falls in the high-efficiency range of broadband PG (Q = 1). Therefore, by solving the Stokes vectors, the theoretical efficiencies are given by

## 3. Linear Stokes Detection

_{1}and S

_{2}of the Stokes vector need to be solved. Mathematically, to solve the two parameters S

_{1}and S

_{2}, two equations are required. Thus, according to Equation (8) or Equation (9), it is necessary to perform the intensity ratio γ measurements twice. corresponding to two phase-retardance conditions, for instance, (θ

_{1}, δ

_{1}) and (θ

_{2}, δ

_{2}). In this case, it is simplified as a linear equation with two variables:

_{1}and S

_{2}can be easily solved. To make the above linear equation solvable, the coefficients must satisfy ${a}_{1}{b}_{2}-{a}_{2}{b}_{1}\ne 0$, which is further simplified as $\mathrm{sin}\left(2{\theta}_{1}-2{\theta}_{2}\right)\ne 0$.

_{1}and δ

_{2}. In other words, to obtain the direction of the incident linear polarization, it is necessary to modulate the fast axis of the retarder. As for the implementation of polarimetry, it can be either a division-of-time or division-of-amplitude type. A cascaded structure, which is typical for the division-of-amplitude type, using 0 order as the input of the next stage is reported and verified for fiber communication applications [23]. However, the intensity distribution among the cascaded stages needs to be delicately designed. As explained in the following sections, for the purpose of the principal-of-operation verification, we used a rotatable QWP to achieve the modulation of fast axis θ for imaging acquisition.

#### 3.1. Optimal Design of Polarization Detection System

^{2}; thus, the EWV value estimated by the Stokes vector can be calculated from Equation (12):

#### 3.2. Linear Stokes Parameter Reconstruction

_{±1}, and the specific experimental results are demonstrated in Figure 6 and Table 1.

_{1}, S

_{2}, AoP and DoLP were 0.004 ± 0.026, −0.970 ± 0.036, −44.872° ± 0.761°, and 0.971 ± 0.019, respectively. Furthermore, the experimental results show that the proposed method could effectively measure the polarization characteristics of the object.

## 4. Full Stokes Detection

_{3}of the object reveals the comprehensive features of the object in addition to its linear components. The LCPGs divide the left-handed and right-handed circular polarization components into ±1 diffraction orders (refer to Equation (2) for details). Therefore, S

_{3}measurement can be performed by directly placing the LCPG in front of the camera and collecting the resulting images. Thus, the full Stokes polarization detection was performed by adding an additional measurement step when the QWP was removed.

#### Full Stokes Parameter Reconstruction

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Structure and properties of broadband PG: (

**a**) LC director view of broadband PG with two chiral layers; (

**b**) diffraction efficiency and ellipticity of broadband PG in visible range.

**Figure 3.**(

**a**) Schematics of generation of exposure beam with continuously changing polarization state using birefringent prism and QWP [26]; (

**b**) photograph of manufactured broadband LCPG.

**Figure 5.**(

**a**) Linear polarized light detection experimental device; (

**b**) diffraction patterns of polarization grating and textures of LCPG observed under polarized microscope with black arrows showing polarizer and analyzer direction.

**Figure 6.**Plots of (

**a**) γ

_{+1}and (

**b**) γ

_{−1}as a function of the direction α of linearly polarized incidence. the theoretical values are represented by solid curves, while the experimental measurements are represented by dots mark.

**Figure 11.**Plots of (

**a**) the experimental and simulation values of S

_{1}, S

_{2}, S

_{3}, and (

**b**) DoLP and DoCP parameters as a function of the fast axis orientation of QWP1 at 632.8 nm measured by the full Stokes polarization detection method.

Direction of Incident Linearly Polarized Light | γ_{−1} | Polarization Parameters | ||||
---|---|---|---|---|---|---|

θ = 0° | θ = 45° | S_{1} | S_{2} | DoLP | AoP/° | |

0° | 0.507 | 0.997 | 0.994 | −0.014 | 0.994 | −0.404 |

30° | 0.067 | 0.748 | 0.496 | 0.866 | 0.998 | 30.099 |

45° | 0.008 | 0.499 | −0.002 | 0.984 | 0.984 | 45.058 |

60° | 0.067 | 0.250 | −0.500 | 0.866 | 1.000 | 60.000 |

90° | 0.499 | 0.001 | −0.998 | 0.002 | 0.998 | 89.943 |

−30° | 0.932 | 0.748 | 0.496 | −0.864 | 0.996 | −30.071 |

−45° | 0.998 | 0.510 | 0.020 | −0.996 | 0.996 | −44.425 |

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**MDPI and ACS Style**

Xuan, Y.; Guo, Q.; Zhao, H.; Zhang, H.
Full Stokes Polarization Imaging Based on Broadband Liquid Crystal Polarization Gratings. *Crystals* **2023**, *13*, 38.
https://doi.org/10.3390/cryst13010038

**AMA Style**

Xuan Y, Guo Q, Zhao H, Zhang H.
Full Stokes Polarization Imaging Based on Broadband Liquid Crystal Polarization Gratings. *Crystals*. 2023; 13(1):38.
https://doi.org/10.3390/cryst13010038

**Chicago/Turabian Style**

Xuan, Yan, Qi Guo, Huijie Zhao, and Hao Zhang.
2023. "Full Stokes Polarization Imaging Based on Broadband Liquid Crystal Polarization Gratings" *Crystals* 13, no. 1: 38.
https://doi.org/10.3390/cryst13010038