Abstract

Stokes polarimetry (SP) is a powerful technique that enables spatial reconstruction of the state of polarization (SoP) of a light beam using only intensity measurements. A given SoP is reconstructed from a set of four Stokes parameters, which are computed through four intensity measurements. Since all intensities must be performed on the same beam, it is common to record each intensity individually, one after the other, limiting its performance to light beams with static SoP. Here, we put forward a novel technique to extend SP to a broader set of light beams with dynamic SoP. This technique relies on the superposition principle, which enables the splitting of the input beam into identical copies, allowing the simultaneous measurement of all intensities. For this, the input beam is passed through a multiplexed digital hologram displayed on a polarization-insensitive Digital Micromirror Device (DMD) that grants independent and rapid (20 kHz) manipulation of each beam. We are able to reliably reconstruct the SoP with high fidelity and at speeds of up to 27 Hz, paving the way for real-time polarimetry of structured light.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Polarization is a remarkable feature of light that evinces its oscillatory wave nature. The study of polarization can be traced back to the 1600s, but it was only after the seminal work by Young, in 1803, that it was linked to the transverse vibrations of the electric field. Many of the greatest minds of the 19th century, including Malus, Brewster, Arago and Fresnel, contributed enormously to the understanding of polarization, but it was Stokes who established the basis for its modern description [1]. The significance of his contribution lies in the introduction of four quantities determined through intensity measurements, known as Stokes parameters, to describe any state of polarization. In essence, the unknown field is projected into a set of three unbiased polarization basis. The intensity reaching the photodetector contains information about the percentage of light in each basis, which is captured by the Stokes parameters. From these, both, the SoP and the wavefront of structured light beams can be measured [2]. Traditionally, the required intensities are measured one by one, a major drawback of present approaches, which do not allow a real-time tracking of the SoP, even though many applications would benefit from it. Previous attempts to ameliorate this have considered amplitude division, which faces the disadvantages of unbalanced distribution of energy, or a mismatch in beam size due to unequal propagation distances [3,4]. An alternative approach to overcome this drawback is wavefront splitting, commonly implemented through static diffraction gratings, see for example [5] and references therein. Modern versions of these have been also implemented with polarization-dependent metasurfaces [6,7] or liquid crystal devices [8] that can only operate in a reduced wavelength range.

Here we put forward an all-digital version of the wavefront-splitting technique for the real-time reconstruction of polarization. Our technique relies on the splitting of a vector beam into four independent beams propagating along different paths, through a computer-controlled Digital Micromirror Device (DMD), of great relevance in the generation of structured light beams [914]. To accomplish this, a digital hologram containing four multiplexed gratings, with unique spatial frequencies, is displayed on the polarization-insensitive DMD. Remarkably, the DMD enables a rapid ($\sim$20 KHz refresh rate) and independent reconfiguration of each beam, its position or intensity, by simply changing the hologram on the DMD. In addition, DMDs can operate in a wide wavelength range, providing with an additional advantage compared to current systems. To demonstrate the capabilities of our technique, we applied this to Vector Beams (VB), non-homogeneous SoP, that are key in a myriad of applications [1522]. Finally, to asses its accuracy, we compared the experimentally reconstructed SoP with numerical simulations.

2. Concept

In VBs the spatial and polarization degrees of freedom are coupled in a non-separable way. These are commonly expressed, in the cylindrical coordinates $(\rho , \phi )$, as [2325],

$$\textbf{E}(\rho,\phi)=\cos(\theta)\exp[i\ell\phi]\hat{e_R}+\sin(\theta)\exp[{-}i\ell\phi]\exp[i\alpha]\hat{e_L},$$
where, the exponential term $\exp [i\ell \phi ]$ is associated to an azimuthal phase distribution, $\ell$ $\in \mathbb {Z}$ is known as the topological charge and $\exp [i\alpha ]$ is an intramodal phase. The amplitude parameter $\theta \in [0, \pi /2]$ allows to change the field $\textbf {E}(\rho , \phi )$, from purely scalar ($\theta =0$ and $\theta =\pi /2$) to vector ($\theta =\pi /4$). The SoP of such light beams can be reconstructed through the Stokes parameters, that can be determined through a minimum of four intensities measurements as [1],
$$S_{0}=I_{0},\quad S_{1}=2I_{H}-S_{0},\quad S_{2}=2I_{D}-S_{0},\quad \textrm{and}\quad S_{3}=2I_{R}-S_{0},$$
where $I_0$ is the total intensity of the beam and $I_H$, $I_D$ and $I_R$ represent the intensity of the horizontal, diagonal and right-handed polarization components, respectively. Figure 1(a) illustrates the traditional way in which the SoP of an input beam is reconstructed. Here, the required intensities are recorded at different times (t$_1$, t$_2$, t$_3$ and t$_4$), limiting its applicability to invariant SoP. In contrast, Fig. 1(b) illustrates our technique, in which the light beam is projected onto a digital hologram displayed on a polarization-insensitive DMD. The hologram consist of four multiplexed diffraction gratings with unique spatial frequencies that enable the splitting of the input beam into four identical copies. Each of this beams is then passed through the necessary optical filters to measure the required intensities. In this way, all intensities can be recorded in a single image, enabling the reconstruction of the SoP in a single shot, thus allowing real-time tracking of the SoP at speeds limited only by the CCD camera. This technique will be of great relevance in the case when the SoP of the beam changes over time, for example, by passing through a polarization-sensitive system, as schematically illustrated in Fig. 1(b).

 figure: Fig. 1.

Fig. 1. (a) In standard polarymetry, the intensities required to reconstruct a SoP are recorded one by one at different times t$_1$, t$_2$, t$_3$ and t$_4$. (b) In our technique, a digital hologram displayed on a Digital Micro Device (DMD), splits the input beam into four identical copies for a simultaneous measurement of the intensities.

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3. Experimental setup

Our experimental setup is depicted in Fig. 2, where a linearly polarized Gaussian beam ($\lambda =532nm$) is converted into a cylindrical vector beam via a q-plate (q=1/2) in combination with a Half Wave-Plate (HWP1) [26]. To reconstruct its SoP , we first split this into four identical copies using a digital hologram displayed on a DMD (DLP Light Crafter 6500 from Texas Instruments). The beams are spatially filtered, to remove higher diffraction orders, and collimated to propagate along parallel paths using lenses L1 and L2 ($f_{1,2}=200$ mm) and a spatial filter (A). We then measure the required intensities $I_0$, $I_H$, $I_D$ and $I_R$ as follows. From path ① we obtain $I_H$ using of a linear polarizer at $\theta =0^0$ (P1). From path ③ we measure $I_D$, using another linear polarizer at $\theta =45^0$ (P2). Path ② is used to obtain $I_R$ by combining a QWP (QWP2) at $\beta =45^0$ and a linear polarizer at $\theta =90^0$. Finally, the beam on path ④ is transmitted unmodified to obtain $I_0$. A third lens (L3, $f_3=200 mm$) is added to focus the beams into a CCD camera (BC106N-VIS from Thorlabs) where the four intensities are measured simultaneously. In principle, there are certain restrictions to the beam’s size, imposed by the dimensions of both, the DMD ($\approx 6$mm) and the CCD ($\approx 4$ mm), as well as their pixel size ($\approx 8 \mu$m). Nonetheless, the use of appropriate telescopes, to increase or decrease the beams’s size, can provide with more flexibility. Importantly, the DMD enables rapid adjustment of each beam’s position and intensity, a feature that makes our system very robust compared to others. Once the system is aligned, everything can be programmed to reconstruct the polarization distribution at the click of a button. With the aim of evaluating the performance of our technique, we inserted the systems represented as E1 (Rotating Half Wave-Plate) and E2 (Quarter Wave-Plate in combination with a non-linear crystal) in the path of the beam, to modify in real-time its SoP prior to its analysis.

To accurately reconstruct the vector beam’s polarization using the experimental setup described above, we first recorded a calibration image [see Fig. 3(a)] that allows to find the center and enclosing area of beams ①, ②, ③ and ④ [see Fig. 3(b)]. These beams are then used to compute the Stokes parameters, as shown in Fig. 3(c), from which, the SOP can be accurately reconstructed, as illustrated in Fig. 3(d), where this is overlapped with the intensity distribution.

 figure: Fig. 2.

Fig. 2. A vector beams is generated from a CW Gaussian beam ($\lambda =532$ nm) using a q-plate (q=1/2) and a Half Wave-Plate (HWP1). The resulting beam is split into four identical copies propagating along parallel paths using a Digital Micromirror Device (DMD) in combination with lenses L1 and L2. The linear polarizers P1 and P2 filter $I_H$ and $I_D$, respectively, while P3 in combination with a quarter wave-plate (QWP2) filters $I_R$. Lens L3 focuses the four beams into a CCD to measure all the intensities simultaneously. The systems E1 (a Rotating Half Wave-Plate: RHWP) and E2 (a QWP in combination with a non-linear crystal: NLC), inserted in the path of the beam, enable real-time evolution of the input beam’s SoP.

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 figure: Fig. 3.

Fig. 3. (a) Calibration image to find the centers of the beams. (b) Example image of the intensities $I_0$, $I_D$, $I_H$ and $I_R$. (c) Computed Stokes parameters. (d) Reconstructed polarization distribution.

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4. Results

As a proof-of-principle, we performed two experiments using the optical systems depicted in the dashed boxes E1 and E2 shown in Fig. 2. Each system was inserted in the path of the vector beam to change dynamically its SoP while we reconstructed in real time the SoP of the emerging beam. In the first experiment we inserted system E1 in the path of a radially polarized vector beam, which introduces an angle-dependant phase delay between both orthogonal polarization components. In this way, as we rotate the RHWP at a constant rate, the SoP of the VBs evolves from radial (at $0^\circ$) to azimuthal (at $45^\circ$) and back to radial (at $90^\circ$), acquiring spiral-like SoP at intermediate angles. Figures 4(a) and 4(b) exhibit four frames, respectively, of a video showing the experimentally reconstructed (see Visualization 1) and numerically simulated (see Visualization 2) SoP, respectively, as the RHWP is rotated from $0^\circ$ to $90^\circ$.

 figure: Fig. 4.

Fig. 4. Extracted frames of the real-time reconstruction of polarization after passing the beam through the system E1. (a) Experiment Visualization 1 and (b) simulation (Visualization 2)

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The second experiment was performed with the idea of showing a potential application of our technique, this in the context of optical metrology, for which we used a temperature-controlled birefringent non-linear crystal (NLC), Potassium titanyl phosphate (KTP). The KTP produces a temperature-dependant phase delay between the horizontal ($\hat {H}$) and vertical ($\hat {V}$) polarization components [27]. Here, we first converted the vector beam from the circular to the linear polarization basis, using a Quarter Wave-Plate (QWP1) at $45^\circ$ (see inset E2 in Fig. 2), and transmit this afterwards through the KTP crystal. In this way, an increase in the crystal’s temperature results in a rotation of the polarization distribution. In the experiment, we increased the temperature from $22 ^\circ C$ to $36 ^\circ C$ while reconstructed the output polarization every $2^ \circ C$. Figure 5 shows the experimentally reconstructed SoP (top row) compared to a numerical simulation of the same (bottom row) where $\hat {H}$ and $\hat {V}$ are represented by black and red ellipses, respectively. Notice how the polarization rotates in a counterclockwise rotation as the temperature increases, indicated by the long arrow along the center of $\hat {H}$. Hence, the temperature of the crystal can be measured through the angle of rotation of the SoP.

 figure: Fig. 5.

Fig. 5. Experimental (top raw) and simulated (botom row) reconstruction of the SoP as the temperature of a non-linear crystal changes from $22 ^\circ C$ to $36^\circ C$. The arrows indicate the rotation of polarization.

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5. Conclusions

Here we have proposed a technique to measure in real-time the SoP of a light beam that takes full advantage of the DMD technology. Even though DMDs have been around for almost four decades, it is only in recent times that they started raising attention into the generation of structured light fields. Nonetheless, their capabilities haven’t been fully exploited yet, as is the case of their polarization independence. Here we propose to use DMDs as a digital reconfigurable grating to split an input beam into four beams propagating along different paths from which, its spatial polarization distribution can be reconstructed in real time. Given that the conversion efficiency of DMDs is quite low (<10%), in the cases of low-power beams, the use of photodetectors with higher sensitivities could be required. Our technique is not intended for static SoP, even though it can be applied to these, but rather for dynamic SoP. That is the case of VBs in the context of optical communications, either in optical fibers or free-space, where due to external perturbations its polarization distribution changes over time. Hence, a system capable to characterize in real-time the evolution of the SoP is highly desirable, since this will allow to perform real-time correction of the transmitted beam. Another example is in optical metrology, where we can assign a one-to-one correspondence of a given property of a system, for example temperature, to a particular vector state. In this way, we can perform remote sensing of the properties of a system using VBs, which could be advantageous in certain scenarios. As such, and in order to evaluate the real capabilities of our system, we devised two experiments where its capabilities were fully tested. We were able to reconstruct the SoP of time-varying vector beams with high fidelity and at speeds limited only by the CCD camera ($\approx$ 27 Hz). Importantly, our technique can be implemented in a very compact and cost effective way paving the way to novel applications, for example in optical metrology, as a polarization-based sensor, or in free-space optical communications with vector beams.

Funding

National Natural Science Foundation of China (61975047, 11934013, 11574065).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. G. G. Stokes, Mathematical and physical papers (Reprinted in Mathematical and Physical Papers, Vol. 3, 233, Cambridge University Press: London, 1901, 1852).

2. A. Dudley, G. Milione, R. R. Alfano, and A. Forbes, “All-digital wavefront sensing for structured light beams,” Opt. Express 22(11), 14031–14040 (2014). [CrossRef]  

3. R. Azzam, “Division-of-amplitude photopolarimeter (doap) for the simultaneous measurement of all four stokes parameters of light,” Opt. Acta 29(5), 685–689 (1982). [CrossRef]  

4. M. Fridman, M. Nixon, E. Grinvald, N. Davidson, and A. A. Friesem, “Real-time measurement of space-variant polarizations,” Opt. Express 18(10), 10805–10812 (2010). [CrossRef]  

5. D. H. Goldstain, Polarized light (CRC Press, 2011).

6. D. Wen, F. Yue, S. Kumar, Y. Ma, M. Chen, X. Ren, P. E. Kremer, B. D. Gerardot, M. R. Taghizadeh, G. S. Buller, and X. Chen, “Metasurface for characterization of the polarization state of light,” Opt. Express 23(8), 10272–10281 (2015). [CrossRef]  

7. N. A. Rubin, A. Zaidi, M. Juhl, R. P. Li, J. B. Mueller, R. C. Devlin, K. Leósson, and F. Capasso, “Polarization state generation and measurement with a single metasurface,” Opt. Express 26(17), 21455–21478 (2018). [CrossRef]  

8. J. A. Davis, I. Moreno, M. M. Sánchez-López, K. Badham, J. Albero, and D. M. Cottrell, “Diffraction gratings generating orders with selective states of polarization,” Opt. Express 24(2), 907–917 (2016). [CrossRef]  

9. X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018). [CrossRef]  

10. Y.-X. Ren, R.-D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015). [CrossRef]  

11. Y. Chen, Z.-X. Fang, Y.-X. Ren, L. Gong, and R.-D. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. 54(27), 8030–8035 (2015). [CrossRef]  

12. K. J. Mitchell, S. Turtaev, M. J. Padgett, T. Čižmár, and D. B. Phillips, “High-speed spatial control of the intensity, phase and polarisation of vector beams using a digital micro-mirror device,” Opt. Express 24(25), 29269–29282 (2016). [CrossRef]  

13. S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22(15), 17999–18009 (2014). [CrossRef]  

14. V. Lerner, D. Shwa, Y. Drori, and N. Katz, “Shaping Laguerre-Gaussian laser modes with binary gratings using a digital micromirror device,” Opt. Lett. 37(23), 4826–4828 (2012). [CrossRef]  

15. H. Rubinsztein-Dunlop, A. Forbes, M. Berry, M. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, and T. Bauer, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017). [CrossRef]  

16. X.-B. Hu, B. Zhao, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “In situ detection of a cooperative target’s longitudinal and angular speed using structured light,” Opt. Lett. 44(12), 3070–3073 (2019). [CrossRef]  

17. B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzmán, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13(4), 397–402 (2017). [CrossRef]  

18. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzmán, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36(2), 292–301 (2018). [CrossRef]  

19. N. Bhebhe, P. A. C. Williams, C. Rosales-Guzmán, V. Rodriguez-Fajardo, and A. Forbes, “A vector holographic optical trap,” Sci. Rep. 8(1), 17387 (2018). [CrossRef]  

20. N. Bhebhe, C. Rosales-Guzman, and A. Forbes, “Classical and quantum analysis of propagation invariant vector flat-top beams,” Appl. Opt. 57(19), 5451–5458 (2018). [CrossRef]  

21. C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20(12), 123001 (2018). [CrossRef]  

22. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]  

23. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full poincaré beams,” Opt. Express 18(10), 10777–10785 (2010). [CrossRef]  

24. 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(15), 2925–2934 (2012). [CrossRef]  

25. E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89(3), 031801 (2014). [CrossRef]  

26. L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef]  

27. Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098–5101 (2014). [CrossRef]  

References

  • View by:

  1. G. G. Stokes, Mathematical and physical papers (Reprinted in Mathematical and Physical Papers, Vol. 3, 233, Cambridge University Press: London, 1901, 1852).
  2. A. Dudley, G. Milione, R. R. Alfano, and A. Forbes, “All-digital wavefront sensing for structured light beams,” Opt. Express 22(11), 14031–14040 (2014).
    [Crossref]
  3. R. Azzam, “Division-of-amplitude photopolarimeter (doap) for the simultaneous measurement of all four stokes parameters of light,” Opt. Acta 29(5), 685–689 (1982).
    [Crossref]
  4. M. Fridman, M. Nixon, E. Grinvald, N. Davidson, and A. A. Friesem, “Real-time measurement of space-variant polarizations,” Opt. Express 18(10), 10805–10812 (2010).
    [Crossref]
  5. D. H. Goldstain, Polarized light (CRC Press, 2011).
  6. D. Wen, F. Yue, S. Kumar, Y. Ma, M. Chen, X. Ren, P. E. Kremer, B. D. Gerardot, M. R. Taghizadeh, G. S. Buller, and X. Chen, “Metasurface for characterization of the polarization state of light,” Opt. Express 23(8), 10272–10281 (2015).
    [Crossref]
  7. N. A. Rubin, A. Zaidi, M. Juhl, R. P. Li, J. B. Mueller, R. C. Devlin, K. Leósson, and F. Capasso, “Polarization state generation and measurement with a single metasurface,” Opt. Express 26(17), 21455–21478 (2018).
    [Crossref]
  8. J. A. Davis, I. Moreno, M. M. Sánchez-López, K. Badham, J. Albero, and D. M. Cottrell, “Diffraction gratings generating orders with selective states of polarization,” Opt. Express 24(2), 907–917 (2016).
    [Crossref]
  9. X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018).
    [Crossref]
  10. Y.-X. Ren, R.-D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
    [Crossref]
  11. Y. Chen, Z.-X. Fang, Y.-X. Ren, L. Gong, and R.-D. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. 54(27), 8030–8035 (2015).
    [Crossref]
  12. K. J. Mitchell, S. Turtaev, M. J. Padgett, T. Čižmár, and D. B. Phillips, “High-speed spatial control of the intensity, phase and polarisation of vector beams using a digital micro-mirror device,” Opt. Express 24(25), 29269–29282 (2016).
    [Crossref]
  13. S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22(15), 17999–18009 (2014).
    [Crossref]
  14. V. Lerner, D. Shwa, Y. Drori, and N. Katz, “Shaping Laguerre-Gaussian laser modes with binary gratings using a digital micromirror device,” Opt. Lett. 37(23), 4826–4828 (2012).
    [Crossref]
  15. H. Rubinsztein-Dunlop, A. Forbes, M. Berry, M. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, and T. Bauer, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
    [Crossref]
  16. X.-B. Hu, B. Zhao, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “In situ detection of a cooperative target’s longitudinal and angular speed using structured light,” Opt. Lett. 44(12), 3070–3073 (2019).
    [Crossref]
  17. B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzmán, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13(4), 397–402 (2017).
    [Crossref]
  18. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzmán, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36(2), 292–301 (2018).
    [Crossref]
  19. N. Bhebhe, P. A. C. Williams, C. Rosales-Guzmán, V. Rodriguez-Fajardo, and A. Forbes, “A vector holographic optical trap,” Sci. Rep. 8(1), 17387 (2018).
    [Crossref]
  20. N. Bhebhe, C. Rosales-Guzman, and A. Forbes, “Classical and quantum analysis of propagation invariant vector flat-top beams,” Appl. Opt. 57(19), 5451–5458 (2018).
    [Crossref]
  21. C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20(12), 123001 (2018).
    [Crossref]
  22. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  23. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full poincaré beams,” Opt. Express 18(10), 10777–10785 (2010).
    [Crossref]
  24. 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(15), 2925–2934 (2012).
    [Crossref]
  25. E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89(3), 031801 (2014).
    [Crossref]
  26. L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96(16), 163905 (2006).
    [Crossref]
  27. Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098–5101 (2014).
    [Crossref]

2019 (1)

2018 (6)

2017 (2)

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzmán, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13(4), 397–402 (2017).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. Berry, M. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, and T. Bauer, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

2016 (2)

2015 (3)

2014 (4)

2012 (2)

2010 (2)

2009 (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

1982 (1)

R. Azzam, “Division-of-amplitude photopolarimeter (doap) for the simultaneous measurement of all four stokes parameters of light,” Opt. Acta 29(5), 685–689 (1982).
[Crossref]

Albero, J.

Alfano, R. R.

Alonso, M. A.

Alpmann, C.

H. Rubinsztein-Dunlop, A. Forbes, M. Berry, M. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, and T. Bauer, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

Andrews, D. L.

H. Rubinsztein-Dunlop, A. Forbes, M. Berry, M. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, and T. Bauer, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

Azzam, R.

R. Azzam, “Division-of-amplitude photopolarimeter (doap) for the simultaneous measurement of all four stokes parameters of light,” Opt. Acta 29(5), 685–689 (1982).
[Crossref]

Badham, K.

Banzer, P.

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Phys. Rev. A (1)

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89(3), 031801 (2014).
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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(16), 163905 (2006).
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Supplementary Material (2)

NameDescription
Visualization 1       All-digital Stokes polarimetry for the real-time reconstruction of the state of polarization of a light beam using a digital micromirror device (simulation)
Visualization 2       All-digital Stokes polarimetry for the real-time reconstruction of the state of polarization of a light beam using a digital micromirror device (experiment)

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Figures (5)

Fig. 1.
Fig. 1. (a) In standard polarymetry, the intensities required to reconstruct a SoP are recorded one by one at different times t $_1$ , t $_2$ , t $_3$ and t $_4$ . (b) In our technique, a digital hologram displayed on a Digital Micro Device (DMD), splits the input beam into four identical copies for a simultaneous measurement of the intensities.
Fig. 2.
Fig. 2. A vector beams is generated from a CW Gaussian beam ( $\lambda =532$ nm) using a q-plate (q=1/2) and a Half Wave-Plate (HWP1). The resulting beam is split into four identical copies propagating along parallel paths using a Digital Micromirror Device (DMD) in combination with lenses L1 and L2. The linear polarizers P1 and P2 filter $I_H$ and $I_D$ , respectively, while P3 in combination with a quarter wave-plate (QWP2) filters $I_R$ . Lens L3 focuses the four beams into a CCD to measure all the intensities simultaneously. The systems E1 (a Rotating Half Wave-Plate: RHWP) and E2 (a QWP in combination with a non-linear crystal: NLC), inserted in the path of the beam, enable real-time evolution of the input beam’s SoP.
Fig. 3.
Fig. 3. (a) Calibration image to find the centers of the beams. (b) Example image of the intensities $I_0$ , $I_D$ , $I_H$ and $I_R$ . (c) Computed Stokes parameters. (d) Reconstructed polarization distribution.
Fig. 4.
Fig. 4. Extracted frames of the real-time reconstruction of polarization after passing the beam through the system E1. (a) Experiment Visualization 1 and (b) simulation (Visualization 2)
Fig. 5.
Fig. 5. Experimental (top raw) and simulated (botom row) reconstruction of the SoP as the temperature of a non-linear crystal changes from $22 ^\circ C$ to $36^\circ C$ . The arrows indicate the rotation of polarization.

Equations (2)

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E ( ρ , ϕ ) = cos ( θ ) exp [ i ϕ ] e R ^ + sin ( θ ) exp [ i ϕ ] exp [ i α ] e L ^ ,
S 0 = I 0 , S 1 = 2 I H S 0 , S 2 = 2 I D S 0 , and S 3 = 2 I R S 0 ,

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