Abstract

Polarization of a light beam is traditionally studied under the hypothesis that the state of polarization is uniform across the transverse section of the beam. In such a case, if the paraxial approximation is also assumed, the propagation of the beam reduces to a scalar problem. Over the last few decades, light beams with spatially variant states of polarization have attracted great attention, due mainly to their potential use in applications such as optical trapping, laser machining, nanoscale imaging, polarimetry, etc. In this tutorial, an introductory treatment of non-uniformly totally polarized beams is given. Besides a brief review of some useful parameters for characterizing the polarization distribution of such beams across transverse planes, from both local and global points of view, several methods for generating them are described. It is expected that this tutorial will serve newcomers as a starting point for further studies on the subject.

© 2020 Optical Society of America

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2019 (7)

N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365, eaax1839 (2019).
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C. H. Krishna and S. Roy, “Generation of inhomogeneously polarized vector vortex modes in few mode optical fiber,” Opt. Quantum Electron. 51, 41 (2019).
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J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Optoelectron. 12, 229–248 (2019).

J. C. Suárez-Bermejo, J. C. G. de Sande, M. Santarsiero, and G. Piquero, “Mueller matrix polarimetry using full Poincaré beams,” Opt. Laser Eng. 122, 134–141 (2019).
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K. Tekce, E. Otte, and C. Denz, “Optical singularities and Möebius strip arrays in tailored non-paraxial light fields,” Opt. Express 27, 29685–29696 (2019).
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G. Piquero, I. Marcos-Muñoz, and J. C. G. de Sande, “Simple undergraduate experiment for synthesizing and analyzing non-uniformly polarized beams by means of a Fresnel biprism,” Am. J. Phys. 87, 208–213 (2019).
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A. Hannonen, H. Partanen, J. Tervo, T. Setälä, and A. T. Friberg, “Pancharatnam-Berry phase in electromagnetic double-pinhole interference,” Phys. Rev. A 99, 053826 (2019).
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2018 (9)

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
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R. A. Terborg, J. P. Torres, and V. Pruneri, “Technique for generating periodic structured light beams using birefringent elements,” Opt. Express 26, 28938–28947 (2018).
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G. Piquero, L. Monroy, M. Santarsiero, M. Alonzo, and J. C. G. de Sande, “Synthesis of full Poincaré beams by means of uniaxial crystals,” J. Opt. 20, 065602 (2018).
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J. Chen, C. Wan, and Q. Zhan, “Vectorial optical fields: recent advances and future prospects,” Sci. Bull. 63(1), 54–74 (2018).
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A. Martnez, “Polarimetry enabled by nanophotonics,” Science 362, 750–751 (2018).
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J. C. G. de Sande, G. Piquero, and M. Santarsiero, “Polarimetry with azimuthally polarized light,” Opt. Commun. 410, 961–965 (2018).
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T. Wakayama, T. Higashiguchi, K. Sakaue, M. Washio, and Y. Otani, “Demonstration of a terahertz pure vector beam by tailoring geometric phase,” Sci. Rep. 8, 8690 (2018).
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N. Bhebhe, P. A. C. Williams, C. Rosales-Guzmán, V. Rodriguez-Fajardo, and A. Forbes, “A vector holographic optical trap,” Sci. Rep. 8, 17387 (2018).
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H. Larocque, D. Sugic, D. Mortimer, A. Taylor, R. Fickler, R. Boyd, M. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 36, 4104–4106 (2018).
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2017 (5)

J. C. G. de Sande, M. Santarsiero, and G. Piquero, “Spirally polarized beams for polarimetry measurements of deterministic and homogeneous samples,” Opt. Laser Eng. 91, 97–105 (2017).
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H. Zhang, J. Li, K. Cheng, M. Duan, and Z. Feng, “Trapping two types of particles using a focused partially coherent circular edge dislocations beam,” Opt. Laser Technol. 97, 191–197 (2017).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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B. Pérez-García, C. López-Mariscal, R. I. Hernández-Aranda, and J. C. Gutiérrez-Vega, “On-demand tailored vector beams,” Appl. Opt. 56, 6967–6972 (2017).
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O. Arteaga, R. Ossikovski, E. Kuntman, M. A. Kuntman, A. Canillas, and E. Garcia-Caurel, “Mueller matrix polarimetry on a Young’s double-slit experiment analog,” Opt. Lett. 42, 3900–3903 (2017).
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2016 (5)

C. Samlan and N. K. Viswanathan, “Generation of vector beams using a double-wedge depolarizer: non-quantum entanglement,” Opt. Laser Eng. 82, 135–140 (2016).
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A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence (invited),” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
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B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
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J. A. Davis, I. Moreno, K. Badham, M. M. Sánchez-López, and D. M. Cottrell, “Nondiffracting vector beams where the charge and the polarization state vary with propagation distance,” Opt. Lett. 41, 2270–2273 (2016).
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D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
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2015 (9)

X. Zheng, A. Lizana, A. Peinado, C. Ramírez, J. L. Martínez, A. Márquez, I. Moreno, and J. Campos, “Compact LCOS–SLM based polarization pattern beam generator,” J. Lightwave Technol. 33, 2047–2055 (2015).
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A. Turpin, Y. V. Loiko, A. Peinado, A. Lizana, T. K. Kalkandjiev, J. Campos, and J. Mompart, “Polarization tailored novel vector beams based on conical refraction,” Opt. Express 23, 5704–5715 (2015).
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W. Zhu, V. Shvedov, W. She, and W. Krolikowski, “Transverse spin angular momentum of tightly focused full Poincaré beams,” Opt. Express 23, 34029–34041 (2015).
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M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
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T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
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C. Wei, D. Wu, C. Liang, F. Wang, and Y. Cai, “Experimental verification of significant reduction of turbulence-induced scintillation in a full Poincaré beam,” Opt. Express 23, 24331–24341 (2015).
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D. Colas, L. Dominici, S. Donati, A. A. Pervishko, T. C. Liew, I. A. Shelykh, D. Ballarini, M. de Giorgi, A. Bramati, G. Gigli, E. del Valle, F. P. Laussy, A. V. Kavolin, and D. Sanvito, “Polarization shaping of Poincaré beams by polariton oscillations,” Light Sci. Appl. 4, e350 (2015).
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J. Kalwe, M. Neugebauer, C. Ominde, G. Leuchs, G. Rurimo, and P. Banzer, “Exploiting cellophane birefringence to generate radially and azimuthally polarised vector beams,” Eur. J. Phys. 36, 025011 (2015).
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R. Martínez-Herrero and F. Prado, “Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre–Gauss beams,” Opt. Express 23, 5043–5051 (2015).
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2014 (5)

Y. Ma and R. Wu, “Characterizing polarization properties of radially polarized beams,” Opt. Rev. 21, 4–8 (2014).
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J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53, 044107 (2014).
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J. C. G. de Sande, G. Piquero, M. Santarsiero, and F. Gori, “Partially coherent electromagnetic beams propagating through double-wedge depolarizers,” J. Opt. 16, 035708 (2014).
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S. Chen, X. Zhou, Y. Liu, X. Ling, H. Luo, and S. Wen, “Generation of arbitrary cylindrical vector beams on the higher order Poincaré sphere,” Opt. Lett. 39, 5274–5276 (2014).
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C. J. R. Sheppard, “Jones and Stokes parameters for polarization in three dimensions,” Phys. Rev. A 90, 023809 (2014).
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2013 (9)

K. Lou, S.-X. Qian, Z.-C. Ren, C. Tu, Y. Li, and H.-T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013).
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T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Front. Optoelectron. 6, 89–96 (2013).
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S. Vyas, Y. Kozawa, and S. Sato, “Polarization singularities in superposition of vector beams,” Opt. Express 21, 8972–8986 (2013).
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F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21, 8815–8820 (2013).
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V. G. Shvedov, C. Hnatovsky, N. Shostka, and W. Krolikowski, “Generation of vector bottle beams with a uniaxial crystal,” J. Opt. Soc. Am. B 30, 1–6 (2013).
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C.-Y. Han, R.-S. Chang, and H.-F. Chen, “Solid-state interferometry of a pentaprism for generating cylindrical vector beam,” Opt. Rev. 20, 189–192 (2013).
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M. Santarsiero, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
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T. Alieva, J. A. Rodrigo, A. Cámara, and E. Abramochkin, “Partially coherent stable and spiral beams,” J. Opt. Soc. Am. A 30, 2237–2243 (2013).
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D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5432–5439 (2013).
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2012 (8)

Y. Li, X.-L. Wang, H. Zhao, L.-J. Kong, K. Lou, B. Gu, C. Tu, and H.-T. Wang, “Young’s two-slit interference of vector light fields,” Opt. Lett. 37, 1790–1792 (2012).
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N. Khilo, T. S. Al-Saud, S. H. Al-Khowaiter, M. K. Al-Muhanna, S. Solonevich, N. Kazak, and A. Ryzhevich, “A high-efficient method for generating radially and azimuthally polarized Bessel beams using biaxial crystals,” Opt. Commun. 285, 4807–4810 (2012).
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G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
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J. C. G. de Sande, G. Piquero, and C. Teijeiro, “Polarization changes at Lyot depolarizer output for different types of input beams,” J. Opt. Soc. Am. A: 29, 278–284 (2012).
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J. C. G. de Sande, M. Santarsiero, G. Piquero, and F. Gori, “Longitudinal polarization periodicity of unpolarized light passing through a double wedge depolarizer,” Opt. Express 20, 27348–27360 (2012).
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E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
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F. Kenny, D. Lara, O. G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express 20, 14015–14029 (2012).
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I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquid crystal spatial light modulator,” Opt. Express 20, 364–376 (2012).
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2011 (3)

W. Han, W. Cheng, and Q. Zhan, “Flattop focusing with full Poincaré beams under low numerical aperture illumination,” Opt. Lett. 36, 1605–1607 (2011).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
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L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
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2010 (5)

2009 (5)

2008 (5)

R. Martínez-Herrero and P. M. Mejías, “Propagation of light fields with radial or azimuthal polarization distribution at a transverse plane,” Opt. Express 16, 9021–9033 (2008).
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M. Erdélyi and G. Gajdátsy, “Radial and azimuthal polarizer by means of a birefringent plate,” J. Opt. A 10, 055007 (2008).
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V. Ramírez-Sánchez and G. Piquero, “The beam quality parameter of spirally polarized beams,” J. Opt. A 10, 125004 (2008).
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E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008).
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R. Martínez-Herrero, P. Mejías, G. Piquero, and V. Ramírez-Sánchez, “Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams,” Opt. Commun. 281, 1976–1980 (2008).
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2007 (2)

2006 (4)

V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006).
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L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
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R. Martínez-Herrero, P. Mejías, and G. Piquero, “Overall parameters for the characterization of non-uniformly totally polarized beams,” Opt. Commun. 265, 6–10 (2006).
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A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal,” Opt. Express 14, 3724–3729 (2006).
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2005 (2)

2004 (3)

G. Piquero and J. Vargas-Balbuena, “Non-uniformly polarized beams across their transverse profiles: an introductory study for undergraduate optics courses,” Eur. J. Phys. 25, 793–800 (2004).
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Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
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R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
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2003 (5)

2002 (8)

2001 (8)

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
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I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
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A. Lapucci and M. Ciofini, “Polarization state modifications in the propagation of high azimuthal order annular beams,” Opt. Express 9, 603–609 (2001).
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L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
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A. Ciattoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
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J. Tervo and J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
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G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18, 1399–1405 (2001).
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G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001).
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2000 (1)

1999 (4)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
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F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
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G. Piquero, J. M. Movilla, P. M. Mejías, and R. Martínez-Herrero, “Degree of polarization of non-uniformly partially polarized beams: a proposal,” Opt. Quantum Electron. 31, 223–226 (1999).
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P. H. Äyräs, A. T. Friberg, M. A. J. Kaivola, and M. M. Salomaa, “Conoscopic interferometry of surface-acoustic-wave substrate crystals,” Appl. Opt. 38, 5399–5407 (1999).
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1998 (2)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941 (1998).
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I. S. Moreno, J. A. Davis, K. D’Nelly, and D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 144–3052 (1998).
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1996 (3)

V. Arrizón, E. Tepichin, M. Ortiz-Gutierrez, and A. Lohmann, “Fresnel diffraction at 1/4 of the Talbot distance of an anisotropic grating,” Opt. Commun. 127, 171–175 (1996).
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M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996).
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V. Bagini, R. Borghi, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, and G. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
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1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Concentric-circle-grating, surface-emitting semiconductor lasers,” Opt. Photon. News 3(12), 41 (1992).
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1990 (2)

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990).
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1989 (1)

R. Simon and N. Mukunda, “Universal SU(2) gadget for polarization optics,” Phys. Lett. A 138, 474–480 (1989).
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1972 (1)

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
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Abramochkin, E.

Alekseeva, L. V.

Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
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Alieva, T.

Al-Khowaiter, S. H.

N. Khilo, T. S. Al-Saud, S. H. Al-Khowaiter, M. K. Al-Muhanna, S. Solonevich, N. Kazak, and A. Ryzhevich, “A high-efficient method for generating radially and azimuthally polarized Bessel beams using biaxial crystals,” Opt. Commun. 285, 4807–4810 (2012).
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Allison, D. B.

I. S. Moreno, J. A. Davis, K. D’Nelly, and D. B. Allison, “Transmission and phase measurements for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 144–3052 (1998).
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Al-Muhanna, M. K.

N. Khilo, T. S. Al-Saud, S. H. Al-Khowaiter, M. K. Al-Muhanna, S. Solonevich, N. Kazak, and A. Ryzhevich, “A high-efficient method for generating radially and azimuthally polarized Bessel beams using biaxial crystals,” Opt. Commun. 285, 4807–4810 (2012).
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Alonso, M. A.

Alonzo, M.

G. Piquero, L. Monroy, M. Santarsiero, M. Alonzo, and J. C. G. de Sande, “Synthesis of full Poincaré beams by means of uniaxial crystals,” J. Opt. 20, 065602 (2018).
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Alpmann, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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Al-Saud, T. S.

N. Khilo, T. S. Al-Saud, S. H. Al-Khowaiter, M. K. Al-Muhanna, S. Solonevich, N. Kazak, and A. Ryzhevich, “A high-efficient method for generating radially and azimuthally polarized Bessel beams using biaxial crystals,” Opt. Commun. 285, 4807–4810 (2012).
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Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Concentric-circle-grating, surface-emitting semiconductor lasers,” Opt. Photon. News 3(12), 41 (1992).
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Andrews, D. L.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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Opt. Laser Eng. (3)

C. Samlan and N. K. Viswanathan, “Generation of vector beams using a double-wedge depolarizer: non-quantum entanglement,” Opt. Laser Eng. 82, 135–140 (2016).
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Figures (18)

Fig. 1.
Fig. 1. Poincaré sphere. Any point on its surface corresponds to a different polarization state.
Fig. 2.
Fig. 2. SPB beam with $ \gamma = \pi /6 $ . Upper row: polarization pattern where the red segments indicate the azimuth of the linear polarization (left) and representation of the corresponding state of polarization by means of red dots on the Poincaré sphere (right). Lower row: Stokes parameters $ {S_1} $ (left) and $ {S_2} $ (right).
Fig. 3.
Fig. 3. Polarization pattern (top) for the FPB given by Eq. (13) with $ \Phi = \pi /4 $ ; two views of such pattern on the Poincaré sphere (middle); and Stokes parameters $ {S_1} $ , $ {S_2} $ , and $ {S_3} $ , from left to right, normalized to the maximum irradiance of the beam (bottom). Red (green) ellipses and dots denote right-handed (left-handed) polarization.
Fig. 4.
Fig. 4. Polarization pattern (top) for the FPB given by Eq. (14) with $ \Phi = \pi /4 $ ; two views of such pattern on the Poincaré sphere (middle); and Stokes parameters $ {S_1} $ , $ {S_2} $ , and $ {S_3} $ , from left to right, normalized to the maximum irradiance of the beam (bottom). Red (green) ellipses and dots denote right-handed (left-handed) polarization.
Fig. 5.
Fig. 5. Experimental scheme for synthesizing SPBs. BE, beam expander; PC, polarization converter; SF, spatial filter; R, rotator; L, lens; PSA, polarization state analyzer composed of a quarter-wave phase plate ( $ \lambda /4 $ ), a dichroic polarizer P, and a CCD camera.
Fig. 6.
Fig. 6. Experimental spirally polarized pattern with $ \gamma { = 20^ \circ } $ . Red (green) ellipses denote right-handed (left-handed) polarization states.
Fig. 7.
Fig. 7. MZI used to synthesize NUTP beams by means of amplitude transmittances, $ {t_i}(r)$ with $ i = 1,2 $ . M’s are mirrors, $ {{\rm BS}_i} $ are beam splitters, $ \lambda /2 $ is a half-wave phase plate, and the subscripts $ s $ and $ p $ denote the polarization of the electric field ( $ s $ perpendicular and $ p $ parallel to the incidence plane). L is a converging lens, which images the synthesized NUTP profile onto the plane $ {P_0} $ .
Fig. 8.
Fig. 8. Theoretical polarization pattern across the transverse section of a NUTP beam at the output of a MZI using only one super-Gaussian transmittance at one arm with $ {n_2} = 2 $ and $ {w_2} = 0.12\,{\rm mm} $ (top) and representation of the states of polarization on the Poincaré sphere (bottom left). Experimentally measured azimuth for the field obtained with the experimental setup in Fig. 7 (bottom right).
Fig. 9.
Fig. 9. Experimental setup. MO, microscope objective; PH, pinhole; $ {{\rm L}_1} $ and $ {{\rm L}_2} $ , lenses; $ {{\rm P}_1} $ and $ {{\rm P}_2} $ , polarizers; B, biprism; PSA, polarization state analyzer composed of a quarter-wave phase plate ( $ \lambda /4 $ ), a polarizer (P), and a CCD camera.
Fig. 10.
Fig. 10. Theoretical (upper left) and experimental (upper right) polarization pattern at the output of the biprism for input light linearly polarized with 45° azimuth with $ \lambda = 632.8\,{\rm nm} $ , $ n = 1.515 $ , and $ \alpha { = 1.5^ \circ } $ . Red (green) ellipses denote right-handed (left-handed) polarization states. Poincaré sphere (bottom) where the states of polarization are represented.
Fig. 11.
Fig. 11. Uniaxial crystal and field decomposition at the output face.
Fig. 12.
Fig. 12. Theoretical polarization pattern at the output of the calcite crystal for input light linearly polarized with $ \pi /2 $ azimuth and $ \lambda = 632.8\,{\rm nm} $ (left) and Poincaré sphere with dots corresponding to the ellipses of polarization shown in the polarization pattern (right). Red (green) ellipses and dots denote right-handed (left-handed) polarization states.
Fig. 13.
Fig. 13. Experimental setup for synthesizing a NUTP beam by means of a calcite crystal. MO is a microscope objective, $ L $ a lens, $ \lambda /4 $ a quarter-wave phase plate, and P a dichroic polarizer.
Fig. 14.
Fig. 14. Experimental polarization pattern at the output of the calcite crystal for input light linearly polarized with $ \pi /2 $ azimuth and $ \lambda = 632.8\,{\rm nm} $ (left), and Poincaré sphere with dots corresponding to the ellipses of polarization shown in the polarization pattern (right). Red (green) ellipses and dots denote right-handed (left-handed) polarization states.
Fig. 15.
Fig. 15. Scheme of a double-wedge anisotropic crystal. Green arrows denote the orientation of the optic axis of each part of the double wedge. Both optic axes are perpendicular to the $ z $ axis, and the angle between them is $ \pi /4 $ . Red and blue arrows denote the linear polarization directions of each exiting beam. The wedge angle is  $ \varphi $ .
Fig. 16.
Fig. 16. Polarization pattern at the exit of a DW when a linearly polarized light along $ y $ axis inputs in it (left) and Poincaré sphere where the states of polarization are represented (right). Red (green) ellipses and dots denote right-handed (left-handed) polarization states.
Fig. 17.
Fig. 17. Experimental setup to generate a NUTP beam by means of a DW.
Fig. 18.
Fig. 18. (a) Measured Stokes parameters at the exit of a DW along the $ x $ direction and (b) the corresponding ellipse of polarization. Red (green) ellipses denote right-handed (left-handed) polarization states.

Equations (36)

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E ( t ) = [ E x ( t ) u x + E y ( t ) u y ] e i ω t ,
E = ( E x E y ) ,
E o u t = T ^ E i n ,
ψ = 1 2 arctan ( 2 R e { E x E y } | E x | 2 | E y | 2 )
χ = 1 2 arcsin ( 2 I m { E x E y } | E x | 2 + | E y | 2 ) ,
P ^ = ( P xx P xy P yx P yy ) ,
P ij = E i ( t ) E j ( t ) ,
S 0 = P xx + P yy , S 1 = P xx P yy , S 2 = 2 R e { P xy } , S 3 = 2 I m { P xy } ,
S 0 = I 0 + I π / 2 S 1 = I 0 I π / 2 , S 2 = I π / 4 I π / 4 , S 3 = I π / 4 I π / 4 ,
s n = S n S 0 , ( n = 1 , 2 , 3 ) .
P = s 1 2 + s 2 2 + s 3 2 .
E S P ( r ) = f ( r ) ( cos ( θ + γ ) sin ( θ + γ ) ) ,
E F P 0 ( r ) = E 0 exp ( r 2 w 0 2 ) ( cos Φ 2 r w 0 e i θ sin Φ ) ,
E F P 1 ( r ) = E 0 exp ( r 2 w 0 2 ) ( r w 0 e i θ cos Φ ( 1 2 r 2 w 0 2 ) sin Φ ) .
P ~ = 1 I T P ( r ) S 0 ( r ) d r ,
I T = S 0 ( r ) d r
σ p 2 = 1 I T [ P ( r ) P ~ ] 2 S 0 ( r ) d r .
ρ c = 1 I T S 3 ( r ) d r
σ c 2 = 1 I T [ S 3 ( r ) S 0 ( r ) ρ c ] 2 S 0 ( r ) d r .
ρ R = 1 2 + 1 2 I T 0 0 2 π cos ( 2 θ ) S 1 ( r , θ ) r d r d θ + 1 2 I T 0 0 2 π sin ( 2 θ ) S 2 ( r , θ ) r d r d θ , ρ A = 1 2 1 2 I T 0 0 2 π cos ( 2 θ ) S 1 ( r , θ ) r d r d θ 1 2 I T 0 0 2 π sin ( 2 θ ) S 2 ( r , θ ) r d r d θ .
ρ R = 1 2 1 2 cos ( γ )
ρ A = 1 2 + 1 2 cos ( γ ) ,
T ^ P G ( r ) = ( cos 2 β x cos β x sin β x cos β x sin β x sin 2 β x ) ,
E P G ( r ) = E 0 2 e i β x ( cos β x sin β x ) .
T ^ A P ( r ) = ( sin 2 θ cos θ sin θ cos θ sin θ cos 2 θ )
E A P ( r ) = i 2 f ( r ) e i θ ( sin θ cos θ ) .
E P C ( r ) = f ( r ) ( sin θ cos θ ) ,
E M Z ( r ) E 0 2 ( t 1 ( r ) t 2 ( r ) ) = E 0 2 ( exp [ ( r w 1 ) 2 n 1 ] exp [ ( r w 2 ) 2 n 2 ] ) ,
E B F ( r ) ( E 0 x e i k α ( n 1 ) y E 0 y e i k α ( n 1 ) y ) ,
Λ = λ 2 α ( n 1 ) ,
E i n = ( E 0 x E 0 y exp ( i ϕ ) ) ,
E e ( r ) = [ E 0 x cos θ + E 0 y sin θ exp ( i ϕ ) ] exp [ i k n ( α ) d e ] u r , E o ( r ) = [ E 0 x sin θ + E 0 y cos θ exp ( i ϕ ) ] exp [ i k n o d o ] u θ ,
t x x ( r ) = cos 2 θ e i δ ( r ) / 2 + sin 2 θ e i δ ( r ) / 2 , t x y ( r ) = i sin 2 θ sin [ δ ( r ) / 2 ] , t y x ( r ) = i sin 2 θ sin [ δ ( r ) / 2 ] , t y y ( r ) = cos 2 θ e i δ ( r ) / 2 + sin 2 θ e i δ ( r ) / 2 ,
δ ( r ) k [ n ( α ) n o ] d k ( n e n o ) r 2 d ,
t x x ( r ) = 1 2 [ 1 + exp ( i δ 2 ( x ) ) ] , t x y ( r ) = 1 2 [ 1 + exp ( i δ 2 ( x ) ) ] exp ( i δ 1 ( x ) ) , t y x ( r ) = 1 2 [ 1 + exp ( i δ 2 ( x ) ) ] , t y y ( r ) = 1 2 [ 1 + exp ( i δ 2 ( x ) ) ] exp ( i δ 1 ( x ) ) .
L = λ | n e ( λ ) n o ( λ ) | tan φ ,

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