# 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

## References

- Rubin, N.A.; D’Aversa, G.; Chevalier, P.; Shi, Z.; Chen, W.T.; Capasso, F. Matrix Fourier Optics Enables a Compact full-Stokes Polarization Camera. Science
**2019**, 365, 43. [Google Scholar] [CrossRef] [PubMed] - Basiri, A.; Chen, X.; Bai, J.; Amrollahi, P.; Carpenter, J.; Holman, Z.; Wang, C.; Yao, Y. Nature-inspired chiral metasurfaces for circular polarization detection and full-Stokes polarimetric measurements. Light Sci. Appl.
**2019**, 8, 78. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dai, M.; Wang, C.; Qiang, B.; Wang, F.; Ye, M.; Han, S.; Luo, Y.; Wang, Q.J. On-chip mid-infrared photothermoelectric detectors for full-Stokes detection. Nat. Commun.
**2022**, 13, 4560. [Google Scholar] [CrossRef] [PubMed] - Wu, W.; Yu, Y.; Liu, W.; Zhang, X. Fully integrated CMOS-compatible polarization analyzer. Nanophotonics
**2019**, 8, 467–474. [Google Scholar] [CrossRef] - Hou, W.; Li, Z.; Wang, J.; Xu, X.; Goloub, P.; Qie, L. Improving Remote Sensing of Aerosol Microphysical Properties by Near-Infrared Polarimetric Measurements Over Vegetated Land: Information Content Analysis. J. Geophys. Res. Atmos.
**2018**, 123, 2215–2243. [Google Scholar] [CrossRef] - Asgarimehr, M.; Hoseini, M.; Semmling, M.; Ramatschi, M.; Camps, A.; Nahavandchi, H.; Haas, R.; Wickert, J. Remote sensing of precipitation using reflected GNSS signals: Response analysis of polarimetric observations. IEEE Trans. Geosci. Remote Sens.
**2021**, 60, 1–12. [Google Scholar] [CrossRef] - Horváth, G.; Barta, A.; Gál, J.; Suhai, B.; Haiman, O. Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection. Appl. Opt.
**2002**, 41, 543–559. [Google Scholar] [CrossRef] - Zhao, H.; Xu, W.; Zhang, Y.; Li, X.; Zhang, H.; Xuan, J.; Jia, B. Polarization patterns under different sky conditions and a navigation method based on the symmetry of the AOP map of skylight. Opt. Express
**2018**, 26, 28589. [Google Scholar] [CrossRef] - Li, Q.; Hu, Y.; Hao, Q.; Cao, J.; Cheng, Y.; Dong, L.; Huang, X. Skylight polarization patterns under urban obscurations and a navigation method adapted to urban environments. Opt. Express
**2021**, 29, 42090. [Google Scholar] [CrossRef] - He, C.; He, H.; Chang, J.; Chen, B.; Ma, H.; Booth, M.J. Polarisation optics for biomedical and clinical applications: A review. Light Sci. Appl.
**2021**, 10, 194. [Google Scholar] [CrossRef] - Kuzyk, A.; Schreiber, R.; Fan, Z.; Pardatscher, G.; Roller, E.; Högele, A.; Simmel, F.C.; Govorov, A.O.; Liedl, T. DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response. Nature
**2012**, 483, 311–314. [Google Scholar] [CrossRef] [PubMed][Green Version] - Myhre, G.; Hsu, W.L.; Peinado, A.; LaCasse, C.; Brock, N.; Chipman, R.A.; Pau, S. Liquid crystal polymer full-stokes division of focal plane polarimeter. Opt. Express
**2012**, 20, 27393–27409. [Google Scholar] [CrossRef] [PubMed] - Gruev, V.; Perkins, R.; York, T. CCD polarization imaging sensor with aluminum nanowire optical filters. Opt. Express
**2010**, 18, 19087–19094. [Google Scholar] [CrossRef] [PubMed] - Töppel, F.; Aiello, A.; Marquardt, C.; Giacobino, E.; Leuchs, G. Classical entanglement in polarization metrology. New J. Phys.
**2014**, 16, 73019–73021. [Google Scholar] [CrossRef] - York, T.; Marinov, R.; Gruev, V. 260 frames-per-second 648x488 resolution division-of-focal-plane polarimeter with structural dynamics and tracking applications. Opt. Express
**2016**, 24, 8243–8252. [Google Scholar] [CrossRef] - Luo, H.; Oka, K.; DeHoog, E.; Kudenov, M.; Schiewgerling, J.; Dereniak, E.L. Compact and miniature snapshot imaging polarimeter. Appl. Opt.
**2008**, 47, 4413–4417. [Google Scholar] [CrossRef] - López-Morales, G.; Sánchez-López, M.; Lizana, Á.; Moreno, I.; Campos, J. Mueller Matrix Polarimetric Imaging Analysis of Optical Components for the Generation of Cylindrical Vector Beams. Crystals
**2020**, 10, 1155. [Google Scholar] [CrossRef] - Chang, J.; Zeng, N.; He, H.; He, Y.; Ma, H. Single-shot spatially modulated Stokes polarimeter based on a GRIN lens. Opt. Lett.
**2014**, 39, 2656–2659. [Google Scholar] [CrossRef] - Kudenov, M.W.; Escuti, M.J.; Dereniak, E.L.; Oka, K. White-light channeled imaging polarimeter using broadband polarization gratings. Appl. Opt.
**2011**, 50, 2283–2293. [Google Scholar] [CrossRef][Green Version] - Sasaki, T.; Hatayama, A.; Emoto, A.; Ono, H.; Kawatsuki, N. Simple detection of light polarization by using crossed polarization gratings. J. Appl. Phys.
**2006**, 100, 63502. [Google Scholar] [CrossRef] - Sasaki, T.; Wada, T.; Noda, K.; Kawatsuki, N.; Ono, H. Merged vector gratings recorded in a photocrosslinkable polymer liquid crystal film for polarimetry. J. Appl. Phys.
**2014**, 115, 23110. [Google Scholar] [CrossRef] - Noda, K.; Momosaki, R.; Matsubara, J.; Sakamoto, M.; Sasaki, T.; Kawatsuki, N.; Goto, K.; Ono, H. Polarization imaging using an anisotropic diffraction grating and liquid crystal retarders. Appl. Opt.
**2018**, 57, 8870–8875. [Google Scholar] [CrossRef] - Escuti, M.J.; Oh, C.; Sánchez, C.; Bastiaansen, C.; Broer, D.J. Simplified spectropolarimetry using reactive mesogen polarization gratings. Imaging Spectrom. XI SPIE
**2006**, 6302, 21–31. [Google Scholar] - Kim, J.; Escuti, M.J. Snapshot imaging spectropolarimeter utilizing polarization gratings. Imaging Spectrom. XIII SPIE
**2008**, 7086, 29–38. [Google Scholar] - Lin, T.; Xie, J.; Zhou, Y.; Zhou, Y.; Yuan, Y.; Fan, F.; Wen, S. Recent Advances in Photoalignment Liquid Crystal Polarization Gratings and their Applications. Crystals
**2021**, 11, 900. [Google Scholar] [CrossRef] - Chen, D.; Zhao, H.; Yan, K.; Xu, D.; Guo, Q.; Sun, L.; Wu, F.; Chigrinov, V.G.; Kwok, H. Interference-free and single exposure to generate continuous cycloidal alignment for large-area liquid crystal devices. Opt. Express
**2019**, 27, 29332. [Google Scholar] [CrossRef] - Zhang, S.; Chen, W.; Yu, Y.; Wang, Q.; Mu, Q.; Li, S.; Chen, J. Twisting Structures in Liquid Crystal Polarization Gratings and Lenses. Crystals
**2021**, 11, 243. [Google Scholar] [CrossRef] - Gao, B.; Beeckman, J.; Neyts, K. Design and Realization of a Compact Efficient Beam Combiner, Based on Liquid Crystal Pancharatnam–Berry Phase Gratings. Crystals
**2021**, 11, 220. [Google Scholar] [CrossRef] - Mu, T.; Chen, Z.; Zhang, C.; Liang, R. Optimal design and performance metric of broadband full-Stokes polarimeters with immunity to Poisson and Gaussian noise. Opt. Express
**2016**, 24, 29691. [Google Scholar] [CrossRef][Green Version] - Letnes, P.A.; Nerbo, I.S.; Aas, L.M.; Ellingsen, P.G.; Kildemo, M. Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm. Opt. Express
**2010**, 18, 23095–23103. [Google Scholar] [CrossRef][Green Version] - Sabatke, D.S.; Descour, M.R.; Dereniak, E.L.; Sweatt, W.C.; Kemme, S.A.; Phipps, G.S. Optimization of retardance for a complete Stokes polarimeter. Opt. Lett.
**2000**, 25, 802–804. [Google Scholar] [CrossRef] [PubMed][Green Version] - Li, X.; Hu, H.; Goudail, F.; Liu, T. Fundamental precision limits of full Stokes polarimeters based on DoFP polarization cameras for an arbitrary number of acquisitions. Opt. Express
**2019**, 27, 31261. [Google Scholar] [CrossRef] [PubMed] - Goldstein, D.H. Polarized Light, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]

**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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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