Abstract

We propose a polarimeter that combines a plasmonic spiral structure and the machine learning algorithm with an ultra-compact footprint of 10*10µm2. Being different from previous similar schemes working only as circular polarization analyzers, arbitrary states of polarization (SOPs) can be retrieved via the spiral structure for the first time to our best knowledge, by analyzing the near-field intensity distribution through machine learning. A 3-layer neural network (NN) is successfully trained to correlate intensity patterns to the SOPs of incident light. Based on simulation, a low estimation error benchmarked by mean-squared error (MSE) of only 1.23e-3 is achieved. In this way, without the conventional bulky optical system or complex nano-structures, SOP detection is achieved via such a simple and ultra-compact spiral. The proposed scheme not only pushes the application limits of the device based on plasmonic spiral structures but also provides a new insight for SOP detection.

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

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References

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  1. A. Schwope and J. Tinbergen, Astronomical Polarimetry (Cambridge University Press, 1996).
  2. G. Vasile, E. Trouvé, J.-S. Lee, and V. Buzuloiu, “Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation,” IEEE Trans. Geosci. Electron. 44(6), 1609–1621 (2006).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  15. Q. Zhang, P. Li, Y. Li, X. Ren, and S. Teng, “A universal plasmonic polarization state analyzer,” Plasmonics 13(4), 1129–1134 (2018).
    [Crossref]
  16. Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).
  17. S. Kiarashinejad, A. Abdollahramezani, and Adibi, “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures.” arXiv preprint arXiv:1902.03865 (2019).
  18. Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
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    [Crossref]
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    [Crossref]

2019 (2)

W. Wu, Y. Yu, W. Liu, and X. Zhang, “Fully integrated CMOS compatible polarization analyzer,” Nanophotonics 8(3), 467–474 (2019).
[Crossref]

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
[Crossref]

2018 (2)

Q. Zhang, P. Li, Y. Li, X. Ren, and S. Teng, “A universal plasmonic polarization state analyzer,” Plasmonics 13(4), 1129–1134 (2018).
[Crossref]

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

2017 (1)

2016 (1)

2015 (3)

2013 (1)

S. A. Hall, M.-A. Hoyle, J. S. Post, and D. K. Hore, “Combined stokes vector and Mueller matrix polarimetry for materials characterization,” Anal. Chem. 85(15), 7613–7619 (2013).
[Crossref]

2012 (1)

2011 (1)

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

2009 (1)

2006 (1)

G. Vasile, E. Trouvé, J.-S. Lee, and V. Buzuloiu, “Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation,” IEEE Trans. Geosci. Electron. 44(6), 1609–1621 (2006).
[Crossref]

1993 (1)

M. F. Møller, “A scaled conjugate gradient algorithm for fast supervised learning,” Neural Netw. 6(4), 525–533 (1993).
[Crossref]

1988 (1)

J. Lee Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42(1), 59–66 (1988).
[Crossref]

1977 (1)

Abdollahramezani, A.

S. Kiarashinejad, A. Abdollahramezani, and Adibi, “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures.” arXiv preprint arXiv:1902.03865 (2019).

Abdollahramezani, S.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
[Crossref]

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).

Adibi,

S. Kiarashinejad, A. Abdollahramezani, and Adibi, “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures.” arXiv preprint arXiv:1902.03865 (2019).

Adibi, A.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
[Crossref]

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).

Berry, H. G.

Bozhevolnyi, S. I.

Buller, G. S.

Buzuloiu, V.

G. Vasile, E. Trouvé, J.-S. Lee, and V. Buzuloiu, “Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation,” IEEE Trans. Geosci. Electron. 44(6), 1609–1621 (2006).
[Crossref]

Capasso, F.

Chen, M.

Chen, W.

Gabrielse, G.

Gao, J.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Gerardot, B. D.

Guo, C.

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

Guo, S.

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

Guo, Z.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Hall, S. A.

S. A. Hall, M.-A. Hoyle, J. S. Post, and D. K. Hore, “Combined stokes vector and Mueller matrix polarimetry for materials characterization,” Anal. Chem. 85(15), 7613–7619 (2013).
[Crossref]

Hemmatyar, O.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
[Crossref]

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).

Hore, D. K.

S. A. Hall, M.-A. Hoyle, J. S. Post, and D. K. Hore, “Combined stokes vector and Mueller matrix polarimetry for materials characterization,” Anal. Chem. 85(15), 7613–7619 (2013).
[Crossref]

Hoyle, M.-A.

S. A. Hall, M.-A. Hoyle, J. S. Post, and D. K. Hore, “Combined stokes vector and Mueller matrix polarimetry for materials characterization,” Anal. Chem. 85(15), 7613–7619 (2013).
[Crossref]

Kiarashinejad, S.

S. Kiarashinejad, A. Abdollahramezani, and Adibi, “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures.” arXiv preprint arXiv:1902.03865 (2019).

Kiarashinejad, Y.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
[Crossref]

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).

Kremer, P. E.

Kumar, S.

Lee, B.

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

Lee, G. Y.

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

Lee, J.-S.

G. Vasile, E. Trouvé, J.-S. Lee, and V. Buzuloiu, “Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation,” IEEE Trans. Geosci. Electron. 44(6), 1609–1621 (2006).
[Crossref]

Lee, K.

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

Lee Rodgers, J.

J. Lee Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42(1), 59–66 (1988).
[Crossref]

Leosson, K.

Li, P.

Q. Zhang, P. Li, Y. Li, X. Ren, and S. Teng, “A universal plasmonic polarization state analyzer,” Plasmonics 13(4), 1129–1134 (2018).
[Crossref]

Li, R.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Li, Y.

Q. Zhang, P. Li, Y. Li, X. Ren, and S. Teng, “A universal plasmonic polarization state analyzer,” Plasmonics 13(4), 1129–1134 (2018).
[Crossref]

Liu, J.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Liu, Q.

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

Liu, W.

W. Wu, Y. Yu, W. Liu, and X. Zhang, “Fully integrated CMOS compatible polarization analyzer,” Nanophotonics 8(3), 467–474 (2019).
[Crossref]

Livingston, A.

Ma, Y.

Miao, J.

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

Møller, M. F.

M. F. Møller, “A scaled conjugate gradient algorithm for fast supervised learning,” Neural Netw. 6(4), 525–533 (1993).
[Crossref]

Mueller, J. B.

Mun, S. E.

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

Nelson, R. L.

Nicewander, W. A.

J. Lee Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42(1), 59–66 (1988).
[Crossref]

Nielsen, M. G.

Pors, A.

Post, J. S.

S. A. Hall, M.-A. Hoyle, J. S. Post, and D. K. Hore, “Combined stokes vector and Mueller matrix polarimetry for materials characterization,” Anal. Chem. 85(15), 7613–7619 (2013).
[Crossref]

Pourabolghasem, R.

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).

Qu, S.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Ren, X.

Schwope, A.

A. Schwope and J. Tinbergen, Astronomical Polarimetry (Cambridge University Press, 1996).

Sung, J.

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

Taghizadeh, M. R.

Teng, S.

Q. Zhang, P. Li, Y. Li, X. Ren, and S. Teng, “A universal plasmonic polarization state analyzer,” Plasmonics 13(4), 1129–1134 (2018).
[Crossref]

Tian, Y.

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

Tinbergen, J.

A. Schwope and J. Tinbergen, Astronomical Polarimetry (Cambridge University Press, 1996).

Trouvé, E.

G. Vasile, E. Trouvé, J.-S. Lee, and V. Buzuloiu, “Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation,” IEEE Trans. Geosci. Electron. 44(6), 1609–1621 (2006).
[Crossref]

Vasile, G.

G. Vasile, E. Trouvé, J.-S. Lee, and V. Buzuloiu, “Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation,” IEEE Trans. Geosci. Electron. 44(6), 1609–1621 (2006).
[Crossref]

Wang, W.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Wang, Y.

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

Wei, S.

Wen, D.

Wu, W.

W. Wu, Y. Yu, W. Liu, and X. Zhang, “Fully integrated CMOS compatible polarization analyzer,” Nanophotonics 8(3), 467–474 (2019).
[Crossref]

Yang, S.

Yang, Z.

Yu, Y.

W. Wu, Y. Yu, W. Liu, and X. Zhang, “Fully integrated CMOS compatible polarization analyzer,” Nanophotonics 8(3), 467–474 (2019).
[Crossref]

Yue, F.

Yun, H.

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

Zandehshahvar, M.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
[Crossref]

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).

Zhan, Q.

Zhang, A.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Zhang, J.

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Zhang, Q.

Q. Zhang, P. Li, Y. Li, X. Ren, and S. Teng, “A universal plasmonic polarization state analyzer,” Plasmonics 13(4), 1129–1134 (2018).
[Crossref]

Zhang, X.

W. Wu, Y. Yu, W. Liu, and X. Zhang, “Fully integrated CMOS compatible polarization analyzer,” Nanophotonics 8(3), 467–474 (2019).
[Crossref]

Zhao, M.

Zhou, Z.

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

Adv. Theory Simul. (1)

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep Learning Reveals Underlying Physics of Light–Matter Interactions in Nanophotonic Devices,” Adv. Theory Simul. 2(9), 1900088–0390 (2019).
[Crossref]

Am. Stat. (1)

J. Lee Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42(1), 59–66 (1988).
[Crossref]

Anal. Chem. (1)

S. A. Hall, M.-A. Hoyle, J. S. Post, and D. K. Hore, “Combined stokes vector and Mueller matrix polarimetry for materials characterization,” Anal. Chem. 85(15), 7613–7619 (2013).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Geosci. Electron. (1)

G. Vasile, E. Trouvé, J.-S. Lee, and V. Buzuloiu, “Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation,” IEEE Trans. Geosci. Electron. 44(6), 1609–1621 (2006).
[Crossref]

Laser Photonics Rev. (1)

K. Lee, H. Yun, S. E. Mun, G. Y. Lee, J. Sung, and B. Lee, “Ultracompact broadband plasmonic polarimeter,” Laser Photonics Rev. 12(3), 1700297 (2018).
[Crossref]

Nanophotonics (1)

W. Wu, Y. Yu, W. Liu, and X. Zhang, “Fully integrated CMOS compatible polarization analyzer,” Nanophotonics 8(3), 467–474 (2019).
[Crossref]

Neural Netw. (1)

M. F. Møller, “A scaled conjugate gradient algorithm for fast supervised learning,” Neural Netw. 6(4), 525–533 (1993).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Optica (2)

Plasmonics (3)

J. Miao, Y. Wang, C. Guo, Y. Tian, S. Guo, Q. Liu, and Z. Zhou, “Plasmonic lens with multiple-turn spiral nano-structures,” Plasmonics 6(2), 235–239 (2011).
[Crossref]

J. Zhang, Z. Guo, R. Li, W. Wang, A. Zhang, J. Liu, S. Qu, and J. Gao, “Circular polarization analyzer based on the combined coaxial Archimedes’ spiral structure,” Plasmonics 10(6), 1255–1261 (2015).
[Crossref]

Q. Zhang, P. Li, Y. Li, X. Ren, and S. Teng, “A universal plasmonic polarization state analyzer,” Plasmonics 13(4), 1129–1134 (2018).
[Crossref]

Other (3)

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge Discovery In Nanophotonics Using Geometric Deep Learning.” arXiv preprint arXiv:1909.07330 (2019).

S. Kiarashinejad, A. Abdollahramezani, and Adibi, “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures.” arXiv preprint arXiv:1902.03865 (2019).

A. Schwope and J. Tinbergen, Astronomical Polarimetry (Cambridge University Press, 1996).

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Figures (5)

Fig. 1.
Fig. 1. (a) The Au spiral structure on a glass substrate illuminated normally by different polarization states. (b) A left hand 4-turn spiral slot penetrated through an Au film. (c) Single turn spiral.
Fig. 2.
Fig. 2. a∼d are simulated near-field intensity distributions for horizontal, 45°, left hand and right hand circularly polarized wave. Intensity patterns illuminated by arbitrary SOPs will be utilized as the training-set. (e) NN architecture with three layers including one hidden layer. Input layer has 2500 nodes that correlate to 2500 pixels of each intensity image and the output layer’s 3 nodes correlate to the 3 Stokes parameters.
Fig. 3.
Fig. 3. (a) The MSE of testing-set for NNs with different number of nodes in the hidden layer. (b) Training process represented by MSE of the training-set and validation-set for every training iteration.
Fig. 4.
Fig. 4. (a) Measurement of the state of polarization ${[{{S_1},{S_2},{S_3}} ]^T}$ of 300 selected polarizations using the spiral polarimeter (orange) with their real Stokes parameters (blue). (b) Comparison between measured Stokes parameters (orange) using our Stokes analyzer and their real values (blue) in a Poincare sphere.
Fig. 5.
Fig. 5. Correlation coefficients (R) between the intensity distribution matrix at wavelength of 980nm and other wavelength conditions.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

r n ( φ ) = r n 0 + φ 2 π λ s p p , for 0 < φ < 2 π n = 1 , 2 , 3 , 4 ,
E s p p ( R , θ ) = e z 2 π E 0 z r 0 e k z z e i k r r 0 J 0 ( k r R )
E s p p ( R , θ ) = e z 2 π E 0 z r 0 e k z z e i k r r 0 e 2 i θ J 2 ( k r R )
[ A B ] = C e i φ 1 [ 2 2 2 2 i ] + C e i φ 2 [ 2 2 2 2 i ]
E s p p = e i φ 2 C e z r 0 e k z Z e i K S P P r 0 J 0 ( K S P P R ) + e i φ 1 C e z r 0 e k z Z e i K S P P r 0 e 2 i θ J 2 ( K S P P R ) = C e z r 0 e k z Z e i K S P P r 0 [ e i φ 2 J 0 ( K S P P R ) + e i φ 1 e 2 i θ J 2 ( K S P P R ) ] ,
I = E s p p E s p p
MSE = 1 N i N j M ( P i [ j ] L i [ j ] ) 2