# Mueller Matrix Polarimetric Imaging Analysis of Optical Components for the Generation of Cylindrical Vector Beams

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Vector Beams in the Stokes formalism

_{0}is the orientation of the polarization state at the azimuth origin $\theta =0$, i.e.,

#### 2.2. Mueller Matrix of an Ideal Radial Polarizer and an Ideal Q-Plate

#### 2.2.1. Radial Polarizer

#### 2.2.2. Q-Plates

#### 2.3. Mueller Matrix Decomposition

## 3. Experimental Arrangement of the Mueller Matrix Polarimeter

_{1}= 75 mm). The LC-PSG is built by a linear polarizer (Thorlabs, LPVISE200A) with its transmission axis (TA) in the vertical direction, followed by two liquid–crystal variable retarders (LCR) from ArcOptix [36], oriented at 45° and 90°, respectively, and a QWP (Thorlabs, AQWP05M600, Newton, NJ, US) with its fast axis (FA) at −45°. The two commercial LCRs employed were calibrated at 488 nm wavelength and their retardance curve as a function of voltage, ${\varphi}_{1}(V)$ and ${\varphi}_{2}(V)$ were obtained. These curves and the driving voltage values of LCR1 and LCR2 that were required to generate the six standard states of polarization—horizontal (H), vertical (V), ±45° linear polarizations, and right- (RCP) and left-handed (LCP) circular polarizations, are given in [28]. The polarization state analyzer (PSA) consists of a second QWP (Thorlabs, WPQ05M488, Newton, NJ, US) with its FA vertically oriented and a Kiralux™ Polarization Camera (Thorlabs, CS505MUP, Newton, NJ, US). This camera has a monochrome CMOS Sensor of 5 megapixels with a wire grid polarizer array that consists of a repeating pattern of polarizers with their transmission axes at 0°, 45°, −45°, and 90°. Therefore, QWP2 is only placed in the PSA when the circular polarization states need to be analyzed. The sample is placed between the LC-PSG and the PSA, and the sample plane is imaged on the camera by a second lens L2 (f

_{2}= 150 mm).

## 4. Results and Discussion

#### 4.1. Imaging Mueller Matrix Results of the Radial Polarizer

#### 4.2. Imaging Mueller Matrix Results of the q-Plate

_{33}. The number of lobes in the Mueller matrix elements depends on the order of the q-plate. For instance, element for m

_{11}the q-plate with q = 1 has four lobes, while for q = 1/2 it only has two lobes.

#### 4.3. Polarization Map of the CVBs Generated by the Radial Polarizer and the q-Plates

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**I**is the 3 × 3 identity matrix, and $\stackrel{\u2322}{D}=\overrightarrow{D}/D$ is the unitary vector along $\overrightarrow{D}$. Once the diattenuation matrix ${M}_{D}$ is calculated, the procedure to calculate the depolarization and retarder parameters depends on the singularity of ${M}_{D}$. In our case of study, the diattenuation matrix of the radial polarizer is a singular matrix, and it is a non-singular matrix when dealing with the q-plate. Therefore, two different approaches must be followed in each case.

#### Appendix A.1. Case of a Singular Diattenuation Matrix

#### Appendix A.2. Case of a Non-Singular Diattenuation Matrix

_{∆}is defined by

## References

- Beeckman, J.; Neyts, K.; Vanbrabant, P.J.M. Liquid-crystal photonic applications. Opt. Eng.
**2011**, 50, 081202. [Google Scholar] [CrossRef] [Green Version] - Bueno, J.M. Polarimetry using liquid-crystal variable retarders: Theory and calibration. J. Opt. A Pure Appl. Opt.
**2000**, 2, 216–222. [Google Scholar] [CrossRef] [Green Version] - Uribe-Patarroyo, N.; Alvarez-Herrero, A.; Heredero, R.L.; Iniesta, J.C.D.T.; Jiménez, A.C.L.; Domingo, V.; Gasent, J.L.; Jochum, L.; Pillet, V.M. IMaX: A polarimeter based on Liquid Crystal Variable Retarders for an aerospace mission. Phys. Status Solidi
**2008**, 5, 1041–1045. [Google Scholar] [CrossRef] - Tyo, J.S.; Goldstein, D.L.; Chenault, D.B.; Shaw, J.A. Review of passive imaging polarimetry for remote sensing applications. Appl. Opt.
**2006**, 45, 5453–5469. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lizana, A.; Foldyna, M.; Stchakovsky, M.; Georges, B.; Nicolas, D.; Garcia-Caurel, E. Enhanced sensitivity to dielectric function and thickness of absorbing thin films by combining total internal reflection ellipsometry with standard ellipsometry and reflectometry. J. Phys. D Appl. Phys.
**2013**, 46, 105501. [Google Scholar] [CrossRef] - Lizana, A.; Van Eeckhout, A.; Adamczyk, K.; Rodríguez, C.; Escalera, J.C.; Garcia-Caurel, E.; Moreno, I.; Campos, J. Polarization gating based on Mueller matrices. J. Biomed. Opt.
**2017**, 22, 056004. [Google Scholar] [CrossRef] - Bueno, J.M.; Hunter, J.J.; Cookson, C.J.; Kisilak, M.L.; Campbell, M.C.W. Improved scanning laser fundus imaging using polarimetry. J. Opt. Soc. Am. A
**2007**, 24, 1337–1348. [Google Scholar] [CrossRef] - Wolfe, J.E.; Chipman, R.A. Polarimetric characterization of liquid-crystal-on-silicon panels. Appl. Opt.
**2006**, 45, 1688–1703. [Google Scholar] [CrossRef] - Lizana, A.; Moreno, I.; Iemmi, C.; Márquez, A.; Campos, J.; Yzuel, M. Time-resolved Mueller matrix analysis of a liquid crystal on silicon display. Appl. Opt.
**2008**, 47, 4267–4274. [Google Scholar] [CrossRef] - Ding, Z.; Yao, Y.; Yao, X.S.; Chen, X.; Wang, C.; Wang, S.; Liu, T. Demonstration of Compact In situ Mueller-Matrix Polarimetry Based on Binary Polarization Rotators. IEEE Access
**2019**, 7, 144561–144571. [Google Scholar] [CrossRef] - Parejo, P.G.; Campos-Jara, A.; Garcia-Caurel, E.; Arteaga, O.; Alvarez-Herrero, A. Nonideal optical response of liquid crystal variable retarders and its impact on their performance as polarization modulators. J. Vac. Sci. Technol. B
**2020**, 38, 014009. [Google Scholar] [CrossRef] - Marc, P.; Bennis, N.; Spadlo, A.; Kalbarczyk, A.; Węgłowski, R.; Garbat, K.; Jaroszewicz, L.R. Monochromatic Depolarizer Based on Liquid Crystal. Crystals
**2019**, 9, 387. [Google Scholar] [CrossRef] [Green Version] - Rosales-Guzmán, C.; Ndagano, B.; Forbes, A. A review of complex vector light fields and their applications. J. Opt.
**2018**, 20, 123001. [Google Scholar] [CrossRef] - Zhan, Q. Cylindrical vector beams: From mathematical concepts to applications. Adv. Opt. Photonics
**2009**, 1, 1–57. [Google Scholar] [CrossRef] - Pachava, S.; Dharmavarapu, R.; Anand, V.; Jayakumar, S.; Manthalkar, A.; Dixit, A.; Viswanathan, N.K.; Srinivasan, B.; Bhattacharya, S. Generation and decomposition of scalar and vector modes carrying orbital angular momentum: A review. Opt. Eng.
**2019**, 59, 041205. [Google Scholar] [CrossRef] [Green Version] - De Sande, J.C.G.; De Sande, J.C.G.; Santarsiero, M.; Piquero, G. Mueller matrix polarimetry using full Poincaré beams. Opt. Lasers Eng.
**2019**, 122, 134–141. [Google Scholar] [CrossRef] - Zhao, B.; Hu, X.-B.; Rodriguez-Fajardo, V.; Zhu, Z.-H.; Gao, W.; Forbes, A.; Rosales-Guzmán, C. Real-time Stokes polarimetry using a digital micromirror device. Opt. Express
**2019**, 27, 31087–31093. [Google Scholar] [CrossRef] - Stalder, M.; Schadt, M. Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters. Opt. Lett.
**1996**, 21, 1948–1950. [Google Scholar] [CrossRef] - Rubano, A.; Cardano, F.; Piccirillo, B.; Marrucci, L. Q-plate technology: A progress review. J. Opt. Soc. Am. B
**2019**, 36, D70–D87. [Google Scholar] [CrossRef] [Green Version] - Cardano, F.; Karimi, E.; Slussarenko, S.; Marrucci, L.; De Lisio, C.; Santamato, E. Polarization pattern of vector vortex beams generated by q-plates with different topological charges. Appl. Opt.
**2012**, 51, C1–C6. [Google Scholar] [CrossRef] [Green Version] - Nersisyan, S.; Tabiryan, N.; Steeves, D.M.; Kimball, B.R. Fabrication of liquid crystal polymer axial waveplates for UV-IR wavelengths. Opt. Express
**2009**, 17, 11926–11934. [Google Scholar] [CrossRef] [PubMed] - Beresna, M.; Gecevičius, M.; Kazansky, P.G.; Gertus, T. Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass. Appl. Phys. Lett.
**2011**, 98, 201101. [Google Scholar] [CrossRef] - Sánchez-López, M.M.; Moreno, I.; Davis, J.A.; Puerto-Garcia, D.; Abella, I.; Delaney, S.W. Extending the use of commercial q-plates for the generation of high-order and hybrid vector beams. Laser Beam Shap. XVIII
**2018**, 10744, 1074407. [Google Scholar] [CrossRef] - Badham, K.; Delaney, S.; Hashimotono, N.; Sánchez-López, M.M.; Kurihara, M.; Tanabe, A.; Moreno, I.; Davis, J.A. Generation of vector beams at 1550 nm telecommunications wavelength using a segmented q -plate. Opt. Eng.
**2016**, 55, 30502. [Google Scholar] [CrossRef] - Quiceno-Moreno, J.C.; Marco, D.; Sánchez-López, M.M.; Solarte, E.; Moreno, I. Analysis of Hybrid Vector Beams Generated with a Detuned Q-Plate. Appl. Sci.
**2020**, 10, 3427. [Google Scholar] [CrossRef] - Moreno, I.; Albero, J.; Davis, J.A.; Cottrell, D.M.; Cushing, J.B. Polarization manipulation of radially polarized beams. Opt. Eng.
**2012**, 51, 128003. [Google Scholar] [CrossRef] - Lu, S.-Y.; Chipman, R.A. Interpretation of Mueller matrices based on polar decomposition. J. Opt. Soc. Am. A
**1996**, 13, 1106–1113. [Google Scholar] [CrossRef] - Morales, G.L.; Sánchez-López, M.D.M.; Soriano, I.M. Liquid-crystal polarization state generator. In Proceedings of the Unconventional Optical Imaging II, Online. Strasbourg, France, 7–10 April 2020; p. 11351. [Google Scholar] [CrossRef]
- Goldstein, D.H. Polarized Light; Marcel Dekker: New York, NY, USA, 2010. [Google Scholar]
- CODIXX. Available online: https://www.codixx.de/en/colorpol-s-patterned/colorpol-s-patterned-polarizer.html (accessed on 2 November 2020).
- THORLABS. Available online: https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=9098 (accessed on 2 November 2020).
- Marco, D.; Sánchez-López, M.M.; García-Martínez, P.; Moreno, I. Using birefringence colors to evaluate a tunable liquid-crystal q-plate. J. Opt. Soc. Am. B
**2019**, 36, D34–D41. [Google Scholar] [CrossRef] - Sánchez-López, M.M.; Abella, I.; Puerto-García, D.; Davis, J.A.; Moreno, I. Spectral performance of a zero-order liquid-crystal polymer commercial q-plate for the generation of vector beams at different wavelengths. Opt. Laser Technol.
**2018**, 106, 168–176. [Google Scholar] [CrossRef] - Morio, J.; Goudail, F. Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices. Opt. Lett.
**2004**, 29, 2234–2236. [Google Scholar] [CrossRef] - Ossikovski, R.; De Martino, A.; Guyot, S. Forward and reverse product decompositions of depolarizing Mueller matrices. Opt. Lett.
**2007**, 32, 689–691. [Google Scholar] [CrossRef] [PubMed] - ArcOptix. Variable Phase Retarder. Available online: http://www.arcoptix.com/variable_phase_retarder.htm (accessed on 2 November 2020).
- Hinds Instruments. Mueller Polarimeters. Available online: https://www.hindsinstruments.com/products/polarimeters/mueller-polarimeter/ (accessed on 4 December 2020).
- Espinosa-Luna, R.; Zhan, Q. Polarization and Polarizing Optical Devices. In Fundamentals and Basic Optical Instruments, Handbook of Optical Engineering; CRC Press: New York, NY, USA, 2017. [Google Scholar]
- Vargas, J.; Uribe-Patarroyo, N.; Quiroga, J.A.; Alvarez-Herrero, A.; Belenguer, T. Optical inspection of liquid crystal variable retarder inhomogeneities. Appl. Opt.
**2010**, 49, 568–574. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Moallemi, P.; Behnampourii, M. Adaptive optimum notch filter for periodic noise reduction in digital images. AUT J. Electr. Eng.
**2010**, 42, 1–7. [Google Scholar] - Smith, M.H.; Woodruff, J.B.; Howe, J.D. Beam wander considerations in imaging polarimetry. In Proceedings of the SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation, Denver, CO, USA, 18–23 July 1999; Volume 3754, pp. 50–54. [Google Scholar] [CrossRef]
- Gil, J.J.; Ossikovski, R. Polarized Light and the Mueller Matrix Approach; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Ju, H.; Ren, L.; Liang, J.; Qu, E.; Bai, Z. Method for Mueller matrix acquisition based on a division-of-aperture simultaneous polarimetric imaging technique. J. Quant. Spectrosc. Radiat. Transf.
**2019**, 225, 39–44. [Google Scholar] [CrossRef] - Ji, W.; Lee, C.-H.; Chen, P.; Hu, W.; Ming, Y.; Zhang, L.; Lin, T.-H.; Chigrinov, V.G.; Lu, Y.-Q. Meta-q-plate for complex beam shaping. Sci. Rep.
**2016**, 6, 25528. [Google Scholar] [CrossRef] [Green Version] - Rafayelyan, M.; Brasselet, E. Laguerre–Gaussian modal q-plates. Opt. Lett.
**2017**, 42, 1966–1969. [Google Scholar] [CrossRef] - Holland, J.E.; Moreno, I.; Davis, J.A.; Sánchez-López, M.M.; Cottrell, D.M. Q-plates with a nonlinear azimuthal distribution of the principal axis: Application to encoding binary data. Appl. Opt.
**2018**, 57, 1005–1010. [Google Scholar] [CrossRef]

**Figure 1.**Different kinds of vector beams of order $\ell =1$. (

**a**) Radial (

**b**) Slanted (

**c**) Azimuthal (

**d**) Arbitrary.

**Figure 2.**(

**a**) Photograph of the radial polarizer under broadband linearly polarized light. Photographs of the q-plates with (

**b**) q = 1 and (

**c**) q = 1/2 placed between linear parallel polarizers and under broadband illumination. (

**d**) Azimuthal orientation of the transmission axis of the radial polarizer. Azimuthal orientation of the optical axis of the q-plate with (

**e**) q = 1 and (

**f**) q = 1/2. The color code indicates the orientation angle of these axis.

**Figure 3.**Schematic representation of the Mueller matrix polarimeter (L: lens, LP: linear polarizer, LCR: liquid-crystal retarder, QWP: quarter-wave plate).

**Figure 4.**Normalized MM of the radial polarizer. (

**a**) Ideal numerical MM (Equation (3)), and (

**b**) experimental measure.

**Figure 5.**Polarimetric parameters of the radial polarizer. (

**a1**–

**a5**) Ideal numerical and (

**b1**–

**b5**) experimental measure of diattenuation, polarizance, depolarizance, retardance, and transmission angle.

**Figure 6.**Normalized MM of the q-plate of q = 1 (first row) and q = 1/2 (second row). (

**a**,

**c**) Ideal numerical MM (Equation (6)) and (

**b**,

**d**) experimental measure.

**Figure 7.**Polarimetric parameters of the q-plate with q = 1. (

**a1**–

**a5**) Numerical and (

**b1**–

**b5**) experimental measure.

**Figure 8.**Experimental polarimetric parameters of the q-plate with q = 1/2. (

**a**) Diattenuation, (

**b**) polarizance, (

**c**) depolarizance, (

**d**) retardance, and (

**e**) orientation of the optical axis.

**Figure 9.**Experimental polarization map of the CVBs generated by (

**a**) the radial polarizer and the q-plate with (

**b**) q = 1 and (

**c**) q = ½, when the input beam is RCP light. The background color corresponds to the values of the Stokes parameter S

_{0}. The drawn polarization ellipses are obtained from the experimental Stokes parameters.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

López-Morales, G.; Sánchez-López, M.d.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.
https://doi.org/10.3390/cryst10121155

**AMA Style**

López-Morales G, Sánchez-López MdM, Lizana Á, Moreno I, Campos J.
Mueller Matrix Polarimetric Imaging Analysis of Optical Components for the Generation of Cylindrical Vector Beams. *Crystals*. 2020; 10(12):1155.
https://doi.org/10.3390/cryst10121155

**Chicago/Turabian Style**

López-Morales, Guadalupe, María del Mar Sánchez-López, Ángel Lizana, Ignacio Moreno, and Juan Campos.
2020. "Mueller Matrix Polarimetric Imaging Analysis of Optical Components for the Generation of Cylindrical Vector Beams" *Crystals* 10, no. 12: 1155.
https://doi.org/10.3390/cryst10121155